optistruct的优化基本理论

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AIAA-2000-4946

AN INTEGRATED APPROACH FOR TOPOLOGY, SIZING AND SHAPE

OPTIMIZATION

M. Zhou, N. Pagaldipti, H.L. Thomas and Y.K. Shyy

Altair Engineering, Inc.

2445 MacCabe Way, Suite 100, Irvine CA92614

zhou@

Topology optimization has become very popular inindustrial applications and most FEM codes haveimplemented certain capabilities of topologyoptimization. However, most codes do not allowsimultaneous treatment of sizing and shapeoptimization during topology optimization phase. Thisposes a limitation on the design space and thereforeprevents finding possible better designs since theinteraction of sizing and shape variables with topologymodification is excluded. In this paper, an integratedapproach is developed to provide the user with thefreedom of combining sizing, shape and topologyoptimization in a single process.

Sizing, shape and topology optimization are the majoringredients of the technology of structural optimization.Sizing and shape optimization capabilities wereavailable since late eighties in some popular FEMsoftware such as MSC/Nastran[1] and ANSYS[2].Specialized structural optimization software such asGENESIS[3] also emerged subsequently, utilizing moreadvanced approximation technology for enhancing theoverall efficiency. This development has lead to asteady increase in industrial application of optimizationtechnology in the past decade. A notable phenomenonseen in recent years is the fast growth of the applicationof topology optimization, especially in the automobileindustry, largely owing to its significant impact increating more efficient design concepts at thepreliminary design stage. Existing structuraloptimization software mentioned above have addedsome basic topology optimization capabilities as acomplementary tool to their existing sizing and shapeoptimization capabilities. At the same time, specialtopology optimization codes such as Altair

Copyright © 2000 by M.Zhou, N. Pagaldipti, H.L.Thomas and Y.K. Shyy

Published by the American Institute of Aeronautics andAstronautics, Inc., with permission.

OptiStruct[4] also appeared in this fast growing field. Ingeneral, specialized optimization codes, althoughequipped with less analysis capabilities than generalFEM codes, offer more features and higher efficiencyfor optimization. The reasons for this are two fold: (1)highly specialized codes are typically smaller andtherefore more flexible for incorporating the latestdevelopments than general codes; (2) for specializedcodes, highest priority is devoted to its core technologyof optimization.

Up to date, topology optimization is performedseparately whilst sizing and shape optimization can becombined in a single process. This separation oftopology optimization may be due to the fact that it isusually used as a tool for finding efficient designconcepts at the early design stage whereas sizing andshape optimization are tools for detailed design at alater stage. However, feedback from industrial usershave shown that, even at the stage of conceptual study,it may be desirable to consider the interaction of somekey sizing and shape parameters with topologyoptimization. For example, one might want to optimizethe thickness of a base plate and simultaneously try tolocate stiffening ribs using topology optimization. Alsothe contour shape of the plate may be optimized as wellduring the optimization of the rib pattern. For suchdesign problems, an integrated approach not onlyallows the freedom of finding better designs by takinginto account the interaction of sizing, shape andtopology variables, but also helps achieve this goalmore efficiently within a single iterative process.In this paper, this integrated optimization problem ismathematically formulated in a general fashion, whichallows the consideration of multiple constraintsinvolving all types of responses. Owing to the largenumber of design variables, local constraints such asstress constraints are not considered for structural partsthat involve topological design variables. Advancedapproximation techniques based on intermediateresponses and intermediate variables are applied in theimplementation of the iterative process. To further

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enhance the efficiency of the design process, theapproximation formulation of the intermediateresponses corresponding to a specific constraint or theobjective function are adapted based on the iterationhistory among linear, reciprocal and convexapproximations. A key objective of this work is also toprovide an assessment of the state-of-the-art advancedapproximation technologies and create a highlyefficient implementation within the commercialstructural optimization code Altair OptiStruct. WhileOptiStruct has been a specialized product for topologyoptimization only, this development extends it into ageneral structural optimization tool with the uniquefeature of allowing users to combine sizing and shapeoptimization with topology optimization.

