optistruct 12.0 user guide

OptiStruct 12.0 User's Guide

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OptiStruct 12.0 User's Guide iAltair Engineering

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OptiStruct 12.0 User's Guide

........................................................................................................................................... 1What is Altair OptiStruct?

............................................................................................................................................... 3Enhancing the Design Process

............................................................................................................................................... 7Capabilities

............................................................................................................................................... 9Features

............................................................................................................................................... 15Formats

............................................................................................................................................... 17Platforms

............................................................................................................................................... 19Running OptiStruct

................................................................................................................................... 23Run Options for OptiStruct

................................................................................................................................... 36OptiStruct GPU

................................................................................................................................... 38OptiStruct SPMD

................................................................................................................................... 53HWSolver Configuration File

................................................................................................................................... 58Expanded Error Message File

................................................................................................................................... 59Memory Limitations

................................................................................................................................... 62Restarting OptiStruct

................................................................................................................................... 63OptiStruct Compression Run

............................................................................................................................................... 65Design Optimization

................................................................................................................................... 66Optimization Problem

................................................................................................................................... 70Responses

................................................................................................................................... 89Topology Optimization

................................................................................................................................... 106Free-size Optimization

................................................................................................................................... 120Topography Optimization

................................................................................................................................... 124Size Optimization

................................................................................................................................... 126Shape Optimization

................................................................................................................................... 128Free-shape Optimization

................................................................................................................................... 148Manufacturing Constraints

................................................................................................................................... 216Optimization of Arbitrary Beam Sections

................................................................................................................................... 218Optimization of Composite Structures

................................................................................................................................... 230Equivalent Static Load Method (ESLM)

................................................................................................................................... 245Gradient-based Optimization Method

................................................................................................................................... 256Global Search Option

............................................................................................................................................... 259Design Interpretation - OSSmooth

................................................................................................................................... 262OSSmooth Parameter File

................................................................................................................................... 268Running OSSmooth

................................................................................................................................... 269Interpretation of Topology Optimization Results

................................................................................................................................... 271Laplacian Smoothing

................................................................................................................................... 273Interpretation of Topography Optimization Results

OptiStruct 12.0 User's Guideii Altair Engineering


................................................................................................................................... 275Shape Optimization Results, Surface Reduction and Surface Smoothing

................................................................................................................................... 276FEA Topology for Reanalysis

................................................................................................................................... 278FEA Topography for Reanalysis

............................................................................................................................................... 281Examples

................................................................................................................................... 282Example Problems for Topology Optimization

................................................................................................................................... 304Example Problems for Topography Optimization

................................................................................................................................... 326Example Problems for Topography Optimization Using Pattern Grouping

................................................................................................................................... 356Example Problems for Size Optimization

................................................................................................................................... 362Example Problems for Shape Optimization

................................................................................................................................... 370Examples for Optimization with ESLM

............................................................................................................................................... 381Optimization References

Altair Engineering OptiStruct 12.0 User's Guide 1


What is Altair OptiStruct?

Altair OptiStruct is an award winning CAE technology for conceptual design synthesis andstructural optimization. OptiStruct uses the analysis capabilities of RADIOSS and MotionSolveto compute responses for optimization.

Structural Design and Optimization

Structural design tools include topology, topography, and free-size optimization. Sizing,shape and free shape optimization are available for structural optimization.

In the formulation of design and optimization problems, the following responses can be appliedas the objective or as constraints: compliance, frequency, volume, mass, moment of inertia,center of gravity, displacement, velocity, acceleration, buckling factor, stress, strain,composite failure, force, synthetic response, and external (user-defined) functions. Static,inertia relief, nonlinear quasi-static (contact), normal modes, buckling, and frequencyresponse solutions can be included in a multi-disciplinary optimization setup.

Topology, topography, size, and shape optimization can be combined in a general problemformulation.

Topology Optimization

Topology optimization generates an optimized material distribution for a set of loads andconstraints within a given design space. The design space can be defined using shell or solidelements, or both. The classical topology optimization set up solving the minimum complianceproblem, as well as the dual formulation with multiple constraints are available. Constraints onvon Mises stress and buckling factor are available with limitations. Manufacturing constraintscan be imposed using a minimum member size constraint, draw direction constraints, extrusionconstraints, symmetry planes, pattern grouping, and pattern repetition. A conceptual designcan be imported in a CAD system using an iso-surface generated with OSSmooth, which ispart of the OptiStruct package.

Free-size optimization is available for shell design spaces. The shell thickness or compositeply-thickness of each element is the design variable.

Topography Optimization

Topography optimization generates an optimized distribution of shape based reinforcementssuch as stamped beads in shell structures. The problem set up is simply done by defining thedesign region, the maximum bead depth and the draw angle. OptiStruct automaticallyprovides the design variable creation and optimization control. Manufacturing constraints canbe imposed using symmetry planes, pattern grouping, and pattern repetition.

Size and Shape Optimization

General size and shape optimization problems can be solved. Variables can be assigned toperturbation vectors, which control the shape of the model. Variables can also be assigned

OptiStruct 12.0 User's Guide2 Altair Engineering


to properties, which control the thickness, area, moments of inertia, stiffness, and non-structural mass of elements in the model. All of the variables supported by OptiStruct can beassigned using HyperMesh. Shape perturbation vectors can be created using HyperMorph.

The reduction of local stress can be accomplished easily using free shape optimization. Shapeperturbations are automatically determined by OptiStruct (based on the stress levels in thedesign) when using this technique.

The layout of laminated shells can be improved by modifying the ply thickness and ply angle ofthese materials.

Finite Elements Analysis

With the exception of frequency response analysis using the direct method and transientresponse analysis (using either the modal or direct approach), all of the solution sequencesoffered through the RADIOSS (Bulk Data) input format may be used to compute responses foran optimization problem and may be used in combination in the same optimization problem.

Multi-body Dynamics Analysis

Different solution sequences for the analysis of mechanical systems are available; theseinclude Kinematics, Dynamics, Static, and Quasi-static solutions.

Flexible bodies can be derived from any finite element model defined in OptiStruct.



Enhancing the Design Process

OptiStruct enhances the design process by:

Accelerating the design process

Shortening the number of design cycles

Increasing the design performance

Providing fast and accurate finite element analysis

Generating optimal design concepts using topology and topography optimization

Providing traditional size and shape optimization to maximize the design performance

The design process can be viewed as an optimization process to find structures, mechanicalsystems, and structural parts that fulfill certain expectations towards their economy,functionality, and appearance. Generally, the design process is an iterative procedureconsisting of the following components:

Conceptual design

Design

Testing

Optimization

Today’s testing ground is usually the computer. Finite element analysis (FEA) and Multi-bodydynamics analysis (MBD) are the most used tools for computational design testing. Theresults of computational analyses are used to determine design improvements.

Changes to the design are introduced in all phases of the process. At a certain stage of thisprocess, changes to the concept become prohibitive. The concept phase plays afundamental role concerning overall efficiency of the design and the cost of the overalldevelopment process.

In the concept phase of a design process, the freedom of the designer is limited only by thespecifications of the design (Figure 1). Today, the decision on how a new design should lookis based largely upon a benchmark design or on previous designs. The decision making isbased on the experience of those involved in the design process. Conceptual design toolssuch as topology and topography optimization can be introduced to enhance the process. The concept can be based on results of a computational optimization rather than onestimations. Using topology and topography optimization, the initial design step is alreadybased on input generated using computational analysis. Topology and topographyoptimization redefine the role of computational analysis and simulation in the design process. Finite element analysis has matured from a testing tool to a design tool.



Figure 1: Decision making in the design process.

Figure 2 compares the design process using topology optimization with the conventionalmethod of leaving the concept entirely to experience and intuition. The overall cost of designdevelopment can be reduced substantially by avoiding concept changes introduced in thetesting phase of the design. This is the major benefit of modifying the design process byintroducing topology and topography optimization.

In the real world, the design process is not as straightforward as described above. Thedesign is not just driven by one performance measure -- it has to be viewed as amultidisciplinary task. Today, the different disciplines work more or less independently. Analysis and optimization is performed for single phenomena such as linear static behavior ornoise, vibration and harshness. Still, the idea persists that if one performance measureimproves, the whole performance improves. A simple example shows that this is not quitetrue. Take the design of a car -- a high stiffness is necessary for good driving and handling,and high deformability is important for the crashworthiness of the design. This shows thatimproving one measure may result in degrading another. Therefore, compromises must go intothe formulation of the optimization problem. The definition of the design problem and of thedesign target is most important. The solution can be left to computational means. Multidisciplinary considerations, especially in the conceptual design, are, in many ways, stillactive research topics and are being covered by future developments of topologyoptimization. However, the inclusion of manufacturing constraints into topology andtopography optimization is already implemented in OptiStruct.



Figure 2: The design process without and with the use of topology optimization.

OptiStruct also provides size and shape optimization to completely support the design processwith finite element based structural optimization. Using the advanced interfacing withHyperMesh, the generation of input data for structural optimization becomes an easy task. This allows structural optimization to be integrated into the design process seamlessly.



Capabilities

OptiStruct can be used to solve and optimize a wide variety of design problems in which thestructural and system behavior can be simulated using finite element and multi-body dynamicsanalysis.

The design and optimization capabilities of OptiStruct allow for the development of preliminarydesign concepts and for the improvement of existing designs based on finite elementanalyses. Some types of optimization problems are listed below:

Two-dimensional truss structure optimization.

Ribbed reinforcement patterns for 3-D shell structures.

Ribbed reinforcements for solid structures.

Spotweld reduction.

Lightening holes for existing 2-D planar and 3-D bending shell problems.

Discrete optimized structures for problems modeled using 3 dimensional solid elementproblems.

Bead (Swages) reinforcements in 3-D shell structures.

Shape modifications for volume parts.

Gage optimization of 3-D shell structures.

Beam cross-section optimization of structures modeled with beam elements .

Layout of laminated shell by modifying ply thickness and ply angle.

Reduction of stress concentrations.

Optimization of mechanisms and mechanical systems to minimize weight and reducestress.



Features

Optimization

General optimization problem formulation for all optimization types

- Response based

- Equation utility

- Interface to external user-defined routines

- Minmax (maxmin) problems

- System identification

- Continuous and discrete design variables

Solution sequences for optimization

- Linear static

- Normal modes

- Linear buckling

- Quasi-static nonlinear (gap/contact)

- Frequency response (modal method with residual vectors)

- Acoustic response

- Random response

- Linear steady-state heat transfer

- Coupled thermo-mechanical

- Multi-body Dynamics

- Fatigue

Responses for optimization

- All optimization types:

- Compliance- Frequency- Compliance index- Volume- Mass- Volume fraction- Mass fraction- Center of gravity- Moments of inertia- Displacement- Velocity- Acceleration- Temperature- Pressure - Stress (global von Mises stress in topology/free-size optimization)- Buckling factor (with limitations in topology/free-size optimization)- Fatigue life/damage- User-defined responses



- Size, shape, free-shape, and topography optimization: (In problems with topology/free-size design domains, these responses can be usedin the non-design domain)

- Strain- Force- Composite stress, strain, and failure (linear static analysis only)

Automatic selection of best optimization algorithm

- Optimality criteria method

- Convex approximation method

- Method of feasible directions

- Sequential quadratic programming

- Advanced approximations

Automatic selection of best method for design sensitivity analysis

- Direct method

- Adjoint variable method

Topology, free-size, topography, size, shape, and free-shape optimization problems canbe solved simultaneously

Multi-disciplinary optimization using combinations of the supported solution sequences

Mode tracking


Generalized optimization problem formulation

Multiple load cases with different solution sequences in combination

Global von Mises stress constraint for static loads

Density method

1-D, 2-D, and 3-D elements in the design space

Non-design space can contain any element type and response

Extensive manufacturing control:

- Minimum member size control to avoid mesh dependent results

- Maximum member size control to avoid large material concentrations

- Draw direction constraints

- Extrusion constraints

- Pattern grouping

- Pattern repetition

- Multiple symmetry planes

Checkerboard control

Discreteness control



Smoothing and geometry generation for 3-D results

Free-Size Optimization

Generalized optimization problem formulation

Multiple load cases with different solution sequences in combination

Global von Mises stress constraint for static loads

Shell element thickness and composite ply-thickness design variables

Non-design space can contain any element type and response

Extensive manufacturing control:

- Minimum member size control to avoid mesh dependent results

- Maximum member size control to avoid large material concentrations

- Draw direction constraints

- Extrusion constraints

- Pattern grouping




Shape optimization for shells with automated design variable definition

Easy set up with one DTPG card

Extensive bead pattern control to allow for manufacturing constraints

- Pattern grouping



- Discreteness control

Size Optimization

Shell, rod, and beam properties can be designed

Spring and concentrated mass properties can be designed

Composite ply thickness and ply angle can be designed

Material properties can be designed

Continuous and discrete design variables

Shape Optimization

Perturbation vector approach

Shape functions are defined through DVGRID cards



Continuous and discrete design variables

Free-shape Optimization

Perturbation vector approach

Automatic generation of perturbation vectors

Reduction of stress concentrations

Structural Optimization in Multi-body Dynamics Systems

Equivalent Static Load (ESL) method

Size, shape, free-shape, topology, topography, free-size, and material optimization offlexible bodies in multi-body dynamics systems

Generalized optimization problem definition

Large number of design variables and constraints

Pre-processing

Fully supported in HyperMesh and MotionView

Nastran type input format

Post-processing

HyperView

- Direct output of H3D format for model and results

- Direct output for iteration history

- Export of iso-density surface in STL format

HyperGraph

- Iteration history graphs

- Sensitivity bar charts

- Complex frequency response displacement, velocity, and acceleration plots for upto 500 nodes

- Random response PSD and auto/cross correlation of displacement, velocity, andacceleration

- Transient response displacement, velocity, and acceleration time history plots forup to 500 nodes

- Bar chart for effective mass

HTML report

- Model summary

- Model and result displayed using HyperView Player

HyperMesh



- Direct binary result file output

Microsoft Excel

- Design sensitivities for size and shape variable approximations

Support of Nastran Punch and OP2 output formats



Formats

OptiStruct supports the following input/output formats:

Formats

Input Nastran Bulk Data Format

Output HyperMesh Result File (Results)

H3D Binary File (Results)

Patran ASCII (Results)

Nastran Output2 (Results)

Nastran Punch File (Results)

OptiStruct 2.0 (Results)

HyperView Format (Iteration history, sensitivities, effective mass)

Microsoft Excel (Sensitivities)



Platforms

OptiStruct runs on the following platforms:

Platform BitsMinimumsystemrequirement

SMP MPP (SPMD)

Linux 32-bit RHEL 4Novell SuSe9.2

Yes Yes

64-bit for ia64SGI ProPack4, RHEL 4, Novell SuSe9.2,SLES 9

Yes Yes

for x86_64RedHatEnterpriseLinux 4, Novell SuSe10.2,SLES 9.1

Yes Yes

Win XP/Vista 32-bit Windows XP/Vista

Yes* No

Win XP/Vista 64-bit Windows XP/Vista x64edition

Yes* No

HP 64-bit HP-UX 11.11for PA-RISC

Yes No

HP-UX 11.23for ia64

No No

IBM 64-bit AIX 5.3 Yes No

SUN 64-bit Solaris 10 forx86_64

Yes No



Platform BitsMinimumsystemrequirement

SMP MPP (SPMD)

SMP– Symmetric Multiprocessing (Multiple processors, singlememory).

MPP (SPMD) Massive parallel processing (Multiple processors eachhaving its own memory).

* Performance gain for SMP runs on Windows platforms ispoor and it is therefore not recommended to use more thanone processor on these platforms.



Running OptiStruct

Note: Your system administrator may need to modify the script to make itcompatible with your system.

This section describes the execution of OptiStruct.

There are several ways to run OptiStruct:

From the script.

From the HyperWorks Solver Run Manager.

From inside the preprocessors HyperMesh.

From inside HyperView and HyperGraph.

In all the above cases, HyperWorks will initialize $PATH and other environment variables

required to run the selected solver, however you are responsible for initializing environment

variables for 3rd party products. In particular, MPI and AMLS/FRF external solvers (if needed)may require PATH and LD_LIBRARY_PATH.

Running OptiStruct from the Script

To run on UNIX from the command line, type the following:

<install_dir>/altair/scripts/optistruct "filename" –option argument

To run RADIOSS from a Windows DOS prompt, type the following:

<install_dir>\hwsolvers\bin\win32\optistruct.bat "filename" –option argument

The options and arguments are described under Run Options.

OptiStruct looks for "filename" in the following manner ("filename" may contain a file path

that is either absolute or relative to the run directory):

First, it checks to see if "filename" exists exactly as input.

If "filename" does not exist exactly as input, and if "filename" does not contain an

extension (i.e. if the actual file name without the path does not contain a period), thenit checks for "filename".parm and then for "filename".fem.

If none of these checks results in a match, OptiStruct reports an error and terminates.

Running OptiStruct from HyperWorks Solver Run Manager

On Windows, a utility to start each solver is provided through Start > Program s > Alta irHyperW orks 12.0 > <solver nam e>. This utility allows you to start multiple solver runs,select options from the menu, and maintains a history of solutions. On UNIX platforms, this



utility can be started from command line as:

<install_dir>/altair/scripts/<solver name> -gui

Running OptiStruct from HyperMesh

If you set up a finite element model in HyperMesh, you can run the simulation directly out ofHyperMesh by going to the OptiStruct panel in the respective user profiles. The panels canbe accessed through the Analysis page, from the Utility menu, or through the Applicationspull down. The panels ask for the file name. After clicking the solver button, the model isexported using the given export options. Then the solver runs the script that is providedlocally on the machine. After solver execution, the results can be viewed in HyperView. Youcan bring up HyperView with the results loaded by clicking the HyperView button.

Note: When running OptiStruct from HyperMesh on UNIX and Linux, a shell is spawnedwith the DISPLAY setting <hostname>:0.0. If this is different from the DISPLAY

setting for HyperMesh, 50 HyperWorks units (in addition to the 21 HyperWorks unitsbeing used for HyperMesh) will be checked out. To avoid the checking out ofadditional units, be sure that the DISPLAY is set to <hostname>:0.0 before starting

HyperMesh.

Running RADIOSS and OptiStruct from HyperView, HyperGraph

If you are in HyperView or HyperGraph, RADIOSS and OptiStruct can be run from the Applications pull down. After picking OptiStruct, the HyperWorks Solver Run Managermain form will appear, which will allow you to pick a file, enter run options, and run thesimulation.

The HwSolver Configuration File

The configuration file hwsolver.cfg may be used to establish default settings for OptiStruct

either system wide, for a particular user, or for a local directory. A full description of thesettings allowed and the usage of the configuration file is provided on the HwSolverConfiguration File page.

Environment Variables

The following environment variable is optional and may be set on either UNIX or PC platforms;however, the preferred way is to define them using the HwSolver Configuration File.

OS_TMP_DIR = path Path – Path name to directory for scratch file storage(Default = directory where the solver is started – can beoverwritten by the definition in the script or input deck).

The following environment variable is optional and may only be set on UNIX platforms;however, the preferred way is to define this using the HwSolver Configuration File.



DOS_DRIVE_$ = path This environment variable allows drive letters to be assignedto UNIX paths. This facilitates copying files which containINCLUDE, TMPDIR, INFILE or OUTFILE definitions containingdrive letters from PC to UNIX on hybrid networks.

$ - Drive letter to be defined (case sensitive).

Path - UNIX path with which you want to replace the driveletter.

Note that after such expansion, the paths are alwaysinterpreted as if there were a ‘\’ immediately after the driveletter in the original PC path.

Memory Allocation

Memory is dynamically allocated for a run. The allocation starts with the initial memory.

The default setting for the memory limit is 1GB for 64-bit solver version and 320MB for 32-bitsolver version (PC and Linux). This setting can be changed by using the SYSSETTING option

OS_RAM, or by defining the –len option in the run script. The script overwrites the

environment variable.

OptiStruct will always attempt to assign enough memory for a minimum core solution.

The initial memory is 10% of the memory limit by default. This setting can be changed byusing the SYSSETTING option OS_RAM_INIT.

A check run can be very helpful in estimating the memory and disk space usage. In a checkrun, the memory necessary is automatically allocated.

The solver automatically chooses an in-core, out-of-core, or minimum core solution based onthe memory allocated. A solution type can be forced by defining the –core option in the run

script; the memory necessary for the specified solution type is then assigned.

Please see the Memory Limitations section for detailed information on the following topics: 32-bit versus 64-bit computations, virtual versus physical memory, and automatic memoryallocation versus fixed memory runs.

Summary Information

OptiStruct always creates an .out file which contains summary information for the job. Thisinformation can be echoed to the screen through the inclusion of the SCREEN I/O option inthe input data or through the use of the -out command line option (see Run Options).

This file also contains memory and disk space estimates. The disk space estimates foreigenvalue analyses (normal modes, linear buckling, modal methods of frequency, transientresponse, and fluid-structure coupling (acoustics)) are sometimes very conservative and canbe three times as much as is truly used. This is because it is not fully predictable how muchdata needs to be saved to scratch files.



The true usage of memory and disk space is reported at the bottom of the file after the solverhas finished.

Should the job be re-run in the same location, the .out file is not overwritten, but is instead

moved to _#.out, where # is the lowest available three digit number that creates a unique

file name.

For example, if filename.fem were run in a directory already containing filename.out, the

existing filename.out would be moved to filename_001.out, and the summary information

for the new job would be written to filename.out. Should the job be repeated again, the

existing filename.out would be moved to filename_002.out, and the summary information

for the latest job would be written to filename.out.

filename.out is the only file that is saved in this manner. All other results files will be

overwritten.

Recommendations

1. Try running OptiStruct with the default setting first (without specification of the –len or

core options).

2. Do a check run before submitting large jobs (>500,000 dof) to NQS to make sure sufficientNQS memory is being provided. The –lM option can be used to change the NQS memory.

Be sure to include at least 12Mb for the executable in addition to the memory necessaryto solve the problem. A check run can also assist in debugging input data without havingto wait in a queue.



Run Options for OptiStruct

Option Argument Description Available on

-acf N/A Option to specify that the input file isan ACF file for a multi-body dynamicssolution sequence.

All Platforms

-amls N/A Invokes the external AMLS eigenvaluesolver. The AMLS_EXE environment

variable needs to point to the AMLSexecutable for this setting to work.

Overrides the PARAM, AMLS setting inthe input file.

(Example: optistruct infile.fem –

amls)

All Platforms

-amlsncpu 1, 2, or 4 Defines the number of CPUs to be usedby the external AMLS eigenvalue solver. This parameter will set theenvironment variable OMP_NUM_THREADS

.

The default value is the current valueof OMP_NUM_THREADS. Note that this

value can be set by the command linearguments –nproc or –ncpu.

OptiStruct and AMLS can be run withdifferent allocations of processors. Forexample, OptiStruct can be run with 1processor and AMLS with 4 processorsin the same run.

Only valid with –amls option or when

PARAM, AMLS is set to YES.

Overrides the PARAM, AMLSNCPUsetting in the input file.

Default: Number of processors used byOptiStruct.


amls –amlsncpu 4)

All Platforms




-analysis N/A Submit an analysis run. This option willalso check the optimization data; thejob will be terminated if any errorsexist.

-optskip will skip checking the

optimization data and the analysis willbe performed.

Cannot be used with -check or -restart


analysis)

All Platforms

-buildinfo N/A Displays build information for selectedsolver executables.

Bulk

-check N/A Submit a check job through thecommand line.

The memory needed is automaticallyallocated.

Cannot be used with –analysis,

-optskip or -restart


check)

All Platforms

-checkel yes, no, full If NO, element quality checks are notperformed, but mathematical validitychecks are performed.

If YES, or if no argument is given, thegeometric quality of each element ischecked. Any violation of the errorlimits is counted as a fatal error andthe run will stop. Any violation ofwarning limits is non-fatal. Error orwarning messages are printed forelements violating the limits along withthe offending property values. Theamount of output is limited to the first3 occurrences for each individual case,plus a summary table of all errors.

All Platforms




If FULL, the same checks are performedas for YES, but the error or warningmessages are printed for all of theelements violating the error or warninglimits.

Default is Yes.

(Example: optistruct infile.fem

–checkel full)

(Example: optistruct infile.fem

–checkel)

-compress N/A Submits a compression run.

Reduces out matching material andproperty definitions. The resulting bulkdata file will be written to a file named <filename>.echo. Cannot be used in

combination with any other option. See OptiStruct Compression Run formore information.


compress)

All Platforms

-core in, out, min in – in-core solution is forced

out – out-of-core solution is forced

min – minimum core solution is forced

The solver assigns the appropriatememory required. If there is notenough memory available, OptiStructwill error out. Overwrites the –len

option.


core in)

All Platforms

-cpu

or-proc

or

Number ofcores

Number of cores to be used for SMPsolution.

(Example: optistruct infile.fem -

All Platforms




-nproc

or-ncpu

or-nt

ncpu 2)

-dir N/A Change directory to the location ofinput file before starting the solver.

All Platforms

-fixlen RAM in MBytes Disables dynamic memory allocation.

OptiStruct will allocate the givenamount of memory and use itthroughout the run. If this memory isnot available, or if the allocated amountis not sufficient for the solutionprocess, OptiStruct will terminate withan error.

To avoid over specifying the memorywhen using this option, it is suggestedfirst to run OptiStruct with the -check

option and use the results of that runto properly define the memory size forthe -fixlen option.

This option allows, on certain platforms,to avoid memory fragmentation andallocate more memory than is possiblewith dynamic memory allocation.

Overwritten by -len and -coreoptions.


fixlen 500)

All Platforms

-gpu N/A Activates GPU Computing All Platforms

-gpuid N/A N: Integer, Optional, Selects the GPUCard. Default = 1.

All Platforms

-h N/A Displays script usage. All Platforms




-len RAM in MBytes Preferred upper bound on dynamicmemory allocation.

The solver automatically chooses an in-core, out-of-core, or minimum coresolution based on the memory required. OptiStruct will attempt to run at leastthe minimum core solution regardless ofhow much memory is designated. Overwritten by the –core option.

Best practices for –len specification:For proper memory allocation whileusing –len in an OptiStruct run, avoid

using the exact reported memoryestimate value (for eg. Using Check).The –len value should be provided

based on the actual memory of thesystem. This would be therecommended memory limit to run thejob, it may not necessarily representthe memory utilized by the job or theactual memory limit. This way, the jobis more likely to run with the bestpossible performance. If the samesystem is shared by multiple jobs, thenthe memory allocation should follow thesame procedure as above; except, thatthe individual maximum memory shouldbe used in place of the total systemmemory. (If a job runs out-of-coreinstead of in-core (it exceeded thememory allocation) it will still run veryefficiently. However, make sure thatthe job does not exceed the actualmemory of the system itself as this willslow the run down by a large factor.The recommended method to deal withthis is to specify –maxlen as the actual

memory of the system to limit themaximum memory that can be used onthe system.

Default = 320Mb.


len 32)

All Platforms




-lic FEA, ADVFEA,HFS,RADIOSSA,RADIOSSB, OPT

FEA - FE analysis only(OptiStructFEA).

All Platforms

ADVFEA - Advanced analysis(OptiStructAdv).

HFS - One-step Metalforming solution(HyperFormSolver)

RADIOSSA - Basic Finite ElementAnalysis (RadiossA)

RADIOSSB - Advanced FiniteElement Analysis(RadiossB)

OPT - Optimization(OptiStruct orOptiStructMulti).

The solver checks out a license of thespecified type before reading the inputdata. Once the input data is read, thesolver verifies that the requestedlicense is of the correct type. If this isnot the case, OptiStruct will terminatewith an error.

No default

(Example: OptiStruct infile.fem -

lic FEA)

-licwait Hours to waitfor a license tobecomeavailable

If present and there are not 50HyperWorks Units available, OptiStructwill wait for up to the number of hoursspecified (default=12) for licenses tobecome available and then start to run. The maximum wait period that can bespecified to wait is 168 hours (a week). OptiStruct will check for availableHyperWorks Units every two minutes.

All Platforms




-manual N/A Launches the online OptiStruct User’smanual.

All Platforms

-maxlen RAM in Mbytes Hard limit on the upper bound ofdynamic memory allocation.

OptiStruct will not exceed this limit.

No default


maxlen 1000)

All Platforms

-mio N/A Enables Modular I/O.

Note: Asynchronous I/O should beenabled on host.


mio)

AIX

-mio_pfcache MIO PF CacheSize inMBytes

Set the PF Cache Size for MIO to use. This memory is allocated in addition tothe memory used by OptiStruct.

Default is 2GB


mio –mio_pfcache 1000)

AIX

-monitor N/AMonitor convergence from an

optimization or nonlinear run. Equivalent

to SCREEN, LOG in the input deck.

All Platforms

-mpi Vendor:

i (Intel MPI),

pl (HP-MPI and

Platform-MPI),

ms (MS-MPI),

sun (Solaris-

Initiate an MPI-based SPMD run onsupported platforms.


mpi –np 4)

All Platforms




MPI),

aix (AIX POE)

-mpipath path Specify the directory containing HP-MPI’s mpirun executable.

Note: This option is useful if MPIenvironments from multiple MPI vendorsare installed on the system. Valid foran MPI run only.

(Example: optistruct infile.fem –mpi –np 4 –mpipath /apps/hpmpi/

bin)

All Platforms

-nlrestart Subcase ID Restart a geometric nonlinear solutionsequence from specified subcase ID.

If Subcase ID is not specified, it willrestart from the first geometricnonlinear subcase ending with error inprevious run.

Note: The geometric nonlinear solutionsequence is a series of geometricnonlinear subcases (ANALYSIS =NLGEOM, IMPDYN or EXPDYN) linked by CNTNLSUB.

All Platforms

-np Number ofprocessors

Number of processors to be used inSPMD analysis.


mpi –np 4)

All Platforms

-op2 littleendian Sets the binary format of the .op2 file

to be little endian so that the .op2 file

can be read by software on PC’s andLINUX computers using x86 basedCPU’s.

AIX




-optskip N/A Submit an analysis run withoutperforming check on optimization data(skip reading all optimization relatedcards).

Cannot be used with –check or –

restart.


optskip)

All Platforms

-out N/A Echos the output file to the screen. This takes precedence over the I/Ooption SCREEN.


out)

All Platforms

-outfile Prefix foroutputfilenames

Option to direct the output files to adirectory different from the one inwhich the input file exists. If such adirectory does not exist, the last partof the path is assumed to be the prefixof the output files. This takesprecedence over the I/O option OUTFILE.


outfile results); here OptiStruct will

output results.out, etc.

All Platforms

-ramdisk Size ofvirtual disk(in MB)

Option to specify area in RAM allocatedto store information which otherwisewould be stored in scratch files on thehard drive.


ramdisk 800)

For a more detailed description, see theRAMDISK setting on I/O option SYSSETTING.

All Platforms

-reanal Densitythreshold

This option can only be used incombination with -restart.

Inclusion of this option on a restart run

All Platforms




will cause the last iteration to bereanalyzed without penalization.

If the "density threshold" given is lessthan the value of MINDENS (default =0.01) used in the optimization, allelements will be assigned the densitiesthey had during the final iteration ofthe optimization. As there is nopenalization, stiffness will now beproportional to density.

If the "density threshold" given isgreater than the value of MINDENS,those elements whose density is lessthan the given value will have densityequal to MINDENS, all others will have adensity of 1.0.


restart -reanal 0.3)

-restart filename.sh Specify a restart run. If no argument isprovided, OptiStruct will look for therestart file, which will have the sameroot as the input file with theextension .sh. If you enter an

argument On PC, you will need toprovide the full path to the restart fileincluding the file name.

Cannot be used with –check, -

analysis or –optskip.


restart); here OptiStruct looks for the

restart file infile.sh.


restart C:\oldrun\old_infile.sh);

here OptiStruct looks for the restart file old_infile.sh.

All Platforms

-rnp Number ofprocessors

Number of processors to be used inRADIOSS SPMD for IMPDYN, EXPDYN,and NLGEOM analysis types.


mpi –rnp 4)

All Platforms




-rnt Number ofcores

Number of cores to be used forRADIOSS SMP for IMPDYN, EXPDYN,and NLGEOM analysis types.


rnt 2)

All Platforms

-rsf Safety factor Specify a safety factor over the limit ofallocated memory.

Not applicable when -maxlen is used.


rsf 1.2)


len 32 –rsf 1.2)


core out –rsf 1.2)

All Platforms

-scr

or-tmpdir

Path,filesize=n,slow=1

Option to choose directories in whichthe scratch files are to be written. filesize=n and slow=1 arguments are

optional. Multiple arguments may becomma separated.

path ; give the path to the directory

for scratch file storage.

filesize=n ; defines the maximum file

size (in GB) that may be written to thatlocation.

slow=1 ; indicates a network drive.

(Example: optistruct infile.fem –scr filesize=2,slow=1,/

network_dir/tmp)

Multiple scratch directories may bedefined through repeated instances of –

tmpdir or –scr.

(Example: optistruct infile.fem –tmpdir C:\tmp –tmpdir filesize=2,

slow=1,Z:\network_drive\tmp)

All Platforms




This overwrites the environmentvariable OS_TMP_DIR, and the TMPDIR

definition in the I/O section of the inputdeck.

For a more detailed description, see theI/O Option TMPDIR.

-scrfmode basic,

buffered,

unbuffer,

smbuffer,stripe,mixfcio

Option to select different mode ofstoring scratch files for linear solver(especially for out-of-core andminimum-core solution modes). Multiplearguments may be comma separated.

(Example: optistruct infile.fem –scrfmode buffered, stripe –

tmpdir C:\tmp)

For a description of the arguments, seethe SCRFMODE setting on I/O option SYSSETTING.

All Platforms

-testmpi N/A Check if MPI is configured properly andif the SPMD version of the OptiStructexecutables is available for this system.

(Example: optistruct infile.fem –mpi –np 4 –mpipath /apps/hpmpi/

bin -testmpi)

All Platforms

-version N/A Checks version and build timeinformation from OptiStruct.

All Platforms

-xml N/A Option to specify that the input file isan XML file for a multi-body dynamicssolution sequence.

All Platforms

Comments

1. Any arguments containing spaces or special characters must be quoted in {} , e.g. -

mpipath {C:\Program Files\MPI}. File paths on Windows may use backward "\" or

forward slash "/" but must be within quotes when using a backslash "\" .

2. The above arguments are processed by solver script(s) and not by the actual executable.If you are developing internal scripts which use the executable directly, you may getspecific information about command line arguments that are accepted by the executable



by looking at the content of the .stat file, where these arguments are listed for each

run, or you can contact [email protected] for more information.



OptiStruct GPU

Introduction

A Graphics Processing Unit (GPU) is a system which can be used to improve the performanceof computationally intensive engineering applications. GPU Computing is a process which usesthe GPU to execute the time consuming sections of the application and the rest of the coderuns on the CPU.

Implementation

Starting from RADIOSS Bulk/OptiStruct version 12.0, the GPU can be used to accelerate thesparse direct equation solver through the NVIDIA CUDA programming model. GPU computing isimplemented by off-loading most of the computation intensive work to the GPU andconcurrently overlapping the communication and data transfer between the CPU cores andthe GPU.

Speedup

A speedup in the equation solver of up to 4 times, and up to 3 times overall when comparedto a Quad-core Intel Nehalem Xeon run, can be achieved. This heterogeneous computingmodel is particularly suitable for jobs dominated by the equation solver. For example: nonlinearstatic analysis on power train structures, topology optimization on blocky structures and soon.

Compatibility

1. GPU computing is available for static analysis/optimization.

2. GPU computing is available in 64-bit Linux platform only.

3. GPU computing is NOT supported in the SPMD module.

4. NVIDIA Fermi and Kepler architecture based Tesla and Quadro graphic cards aresupported. Tesla C2050/C2070/M2090/K10/K20, Quadro 6000/K5000/K6000 cards arerecommended for computing by NVIDIA.

Activating RADIOSS Bulk /OptiStruct GPU

Command option “-gpu” is used to activate RADIOSS Bulk/OptiStruct GPU. Currently, only one

graphics card is supported, and “–gpuid” can be used to pick the desired graphic card for

computation when multiple cards are present. Compatible drivers for the graphics card needsto be installed by the user prior to launching RADIOSS Bulk/OptiStruct GPU using the option “-



gpu”.

Command LineOption

Value Action

-gpu Activates GPU computing

-gpuid N Integer: Optional, selects the GPU cardDefault = 1

Note: I/O usually accounts for an appreciable percentage of the totalsolution time in RADIOSS Bulk/OptiStruct for an out-of-core or min-core run. This cannot be addressed or improved through GPUcomputing. Therefore, -core in (at least –core out) is

recommended when the memory in system is large enough.

Recommended Telsa GPU Computing Processor List for RADIOSS Bulk/OptiStruct

The following table lists the recommended Telsa graphic boards for use with the AltairHyperWorks Solver suite of applications for high-powered GPU computing.

Manufacturer Adaptor TypeDriver Version

(minimum or higher)

NVIDIA

(Telsa C-CLASS series)

C2070

C2075

Linux (32-bit/64-bit):

295.59

NVIDIA

(Telsa M-CLASS seriesM2090

Linux (32-bit/64-bit):

295.59

NVIDIA

(Tesla Kepler)K20 Linux (32-bit/64-bit)

Note: The most recent vendor/manufacturer drivers should be used and alldriver support for these cards should be addressed to theappropriate manufacturer of the graphic board.



OptiStruct SPMD

SPMD is a parallelization technique in computing that is employed to achieve faster results bysplitting up tasks and running them simultaneously on multiple processors. SPMD typicallyimplies running the same process or program on different machines (Nodes) with differentinput data for each individual task. However, SPMD can sometimes be implemented on a singlemachine with multiple processors depending upon the program and hardware limitations/requirements.

(T his doc um ent is only for Opt iSt ruc t SPMD or RADIOSS (Bulk Form at ) SPMD. Please refer tothe HyperWorks Insta llat ion Guide for inform at ion about SPMD in RADIOSS (Bloc k Form at ). Allre ferences to RADIOSS or RADIOSS SPMD in this docum ent im ply RADIOSS (BulkForm at) or RADIOSS (Bulk Form at) SPMD respective ly ).

OptiStruct/RADIOSS (Bulk Format) SPMD

SPMD can be implemented when you need to parallelize programs in RADIOSS/OptiStruct.SPMD in RADIOSS/OptiStruct is used when a program can be distributed into parallel tasks, asshown in Figure 1. The schematic shown in Figure 1 is applicable to SPMD run on multiplemachines.



Figure 1: Overview of the SPMD implementation in RADIOSS/OptiStruct

You can split a Program/FE Model for Analysis/Optimization into several tasks as shown inFigure 1. The tasks are assigned to various nodes that run in parallel. Ideally, if you need tosplit the program into N parallel tasks, then (N+1) nodes/machines would be required formaximum efficiency. (This is dependent on various other factors like: type of tasks,processing power of the nodes, memory allocation at each node and so on. During SPMDimplementation, using more than (N+1) nodes for N parallelizable tasks would not increaseefficiency.) The extra node is known as the Manager Node. The manager node decides thenature of data assigned to each node and the identity of the Master Node. The manager nodealso distributes multiple input decks and tasks to various nodes. It does not need a machinewith high processing power as no analysis or optimization is run on the manager node. TheMaster Node needs a higher amount of memory since it contains the main input deck and italso collects all results and performs all processes that cannot be parallelized. Optimization isrun on the Master Node. The Platform Dependent Message Passing Interface (MPI) helps inthe communication between various nodes and also between the Master Node and the SlaveNodes.

Note: A Task is a minimum distribution unit used in parallelization. Each buckling analysissubcase is one task. Each Left-Hand Side (LHS) of the static analysis subcases isone task. Typically, the static analysis subcases sharing the same SPC (Single PointConstraint) belong to one task. Note that not all tasks can be run in parallel at the



same time (For example: A buckling subcase can not start before the execution of itsSTATSUB subcase).

Checklist for the User (OptiStruct/RADIOSS SPMD)

Please look at the following information before deciding the need and the extent ofparallelization for a specific program.

Supported Platforms

Supported Solution Sequences

The Number and Type of nodes available for parallelization

Setting up SPMD for a program

Frequently Asked Questions (FAQ)

Supported Platforms

All the platforms that are supported by RADIOSS SPMD and OptiStruct SPMD are listed inTable 1.

Application

Version

SupportedPlatforms

MPI

OptiStruct SPMD

andRADIOSS

SPMD

12.0

Linux (64-bit)

RequiresHP-MPI

(Version02.02.05.01)

;(or)

Platform-MPI(Version

7.1);(or) Intel

MPI(Version3.2.011)

Windows(64-bit)

Requires HP-MPI

(Version2.0);(or)

Platform-MPI(Version

7.1);(or) Intel

MPI(Version:Windows



Application

Version

SupportedPlatforms

MPI

HPC)

Table 1: Supported Platforms for OptiStruct SPMD and RADIOSS SPMD

Supported Solution Sequences

RADIOSS and OptiStruct can handle a variety of solution sequences as seen in the overview.However, all solution sequences do not lend themselves to parallelization. In general, manysteps in a program execution are not parallelized. Steps like Pre-processing and MatrixAssembly are repeated on all nodes, while response recovery, screening, approximation,optimization and output of results are all executed on the Master Node.

Solution Sequencesthat Support

Parallelization

Parallelizable Steps Non-Parallelizable Steps

Static Analysis1. Two or more static

Boundary Conditions areparallelized (MatrixFactorization is the stepthat is parallelized since itis computationallyintensive.)

2. Sensitivities are parallelized(Even for a single BoundaryCondition as analysis isrepeated on all slavenodes).

1. Iterative Solution is notparallelized. (Only DirectSolution is parallelized).

Buckling Analysis1. Two or more Buckling

Subcases are parallelized.

2. Sensitivities areparallelized.

Direct FrequencyResponse Analysis

1. Loading Frequencies areparallelized.

2. No Optimization.

Modal FrequencyResponse Analysis

1. Loading Frequencies areparallelized.

1. Modal FRF pre-processingis not parallelized.

2. Sensitivities are notparallelized.



Table 2: OptiStruct/RADIOSS SPMD- Parallelizable Steps for various solution sequences

In HyperWorks 11.0 and beyond, the presence of non-parallelizable subcases WILL NOT makethe entire program non-parallelizable. The program execution will continue in parallel and thenon-parallelizable subcase will be executed as a serial run.

Number and Type of Nodes available for Parallelization

The types and functions of the nodes that are used in OptiStruct/RADIOSS SPMD areindicated in Table 3. The first node is automatically selected as the manager, the secondnode is the master node and the rest are slave nodes.

Node Type Functions

Master Node

(1 Node)

Runs all non-parallelizable tasks

Optimization is run here

Slave Node

(N-2 Nodes)

Runs all parallelizable tasks

Input deck copies are provided

Manager Node

(1 Node)

No tasks are run on this node, it manages the way nodesare assigned tasks.

Manager makes multiple copies of the input deck andsends them to the slave nodes.

Table 3: Types and functions of the Nodes

This assignment is based on the sequence of nodes that the user specifies in the appfile.

The appfile is a text file which contains process counts and the list of programs. Nodes can

be repeated in the appfile, multiple cores of the repeated nodes will be assigned parallel jobs

in the same sequence discussed here.

