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AIAA 2002–1676 AIAA 2002-1676 Structural Design Using Optimality Based Cellular Automata 43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Con 22-25 April 2002, Denver, Colorado Mostafa M. Abdalla ∗ Department of Aerospace and Ocean Engineering, Virginia Polytechnic institute and state university, Blacksburg, VA 24061 Zafer Gürdal † Departments of Aerospace and Ocean Engineering, and Engineering Science and Mechanics, Virginia Polytechnic institute and state university, Blacksburg, VA 24061 Cellular Automata (CA) is an emerging paradigm for the combined analysis and design of complex systems using local update rules. An implementation of the paradigm has recently been demonstrated successfully for the design of truss and beam structures. In the present paper, CA is applied to two-dimensional continuum topology optimization problems. The topology problem is regularized using the popular SIMP approach. The design rules are derived based on continuous optimality criteria interpreted as local Kuhn-Tucker conditions. Both CA based and conventional finite element analyses are considered. Numerical experiments with the proposed algorithm indicate that the approach is quite robust and does not suffer from checkerboard patterns, mesh-dependent topologies, or numerical instabilities. Introduction U SE of Cellular Automata (CA) paradigm for modeling complex systems is finding widespread applications in science and engineering. CA uses a lattice of regularly spaced cells to model the physical domain (e.g., a continuum structure). Each cell contains all the information needed to update its state. This 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 the adjacent cells, which along with the cell forms a neighborhood. The attraction of CA comes from two facts. First, the application of simple local CA rules can be used to generate global complex behavior. Second, by limiting computations to neighborhoods and using identical update rules for cell variables in the entire lattice, CA proves to be an inherently massively parallel algorithm. For the solution of heat transfer equations, Lowekamp et. al.1 showed that CA scales very well with the number of available processors and concluded that CA is ideally suited for distributed memory systems. For structural analysis, Hajela and Kim2 use a genetic algorithm to find CA analysis rules for twodimensional isotropic continuum. ∗ Research assistant. Member, AIAA. Associate fellow, AIAA. † Professor. c 2002 by Mostafa M. Abdalla and Zafer Gürdal. Copyright ° Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. Previous work on applying CA to structural design problems relied mainly on heuristic stress based rules. Kita and Toyoda3 construct CA rules to obtain two-dimensional topologies based on the Evolutionary Structural Optimization (ESO) method of Xie and Steven,4 where the analysis of the entire structure is still performed using the finite element method. The possibility of simultaneously updating both the field and the design variables, led to the application of the CA paradigm to the combined analysis and design of structural systems. Such an approach greatly reduces the computational effort required to reach an improved design over the current approaches, since most traditional design methods rely on repetitive analyses of the entire structure as the design modifications are performed. Gürdal and Tatting,5 and Tatting and Gürdal6 were the first ones to use the CA paradigm to perform the combined analysis and design, and used Stress Ratio (SR) method for the design of trusses and two dimensional continua modeled as an equivalent truss. These formulations lack the more formal basis for the optimality, and might lead to significantly suboptimal designs.7 In a previous paper,8 the present authors extended CA to treat design problems for eigenvalue requirements. Buckling critical design of elastic EulerBernoulli columns under compressive load for weight minimization was considered. The CA algorithm presented therein converges to the analytic optimal solu- 1 of 8 American Institute of Aeronautics and Astronautics Paper 2002–1676 Copyright © 2002 by the author(s). Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. tion for different boundary conditions and geometric constraints. The design formulation is again heuristic, and depends on the use of stress constraints for cross sectional sizing. Lack of analytic justification of the excellent results obtained using this formulation limits the confidence in the algorithm for more complex problems. Development of local design rules based on a sound theoretical basis is deemed to be essential for the success of CA application to structural design. In this paper, the local design rules are formulated on the basis of fundamental optimality criteria using the methods of calculus of variations. The continuous optimality conditions are interpreted as local Kuhn-Tucker conditions. This approach naturally leads to rigorous local design rules that satisfy optimality. Two-dimensional minimum compliance topology design is considered as an application. Since the original work of Bendsφe and Kikuchi,9 many numerical and theoretical approaches to topology optimization were attempted (see Rozvany10 and the references therein). Many of these methods suffer from numerical instability problems in the form of checkerboard patterns and mesh dependency.11 In the present paper, The design rules are developed by adopting the Simple Isotropic Microstructure with Penalization (SIMP)10 approach as a regularization of the topology problem. Closed form analytic expressions for the local design rules are derived. The proposed technique of deriving design rules can be easily extended to a wide variety of elastic problems. The algorithm presented in this paper avoids the need for sensitivity analysis or the solution of large-scale mathematical programming problems. The CA design rules will be shown to produce checkerboard-free mesh-independent optimal topologies. CA analysis update rules for two-dimensional linearly elastic continuum are derived from the principle of minimum total potential energy. Alternatively, CA rules can be used only for design while conventional finite elements are used for analysis. Both CA analysis and finite elements are theoretically equivalent, but differ in computational aspects. In applying CA analysis rules, different iteration schemes are considered, and compared based on their numerical efficiency and suitability for parallel implementation. The formulation of local design rules is given first, followed by the details of CA implementaion. CA design rule for FEM implementation is described, follwed by numerical results. topology design problem, artificial density variables are introduced to determine the local contribution of the material to stiffness. This approach converts the discrete topology problem into a continuous sizing problem. By using a suitable relation between local structural stiffness and density variables (material interpolation scheme), the design is driven to what is called a black/white topology, where the distribution of the material is visualized graphically by black and white regions in the domain. In the following first the formulation of the continuous sizing problem for minimum compliance design is discussed, followed by its specialization to the SIMP topology design. The minimum compliance design problem can be stated as: Z Minimize: Φe dΩ Ω Z ρ(d) dΩ = η VΩ , and g(d) ≤ 0 Subject to: Ω where Ω is the prescribed design domain, VΩ is its volume, d is a vector of design functions, ρ is the local density measure distribution, η is a prescribed volume fraction, g are local side constraints, and Φe is the strain energy density, given by: Φe = 1 Γ · D(d) · Γ 2 (1) where Γ is the strain measure vector, and D is the elasticity tensor. The local density measure ρ should ideally be either 1 or 0 at any given point. The functional dependence of the density measure ρ and the elasticity tensor D on the design variable vector d should be chosen so as to penalize intermediate densities. Following the standard procedure of calculus of variations we form the Lagrangian: Z £ ¤ L= Φe − γ (ρ − η) + λ · (g + s2 ) dΩ (2) Ω where λ is a vector of Lagrange multiplier functions, s is a vector of slack variables functions, and γ is a constant Lagrange multiplier associated with the volume constraint. Setting the first variation