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Solving three-dimensional layout optimization problemsusing fixed scale wavelets

G. C. A. DeRose Jr., A. R. DõÂaz

Abstract The layout optimization problem in three-dimensional elasticity is solved with a meshless, wavelet-based solution scheme. A ®ctitious domain approach isused to embed the design domain into a simple regulardomain. The material distribution and displacement ®eldare discretized over the ®ctitious domain using ®xed-scale,shift-invariant wavelet expansions. A discrete form of theelasticity problem is solved using a wavelet-Galerkintechnique during each iteration of the layout optimizationsequence. Approximate solutions are found with an ef®-cient preconditioned conjugate gradient (PCG) solverusing non-diagonal preconditioners which lead to con-vergence rates that are insensitive to the level of resolu-tion. The convergence and memory managementproperties of the PCG algorithm make the analysis oflarge-scale problems possible. Several wavelet-based lay-out optimization examples are included.

1IntroductionThe goal of a typical layout or topology optimizationproblem is to determine the layout of the most rigidstructure capable of supporting a given load, constrainedby the amount of material available and restricted spatiallyto be within a prescribed package space. An importantbene®t of this formulation is that the general shape andconnectivity of the design are not speci®ed a priori. Be-cause of this, layout optimization has become an integralpart of conceptual design in several engineering contexts.Engineers use layout optimization to aide in the design ofmechanical components creating stronger, lighter, andmore cost effective designs.

Layout optimization in three dimensions often requiresthe use of large-scale models solved using iterative opti-mization algorithms. During each iteration of the optimi-zation sequence the governing equations (e.g., elasticity)are solved to determine the performance of the currentdesign. Typically, ®nite element methods are used for thispurpose and one ®nite element is used per ``shape'' designvariable. Thus, when a high resolution representation of

the shape is desired, a very large number of elements isrequired. This implies that when layout optimization isused in a detailed design setting, very large-scale ®niteelement models must be solved numerous times (at leastonce per optimization iteration). As the problem size in-creases, the memory requirements and processing timenecessary to obtain ®nite element solutions dominate theoverall process. For large scale problems, high computa-tional demands often make the problems impractical tosolve using reasonable computational resources.

Iterative solvers (e.g., based on conjugate gradients) arerequired to solve the discretized elasticity equations aris-ing in layout optimization problems of typical size. Un-fortunately, when ®nite elements are used, the conditionnumber of the system matrices often increases with thesize of the problem. When this happens, as the problemsize becomes very large, iterative schemes may fail toconverge due to poor conditioning (Stephane, 1992). Thisproblem becomes particularly acute when the materialdistribution in the domain is very heterogeneous, as is thecase in layout optimization problems. In such cases, theamount of shape detail available in the design domain islimited by the analysis technique. This paper presents analternative method to solve layout optimization problemswhich replaces ®nite element analysis with meshless,wavelet-based techniques that are specially tailored tosolve these problems ef®ciently.

1.1Outline of solution strategyThe recent literature suggests a number of meshless meth-ods that may be used address the issue at hand. For example,the smooth particle hydrodynamics method (Lucy, 1977;Gingold and Monagham, 1977); the diffuse element method(Nayroles et al., 1992); the element-free Galerkin method(Belytschko et al., 1994; Belytschko et al., 1995; Krysl andBelytschko, 1995); the Petrov-Galerkin diffuse elementmethod (Krongauz and Belytschko, 1997); the reproducingkernel particle method (Liu et al., 1996; Liu et al., 1997); h-pclouds (Duarte and Oden, 1996) and the meshless localPetrov-Galerkin method (Atluri and Zhu, 1998a, b; Zhu andAtluri, 1998) are a few possible choices. Our approach fol-lows the previous work by DõÂaz (1999) and DeRose and DõÂaz(1999) and relies heavily on the work of Wells and Zhou(1992), Glowinski et al. (1995, 1996a, b), Kunoth (1995),Dumont and Lebon (1996), and Barsch et al. (1997).

