Effects of Microscale Material Randomness on the Attainment Of

Research paper Struct Multidisc Optim 25, 1–10 (2003) DOI 10.1007/s00158-003-0304-9

Eﬀects of microscale material randomness on the attainment of optimal structural shapes

T. Liszka, M. Ostoja-Starzewski

Abstract Classical procedures of shape optimization of fectly homogeneous continuum. Consequently, the pro- engineering structures implicitly assume the existence of cess of optimization does not recognize the presence of a hypothetical perfectly homogeneous continuum – they any microscale material randomness, although the lat- do not recognize the presence of any microscale mate- ter becomes progressively more relevant as the mesh rial randomness. By contrast, the present study inves- is refined. A study of this effect conducted recently on tigates this aspect for the paradigm of a Michell truss the paradigm of the Michell truss (Michell 1904) made with minimum compliance (maximum stiffness) that has of elastic–plastic members showed that the random mi- a prescribed weight. The problem involves a stochastic crostructure prevents the attainment of the classically generalization of the topology optimization method im- predicted optimal shape using the homogeneous con- plemented in the commercial Altair’s OptiStruct com- tinuum (Ostoja-Starzewski 2001). The aforementioned puter code. In particular, this generalization allows for investigation was carried out under the assumption of the dependence of each finite element’s stiffness matrix fixed load on the structure and the search was con- on the actual microstructure contained in the given ele- ducted for a structure of optimal (i.e., minimal) weight. ment’s domain. Contrary to intuition, stochastic material The study considered the effect on the ability to at- properties may improve the compliance of optimal design. tain the classically expected minimum of a spatially dis- This is because the optimization is performed on a given ordered (non-periodic) material microstructure of which random distribution, so that the design process has an op- the Michell truss-like continuum would be manufactured: portunity to choose ‘stiffer’ cells and discard those with the denser the truss (i.e., the finer its mesh spacing), the weaker material. The paper does not aim for a robust de- more significant the effect of microstructural fluctuations sign process, but tries to answer a simpler intermediate on the plastic limit σ0. This led to the question of the question: how the random fluctuation of material prop- sensibility of the ad infinitum refinement of the Michell erties influences a structure that has been designed using truss, and hence to the open issue of the plausibility of its classical continuum-based optimization algorithms. attainment. In general, the planar material of which the truss is Key words optimal structures, random microstructure manufactured is a random medium:asetB = {B(ω); ω ∈ Ω},whereeachB(ω) is a deterministic, albeit spatially inhomogeneous, body with Ω being a sample space. More specifically, each B(ω) is described by a scalar- valued planar field of plastic limit k,sothatwehave 1 the random field k = k(ω,x); ω ∈ Ω, x ∈ R2 specifying Background B. The actual setup of the field k depends on a non- dimensional scale parameter δ = L/d > 1, relating the Problems of shape optimization of engineering structures mesh spacing L to the size of the microheterogeneity d implicitly assume the existence of a hypothetical per- (e.g., crystal diameter). Micromechanics of random heterogeneous media (Jeulin and Ostoja-Starzewski 2001), Received: while only recently making inroads into plasticity (Jiang Published online: 2003 et al. 2001), suggests that the following two aspects  Springer-Verlag 2003 should be included in any physically acceptable model: 1, u 2 T. Liszka , M. Ostoja-Starzewski (i) the scatter in k (as described by its standard devia- 1 tion σk) should grow as δ decreases, and (ii) the statistical Altair Engineering, Inc., 7800 Shoal Creek Blvd, Austin, TX, 78757, USA average k may be taken to be constant as δ decreases: e-mail: [email protected] 2 Department of Mechanical Engineering, McGill University, 1 k =const,σk(δ) ∼ . (1) Montreal, Que. H3A 2K6, Canada δ 2

