A New Approach for Sizing,Shape and Topology

Optimization

1996 SAE International Congress and ExpositionDetroit, Michigan USA, February 26-29

Nima BakhtiaryThe MacNeal-Schwendler Corporation, Los Angeles

Allinger, Friedrich, Müller, Mulfinger, Puchinger, SauterFE-DESIGN and University of Karlsruhe, Germany

Authors:

Nima Bakhtiary MSC Email: nima.bakhtiary@macsch.comThe MacNeal-Schwendler Corporation815 Colorado Boulevard,Los Angeles, CA 90041

Peter Allinger FE-DESIGNMatthias Friedrich FE-DESIGNFritz Mulfinger FE-DESIGNJürgen Sauter FE-DESIGN Email: juergen.sauter@mach.uni-karlsruhe.de

FE-DESIGN SauterSchützenstraße 6976137 Karlsruhe/GermanyTelefon 0049-721-608-2376Telefax 0049-721-608-6053

Ottmar Müller University of Karlsruhe Email: ottmar.mueller@mach.uni-karlsruhe.deMartin Puchinger University of Karlsruhe Email: martin.puchinger@mach.uni-karlsruhe.de

Institut für Maschinenkonstruktionslehre und KraftfahrzeugbauUniversität Karlsruhe (TH)Kaiserstraße 1276128 Karlsruhe/GermanyTelefon 0049-721-608-2376Telefax 0049-721-608-6053

Abstract

This paper for the first time presents the new inter-face between CAOSS and MSC/NASTRAN.CAOSS® (Computer Aided Optimization SystemSauter) is an optional finite element module for theefficient sizing, shape and topology optimization.The introduction gives a survey on the program itselfand on its history. To begin with, the first chaptergives an overview of the optimization type (sizing,shape and topology) and the state-of-technology. Inaddition the principal differences between the math-ematical programming methods and the methodsbased on optimum criteria are explained. A compre-hensive chapter reveals the theoretical backgroundsof the optimum criteria for shape and topology opti-mization purposes as well as the especially devel-oped controllers created by the authors. Theinteraction of CAOSS and MSC/NASTRAN and theoptimization options new to the MSC/NASTRANuser in conjunction with CAOSS are emphasized.Finally industrial application examples give animpression of the capabilities.

Table of Contents

1. Introduction

2. Optimization Types2.1 Sizing Optimization2.2 Shape Optimization2.3 Topology Optimization

3. Optimization strategies3.1 Mathematical Programming Methods3.2 Optimum Criteria

4. A new „Control“ Method for Shape and TopologyOptimization4.1 Origin and Method Development4.2 The Shape Optimization Theory

5. A New „Control“ Method

5.1 Sizing Optimization5.2 Shape Optimization5.3 Topology Optimization

6. CAOSS and MSC/NASTRAN6.1 Optimization with MSC/NASTRAN6.2 Optimization using CAOSS and MSC/NASTRAN6.3 Communication between CAOSS and MSC/

NASTRAN6.4 Sample-Listing

7. Examples7.1 Sizing Optimization7.2 Shape Optimization7.3 Topology Optimization

8. References

1. Introduction

In the tough international competition, companiescan only survive if, besides highly innovative powerthey can provide strongly cost optimized products.Therefore in new procedures like the SimultaneousEngineering, the calculation engineer is already inte-grated into the concept phase of the product develop-ment process. Efficient methods of working requirepowerful optimization algorithms to be provided inaddition to the discrete methods (FEM/BEM) provedworth while to support the calculation engineer inthe draft and design phase.In recent years many optimization approaches havebeen integrated into commercial FE programs. Inindustrial applications only few optimizationmethods are established partially. The problems to besolved in industry have to be abstracted dramatically.The further development of the optimizationmethods regarding application, integration andnumerics is necessary and pushed ahead intensively.In this context new optimization criteria and controlstrategies for sizing, shape and topology optimiza-tion [1, 57, 59, 61] were created at the University of

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Karlsruhe, Germany. Based on these new strategiesthe computer program CAOSS was developed byengineers for engineers. Considering the hypothesisformulated in [57, 59] relating to the strain minimum,very fast and efficient control algorithms could bedeveloped for sizing, shape and topology optimiza-tion based on FE models. CAOSS is a moduleextending FE programs for optimization purposes.Interfaces exist to various FE programs. In this paperthe interface to MSC/NASTRAN is presented for thefirst time.CAOSS was first presented to a wide public at theCOMETT Optimization Conference of the EuropeanUnion in May 1992. Both representatives of theindustry and participating researchers from universi-ties were impressed by the results. Since 1993 CAOSShas been distributed by CAE Partner (a company ofthe MSC Group) and has proved worth while inmany companies. Many serial components havebeen optimized [17, 22, 31, 32].In 1994 the program was awarded with the EuropeanAcademic Software Award 1994 of the EuropeanCommission for the best academic software in thefield of mechanics.

CAOSS has the following advantages:

• CAOSS is easy to use, flexible and requires onlyvery little computing time.

• For optimization purposes, only an existing FEmodel is required (Bulk Data Deck), which neednot be parametric thus old models can subse-quently be optimized, too.