of the Lagrangian to zero, we obtain the following optimality conditions: 1. local stationarity ∂ Φe ∂ρ ∂g −γ +λ· ∂d ∂d ∂d (3) 2. local side constraints Formulation Minimum compliance topology design attempts to find the optimal distribution of material in a given domain to minimize the compliance of the structure under given loads with a prescribed volume constraint to limit available material. Instead of solving the discrete g(d) ≤ 0 (4) 3. switching conditions λk sk = 0, no sum over k 2 of 8 American Institute of Aeronautics and Astronautics Paper 2002–1676 (5) Objective function Hence, the local optimization problem (9) at a specified point in the design domain reduces to: Φ̂e +γρ ρp Subject to: ² ≤ ρ ≤ 1 Minimize: (13) where ² is a small value selected to avoid numerical ill-conditioning, and Φ̂e is given by: ε 0 Fig. 1 ρ 1 Φ̂e = Local optimization problem. In addition to the volume constraint. Using (1), the first term in the stationarity condition evaluates to: 1 ∂D ∂ Φe = Γ· ·Γ (6) ∂d 2 ∂d Furthermore, by defining the generalized stress measures to be: ∂ Φe σ= =D·Γ (7) ∂Γ the sensitivity of the strain energy density can be written, after algebraic manipulations, in terms of the generalized stress measures as: 1 ∂ D−1 ∂ Φe =− σ· ·σ (8) ∂d 2 ∂d Thus, we can interpret the continuous optimality conditions given above as local Kuhn-Tucker conditions for the problem: Minimize: Φ∗e + γ ρ(d) Subject to: g(d) ≤ 0 (9) Φ∗e where is the complementary strain energy density given by: 1 Φ∗e = σ · D−1 · σ (10) 2 Thus, the minimum compliance problem is reduced locally to a mathematical optimization problem. This local optimization problem can be either solved analytically, or using standard numerical techniques.12 Specialization to SIMP In the SIMP approach, the elasticity tensor is interpolated by: D = ρp D̃ (11) where D̃ is the elasticity tensor of the base material. In the present approach the local density measure distribution (0 < ρ(d) < 1) is used as the design variable. To steer the design to a black and white topology the penalization parameter p ≥ 1 is chosen high enough to penalize intermediate local density measure without leading to numerical difficulties; typically p ≥ 3 is used. With this definition, the local stress measure is given by: σ = ρp D̃ · Γ (12) ρ2p Γ · D̃ · Γ 2 (14) Note that Φ̂e is calculated holding the generalized stresses σ constant. This justifies the appearance of ρ in (14), while Φ̂e is considered indepebdent of ρ in (13). Figure 1 illustrates this one-dimensional problem for three different values of a parameter ρ̂ which is given by: 1 Ã ! p+1 Φ̂e ρ̂ = (15) γ̄ where γ̄ = γ/p is a modified Lagrange multiplier that has units of energy density. It is clear that the problem is convex, and depending on the value of the parameter ρ̂ the solution of the local optimization problem is given by: ε < ρ̂ < 1 ρ̂ ε ρ̂ < ε (16) ρ= 1 ρ̂ > 1 The Lagrange multiplier γ̄ can be loosely interpreted as an average strain energy density in the structure. So, instead of pre-specifying the volume fraction η, γ̄ may be specified as input. This eliminates the need to iteratively determine the Lagrange multiplier to satisfy the volume constraint. CA Implementation In the previous section, optimality based local rules for updating the material density were derived. We now turn our attention to the discretization of a two-dimensional structural domain using CA. Figure 2 depicts a generic two-dimensional topology design problem. We use a lattice of regularly spaced cells with the same spacing h in both x and y directions. Traditional Moore neighborhood is used to define the connectivity of the lattice as shown in the inset of Fig. 3. Each cell (C) has eight neighboring cells (N, S, E, W, N W, N E, SW, SE), and the neighborhood is split into four quadrants. The state of a cell i is denoted as φki where k is the iteration number. For topology design in two dimensions we define a cell, i, as:5 φi = {(ui , vi ) , (fxi , fyi ) , ρi } (17) where ui and vi are the cell displacements in x and y directions, respectively, and fxi and fyi are the external 3 of 8 American Institute of Aeronautics and Astronautics Paper 2002–1676 where ρi ’s are the density measures of the four cells surrounding the