As is typically done in mechanics of random media, it is where convenient to split the field k into its constant mean and the randomly fluctuating part. We can then write aC(u, v)= ε(u)C(x)ε(v)dx (4) V k(ω)=k + k(ω) , k =0, (2) is the bilinear form of the energy, and where k is the zero-mean noise in k. In setting up the model, we also assumed: (i) the truss spacing L of inter- L(u)= fvdx+ tvdx (5) est to us is greater than the grain size d,sothatk may be V ∂Bt treated as a field of independent random variables when entering the finite difference formulation generalized to is the linear form of the load. That is, we seek the opti- ≡ inhomogeneous media; and (ii) the underlying material mal choice of the stiffness tensor C( Cijkl)insomegiven microstructure is space-homogeneous and ergodic. set of admissible tensors Uad. C are generally fields over R2 ∈ ∞ 6 Since the reference problem of this Michell truss is ,sothatUad (L (V )) , corresponding to the six in- governed by a quasi-linear hyperbolic system (Hegemier dependent elements of the in-plane stiffness tensor. In (3), and Prager 1969), ours was described by a stochastic gen- by ‘design constraints’, we mean constraints on stresses, eralization thereof (Ostoja-Starzewski 2001). The term strains, displacements, etc., while sizing constraints, vol- ‘stochastic’ refers to an ensemble of deterministic prob- ume constraints, etc., are accounted for in the choice of lems that exist on spatially inhomogeneous realizations of Uad. Finally, U is the space of kinematically admissible the random field k. Due to the randomness of k on scales displacement fields. δ<∞, the net of characteristics display a statistical scat- In the case of optimal shape design, elements C of Uad ter that increases with decreasing δ, which translates into take the form a trend to use more structural material to carry the pre- C (x)=χ(x)C , (6) scribed loading. This, of course, is an opposing tendency ijkl ijkl to the convergence to the Michell truss with mesh refine- where Cijkl is the constant stiffness tensor for the ma- ment, and, beyond a certain mesh refinement, the solu- terial employed for the construction of the mechanical tion to the governing hyperbolic system breaks down. To element, and χ(x) is the indicator function. sum up, there is a limitation to this ideal optimality. In the case of sizing problems, like the design of sheets Therefore, the above method cannot be used for in- of variable thickness, the admissible Cijkl’s have the form vestigation of systems with too fine a mesh for a given microstructural randomness, nor can it be employed for Cijkl(x)=h(x)Cijkl , (7) elastic structures. These restrictions suggested another ∈ ∞ approach – in fact, much more in line with the conven- where again Cijkl is a constant tensor, and h(x) L (V ) tional methodology of shape optimization of engineering is the sizing function. structures (Rozvany et al. 1995; Bendsøe and Kikuchi The discretized formulation of the topology optimiza- 1988) – which seeks the shape of an elastic structure with tion problem (3) can be stated as follows: minimum compliance (maximum stiffness) that possesses min f(ρ) a prescribed weight. Note, however, that this is differ- ∗ ent from the problem considered in (Ostoja-Starzewski s.t. V = ρjνi ≤ V , (8)

2001), which dealt with a plastic structure of minimal η ≤ ρi ≤ 1,i=1,...,N. weight under a fixed load. The present paper reports where f represents the objective function, ρi and νi are a stochastic generalization of the topology optimization ∗ method implemented in a commercial Altair’s OptiStruct element densities and volumes, respectively, V is the computer code, which takes into account a material mi- target volume, N is the total number of elements, and crostructure of random chessboard type. η is a small number that prevents the stiffness matrix from being ill-conditioned. The common objective function is the weighted sum of the compliance under all load 2 cases. Note that the problem in (8) is a relaxation for- Problem formulation mulation of the topology problem, in which the density should only take the values 0 or 1. To enforce the design to be close to a 0/1 solution, a penalty is introduced to re- The minimum compliance problem for a planar (respec- duce the efficiency of intermediate density elements. Here tively, 3-D) body of volume V in R2 (R3) subjected to the penalization is achieved by the following power-law body forces f and tractions t takes the form: formulation: min L(u) ∗ p Ki (ρ)=ρi Ki , (9) Cijkl ∈ Uad ∈ ∗ subject to aC(u, v)=L(v), all v U, design constraints, where Ki and Ki represent the penalized and the real (3) stiffness matrix of the i-th element, respectively, and p is 3 the penalization factor, which is bigger than 1 (Bendsøe and Kikuchi 1988). A more general formulation of the topology optimization problem can be stated as follows: min f(ρ) − ∗ ≤ s.t. gj(ρ) gj 0,j=1,...,m, (10)

