Checkerboard patterns in layout optimization

Abstract

A b s t r a c tEffective properties of arrangements of strong and weak materiais in a checkerboard fashion are computed. Kinematic constraints are imposed so that the displacements are consistent with typical finite element approximations. It is shown that when four-node quatrilaterai elements are involved, these constraints result in a numerically induced, artificially high stiffness. This can account for the formation of checkerboard patterns in continuous layout optimization problems of compliance minimization. I n t r o d u c t i o nThe popularity of layout optimization methods in structural design has increased rapidly since the publication of the paper by Bendsee and Kikuchi (1988) triggered a renewed interest in the topic. Since then several different versions of the problem have been developed and a fair amount of debate has taken place in the literature and at specialized meetings regarding the advantages and disadvantages of the different approaches. However, while there are indeed fundamental differences that distinguish the more popular methods in use today, experiments have shown that most methods have in common one undesirable feature: they may result in solutions where material is distributed in a checkerboard pattern. In layout problems where the amount of material present at a location x is measured by the scalar density function p(x), a checkerboard pattern is defined as a periodic pattern of high and low values of p(z) arranged in the fashion of a checkerboard, as illustrated in Fig. 1. This behaviour is undesirable as it is the result of a numerical instability and does not correspond to an optimal distribution of material. In this paper we discuss the reasons for the formation of such patterns.Four element checkerboard patch with average density 9=1/2 Fig. 1. Solution displaying checkerboard patternsThe literature offers only little discussion on the formation of checkerboards in layout optimization problems (Belldsee et al. Jog et al. 1992;Bends~e 1994;Jog and Haber 1994), although similar patterns affecting the finite element solution of mixed variational problems have been studied extensively (see e.g. Brezzi and Fortin 1991). In such problems the fo> mation of checkerboards is related to the violation of the so-called Babuska-Brezzi or LBB condition. This similarity was used by Jog et al. (1992) and :log and Haber (1994) to attribute the patterns in the layout problem to an LBB type instability. In order to pursue this argument, the authors interpret the layout optimization problem as a mixed variational problem in the density variable p and the displacement field u. To avoid the formation of checkerboard patterns, the authors suggest that different functions be used to interpolate p and u, in a fashion similar to that suggested by the LBB condition in other mixed problems. Unfortunately, the conditions under which the standard Babuska-Brezzi arguments are applied to mixed variational problems are not met by the layout optimization problem (Bends,ae 1994). If indeed a Babuska-Brezzi typ...

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