01-introduction to finite element method

5/27/2018 01-Introduction to Finite Element Method

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Introduction to Finite ElementMethod

By

S. Ziaei-Rad


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Where the Course Fits

The field of Mechanics can be subdivided into 3 major

areas:

Mechanics

Theoretical

Applied

Computational


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Computational Mechanics

Nano and Micromechanics

Continuum Mechanics

Computational

Mechanics

Branches of Computational Mechanics can be distinguisheAccording to the physical focus of attention

Solid & Structures

Fluids

Multiphysics


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Computational Solid andStructural Mechanics

A convenient subdivision of problems in ComputationalSolid and Structural Mechanics (CSM) is

ComputationalSolid and Structural

Mechanics (CSM)

Statics

Dynamics

Linear

Non-Linear


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CSM Linear Statics

For the numerical simulation on the computer we must nowchose a spatial discretization method:

CSM Linear Statics

Finite Element Method

Finite Difference Method

Boundary Element Method

Finite Volume Method

Spectral Method

Mesh-Free Method


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CSM Linear Statics by FEM

Having selected the FEM for discretization, we must nextpick a formulation and a solution method:

Formulation of FEM Model

Solution of FEM Model

Direct Method

Variational Method

Weighted Residuals

StiffnessFlexibility

Mixed


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Formulation of FEM Model

1- The Direct Method- Limited to very simple element- It worth studying because it enhances the physicalmeaning.

2- The Variational Method- Applicable to problems that can be stated by certainintegral expression.

3- Weighted Residual Methods- Applicable to problems for which differential equationsare known but no variational statement is available.


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Basic Concepts

The finite element method (FEM), or finite element analysis(FEA), is based on the idea of building a complicated object wi

simple blocks, or, dividing a complicated object into small and

manageable pieces. Application of this simple idea can be fou

everywhere in everyday life as well as in engineering.

Examples:

Lego (kidsplay)

Buildings

Approximation of the area of a circle:


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Basic ConceptsArchimedes' problem (circa 250 B.C.): rectification of the

circle as limit of inscribed regular polygons

Area of one triangle:

Area of the circle:

where N = total number of triangles (elements).

)2/sin)(2/cos( iii RRS


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Computing "byArchimedes FEM"


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Why Finite ElementMethod?

Design analysis: hand calculations, experiments,and

computer simulations

FEM/FEA is the most widely applied computersimulation method in engineering

Closely integrated with CAD/CAM applications


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Applications of FEM inEngineering

Mechanical/Aerospace/Civil/Automobile Engineering Structure analysis (static/dynamic, linear/nonlinear) Thermal/fluid flows

Electromagnetics Geomechanics BiomechanicsFluid/solid Interactions

Fluid/thermal/solid Interactions


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A Brief History of the FEM

1941 ----- HrennikoffUsed 1D element (bars and beams) for the solution of stres

continuous solids.

1943 ----- Courant (Variational methods)

First to propose the FEM as we know today, he used princi

of stationary potential energy.

1956 ----- Turner, Clough, Martin and Topp (Stiffness)

Stiffness equations in matrix format and solved equationsdigital computers. (100 DOFs)

1960 ----- Clough (2D Finite Element, plane problems)

Triangular plane stress element to model skin of a delta win


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A Brief History of the FEM

1961 ----- Martin (3D tetrahedral elements)

1962 ---- Callagher, Padlog and Bijlaard (3D elements)

1963, 1964 ----- Melosh and Argyris (3D elements)

1965 ---- Clough and Rashid , Wilson (Axisymmetric solid)

1970s -----Applications on mainframe computers 1980s ----- Microcomputers, pre- and postprocessors

1990s -----Analysis of large structural systems


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A Brief History of the FEMFE DOFs

1950s ----- 100 DOFs1960s ----- 1000 DOFs1980s ----- 10000 DOFs1990s ----- 100000 DOFs2000s ----- 500000-Several millions DOFs

Papers Published in FEM

1961 ----- 10

1966 ----- 1341971 ----- 8441976 ----- 70001986 ----- 20000

..


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FEM in Structural Analysis

Procedures: Divide structure into pieces (elements with nodes). Describe the behavior of the physical quantities oneach

element,.

Connect (assemble) the elements at the nodes to forman

approximate system of equations for the wholestructure.

Solve the system of equations involving unknownquantities at the nodes (e.g., displacements).

Calculate desired quantities (e.g., strains and


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Computer Implementations

Preprocessing (build FE model, loadsand constraints)

FEA solver (assemble and solve the

system of equations)

Postprocessing (sort and display the

results)


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Available Commercial FEMSoftware Packages

ANSYS (General purpose, PC and workstations)

NISA (PC and workstation)

SDRC/I-DEAS (Complete CAD/CAM/CAE package)

NASTRAN (General purpose FEA on mainframes)ABAQUS (Nonlinear and dynamic analyses) COSMOS (General purpose FEA)ALGOR (PC and workstations)

PATRAN (Pre/Post Processor) HyperMesh (Pre/Post Processor)LsDyna, Dyna-2D, Dyna-3D (Crash/impact analysis)

Pro-Mechanica (PTC Company)

..