In our approach, discretized models are constructedstarting from an image of the component embedded into asimple ®ctitious domain, i.e., a square in two dimensions

Computational Mechanics 25 (2000) 274±285 Ó Springer-Verlag 2000

274

G.C.A. DeRose Jr., A.R. DõÂaz (&)Department of Mechanical Engineering,Michigan State University,East Lansing, MI 48824, USA

This work is supported by Grant DMI 9532066 form theUS National Science Foundation. This support is gratefullyacknowledged.

and a cube in three dimensions. In two dimensions, thisprocess corresponds to a pixel level discretization of thecomponent, while in three dimensions the discretizationcorresponds to a voxel discretization (the three-dimen-sional equivalent to pixels). Given the image discretization,wavelet bases de®ned at the resolution of the componentimage are introduced in a standard Galerkin scheme. Aniterative, preconditioned conjugate gradient solver is usedto solve the resulting linear equations. A special precon-ditioner that results in convergence rates that are insen-sitive to the resolution of the image discretization is used.This approach is easily integrated within traditional layoutoptimization techniques and requires minimal modi®ca-tions to standard layout optimization algorithms.

2Problem statementIn a typical layout optimization problem, the goal is todetermine the optimal material distribution within a designdomain given a ®xed amount of material. Figure 1 illus-trates the setting of a typical problem. Here, x representsthe design domain or package space where the structure isto be laid, Cu � ox is the section of the boundary wheredisplacements are constrained and Ct � ox is the sectionof the boundary where loads (tractions) are applied. Asusual, Cu [ Ct � ox and Cu \ Ct � ;.

A typical formulation of the layout optimization prob-lem may be stated as

Find q�x� that

minimizes

ZCt

tux dx

subject to

Zx

q�x�dx � V �meas�x�0 < q�x� � 1; x 2 x �1�

where q�x� describes the material distribution in thepackage space x 2 <n; V , 0 < V < 1, limits the amount ofavailable material; t is the applied traction; and ux is theequilibrium displacement ®eld, i.e., the solution to thefollowing elasticity problem

Find ux 2 Kx that

minimizes Px�ux� � 1

2

Zx

Exe�ux�e�ux�dx

ÿZ

Cttux dC �2�

In (2),

Kx � fuxi2 H1�x�: uxi

� 0 on Cu � oxg �3�is the set of all kinematically admissible solutions. Forthree-dimensional problems, ux � �ux1

; ux2; ux3�,

t � �t1; t2; t3�, and Ex�x; y; z� is the tensor of elasticproperties which depend the current design q�x�. As intypical in layout optimization problems (e.g., Bendsùe andKikuchi, 1988) q�x� represents the effective density ofmaterial assigned to a point x in the package space x. Notethat the notations �x; y; z� and �x1; x2; x3� are used inter-changeably.

2.1Fictitious domain solution methodTo solve problem (2) using a ®ctitious domain approachwe ®rst embed the package space x into a simpli®ed do-main �X � �0; d� � �0; d� � �0; d�, as shown in Fig. 2 (shownin two dimensions for simplicity). The domain X, calledthe ®ctitious domain, becomes the domain in a newX-periodic elasticity problem,

Find u 2 KX that

minimizes PX�u� � 1

2

ZX

EXe�u�e�u�dXÿZ

Xfu dX

�4�In (4),

KX � fui 2 VX : Bui � 0 on Cu � oxg �5�and

VX � fui 2 H1�X�: ui is X-periodicg �6�where B is a linear operator on VX, u � �u1; u2; u3�,f � �f1; f2; f3�, and EX�x; y; z� is the material tensor overthe entire ®ctitious domain X. Due to the periodic settingof problem (4), the material tensor EX and forcing functionf must be X-periodic. To recast the original elasticityproblem (2) into the ®ctitious domain problem (4), onemust:

(i) De®ne an appropriate material tensor EX over theentire domain X. Typically, EXnx � Ex, i.e., the ®cti-tious material is very compliant

Fig. 1. The setting of a typical layout optimization problem Fig. 2. The package space embedded into a ®ctitious domain

275

(ii) Approximate the applied tractions ti by body forces fi

(iii) De®ne KX in a way such that restriction of u to xsatis®es the original boundary conditions on ux.

As discussed in DeRose and DõÂaz (1999), with reasonablechoices of the material tensor EX, the body force f , and theconstraint set KX, the solution of the ®ctitious domainproblem (4) approaches the solution of the originalproblem (2) in the domain of interest, i.e., u approachesux in x. For a more complete discussion of the ®ctitiousdomain approach, see Glowinski et al. (1996a, b) andBakhvalov and Knyazev (1994).

