Multigrid Accelerated Cellular Automata for

Structural Design Optimization: A 1-D

Implementation

Sunwook Kim ∗ Mostafa M. Abdalla ∗ Zafer Gurdal †

Mark Jones ‡

Virginia Polytechnic Institute and State University, Blacksburg, VA, 24060, USA

Multigrid acceleration is typically used for the iterative solution of partial differentialequations in physics and engineering. A typical multigrid implementation uses a base dis-cretization method, such as finite elements or finite differences, and a set of successivelycoarser grids that is used for accelerating the convergence of the iterative solution on thebase grid. The proposed paper extends the use of multigrid acceleration to the designoptimization of a sample structural system and demonstrates it within the context of re-cently introduced Cellular Automata paradigm for design optimization. Within the designcontext, the multigrid scheme is not only used for accelerating the analysis iterations, butis also used to help refine the design across multiple grid levels to accelerate the designconvergence. A comparison of computational efficiencies achieved by different multigridimplementations, including the multigrid accelerated nested design iteration scheme, ispresented. The method is described in its generic form which can be applicable not onlyto the Cellular Automata paradigm but also to more general finite element analysis baseddesign schemes as well.

Nomenclature

a Height of a beamEI Bending rigiditye Correction vectorF Prescribed concentrated loadfC Vector of concentrated force and moment on the center cellfg Influence of neighboring cellsh Cell spacingIh2h Prolongation operator

I2hh Restriction operator

L Length of a beamM Bending momentN Number of cellsNa Number of analysis updatesNa Normalized number of analysis updatesNd Number of design updatesNg Number of multigrid cyclesNt Total number of cell updatesNt Normalized total number of cell updatesp Prescribed distributed load

∗Research assistant, Department of Aerospace and Ocean Engineering. Student member AIAA.†Currently Aerospace Structures Chair, Delft University of Technology, The Netherlands. Associate Fellow, AIAA.‡Associate Professor, Bradley Department of Electrical and Computer Engineering. Member AIAA.

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45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference19 - 22 April 2004, Palm Springs, California

AIAA 2004-1644

Copyright © 2004 by Zafer Gurdal. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

r Residual vectorS Number of smoothing iterationsw Transverse deflectionx Beam coordinate

SubscriptsC Center celli Cell numberL Left cellR Right cell

Symbolsα Proportionality constant in the design ruleθ Rotationξ Non-dimensional coordinate

I. Introduction

CellularC Automata(CA)paradigm has recently been successfully applied to structural analysis anddesign.1,2 CA uses a lattice of regularly spaced cells to model a physical phenomenon. Each cell

contains all the information needed to update its own state. This information includes both field variables(e.g., displacements or stresses) as well as local design variables (e.g., local cross-sectional area or thickness).The only external information to the cell comes directly from adjacent cells, which along with the cell itselfform the cell neighborhood. By limiting computations to local neighborhoods and using identical updaterules for cell variables in the entire lattice, the CA proves to be an inherently parallel algorithm. Moreover,since both field and design variables can be simultaneously updated, CA allows combined analysis and design.

In earlier studies, it has been observed1,2 that the CA convergence rate deteriorates considerably asthe lattice is refined. It has been recognized that this deterioration of the convergence rate is due toslow propagation of information across the domain; that is, as the cell spacing is reduced, more iterationsare needed to update all the cells to achieve full convergence in the domain. In the following, we willrepresent an implementation of a Multigrid (MG) acceleration scheme for CA algorithms. Multigrid isa traditional approach for accelerating the convergence of relaxation methods relying on local stencils3,4

through accelerating information transfer across the domain. The application of multigrid acceleration tothe solution of field problems is a well-researched area.3,4 In this work, a multigrid algorithm is proposedthat incorporates both analysis and design.

The implementation presented in this paper is aimed at demonstrating the MG accelerated CA fordistributed parameter optimization problems. This class of problems involves the determination of thedistribution of design parameters over a given domain. Examples of distributed parameter optimizationinclude topology optimization, variable fiber angle optimization, and beam shape design. The shape designof Euler-Bernoulli beams for minimum compliance is selected for this demonstration for two reasons. First,the implementation of CA and MG is fairly straightforward for this one-dimensional problem. Second, forthis problem, exact solutions can be easily obtained and provide a systematic way to check convergence.