• The choice of design variables need not be takeninto consideration to begin with. The number ofiterations until a solution is found is very small andindependent of the number of design variables.

• CAOSS is a module in addition to the already exist-ing FE environment working largely in batchmode, thus ensuring a quick and easy lead-ing tooptimization.

• The optimization procedure is clear and compre-hensible. The approach corresponds to the way anengineer works.

• CAOSS can optimize structures of any size for vari-ous load cases at the same time.

• For shape optimization purposes a CAOSS internalmesh correction module adapts the mesh topologyto the modified geometry. Thus remeshing with tet-raeder elements is avoided.

• CAOSS runs under UNIX and DOS on worksta-tions, mainframes and PC’s.

2. Optimization Types

The following gives an overview of the main differ-ences among sizing, shape and topology optimiza-tion. Both the numerical and the user-specificcharacteristics are discussed shortly.

2.1. Sizing Optimization

For optimization purposes using „Sizing Variables“i.e. cross sections and thicknesses of finite elements,many mathematical programming approaches havebeen tested and implemented into finite elementprograms (e.g. MSC/NASTRAN, PERMAS,COSMOS/M) and special optimization programs(e.g. MBB-LAGRANGE, STARS, ADS). During theoptimization, the element properties are modified onthe FE level. Due to the easy calculation of the sensi-tivities for sizing optimization purposes even real-istic problems can be handled. Today theseapproaches can be considered as state-of-the-art.

2.2. Shape Optimization

Compared with the sizing optimization the shapeoptimization is more complex. For the shape optimi-zation two approaches are used:

a.Shape optimization based on FE modelsThe coordinates of the surface nodes are regardedas design variables which will be modified duringthe optimization. This usually leads to a large num-ber of design variables which might cause consi-derable mathematical difficulties. Using suitablecouplings of node displacements to define basis

VariationArea

TopologyOptimization

ShapeOptimization

Figure 1: Integrated Design Concept using CAOSS

Figure 2: Sizing Optimization of a Shell Structure(For the Visualization, the PropertyID’s are shown as Solids)

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vectors, complex geometry changes can be descri-bed in the Solution 200 of MSC/NASTRAN withonly few design variables [43, 50].

b.Shape optimization based on geometry modelsUsing the shape optimization method based ongeometry models, the linkage of an FE model and ageometry model is maintained. As in this case theparameters of the geometry model are the designvariables, the geometry model has to be fully para-metric. Therefore the use of an efficient solid mode-ler is necessary. Each parameter modification of thegeometry model also results in changes of the FEmodel. Within each optimization loop the entire FEmodel has to be set up anew according to the modi-fications of the geometry model parameters consi-dering the boundary conditions. This method isused e.g. in the programs ANSYS, COSMOS/Mand IDEAS. Many interfaces are put on the integra-tion of an FE solver and a CAD system (e.g. ProEn-gineer). The selection of the design variables is leftto the user. In general they differ due to his experi-ences and creativity. The results of the optimizationessentially depend on the number and selection ofthe design variables. If free form surfaces are allo-wed, the selection of the design variables is verydifficult. Using mathematical programmingmethods, the number of design variables must notbe to large too, because this causes the numericalefforts to increase drastically.

The main difficulty with shape optimization is totransfer the surface changes to the FE mesh. Mostprograms avoid this transfer by an automaticremeshing in each optimization loop. Hence the orig-inal element topology (meshing) is destroyed andoften models with only tetraeder elements arecreated. Only few programs are capable of such atransfer in a way that the mesh modification isanalyzed starting from the modification of the

surface while maintaining the element topology. Toavoid the difficulties with remeshing some programsuse p-elements for the shape optimization [49]. Stillthere is the problem of selecting suitable design vari-ables. For large models the numerical efforts areextremely high.

2.3. Topology Optimization

Both for sizing and shape optimization a first designproposal, which is used as the start design, exists.The objective of general structural optimizationmethods is to compute even this first designproposal. Therefore an area (2D or 3D) with a homo-geneous material distribution is used. Subsequentlythe functionally required boundary conditions (e.g.node constraints, nodal loads) are applied. Theefforts for the modelling and preparation isextremely low. The optimum structural shape withthe appropriate topology is issued as designproposal. The originally homogeneous materialdistribution becomes highly inhomogeneous. Areasarise with no more mass at all (openings and holes)and areas which contain high density mass (bars andstruts).

Compared with the sizing and shape optimizationthe numerical efforts strongly increases. The numberof design variables is typically between 5.000 and100.000. Therefore large efforts have to be put intothe sensitivity analylsis up to now. So far no method[8, 9, 10, 11, 33, 34, 40, 41, 54, 55] can be considered asa standard for calculating the optimum topology dueto the above mentioned difficulties regarding themathematical approaches. Essentially because of theexpensive calculation efforts these approaches onlycan handle extremely simplified models. Commer-cially only few programs are available [2, 63]. ManyFE developers work in this field [13, 69]. Because ofthe development of powerful iterative solvers andthe more and more increasing computer capacities,topology optimization will be state-of-the-art verysoon.