Equation (9) presents a method for solving two-di-mensional layout optimization problems using a ®ctitiousdomain approach. This section describes the extension ofthis method into three dimensions.

2.2Geometric discretizationTo facilitate the discretization, the ®ctitious domain X is®rst treated as a cube of dimensions N � N � N whereN � 2j for some integer j > 0. We then designate each unitcube X�k; l;m� � �k; k� 1� � �l; l� 1� � �m;m� 1� in �Xas a voxel. This effectively yields a level j, N � N � N ,raster description of the domain X. The boundary of thedesign domain ox is approximated by

Dx � [�k;l;m�2SxX�k; l;m� �7�

where Sx is the set of all voxels that intersect ox (surfacevoxels of x). Approximations of Cu and Ct, DCu and DCt,respectively, follow the same simpli®ed analysis. Wenote that this simpli®ed description of the geometry issuf®cient in most instances for analysis of layout optimi-zation problems, as the shape of the package space x istypically made of simple polyhedral shapes, e.g., rectan-gular prisms.

2.3Material tensor discretizationWe assume that the material properties over each voxel inX are constant, and use three-dimensional Haar scalingfunctions /haar

jklm�x; y; z� in the representation of the materialtensor, which may be expressed by the sum

Epqrs�x; y; z� �X2jÿ1

k�0

X2jÿ1

l�0

X2jÿ1

m�0

�Epqrs�jklm/haarjklm�x; y; z� �8�

The reader will ®nd the book by Resnikoff and Wells(1998) a very useful reference on wavelets and waveletanalysis. We shall use a simpli®ed material model (Mlejnikand Schirrmacher, 1993) to reduce storage requirements.The material tensor at any location �x; y; z� 2 X is

E�x; y; z� � q�x; y; z�pE0; p > 1 �9�that is, E is a scaled version of a ®xed material tensor E0,an isotropic material tensor. This reduces the representa-tion of the material distribution to a single expansion ofthe effective density, q, as

q�x; y; z� �X2jÿ1

k�0

X2jÿ1

l�0

X2jÿ1

m�0

qjklm/haarjkl �x; y; z� �10�

2.4Displacement field discretizationThe discretization of the displacement ®eld is of theform

u�x; y; z� �X2jÿ1

k�0

X2jÿ1

l�0

X2jÿ1

m�0

ujklm/ujklm�x; y; z� �11�

where ujklm are expansion coef®cients and /ujklm are scaling

functions (wavelets) that possess the necessary smooth-ness. Some possible choices are Daubechies D6 wavelets(Daubechies, 1988) or Cardinal B-spline wavelets (Chui,1997).

2.5The discrete system of equationsUpon the approximation of the material distribution,displacement ®eld, and boundary and loading conditionswe obtain the discretized problem

Find U 2 <3N3

that

minimizes 12UTAUÿ UTF

subject to BU � 0

�12�

where U contains the wavelet expansion coef®cients of thedisplacement ®eld. Stationary conditions of Eq. (12) resultin the linear system

A BT

B 0

� �Uk

� �� F

0

� ��13�

where k contains the wavelet expansion coef®cients ofLagrange multipliers associated with the kinematic con-straints BU � 0.

Remark. For clarity of exposition, it is convenient torepresent the discrete system in a partitioned form asshown in Eq. (14).

A �A11 A12 A13

A21 A22 A23

A31 A32 A33

264375; U �

u1

u2

u3

8><>:9>=>;

k �k1

k2

k3

8><>:9>=>; and F �

f1

f2

f3

8><>:9>=>;

�14�

2.5.1Constructing stiffness matrices associatedwith a heterogeneous material distributionThe stiffness matrix A is of dimension 3N3 � 3N3 (N � 2j)and has entries of the form

A11� ��abc��klm� �Xp;q;r

E1111� �jpqrC�1;0;0;1;0;0�jpqrabcklm

� E1212� �jpqrC�0;1;0;0;1;0�jpqrabcklm


276







�15�











�16�











�17�

where

C�a;b;c;f;g;h�jpqrabcklm �

ZXx

/haarjp �x�Da/u

ja�x�Df/ujk�x�dXx

� �

�Z

Xy

/haarjq �y�Db/u

jb�y�Dg/ujl�y�dXy

!