Setting up OptiStruct/RADIOSS SPMD

The user can follow the steps detailed in this section to install and launch OptiStruct/RADIOSS SPMD.

Step 1: Install OptiStruct/RADIOSS SPMD on Linux Machines

System Requirements



Operating system: Linux64

The MPI library: HP-MPI, Platform-MPI or Intel MPI must be installed and accessiblefrom every machine in the cluster.

Installing Software and activating the License

1. OptiStruct/RADIOSS SPMD is included with the OptiStruct/RADIOSS solver package and theSPMD executables are included in the installation

2. Test if OptiStruct/RADIOSS SPMD is able to run in serial mode.

Configuring the machines

1. The account used to run OptiStruct/RADIOSS SPMD must exist on every cluster'snode.

2. OptiStruct/RADIOSS SPMD executables must be accessible to all cluster nodes.

3. The local scratch directories must be the same (same path).

4. It is highly recommended to have all the computation (working) directories at the samelocation on all cluster nodes.

SSH installation

1. HP-MPI default installation uses ssh to launch OptiStruct/RADIOSS SPMD on differentnodes.

2. ssh should be configured to permit the connection on all hosts of the cluster withoutthe need to type passwords.

3. Refer to ssh man pages to generate and install rsa keys (ssh-keygen tool).

To check the functionality of ssh, the following test can be performed on the differentnodes:

[optistruct@host1] ssh host1 ls

[optistruct@host1] ssh host2 ls

...

[optistruct@host1] ssh host[n] ls

RSH (An alternative to SSH)

1. It is also possible to use rsh instead of ssh to launch OptiStruct/RADIOSS SPMD.

2. Computation nodes need to be accessible to all the other nodes without need for apassword.

3. Refer to the rsh manpages for installation instructions. Please check with HP-MPImanual for instructions on how to use RSH instead of SSH.

4. To check the functionality of rsh, the following test can be performed on the differentnodes:



[optistruct@host1] rsh host1 ls

[optistruct@host1] rsh host2 ls

...

[optistruct@host1] rsh host[n] ls

HP-MPI installation

1. HP-MPI needs to be accessible to all computation nodes on which OptiStruct/RADIOSSSPMD will be launched.

2. Download the HP-MPI images for the platform you desire to use from the respective

vendor.

3. Install the HP-MPI package on each node.

[root@host[i]]rpm -ivh hpmpi-2.02.05.01-20070708r[platform].rpm

4. No license is required for HP-MPI to run OptiStruct/RADIOSS SPMD.

5. Add $MPI_ROOT/hpmpi/lib/[platform] into LD_LIBRARY_PATH, if needed.

Platform-MPI installation

1. Platform-MPI needs to be accessible from all computation nodes on which OptiStruct/RADIOSS SPMD will be launched.

2. Download the Platform-MPI images for the platform you desire to use from the

respective vendor.

3. Install the Platform-MPI package on each node.

[root@host[i]]rpm -ivh platform_mpi-7.01.00-20091019r.[platform].rpm

4. No license is needed for Platform-MPI to run OptiStruct/RADIOSS SPMD.

5. Add $MPI_ROOT/platform_mpi/lib/[platform] into LD_LIBRARY_PATH, if needed.

Intel-MPI installation

1. Intel-MPI needs to be accessible from all computation nodes on which OptiStruct/RADIOSS SPMD will be launched.

2. Download the Intel-MPI images for the platform you desire to use from the respective

vendor.

3. Install the Intel-MPI package on each node.

[root@host[i]] ./install.sh

A license is needed to install Intel-MPI libraries. (However, OptiStruct/RADIOSS SPMDdoes not require a separate license).

4. Add $MPI_ROOT/intel/impi/3.2.2.006/lib into LD_LIBRARY_PATH, if needed.



Step 2: Launch OptiStruct/RADIOSS SPMD on Linux Machines

There are several ways to launch parallel programs with HP-MPI. Below are some typicalways to launch OptiStruct/RADIOSS SPMD. Remember to propagate environment variableswhen launching OptiStruct/RADIOSS SPMD, if needed. Please refer to HP-MPI’s manual formore details.

The parallel run of OptiStruct/RADIOSS SPMD requires an additional command-line option “-

mpimode”. If this option is not used, there will be no parallelization and the entire program

will be run on each node.

Note:

1. A minimum of three processes are required to launch OptiStruct/RADIOSS SPMD.

2. OptiStruct/RADIOSS SPMD must match the MPI implementation you use.

Using Solver Scripts

On a single host (for HP-MPI, Platform-MPI and Intel MPI) using solver script

[optistruct@host1~]$ $ALTAIR_HOME/scripts/optistruct.bat –mpi [MPI_TYPE]–np [n] [INPUTDECK] [OS_ARGS]

Where,

optistruct_script: is the OptiStruct script

[MPI_TYPE]: is the mpi implementation used:

pl for HP-MPI and Platform-MPI

i for Intel MPI

[n]: is the number of processors

[INPUTDECK]: is the input deck file name

[OS_ARGS]: lists the arguments to OptiStruct SPMD

Note:

1. Adding command line option “-testmpi” verifies your MPI installation and setup.

2. OptiStruct/RADIOSS SPMD can also be launched using the Run Manager GUI. (Pleaserefer to HyperWorks Solver Run Manager)

3. It is also possible to launch OptiStruct/RADIOSS SPMD without the GUI/ Solver Scripts.(Please refer to the Appendix)

Step 3: Install OptiStruct/RADIOSS SPMD on Windows Machines



System Requirements

Operating system: Windows Xp/Vista/7 64-bit

The MPI library: HP-MPI, Platform-MPI, Intel MPI or MS-MPI must be installed andaccessible to each machine in the cluster.

Software Installation and License Activation

1. OptiStruct/RADIOSS SPMD is included with the OptiStruct/RADIOSS solver package andthe SPMD executables are included in the installation.

2. Test if the OptiStruct/RADIOSS SPMD be able to run in serial mode.

Machine Configuration

1. The account used to run OptiStruct/RADIOSS SPMD must exist on every node in acluster.

2. OptiStruct/RADIOSS SPMD executables must be accessible to all cluster nodes. Thelocal scratch directories must be the same (same path). It is highly recommended that

the computation (working) directories also remain the same on all nodes in a cluster.

HP-MPI, Platform-MPI, MS-MPI installation

On windows it is quite straightforward to install the MPI implementation. Please refer totheir respective installation guides for further assistance.

Step 4: Launch OptiStruct/RADIOSS SPMD on Windows Machines

There are several ways to launch parallel programs with each MPI. Below are some typicalways to launch OptiStruct/RADIOSS SPMD. Remember to propagate environment variableswhen launching OptiStruct/RADIOSS SPMD, if needed. Please refer to corresponding MPI’smanual for more details.

The parallel run of OptiStruct/RADIOSS SPMD requires an additional command-line option “-

mpimode”. If this option is not used, there will be no parallelization and the entire program

will be run on each node.

Note:

1. A minimum of three processes are required to launch OptiStruct/RADIOSS SPMD.

2. OptiStruct/RADIOSS SPMD must match the MPI implementation you use.

Using Solver Scripts

On a single host using solver script (for HP-MPI, Platform-MPI, Intel-MPI and MS-MPI)

[optistruct@host1~]$ $ALTAIR_HOME/hwsolvers/scripts/optistruct.bat –mpi



[MPI_TYPE] –np [n] [INPUTDECK] [OS_ARGS]

Where,

[MPI_TYPE]: is the mpi implementations used:

pl for HP-MPI and Platform-MPI

i for Intel MPI

ms for MS-MPI



[OS_ARGS]: lists the arguments to OptiStruct SPMD

Note:

1. Adding command line option “-testmpi” verifies your MPI installation and setup.

2. OptiStruct/RADIOSS SPMD can also be launched using the Run Manager GUI. (Pleaserefer to HyperWorks Solver Run Manager)

3. It is also possible to launch OptiStruct/RADIOSS SPMD without the GUI/ Solver Scripts.(Please refer to the Appendix)

Frequently Asked Questions

How many nodes should I use?

The parallelization of OptiStruct/RADIOSS SPMD is based on task distribution. If the maximumnumber of tasks which can be run at the same time is N, then using (N+1) nodes is ideal (theextra one node for the manager distributing tasks). Using more than (N+1) nodes will notimprove the performance.

The .out file suggests the number of nodes you can use, based on your model.

When there are only M physical nodes available (M < N), then the correct way to start thejob is to use M+1 nodes in the hostfile/appfile. The manager node requires only a smallamount of resources and can be safely shared with the master: The way to assign such adistribution is repeating the first physical host in the hostfile/appfile. For example: Hostfile forIntel MPI can be:

Node1

Node1

Node2

Node3

…….



Note: For Frequency Response Analysis any number of nodes may be used. (up to the number ofloading frequencies in the model.)

How to run OptiStruct/RADIOSS SPMD on a dual/quad CPU’s/Cores machine?

Follow the instructions to run OptiStruct/RADIOSS SPMD on a single machine. The idealnumber of nodes is m in(N+1, M), where N is the maximum number of tasks that can be run atthe same time, and M is the number of CPU’s/Cores.

Note: For dual/quad code machines it may be more efficient to run OptiStruct in serial + SMPmode. (i.e. use –nt argument in the solver script) .

How to run OptiStruct/RADIOSS SPMD over LAN?

It is possible to run OptiStruct/RADIOSS SPMD over LAN. Follow the HP-MPI manual to setupdifferent working directories of each node the OptiStruct/RADIOSS SPMD is launched.

Is it better to run on cluster of separate machines or on shared memory machine(s)with multiple CPU’s?

There is no easy answer to this question. If the computer has enough memory to run all tasksin-core, then we can expect faster solution times as MPI communication is not slowed downby the network speed. But if the tasks have to run out-of-core, then computations areslowed down by disk read/write delay. Multiple tasks on the same machine may compete fordisk access, and (in extreme situations) even result in wall clock time slower than that forserial (non-MPI) runs.

Will OptiStruct/RADIOSS SPMD use less memory on each node than in the serial run?

No, Memory estimates for serial runs and parallel runs on each node are the same. They arebased on the solution of a single (most demanding) subcase.

Will OptiStruct/RADIOSS SPMD use less disk space on each node than in the serial run?

Yes. Disk space usage on each node will be smaller, because, only temporary files related totask(s) solved on this node will be stored. But the total amount of disk space will be largerthan that in the serial run and this can be noticed, especially in parallel runs on a shared-memory machine.

I have a cluster with N nodes each with M cores. What is the most efficient way I canuse the resources that I possess?

1. When each host has sufficient RAM to execute only a single serial OptiStruct run, thenuse multiple cores to activate SMP on each node. (using more than four cores is usuallynot effective). For example: on a 4 host cluster, each with 8 cores, you can run:

optistruct <inputfile> -mpi <mode> -np 5 –nt 4 –hostfile…



2. When each host has sufficient RAM to efficiently execute more than one serial run, thenyou can assign multiple MPI nodes to each host. For example:

optistruct <inputfile> -mpi <mode> -np 9 –nt 4 –hostfile…

Appendix

How to Launch OS/RD SPMD on Linux Machines using Direct calls toExecutable

On a single host (for HP-MPI, Platform-MPI and Intel MPI)

[optistruct@host1~]$ mpirun -np [n] $ALTAIR_HOME/hwsolvers/optistruct/bin/linux64/optistruct_spmd [INPUTDECK] [OS_ARGS] -mpimode

Where,

optistruct_spmd is the OptiStruct SPMD binary



[OS_ARGS]: lists the arguments to OptiStruct SPMD other than –mpimode

On a Linux cluster (for HP-MPI and Platform-MPI)

[optistruct@host1~]$ mpirun –f [appfile]

Where,

[appfile]: is a text file which contains process counts and a list of programs.

Line format is as follows:

-h [host i] -np [n] $ALTAIR_HOME/hwsolvers/optistruct/bin/linux64/optistruct_spmd [INPUTDECK] -mpimode

Example: 4 CPU job on 2 dual-CPU hosts (the two machines are named: c1 and c2)

[optistruct@host1~]$ cat appfile

-h c1 –np 2 $ALTAIR_HOME/hwsolvers/optistruct/bin/linux64/optistruct_spmd [INPUTDECK] -mpimode

-h c2 –np 2 $ALTAIR_HOME/hwsolvers/optistruct/bin/linux64/optistruct_spmd [INPUTDECK] –mpimode

On a Linux cluster (for Intel-MPI)



[optistruct@host1~]$ mpirun –f [hostfile] -np [n] $ALTAIR_HOME/hwsolvers/optistruct/bin/linux64/optistruct_spmd [INPUTDECK] [OS_ARGS] -mpimode

Where,

[hostfile]: is a text file which contains the host names.


[host i]

Note: One host needs only one line.


[optistruct@host1~]$ cat hostfile

c1

c2

How to Launch OptiStruct/RADIOSS SPMD on Windows Machinesusing Direct calls to Executable

On a single host (for HP-MPI and Platform-MPI)

[optistruct@host1~]$ mpirun -np [n]

$ALTAIR_HOME/hwsolvers/optistruct/bin/win64/optistruct_spmd [INPUTDECK][OS_ARGS] -mpimode

Where,

[PLATFORM] is the platform that the SPMD module is run on.

optistruct_spmd is the OptiStruct SPMD binary




On a single host (for Intel-MPI and MS-MPI)

[optistruct@host1~]$ mpiexec -np [n]

$ALTAIR_HOME/hwsolvers/optistruct/bin/win64/optistruct_spmd [INPUTDECK][OS_ARGS] -mpimode

Where,



[PLATFORM] is the platform that the SPMD module is run on.

optistruct_spmd is the OS SPMD binary




On Multiple Windows Hosts (for HP-MPI and Platform-MPI)

[optistruct@host1~]$ mpirun –f [appfile]

Where,

[appfile]: is a text file which contains process counts and a list of programs.


-h [host i] -np [n] $ALTAIR_HOME\optistruct [INPUTDECK] -mpimode


[optistruct@host1~]$ cat appfile

-h c1 –np 2 $ALTAIR_HOME/hwsolvers/optistruct/bin/win64/optistruct_spmd [INPUTDECK] -mpimode

-h c2 –np 2 $ALTAIR_HOME/hwsolvers/optistruct/bin/win64/optistruct_spmd [INPUTDECK] –mpimode

On Multiple Windows Hosts (for Intel-MPI and MS-MPI)

[optistruct@host1~]$ mpiexec –configfile [config_file]

Where,

[config_file]: is a text file which contains the command for each host. Line format is asfollows:

-host [host i] –n [np] $ALTAIR_HOME/hwsolvers/optistruct/bin/win64/optistruct_spmd [INPUTDECK] -mpimode

Note: One host needs only one line.


[optistruct@host1~]$ cat hostfile

-host c1 –n 2 $ALTAIR_HOME/hwsolvers/optistruct/bin/win64/optistruct_spmd [INPUTDECK] -mpimode

-host c2 –n 2 $ALTAIR_HOME/hwsolvers/optistruct/bin/win64/optistruct_spmd



[INPUTDECK] –mpimode



HWSolver Configuration File

HWSolver configuration files may be used to establish default settings for the HyperWorkssolvers. Currently, only input with Bulk Data Format is supported. Configuration files must benamed hwsolver.cfg. Any errors or warnings caused by the content of these files will beechoed to the screen only.

Any default setting will be ignored if it is defined by an environment variable (see RunningOptiStruct), bulk data input deck entry (see The Input File), or command line argument (seeRun Options for OptiStruct).

File Location

The configuration file allows default settings to be established on five different levels:

1. System level

If a configuration file is located in the ${ALTAIR_HOME}/hwsolvers directory, the

default settings defined in this file apply to all solver runs.

A configuration file is included in the installation at this location. This configuration filecontains all of the configuration file options in a comment format. Uncommenting anoption will activate it. This file may be used as a template for all configuration files.

If the ${ALTAIR_HOME} environment variable is not set, then the configuration file at

this location will not be used. This variable is automatically set when therecommended HyperWorks installation and execution procedures are followed.

2. Corporate level

If a configuration file is located in the ${HW_CORPORATE_CUSTOMIZATION_DIR}

directory, the default settings defined in this file are added to the system defaults.

If a default setting, which was defined at the system level, is redefined at this level,the redefined setting is used.

If the ${HW_CORPORATE_CUSTOMIZATION_DIR} environment variable is not set, then

the configuration file at this location will not be used.

3. Group level

If a configuration file is located in the ${HW_GROUP_CUSTOMIZATION_DIR} directory,

the default settings defined in this file are added to the system and corporatedefaults.

If a default setting, which was defined at the system or corporate level, is redefinedat this level, the redefined setting is used.

If the ${HW_GROUP_CUSTOMIZATION_DIR} environment variable is not set, then the

configuration file at this location will not be used.

4. User level

If a configuration file is located in the ${HOME} directory, the default settings defined

in this file are added to the system, corporate and group defaults.



If a default setting, which was defined at the system, corporate or group level, isredefined at this level, the redefined setting is used.

If the ${HOME} environment variable is not set, then the configuration file at this

location will not be used. This variable is normally set for UNIX and Linux operatingsystems to point to a user's home directory, but it may vary for different versions ofthe Windows operating system.

5. Local level

If a configuration file is located in the current directory, the default settings defined inthis file are added to the system, corporate, group and user defaults.

If a default setting, which was defined at the system, corporate, group or user level,is redefined at this level, the redefined setting is used.

Entries and Format

Below is a list of entries recognized in the configuration file. As stated above, theconfiguration file contained in the installation may be used as a template for otherconfiguration files.

The format of the entries (with the exception of ELEMQUAL) is similar to the format of the I/O

Options in the input deck, namely:

ENTRY = Argument

Comments may be inserted using the $ character; which indicates that everything which

follows on that line is a comment.

ELEMQUAL is a recognized entry in the configuration file. It is used as described in the bulk

data entry description ELEMQUAL with the condition that it must be written in free format(see Guidelines for Bulk Data Entries).

Entry Argument Description

DOS_DRIVE_$ Path Same as DOS_DRIVE_$ environment

variable (see Running OptiStruct).

SYNTAX <ALLOWINT, STRICT> Same as SYNTAX setting on the I/O

option SYSSETTING.

SPSYNTAX <STRICT, CHECK,

MIXED>

Same as SPSYNTAX setting on the I/O

option SYSSETTING.

CORE <OFF, AUTO, IN, OUT,

MMIN>

Same as the –core run option in the

Running RADIOSS section.




SAVEFILE <ALL, OUT, NONE> Same as SAVEFILE setting on the I/O

option SYSSETTING.

RAMDISK Integer Same as RAMDISK setting on the I/O

option SYSSETTING.

SKIP10FIELD <CHECK, WARN> Same as SKIP10FIELD setting on the I/

O option SYSSETTING.

CARDLENGTH Integer Same as CARDLENGTH setting on the I/O

option SYSSETTING.

TABSTOPS Integer Used to change the default tab lengthfrom 8 to another value.

MAXLEN Integer Used to define the initial memoryallocation in MB. The default is 10% of OS_RAM.

MENLEN Integer Used to define the maximum allowableamount of memory to be used in MB. There is no default.

BUFFSIZE IntegerDefault = 16832

The maximum size in 8 byte words ofthe records of data written to the .op2

file. Use -1 to turn off buffering.

MSGLMT Various See MSGLMT setting on the I/O options

section MSGLMT.

ASSIGN, UPDATE,filename

Various See ASSIGN in the I/O options section

LOADTEMP <SHAREID> Same as LOADTEMP setting on the I/O

option SYSSETTING.

OS_RAM RAM in Mbytes Same as SYSSETTING option OS_RAM.

OS_RAM_INIT RAM in Mbytes Same as SYSSETTING option

OS_RAM_INT.




PLOTELID <UNIQUE, ALLOWFIX> Same as SYSSETTING option PLOTELID.

RAM_SAFETY_FACTOR Multiplier Same as -rsf option for running from

the script (see Run Options).

FORMAT <HM, H3D, ASCII,

OPTI, OS, NASTRAN,

PUNCH, O2, OUT2,

OUTPUT2, PATRAN,

APATRAN>

Same as I/O option FORMAT .

SCREEN <OUT, LOG, NONE> Same as I/O option SCREEN .

TMPDIR Path Same as I/O option TMPDIR .

SCRFMODE <BASIC, BUFFERED,

UNBUFFER, STRIPE,

MIXFCIO>

Same as SCRFMODE setting on the I/O

option SYSSETTING.

CHECKEL <YES, NO, FULL> Same as CHECKEL option for bulk data

entry PARAM.

OutputDefault <AUTO, NONE> The OutputDefault entry allows

default outputs to be disabled. Thisentry controls output for subcases forwhich there is no output requested.

AUTO: Output is automatically generated

for certain solution sequences.

NONE: No output that is not specifically

requested is output.

CHECKMAT <YES, NO, FULL> Same as CHECKMAT option for bulk data

entry PARAM.

COUPMASS <-1, 0, 1, YES, NO> Same as COUPMASS option for bulk data

entry PARAM.

EFFMASS Integer Same as EFFMASS option for bulk data

entry PARAM.




PRGPST <YES, NO> Same as PRGPST option for bulk data

entry PARAM.

KGRGD <YES, NO> Same as KGRGD option for bulk data

entry PARAM.

WTMASS Real > 0.0 Same as WTMASS option for bulk data

entry PARAM.

MBDH3D <NODAL, MODAL,

BOTH, NONE>

Same as MBDH3D option for bulk data

entry PARAM.

FLEXH3D <AUTO, YES, NO> Same as FLEXH3D option for bulk data

entry PARAM.



Expanded Error Message File

Please see Expanded Error Message File in the RADIOSS User's Guide.



Memory Limitations

32-bit Versus 64-bit Computations

On 32-bit operating systems, OptiStruct can, in most cases, make use of no more than 2.0 GBof memory (depending on the operating system and other software installed on the system,this limit is often lower). The typical memory limit for Windows 32-bit operating system is 1.9GB. Note that use of 64-bit hardware (EM64T or AMD64: 64-bit capable CPU) does not affectthis limit. Currently, the only versions of OptiStruct supported on 32-bit hardware arecompiled for Windows and Linux.

On certain 32-bit operating systems, OptiStruct can benefit in a limited way from additionalphysical memory (above 2GB RAM) in a number of ways:

1. The operating system and other programs running alongside OptiStruct (HyperMesh, forexample) can make use of the additional memory.

2. OptiStruct requires that its memory be allocated in one continuous chunk. The mainexception is when it launches a bandwidth minimizer which requires an additional smallamount of memory. On Windows platforms, the additional memory used by the bandwidthminimizer may be allocated above the 2GB limit, thus allowing the solution of slightly biggerjobs. In order to allow this, Windows must be started with the "/3GB" switch in the 'boot.

ini' file. This option is available on most versions of Windows 2000, and Windows XP

(requires SP2 on XP), but it is not 'officially' documented by Microsoft because on somehardware configurations it can cause unexpected crashes. In any case, the /3GB switchdoes not increase the amount of continuous memory allocation, thus it only marginallyhelps in solving large problems.

3. A 32-bit Linux operating system is the only one which allows more than 2GB runs. Onmost current versions of Linux, OptiStruct can allocate up to 4GB for each task (ifavailable). In general, if you use kernal 2.4 or older, you should upgrade to a newerversion of Linux as it does not allow to use all 4GB memory. Consult your Linuxdocumentation to find what specific options may be available. Linux command ‘uname -

a’ will show the kernal version.

On 64-bit machines, when using a 64-bit compiled version of OptiStruct and a 64-bitoperating system, OptiStruct can use all memory available in the system (real RAM and virtualmemory). There are still size limitations resulting from the limited size of standard integervariables, but they should happen in very limited cases. Currently OptiStruct is built with twoversions of linear solvers: one using 32-bit integers, and the other one using 64-bit integers. By default, the 32-bit solver is used (as it requires less memory/disk and runs visibly faster),but the 64-bit solver is automatically selected when the size of a problem requires it.

OptiStruct automatically takes into account all internal memory limits and tries to use as muchmemory as possible if it makes sense. When printing estimates for the memory usage,OptiStruct does not show results if they exceed these limits (irrespective of the amount ofmemory actually available – thus, if there is no estimate for memory requirements for in-coresolution, this means that OptiStruct will not be able to solve in-core, even on a computerwith unlimited memory).



Virtual Versus Physical Memory

OptiStruct can use more memory than is actually installed on a given system (i.e. more thanthe installed RAM). This is what the virtual memory (swap space) is for. OptiStruct is moreefficient, however, if it uses only actual RAM (remember to allow some RAM to be used by theoperating system and other codes running at the same time). When more memory isrequested than actual available RAM, OptiStruct will run much slower due to swapping. Youwill hear disks working constantly with little CPU being used, and there will be a significantdifference between the elapsed time and the CPU time.

Memory specification for OptiStruct (using –len command line option) is actually only giving

OptiStruct a hint about the amount of physical RAM available for the run (i.e. it should specifythe amount of physical memory not used by the operating system and other runningprograms, and as explained above, always less than the total amount of RAM in thecomputer). Based on this information, OptiStruct will automatically determine if an in-core runis desirable and switch to out-of-core, or even minimum-core mode, if needed. Note that ifminimum-core run requires more memory than indicated by the -len option, then OptiStruct

will not be limited by this value and will try to use as much memory as needed. In otherwords, it is incorrect to overestimate memory size for -len option; it will not help run larger

jobs, and may actually slow down turnaround time because of excessive swapping.

On most machines OptiStruct asks operating systems for information about available memory.This information is printed in the header of the .out file, and can be used to issue a warning

when it is possible that the run may fail because of lack of this resource. This information isdynamic (changes with other programs running at the machine) and therefore is never usedinside OptiStruct – user supplied information (e.g. with –len argument or from the config file)

is used instead.

Automatic Memory Allocation Versus Fixed Memory Runs

In standard modes of operation, OptiStruct automatically estimates the amount of memoryrequired, and this memory is requested in successive steps from the operating system. Sometimes the memory could be used more efficiently if requested at once and not inincrements (especially on 32-bit operating systems). This can be done using the "-fixlen"

command line option (see Run Options). When using the "-fixlen" option, OptiStruct may

start to run, but fail after some time with a memory allocation error. This can happen whenalmost all available memory is requested by the "-fixlen" argument, because in addition to

the memory required by the OptiStruct solver, OptiStruct launches a bandwidth minimizer,which uses an additional small amount of memory. Requesting slightly less memory with the "-

fixlen" option is a possible solution. Again, we see that it is incorrect to assign too much

memory to OptiStruct.

Additional Control of Used Memory



OptiStruct has new command line argument "-maxlen" which can be used in automatic mode.

This switch can be useful in some batch scheduling installations, as it will not allowOptiStruct to use more than a given amount of memory. Note that for models which requiremore memory than allowed by this argument, OptiStruct will abort during the solution,potentially after spending some time in computations.



Restarting OptiStruct

It is possible to restart an OptiStruct optimization by using the command line option -restart(see Run Options), by adding the I/O option RESTART to the input file, or from the OptiStructpanel in HyperMesh.

To restart an optimization, you will need information about the final iteration of the previousoptimization run. This information is stored in the .sh file.

The DESMAX entry on the DOPTPRM card, in the .fem file, specifies the maximum number of

additional iterations. To perform an analysis on the optimized structure, restart with DESMAX

set to 0. If DESMAX is not defined, then the default value of DESMAX is assumed (30 iterations

is the default value for DESMAX unless topology manufacturing constraints are used, in which

case the default is 80 iterations).

There are a number of conditions that must be observed when restarting an optimization:

The number of design variables or design elements cannot be changed.

It is invalid to restart with minimum member size control removed if it was present inthe original run.

It is invalid to restart with checkerboard control turned on if it was not activated inthe original run. It is, however, acceptable to deactivate checkerboard control in therestart if it was activated in the original run.

It is invalid to restart with manufacturing constraints that differ from those of theoriginal run.

The purpose of the restart functionality is for restarting with unconverged optimization runs oroptimization runs that were terminated before completion (due to a power outage, etc.).

Output files from a restart run are appended with the extension _rst#, where # is a 3 digit

number indicating the starting iteration for the restart run. For example, filename_rst030.

out is the .out file created when restarting filename.fem from iteration 30.

Iterations for the restart are numbered starting with the iteration number in the .sh file (the

last iteration from the previous run).

You may manually append new .dens, .disp, and .strs files to old ones and post-process

the combined files.



OptiStruct Compression Run

The OptiStruct solver can be used as a simple input deck preprocessor, intended to reduceout matching material and property definitions. It is useful for models (produced on somesimpleminded mesh generator tools) which sometimes have unique property IDs and uniquematerial IDs assigned to each element, even in cases where actual data is identical. Suchmodels (although technically correct) will be expensive to solve, or slow to process inHyperMesh or HyperView. In this special run mode, OptiStruct will simply read the model,compare all material and property data, and remove redundant data.

Example of Run

optistruct infile.fem -compress

will produce a new bulk data file named:

infile.echo

which will contain a new model with all duplicate materials and properties deleted. Allreferences to removed data will be replaced with the remaining ones, so for all practicalpurposes the model should yield identical results.

The additional argument to -compress represents the tolerance value in percent. All floating

point values in material and property data are compared using that tolerance. Using tolerancemay increase significantly run time.

Restrictions

1. Comparison is performed exactly (meaning all data are compared without allowing for anytolerance or round-off). If optional tolerance value is specified, then the run is performedin two passes: exact matches are removed first, then all remaining materials andproperties are compared with each other using following formula:

(2 * abs(value1-value2)) / (abs(value1)+abs(value2)) < tolerance *0.01.

2. Optimization data, nonlinear data, and thermal materials are not processed. If such dataare present they may reference removed entities, but a compress run will not adjustreferences. The resulting file (<filename>.echo) may not be valid.

3. Cards which extend or modify Materials or Properties (such as MATT1, MATX02, MATS1, orPSHELLX) are not used in comparison, and can also be left orphaned as a result of acompress run.

4. SETs referencing Materials or Properties are not processed. This will not result in a baddeck because SETs are allowed to reference non-existent IDs, however SETs in theoutput file may be different from the input file.

5. After the .echo file is produced, OptiStruct terminates the run, therefore -compress

cannot be combined with any other option.

The following cards are processed:



1. Properties

PBEAM, PBAR, PBEAML, PBARL, PSOLID (also fluid), PCOMP, PSHELL, PROD, PWELD, andPSHEAR,

as well as properties not referring to materials:

PELAS, PBUSH, PVISC, PDAMP, PGAP, PCONT, PAABSF, and PACABS

2. Materials

MAT1, MAT2, MAT4, MAT5, MAT8, MAT9, and MAT10

3. All elements referencing properties or materials, including PRBODY and PFBODY.

Any other cards present in the deck are allowed only if they do not reference materials orproperties; however OptiStruct does not verify this assumption. If such card is present in thedeck (e.g. DTPG referring to a list of properties), it may be printed with negative IDs forremoved entities.

The resulting file (<filename>.echo) is produced using the same routines which produce

ECHO; all restrictions present for ECHO will affect a -compress run; in particular:

Some optimization cards are currently known to produce incorrect ECHO, meaning anECHO of these cards cannot be read back into OptiStruct.

Results are formatted in fixed format, irrespective of the format used in the input file. This limits the accuracy of most coefficients because of 8 character fields. Currentformatting preserves as many decimal places as possible within 8 characters, but forvalues which require an exponential form, it is sometimes possible to retain accuracyto only 3-4 decimal places. Exceptions: GRID and DMIG cards are printed in freeformat with accuracy to at least 10 decimal places.

Only bulk data is printed to .echo file (i.e. no i/o or control sections).

Some cards are not printed: in particular, PARAM and DOPTRM do not appear in ECHOfiles.



Design Optimization

The following features can be found in this section:

Optimization Problem

Responses


Free-size Optimization


Size Optimization

Shape Optimization


Manufacturing Constraints

Optimization of Arbitrary Beam Sections

Optimization of Composite Structures

Equivalent Static Load Method (ESLM)

Gradient-based Optimization Method

Global Search Option



Optimization Problem


Minimize Objective Function

Minmax Objective Function

System Identification



Minimize Objective Function

OptiStruct solves the following structural optimization problem:

The objective function f(x) and the functions g(x) in the constraint function are structuralresponses obtained from a finite element analysis. A constraint is considered active if it issatisfied exactly (g = 0); it is considered inactive if g < 0; it is considered violated if g > 0.

The selection of the vector of design variables x depends on the type of optimization beingperformed. In topology optimization, the design variables are element densities (see DesignVariables for Topology Optimization). In size optimization (including free-size), the designvariables are properties of structural elements (see Design Variables for Size Optimization). Intopography and shape (including free-shape) optimization, the design variables are thefactors in a linear combination of shape perturbations (see Design Variables for TopographyOptimization and Design Variables for Shape Optimization).

The objective function is defined using a DESOBJ entry in the subcase information section.DESOBJ references a response defined by either the DRESP1, DRESP2, or DRESP3 bulk dataentry. Depending on the type of response, DESOBJ is located inside or outside of a SUBCASE. The constraints are defined using a DESSUB or DESGLB entry in the subcase informationsection, depending on if the type of response is subcase related or global, respectively. DESSUB and DESGLB refer to DCONSTR or DCONADD bulk data entries. DCONSTR relates theconstraint value or bound to a response defined by DRESP1, DRESP2, or DRESP3.



Minmax Objective Function

The minmax optimization problem is given as:

The reference values can take different values for positive or negative objective functions. These problems are solved using the Beta-method. In this method, the problem istransformed into a regular optimization problem through the introduction of an additionaldesign variable such that:

The functions fi(x) and the functions g(x) in the constraint function are structural responsesobtained from a finite element analysis. A constraint is considered active if it is satisfiedexactly (g = 0); it is considered inactive if g < 0; it is considered violated if g > 0.


The objective function of a minmax problem is defined using MINMAX or MAXMIN statements inthe subcase information section. MINMAX or MAXMIN references a DOBJREF statement in thebulk data section, which again refers to a DRESP1, DRESP2, or DRESP3 response definition.The reference values are defined on the DOBJREF entry. The constraints are defined asstated above. The constraints are defined using a DESSUB or DESGLB entry in the subcaseinformation section, depending on if the type of response is subcase related or global,respectively. DESSUB and DESGLB refer to DCONSTR or DCONADD bulk data entries.DCONSTR relates the constraint value or bound to a response defined by DRESP1, DRESP2, orDRESP3.



System Identification

For system identification, OptiStruct solves the following two structural optimization problems:

or

The functions fi(x) and the functions g(x) in the constraint function are structural responses

obtained from a finite element analysis. A constraint is considered active if it is satisfiedexactly (g = 0); it is considered inactive if g < 0; it is considered violated if g > 0. The valuesT i are the target value for the particular response, Wi is a weighting factor.


The objective function is defined using a DESOBJ entry or a MINMAX, MAXMIN entry in thesubcase information section. DESOBJ, MINMAX, or MAXMIN reference a DSYSID entry thatdefines target values for responses defined by either a DRESP1, DRESP2, or DRESP3 bulk dataentry. The constraints are defined using a DESSUB or DESGLB entry in the subcaseinformation section, depending on if the type of response is subcase related or global,respectively. DESSUB and DESGLB refer to DCONSTR or DCONADD bulk data entries.DCONSTR relates the constraint value or bound to a response defined by DRESP1, DRESP2, orDRESP3.



Responses

The following responses can be found in this section:

Internal Responses

External Responses



Internal Responses

OptiStruct allows the use of numerous structural responses, calculated in a finite elementanalysis, or combinations of these responses to be used as objective and constraint functionsin a structural optimization.

Responses are defined using DRESP1 bulk data entries. Combinations of responses are definedusing either DRESP2 entries, which reference an equation defined by a DEQATN bulk dataentry, or DRESP3 entries, which make use of user-defined external routines identified by theLOADLIB I/O option. Responses are either global or subcase (loadstep, load case) related.The character of a response determines whether or not a constraint or objective referencingthat particular response needs to be referenced within a subcase.

Subcase Independent

Mass, Volume

Both are global responses that can be defined for the whole structure, for individual properties(components) and materials, or for groups of properties (components) and materials.

It is not recommended to use mass and volume as constraints or objectives in a topographyoptimization. Neither is very sensitive towards design modifications made in a topographyoptimization.

In order to constrain the mass or volume for a region containing a number of properties(components), the SUM function can be used to sum the mass or volume of the selectedproperties (components), otherwise, the constraint is assumed to apply to each individualproperty (component) within the region. Alternatively, a DRESP2 equation needs to bedefined to sum the mass or volume of these properties (components). This can be avoided byhaving all properties (components) use the same material and applying the mass or volumeconstraint to that material.

Fraction of Mass, Fraction of Design Volume

Both are global responses with values between 0.0 and 1.0. They describe a fraction of theinitial design space in a topology optimization. They can be defined for the whole structure,for individual properties (components) and materials, or for groups of properties (components)and materials.

The difference between the mass fraction and the volume fraction is that the mass fractionincludes the non-design mass in the fraction calculation, whereas the volume fraction onlyconsiders the design volume.

Formulation for volume fraction:Volume fraction = (total volume at current iteration – initial non-design volume)/initial designvolume

Formulation for mass fraction:Mass fraction = total mass at current iteration/initial total mass



If, in addition to the topology optimization, a size and shape optimization is performed, thereference value for the volume fraction (the initial design volume) is not altered by size andshape changes. This can, on occasion, lead to negative values for this response. Therefore,if size and shape optimization is involved, it is recommended to use the Volume responsesinstead of the Volume Fraction response.

In order to constrain the volume fraction (or mass fraction) for a region containing a numberof properties (components), a DRESP2 equation needs to be defined to sum the volume (ormass) fraction of these properties (components), otherwise, the constraint is assumed toapply to each individual property (component) within the region. This can be avoided byhaving all properties (components) use the same material and applying the volume fractionconstraint to that material.

These responses can only be applied to topology design domains. OptiStruct will terminatewith an error if this is not the case.

Center of Gravity

This is a global response that may be defined for the whole structure, for individual properties(components) and materials, or for groups of properties (components) and materials.

Moments of Inertia

This is a global response that may be defined for the whole structure, for individual properties(components) and materials, or for groups of properties (components) and materials.

Weighted Compliance

The weighted compliance is a method used to consider multiple subcases (loadsteps, loadcases) in a classical topology optimization. The response is the weighted sum of thecompliance of each individual subcase (loadstep, load case).

This is a global response that is defined for the whole structure.

Weighted Reciprocal Eigenvalue (Frequency)

The weighted reciprocal eigenvalue is a method to consider multiple frequencies in a classicaltopology optimization. The response is the weighted sum of the reciprocal eigenvalues ofeach individual mode considered in the optimization.

This is done so that increasing the frequencies of the lower modes will have a larger effect onthe objective function than increasing the frequencies of the higher modes. If thefrequencies of all modes were simply added together, OptiStruct would put more effort intoincreasing the higher modes than the lower modes. This is a global response that is defined



for the whole structure.

Combined Compliance Index

The combined compliance index is a method to consider multiple frequencies and staticsubcases (loadsteps, load cases) combined in a classical topology optimization. The index isdefined as follows:

This is a global response that is defined for the whole structure.

The normalization factor, NORM, is used for normalizing the contributions of compliances andeigenvalues. A typical structural compliance value is of the order of 1.0e4 to 1.0e6. However, a typical inverse eigenvalue is on the order of 1.0e-5. If NORM is not used, thelinear static compliance requirements dominate the solution.

The quantity NORM is typically computed using the formula:

where is the highest compliance value in all subcases (loadsteps, load cases) and is the lowest eigenvalue included in the index.

In a new design problem, the user may not have a close estimate for NORM. If this happens,OptiStruct automatically computes the NORM value based on compliances and eigenvaluescomputed in the first iteration step.

Von Mises Stress in a Topology or Free-Size Optimization

Von Mises stress constraints may be defined for topology and free-size optimization throughthe STRESS optional continuation line on the DTPL or the DSIZE card. There are a number ofrestrictions with this constraint:

The definition of stress constraints is limited to a single von Mises permissible stress. The phenomenon of singular topology is pronounced when different materials withdifferent permissible stresses exist in a structure. Singular topology refers to theproblem associated with the conditional nature of stress constraints, i.e. the stressconstraint of an element disappears when the element vanishes. This creates anotherproblem in that a huge number of reduced problems exist with solutions that cannotusually be found by a gradient-based optimizer in the full design space.

Stress constraints for a partial domain of the structure are not allowed because theyoften create an ill-posed optimization problem since elimination of the partial domainwould remove all stress constraints. Consequently, the stress constraint applies tothe entire model when active, including both design and non-design regions, and stressconstraint settings must be identical for all DSIZE and DTPL cards.

The capability has built-in intelligence to filter out artificial stress concentrationsaround point loads and point boundary conditions. Stress concentrations due to



boundary geometry are also filtered to some extent as they can be improved moreeffectively with local shape optimization.

Due to the large number of elements with active stress constraints, no element stressreport is given in the table of retained constraints in the .out file. The iterative

history of the stress state of the model can be viewed in HyperView or HyperMesh.

Stress constraints do not apply to 1-D elements.

Stress constraints may not be used when enforced displacements are present in themodel.

Bead Discreteness Fraction

This is a global response for topography design domains. This response indicates the amountof shape variation for one or more topography design domains. The response varies in therange 0.0 to 1.0 (0.0 < BEADFRAC < 1.0), where 0.0 indicates that no shape variation hasoccurred, and 1.0 indicates that the entire topography design domain has assumed themaximum allowed shape variation.

Subcase Dependent

Linear Static Analysis

Static Compliance

The compliance C is calculated using the following relationship:

The compliance is the strain energy of the structure and can be considered a reciprocalmeasure for the stiffness of the structure. It can be defined for the whole structure, forindividual properties (components) and materials, or for groups of properties (components)and materials. The compliance must be assigned to a linear static subcase (loadstep, loadcase).

In order to constrain the compliance for a region containing a number of properties(components), the SUM function can be used to sum the compliance of the selectedproperties (components), otherwise, the constraint is assumed to apply to each individualproperty (component) within the region. Alternatively, a DRESP2 equation needs to bedefined to sum the compliance of these properties (components). This can be avoided byhaving all properties (components) use the same material and applying the complianceconstraint to that material.



Static Displacement

Displacements are the result of a linear static analysis. Nodal displacements can be selectedas a response. They can be selected as vector components or as absolute measures. Theymust be assigned to a linear static subcase (loadstep, load case).

Static Stress of Homogeneous Material

Different stress types can be defined as responses. They are defined for components,properties, or elements. Element stresses are used, and constraint screening is applied. It isalso not possible to define static stress constraints in a topology design space (see above). This is a linear static subcase (loadstep, load case) related response.

Static Strain of Homogeneous Material

Different strain types can be defined as responses. They are defined for components,properties, or elements. Element strains are used, and constraint screening is applied. It isalso not possible to define strain constraints in a topology design space. This is a linearstatic subcase (loadstep, load case) related response.

Static Stress of Composite Lay-up

Different composite stress types can be defined as responses. They are defined for PCOMP(G) components or elements, or PLY type properties. Ply level results are used, andconstraint screening is applied. It is also not possible to define composite stress constraintsin a topology design space. This is a linear static subcase (loadstep, load case) relatedresponse.

Static Strain of Composite Lay-up

Different composite strain types can be defined as responses. They are defined for PCOMP(G)components or elements, or PLY type properties. Ply level results are used, and constraintscreening is applied. It is also not possible to define composite strain constraints in atopology design space. This is a linear static subcase (loadstep, load case) related response.

