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 engineering structures implicitly assume the existence of a hypothetical perfectly homogeneous continuum – they do not recognize the presence of any microscale material randomness. By contrast, the present study investigates this aspect for the paradigm of a Michell truss with minimum compliance (maximum stiffness) that has a prescribed weight. The problem involves a stochastic generalization of the topology optimization method implemented in the commercial Altair’s OptiStruct computer code. In particular, this generalization allows for the dependence of each finite element’s stiffness matrix on the actual microstructure contained in the given element’s domain. Contrary to intuition, stochastic material properties may improve the compliance of optimal design. This is because the optimization is performed on a given random distribution, so that the design process has an opportunity to choose ‘stiffer’ cells and discard those with weaker material. The paper does not aim for a robust design process, but tries to answer a simpler intermediate question: how the random fluctuation of material properties influences a structure that has been designed using classical continuum-based optimization algorithms. Key words optimal structures, random microstructure
1 Background Problems of shape optimization of engineering structures implicitly assume the existence of a hypothetical perReceived: Published online: 2003 Springer-Verlag 2003 T. Liszka1, u , M. Ostoja-Starzewski2 1 Altair Engineering, Inc., 7800 Shoal Creek Blvd, Austin, TX, 78757, USA e-mail:
[email protected] 2 Department of Mechanical Engineering, McGill University, Montreal, Que. H3A 2K6, Canada
fectly homogeneous continuum. Consequently, the process of optimization does not recognize the presence of any microscale material randomness, although the latter becomes progressively more relevant as the mesh is refined. A study of this effect conducted recently on the paradigm of the Michell truss (Michell 1904) made of elastic–plastic members showed that the random microstructure prevents the attainment of the classically predicted optimal shape using the homogeneous continuum (Ostoja-Starzewski 2001). The aforementioned investigation was carried out under the assumption of fixed load on the structure and the search was conducted for a structure of optimal (i.e., minimal) weight. The study considered the effect on the ability to attain the classically expected minimum of a spatially disordered (non-periodic) material microstructure of which the Michell truss-like continuum would be manufactured: the denser the truss (i.e., the finer its mesh spacing), the more significant the effect of microstructural fluctuations on the plastic limit σ0 . This led to the question of the sensibility of the ad infinitum refinement of the Michell truss, and hence to the open issue of the plausibility of its attainment. In general, the planar material of which the truss is manufactured is a random medium: a set B = {B(ω); ω ∈ Ω}, where each B(ω) is a deterministic, albeit spatially inhomogeneous, body with Ω being a sample space. More specifically, each B(ω) is described by a scalarvalued planar field of plastic limit k, so that we have the random field k = k(ω, x); ω ∈ Ω, x ∈ R2 specifying B. The actual setup of the field k depends on a nondimensional scale parameter δ = L/d > 1, relating the mesh spacing L to the size of the microheterogeneity d (e.g., crystal diameter). Micromechanics of random heterogeneous media (Jeulin and Ostoja-Starzewski 2001), while only recently making inroads into plasticity (Jiang et al. 2001), suggests that the following two aspects should be included in any physically acceptable model: (i) the scatter in k (as described by its standard deviation σk ) should grow as δ decreases, and (ii) the statistical average k may be taken to be constant as δ decreases: k = const ,
σk (δ) ∼
1 . δ
(1)
2 As is typically done in mechanics of random media, it is convenient to split the field k into its constant mean and the randomly fluctuating part. We can then write k(ω) = k + k (ω) ,
k = 0 ,
where
aC (u, v) =
ε(u)C(x)ε(v) dx
(4)
V
(2)