�Z

Xz

/haarjr �z�Dc/u

jc�z�Dh/ujm�z�dXz

� ��18�

and Di denotes the derivative operator.Terms C

�a;b;c;f;g;h�jpqrabcklm in Eqs. (15)±(18) are products of

connection coef®cients and depend only on the choice ofbasis functions used to approximate Epqrs and u. Connec-tion coef®cients for each type of wavelet may be computedonce for all problems and stored as data. The entries in A

are computed by a material weighted sum of connectioncoef®cients, eliminating the need for the numerical inte-gration. Due to the compact support of wavelets and theirderivatives, the number of non-zero terms in the sum-mation is only dependent on the type of wavelet used. Inaddition, one may use of several accurate and ef®cientmethods for computing the connection coef®cients, seeMaday (1992) and Dahmen and Micchelli (1993) for de-tails.

Remark. Matrix A is positive semi-de®nitive (boundaryconditions are imposed via Lagrange multipliers) and hasthree zero eigenvalues with eigenvectors

p �1 � � � 1 0 � � � 0 0 � � � 00 � � � 0 1 � � � 1 0 � � � 00 � � � 0 0 � � � 0 1 � � � 1

0@ 1AT

�19�

2.5.2Constructing stiffness matrices associatedwith homogeneous material distributionsThe stiffness matrix associated with a homogeneous ma-terial distribution in X (i.e., q in (9) is constant) hasspecial properties that will be exploited later. The homo-geneous material stiffness matrix, Ah, is partitioned in asimilar manner as the stiffness matrix, A, namely

Ah �Ah

11 Ah12 Ah

13

Ah21 Ah

22 Ah23

Ah31 Ah

32 Ah33

24 35 �20�

Entries in Ah are of the form

Ah11

ÿ ��abc��klm� � E1111� �C�1;0;0;1;0;0�jabcklm � E1212� �C�0;1;0;0;1;0�jabcklm

� E1313� �C�0;0;1;0;0;1�jabcklm

Ah12


Ah13


�21�Ah

21


Ah22


� E2323� �C�0;0;1;0;0;1�jabcklm

Ah23


�22�Ah

31


Ah32


Ah33


� E3333� �C�0;0;1;0;0;1�jabcklm �23�

277

where

C�a;b;c;f;g;h�jabcklm �

ZXx

Da/uja�x�Df/u

jk�x�dXx

� ��

ZXy

Db/ujb�y�Dg/u

jl�y�dXy

!

�Z

Xz

Dc/ujc�z�Dh/u

jm�z�dXz

� ��24�

Remark. Each of the sub-matrices of Ah, Ahij, is two-block

circulant. Because block circulant matrices are diagonalin the Fourier basis (Chao, 1988), each sub-matrix Ah

ijmay be inverted very easily using Fourier transforms.

Remark. As is the case with A, Ah has three zero eigen-values associated the rigid body modes in Eq. (19).

2.5.3Constructing the constraint matrix and load vectorThe constraint matrix B is used to impose the essentialboundary conditions. Entries in B are determined by ex-amining the displacement ®eld along the discretized boun-dary DCu and result from constraining the averagedisplacement over a given voxel in DCu. They are of the form

B�abc�;�klm� �Z

DCu/haar

jabc /ujklm dX �25�

Tractions are approximated by body forces and the loadvector F is constructed based on a Haar discretization ofthe body force f i� �jklm,

Fi� ��abc��X2jÿ1

k�0

X2jÿ1

l�0

X2jÿ1

l�0

f i� �jklmCjklmabc �26�

where

Cjklmabc �Z

X/haar

jklm/ujabc dX �27�

2.6Alternative fictitious domain problem formulationIn this section we discuss an alternative form of the dis-crete problem, introduced to facilitate its solution in a PCGalgorithm. We decompose U and F as

U � Uÿ pa

F � Fÿ pb�28�

where p are the rigid body modes in Eq. (19) and U and Fhave mean value zero. Vectors U, F, a and b are of the form

