Review article Struct Multidisc Optim 23, 97110 Springer-Verlag 2002

Issues of commercial optimization software developmentH. Thomas, M. Zhou and U. Schramm

Abstract Commercial optimization software develop-ment has a different set of goals and constraints thanthe development of academic or industrial research codes.Commercial codes must be all things to all people. Theymust contain a wide range of analysis options and be ableto handle large, real world, industrial analysis models. Asmost of the users of the software in industry come fromanalysis, rather than design optimization backgrounds,the codes must perform in a robust manner. Inconsis-tent input data must be detected. Optimization methodsmust be automatically chosen by the program. Optimiza-tion parameters need to be adjusted automatically by theprogram. Another very important aspect is ease of use.A very intuitive and easy to use GUI (Graphical User In-terface) should be developed. This work describes someof the development objectives and concerns that are es-sential to the development of commercial optimizationsoftware products.

Key words optimization, commercial software, sizing,shape and topology optimization

1Background

The development of commercial optimization softwarehas a different set of objectives and constraints than thedevelopment of academic or industrial research codes.Codes developed in academia are, in general, experimen-tal in nature and do not address issues such as ease ofuse, robustness and documentation. They usually focuson a specific subset of applications and are typically writ-ten to prove a point of view. Codes developed in the in-dustrial research labs must be more robust so that they

Received December 30, 2000

H. Thomas, M. Zhou and U. Schramm

Altair Engineering, Inc., 2445 McCabe Way, Suite 100, Irvine,CA 92614, USAe-mail: thomas@altair.com

can be transferred to a production environment. How-ever, they still tend to focus only on the applications intheir specific industry. Commercial codes must be moregeneral and target users from diverse fields of applica-tions. These codes must contain a wide range of analysisoptions and be able to handle large, real world, indus-trial analysis models. As most of the industrial users ofcommercial optimization software are more experiencedin analysis rather than design optimization, the codesmust be robust and easy to use. Illogically set up prob-lems must be detected as the input data is read in, ratherthan wasting many CPU hours to come up with a uselessdesign. There can be no tweaking and tuning of optimiza-tion control parameters, as the user does not have theknowledge to adjust, or even understand the impact of,these parameters.

Structural optimization is typically performed basedon FEM analysis. Academic codes, which are usuallywritten to prove that an optimization methodology is su-perior, are not concerned with the underlying analysisusefulness. Typically, academic codes are based on 2Dtruss or membrane finite elements. Simple beam elementsare also sometimes used. While some academic codes canhandle more than a single load case, most do not allowmore than a single boundary condition. While industrialcodes can be slightly more general, they still tend to fo-cus on a specific application, and therefore tend to havea limited set of element types. In addition to being able tohandle multiple boundary conditions, commercial codesneed to be able to handle simultaneous analysis types,such as linear statics, eigenvalue extraction, heat trans-fer, and frequency response. This is because products un-dergo many different loadings during their lifetime, andthese must all be taken into account when the product isdesigned.

A general-purpose commercial code must containa wide range of finite element types such as spring, truss,beam, triangular and quadrilateral plate, hexa, penta,and tetra solid, as well as concentrated mass elements,as a minimum set. The elements themselves tend to bequite complicated; beam elements must include shear de-formation, node point offsets, and unconnected degreesof freedom (hinges). 2D elements must include mem-brane, bending, shear deformation, bending/membrane

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coupling, grid point offsets, nonstructural mass, and evencomposite layups. The 2D and 3D solid elements are notstandard isoparametric formulations, but usually includesome type of reduced integration and fictitious modes.These complicated element formulations lead to difficul-ties during sensitivity analysis. In addition to a selectionof elastic elements, there must be a good selection ofboundary conditions and load specifications, as well asrod, bar, and general rigid elements. Physical loads suchas point, pressure, thermal, and body forces, as well asa wide range of materials, such as isotropic, orthotropicand general anisotropic materials must be available. Realworld problems tend to be quite large in terms of thenumber of elements and the number of degrees of free-dom. A typical problem has 50000 to 100000 degrees offreedom. Models with 500000 are not uncommon. Thespeed of the linear equation solver is evenmore critical foroptimization codes than for analysis-only codes, becausethe optimization process is iterative in nature and thenumber of analyses can be quite large. In addition, spe-cial care must be taken to make the sensitivity analysis asefficient as possible.

