Eigenfrequency optimization in linearized elasticity: A novel numerical method for two-dimensional and three-dimensional problems

We present a numerical method for eigenfrequency optimization in two-dimensional and three-dimensional linearized elasticity, based on necessary optimality conditions where the optimal stiffness tensor saturates the upper Hashin–Shtrikman bound. The approach, grounded in the homogenization method, exploits recently derived explicit Hashin–Shtrikman bounds and corresponding optimal microstructures in three space dimensions for mixtures of two non-void elastic phases, enabling an efficient computational realization of the optimality criteria framework. It is capable of identifying global optimizers in most cases. Benchmark examples, including cantilever and bridge problems, demonstrate that true composite designs are obtained as optimal. A penalization procedure is then applied to recover mostly classical designs, with only a slight decrease in the objective functional. Finally, the proposed method is compared with the solid isotropic material with penalization method and is shown to yield improved performance.