Static Failure in a Composite Lay-up

Different composite failure criterion can be defined as responses. They are defined for PCOMP(G) components or elements, or PLY type properties. Ply level results are used, andconstraint screening is applied. It is also not possible to define composite failure criterionconstraints in a topology design space. This is a linear static subcase (loadstep, load case)related response.

Static Force

Different force types can be defined as responses. They are defined for components,



properties, or elements. Constraint screening is applied. It is also not possible to define forceconstraints in a topology design space. This is a linear static subcase (loadstep, load case)related response.

Linear Heat Transfer Analysis

Temperature

Temperatures are the result of a heat transfer analysis, and must be assigned to a heattransfer subcase (loadstep, load case). Temperature response cannot be used in topology orfree-size optimization.

Normal Modes Analysis

Frequency

Natural frequencies are the result of a normal modes analysis, and must be assigned to thenormal modes subcase (loadstep, load case). It is recommended to constrain the frequencyfor several of the lower modes, not just of the first mode.

Linear Buckling Analysis

Buckling Factor

The buckling factor is the result of a buckling analysis, and must be assigned to a bucklingsubcase (loadstep, load case). A typical buckling constraint is a lower bound of 1.0,indicating that the structure is not to buckle with the given static load. It is recommended toconstrain the buckling factor for several of the lower modes, not just of the first mode.

Frequency Response Function (FRF Analysis)

Frequency Response Displacement

Displacements are the result of a frequency response analysis. Nodal displacements, i.e.translational, rotational and normal, can be selected as a response. They can be selected asvector components in real/imaginary or magnitude/phase form. They must be assigned to afrequency response subcase (loadstep, load case).



Frequency Response Velocity

Velocities are the result of a frequency response analysis. Nodal velocities, i.e. translational,rotational and normal, can be selected as a response. They can be selected as vectorcomponents in real/imaginary or magnitude/phase form. They must be assigned to afrequency response subcase (loadstep, load case).

Frequency Response Acceleration

Accelerations are the result of a frequency response analysis. Nodal accelerations, i.e.translational, rotational and normal, can be selected as a response. They can be selected asvector components in real/imaginary or magnitude/phase form. They must be assigned to afrequency response subcase (loadstep, load case).

Frequency Response Stress

Different stress types can be defined as responses. They are defined for components,properties, or elements. Element stresses are not used in real/imaginary or magnitude/phaseform, and constraint screening is applied. Von Mises stress for solids and shells can also bedefined as direct responses. It is not possible to define stress constraints in a topologydesign space. This is a frequency response subcase (loadstep, load case) related response.

Frequency Response Strain

Different strain types can be defined as responses. They are defined for components,properties, or elements. Element strains are used in real/imaginary or magnitude/phase form,and constraint screening is applied. Von Mises strain for solids and shells can also be definedas direct responses. It is not possible to define strain constraints in a topology design space. This is a frequency response subcase (loadstep, load case) related response.

Frequency Response Force

Different force types can be defined as responses. They are defined for components,properties, or elements in real/imaginary or magnitude/phase form. Constraint screening isapplied. It is also not possible to define force constraints in a topology design space. This isa frequency response subcase (loadstep, load case) related response.

Random Response Analysis

PSD and RMS Responses

PSD displacement, PSD velocity, PSD acceleration, PSD acoustic pressure, PSD stress, PSDstrain, RMS displacement, RMS velocity, RMS acceleration, RMS acoustic pressure, RMS stressand RMS strain responses are available.



Coupled FRF Analysis on a Fluid-structure Model (Acoustic Analysis)

Acoustic Pressure

Acoustic pressures are the result of a coupled frequency response analysis on a fluid-structure model. This response is available for fluid grids. It must be assigned to a coupledfrequency response subcase (loadstep, load case) on a fluid-structure model.

Multi-body Dynamics Analysis

Flexible Body Responses

For Multi-body Dynamics problems, the Mass, Center of gravity, and Moment of Inertia of oneor more flexible bodies are available as responses. This is in addition to other usual structuralresponses.

MBD Displacement

MBD displacements are the result of a multi-body dynamics analysis. They must be assignedto a multi-body dynamics subcase (loadstep, load case).

MBD Velocity

MBD velocities are the result of a multi-body dynamics analysis. They must be assigned to amulti-body dynamics subcase (loadstep, load case).

MBD Acceleration

MBD acceleration are the result of a multi-body dynamics analysis. They must be assigned toa multi-body dynamics subcase (loadstep, load case).

MBD Force

MBD forces are the result of a multi-body dynamics analysis. They must be assigned to amulti-body dynamics subcase (loadstep, load case).

MBD Expression

MBD expression responses are the result of a multi-body dynamics analysis. They are theresult of the evaluation of an expression. They must be assigned to a multi-body dynamics



subcase (loadstep, load case).

Fatigue

Life/Damage

Life and Damage are results of a fatigue analysis. They must be assigned to a Fatiguesubcase.

Dynamic/Nonlinear Analysis

Equivalent Plastic Strain

Equivalent plastic strain can be used as an internal response when a nonlinear responseoptimization is run using the equivalent static load method. This is made possible through theuse of an approximated correlation between linear strain and plastic strain, which arecalculated in the inner and outer loops respectively, of the ESL method.

User Responses

Function

A function response is one that uses a mathematical expression to combine design variables,grid point locations, responses, and/or table entries. Whether the function is subcase(loadstep, load case) related or global, is dependent on the response types used in theequation.

External

An external response is one that uses an external user-defined routine to combine designvariables, grid point locations, eigenvectors, responses, and/or table entries. Whether thefunction is subcase (loadstep, load case) related or global is dependent on the responsetypes used in the routine. Please refer to the External Responses section for moreinformation.



External Responses

The DRESP3 bulk data entry, in combination with the LOADLIB I/O option entry, allows for thedefinition of responses through user-defined external functions. The external functions maybe written in HyperMath Language (HML), FORTRAN or C. The resulting libraries should beaccessible by OptiStruct regardless of the coding language, providing that consistent functionprototyping is respected, and adequate compiling and linking options are used.

Writing External Functions

The OptiStruct installation provides "barebone" functions for FORTRAN (dresp3_barebone.F)

and for C (dresp3_barebone.c) with proper function definition, arguments, and compilation

directives. These files can be used as starting points to write your own functions.

HyperMath Language (HML) functions are defined as follows:

integer function myfunct(iparam, rparam, iresp, rresp, userdata)

If sensitivities need to be requested, then the following external function can be used.

integer function myfunct(iparam, rparam, iresp, rresp, dresp, isens

userdata)

FORTRAN functions are defined as follows:

integer function myfunct(iparam, rparam, nparam,

iresp, rresp, nresp,

userdata)


integer function myfunct(iparam, rparam, nparam,

iresp, rresp, dresp, nresp, isens

userdata)

character*32000 userdata

integer nparam, nresp

integer iparam(nparam), iresp(nresp)

double precision param(nparam), rresp(nresp), dresp(nparam,nresp)

C functions are defined as follows:

int myfunct(int* iparam, double* rparam, int* nparam,

int* iresp, double* rresp, int* nresp,

char* userdata)




int myfunct(int* iparam, double* rparam, int* nparam,

int* iresp, double* rresp, double* dresp, int* nresp, int*

isens, char* userdata)

Note that the functions' arguments are identical in both languages so as to preservecompatibility. However, since FORTRAN always passes arguments by address, it is importantto understand that external C functions receive pointers instead of variables.

In order to ensure portability, the following must be adhered to:

Function names should be written using either all lower-case or all upper-casecharacters.

Only alphanumeric characters should be used.

Underscore characters are prohibited.

Names cannot be longer than eight characters.

Regarding implementing of external, user-defined routines using HyperMath, please refer tothe online documentation for HyperMath for writing scripts. HyperMath is supported on theWindows and Linux operating systems only.

Function Return Values

External functions should return 0 or 1 for successful completion, where 1 indicates that auser-defined information message should be output by OptiStruct. External functions shouldreturn -1 in case of fatal error, in which case OptiStruct will terminate after outputting auser-defined error message. See below for more information about error and informationmessages.

Function Arguments

The following table briefly describes the arguments which are passed from OptiStruct to theexternal functions.

Argument Type Input / Output Description

iparam integer (table) Input Input parameters types (optional use)

rparam double (table) Input Input parameters values

nparam integer Input Number of parameters

iresp integer (table) Input Output responses requests (optional use)

rresp double (table) Output Output responses values



Argument Type Input / Output Description

dresp double (table) Output Output sensitivity values

nresp integer Input Number of responses

isens integer Input Sensitivity output flag

userdata string Input / Output User data / Error or informationmessage

Parameters:

nparam is the number of input parameters that were defined on the DRESP3 card.

rparam(nparam) contains the values of the input parameters as evaluated byOptiStruct.

iparam(nparam) indicates the types of the input parameters as described below.

Parameter values are passed in the exact order in which they were defined on theDRESP3 card, regardless of their type. Using the parameter types table is optional, forinstance to perform verifications or code-branching.

The following types are currently supported:

Parameter type iparam value

DESVAR 1

DTABLE 2

DGRID/DGRIDB 3

DRESP1 4

DRESP2 5

DRESP1L 6

DRESP2L 7

DVPREL1 8

DVPREL2 9

DVMREL1 10

DVMREL2 11

DVCREL1 12

DVCREL2 13

DVMBRL1 14



Parameter type iparam value

DVMBRL2 15

DEIGV 16

DGRIDB 18

Responses:

nresp is the maximum number of responses which the function is able to compute, asdefined on the MAXRESP field of the DRESP3 card.

rresp(nresp) returns the values of the responses as evaluated by the externalfunction.

iresp(nresp) contains the responses requests as described below.

The responses requests table indicates which of the available responses are actuallyneeded by OptiStruct. Entries in iresp(nresp) are flagged as 1 for requestedresponses and as 0 otherwise. Using that information is optional, and allows for savingcomputational effort by not evaluating responses which OptiStruct does not need.

Userdata String

Upon entering the function, the userdata string contains data as defined in the USRDATA fieldof the DRESP3 card. It provides a convenient mechanism to pass constants or any otherrelevant information to the function. There are no restrictions regarding the contents of thestring, but its length must be 32,000 characters at most.

Upon exiting the function, the string may contain a user-defined error or information message. The updated string is then returned to OptiStruct, where it is printed to the standard output(.out file and/or screen). Here again, the contents of the string are not restricted as long as

its length does not exceed 32,000 characters.

The error or information messages may be formatted by using the character "|" as a line-break indicator. Standard C escape sequences are supported as well. It is advised, but notnecessary, to format messages in such a way that each line does not exceed 80 characters,since the same convention is used in OptiStruct's output files.

Sensitivity Flag

isens indicates whether sensitivities are requested in the code. It is recommended toskip the calculation of sensitivities when isens is turned off. This will avoidunnecessary computations.

Building External Libraries



Windows Systems with Microsoft Developer Studio

Creating dynamic libraries under Windows with Microsoft Developer Studio is an extremelyeasy task. Simply start a new project and select either "FORT RAN Dynam ic Link Library" forFORTRAN, or "Win32 Dynam ic -Link Library" for C.

For FORTRAN libraries, you need to change the argument passing conventions in the projectsettings. Under the "FORTRAN" tab, select the category "External Procedures" and thenchange the "Argument Passing Conventions" to "C, By Reference".

On Windows systems, %PATH% must be set correctly to ensure that the right compiler DLLs are

picked up at runtime. Note that 32-bit dll works with 32-bit OptiStruct and 64-bit dll workswith 64-bit OptiStruct.

UNIX Systems

Under UNIX, the general syntax to build a shared library starting from a FORTRAN or C file is:

FC [options] -c myfile.F -o myfile.o (for FORTRAN)

CC [options] -c myfile.c -o myfile.o (for C)

LD [options] myfile.o -o mylib.so

where FC refers to the FORTRAN compiler (for instance f77), CC refers to the C compiler (forinstance c c or gc c), and LD refers to the linker (for instance ld) installed on your computer.Please refer to your system's manuals for more information.

The compiler and linker options provide information about the platform you are building thelibrary for. For instance, they may indicate whether the library should be built in 64-bits or32-bits mode. The linker options also specify that you are building a shared library. Otheroptions, such as code optimization parameters, are left to your discretion and should notusually affect the compatibility with OptiStruct.

The following table defines options for each of OptiStruct's release platforms, which havebeen verified to work correctly on various systems. Keep in mind that these options mightchange depending on the compilers and linker installed on your computer, so please refer toyour operating system manual for further information. In most cases GNU compilers can beused in place of Intel compilers. Please use the appropriate compiler linker options to create ashared library with the compiler of your choice. The compilers and versions in the followingtable are the ones used to build OptiStruct.



Platform

FORTRANCompiler Version

FORTRANCompiler Options

C Compiler Version

CCompiler Options

Linker Options

Win32 Intel FORTRAN10.0.027

/iface:default /libs:dll /threads

Visual Studio2005

/MD /dll

Win64 Intel FORTRAN10.0.027

/iface:default /libs:dll /threads

Visual Studio2005

/MD /dll

solarisx64

Sun FORTRAN95 8.3

-m64 -fPIC Sun C 5.9 -m64 -fPIC

-dy -G

aix64 IBM XL FORTRAN EnterpriseEdition V10.1

f77 -q64 -qxlf77=LEADZERO

IBM XL C/C++ EnterpriseEdition V8.0

cc -q64 -b64 -bshared -bexpall -G

linux32 Intel FORTRAN 10.0 20070809

-fPIC Intel C++ 10.0. 20070809

-fPIC -shared

linuxia64

Intel FORTRAN 10.0.026

-ftz –fPIC Intel C++compiler10.0.026

-ftz -fPIC -shared

linux64 Intel FORTRAN 10.0 20070809

–fPIC Intel C++ 10.0 20070809

-fPIC -shared

Once your library has been built, you can verify that the functions have been exportedcorrectly by using nm mylib.so on UNIX systems and dumpbin /exports mylib.dll on

Windows systems with Microsoft Developer Studio. This command will display the list ofsymbols found in the library, among which you should recognize the function(s) which youhave written.

Note that some FORTRAN compilers convert function names to lower-case or upper-casesymbols, and some compilers also append an underscore to these names. However, in yourinput decks, you do not have to worry about the exact symbol name. Simply use the functionname as it is defined in your code, and OptiStruct will automatically locate the appropriatesymbol.

Using External Libraries

All files referenced here are located in the HyperWorks installation directory under<install_directory>/demos/os/manual/.

To locate the HyperWorks installation directory, <install_directory>, use the following



approach:

From the permanent menu, select the global panel and review the path next to template file:<install_directory> is the portion of the path preceding the templates/ directory on PC,

and the hm/ directory on UNIX.

HyperMath Example

Please refer to the OptiStruct tutorial, OS-4095 (Size Optimization using External Responses(DRESP3) through HyperMath), for information on using DRESP3 with HyperMath to implementexternal, user-defined routines.

Simple Example

The files dresp3_simple.F and dresp3_simple.c contain source code for simple examples of

external functions written in FORTRAN and C, respectively. Both functions are named m ysumand compute two responses – the sum of the parameters and the averaged sum of theparameters.

The input deck dresp3_simple.fem contains an example problem calling both of these

external functions. We have defined two LOADLIB cards referring to the FORTRANand Clibraries:

LOADLIB DRESP3 FLIB dresp3_simple_f.dllLOADLIB DRESP3 CLIB dresp3_simple_c.dll

We have created four DRESP3 cards, which are pointing to the FORTRAN and C functions andrequesting the first and second responses in each of those functions. Two DRESP1 responsesare used as parameters:

DRESP3 6 SUMF FLIB MYSUM 1 2+ DRESP1 2 3DRESP3 7 AVGF FLIB MYSUM 2 2+ DRESP1 2 3DRESP3 8 SUMC CLIB MYSUM 1 2+ DRESP1 2 3DRESP3 9 AVGC CLIB MYSUM 2 2+ DRESP1 2 3

For verification purposes, we have also defined two DRESP2 cards that are pointing to twosimple equations which evaluate the sum and the averaged sum of their parameters:

DEQATN 1 F(x,y) = x+yDEQATN 2 F(x,y) = avg(x,y)

DRESP2 4 SUME 1+ DRESP1 2 3DRESP2 5 AVGE 2+ DRESP1 2 3

Running this input deck through OptiStruct shows that the FORTRAN external functions, the C



external functions and the internal equations always return the same values, and are updatedsimultaneously throughout the optimization process.

Advanced Example

The file dresp3_advanced.F contains the FORTRAN source code of the second example, in

which we are making use of advanced features of the DRESP3 functionality.

The external function is able to compute the von Mises and maximum principal stresses(strains) of an element based on its stress (strains) components. Either 3 or 6 componentscan be passed as parameters – 3 components for a shell element and 6 components for asolid element. The following features are used:

The USRDATA string is parsed to determine whether stresses or strains are requested,and an error message is returned otherwise.

The number of parameters is used to determine whether a shell or solid element istreated, and an error message is returned if that number is not equal to 3 or 6.

An error message is returned if the parameters are not of type DRESP1 or DRESP1L,since stress or strain components are expected.

Even though the function is able to compute two different responses, only theresponse(s) actually requested by OptiStruct are computed when the function iscalled.

An information message is returned indicating which responses were evaluated.

The input deck dresp3_advanced.fem gives a simple example of problem making use of this

external function, for analysis only.

The DRESP1 responses 10-12 and 13-18 correspond to the stress components of a 2-D and a3-D element, respectively. The DRESP1 responses 20-23 evaluate the von Mises stress andthe maximum principal stress of the same two elements:

DRESP1 10 SXX2D STRESS ELEM SX1 100DRESP1 11 SYY2D STRESS ELEM SY1 100DRESP1 12 SXY2D STRESS ELEM SXY1 100DRESP1 13 SXX3D STRESS ELEM SXX 50DRESP1 14 SYY3D STRESS ELEM SYY 50DRESP1 15 SZZ3D STRESS ELEM SZZ 50DRESP1 16 SXY3D STRESS ELEM SXY 50DRESP1 17 SXZ3D STRESS ELEM SXZ 50DRESP1 18 SYZ3D STRESS ELEM SYZ 50

DRESP1 20 SVM2D-1 STRESS ELEM SVM1 100DRESP1 21 SMP2D-1 STRESS ELEM SMP1 100DRESP1 22 SVM3D-1 STRESS ELEM SVM 50DRESP1 23 SMP3D-1 STRESS ELEM SMP 50

In addition, we have defined DRESP3 cards which compute the same stress results throughour external library. We are also using the SLAVE feature to clone the parameters of similarcards:



DRESP3 30 SVM2D-3 STRLIB GETSTR 1 2+ DRESP1 10 11 12 + USRDATA STRESSDRESP3 31 SMP2D-3 STRLIB GETSTR 2 2+ SLAVE 30DRESP3 32 SVM3D-3 STRLIB GETSTR 1 2+ DRESP1 13 14 15 16 17 18+ USRDATA STRESSDRESP3 33 SMP3D-3 STRLIB GETSTR 2 2+ SLAVE 32




Topology Optimization is a mathematical technique that produces an optimized shape andmaterial distribution for a structure within a given package space. By discretizing the domaininto a finite element mesh, OptiStruct calculates material properties for each element. TheOptiStruct algorithm alters the material distribution to optimize the user-defined objectiveunder given constraints. Convergence occurs in line with the description provided on the Iterative Solution page.

The following responses (see Responses for a description) are currently available as theobjective or as constraint functions:

Mass Volume Volume or Mass Fraction

Center of Gravity Moment of Inertia Static Compliance

Static Displacement Natural Frequency Von Mises Stress on EntireModel (only as constraint)

Buckling Factor (specialcase)

Frequency ResponseDisplacement,Velocity, Acceleration

Temperature

Weighted Compliance Weighted Frequency Combined ComplianceIndex

Function




The capability has built-in intelligence to filter out artificial stress concentrationsaround point loads and point boundary conditions. Stress concentrations due toboundary geometry are also filtered to some extent as they can be improved more



effectively with local shape optimization.





The buckling factor can be constrained for shell topology optimization problems with a basethickness not equal to zero. Constraints on the buckling factor are not allowed in any othercases of topology optimization.

The following responses are currently available as the objective or as constraint functions forelements that do not form part of the design space:

Static Stress Static Strain Static Force

Composite Stress Composite Strain Composite Failure Criterion

Frequency ResponseStress

Frequency ResponseStrain

Frequency Response Force

If an element is in the topology design region, its individual stress/strain or force criterionvalue cannot be constrained.



Generating and Evaluating a Design using TopologyOptimization

The following example illustrates how OptiStruct is used to generate a design for a controlarm and how engineering analysis is used to evaluate the design.

1. The package space for the control arm is filled with a finite element mesh.

2. Parts of the mesh are designated as nondesign, and the elements that make up theseareas are placed in a nondesign component.

The darker elements represent attachment points for the frame, shock, spring seat,stabilizer bar, and spindle. Nondesign elements are placed where loads and constraintsare applied to the model.

Nondesign and design space of FEM model.

3. Loads and constraints are applied to the finite element model.

Three load cases are applied at the spindle and stabilizer bar attachment points.

Constraints are applied at the frame connections and shock point.

The model and parameters are submitted to OptiStruct for topology optimization.

4. Elements with material densities below 60% are masked out.



Density plot of control arm with elements below 60% material density removed from the display.

5. A finite element model of the control arm using the suggested layout as a guide isgenerated.

Finite element model of control arm design based on OptiStruct results.

6. Stress analysis is performed on the model using the loads and boundary conditions fromthe topology optimization run.



Stress contour plot of control arm model during braking load.

7. The performance of the part is evaluated.

Subsequent size and shape optimization is performed to minimize the mass while meetingstress and deflection criteria.

Final design with shape-optimized structural member.



Design Variables for Topology Optimization

OptiStruct solves topological optimization problems using the density method, also known asthe SIMP method in the research community.

Under topology optimization, the material density of each element should take a value ofeither 0 or 1, defining the element as being either void or solid, respectively. Unfortunately,optimization of a large number of discrete variables is computationally prohibitive. Therefore,representation of the material distribution problem in terms of continuous variables has to beused.

With the density method, the material density of each element is directly used as the designvariable, and varies continuously between 0 and 1; these represent the state of void andsolid, respectively. Intermediate values of density represent fictitious material. The stiffnessof the material is assumed to be linearly dependent on the density. This material formulationis consistent with our understanding of common materials. For example, steel, which isdenser than aluminum, is stronger than aluminum. Following this logic, the representation offictitious material at intermediate densities does reflect engineering intuitions.

In general, the optimal solution of problems involves large gray areas of intermediate densitiesin the structural domain. Such solutions are not meaningful when we are looking for thetopology of a given material, and not meaningful when considering the use of differentmaterials within the design space. Therefore, techniques need to be introduced to penalizeintermediate densities and to force the final design to be represented by densities of 0 or 1for each element. The penalization technique used is the "power law representation ofelasticity properties," which can be expressed for any solid 3-D or 2-D element as follows:

where K and K represent the penalized and the real stiffness matrix of an element,respectively, is the density and p the penalization factor which is always greater than 1.

In OptiStruct, the DISCRETE parameter corresponds to (p - 1). DISCRETE can be defined on

the DOPTPRM bulk data entry. p usually takes a value between 2.0 and 4.0. For example,compared to the non-penalized formulation (which is equivalent to p=1) at =0.3, p=2reduces the stiffness of the element from 0.3 to 0.09 times the stiffness of the fully denseelement. The default DISCRETE is 1.0 for shell dominant structures, and 2.0 for solids

dominant structures (the dominance is defined by the proportion of number of elements). Anadditional parameter, DISCRT1D, can also be defined on the DOPTPRM bulk data entry.

DISCRT1D allows 1-D elements to use a different penalization to 2-D or 3-D elements.

When minimum member size control is used, the penalty starts at 2 and is increased to 3 forthe second and third iterative phases. This is done in order to achieve a more discretesolution. For other manufacturing constraints such as draw direction, extrustion, patternrepetition, and pattern grouping, the penalty starts at 2 and increases to 3 and 4 for thesecond and third iterative phases, respectively. Obviously, due to the existence of semi-dense elements, the analysis results may change dramatically when the design process entersa new phase using a different penalization factor.

Three types of finite elements can be defined as topology design elements in OptiStruct: Solidelements, shell elements, and 1-D elements (including ROD, BAR/BEAM, BUSH, and WELDelements).



Solid Elements

The SIMP method (Solid Isotropic Material with Penalty) is used in OptiStruct. In the SIMPmethod, a pseudo material density is the design variable, and hence it is often called densitym ethod as well. The material density varies continuously between 0 and 1, with 0representing void state and 1 solid state. The SIMP method applies a power-law penalizationfor stiffness-density relationship in order to push density toward 0/1 (void/solid) distribution:

Where,

K is the penalized stiffness matrix of an element.

K is the real stiffness matrix of an element.

is the density.

p is the penalization factor (Always greater than 1, with default penalty at 3.0 if no manufacturingconstraints are applied).



Shell Elements

The SIMP method (Solid Isotropic Material with Penalty) is used in OptiStruct. In the SIMPmethod, a pseudo material density is the design variable, and hence it is often called densitym ethod as well. The material density varies continuously between 0 and 1., with 0representing void state and 1 solid state. The SIMP method applies a power-law penalizationfor stiffness-density relationship in order to push density toward 0/1 (void/solid) distribution.

Where,

K is the penalized stiffness matrix of an element.

K is the real stiffness matrix of an element.

is the density.

p is the penalization factor (Always greater than 1)

For isotropic material a non-zero base plate thickness can be defined. For a composite plateor a plate with anisotropic material, the base plate thickness must be zero (the limitation ofthe current development).

Topology optimization of composites has certain unique characteristics and is discussed in Composite Topology and Free-size Optimization.



1-D Elements

Only the density method is implemented for topology optimization of 1-D elements. Currentlyavailable elements include ROD, BAR/BEAM, BUSH, and WELD elements. Each element iscontrolled by a single design variable that is the material density of this element that variesbetween 0 (numerically a small value is used) and 1.0. In essence, 0 represents nonexistenceand 1.0 represents full existence of the corresponding element. The following power lawrepresentation of elastic properties is used to penalize intermediate density:

where K and K represent the penalized and the real stiffness matrix of an element,respectively, p is the penalization factor which is always bigger than 1. The penalty iscontrolled by the DISCRETE or DISCRT1D parameters, the value of these parameters

correspond to (p - 1).



Topology Optimization (Level Set Method)

A new topology optimization algorithm based on the level set method has been implemented in

OptiStruct 12.0. In the level set method, the boundary of the design is implicitly represented

as the isosurface (the zero level set) of a function defined on the finite element mesh,as shown in Figure 1. In the level set model, the domain is defined based on the value of thelevel set function:

Where, D denotes the design domain; represents the material region, stands for theboundary, and D/ denotes the region with no material. The dynamic motion of the boundaryis governed by the so-called, level set equation:

Where, Vn is the normal velocity and is the norm of the gradient of the level set

function. The basic idea of the level set equation is to map the boundary evolution into an

evolution of the level set function .

Figure 1: A 2-D Design and its corresponding Level Set representation

Level-set based topology optimization can be considered as advanced shape optimization. Itworks in a way like conventional shape optimization, where the design is changed by movingthe boundary, while at the same time topological changes such as boundary emerging andsplitting can be handled naturally.



Composite Topology and Free-size Optimization

Input Definition

For composite structures, topology and free-size optimization are defined through the DTPLand DSIZE bulk data entries, respectively. Both are supported in the HyperMeshoptimization panel. Features available include: minimum member size control, symmetry,pattern grouping and pattern repetition. Stress or failure constraints are not supported atthis stage.

Prior to OptiStruct 8.0, composite topology optimization was based on the notion that thehomogenized properties of an element remain unchanged. This construct does not allow thefreedom for material redefinition. However, if this is indeed a preferred assumption, the HOMOoption can be set on the MAT line of the DTPL card. Otherwise, an individual ply-basedformulation (discussed below) will be the default option.

Topology and free-size methods target a system level composite design where laminate familydefinition is the objective. Therefore, the PCOMP model should not reflect a detailed stackingof plies of the same orientation. For example, even though 10 layers of 0 degree graphitecloth might be separated in the stacking of the final structure, the modeling for a conceptstudy using topology and free-size should group them together in one ply in the PCOMP sothat the optimal total thickness distribution of a 0 degree ply is optimized throughout thestructure.

Involving both topology and free-size in the same optimization problem is not recommendedsince the penalization on topology components creates a bias that could lead to sub-optimalsolutions.

Problem Formulation

For a composite shell element (shown in the figure below), the thickness ti of each ply is avariable between 0 and Ti defined on the PCOMP card.

Composite element

The only difference between topology and free-size here is that the former targets a discrete



final solution of 0 (or Ti) for ti, while free-size allows ti to vary freely between 0 and Ti. Thediscrete solution is achieved by penalizing intermediate thickness. Most generalcharacteristics of regular shell topology and free-size optimization also apply to composite. Itis recommended that users become familiar with free-size before proceeding. The majordifferences between topology optimization and free-size can be illustrated through a simpleexample.

Example: Cantilever Plate

The cantilever plate is shown in the following figure. A symmetric lay-up of (0, +45, -45, and90) degree plies are used. The optimization problem is stated as:

Minim ize ComplianceSubjec t to Volume fraction < 0.3

Composite cantilever plate

For topology optimization, the thickness distribution of individual plies in the final design isshown in the following figure.



Topology result – thickness of individual plies

The total thickness of the laminate is shown next.

Topology result – total thickness of the laminate

It can be seen that a rather discrete thickness for each ply is obtained. Note that while littleoverlapping of different orientations is shown in this result, it should be expected thatoverlapping of plies of different angles might be more pronounced when multiple load casesexist.

The thickness distribution of free-size optimization for this example is shown below.



Free-size result – thickness of individual plies

Free-size result – total thickness of the laminate

As expected, free-size created a design in which variable ply thickness appears in a large areaof the structure. The compliance of both designs are compared in the figure below. It is notsurprising to see that the free-size design outperforms the topology design in terms ofcompliance since a continuous variation of thickness offers more design freedom.



Compliance of topology and free-size results

While ply angles are not variables for topology and free-size optimization, thicknessoptimization of plies indirectly leads to a discrete optimization of angles. The available anglesin the PCOMP can be interpreted as discrete angle variables. Also, while free-size oftencreates variable thickness distribution without extensive cavity, it does not prevent cavity ifthe optimizer demands it. For this example, we can see cavity in the free-size results in the45 degree region, adjacent to the support, and in the upper and lower corners of the freeend.

Comparing Design Characteristics of Topology and Free-size

Most of the characteristics for general shell discussed in free-size section also apply tocomposite structures. One important exception is that the manufacturing cost is no longer arestrictive factor for composites. The reason for this is that a composite structure ismanufactured by laying very thin plies of fiber cloth over each other and binding them with amatrix material, like epoxy resin. Therefore, an almost continuous change of laminatethickness can be achieved seamlessly by dropping/adding plies freely.

Characteristics of Composite Topology vs. Free-size

Composite Topology Composite Free-size

Angle opt im ized indirec t ly . Angle opt im ized indirec t ly .

Goal – 0/Ti discrete thickness of individualplies

-> Restricted freedom

Goal – variable thickness of individual plies-> "Free" under upper bound Ti

Results – Truss-like design concepts. Variable thickness panel likely for in-planeloading, 0/1 thickness likely when bending isdominant.



Not useful compared to Free-size? Always better design?

Manufacture – not a factor for compositeunless pre-manufactured laminate is used.

Manufacture – naturally achieved with noadditional cost.

Concentrated full thick members arestronger against out of plane buckling.

Spread thin shell could be prone to buckling.

Functionality may need holes for other non-structural components or for passing lines/pipes.

Cavity is controlled by optimality, and isusually not extensive under in-plane loading.

Interpreting Topology and Free-size Results

Interpretation of topology results is rather straight-forward. For free-size, the change inthickness of individual plies provides insight for ply dropping/adding zones. The thickness ofeach ply in each individual zone can then be defined as a design variable in a detailed sizeoptimization. At this stage, discrete variables can be used to reflect the discrete nature ofply thickness change. Overlapping all zones of individual plies can than help to generatePCOMP zones, where a ply traveling through different zones can be defined using PCOMPG.



Free-size Optimization

Input Definition

Free-size optimization is defined through the DSIZE bulk data entry that is supported in theHyperMesh optimization panel. Features available for free-size include: minimum membersize control, symmetry, pattern grouping and pattern repetition, and stress constraintsapplied to von Mises stresses of the entire structure.

Involving both topology and free-size in the same optimization problem is not recommendedsince penalization on topology components creates a bias that could lead to sub-optimalsolutions.

Problem Formulation

For a shell cross-section (shown below), free-size optimization allows thickness t to varyfreely between T and T0 for each element; this is in contrast to topology optimization whichtargets a discrete thickness of either T or T0. The differences of topology optimization andfree-size can be illustrated through a simple example.

Shell cross-section


The cantilever plate is shown in the following figure. Base-plate thickness T0 is zero. Theoptimization problem is stated as:




Cantilever plate

The next figure shows the final results of topology and free-size optimization as performed onthis plate, side by side. As expected, the topology result created a design with 70% cavity,while the free-size optimization arrived at a result with a zone of variable thickness panel.

Topology result Free-size result

The compliance of both designs are compared in the following figure.

It is not surprising to see that the free-size design outperforms the topology design in termsof compliance since continuous variation of thickness offers more design freedom.

It should be emphasized that free-size offers a concept design tool alternative to topologyoptimization for structures modeled with 2-D elements. It does not replace a detailed size



optimization that would fine tune the size parameters of an FEA model of the final product. To illustrate the close relationship between free-size and topology formulation, consider a 3-Dmodel of the same cantilever plate shown previously. The thickness of the plate is modeled in10 layers of 3-D elements.

Cantilever plate – 3-D model

3-D topology result

The topology design of the 3-D model shown above looks similar to the free-size resultsshown previously. This should not be surprising because when the plate is modeled in 3-D, avariable thickness distribution becomes possible under the topology formulation that seeks adiscrete density value of either 0 or 1 for each element. If infinitely fine 3-D elements areused, a continuous variable thickness of the plate can be achieved via topology optimization.The motivation for the introduction of free-size is based on the conviction that limitations dueto 2-D modeling should not become a barrier for optimization formulation. In regards to the 3-D modeling of shell, topology optimization is equivalent to the application of extrusionconstraint(s) in the thickness direction of a 3-D modeled shell.

It is important to point out that while free-size often creates variable thickness shells withoutextensive cavity, it does not prevent cavity if the optimizer demands it. For the examplealready shown, we can see cavity in the free-size result in the 45 degree region, adjacent to



the support, and in the upper and lower corners of the free end.

If a plate is predominantly under a bending load, free-size design can converge to a discrete0/1 thickness distribution similar, or even identical to, the result of a topology optimization. The reason is that bending stiffness is a function of t3 and, therefore, maximum thickness isheavily favored. In other words, intermediate thickness is naturally penalized for bendingperformance. In the following figure, the free-size result of a plate under bending clearlydemonstrates this behavior.

Free-size result of a plate under bending

Stress Constraints for Free-size Optimization




The capability has built-in intelligence to filter out artificial stress concentrationsaround point loads and point boundary conditions. Stress concentrations due toboundary geometry are also filtered to some extent as they can be improved moreeffectively with local shape optimization.








The differences in the characteristics of topology and free-size are summarized in thefollowing table. It is important to note that while the free-size design concept generallyachieves better performance when buckling constraints are ignored, the topology conceptcould outperform free-size if buckling constraints become the driving criteria during the sizeand/or shape optimization stage. The reason for this is that topology optimization eliminatesintermediate thicknesses, which leads to a more concentrated material distribution and a shellthat is stronger against out-of-plane buckling. The performance of topology and free-size iscompared in the next practical example. Since it is usually not possible to know what criteriaare most critical for a given structure, it is recommended to follow both design concepts untildetailed size and shape optimization is complete and can be evaluated. If it is not possible toderive two designs for every structural component, a benchmark of the relative performanceof both concepts for every type of commonly evaluated structure should be established sothat general guidelines can be used for reference.

Manufacturing and functional considerations may favor topology optimization. Two cases inwhich free-size may not be the best choice from the start include those in which:

(1) A variable thickness shell is typically far more expensive to manufacture and may not be aviable choice; as with most shell structures of an automobile that are manufactured usingstandard sheet metal, for example. (2) The functionality of the structure might require extensive cavity in the design; as with anairplane fuselage floor supporting beam which may need a significant amount of cavity toallow for the pass-through of wires, pipes or other equipment.

Characteristics of Shell Topology vs. Free-size

Shell Topology Optimization Free-size

GOAL: 0/1 thickness-> Restricted freedom

GOAL: variable thickness-> "Free" under upper bound T

Results: Truss-like design concepts. Variable thickness panel likely for in-planeloading, 0/1 thickness likely when bending isdominant.

Equivalent to ext rusion c onst ra int s w henshell is m odeled in infinit e ly f ine 3-Delem ents.

Equivalent to m odel w ith infinit e ly f ine 3-Delem ents.



Not useful compared to free-size? Always better design?

Manufacturing constraint – punched sheetmetal of constant thickness.

Manufacture – expensive and only used inindustries less sensitive to cost.


Spread thin shell could be prone tobuckling.


Cavity is controlled by optimality, and isusually not extensive under in-planeloading.

Interpreting Free-size Results

In most cases, variable thickness of a shell structure is achieved through step-wise change ofthickness. Free-size results provide a different concept about how the zones of differentthicknesses should be designed. Detailed size optimization can then be performed to fine tunethe final design. This process is illustrated in the following example.

Example: Supporting Beam of an Airplane Door Structure

This example was discussed in a paper by Cervellera, Zhou and Schramm in 2005 (Proceedingsof 6th World Congresses of Structural and Multidisciplinary Optimization, Rio de Janeiro, 30May - 03 June 2005, Brazil). Free-size optimization is applied to improve the traditional beamdesign consisting of an "I" cross section with circular cut-outs. A model representing thedesign space of a beam component has been generated, in which a portion of outer skin andvertical frames is included (shown in following image).

Supporting beam of an airplane door structure

The design areas include the upper flange and the web, while the lower flange and theattachment ribs of vertical frames remain unchanged. Free-size optimization allows element



thickness to vary between 0.05 mm and 10.0 mm. The design problem is to minimize the masssubject to a beam center deflection of 3 mm. The free-size result is shown on the left in thefigure below. This result is interpreted into zones of different thicknesses as shown with thedifferent colors on the right in the figure.

Left: Free-size result; Right: Interpreted zones of constant thickness

For comparison, topology optimization is applied to the same problem for shell thickness of 5mm in the design area. The result and its interpretation is shown below.

Left: Topology result; Right: Interpreted zones of constant thicknesses

Detailed size optimization is then carried out for both concepts, allowing all shell thickness tovary between 1.6 mm and 20 mm. The optimization problem is formulated as minimization ofthe beam mass subject to the following constraints:

Maximum deflection of the beam < 3.0 mm.Maximum von Mises stress in the beam design area < 300 MPa.Buckling load factors > 1.0.

In order to study the behavior of the design concepts under different design criteria, sizeoptimization is carried out for different permissible deflection constraints (1.5 mm, 2.0 mm, 3mm, 4.0 mm, and 5.0 mm). The results are summarized with the figure below, in which criticalconstraints are highlighted in red numbers. The figure also shows the optimum mass of thetwo concepts with respect to the maximum displacement. Note that the plate design is more



efficient than the truss-like concept if high stiffness is required, while it is less efficient ifstability and strength requirements dominate the final designs. More details of this exampleand additional discussions about free-size can be found in the paper by Cervellera, Zhou andSchramm in 2005.

Comparison of results for different deflection constraints



Composite Topology and Free-size Optimization

Input Definition

For composite structures, topology and free-size optimization are defined through the DTPLand DSIZE bulk data entries, respectively. Both are supported in the HyperMeshoptimization panel. Features available include: minimum member size control, symmetry,pattern grouping and pattern repetition. Stress or failure constraints are not supported atthis stage.

Prior to OptiStruct 8.0, composite topology optimization was based on the notion that thehomogenized properties of an element remain unchanged. This construct does not allow thefreedom for material redefinition. However, if this is indeed a preferred assumption, the HOMOoption can be set on the MAT line of the DTPL card. Otherwise, an individual ply-basedformulation (discussed below) will be the default option.

Topology and free-size methods target a system level composite design where laminate familydefinition is the objective. Therefore, the PCOMP model should not reflect a detailed stackingof plies of the same orientation. For example, even though 10 layers of 0 degree graphitecloth might be separated in the stacking of the final structure, the modeling for a conceptstudy using topology and free-size should group them together in one ply in the PCOMP sothat the optimal total thickness distribution of a 0 degree ply is optimized throughout thestructure.

Involving both topology and free-size in the same optimization problem is not recommendedsince the penalization on topology components creates a bias that could lead to sub-optimalsolutions.

Problem Formulation

For a composite shell element (shown in the figure below), the thickness ti of each ply is avariable between 0 and Ti defined on the PCOMP card.

Composite element

The only difference between topology and free-size here is that the former targets a discretefinal solution of 0 (or Ti) for ti, while free-size allows ti to vary freely between 0 and Ti. The



discrete solution is achieved by penalizing intermediate thickness. Most generalcharacteristics of regular shell topology and free-size optimization also apply to composite. Itis recommended that users become familiar with free-size before proceeding. The majordifferences between topology optimization and free-size can be illustrated through a simpleexample.


The cantilever plate is shown in the following figure. A symmetric lay-up of (0, +45, -45, and90) degree plies are used. The optimization problem is stated as:


Composite cantilever plate

For topology optimization, the thickness distribution of individual plies in the final design isshown in the following figure.



Topology result – thickness of individual plies

The total thickness of the laminate is shown next.

Topology result – total thickness of the laminate

It can be seen that a rather discrete thickness for each ply is obtained. Note that while littleoverlapping of different orientations is shown in this result, it should be expected thatoverlapping of plies of different angles might be more pronounced when multiple load casesexist.

The thickness distribution of free-size optimization for this example is shown below.



Free-size result – thickness of individual plies

Free-size result – total thickness of the laminate

As expected, free-size created a design in which variable ply thickness appears in a large areaof the structure. The compliance of both designs are compared in the figure below. It is notsurprising to see that the free-size design outperforms the topology design in terms ofcompliance since a continuous variation of thickness offers more design freedom.



Compliance of topology and free-size results

While ply angles are not variables for topology and free-size optimization, thicknessoptimization of plies indirectly leads to a discrete optimization of angles. The available anglesin the PCOMP can be interpreted as discrete angle variables. Also, while free-size oftencreates variable thickness distribution without extensive cavity, it does not prevent cavity ifthe optimizer demands it. For this example, we can see cavity in the free-size results in the45 degree region, adjacent to the support, and in the upper and lower corners of the freeend.


Most of the characteristics for general shell discussed in free-size section also apply tocomposite structures. One important exception is that the manufacturing cost is no longer arestrictive factor for composites. The reason for this is that a composite structure ismanufactured by laying very thin plies of fiber cloth over each other and binding them with amatrix material, like epoxy resin. Therefore, an almost continuous change of laminatethickness can be achieved seamlessly by dropping/adding plies freely.