Figure 3: Shape Optimization based on a FE-Model(approx. 100 Design Variables) [67]

Figure 4: Shape Optimization based on a Geometry Model(6 Design Variables) [12]

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3. Optimization strategies

Two approaches to the solution of a problempredominate over optimization. On the one handthese are the mathematical programming methods.They replace the mechanical model by a parametricmathematical substitute model which will then bestudied with strictly mathematical methods. Theother approach is based on the optimum criteriamethods. For that reason requirements are formu-lated which are valid for the optimum design. If theoptimum criteria can be applied to the certain task,the solution converges rapidly.

3.1. Mathematical Programming Methods

The structural optimization problem will be openedto the mathematical programming methods byformulating a substitute problem. The problemformulation [62] looks as follows.

In this case f(x) is the target function (e.g. structuralweight, deformation, stress, expansion, eigenfre-quencies, flutter/wobble rate, etc.). The function gj(x)represents mostly nonlinear boundary conditions,which constrain the solution.The calculation of the solution is done in two steps,the calculation of the search direction and theincrement γ. With this the vector is deter-mined, which leads from the current vector to thenext solution . For the calculation ofthe search direction and the increment manyapproaches exist, according to the various types oftarget functions and boundary conditions. Finally thesolution produced by the analysis program must beverified. If the start values are „disadvantageous“, itmight occur that the optimizer only identifies a local

minimum as the solution. Therefore it is alwaysrecommendable to modify the start values. As theinitial problem is transformed into a substitutemodel, the mathematical programming methods cangenerally be used. But they require an enormousamount of computing time which will even increaseif the number of design variables and active restric-tions is increased [62].

3.2. Optimum Criteria

In contrast to the mathematical programmingmethods, the optimum criteria methods take advan-tage of the knowledge on the physics and mechanicsof the respective problem set. Theses will be postu-lated describing the optimum.A well-known and ascertained physical law relatingto structural mechanics is for instance the FullyStressed Design which can actually only be appliedto statically determined structures. An importantmathematical optimum criterion is the Kuhn-Tuckercondition which is for convex optimization purposesfulfilled necessarily and adequate in the optimum.The theses on stress homogenization and stress mini-mization are optimum criteria, too [59, 60].Regarding the optimum criteria methods, thesecriteria and the response behaviour of modificationsof the physical model are implemented into the algo-rithm. With suitable redesign rules, a convergencebehaviour is achieved which cannot be attained withmathematical optimizers. Applying this particularphysical and mechanical knowledge, the optimumcriteria methods remain limited to the certain appli-cation areas. Applying this knowledge makes theindividual optimization steps comprehensible. Theremaining potential to the optimum, which is knownfrom the optimum criteria, can easily and exactly beestimated.The optimum criteria are particularly well proven forshape and topology optimization where a largenumber of design variables is required. The conver-gence speed is independent of the number of designvariables.

4. A new „Control“ Method for Shape andTopology Optimization

4.1. Origin and Method Development

In 1991 Sauter of the „Institut für Maschinenkon-struktionslehre und Kraftfahrzeugbau“ the „MachineDesign Institute“ of the University of Karlsruhedeveloped a theory [57] describing the optimumcontour for components leading to minimum strainmaxima. The theory is based on works of Baud [6, 7]and Neuber [47, 48] as well as extended statements ofSchnack [64, 65, 66]. Objective is to minimize themaximum strain, modifying a given shape and givenvariation areas (areas where the surface is to bemodified in whole or in part).Significant changes were made w.r.t. the control ofthe surface compared with Schnacks’ gradientless

Figure 5: Topology of different Load Cases

min f x( )

gi x( ) 0 j; 1, ... , me= =

x IRn

j;∈ me 1 … m, ,+=

xi x xu≤ ≤

dνγdν

xνxν 1+ xν γdν+=

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shape optimization [30, 64, 65, 67] and the relatedoptimization strategy according to the rules of natureby Mattheck [36, 38]. Thus the convergence speedand the handling were highly improved. An essentialextension was made for areas not to be modified sothat now even contact problems can be optimized.The incorporation of user-optimized functionalitieswas of special importance.In 1992 the same research team expanded theoptimum criteria w.r.t sizing and topology optimiza-tion purposes of 2D and 3D structures, loaded by oneor several load cases.The entire description of the theory of theseapproaches would go beyond this paper. The authorspublished detailed papers on shape optimization [56,57, 59]. Therefore only the first two theses areexplained. For further details, please refer to [60].

4.2. The Shape Optimization Theory

The formulation of the shape optimization problemis:

An area with the edge δV is given,whereby V resp. δV are defined by the componentresp. its edge.The maximum load stress Bmax resulting from aprescribed component load shall be minimized by an„optimum edge“ between two given edge points A

and B (A, B δV).

The optimum edge (Γ δV) is required lying within

a specific variation area x (Γ Γ *) defined by thedesign boundary conditions so that the load stressmaximum is minimized in V.The maximum load stress derives from one or moreload cases.