quadrant, and Ncell is the number of cells defining the quadrant. For the chosen neighborhood structure Ncell = 4. This interpolation scheme is chosen so that any cell with a density measure below the threshold value ² (or zero) would turn-off all four quadrants in which that cell participates. This makes the effect of cells in white (void) regions to have a negligible (or no) effect in the equilibrium equations of cells in the black regions. Also, by smoothly interpolating cell densities, checkerboard patterns are automatically suppressed.13 The strain energy in the cell neighborhood is calculated by summing the contribution of each quadrant. Since the thickness of each quadrant is assumed to be constant, the strain energy can be written in terms of the strain energy of the base material as: y x ? Fig. 2 NW Sample domain for topology design. N Uq = ρ̄p Ũ NE where, II W I E C III SW Z 1 Ũ = 2 Γ · D̃ · Γ dxdy (20) quadrant is expressed as a quadratic form in cell displacements. The coefficients of this quadratic form are functions of material parameters and base sheet thickness t. For isotropic materials, the elasticity tensor can be written in terms of Young’s modulus E, Poisson’s ratio ν, and the sheet thickness as: 1 ν 0 0 D̃ = t ν 1 (21) 0 0 1−ν IV S (19) SE Moore Neighborhood and the small-strain tensor is given by: Fig. 3 CA lattice and Moore neighborhood. Γ = (εx , εy , εxy ) forces acting on the i-th cell in the respective directions. Note that each cell has its own density measure ρi at the cell point independent of the thickness of the quadrants that are used to define the neighborhood. The optimization formulation described in the previous section for the derivation of the local optimum material density (16) is valid for both linear and nonlinear analysis. For geometrically nonlinear analysis Γ is the Green-Lagrange strain tensor. To handle material nonlinearity, the material elasticity tensor D̃ can be related to the generalized stresses σ. In the present work we only consider linear elastic behavior of an isotropic material, thus the material elasticity tensor D̃ is considered to be the familiar Hooke’s law for plane stress and infinitesimal strain-displacement relations are used. Each quadrant is assumed to have a constant density ρ̄ given by: 1 X 1 1 = (18) p ρ̄ Ncell ρpi cells where ∂u ∂v , εy = , ∂x ∂y µ ¶ 1 ∂u ∂v = + 2 ∂y ∂x εx = εxy (22) (23) Displacement update On each quadrant (I to IV ), each displacement component is expressed in terms of cell values using bilinear interpolation. The approximate equilibrium equations are found by minimizing the total potential energy over the cell neighborhood with respect to cell displacements, this gives the following form of equilibrium equations: · ¸ ½ ¾ ½ ¾ fx + fxe A B uC E · = (24) fy + fye B A vC where A and B are parameters that depend on Poisson’s ratio and thicknesses. fxe and fye are elastic forces that depend on thicknesses and material parameters 4 of 8 American Institute of Aeronautics and Astronautics Paper 2002–1676 and linearly on neighbor displacements. These forces arise because neighbor displacements are assumed to be fixed at their values in the previous iteration, while cell displacements and/or forces are updated to restore equilibrium during the new iteration. Closed form expressions for these quantities are generated using MATHEMATICATM . This 2 × 2 system of equations is solved for cell displacements, cell forces or a mixture of both according to the type of constraint or the lack thereof. Design update Cell densities are updated using (16). To this end, the cell strain energy density Φ̂e is calculated by averaging over cell neighborhood. Since the CA algorithm should handle irregular domains, some cells will have shadow neighbors that lie outside the computational domain. Shadow cells are treated by setting their density ρ to zero. Due to the density interpolation scheme (18), these cells automatically decouple from the solution.. The area of the quadrants corresponding to shadow cells is not considered in averaging. In summary, we have: Φ̂e = IV 1 X 2p ρ̄i Ũi n h2 (25) I where n is the number of quadrants with nonzero density. Fig. 4 Checkerboard ordering for Gauss-Seidel iteration. checkerboard