The use of multigrid methods for structural optimization is rather recent, with applications primarilydirected towards topology optimization. In the standard approach to topology optimization, design variablesare associated with individual finite elements. To obtain a crisp, black/white topology design, a fairly finemesh is required, leading to a large-scale optimization problem. Due to this large-scale nature, and inherentnumerical instabilities, such as checkerboards and mesh dependency of the solution, topology optimizationseems to be a natural candidate for convergence enhancement and acceleration studies.

The first application of multigrid methods to large-scale structural optimization problems is the workof Maar and Schulz,5 where the multigrid method is incorporated to accelerate the convergence of nonlin-ear interior point optimization methods. The effectiveness of his method was demonstrated through two-dimensional topology design example. Kim and Yoon6 introduced a new approach to topology optimization,Multi-resolution multi-scale topology optimization (MTOP). They transformed density design variables of aselected 2-D problem into a multi-scale of design domains in the wavelet space and utilized a design capturedat a coarse resolution for the design of a finer resolution. As a result, the rate of convergence is successfullyenhanced, and numerical instabilities were significantly suppressed. Kwon et al7 incorporated the multigrid

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method into MTOP to improve the efficiency of the numerical computations. The study showed the potentialadvantage of implementation of the multigrid method in the multiscale topology optimization.

The proposed MultiGrid accelerated CA algorithm shares the same motivation as previous studies usingmultigrid in topology optimization. However, the proposed CA approach is distinguished by its emphasis onlocalized computations for both analysis and design, and the use of rigorous optimality criteria. In addition,we present detailed convergence studies and demonstrate the ability of MG acceleration to achieve optimalcomputational complexity for combined analysis and design. The MG approach can be further acceleratedby applying the design rule on successively refined grids. The design process on the finer grid is initializedusing the converged solution (both field and design variables) from the coarse grid, which is usually referredto as Full MultiGrid.

The rest of the paper is arranged as follows. Minimum compliance design of Euler-Bernoulli beamsis described next and the optimality criterion is derived. CA Formulation for Euler-Bernoulli beams ispresented in Sec. III. Multigrid acceleration is described in Sec. A, followed by numerical results andconvergence studies in Sec. V. Finally we present conclusions and suggestions for future work.

II. Minimum Compliance Design

The governing equation for an Euler-Bernoulli beam is,

d2

dx2

(EI(x)

d2w

dx2

)= p(x), 0 ≤ x ≤ L, (1)

where w(x) is the transverse deflection, E is the modulus of elasticity, I(x) is the principal moment of inertia,and p(x) is the distributed loading.

The formulation of the design optimization problem is posed as a minimum compliance design. Forsimplicity, we assume that the cross section is rectangular with a constant width (assumed unity), while theheight of the cross section is used as the design variable. The minimum compliance problem is formulatedas,

minh(x)

12

∫ L

0

EI(x)w′′2dx, (2)

subject to:

V =∫ L

0

a(x)dx ≤ Vo.

Using the definition of the bending moment,

M = EI(x)d2w

dx2, (3)

the Lagrangian is written as,

L =12

∫ L

0

[M2

EI(x)+ λ(a(x)− ao)

]dx, (4)

where a(x) is the cross sectional area of the beam, M is the bending moment and λ is the Lagrange multiplierassociated with the volume constraint. Stationary condition for the Lagrangian is the first variation of (4)about the design variable, a(x), must be zero. The bending stiffness EI can be expressed as a3/12. Then,the condition that variation of the Lagrangian,δL, is zero can be written as,

δL =∫ L

0

[−36M2

a4+ λ

]δadx = 0, (5)

and since δa is arbitrary, the coefficient of δa in (5) should be set to zero. Thus, the optimality condition isobtained that yields the following design update rule ,

a(x) = α√|M(x)|, (6)

where α is a constant. Additionally, it can be easily shown that (6) is basically equivalent to the fullystressed design using the flexural formula. We also set a lower bound constraint on the beam height to avoidnumerical problems.

