Element exchange method for topology optimization

Transcription

1 Struct Multidisc Optim DOI /s RESEARCH PAPER Element exchange method for topology optimization Mohammad Rouhi Masoud Rais-Rohani Thomas N. Williams Received: 5 December 2008 / Revised: 22 January 2010 / Accepted: 27 January 2010 c Springer-Verlag 2010 Abstract This paper presents a stochastic direct search method for topology optimization of continuum structures. In a systematic approach requiring repeated evaluations of the objective function, the element exchange method (EEM) eliminates the less influential solid elements by switching them into void elements and converts the more influential void elements into solid resulting in an optimal 0 1 topology as the solution converges. For compliance minimization problems, the element strain energy is used as the principal criterion for element exchange operation. A wider exploration of the design space is assured with the use of random shuffle while a checkerboard control scheme is used for detection and elimination of checkerboard regions. Through the solution of multiple two- and three-dimensional topology optimization problems, the general characteristics of EEM are presented. Moreover, the solution accuracy and efficiency of EEM are compared with those based on existing topology optimization methods. Keywords Topology optimization Element exchange method EEM Stochastic Non-gradient Binary M. Rouhi M. Rais-Rohani (B) Department of Aerospace Engineering, Mississippi State University, Mississippi State, MS 39762, USA masoud@ae.msstate.edu M. Rouhi rouhi@cavs.msstate.edu M. Rouhi M. Rais-Rohani T. N. Williams Center for Advanced Vehicular Systems, Mississippi State University, Mississippi State, MS 39762, USA 1 Introduction Topology optimization of continuum structures is aimed at finding the optimum distribution of a specified volume of material over a selected design domain that would push a desired objective function toward its extreme value. Although optimum topology could be defined by such criteria as displacement or stress, it is commonly based on the minimization of structural compliance or strain energy resulting in an optimal load path between the loading points and the structural supports. Since the general topology optimization problem with binary density function (i.e., ρ = 0 or 1) is ill-posed, various methods have been developed to solve the modified problem with a continuous density function (i.e., 0 < ρ 1). For the most part, these methods have been based on relaxation through homogenization (Bendsoe and Kikuchi 1988; Diaz and Bendsoe 1992), where the geometry and orientation of anisotropic hole-in-cell microstructure are applied as continuous design variables, or with a continuous density function, where the intermediate-density elements are penalized (Bendsoe 1989; Zhou and Rozvany 1991; Rozvany et al. 1992) to yield the desired 0 1 (void solid) topology. Zhou and Rozvany (1991) showed that the use of non-optimal microstructures homogenized into an anisotropic continuum introduces a penalty for perforated (grey) regions into shape optimization. Their use of an isotropic microstructure with a suitable penalty function coupled with a gradient-based optimization approach resulted in the elimination of intermediate-density elements in generalized shape optimization problems. As it came to be known by its acronym SIMP (Rozvany et al. 1992), solid isotropic microstructure with penalty has become a popular approach partly because of its accuracy and computational

2 M. Rouhi et al. efficiency, as well as ease of integration with generalpurpose finite element analysis (FEA) codes. Besides the existence of multiple local minima, topology optimization problems can also suffer from mesh dependency and the formation of checkerboard regions. Some approaches to combat the latter two problems have included the use of heuristic mesh-independent filtering (Sigmund and Peterson 1988), higher-order finite elements (Jog and Haber 1996), perimeter control (Haber et al. 1996), alternative density stiffness interpolation schemes (Guo and Gu 2004), hyperbolic sine functions for the intermediate densities (Bruns 2005), and techniques for producing better topologies with sharper solid void solutions having greater stiffness (Fuchs et al. 2005) as well as less checkerboards (Zhou et al. 2001; Poulsen 2002; Pomezanski et al. 2005). Research efforts in non-gradient based topology optimization have led to the development and application of such methods as simulated biological growth (SBG; Mattheck andburkhardt1990), particle swarm optimization (PSO; Fourie and Groenwold 2001), evolutionary structural optimization (ESO; Xie and Steven 1993), bidirectional ESO (BESO; Querin et al. 1998), and metamorphic development (MD; Liu et al. 2000). Some of these methods, together with other algorithms mimicking biological systems such as genetic algorithms (GA; Goldberg 1989) and cellular automata (CA; Kita and Toyoda 1999), have also been used in the solution of sizing and shape optimization problems. The use of binary design variables enables these methods to produce a black white (solid void) optimal topology that excludes any gray (i.e., fuzzy or intermediate density) regions without using penalization. Another advantage of the stochastic direct search methods is their non-local search algorithms that can lead to a better solution than the local optimum in the vicinity of the initial design point. However, due to the need for a large number of function evaluations for the multitude of candidate designs, the direct search methods tend to be computationally inefficient (Fourie and Groenwold 2001; Werne2006; Jakiela et al. 2000; Mei et al. 2007). To remedy the checkerboard problem, non-gradient based methods also resort to using mostly heuristic schemes. Zhou and Rozvany (2001) discussed some of the shortcomings of non-gradient based methods such as ESO, and more recently, Rozvany (2009) offered a detailed critical review of SIMP and ESO by examining their mathematical foundations and highlighting their differences in terms of solution accuracy and computational efficiency. In this paper, we introduce a new non-gradient based topology optimization method that has many of the same advantages and some of the shortcomings of the other stochastic direct search methods but with noticeably better computational efficiency. Named after the principal operation in the topology optimization strategy, the element exchange method (EEM) falls under the same category as BESO and GA due to the use of heuristic relationships, but it has certain features that are quite distinct from the other two methods. In the remaining portion of the paper, we provide details of the EEM and describe the element exchange strategy, checkerboard control procedure, convergence criteria, and the algorithmic parameters used in conjunction with different operations in EEM. The convergence properties of EEM are illustrated along with a comparison to a known Michell truss structure. Moreover, the results for several two- and three-dimensional problems of varying complexity are presented while making comparisons with the solutions found using other methods as reported in the literature. 2 General principle of element exchange The general principle of element exchange for the case of compliance minimization is explained using a simple structural system that is idealized by a combination of four linearly elastic springs and associated boundary conditions as shown in Fig. 1. The total strain energy, E t stored in (a) (b) Fig. 1 Spring system a before and b after element exchange operation