where k is the zero-mean noise in k. In setting up the model, we also assumed: (i) the truss spacing L of interest to us is greater than the grain size d, so that k may be treated as a field of independent random variables when entering the finite difference formulation generalized to inhomogeneous media; and (ii) the underlying material microstructure is space-homogeneous and ergodic. Since the reference problem of this Michell truss is governed by a quasi-linear hyperbolic system (Hegemier and Prager 1969), ours was described by a stochastic generalization thereof (Ostoja-Starzewski 2001). The term ‘stochastic’ refers to an ensemble of deterministic problems that exist on spatially inhomogeneous realizations of the random field k. Due to the randomness of k on scales δ < ∞, the net of characteristics display a statistical scatter that increases with decreasing δ, which translates into a trend to use more structural material to carry the prescribed loading. This, of course, is an opposing tendency to the convergence to the Michell truss with mesh refinement, and, beyond a certain mesh refinement, the solution to the governing hyperbolic system breaks down. To sum up, there is a limitation to this ideal optimality. Therefore, the above method cannot be used for investigation of systems with too fine a mesh for a given microstructural randomness, nor can it be employed for elastic structures. These restrictions suggested another approach – in fact, much more in line with the conventional methodology of shape optimization of engineering structures (Rozvany et al. 1995; Bendsøe and Kikuchi 1988) – which seeks the shape of an elastic structure with minimum compliance (maximum stiffness) that possesses a prescribed weight. Note, however, that this is different from the problem considered in (Ostoja-Starzewski 2001), which dealt with a plastic structure of minimal weight under a fixed load. The present paper reports a stochastic generalization of the topology optimization method implemented in a commercial Altair’s OptiStruct computer code, which takes into account a material microstructure of random chessboard type.
2 Problem formulation The minimum compliance problem for a planar (respectively, 3-D) body of volume V in R2 (R3 ) subjected to body forces f and tractions t takes the form: min L(u) Cijkl ∈ Uad subject to aC (u, v) = L(v), all v ∈ U, design constraints, (3)
is the bilinear form of the energy, and L(u) = fv dx+ tv dx V
(5)
∂Bt
is the linear form of the load. That is, we seek the optimal choice of the stiffness tensor C(≡ Cijkl ) in some given set of admissible tensors Uad . C are generally fields over R2 , so that Uad ∈ (L∞ (V ))6 , corresponding to the six independent elements of the in-plane stiffness tensor. In (3), by ‘design constraints’, we mean constraints on stresses, strains, displacements, etc., while sizing constraints, volume constraints, etc., are accounted for in the choice of Uad . Finally, U is the space of kinematically admissible displacement fields. In the case of optimal shape design, elements C of Uad take the form Cijkl (x) = χ(x)C ijkl ,
(6)
where C ijkl is the constant stiffness tensor for the material employed for the construction of the mechanical element, and χ(x) is the indicator function. In the case of sizing problems, like the design of sheets of variable thickness, the admissible Cijkl ’s have the form Cijkl (x) = h(x)C ijkl ,
(7)
where again C ijkl is a constant tensor, and h(x) ∈ L∞ (V ) is the sizing function. The discretized formulation of the topology optimization problem (3) can be stated as follows: min f (ρ)
s.t. V = ρj νi ≤ V ∗ , η ≤ ρi ≤ 1, i = 1, . . . , N .
(8)
where f represents the objective function, ρi and νi are element densities and volumes, respectively, V ∗ is the target volume, N is the total number of elements, and 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 cases. Note that the problem in (8) is a relaxation formulation 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 reduce the efficiency of intermediate density elements. Here the penalization is achieved by the following power-law formulation: K∗i (ρ) = ρpi Ki ,
(9)
where K∗i and Ki represent the penalized and the real 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, 0 ≤ ρi ≤ 1,
j = 1, . . . , m ,
(10)
i = 1, . . . , N ,
where gj and gj∗ represent the j-th constraint and its upper bound, respectively, and m is the total number of 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 ,
Fig. 1 One-dimensional tension problem on a square domain
(11)
where dist(i, k) denotes the distance between adjacent elements i and k, and ρmin , say 0.1, 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 , with n bigger than 2, additional linear constraints and makes the direct solution of this formulation computationally prohibitive. Our code Altair OptiStruct implements a different algorithm (Zhou et al. 1999) that is very efficient for these types of constraints. For a survey of historical developments and a summary of the theory and techniques of topology optimization, we refer the reader to (Rozvany et al. 1995), while for a background on penalization formulation to (Allaire and Kohn 1993).