Characteristics of Composite Topology vs. Free-size

Composite Topology Composite Free-size

Angle opt im ized indirec t ly . Angle opt im ized indirec t ly .

Goal – 0/Ti discrete thickness of individualplies-> Restricted freedom

Goal – variable thickness of individual plies-> "Free" under upper bound Ti

Results – Truss-like design concepts. Variable thickness panel likely for in-planeloading, 0/1 thickness likely when bending isdominant.



Not useful compared to Free-size? Always better design?

Manufacture – not a factor for compositeunless pre-manufactured laminate is used.

Manufacture – naturally achieved with noadditional cost.


Spread thin shell could be prone to buckling.


Cavity is controlled by optimality, and isusually not extensive under in-plane loading.

Interpreting Topology and Free-size Results

Interpretation of topology results is rather straight-forward. For free-size, the change inthickness of individual plies provides insight for ply dropping/adding zones. The thickness ofeach ply in each individual zone can then be defined as a design variable in a detailed sizeoptimization. At this stage, discrete variables can be used to reflect the discrete nature ofply thickness change. Overlapping all zones of individual plies can than help to generatePCOMP zones, where a ply traveling through different zones can be defined using PCOMPG.




Topography optimization is an advanced form of shape optimization in which a design regionfor a given part is defined and a pattern of shape variable-based reinforcements within thatregion is generated using OptiStruct. The approach in topography optimization is similar tothe approach used in topology optimization, except that shape variables are used rather thandensity variables. The design region is subdivided into a large number of separate variableswhose influence on the structure is calculated and optimized over a series of iterations. Thelarge number of shape variables allows the user to create any reinforcement pattern withinthe design domain instead of being restricted to a few.


Mass* Volume* Center of Gravity

Moment of Inertia Static Compliance Static Displacement

Natural Frequency Buckling Factor Static Stress, Strain,Forces

Static Composite Stress, Strain, Failure Index

Frequency ResponseDisplacement, Velocity,Acceleration

Frequency ResponseStress, Strain,Forces

Weighted Compliance Weighted Frequency CombinedCompliance Index

Function Bead discreteness fraction Temperature

* Mass and Volume are not recommended for use as objectives or constraints sincemass and volume are not very sensitive to design changes in topography optimization.



Design Variables for Topography Optimization

OptiStruct solves topography optimization problems using shape optimization with internallygenerated shape variables. One or more design domains are defined using the DTPG card.These cards must, in turn, reference PSHELL, PCOMP or DESVAR definitions. If a DESVARdefinition is referenced, it must be a shape design variable, meaning that it must, in turn, bereferenced by one or more DVGRID cards. If a PSHELL or PCOMP definition is referenced,OptiStruct generates shape variables using the parameters defined on the DTPG card,creating internal DVGRID data for the nodes associated with the PSHELL or PCOMP definitions. In both cases, the end result is that each DTPG card references a single shape variable. This shape variable then gets converted into topography shape variables.

Basic topography shape variables follow the user-defined parameters on the DTPG card(minimum bead width, and draw angle), they are circular in shape, and they are laid outacross the design domain in a roughly hexagonal distribution. Each topography shape variablehas a circular central region of diameter equal to the minimum bead width. Grids within thisregion are perturbed as a group, which prevents the formation of any reinforcement bead ofless than the minimum bead width. Grids outside of the central circular region of thetopographical variables are perturbed as the average of the variables to which they arenearest. This results in smooth transitions between neighboring variables. If two adjacentvariables are fully perturbed, all of the nodes between them will be fully perturbed. If onevariable is fully perturbed and its neighbor is unperturbed, the nodes in between will form asmooth slope connecting them at an angle equal to the draw angle. The spacing of thevariables is determined by the minimum bead width and the draw angle in such a way that nopart of the bead reinforcement pattern forms an angle greater than the draw angle.

Pattern grouping options link topographical variables together in such a way that the desiredreinforcement patterns are formed. Linear, planar, circular, radial, etc. shaped reinforcementsare controlled by single variables, ensuring that the reinforcements follow the desired pattern. One-plane, two-plane, three-plane and cyclical symmetry pattern grouping options also use asimilar approach to ensure that symmetry is created in the solution.

Although topography optimization is primarily a tool for creating bead type reinforcements inshell elements, it can accommodate solid models as well. Many pattern grouping options(such as planar and cylindrical) are intended to be used with solid models since theyeffectively reduce 3-D problems into 2-D ones.

Variable Generation

There are three methods of automatically generating shape variables for topographyoptimization using the DTPG card. The first two, element normal and draw vector, areperformed entirely in OptiStruct. The third (user-defined) requires that the input data containone or more shape design variables that are used as the design domain.

Method Description

Element normal This method is the easiest one to use. When norm is entered forthe draw direction, the normal vectors of the elements are used todefine the draw vector for the shape variables. This method isespecially effective for curved surfaces and enclosed volumes where



Method Description

the beads are intended to be drawn normal to the surface.

Beads created using the element normal method of determining draw vector.

Draw vector This method allows you to define the draw vector that is used forgenerating the shape variables. The X, Y, and Z components of thedraw vector in the nodal coordinate system are entered. Thismethod is useful when all beads must be drawn in the samedirection. Note that the draw angle may not be maintained whileusing this method.

Beads created using the Draw vector method of determining draw vector.

User-defined This method allows you to set up the vectors and heights for thetopography optimization. A DESVAR card is referenced in place of aPSHELL or PCOMP card. All of the grids with DVGRID cards



Method Description

associated with that DESVAR card are considered part of the designdomain. The DESVAR and DVGRID entries are redefined to reflectthe minimum bead width and draw angle parameters that have beenset by the user. The vectors and magnitudes of the displacementvectors on each DVGRID card for each grid are retained, so theseentries must be left blank on the DTPG card. This allows you tocreate a design domain in which each node can have its own drawvector and draw height.

Multiple Topography Design Regions

OptiStruct generates topography shape variables for each design domain defined by a DTPGcard. It allows for overlapping of design domains. A grid that is in more than one designdomain will be a part of shape variables for each design domain. For automatically generatedbead variables, the draw height is divided by the number of bead variables acting on that grid. Thus, if a grid is a part of two DTPG cards that have draw heights of 3.0mm and 5.0mm, thedraw heights become 1.5mm and 2.5mm. If this is not desired, simply make sure that no gridis in more than one design domain. In cases where two design components touch each otherand the design domains are not user-defined (i.e. PSHELL or PCOMP definitions arereferenced), a row of non-design elements needs to be inserted between them to preventaveraging. If the bead variables are user-defined (i.e. DESVAR definition is referenced), noaveraging will be performed. It is assumed that the user intends to have the shape variablesoverlap, which will result in the grid deflection being cumulative between multiple influencingbead cards.

Bead Discreteness Fraction

The bead discreteness fraction is a response that can be used to control the amount ofshape variation for topography design domains. This response indicates the amount of shapevariation for one or more topography design domains. The response varies in the range 0.0 to1.0 (0.0 < BEADFRAC < 1.0), where 0.0 indicates that no shape variation has occurred, and1.0 indicates that the entire topography design domain has assumed the maximum allowedshape variation.



Size Optimization

OptiStruct has the capability of performing size optimization. Size optimization can beperformed simultaneously with the other types of optimization.

In size optimization, the properties of structural elements such as shell thickness, beamcross-sectional properties, spring stiffness, and mass are modified to solve the optimizationproblem.

Defining size variables in OptiStruct is done very similarly to other size optimization codes. Each size variable is defined using a DESVAR bulk data entry. If a discrete design variable isdesired, a DDVAL bulk data entry needs to be referenced for the design variable values. TheDESVAR cards are related to size properties in the model using a DVPREL1 or DVPREL2 bulkdata entry. Each DVPREL bulk data entry must reference at least one DESVAR bulk dataentry to be active during the optimization. HyperWorks includes a pre-processor calledHyperMesh that can be used to set up any number of size variables for the properties.


Mass Volume Center of Gravity



Static Composite Stress, Strain, Failure Index

Frequency ResponseDisplacement,Velocity, Acceleration

Frequency ResponseStress, Strain, Forces


Function Temperature



Design Variables for Size Optimization

In finite elements, the behavior of structural elements (as opposed to continuum elements),such as shells, beams, rods, springs, and concentrated masses, are defined by inputparameters, such as shell thickness, cross-sectional properties, and stiffness. Thoseparameters are modified in a size optimization. Some structural elements have severalparameters depending on each other; like beams in which the area, moments of inertia, andtorsional constants depend on the geometry of the cross-section.

The property itself is not the design variable in size optimization, but the property is definedas a function of design variables. The simplest definition, as defined by the design-variable-to-property relationship DVPREL1, is a linear combination of design variables defined on aDESVAR statement such that

where is the property to be optimized, and are linear factors associated to the design

variable .

Using the equation utility DEQATN, more complicated functional dependencies using eventrigonometric functions can be established. Such design-variable-to-property relations arethen defined using the DVPREL2 statement.

For a simple gage optimization of a shell structure, the design-variable-to-propertyrelationship turns into

where the gage thickness, t is identical to the design variable.

If a discrete design variable is desired, a DDVAL bulk data entry needs to be referenced onthe DESVAR bulk data entry for the design variable values.



Shape Optimization

OptiStruct has the capability of performing shape optimization. In shape optimization, theouter boundary of the structure is modified to solve the optimization problem. Using finiteelement models, the shape is defined by the grid point locations. Hence, shape modificationschange those locations.

Shape variables are defined in OptiStruct in a way very similar to that of other shapeoptimization codes. Each shape variable is defined by using a DESVAR bulk data entry. If adiscrete design variable is desired, a DDVAL bulk data entry needs to be referenced for thedesign variable values. DVGRID bulk data entries define how much a particular grid pointlocation is changed by the design variable. Any number of DVGRID bulk data entries can beadded to the model. Each DVGRID bulk data entry must reference an existing DESVAR bulkdata entry if it is to be a part of the optimization. The DVGRID data in OptiStruct containsgrid location perturbations, not basis shapes.

The generation of the design variables and of the DVGRID bulk data entries is facilitated bythe HyperMorph utility, which is part of the HyperMesh software.


Mass Volume Center of Gravity



Static Composite Stress,Strain, Failure Index

Frequency ResponseDisplacement, Velocity,Acceleration

Frequency ResponseStress, Strain, Forces


Function Temperature



Design Variables for Shape Optimization

In finite elements, the shape of a structure is defined by the vector of nodal coordinates (x).In order to avoid mesh distortions due to shape changes, changes of the shape of thestructural boundary must be translated into changes of the interior of the mesh.

The two most commonly used approaches to account for mesh changes during a shapeoptimization are the basis vector approach and the perturbation vector approach. Bothapproaches refer to the definition of the structural shape as a linear combination of vectors.

Using the basis vector approach, the structural shape is defined as a linear combination ofbasis vectors. The basis vectors define nodal locations.

where x is the vector of nodal coordinates, is the basis vector associated to the design

variable .

Using the perturbation vector approach, the structural shape change is defined as a linearcombination of perturbation vectors. The perturbation vectors define changes of nodallocations with respect to the original finite element mesh.

where x is the vector of nodal coordinates, is the vector of nodal coordinates of the initial

design, is the perturbation vector associated to the design variable .

The initial nodal coordinates are those defined with the GRID entity. The perturbation vectorsare defined on the DVGRID statement, which is referenced by the design variable entityDESVAR.

If a discrete design variable is desired, a DDVAL bulk data entry needs to be referenced onthe DESVAR bulk data entry for the design variable values.

Note: In OptiStruct, only the perturbation vector approach is available. The DVGRIDcards must contain perturbation vectors.




Free-shape optimization uses a proprietary optimization technique developed by AltairEngineering Inc., wherein the outer boundary of a structure is altered to meet with pre-defined objectives and constraints. The essential idea of free-shape optimization, and whereit differs from other shape optimization techniques, is that the allowable movement of theouter boundary is automatically determined, thus relieving users of the burden of definingshape perturbations.

Free-shape design regions are defined through the DSHAPE bulk data entry. Design regionsare identified by the grids on the outer boundary of the structure (the edge of a shellstructure or the surface of a solid structure). These grids are listed on the DSHAPE entry.

Free-shape optimization allows these design grids to move in one of two ways:

1. For shell structures; grids move normal to the surface edge in the tangential plane.

2. For solid structures; grids move normal to the surface.

During free-shape optimization, the normal directions change with the change in shape of thestructure, thus, for each iteration, the design grids move along the updated normals.



Defining Free-shape Design Regions

Ideally, free-shape design regions should be selected where it can be assumed that the shapeof the structure is most sensitive to the concerned responses. For example, it would beappropriate to select grids in a high stress region when the objective is to reduce stress.

Free-shape design regions should be defined at different locations on the structure where it isdesired for the shape to change independently. For solid structures, feature lines oftendefine natural boundaries for free-shape design regions. Containing any feature lines inside afree-shape design region should be avoided unless the intention is to smooth the feature linesduring an optimization. Likewise for a shell structure, sharp corners should not be containedinside a free-shape design region unless the intention is to smooth out such corners.

The DSHAPE card identifies the design region through the GRID continuation card, shownhere:

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GRID GID1 GID2 GID3 GID4 GID5 GID6 GID7

GID8 GID9 … …

A free-shape design region is defined on the curved edgeof the plate by selecting the edge grids; the grids are freeto move in the normal direction on the tangential plane.

A free-shape design region is defined on asurface of the solid structure by selecting theface surface grids; the grids are free to movenormal to the surface.



Free-shape Parameters

The five parameters that affect the way in which the free-shape design region deforms arethe direction type, the move factor, the number of layers for mesh smoothing, the maximumshrinkage, and the maximum growth.

Direction Type

This provides a general constraint on the direction of the movement of the free-shape designregion. It is defined on the PERT continuation line of the DSHAPE entry in the DTYPE field, asshown:

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PERT DTYPE MVFACTOR

NSMOOTH

MXSHRK MXGROWTH

DTYPE has three distinct options:

1. GROW – grids cannot move inside of the initial part boundary.

2. SHRINK – grids cannot move outside of the initial part boundary.

3. BOTH – grids are unconstrained.

GROW SHRINK BOTH

Undeformed

Deformed

Move Factor

The maximum allowable movement in one iteration of the grids defining a free-shape designregion is specified as:



MVFACTOR* mesh_size

where "mesh_size" is the average mesh size of the design region defined in the same DSHAPEcard.

MVFACTOR is defined on the PERT continuation line of the DSHAPE entry.

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PERT DTYPE MVFACTOR NSMOOTH MXSHRK

MXGROWTH

The default value of MVFACTOR is 0.5. A smaller MVFACTOR will make free-shapeoptimization run slower but with more stability. Conversely, a larger MVFACTOR will makefree-shape optimization run faster but with less stability.

MVFACTOR affects the maximum movement in one iteration.

Undeformed shape

Shape at iteration 1 with MVFACTOR = 0.5 (default)

Shape at iteration 1 with MVFACTOR = 1.0

Number of Layers for Mesh Smoothing

With free-shape optimization, internal grids adjacent to those grids defining the design regionare moved to avoid mesh distortion. The number of layers of grids to be included in the meshsmoothing buffer may be defined by the NSMOOTH field on the PERT continuation line of theDSHAPE entry.



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PERT DTYPE MVFACTOR NSMOOTH MXSHRK

MXGROWTH

The default value of NSMOOTH is 10. A larger NSMOOTH will give a larger smoothing buffer,and consequently will work better in avoiding mesh distortion; however, it will result in aslower optimization.

NSMOOTH=5, 5 layers of grids move along withthe design boundary.

NSMOOTH=1, only 1 layer of grids move alongwith the design boundary.

Maximum Shrinkage and Growth

The maximum shrinkage and growth provide a simple way to limit the total amount ofdeformation of the free-shape design region. These parameters are defined on the PERTcontinuation line of the DSHAPE entry.

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PERT DTYPE MVFACTOR NSMOOTH MXSHRK MXGROWTH

The design region is offset to form two barriers; MXSHRK is the offset in the shrinkagedirection and MXGROWTH is the offset in the growth direction. The design region is thenconstrained to deform between these two barriers.



Deformation space defined by the maximum growing/shrinking distance

For more details and an example, please refer to the section on the Mesh Barrier Constraintbelow.

Constraints on Grids in the Design Region

It is possible to identify additional constraints on certain grids in free-shape design regions. Three types of constraints are available for specified grids as defined by CTYPE# on theGRIDCON continuation line of the DSHAPE entry:

1. FIXED – grid cannot move due to free-shape optimization.

2. VECTOR – grid is forced to move along the specified vector.

3. PLANAR – grid is forced to remain on a plane for which the specified vector defines thenormal direction.

Note: VECTOR is used to constrain a grid to move along a line, thus it makes nodifference by rotating the vector by 180 degrees.

Constraints are defined on the GRIDCON continuation line as follows:

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GRIDCON GCMETH GCSETID1/ GDID1

CTYPE1 CID1 X1 Y1 Z1



GCMETH GCSETID2/ GDID2

CTYPE2 CID2 X2 Y2 Z2

Example Showing CTYPE = VECTOR

This example demonstrates a simple case where it is necessary to use the "DIR" constrainttype to force grids to move in a predefined direction.

A free-shape optimization is performed on a quarter model of a rectangular plate with a hole,shown here:

The curved edge is the free-shape design region. Without any constraints on the free-shapedesign region, the grids at the ends of the curved edge do not move exactly along the line ofthe straight edge, but move slightly outward, as shown here:

In order to prevent this phenomenon, the grids at the ends of the curved edge (shown inyellow below) are both constrained to move along the vector indicated by the red arrows.



Using these constraints - corner grids moving along the constrained direction - the grids atthe ends of the curved edge now move as desired, along the line of the straight edge, asshown here:

Example showing CTYPE = PLANAR

In this example, the total volume of a cantilever beam is to be minimized subject to adisplacement constraint in the loading direction at the free-end of the beam. The model isshown here:



Two free-shape design regions are defined in this example. Both of the vertical sides of thebeam are selected as design regions and a free-shape optimization is performed.

Without any constraints on the free-shape design region, the top and bottom surfaces of thebeam do not remain strictly on the X-Z plane.



To ensure that the top and bottom surfaces remain on the X-Z plane, the grids along theedges of the design regions DSHAPE1 and DSHAPE2 are constrained to move only on the x-zplane.

Using these constraints – constrained grids moving only on the X-Z plane – the top andbottom surfaces of the beam remain on the X-Z plane as desired.



1-plane Symmetry Constraint

It is often desirable to produce a symmetric design. Even if the loads and boundaryconditions are perfectly symmetric, there is no guarantee that the resulting design will beperfectly symmetric. In order to ensure a symmetric design, a symmetry constraint must bedefined. An additional advantage of this constraint is that it will produce symmetric designsregardless of the initial mesh, loads or boundary conditions.

The 1-plane symmetry constraint is defined on the PATRN continuation line:

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PATRN TYP AID/XA YA ZA FID/XF YF ZF

Example Showing 1-plane Symmetry Constraints in 2-dimensions

In this example, the objective is to minimize the total volume subject to a stress constraintusing free-shape optimization. Results are shown with and without symmetry constraints.



2-D model showing free-shape design grids

Result without symmetry constraint Result with symmetry constraint (XZ plane)

Example Showing 1-plane Symmetry Constraints in 3-dimensions

In this example, the objective is to minimize the compliance subject to a volume constraintusing free-shape optimization. Results are shown with and without symmetry constraints.

3-D Model showing free-shape design grids



Result without symmetry constraint Result with symmetry constraint (XZ plane)

Extrusion Constraint

It is often desirable to produce a design with a constant cross-section along a given path,particularly in the case of parts manufactured by an extrusion process. By using extrusionmanufacturing constraints with free-shape optimization, constant cross-section designs canbe attained for solid models (regardless of the initial mesh, loads or boundary conditions).

The extrusion constraint is defined on the EXTR continuation line:

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EXTR ECID XE YE ZE

Two types of extrusion path are available for free-shape optimization – straight line andcircular.

Example Showing Extrusion Constraint Along a Straight Line

The FE model, optimization problem and design variables definition are the same as in theprevious example, so the result without the extrusion constraint is the same as shown above.The result with the extrusion constraint (straight line) is shown here.



Result with extrusion path (along x-axis)

Example Showing Extrusion Constraint Along a Circular Path

In this example, the objective is to minimize the von Mises stress subject to a volumeconstraint using free-shape optimization. A circular extrusion path is defined using acylindrical coordinate system ( direction). Results are shown with and without extrusionconstraints (circular).

Model showing free-shape design grids



Result without extrusion path

Result with extrusion path (circular)

Draw Direction Constraint

In the casting process, cavities that are not open and lined up with the sliding direction ofthe die are not feasible. Draw direction constraints may be defined for the design region sothat the optimized shape will allow the die to slide in a prescribed direction. Only a single dieis considered for each design region (defined in each DSHAPE card), and non-design regionswill not be considered for this constraint.

The draw direction constraint is defined on the DRAW continuation line:

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DRAW DTYP DAID/XDA

YDA ZDA DFID/XDF

YDF ZDF

Example Showing Draw Direction Constraint

The FE model, optimization problem and design grids definition are the same as those in theexample showing 1-plane symmetry constraints in 3 dimensions. Results with draw directionconstraint are shown here.



Result with draw direction constraint (along Y-axis)

Example Showing Combination of 1-plane Symmetry and Draw Direction Constraints

The FE model, optimization problem and design grids definition are the same as those in theexample showing 1-plane symmetry constraints in 3 dimensions. Results with 1-planesymmetry and draw direction constraints are shown here.

Result with both draw direction constraint (Y-axis) and 1-plane symmetry constraint (XY-plane)

Side Constraints

Similar to the maximum shrinkage and growth parameters as defined on the PERT continuationline, it is possible to limit the extent of the total deformation of the design region by way ofside constraints. Side constraints allow the deformation space to be defined as a coordinaterange; i.e. between (x1, y1, z1) and (x2, y2, z2). These ranges may be with reference to

rectangular, cylindrical or spherical systems.



Example Showing Side Constraints

In this example, the objective is to minimize the von Mises stress subject to a volumeconstraint using free-shape optimization. Results are shown with and without side constraints.

Model showing side constraints defined by the radii R1 and R2 (1-direction of cylindrical system)

Result without side constraints Result with side constraints

Mesh Barrier Constraint

Aside from shrinkage and growth parameters and side constraints, a more general capability tolimit the extent of the total deformation of the design region is available by way of defining amesh barrier constraint. The mesh barrier is composed of special shell elements (BMFACE),and in order to keep computational effort to a minimum, as few elements as possible should be



used in its definition.

The mesh barrier is defined on the BMESH continuation line.

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BMESH BMID

Example Showing Mesh Barrier Constraint

The FE model, optimization problem and design grids definition are the same as those in theexample showing 1-plane symmetry constraints in 3 dimensions. A mesh barrier is added(large red tria elements).

Model with mesh barrier

Result without mesh barrier constraint Result with mesh barrier constraint



Result combining mesh barrier constraint and 1-plane symmetry constraint

From the results, we can see how the mesh barrier constrains the model deformation, but ifthe mesh barrier is not big enough, the design region deformation is unconstrained beyond itslimits.

Example Showing Maximum Shrinkage and Growth Parameters

The FE model, optimization problem and design grids definition are the same as those in theexample showing 1-plane symmetry constraints in 3 dimensions. In addition, maximumshrinkage and growth parameters (2.0) and a 1-plane symmetry constraint (XZ-plane), aredefined.

Result with shrinkage and growth parameters.Max. growth distance = 2.0Max. shrinkage distance = 2.0

Result with shrinkage and growth parameters and 1-plane symmetry constraint (XZ-plane).Max. growth distance = 2.0Max. shrinkage distance = 2.0

Additional Comments



1. In the case where multiple constraints are defined for the same design region, while theoptimizer tries its best to satisfy all the different constraints, it is possible that it may notbe able to coordinate all these constraints.

2. It should be pointed out that if constraints like mesh barrier, maximum growth andshrinkage, or side constraints are applied to avoid of interference between structuralparts, the user should define the constraints in such a way that clearance is guaranteedunder manufacturing tolerance and structural deformation. In other words, the barriersurface should contain an offset from the potentially interfering parts.



Manufacturing Constraints

The following optimization features can be found in this section:

Manufacturability for Topology Optimization

Manufacturability for Topography Optimization

Manufacturability for Free-size Optimization



Manufacturability for Topology Optimization

A concern in topology optimization is that the design concepts developed are very often notmanufacturable. Another problem is that the solution of a topology optimization problem canbe mesh dependent, if no appropriate measure is taken.

OptiStruct offers a number of different methods to account for manufacturability whenperforming topology optimization:

Member Size Control

Draw Direction Constraints

Extrusion Constraints

Pattern Repetition

Pattern Grouping



Member Size Control for Topology Optimization

Member size control allows you some control over the member size in the final topology andthe resulting degree of simplicity of the final design. This feature may be added one of thetwo ways described below.

1. The DOPTPRM card:

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DOPTPRM MINDIM VALUE

Here, only the preferred minimum diameter (width in 2-D) of members may be defined asthe VALUE field, following the MINDIM keyword. A global minimum member size is definedin this way.

2. The DTPL card:

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DTPL ID PTYPE PID1 PID2 PID3 PID4 PID5 PID6

PID7 … … …

MEMBSIZ MINDIM MAXDIM MINGAP

Here, both the preferred minimum, MINDIM, and the maximum, MAXDIM, diameter ofmembers may be defined on the MEMBSIZ continuation line. Member size dimensions canbe defined differently for each DTPL in this way.

Minimum Member Size Control

Although minimum member size control penalizes the formation of small members, results thatcontain members significantly under the prescribed minimum member size can still be obtained. This is because a small member in the structure can be very important to the loadtransmission and may not be removed by penalization. Minimum member size control functionsmore as a quality control than a quantity control.

A discrete solution is achieved in two iterative steps. The first step converges to a solutionwith a large number of semi-dense elements. The second step tries to refine this solution toa solution with fully dense members. Each step consists of a number of iterations. The firststep consists of two entire convergence phases - the first run with the initial discretenessvalues (defined by DISCRETE and DISCRT1D parameters on the DOPTPRM bulk data entry),followed by a run with the discreteness values increased by 1.0. This procedure isimplemented in order to achieve a solution with clearly defined members. If this step couldnot create a solution with clearly defined members, the preferred minimum member size will



not be preserved in the second step. In which case, the user needs to increase thediscreteness parameters and/or reduce the convergence tolerance (defined by the OBJTOLparameter on the DOPTPRM bulk data entry) to improve the solution of the first phase. Thedefault discreteness is set to 1.0 for 1-D elements, plates and shells, and 2.0 for 3-D solids.

In general, once MINDIM is activated, checkerboarding is controlled by the methods appliedfor this feature, eliminating the need for the CHECKER parameter. In rare circumstances,checkerboards may still be introduced in the second phase described above for 3-D solids. Ifthis happens, an additional checkerboard control algorithm can be activated with theMMCHECK parameter. (The CHECKER and MMCHECK parameters are defined using theDOPTPRM bulk data entry).

The use of this card will assure a checkerboard-free solution, although with the undesired sideeffect of achieving a solution that involves a large number of semi-dense elements, similar tothe result of setting CHECKER equal to 1. Therefore, use this card only when it is necessary.

It is recommended that MINDIM be at least 3 times, and no greater than 12 times, theaverage element size for all elements referenced by that DTPL (or all designable elementswhen defined on DOPTPRM). The average element size for 2-D elements is calculated as theaverage of the square root of the area of the elements, and for 3-D elements, as the averageof the cubic root of the volume of the elements.

This recommendation is enforced when combined with other manufacturing constraints, and ifthe defined MINDIM is less than this value, it will be reset to a default value equal to 3 timesthe average element size. Similarly, if the defined MINDIM is larger than 12 times the averageelement size, it will be reset to a value equal to 12 times the average element size. This limithas been set to trim memory requirements that can become too large as a result of having tokeep track of a much larger number of elements needed to satisfy the MINDIM constraint. Instructures where the mesh is aligned with the draw or extrusion direction, setting MTYP asALIGN on the MESH continuation line of the DTPL card may circumvent this constraint.

The following examples demonstrate significant improvement in the manufacturability ofresults through the use of minimum member size control:

Michell-truss Example

MBB-Beam Example

Arch Example

3-D Bridge Model Example

Maximum Member Size Control

Maximum member size control penalizes the formation of large members. The control is notdirectional, meaning that if the thickness of a member is less than MAXDIM in any direction,this constraint is satisfied. This reflects the need to control the rib thickness of castingparts.

MAXDIM must be at least twice MINDIM, and hence the minimum mesh requirement is thatMAXDIM has to be at least 6 times the average element size for all elements referenced by



that DTPL. This constraint is strongly enforced and an error termination will occur when thiscriteria is not met. In addition, MAXDIM should be less than half the width of the thinnestpart of the design region.

Based on the constraints mentioned above, a fine mesh is required to achieve good resultswith this manufacturing constraint.

It is to be noted that use of the maximum member size control induces further restriction ofthe feasible design space and should therefore only be used when it is truly desirable. Alsonote that this feature is a new research development, and the techniques are still undergoingimprovement. An undesired side effect that has been noticed for some examples is that itmight result in more intermediate density in the final solution. Therefore, it is recommendedthat this feature be used sparingly until the technology becomes more robust.

While MAXDIM also enforces a spacing of members of the same dimension, the maximumreachable volume fraction is 0.5. For problems involving constraints on structural responses,this could interfere with constraint satisfaction. It is strongly recommended that the behaviorof the design problem be studied without MAXDIM first in order to determine if the use ofMAXDIM would be advantageous, and if the target volume allows for it to be applied.

The following examples demonstrate the impact of maximum member size control to the designoutcome.

Example 1: Engine Bracket

Engine Bracket Design with Draw Direction Constraint



Engine Bracket Design with Draw Direction and Maximum Member Size Constraints

Example 2: Steering Wheel Bracket



Steering Wheel Bracket Design with Draw Direction Constraint



Steering Wheel Bracket Design with Draw Direction and Maximum Member Size Constraints



Draw Direction Constraints for Topology Optimization

In the casting process, cavities that are not open and lined up with the sliding direction ofthe die are not feasible. Designs obtained by topology optimization often contain cavitiesthat are not viable for casting. Transformation of such a design proposal to a manufacturabledesign could be extremely difficult, if not impossible. In some cases where this transformationis made, the likelihood of severely affecting the design optimality is high.

OptiStruct allows you to impose draw direction constraints so that the topology determinedwill allow the die to slide in a given direction. These constraints are defined using the DTPLcard. Different constraints can be applied to different structural parts, specified by PSOLIDIDs. There are two DRAW options available. The option 'SINGLE' assumes that a single diewill be used and it slides in the given drawing direction. The bottom surface of the consideredcasting part is the predefined contra part for the die. The option 'SPLIT' implies that two diessplitting apart in the given draw direction will be used to cast the part described in this DTPLcard. The splitting surface of the two dies is optimized during the optimization process.

It is often a requirement of certain designs that no through – holes exist. These holes can beprevented from forming in the direction of the draw through use of the ‘NO HOLE’ option. Thisparameter is also defined on the DTPL card. With ‘NO HOLE,’ the topology can only evolvegradually from the boundary one layer at a time, and in certain cases, it may take severaliterations to remove one layer.

Available with the ‘SINGLE’ draw option is a stamping or sheet metal manufacturing constraint. This option forces the evolution of 3-D shell interpretable structure from a 3-D designdomain. This allows the design of 2-D shell or stamped parts from a 3-D design domainallowing greater design flexibility. The ‘STAMP’ option is also specified on the DTPL card alongwith a thickness value that represents the desired thickness of the resulting shell or stampedpart.

A casting may contain a non-designable region in addition to a designable region. These non-designable regions must be defined as obstacles for the casting process on the same DTPLcard. This preserves the casting feasibility of the final structure.

Also note that there is a default minimum member size for use with draw direction constraints. This is determined internally to be three times the average mesh size of the relevantcomponents. Therefore, the mesh density of the model and the target volume fraction shouldbe chosen so that enough material is available to fill members of the default minimum size. The user can specify a desired minimum member size for each design part defined by a DTPLcard. This value must be bigger than the default value or else it will be replaced by thedefault value.

Example 1: Beam under Torsion

The considered beam is clamped on one end and loaded with a pair of twisting forces on thefree end. The finite element model is shown in Figure 1.1. The design problem is to minimizethe compliance with a volume fraction constraint of 0.3. The final design without drawdirection constraints is shown in Figure 1.2. The chosen draw direction is along the Z-axis. The designs with the options 'SINGLE’ and 'SPLIT' for draw direction constraint are shown inFigures.1.3 and 1.4, respectively.



Figure 1.1: Finite element model of the beam under torsion

Figure 1.2: Design without draw direction manufacturing constraint



Figure 1.3: Design with draw direction Z and die option 'SINGLE'

Figure 1.4: Design with draw direction Z and die option 'SPLIT'

As expected, the result without manufacturing constraints is a tube-like structure that isindeed the optimal topology for torsion load. However, this design does not permit the slidingof the die in the Z direction. The result that allows the sliding of the die for casting is notvery intuitive, it forms a periodical X pattern to cross the pair of twist loads until they reachthe supported end. Significantly more material at the crossing point reflects the doubledshear force at this point. Compared to an upward facing C channel solution, the crosspattern has the advantage that the stress is periodical in every X cell, thus eliminating higherorder influence of the span of the beam for any solution that has system level bending action.



Example 2: Engine Bracket

The example shown below is an engine bracket model of a car. The finite element model ofthe design domain is shown in Figure 2.1, in which 9046 elements are used and the designdomain is shown in blue color. Six load cases were considered, which reflect the followingdriving and service status: 1) start; 2) backwards; 3) into a pothole; 4) out of a pothole; 5)loads from an attaching part; and 6) loads during engine transport. The final topology thatallows a single die sliding upwards is shown in Figure 2.2. The design that allows two dies toslide up and down, respectively, is shown in Figure 2.3.

Figure 2.1: Finite element model of a engine bracket

Figure 2.2: Design with draw direction Z and die option 'SINGLE'



Figure 2.3: Design with draw direction Z and die option 'SPLIT'

Example 3: Compressor Mounting Bracket

In this example, a topology optimization is run on a compressor mounting bracket, and theeffects of the ‘NO HOLE’ manufacturing constraint will be showcased. The finite elementmodel is shown in Figure 3.1. Regions in red indicate non-design space and the region in blueindicates design space. Four different loading conditions were considered representingoperating conditions, and the model was built up with approximately 90,000 elements. Thedesign problem was formulated to minimize the compliance as the objective function, with aconstraint on the design volume fraction.

Figure 3.2 shows the design proposal for the compressor mounting bracket without the use ofthe ‘NO HOLE’ manufacturing constraint. As can be seen from the image, the design containsthrough-holes.

Figure 3.3 on the other hand, shows the design proposal of the bracket with the use of the‘NO HOLE’ constraint. In this case now, the resulting design proposal does not contain anythrough-holes.



Figure 3.1: Finite element model representing the design and non-design spaces of the compressormounting bracket

Figure 3.2: Design proposal without the ‘NO HOLE’ option



Figure 3.3: Design proposal with the ‘NO HOLE’ option

Example 4: Automotive Bracket

The bracket in this example was designed incorporating the ‘NO HOLE’ constraint along withthe ‘STAMP’ constraint. The goal of the study was to generate a design proposal without anythrough-holes, while at the same time being interpretable as a sheet metal design. The‘STAMP’ constraint here forces the formation of a 3-D shell interpretable structurerepresenting a sheet metal design. Four subcases representing four different loadingconditions were considered and the model was comprised of approximately 20,000 elements. The optimization problem was formulated to minimize the weighted compliance (onecompliance value per subcase) for a constrained design volume fraction. This essentiallyresults in evolving the stiffest design for a given amount of material.

Figure 4.1 represents the finite element model of the design and non-design spaces of thebracket. The regions in red are the non-design regions and the region in blue is the designspace.

Figure 4.2 is the design proposal without the consideration of the ‘STAMP’ and ‘NO HOLE’manufacturing requirements. While it is possible to cast such a design, stamping such a partis not feasible.

Figures 4.3 and 4.4 showcase the design proposal after incorporating the ‘STAMP’ and ‘NOHOLE’ manufacturing constraints. The ‘NO HOLE’ constraint prevents the formation of anythrough-holes in the design space and helps keep a continuous shell layout, while the ‘STAMP’constraint leads to the formation of a uniform thickness design proposal. The thickness isdefined along with the ‘STAMP’ constraint and represents the desired thickness of the sheetmetal part. From this proposal, it is now possible to interpret the design as a sheet metal orstamped part.



Figure 4.1: Finite element model representing the design and non-design spaces of the bracket

Figure 4.2: Design proposal without ‘STAMP’ and ‘NO HOLE’

Figure 4.3: Design proposal with ‘STAMP’ and ‘NO HOLE’ (view 1)



Figure 4.4: Design proposal with ‘STAMP’ and ‘NO HOLE’ (view 2)



Extrusion Constraints for Topology Optimization

In some cases, it is desirable to produce a design characterized by a constant cross-sectionalong a given path, particularly in the case of parts manufactured through an extrusionprocess. By using extrusion manufacturing constraints in topology optimization, constantcross-section designs can be attained for solid models – regardless of the initial mesh,boundary conditions or loads.

Extrusion constraints can also be used for the conceptual design study of structures that donot specifically need to be manufactured using an extrusion procedure. Those requirementscan be regarded as specific geometric constraints and can be used for any design thatdesires such characteristics. For instance, it might be desirable to have ribs going throughthe entire depth of a solid domain.

As with other manufacturing constraints, extrusion constraints can be applied on a componentlevel, and can be defined in conjunction with minimum member size control using the DTPLcard.

Setting up the Problem

Extrusion constraints can be applied to domains characterized by non-twisted cross-sections(left figure) or twisted cross-sections (right figure) by using the NOTWIST or TWISTparameters respectively in the ETYP field. The structure is non-twisted when the localcoordinates systems associated with each cross-section, projected onto a reference plane,remain parallel to each other.

Defining the Extrusion Path

It is necessary to define a ‘discrete’ extrusion path by entering a series of grids in the EPATH1field. The curve between these grids is then interpolated using parametric splines. Theminimum amount of grids depends on the complexity of the extrusion path. Only two grids arerequired for a linear path, but it is recommended to use at least 5-10 grids for more complexcurves.



In the example above, four grids are used to define the extrusion path (left figure). As wecan see, the path computed by OptiStruct is inaccurate. To obtain a more accurateapproximation, more grids are included in the extrusion path (right figure).

For twisted cross-sections, a secondary extrusion path needs to be defined in a similarmanner through the EPATH2 field.

Example 1

In this example, a curved beam is considered to be a rail over which a vehicle is moving. Bothends of the beam are simply supported. A point load applied over the length of the rail as fiveindependent load cases simulates the movement of the vehicle. The objective is to minimizethe sum of the compliances, and the material volume fraction is constrained at 0.3. The railshould be manufactured through extrusion. The 13 grids represented as black dots on theright figure define the extrusion path.

The optimized topologies without and with extrusion constraints are shown below. Reanalyzing the final designs without penalty for intermediate density, the compliances forthese two designs are 29.9396 and 37.4377 respectively, which implies a 20% loss in



performance due to extrusion constraints. The extruded design represents a clean proposalthat requires little refinement. On the other hand, the design obtained without manufacturingconstraints may require significant modifications that could cause efficiency loss inperformance.

Example 2

In this example, a "stairs" shaped structure is submitted to two lateral pressure loads definedin two separate subcases. The objective is to minimize the sum of the compliances underboth load cases. The extrusion path is defined as a straight line parallel to the global Y-axis.The cross-section of the finite elements model along that path is not constant.

Clearly, this type of structure is not suitable to be manufactured through an extrusionprocess. However, extrusion constraints can be applied to obtain a manufacturable designcharacterized by ribs going through the entire depth of the structure. The optimized design



gives a good idea of the layout of the resulting stiffening panels.

Example 3

This example illustrates how extrusion constraints can be used to develop commoncomponents in different areas of a structure. The extrusion path can be defined through asolid mesh that is not continuous.



Pattern Repetition for Topology Optimization

Pattern repetition is a technique that allows different structural components to be linkedtogether so as to produce similar topological layouts.

To achieve this goal, a master DTPL card needs to be defined, followed by any number ofslave DTPL cards which reference the master. The master and slave components are relatedto each other through local coordinate systems, which are required, and through scalingfactors, which are optional.

Other manufacturing constraints, such as minimum or maximum member size, draw directionconstraints or extrusion constraints, can be applied to the master DTPL card. Theseconstraints will then automatically be applied to the slave DTPL card(s) as described in thenext sections.

The following procedure should be followed to set up pattern repetition:

1. Create a master DTPL card.

2. Apply other manufacturing constraints as needed.

3. Define the local coordinate system associated to the master DTPL card.

4. Create a slave DTPL card.

5. Define the local coordinate systems associated to the slave DTPL card.

6. Apply scaling factors as needed.

7. Repeat steps 4-6 for any number of slave DTPL cards.

Local Coordinates Systems

Local coordinates systems are generated by providing four points. These points can bedefined either by entering explicit coordinates or by referencing existing grids, as follows:

1. CAID defines the anchor point for the local coordinates system.

2. CFID defines the direction of the X-axis.

3. CSID defines the XY plane and indicates the positive sense of the Y-axis.

4. CTID indicates the positive sense of the Z-axis.

The definition of the fourth point allows for both right-handed and left-handed coordinatesystems, which facilitates the creation of reflection patterns.



Right-handed coordinates system Left-handed coordinates system

Alternatively, local coordinate systems can be defined by referencing an existing rectangularcoordinate system in the CID field, and by defining an anchor point in the CAID field.

Note that if the fields defining CFID, CSID, CTID, and CID are left blank, then the globalcoordinates system is used by default. The anchor point CAID, however, is always required.

Scaling Factors

Scaling factors in the X, Y, and Z directions can be defined for each slave DTPL card. Thesefactors are always related to the local coordinate system. By playing with the localcoordinate systems and the scaling factors, a wide range of effects can be obtained asillustrated with the figure below.



Pattern Repetition with Draw Direction Constraints

Draw direction constraints can be applied simultaneously with pattern repetition. To achievethis, simply define the draw direction for the master DTPL card, and the draw direction for theslave(s) will automatically be generated based on the local coordinate system.

Even if some components are not naturally identical, the optimized design for each componentwill still satisfy the draw direction constraints. In particular, if different components containdifferent obstacles, the combination of all obstacles will always be considered.

Pattern Repetition with Extrusion Constraints

Extrusion constraints can also be used in conjunction with pattern repetition. This allows forcreating parts which have identical cross-sections. The components do not need to beidentical in a three-dimensional sense; each part can have its own extrusion path.

If the components have different extrusion paths, these paths have to be defined explicitly oneach DTPL card. However, if the components have identical extrusion paths, the paths forthe slave(s) will automatically be computed based on the master's extrusion path.

Example 1

This example shows how pattern repetition may be used to generate the same topology indifferent parts. The first figure shows two similar blocks loaded in two different ways. Theoptimization problem is to minimize the compliance with 30 percent volume fraction.



If we do not use pattern repetition, we can clearly see that the optimized topologies aredifferent, as shown in the figures below:

Same view as in the first figure.

Viewed from behind showing that the turquoise block is hollowed out.