After the requirements for an efficient general-purposeanalysis code are satisfied, an efficient and robust op-timization methodology must be incorporated. It mustbe efficient so that the number of repeated analyses issmall. It has to be robust so that the novice user canget good results. Poorly specified design problems shouldbe detected and appropriate corrections suggested. Opti-mization control parameters must be adjusted internally,rather than forcing the user to change themmanually andrerun the problem.

The issues mentioned above have been given the high-est attention and emphasis through out the developmentof the commercial software Altair OptiStruct. The his-tory of this software has undergone several stages of evo-lution towards its maturity today. The first version ofthis code was a research code that was developed inuniversity research labs by a group led by Profs. Diazand Kikuchi in 1991. It performed topology optimiza-tion based on a homogenization method (Bendse andKikuchi 1988). The only problem that was solved wasthe minimization of weighted compliance and/or eigen-frequencies. Altair started to market it to industrial usersas Altair OptiStruct 1.0 in 1993. It soon became ap-parent that the following two major limitations existed.First, while it represented an extremely appealing newtechnology for generating efficient design concepts at theearly design stage, its FE analysis capabilities were notsufficient for common industrial applications. Secondly,it was not robust enough, in terms of both the linearequation solver and the optimization process. As a con-sequence Altair Engineering founded a development teamto address the issues of maintenance and further devel-opment of this software. The first version of this soft-ware developed by Altair was released in 1996 as Op-tiStruct 3.0. In this version some commonly used finiteelements, such as rigid elements, were added. Also, a more

robust linear equation solver replaced the one in the ori-ginal code. The release of OptiStruct 3.5 in 1998 signifi-cantly enhanced the optimization capability of the codeby allowing a general setup of the optimization prob-lem. A general multiple constrained optimization prob-lem could be solved using a dual optimization algorithm(Fleury and Braibant 1986). Responses that could beused either as objective function or constraints involvedmass, volume, displacements, compliance and eigenfre-quencies. Shape optimization using the basis vector ap-proach was also added. Another popular capability re-leased in OptiStruct 3.5 is called topography optimiza-tion, which allows the user to optimize the bead patternof shell structures. The user only needs to use a singleline of data to specify the desired minimum bead width,maximum bead height and the draw angle. The codethen generates corresponding shape design variables au-tomatically. It is worth noting that both topology andtopography optimization are extremely easy to use, whichmay have been one of the keys for the early success ofthis software. OptiStruct 3.6 released in 1999 added an-other key feature that allows the user to control the mini-mum member size for topology optimization (Zhou et al.1999). The latest major release OptiStruct 5.0, releasedin 2001, represents another milestone in the developmentof this code. In this release, full sizing and shape op-timization capabilities are added. Stress responses aremade available. An equation utility is provided to allowthe user to set up custom responses using available in-formation. A unique feature of OptiStruct 5.0 is that itallows combinations of different types of design variables.This enlarges the design space and increases the opti-mization potential (Zhou et al. 2000). Another notableadvance within this release is a fully functional GUI thathas been implemented in Hypermesh (Altair Engineering,Inc. 2001b). This graphical interface allows the user tosetup the optimization problem seamlessly and start theexecution within the GUI panel. The purpose of this pa-per is to share our experiences from the development ofAltair OptiStruct, and discuss some general issues thatare critical for commercial software. Some of the key tech-nical details that address such general aspects are alsodiscussed.