0 ≤ ρi ≤ 1,i=1,...,N, ∗ where gj and gj represent the j-th constraint and its up- per bound, respectively, and m is the total number of Fig. 1 One-dimensional tension problem on a square domain constraints. The ‘minimum member size’ constraint, implemented in Altair OptiStruct is based on constraining the discrete slope of the density. To achieve a predetermined minimum member size of radius rmin = dmin/2, the slope constraint can be formulated for a general irregular ﬁnite element mesh as follows:

|ρi − ρk|≤(1.0 − ρmin)dist(i, k)/rmin , (11) where dist(i, k) denotes the distance between adjacent elements i and k,andρmin,say0.1, is the threshold that is interpreted as void in the final solution (Zhou et al. 1999). This condition guarantees that whenever an element j reaches a density of 1.0, the member connected to this element has a diameter of at least dmin. This condition introduces n · N,withn bigger than 2, additional linear constraints and makes the direct solution of this for- Fig. 2 Optimal solution for the uniform unidirectional prob- mulation computationally prohibitive. Our code Altair lem (dark color represents the converged optimal shape OptiStruct implements a different algorithm (Zhou (density = 1.0), and void areas are shown as white; the few et al. 1999) that is very efficient for these types of con- remaining cells with intermediate densities are shown with straints. For a survey of historical developments and different colors on a gray scale) a summary of the theory and techniques of topology optimization, we refer the reader to (Rozvany et al. 1995), generated for an isotropic material, but any distribu- while for a background on penalization formulation to tion of horizontal trusses will satisfy the optimization (Allaire and Kohn 1993). problem. Two predefined (non-random) distributions of material density were tested: each consisted of regular bands 3 of alternating materials parallel to the axis. For the ho- Numerical results rizontal distribution (Fig. 3a), the optimization process placed the trusses inside the stiffer material (gray and 3.1 white bands represent hard and soft material, while black Preliminary tests represents the optimal shape). For the vertical distribution (Fig. 3b – material types not shown in the figure) Before proceeding with the solution of the Michell truss, the trusses were generated with varying thickness, and we start with a very simple example of a unidirectional smaller thicknesses coincided with bands of stiffer mate- stress state in a square domain of size 1×1. The boundary rial. Although both cases are clearly contrived, the solu- conditions are ux = uy =0atx =0,andux =0.1,uy =0 tion in Fig. 3a proves that it is possible to obtain a better at x = 1 (see Fig. 1). We are trying to find the optimum optimal shape (lower compliance) in the presence of non- distribution of material with total volume V ∗ =0.1. Be- uniform material properties. cause the problem is ill-posed, the results of the simula- Finally, Fig. 4 shows the optimal shape in the presence tion exhibit poor convergence or numerical artifacts (e.g. of very strong noise (s =0.9). By comparing the mate- splits at each member near the edges of the boundary). rial distribution (Fig. 4b), representing material strength However, this (as expected) provides an optimization al- (black-gray-white scale, black being the soft material) gorithm with significant “freedom” that results in a vis- with the optimal shape shown in Fig. 4a, one can verify ible tendency of the “optimal” structure toward harder that the optimization process really avoids soft cells in the cells in the design space. The solution in the Fig. 2 was model. 4

Fig. 3 Optimal solution for the unidirectional problem with “banded” material properties: a) horizontal, and b) vertical distribution of material properties

Fig. 4 Optimal solution for the non-uniform unidirectional problem (noise = 0.9): a) optimal shape, b) distribution of material strength, c) and d), zoom of the lower part of the domain

3.2 Michell truss problem

The formulation outlined above was applied to the classical problem of finding an optimal structure supported by a circular foundation boundary F and subjected to A a loading condition at A (see Fig. 5). First, for the sake F of reference, in Fig. 6a we display the truss found by car- rying out an optimization on a 27 × 27 mesh of square- shaped finite elements, all with identical properties. Next, we considered the optimal structure problem under the same support and loading conditions, but defined on 7 7 a2 × 2 mesh of a random chessboard with single elem- Fig. 5 Michell truss problem: design space and boundary ents being isotropic and linearly elastic: conditions 5

Cijkl = λδij δkl + µ(δjkδil + δjlδik) . (12) As an example, for the noise scale factor s =0.3, the Young modulus was varied from 0.7 to 1.3. The Pois- The material randomness is introduced via a statement son’s ratio was ν =0.3. All the elements are generated ac- analogous to (4): cording to the same process, independent of their spatial location, so that we effectively have a binomial random 7 7 C(ω)= C + C (ω) C (ω)=0 (13) field on the 2 ×2 mesh. All presented examples used the same random field r with different scale factors s. ≡ and generating C (ω)( Cijkl (ω)) by multiplying each ≡ element’s mean part C ( Cijkl ) by a random num- 3.3 − ber r sampled from a uniform distribution [ 1, 1] multi- Experiment 1 plied by a constant scale factor s:

In this experiment we carried out a parametric study C (ω)=rs C . (14) of the optimization problem by considering three cases

Fig. 6 Optimal solutions for the Michell problem, case 1: r =0.09 (noise = 0.0, 0.1, 0.15, 0.2, 0.25, 0.3, 0.6, 0.9) 6

Fig. 7 Optimal solutions for the Michell problem, case 2: r =0.10 (noise = 0.0, 0.1, 0.15, 0.2, 0.25, 0.3, 0.6, 0.9) of the radius of foundation: 0.09, 0.1, and 0.15. In tions (Noise = 0.3), and also two different distributions each case we first display the homogeneous medium usingameshof256× 256 cells (Fig. 9). case (a), and then generate consecutive realizations of Tables 1–3 summarize the compliance results for the the random field with increasing scale factor. In par- optimal structure. Although the numerical noise1 of the ticular, figures of the sequence (b) through (j) corres- results is noticeable in the results, the overall trend to pond, respectively, to the multipliers s =0.1, 0.15, 0.2, produce deteriorating (softer) structure is very clear. In- 0.25, 0.3, 0.6, 0.9. In all the figures, immediately evi- terestingly in some cases the results are actually better, dent is the breakdown of global symmetry of the truss which indicates that the optimization process was able to structure due to the spatial non-uniformity of material select stiffer paths through the structure. (Figs. 6–8). All results were obtained using the same random dis- 1 Topology optimization is realized as a highly nonlinear tribution, mutiplied by a varying scalar factor (column 0, iterative process with a very shallow local minimum. The non- “Noise” in all tables). In addition Table 1 (sections (b) linearity of the process is additionally compounded by the and (c)) contains results for different random distribu- minimum member control algorithm. 7 N N N To better estimate that the optimization process in- 1 1 1 AnBn − An Bn deed “favors” stiffer paths through the structure, the fol- N N N ρ = n =1 n=1 n=1 , lowing quantities were computed: AB N N 2 • The average material stiffness in the optimal structure 1 2 − 1 × N An N An (normalized to the range 0.0–2.0, such that 1.0 corres- n=1 n=1 ponds to the structure having the same average stiff- N N 2 nessastheentiredesignspace, and 2.0 corresponds to 1 2 − 1 N Bn N Bn the structure consisting entirely of the single material n=1 n=1 (15) at the maximum level of the random distribution). The stiffness is summarized in column 3 (Avg_material)of where An and Bn represent two series. all tables. It is easy to note that for this particular experiment, • Correlation between the material stiffness and the ρAB is not really normalized (the maximum value is sig- density of the design (column 4). The correlation is nificantly less than 1.0). In particular it is easy to show computed using the standard formula that if it were possible for the structure to select only

Fig. 8 Optimal solutions for the Michell problem, case 3: r =0.15 (noise = 0.0, 0.1, 0.15, 0.2, 0.25, 0.3, 0.6, 0.9) 8

Table 1 (a) Configuration 1, fixed random distribution; (b) different random distributions; (c) results for fine model

Noise Compliance Avg_material Correlation Compl1 Compl2

(a) 0.00 114.99 – 0.010795 – – 0.05 115.69 1.0340 0.022403 114.87 115.90 0.10 115.98 1.0439 0.028971 114.90 116.39 0.15 114.84 1.0529 0.034953 115.08 115.69 0.20 114.06 1.0639 0.042046 115.42 115.16 0.25 115.27 1.0459 0.036870 115.93 116.61 0.30 116.10 1.0670 0.044367 116.61 117.83 0.60 118.54 1.0917 0.060332 125.56 121.21 0.90 123.56 1.1334 0.087990 156.01 126.72 (b) 0.30 115.59 1.0424 0.028058 117.10 116.08 0.30 117.18 1.0705 0.047003 117.06 118.68 0.30 117.09 1.0573 0.038294 118.17 117.56 0.30 118.84 1.0725 0.048413 118.44 120.22 0.30 114.94 1.0704 0.046924 118.01 116.29 0.30 116.86 1.0521 0.034725 118.67 117.29 0.30 116.81 1.0383 0.025495 117.38 117.35 (c) 0.00 105.98 – – – – 0.30 107.52 1.0496 0.032621 107.69 107.88 0.30 108.70 1.0511 0.033657 107.63 109.12