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1969 --- Pedro Marcal taught at Brown University for a time butHe set up a firm to market the first nonlinear commercial FEprogram called MARC.1969 --- John Swanson was developing a NFE program at

Westinghouse for Nuclear applications. He left Westinghouseto market program ANSYS.1972 --- David Hibbit who worked for Marcal until 1972 andThen co-founded HKS which markets ABAQUS. The program

was the first to introduce gateways for researchers to addelements and material models.1970s--- Bathe launched his program after completing his PhDUnder supervision of Wilson at MIT. ADINA was an outgrowth

Of NONSAP.


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1975 ---A milestone in the advancement of explicit FE wasJohn Halliquists work at Lawrence Livermore. He released

his code called DYNA in 1976.His success was the development of contact-impact interfaces

with Dave Benson and the resulting codes DYNA-2D andDYNA-3D.The DYNA code first commercialized by French firm ESI in 1980sand called PAMCRASH with many routines from WHAMS.-John Halliquist left Livemore and started his own firm todistribute LSDYNA, a commercial version of DYNA.


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Advantages and Disadvantages ofgeneral-purpose programs

Advantages1- The input is well organized.2- The are large systems and can solve many types of problemsof large or small size.3- Many programs have the ability for adding new modules for

new kinds of problems or new technology with minimum efforts.4- Many of them can run on PCs.5- Many of them have become very attractive in price and cansolve a wide range of problems.

Disadvantages1- General-purpose programs are less efficient than special-purposeprograms.2- The initial cost of developing general-purpose programs is high.

3- The user has little access to the logic of the program.


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Objectives of This FEMCourse

Understand the fundamental ideas of the FEM Know the behavior and usage of each type of elementscovered in this course

Be able to prepare a suitable FE model for given problems

Can interpret and evaluate the quality of the results (knowthe physics of the problems)

Be aware of the limitations of the FEM (dont misuse the

FEM - a numerical tool)


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Course Coverage

Finite Element DiscretizationConcepts

Formulation of Finite Elements

Computer Implementation of FEM

What do we need?1- ANSYS

2- MATLAB


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Examples:Boot SealBoot seals are used to protect steering mechanisms in automobiles.

These flexible components must accommodate the motions associatedwith angulation of the steering mechanism. Some regions of the boot

seal are always in contact with an internal metal shaft, while other areas

come into contact with the metal shaft during the angulation. In addition,

the boot seal may also come into contact with itself, both internally andexternally. The contacting regions affect the performance and longevity

of the seal.


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Boot Seal

Deformed configuration at 20

degrees rotation of shaft.Contours of maximum

principal stress in boot.


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Exhaust Manifold Assembly

The assembly considered (Fig. 1) consists of a four-tube exhaust manifold

fastened to a partial section of an engine head by seven bolts acting on threeflanges. The analysis consists of three steps. First, prescribed bolt loads fasten

the manifold to the head. Then, the assembly is heated to a steady- state thermal

operating condition, shown in Fig. 2. Finally, the assembly is cooled to a uniform

ambient temperature. The variation of the bolt loads is monitored as the bolts

respond to the thermal loading of the assembly.

Fig. 1 Exhaust manifold assembly


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Exhaust Manifold AssemblyThe base of the engine head is constrained vertically. Furthermore, it is

assumed that the bolts are threaded tightly into the head so that thebottoms of the bolt shanks share nodes with the surrounding head

elements and consequently are constrained vertically. The bolts and

engine head are modeled as elastic materials; the manifold is modeled

as an elastic-plastic, temperature-dependent material.

Fig. 2 Steady-state temperature distribution.


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Gear Meshing

Gears of various types are commonly used in modern machinery.

Historically, gear design has been based largely on textbook formulas,extensive testing, and previous design experience. This application brief

describes the simulation of gear meshing to predict gear tooth stresses

and overall gear performance during operation.

Contours of maximum principal stress


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Rail Crush

Crash simulations are performed on entire vehicle models, but the design

of individual components often requires their study on a stand-alone basis.This application brief describes a rail crush calculation.

The rectangular, box-section rail has an init ial velocity of 160 km/h and

impacts a rigid wall. Because of symmetry only half of the rail needs to be

modeled. The rail is made of an elastic-plastic, material. Its initial geometry

is designed to induce a collapse mechanism that wil l maximize energy

absorpt ion. The shell elements account for finite membrane strain, which

is required for accurate simulation of this crushing process


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Rail Crush

The program accounts for self contact throughout the simulation,including the effects of changing shell thickness, as points come into

contact and surfaces slide along one another.