The general optimization problem can be statedmathematically as follows

Z(X)=Z0

+

i=1

∑xiPVi

K

(2)

where Zis the vector of nodal coordinates, Z0is thevector of nodal coordinates at the initial design, PViisthe i-th grid perturbation vector. K is the total numberof shape design variables. Note that the vector Zmustalso include internal nodes of the finite element mesh inorder to avoid mesh distortion. This approach is easy toimplement since it neither needs remeshing capabilitynor mesh smoothing algorithm during the iterativeprocess. However, it may encounter mesh distortion forlarge shape changes. Literature on shape optimization isvery extensive and reviews can be found in surveyarticles and some recent papers (see, e.g., Haftka andGrandhi[5], Ding[6], Kikuchi[7], Chang and Choi[8],Yang et al.[9], Schramm and Pilkey[10], Schleupen etal.[11]). An overview of sizing optimization can befound in textbooks and review articles (see, e.g.,Schmit[12], Vanderplaats[13], Haftka and Gürdal[14],Kirsch[15]).

Topology design variables are parameters ofmicrostructures in the homogenization approach (see,e.g., Bendsøe Kikuchi[16], Allaire and Kohn[17] andOlhoff et al.[18]) or the material density ρi of eachelement in the density approach termed SIMP (see, e.g.,Bendsøe[19], Zhou and Rozvany[20]). An overview oftopology optimization can be found in the book byBendsøe[21] and the review article by Rozvany etal.[22]. In order to achieve a 0/1 density distribution,the following power law penalization is used for thedensity approach

Minimize f(X)

Subject to gj(X) gj≤0, j=1,...,M xi≤xi≤xi, i=1,...,N

Where

L

UU

(1)

f(X) represents the objective function,

U

gj(X)and gj represent the j-th constraint responseand its upper bound, respectively. M is the total numberof constraints; xiis the i-th design variable, xi and

L

xi represent its lower and upper bounds, respectively.The total number of design variables is N. In theproblem considered in this paper, the design variablesinclude: (1) sizing variables that define the cross-sectional dimensions of 1-D elements (rods and beams)and 2-D elements (plates and shells); (2) shapevariables that define the shape variation of existingboundaries; and (3) topology variables that define thegeneralized material distribution allowing topologicalchanges to the structure. The objective function anddesign constraints can be any of the followingresponses: volumes or weights of structural parts,compliance, eigenfrequencies, displacements, stresses.An equation utility has also been developed that allowsusers to formulate any custom response using thesupported responses and design variables[4]. Owing tonumerical difficulties, stress constraints could not beapplied to the structural domain for topologyoptimization.

Shape variations in this work are defined as a linearcombination of predefined vectors of shapeperturbation:

U

i(?)=ρiKi

p

(3)

Where i and Ki represent the penalized and non-penalized stiffness matrix, respectively. The parameterp is the penalization factor, which typically takes valuesbetween 2 and 4. Special numerical difficultiesassociated with topology optimization such ascheckerboarding and mesh dependency have beenaddressed in the literature (for an overview seeSigmund and Petersson[23]). An efficient approach forcheckerboard and minimum member size control hasbeen developed and implemented recently by Zhou etal.[24] in Altair OptiStruct[4]. This technique highlyenhances the manufacturability of solutions.

The general approach for the optimization problem ineq.(1) is the approximation concept approach pioneeredby Schmit and Farshi[25]. In this approach, the

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optimization problem is solved by solving a serious ofexplicit approximate problems. The overall efficiencyof this approach is determined by the accuracy of theapproximation. Typical approximation formulationsused in structural optimization are linear approximationshown in eq.(4), reciprocal approximation in eq.(5)(Schmit and Farshi[25]) and convex approximation ineq.(6) (Haftka and Starnes[26]):

~(X)=~ggj(Y(X))=gj0 ∑j

N

gj

i

i=1 y

yi0(

2

1yi

1yi0

)(7)

It can be shown that the above approximation is exactfor statically determinate structures. For stressesconstraints, the relevant element forces, termedintermediate responses herein, are approximated asfollows:

N F~~

Fk(X)=Fk(Y(X))=Fk0+∑k(yi yi0)

i=1 yi

~(X)=g+∑gjj0~(X)=g ∑gjj0~(X)=g+∑gjj0with cji=1 if

N

N

gj

i

i=1 xN

(xi xi0)xi0(

2

(4)