2Optimization problem

The optimization problem can be stated as

min f(X)

subject to

gj(X) gj , j = 1, . . . ,M ,

XLi Xi XUi , i= 1, . . . , N . (1)

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where f(X) is the objective function, g(X) are theconstraints, both of which are functions of the de-sign variables. There are M constraints and N designvariables. The type of design variables can be size,shape/topography, and topology. Examples of size de-sign variables are plate thickness and rod area. Shape/topography design variables control the locations of thefinite element nodes in the model. Topography designvariables control the element material properties of indi-vidual elements.

The objective function and constraints can be anyglobal quantity such as mass, volume, compliance, fre-quency, and displacement, or a local quantity such asstress. In addition, responses that are combinations oflocal or global responses combined through the use of userdefined equations can be used as the objective function orconstrained.

3Sensitivity analysis

For basic linear analysis types, gradient based optimiza-tion is the most efficient approach. If not performed ef-ficiently, the sensitivity analysis can actually take longerthan the original response analysis. Two approaches tosensitivity analysis, the direct and adjoint methods, willbe discussed here (Haftka and Adelman 1989; Haftka andGurdal 1993). Given the equation of analysis

[K]{U}= {P} , (2)

and its derivative with respect to design variableX

[K]

X{U}+[K]

[U]

X=

{P}X

. (3)

One can calculate the sensitivity of the displacementvector {U} as

{U}X

= [K]1({P}X

[K]

X{U}

). (4)

The constrained quantity, g, is calculated from the dis-placements as

g = {Q}T{U} , (5)

then the gradient of the constraint is

g

X=

QT

X{U}+QT

{U}X

. (6)

If (6) is used to calculate gradient, then the largestcost in this calculation is the forward backward substitu-tion required for the calculation of the derivative of thedisplacement vector with respect to the design variable.

This is called the direct method. One forward backwardsubstitution is required for each design variable. If con-straints are active in more than one load case, the setof forward and backward substitutions need to be per-formed for each active load case.

For the adjoint method of sensitivity analysis, the ad-joint displacement vector {E} calculated as

[K]{E}= {Q} , (7)

is introduced. Then the derivative of the constraint can becalculated as

g

X=

QT

X{U}+{E}T

({P}X

[K]

X{U}

). (8)

When the adjoint method for sensitivity analysis is used,then a single forward backward substitution is need foreach retained constraint. This forward backward substi-tution is needed to calculate {E}.

In shape and sizing optimization, there are typicallya small number of design variables (say 5 to 50) anda large number of constraints. The large number of con-straints comes from stress constraints. If there are 20 000elements, each with a single stress constraint, and 10load cases, there are a total of 200 000 possible stressconstraints. In topology optimization there is typicallya large number of design variables (between one and threeper element) and a small number of constraints. Thisis because stress constraints are usually not consideredin topology optimization. Therefore, it makes sense thatthe adjoint method of sensitivity analysis be used fortopology optimization in order to reduce computationalcosts.

For shape and sizing optimization, it usually makessense to use the direct method for sensitivity analysis.However, in some cases, when there is a large numberof design variables and a small number of constraints,the adjoint method should be used. The number ofconstraints whose gradients need to be calculated forcan be reduced using constraint screening. With con-straint screening, constraints that are not close to be-ing violated are ignored. Only constraints that are vi-olated, or nearly violated, are retained. In addition, ifthere are many stress constraints that are retained ina small region of the structure, say at a stress concen-tration, then only a few of the most critical need beretained.

A commercial optimization package needs to performconstraint screening based on both constraint value andconstraint region. This reduces the number of active con-straints, and therefore the sensitivity costs. After con-straint screening is performed, the code should auto-matically decide whether the direct or adjoint sensitiv-ity method should be used for sensitivity analysis. Thischoice is made based on the number of design variables,active load cases and boundary conditions, and the num-ber of retained constraints.