Intermediate deformed configurations. Plastic dissipation history in rail.


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Dynamic Analysis of a Jack-Up PlatformMobile jack-ups play an important role in the initial development of

shallow-water oil reserves. They must be designed to withstand severeand random ocean wave, wind, and current loading caused by storm

conditions.

Figure 1: Elevated jack-up platform


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Dynamic Analysis of a Jack-Up Platform

The next phase of the investigation involves a geometrically nonlinear,

transient dynamic simulation of the jack-up subjected to prescribed waveand current loadings. Gravity, buoyancy, fluid inertial, drag, and structural

and hydrodynamic damping effects should all be modeled.

Partial time history of the wave trace. Partial time history of hull sidesway.


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Thermal Fatigue of aSurface Mount AssemblyLow-cycle fatigue is a common failure mechanism in solder joints of surface

mount assemblies in the electronic packaging industry. Cyclic thermal loadingcombined with di fferences in thermal expansion properties for the various

components of the assembly lead to stress reversals and the accumulation of

inelastic strain in the joints. Predictions of fatigue life in solder joints require a

thorough understanding of the deformation and failure mechanisms of the

solder alloy and an accurate calculation of the stresses and strains in the joint.


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Thermal Fatigue of aSurface Mount Assembly

The analysis consists of a single superelement generation step and three

cycles (12 steps) of thermal loading. The automatic t ime stepping scheme

uses a combination of implicit and explicit time integration techniques to

maximize solution efficiency for problems involving creep behavior.

Deformation of corner legs at the

end of the first holding period.

120 C 0 C

Equivalent creep strain

distribution in the solder joint

after three thermal cycles


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Thermal Fatigue of aSurface Mount Assembly

The corresponding Mises stress history for point A is plotted in Figure 1 (right).

The second and third cycles appear to be the same because the init ial stressstate condit ions of the second and subsequent cycles are similar. The initial

dip and subsequent peak in stress during the heating stage of the second and

third cycles are due to the combination of the initial stress state and the

competing effects of creep relaxation and CTE mismatch between the PCB

and chip.

Figure 1: Strain and stress histories at point A.


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Hydroforming of a SquareBox

Hydroforming of sheet metal components is widely used in several

industries. While numerous variations of hydroforming exist, the basic

principle remains the same: utilize fluid pressure to form a component.


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A critical parameter in hydroforming is the chamber pressure magnitude,which typically varies as a function of punch displacement. Excessive

pressure may lead to tearing of the blank, while insufficient pressure may

result in wrinkling.

Hydroforming specifications rely heavily on the intuition and experience ofdesign engineers. Iterative cut-and-try development cycles are cost ly and

time-consuming. As demonstrated by the square box hydroforming

problems described here, use of an explicit FE is an accurate and efficient

simulation tool that reduces time and costs associated with physical

cut-and-try methods.


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Initial configuration for tr ial 1. Initial configuration for trial 2.


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Hydroforming of a Square

Box

Final configuration of the box. Contours of box wall thickness.

To suppress wrinkl ing of the box, a rigid draw cap is added to the model, as

shown in Figure trial 1. The position of the draw cap (shown in orange) in theactual hydroforming process is depicted in the inset to Figure trial 1.

For modeling purposes only the surfaces of the draw cap that contact the blank

are required


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Continuous Casting

Continuous casting simulation.

In a continuous casting process liquid aluminum (Al-0.7% Mg) is passed

through a water-chilled mold to initiate the solidif ication process on the

outer skin of the liquid. Water is then sprayed on the top and bottom of

the casting to continue the cooling process. The objective of this analysis

is to determine the shape and location of the freeze front under steady-state

conditions.


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Continuous Casting

Steady-state temperature distribution

The upstream boundary condition consists of a fixed inlet temperature, while

the downstream condition allows convection but no axial conduction. Heattransfer between the aluminum and the mold and between the aluminum and

the water spray is modeled with " surface-based" thermal interaction. A thermal

boundary layer is simulated along this interface, with surface heat transfer

coefficients and cooling fluid temperatures specified as functions of axial

position.


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Continuous Casting

Steady-state temperature distribution


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Hip Implant

While total hip replacement has become a common surgical practice, thereare continuing efforts to optimize further the implant design and to extend

the durability and life of the joint. This application br ief examines the

interaction between the implant and femur resulting from an initial interference

fit and subsequent service loads.


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Hip Implant

The analysis is conducted in two steps. In the first step an interference

fit between the implant and femur, simulating an implant using press-fitfixation, is resolved.