3 Element exchange method for topology optimization the system is simply the sum of energy stored in individual springs found as E t = 4 E i = 1 2 i=1 4 K i δi 2 (1) i=1 where E i is the energy in the ith spring defined in terms of the corresponding stiffness, K i and elongation, δ i. Assuming that only two springs can be used for minimizing the strain energy of the system in this example, the problem becomes one of deciding which two springs to keep and which ones to eliminate. The two springs that are kept create an optimal load path between the loaded and supported points of the system. For simplicity, a solid spring is assumed to have a stiffness of K s while a void spring has a stiffness of K v = K s. Using the initial distribution of springs shown in Fig. 1a and recognizing that springs 1 and 2 are under equal axial force, elongations of springs 1 through 4 can be shown to be: δ 1 = δ 1,001 δ 1,000 ; δ 2 = 1,000 1,001 δ δ; δ 3 = δ 4 = δ. Thus, the substitution of appropriate values into (1) givese t = K s [ ] δ K sδ 2 = 1 2 F 2 K s. Since spring 1 is a solid spring with the lowest strain energy between the two solid springs, it will be converted into a void spring in the next iteration while spring 4 representing a void spring with the highest strain energy between the two void springs will be converted into a solid spring. Figure 1b showsthe updatedlayoutafter the element exchange operation is performed. Now, the total strain energy stored in the system can be shown to be E t = K s [ ] δ 2 K s δ 2 = 1 4 F 2 K s. While the number of solid springs is kept constant, the total strain energy of the system is reduced by 50%, signifying greater stiffness and smaller compliance. Hence, by identifying and switching the less influential solid spring into a void spring and the more influential void spring into a solid spring, a better (more efficient) load path is created. By extending the problem to a continuum domain represented by a finite element mesh, it would be possible to use the element exchange as part of a more general algorithm and solution procedure for finding the optimal topology. 3 EEM algorithm Here the EEM algorithm is discussed with focus on compliance minimization problems. With a continuum structure represented by a discretized domain of finite elements and associated boundary conditions, the compliance minimization problem is one of finding the optimal distribution of solid and void elements that would min s.t. f (ρ) = u T Ku = M u T j K ju j = M 2E j M ρ j V j V 0 j=1 ρ min ρ 1.0 j=1 j=1 where f (ρ) represents the total strain energy, ρ the vector of non-dimensional element densities treated as design variables, u the vector of global generalized nodal displacements, K the global stiffness matrix, M the total number of finite elements, with u j, K j and E j as the displacement vector, stiffness matrix and strain energy of the jth element, respectively. With ρ j and V j representing the nondimensional density and volume of the j th element, the constraint in (2) imposes an upper bound on the acceptable volume fraction of solid elements in the design domain. To avoid ill-conditioning of stiffness matrix, the void elements are assumed to have a density equal to ρ min, with a very small positive value. In EEM, stiffness of the jth element represented by E j is linearly related to its non-dimensional density (i.e., E j = ρ j E,whereE is the Young s modulus of the solid material), with ρ j treated as a discrete design variable ρ j {ρ min, 1.0}, where ρ min = For a uniformly discretized domain of identical elements, the volume fraction constraint in (2) becomes a strict equality that is satisfied in every iteration. However, for non-uniform meshes, there can be a small fluctuation of volume around the specified limit, V 0. The EEM algorithm, as depicted by the flowchart in Fig. 2 applies to both single- and multiple-load case problems. Once the domain is discretized into a uniform finiteelement mesh, the number of solid elements found as N s = MV 0 is randomly distributed throughout the design domain. If the mesh is non-uniform, then N s would need to be adjusted accordingly to satisfy the specified volume fraction. All void elements are given a non-dimensional material density of The EEM parameters and their recommended values appear later in the paper. Because of the stochastic operations that occur at different stages of EEM together with the fact that the initial distribution is selected in random, EEM procedure can take different solution paths for the same topology optimization problem. As will be shown later, if a problem has multiple local optima with nearly equal objective function values (Kutylowski 2002), then it is possible for the EEM solution to converge to any one of these locations, which may have different solid void material distributions. However, the randomly selected initial design and the stochastic operations that occur at different stages of the solution process (2)