Table 2 Conﬁguration 2, ﬁxed random distribution

Noise Compliance Avg_material Correlation Compl1 Compl2

0.00 99.31 – 0.010576 – – 0.05 99.69 1.0266 0.017573 99.29 99.80 0.10 99.27 1.0293 0.019326 99.39 99.46 0.15 98.56 1.0318 0.020874 99.63 98.76 0.20 98.50 1.0416 0.027470 100.00 98.78 0.25 98.97 1.0370 0.024382 100.52 99.38 0.30 99.94 1.0491 0.032404 101.20 100.46 0.60 99.01 1.0834 0.054779 109.65 100.06 0.90 107.01 1.1208 0.079803 137.44 107.03

Table 3 Conﬁguration 3, ﬁxed random distribution

Noise Compliance Avg_material Correlation Compl1 Compl2

0.00 60.51 – 0.017407 – – 0.05 60.44 1.0354 0.023236 60.45 60.57 0.10 59.66 1.0348 0.025773 60.45 59.87 0.15 59.39 1.0469 0.030858 60.53 59.73 0.20 59.60 1.0506 0.033297 60.68 60.01 0.25 58.72 1.0673 0.044165 60.91 59.38 0.30 59.63 1.0684 0.045002 61.22 60.30 0.60 60.71 1.0818 0.053469 65.34 61.28 0.90 63.68 1.1346 0.088292 78.79 64.95 9

3.4 Experiment 2

In this experiment we tested how much the optimality of the design is affected by material properties different from the ones assumed during optimization process. With the shapes obtained in Experiment 1 we computed the compliance with different material properties. Two cases were tested: • the shape optimized for uniform material properties (“symmetric design”) was analyzed with random material properties (sr), and • the shape optimized for the random distribution (sr) was analyzed with uniform material properties (s =0). The results are summarized in Tables 1–3 (columns 5 and 6, respectively). Please note that this experiment did not introduce numerical noise2. As expected the results consistently contain lower values, which shows that stochastic material properties indeed decrease the “optimality” of the design. However the results from Ex- periment 1 (column 2) are better, and sometimes significantly, which shows that the optimization process is able to compensate for varying material properties.

4 Closure

In the language of stochastic mechanics (e.g. Jeulin and Ostoja-Starzewski 2001), our equation (8) falls into a general class of problems written as Fig. 9 Optimal solutions for the Michell problem, case 1: L ∈ r =0.09, 256 × 256 fine mesh (noise = 0.0, 0.30, 0.30–twodif- (ω)φ = fωΩ, (17) ferent distributions) where L(ω) is the differential operator of elasto-statics having random (rather than deterministic) stiffness coef- the cells of the maximum stiffness, then the value of ficients, φ is the field that is sought (such as the displace- this correlation would be ment u), f is the forcing, and Ω is the sample space of elementary outcomes ω (realizations of the random field). ρmax =1/ (1/v − 1) , (16) The conventional route of phenomenological deterministic continuum mechanics without regard for microstruc- where v represents the volume constraint. The above tural randomness is to average (17) directly so as to ob- formula was derived using two binary series (i.e. contain taining only two values, 0.0 and 1.0). The best possible L φ = f. (18) case is if An is uniformly distributed (N/2 zero’s and N/2 one’s) and series B contains Nv (v<0.5) 1’s co- n The correct average solution φ, however, would, in inciding with the 1’s in A . n principle, be obtained from For the conditions of the experiment, this value is −1 ρmax =0.36664 (to speed up computations the domain L−1 φ = f, (19) was trimmed from the top and bottom and v was ad- justed accordingly to 0.1185). which almost always would be different from φ solved in Although both values are significantly lower than the (18). Indeed, depending on the case, we get an effect- theoretical maximum, it is clear that the maximum can- ive compliance either worse or better than that of the not be attained because the optimization algorithm has reference homogeneous material case, which replaces the to select neighboring cells so as to obtain a continuous design. The observed values are consistent enough and large 2 above the noise already existing in the optimal shapes ob- enough to indicate the desired behavior. tained from experiment 2. 10