(8)

gj gj xi

1xi

i=1 xi

1xi0

(5)

i=1

cji(xi xi0)

xi0xiif

gj xi

(6)

Then the approximate stresses are recovered using

exact stress recovery relationships:

gj xi

~~(X)=g~(X,Fg(Y(X)))jj

(9)

≥0; cji=

<0

The formulation in terms of mixed variables in eq.(6) is

also termed conservative approximation since it hasbeen shown by Haftka and Starnes[26] that thisformulation gives a more conservative approximationof the constraint compared with both linear andreciprocal approximations. Because this approximationis convex and separable, it is used to create an efficientdual method by Fluery and Brabaint[27].

Advanced approximation techniques developed in thelate 1980s utilizes the use of intermediate variables andintermediate responses to enhance the quality ofapproximation (Vanderplaats and Salajeh[28], Zhouand Xia[29], Zhou[30], Canfield[31], Vanderplaats andThomas[32], Zhou and Thomas[33]). For sizingproblems, the intermediate variables are the followingcross-sectional properties:

Note that alternative stress approximation of equalquality can be found in references [29] and [33]. Suchapproximation formulations have been shown to highlyenhance the approximation quality. It is easy to see thatthis approximation is exact for statically determinatestructures since no force redistribution occurs.

Canfield[31] showed that the quality of theapproximation of an eigenvalue can be enhanced byapproximating its modal strain energy and modalkinetic energy in the Rayleigh’s quotient asintermediate responses:

~U~(X)=(ω~(X))=k(Y(X))µkk

Tk(Y(X))

2

(10)

Canfield applied reciprocal approximation for themodal strain energyUkand linear approximation for themodal kinetic energy Tk.

General trend for appropriate selection of linear orreciprocal variable space for the approximation of aresponse type can be observed by studying the behaviorof the analysis equations. For example, reciprocalintermediate variables are most suited for theapproximation of displacements. However, since asimple separable approximation cannot fully capture thehigh nonlinearity and complexity of the exact responsefunction, such general rules may not applied to aspecific response. Therefore, the choice of the “best”approximation formulation among equations (4)-(6)could be quite heuristic for an individual response.Thomas[34] introduced an adaptive approach to selectthe appropriate approximation in the form of equations

Yi=(A,I1,I2,J,NSMYi=(t,D,ts,NSM)i

T

T)i

for beamsfor shells

where A is the cross-sectional area, I1, I2, J aremoments of inertia of the i-th beam properties. NSMstands for non-structural mass. t, D and ts are thethickness, bending stiffness and shear thickness of the i-th shell properties, respectively. The above intermediatevariables Y can be explicitly expressed as functions ofsizing variables X, i.e., Y=Y(X). The quality of theapproximation of displacements has been shown to behighly enhanced when reciprocal approximation isformulated in terms of intermediate variables Y:

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(4)-(6) along with the iterative process. This approachhas been shown to further improve the efficiency of theoptimization process. Note that for shape designvariables, no intermediate variables are used in thiswork.

The advanced approximation techniques summarizedherein are implemented in the Altair OptiStruct code forthe integrated problem covering sizing, shape andtopology optimization. Note that many approximationapproaches that are well suited for other types ofoptimization problems are not mentioned in this paperFor a review, see the paper by Barthelemy andHaftka[35].

Discrete sensitivity analysis that is directly formulatedon the basis of the discrete finite element formulation isused. An overview of sensitivity analysis can be foundin textbooks and review papers(see, e.g., Haftka andAdelman[36], Haftka and Gürdal[13], Kirsch[14]). Forstatic analysis, responses such as displacements,stresses and forces can be expressed as a function of thedisplacement vector U as follows:

Substitution of eq.(14) into eq.(12) yields the followingexpression:

Rj yiwith

=

Qj yi

T

U

+j

y i yi

(15)

Kj=Qj

(16)

The method using equation (15) for sensitivity analysisis called the adjoint method. The vectors jand

Qjare called the adjoint displacement vector and the

adjoint load vector, respectively. The solution of one

adjoint vector is needed for calculation the derivativesof each response. For NR responses involved in theapproximate problem, the total number of adjoint loadvectors is NR, which is independent of the number ofdesign variables.