Preconditions are that:1. the law of stress decay is in effect,.2. there is a maximum load stress on the component

edge δV.If the maximum load stress exists on the edge δVwithin the variation area Γ*, the following thesesapply to a load case:

Thesis 1a.The load stress on an edge Γ between two given

points A and B is minimal, if the load stress onthe edge Γ is constant.

b.A constant load stress on the edge Γ exists, if theedge Γ between the limiting points A and B doesnot adjoin to the border of the variation area Γ*.

Thesis 2a.If the edge Γ adjoins to the border of the variation

area, the load stress on the portion Γf between thetransition points A' and B' is constant and theload stress on the edges Γga (AA') and Γgb (BB') issmaller than on Γf.

b.The longer the edge Γf is, the smaller the maxi-

mum load stress is on Γf. If Γf is equal to Γ, theload stress is minimal (if the variation limits Aand B are fixed) and Thesis 1 is applicable.

Example relating to Thesis 1 and 2 (shaft shoulder):

The example is a shaft shoulder loaded with tensilestress. The axial cross-section shows the contour ofthe start design and the variation area Γ∗. The verticallongitudinal axis is the rotational axis, the force actsin vertical direction. The complete load stresshomogenization results from an increase in materialat the transition between shaft and radius. The varia-tion area has been defined in a way that no surface

V IRq

(q=2, 3)⊂

∈

⊂⊂

Figure 6: Load Stress Minimum in Case EdgeΓ adjoins

to the Borders of the Variation AreaΓ*

Figure 7: Shaft Shoulder with Variation Area Γ∗

Page 5

displacement is allowed into the component. In 1934Baud [6] found a similar contour for flat bars loadedwith tensile stress.

The reference stress curve according to von Misesalong the surface (being the running coordinatealong the surface) shows the complete load stresshomogenization in the area where the edge to beoptimized is not limited by the variation area. Themaintained constancy in this profile can vividly bedescribed by illustrating that the edge effect for allcross-sections is exactly compensated by its cross-sectional increase. This directly causes the curvatureto constantly grow. The arc of the edge AA’ lies onthe edge of the variation area. In this area, the loadstress is not constant, but smaller than on the edgelying within the variation area. This example provesThesis 2a. If the variation area is reduced, theconstant load stress level of the free edge willincrease.

4.3. The Topology Optimization Theory

Actually this is an applied energy equation. Theenergy equation says that for elastic systems theouter work Wa is preserved without loss as shapechange energy in the deformed system. Therefore thedisplacement depends on the accumulated strainenergy. According to the energy equation the shapechange energy U yields from the identity with the

external shape change energy Wa which results fromthe n applied generalized forces Fi.

For maximizing the stiffness, the minimum of thestrain energy has to be found:

This corresponds to the equation of Menabrea, i.e. astatically undetermined system, which is stresslesswithout external loads, has a stationary minimalshape change energy value. In accordance with thehypothesis of Beltrami this also leads to a strainminimum.For the optimization, the shape change energy U inthe variation area is homogenized and minimized byspecific changes of the compliance matrix. Intro-ducing a fictive porosity and with a defined masslimit (mean porosity) the stiffness variation can betransformed into a density variation. The similarityto the homogenization approach of Bendsoe andKikuchi and the method of Mlejnek is obvious [9, 10,11, 34, 40].

5. A New „Control“ Method

The use of optimum criteria requires the formulationof redesign rules which, in dependence on the statevariables (load resp. energy), moves rapidly towardsan optimum component via feedback, thus avoidingthe problem of the highly computer-bound sensi-tivity analysis. Using this approach, the importanttask is actually to find an appropriate controller. Acontroller can be designed the more precisely, themore exactly the system behaviour is known.For mechanically loaded components the authorsderived redesign rules for sizing, shape and topologyoptimization purposes from the optimum criteria,discussing the principal static and not the dynamicaldamper and memo characteristics nor the adaptivityand „learning capabilities“ of the controllers. Forpractical use the integration of restrictions and func-tionalities and various particular cases have to beobeyed.The description assumes that the optimization is

Figure 8: Profile of the Shaft Shoulder loaded with TensileStress before and after the Optimization

Start Design

OptimizedDesign

0

50

100

150

200

87 97 107 117

Startdesign

opt. Design

SIGE [MPa]

S

Figure 9: Reference Stress along the Surface of the ShaftShoulder loaded with Tensile Stress

Start Design

OptimizedDesign

U Wa12--- Fi wi⋅

i 1=

n∑==

min U( ) min FTH F =

Page 6

based on finite element models.

5.1. Sizing Optimization

The input parameters for the controller are theelement-specific characteristics (Property-ID’s, Shellthicknesses, cross sections, etc.) and the local elementloads, the output parameters are the modification ofthe element-specific characteristics. Dependent onthe strain level the thicknesses resp. cross sections areincreased or reduced so that the strains seek homoge-neous values. Basically this complies with the FullyStressed Design rules.

5.2. Shape Optimization

The input parameters for the controller are the localnode coordinates and the local node strains. Theoutput parameters are the local modifications of thenode coordinates. The controller reduces the surfacecurvature by applying mass at points with highstrains. For low strains the surface curvature isincreased by removing mass at these points.

