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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A b s t r a c t A formulation for shape optimization of elastic structures subject to multiple load cases is presented. The problem is solved using a homogenization method. When compared to the single load solution strategy, it is shown that the more general formulation can produce more stable designs while it introduces little additional complexity. I n t r o d u c t i o nIn a paper by BendsCe and Kikuchi (1988) and in several papers that followed by the same authors and others (Bendsce 1989; Suzuki and Kikuchi 1991;Dfaz and Belding 1990; O1-hoff et al. 1991), a new approach to shape optimization in structural design, based on a homogenization method, was introduced. The strength of the new approach is based primarily on the versatility with which arbitrary topologies can be represented. In a typical problem the optimum shape is found without the specification of more geometric details than the amount of material used and the location of the supports and points of application of the loads, as illustrated in Fig. 1. The method does not require that the topology (e.g. the connectivity or the number of holes present in the structure) be prescribed a priori, a significant advantage over traditional methods that use parametric boundary representations of the shape.

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The transmission characteristics of a folded surface decorated with a periodic arrangement of split-ring resonators is investigated. The folding pattern has one displacement degree of freedom, allowing motion that can be used to adjust the separation between the rings. When the geometry of the folded surface is varied by mechanical means, the change in spacing between the rings causes a shift in resonance frequency, making the surface mechanically tunable.