It is easy to see that for a specific load case, the adjointmethod is more efficient than the direct method if NRrelated to this load case is smaller than Ny, and viceversa. Both methods are implemented in OptiStruct andthe favorable one is automatically selected according tothis rule.

For sensitivity with respect to shape design variables,the so-called semi-analytical method is used. In thisapproach, the derivatives of the stiffness matrix iscalculated using central finite differences as follows:

Rj(Y)=QjU

T

(11)

The derivatives of the response can be expressed as:

Rj yi

=

Qj yi

T

U+Qj

T

U yi

(12)

From the stiffness equation

KU=P(13)

K xi

=

K(xi+ xi) K(xi xi)

2 xi

(17)

where K is the stiffness matrix and P is the load vector,the following equation can be derived for thecalculation of the displacement sensitivities.

K

U yi

=

P yi

K yi

U(14)

It has been shown that very large errors can occur whenthis method is used. This phenomenon has stimulatedintensive research effort in revealing the reasons for theerrors and developing methods to eliminate them (see,e.g., Barthelemy and Haftka[35], Ollhoff et al.[36]).The overall iterative scheme is shown in the flowchartin Fig.1. For the optimization of the explicitapproximate problem, two optimizers are used inOptiStruct. The optimizer CONMIN is animplementation of the method of feasible directions byVanderplaats[37] and the optimizer CONLIN is animplementation of the dual method based on convexseparable approximations developed by Fluery[38].Because of the use of intermediate variables for the

In eq.(14), the vector U/ yi can be interpreted as thedisplacement vector corresponding to a load vector

P Ki= U, where i is termed the pseudo load

yi yivector. For Ny intermediate variables, Ny pseudo loadvectors need to be solved for each loading case in orderto calculate the derivatives of any number of responses.

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approximation formulation, the approximate probleminvolves, in general, functions that are none convex andnone separable. Therefore, an inner iteration loop has tobe implemented to solve the approximate problemiteratively using the CONLIN optimizer, as suggestedby Zhou[29][30].

Two numerical examples are presented to illustrate thecapabilities of the optimization procedure.

dimension 50 mm x 100 mm is fixed at two adjacentcorners (short side) and loaded with two equal andopposite out-of-plane forces (of magnitude 1 N) at theother two corners. The Young’s modulus E is 200000N/mm2. The plate is modeled using 5000 uniform,square shell elements of dimension 1 mm x 1 mm. Thecompliance for this loading condition is minimized.Seven optimization studies were conducted.

The first optimization study is a pure shapeoptimization of the twist plate. The shape designvariables are automatically generated by the beadgeneration capability of OptiStruct[4][39]. The beadparameters are: bead minimum width 15 mm, drawangle 60 degrees and draw height 5 mm. The platethickness is fixed at 1 mm. The optimal shape is shownin Fig. 2. The optimizer successfully reduces thecompliance from its initial value of 0.09636 to 0.01314in 21 iterations. The volume of the plate increases from5000 mm3 to 5606 mm3. However, the compliance hasbeen reduced to only 13.6% of that of the flat plate. Inthe next six optimization studies, this final volume isimposed as an upper bound constraint.

Fig. 1 Flowchart of the optimization process

The method of feasible directions is very robust,however, not efficient for problems with a large numberof design variables. Since the number of designvariables is usually large for design problems involvingtopology and/or topography optimization[39][4], theCONLIN optimizer is selected a priori for such type ofproblems. Otherwise, CONLIN optimizer is onlyselected if the number of constraints is significantlysmaller than the number of design variables.Topography optimization is a special shapeoptimization tool developed by Voth[39] for optimizingbead patterns of shell structures. Shape design variablesare generated in an automatic fashion using the beadminimum width, draw height and draw angle definedby the user.

Fig.2 Contour of the twist plate – pure contouroptimization

The second optimization study is a pure topologicaloptimization of the twist plate. The topological designvariables are the material densities of the elements inthe plate model. The maximum plate thickness is set to3 mm and the base plate thickness is zero.Checkerboard control[4] is turned on during theoptimization. The optimal topology is shown in Fig. 3.The optimizer reduces the compliance to a final valueof 0.01217 in 23 iterations. The upper bound volumeconstraint is active.