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4Approximation concepts

Simply using an optimization algorithm in conjunctionwith a finite element code for optimization will lead to anexcess number of finite element analyses. A more efficientapproach is to use approximation concepts. In the ap-proximation concepts approach (Schmit and Farshi 1973;Schmit and Miura 1976), the calculated values of the ob-jective function and retained constrained responses andtheir gradients with respect to the design variables areused to create a linear Taylor series expansion for each ofthe responses.

gj(X) = gj0Ni=1

gjXi

(XiXi0) , (9)

These expansions can also be made with respect to thereciprocal of the design variables as follows:

gj(X) = gj0Ni=1

gjXi

X2i0

(1

Xi

1

Xi0

). (10)

This makes sense in cases where it is expected thatthe response will be a more linear function of the re-ciprocal of the design variable. A good example of thisis the displacement of a truss structure with respect tothe areas of the individual truss members. For a stat-ically determinate structure, the exact displacement isa separable linear function of the reciprocals of the cross-sectional areas. Therefore, the approximation formula-tion in (10) yields exact displacements for statically de-terminate trusses. On the other hand, the mass of a trussstructure is a linear function of the areas of the mem-bers. Therefore, the decision to use a direct or reciprocaldesign variable based approximation should be made re-sponse by response. A mixed variable formulation of theTaylor expansion has been used extensively in structuraloptimization

gj(X) = gj0+Ni=1

gjXi

cji(XiXi0) ,

with

cji = 1 ifgjXi

0 , cji =Xi0Xi

ifgjXi

< 0 . (11)

This formulation is also termed conservative approxima-tion since it has been shown by Haftka and Starnes (1976)that this formulation gives a more conservative approxi-mation of the constraint compared with both linear andreciprocal approximations. Because this approximation isconvex and separable, it is used to create an efficient dualmethod by Fleury and Braibant (1986).

The Taylor series expansions are used to constructan approximate analysis system. While the calculatedapproximate stresses, frequencies, mass, displacements,etc. are only accurate near the original analysis point,

the cost of calculating these quantities is many ordersof magnitude less than a full finite element analysis.The optimization algorithm is used in conjunction withthis approximate analysis problem to determine an ap-proximate optimum design. This design is then analyzedusing a full finite element analysis, which is used asthe basis for the next approximate optimization prob-lem. This process repeats until the approximate optimumdesign converges to the actual optimum design. Usingthis approach, the number of full finite element ana-lyses is significantly reduced. The overall iterative schemeof the approximation concept approach is illustrated inFig. 1.

Fig. 1 Iteration flowchart

The commercial software should determine whethera direct, reciprocal or conservative approximation is usedfor each response, based on the response type and de-sign variable type. If it is not clear what the relation-ship is between the response and the design variables,the commercial software should be able to determinewhich is more accurate (direct, reciprocal, or conserva-tive) following the full analysis of an approximate opti-mum design. This is done by choosing an approximationfor each type of response based on general knowledge forthe first approximate problem. After the optimum of theapproximate problem is obtained, the response values atthis design are calculated for all three approximations.After the exact analysis, the three approximate values

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are compared to the true response values, and the moreaccurate approximation is used in the next approximateproblem.

5Advanced approximation concepts

Better (more accurate) approximate analyses can be con-structed using the concepts of intermediate design vari-ables and intermediate response quantities (Vanderplaatsand Salajegheh 1989; Zhou and Xia 1990; Canfield 1990).As an example of intermediate design variables, considerthe bending stiffness parameter of a plate: D = t3/12. Ifthe approximation for the bending displacement of theplate is formed in terms of the reciprocal ofD as shown in(12), it is exact for statically determinate structures,

gj(t) = gj0Ni=1

gjDi

D2i0

(1

Di

1

Di0

). (12)

In the approximate analysis, the design variable t is usedto calculate the intermediate design variable D. Then Dis used to evaluate the approximate displacement usinga Taylor series expansion with respect to the reciprocalofD.