heterogeneous microstructure by averaging (14) so that – since the particular function form is decisive with C = C . Although the double inversion is not possible respect to the choice of an optimal shape, the sen- analytically in the optimal shape problems – and, in fact, sitivities need to be modified so as to account for in most other problems of mechanics – computational me- a characteristic of spatial material randomness and chanics offers a viable route, such as followed in Sect. 3 a statistical goal function. above. Let us end with a summary of principal conclusions: Acknowledgements This material is based upon work sup- • The classical procedures of shape optimization for ported by the Altair Engineering, Inc., and the Canada Re- a structure of minimum compliance that has a pre- search Chairs program. scribed weight has been generalized to bodies with Altair OptiStruct is a registered trademark of Altair random microstructure. The process assumes that the Engineering, Inc. (http://www.altair.com) actual stochastic realization (not only stochastic properties) are known a priori during the optimization process. This does not correspond to a typical engineering process, but rather represents an “organic”-type opti- References mization, like the growth of veins in leaves on a tree. • In general, the introduction of stochastic properties reduces the quality of the results, but in some cases Allaire, G.; Kohn, R.V. 1993: Topology optimization and optimal shape design using homogenization. In: Bendsøe, M.P.; stochastic material properties may improve the com- Mota Soares, C.A. (eds.) Topology Design of Structures, pp. pliance of the optimal design because the optimization 207–218 is performed on a given random distribution, so that the design process has an opportunity to choose ‘stiffer’ Altair OptiStruct 1998–2001: User’s Manual, Altair En- cells and discard those with weaker material. gineering, Inc. • The present study was based on a very simple assumption of randomness and scaling as expressed by (12). Bendsøe, M.P.; Kikuchi, N. 1988: Generating optimal topolo- This equation, however, cannot represent smoothing gies in structural design using homogenization method. Com- of a heterogeneous material microstructure with vari- put. Methods Appl. Mech. Eng. 71, 197–224 ability on a smaller scale – say, ten times smaller than Hegemier, G.A.; Prager, W. 1969: On Michell trusses. Int. J. a single finite element employed here. If that were Mech. Sci. 11, 209–215 attempted, we would have to proceed according to the mechanics of random media, which dictates that Jeulin, D.; Ostoja-Starzewski, M. (eds.) 2001: Mechanics of the stiffness of a heterogeneous material should be Random and Multiscale Microstructures, CISM Courses and derived from one of three boundary value problems: Lectures. Wien, New York: Springer either under uniform kinematic or uniform traction boundary conditions, or an orthogonal combination of Jiang, M.; Ostoja-Starzewski, M.; Jasiuk, I. 2001: Scale- these (Ostoja-Starzewski 2001). An optimization study dependent bounds on effective elastoplastic response of ran- based on these concepts is presently in progress. dom composites. J. Mech. Phys. Solids 49, 655–673 • The computational time involved in the generation Michell, A.G.M. 1904: The limits of economy in frame struc- of one elasto-plastic truss by a hyperbolic system tures. Philos. Mag. 8, 589–597 (Hegemier and Prager 1969; Ostoja-Starzewski 2001) is a fraction of a second on a modern personal computer. Ostoja-Starzewski, M. 2001: Michell trusses in the presence While the present study pertains to an elliptic prob- of microscale material randomness: limitation of optimality. lem, the nature of the numerical algorithms employed Proc. R. Soc. Lond. A 457, 1787–1797 results in any one of the trusses shown in Fig. 9 using at least 4 hours of CPU time on an identical computer. Ostoja-Starzewski, M. 2001: Mechanics of random materi- This obviously sets limits on the extent of analysis of als: stochastics, scale effects, and computation, in mechanics of random and multiscale microstructures. In: Jeulin, D.; any stochastic problem by a Monte Carlo-type gener- a ation of many samples from an ensemble B of random Ostoja-Starzewski, M. (eds.) CE 2001 bodies {B(ω); ω ∈ Ω}. • Rozvany, G.I.N.; Bendsøe, M.; Kirsch, U. 1995: Layout opti- Our study is a preparation for the robust design itself. mization of structures. Appl. Mech. Rev. 41, 48–119 In particular, we propose and plan the following course b of action: Zhou, M.; Shyy CE , Y.K.; Thomas, H.L. 1999: Checkerboard and minimum member size control in topology optimization. – in the course of an optimization procedure, the In: Proc. 3rd World Congress of Structural and Multidisci- choice of a goal function in terms of local variabilities plinary Optimization (held in Buffalo, New York, USA); See will represent an intermediate step; also Struct. Mult. Optim. 21, 152–158

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