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Fig.3 Density distribution of the twist plate – puretopology optimization (base plate thickness = 0)The third optimization study is a small variant of thesecond in that the base plate thickness during thetopology optimization is set to 0.3 units. The optimaltopology is shown in Fig. 4. This topology is verysimilar to the one corresponding to a base platethickness of zero obtained from the previous study.This is because the optimal load path for the twist platefor this particular loading is clearly determined. Thefinal compliance value is 0.01557 (24 iterations), about28% higher than the final compliance value from thezero base plate thickness optimization. This is becausethe rib size in this case is thinner than that obtained inthe previous study, owing to lesser material being

available for forming ribs.

study. Fig. 6 shows the final density distribution andcontour corresponding to the fifth optimization studywith minimum member size control. The finalcompliance values are 0.00778 (30 iterations) and0.00690 (51 iterations) respectively. Clearly, theintegrated topology and contour optimization yieldsstiffer designs than the pure contour and puretopological optimization studies. The minimummember size control takes more iterations to convergebut consolidates the rib patterns and eliminates some ofthe thinner members obtained in the study without thiscontrol, thereby improving the stiffness as well as themanufacturability of the final design. Minimum sizecontrol does not affect the final contours significantly

as seen from the similarity of Figs. 5 and 6.

Fig.4 Density distribution of the twist plate – puretopology optimization (base plate thickness = 0.3 mm)The fourth and the fifth optimization studies are acombined contour and topology optimization of thetwist plate. The base plate thickness is set to zero andthe maximum plate thickness is 3 units. The fifthoptimization study has minimum member sizecontrol[24][4] turned on. The minimum member size isset to 5 units (size of 5 element widths in this model).Fig. 5 shows the final density distribution and contourof the plate corresponding to the fourth optimization

Fig.5 Density distribution (top) and Contour (bottom)of the twist plate – contour + topology optimization(base plate thickness = 0, no minimum member sizecontrol for topology optimization)

The sixth and the seventh optimization studies differfrom the fourth and the fifth in that the base platethickness is set to 0.3 units. Fig. 7 shows the finaldensity distribution and contour of the platecorresponding to the sixth optimization study. Fig. 8illustrates the final topology and contour correspondingto the seventh optimization study with minimummember control. The final compliance values are0.00862 (24 iterations) and 0.00847 (62 iterations)respectively. Several important observations have to be

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Fig.6 Density distribution (top) and Contour (bottom)of the twist plate – contour + topology optimization(base plate thickness = 0, with minimum member sizecontrol for topology optimization)

made from these results. The final contours obtainedfrom these two studies (Figs. 7 and 8), though similar toeach other, are quite different from those obtained inthe previous two studies (Figs. 5 and 6). Indeed, thefinal contours are closer to that obtained in the firststudy (Fig. 2). This difference in contours is attributedto the structural stiffness obtained from the presence ofthe base plate. The density distributions (Figs. 7 and 8)are quite similar, unlike the previous two studies (Figs.5 and 6). This is because the base plate takes up aboutone-third of the available material (based on the upperbound volume constraint). Because of the limitedmaterial available, thinner ribs are not formed andminimum member control does not change the topologysignificantly.

to collect the engine oil from an automobile, is designedfor minimum volume of the structure. Lower boundconstraints of 10 Hz, 10 Hz, 12 Hz and 15 Hz areimposed on the first four eigenfrequencies, respectively,for the purpose of noise reduction. The pan is 200 mmwide, 500 mm long and has two levels of depths, 75mm and 150 mm, respectively. The Young’s modulusE is 200000 N/mm2 and the density is 7.87 x 10-6Kg/mm3

. The oil pan is modeled using 2392 shell

Fig.7 Density distribution (top) and Contour (bottom)of the twist plate – contour + topology optimization(base plate thickness = 0.3 mm, no minimum member

size control for topology optimization)

Fig.8 Density distribution (top) and Contour (bottom)of the twist plate – contour + topology optimization(base plate thickness = 0.3 mm, no minimum membersize control for topology optimization)

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elements and is fixed at eight points along the lip of thepan (see Fig. 9). The lip has a thickness of 0.7 mm andis defined as non-design domain. The initial design hasa shell thickness of 0.18 mm and a volume of 56434.6mm3. The four first eigenfrequencies of the initialdesign are 1.48 Hz, 1.499 Hz, 1.631 Hz and 1.973 Hz,respectively, which implies that the frequencyconstraints are severely violated in the initial design.