5.3. Topology Optimization

For topology optimization the assignment of the FEdata to the control parameters is not as clear as forsizing and shape optimization. The input parametersare the material distribution (Property-ID’s, ElementProperty-ID’s) and the local element strains. Theredesign rule says how the required resp. allowedmaterial has to be distributed regarding theboundary conditions and the set target mass so thatthe remaining mass will than be equally loadedconsidering all load cases. Starting from a homoge-neous material distribution, mass will be compressedin areas of high energy density and diluted in areasof low energy density. To simulate the inhomoge-neous material distribution for shell structures, thethickness of the shells (Property-ID of the finiteelement) resp. Young’s modulus are well suited. Forsolid structures the inhomogeneous material distri-bution is easily simulated by using Young’s modulus.Hence follows that the output parameters are newassignments of the element properties includingmaterial modifications.

6. CAOSS and MSC/NASTRAN

6.1. Optimization with MSC/NASTRAN

For many years MSC/NASTRAN has comprisedpowerful sizing optimization solutions which arebased on mature mathematical algorithms [43]. Inaddition MSC/NASTRAN provides sensitivities formany areas. With the Solution 200, which is based on amathematical approach, an efficient shape optimiza-tion module has been implemented in Version 68 [50,52]. Hereby complex restrictions can be defined andthe combination of different analysis types is possible.The proper selection and definition of the basis vectorsmainly depends on the experiences and creativity ofthe user. The efforts the user must take is very muchinfluenced by the used Pre- and Postprocessor.

6.2. Optimization using CAOSS and MSC/NASTRAN

With the integration of CAOSS into MSC/NASTRANfurther opportunities are opened to the user. In avery easy way shape optimizations can beperformed. The more flexible the surfaces are themore advantageous are the optimization results. Ifthe optimum design has to fulfill complex restrictionswhich are not included in the optimum criteria,preoptimizations can be performed with CAOSSvery easily providing the basis vectors for the Solu-tion 200. With the CAOSS topology optimizationcapabilities the MSC/NASTRAN functionalities areconsiderably improved.

6.3. Communication between CAOSS and MSC/NASTRAN

The CAOSS concept ensures that the MSC/NASTRAN user may continue working in his wellknown FE environment. He need not learn furtherpre- or postprocessors and may use the MSC/NASTRAN solver. Both the FE model for the FEcalculation (FE bulk data deck) and the optimizationmodel for the optimization (optimization bulk datadeck) are created using e.g. MSC/PATRAN or MSCfor Windows. The FE bulk data deck contains the

Figure 10: Optimization based on the Optimum Criteria

boundaryconditions

geometry

strain (Loading)

strain (Loading)

System(Component)

Controller(Optimization)

Figure 11: Optimization of a Flyweel Clutch usingMSC/NASTRAN, MSC/PATRANand CAOSS

Shape OptimizedArea

Page 7

input for the solver containing all FE-specific modeldata and boundary conditions. Besides the node andelement information, the optimization bulk data deckcontains further optimization boundary conditionsfor the optimization run. These might be simple noderestraints or e.g. shell structures can be defined usedas displacement limits during the shape optimiza-tion. This is a way to define complex geometric opti-mization boundary conditions very easily.

The interactive CAOSS preprocessor (CAOSS_PREP)reads the optimization bulk data deck and generatesa file controlling the optimization. Additionallyrequired resp. possible options are set for the optimi-zation (e.g. definition of design variables, optimiza-tion type, load stress hypothesis, variation andrestricted areas, node restraints, stop criteria) whichare not contained in the optimization model directly.

The optimization loop can be started as soon as theFE model, the optimization model and the control filehave been created. In case of an identical FE modeland optimization model (e.g. for the topology optimi-zation) all information (CAOSS control file included)can be written to a MSC/NASTRAN input file. Inthis case all CAOSS-specific commands begin with$CAOSS. The CAOSS optimization itself runs inbatch mode. In the first iteration the original FE bulkdata deck is calculated with MSC/NASTRAN. Theoptimization module (CAOSS_OPT) directly accesses

the XDB data base of MSC/NASTRAN to read theanalysis results. For the postprocessing of the resultswith MSC/PATRAN, a DMAP sequence creates anOUTPUT2 for each iteration. The load stress quanti-ties of the structure are evaluated automatically byCAOSS_OPT to determine the modifications of theFE model obeying the defined optimizationboundary conditions. CAOSS_OPT modifies themodel in the FE bulk data deck which is the input forthe solver during the next iteration. In detail these areproperty and element modifications for the sizingoptimization and grid location modifications for theshape optimization. For the topology optimizationthe material cards, property ID’s and elements aremodified. The new results deriving from the MSC/NASTRAN analysis are checked by CAOSS_OPT forthe stop criterion.

For the shape optimization an adaptive mesh correc-tion algorithm is implemented in CAOSS which runswithout an additional FE analysis. The original meshtopology is maintained thus not destroying the elab-orate mesh. This mesh adaption preserves the qualityof the mesh. Large element distortions only appearfor extremely large shape modifications.