3 Numerical results 3.1 Preliminary tests Before proceeding with the solution of the Michell truss, we start with a very simple example of a unidirectional stress state in a square domain of size 1 × 1. The boundary conditions are ux = uy = 0 at x = 0, and ux = 0.1, uy = 0 at x = 1 (see Fig. 1). We are trying to find the optimum distribution of material with total volume V ∗ = 0.1. Because the problem is ill-posed, the results of the simulation exhibit poor convergence or numerical artifacts (e.g. splits at each member near the edges of the boundary). However, this (as expected) provides an optimization algorithm with significant “freedom” that results in a visible tendency of the “optimal” structure toward harder cells in the design space. The solution in the Fig. 2 was
Fig. 2 Optimal solution for the uniform unidirectional problem (dark color represents the converged optimal shape (density = 1.0), and void areas are shown as white; the few remaining cells with intermediate densities are shown with different colors on a gray scale)
generated for an isotropic material, but any distribution of horizontal trusses will satisfy the optimization problem. Two predefined (non-random) distributions of material density were tested: each consisted of regular bands of alternating materials parallel to the axis. For the horizontal distribution (Fig. 3a), the optimization process placed the trusses inside the stiffer material (gray and white bands represent hard and soft material, while black represents the optimal shape). For the vertical distribution (Fig. 3b – material types not shown in the figure) the trusses were generated with varying thickness, and smaller thicknesses coincided with bands of stiffer material. Although both cases are clearly contrived, the solution in Fig. 3a proves that it is possible to obtain a better optimal shape (lower compliance) in the presence of nonuniform material properties. Finally, Fig. 4 shows the optimal shape in the presence of very strong noise (s = 0.9). By comparing the material distribution (Fig. 4b), representing material strength (black-gray-white scale, black being the soft material) with the optimal shape shown in Fig. 4a, one can verify that the optimization process really avoids soft cells in the 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 loading condition at A (see Fig. 5). First, for the sake of reference, in Fig. 6a we display the truss found by carrying out an optimization on a 27 × 27 mesh of squareshaped finite elements, all with identical properties. Next, we considered the optimal structure problem under the same support and loading conditions, but defined on a 27 × 27 mesh of a random chessboard with single elements being isotropic and linearly elastic:
A F
Fig. 5 Michell truss problem: design space and boundary conditions
5 C ijkl = λδij δkl + µ(δjk δil + δjl δik ) .
(12)
The material randomness is introduced via a statement analogous to (4): C(ω) = C + C (ω) C (ω) = 0
(13)
As an example, for the noise scale factor s = 0.3, the Young modulus was varied from 0.7 to 1.3. The Poisson’s ratio was ν = 0.3. All the elements are generated according to the same process, independent of their spatial location, so that we effectively have a binomial random field on the 27 × 27 mesh. All presented examples used the same random field r with different scale factors s.
and generating C (ω) (≡ C ijkl (ω)) by multiplying each element’s mean part C (≡ C ijkl ) by a random number r sampled from a uniform distribution [−1, 1] multiplied by a constant scale factor s:
3.3 Experiment 1
C (ω) = rs C .
In this experiment we carried out a parametric study of the optimization problem by considering three cases
(14)
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 each case we first display the homogeneous medium case (a), and then generate consecutive realizations of the random field with increasing scale factor. In particular, figures of the sequence (b) through (j) correspond, respectively, to the multipliers s = 0.1, 0.15, 0.2, 0.25, 0.3, 0.6, 0.9. In all the figures, immediately evident is the breakdown of global symmetry of the truss structure due to the spatial non-uniformity of material (Figs. 6–8). All results were obtained using the same random distribution, mutiplied by a varying scalar factor (column 0, “Noise” in all tables). In addition Table 1 (sections (b) and (c)) contains results for different random distribu-
tions (Noise = 0.3), and also two different distributions using a mesh of 256 × 256 cells (Fig. 9). Tables 1–3 summarize the compliance results for the optimal structure. Although the numerical noise1 of the results is noticeable in the results, the overall trend to produce deteriorating (softer) structure is very clear. Interestingly in some cases the results are actually better, which indicates that the optimization process was able to select stiffer paths through the structure. 1
Topology optimization is realized as a highly nonlinear iterative process with a very shallow local minimum. The nonlinearity of the process is additionally compounded by the minimum member control algorithm.