4 M. Rouhi et al. Fig. 2 Flowchart of EEM make it more likely for EEM to find a better solution for a more general problem. With the initial design domain and boundary conditions specified, a static FEA is performed to find the strain energy distribution among the elements as well as the total strain energy of the structure as a whole. At the next step, a subset of solid elements with the lowest strain energy density amongst the solid elements are converted into void elements while a volumetrically equivalent number of void elements with the highest strain energy density amongst the void elements are converted into solid elements such that the volume fraction remains fixed. In the case of a uniform mesh, all elements are geometrically identical; hence, volume fraction remains constant by simply setting the number of solid elements converted into voids and vice versa equal to each other. On the other hand, if the mesh is non-uniform, then for the specified exchange volume the solid elements with the lowest strain energy density are converted one by one into void, and similarly the void elements with the highest strain energy density are converted into solid until the exchange volume is balanced. Due to the use of discrete density and variation in element geometry, it is possible to encounter a small difference in volume between the two sets of exchanged elements, which requires the relaxation of the volume fraction constraint. Since the type of mesh used does not change the basic framework of EEM, henceforth, the mesh is assumed to be uniform. The new layout is analyzed for strain energy, and the element exchange operation is repeated. This procedure is continued for a specified number of iterations before the checkerboard control (detection and elimination) operation is performed, with further details provided later in the paper. After completion of several iterations, a subset of the solid elements are randomly scattered in the void regions. This so-called random shuf f le is similar to the mutation operation in GA and serves a similar purpose in that it enhances the chance of finding a better solution to the topology optimization problem by exploring other regions of the design space. For a given design problem, the principal operations (i.e., FEA, element exchange, checkerboard control, and random shuffle) are repeated at different intervals until a convergence criterion is satisfied. As in the case of the other stochastic methods, a limit is imposed on the number of iterations in order to stop the program when the selected convergence criterion is too tight. Figure 3a illustrates the evolution in topology from the initial to optimum (minimum compliance) design point for a two-dimensional domain using the EEM. Here, convergence is defined as a nearly stationary topology with changes in the strain energy below the specified threshold. The strain energy history plot in Fig. 3a shows the general convergence pattern in EEM. The spikes that appear at different intervals are mostly due to the random shuffle operation, although it is also possible to see an abrupt change in strain energy during a routine element exchange operation. In EEM, unlike ESO (Xie and Steven 1993;Querinetal. 1998), void elements can be converted into solid and vice versa. Note that in EEM, the void elements have small but nonzero stiffness and density. Even if the initial random distribution of solid elements gives an appearance of a discontinuous load path (infeasible topology), EEM gradually connects all the solid elements in its search for an optimum topology. Furthermore, in EEM the solid and void elements participating in the conversion operation are not limited to any specific region of the design domain as is the case with BESO (Querin et al. 1998). These characteristics together with the random shuffle and overall topology optimization scheme help distinguish EEM from both ESO and BESO. The EEM algorithm is readily applicable to any two- or three-dimensional domain and boundary conditions, irrespective of its geometric or loading complexity.

5 Element exchange method for topology optimization EEM implementation is very straightforward for compliance minimization problems since the total strain energy of the system is a simple summation of energy stored in individual elements distributed in the design domain. As such, it is possible to use the element strain energy to measure the relative influence of a solid or void element with respect to other elements in the same group as well as on the objective function. If a different objective function is chosen or a new design constraint is added to the topology optimization problem, then it is necessary to use a different metric to determine which solid elements need to be convertedinto void and vice versa. In the remaining discussions the focus is on compliance minimization problems. 3.1 Element exchange Experience shows that if the number of exchange elements, M EE is kept constant, EEM may have an oscillatory behavior and have difficulty converging. This characteristic is not unique to EEM, however, and has been reported for other methods as well (Huang and Xie 2007). To overcome this problem, M EE is gradually decreased from its maximum value, M EE max at the first iteration (k = 0) to its minimum value, M EE min as the solution approaches convergence (i.e., k = N max ). For a linear reduction scheme, M EE is found as ( )] M [M k = int MEE max M EE min EE EE max k (3) N max where k and N max denote the iteration counter and the prescribed maximum number of EEM iterations, respectively. If the value of N max is set too low (<100), EEM may not have sufficient opportunity to explore the design space in search of the optimum topology. On the other hand if it set too high (>1,000), the computational time will significantly increase without commensurate improvement in the optimal topology. For the multitude of benchmark problems discussed later in this paper, we have found that N max = 500 is acceptable. Moreover, for the total number of finite elements, M and volume fraction, V 0 specified in the problem, EEM parameter values in the range of 5% MV 0 M EE max 10% MV 0 and 0.2% MV 0 M EE min 0.4% MV 0 are found to be effective in producing converged solutions. Generally speaking, these parameters have a wide range of acceptable values. Although they can affect the convergence property of EEM, there is practically no significant influence on the optimum design. Fig. 3 Results of EEM for a tip-loaded cantilevered beam with a hole showing a the evolution in topology and the strain energy convergence history, and b standard deviation of element strain energy distribution Fig. 4 a Solid checkerboard. b Void checkerboard. c Topology before checkerboard control. d After solid checkerboard elimination. e After void checkerboard elimination