Six optimization studies were conducted.

density distribution of the final design is shown in Fig.11. The four first eigenfrequencies of the final designare 9.99 Hz, 10.01 Hz, 12.26 Hz, 14.98 Hz,respectively. Constraints 1 and 4 are marginallyviolated (0.1% each). The volume of the final design is125373 mm3.

Fig.11 Density distribution of the oil pan – puretopology optimization

Fig.9 Finite element model of the oil pan

The first optimization study is a pure contour shapeoptimization. The bead parameters are: bead minimumwidth 125 mm, draw angle 60 degrees and draw height10 mm[4]. The plate thickness is fixed at 0.18 mm. Theiterative process converged after 77 iterations. The finalcontour shape is shown in Fig. 10. The four firsteigenfrequencies of the final design are 10.00 Hz, 10.25Hz, 12.09 Hz, 14.483 Hz, respectively. Only the fourthconstraint has a 1.1% constraint violation. The volumeof the final design is 55688.4 mm3. The initial designhas highly violated constraints and much of theoptimization effort goes into recovering these violated

constraints.

The third optimization study is a combined contour andsizing optimization of the oil pan. The plate thicknessof the shell elements in the design domain is used as thesizing variable. Its initial value is 0.18 mm and its lowerand upper bounds are set to 0.1 mm and 2.0 mm,respectively. The bead parameters are the same asthose used in the first study. The iterative processconverged after 68 iterations. The final thickness is0.1424 mm. The final contour shape is shown in Fig.12. The four first eigenfrequencies of the final designare 10.46 Hz, 10.53 Hz, 12.01 Hz, 14.95 Hz,respectively. Only the fourth frequency constraint isviolated by 0.3%. The volume of the final design is48149.1 mm3. This particular case study illustrates theadvantage of having the freedom of integrating sizingand shape variables in the optimization loop over a pureshape optimization.

Fig.10 Contour shape of the oil pan – pure contouroptimization

The second optimization study is a pure topologyoptimization. The maximum plate thickness is set to3.0 mm and the base plate thickness is 0.1 mm. Theiterative process converged after 35 iterations. The

Fig.12 Contour shape of the oil pan – contour + sizing

optimization

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The fourth optimization study is a combination ofsizing and topology optimization. As in the third study,the plate thickness of the elements in the design domainis used as the sizing design variable. Its initial value is0.18 mm and its lower and upper bounds are set to 0.1mm and 3.0 mm, respectively. The iterative processconverged after 36 iterations. The density distributionof the final design is shown in Fig. 13. The four firsteigenfrequencies of the final design are 9.98 Hz, 10.02Hz, 12.34 Hz, 14.93 Hz, respectively. The firstconstraint and the fourth constraint are marginallyviolated (0.2% and 0.5% respectively). The volume ofthe structure is 128280 mm3. Note that this design issomewhat similar to the design from the second study.

Fig.14 Density distribution (top) and contour shape(bottom) of the oil pan – contour + topologyoptimization

The sixth optimization study is combination of sizing,contour and topology optimization. The iterativeprocess converged after 80 iterations. The density andcontour distributions are shown in Fig. 15. The firstfour eigenfrequencies of the final design are 11.45 Hz,11.47 Hz, 13.87 Hz, 14.99 Hz, respectively. Allconstraints are successfully recovered with only thefourth constraint having a marginal 0.1% constraintviolation. The volume of the final design is 57078.3mm3

and the final plate thickness is 0.543 mm.

Fig.13 Density distribution of the oil pan – topology +sizing optimization

The fifth optimization study is a combination of thecontour optimization and the topology optimizationdescribed in studies 1 and 2, respectively. The iterativeprocess converged after 68 iterations. The densitydistribution and the contour shape of the final designare shown in Fig. 14. The four first eigenfrequencies ofthe final design are 10.91Hz, 11.32 Hz, 12.99 Hz, 14.98Hz, respectively. Except for the fourth frequencyconstraint which is marginally violated (0.1%), all theconstraints are well satisfied. The volume of the finaldesign is 45080mm3.