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Figure 1. 1-D Cell neighborhood.

III. Cellular Automata (CA) Formulation

CA update rules are derived by defining a certain relationship between a cell and its neighboring cells.Therefore, the update rules become uniformly applicable to every cell in the lattice. The neighborhoodstructure is mainly dependent on the nature of a physical model that is being solved. In this paper, ananalysis update rule is sought by the static equilibrium of an Euler-Bernoulli beam and a design update ruleis by the optimality criterion of the minimum compliance.

The cell definition for structural problems typically includes field variables, applied loads, and designvariables. A one-dimensional cell neighborhood and the field variables associated with the cells for an Euler-Bernoulli beam problem are shown in figure 1. The neighborhood comprises of a center cell (C) with twoneighbors, the left (L) and the right (R) cells. Each cell has two kinematic variables, w and θ, where w isthe transverse deflection and θ is the rotation, and externally acting force, F , and moment, M . Thus, thestate of a cell can be represented as,

φC = w, θ, F, M, a, (7)

where the subscript, C denotes a particular cell in the domain.

A. Analysis Update

The analysis update rule is derived by approximating the equilibrium equations using the minimization ofthe total potential energy inside the control volume in Fig. 1. The displacement field for the Euler-Bernoullibeam between two cells is approximated as,

w = wiH1(ξ) + hθiH2(ξ) + wjH3(ξ) + hθjH4(ξ), (8)

where ξ is a non-dimensional variable defined by,

ξ =x

hcs, (9)

Hi are Hermite cubic interpolation functions, and h is the cell spacing. The strain energy of the beam insidethe control volume is given by,

U =12

∫

CV

EI(x)(

d2w

dx2

)dx, (10)

The integration is evaluated by substituting the displacement field (8) into (10), and using an averagevalue of the flexural rigidity EI, for the integration over the left and right portions, similar to finite volumediscretization. The strain energy is re-written as follows,

U =EI∗L2h3

∫w′′2L dξ +

EI∗R2h3

∫w′′2R dξ, (11)

where,

EI∗L =12(EIL + EIC), and EI∗R =

12(EIC + EIR).

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The equilibrium equations are obtained by minimizing the total potential with respect to the displace-ments of the center cell and have the form,

[k11 k12 k13

]

uL

uC

uR

= fC , (12)

fC = [F |M ]T , (13)

where fC is the vector of concentrated forces and moments, and kij are 2× 2 stiffness matrices given by,

k11 =∂2U

∂uCuC, k12 =

∂2U

∂uCuL, k13 =

∂2U

∂uCuR. (14)

Variables of the center cell are solved for in terms of its neighbors from (12) to form the relation,

k11 · uC = fC + fg, (15)

where fg accounts for the influence of neighboring cells on the center cell, and is given by,

fg = −k12 · uL − k13 · uR. (16)

Thus, the local equilibrium update rule for the beam, (15), can be shown to be in the following matrixform,

1h3

[12(EI∗L + EI∗R) −6(EI∗L − EI∗R)−6(EI∗L − EI∗R) 4(EI∗L + EI∗R)

][wC

θC

]= fg +

FC

1h

MC

, (17)

where fg can be simplified to,

fg = −EI∗Lh3

[−12 −66 2

][wL

hθL

]− EI∗R

h3

[−12 6−6 2

] [wR

hθR

]. (18)

1. Residual-Correction Form

The equilibrium update rule is recast in the residual-correction form. This form of the update rule will beuseful later in developing the multigrid acceleration scheme. In the residual-correction approach, the residualforces at the cell is first computed, and then used to obtain the needed correction of cell displacements torestore equilibrium. The residual, r is,

r = fg + fC − k11 · uC , (19)

then the correction is calculated from,e = k11

−1 · r, (20)

Hence, we get for the residual correction equation,

uCk+1 = uC

k + e. (21)

In the residual/correction form, the successive computation of the three equations is the equilibrium updaterule.