6 M. Rouhi et al. 3.2 Checkerboard control Checkerboard patterns are generally undesirable and depending on the topology optimization methodology used, different strategies are employed to eliminate them (Sigmund and Peterson 1988; Zhou et al. 2001; Poulsen 2002; Pomezanski et al. 2005). In EEM, a solid checkerboard element is defined as a solid element whose edges are shared with void elements as shown in Fig. 4a, whereas a void checkerboard element is the exact opposite as illustrated in Fig. 4b. Whether the dashed elements shown in Fig. 4a and b are solid or void will not change the checkerboard condition. Since in EEM the initial topology is a random distribution of solid elements per the specified volume fraction, it is natural to immediately encounter multiple checkerboard regions as shown in Fig. 3a. However, at the beginning, several element exchange iterations (N CI = 5 10% N max ) are allowed to proceed before actively searching for checkerboard patterns. To eliminate checkerboard regions, first the solid checkerboard elements are identified and converted into void elements as shown in Fig. 4c and d, and then the void checkerboard elements are converted into solid elements (Fig. 4d, e). The checkerboard search and elimination step is repeated every N CC = 1 5% N max iterations. To maintain the specified volume fraction, the difference between the numbers (volumes) of the switched solid and void elements is randomly redistributed in the design domain. It is possible for this random redistribution of the difference to result in the creation of small checkerboard region(s). However, as EEM procedure is continued, these regions tend to gradually diminish before the final topology emerges. The checkerboard elimination procedure in EEM is heuristic and checkerboardelementsare removed regardless of their impact on the overall compliance of the structure. It is important to note the interaction between checkerboard and the mesh size. As shown in exact analytical solutions for Michell truss structures (Lewinski and Rozvany 2008; Rozvany et al. 2006; Lewinski et al. 1994a), the optimal layout is one with many narrow connecting members or branches. However, if the mesh is relatively coarse, the elements in some of these branches would have a pixilated appearance and, hence, marked as checkerboard elements to be eliminated. As will be shown in the example problems later, there is a greater chance for EEM to produce a topology that approaches the analytical solution through mesh refinement. Although the current checkerboard control algorithm is fairly effective, it does have some limitations in that it may not recognize checkerboard patterns that do not perfectly match the models shown in Fig. 4aandb. 3.3 Random shuffle Although both element exchange and checkerboard control are effective tools in helping the EEM algorithm push toward the optimum topology, they are not sufficient. Therefore, an additional operation (i.e., random shuffle) is introduced. A random shuffle involves the selection of a subset of solid elements and their redistribution in void regions of the domain. This action is analogous to the mutation operation in GA (Goldberg 1989) or craziness in PSO (Fourie and Groenwold 2001), and is used for the same principal reason, i.e., it helps prevent premature convergence or insufficient exploration of the design domain in search of optimum design. While preserving the specified volume fraction, random shuffle can also help with convergence by alleviating the occasional back and forth oscillation (oscillatory exchange) in a subset of elements from solid to void back to solid in successive element exchange operations. Random shuffle will change the topology of the structure by introducing a random replacement of a group of solid elements thereby alleviating the oscillatory exchange. The effect of random shuffle on the solution is shown later in the paper. Two things can generally happen as a result of the random shuffle operation: (1) an abrupt change in stiffness (Rouhi and Rais-Rohani 2008) and total strain energy as shown by the spikes in Fig. 3a, and (2) creation of checkerboard regions. However, neither one of these side effects is detrimental as both are corrected by the element exchange and checkerboard control operations of EEM. Random shuffle is third in the sequence of operations in EEM and occurs at N RS = 2 10% N max iterations until the optimal topology is found. The number of elements participating in the random shuffle, M RS varies from its maximum value, M RS max at the beginning (k = 0) to its minimum value, M RS min as the solution approaches convergence. The value of M RS is found using the expression ( )] M [M k = int MRS max M RS min RS RS max k N max where k and N max denote the iteration counter and the prescribed maximum number of EEM iterations, respectively. For the benchmark problems discussed later in this paper, M RS max = M EE max, M RS min = M EE min have been found to be effective in producing converged solutions. 3.4 Passive elements Some continuum structures may contain solid and/or void sub-regions whose geometry and locations cannot be altered during topology optimization. As a matter of convenience and meshing simplicity, the permanent voids (or solids) may (4)

7 Element exchange method for topology optimization be included in the finite element mesh but represented by a series of passive elements with ρ pi = ρ min (or ρ pi = 1.0) that will not be exchanged during the EEM solution process. The hole region in the structure shown in Fig. 3 was modeled using passive elements. 3.5 Convergence criteria Increasing the number of iterations in EEM will usually lead to a more refined optimal topology but at the expense of more function calls (i.e., additional FE solutions). Besides imposing a limit on the maximum number of iterations, two additional criteria are also used to establish a two-part convergence condition in EEM-based topology optimization. The first convergence criterion considers the relative difference in the element strain energy distributions in two consecutive elite topologies. Here, elite topology refers to the topology with the lowest strain energy obtained prior to the current iteration in the EEM procedure. Since it is possible for two distinctly different topologies to have almost equal total strain energies (as noted in the SIMP based results in Fig. 5), it is necessary to compare the element strain energy distribution, as represented by the vector E, for two consecutive elite topologies as E ce E pe E pe ε E (5) where subscripts ce and pe refer to the current and previous elite topologies within N max iterations, respectively. The second convergence criterion examines the density distribution in two consecutive elite topologies. The domain topology is defined by vector D whose individual terms have binary values depending on the solid (1) or void (0) property of the corresponding elements. Based on this definition, the convergence criterion is defined as Dce Dpe ε t (6) Dpe Fig. 5 Two different topologies with nearly identical strain energy values Figure 3a shows how the strain energy in EEM converges to its minimum value for a cantilevered beam with a fixed hole. Strain energy starts from an extremely large value because of the randomly distributed elements in the initial step. However, after a few iterations, it reduces to roughly the same order of magnitude as its minimum value. The continuation of the element exchange will refine the topology toward its minimum strain energy as shown in Fig. 3a. Although there are occasional jumps in total strain energy (due to element exchange or random shuffle operation), the overall trend shows a gradual convergence. It should be noted that similar spikes in the strain energy convergence plots have also been observed when using BESO (Querin et al. 1998). Figure 3b shows the plot of the standard deviation of strain energy in individual solid elements at different stages of the solution process. The trend indicates that the strain energy density field is approaching a more uniform state, resembling the fully stressed design in optimality criteria (Bendsoe and Sigmund 2002; Tanskanen 2002), as it makes the most efficient use of available material in the design domain. Another point that needs to be mentioned here is that, in its current implementation, when EEM identifies an elite topology, no check is made whether any of the solid elements that were previously redistributed as a result of random shuffle operation still remain in the void regions. As such, a few floating solid elements may appear as specks in the void regions of the final topology in different example problems. Since the few floating elements have no significant impact on the final topology, no attempt is made to remove them in any of the presented solutions. Due to the stochastic nature of EEM, another solution to the same optimization problem may show different floating elements. 4 Results for two-dimensional problems Several benchmark problems are used to evaluate the performance of EEM and to compare its solutions with those obtained using some other methods. Each two-dimensional design domain is defined according to n x, n y, V 0 representing the number of finite elements in the x and y directions and the limit on volume fraction, respectively. Hereafter, strain energy refers to the non-dimensional strain energy since the nodal displacements and element stiffness are normalized with respect to element size and material Young s modulus. The reported number of iterations (N) for the EEM results coincides with the number of FE analyses performed in the solution process. Since most of the computational time in each iteration, regardless of the method used, is spent on the compliance