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become highly non-convex, resulting in the existence ofa large number of local optima. Therefore, the resultingdesign should be assessed with a critical view. Ingeneral, experimenting with different combinations ofdesign variables could help understand the behavior ofa specific design problem.

Note also that a user-friendly graphic interface forOptiStruct has been developed with AltairHyperMesh[40]. Within this modeling environment, theuser can interactively set up the details of theoptimization problem such as design variable selection,objective function and design constraint definitions.1.

Fig.15 Density distribution (top) and contour shape(bottom) of the oil pan – contour + topologyoptimization

Comparing the six studies, the combination of contourand topology optimization gave the best result.Although the combination of sizing, contour andtopology optimization represents the maximum designfreedom, its final volume is higher than the studywithout sizing variable. The reason is that, in general,there exist a large number of local optima for suchcomplicated problem formulation. Since the gradientbased optimization algorithms could only converge to alocal optimum, the final results should always beassessed with a critical view.

The integration of sizing, shape and topologyoptimization in a single iterative process has beenstudied in this work. This unique capability has beencreated within the commercial code Altair OptiStruct,during the process of extending it into a generalstructural optimization tool that provides sizing, shapeand topology optimization capabilities. Emphasis hasbeen placed on integrating state-of-the-art techniquesthat enhance the overall efficiency of the optimizationprocess. Advanced approximation techniques play acentral role in this regard, which have been discussed indetail. The feasibility of combining topologyoptimization with sizing and shape optimization hasbeen studied with numerical examples. Promisingresults have been obtained for the examples considered.The ability to combine different types of optimizationvariables provides increased freedom for concept study.However, caution needs be used when different types ofdesign variables are considered simultaneously. Sincethe freedom of design choice increases under theintegrated treatment of different types of designvariables, the feasible domain of the design space may

2. 3. 4. 5. 6. 7.

MSC/NASTRAN 1999: Users manual. MSCSoftware, Inc., Los Angeles, CA

ANSYS 1999: Users manual. ANSYS, Inc.,Houston, PA

GENESIS 1999: Users manual. VMA Engineering,Inc., Colorado Springs, CO

Altair OptiStruct 1999: Users manual. AltairEngineering, Inc., Troy, MI

Haftka, R.T.; Grandhi, R.V. 1986: Structural shapeoptimization – a survey. Comp. Meth. Appl. Mech.Engrg., 57, 91-106

Ding, Y. 1986: Shape optimization of structures – aliterature survey. Comp. Struct., 24, 985-1004

Kikuchi, N.; Chung, K.Y.; Torigaki, T.; Taylor, E.1986: adaptive finite element methods for shapeoptimization. . Comp. Meth. Appl. Mech. Engrg.,57, 67-89

Chang, K.H.; Choi, K.K. 1992: A geometry-basedparameterization method for shape design of elasticsolids. Mech. Struct. & Mach., 20, 215-252

Yang, R.J.; Lee, A.; McGeen, D.T. 1992:Application of basis function concept to practicalshape optimization problems. Struct. Optim., 5, 55-63

Schramm, U.; Pilkey, W.D. 1993: The coupling ofgeometric descriptions and finite elements usingNURBs - a study in shape optimization. FiniteElements in Analysis and Design, 15, 11-34

Schleupen, A.; Maute, K.; Ramm, E. 1995:Adaptive FE-procedures in shape optimization.Struct. Optim., 19, 282-302

Schmit, L.A. 1981: Structural synthesis – itsgenesis and development. AIAA J., 19, 1249-1263Vanderplaats, G.N. 1982: Structural optimization –past, present and future. . AIAA J., 20, 992-1000Haftka, R.T.; Gürdal, Z. 1992: Elements ofstructural optimization. Kluwer Acad. Publ.,Dordrecht.

Kirsch, U. 1993: Structural optimization:fundamentals and applications. Springer-Verlag,

Berlin

8. 9.

10.

11. 12. 13. 14. 15.