As an example of an intermediate response quantity,consider the bending moment in a beam. Instead of ap-proximating the stress in the beam element, the momentcan be approximated, and the approximate stress cal-culated using the approximate moment and the currentvalues of the section properties

= M c/I , (13)

M(I) =M0+Ni=1

M

Ii(Ii Ii0) , (14)

where c is the fiber distance of the stress recovery pointand I is the moment of inertia. Note that the stress in thebeam changes due to the force redistribution in the struc-ture as the beam stiffness changes and due to the changesin the section properties. If the stress alone is approxi-mated by a linear Taylor series, the interaction of the twoeffects on the stress in the beam is lost. If, however, theforce is approximated and the stress calculated from thisapproximate force and the current section properties, theinteraction of the two effects is captures. Note that fora statically determinate structure, in which the forces donot change, the intermediate response quantity approxi-mation is exact.

For a 3D beam element, the intermediate variablesare Z = (A, I1, I2, J,NSM)T with A as the cross-sec-tional area, I1, I2, J as moments of inertia, and NSMas the nonstructural mass. For shell elements, the inter-mediate variables are with as shear thickness. The inter-mediate responses are displacements, element forces. Forthe approximation of eigenfrequency, the modal strain

energy and modal kinetic energy in the Rayleighs quo-tient expression of eigenvalue are used as intermediateresponses (Canfield 1990). The use of the advanced ap-proximation concepts of intermediate design variablesand intermediate response quantities can further re-duce the number of full analyses needed to reach theoptimum design from the range of 1020 to 510. Anadded benefit is that the optimization process itself be-comes more robust. Since the approximations themselvesare more accurate, less tuning of optimization param-eters is required to make the convergence smooth andrapid.

6Move limits

As the design moves away from its initial point in theapproximate optimization problem, the approximatevalues become less accurate. This can lead to slow over-all convergence, as the approximate optimum designsare not near the actual optimum design. Move limits onthe design variables, and/or intermediate design vari-ables, are used to protect the accuracy of the approx-imations. Small move limits lead to smoother conver-gence. However, many iterations may be required dueto the small design change during each iteration. Largemove limits may lead to oscillations between infeasi-ble designs as critical constraints are calculated inaccu-rately. If the approximations themselves are accurate,then large move limits can used. Typical move limitsin the approximate optimization problem are 20% ofthe current design variable value. If advanced approx-imation concepts are used, move limits up to 50% arepossible.

Even with advanced approximation concepts, it ispossible to have poor approximations of the actual re-sponse behaviour with respect to the design variables. Itmakes sense to use larger move limits for accurate ap-proximations and smaller move limits for those that arenot so accurate. Note that the same set of design vari-able move limits must be used for all the response ap-proximations. Therefore, it is important to look at theapproximations of the responses that are driving the de-sign. These are the objective function and most criticalconstraints. If the objective function moves in the wrongdirection, or critical constraints become even more vi-olated, it is a sign that the approximations are not soaccurate. In this case, it makes sense to reduce all thedesign variable move limits. However, if the move limitsbecome too small, convergence may be slowed, as designvariables that are a long way from the optimum designare forced to change slowly. Therefore, it makes sense toincrease the move limits on the individual design vari-ables that keep hitting the same upper or lowermove limitbound.

An example of convergence history with and withoutmove limit adjustment is shown below in Fig. 2.

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Fig. 2 Effect of move limit adjustment strategy

Since move limit adjustment is very problem depen-dent, this must be done automatically in commercial op-timization software (Thomas et al. 1992). The use of movelimit adjustment and advanced approximation conceptsis very important in commercial optimization software.Not only is it important to reduce the overall run time, inwhich a single full finite element analysis may take hours,it is evenmore important to make the convergence robust.This robustness reduces the need to adjust optimizationparameters. In academic codes, the optimization param-eters are usually adjusted by trial and error rerunning ofthe same example problem. A commercial customer hasneither the time nor expertise to rerun production opti-mization problems.