Using pattern grouping, both of the loads on the master (the left hand block in first figure)



and the loads on the slave are taken into account, and the optimized topology is repeated forboth blocks, as shown below:

Example 2

This example shows a simplified wing model.



The internal wing structure consists of 2 spars and 11 ribs. In this example, each rib issubdivided into three sections; the nose section, the center section and the tail section, andeach of these sections is chosen as a topology design region.



The optimization problem is to minimize compliance for 30 percent of the design volumefraction. Here we see the optimized topology when each region is independent.

Pattern repetition is used to group all of the noses together, all of the centers together andall of the tails together, resulting in 3 master pattern definitions, each with 10 slavedefinitions. Notice how different meshes are used for each rib; pattern repetition is meshindependent. Also the wing tapers, so the outboard ribs are shorter and thinner than theinboard ribs, scaling is defined for the slaves so that the pattern fits in the design space.



The optimized topology achieved using pattern repetition is shown below, and we can clearlysee how the same topological layout is repeated for each rib.



Pattern Grouping for Topology Optimization

Pattern grouping is a feature that allows you to define a single part of the domain that shouldbe designed in a certain pattern.

Planar Symmetry

It is often desirable to produce a design that has symmetry. Unfortunately, even if thedesign space and boundary conditions are symmetric, conventional topology optimizationmethods do not guarantee a perfectly symmetric design.

By using symmetry constraints in topology optimization, symmetric designs can be attainedregardless of the initial mesh, boundary conditions, or loads. Symmetry can be enforcedacross one plane, two orthogonal planes, or three orthogonal planes. A symmetric mesh isnot necessary, as OptiStruct will create variables that are very close to identical across theplane(s) of symmetry.

To define symmetry across one plane, it is necessary to provide an anchor grid and areference grid. The first vector runs from the anchor grid to the reference grid. The plane ofsymmetry is normal to that vector and passes through the anchor grid.

To define symmetry across two planes, a second reference grid needs to be provided. Thesecond vector runs from the anchor grid to the projection of the second reference grid ontothe first plane of symmetry. The second plane of symmetry is normal to that vector andpasses through the anchor grid.



To define symmetry across three planes, no additional information is required, other than toindicate that a third plane of symmetry is to be used. The third plane of symmetry isperpendicular to the first two planes of symmetry, and also passes through the anchor grid.

Uniform Element Density

Pattern grouping also provides the possibility to request a uniform element density throughoutselected components.



This pattern group ensures that all elements of selected components maintain the sameelement density with respect to one another.

Cyclical Symmetry

Cyclical symmetry can also be defined through the use of pattern grouping.

With cyclical pattern grouping, the design is repeated about a central axis a number of timesdetermined by the user. Furthermore, the cyclical repetitions can be symmetric withinthemselves. If that option is selected, OptiStruct will force each wedge to be symmetricabout its centerline.

To define cyclical symmetry, it is necessary to provide an anchor grid and a reference grid. The axis of symmetry runs from the anchor grid to the reference grid. It is also necessary tospecify the number of cycles; the repetition angle will be automatically computed.

To add planar symmetry within each wedge, a second reference grid needs to be provided. The plane of symmetry is determined by the anchor grid and the two reference grids.



Pattern Grouping with Draw Direction Constraints

Draw direction constraints can be combined with pattern grouping.

As illustrated below for one-plane symmetry, the sense of the vector defining the symmetryalso determines the primary side of the design space to which the draw direction applies. Thedraw direction for the secondary side of the design space is automatically computed.



Vector defining the symmetry Vector defining the draw direction Primary domain Plane of symmetry Secondary

domain

Vector defining the draw direction for the secondary domain (automatically created)

The same reasoning applies for two- and three-plane symmetries, as well as for cyclicalsymmetry.

Caution should be used in order to achieve manufacturable designs. With cyclical symmetry,for instance, the draw direction should be parallel to the axis of symmetry.

Pattern Grouping with Extrusion Constraints

Currently extrusion constraints cannot be used simultaneously with pattern grouping.

Example 1

In this example, a solid block of material is used. The grids located on the rear of the block(in the YZ plane) are fully constrained. Axial loading is applied to the block's upper edge inthe direction of the negative Y-axis. The objective is to minimize the compliance with aconstraint on the volume fraction. Single-die draw direction constraints are applied in thedirection of the positive Z-axis.

The figures below illustrate the results obtained for various symmetry combinations. As the



loading is not symmetric with respect to the XY and YZ planes, the design is not symmetricalabout these planes when symmetry constraints are not prescribed. Enforcing symmetryconditions about the XY or YZ planes yields significantly different results.

Example 2

Here, solid elements are used to model a car wheel. The outer layers as well as the bolts arenon-designable. Twenty load cases are considered. The objective is to minimize theweighted compliance with a constraint on the volume fraction. Split-die draw directionconstraints are applied in the direction of the X-axis. Cyclical pattern grouping is defined withplanar symmetry within each cycle.



As the results show, a clean and reasonably manufacturable design is achieved. Cyclicalsymmetry is obtained even though the loading is not symmetric.



Combining Pattern Repetition and Pattern Grouping withother Manufacturing Constraints for

Pattern repetition is a feature that allows you to define multiple parts of the model thatshould be characterized by identical or similar designs. Pattern grouping is a feature thatallows you to define a single part of the model that should be designed in a certain pattern.

Pattern repetition and pattern grouping can be used with solid and shell elements. They canalso be applied in conjunction with minimum and maximum member size constraints, with drawdirection constraints, and (to some extent) with extrusion constraints. The followingcombinations are allowed:

Shells

Solids

Simple Draw Extrusion

Pattern repetition

Without scaling Yes Yes YesYes

With scaling Yes Yes YesNo

Pattern grouping

1-plane symmetry Yes YesYes

No

2-planessymmetry

Yes YesYes

No

3-planessymmetry

Yes YesYes

No

Cyclical symmetry Yes YesYes

No

Pattern grouping can be combined with draw direction constraints, but you should usecaution in order to achieve manufacturable designs.

For extrusion, pattern repetition generates identical cross-sections for differentcomponents. Therefore, scaling is not supported for this combination.



Manufacturability for Topography Optimization

Manufacturing methods can place constraints on the types of reinforcement patternsavailable for a given part. Some examples of this are: channels, which must have acontinuous cross-section; discs, which must be turned on a lathe; and stampings, whichcannot have the die lock conditions.

These constraints can be accounted for in topography optimization by using Pattern GroupingOptions, and a design with a manufacturable reinforcement pattern can be generated.

Pattern repetition is a technique that allows different structural components to be linkedtogether so as to produce similar topographical layouts.

To achieve this goal, a master DTPG card needs to be defined, followed by any number ofslave DTPG cards which reference the master. The master and slave components are relatedto each other through local coordinate systems, which are required, and through scalingfactors, which are optional.

Other manufacturing constraints, such as pattern grouping, can be applied to the masterDTPG card. These constraints will then automatically be applied to the slave DTPG card(s).


1. Create a master DTPG card.


3. Define the local coordinate system associated to the master DTPG card.

4. Create a slave DTPG card.

5. Define the local coordinate systems associated to the slave DTPG card.


7. Repeat steps 4-6 for any number of slave DTPG cards.







The definition of the fourth point allows for both right-handed and left-handed coordinatesystems, which facilitates the creation of reflected patterns.





Scaling Factors

Scaling factors in the X, Y, and Z directions can be defined for each slave DTPG card. Thesefactors are always related to the local coordinate system. By playing with the localcoordinate systems and the scaling factors, a wide range of effects can be obtained asillustrated with the figure below.



There are over 70 pattern grouping options and variations available for topographyoptimization. A summary of the major categories is shown below:

Variablegroupingpattern

Pattern Option Type #RequiredVectorDefinitions

Description

None - 0 - Variables grouped as points.

One planesymmetry

- 10 One Reflection of variables acrossone plane normal to firstvector.

Two planesymmetry

- 20 Two Reflection of variables acrosstwo planes, one normal to thefirst vector and one normal tosecond vector.

Threeplanesymmetry

- 30 Two Reflection of variables acrossthree planes, one normal tofirst vector, one normal to thesecond vector and oneperpendicular to both vectors.

Linear - 1 One Variables grouped as linesextending in direction of firstvector.

+1 plane 21 Two Reflection of variables acrossone plane normal to secondvector.

+2 planes 31 Two Reflection of variables acrosstwo planes, one normal tosecond vector and oneperpendicular to both vectors.

Circular - 2 One Variables grouped as circlesaround anchor node lying in aplane normal to first vector.

+1 plane 12 One Reflection of variables acrossone plane normal to firstvector.

Planar - 3 One Variables grouped as planes





Description

extending normal to firstvector.

+1 plane 13 One Reflection of planes acrossplane normal to first vector.

Radial 2-D - 4 One Variables grouped as raysextending radially and normalto first vector.

+1 plane 14 One Reflection of rays across planenormal to first vector.

+2 planes 24 Two Reflection of rays across twoplanes, one normal to the firstvector and one normal tosecond vector.

+3 planes 34 Two Reflection of rays across threeplanes, one normal to firstvector, one normal to thesecond vector and oneperpendicular to both vectors.

Cylindrical - 5 One Variables grouped as endlesscylinders extending along andcentered around first vector.

Radial 2-D& Linear

- 6 One Variables grouped as acombination of radial and linearpatterns.

+1 plane 26 Two Reflection of radial planesacross plane normal to secondvector.

+2 planes 36 Two Reflection of radial planesacross plane normal to bothfirst and second vectors.

Radial 3-D - 7 - Variables grouped as rays





Description

extending radially outwardfrom anchor node.

+1 plane 17 One Reflection of rays across planenormal to first vector.

+2 planes 27 Two Reflection of rays across twoplanes, one normal to the firstvector and one normal tosecond vector.

+3 planes 37 Two Reflection of rays across threeplanes, one normal to firstvector, one normal to thesecond vector and oneperpendicular to both vectors.

Vectordefined

- 8 - Variables grouped alongvectors defined by the drawvectors of the individualnodes.

+1 plane 18 One Reflection of variables acrossplane normal to first vector.

+2 planes 28 Two Reflection of variables acrosstwo planes, one normal to thefirst vector and one normal tosecond vector.

+3 planes 38 Two Reflection of variables acrossthree planes, one normal tofirst vector, one normal to thesecond vector and oneperpendicular to both vectors.

Cyclical* - 40,41 Two Cyclical repetition of variablesabout axis defined by firstvector.

+1 plane 50,51 Two Reflection of variables across





Description

one plane normal to firstvector.

+ linear 60,61 Two Cyclically repeated variablesgrouped as lines extending indirection of first vector.

+ radial 70,71 Two Cyclically repeated variablesgrouped as rays extendingradially and normal to firstvector.

+ radial & linear 80,81 Two Cyclically repeated variablesgrouped as a combination ofradial and linear patterns.

* For cyclical symmetry, the UCYC parameter (field 30) controls the number of repetitions(and thus the repetition angle) for the cycles. If the TYP option selected for cyclicalsymmetry is 40, 50, 60, 70, or 80, the cyclical repetition pattern will be non-reflective. If theTYP option selected for cyclical symmetry is 41, 51, 61, 71, or 81, the cyclical repetitionpattern will be reflective.

These options can be used with shell and solid models to create reinforcement patterns thatobey manufacturing constraints and which conform to the shapes of the parts. Examples ofpattern grouping options are given in the following sections:

Cross-section Optimization of a Spot Welded Tube

Optimization of the Modal Frequencies of a Disc Using Constrained Beading Patterns

Multi-plane Symmetric Reinforcement Optimization for a Pressure Vessel

Shape Optimization of a Stamped Hat Section

Shape Optimization of a Solid Control Arm

Using Topography Optimization to Forge a Design Concept Out of a Solid Block

If no variable grouping pattern is selected, OptiStruct will automatically generate circular beadvariable definitions throughout the design variable domain as shown below:



TYP = 0: No symmetry

For two planes of symmetry (TYP = 20), the planes of symmetry are defined normal to boththe first and second vectors as shown below. Note that the second grid does not have to bein the plane defined by the first vector, OptiStruct will calculate the second vector byprojecting the second grid (or vector) onto the plane defined by the first vector.

TYP = 20: Two planes of symmetry

For three planes of symmetry (TYP = 30), the symmetry plane definitions are identical tothose for two planes of symmetry with the third plane being placed perpendicular to the firsttwo and located at the anchor node as shown below:

TYP = 30: Three planes of symmetry



For a single plane of symmetry (TYP = 10), the plane is defined normal to the first vector andis located at the anchor node as shown below:

TYP = 10: One plane of symmetry

Linear pattern grouping allows you to force OptiStruct to create beads in a given directionalong the entire length of the part. This can be very useful for optimizing the shape ofextruded parts which must maintain a constant cross section. It is also very useful whenoptimizing the side walls of stamped plates whose beads must run from the top to the bottomso that the part can be drawn from the die. In solid models, where the variable needs tocontrol the movement of all grids through the thickness, linear pattern grouping is also veryuseful.

For linear pattern grouping (TYP = 1, 21, or 31), OptiStruct generates shape variables thatrun along a line parallel to the first vector. These shape variables have a width equal to theminimum bead width parameter but have no limit on length. For simple linear pattern grouping,the anchor point and first vector can be located anywhere as shown below:

TYP = 1: Linear pattern grouping

For one and two plane linear symmetry, the anchor point locates the plane(s) of symmetry. For one plane linear symmetry (TYP = 21), the second vector defines the symmetry plane(since the first vector has been used to define the direction of the pattern).



TYP = 21: One plane linear symmetry

For two plane symmetry (TYP = 31), the symmetry planes are defined by the second vectorand the cross product of the first and second vectors as shown below. There is no threeplane linear pattern grouping since the pattern is automatically symmetric in the direction ofthe first vector.

TYP = 31: Two plane linear symmetry

Circular pattern grouping allows you to force OptiStruct to create beads that form concentriccircles around a user-defined axis. This can be very useful for optimizing the shape of circularparts that must have a circular reinforcement pattern such as a part turned on a lathe.

For circular pattern grouping (TYP = 2 or 12), OptiStruct generates shape variables that formcircles about an axis defined by the first vector. These circular beads have a width equal tothe minimum bead width parameter. The anchor point can be located anywhere, but the firstvector must be collinear with the desired central axis for the circular beads. The simplecircular pattern grouping (TYP = 2) is shown below:



TYP = 2: Circular pattern grouping

For one plane circular pattern grouping (TYP = 12), the circular patterns are reflected about aplane located at the anchor node and defined by the first vector. One plane circularsymmetry ensures that nodes equal distances above and below the plane of symmetry will begrouped into the same variables. See below:

TYP = 12: One plane circular symmetry

Planar pattern grouping allows you to force OptiStruct to create variables that consolidatethe perturbations of active grids in a given plane. This can be very useful in forming beadsthat run in a fixed direction across an uneven part or in solid models to control the changes inthe shape of a cross section.

For a planar pattern grouping (TYP = 3 or 13), OptiStruct generates a series of parallel planarshape variables that are defined by the first vector. These shape variables have a widthequal to the minimum bead width parameter, but have no limit on length. Beads formed usingplanar pattern grouping can turn vertical corners. For simple planar pattern grouping, theanchor point and first vector can be located anywhere as shown below:



TYP = 3: Simple planar pattern grouping

For one plane planar symmetry (TYP = 13), the planes are symmetric about a plane located atthe anchor point as shown below. There is no need for two and three plane planar symmetry.

TYP = 13: One plane planar symmetry

Radial (2-D) pattern grouping allows you to force OptiStruct to create beads in a radialdirection extending outward from a central axis. This can be very useful for optimizing circularparts in which radial reinforcements are desired.

For radial (2-D) pattern grouping (TYP = 4 , 14, 24, and 34), OptiStruct generates shapevariables that run radially away from a central axis defined by the first vector. Radial beads,at their closest point to the central axis, have a width equal to the minimum bead widthparameter. The width of the beads increases with distance from the center. There is no limiton the bead length. The anchor point can be located anywhere, but the first vector must becollinear with the desired central axis for the radial beads. The simple radial (2-D) patterngrouping (TYP = 4) is shown below:



TYP = 4: Simple radial (2-D) pattern grouping

For one plane radial (2-D) pattern grouping (TYP = 14), the radial patterns are reflectedabout a plane located at the anchor node and defined by the first vector. One plane radialsymmetry ensures that nodes equal distances above and below the plane of symmetry will begrouped into the same variables. See below:

TYP = 14: One plane radial pattern grouping

For two and three plane radial (2-D) pattern grouping (TYP = 24 and 34), two symmetryplanes are determined by the first and second vectors as shown below.



TYP = 24: Two plane radial (2-D) pattern grouping

TYP = 34: Three plane radial (2-D) pattern grouping

Cylindrical pattern grouping allows you to force OptiStruct to create variables thatconsolidate the perturbations of active grids along the surface of a cylinder. This can beuseful in assigning a circular pattern grouping through the thickness of a solid model.

For a cylindrical pattern grouping (TYP = 5), OptiStruct generates a series of concentriccylinders that run parallel to and are positioned about the first vector. The cylindrical patterngrouping is essentially the linear pattern grouping combined with the circular pattern grouping. The anchor point can be located anywhere, but the first vector must be collinear with thedesired central axis for the cylinders.



Radial linear pattern grouping allows you to force OptiStruct to create variables thatconsolidate the perturbations of active grids along planes running radially from a central axis. This can be useful for assigning radial pattern groupings through the thickness of a solidmodel.

For a radial linear pattern grouping (TYP = 6), OptiStruct generates a series of planes that runradially away from, and in the same plane as, the first vector. The radial linear patterngrouping is essentially the linear pattern grouping combined with the radial pattern grouping. The anchor point can be located anywhere, but the first vector must be collinear with thedesired central axis for the radial planes.

One and two planes of radial linear pattern grouping (TYP = 26 and 36) can be created byusing the second vector to define the planes of symmetry. The symmetry planes areassigned in a manner similar to that for two and three plane radial symmetry.

Radial (3-D) pattern grouping allows you to force OptiStruct to create variables thatconsolidate the perturbations of active grids in a radial direction away from a central point. This can be very useful for optimizing spherical models with solid elements.

For radial (3-D) pattern grouping (TYP = 7, 17, 27, and 37), OptiStruct generates shapevariables that run radially away from a central point defined by the anchor node. Radialbeads, at their closest point to the central axis, have a width equal to the minimum beadwidth parameter. The width of the beads increases with distance from the center. There isno limit on the bead length. The anchor point can be located anywhere, but is ideally locatedat the center of a sphere.

The planes for one, two, and three plane radial (3-D) symmetry are established in a manneridentically to one, two and three plane symmetry without radial (3-D) pattern grouping (TYP =10, 20, and 30).

Vector defined pattern grouping allows you to force OptiStruct to create variables which aregrouped according to the direction and magnitude of the individual draw vectors of the grids.This pattern grouping option is similar to the linear pattern grouping option except that thelinear vector is not constant for the entire model. Instead, the direction of the draw vectorfor each grid is used to determine the variable groupings in place of a global linear vector. Additionally, unlike the linear pattern grouping option, the lengths of the beads are notinfinite. The lengths of the beads are equal to the magnitude of the draw vectors for thegrids. This pattern grouping option can be very effective for optimizing the shape ofamorphous solid models.

For vector defined pattern grouping (TYP = 8, 18, 28, and 38), OptiStruct generates shapevariables by consolidating the perturbations of active grids that are within a cylindrical regionabout evenly spaced grids in the model. For a selected grid, a cylindrical zone of influence iscreated around it which has a radius defined by the minimum bead width and draw angleparameters, a length defined by twice the draw vector for the selected grid, and is oriented inthe direction of the draw vector. See the figure below:



Note that for solid models, the internal grids will move along with the surface grids providedthat the internal grids have draw vectors associated with them. This allows for large scaleperturbations both inward and outward from the surface of a solid part while maintaining anacceptable mesh quality.

Users must create perturbations for all of the grids in the model to effectively use vectordefined pattern grouping. If perturbations are defined for the surface grids only, those gridsmay end up passing through the second layer of grids if the variables are perturbed inward. The best way to use this pattern grouping option is to create a single shape variable byuniformly collapsing all of the grids in a solid model towards the center and then creating aDTPG card which points at the DESVAR for that shape variable.

The cyclical pattern grouping allows you to force OptiStruct to create a series of symmetricshape variables about a central axis that repeat a number of times determined by you (withthe UCYC field). This can be useful in assigning a reinforcement pattern in a circular platethat matches an angularly repeated load in a symmetric fashion.

For cyclical pattern groupings (TYP = 40 and 41), OptiStruct generates a series of symmetricshape variables about an axis defined by the cross product of the first and second vectors. The axis of rotation is positioned at the anchor point. The first vector defines a planeestablishing one side of the cyclical wedge. The other side of the cyclical wedge is definedby the angle of repetition. The figure below shows cyclical pattern grouping for three"wedges".



TYP = 40: Cyclical pattern grouping for 3 repetitions

OptiStruct allows any number of repeated cyclical wedges. The user enters the number ofdesired wedges into field 30 (UCYC). OptiStruct internally calculates the repetition angleaccording to the formula 360.0 / UCYC. For example, setting UCYC to three results in threewedges of 120.0 each, and setting UCYC to 6 results in six wedges of 60.0 each.

You can also control whether the cyclical repetitions will be symmetric within themselves. This is done by choosing one of the cyclical TYP options ending in ‘1’ (41, 51, 61, 71, and 81). If the symmetric wedge option is selected, OptiStruct will force each wedge to be symmetricabout its centerline. Selecting one of the cyclical options ending in ‘0’ (40, 50, 60, 70, and80) will result in the wedges being non-symmetric. See the figures below:

TYP = 40 (non-symmetric) with UCYC = 5 TYP = 41 (symmetric) with UCYC = 3

Other Forms of Cyclical Pattern Grouping

OptiStruct supports the combination of cyclical pattern grouping with one plane symmetry,



linear pattern grouping, radial pattern grouping, and radial linear pattern grouping. Eachoption is specified with a different base TYP number to which the number denoting therepetition angle is added.

For one plane cyclical pattern grouping (TYP = 50 and 51), the cyclical patterns are reflectedabout a plane located at the anchor node and defined by the cross product of the first andsecond vectors. One plane cyclical symmetry ensures that nodes equal distances above andbelow the plane of symmetry will be grouped into the same variables. See below:

TYP = 50: One plane cyclical symmetry with 3 wedges

For linear cyclical symmetry (TYP = 60 and 61), OptiStruct generates shape variables that runalong a line parallel to the cross product of the first and second vectors. These shapevariables have a width equal to the minimum bead width parameter but have no limit onlength. The full lengths of the linearly drawn shape variables will be cyclically symmetric:

TYP = 60: Linear cyclical pattern grouping with 3 wedges

For radial cyclical pattern grouping (TYP = 70 and 71), OptiStruct generates shape variablesthat run radially away from a central axis defined by the cross product of the first and secondvector. Radial beads have a width equal to the minimum bead width parameter but have no



limit on length. The width of the beads does not change depending on the distance from thecenter. The full lengths of the radially drawn shape variables will be cyclically symmetric:

TYP = 70: Radial cyclical pattern grouping with 3 wedges

For radial linear cyclical pattern grouping (TYP = 80 and 81), OptiStruct generates a series ofplanes that run radially away from and in the same plane as the first vector. The radial linearcyclical pattern grouping is essentially the linear cyclical pattern grouping combined with theradial pattern grouping. The full lengths of the radially drawn shape variables will be cyclicallysymmetric.



Manufacturability for Free-size Optimization

A concern in free-size optimization is that the design concepts developed are very often notmanufacturable. Another problem is that the solution of a free-size optimization problem canbe mesh dependent, if no appropriate measure is taken.

OptiStruct offers a number of different methods to account for manufacturability whenperforming free-size optimization:

Member Size Control

Pattern Repetition

Pattern Grouping



Member Size Control for Free-size Optimization

Member size control allows you some control over the member size in the final free-size designand the resulting degree of simplicity therein. This feature may be added one of the twoways described below.

1. The DOPTPRM card:

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

DOPTPRM MINDIM VALUE

Here, only the preferred minimum diameter (width in 2-D) of members may be defined asthe VALUE field, following the MINDIM keyword. A global minimum member size is definedin this way.

2. The DSIZE card:

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

DTPL ID PTYPE PID1 PID2 PID3 PID4 PID5 PID6

PID7 … …

MEMBSIZ MINDIM MAXDIM

Here, both the preferred minimum, MINDIM, and the maximum, MAXDIM, diameter of membersmay be defined on the MEMBSIZ continuation line. Member size dimensions can be defineddifferently for each DSIZE entry in this way.

Minimum Member Size Control

Although minimum member size control penalizes the formation of small members, results thatcontain members significantly under the prescribed minimum member size can still be obtained. This is because a small member in the structure can be very important to the loadtransmission and may not be removed by penalization. Minimum member size control functionsmore as a quality control than a quantity control.

A discrete solution is achieved in two iterative steps. The first step converges to a solutionwith a large number of semi-dense elements. The second step tries to refine this solution toa solution with fully dense members. Each step consists of a number of iterations. The firststep consists of two entire convergence phases - the first run with the initial discretenessvalues (defined by DISCRETE and DISCRT1D parameters on the DOPTPRM bulk data entry),followed by a run with the discreteness values increased by 1.0. This procedure isimplemented in order to achieve a solution with clearly defined members. If this step could



not create a solution with clearly defined members, the preferred minimum member size willnot be preserved in the second step. In which case, the user needs to increase thediscreteness parameters and/or reduce the convergence tolerance (defined by the OBJTOLparameter on the DOPTPRM bulk data entry) to improve the solution of the first phase. Thedefault discreteness is set to 1.0 for 1-D elements, plates and shells, and 2.0 for 3-D solids.

In general, once MINDIM is activated, checkerboarding is controlled by the methods appliedfor this feature, eliminating the need for the CHECKER parameter. In rare circumstances,checkerboards may still be introduced in the second phase described above for 3-D solids. Ifthis happens, an additional checkerboard control algorithm can be activated with theMMCHECK parameter. (The CHECKER and MMCHECK parameters are defined using theDOPTPRM bulk data entry).

The use of this card will assure a checkerboard-free solution, although with the undesired sideeffect of achieving a solution that involves a large number of semi-dense elements, similar tothe result of setting CHECKER equal to 1. Therefore, use this card only when it is necessary.

It is recommended that MINDIM be at least 3 times the average element size for all elementsreferenced by that DSIZE (or all designable elements when defined on DOPTPRM). Theaverage element size for 2-D elements is calculated as the average of the square root of thearea of the elements, and for 3-D elements, as the average of the cubic root of the volume ofthe elements.

This recommendation is enforced when combined with other manufacturing constraints, and ifthe defined MINDIM is less than this value, it will be reset to a default value equal to 3 timesthe average element size.

Maximum Member Size Control

Maximum member size control penalizes the formation of large members. The control is notdirectional, meaning that if the thickness of a member is less than MAXDIM in any direction,this constraint is satisfied.

MAXDIM must be at least twice MINDIM, and hence the minimum mesh requirement is thatMAXDIM has to be at least 6 times the average element size for all elements referenced bythat DTPL. This constraint is strongly enforced and an error termination will occur when thiscriteria is not met. In addition, MAXDIM should be less than half the width of the thinnestpart of the design region.

Based on the constraints mentioned above, a fine mesh is required to achieve good resultswith this manufacturing constraint.

It is to be noted that use of the maximum member size control induces further restriction ofthe feasible design space and should therefore only be used when it is truly desirable. Alsonote that this feature is a new research development, and the techniques are still undergoingimprovement. An undesired side effect that has been noticed for some examples is that itmight result in more intermediate density in the final solution. Therefore, it is recommendedthat this feature be used sparingly until the technology becomes more robust.



Pattern Repetition for Free-size Optimization

Pattern repetition is a technique that allows different structural components to be linkedtogether so as to produce similar topological layouts.

To achieve this goal, a master DSIZE card needs to be defined, followed by any number ofslave DSIZE cards which reference the master. The master and slave components are relatedto each other through local coordinate systems, which are required, and through scalingfactors, which are optional.

Other manufacturing constraints, such as minimum or maximum member size, can be applied tothe master DSIZE card. These constraints will then automatically be applied to the slaveDSIZE card(s) as described in the next sections.


1. Create a master DSIZE card.


3. Define the local coordinate system associated to the master DSIZE card.

4. Create a slave DSIZE card.

5. Define the local coordinate systems associated to the slave DSIZE card.


7. Repeat steps 4-6 for any number of slave DSIZE cards.







The definition of the fourth point allows for both right-handed and left-handed coordinatesystems, which facilitates the creation of reflected patterns.





Scaling Factors

Scaling factors in the X, Y, and Z directions can be defined for each slave DSIZE card. Thesefactors are always related to the local coordinate system. By playing with the localcoordinate systems and the scaling factors, a wide range of effects can be obtained asillustrated with the figure below.



Pattern Grouping for Free-size Optimization

Pattern grouping is a feature that allows you to define a single part of the domain that shouldbe designed in a certain pattern.

Planar Symmetry

It is often desirable to produce a design that has symmetry. Unfortunately, even if the designspace and boundary conditions are symmetric, conventional free-size optimization methods donot guarantee a perfectly symmetric design.

By using symmetry constraints in free-size optimization, symmetric designs can be attainedregardless of the initial mesh, boundary conditions, or loads. Symmetry can be enforcedacross one plane, two orthogonal planes, or three orthogonal planes. A symmetric mesh isnot necessary, as OptiStruct will create variables that are very close to identical across theplane(s) of symmetry.

To define symmetry across one plane, it is necessary to provide an anchor grid and areference grid. The first vector runs from the anchor grid to the reference grid. The plane ofsymmetry is normal to that vector and passes through the anchor grid.

To define symmetry across two planes, a second reference grid needs to be provided. Thesecond vector runs from the anchor grid to the projection of the second reference grid ontothe first plane of symmetry. The second plane of symmetry is normal to that vector andpasses through the anchor grid.



To define symmetry across three planes, no additional information is required, other than toindicate that a third plane of symmetry is to be used. The third plane of symmetry isperpendicular to the first two planes of symmetry, and also passes through the anchor grid.

Uniform Element Thickness

Pattern grouping also provides the possibility to request a uniform element thicknessthroughout selected components.



This pattern group ensures that all elements of selected components maintain the sameelement thickness with respect to one another.

Cyclical Symmetry

Cyclical symmetry can also be defined through the use of pattern grouping.

With cyclical pattern grouping, the design is repeated about a central axis a number of timesdetermined by the user. Furthermore, the cyclical repetitions can be symmetric withinthemselves. If that option is selected, OptiStruct will force each wedge to be symmetricabout its centerline.

To define cyclical symmetry, it is necessary to provide an anchor grid and a reference grid. The axis of symmetry runs from the anchor grid to the reference grid. It is also necessary tospecify the number of cycles; the repetition angle will be automatically computed.

To add planar symmetry within each wedge, a second reference grid needs to be provided. The plane of symmetry is determined by the anchor grid and the two reference grids.



Optimization of Arbitrary Beam Sections

Optimization of Thin-walled Sections

Optimization of thin-walled sections is facilitated through the use of the dimension fields onthe PBARL and PBEAML bulk data entries and the DIM entry within the section definition.

The DIM entry is used in the section definition to link either the thickness of a PSEC entry orthe Y or Z coordinate of a GRIDS entry to the value of the corresponding DIM field of thereferencing PBARL or PBEAML. Should the DIM field on the referencing PBARL or PBEAML bereferenced by a DVPREL, the thickness or grid coordinate in the section definition will beaffected by changes in the design variable.

Solid sections are not designable at this time.

The following is an example of a thin-walled section, where the wall thickness and the ycoordinate of G2 are designable:

$

DESVAR,1,THK,0.1,0.05,0.15

DESVAR,2,G2Y,1.0,0.5,1.5

$

DVPREL1,1,PBARL,1,DIM1,,,0.0

,1,1.0

DVPREL1,2,PBARL,1,DIM2,,,0.0

,2,1.0

$

PBARL,1,1000,HYPRBEAM,SQUARE

,0.1,1.0

$

MAT1,1000,2e5,,0.33,7.85e-9

$

BEGIN,HYPRBEAM,SQUARE

$

GRIDS,1,0.0,0.0

GRIDS,2,1.0,0.0

GRIDS,3,1.0,1.0

GRIDS,4,0.0,1.0

$

CSEC2,10,100,1,2

CSEC2,20,100,2,3

CSEC2,30,100,3,4

CSEC2,40,100,4,1

$

PSEC,100,1000,0.1



$

DIM,1,T,100

DIM,2,G,2,Y

$

END,HYPRBEAM

$



Optimization of Composite Structures

OptiStruct offers a comprehensive optimization solution aimed at guiding and simplifying thedesign of laminate composite structures. The solution includes the following optimizationphases and associated techniques:

Phase I – Concept. Free-sizing optimization is used to generate design concepts, while considering globalresponses and optional manufacturing constraints.

Phase II – System. Sizing optimization – with ply-based modeling – is performed to control the thickness ofeach ply bundle, while considering all design responses and optional manufacturingconstraints.

Phase III – Detail. Ply-stacking optimization is applied to determine the detailed stacking sequence, againwhile considering all behavioral responses, manufacturing constraints and various plybook rules.

While these techniques can be applied independently, it is recommended to use them togetheras a three-phase integrated process guiding the design from concept to completion. This isparticularly important when manufacturing constraints are involved. In order to satisfy suchconstraints at the finishing stage, they should be incorporated at the beginning so that thedesign concept can be carried forward. Automated tools are provided to facilitate thetransition between the design optimization phases.



Composite Structures - Free-sizing Optimization

The purpose of composite free-sizing optimization is to create design concepts that utilize allthe potential of a composite structure where both structure and material can be designedsimultaneously. By varying the thickness of each ply with a particular fiber orientation forevery element, the total laminate thickness can change ‘continuously’ throughout thestructure, and at the same time, the optimal composition of the composite laminate at everypoint (element) is achieved simultaneously. At this stage, a super-ply concept should beadopted, in which each available fiber orientation is assigned a super-ply whose thickness isfree-sized. In other words, a super-ply is the total designable thickness of a particular fiberorientation. In addition, in order to neutralize the effect of ply stacking sequence, the SMEARoption is usually a good choice for this design phase unless it is intended to follow throughwith the stacking preference of the super-ply laminate model.

In OptiStruct, additional manufacturing constraints can be defined for free-sizing optimization. As a composite laminate is typically manufactured through a stacking and curing process,certain manufacturing requirements are necessary in order to limit undesired side effectsemerging during this curing process. For example, one such typical constraint for carbon fiberreinforced composites is that plies of a given orientation cannot be stacked successively formore than 3 or 4 plies. This implies that a design concept that contains areas ofpredominantly a single fiber orientation would never satisfy this requirement. Therefore, toachieve a manufacturable design concept, manufacturing requirements for the final productneed to be reflected during the concept design stage. For the particular constraintmentioned above, for instance, the design concept would offer enough alternative plyorientations to break the succession of plies of the same orientation if the percentage of eachfiber orientation is controlled (e.g. no ply orientation should drop below 15%). In addition,balancing of a pair of ply orientations could be useful for practical reasons. For example,balancing 45° and -45° plies would eliminate twisting of a plate under bending along the 0axis. In order to address these needs, the following manufacturing constraints are madeavailable for composite free-sizing:

Lower and upper bounds on the total thickness of the laminate.

Lower and upper bounds on the thickness of individual orientations.

Lower and upper bounds on the thickness percentage of individual orientations.

Constant thickness of individual orientations.

Thickness balancing between two given orientations.

As for the constraints on laminate thickness, ply thickness and thickness percentage, thesecan be applied locally through the definition of element sets. Therefore, multiple instances ofthese constraints are supported. Advantages include being able to allow different constraintsin different regions while preserving the continuity of plies. For example, different thicknessrequirements in critical regions, such as bolted areas, can be factored into the designprocess. Additionally, zone based free-sizing optimization can be performed. Zones aredefined through groups of elements and there can be elements that do not belong to anyzone. Zones are typically defined to simplify the design interpretation process and improvemanufacturability. Instead of having to interpret manufacturable zones from the solution of afree-sizing optimization, the optimization is run based on pre-defined zones. While theinterpretation process is simplified, there is a loss in design freedom as now the optimization isrestricted to some extent due to the defined zones.



Please refer to the DSIZE card for detailed information regarding composite free-sizingoptimization and its associated manufacturing constraints. Note also that other genericmanufacturing constraints, such as pattern grouping or member size control, can be activatedfor composite free-sizing as well.

The standard result from a free-sizing optimization is the thickness contribution of eachorientation defined on the PCOMP(G) or STACK card referenced by the DSIZE card in theoptimal laminate design. But, using free-sizing as part of the three-phase composite designand optimization process, and the mechanism to automatically generate an input file for phasetwo of this process, an additional level of detail/results can be drilled down to in terms of thethickness contributions per orientation. The automatically generated input file for phase twocontains ply bundle data that can be reviewed in HyperMesh.

A ply bundle is a continuous stack of plies of the same shape (or coverage area). Eachsuper-ply results in the formation of 4 ply bundles. This is the default behavior and can bechanged, i.e. a different number of ply bundles can be requested from the super-plies. However, in most cases, it is recommended to use the default approach.

As described above, multiple ply shapes per orientation (through ply bundles) can bedetermined and generated from a free-sizing optimization.



Composite Structures - Ply-based Sizing Optimization

Ply-based Laminate Modeling

Complimentary to the conventional property-based composite definition, a new ply-basedmodeling technique was introduced in OptiStruct 9.0. In this format, laminates are defined interms of ply entities and stacking sequences, which reflect the native language of ‘ply-book’standards to composite laminate modeling and manufacturing. This is also analogous withhow a laminate composite is manufactured. The PLY card specifies the thickness, orientationand material data for each ply, as well as its layout in the structure. The STACK card ‘glues’the PLYs together to produce the laminate structure. Properties of every zone of uniquelaminate lay-ups are uniquely, albeit implicitly, defined. This allows the user to simply focuson the physical buildup of the composite structure and eliminates the burden associated withidentifying patches (PCOMPs) of unique lay-ups, which can be especially complicated for afree-sizing generated design.

Ply-based Optimization

In property-based sizing optimization, the designable entities are the ply thicknessesassociated with the PCOMP(G) properties. In ply-based sizing optimization, the PLYthicknesses are directly selected as designable entities. This approach greatly simplifies thedesign variables definition, since ply continuity across patches is automatically taken intoaccount.

As with free-sizing optimization, several composite manufacturing constraints are available tocontrol the thickness of the laminate or the thicknesses of specific orientations. Theseconstraints are defined on the DCOMP card and should generally be inherited from theconcept phase. A mechanism exists whereby the composite manufacturing constraintsdefined in the free-sizing phase are automatically carried over into the ply sizing optimizationphase. This is part of the same mechanism that also generates the input file for the plybased sizing optimization (phase 2), containing ply bundles as explained in the section on Phase 1: Free-Sizing Optimization. Through this, the ply bundles are automatically set up foroptimization with the necessary DESVAR and DVPREL cards defined. The ply bundles are nowready to be sized to determine the optimum thickness per bundle per fiber orientation.

In addition, discrete optimization is automatically activated when TMANUF, the thickness ofthe basic manufacturable ply, is specified for the PLY associated with a given design variable.This feature forces ply bundles to reach thicknesses reflecting a discrete number of physicalplies.

Therefore, from a ply bundle sizing optimization, the number of plies required per orientationcan be established.

Typically, additional behavioral constraints such as failure, strain, etc. are added to theproblem formulation at this stage.

To proceed to the final phase, an input file for phase 3 can be automatically generated fromrunning phase 2, i.e. ply bundle sizing optimization.



Composite Structures - Stacking Sequence Optimization

Composite plies are shuffled to determine the optimal stacking sequence for the given designoptimization problem while also satisfying additional manufacturing constraints, defined on the DSHUFFLE card, such as:

The stacking sequence should not contain any section with more than a given numberof successive plies of same orientation.

The 45° and -45° orientations should be paired together.

The cover and/or core sections should follow a predefined stacking sequence.

Using the three-phase process, composite plies are generated from running a discrete plybundle sizing optimization (through TMANUF) in phase 2. Additionally, an input file for phase3, i.e. stacking sequence optimization, is automatically generated from phase 2.

An efficient proprietary technique is developed to allow the process to evaluate a hugenumber of stacking combinations from both performance and manufacturability perspectives.



Phase Transitions in the Optimization of CompositeStructures

In order to simplify the transition between the three design phases, OptiStruct is able toautomatically generate input decks after the free-sizing optimization or the sizing optimizationstages have converged.

OUTPUT,FSTOSZ (free-sizing to sizing) is an output request that can be used during the free-sizing optimization phase to write a ply-based sizing optimization input deck. For eachorientation, the composite interpreter identifies regions of similar thickness and creates PLYsfor these regions. The resulting deck contains PCOMPP, STACK, PLY, and SET cardsdescribing the ply-based composite model, as well as DCOMP, DESVAR, and DVPREL cardsdefining the optimization data. Manufacturing constraints are transferred from the DSIZE cardto the DCOMP card. Typically, additional design responses such as stress/failure constraintswould be introduced at this optimization stage.

OUTPUT,SZTOSH (sizing to shuffling) is an output request that can be used during the ply-based sizing optimization stage to write a ply stacking optimization input deck. Each PLYbundle is divided into multiple PLYs whose thickness is equal to the manufacturable thicknessTMANUF, and the STACK card is updated accordingly. The DESVAR and DVPREL cards fromthe previous stage are removed, and a bare DSHUFFLE card is introduced. As required by thedesign, additional ply-book rules or manufacturing constraints can be defined on theDSHUFFLE card.



Example of Composite Structure Optimization -Rectangular Composite Panel

In this example, a rectangular composite panel is clamped on one side and subjected to a tipload on the other side. This simple model can be used to demonstrate how the three phasesof the composite optimization package interact with each other to ultimately generate amanufacturable design.

Free-sizing Optimization

During the concept phase, the composite panel is modeled with four orientations (0°, 90°, 45°and -45°) of uniform thickness, and the SMEAR option is applied to eliminate stack biasing. Atthis stage, we are minimizing the compliance of the structure while maintaining its volumefraction below 30%. Manufacturing constraints are introduced to limit the total thickness ofthe panel and to ensure that each orientation accounts for at least 10% of the totalthickness. In addition, the thicknesses of the 45° and -45° orientations should be balanced. The resulting DSIZE card is:

DSIZE 1 PCOMP 1

+ COMP LAMTHK 3.2

+ COMP PLYPCT ALL 0.10

+ COMP BALANCE 45.0 -45.0

The optimization converges in seven iterations, after which the ply-based interpreteridentifies thickness zones and generates a ply-based input deck. As illustrated by thefollowing image (Figure 1), four ply bundles are created for each orientation. Figure 2 showsthe thickness of each orientation after free-sizing optimization, while Figure 3 shows theequivalent thicknesses after going through the ply-based interpreter. As expected, thethickness zones are considerably more discrete in the interpreted design.



Figure 1: Ply bundles after ply-based interpretation

Figure 2: Thicknesses after free-sizing optimization



Figure 3:Thicknesses after ply-based interpretation

Ply-based Sizing Optimization

Once a concept design has been established, additional performance measures should beintroduced. In this example, we are changing the formulation to minimize the volume whileconstraining the maximum principal stress in every ply. Also, the composite manufacturingconstraints from the previous stage are preserved and transferred to the DCOMP card.