The interaction between the CAOSS modules on theone hand and MSC/NASTRAN on the other hand iscontrolled by a control file on the operating systemlevel (UNIX-Shell). Modifications for special prob-

Page 8

FE-Model

Optimization-Model

CAOSS-Preprocessor

MSC/NASTRAN

CAOSS

CAOSSData Base

MSC/PATRAN,MSC/NASTRAN for Windows

FE-Bulk Data

Mod

ified

FE

-Bul

kD

ata

Opt

imiz

atio

nB

ulk

Dat

a

MSC/NASTRANData Base

(XDB)

OptimizationControl File

Figure 12: Schematic Flow of the Optimization using CAOSS and MSC/NASTRAN

lems or company-specific peculiarities can beperformed easily.

6.4. Sample-Listing

The following shows an MSC/NASTRAN input filefor topology optimization purposes. Both the BulkData Section from the FE model and the Bulk DataSection of the optimization model are identically forthis problem. CAOSS_PREP reads the entire MSC/NASTRAN input file including the special CAOSScommands ($CAOSS ... ). MSC/NASTRAN inter-prets these lines as comments. The shown CAOSScommands are sufficient for the topology optimiza-tion of a small problem. Totally, CAOSS providesapprox. 20 different commands.

_____________________________________________________

$ FILE MANAGEMENT SECTION (FMS)

ASSIGN output2=’onehole.op2’ status=unknown unit=12

$

$____________________________________________________

$ EXECUTIVE CONTROL SECTION (ECS)

SOL SESTATICS

TIME 10000

$ DMAP to create two data bases: OUTPUT2 and XDB

INCLUDE ’XDBAOUT.DAT’

CEND

$____________________________________________________

$ CASE CONTROL SECTION (CCS)

ECHO = NONE

DISPLACEMENT = ALL

SPCFORCE = ALL

OLOAD = ALL

FORCE = ALL

STRESS = ALL

SPC = 1

STRFIELD = ALL

SUBCASE 1

LOAD = 1

SUBCASE 2

LOAD = 2

SETS DEFINITIONS

SET 1 = ALL

SURFACE 1 SET 1 SYSTEM BASIC NORMAL X3

VOLUME 1 SET 1 SYSTEM BASIC

$

$____________________________________________________

$ BULK DATA SECTION (BDS)

BEGIN BULK

PARAM,DBDICT,2

PARAM,POST,0

INCLUDE ’MODELL.BDF’

Page 9

Figure 13: Comparison of the Mesh Topology before and after Optimization

ENDDATA

$ _____________________________________________________

$ CAOSS CONTROL DATA SECTION (CCDS)

$CAOSS $ This is a short example forCAOSS_PREP input for topology

$CAOSS SEL,NODE, S, LOC_Y, 39, 100

$CAOSS SEL,NODE, R, LOC_X, -1, 50

$CAOSS SEL,ELEM, S, ND_ALL

$CAOSS GROUP,ELEM, TOPO_ELGR

$CAOSS DV_DEF, TOPO, TOPO_ELGR, 30

$CAOSS OPT_CONT, TOPO_ELGR, START_DEATH, 30

$CAOSS OPT_CONT, TOPO_ELGR, SPEED_TOPO, FAST

$CAOSS SAVE $ create CAOSS data base

_____________________________________________________

7. Examples

In the following, typical structures for sizing, shapeand topology optimizations are shown which havebeen all computed with CAOSS.

7.1. Sizing Optimization

The sizing optimization will exemplarily be shownoptimizing a small structure. An axisymmetric bottleis modelled over 120° (a third segment). In the startdesign all 1.500 shell elements have the same thick-ness. The bottle is loaded with a vertical load distri-bution at the top. The bottle ground is fixed ring-likein axial direction. 1.500 design variables (one for eachelement) are defined altogether. The shell elementswere allowed to change in discrete steps, with 40different thicknesses defined. The shell thicknesses ofthe bottle top elements were coupled. Further optimi-zation boundary conditions were defined withminimum and maximum shell thicknesses for certainelement groups. The optimized model, which needed3 iterations to be computed, shows an extensive areawith a shell thickness distribution according to ahomogeneous strain distribution. For the visualiza-tion, the Property-ID’s of the shell thicknesses areshown as solids. The coupling restrictions of the shellthicknesses of the bottle top elements is apparent.Obviously the shell thicknesses of elements of certainelement groups meet the maximum or minimumlimit, e.g. at the transition from the bottle ground to

the bulb or from the bulb to the neck.

This example was calculated within approx. 5 min ona PC. It shows that in particular cases sizing optimi-zation based on the optimum criteria can be usedefficiently. The problem can be calculated with theSolution 200 of MSC/NASTRAN, too. If bucklingand stability criteria or dynamical characteristicshave to be considered for the optimization, CAOSScan quickly compute a preoptimized model, whichprovides the start design for the following optimiza-tion with Solution 200.

7.2. Shape Optimization

Example 1 - Primary FlywheelThis is the shape optimization of a completeassembly of a primary side of a damped flywheelclutch by the car supplier LuK, Bühl (Germany) [22,31]. The entire 3D-model has approx. 30.000 Degreesof Freedom (DOF) and considers both the geomet-rical nonlinear structural behaviour and several load

Optimized ModelStart Model

Figure 14: FE-Model of the Bottle

Page 10

cases.