关于优化的理论,内容不错,

16. Bendsøe, M.; Kikuchi, N. 1988: Generating

optimal topologies in optimal design using ahomogenization method. Comp. Meth. Appl. Mech.Engrg. 71, 197-224

17. Allaire, G.; Kohn, R.V. 1993: Topology

optimization and optimal shape design usinghomogenization. In: Bendsøe, M.P.; Mota Soares,C.A. (eds) Topology design of structures, 207-21818. Olhoff, N.; Rønholt, E.; Scheel, J. 1998: Topology

optimization of plate-like structures using 3-Delasticity and optimum 3-D microstructures. Proc.7th AIAA/USAF/NASA/ISSMO Symposium onMultidisciplinary Analysis and Optimization, St.Louis, Missouri, 1853-1863

19. Bendsøe, M. 1989: Optimum shape design as a

material distribution problem. Strrut. Optim., 1,193-202

20. Zhou, M.; Rozvany, G.I.N. 1991: The COC

algorithm, Part II: topological, geometry andgeneralized shape optimization. Comp. Meth. Appl.Mech. Engrg., 89, 197-224

21. Bendsøe, M. 1995: Optimization of structural

topology, shape and material. Springer-Verlag,Berlin

22. Rozvany, G.I.N.; Bendsøe, M.; Kirsch, U. 1995:

Layout optimization of structures. Appl. Mech.Rev., 48, 41-119

23. Sigmund, O.; Petersson, J. 1998: Numerical

instabilities in topology optimization: A survey onprocedures dealing with checkerboards, mesh-dependencies and local minima. Struct. Optim., 16,68-75

24. Zhou, M.; Shyy, Y.K.; Thomas, H.L. 1999:

Checkerboard and Minimum Member Size Controlin Topology Optimization. Proc. 3rd WorldCongress of Structural and MultidisciplinaryOptimization, Buffalo, New York

25. Schmit, L.A.; Farshi, B. 1974: Some

approximation concepts for structural synthesis.AIAA J., 12, 692-699

26. Haftka, R.T.; Starnes, J.H. 1976: Application of a

quadratic extended interior penalty function forstructural optimization. AIAA J., 14, 718-724

27. Fleury, C.; Braibant, V. 1986: Structural

optimization: a new dual method using mixedvariables. Int. J. Num. Meth. Engrg., 23, 409-42828. Vanderplaats, G.N.; Salajegheh, E. 1989: A new

approximation method for stress constraints instructural synthesis. AIAA J., 27, 252-358

29. Zhou, M.; R.W. Xia 1990: Two-level

approximation concept in structural synthesis. Int.J. Num. Meth. Engrg., 29, 1681-1699

30. Zhou, M. 1989: Geometrical optimization of

trusses by a two-level approximation concept.Struct. Optim., 1, 235-240

31. Canfield, R.A. 1990: High quality approximation

of eigenvalues in structural optimization. AIAA J.,28, 1143-1149

32. Vanderplaats, G.N.; Thomas, H.L. 1993: An

improved approximation for stress constraints inplate structures. Struct. Optim., 6, 1-6

33. Zhou, M.; Thomas, H.L. 1993: Alternative

approximation for stresses in plate structures. AIAAJ., 31, 2169-2174

34. Thomas, H.L. 1996: Optimization of structures

designed using nonlinear FEM analysis. Proc. 7thAIAA/ASME/ASCE/AHS/ASC 37th Structures,Structural Dynamics and materials conference,April, Salt Lake City, UT

35. Barthelemy, J.M.; Haftka, R.T. 1993: Recent

advances in approximation concepts for optimumstructural design. Struct. Optim., 12, 129-144

36. Ollhoff, N.; Rasmussen, J.; Lund, E. 1993: Method

of “exact” numerical differentiation for errorelimination in finite element based semi-analyticalsensitivity analysis. Mech. Struct. Mach., 21, 1-6637. Vanderplaats, G.N. 1973: CONMIN – a Fortran

program for constrained function minimization:user’s manual. NASA TM X-62282

38. Fluery, C. 1989: CONLIN: an efficient dual

optimizer based on convex approximationconcepts. Struct. Optim., 1, 81-89

39. Voth, B. 1999: Using automatically generated

shape variables to optimize stamped plates.Technical Memorandum, Altair Engineering, Inc.,Troy, MI

40. Altair HyperMesh 2000: Users manual. Altair

Engineering, Inc., Troy, MI

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