ordering is used for the analysis update, and Jacobi mehod is used for the design update. In previous work on CA,5, 6 the analysis and design are nested. A fixed number of analysis update Na is performed followed by a design update. The design changes were damped to prevent divergence of the nonlinear iteration. Fixing the number of analysis updates is not completely satisfactory, since the convergence rate of the Gauss-Seidel method depends on the number of cells.14 In this paper, the analysis and design are likewise nested, but instead of using a fixed number of iterations, the analysis updates are performed until the norm of the residual forces (unbalanced equilibrium) reaches a prespecified tolearnce. Instead of damping design changes, a minimum value of cell density ρmin is used: ρmin = max {ε, (1 − α) ρk } Iterative scheme The CA analysis and design rules are applied at each cell, keeping the values of other cells fixed. Cells can be either updated simultaneously which corresponds to a Jacobi iteration, in which case: φk+1 = f (φkC , φkN , φkS , φkE , φkW , φkN W , φkN E , φkSW , φkSE ) C (26) or sequentially which corresponds to a Gauss-Seidel iteration. In order to preserve the symmetry of solutions (when the domain and the loading are symmetric), a variant of the Gauss-Seidel method is used where the cells are updated in a checkerboard ordering (see fig. 4). The update rule for black cells takes the form: φk+1 = f (φkC , φkN , φkS , φkE , φkW , φkN W , φkN E , φkSW , φkSE ) C (27) and for white cells: k+1 k+1 k+1 φk+1 = f (φkC , φkN , φkS , φkE , φkW , φk+1 C N W , φN E , φSW , φSE ) (28) The CA analysis updates correspond to block Jacobi or block Gauss-Seidel mehod applied to the finite element equations written for a mesh of bilinear quadrilateral elements. Since the stiffness matrix of the finite element method is symmetric positive definite, Gauss-Seidel method is proven to converge.14 On the other hand Jacobi update may or may not converge. In the present paper, Gauss-Seidel method with (29) where α is a prescribed move limit. The overall iteration is terminated when the maximum change in cell densities is less than a perspecified tolearnce. CA Design Rules for FEM Implementation The local CA analysis rules for two-dimensional continuum are derived so as to be consistent with a finite element mesh of square bilinear elements. Thus, the analysis can be directly carried out using matrix techniques as well as the CA displacement update rules derived earlier. In this section, we formulate the CA design rule in a manner independent of mesh structure. In this fashion, the CA design rule can be used as a post-processor for arbitrary finite element meshes. First, the CA cells are identified with finite element nodes, whereas in Kita and Toyoda3 CA cells were identified with elements. Figure 5 depicts a general cell (node) in an unstructured FEM mesh. Before a FEM analysis, element material properties are assigned. Poisson’s ratio is kept fixed while the element Young’s modulus Ē is altered according to: Ē = ρ̄p E (30) where ρ̄ is evaluated from (18), and the summation is extended over each element’s nodes. 5 of 8 American Institute of Aeronautics and Astronautics Paper 2002–1676 Fig. 6 Symmetric cantilever problem. Symmetric cantilever (FEM analysis) Fig. 5 mesh. Typical node in an unstructured FEM After the FEM analysis is performed, element strain energy Ue and element volumes Ve are reported. The density at each cell is updated according to (16), where the cell strain energy density Φ̂e is calculated by: P ρ̄pe Ue elements P Φ̂e = (31) Ve elements and the summation is extended over all elements having the cell as a node. Updating the Lagrange Multiplier When the volume fraction is pre-specified, the Lagrange multiplier γ̄ needs to be determined. Applying approximate Newton method leads to very simple update rule for the Lagrange multiplier. The volume constraint is approximated as: X X ρ̄e Ve − η Ve = 0 (32) mesh Geometry of a symmetric cantilever beam like domain is depicted in Figure 6. Poisson’s ratio is taken as 0.3 for all subsequent examples. The aspect ratio of the domain is 4, and the volume fraction is set to 0.5. Since the analysis is linearly elastic, Young’s