This formulation is also significantly advantageous for CA implementation on hardware capable of variableprecision computations such as Field Programmable Arrays (FPGA). The residual can be calculated in higherprecision, while the correction can be calculated at a lower precision without compromising the convergencerate.8 A parallel effort to this paper is in progress , where the authors are studying the feasibility of designinga configurable computing hardware for a massively parallel implementation of CA using FPGA.

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Figure 2. CA design algorithm.

B. Design Update Rule

As established in Sec. II, the beam height at any given point will depend on the bending moment as in (6).The bending moment at a cell is recovered by differentiating the interpolation functions and it simplifies to,

M−C =

EI∗Lh2

([6 2]

[wL

hθL

]+ [−6 4]

[wC

hθC

]), (22)

M+C = −EI∗R

h2

([6 4]

[wC

hθC

]+ [−6 2]

[wR

hθR

]). (23)

where M−C and M+

C are the bending moments just to the left and just to the right of the cell, respectively.M−

C and M+C can be different if a concentrated moment is applied at a cell. The design at each cell is updated

using the maximum of the absolute values of M−C and M+

C in (6).

C. CA Design Algorithm

In a nested analysis and design algorithm, a design step is performed after an analysis step until certainconvergence criteria are satisfied. The CA design algorithm adopts a modified version of this scheme. Theflowchart of the design algorithm is presented in figure 2. The innermost loop indicates that the field variablesof all the N cells of the grid is updated for a certain number of times, Na, so that a cell reaches a reasonableequilibrium state. This is done to ensure that the calculated cell bending moment is realistic and ensuresthe convergence of the design iteration. Then the design is updated for all the cells at the grid. After thedesign update, the infinity norm of correction to field variables between two successive design steps is usedas a stopping criterion for the algorithm.

The CA algorithm presented here differs from traditional optimization methods in two significant features.First, the CA algorithm does not require a fully converged analysis (determination of displacements) beforethe design is updated. The analysis is converged within the design iterations. Second, the design step, wherethe design variables are updated, is directly based on the latest analysis results, and is applied locally.

IV. Multigrid Acceleration

In earlier studies, it was observed that the computational effort associated with CA iterations increasesubstantially as the number of cells is increased. The deterioration of the convergence rate of CA algorithms

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is due to slow propagation of information across the domain. As the lattice is refined, the cell spacing becomessignificantly less than the characteristic length of the problem. Given that cells can only communicate locally,relevant information is slowly transferred across the domain leading to the observed poor convergence.

In formal mathematical description, long wavelength component of the solution are poorly convergedwhen the lattice is fine. A traditional solution to this problem is to use Multigrid/Full Multigrid acceleration.Even though a one-dimensional example is mainly discussed in the paper, the acceleration algorithms canbe easily extended to higher dimensional problems.

A. Multigrid (MG) Analysis Acceleration

As the name implies, the basic idea of Multigrid is to define a hierarchy of lattices from the finest to thecoarsest, see for example Fig. 3. The residual is regarded as a superposition of Fourier components of differentwavelengths. The short wave length components of the residual can be effectively eliminated using localiterative techniques, while long wavelength components show a slower decay since their elimination requiresthe information to travel between a large number of cells. Long wavelength components of residual becomeshort wavelength components when the lattice is coarsened, and can thus be more effectively eliminated.The Multigrid principle is to approximate and reduce the long wavelength error of a solution on a coarsegrid, while reducing the short wavelength components on the fine grid. Consequently, all wavelength modesbecome to converge at the same rate.

This fundamental principle of Multigrid algorithms can be applied to accelerate the convergence of CAalgorithms. In implementing MG algorithm for CA, the equilibrium update rule is applied to the fine latticefor a fixed number of times, S, to smooth the residual by eliminating short wavelength components. Thesmoothed residual on the fine lattice is mapped to the next coarser lattice through a restriction operator.The correction for the residual is obtained on the coarse lattice. Then, the correction from the coarse latticeis mapped to the fine lattice through a prolongation operator. This idea can be applied recursively usingmultiple levels of lattices. For the beam problem, we use a very simple coarsening scheme in which alternatecells are dropped to form a coarser lattice. Thus, the lattice size is reduced roughly by a factor of two eachtime the lattice is coarsened.