U �u1

u2

u3

24 35; F �f1

f2

f3

264375; a �

a1

a2

a3

24 35 and b �b1

b2

b3

24 35�29�

A similar decomposition is used to represent the con-straint matrix B, i.e.,

B � B� BQ �30�where Q � ppT. Since ui and fi have mean value zero,pTU � 0 and pTF � 0. Also, since p contains the eigen-vectors of A associated with zero value eigenvalues,Ap � 0. In addition,

BU � BU� Bpa � BU� ga �31�where g � Bp. This reduces Eq. (12) to the equivalentproblem

Find U 2 <3N3

and a 2 <3 that

minimizes 12 UTAUÿ UTFÿ aTb

subject to BU� ga � 0

�32�

Finally, we add the penalty term

P � 12 jUTQU� 1

2 ��BU�T�BU� �33�to the objective function of Eq. (32), where j and � > 0 are®xed scalars. This results in the problem

Find U 2 <3N3

and a 2 <3 that

minimizes 12 UTAUÿ UTFÿ aTb� P

subject to BU� ga � 0

�34�

which is equivalent to Eq. (32), since at a solution ofEq. (34) QU � 0 and BU � 0 and the penalty term Pvanishes. The stationary conditions of the Lagrangianfunction associated with problem Eq. (34) yield the dis-crete system of equations

A� jQ� �BTB �BTg BT

�gTB �gTg gT

B g 0

24 35 Uak

8<:9=; � F

b0

8<:9=; �35�

At the solution gTk � b and after some algebra one ®nds

�A� jQ� �BTBÿ �BTg�gTg�ÿ1gTB� BT ÿ BTg�gTg�ÿ1gT

Bÿ g�gTg�ÿ1gTB cggT ÿ 1� g�gTg�ÿ1gT

" #

� U

k

( )� Fÿ BTg�gTg�ÿ1b

cgbÿ 1� g�gTg�ÿ1b

( )�36�

where c is an arbitrary scalar.

Remark. System (36) is equivalent to (12) with arbitraryscalars j and c, provided � > 0 . However, this systemresults in more ¯exibility and better performance in theimplementation of the PCG algorithm.

3A preconditioned conjugate gradient algorithmWe investigate the solution of system (36) using the pre-conditioned conjugate gradient (PCG) algorithm ofBramble and Pasciak (1988). The algorithm is designed tosolve problems of the form

�A �BT

�B ÿ�C

� ��Uk

� �� F

�G

� ��37�

278

where �A is positive de®nite and �C is positive semi-de®nite.In Bramble and Pasciak (1988) it is shown that a precon-ditioner of the form

P � Aÿ10 0

�BAÿ10 ÿI

� ��38�

leads to systems of equations with condition numbersapproaching unity for an appropriate choice of A0. A``good'' A0 satis®es the condition

0 < ��Aÿ A0��U;U�� g� �AU;U� for all U 6� 0; U 2 <3N3 �39�

for some constant g. In general, A0 will be ``close'' tosatisfying Eq. (39) if

(i) A0 > 0(ii) �Aÿ A0 � 0(iii) �BAÿ1

0�BT > 0

In addition, for the preconditioner to be useful incomputations we require that:

(iv) The spectrum of A0 closely approximates the spectrumof �A

(v) A0 be easily invertible.

A preconditioner involving a stiffness matrix Ah arisingfrom a homogeneous isotropic material distribution suchthat Ah < A is a good candidate, as Ah and A have similarstructure and Ah is easy to invert. We choose

A0 � Ah � jQ� �BTBÿ BTg�gTg�ÿ1gTB �40�This choice of A0 satis®es the conditions (i)±(v) above andresults in an ef®cient preconditioner.

The introduction of the preconditioner P results in thefollowing system of equations:

MUk

� �� ~F �41�

where

M � Aÿ10 0

�BAÿ10 ÿI

" #�A �BT

�B ÿ�C

� �

� Aÿ10

�A Aÿ10

�BT

�BAÿ10

�Aÿ �B �BAÿ10

�BT � �C

" #�42�

and

~F � Aÿ10

�F�BAÿ1

0�Fÿ �G

� ��43�

Note that M is not symmetric, but it is positive de®nite andsymmetric with respect to the weighted inner product

wx

� �;

yz

� �� Aÿ A0�w; y� �<2��x; z�<2 �44�

System (41) has a condition number that is independent ofresolution (Bramble and Pasciak, 1988). Therefore, sys-tem (41) could be solved using a PCG algorithm based on

the weighted inner product de®ned in Eq. (44) at con-vergence rates independent of scale.