modulus, sheet thickness and load value do not influence the optimal topology. Figure 7 shows the evolution of the topology on a 161 × 41 cells mesh. The minimum density ε is set to 10−3 , and the penalization parameter is chosen as p = 3. The value of the Lagrange multiplier was calculated at each re-design step to satisfy the volume constraint within 10−3 . The CA design algorithm converges to an almost black/white topology in 80 re-design steps. The figure clearly indicates that no checkerboard patterns are encountered. a) after 20 iterations. mesh We assume that the optimal density is determined by ρ̂, and consider (32) as a function of γ̄. The derivative of (32) with respect to γ̄ is obtained by applying the chain rule to (18) and (16). The corresponding update rule of the Lagrange multiplier is: γ̄ k+1 = γ̄ k (1 + (p + 1) ξ) where, b) after 40 iterations. (33) P Ve mesh P ξ =1−η ρ̄e Ve (34) c) after 60 iterations. mesh Results The CA based iterative local analysis and design update formulation described in the preceding sections was implemented using a Fortran90 code, while a MATLABTM code was developed for the finite element analysis based approach. Both approaches are tested on a single processor Pentium III machine. Example results for both analysis and design are provided to illustrate the possibilities of the CA combined analysis and design, and the CA design rule for FEM implementation. d) after 80 iterations. Fig. 7 cells). Evolution of cantilever topology (161 × 41 To check whether mesh independent results are obtained, the same topology problem with the same pa- 6 of 8 American Institute of Aeronautics and Astronautics Paper 2002–1676 rameters as above is solved on three successively finer meshes. Figure 8 shows the converged optimal topologies for the three meshes. We note that the design features are the same in all three solutions. As the number of cells is increased, the grey areas encountered on coarser meshes disappear giving a more crisp solution. The run time for the MATLABTM implementation ranges from 226 seconds for the coarsest mesh to about 2000 seconds for the finest mesh. 89 617 05 34 2 /10 !#"%$'&)(!*)+&,$-. Fig. 10 Converged optimal topology using CA combined analysis and design algorithm. a) 81 × 21 cells b) 121 × 31 cells c) 161 × 41 cells Fig. 8 (p = 3). Cantilever topology after 100 re-designs The effect of changing the value of the penalization parameter p is shown in fig. 9. The converged designs are similar for the cases p = 4 and p = 5 to the design for the case p = 3. a) 161 × 41 cells, p = 4 the convergence behavior. The convergence history is shown in fig. 10. It is clear that the Gauss-Seidel iteration achieves considerable reduction in residue initially and then, the convergence rate deteriorates. The combined analysis and design algorithm was run with p = 3 and move limit α = 5%. The Lagrange multiplier was adjusted to produce a volume fraction of approximately 0.5. The algorithm converged in a total of 294,414 analysis updates. The run time on a Pentium III machine is about 908 seconds. The predicted minimum compliance (non-dimensionalized by P 2 /E L, where P is the applied load) is 440.5 as compared to 410.6 predicted by the FEM solution of the previous section. This discrepancy (≈ 7%) is attributed to the way the Lagrange multiplier is handled. While in the FEM based implementation the Lagrange multiplier is updated after each analysis iteration, the CA algorithm uses a user-supplied value of the Lagrange multiplier. Notwithstanding this discrepancy, the converged topology as shown in fig. 11 compares very well with FEM results as shown in fig. 8-a. Fig. 11 Converged optimal topology using CA combined analysis and design algorithm. Unsymmetric cantilever (FEM analysis) b) 161 × 41 cells, p = 5 Fig. 9 Effect of penalization parameter on converged cantilever topology. To further illustrate the robustness of the CA design rule, we consider an unsymmetric cantilver of aspect ratio 4 as shown in fig. 12. The algorithm was run on a 161 × 41 cells mesh and p = 3 for a volume fraction η = 0.5. The converged topology as shown in fig. 13 is close to the optimal layout calculated by.13 Symmetric Cantilever (CA analysis) Michell truss (FEM analysis) The