8 M. Rouhi et al. calculation via FEA, the total number of function calls (FE solutions) gives a fairly accurate measure of computational efficiency. We purposefully avoided a time-based comparison because we did not want to improperly attribute inefficiencies in the implementation of FEA to that of the different optimization algorithms considered. In EEM or SIMP, the required computational time is proportional to the number of iterations whereas in population based methods such as GA or PSO it is proportional to the number of iterations times the population size. 4.1 Simply-supported beams Table 2 Comparison of EEM results with Level Set and BESO solutions for model A2 Design domain & Boundary conditions EEM Y F Level Set 38 X BESO 18 t = 27.3, N = 127 t = -, N = 140 t = -, N = 47 a Model A1: Single mid-span force applied on top The simply-supported MBB (Olhoff et al. 1991; Lewinski et al. 1994b) beam model with the loading shown in Table 1 is optimized for minimum compliance. Due to the overall symmetry, the computational model represents only one half of the physical domain. For (n x, n y, V 0 ) = (90, 30, 0.5),the final topology along with the number of iterations (N) and the total strain energy (E t ) are given in Table 1. The EEM results represent three converged solutions using the same Table 1 Comparison of EEM and SIMP results for model A1 Design domain & Boundary conditions F EEM Y (n x, n y, V 0 ) = (90, 30, 0.5) SIMP a SIMP b t = 187, N = 210 t = 190, N = 130 t = 204, N = 29, r = 2 t = 191, N = 227 t = 192, N = 200 t = 195, N = 45, r = 1.2 t = 201, N = 192 t = 193, N = 123 t = 201, N = 33, r = 1 a Based on different initial designs with equal volume fraction and filtering radius b Based on different filtering radii with equal volume fraction and initial design X a The reported steady state number is shown as the total number of FE solutions was not specified values for EEM algorithmic parameters. The difference is caused by the existence of multiple local minima for this problem. For comparison purposes, the SIMP based solutions based on the algorithm provided by Sigmund (2001), are also shown in Table 1 while considering the effect of initial design and filter radius on the optimum topology. Generally, the closer the topology to the ideal Michell truss structure the lower the strain energy. This resemblance (Lewinski et al. 1994a) is more visible in the models with the lowest strain energy than the rest in Table 1. Model A2: Single mid-span force applied at the bottom with modified boundary conditions The roller support at the left side of the beam in model A1 is replaced by a pin support, preventing the beam from having any horizontal displacement at the supports. The load is also moved to the bottom of the beam. The change in the boundary condition affects the optimum topology as shown in Table 2. The EEM solution for (n x, n y, V 0 ) = (61, 62, 0.31) is compared with those reported in the literature by Wang and Wang (2003) using the Level Set method for (n x, n y, V 0 ) = (61, 62, 0.31) andbyquerinetal.(1998) using BESO for (n x, n y, V 0 ) = (31, 32, 0.25). The value shown for the number of iterations in BESO is the value reported (Querin et al. 1998) forthesteady state number, which can be considerably less than the total number of FE solutions. The optimal strain energy values cannot be compared due to the lack of data in the cited references. The optimal