7 To better estimate that the optimization process indeed “favors” stiffer paths through the structure, the following quantities were computed: • The average material stiffness in the optimal structure (normalized to the range 0.0–2.0, such that 1.0 corresponds to the structure having the same average stiffness as the entire design space, and 2.0 corresponds to the structure consisting entirely of the single material at the maximum level of the random distribution). The stiffness is summarized in column 3 (Avg_material ) of all tables. • Correlation between the material stiffness and the density of the design (column 4). The correlation is computed using the standard formula
1 N
ρAB =
N
An Bn −
1 N
N
An
1 N
N
Bn
n=1 n=1
2 N N 1 × A2n − N1 An N n=1 n=1 2 N N 1 1 2 Bn − N Bn N
n=1
n=1
n=1
,
(15)
where An and Bn represent two series. It is easy to note that for this particular experiment, ρAB is not really normalized (the maximum value is significantly less than 1.0). In particular it is easy to show 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 0.05 0.10 0.15 0.20 0.25 0.30 0.60 0.90
114.99 115.69 115.98 114.84 114.06 115.27 116.10 118.54 123.56
– 1.0340 1.0439 1.0529 1.0639 1.0459 1.0670 1.0917 1.1334
0.010795 0.022403 0.028971 0.034953 0.042046 0.036870 0.044367 0.060332 0.087990
– 114.87 114.90 115.08 115.42 115.93 116.61 125.56 156.01
– 115.90 116.39 115.69 115.16 116.61 117.83 121.21 126.72
(b) 0.30 0.30 0.30 0.30 0.30 0.30 0.30
115.59 117.18 117.09 118.84 114.94 116.86 116.81
1.0424 1.0705 1.0573 1.0725 1.0704 1.0521 1.0383
0.028058 0.047003 0.038294 0.048413 0.046924 0.034725 0.025495
117.10 117.06 118.17 118.44 118.01 118.67 117.38
116.08 118.68 117.56 120.22 116.29 117.29 117.35
(c) 0.00 0.30 0.30
105.98 107.52 108.70
– 1.0496 1.0511
– 0.032621 0.033657
– 107.69 107.63
– 107.88 109.12
Table 2 Configuration 2, fixed random distribution Noise
Compliance
Avg_material
Correlation
Compl1
Compl2
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.60 0.90
99.31 99.69 99.27 98.56 98.50 98.97 99.94 99.01 107.01
– 1.0266 1.0293 1.0318 1.0416 1.0370 1.0491 1.0834 1.1208
0.010576 0.017573 0.019326 0.020874 0.027470 0.024382 0.032404 0.054779 0.079803
– 99.29 99.39 99.63 100.00 100.52 101.20 109.65 137.44
– 99.80 99.46 98.76 98.78 99.38 100.46 100.06 107.03
Compl2
Table 3 Configuration 3, fixed random distribution Noise
Compliance
Avg_material
Correlation
Compl1
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.60 0.90
60.51 60.44 59.66 59.39 59.60 58.72 59.63 60.71 63.68
– 1.0354 1.0348 1.0469 1.0506 1.0673 1.0684 1.0818 1.1346
0.017407 0.023236 0.025773 0.030858 0.033297 0.044165 0.045002 0.053469 0.088292
– 60.45 60.45 60.53 60.68 60.91 61.22 65.34 78.79
– 60.57 59.87 59.73 60.01 59.38 60.30 61.28 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 Experiment 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: r = 0.09, 256 × 256 fine mesh (noise = 0.0, 0.30, 0.30 – two different distributions)
the cells of the maximum stiffness, then the value of this correlation would be ρmax = 1/ (1/v − 1) , (16) where v represents the volume constraint. The above formula was derived using two binary series (i.e. containing only two values, 0.0 and 1.0). The best possible case is if An is uniformly distributed (N/2 zero’s and N/2 one’s) and series Bn contains N v (v < 0.5) 1’s coinciding with the 1’s in An . For the conditions of the experiment, this value is ρmax = 0.36664 (to speed up computations the domain was trimmed from the top and bottom and v was adjusted accordingly to 0.1185). Although both values are significantly lower than the theoretical maximum, it is clear that the maximum cannot be attained because the optimization algorithm has to select neighboring cells so as to obtain a continuous design. The observed values are consistent enough and large enough to indicate the desired behavior.