symmetric cantilever problem analyzed previously is re-solved using the CA combined analysis and design algorithm. First the problem was run on analysis mode only on a 81 × 21 cells lattice to study The problem under study is a rectangular ground structure supported at the lower corners, with a downward force applied at the center of the bottom edge 7 of 8 American Institute of Aeronautics and Astronautics Paper 2002–1676 Fig. 12 Fig. 13 terpreted as local Kuhn-Tucker condition and using the SIMP approach. The effectiveness of the design algorithm in suppressing numerical instabilities such as checkerboard patterns or mesh-dependency comes from using continuous interpolation of density. The analysis is carried out using either traditional finite element analysis or using CA analysis rules. Initial results indicate that the CA combined analysis and design algorithm performs satisfactorily even on a serial machine. Further numerical experiments on parallel architectures are required to determine the relative merits of using matrix finite element techniques versus CA local analysis rules. Unsymmetric cantilever problem. Unsymmetric cantilever optimal topology. Fig. 14 Michell truss, domain and optimal layout. (see fig. 14). This problem roughly corresponds to the Mitchell truss, a classical topology optimization problem. The ideal solution is shown superimposed on the ground structure. The CA design algorithm is run on a 81 × 81 cells modeling half the domain by using symmetry. The converged design for p = 3 and a volume fraction η = 0.3 is shown in fig. 15. The converged topology corresponds reasonably well to the exact solution. It is noteworthy that the Michell truss solution can be approached only in the limit of vanishingly small volume fraction. Fig. 15 Michell truss optimal topology. Conclusion References 1 Lowekamp B.B., Watson L.T., Cramer M.S., “The Cellular Automata Paradigm for the Parallel Solution of Heat Transfer Problems”, parallel algorithms and applications, 9, 1996, 119130. 2 Hajela P., and Kim B., “On the use of energy minimization for CA based analysis in elasticity”, Struct Multidisc Optim, 23, 2001, 2433. 3 Kita, E., Toyoda, T., “Structural Design Using Cellular Automata”, Struct Multidisc Optim, 19, 2000, 64-73. 4 Xie, Y. M., Steven, G. P., “Shape and Layout Optimization via an Evolutionary Procedure”, Proc. Int. Conf. Comp. Engng., Hong Kong University, 1992. 5 Gürdal, Z., Tatting B., “Cellular Automata for Design of Truss Structures with Linear and Nonlinear Response”, 41st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference and Exhibit, 2000, Atlanta, GA. 6 Tatting B., Gürdal, Z, “Cellular Automata for Design of Two-Dimensional Continuum Structures. 8th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Long Beach, 2001, CA. 7 Rozvany G.I.N., “Stress ratio and compliance based methods in topology optimization-a critical review”, Structural and Multidisciplinary Optimization, 21, 2001, 109-119. 8 Abdalla M. M., Gürdal Z., “Structural Design using Cellular Automata for Eigenvalue Problems”, 6th US National Congress on Computational Mechanics, 2001, Dearborn, MI. 9 Bendsφe, M. P., Kikuchi, N., “Generating optimal topologies in optimal design using a homogenization approach”, Comp. Meth. Appl. Mech. Engng., 71, 1988, 197-224. 10 Rozvany, G.I.N., “Aims, Scope, Methods, History and Unified Terminology of Computer-Aided Topology Optimization in Structural Mechanics”, Structural and Multidisciplinary Optimization, 21, 2001, 90-108. 11 Sigmund O., Petersson J., “Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima”, Structural Optimization, 16, 1998, 68-75. 12 Haftka R. T., Gürdal Z., “Elements of Structural Optimization”, Third expanded edition, Kluwer Academic Publishers, 1993. 13 Maar O., Schulz V., “Interior point multigrid methods for topology optimization”, Structural and Multidisciplinary Optimization, 19, 2000, 214-224. 14 Golub G.; Ortega J. M., “Scientific Computing, an Introduction with Parallel Computing”, Academic Press, 1993. A cellular automata (CA) design algorithm is presented for two-dimensional minimum compliance design based on the continuous optimality criteria in8 of 8 American Institute of Aeronautics and Astronautics Paper 2002–1676