The MG algorithm starts from the finest lattice and visits all coarse lattices. The order in which latticesare visited is called the multigrid schedule. According to the shape of the multigrid algorithm diagrams,algorithms are called V-, W-, and F-cycle and so on. For instance, V- and W-cycle are illustrated in figure4 where an upward arrow indicates a prolongation operation and a downward arrow indicates a restrictionoperation.

B. Prolongation and Restriction Operators

The key components of the multigrid algorithm are the prolongation and the restriction operator, so-calledtransfer operators, acting between fine and coarse lattices. In particular, the prolongation operator mapscorrections of the solution from a coarse to a fine lattice,

eh = Ih2he2h, (24)

in which the subscript 2h indicates a coarse grid quantity and the superscript h indicates a fine grid quantity.The restriction operator is needed for transferring the residual from a fine lattice to a coarse lattice,

r2h = I2hh rh. (25)

Figure 3. Multi-level lattices for MG.

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Figure 4. V- and W-cycles.

Figure 5. Prolongation.

To retain the symmetry of the multigrid formulation,9 the restriction operator is defined in terms of theprolongation operator as,

I2hh = (Ih

2h)T . (26)

For the prolongation operator, the correction from the coarse lattice cells is either passed onto the cellsdirectly above them or to the cells that do not appear in the coarse lattice, see figure 5. The correction of acell on a coarse lattice is projected unchanged to a facing cell on a fine lattice,

[wh

2i−1

hθh2i−1

]=

[1 20 1

2

][w2h

i

2hθ2hi

], (27)

where, for notational simplicity, only in this section w and θ represent the components of the correctionvector e. However, the correction of a cell on a fine lattice, but not on a coarse lattice, is approximated usingthe shape functions on the coarse lattice using the following relations,

[wh

2i

hθh2i

]=

[12

18

34 − 1

8

] [w2h

i

2hθ2hi

]+

[12

18

− 34 − 1

8

][w2h

i+1

2hθ2hi+1

]. (28)

The prolongation operator is therefore expressed in matrix form as,

Ih2h =

[ [12

34

− 18 − 1

8

] [1 00 1

2

] [12 − 3

418 − 1

8

] ]T

. (29)

C. MG Design Algorithm

The MG design algorithm shown in figure 6 is similar to the CA design algorithm described in figure 2.The main difference is that a multigrid cycle is employed for the analysis update and repeated for Ng times.The use of multigrid analysis introduces one more algorithm control parameter: the number of smoothingiterations, S, which needs to be prescribed.

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Figure 6. MG design update scheme.

D. Nested Iteration (Full MultiGrid, FMG)

Nested Iteration is a numerical technique where the solution is sought on a set of successively finer grids.The converged solution at a given grid level is used as an initial solution for the next finer grid. Whenapplied to multigrid algorithms, nested iteration is referred to as FMG. The basic idea of FMG algorithmis that coarse lattices requires less computational time to converge so that the computation starts from thecoarsest lattice and then successively interpolate the needed information to the next finer lattice. Both fieldand design variables are interpolated to a finer lattice after obtaining a converged design on the previouslevel of lattices. The design process is carried out on every level successively starting from the coarsest grid,while for MG the design process is carried out only at the finest grid level. The FMG algorithm achievessuperior performance to the MG algorithm because FMG accelerates both analysis and design while MGaccelerates analysis only.

The pseudo code in figure 7 illustrates the design algorithm for FMG, where k is the grid level (k = 1denotes the coarsest grid and k = K denotes the finest grid), S is the smoothing number, and Ng is thenumber of multigrid cycles between design updates. A multigrid schedule is, too, selected before applyingthe algorithm. The parametric studies in the result section give guidelines for the choice of Ng and S formaximum numerical efficiency.