3.1Determining constants j, e and cThe constants j, � and c are selected as follows:

(i) Select � and c so the matrix �C � ÿcggT � 1� g�gTg�ÿ1gT

is positive semi-de®nite.(ii) Given � and c, select j so the matrices �A � A� jQ�

�BTBÿ �BTg�gTg�ÿ1gTB and A0 � Ah � jQ� �BTBÿ�BTg�gTg�ÿ1gTB are positive de®nite.

When E0 is constructed with Young's Modulus E � 1:0and the ®ctitious (isotropic) material is constructed withYoung's Modulus E � 10ÿ5, � � 1:0, c � 0:1, and j � 1:0satisfy (i) and (ii) above.

3.2Inverting A0

The matrix A0 must be inverted to build the precondi-tioner P. This is done using the Sherman-Morrison-Woodbury formula Axelson (1996).

�A� UVT�ÿ1 � Aÿ1 ÿ Aÿ1U�I � VTAÿ1U�ÿ1VTAÿ1

�45�for rank-n updates of a matrix A. Matrix A0 may be writtenas

A0 � Ah � jQ� � � ��BTR��STB� �46�where

RST � I ÿ g�gTg�ÿ1gT �47�

Remark. The matrix Ah � jQ is two-block circulant andpositive de®nite and can be inverted very ef®ciently usingFFT algorithms.

4ExamplesThis section includes examples that illustrate the perfor-mance of the wavelet-based ®ctitious domain solutionstrategy. In the following examples we use a simpli®edmaterial model with p � 2:5, and E0 is built using Young'sModulus E0 � 1:0 and Poisson's ratio m0 � 0:3. The ®cti-tious material is isotropic with Young's modulus E � 10ÿ5

and Poisson's ratio m � 0:3. This material is used to buildAh. The outline below describes the PCG convergencecriterion and displacement ®eld approximation used in thefollowing examples.

� PCG convergence criterion: PCG convergence is basedon the normalized residual norm �R;R�=�~F; ~F� whereR � MX�i� ÿ ~F and X�i� is the ith guess of the solutionvector U; kf gT.

� PCG initialization: The displacement ®eld and Lagrangemultipliers from the ith layout optimization iterationare used to initialize the �i� 1�th layout optimizationPCG solver.

� Displacement ®eld approximation: Possible choices forbases /u

jklm in the approximation of the displacement

279

®eld are the Daubechies D6 scaling function, the qua-dratic spline scaling function, or the linear spline scal-ing function. Based on results in two dimensions(DeRose and DõÂaz, 1999) we select here linear splinescaling functions, e.g., Cardinal B-Splines (Chui, 1997;Ueda and Lodha, 1995). These wavelets are selectedbecause of their symmetry and small support. Theyresult in very sparse stiffness matrices and reductions incomputations. Figure 3 displays a one-dimensionallinear spline scaling function and wavelet. Three-di-mensional wavelets for our analysis are constructedfrom a tensor product, i.e.,

/ujklm�x; y; z� � /u

jk�x�/ujl�y�/u

jm�z� �48�

4.1Performance of the fictitious domain approachA layout optimization problem with design domain asdisplayed in Fig. 4 and volume constraint parameterV � 0:25 is solved at four resolution levels: j � 4, j � 5,j � 6 and j � 7. The number of degrees of freedom in eachdiscretization are displayed in Table 1, where x is thedesign domain and X is the ®ctitious domain. (The do-main dimensions in Table 1 correspond to discretizationsof the design domain in voxels). The optimal topologiesfor each of the discretizations listed in Table 1 are dis-played in Fig. 5. We note from the optimal topologies thatthe results appear to be converging to the same structurerepresented at different resolutions.

The performance of the PCG solver during the layoutoptimization sequence is shown in Fig. 6. Although the

total number of degrees of freedom varies greatly in thefour models, the performance of the PCG solver remainsessentially the same, supporting the claim of convergenceinsensitive to problem size.

