Numerical Methods For Fourth-Order Elliptic Eigenvalue Shape Optimization Problems

Optimal shape design problems have important backgrounds for applications in computer science and engineering.In this paper,we consider to solve second-order elliptic eigenvalue optimization problems by numerical methods.Two kinds of model problems,area constrained and area unconstrained formulations,are studied.We present two kinds of expressions of Eulerian derivatives,the volume type and boundary type.We show numerical examples based on finite element discretizations of eigenvalue problems and gradient descent flows.We consider numerical methods for solving two kinds of fourth-order elliptic eigenvalue problems.We use the nonconforming finite element method to approximate the fourth order problems.Then,we propose algorithms with gradient descent flows and present numerical examples for fourth-order clamped plate eigenvalue and fourth-order buckling plate eigenvalue problems.