7Optimization methods

There are two common methods used to solve the ap-proximate optimization problem. The first is the directmethod. In the direct method, a search direction is de-termined based on the gradients of the objective functionand critical constraints, and then a line search is per-formed along this search direction. Examples of directionmethods are the method of feasible directions, sequentialquadratic programming, and the generalized reduced gra-dient method. The CPU time cost for the direct methodsis proportional to the number of design variables. This isdue to the direction finding problem. Problems with largenumbers of design variables tend to fail to converge dueto numerical difficulties in the direction finding problem.Although the number of design variables must remain lessthan typically 500, a large number of constraints can behandled. This is because the direction finding problemis only concerned with the most critical constraints, and

this number is usually of the order of the number of de-sign variables or much less. The direct method works wellfor size and shape optimization, in which there are usu-ally few design variables, say less than 100, and manyconstraints. The large number of constraints is typicallydue to stress constraints on many elements in many loadcases.

The second optimization method is called the dualmethod. In this method, a convex separable approxima-tion is used to form the dual optimization problem. In thedual problem, the move direction is determined by the ac-tive constraint set. The dual method is very efficient forproblems that have a large number of design variables,but only a small number of constraints, say less than 50.The dual method works well for topology and topographyoptimization problems, in which there can be hundredsof thousands of design variables, and only a few, global(compliance, frequency, mass, etc.) constraints.

Since commercial optimization software must be verygeneral, both the direct and dual optimization methodsmust be available. The software needs to automaticallychoose which method to use, based on the current prob-lem description.

8Topology optimization

There are two main approaches to defining topologyoptimization design variables. The first, termed SIMPmethod (called the density approach herein) (see, for ex-ample Rozvany and Zhou 1990), has the stiffness of eachelement be a function of the, usually penalized, density ofthe element. The density is the design variable in this ap-proach. The second approach (see, for example Bendseand Kikuchi 1988), based on homogenization theory, has

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Fig. 3 (a) Coarse mesh result and (b) fine mesh result

the stiffness and density of the element determined bythe parameters of the microstructure of a fictitious ma-terial. One early microstructure introduced by Bendseand Kikuchi (1988) is a union of rectangular micro voidsthat defines a fictitious porous material. The design vari-ables in this case are the two dimensions of the void andits orientation angle. The void is used to formulate an or-thotropic material matrix for the element. Usually, thisstiffness is penalized (made less) for intermediate densityelements. Note that the material becomes isotropic in thecase of no void or fully filled with voids (no material).

Each approach has its advantages and disadvantages.The homogenization approach had the advantage thatthe design can form rapidly along the lines of the forcetransmission path. This is due to the angular orthotropyof the intermediate dense elements. Some of the disad-vantages of this method are that there are more designvariables and that it is difficult to determine the angle ofthe void for multiplied constrained problems. The advan-tages of the densitymethod are that it is more general andrequires less design variables.

Commercial topology optimization software can haveboth methods. The appropriate method can be deter-mined based on the objective function type and the num-ber and type of constraints.

9Manufacturing constraints in topology optimization

Mesh dependency of designs generated by topology op-timization is shown in Figs. 3a and b. The problem con-sidered is the well-known MBB beam where the simplysupported beam is loaded in center. Due to symmetry,only half of the beam is modelled.

Using a manufacturing constraint consisting of a limiton the minimum size of structural members, the mesh de-pendency of the optimum design can be removed (Zhouet al. 1999). Figures 4a and b show that the same de-sign is obtained for different mesh densities using thesame minimum member dimension specification. More-over, the solutions in Fig. 4 are simpler than those inFig. 3. This allows the designs to be manufactured moreeasily.