9 Element exchange method for topology optimization Table 3 Comparison of EEM and SIMP results for model B1 Design domain & Boundary conditions Y (n x, n y, V 0 ) = (32, 20, 0.4) EEM SIMP t = 53.6, N = 178 t = 57.4, N = 71 (n x, n y, V 0 ) = (64, 40, 0.4) EEM SIMP t = 57, N = 174 t = 55.7, N = 57 topologies are fairly similar with both EEM and BESO solutions showing one extra member than that in the Level Set solution. 4.2 Cantilevered beams Model B1: F Single tip force applied at the bottom Table 3 shows the beam model and loading condition along with the results of EEM and SIMP for two different mesh sizes at the same volume fraction. In the case of the SIMP, the results are based on the filtering radius of 1.2. While the optimal strain energy values are comparable, the total iteration numbers are different. As a result of mesh refinement, the optimal topology changes with minimal change in the final strain energy. By increasing the mesh size, both EEM and SIMP solutions move toward Michell truss topology. At first, it appears counterintuitive for the more Michell like structure (Lewinski et al. 1994a) associated with the fine-mesh solution of EEM to have a strain energy that is higher than that of the coarse mesh. This does not imply that the fine-mesh solution is inferior. On the contrary, the apparent discrepancy can be explained by the fact that the FEA solution (predicted strain energy) using the coarse mesh is not as accurate as that for the fine mesh. If the optimal layout (ground elements) in each case were discretized so as to increase the accuracy of post-optimum FEA solution, then X the more Michell like topology would have smaller strain energy for the same volume fraction. Model B2: Single tip force applied at mid height The beam model and the corresponding topology optimization results are shown in Table 4. Two different mesh densities are used for EEM solutions at the same volume fraction. Both solutions show a fairly similar trend for material distribution, although the fine-mesh solution is more accurate. The results reported by Wang et al. (2006) based on the enhanced GA approach are also shown in Table 4 for comparison. Although the final geometry and strain energy values are nearly the same, the EEM solution converges 160 times faster. Jakiela et al. (2000) state that, in general, GA based solutions may require 10 to 100 times more function evaluations than would be required by homogenization based solutions. It is notable that the number of function calls is in the order of the number of iterations multiplied by the population size in both GA and PSO as will be shown later. A qualitative and quantitative comparison of the performance of EEM, enhanced GA and SIMP in finding the optimum topology for this model is shown in Fig. 6. Although EEM is not as computationally efficient as SIMP, Table 4 Comparison of EEM results with enhanced GA solution for model B2 Design domain & Boundary conditions Y (n x, n y, V 0 ) = (24, 12, 0.5) EEM Enhanced GA 39 t = 66.1, N = 150 t = 64.4, N = 4x10 4 (n x, n y, V 0 ) = (48, 24, 0.5) EEM t = 63.5, N = 250 F X

10 M. Rouhi et al. Model B3: Model B2 with modified dimensions Compliance Number of iteration (a) The dimensions of the cantilevered beam in model B2 are modified such that the beam s height is greater than its length. In Table 5, the results of EEM for two different mesh sizes are compared with the PSO based solutions reported by Fourie and Groenwold (2001). While the topologies for the fine mesh are nearly identical, the EEM optimum topology for the coarse mesh is better than that produced by PSO. Although PSO is a population-based method requiring multiple FEA in every iteration, the results still show that the EEM solution can converge 10 to 1000 times faster than PSO with no loss of accuracy. Model B4: Multiple load cases Compliance Compliance Number of iteration (b) Number of iteration (c) Fig. 6 Compliance convergence history and final topology for a enhanced GA (Wang et al. 2006), b SIMP and c EEM The cantilevered beam model in Table 6 is optimized for two separate load cases. In one load case, only force F 1 is applied at the tip whereas in the other only F 2 is applied. Forces F 1 and F 2 have equal magnitudes and opposite directions. For EEM solution, the strain energy in each element is the sum of that for each load case. As a result, the additive form of the objective function is retained and the relationship between element strain energy and element exchange operation remains unchanged. Thus, the solid elements with the lowest strain energy sum (from the two load cases combined) are converted into void elements whereas the void elements with the highest strain energy sum are converted into solid elements. The results for EEM and SIMP (with r = 1.2) are compared in Table 6. Because both F 1 and F 2 have equal magnitudes, the structure tends to have a symmetric layout. The EEM and SIMP topologies have some distinct differences. The two horizontal (top and bottom) members in SIMP solution appear as slanted in the EEM layout and the vertical member in SIMP layout is absent in the EEM topology. The dark specks seen in the EEM topology are the residue or the floating elements from the last random shuffle operation as discussed earlier in the paper. 4.3 L-shaped domain Model C1: Distributed force applied along one boundary it is considerably more efficient than GA. It is also worth noting that for GA the number of iterations times the population size gives the total number of function calls (Wang et al. 2006). An L-shaped domain with clamped boundary conditions along the top edge and a distributed force along the middle third section of the right edge as shown in Fig. 7a is optimized for minimum compliance. For simplicity, the problem is modeled as a square domain with elements located in the upper right quadrant treated as passive elements. Using

11 Element exchange method for topology optimization Table 5 Comparison of EEM results with PSO solutions for model B3 Design domain & Boundary conditions X Y F (n x, n y, V 0 ) = (20, 47, 0.5) (n x, n y, V 0 ) = (40, 94, 0.5) EEM PSO 16 EEM PSO 16 t = 3.0, N = 100 t = -, N = 10 5 t = 5.1, N = 103 t = -, N = 10 3 Continuous Density Binary Density (n x, n y, V 0 ) = (90, 90, 0.35), EEM finds a minimum compliance solution after 167 iterations with the strain energy of The optimal EEM topology in Fig. 7c is compared with that of the Neighborhood Search method (Svanberg and Werme 2005) infig.7b. For the same mesh size, the solution based on Neighborhood Search method takes 378 iterations. In both cases, the larger members show the average orientation for the more finely distributed members over the same domain (Lewinski and Rozvany 2008). Table 6 Comparison of EEM results with SIMP solution for model B4 Design domain & Boundary conditions F 1 =F Fig. 7 a L-shaped domain with distributed tip load and optimal layouts based on b Neighborhood Search method (Svanberg and Werme 2005) and c EEM (a) F 2 =F (F 1 and F 2 represent separate load cases) (n x, n y, V 0 ) = (50, 50, 0.4) EEM SIMP t = 60.9, N = 104 t = 61.3, N = 60 (b) (c)