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Creep of a Pipe Intersection

Creep is the permanent elongation of a component under a staticload maintained for a period of time. Most metals and their alloyscreep only at elevated temperatures, but several materials such asthermoplastics and rubbers do so at room temperature. Designersestimating the service life and structural integrity of componentsmust account for creep effects in their designs.

This model represents the intersection of a pipe with a cylindricalpressure vessel. The system operates at an elevated temperatureand carries internal pressure. The calculation consists of two steps.

In the first step a static analysis is performed, during which theinternal pressure is applied.In a second step a transient analysis is carried out to determinethe creep of the pressurized vessel.


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Creep of a Pipe Intersection

The one-quarter model shownconsists of 904 second-order brickelements. Symmetry boundaryconditions are enforced on theappropriate sectioned surfaces(displayed in red).The remaining sectioned surfaces(displayed in blue) are undertensile load and are constrainedto remain planar.A single node is constrainedin the vertical direction toeliminate rigid body motion.


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Creep of a Pipe IntersectionIn this case a power law creep model is used. The automatic time

stepping scheme uses a combination of implicit and explicit timeintegration techniques to maximize solution efficiency for problemsinvolving creep behavior.


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Creep of a Pipe IntersectionStress and strain histories at point A.


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Circuit Board Drop TestElectronic components frequently have drop test requirements. Thisapplication brief describes the simulation of a circuit board dropped ontoa flat, rigid floor. The objective of the simulation is to assist in the designof the packaging material (a crushable foam), which is intended to protectthe circuit board from damage in such an event.

The analysis, performed with Explicit FE, uses the crushable foam model

for the packaging material. This inelastic material model has beenimplemented for use with foamed, lightweight, energy absorbing materials.Strain-rate-dependent effects, as well as high volumetric compressibility,are included in the model. The packaging is modeled with 1200 brickelements. The circuit board is treated as a rigid body with appropriate massand rotary inertia: we are interested in the accelerations experienced by the

circuit board but not in the details of any deformation in this component.The circuit board and foam assemblage is dropped at an angle onto a flat,rigid surface from a height of 1 meter. The illustrations shown correspond

to deformed shapes at separate points in time. An acceleration history ofthe center of gravity of the circuit board is also included. The simulationshows that part of the board disengages from the packaging materialduring the impact event.


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Circuit Board Drop Test

f


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Three configurationsduring impact

Acceleration history of the


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ycenter of gravity of the circuit

board (g's).

R lli f Thi k Pl t


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Rolling of Thick Plates

Hot rolling is a basic manufacturing technique that is used to

transform preformed shapes into forms that are suitable forfurther processing. Important aspects of the manufacturingoperation are elongation and spread of the material during therolling process.

Friction plays a key role in the simulations since it provides themechanism by which the plate is pulled through a roller. When apoint on the surface of the plate has just made contact with a roller,the roller surface is moving faster than the point on the workpiece,and there is relative slip between the two surfaces. As the point onthe plate is drawn into the process zone under the roller, it movesfaster and, after a certain distance, sticks to the roller. As the pointon the workpiece is pushed out of the process zone, it speeds upand moves faster than the roller, causing slip in the oppositedirection before separation takes place.


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A single-roller operation

A refined mesh is used for the single roller isothermal simulation.There are 2944 8-node brick elements used for the workpieceand 89 rigid elements used for the roller.


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A two-roller operation

A coarse mesh is used for the two-roller simulation, which includesthe effects of adiabatic heating due to plastic work. There are80 8-node brick elements used for the workpiece and 60 rigidelements used for each of the rollers.


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A two-roller operation


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Airplane


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References

1-An Introduction to the Finite Element Method

J. N. Reddy, McGraw-Hill, 19932- The Finite Element Method Linear Static and Dynamic Finite Element Analysis

Thomas J. R. Hughes, Prentice-Hall, 19873-Finite Element Modeling for Stress Analysis

R. D. Cook, John Wiley & Sons, 19954- Building Better Products with Finite Element Analysis,Vince Adams & Abraham Askenazi, OnWord Press, 19985- The Finite Element Method: Volume 1, Basic Formulation

and Linear Problems

O. C. Zienkiewicz and R. L. Taylor, Fourth Edition, McGraw Hill, 19756- Introduction To Finite Elements in Engineering,T. Chandrupatla and A. D. Belegundu, Prentice Hall, 19977-A First Course in the Finite Element Method,

D. L. Logan, PWS-Kent, 1986


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References

8-Numerical Methods using MATLAB

J. Penny & G. Lindfield, Ellis Horwood Limited, 19959- Programming the Finite Element MethodSmith & Griffiths, John Wiley and Sons, 199210- Finite Element for Analysis and Design

J. E. Akin, Academic press, 199411- The Finite Element Method using MATLABY. W. Hown, H. Bang, CRC Press, 1996

01-introduction to finite element method

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