V. Results

To demonstrate the potential of MG and FMG accelerated CA algorithms, several numerical experimentshave been completed. The example problem is a cantilevered-pinned beam with a constant distributed load,p, over half-span as shown in figure 8. For the numerical experiments, the load level p is set to 1.0, the lengthL is set to 1.0, and the minimum height is set to be 0.1. In addition, the convergence tolerance is chosen tobe 10−6.

A. CA implementation

First, we examine the convergence characteristics of the CA algorithm for analysis and design using a directmatrix solver. The convergence of the CA analysis update rule is calculated by explicitly constructing theiteration matrices and calculating the spectral radius, ρ. The convergence rate, (1-ρ), is plotted as a functionof the number cells on log-log scale in figure 9(a). The slope on the plot is constant at −4. Thus, the analysisconvergence rate is inversely proportional to the number of cells raised to the fourth power. The norm of thedesign variable correction is plotted the design iteration number in figure 9(b). The convergence behavior ofthe design rule is independent of the number of cells.

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k is the level of a grid

Choose initial guess 1~u when k = 1 Complete the design of

1 within tolerance for k = 2:K

Interpolation: ( )12

−Ω←Ω khh

k I (both field and design variables)

while Design update

k

for i = 1:Ng

MG(k, S) cycle and get solution, ku end

Set kk uu =~ Check for convergence

end end

Figure 7. Pseudo-code for the FMG design update scheme.

Figure 8. A cantilevered-pinned beam with a distributed load

The CA convergence behavior when analysis and design are combined is shown in figure 10 where thetotal number of cell updates, Nt, is shown as a function of the number of analysis updates between thedesign updates, Na. Motivated by the convergence characteristics of CA in analysis and design, the figureuses non-dimensional variables. The vertical axis is the total number of cell updates normalized by thenumber of cells to the fifth power, Nt = Nt/N

5. The horizontal axis is the number of analysis updatesbetween the design updates normalized by the number of cells to the fourth power, Na = Na/N4.

From this figure, it is clear that the total number of cell updates to convergence is large when the numberof the analysis updates between design updates is small. This is due to poor convergence of the displacementfield. On the other hand, the total number of cell updates to convergence is also large when the number of theanalysis updates between design updates is large. Beyond a certain values N∗

a , the displacement field is fullyconverged and increasing the value of Na increases the computational cost without improving convergencerate. There exists an optimal value N∗

a for which the total number of cell updates is minimum, and theminimum value of Nt is nearly independent of the number of cells. Thus, the optimal total number of cellupdates is proportional to the number of cells raised to the fifth power. Thus, CA is polynomial boundedwith the rather high computational complexity of O(N5).

B. MG acceleration

Convergence of the combined analysis and design MG accelerated CA algorithm is illustrated in figure 11along with the standard CA algorithm and FMG results (to be discussed later). Results were generated for11 grid levels; the coarsest is 5 cells and the finest is 4097 cells, the number of smoothing iterations, S = 5,and the number of multigrid cycles, Ng = 2. The vertical axis represents the total number of cell updates, Nt,and the horizontal axis the number of cells, N . The plot of MG results shows a linear asymptotic variationon the log-log scale having a unit slope. Thus, the computational cost is proportional to the number ofcells which is the optimal computational complexity. Therefore, this result verifies that MG accelerationalgorithm successfully remedies the deterioration of CA convergence.

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(a) Analysis (b) Design

Figure 9. CA convergence.

Figure 10. CA convergence combining analysis and design

C. FMG acceleration

The results of FMG acceleration are shown in figure 11 in comparison to MG and the standard CA results.FMG accelerated CA algorithm performs better than MG accelerated CA especially as the size of a latticegrows. The striking fact about FMG performance is that as the lattice is refined, the computational costalmost remains constant.