The assembly is welded from three parts. Theprimary side and the cover are deep drawing parts,the gear rim is machined. The primary flywheel isloaded alternately. The rotation of the componentleads to centrifugal forces due to both the mass of theassembly itself and secondary parts within theassembly. Furthermore a hydrostatic pressure iscaused through the fluid between the primary sideand the cover. In addition these forces are overlaid byan alternating torque. As the maximum alternatingload appears in the ventilation holes, their form is tobe optimized. For functional reasons a bore hole aslarge as possible is desired. With regard to theproduction and function the following restrictionshave to be obeyed:• The primary side must be produced as a deep

drawing part.• The ventilation bore holes must have the same

geometry.• The outer contour of the primary side must be

maintained.The objective of this optimization was the modifica-tion of the bore hole shape to minimize the alter-nating strains caused by the two load cases. Duringthe optimization, all boundaries of the ventilationholes were allowed to move. The demand for asymmetric geometry of the ventilation hole was

(v. Mises Stresses)

[N/mm ]2

102030405060708090

100

Figure 15: Comparison of the Stresses in the VentilationHole

Page 11

Page 12

With authorization by LuK GmbH & Co, Bühl, Germany

Figure 16: FE-Model of the Flywheel

With Authorization by LuK GmbH & Co, Bühl, Germany

easily met bycoupling the related nodes at the respre-ctive holes. Furthermore the production requirement„constant component thickness“ could be fulfilled bycoupling the node displacement over the thickness.

In 4 iterations the maximum stress was reduced byapprox. 22%. Although the analysis along the rim ofthe bore hole shows the reduction of the stresses ofthe original structure, an entire homogenization hasnot yet been achieved. Especially the area close to theaxle is lowloaded

.

Example 2 - Rocker Arm

CAOSS was used for the shape optimization of anexhaust valve rocker arm by MTU Friedrichshafen(Germany), manufacturer of high performance dieselengines [17]. Take advantage of the symmetries, the3D model has approx. 15.000 DOF. Main loads resultfrom the superposition of a bending moment, causedby tappet forces, and a force fit.

For the shape optimization of the rocker arm thefollowing production and functional restrictions hadto be obeyed:

• For the holding fixture, two planes parallel to eachother must lie at the upper belts. Their position isoptional.

[N/mm²]

Path [mm]

Start DesignOptimized Design

0

50

100

150

200

250

300

350

0 20 40 60 80 100 120

Figure 17: Stress Curves of the Bore before and after theOptimization

With Authorization by MTU-Friedrichshafen, Germany

Figure 18: FE-Model of the Rocker Arm

Threat Bolt (M16)

Force Fit

VerticalLoad

HorizontalLoad

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• To further ensure forging these components, norelievings in the outer and inner contours areallowed.

• For an easy machining of the face plane of the forcefit, a run-out and a relieving for the cutter have to beensured.

• The inner contour of the bores and the adjacentplanes must not be changed

.

The objective was to reduce the local strain peaksmeeting the geometric boundary conditions. The stiff-ness of the component should be maintained or evenincreased. Coupling the certain node displacements,the requirements „parallelity“ and „forgeability“ werefulfilled. With constrained areas (areas where thecomponent must not move into) the productionrestrictions were fulfilled, tooAfter 5 iterations the maximum strain of the compo-

nent was reduced approx. by 35%, increasing the stiff-ness approx. by 10%. The reference stress in the criticalarea clearly shows a more homogeneous curve.

Example 3 - Brake Carrier

The brake carrier of a car brake by the car supplierAutomotive Lucas, Koblenz (Germany) [32] has morethan 30.000 DOF, too. The maximum strain could bereduced about 30% due to shape modifications at thecritical location by only very few iterations.

Example 4 - Contact Surfaces

In the next example the pressure-loaded contactsurface between cylinders with a symmetry axis incommon is shape optimized.

0

10

20

30

40

50

60

70

80

90

100

2985 15007 15005 15003 15001 11629 5623 3920 3918

node number

Str

ess

[%]

Figure 19: Stress Curves at the critical Location before andafter Optimization

Optimized Design

Start Design

Node Number

Stre

ss[%

]

Whole-Model

Range forAnalysis

Figure 21: FE-Model (Three Cylinders)

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With authorization by Lucas Automotive, Koblenz, Germany

Figure 20: FE-Model of the Brake Carrier

With Authorization by Lucas Automotive, Koblenz, Germany

The cylinder with the smaller diameter iscompressed between the larger cylinders. In theinitial state the different diameters lead to a stresspeak at the edge of the common contact surface. Thenominal stress in the contact surface is 100 MPa. Dueto the symmetry, only a quarter of the cross sectionhas to be modelled. The compression curve in thecontact surface increases with rising radii. Singularityproblems were encountered at the outer edge of thecylinder so that the compression level cannot bedetermined exactly (approx. 240 MPa

To analyze compression-loaded contact surfaces,special algorithms have been developed and imple-mented into CAOSS. Material increase at locationswith high contact stresses would not lead to properresults.).