12 M. Rouhi et al. Fig. 8 a L-shaped domain with single tip load and optimal layouts based on b exact analytical solution (Lewinski and Rozvany 2008). c EEM with coarse mesh. d EEM with fine mesh the analytical case. However, the trend captured in the EEM solutions appears to be in reasonable agreement with the analytical solution. Since there is some randomness in the EEM procedure, the number of lumped bars and their orientation may vary from one solution to another for the same problem. (a) 5 Results for three-dimensional problems Each three-dimensional design domain is defined according to n x, n y, n z, V 0 values representing the number of finite elements in the x, y, andz directions and the limit on volume fraction, respectively. As in the previous section, the EEM results are compared with those reported in the literature. (b) 5.1 Cubic domain A cubic domain is simply supported at its four bottom corners and is loaded by four concentrated vertical forces acting at the top surface as shown in Fig. 9a. Using the EEM procedure with (n x, n y, n z, V 0 ) = (20, 20, 20, 0.08), the optimum (c) Fig. 9 a Design domain and boundary conditions with optimum topology based on b Optimum Microstructures (Olhoff et al. 1998) andc EEM (a) (d) Model C2: Single tip force applied at the bottom The problem is similar to that in model C1 except for the applied load as shown in Fig. 8a. Here the EEM solutions in Fig. 8c anddfor(n x, n y, V 0 ) = (90, 90, 0.35) and (n x, n y, V 0 ) = (250, 250, 0.35), respectively, are compared to the exact analytical solution (Lewinski and Rozvany 2008; Rozvany et al. 2006) infig.8b. It should be noted that the unlimited number of bars in the analytical solutions would be very difficult to capture in an FE solution with a limited discretization, especially in this case where the boundary conditions along the top edge are not exactly the same as (b) (c)

13 Element exchange method for topology optimization (a) (c) (b) (d) Fig. 10 a Design domain and boundary conditions with optimum topology based on b EEM, V 0 = 0.3, c EEM, V 0 = 0.08 and d Optimum Microstructure (Olhoff et al. 1998), V 0 = 0.3 (elements with densities less than 0.5 filtered out) Altair / HyperWorks / ostutorials / ostut. htm # os2010. htm). While the triangular region in the middle can be altered through topology optimization, the three corner regions (knuckles) are held fixed with the specified boundary conditions shown in Fig. 11a. The EEM solution based on (n x, n y, n z, V 0 ) = (26, 40, 12, 0.1) is shown in Fig. 11b with a final strain energy of Since the design domain is symmetric, only the upper half of the final topology is considered and shown for clarity. The optimization result in Fig. 11c is obtained using a commercial software code (OptiStruct; reported in /Altair /HyperWorks /ostutorials/ostut. htm#os2010.htm). In Fig. 11c, the elements with density less than 0.15 are removed and the resulting geometry is post-processed to obtain a smoother shape. YZ DOF fixed Z DOF fixed topology in Fig. 9b is obtained after 178 iterations with a strain energy of For comparison, the results obtained by Olhoff et al. (1998) using the Optimum Microstructure (OM) method is shown in Fig. 9c. The gray regions in the Fig. 9c imply intermediate density since the OM method uses a continuous density function. Also, elements with density less than a threshold value are filtered out in the OM method to arrive at the final topology. Therefore, the final topology may not match the pre-specified volume fraction. XYZ DOF fixed Z X Y (a) F y F x F z 5.2 Clamped-clamped beam A clamped-clamped beam is loaded in the middle by a concentrated bending moment as shown in Fig. 10a. The EEM results for (n x, n y, n z, V 0 ) = (50, 10, 10, 0.3) and (n x, n y, n z, V 0 ) = (50, 10, 10, 0.08) are shown in Fig. 10b andc. For the OM based solution shown in Fig. 10d, the elements with density less than 0.5 are removed. For the same reason stated earlier, the actual volume fraction is less than the specified value of 0.3. Figure 10c shows that EEM result is sensitive to the direction of the applied moment. It is not clear if a similar sensitivity also exists in the OM based solutions at a lower volume fraction values. For the solutions in Fig. 10b and c, the number of iterations is found to be 197 and 206, respectively. (b) 5.3 Automobile control arm (c) The geometry shown in Fig. 11a is a generic model of an automobile control arm (discussed in Fig. 11 a Control arm model with optimum topology based on b EEM and c OptiStruct ( ostutorials/ostut.htm#os2010.htm) software