10Topography optimization

In topography optimization, finite element nodes aremoved normal to the surface of plate like structures in

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Fig. 4 (a) Optimum design with coarse mesh and member size control and (b) optimum design with fine mesh and member sizecontrol

order to generate reinforcement patterns of beads orswadges. In this optimization problem statement, thereare a large number of design variables and the adjointvariable method of sensitivity analysis must be used incombination with the dual optimization problem solver.An advanced technique has been developed to automat-

Fig. 5 Engine stiffening panel design

ically generate shape design variables based on beadconfiguration parameters like height, width, and drawangle.

An example of an automotive engine stiffening panel(Meyer-Prussner 1999) is shown in Fig. 5. Note that thecolour contours represent the height of the beads. The

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final design of the stiffening panel is generated by inter-preting the results of the topography optimization.

11Graphical user interface development

It is important that the user can easily set up the fulldesign optimization problem. This includes not only theunderlying analysis data, but also the definition of designvariables, the objective function, and constraints. Whilean incredible amount of progress has been made in thelast 20 years in the generation of the analysis data (inter-active graphical mesh generation, automatic mesh gener-ation, material databases, etc.), most optimization datais still generated by hand in a text editor. Intuitive andeasy to use GUIs are starting to be developed to generatethe optimization data.

The GUI should not be configured so that the useressentially just types in the data as he would witha text editor. For example, the data fields for element

Fig. 6 GUI Panel for generation of displacement response data

thickness should include, in addition to its value, up-per and lower bounds. If these are filled in, the GUIshould automatically generate design variables and linkthese design variables to the properties. The same ap-proach can be used for material properties. If, in add-ition to the stiffness properties, stress limits are given,then the GUI should automatically create stress con-straints for the elements composed of the material. InFig. 6, a panel used to specify a displacement responseis shown. The user simply has to click on the node ofinterest to generate the response. Without an intuitiveGUI, the user would have to determine the node num-ber and hand edit this number into the input data file.The user can also specify the specific degree of freedomsto be constrained, as well as easily assign a name to theresponse.

For the automotive industry, special attention mustbe paid to the modelling and optimization of multi-celled thin walled beam cross sections. Many of thestructural members of an automobile are composed ofbeam elements formed by spot welding together twoor three sheet metal components. The design of these

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beams is critical for the stiffness of the entire automo-bile. Both the shape of the cross-section and the thick-ness of the sheet metal components need to be designed.In addition, certain components of one beam need tobe linked to components of other beams in the auto-mobile frame. This linking can be quite complicatedfor an entire frame. Innovative part definitions must bedeveloped.

In the area of graphical post processing of optimiza-tion output, simple XY graphs of design variable, ob-jective function, and constrained response histories areuseful. For size optimization results, element dimensionsneed to be plotted by colour coding the elements. Shapeoptimization results can be plotted and the shape designhistory can be viewed as animation. In order to post pro-cess topology results, it is desirable to be able to drawiso-density surfaces for solid models.

The time spent creating FEM meshes is far greaterthan the time spent running the analysis and even the op-timization. This is especially true if the engineers time isconsidered, as he can perform other tasks while the com-puter job is running. Creating new CAD data or a newFEM mesh from topology optimization results is verytime consuming. Commercial topology optimization soft-

Fig. 7 Original (left) and final (right) bracket designs

Fig. 8 Bracket performance characteristics

ware needs to have the ability to automatically createCAD data and FEM mesh data from the topology opti-mization results. This includes both solid and shell modeloptimization results.

12Examples

The first example is of the topology optimization of anautomotive radiator bracket. This example is courtesyof TECOSIM GmbH. The original design failed due tohigh fatigue stresses. The improved design reduced themaximum stress by 64% and at the same time reducedthe mass of the bracket by 31%. The deformed originaland final brackets are shown in Fig. 7. The performancecharacteristics of the original and improved designs arecompared in Fig. 8.

