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Filters in Topology Optimization INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2001; 50:2143–2158 Filters in topology optimization ; Blaise Bourdin∗ † Department of Mathematics; Technical University of Denmark; DK-2800 Lyngby; Denmark SUMMARY In this article, a modiÿed (‘ÿltered’) version of the minimum compliance topology optimization problem is studied. The direct dependence of the material properties on its pointwise density is replaced by a regularization of the density ÿeld by the mean of a convolution operator. In this setting it is possible to establish the existence of solutions. Moreover, convergence of an approximation by means of ÿnite ele- ments can be obtained. This is illustrated through some numerical experiments. The ‘ÿltering’ technique is also shown to cope with two important numerical problems in topology optimization, checkerboards and mesh dependent designs. Copyright ? 2001 John Wiley & Sons, Ltd. KEY WORDS: topology optimization; regularization method; convolution; ÿnite element approximation; existence of solutions 1. INTRODUCTION Topology optimization problems in mechanics, electro-magnetics and multi-physics settings are well known to be ill-posed in many typical problem settings, if one seeks, without restriction, an optimal distribution of void and material with prescribed volume (see, e.g. References [1–5]). This shows up through the possibility of building non-convergent min- imizing sequences for the considered problem, the limiting solution achieved as a micro perforated material. A consequence of this non-existence is that even if each discretization (by, e.g. ÿnite elements) of the unrestricted problem is well-posed, these solutions do not converge to a macroscopic design when the discretization parameter tends to 0, and smaller and smaller patterns are exhibited. This phenomenon is often also refered to in the literature as mesh dependency. The methods currently used to provide solutions to these non-existence issues and associated numerical side-e"ects can be sorted in three categories. In the homogenization method, one enlarges the set of admissible domains with the limits (in the sense of homogenization) of any distribution of void and material and characterize these limits as micro structures that depend on a number of parameters (see, for example, Reference [6] or [7]). In another approach, one adds extra constraints on the set of admissible designs in order to ensure existence, for ∗Correspondence to: Blaise Bourdin, Courant Institute of Mathematical Sciences, New York University, 251 Mercer St, New York, NY 10012, U.S.A. †E-mail: [email protected] Received 16 December 1999 Copyright ? 2001 John Wiley & Sons, Ltd. Revised 15 May 2000 2144 B. BOURDIN example by imposing an upper bound on the perimeter of the resulting design. The ‘void and material problem’ is then solved directly (see Reference [8] for the existence result and Reference [9] for a numerical implementation). The third approach to continuum topology design in a well-posed setting, can be referred to as a ‘ÿltered, penalized artiÿcial material’ method. This is the type of setting studied here, in a structural optimization context. In this method one deÿnes at each point of the domain a density of material ! that varies continuously between 1 and 0, with density 1 characterizing the material and 0 the void (no material). The elastic properties for intermediate densities are expressed in terms of the function !, for example, using a simple power-law interpolation which, through optimization, is known to lead to designs without intermediate densities. Even though such a method is often labeled as an artiÿcial power law method, one can actually give a physical interpretation in terms of sub-optimal, isotropic micro-structures (see Reference [10]). In order to ensure existence of solutions for this method, one is normally also in this case required to add some extra constraints on the admissible designs, i.e., on the admissible densities !. This can take the form of a constraint on the total variation of the density (that can be interpreted as the perimeter, in case the design is a bi-level function whose level sets are regular enough) or on the gradient of the density (see References [11; 12] for the former method, and Reference [13] for the latter). An alternative to these approaches has been proposed in References [14; 15] and consists of a ÿltering technique implemented in the optimization algorithm. This gives mesh-independent designs at a moderate computational cost. Filtering techniques is the subject of the present study, implemented in our case in a form where ÿltering is performed on the interpolation of the elastic properties, along the lines proposed in Reference [16]. Here we prove existence of solutions and show that one obtains, with a ÿxed ÿlter function, convergence of ÿnite element discretized forms of the problem. Numerical