The optimization converges in 19 iterations, at which point the objective function has beenslightly reduced while satisfying the design constraints and the manufacturing constraints. Figure 4 shows the thickness of each orientation after convergence has been achieved. Notethat, due to the implicit discrete variables formulation, each ply bundle’s thickness is equal toa multiple of TMANUF=0.05.



Figure 4: Thicknesses after ply-based sizing optimization

Ply Stacking Optimization

At this stage, we are keeping the formulation that was introduced in the previous phase,while adding detailed stacking constraints. Specifically, we are requesting that no more thanfour successive plies of same orientation be present in the stack, and that -45° and 45°orientations be paired together in the reverse manner to minimize angle changes. Theresulting DSHUFFLE card is:

DSHUFFLE 1 STACK 1

+ MAXSUCC ALL 4

+ PAIR 45.0 -45.0 REVERSE

The optimization converges in seven iterations, and the resulting stacking sequence strictlysatisfies all constraints.

The image below illustrates how the ply-based sizing optimization and the ply stackingoptimization phases work together once free-sizing optimization has been performed. Figure(a) shows the initial stack for the sizing optimization phase; it consists of four ply bundles foreach orientation as determined by the ply-based interpreter. Figure (b) shows the optimizedstack; the thickness of each ply bundle has changed, and the total thickness has beenslightly reduced to satisfy the manufacturing constraints. Figure (c) shows the initial stackingsequence for the shuffling optimization phase, where the ply bundles have been converted toactual plies. Figure (d) shows the optimized stacking sequence, which now satisfies thedetailed manufacturing constraints as well as the design constraints.



Figure 5: Stacking sequences during sizing and shuffling optimization

Note that, because most plies only cover part of the laminate structure, the stackingsequence for each zone of unique lay-ups is different from the one illustrated above. However, the manufacturing feasibility is evaluated for every individual zone.



Equivalent Static Load Method (ESLM)

The equivalent static load method, originally published by Dr. Park, Hanyang University, is atechnique suitable for optimization of designs undergoing dynamic loads. The method has beenimplemented for the optimization of the following solutions:

Multi-body dynamics problems including flexible bodies.

Nonlinear responses from implicit static analysis, implicit dynamic analysis and explicitdynamic analysis.

The equivalent static load method takes advantage of the well established static responseoptimization capabilities of OptiStruct.

Equivalent Static Load

Figure 1.1: Displacement – time history response

The equivalent static load is that load which creates the same response field as that of thedynamic/nonlinear analysis at a given time step. From Figure 1.1, an equivalent static load iscalculated corresponding to each time step in the time history of the solution, therebyreplicating the dynamic/nonlinear behavior of the system in a static environment.

The calculated equivalent static loads from the analysis (as explained above) are consideredas separate load cases, and these multiple load cases are used in the linear responseoptimization loop. An updated design from the optimization loop is then passed back to theanalysis for validation and overall convergence. The design is validated against the originaldynamic/nonlinear analysis. Based on the outcome of this validation, the solution converges oran updated set of equivalent static loads is calculated for the updated geometry, and theentire process is repeated till convergence. Figure 1.2 is a graphic description of theequivalent static load method for optimization.

Figure 1.2: Equivalent static load method



The equivalent static load method is completely automated in OptiStruct and is an efficientapproach for the optimization of responses from transient, dynamic and nonlinear solutions.

Apart from others, the equivalent static load method offers the following benefits:

It can be applied at the concept design phase as well as design fine tuning phase, i.e.it can be used with topology, free-sizing, topography, size, shape and free shapeoptimization.

A design is optimized for updated loads due to an updated design during theoptimization process.



ESLM for MBD

The equivalent static load method has been implemented for the solution of multi-bodydynamics problems that include flexible bodies. Size, shape, free-shape, topology,topography, free-size, and material optimization can be applied to flexible bodies. Responsesare mass, volume, center of gravity, moment of inertia, stress, strain, compliance (strainenergy), and displacement of the flexible bodies. Displacement responses are taken intoaccount with respect to local boundary conditions defined on the flexible body (see definitionbelow). The optimization problem setup follows the setup typical for size, shape, free-shape,topology, topography, free-size, and material optimizations in OptiStruct. Responses aredefined using DRESP1, DRESP2, and DRESP3 bulk data entries. Responses can only be definedon flexible bodies (PFBODY) using the properties, elements and grid points included in suchbodies as reference. Design variables are defined using DESVAR, DVPREL1, DVPREL2,DVMREL1, DVMREL2, DSHAPE, DTPL, DTPG, DSIZE, and DVGRID bulk data entries. Free-sizeoptimization is currently only available when PTYPE=PSHELL on the DSIZE card. Designvariables can only be defined on flexible bodies (PFBODY) using the properties included in suchbodies as reference. Constraints are defined using DCONSTR, DCONADD and DOBJREF bulkdata entries. Constraints and objectives are referenced in the multi-body dynamics subcaseor globally through DESOBJ, DESSUB, DESGLB, and MINMAX, respectively.

ESLM specific parameters can be set through the DOPTPRM bulk data entry (see Parametersfor the ESLM).

The Method

An optimum solution can be found through a series of static response optimizations with theequivalent static load set, i.e.:

where , u, a, v, and f are the equivalent static load, deformation, acceleration, velocity,and external load, respectively. The steps involved in the ESLM can be summarized as:

Step 1 Initial design.

Step 2 Dynamic analysis.

Step 3 Static response optimization with the multiple subcases that consist of multipleequivalent static load sets.

Step 4 If design converged, Stop. Otherwise go to Step 2.

The iteration at the third step is referred to as an inner iteration; Steps 2 through 4 form theouter loop. The converged solution at Step 3 is the starting point of the next outer loop (ifthe design does not converge at the current outer loop). Note that if there are time steps inthe multi-body dynamics analysis at Step 2, subcases in static response optimization aregenerated at Step 3 (provided that the time step screening option is deactivated). See



Parameters for the ESLM.

Boundary Conditions for Structural Analysis and Optimization

To perform structural optimization with ESL, you must specify boundary conditions for eachflexible body. In the solution of the dynamic analysis, the flexible and rigid bodies areconnected by joints to form a multi-body system. When performing ESLM on the flexiblebodies, these joints are not included in this static subcase based structural optimization. Thismeans that each flexible body will have 6 rigid body modes. The 6 rigid body modes of eachflexible body must be removed for structural analysis. Exactly 6 degrees of freedom (DOF) ofeach flexible body must be fixed to remove the 6 rigid body modes. If more than 6 DOF arefixed in a flexible body, the additional fixed DOF become the constraint of the flexible body,which may not result in an optimal solution and consequently increases the required ESLMouter loops.

Due to the way ESL is calculated, each flexible body is in its equilibrium at the 0-th inneriteration of each outer loop. This is why we can fix an arbitrary 6DOF (usually single node) toget rid of rigid body modes in order to do static analysis with ESL. Reaction force at the fixednode is zero at the 0-th inner iteration. Thus, no additional load is applied to a flexible bodyalthough 6DOF of the flexible body are fixed. However, design changes occur as the inneriteration goes on. This means the original configuration that maintains equilibrium alsochanges. As a result, the equilibrium status does not hold true anymore from the 1st inneriteration in each outer loop. An undesirable effect caused by a broken equilibrium status(disequilibrium status) is the reaction force at the fixed point is not zero anymore, whichmeans additional load is applied to that fixed point. This effect, due to shape/sizechanges, can be minimized by fixing a proper node, as explained below.

A common way to remove the 6 rigid body modes is to fix all 6 DOF of a node. When lookingfor a node to fix, choose one that is not in a high stress region. If the fixed point is locatedin a high stress region, the optimization can be very slow. It is highly recommended that the6 DOF of an independent node of a spider or rigid element (usually used to model joints) beselected to be fixed. On solid models, where the nodes do not have rotational stiffness; if asolid model does not have a rigid element to represent a joint, the next best way to removethe 6 rigid body modes is to fix 3 translational DOF (123) of one node, 2 translational DOF(23) of another node, and 1 translational DOF (3) of a third node. However, still make surethat the 3 nodes are not in a high stress region, otherwise this method of removing 6 rigidbody modes does not work. One way or another, all of the 6 rigid body modes of each flexiblebody must be removed. SPC, SPC1, or SPCADD (referenced in the subcase informationsection) can define the fixed DOF.

The example that follows shows a solid model with joints at the centers of two holes.



Stress contours of this model at two time steps are shown in the following images.



Nodes A and B are the locations of joints. The best option here is to fix 6 DOF of either node(A or B) in order to remove 6 rigid body motions in this model. To fix alternative nodes otherthan node A or B, it would work to fix 3 DOF (123) of node E1, 2 DOF (23) of node E2, and 1DOF (3) of node E3. These three nodes are located in a relatively low stress region. In thismodel, nodes C1, C2, and C3 or nodes D1, D2, and D3 would take a long time to converge iffixed. Again, the best and simplest way to remove 6 rigid body modes of each flexiblebody is to fix one of the joint locations in each flexible body.

When the boundary conditions are properly defined, displacement constraints in anoptimization can be applied to limit the deformation of the flexible body. An example is tooptimize a rotating cantilever. You could fix either the left end or the middle point to retrievedeformation as long as the point has nothing or little to do with the shape perturbationvectors. If a relative displacement at the right end with respect to the left end is to beconstrained, it would be most convenient to fix the 6 DOF of the left end to measure therelative displacement. Constraining a relative displacement of the left end with respect to themiddle point of the cantilever, it would be best to fix the 6 DOF of the middle point. Regardless of whether the left end or the middle point (or even the right end) is fixed, thestress is the same. If the optimization problem is simply to constrain stress or minimize themaximum stress of a flexible body, all that is needed is to fix one proper node of the flexiblebody.

Rotating cantilever

Deformation when the left end is fixed



Deformation when the middle point is fixed

Output Files Generated by the Optimization Process

Once ESL optimization converges, the following output files are available:

.eslout This text file contains brief and useful information about the optimizationprocess. Open this file first to find out what occurred during the ESLoptimization. This file is often enough to understand the overalloptimization process. The following information is stored in this file:

Involved time steps and corresponding subcases.

The number of involved time steps.

Design results.

The number of inner iterations.

CPU time.

_mbd_#.h3d This binary output file contains the MBD analysis results of the #-th outer

loop. Displacement, stress, and deformation are available in this file. Thisfile is a modal h3d format (PARAM,MBDH3D,MODAL), which is only available

format in ESL optimization. HyperView can display this file.

_des_#.h3d This binary output file contains design change of the #-th outer loop. By

selecting the contour button, you can see the design change. HyperViewcan display this file.

_mbd_#.abf This binary output file contains information about rigid body dynamics andmodal participation factors of the #-th outer loop. HyperGraph can displaythe data stored in this file.

Other than above output files, .desvar, .prop, .grid, .fsthick, and .oss will be found, if

available.



Convergence Enhancement for the ESLM

It is important to catch the critical responses of a structure properly when optimizing thestructure. Generally, it is a good practice to define many time steps in the time intervalswhere the critical responses of interest show up. By doing this, the optimizer can considermore precise responses at the critical time intervals, resulting in a decreased probability thatthe optimizer will go in the wrong direction in the current outer loop.

Here is an example that shows how to refine specific time intervals. Suppose the analysistime interval is from 0 seconds to 1.0 seconds. In order to find out at which steps the criticalresponses of interest show up, you could use equi-spaced time steps first,

MBSIM 4 TRANS END 1.0 NSTEPS 100

The above card divides the time interval from 0 seconds to 1.0 seconds into 100 time steps,which is an equi-spaced time step. After analyzing with the above card, you can obtain thefollowing time history for stress: At the time interval of around 0 seconds to 0.02 secondsand 0.60 seconds to 0.63 seconds, critical values of the stress show up. Generally speaking,these critical responses are dominant responses that control the optimization process. It isdesirable to consider more critical responses at around these time intervals. In order toconsider more critical responses at around these time intervals, increase the number of timessteps at these time intervals, i.e.,

$more time steps from 0.0 sec to 0.02 sec.


+ VSTIFF


+ VSTIFF

$more time steps from 0.61 sec to 0.63 sec.


+ VSTIFF


$

MBSEQ 10 1 2 3 4



Time history of a response

In the above MBSIM cards, time steps in the time interval 0 to 0.02 seconds and time interval0.61 and 0.63 seconds have been increased. Generally, the more information the optimizerknows, the better the solution it can provide. It is beneficial to define many time steps;enough to catch precisely the critical responses in the time intervals where critical responseof interest show up. This will enhance convergence of the optimization process.



Parameters for the ESLM

Control the maximum number of outer loops executed in the ESLM by using the followingparameter:

DOPTPRM, ESLMAX

In this case, the ESLMAX is an integer value, greater than or equal to 0. If the ESLMAX is

equal to 0, then optimization process is not activated. The default value for the ESLMAX is

30.

The ESLM has a feature that screens out the time steps that are not dominant during theoptimization. In the most cases, this feature is very helpful to improve the efficiency of theESL optimization process in terms of CPU time. This feature can be activated or deactivatedby using:

DOPTPRM, ESLSOPT

The ESLSOPT can be either 0 or 1. If ESLSOPT is 0, then the screening is deactivated. If

ESLSOPT is 1, then this screening is activated. The default value is 1. Unless deactivated

by DOPTPRM, ESLSOPT, 0, this feature is always activated.

The time step screening feature is different from the constraint screening of the staticresponse optimization in Step 3, although the concepts of both screening capabilities aresimilar. The time step screening is performed at time Step 2, which is right before enteringthe static response optimization phase. By performing the time step screening beforeentering the static response optimization phase, the huge amounts of CPU time required forstructural analysis and subsequent constraint screening strategy can be saved at the 0-thiteration in the static response optimization phase.

Once ESLSOPT is activated, the number of time steps that should be screened out can be

controlled by using:

DOPTPRM, ESLSTOL

ESLSTOL is a real number between 0.0 and 1.0. The smaller the value of ESLSTOL, the fewer

the number of time steps involved in the optimization process. If ESLSTOL is 1.0, all of the

time steps in multi-body dynamic analysis will be involved in the optimization process. Obviously, the smaller the ESLSTOL value, the less total CPU time will be used. Note,

however, that assigning too small a value to the ESLSTOL could cause the optimization

process to diverge. Therefore, if the number of time steps retained by ESLSTOL is less than

10, the 10 most dominant time steps will be involved in the optimization process. The defaultvalue for the ESLSTOL is 0.3.



Output Requests and ESLM

Users are not allowed to control analysis output in ESL optimization. All information from theMBD analysis is already available in _mbd_#.h3d, including displacement, stress, and

deformation. All analysis output requests are invalidated during ESL optimization except thatOutput Request = option is allowed above the first subcase. For example, STRESS = ALL or

DISPLACEMENT = SID is allowed above the first subcase.

Optimization output is not controllable in ESL optimization. See Equivalent Static LoadMethod (ESLM) for more details.

Output and format in output format controls are not controllable in ESL optimization. Allrequests on output and format are invalidated during ESL optimization, except for FSTHICKand OSS.



MBD System Level Response Optimization

The original ESLM implemented in OptiStruct is for the optimization of flexible bodies in MBDsystems through the control of structural responses, such as stress or deformation. Theoriginal ESLM is not capable of handling system level responses, such as joint force of a jointor velocity of a node of a body. System-level responses referred to in this section can bedisplacement, velocities, acceleration, joint force, and functions of those 4 quantities definedby MBVAR. In addition to the original ESLM, MBD system level response optimization is alsoavailable in OptiStruct. ESLM and MBD system level optimization can be combined together tooptimize MBD systems. One of typical optimization formulations that can be solved byOptiStruct could be:

Minimize joint force of a joint

subject to stress < allowable value

deformation < allowable value

velocities of a node < allowable value

MBD system level responses can be controlled by the properties of PRBODY, CMBUSH(M),CMBEAM(M), and CMSPDP(M) as well as usual design variables in structural optimization suchas thickness of bodies, shape of bodies, etc. DVMBRL1 and DVMBRL2 are available to definerelationships between design variables and properties of PRBODY, CMBUSH(M), CMBEAM(M),and CMSPDP(M).

Large Number of Design Variables

MBD system level responses are approximated by adaptive response surfaces. The basicsolution strategy for MBD system level optimization is the same as that implemented inHyperStudy. In general, response surface based optimization cannot handle large number ofdesign variables. This issue has been resolved in OptiStruct through the introduction ofintermediate design variables for bodies. All the design variables defined on rigid/flexiblebodies are converted to some predefined intermediate design variables. Adaptive responsesurfaces are built based on those intermediate design variables. As a result, even if there arehundreds of design variables defined on bodies, adaptive response surfaces for MBD systemlevel responses can be built with only several intermediate design variables. This way,OptiStruct can handle MBD system level responses with large number of design variables.

Change of Length of Bodies as Shape Optimization

When users try to achieve a specific velocity value less than an allowed value, changing thelength of bodies is an efficient way to achieve it. Users can set up shape optimizationproblem to change the length of bodies. The designable bodies can be either rigid bodies orflexible bodies. When defining shape perturbation vectors with DVGRID, users have to makesure that each perturbation vector does not break down the original joint configuration as thedesign changes. For example, a revolute joint must have two coincident nodes attached totwo different bodies. Be aware while defining shape perturbation vectors, that theperturbation vector could cause a location change for only one of the two coincident nodesso that the revolute joint definition would not be valid after one design iteration.



Limitations of MBD System Level Response Optimization

Design variables associated with PRBODY, CMBUSH(M), CMBEAM(M), and CMSPDP(M) cannotcontrol structural responses, such as deformation or stress. They can only control MBDsystem level responses such as MBDIS, MBVEL, MBACC, MBFRC, and MBEXPR. Designvariables associated with structures such as length, shape, and thickness can control MBDsystem level responses. Thus, if interaction between structural responses and MBD systemlevel responses are mutually strong, this feature is not applicable.

Rigid bodies cannot have structural design variables (shape, properties, thickness, etc.) andthe design variables associated with DVMBRL1/2 with TYPE=PRBODY at the same time.

Currently, system level responses must be scalar values. OptiStruct provides an option topick up the maximum, minimum, maximum absolute, or minimum absolute value ofdisplacement/velocity/acceleration/ joint force when defining MBD system level responsesusing the DRESP1 card. If a system level response is an objective function, and the timewhen the objective function is picked up jumps around during the optimization process,convergence could be slow or even diverge in some cases.



ESLM for Nonlinear Response Optimization

The equivalent static load method has been extended to support nonlinear responseoptimization. The following geometric nonlinear analyses are supported for optimization inOptiStruct:

Implicit (quasi-) static analysis (NLGEOM)

Implicit dynamic analysis (IMPDYN)

Explicit dynamic analysis (EXPDYN)

Refer to the Geometric Nonlinear Analysis section for more details on the supported analysistypes.

This method is implemented to support various types of responses that are available in a usualstatic response optimization problem, i.e. displacement, strain, stress, etc., for example. Onlyresponses defined by DRESP1 are currently supported.

Concept level design techniques such as topology, free-sizing and topography optimization,and design fine tuning techniques such as size, shape and free-shape optimization aresupported for nonlinear response optimization using ESLM. In the current implementation, fortopology optimization, PSHELL and PSOLID are supported and PSHELL is supported for free-size optimization.

The optimization setup using ESLM for nonlinear response optimization in terms of designvariables, responses, constraints and objective function is the same as the setup for a typicalstatic response optimization problem for topology, topography, free-size, size, shape andfree-shape optimization.

Output Files Generated by the Optimization Process

In addition to the standard optimization outputs, the following files are output:

[model]_#.h3d: Analysis results for each #th outer loop.

[model].eslout: Outer loop iteration history summary.



Parameters for ESLM

Several parameters are available to control the optimization process:

DOPTPRM, ESLMAX

This parameter controls the maximum number of outer loops.

DOPTPRM, DESMAX

This parameter controls the maximum number of inner loop iterations.

DOPTPRM, NESLNLGM

This parameter controls the number of ESLs generated for each NLGEOM subcase.

DOPTPRM, NESLIMPD

This parameter controls the number of ESLs generated for each IMPDYN subcase.

DOPTPRM, NESLEXPD

This parameter controls the number of ESLs generated for each EXPDYN subcase.

For more details, refer to DOPTPRM parameters.



Gradient-based Optimization Method


Iterative Solution

Sensitivity Analysis

Move Limit Adjustments

Constraint Screening

Discrete Design Variables



Iterative Solution

OptiStruct uses an iterative procedure known as the local approximation method to solve theoptimization problem. This method determines the solution of the optimization problem usingthe following steps:

1. Analysis of the physical problem using finite elements.

2. Convergence test; whether or not the convergence is achieved.

3. Response screening to retain potentially active responses for the current iteration.

4. Design sensitivity analysis for retained responses.

5. Optimization of an explicit approximate problem formulated using the sensitivityinformation. Back to 1.

To achieve a stable convergence, design variable changes during each iteration are limited toa narrow range within their bounds, called move limits. The biggest design variable changesoccur within the first few iterations and, due to an advanced formulation and other stabilizingmeasures, convergence for practical applications is typically reached with only a small numberof FE analyses.

The design sensitivity analysis calculates derivatives of structural responses with respect tothe design variables. This is one of the most important ingredients for taking FEA from asimple design validation tool to an automated design optimization framework.

The design update is generated by solving the explicit approximate optimization problem,based on sensitivity information. OptiStruct has two classes of optimization methodsimplemented: dual method and primal method. The dual method solves the optimizationproblem in the dual space of Largrange multipliers associated with active constraints. It ishighly efficient for design problems involving a very large number of design variables but muchfewer constraints (common to topology and topography optimization). The primal methodsearches the optimum in the original design variable space. It is used for problems thatinvolve equally as many design constraints as design variables, which is common for size andshape optimizations. The choice of optimizer is made automatically by OptiStruct, based onthe characteristics of the optimization problem.

Regular or Soft Convergence

Two convergence tests are used in OptiStruct and satisfaction of only one of these tests isrequired.

Regular convergence (design is feasible) is achieved when the convergence criteria aresatisfied for two consecutive iterations. This means that for two consecutive iterations, thechange in the objective function is less than the objective tolerance and constraint violationsare less than 1%. At least three analyses are required for regular convergence, as theconvergence is based on the comparison of true objective values (values obtained from ananalysis at the latest design point). An exception (design is infeasible) is when theconstraints remain violated by more than 1%, and for three consecutive iterations the changein the objective function is less than the objective tolerance and the change in the constraintviolations is less than 0.2%. In this case, the iterative process will be terminated with aconclusion ‘No feasible design can be obtained.’



Soft convergence is achieved when there is little or no change in the design variables for twoconsecutive iterations. It is not necessary to evaluate the objective (or constraints) for thefinal design point, as the model is unchanged from the previous iteration. Therefore, softconvergence requires one less iteration than regular convergence.



Sensitivity Analysis

The response quantity, g, is calculated from the displacements as:

The sensitivity of this response with respect to the design variable x, or the gradient of theresponse, is:

Two approaches to sensitivity analysis, the direct and adjoint variable method, are possible. Given the equation of motion:

and its derivative with respect to design variable x,

one can calculate the sensitivity of the displacement vector u as:

Using this equation, the largest cost in the calculation of the response gradient is theforward-backward substitution required for the calculation of the derivative of thedisplacement vector with respect to the design variable. This is called the direct method. One forward-backward substitution is required for each design variable.

If constraints are active in more than one load case, and the load is a function of the designvariable (say body force or pressure loads for shape optimization), then the set of forward-backward substitutions must be performed for each active load case. If the loads are not afunction of the design variables, but there are active load cases with multiple boundaryconditions, then the set of forward-backward substitutions must be performed for each activeboundary condition.

For the adjoint variable method of sensitivity analysis, the vector (adjoint variable) a isintroduced, which is calculated as:

Then the derivative of the constraint can be calculated as:

When the adjoint variable method for sensitivity analysis is used, a single forward-backwardsubstitution is needed for each retained constraint. This forward-backward substitution is



needed to calculate the vector a.

There are typically a small number of design variables in shape and size optimization (say 5 to50) and a large number of constraints. The large number of constraints comes from stressconstraints. If there are 20,000 elements, each with a single stress constraint, and 10 loadcases, there are a total of 200,000 possible stress constraints.

There are typically a large number of design variables in topology optimization (between 1 and3 per element) and a small number of constraints. Because stress constraints are not usuallyconsidered in topology optimization, it makes sense that the adjoint variable method ofsensitivity analysis be used for topology optimization (in order to reduce computationalcosts).

For shape and sizing optimization, it is often beneficial to use the direct method for sensitivityanalysis. However, in some cases, when there are a large number of design variables and asmall number of constraints, the adjoint variable method should be used. For example, in atopography optimization, the number of constraints that gradients need to be calculated forcan be reduced using constraint screening. With constraint screening, constraints that arenot close to being violated are ignored. Only constraints that are violated, or nearly violated,are retained. Also, if there are many stress constraints that are retained in a small region ofthe structure, say at a stress concentration, only a few of the most critical need to beretained.

The sensitivities of responses with respect to design variables can be exported to an Excelspreadsheet (see OUTPUT, MSSENS) or plotted in HyperGraph (see OUTPUT, HGSENS). Forcontouring in HyperView, the sensitivities of topology, free-sizing and gauge design variablescan be exported to H3D format (see OUTPUT, H3DTOPOL and OUTPUT, H3DGAUGE,respectively). Sensitivity output in ASCII format for topology and free-sizing variables can berequested through OUTPUT, ASCSENS.

The Excel spreadsheet allows the modification of design variables and then computesapproximated responses. This can be used to make design studies without running OptiStructagain. See the image below.

Example spreadsheet output showing that modification of field C10 yields approximate results in thelower right of the spreadsheet, identified by a border surround here.



Move Limit Adjustments

As the design moves away from its initial point in the approximate optimization problem, theapproximate values become less accurate. This can lead to slow overall convergence, as theapproximate optimum designs are not near the actual optimum design. Move limits on thedesign variables, and/or intermediate design variables, are used to protect the accuracy ofthe approximations. They appear as:

Small move limits lead to smoother convergence. Many iterations may be required due to thesmall design changes at each iteration. Large move limits may lead to oscillations betweeninfeasible designs as critical constraints are calculated inaccurately. If the approximationsthemselves are accurate, large move limits can be used. Typical move limits in theapproximate optimization problem are 20% of the current design variable value. If advancedapproximation concepts are used, move limits up to 50% are possible.

Even with advanced approximation concepts, it is possible to have poor approximations of theactual response behavior with respect to the design variables. It is best to use larger movelimits for accurate approximations and smaller move limits for those that are not so accurate.

Note that the same set of design variable move limits must be used for all of the responseapproximations. It is important to look at the approximations of the responses that are drivingthe design. These are the objective function and most critical constraints. If the objectivefunction moves in the wrong direction, or critical constraints become even more violated, it isa sign that the approximations are not accurate. In this case, all of the design variable movelimits are reduced. However, if the move limits become too small, convergence may beslowed, as design variables that are a long way from the optimum design are forced to changeslowly. Therefore, the move limits on the individual design variables that keep hitting thesame upper or lower move limit bound are increased. Move limits are automatically adjusted byOptiStruct.



Constraint Screening

At each iteration of the optimization process, the objective function(s) and all constraints ofthe design problem are evaluated. Retaining all of these responses in the optimization problemhas two potential disadvantages:

1. This can result in a big optimization problem with a large number of responses and designvariables. Most optimization algorithms are designed to handle either a large number ofresponses or a large number of design variables, but not both.

2. For gradient-based optimization, the design sensitivities of these responses need to becalculated. The design sensitivity calculation can be very computationally expensivewhen there are a large number of responses and a large number of design variables.

Constraint screening is the process by which the number of responses in the optimizationproblem is trimmed to a representative set. This set of retained responses captures theessence of the original design problem while keeping the size of the optimization problem at anacceptable level. Constraint screening utilizes the fact that constrained responses that are along way from their bounding values (on the satisfactory side) or which are less critical (i.e.for an upper bound more negative and for a lower bound more positive) than a given numberof constrained responses of the same type, within the same designated region and for thesame subcase, will not affect the direction of the optimization problem and therefore can beremoved from the problem for the current design iteration.

Consider the optimization problem where the objective is to minimize the mass of a finiteelement model composed of 100,000 elements, while keeping the elemental stresses belowtheir associated material's yield stress. In this problem, we have 100,000 constraints (thestress for every element must be below its associated material's yield stress) for eachsubcase. For every design variable, 100,000 sensitivity calculations must be performed foreach subcase, at every iteration. Because design variable changes are restricted by movelimits, stresses are not expected to change drastically from one iteration to the next.Therefore, it is wasteful to calculate the sensitivities for those elements whose stresses areconsiderably lower than their associated material's yield stress. Also the direction of theoptimization will be driven primarily by the highest elemental stresses. Therefore, the numberof required calculations can be further reduced by only considering an arbitrary number of thehighest elemental stresses.

Of course there is trade-off involved in using constraint screening. By not considering all ofthe constrained responses, it may take more iterations to reach a converged solution. If toomany constrained responses are screened, it may take considerably longer to reach aconverged solution or, in the worst case, it may not be able to converge on a solution if thenumber of retained responses is less than the number of active constraints for the givenproblem.

Through extensive testing it has been found that, for the majority of problems, usingconstraint screening saves a lot of time and computational effort. Therefore, constraintscreening is active in OptiStruct by default. The default settings consider only the 20 mostcritical (i.e. for an upper bound most positive and for a lower bound most negative)constraints that come within 50 percent of their bound value (on the satisfactory side) foreach response type, for each region, for each subcase.

The DSCREEN bulk data entry controls both the screening threshold and number of retainedconstraints. Different DSCREEN settings are allowed for all of the response types supportedby the DRESP1 bulk data entry. Responses defined by the DRESP2 bulk data entry are



controlled by a single DSCREEN entry with RTYPE = EQUA. Likewise, responses defined by theDRESP3 bulk data entry are controlled by a single DSCREEN entry with RTYPE = EXTERNAL. Itis important to ensure that DRESP2 and DRESP3 definitions that use the same region identifieruse similar equations. (In order for constraint screening to work effectively, responses withinthe same region should be of similar magnitudes and demonstrate similar sensitivities, theeasiest way to ensure that is through the use of similar variable combinations).

In order to reduce the burden on the user, it is possible to allow the screening criteria to beautomatically and adaptively adjusted in an effort to retain the least number of responsesnecessary for stable convergence. Setting RTYPE=AUTO on the DSCREEN bulk data entry willenable this feature. Region definition is also automated with this setting. This setting isuseful for less experienced users and can be particularly useful when there are many localconstraints. However, there are some drawbacks; experienced users may be able to achievebetter performance through manual definition of screening criteria, more memory may berequired to run with RTYPE=AUTO, and manual under-retention of constraints will require lessmemory and may, therefore, be desirable for very large problems (even with compromisedconvergence stability and optimality).

Regions and Their Purpose

In OptiStruct, a region is a group of responses of the same type.

Regions are defined by the region identifier field on the DRESP1, DRESP2, and DRESP3 bulkdata entries used to define the responses. If the region identifier field is left blank or set to0, then each property associated with the response forms its own region. The same regionidentifier may be used for responses of different types, but remember that because they arenot of the same type they cannot form the same region.

Example 1

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

DRESP1 1 label STRESS PSHELL SMP1 1

2 3

DRESP1 with ID 1 defines stress responses for all the elements that reference the PSHELLdefinitions with PID 1, 2, or 3. As no region identifier is defined, the stress responses for eachPSHELL form their own regions. So, all of the stress responses for elements referencingPSHELL with PID1 are in a different region than all of the stress responses for elementsreferencing PSHELL with PID2, which in turn are in a different region than all of the stressresponses for elements referencing PSHELL with PID3. If this response definition isconstrained in an optimization problem, and the default settings for constraint screening areassumed, then 20 elemental stresses are considered for each of the three PSHELL definitions,i.e. 20 for each region, giving a total of 60 retained responses.



Example 2

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

DRESP1 2 label STRESS PSHELL 1 SMP1 1

2

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

DRESP1 3 label STRESS PSHELL 1 SMP1 3

All of the stress responses defined in the DRESP1 entries above form a single region - noticethe non-zero entries in field 6 (0 is equivalent to leaving it blank). Now, if these responsedefinitions (which are of the same type (STRESS), with the same non-zero entry in field 6)are constrained in an optimization problem (assuming the default settings for constraintscreening), then 20 elemental stresses are considered in total for the three PSHELL definitionsbecause they form a single region.



Discrete Design Variables

OptiStruct uses a gradient-based optimization approach for size and shape optimization. Thismethod does not work well for truly discrete design variables, such as those that would beencountered when optimizing composite stacking sequences. However, the method has beenadopted for discrete design variables where the discrete values have a continuous trend,such as when a sheet material is provided with a range of thicknesses. The adopted methodworks best when the discrete intervals are small. In other words, the more continuous-likethe design problem behaves, the more reliable the discrete solution will be. For example,satisfactory performance should not be expected if a thickness variable is given two discretevalues 0 and T.

It is known that rigorous methods such as branch and bound are very time consumingcomputationally. Therefore, we developed a semi-intuitive method that is targeted at solvingrelatively large size problems efficiently. It is recommended to benchmark the discrete designagainst the baseline continuous solution. This helps to quantify the trade-off due to discretevariables and to understand whether the discrete solution is reasonable. As local optima arealways a barrier for none convex optimization problems, and discrete variables tend toincrease the severity of this phenomenon, it could be helpful to run the same design problemfrom several starting points, especially when the optimality of a solution is in doubt.

It is also possible to mix these discrete variables with continuous variables in an optimizationproblem.

Discrete design variables are activated by referencing a DDVAL entry on a DESVAR card.

The DDVOPT parameter on the DOPTPRM card allows you to choose between a fully discreteoptimization or a two phased approach where a continuous optimization is performed first, anda discrete optimization is started from the continuous optimum.



Global Search Option

A common discussion that arises when an optimization problem is solved is whether or not theobtained optimum is a local or global optimum. Local approximation based methods (gradient-based optimizations) are more susceptible to finding a local optimum, while globalapproximation methods (response surface methods) and exploratory techniques (geneticalgorithms) are less susceptible than local approximation based methods to finding a localoptimum. In other words, these techniques improve the chances of finding a more globaloptimum. However, no algorithm can guarantee that the optimum found is in fact the globaloptimum. An optimum can be guaranteed to be the global optimum only if the optimizationproblem is convex. For a convex optimization problem, the objective function and feasibledomain need to be convex. Unfortunately, in reality, most engineering problems being solvedcannot be shown to be convex. Therefore, for practical problems, a global optimum remainselusive. Different algorithm types simply alter one’s chances of finding a more global optimum,not guarantee it. With that consideration, it is important to keep in mind that algorithmswhich improve the chances come at a computational cost. And most often this can besignificant.

The following image illustrates the concept of a convex problem as discussed above. Aconvex optimization problem has just one minimum (or maximum). This minimum (Point A inthe image) is the global minimum.

Convex function, f(x)

In the case of non-convex problems solved using gradient-based techniques, the inherentbehavior is that the optimized result obtained is dependent on the initial design starting point. This makes these types of algorithms all the more susceptible to finding local optimum. Recently implemented in OptiStruct version 11.0, is a new global search algorithm – an



extension to the gradient-based optimization approach. The approach is called MultipleStarting Points Optimization. This global search algorithm performs an extensive search of thedesign space for multiple starting points to improve the chances of finding a more globaloptimum. Being dependant on the initial design starting point, n different design startingpoints could potentially result in n different optimum solutions. It is also highly likely thatdifferent design starting points could result in the same optimum solution. However, this doesnot mean that the optimum solution found is the global optimum. This concept is illustrated inthe following image.

Non-convex function, f(x)

Consider the non-convex function, f(x), bounded by –a < x < b. Optimizing a design fromdesign starting point A will result in the optimum solution, P. Similarly, optimizing a designfrom starting point B will result in the same optimum solution, P. On the other hand,optimizing a design from initial design starting point C will result in the optimum solution, Q. From this, it can be seen that through the multiple starting points approach, a global optimumcannot be guaranteed (as with any other algorithm), but at the same time, the chances offinding a more global optimum are improved.

As of version 11.0, the Global Search Option (GSO) in OptiStruct supports those optimizationdisciplines with user-defined variables.

Identifying a global search optimization study is done through the DGLOBAL entry in the I/Osection of the input deck, and the parameters required to setup and run a GSO are defined onthe DGLOBAL bulk data entry.



Design Interpretation - OSSmooth

OSSmooth is a semi-automated design interpretation software, facilitating the recovery of amodified geometry resulting from a structural optimization, for further use in the designprocess and FEA reanalysis. The tool has two incarnations: a standalone version that comeswith the OptiStruct installation, and a dependent version that is embedded in HyperMesh.

OSSmooth can be used in three different ways: OSSmooth for geometry, FEA topologyreanalysis, and FEA topography reanalysis.

OSSmooth (for geometry) has several uses and can be used to:

Interpret topology optimization results, creating an iso-density boundary surface (Iso-surface).

Interpret topography optimization results, creating beads or swages on the designsurface.

Recover and smooth geometry resulting from a shape optimization.

Reduce the amount of surface data from a given set of triangular patches bycombining smaller patches.

Smooth surface data given as triangular patches.

For FEA topology reanalysis and FEA topography reanalysis, OSSmooth can be used to:

Preserve component boundaries for multiple design components.

Recover geometry with or without an artificial layer of elements around a non-designspace optionally.

Tetramesh Iso-surfaces ‘by property’.

Preserve boundary conditions upon geometry recovery to enable quick reanalysis.

The following flowchart provides an overview of how OSSmooth works to interpretoptimization results from OptiStruct:



Each of the three applications of OSSmooth has a corresponding sub-panel in the OSSmoothpanel in HyperMesh. OSSmooth (for geometry) is generally used to recover geometry byinterpreting topology, topography, and shape optimization results, while FEA topology and FEAtopography are used to generate recovered geometry with boundary conditions for FEAreanalysis.

OSSmooth (for geometry) requires a parameter file (generally has the file extension .oss) to

run. This parameter file may be generated from the OSSmooth panel in HyperMesh, or it maybe generated manually through a text editor. At the completion of an optimization run,OptiStruct automatically exports an OSSmooth parameter file <prefix>.oss with certain

default settings depending on the type of optimization run.

In addition to the parameter file, OSSmooth (for geometry) also requires the input file (<prefix>.fem), the shape file (<prefix>.sh), and/or the grid file (<prefix>.grid) from an

OptiStruct run. The grid file <prefix>.grid contains the grid point locations after a

topography or shape optimization and is output at the end of a topography or shapeoptimization run. The shape file, <prefix>.sh, contains the element density information of a

topology optimization and is output at the end of a topology optimization run.

FEA topology requires the input model (<prefix>.fem) to be loaded into HyperMesh before

running, which is different from OSSmooth (for geometry). It also requires the shape file (<prefix>.sh) generated by a topology optimization. For processing of the non-design



elements, two options (Keep sm ooth narrow layer around and Split a ll quads) areprovided to recover geometry.

FEA topography requires a grid file (<prefix>.grid) to run. Similar to FEA topology, it also

requires that the input model (<prefix>.fem) be loaded into HyperMesh first, with the option

for iso-surface that performs the same functionality as FEA topology.

Note: OSSmooth currently does not recognize OptiStruct long-format input data. A possiblework-around for this problem is to import the long-format input file into HyperMesh andexport it using the regular OptiStruct template before running OSSmooth.

The interpreted design from OSSmooth can be exported as a finite element mesh in thebulk data format, as IGES surfaces, as a stereolithography file, or as a Hyper3D file.



OSSmooth Parameter File

The OSSmooth parameter file is composed of a number of parameter statements, each ofwhich has the following format:

parameter_name arg1,arg2,...,argn

The parameter_name and arguments can be separated either by spaces or commas. The file

is not case sensitive.

Comment lines in the OSSmooth parameter file should start with either ‘#’ or ‘$’.

The following is a list of allowable parameters and their respective arguments:

Parameter Description

input_file Identifies the files to be interpreted by OSSmooth.

arg1 The file name (without extension) of the OptiStruct .fem, .sh,

and/or .grid files to be interpreted by OSSmooth.

output_file Name of the file to be output by OSSmooth.

arg1 Full name of the file output by OSSmooth.

output_code Identifies the type of output.

Argument

Description

arg1 Output format for iso-surface:

1 – Bulk data trias2 – IGES patches3 – STL trias [default]4 – H3D trias

arg2 Output control for tet-meshing of volume enclosed by the iso-surface.

[Default: no tet-meshing]

1 – Tetra4 + Tria3 elements2 – Tetra10 + Tria6 elements3 – Tetra4 elements4 – Tetra10 elements

The tet-mesh is written in OptiStruct input format to a file <input_file>_mesh.fem. In this file, the design space of the

file <input_file>.fem is replaced with a tet-mesh for

immediate reanalysis. Loads, boundary conditions, and non-design elements are carried over from the input file.



units Defines output units for IGES format. This information gets written tothe header of the IGES file and may be recognized by your CAD system.

1 – inch [default]2 – mm4 – foot6 – m10 – cm

autobead Improves the recovered geometry from a topography optimization byapplying automatic geometry creation.

Argument

Description

arg1 Operation flag [integer]:

0 – autobead off1 – autobead on [default]

arg2 Threshold value for creating autobead.

[real, between 0.0 and 1.0, default = 0.3]

arg3 Bead layer [integer]:

1 – create 1 layer bead [default]2 – create 2 layers bead

isosurface Generate threshold surface from a topology optimization by applyingautomatic geometry creation.

Argument

Description

arg1 Operation flag [integer]:

0 – isosurface off1 – isosurface on [default]

arg2 Type of surface created [integer]:

0 – isosurface only1 – isosurface with Optimization-based smoothing2 – Element threshold surface3 – isosurface with Laplacian smoothing [default]

For topology results, smoothing maintains the topology assuggested by OptiStruct, but it can deviate from the givendensity distribution. If option 1 or 3 is used, check themaximum and average smoothing error output by OSSmooth.



arg3 Density threshold for creating isosurface.

[real, between 0.0 and 1.0, default = 0.3]

opti_smoothing Optimization-based smoothing [Only used if isosurface C2=1].

Argument

Description

arg1 Unit-less surface distance coefficient

[real, default = 0.0]

Defines closeness of the smooth surface from the thresholdsurface. The effect of this coefficient varies for differentinput meshes. Higher magnitudes (both positive and negative)give smoother results, but the surface deviates more from theoriginal density distribution. The recommended range is from -50 to 50. When the coefficient is set between 0 and 50, thesurface usually tends to smooth and shrink. When thecoefficient is set between 0 and -50, the surface usuallytends to smooth and expand.

arg2 Smooth isosurface boundary flag [integer]:

0 – boundary not included in smoothing [default]1 – boundary included in smoothing

laplacian_smoothing

Laplacian smoothing [Only used if isosurface C2=3].

Argument

Description

arg1 Number of iteration for Laplacian smoothing

[integer > 0, default = 10]

arg2 Feature angle threshold in degrees


The feature angle is defined as the angle of normal betweentwo intersected element planes. All corners with a featureangle larger than the threshold will be preserved in thesmoothing process.

arg3 Smooth isosurface boundary flag [integer]:

0 – boundary not included in smoothing1 – boundary included in smoothing [default]

remesh Remesh autobead surface and/or isosurface flag [integer]:



0 – remesh off [default]1 – remesh on

Remesh detects 2-layer elements around bead shape and/or boundary ofisosurface. Use mixed type remesh if input mesh contains any QUADelements, otherwise remesh with TRIA elements.

surface_reduction

Reduces the number of surfaces representing the geometry. Can reducethe number of surfaces by up to 80%.