Another way to compare the effect of discretization levelon convergence is to examine the variation of PCG re-siduals during the initial iteration of the layout optimiza-tion sequence. This corresponds to the analysis of thedeformation of a solid, isotropic parallelopiped of di-mensions and boundary conditions as in Fig. 4. Theanalysis is performed at all four discretization levels. Theresidual iteration history for the four models is displayedin Fig. 7. Although the four models have very differentnumbers of degrees of freedom the PCG performance forthe four discretizations is rather similar. This again ex-hibits PCG performance that is insensitive to the discret-ization level.

The performance of the PCG algorithm is also com-pared to the performance of standard ®nite element PCGmethods using a diagonal preconditioner. Finite elementdiscretizations were generated by treating each voxel in xas an 8-noded hexahedron element; hence, the number ofelements at each resolution corresponds to the number ofdesign voxels in x listed in Table 1. The performance of atraditional PCG solution using a diagonal preconditioneris shown in Fig. 8. Convergence of the traditional ®niteelement approach is much slower and, unlike the wavelet-based approach, it is affected by discretization level.

4.2Additional examplesThe section includes additional examples to furtherillustrate the effectiveness of the wavelet-based solution

Fig. 3a, b. Linear spline scaling function and wavelet.a Linear spline scaling function, b linear spline waveletfunction

Table 1. Discretization levels for analysis of PCG performance

Level Domaindimensions

Design voxelsin x

Total voxelsin X

Total DOFin X

4 12� 6� 6 432 4,096 12,2885 24� 12� 12 3,456 32,768 98,3046 48� 24� 24 27,648 262,144 786,4327 96� 48� 48 221,184 2,097,152 6,291,456

Fig. 4. Example problem statement

280

strategy. Here, the PCG convergence tolerance in the pa-rameter �R;R�=�~F; ~F� is set to 10ÿ3.

4.2.1Truss ExampleFigure 9 illustrates the setting of the Truss Example. Theproblem is solved at discretization level j � 6 with avolume constraint parameter V � 0:35. This discretiza-

tion results in a design domain with dimensions (invoxels) of 48� 30� 12 and 17,280 design voxels. Theoptimal layout and performance of the PCG solver areshown in Fig. 10.

The solution method generates a highly detailed, sym-metric optimal layout. We also note that the PCG solverrequires the highest number of iterations in the ®rst fewlayout optimization iterations and fewer as the optimiza-

Fig. 5a±d. Optimal topologiesfound for various discretiza-tions. a Level j � 4, b levelj � 5, c level j � 6, d level j � 7

Fig. 6. PCG convergence prop-erties for various discretizations

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tion sequence progresses. This behavior is explained bythe fact that the starting guess of the PCG scheme becomesmore accurate as the shape is formed.

4.2.2Bridge ExampleA multiple load case problem is shown in Fig. 11, wherean inner rectangular prism is prescribed as void material.This is modeled by simply denoting the inner rectangularprism as part of the ®ctitious domain. This problem isalso discretized at level j � 6. At this discretization, thelarge domain has voxel dimensions 56� 28� 28 and theinner domain has voxel dimensions 56� 12� 12. Thiscon®guration leads to 35,840 design voxels. Figure 12displays the results of the layout optimization for thisexample.

Fig. 7. PCG convergence forthe initial layout optimizationiteration

Fig. 8. PCG convergence using a traditional ®nite element approach

Fig. 9. Problem geometry for the Truss Example

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5ConclusionsThe wavelet-based PCG algorithm is effective in solvinglarge-scale problems de®ned on simple domains. The ex-amples have exhibited convergence rates insensitive toproblem size. In addition, for large-scale problems thesolution method is a substantial improvement (in terms ofcomputational effort) over traditional iterative ®nite ele-ment techniques with diagonal preconditioners.

The PCG algorithm derives it special features from theperiodic setting of the ®ctitious domain problem. Theuse of other non-wavelet bases may possess similarperformance. However, the use of wavelets facilitatesmulti-resolution analysis a natural next step for provid-ing region discretization re®nement for analysis anddesign.

Fig. 10a±c. Truss Example:results. a Optimal topology,b performance of PCG solver,c compliance history

Fig. 11. Problem geometry for the Bridge Example

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