In the following two possible solutions are discussed.

The first possibility is to make the end face of thesmaller cylinder convex (Variant A). This corre-sponds to the principle of the outer contouring ofrolling elements. Secondly an appropriate relievingcan make the smaller cylinder pliable above the edge(Variant B). This corresponds to the principle of theinner contouring of rolling elements.

The compression curve in the contact surface showssignificant strain reductions for both possibilities.The modification of the outer contour corresponds tothe solution, known from the bearing technology[16]. The contour of Variant A considerably dependson Young’s Modulus of the applied materials. Tovisualize the convexity, Young’s Modulus of thematerial is set to only 2.100 MPa. For the inner-contouring the stress singularity disappears for anedge angle of 80°.

Pressure stress

contact surface cylinder casing

contact surface optimized

cylinder casing optimized

start model

Figure 22: Compression Curve in the Contact Surface

variant B

variant A

Geometry modification

start model

contact surface

cylin

der

casi

ng

contact surface optimized

cylinder casing optimized

Figure 23:Contour of both Possibilities

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With Authorization by MTU-Friedrichshafen, Germany

Figure 24: Stress Plot of the Rocker Arm

7.3. Topology Optimizatio

Example 1 - Railway BridgeThe applied loads and boundary conditions of thefirst example correspond to a simple railway bridge.A rectangular solid with the left surface fixed andtwo translational DOF at the right is used as the startdesign. This arrangement is required from a technicalpoint of view, to compensate length expansions of thebridge. A further demand is to define a lane at thebottom of the solid, achieved by a thin horizontalrated area (area which will be formed as solid mate-rial in every case). A constant surface load is appliedto the top side. The shown topology results from atarget mass of 42% of the initial volume (i.e. 42% ofthe variation area are formed as solid material at theend of the optimization) and is formed as one solidbend with attached tensile struts. The attachment ofthe struts to the bend and the lane show slightly theformation of radii. The high symmetry of theresulting structure, even with the unsymmetricboundary conditions, is of special interest.

Example 2 - Car BodyThe second example is a car body which undergoes atopology optimization with regard to six differentload cases [24]. To show explicitly that the optimiza-tion of 3D structures causes no further efforts, the carbody is designed with approx. 5.000 3D solidelements. Hence follow 30.000 DOF for the FE calcu-lation and 5.000 design variables for the optimiza-tion. For the car front, forces in several directionswere assumed, resulting e.g. from crash situations.At the rear end only an axial force is assumed.Element rows are „frozen“ at the front and rear end.The shown structure comes up with a target mass of43%. Decisive solution parameters are bendingmoments, caused by the lateral effective loads.Slender frameworks were formed at the front end,due to the immediate affect of the loads applied fromvarious directions.

Example 3 - StarThe initial design consists of a ring, stiffened with120° symmetric struts. The model is designed with7.000 3D solid elements and has approx. 45.000 DOF.In the center of the struts, the star is loaded with anaxial force. The model is fixed axially on the ringbetween the struts. For the struts the load resultsmainly in bending moments and for the ring in asuperposition of bending and torsional moments.The optimization with a defined target mass of 50%leads to the shown structure.

Remarks on the Topology OptimizationThe computer simulation results are very schematicand simplified, but show that the results often lead tounconventional or unexpected designs, whichnormally are not taken into consideration. This way,the topology optimization supports the engineerscreativity in the draft and design process.For low target masses, mostly lean structures areappear. If these structures are interpreted as frame-works with tensile and compression struts nostability conditions (bending, buckling) are consid-ered [29].For industrial applications, the structures often are socomplicated that they require equations with 20.000to 200.000 DOF and even more which have to besolved. For this iterative solvers would lead to lowercomputation efforts of which the preconditioning isderived from each previous iteration [25].

Figure 25: Topology Optimized Model

1.2.3.4.

5.

6.

Figure 26: Geometry and Loads (6 Load Cases)

Figure 27: FE-Model of the Start Design

Figure 28: Topology Optimized Model

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8. References

1 Allinger, P.:Untersuchung und Implementierung von verschie-denen Algorithmen zur Topologie- und Schalen-dickeoptimierung auf der Basis vonRegelstrategien und Optimalitätskriterien in dasProgrammsystem CAOSS, Studienarbeit am Insti-tut für Maschinenkonstruktionslehre der Universi-tät Karlsruhe, 1994

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Figure 29: FE-Model with Boundary Conditions

Figure 30: Topology Optimized Model

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17 Enkelmann, F.:Untersuchung und Formoptimierung eines Kipp-hebels mit Hilfe der Methode der Finiten Elementeund CAOSS, Diplomarbeit am Institut für Maschi-nenkonstruktionslehre und Kraftfahrzeugbau, Uni-versität Karlsruhe, 1995

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neuen Optimalitätskriterien, Studienarbeit amInstitut für Maschinenkonstruktionslehre, Univer-sität Karlsruhe, 1992

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CAOSS is a registered trademark of FE-DESIGN.

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