Argument

Description

arg1 Surface reduction flag [integer]:

0 – no surface reduction [default]1 – do surface reduction



The feature angle is defined as the angle formed by thesurface normal of two adjacent elements. The surfacereduction will be performed on any two adjacent elements inwhich the feature angle between the two elements is smallerthan the threshold. The greater the threshold, the moresurface reduction will be conducted. The valid range of thethreshold is [1.0, 80.0].

pure_surf_smoothing

Surface smoothing only.

Argument

Description

arg1 Pure surface smoothing flag [integer]:

0 – no surface smoothing [default]1 – Optimization-based smoothing2 – Laplacian smoothing

arg2 Number of iteration [Only used if G1=2]

[integer > 0, default = 10]

arg3 Feature angle threshold in degrees [Only used if G1=2]


The feature angle is defined as the angle of normal betweentwo intersected element planes. All corners with a featureangle larger than the threshold will be preserved in thesmoothing process.



pure_surf_reduction

Surface reduction only.

Argument

Description

arg1 Pure surface reduction flag [integer]:

0 – no surface reduction [default]1 – do surface reduction



The feature angle is defined as the angle formed by thesurface normal of two adjacent elements. The surfacereduction will be performed on any two adjacent elements inwhich the feature angle between the two elements is smallerthan the threshold. The greater the threshold, the moresurface reduction will be conducted. The valid range of thethreshold is [1.0, 80.0].

OSS Example Input File

Parameter Description

input_file example Identifies the root of the input files as example, soOSSmooth will look for the files example.fem, example.grid

, and example.sh.

output_file example.stl The resulting output will be example.stl.

output_code 3 The output will be in stereolithography format.

Autobead 1 0.3 1 Topography results will be interpreted using the autobeadfeature with a threshold value of 30% creating single depthbeads.

Isosurface 1 3 0.3 Topology results will be interpreted by creating an iso-density boundary surface with at a density value of 30%and smooth using laplacian smoothing.

laplacian_smoothing 10 30 1

The Laplacian smoothing will run for 10 iterations, consider afeature angle of 30-degrees and including the boundary inthe smoothing.

Remesh 1 The two rows of elements around the recovered geometrywill be remeshed in an attempt to smooth the meshtransition.



Running OSSmooth

To run OSSmooth from the command line, type:

ossmooth <prefix>.oss

To run OSSmooth from the HyperMesh solver panel:

1. Select Solver from the Tools menu.

2. Click the switch and select OSSm ooth.

3. After input file=, enter <prefix>.oss.

4. Click solve.

Note: OSSmooth standalone, which can also be invoked from the solver panel inHyperMesh, checks out 50 HyperWorks Units.

To run OSSmooth from the HyperMesh ossmooth panel:

1. Select the ossm ooth panel on the post page.

2. Choose either OSSm ooth (for geom etry), FEA topology or FEA topography.

3. Select the OptiStruct input file (<prefix.fem> and/or <prefix.sh> and/or <prefix.

grid>) using the file= browser.

4. Edit the OSSmooth input data by making selections on the screen.

5. Click ossm ooth.

6. OSSmooth (for geometry) will write a new <prefix.oss> file with the screen settings and

load the geometry recovered into HyperMesh if the data format is IGES, STL, or Nastran.FEA topology and FEA topography will update the model in HyperMesh without outputtingthe result; if required, data can be exported from HyperMesh.

Note: OSSmooth invoked from the ossmooth panel in HyperMesh checks out 42HyperWorks Units (21 leveled and 21 stacked).

To run OSSmooth from the HyperWorks Run Manager:

1. Select the .oss file using the Input file(s) browser.

2. The Options field must be empty.

3. Click Run.



Interpretation of Topology Optimization Results

The purpose of this functionality is to provide an iso-density surface based on the volumetricdensity information of a topology optimization, which is conducted using OptiStruct.

OSSmooth can handle both shell and solid elements with the same parameter setting. Oneexample of post-processing of shell element topology optimization is shown below with thefollowing parameter setting in the OSSmooth parameter file:

#general parameters

input_file mattel

output_file mattel.stl

output_code 3

#specific parameters

isosurface 1 3 0.300

laplacian_smoothing 10 30.000 1

surface_reduction 1 10.000

Surface reconstruction of shell element topology optimization.

The parameter laplacian_smoothing is used for additional smoothing. In most cases, the

threshold surface (isosurface with second argument 0) already creates a smooth shape.

Additional smoothing (isosurface with second argument 3) maintains the topology as

suggested by OptiStruct, but it can deviate from the given density distribution. If this option



is used, the maximum and average smoothing error output by OSSmooth should be checked. The surface_reduction parameter is used to reduce the number of elements.



Laplacian Smoothing

Laplacian smoothing can be used in the smoothing of the results of topology optimization. The laplacian_smoothing statement controls the iteration number of when the Laplacian

smoothing will be performed and the feature angle threshold to preserve normal discontinuityat corners. One smoothing result is shown below with the following parameter setting in theOSSmooth parameter file.

#general parameters

input_file surf

output_file surf.stl

output_code 3

isosurface 1 3 0.300


laplacian_smoothing 10 30.000 1

Laplacian smoothing creates smooth boundary iso-surface by entering 1 as the 3rd argumentof the laplacian_smoothing parameter statement. The comparison of the following two

figures shows that the second figure is almost ready for casting.

Fix boundary of iso-surface.



Smooth boundary of iso-surface.

The advantages of the laplacian_smoothing statement in OSSmooth include:

The flexibility of controlling the number of smoothing iterations to obtain differentdegrees of smoothing (possibly a smoothing quality ready for casting). Normally, theiteration number ranges from 5 to 20.

Smooth boundary of iso-surface with feature angle constrain are seamlesslyincorporated into the smoothing process, which is more challenging in a pure CADsystem.



Interpretation of Topography Optimization Results

The autobead feature of OSSmooth allows OptiStruct topography optimization results to beinterpreted as one or two level beads. The following figure shows the level of detail capturedin both cases; while the 2-level approach captures more details, it is more complicated tomanufacture than the 1-level interpretation, often without significant performance gain.

Autobead interpretation of topography optimization result.

One example of post-processing of topography optimization is shown below with the followingparameter setting in the OSSmooth parameter file:

#general parameters

input_file decklid

output_file decklid.fem

output_code 1


autobead 1 0.300 1

remesh 1



Autobead result from topography optimization.

Some topography performances are relying on the half translation part. OSSmooth caninterpolate topography optimization results to 2-layer autobead (autobead third argument 2).Here is one example of creating 2-layer autobead with the following parameter setting in theOSSmooth parameter file:

#general parameters

input_file decklid

output_file decklid.nas

output_code 1


autobead 1 0.300 2

2-layer autobead result from topography optimization.



Shape Optimization Results, Surface Reduction andSurface Smoothing

OSSmooth may also be used to reduce and smooth surfaces or the surfaces of a domain. Theparameter statements pure_surf_reduction and pure_surf_smoothing may be used for

this purpose.

The file defined by input_file must be in the OptiStruct bulk data format, and OSSmooth

can smooth the surface or the surfaces of a domain of the model.

The usage of in the OSSmooth parameter file is as follows:

#general parameters

input_file surf

output_file surf.stl

output_code 3


pure_surf_smoothing 2 10 30.000

pure_surf_reduction 1 10.000



FEA Topology for Reanalysis

The purpose of this functionality is to provide an iso-density surface based on the volumetricdensity information from a topology optimization. Through tetrameshing for 3-D models andinheriting boundary conditions, the results from FEA topology can be used for quick reanalysis.

FEA topology can handle both shell and solid elements. For 3-D models, the recovered iso-surface can be tetrameshed-by-property automatically. FEA topology provides two optionsfor the processing of non-design elements: Keep smooth narrow layer around and Split allquads. Keep smooth narrow layer around will retain an artificial layer of elements aroundthe non-design space in the interpretation, and Split all quads will split quad elements in thenon-design space, if present, to generate a tetra connection between design and non-designregions. Finally, FEA topology preserves boundary conditions by inheriting them from theoriginal model (<prefix>.fem). Those boundary conditions unattached to nodes/elements

after geometry recovery are deleted to ensure reanalysis.

An example of FEA topology for reanalysis is shown below with the following input datadefinition:

file block

density threshold 0.300

Keep smooth narrow layeraround

off

Split all quads on



Result of FEA topology for reanalysis

The same model is run, this time with Keep smooth narrow layer around on, and Split allquads off. This approach creates a layer of elements around the non-design region andpyramids around the quad elements, if quads exist, to connect to the design spacetetrahedral elements.

Result of FEA topology with a layer of elements around non-design space

Tetramesh will be applied on the iso-surface result if there is one close volume at least. Theadvantages of the tetramesh in FEA topology include:

Tetramesh can be performed by property.

The flexibility of controlling the number of tetramesh retries by perturbing the densitythreshold value, in cases where tetramesh sometimes fails.



FEA Topography for Reanalysis

The FEA topography option in OSSmooth allows the results from an OptiStruct topographyoptimization to be interpreted as one or two level beads and recover boundary conditionsupon geometry extraction. An option for iso surface is also provided for combined use, whichperforms the same functionality as FEA topology, with FEA topography. The following figureshows the level of detail captured in a 1-level bead and 2-level bead case while preservingboundary conditions for quick reanalysis.

The input data definition for a 1-level Autobead extraction is as follows:

Grid file brkt

Threshold 0.300

Layers 1

1-level autobead result from FEA Topography

A 2-level Autobead extraction is activated with the following input data:

Grid file brkt

Threshold 0.300

Layers 2



2-level autobead result from FEA Topography



Examples

The following examples can be found in this section:

Example Problems for Topology Optimization

Example Problems for Topology Optimization Using Minimum Member Size Control

Example Problems for Topography Optimization

Example Problems for Topography Optimization Using Pattern Grouping

Example Problems for Size Optimization

Example Problems for Shape Optimization

Examples for Optimization with ESLM



Example Problems for Topology Optimization

This section presents optimized topology examples generated using OptiStruct. Each exampleuses a problem description, execution procedures, and results to demonstrate how OptiStructis used as a design concept tool.

Introductory Example

Two-dimensional Michell-truss

Suspension Bridge

Cantilever Beam in Bending

Air Conditioner Bracket



Introductory Example of Topology Optimization

To develop a new design for a socket wrench, you are given:

The design space

The maximum forces that may develop during use

A mass target

A finite element model representing the design space is created and the loads and boundaryconditions are applied.

Design Space.



FE model of design space with loads and constraints applied.

The end of the handle is assumed to be fixed. Forces simulating service conditions areapplied at the tip. The mass of the optimized design and other parameters are specifiedbefore topology optimization. OptiStruct initializes the design space by distributing theavailable material evenly to all elements.

OptiStruct uses an iterative process to determine the optimum distribution of material usingthe Solid Isotropic Material with Penalization (SIMP) method. In the density method, the rho-mat is used directly as the density design variable. The material density varies continuouslybetween 0 and 1. These represent the void and solid states respectively for each element.The stiffness of the material is assumed to be linearly dependent on its density.

Areas of intermediate density are fictitious when looked at from a practical standpoint. This isdue to the fact that the presence or absence of material at a particular region is deducedfrom the density being either 1 or 0, respectively. All other values of density are practicallymeaningless. Therefore, the intermediate densities are penalized to force the final design tobe represented by densities of only 0 or 1 for each element. Ideally, on convergence, thefinite element model is transformed into a structure with elements having either a density near1.0 (representing material) or 0.0 (representing no material). The load paths in the structureare prominently displayed at the end of the optimization.

Structure of the socket wrench after 22 iterations.

A design for the socket wrench is generated using the OptiStruct solution. The solutionshows major load paths running from the tip, which is loaded with a torque and surfacepressure, to the rigidly held areas farthest on the handle. Translators provided withOptiStruct can be used to generate IGES surfaces which can be imported into any CADsystem for developing a complete design concept. A finite element model of the concept canalso be developed for analysis and, if necessary, further optimization using size and shapeoptimization methods can be performed.



Two-dimensional Michell-truss

The two-dimensional Michell-truss is an optimal topology structure generated under bending. The design space for this problem consists of a rectangle with a single vertical load at thefree end. A circular cut-out is constrained in all translational degrees-of-freedom on theinside free edge. This is a compliance minimization problem with a material volume fractionconstraint of 20%. CQUAD4 (4-noded isoparametric) elements are used in a design spacedefined by a rectangular region with an aspect ratio approximately equal to one quarter of theshort edge located closer to the edge away from the load.

Subcase Section

The objective function (compliance) is a subcase dependent response, therefore the responsereference is part of the subcase definition. The constraint (volume fraction) is a globalresponse, therefore the reference is outside the subcase.

DESGLB = 2

$

SUBCASE 1

SPC = 1

LOAD = 2

DESOBJ = 1

Bulk Data Section

The responses and constraints are defined in the bulk data section. Two responses aredefined here, the compliance (which is referenced by the objective function), and the volumefraction, referenced by the constraint statement to put up an upper bound of 0.2 (20% of thedesign space volume). The constraint statement is then referenced as a global constraint inthe subcase section.

BEGIN BULK

$

DRESP1,1,comp,COMP

DRESP1,2,volfrac,VOLFRAC

DCONSTR,2,2,,0.2



Finite element mesh of the design space for the two-dimensional Michell-truss.

Execution

This example is analyzed using the one-file setup with the file, michell.fem. The OptiStruct

batch job is submitted using the command shell script, % optistruct michell.

Results

The optimization converges in 29 iterations. The results are requested in HyperMesh binaryformat and written to the file, michell.res. The shape of the solution at the final iteration

is visualized by creating a contour plot of the density results at the 29th iteration in theHyperMesh contour panel.



Contour plot of density results for the two-dimensional Michell-truss.

For the input file sample, see <install_directory>/demos/hwsolvers/optistruct/

michell.fem.



Suspension Bridge

The suspension bridge topology is an optimal structure generated under a distributed load. Afine mesh is generated to simulate the design space and loads are applied. The distributedload forms a single load case.

Subcase Section

The objective function (compliance) is a subcase dependent response, therefore the responsereference is part of the subcase definition. The constraint (volume fraction) is a globalresponse, therefore the reference is outside of the subcase.

DESGLB = 2

$

SUBCASE 1

SPC = 1

LOAD = 2

DESOBJ = 1

Bulk Data Section

The responses and constraints are defined in the bulk data section. Two responses aredefined here: the compliance, which is referenced by the objective function, and the volumefraction, which is referenced by the constraint statement to put up an upper bound of 0.2(20% of the design space volume). The constraint statement is then referenced as a globalconstraint in the subcase section.

BEGIN BULK

$

DRESP1,1,comp,COMP


DCONSTR,2,2,,0.2

Loads and constraints for suspension bridge.

Execution

This example is analyzed in the one-file setup with the file, bridge.fem. The OptiStruct

batch job is submitted using the command shell script, % optistruct bridge.



Results

The optimization converges in 24 iterations. The solution is well defined with discrete trussmembers connecting the load carrying arch to the load applied points. The results arerequested in HyperMesh binary format and written to the file, bridge.res. The shape of the

solution at the final iteration is visualized by creating a contour plot of the density results atthe 24th iteration in the HyperMesh contour panel.

Design topology for suspension bridge with all loads weighted equally.

For the input file sample, see <install_directory>/demos/hwsolvers/optistruct/bridge.

fem.



Cantilever Beam in Bending

This example demonstrates how OptiStruct is used to create optimized design concepts froma solid block of material. The design space consists of a cantilever beam loaded at the mid-section of the free end.

Subcase Section

The objective function (compliance) is a subcase dependent response, therefore the responsereference is part of the subcase definition. The constraint (volume fraction) is a globalresponse, therefore the reference is outside the subcase.

DESGLB = 2

$

SUBCASE 1

SPC = 1

LOAD = 2

DESOBJ = 1

Bulk Data Section

The responses and constraints are defined in the bulk data section. Two responses aredefined here: the compliance, which is referenced by the objective function, and the volumefraction, which is referenced by the constraint statement to put up an upper bound of 0.25(25% of the design space volume). The constraint statement is then referenced as a globalconstraint in the subcase section.

BEGIN BULK

$

DRESP1,1,comp,COMP


DCONSTR,2,2,,0.25

The volume fraction constraint is set to 25% of the total design material. In the beam.fem

file, the following PSOLID entry is used:

(1) (2) (3) (4) (5) (6) (7) (8) (9)

PSOLID 1 1 1

The 1 in the 9th field denotes that the component is designated as design material. Theexample is run for 20 iterations.

By running the file, beam.HM.comp.cmf, as a command file in HyperMesh, the elements are

grouped into sets according to their final material density values. The set labeled "0.0 - 0.1"contains all of the elements in which densities range from 0% to 10%. The set labeled "0.1 -0.2" contains all of the sets in which densities range from 10% to 20%. The elements in thesets that have material densities less then 30% are masked so that the solution is easier tovisualize. The material densities of the remaining elements are plotted as shown.



Finite element model of a cantilever beam.

Execution

This example is analyzed in the one-file setup with the file, beam.fem. The OptiStruct batch

job is submitted using the command shell script, % optistruct beam.

Results

The optimization runs for 20 iterations. The results are requested in HyperMesh binary formatand written to the file, beam.res. The shape of the solution at the final iteration is visualized

by creating an assign plot of the density results at the 20th iteration in the HyperMesh contour panel. By removing components labeled "0.1 - 0.2", "0.2 - 0.3", "0.3 - 0.4", and "0.4- 0.5" from the display, a concept of the optimized beam can be visualized.



OptiStruct results thresholds at 25%.

For the input file sample, see <install_directory>/demos/hwsolvers/optistruct/beam.

fem.



Air Conditioner Bracket

The air conditioner bracket is an optimal topology structure generated under both linear staticstiffness and modal frequency response. Shell elements are used to ensure that the bracketis manufacturable using a casting process.

A concentrated mass, M, is used to model the air conditioner unit connected to the mountingbolts by beam elements. Shell elements close to the mounting bolt and engine boltconnections (shown as darker elements) are moved to a nondesign component. The model isconstrained at two engine mounting bolts that are fixed for all displacements and rotationsexcept for the rotation perpendicular to the plane of the bracket. The belt tension load isapplied at the node representing the air conditioner bracket.

Subcase Section

Since the objective function (weighted combination of compliance and frequencies) is a globalresponse, the response reference is outside of the subcase definition. The constraint (volumefraction) is a global response too, therefore the reference is outside the subcase. The weightfactors are defined within the load cases.

DESOBJ = 1

DESGLB = 2

$

SUBCASE 1

SPC = 1

LOAD = 2

$

WEIGHT = 1.0

$

SUBCASE 2

SPC = 2

METHOD = 2

$

MODEWEIGHT 1 1.0

MODEWEIGHT 2 1.0

MODEWEIGHT 3 1.0

$

NORM = 40000.0

Bulk Data Section

The responses and constraints are defined in the bulk data section. Two responses aredefined here, the weighted combination of compliance and reciprocal frequencies (referencedby the objective function), and the volume fraction, which is referenced by the constraintstatement to put up an upper bound of 0.3 (30% of the design space volume). Theconstraint statement is then referenced as a global constraint in the subcase section.

BEGIN BULK

$

DRESP1,1,freqstat,COMB




DCONSTR,2,2,,0.3

The same problem definition can also be achieved by using the equation utility DEQATN todefine the weighted combination.

Finite element model of air conditioner bracket.

The OptiStruct output for the initial iteration appears as follows:

ITERATION 0

Subcase Weight Compliance Weight*Comp.

1 1.000E+00 2.573592E+01 2.573592E+01

----------------

Sum of Weight*Compliance 2.573592E+01

Subcase Mode Weight Frequency Eigenvalue Weight/Eigen

2 1 1.000E+00 3.129449E+00 3.866299E+02 2.586453E-03

2 2 1.000E+00 4.035579E+01 6.429414E+04 1.555352E-05

2 3 1.000E+00 8.710232E+01 2.995154E+05 3.338726E-06

2 4 0.000E+00 1.203149E+02 5.714770E+05 0.000000E+00

2 5 0.000E+00 1.513964E+02 9.048794E+05 0.000000E+00

-----------------



(Sum of Weight/Eigenvalue) / Sum of Weights 8.684483E-04

Mode Normalization Factor x 4.000E+04

-----------------

Eigenvalue total weight 3.473793E+01

Compliance total weight 2.573592E+01

-----------------

Objective Function 6.047386E+01

Execution

This example is analyzed in the one-file setup with the file, acbrack.fem. The OptiStruct

batch job is submitted using the command shell script, % optistruct acbrack.

Results

The optimization converges after 22 iterations. The results are requested in HyperMeshbinary format and written to the file, acbrack.res. The shape of the solution at the final

iteration is visualized by creating an assign plot of the density results at the 22nd iteration inthe HyperMesh contour panel.

Density plot of air conditioner bracket for combined eigenvalue and compliance objective.

Two thick ribs extend from the engine bolts to the lower air conditioner attachment and onethin rib extends from the middle of the upper main rib to the upper air conditioner attachment. There is webbing between the two main ribs and a wide, half-height rib running through theupper half of the design space.



Many elements did not converge to either 100% dense or 0% dense. These elements can beforced toward 0% and 100% by increasing the discreteness parameter. This solution isacceptable because webbings and half-height ribs are manufacturable within a castingprocess.

Optimized models for linear static and frequency are analyzed separately.

The two main ribs from the combined solution are present in both solutions although they varyin thickness between the cases. The third rib, which is full height in the combined load case,appears only in the static solution and is thinner in the combined solution. The webbingbetween the two main ribs appears as several discrete crossribs in both separate solutions. The upper rib is half-height in the combined solution and appears in both the static andeigenvalue solutions. This rib is in a different position in each analysis. The thicker half-height rib in the combined solution is a compromise between the two.

Density plot for linear static load only.



Density plot for eigenvalue subcase only.


acbrack.fem.



Example Problems for Topology Optimization UsingMinimum Member Size Control

The examples in this section demonstrate how topology optimization is used with minimummember size control.


MBB Beam Example

Arch Example





For a Michell-truss type solution, a point load is applied at the right hand tip, and nodes alongthe circle are fixed. The upper half of the system is modeled because of the symmetry(Dimension of the FE model = 100*30).

CHECKER,2.

MINMEMB,6.0,1.

MINMEMB,6.0,2.



MBB Beam Example

In this example, an MBB-Beam with a point load in the upper middle point and two roller pointsupports at the two lower corners is used. The right half of the beam is modeled (dimensionof design domain = 900*300).

DOPTPRM CHECKER 2

DOPTPRM MINDIM 60.0 MINMETH 1




Arch Example

In this example, an arch with three single point loads as three individual load cases on thelower edge is used. Two roller supports are placed at the two lower corners. The dimensionof the domain equals 500*250.

CHECKER, 2.

MINMEMB,40.0,1.




In this example, a 3-D bridge model with a fixed support at both ends is used. Three loadcases: a uniform load p=1 on the entire surface of the bridge and a uniform load p=0.3 on thelanes in each direction separately. The dimension of the design domain: length/width/depth =500/100/100. Target volume fraction = 0.2.

CHECKER, 1.

MINMEMB, 20.0,1.

The reason for running CHECKER=1 in two stages with DISCRETE=1, and then 2, is that afterthe convergence with DISCRETE = 1, the solution is far from DISCRETE. The two-stagesolution provides a fair comparison to results with minimum member control since the lattercontains an automatic increase of the DISCRETE value by an increment of 1.



Example Problems for Topography Optimization

The examples in this section demonstrate how topography optimization is used to generateboth bead reinforcements in stamped plate structures and rib reinforcements for solidstructures.

Plate in Torsion

Optimization of a Rectangular Pressure Vessel

Optimization of a Two-Layer Stamped Control Arm

Eigenvalue Maximization with Topography Optimization

Surface Optimization of a Control Arm

Shape and Size Optimization of a Plate in Bending



Plate in Torsion

The following example demonstrates the power of topography optimization.

For this example it is assumed that:

A specific part of the structure is to be loaded in torsion (see Figure 1.1).

The part is to be formed using a stamping process.

Only the shape of the plate can be changed, the thickness cannot be changed.

Figure 1.1: Loads and Constraints.

Topography optimization divides the design region into smaller areas, each with its own shapevariable. OptiStruct performs this process using the parameters defined by the user. In thisexample, the design space consists of the entire plate minus the areas near the loads andconstraints. The smaller areas can each move upward. The inner portions of these smallerareas are shown fully deflected in Figure 1.2.



Figure 1.2: Inner portions of the shape variables for the plate.

The potential reinforcement pattern can be any combination of these variables deflected atany height between zero and the user-defined maximum height. OptiStruct creates areinforcement pattern of any shape by manipulating the 174 discrete shape variables. Thepattern could resemble an X, an oval, a series of straight beads, or any number of the millionsof potential designs. By setting various parameters, the user can ensure that any designOptiStruct creates is manufacturable.

Once OptiStruct generates the shape variables, it begins the optimization of the plate. Theobjective in this example is simple, the stiffness of the plate under the given torsion load is tobe maximized. OptiStruct performs a series of analysis runs to evaluate the stiffness of theplate, determines what variable value changes will improve the stiffness of the plate, andapplies those changes to the model. After much iteration, OptiStruct reaches the maximumdesign at a point where the stiffness can no longer be improved. The solution is shown inFigure 1.3 where the colors display the heights of the bead reinforcement patterns. A finiteelement model based on that solution is shown in Figure 1.4. The symmetry of the solution isdue to the use of an OptiStruct design symmetry plane feature.



Figure 1.3: OptiStruct bead reinforcement pattern for a plate in torsion.

Figure 1.4: Finite element model built from OptiStruct results.

The face of the plate is covered with X-shaped cross-beads that work well in torsion. Noneof the beads run completely across the plate in a straight line which would reduce theireffectiveness. Finite element analysis of the plate revealed a well distributed stress patternand low deflection at the load point.

The traditional method is to design a reinforcement pattern in the form of raised "beads"across the surface of the plate, test the stiffness of the plate, and increase the stiffness ofthe plate until it meets the design requirements. The efficiency of the plate in terms of costand weight is strongly dependent upon how good the reinforcement pattern is, so it is criticalto generate a good one. Two examples of conventional bead reinforcement patterns for theplate in twisting model are shown in Figure 1.5. These patterns are ones which wouldcommonly be found in commercial products.



Figure 1.5: Common bead reinforcement patterns for a plate in torsion.

The plate generated by OptiStruct using topography optimization is far stiffer than both ofthe conventional plates shown in Figure 1.5. Peak deflection for the topography plate is0.83mm. For the conventional plates, the peak deflections are 1.27mm for the one on the leftand 6.47mm for the one on the right. The plate developed with OptiStruct is 35% stiffer thana good conventional design and far better than a poor one. The poor design, while followingthe conventional wisdom of using an X-shaped reinforcement pattern, uses beads that runcompletely across the plate in a straight line which are susceptible to kinking when loaded. Such design mistakes are caught and corrected by OptiStruct during the optimization processallowing it to yield a superior design.


twistplate.fem.



Optimization of a Rectangular Pressure Vessel

This example involves a rectangular, thin-walled container used for storing fluid. Theobjective is to minimize the outward bulging of the sides of the container caused by thepressure of its contents. Additionally, the maximum outward displacement of the side panelsmust be below a given value.

The model is constrained for displacement in all directions at the four lower corners, but isfree to rotate about those constraints. The loading is a distributed pressure through the areashown in green. The pressure is higher at the bottom of the vessel.

Loads and constraints on rectangular pressure vessel.

The design domain includes the entire box with the exception of the filling hole on the top(shown in red). All of the elements in the design domain are placed in the same componentand reference the same material property. The normal vectors for all of the elements in thedesign domain are pointing outward. The topology variables are set up with the followingDTPG card:

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

DTPG 1 PSHELL 1

+ 15.0 60.0 NO 4.5 NORM NONE



PATRN 10 7229 7209

The minimum bead width is set at 15.0mm, which is roughly the size of three elements. Thedraw angle is set at 60 degrees, and the draw depth is set at 7.5mm. The shape variabledraw vectors are determined according to the element’s normal direction. No buffer zone hasbeen selected between the filling cap and the rest of the model. No grids are to be skipped. Therefore, the nodes (where the constraints are applied) are associated with shape variablesand are free to move.

The third line of the DTPG card defines the pattern grouping option, in this case it is a planeof symmetry. For this model, symmetry was enforced by dividing the box in half lengthwise. The vector that defines the plane of symmetry was created pointing away from the side withthe non-design filler cap to prevent the absence of design variables from being reflected tothe design side.

Optimization objective function and constraints are set up as follows.

In the header, the following cards are added inside the subcase definition:

DESOBJ(MIN) 1

DESSUB 101

In the bulk data the following cards are added:

DRESP1 1 obj COMP

DRESP1 2 swall1 DISP 7 7332

DRESP1 3 swall2 DISP 7 9783

DRESP1 4 fwall DISP 7 9162

DRESP1 5 bwall DISP 7 8813

DRESP1 6 bottom DISP 7 11028

DCONSTR 101 2 10.0

DCONSTR 101 3 10.0

DCONSTR 101 4 1.0

DCONSTR 101 5 1.0



DCONSTR 101 6 5.0

The objective is to minimize compliance for the pressure load case, which is the same asminimizing the strain energy of the entire model. The displacement of the center point ofeach of the five loaded surfaces was constrained to be less than a given value.

OptiStruct generated the following solution for the problem:

OptiStruct topography solution for pressure vessel, front and rear views.

All of the optimization constraints are met for the model. The red areas represent the beadreinforcements that OptiStruct created to increase the stiffness of the model. Circular oroval reinforcement beads are generated for the large side panels and the bottom panel of thebox.

Circular and oval beads are very effective in stiffening the panels against a distributed orcentral load. This is due to the fact that bending in the central areas of the panels isoccurring in two directions, both vertically and horizontally. Straight beads provide stiffnessfor bending in one direction, but would be vulnerable to kinking in the other. Round beadsprovide stiffness without kinking. Bulbous beads are created at the eight corners of themodel, anchoring the sides of the box together and allowing each side of the box to gainsupport from the adjacent sides.

By changing the draw height on the DTPG card from 4.5 to –4.5, the direction of the beadscan be changed from outward to inward.



OptiStruct topography solution for pressure vessel, reversed bead directions.

The bead pattern for the inward bead model is almost a complete reflection of the one thatOptiStruct created for the outward bead model. The areas that are not pushed out in theoutward bead model are pushed in for the inward bead model, generating the same basicstructure.


pressurebox.fem.



Optimization of a Two-Layer Stamped Control Arm

In this example, an automobile control arm is manufactured by using two stamped platesjoined at the seam. The shape of the bead reinforcements on the plates is optimized towithstand the applied loads. The part is modeled as two layers of shell elements (green andblue) connected with fringe elements (red), as shown in the figure below. The model ispinned, but is free to rotate about an axis at the frame attachments. A vertical constraint isplaced at the shock absorber attachment point. This point is connected to the fringeelements with rigid bars. Vertical, lateral, and twisting loads are applied to the spindleattachment.

Loads and constraints for the stamped control arm model.

The elements shown in green are included in the design space. The nodes in the top layercan move upward and the nodes on the bottom layer can move downward. Symmetry is usedto force the bead pattern on the top to match the one on the bottom. The DTPG card isgenerated as follows:

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

DTPG 2 DVGRID 1

+ 15.0 60.0 YES

PATRN 10 4184 1211

The plane of symmetry is normal to the vertical axis and is positioned running through thecenter plane of the model.

The optimization objective function is simply defined as minimizing the sum of the weightedcompliance of all three load cases:

The solution for the model is shown in the figures below. The first figure shows the symmetry



of the solution about the vertical axis.

OptiStruct solution for the stamped control arm model, side view.

OptiStruct solution for the stamped control arm, top view.

The solution shows the importance of adding vertical bending stiffness in the area around theshock absorber attachment point. OptiStruct creates a large bead running from the upperframe attachment point, past the shock attachment, and up to the spindle attachment(second figure). This bead supports most of the bending load. In addition, there is verticalbending which runs in the perpendicular direction. OptiStruct creates a bead running from theshock attachment point to the lower frame attachment (second figure). This second bead isnot as pronounced because there is less bending in that direction compared to the primarydirection.


stampedarm.fem.



Eigenvalue Maximization with Topography Optimization

In this example, an irregularly shaped plate is optimized to increase its natural frequencies. The plate is supported at ten bolt locations around its perimeter. The edge of the plate isturned downward to add stiffness.

Cover plate model with constraints shown.

The red areas are excluded from the design domain. The blue area is open for OptiStruct toadd a bead reinforcement pattern. The bead is drawn upward with respect to the plateorientation. The DTPG card used is as follows. Four different runs were made using differentvalues for the draw height. The first run was made with a draw height of 20mm, the secondwith 40mm, the third with 60mm, and the fourth with 80mm.

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

DTPG

1 PSHELL

5

20.0 60.0 YES 65.0 NORM

NONE

The optimization is set up to maximize the frequencies of the first six modes (minimizing thesum of the weighted inverse eigenvalues) and to ensure that the first three modes wereabove certain design constraints. This is accomplished by placing the following cards in thesubcase definition:

DESOBJ(MIN) 1

DESSUB 101

The following cards are placed in the bulk data section:



DRESP1 11 freq1 FREQ 1



DRESP1 1 wfreq WFREQ

DCONSTR 101 11 400.0

DCONSTR 101 12 500.0

DCONSTR 101 13 600.0

DRESP2 1 wfreq 900

Setting constraints on the first three modes results in separations between the frequencyvalues of the modes and prevents OptiStruct from falling into local minimums when optimizingthe modes. This approach ensures that a minimum performance criterion is satisfied. Notethat for the 40mm, 60mm, and 80mm draw height runs, the constrained frequencies are higherthan those shown above.

The solutions generated for the plate runs are shown in the following figures:

Solution for plate with draw height equal to 20mm.




The reinforcement patterns have a similar shape, but runs with a higher maximum draw heightuse more levels of draws throughout the plate. All of the solutions made good engineeringsense, connecting the weak areas of the plate with beads running primarily across the shortspan of the plate. These beads were fluidly connected together across the long span of theplate allowing the beads to reinforce each other.


eigenplate.fem.



Surface Optimization of a Control Arm

Topography optimization has applications beyond creating beads in shell surfaces. Since thebasic topography approach can be applied to any model containing large fields of shapevariables, it lends itself to solid model applications as well. In the following example, globalshape optimization of a solid part is performed by generating shape variables using the DTPGcard.

The solid control arm model is based on the topology optimization results as shown in thefigure below. After a basic material layout is generated using topology optimization, thesolution can be refined to meet design specifications using shape optimization. However, itcan be a very time consuming procedure if done by hand. By automating the procedure ofgenerating shape variables over the entire surface of a solid part, OptiStruct allows users toquickly set up the shape optimization model.

Loads and constraints for solid control arm.

To use topography optimization to optimize the surface of a solid model, the areas to beoptimized must first be covered with shell elements (see gray elements in figure below).

Solid control arm model partially covered in shell elements (gray areas).



The side surfaces of the structural members are optimized in this example.

Shells are attached to the sides of the members. Define the normal vectors of thoseelements to point inward. Set the value of the draw height on the DTPG card so that thesolid elements do not get turned inside out during shape optimization. If this occurs,OptiStruct gives an error and terminates the optimization. The lower bound on the DTPG cardwas set to –1.0 and the upper bound was set to 1.0, which allows the control arm to becomelarger or smaller.

The objective was set as minimizing the weighted sum of the compliance of the load caseswhile holding the mass of the part constant.

OptiStruct reduces the thickness of the rear crossbar and shapes the struts on either side toresemble C-sections. The front member is tapered to be smaller the closer it is to the area ofthe applied loads.

OptiStruct solution for surface optimized control arm, areas to be thinned are red.

OptiStruct undeformed model for surface optimized control arm, top view.



OptiStruct solution for surface optimized control arm, top view.


solidarm.fem.



Shape and Size Optimization of a Plate in Bending

Shape variables are used in this example to vary both the thickness and the shape of a plateconstructed from solid elements. Topography optimization allows discrete areas of the plateto change thickness and/or height, providing a greater range of possible solutions than withshape or size optimization alone.

A thin plate is built using two layers of solid elements. A uniform pressure is applied to thecenter portion of the plate (gray area) and the plate is simply supported at the corners.

Loads and constraints for a plate to be optimized for both shape and size.

The design shape variables are generated externally to OptiStruct. Size variation is definedby creating a single design variable that controls the thickness of the plate. Shape variationis defined by creating a single design variable that controls the height of the plate relative tothe constrained nodes.

Models showing the size and shape variations at maximum thickness and height.

The two design variables are included in the finite element deck submitted to OptiStruct. Each variable has a single DESVAR card and a DVGRID for each node in the model affected bythat variable. DTPG cards added to the deck reference the ID number of the DESVAR cards



as follows:

DESVAR 1 DV001 0.25 0.0 0.25

DESVAR 2 DV002 0.75 0.0 0.75

DTPG 3 DVGRID 1

3.0 60.0 YES

DTPG 4 DVGRID 2

3.0 60.0 NO

DVGRID* 1 4 0 1.0

* 0.000000000E+00 9.9999999748E-07 5.0000000000E+00

DVGRID* 1 5 0 1.0

* 0.000000000E+00 9.9999999748E-07 5.0000000000E+00

…

Entries in fields 15 through 18 are not required because the draw vectors are already definedfor each node on the DVGRID cards. A buffer zone is requested for the shape variable thatensures a smooth transition between the constraints and the rest of the structure. Lengthwise symmetry is enforced (the left side of the plate mirrors the right side).

The compliance of the part was minimized with the mass of the part constrained to be belowa given value.

The solution is shown in the figures below.



OptiStruct solution for shape and size optimization of plate in bending, top view.

OptiStruct solution for shape and size optimization of plate in bending, bottom view.

OptiStruct creates a thick rib around the perimeter of the plate, providing a solid foundationfor the attachment of the remaining design features. The highest degree of bending occursat the center of the part, with the largest bending component running in the longitudinaldirection. The plate is stiffened in this direction when OptiStruct creates a central rib and aW-shaped reinforcement pattern by raising the thick areas of the plate and lowering the thinareas of the plate. This shape employs the entire structure, not just the parts with ribs.



Cross-sections of the optimized shape and size of the plate in bending.


sizeandshape.fem.



Example Problems for Topography Optimization UsingPattern Grouping

The examples in this section demonstrate how topography optimization is used with patterngrouping.


Optimization of the Modal Frequencies of a Disc Using Constrained Beading Patterns

Multi-plane Symmetric Reinforcement Optimization for a Pressure Vessel



Using Topography Optimization to Forge a Design Concept Out of a Solid Block




A tube made of two sheet metal pieces is intended to carry a load in both bending andtorsion. The cross-section of the tube may be of any shape, but due to manufacturingrequirements, it must remain constant through the entire length. Conventional shapeoptimization can be used to optimize the cross-section, but setting up the variables is timeconsuming. With topography optimization and pattern grouping, cross-section shapeoptimization can be performed in a fraction of the set up time. All optimization set up is doneusing the optimization panel and its subpanels in HyperMesh.

The loads and constraints are applied to the flanges at the ends of the tube, and are shownin Figure 1.1. Spot welds connect the two pieces of the tube along the flanges at regularintervals.

Figure 1.1: Loads and constraints for the spot welded tube.

The initial cross-section of the tube is arbitrarily chosen to be roughly circular. Two similartopography variables are defined to allow the shape of the tube to change. The twovariables assign the blue and green pieces to be in the design domain while the flanges remainin their original shape. The topography variables are in the bulk data section using the DTPGcard as shown below.

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

DTPG 1 PSHELL 1

+ 2.5 85.0 NO 5.0 NORM NONE

+ PATRN 1 0.0 0.0 0.0 1.0 0.0 0.0



The draw height of 5.0, combined with the upper and lower bounds of 2.0 and –2.0, allow abead height of 10.0 model units in both directions (inward and outward) and a bead growthdirection normal to the surface of the tube. A linear type pattern grouping is applied for thisproblem. This means that the beads formed during the topology optimization will be constantalong a line parallel to the direction defined -- the central axis (X-axis) of the tube, in thiscase. By setting this pattern grouping option, the cross-section of the tube will be allowed tochange, but will remain constant through the length of the tube.

The objective was set to minimize the weighted compliance of both load cases. In this case,both load cases were weighted equally. The mass of the tube was constrained to be belowthe initial value. OptiStruct generated the following solution for the model. See Figure 1.2.

Figure 1.2: Optimized cross-section for the tube.

The solution is not completely smooth, but the basic shape of the tube is clear. The lowerhalf of the tube has been lowered to increase the bending stiffness of the section while theupper half of the tube runs directly from one flange to the other to support the shear forcegenerated by the twist load. The cross-sections of the model before and after optimizationare shown in Figure 1.3.



Figure 1.3: Cross-sections of the tube before and after optimization.

To verify the OptiStruct solution, a smooth model of the tube is built based upon the results.This model is shown in Figures 1.4 and 1.5. Static analysis shows the optimized cross-sectionafter smoothing to have a 35% lower peak deflection for torsion and an 18% lower peakdeflection for bending. Additionally, the mass of the part was reduced by 2.2%.

Figure 1.4: Smoothed cross-section of the tube.



Figure 1.5: Finite element model of the smoothed optimized tube.

For the input file sample, see <install_directory>/demos/hwsolvers/optistruct/tube.

fem.



Optimization of the Modal Frequencies of a Disc UsingConstrained Beading Patterns

Finding a good reinforcement pattern for a single modal frequency is difficult when dealingwith beaded plates since adding stiffness in one direction often reduces stiffness in anotherdirection. The problem posed by finding a good reinforcement pattern for four modalfrequencies simultaneously is more than four times more difficult. Add to that the difficultyconstraints on the variety of reinforcement patterns, and the problem becomes a formidabletask. By implementing topography optimization and pattern grouping, the task is greatlysimplified and good quality results are quickly generated. All optimization set up is done usingthe optimization panel and its subpanels in HyperMesh.

The first four modal frequencies of a thin metal disc are to be optimized using beadreinforcement patterns. A variety of different manufacturing methods are being consideredwhich place limitations on the bead reinforcement patterns allowed. The metal disc has ahole in the center where it is constrained (see Figure 2.1).

Figure 2.1: Constraints on the disc model.

The objective for the model is to increase the sum of the frequencies of the first four normalmodes of the disc. This is achieved by using the WFREQ response type. The WFREQresponse is the sum of the inverse eigenvalues of the chosen frequencies. This is done inorder to assign higher weightage to the earlier modes than the latter ones.

The first manufacturing method to be considered is turning the disc on a lathe. This restrictsthe bead reinforcements to being circular. A circular pattern grouping type is define. TheDTPG card for this card is shown below.

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

DTPG 1 PSHELL 1

+ 5.0 60.0 YES 2.5 NORM SPC



+ PATRN 2 0.0 0.0 0.0 0.0 0.0 1.0

The anchor node is set at the center of the disc and the first vector points in the z direction. The nodes in the center ring are removed from the design region by choosing "Spc" in the"Boundary skip" option in the topography subpanel. No other additions to the finite elementmodel were required.