The second example is the design of a lightweight citybus from preliminary design to final structural design.This example is courtesy of Altair Engineering, Inc. Atfirst, topology optimization is used to generate the opti-mum structural layout concept. The design package space

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Fig. 9 Bus design package space (in gray)

Fig. 10 Topology optimization result

Fig. 11 CAD model interpretation of topology optimization design proposal

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is shown in Fig. 9 as the gray coloured panels. In Fig. 10,the results of the topology optimization are shown. Notethat the structure of the roof of the bus is not symmet-ric. This is because there are doors on only one side ofthe bus. Figure 11 shows the CAD representation of theinterpretation of the results of the topology optimiza-tion. Then sizing optimization is performed on the hol-low rectangular bar members of the bus structure. Thecolour-coded bar sizes are shown in Fig. 12. In Fig. 13,the final bus design is shown. Note that the shape ofthe windows was decided by the results of the structuralneeds identified by the topology optimization. It seems

Fig. 12 Results of sizing optimization

Fig. 13 Final bus design

to make the optical appearance of the bus even moreattractive.

13Future developments

One thing that is missing from most optimization codesis the ability to include manufacturing constraints. Themanufacturing engineer commonly says of optimizationresult, great design, but I cannot build it. Designsgenerated manually do not usually have this problem,

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as the designer, either consciously or subconsciously, in-cludes the manufacturing constraints in the design pro-cess. A simple manufacturing constraint is a lower boundon element or shape dimensions.

In topology optimization, it is very easy to generatestructures that cannot be easily manufactured. One prob-lem is that very thin members can be generated. A con-straint on the minimum size of members in the optimumtopology is a very useful manufacturing constraint. An-other very important manufacturing constraint for top-ology optimization is the stamping draw die, or castingmold removal, constraint. Resulting topologies that haveinternal members or voids cannot be stamped or cast.A constraint that does not allow internal voids to be gen-erated in the optimum topology is required for parts thatare to be stamped or cast.

Another important feature that needs to be addedto commercial codes is the ability to handle discrete de-sign variables. This is very important for civil engineeringproblems, as members come in fixed sizes. It is also im-portant for smaller companies, which must use off theshelf, rather than special order, components.

14Conclusions

Commercial optimization software has to be written withthe user and potential applications in mind. It is veryimportant to remember that the final goal is to makea profit by selling many copies of the software. In orderto develop a big and loyal customer base, the softwarehas to be easy to use and capable of solving practical de-sign problems in diversified industrial fields. In addition,the software must address a range of issues of structuraldesign from structural reliability through manufacturingrequirements. User friendly graphical pre and post pro-cessing is very important, as is the robustness of the code.Users must feel comfortable using the code and musthave faith in the results. Most commercial optimizationsoftware is leased on a yearly basis, and if the user is notsatisfied with the code, he or she will not renew the leasein the upcoming year. Since the FEM models of mostoptimization problems usually come from existing analy-sis models, commercial optimization software must havenearly all the FEM features of existing mainstream FEMcodes. This puts a huge burden on the optimization codedevelopers, as they must support a large number of FEMfeatures. Finally, it is very important to be able to deter-mine which features to add to the code for each release.Sales people and customers typically want features thathave to do with the analysis model capabilities of thecode. However, it is not possible to add all the featuresof a commercial FEM code in a short time, nor does itmake sense to do so. New optimization capabilities areimportant for the long-term sales of the code. However,customers are usually not asking for these features be-

cause, usually, they have not considered how optimizationcould be applied to the design process. They typically donot know enough about optimization technology to knowwhat to ask for. New optimization capabilities need to bedeveloped and then shown to the users. Finding the cor-rect mix of short- term and long-term features to add tothe code for each release is very critical in a competitiveindustry.

Acknowledgements The authors wish to thank TECOSIM

GmbH for the permission to use the radiator bracket example

and Volkswagen AG for permission to use the engine stiffeningpanel example.

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