In order to study the convergence characteristics of the MG and FMG schemes, the norm of the designvariable correction is plotted as a function of the number of design updates, Nd for different values of N infigure 13(a) and 13(b), respectively. The design convergence of MG acceleration is shown in figure 13(a),and is independent of a number of cells. This is to be expected since MG accelerates analysis convergenceonly. In case of FMG acceleration, figure 13(b), we can draw out a relatively simple account for the superiorperformance of FMG. The initial height correction for a finer lattice is less than that of coarser lattices, hence,a fewer number of design updates are needed as the lattice is refined. Since the largest computational costis incurred on fine lattices, this leads to a significant decrease of the overall computational cost of the FMGalgorithm. The marked decrease in the number of design steps to convergence as the lattice is refined canalso be explained by the fact that the solution to the design problem becomes grid-independent as the latticeis refined as shown in figure 12 where the design progress of FMG for finer and finer lattices is compared tothe analytic solution.

This result is also significant by the fact that traditional wisdom in structural optimization suggests

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Figure 11. Comparison of CA, MG and FMG accelerated results.

Figure 12. Comparison of FMG design progression and analytic solution.

that as the number of design variables is increased, the computational cost increases exponentially; a factcommonly expressed as ”curse of dimensionality”. Since in this particular problem the number of cells isproportional to the number of design variables, it may be suggested that the ”curse of dimensionality” isbroken possibly for the first time.

Results for different combinations of control parameters are shown in Fig. 14(a), where the vertical axisis in log scale and the total number of cell updates is plotted for different numbers of multigrid cycles, Ng

whereas S is unchanged. Results for different smoothing iteratons (S) are shown in Fig. 14(b); the verticalaxis is also in log scale and W cycle is only considered. It can be concluded that a minimal numbers of thesmoothing and a multigrid cycle are sufficient to obtain converged design. In case of S = 1, it was observedthat the convergence rate deteriorated considerably and for many cases resulted in lack of convergence.

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(a) MG (b) FMG

Figure 13. Norm of height correction using MG and FMG acceleration.

(a) with number of multigrid cycles Ng (b) with number of smoothing iterations S

Figure 14. Variation of the total number of cell updates Nt for FMG.

VI. Conclusion

This study has demonstrated that MG and FMG accelerations successfully accelerate CA convergencefor combined analysis and design problems. Numerical experiments prove that MG acceleration improvesthe analysis convergence and that FMG acceleration improves both the analysis and design convergence.MG accelaration achieves optimal complexity, and the FMG algorithm achieves even greater computationalsavings for a given convergence tolerance. Even though the potential of MG and FMG accelerations instructural optimization is underlined with the relatively simple beam problem, the presented multigridacceleration schemes are expected to be applicable to more general structural optimization problems. Furtherinvestigation into the application of FMG accelerated CA, especially on FPGA hardware, will potentiallyprovide powerful computaional tools for large scale computationally demanding problems.

VII. Acknowledgement

The authors would like to express their appreciation to Dr. J. Sobieski of NASA Langley for valuableadvice and discussions throughout the progress of this work. Part of this work was supported under theVirginia Tech ASPIRES program. This funding is gratefully acknowledged by the authors.

References

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2Abdalla, M. and Gurdal, Z. Structural design using cellular automata for eigenvalue problems. In Structural and Multi-

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disciplinary Optimization, Vol 26, No. 3-4, pp. 200-208., 2004.3Hackbush, W. Multigrid methods and applications. Springer., 1985.4Wesseling. An Introduction to Multigrid Methods. Wiley., 1992.5Maar, B. and Schulz., V. Interior point multigrid methods for topology optimization. technical report 98-57. In IWR,

University of Heidelberg, 1998.6Kim, Y. Y. and Yoon, G. H. Multi-resolution multi-scale topology optimization - a new paradigm. In Int. J. Solids and

Structure, 37, 5529-5559, 2000.7Kwon, K., Kim, J. E., and Jang, G. W.and Kim Y.Y. Multigrid approach for efficient integrated multiscale analysis and

design optimization. In 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization 4-6, 2000.8Jones, M., Plassmann, P., and Gurdal, Z. Efficient algorithms for iterative refinement based on configurable precision

arithmetic. In Submitted for publication in SIAM Journal On Matrix Analysis and Applications.9Briggs, W. L. A Multigrid Tutorial. SIAM., 1987.

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