Minimizing Eigenvalues for Inhomogeneous Rods and Plates

Optimizing eigenvalues of biharmonic equations appears in the frequency control based on density distribution of composite rods and thin plates with clamped or simply supported boundary conditions. In this paper, we use a rearrangement algorithm to find the optimal density distribution which minimizes a specific eigenvalue. We answer the open question regarding optimal density configurations to minimize k-th eigenvalue for clamped rods and analytically show that the optimal configurations are distinct for clamped rods and simply supported rods. Many numerical simulations in both one and two dimensions demonstrate the robustness and efficiency of the proposed approach.