L(ω)φ = f
ω∈Ω,
(17)
where L(ω) is the differential operator of elasto-statics having random (rather than deterministic) stiffness coefficients, φ is the field that is sought (such as the displacement u), f is the forcing, and Ω is the sample space of elementary outcomes ω (realizations of the random field). The conventional route of phenomenological deterministic continuum mechanics without regard for microstructural randomness is to average (17) directly so as to obtain L φ = f .
(18)
The correct average solution φ, however, would, in principle, be obtained from −1 −1 L φ = f , (19) which almost always would be different from φ solved in (18). Indeed, depending on the case, we get an effective compliance either worse or better than that of the reference homogeneous material case, which replaces the 2 above the noise already existing in the optimal shapes obtained from experiment 2.
10 heterogeneous microstructure by averaging (14) so that C = C . Although the double inversion is not possible analytically in the optimal shape problems – and, in fact, in most other problems of mechanics – computational mechanics offers a viable route, such as followed in Sect. 3 above. Let us end with a summary of principal conclusions: • The classical procedures of shape optimization for a structure of minimum compliance that has a prescribed weight has been generalized to bodies with random microstructure. The process assumes that the 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 optimization, 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 stochastic material properties may improve the compliance of the optimal design because the optimization is performed on a given random distribution, so that the design process has an opportunity to choose ‘stiffer’ cells and discard those with weaker material. • The present study was based on a very simple assumption of randomness and scaling as expressed by (12). This equation, however, cannot represent smoothing of a heterogeneous material microstructure with variability on a smaller scale – say, ten times smaller than a single finite element employed here. If that were attempted, we would have to proceed according to the mechanics of random media, which dictates that the stiffness of a heterogeneous material should be derived from one of three boundary value problems: either under uniform kinematic or uniform traction boundary conditions, or an orthogonal combination of these (Ostoja-Starzewski 2001). An optimization study based on these concepts is presently in progress. • The computational time involved in the generation of one elasto-plastic truss by a hyperbolic system (Hegemier and Prager 1969; Ostoja-Starzewski 2001) is a fraction of a second on a modern personal computer. While the present study pertains to an elliptic problem, the nature of the numerical algorithms employed results in any one of the trusses shown in Fig. 9 using at least 4 hours of CPU time on an identical computer. This obviously sets limits on the extent of analysis of any stochastic problem by a Monte Carlo-type generation of many samples from an ensemble B of random bodies {B(ω); ω ∈ Ω}. • Our study is a preparation for the robust design itself. In particular, we propose and plan the following course of action: – in the course of an optimization procedure, the choice of a goal function in terms of local variabilities will represent an intermediate step;
a
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b
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– since the particular function form is decisive with respect to the choice of an optimal shape, the sensitivities need to be modified so as to account for a characteristic of spatial material randomness and a statistical goal function. Acknowledgements This material is based upon work supported by the Altair Engineering, Inc., and the Canada Research Chairs program. Altair OptiStruct is a registered trademark of Altair Engineering, Inc. (http://www.altair.com)
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