14 M. Rouhi et al. The possibility of sub-structuring the computational domain to reduce the number of FE calculations was investigated in Rouhi and Rais-Rohani (2008).Theideawasto limit the FE analysis to the parts of the domain that participate in the element exchange operations. The participation of a large number of elements in different regions of the structure in the various EEM operations showed that such a sub-structuring may not be possible. The requirement for a relatively large number of FE solutions is the main drawback in all stochastic approaches including EEM. Y (a) F X 6 Effects of special operations in EEM on the solution As noted previously, there are two special operations in EEM in the form of checkerboard control and random shuffle. The influence of each of these operations on the optimal topology is examined here with the help of the model shown in Fig. 12. A tip-loaded cantilevered beam as shown in Fig. 12ais optimized based on theinitial layoutof material in Fig. 12b. To limit the list of variables involved, the solid elements are distributed uniformly in one portion of the domain per the specified volume fraction. With both the checkerboard control and random shuffle operations preserved, the optimum topology is that shown in Fig. 12c. Without using the checkerboard control operation, the optimum solution is that in Fig. 12d. The strain energy in the final topology is partly due to the numerically induced (artificial) high stiffness of the checkerboard regions. When the random shuffle operation is disabled, the final topology is that shown in Fig. 12e with both the final layout and the strain energy value indicating a local optimum design point that is not as good as the one in Fig. 12c. (b) (c) (d) 7 Effects of EEM parameters on the solution As mentioned previously, the desired values for the EEM input parameters must be specified before starting the solution process. These parameters are: V 0 (the volume fraction), M (the number of finite elements in the model), N max (the maximum allowable number of iterations in EEM solution), ε E (total strain energy convergence parameter), ε t (element topology or density distribution convergence parameter), M EE max (the maximum number of exchange elements), M EE min (the minimum number of exchange elements), M RS max (the maximum number of solid elements participating in the random shuffle), M RS min (the minimum number of solid elements participating in the random shuffle), N CI (the number of completed iterations before starting checkerboard control), N CC (the number of iterations or interval between checkerboard control operations), (e) Fig. 12 a The design domain and BCs. b Initial design topology. c EEM result, E t = 54. d EEM result with no checkerboard control operation, E t = e EEM result with no random shuffle operation, E t = 58.5 and N RS (the number of iterations or interval between random shuffle operations). All other parameters not listed above are calculated from these input parameters.

15 Element exchange method for topology optimization The convergence parameters (N max, ε E and ε t ) control the solution accuracy. Generally speaking, the EEM search for optimum topology becomes more rigorous by increasing the value of N max and decreasing the values of ε E and ε t at the expense of increased computational time. As with the other stochastic methods, if N max is too small, EEM may not be able to find an optimum solution. On the other hand, if ε E and ε t are too small, convergence may be hard to achieve. As shown in Fig. 3, EEM finds the basic layout of the optimum topology in about 20 iterations with the remaining iterations devoted to refinement of the topology. However, when the design domain includes multiple local optima, then it is possible for the topology to vary widely during the solution sequence. Since element exchange is the main operation in EEM, the value selected for M EE is important. As noted previously, M EE cannot be treated as a constant and must be gradually reduced from its prescribed maximum value (M EE max ) at the very beginning to its imposed minimum (M EE min ) at the end. Based on the multitude of problems examined, choosing M EE max 5 10% and M EE min % of the solid elements (MV 0 ) would be appropriate. Generally speaking, selecting a relatively large value for M EE max will lead to the formation of the main load path in the early stages of the optimization process. However, it may also reduce the computational efficiency due to participation of a large number of elements in oscillatory exchange phenomenon discussed earlier. Likewise, choosing a relatively small value for M EE min makes the solution easier to converge with a more refined final topology. Similar to element exchange, the number of elements (M RS ) and the interval (N RS ) selected for random shuffle are crucial to the success of EEM. Equation (4) provides an acceptable reduction scheme for M RS in the range M RS min M RS M RS max. While starting with M RS = M RS max widens the domain of exploration for optimum design (similar to a larger coefficient for the particle s velocity in PSO (Fourie and Groenwold 2001), it may reduce the computational efficiency by exchanging a large number of elements in a random fashion. On the other hand, small M RS min value helps with convergence while enhancing the final topology. For problems with a large number of solid elements (i.e., large volume fraction) or large MV 0, EEM is less likely to get trapped in a loop (see Section IIIC) and the step size for random distribution can be increased to improve the computational efficiency. Choosing N RS 2 5% N max is found to be sufficient to help EEM not to get trapped in a local optimum while reducing the number of elements involved in oscillatory exchange. As shown in (3),alargevalueforN max decreases the rate of reduction in M EE, which increases the number of EEM iterations. Although it delays the finding of the final topology, it makes entrapment in a local minimum less likely. The value selected for N CI should provide sufficient opportunity for element exchange operation to improve upon the initial topology (random distribution of solid elements), which may include many checkerboard regions. Similarly, N CC should not coincide directly with N RS,as random shuffle is likely to cause the creation of checkerboard elements. Choosing N CI 5 10% N max and N CC 1% N max are found to be sufficient to allow both the element exchange and random shuffle operations to improve the design before the accumulated checkerboard elements are identified and eliminated by the checkerboard control procedure. All of these parameters are flexible and reasonable deviations from the suggested values may not dramatically affect the final results. The values selected for the EEM parameters in some of the example problems presented earlier are shown in the Appendix. 8 Concluding remarks A stochastic direct search topology optimization method called element exchange method (EEM) was presented with application to compliance minimization problems subject to a volume fraction constraint. The non-dimensional density of each finite element was treated as a binary design variable (ρ void = and ρ solid = 1) with a linear element density-stiffness relationship. The basic premise of the proposed method is that by systematically converting the less critical solid elements into void and the more critical void elements into solid, an optimum topology will emerge. The proper selection of EEM parameters assures convergence. However, depending on the selected mesh density and the desired level of clarity in the final topology, the number of iterations required for convergence may vary. Through the solution of several two- and threedimensional example problems for compliance minimization, the accuracy and efficiency of EEM were examined and compared with different gradient based (e.g., SIMP) and non-gradient based (e.g., GA, EBSO) methods as reported in the literature. The results show that EEM is reasonably accurate in finding an optimum topology with clear solid-void layout of material in the design domain. As for computational efficiency, EEM is shown to be inferior to gradient based methods (e.g., SIMP) but superior to non-gradient based methods such as GA and PSO. Acknowledgments This material is based upon work supported by the Department of Energy under Award Number DE-FC26-06NT The authors are also grateful to Prof. Rozvany for valuable discussions on topics related to topology optimization.