experiments are also reported, and the possibilities and complications of using alternatives such as design spaces of ÿltered density functions are also discussed. The method is applicable to a range of problems and application areas, but the scope of this paper is to focus on a simple two-dimensional minimum compliance optimization problem and to give a full mathematical justiÿcation of the ÿltering technique in this setting. The use of ÿlters in numerical methods in order to ensure regularity or existence of solutions to a problem has been used for many years in various domains of applications. The basic idea is to replace a (possibly) non-regular function by its regularization obtained by the convolution with a smooth function. An overview of the paper is as follows. In Section 2, we ÿx notations and deÿne the ÿltered version of the minimum compliance topology optimization problem, depending on a ÿlter function, F, inspired by Bruns and Tortorelli [16] and Sigmund [14; 15]. In Section 3, it is shown that if the ÿlter function F veriÿes some (weak) regularity assumptions, the ÿltered topology optimization problem admits at least one, possibly non-unique, solution. An approximation result in terms of ÿnite elements is then given in Section 4. The numerical implementation is detailed in Section 5.1 and illustrated by some numerical experiments in Section 5.2. Finally, some further extensions of the method are discussed in Section 6. 2. NOTATIONS AND STATEMENT OF THE PROBLEM In the following we will deÿne a ÿltered version of the minimum compliance topology design problem based on the power-law interpolation approach. The technique is to replace the Copyright ? 2001 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2001; 50:2143–2158 FILTERS IN TOPOLOGY OPTIMIZATION 2145 dependence of the elastic properties on the density of material by a dependence of a ÿltered version of the density function. This means that rapid variations in material properties are not allowed by the problem statement. Loosely speaking, this ensures existence of solutions. However, the proof involves some technicalities. In the standard framework of material distribution methods for topology design, we work, throughout this paper, in a ÿxed domain # R2, and the optimal design is generated referring to this ‘ground-structure’. Here, this domain⊂ is a Lipschitz bounded and open domain. We denote by (m; p; #) and (m; p; #) the usual norms and semi-norms over the Sobolev space m; p 2#·# |·| W (#; R ) and for the sake of simplicity abbreviate the notation for (m; 2; #) and (m; 2; #) #·# |·| with m and m (we refer to Reference [17] for actual deÿnitions and properties of the Sobolev#·# spaces).|·| What will be called a ÿlter of characteristic radius R¿0, is a function F verifying the following properties: 1; 2 F W ∞(R ) ∈ Supp F BR ⊂ F¿0 a:e: in BR F dx = 1 !BR where BR denotes the open ball of centre 0 and radius R. The ÿltering operation is achieved by mean of the convolution product of the ÿlter and the density (F !)(x) = F(x y)!(y) dy ∗ 2 − !R Loosely speaking, we replace at each point the density ÿeld by a weighted average of its values. One consequence of this operation is that the ÿltered density is then a smooth and di"erentiable function, among other properties [see e.g. Reference 18, IV 6, p. 66]. Remark that this deÿnition requires to extend the density ÿeld ! to the whole space R2. Some ways to address this technicality are detailed in Section 5. In the following we will thus work with a parametrization of design through a density function !, while the material properties of the equilibrium equation will depend on the ÿltered density F !. Thus, the ‘ÿltered’ version of the minimum compliance topology optimization problem (MC∗ F ) in structural optimization is deÿned as (for further details on the minimum compliance topology optimization problem, refer to Reference [20]) inf l(u) ! H F ∈ (MC ): subject to (u; !) EF (#) ∈ where l(u) = fu dx !# is the compliance of the design given by the density ÿeld ! subject to the body load f L2(#) ∈ and where EF denotes the set of densities and related displacements under the given load, Copyright ? 2001 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2001; 50:2143–2158 2146 B. BOURDIN the exponent F signifying that the displacement is computed from a ÿltered version of the density. In order to rigorously deÿne the set of admissible couples (u; !) EF , one needs some ∈ extra deÿnitions: by U; one denotes the set of kinematically admissible displacements, i.e. 1; 2 U := u W (#); u = 0 on @#D { ∈ } while H is the space of feasible designs: 2 1 2 H:= ! L∞(R ) L (R ); 0¡!6 ! 6 1 a:e: on #; ! dx 6 V ∈ ∩ 2 & !R ' for any given 0¡! and 0¡V . A displacement ÿeld u U is said to verify the equilibrium condition for the density ! H if one has ∈ ∈ (F !)pE"(u) : "(v) dx = fv dx v U (1) ∗ ∀ ∈ !# !# where we use the ÿltered density.
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