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Description
Structural optimization involves strategically arranging certain materials within a structure to enhance its properties with respect to some optimality criteria. This optimization typically entails minimizing or maximizing an integral functional, which depends on the rearrangement of materials within the domain, and the solution of a partial differential equation that models the underlying physics.
We present a novel optimality criteria method for optimal design problems in the setting of linearized elasticity. More precisely, we derive a method for maximizing the first eigenvalue in a mixture of two isotropic elastic materials in a bounded domain, with a volume constraint for the most rigid material. The algorithm is based on necessary conditions of optimality for problem which was obtained by relaxing the original one via the homogenization method. Since the method relies on explicit expressions for the lower Hashin–Shtrikman bound on the complementary energy and information on the microstructure that saturates the bound, implementing the algorithm in three space dimensions was not feasible until a recent explicit calculation was done by Burazin, Crnjac and Vrdoljak (2024). We demonstrate the method on a number of examples of two- and three-dimensional eigenfrequency maximization problems.