The solution for the circular reinforcement pattern method is shown in Figure 2.2. OptiStructgenerated a single wide circular bead running from the inner ring to about three-quarters ofthe way to the outer edge of the disc. The bead is gently sloped throughout its width. Thisreinforcement pattern more than doubled the frequencies of the first three modes andincreased the fourth and fifth modes by over 75%.

Figure 2.2: Solution for the circular bead pattern method.

The second manufacturing method to be considered is stamping with radial beads only. Thetopography set up for this approach is almost identical to that of the one for the circularapproach with the only change being the radial pattern grouping type. The DTPG card for thisconfiguration is shown below.

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

DTPG 1 PSHELL 1

+ 5.0 60.0 YES 2.5 NORM SPC

+ PATRN 4 0.0 0.0 0.0 0.0 0.0 1.0

The solution for the radial reinforcement pattern is shown in Figure 2.3. OptiStruct generateda series of eight evenly spaced wide radial beads. Note that the beads are not drawn to theirfull height. This demonstrates that full height beads are not necessarily optimal. Thisreinforcement pattern also doubled the frequencies of the first three modes and increased thefourth and fifth frequencies by more than 70%, but was not quite as efficient as the circular



pattern.

Figure 2.3: Solution for the radial bead pattern method.

The third manufacturing method to be considered was stamping with no constraints on thebead shapes. No pattern grouping was defined for this approach. The DTPG card for this setup is shown below.

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

DTPG 1 PSHELL 1

+ 5.0 60.0 YES 2.5 NORM SPC

The solution for this setting is shown in Figure 2.4. OptiStruct generated a reinforcementpattern that appears to be a combination of the circular and radial reinforcement patterns. The inside of the disc has a series of radial beads while the outside has a roughly circularreinforcement. This reinforcement pattern had the best increases in the modal frequenciesfor all of the patterns compared.



Figure 2.4: Solution for the default pattern method.

Since the default reinforcement pattern is almost symmetric, a cyclically symmetric patterncan be used to clean up the solution. Choosing four wedges for cyclical symmetry appears tobe very close to the pattern created with the default settings. The cyclical symmetrypattern grouping type is applied. The DTPG card for the cyclical symmetry method is shownbelow.

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

DTPG 1 PSHELL 1

+ 5.0 60.0 YES 2.5 NORM SPC

+ PATRN 41 0.0 0.0 0.0 0.0 1.0 0.0

+ PATRN2 4 1.0 0.0 0.0

For cyclical symmetry, both the first vector (fields 12, 13, and 14) and second vector aredefined in the same plane as the disc. The anchor node is located at the center of the disc. This cyclical pattern is set up to have four symmetric wedges with each wedge beingsymmetric about itself.

Figure 2.5: Cyclical symmetry pattern for 4 symmetric wedges.

The solution for the cyclical pattern grouping option is shown in Figure 2.6. OptiStruct



generated a reinforcement pattern very similar to the default pattern grouping pattern butwith fourfold symmetry. The modal frequencies were slightly less than those for the defaultpattern.

Figure 2.6: Solution for the cyclical pattern method with 4 symmetric wedges.

The frequency results (in Hz) for the first six normal modes for the baseline model and fourreinforcement patterns are shown in the following table:

Mode # Baseline Circular Radial Default Cyclical

1 18.7 47.3 45.3 51.1 50.4

2 18.7 47.3 45.4 52.7 50.4

3 20.1 73.0 43.8 69.1 68.5

4 34.8 61.5 60.2 66.3 66.7

5 34.8 61.5 60.6 68.4 67.5

6 74.4 113.0 93.0 115.0 117.0


disc_radial.fem and <install_directory>/demos/hwsolvers/optistruct/disc_cyclic.

fem.



Multi-plane Symmetric Reinforcement Optimization for aPressure Vessel

A rectangular thin-walled box is to be used to store fluid. The outward bulging of the sides ofthe container (due to the pressure of the contents) is to be minimized. Additionally, themaximum outward displacement of the side panels must be below a given value. The model isshown in Figure 3.1. All optimization set up is done using the optimization panel and itssubpanels in HyperMesh.

The model is constrained for displacement in all directions at the four lower corners but is freeto rotate about those constraints. The loading is a distributed pressure through the areashown in green. The pressure is higher at the bottom of the vessel.

Figure 3.1: Loads and constraints on rectangular pressure vessel.

The entire box is to be used as the design domain with the exception of the filling hole on thetop shown in red. All of the elements in the design domain are placed in the same componentand reference the same material property. The normal vectors for all of the elements in thedesign domain are pointing outward. The topology variables are set up with the followingDTPG card:

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

DTPG 1 PSHELL 1



+ 7.0 60.0 YES 5.0 NORM NONE

+ PATRN 30 50.0 100.0 75.0 0.0 1.0 0.0

+ PATRN2 0 0.0 0.0 1.0

+ BOUNDS 0.0 1.0

Three orthogonal planes of symmetry are defined (see Figure 3.2). The anchor node is placedat the center of the box. The first and second vectors are defined parallel to the X and Yaxes. The first vector is defined pointing away from the filler cap. This ensures that theautomatically generated variables will cover the entire surface of the box. If the first vectorwas pointing in the other direction, the symmetry method would reflect the lack of variables inthe area of the filler cap across all three planes of symmetry.

Figure 3.2: Orthogonal symmetry planes for the pressure box.

The objective is to minimize compliance for the pressure load case, which is the same asminimizing the strain energy of the entire model. The displacement of the center point ofeach of the five loaded surfaces was constrained to be less than a given value.

OptiStruct generated the following solution for the pressure box shown in Figure 3.3.



Figure 3.3: OptiStruct solution for the pressure box.

The OptiStruct solution met all of the optimization constraints and yielded a good design. Theareas shown in red (In Figure 3.3) are the bead reinforcements that OptiStruct created toincrease the stiffness of the model. The solution is unconventional, but makes a good deal ofengineering sense. For the large side panels and the top and bottom panels of the box,OptiStruct has generated large, rounded, centrally located reinforcement beads. These typesof beads are very effective in stiffening the panels against a distributed or central load. Thisis due to the fact that bending in the central areas of the panels is occurring in twodirections, both vertically and horizontally. The rounded beads create stiffness in bothdirections and are weak in neither. At the eight corners of the model, OptiStruct createdbeads that anchor the sides of the box together allowing each side of the box to gain supportfrom the neighboring sides.

A finite element model was created from this reinforcement pattern (see Figure 3.4).



Figure 3.4: Finite element model of the OptiStruct solution for the pressure box.

This pattern was compared to two other bead reinforcement designs shown in Figure 3.5.

Figure 3.5: Conventional reinforcement patterns for the pressure box.

The OptiStruct model is superior in stiffness to both of the two conventional models. Themaximum deflection of the OptiStruct model was 30% less than the lightly reinforcedconventional model (the one on the left), and 46% less than the heavily reinforced model (theone on the right). The lightly reinforced model was stiffer than the heavily reinforced model,which goes against the assumption that more reinforcements result in increased stiffness. With bead type reinforcements that assumption is not always true, which demonstrates theeffectiveness of topography optimization. OptiStruct delivers an optimized first design,eliminating the need to do a series of re-designs where the second, third, fourth, etc., modeldoes not always result in an improvement.

Manufacturing constraints can be accounted for in the pressure box model using other patterngrouping options. The set up is done easily through the HyperMesh interface. In order tomanufacture the pressure box using a two piece die mold, bead reinforcements that runlaterally would need to be eliminated or else they would cause a die lock condition. Topography optimization can be used to generate reinforcement beads on the sides of thebox that run vertically only. This is done by using separate topography variables for the sidewalls, front and back panels, and the top and bottom panels. Topography variables withplanar symmetry with one plane symmetry grouping type are defined for the side panels andthe front/back panels, respectively. As in the earlier case, a three plane symmetrytopography variable is assigned for the top/bottom panels. The two cards are shown belowand differ only in the direction of the first vector.

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

DTPG 1 PSHELL 1



+ 7.5 60.0 YES 5.0 NORM NONE

+ PATRN 13 50.0 100.0 75.0 0.0 1.0 0.0

+ BOUNDS 0.0 1.0

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

DTPG 1 PSHELL 1

+ 7.5 60.0 YES 5.0 NORM NONE

+ PATRN 13 50.0 100.0 75.0 1.0 0.0 0.0

+ BOUNDS 0.0 1.0

The planes for the planar pattern grouping run vertically and perpendicular to the Y-axis,which causes OptiStruct to generate vertical beads. Because the planes run through bothsides of the box, there will be symmetry between opposing sides. Also, symmetry on eitherside of the anchor node in the direction of the first vector (Y-axis) is forced with the oneplane symmetry option (pattern grouping option 13). See Figure 3.6.

Figure 3.6: Planar pattern grouping planes for sides of pressure box.

With planar symmetry enforced for the sides of the pressure box and three-plane symmetryenforced on the top and bottom of the box, OptiStruct generated the solution shown in Figure3.7. Even without the presence of lateral beads on the sides of the box, the OptiStruct



solution shown below had a maximum deflection 6% less than the lightly reinforcedconventional model.

Figure 3.7: Pressure box solution for combined three-plane and planar symmetry options.


pressure.fem and <install_directory>/demos/hwsolvers/optistruct/pressure13.fem.




Pattern grouping and shape variables can be used to optimize stamped plates that are difficultto deal with due to corners and sharp edges. Automatic generation of topography variablesdoes not take into account situations in which elements can be folded inside out when thevariables are fully perturbed. This limits the draw depth that can be used with thattechnique. The following example demonstrates how this problem can be avoided with user-defined variables. All optimization set up is done using the optimization panel and itssubpanels in HyperMesh.

A hat section is centrally loaded, creating both bending and torsional loads (see Figure 4.1). The hat section is constrained at either end by four bolts.

Figure 4.1: Loads and constraints for the stamped hat section.

It is preferable to have the size of the reinforcements able to run deeper than the height of asingle element. To ensure that this will not cause a problem with the element mesh, threeshape variables are created using HyperMesh and are added to the deck. The shape variablesfor the face and top side of the hat are shown in Figure 4.2.



Figure 4.2: User-defined shape variables for the hat section.

Note in Figure 4.2 that the first three rows of elements adjacent to the elements being fullydeflected are a part of the user-defined shape variable for that side. Also note that the drawdepth is equal to one and a half times the average element size.

It is desired to create this hat section using a stamping process which means that reinforcingfeatures on the sides of the hat must be constant (from top to bottom), or else a die lockcondition will occur. Pattern grouping can be used to create variables that ensuremanufacturability. For the three variables created for the hat section optimization, planarpattern grouping was selected with the planes running perpendicular to the length of thesection (X-axis). The DTPG card and associated DESVAR card for one of the variables areshown below.

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

DTPG 4 DVGRID 1

+ 20.0 60.0 YES

+ PATRN 13 500.0 0.0 0.0 1.0 0.0 0.0



(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

DESVAR 1 DV001 0.0 0.0 1.0

Additionally, a plane of symmetry was used to force both halves of the section to be thesame.

Figure 4.3 shows the symmetry plane for the hat section.

Figure 4.3: Plane of symmetry for the hat section model.

OptiStruct generates variables which allow for great flexibility in the reinforcementpossibilities, but which prevent a die lock condition as shown in Figure 4.4. Note that thearea where the load is applied is left out of the face variable.



Figure 4.4: Variables generated for the hat section.

The objective is to minimize the compliance for the applied load. OptiStruct generated theshape shown in Figure 4.5.

Figure 4.5: OptiStruct solution for the hat section optimization.

The solution generated by OptiStruct is manufacturable using a stamping process. Also, thesolution is very well behaved and needs little refinement to turn it into a production-readydesign.

The optimized hat section increases the stiffness of the part by more than eightfold from theinitial condition with no beads. The eight ‘square’ beads for the hat section, especially thefour at the ends of the beam, are the key to bolstering the beam against shear collapse. Those beads also serve to prevent the flanges from folding under the bending load. OptiStruct has generated a strong design that supports both torsion and bending withrestricted reinforcement possibilities. The shape and placement of the reinforcements areoptimized resulting in a very efficient solution.

For the input file sample, see <install_directory>/demos/hwsolvers/optistruct/hat.

fem.




Topography optimization has applications beyond creating beads in shell surfaces. Since thebasic topography approach can be applied to any model containing large fields of shapevariables, it lends itself to solid model applications. The following example demonstrates howtopography can be used in conjunction with user-defined shape variables to do global shapeoptimization of a solid part. All optimization set up is done using the optimization panel andits subpanels in HyperMesh.

A solid control arm model is built based around OptiStruct topology optimization results (seeFigure 5.1). Topology optimization is very effective at generating a basic material layout, butthe solution is generally not well refined. Refining the solution to meet design specificationscan be done using shape optimization. The generation of shape variables over the entiresurface of a solid part can be a very time consuming procedure if done by hand. WithOptiStruct, this procedure is automated and the shape optimization model can be set up inminutes.

Figure 5.1: Loads and constraints for solid control arm.

The first step is to generate shape variables for the part. HyperMesh is an excellent tool fordoing this. For this model, four basic shape variables were created which control the sizes ofthe two legs and rear beam of the control arm. These are shown below in Figures 5.2 and5.3. Care was taken to ensure that the final design would have no internal cavities whichwould prevent it from being manufacturable.



Figure 5.2: Shape variables controlling the thickness and height of the control arm legs.



Figure 5.3: Shape variables controlling the thickness and height of the control arm rear beam.

Topography optimization will divide these shape variables into smaller variables that controlthe height and thickness of sections of each leg. The type of pattern grouping selected foreach variable is used to control the way that the variables get divided. Pattern groupingenhances the ability of topography optimization to affect the shape of solid models. Withoutpattern grouping, the distance that a grid is allowed to move must be less than the distanceto the neighboring grid or else the elements will get turned inside out. With pattern grouping,the movement of several grids can be linked together so that those grids can move farbeyond their original positions while still maintaining a reasonable element mesh (asdemonstrated in this example problem).

Linear pattern grouping is applied to the first variable which links the movement of all of thegrids through the thickness of both legs together. This allows the thickness of the legs tochange at many points across the sides of the legs, which gives OptiStruct a high degree offlexibility in influencing their shape without causing problems with the finite element mesh.

Planar pattern grouping is applied to the second variable which links the movement of all ofthe grids through both the thickness and height of both legs together. This allows the heightof entire cross-sections of the legs of the control arm to change. One of these planes isshown in Figure 5.4 cutting through both legs of the control arm.



Figure 5.4: Pattern grouping plane for variable #2.

Figure 5.5 shows the variables created from the planar pattern grouping option. Note thatFigure 5.5 only shows the deflections of the centers of the variables. Because of the waythat topography optimization works, if all three variables were fully deflected, the legs of thecontrol arm would be uniformly at the minimum height.

Figure 5.5: Planar variable dispersion pattern for the height of the legs of the control arm.

Planar variable pattern grouping was also used for the height and thickness of the rear beam.For both variables cards, TYP = 13 was used to ensure a beam shape that was symmetricabout the center. The DTPG card for variable #4 (beam height) and the associated DESVARcard are shown below.

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)



DTPG 8 DVGRID 4

+ 15.0 60.0 NO

+ PATRN 13 0.0 0.0 0.0 1.0 0.0 0.0

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

DESVAR 4 DV004 0.0 -1.5 1.0

Since shape variables are being assigned to topography, the height and the bounds in theDTPG card do not need to be defined, as they are controlled by the shape variable. Even ifthese values are defined, OptiStruct will ignore them.

The objective for the optimization was to minimize the mean compliance of the control arm forall three load cases combined. The mass of the control arm was constrained to be less than10% below its initial mass. OptiStruct generated the following solution (see Figure 5.6). Thered shows areas where material was added, the blue shows areas where material wasremoved.

Figure 5.6: Optimized control arm using topography and user-defined shape variables.

OptiStruct optimizes the shape of the legs by shifting mass from the centers to the top andbottom, creating C-shaped sections. These sections add vertical bending stiffness whileleaving the shear and axial stiffness intact. Additionally, the height of the legs was increasedalong their length to further increase the bending stiffness of the sections. OptiStruct greatlyreduced the size of the rear beam in both height and thickness, indicating that it wasoversized or perhaps even unnecessary. Overall, the maximum deflection for all three loadcases was reduced by 5% while the mass was reduced by 10%. Figure 5.7 shows a refined



finite element model of the OptiStruct solution.

Figure 5.7: Finite element model of the OptiStruct solution.


controlarm.fem.



Using Topography Optimization to Forge a DesignConcept Out of a Solid Block

Pattern grouping lends itself very well to applications where manufacturing conditions must bemet. In the following example, topography optimization is used to form a design concept outof a solid block. Manufacturing the design concept using a casting method is preferable. Alloptimization set up is done using the optimization panel and its subpanels in HyperMesh.

A solid rectangular block is fixed at both ends and loaded in the center (see Figure 6.1).

Figure 6.1: Loads and constraints for the solid block model.

Two shape variables are generated using HyperMesh to control the height and width of theblock. These are shown in Figures 6.2 and 6.3.

Figure 6.2: User-defined variable #1.



Figure 6.3: User-defined variable #2.

It is preferable to manufacture the resulting part using a casting process. This can beaccomplished by using a linear pattern grouping in the casting draw direction and a planarpattern grouping perpendicular to the draw. This will ensure that there are no cavities thatwould create a die lock situation.

Thinking ahead, it is predictable that the cross-section of the solution will be roughly an I-shaped section with the web running vertically. This prediction is used to establish the drawdirection as being horizontal, which corresponds to variable #1 (block width), thus variable #1will be split using linear pattern grouping and variable #2 will be split using planar patterngrouping. The DTPG cards and associated DESVAR cards are shown below:

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

DTPG 3 DVGRID 1

+ 2.0 60.0 NO

+ PATRN 21 50.0 250.0 50.0 0.0 0.0 1.0

+ PATRN2 0.0 1.0 0.0

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

DESVAR 1 DV001 0.0 0.0 1.0



(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

DTPG 4 DVGRID 2

+ 20.0 60.0 NO

+ PATRN 13 50.0 250.0 50.0 0.0 1.0 0.0

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

DESVAR 2 DV002 0.0 0.0 1.0

The linear variable dispersion pattern for variable #1 allows OptiStruct to control thethickness of the block at numerous points across its side giving the solution a great deal offlexibility. The planar variable dispersion pattern for variable #2 allows OptiStruct to controlthe height of the cross-sections along the length of the block. The objective was to minimizethe mean compliance of the block under the given load. The mass was constrained to bebelow one fourth of the initial mass of the block. OptiStruct generated the solution, shown inFigure 6.4.

Figure 6.4: OptiStruct solution for the solid block.

The cross-section of the block is roughly I shaped, concentrating the material at the top andbottom of the end and center areas where the bending moment is the greatest. The design isflat and tall in areas where shear is dominant. The solution is manufacturable by use of acasting process since there are no cavities or die lock conditions. The efficiency of thesolution can be seen by looking at the stress contours. The nearly uniform stress distribution,shown in Figure 6.5, indicates that almost every part of the structure is being used to itsfullest potential.



Figure 6.5: Stress contours for the OptiStruct solution.

The maximum dimensions of the block were reduced by 2.5 times and a second topographyoptimization was performed. The solution OptiStruct produced is shown in Figure 6.6.

Figure 6.6: OptiStruct solution for solid block with 2.5 times smaller cross-section.

The basic shape of the block is the same in the reduced dimension model, but has morepronounced features. The I shaped cross-sections in the center and at the ends have widerflanges, and the shear carrying areas in between are thinner. This makes sense consideringthe smaller dimensions increase the need for bending stiffness more than the need for shearstiffness.

For the input file sample, see <install_directory>/demos/hwsolvers/optistruct/block.

fem and <install_directory>/demos/hwsolvers/optistruct/blocklong.fem.



Example Problems for Size Optimization

This section presents size optimization examples solved using OptiStruct. Each example usesa problem description, execution procedures, and results to demonstrate how OptiStruct isused in size optimization.

Introductory Example of Size Optimization

Size Optimization of a Ten Bar Truss

Size Optimization of a Cantilever Beam using Equations



Introductory Example of Size Optimization: ShellCantilever

This example demonstrates how size optimization can be used.

For the cantilever beam below, a thickness distribution is sought that minimizes the structuralmass and allows a limited vertical deflection of the loading end. The displacement is limited to2.0mm.

Structural Model

In this example, the thickness for each element is varied. Therefore, five variables aredefined in the analysis deck. The initial thickness of all elements is 2mm. All thicknessvariables have a lower bound of 1mm and an upper bound of 3mm.

The optimization problem, objective and constraint functions, is defined in the same manneras for the other types of structural optimization. OptiStruct then goes through an iterationprocess to solve the optimization problem. OptiStruct converged after 3 iterations. The tablebelow shows the optimization results of the cantilever beam, which met the constraints andminimizes the mass (decrease of 6.6%).

Element1

Element2

Element3

Element 4 Element5

Disp Volume

Initial 2.0mm 2.0mm 2.0mm 2.0mm 2.0mm 2.5mm 400.00mm3

Final 2.6mm 2.3mm 1.9mm 1.5mm 1.0mm 2.0mm 373.204mm3

The input file can be found in <install_directory>/demos/hwsolvers/optistruct/size1.fem.



Size Optimization of a Ten Bar Truss

This example demonstrates how size optimization can be used.

For the ten bar truss structure, a cross-sectional area distribution is sought that minimizesthe structural mass and allows limited stress values in the elements. The stress limit is25000. The y displacements are limited to 2.

Structural Model

In this example, the cross-sectional area for each element is varied. Therefore, 10 variablesare defined in the analysis deck. The initial cross-sectional area of all the elements is 20,with a lower bound of 0.2 and an upper bound of 200.

The optimization problem, objective and constraint functions, is defined in the same manneras for the other types of structural optimization. OptiStruct then goes through an iterationprocess to solve the optimization problem. OptiStruct converged after thirteen iterations. The table below shows the optimization results of the ten bar truss structure, which met theconstraints and minimizes the mass (decrease of 38.7%).

Bar1 Bar2 Bar3 Bar4 Bar5 Bar6 Bar7 Bar8 Bar9 Bar10 Volume

Initial 20 20 20 20 20 20 20 20 20 20 83929

Final 30.4 38.4 24.5 15.4 0.2 0.2 8.8 21.5 20.1 0.4 51448

The input file for this example can be found in <install_directory>/demos/hwsolvers/

optistruct/tenbar.fem.



Size Optimization of a Cantilever Beam using Equations

This example demonstrates how to use the equation utility to define cross-sectionalproperties in a size optimization.

The structure is a cantilever beam of a length of 55 modeled with five CBAR elements. Thecross-section of the beam is a solid circle. In each element, the diameter of the section isthe design variable. Hence, there are five design variables. The cross-sectional propertiessuch as area, moment of inertia, and torsional constant are calculated using the explicitformulas for a circle. In terms of the DEQATN card, they appear as:

$AREA

DEQATN,111,A(D)=PI(1)*D**2/4

$MOMENT OF INERTIA

DEQATN,122,I(D)=PI(1)*D**4/64

$TORSIONAL CONSTANT

DEQATN,133,J(D)=PI(1)*D**4/32

Using these equations, DVPREL2 statements are used to assign each design variable to therespective PBAR property. The statements that assign the diameter of the first bar elementto the cross-sectional area of that element look like:

DESVAR,1,Diam1,10,1,20,0.5

DVPREL2,11,PBAR,1,4,,,111

+,DESVAR,1

The optimization problem to be solved is the minimization of the tip displacement with avolume constraint of 4000. Convergence was achieved after four iterations.

The result is given in the following table.

Diameter 1 Diameter 2 Diameter 3 Diameter 4 Diameter 5 Tip-Displ

Initial 10.00 10.00 10.00 10.00 10.00 40.42

Final 12.48 11.49 10.28 8.70 6.29 26.51

The input file for this example can be found in <install_directory>/demos/hwsolvers/

optistruct/bar.fem.



Solving an Optimization Problem that is not defined by aFinite Element Model

This example involves the optimization of a box defined entirely by equations (there is nofinite element model in the solution).

The optimization problem is defined as:

Objective: Maximize the volume of a cube AxBxC

Constraint: The surface of the cube should be between 2.0 and 3.0

Design Variables: A, B, C

The volume and surface are defined as equations using DRESP2 and DEQATN:

$

$ VOLUME

$

DEQATN 1 VOL(W,L,H)=W*L*H

$

$ SURFACE

$

DEQATN 2 AREA(W,L,H)=2.0*(W*H+L*H+W*L)

$

DRESP2 1 VOLUME 1

DESVAR 1 2 3

DRESP2 2 SURFACE 2

DESVAR 1 2 3

$

DESVAR 1 W 1.1 0.1 10.0

DESVAR 2 L 0.9 0.1 10.0

DESVAR 3 H 2.0 0.1 10.0

$

Then, in the optimization problem, the objective and constraint are global responses (forexample, DESOBJ and DESGLB are used outside of a SUBCASE).

To trick OptiStruct into solving this problem, a dummy finite element model must be provided.Here, a single shell element with some load is used.

As expected, the solution yields a cube with even sides of about 0.707, a surface of 3.0, anda volume of 3.53.

The input file can be found in <install_directory>/demos/hwsolvers/optistruct/box.

fem.



Example Problems for Shape Optimization

This section presents shape optimization example problems, solved using OptiStruct. Eachexample uses a problem description, execution procedures and results to demonstrate howOptiStruct is used in shape optimization.

Introductory Example of Shape Optimization

Optimization of a Cantilever Beam Modeled with Solid Elements



Introductory Example of Shape Optimization

This example demonstrates how shape optimization can be used.

For the cantilever beam below, a shape is sought that minimizes the structural mass andallows a limited vertical deflection at the lower right corner.

Structural Model.

The shape of the beam is defined using a linear combination of the two basis shapes below. The linear factors are the design variables in the optimization problem.

Basis Shape 1.



Basis Shape 2.

The basis shapes are defined using shape basis vectors. These can be generated usingAutoDV, which is part of HyperMesh. The output file of AutoDV contains the definition of theDESVAR and DVGRID cards. This file can then be included in the OptiStruct input file, bulkdata section, via the INCLUDE statement.

The optimization problem, objective and constraint functions, is defined in the same manneras the other types of structural optimization. OptiStruct then goes through an iterationprocess to solve the optimization problem. The figures below show the optimization result ofthe cantilever beam.

Final Shape.



Final Shape, deformed with von Mises stresses.


shape1.fem.



Optimization of a Cantilever Beam Modeled with SolidElements

The objective of this example is to minimize the volume of a prismatic cantilever beam. Themaximum displacement at the beam tip is limited, and the 1st and 2nd eigen frequencies havea lower bound. Two subcases are defined; subcase 1 is the static load case, subcase 2 isthe eigenmode analysis.

Figure 1: Cantilever beam. Loads and boundary conditions.

The design domain is subdivided into two design elements; the web and the flange. Sixdesign variables are defined using the design elements and vectors (Fig.2). For shapeoptimization, the shape of the beam is defined using the nodal positions of the original shape

and a linear combination of the six shape perturbations associated with

the design variables. The linear factors are the design variables in theoptimization problem. The shape of the beam appears as:

.

Figure 3 shows the shape of the beam perturbed by the first design variable, which is a linearperturbation. Figure 4 shows the quadratic perturbation caused by design variable 4.



Figure 2: Cantilever beam. Design elements and design variables.

Figure 3: Cantilever beam. Perturbed shape number 1.



Figure 4: Cantilever beam. Perturbed shape number 4.

The perturbation vectors need to be provided in the format of the DVGRIDcards using AutoDV (part of HyperMesh). These cards can be generated automatically. Theoutput of AutoDV also includes the design variable definition DESVAR. The output file Beam_shape.dat can be incorporated into the bulk data section of the OptiStruct input deck

via an include statement.

The definition of the optimization problem is included in the case control section of the inputdeck. Figure 5 shows the section of the OptiStruct input file that includes the definition ofthe optimization problem and the inclusion of the AutoDV output.

All optimization constraints are met for the model. The final shape is shown in Fig.5.

Cantilever beam. input data

$-----------------------------------------------------------------

$

$ Case Control Cards

$

$-----------------------------------------------------------------

$

DESOBJ(MIN) = 1

$

$HMNAME LOADSTEPS 1Static

$

SUBCASE 1

LOAD = 2

SPC = 3



DESSUB = 101

$

$HMNAME LOADSTEPS 2Eigenvalues

$

SUBCASE 2

SPC = 3

METHOD = 4

DESSUB = 201

$

BEGIN BULK

INCLUDE Beam_shape.dat

$

$ LOAD cards

$

EIGRL, 4, , , 10

$

DRESP1, 1, vol, VOLUME

DRESP1, 2, disp, DISP,,,2,,29530

DCONSTR, 101, 2, -0.01

DRESP1, 3, f1, FREQ,,,1

DRESP1, 4, f2, FREQ,,,2

DCONSTR, 202, 3, 2600.0

DCONSTR, 203, 4, 3000.0

DCONADD, 201, 202, 203

Figure 5: Cantilever beam. Final shape.


beam_shape.fem.



Examples for Optimization with ESLM

The examples in this section demonstrate how the Equivalent Static Load Method (ESLM) canbe used for the optimization of flexible bodies in multi-body systems.

Rotating Bar Example

Connecting Rod of a Slider Crank Example

Rotating Shell Example

Optimization of MBD System Level Response Optimzation Example



Rotating Bar Example

This is an introductory example of the optimization of a multi-body dynamics system. Theobject of the rotating bar problem is to minimize the maximum stress of the bar. Thestructure consists of 5 bar elements. The driving motion has a velocity of sin(2t), which isapplied to the left end of the structure. Sections of the bar elements are solid circles. Design variables are the radii of the sections. The mass of the structure should be less than10kg.

Rotating bar

A portion of the input file:

DESGLB = 4

MINMAX = 14

STRESS = ALL

SUBCASE 1

MBSIM = 1

MOTION = 1

SPC = 10

:

:

$

DESVAR 1 RAD1 10.00 0.05 100.0

DESVAR 2 RAD2 10.00 0.05 100.0

DESVAR 3 RAD3 10.00 0.05 100.0

DESVAR 4 RAD4 10.00 0.05 100.0

DESVAR 5 RAD5 10.00 0.05 100.0

$

DVPREL1 10 PBARL 1 DIM1 0.0

+ 1 1.0


+ 2 1.0


+ 3 1.0


+ 4 1.0


+ 5 1.0

$

DRESP1 33 STRESS STRESS PBARL SNMAX 1

+ 2 3 4 5



DRESP1 100 MASS MASS

$

DOBJREF 14 33 1 -1.0 1.0

$

DCONSTR 4 100 2.5

:

:

SPC1 10 123456 1

:

:

ENDDATA

The input file is just like one for an ordinary min-max problem. The maximum normal stress ofthe flexible body of the subcase 1 is to be minimized by using DOBJREF and MINMAX.

Notable points include:

Because the stress in multi-body dynamics systems is a time variant quantity, theminimization of stress in multi-body dynamics analysis subcases should be a min-maxproblem.

The SPC1 card fixes only 6 DOF of node 1 of the flexible body in order to remove 6rigid body motions. If you fix more than 6 DOF of the flexible body, the additional fixedDOF become constraints of the flexible body.

Stress output request is placed above the first subcase. If you place output requestinside subcase, your output request will be ignored.

The input file can be found in <install_directory>/demos/hwsolvers/optistruct/

rotating_bar_design.fem.



Connecting Rod of a Slider Crank Example

This example involves the shape optimization of the connecting rod of a slider crank. Themass of the connecting rod is to be minimized. Initial velocity is applied to the block of themodel. The stress of all connecting rod elements must be less than an allowable value.

Slider crank

Four design variables follow:

Input decks for the design are as follows.

SUBCASE 1

MBSIM = 1

INVEL = 2

DESSUB = 10

SPC = 10

$

BEGIN BULK

$

DESVAR 1 UPPER0.0 -1.0 2.0

DESVAR 2 LOWER0.0 -1.0 2.0



DESVAR 3LEFT_OUT0.0 -1.0 1.0

DESVAR 5RIGHT_OU0.0 -1.0 1.0

$

DRESP1 1 MASS MASS PSOLID 18

DRESP1 2 STRESS STRESS PSOLID SVM 18

$

DCONSTR 10 2 -100.E1 100.E1

DVGRID* 1 9925511 0 1.0

* -1.568286121E-02-3.884642199E-02-6.483015791E-10

:

:

The final design should look appear as follows:

The optimum design


sc_design.fem.



Rotating Shell Example

In some models, the rate of change for the response value with respect to time is very high,and the response is dominant in the design process; this is the case in the following example.The model is a rotating shell structure. A lumped mass is attached to the center of the righthole. The mass of the structure is to be minimized. The driving motion is a rotational velocityat the center of the left hole. Its profile is shown below. The movement of the structure islike the second hand of a watch. Stress of all the elements must be less than an allowablevalue. Four shape design variables are controllable.

Rotating shell

Profile of the driving motion.

After analyzing the initial model, the time history of stress using HyperView can be seen.

Analysis result of the initial design



According to the analysis result image shown above, the peaks of stress are at around 0.3seconds. Since the values of these peaks are the largest, you can expect that the responsesat around 0.3 seconds will be dominant in the design process. It is a good practice to "zoomin" on the time period around 0.3 seconds so that the optimization process can consider moreprecise responses. General steps to address the process in this case follow:

1. Run an analysis model with reasonable number of steps in MBSIM card.

ANALYSIS

$

DESOBJ(MIN)=1

$

SUBCASE 1

MBSIM = 10

MOTION = 11

DESSUB = 11

SPC = 10

:

:


Now you can find out the behavior of the structure as in the analysis result image.

2. According to the results post-processed by HyperView, the maximum stresses aredeveloped at around 0.3 seconds. Increase the number of time steps around 0.3 seconds. That is, divide the time period of 0.28 seconds – 0.34 seconds into 200 steps. Increasingthe number of time steps in this period provides the optimizer with more information.

Note that the element that has the maximum stress and the corresponding time can bechanged as the design changes. Thus, Step 2 does not always work. If the time whenmaximum stress is developed and corresponding element are expected to changedramatically as the design changes, it is best to consider the changed peak time andcorresponding element as much as possible.

3. Replace the previous single MBSIM card with multiple MBSIM cards as the following.


+ VSTIFF


+ VSTIFF


+ VSTIFF




+ VSTIFF

MBSEQ 10 1 2 3

4. Run the optimization problem by removing ANALYSIS command.

$ANALYSIS

DESOBJ(MIN)=1

$

SUBCASE 1

MBSIM = 10

MOTION = 11

DESSUB = 11

SPC = 10

:

:


+ VSTIFF


+ VSTIFF


+ VSTIFF

MBSEQ 10 1 2 3

Using the above steps, the design process convergence can be enhanced.


rotating_shell_design.fem.



Optimization of MBD System Level Response OptimizationExample

MBD system level responses can be displacement, velocity, acceleration, force, or somepredefined function of those four quantities. MBD system level responses can be optimized/constrained along with other structural responses of flexible bodies in MBD systems.

The example shown here is intended to minimize the mass of a system. Constraints areimposed on the stress of a flexible body and the velocity of a point. Notable points are shownbelow.

System Level Responses

In order for the system level responses to be made available to the optimization process, MBREQE or MBREQM entries must be defined so that the MBD solver can generate theappropriate output.

$------|--MID--|--GID--|

MARKER 55 9929617

$------|--RSID-|--RID--|--ITEM-|--MID--|

MBREQM 99 999 VEL 55

Then RID (999 in this case) must be referenced in DRESP1.

$------|-------|-------|-------|-------|-------|-------|-------|-------|

DRESP1 3 VELO MBVEL MBREQM TX MIN 999

It is not necessary to reference the requests defined by MBREQM/E in the subcase to makethe requested responses available for the optimization. RSID (99 in this case) of MBREQM canbe an arbitrary positive integer.

DVGRID Including Joint Location

Changing the length of a body is a common method of achieving desired system levelresponses. This change in shape can be defined using DVGRIDs. In this case, the DVGRIDsneed to be defined carefully, as it is necessary to ensure that the joints remain in a validconfiguration while changing the length of the bodies.



A common mistake in defining shape change that involves change of joint locations

In the figure above, Nodes A and B must remain coincident after applying the shapeperturbation.

The correct configuration after applying shape perturbation vector should be:

Correct way of defining shape change that involves change of joint locations

The input file for this problem can be found at <install_directory>/altair/demos/

hwsolvers/optistruct/mbdsystemlvlopt.fem.

The optimization result should look like the following. The opaque orange was the base linemodel and blue one is optimized model.



Optimization References

Design Optimization – Books

Arora, J., Int roduc t ion to Opt im um Design (McGraw-Hill, 1989).

Bendsoe, M.P., and Sigmund, O., T opology Opt im izat ion - T heory, Methods and Applic at ions(Springer, 2003).

Haftka, R.T., and Guerdal, Z., Elem ents of St ruc tura l Opt im izat ion (Kluwer AcademicPublishers, 1996).

Rozvany, G.I.N., T opology Opt im izat ion in St ruc tura l Mec hanic s (Springer, 1997).

Design Optimization – Papers

Bendsøe, M., and Kikuchi, N., Generating Optimal Topologies in Optimal Design using aHomogenization Method. Com puter Methods in Applied Mec hanic s and Engineering, 71 (1988),197-224.

Canfield, R.A., Design of Frames Against Buckling using a Rayleigh Quotient Approximation. AIAA Journal, 31 (1993) 1143-1149.

Canfield, R.A., High-quality Approximation of Eigenvalues in Structural Optimization. AIAAJournal, 28 (1990) 1116-1122.

Choi, W.S., and Park, G.J. Structural Optimization Using Equivalent Static Loads at All theTime Intervals. Com puter Methods in Applied Mec hanic s and Engineering, 191 (2002) 2105-2122.

Duysinx, P., and Bendsøe, M., Topology Optimization of Continuum Structures with LocalStress Constraints. DCAMM Report, Technical University of Denmark, 1996.

Duysinx, P., and Sigmund, O., New Developments in Handling Stress Constraints in OptimalMaterial Distribution. Proc eedings of the 7th AIAA/USAF/NASA/ISSMO Sym posium onMult idisc iplinary Analysis and Opt im izat ion, St. Louis, Missouri, 1997, 1501-1509.

Fleury, C., Structural Weight Optimization by Dual Methods of Convex Programming. Internat ional Journal for Num eric a l Methods in Engineering, 14 (1979) 1761-1783.

Fleury, C., and Braibant, V., Structural Optimization: A New Dual Method using MixedVariables. Internat ional Journal for Num eric a l Methods in Engineering, 23 (1986) 409-428.

Kang, B.S., Park, G.J., and Arora, J.S. Optimization of Flexible Multibody Dynamic SystemsUsing the Equivalent Static Load Method. AIAA Journal, 43 (4) (2005) 846-852.

Kidd, B.J.G., Kemp, M.D., and Jones, R.D., Application of Topology Optimization Technology toan Automotive Bracket Component. Engineering Design Opt im izat ion: Produc t and Proc essIm provem ents, 1st ASMO UK Conferenc e, West Yorkshire, 1999.

Kirsch, U., On Singular Topologies in Optimum Structural Design. St ruc tura l Opt im izat ion, 2



(1990) 133-142.

Meyer-Prüßner, R., Prozeßkette Topologieoptimierung. Beit räge zum NAFEMS Sem inar zurT opologieopt im ie rung, Aalen, GY, 1997, 12.1 - 12.5.

Park, G.J., and Kang, B.S. Validation of a Structural Optimization Algorithm TransformingDynamic Loads into Equivalent Static Loads. Journal of Opt im izat ion T heory and Applic at ions,119 (1) (2003) 191-200.

Rozvany, G.I.N., A Critical Investigation of Industrially used Structural Topology OptimizationMethods, 7th World Congress on St ruc tura l and Mult i-disc iplinary Opt im izat ion, Seoul, Korea,2007.

Schmit, L.A., Structural Design by Systematic Synthesis, Proc . of the 2nd Conferenc e onElec t ronic Com putat ion, ASCE, (1960), New York, 105-132.

Schmit, L.A., and Farshi, B., Some Approximation Concepts for Structural Synthesis, AIAAJournal, 12 (1974), 692-699.

Schmit, L.A., and Fleury, C., Structural Synthesis by Combining Approximation Concepts andDual Methods, AIAA Journal, 18 (1980) 1252-1260.

Schramm, U., Structural Optimization – An Efficient Tool in Automotive Design. AT Z –Autom obilt ec hnisc he Zeit sc hr ift, 100 (1998) Part 1: 456-462, Part 2: 566-572. (In German,English in AT Z Worldw ide).

Schramm, U., Thomas, H.L., Zhou, M., and Voth, B., Topology Optimization with AltairOptiStruct, Proc eedings of the Opt im izat ion in Indust ry II Conferenc e, Banff, CAN, 1999.

Sigmund, O., and Petersson, J., Numerical Instabilities in Topology Optimization: A Survey onProcedures Dealing with Checkerboards, Mesh-dependencies and Local Minima. St ruc tura lOpt im izat ion, 16 (1998), 68-75.

Svanberg, K., Method of Moving Asymptotes – A New Method for Structural Optimization. Internat ional Journal for Num eric a l Methods in Engineering, 24 (1987) 359-373.

Thomas, H. L., Vanderplaats, G. N., and Shyy, Y.-K., A study of move limit adjustmentstrategies in the approximation concepts approach to structural synthesis, Proc eedings of the4th AIAA/USAF/NASA/OAI Sym posium on Mult idisc iplinary Design Opt im izat ion, Cleveland, OH,1992, 507-512.

Vanderplaats, G.N., and Salajehgeh, E., A New Approximation Method for Stress Constraints inStructural Synthesis. AIAA Journal, 27 (1989) 352-358.

Vanderplaats, G.N., and Thomas, H.L., An Improved Approximation for Stress Constraints inPlate Structures. St ruc tura l Opt im izat ion, 6 (1993) 1-6.

Zhou, M., Difficulties in Truss Topology Optimization with Stress and Local BucklingConstraints. St ruc tura l Opt im izat ion, 11 (1996) 134-136.

Zhou, M., Geometrical Optimization of Trusses by a Two-Level Approximation Concept. St ruc tura l Opt im izat ion, 1 (1989) 47-72.



Zhou, M., and Haftka, R.T., A Comparison Study of Optimality Criteria Methods for Stress andDisplacement Constraints. Com puter Methods in Applied Mec hanic s and Engineering, 124(1995), 253-271.

Zhou, M., and Rozvany, G.I.N., DCOC: An Optimality Criteria Method for Large Systems, PartI: Theory; Part II: Algorithm. St ruc tura l Opt im izat ion, 5 (1992) 12-25, 6 (1993) 250-262.

Zhou, M., Shyy, Y.K., and Thomas, H.L., Checkerboard and Minimum Member Size Control inTopology Optimization. Proc eedings of the 3rd WCSMO, Buffalo, NY, 1999.

Zhou, M., and Thomas, H.L., Alternative Approximation for Stresses in Plate Structures. AIAAJournal, 31 (1993) 2169-2174.

Zhou, M., and Xia, R.W., Two Level Approximation Concept in Structural Synthesis. Internat ional Journal for Num eric a l Methods in Engineering, 29 (1990) 1681-1699.

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