Model order reduction using Karhunen-Loe`ve expansion in shape optimization analysis of hyperelastic body

This paper presents a method for reducing the computational time required to solve the shape optimization problem of a hyperelastic body. In a previous report, the authors presented a method for applying a model order reduction technique based on the Karhunen-Loeve expansion (KLE) in a shape optimization problem of a linear elastic body. In the investigated case, we applied this method solely to solve a linear state determination problem. In this paper, we show that this idea is applicable not only to the adjoint problem and to the problems seeking domain variations through the H1 gradient method, but also to a nonlinear state determination problem. From a theoretical perspective, we assume that the solutions to these three problems for the domain obtained using a conventional shape optimization method at a prescribed number of iterations are random variables, and we apply the idea of KLE to such variables using the solutions obtained in the steps prior to the prescribed number of iterations through a conventional method as a sampling dataset. The orthonormal bases of KLE are defined as the eigenfunctions of the eigenvalue problems obtained as the optimality conditions of the variance maximization problems for the random variables. In the case of a nonlinear state determination problem, we use the incremental solutions instead of the solutions themselves. When using the finite element method to solve the three problems, these eigenfunctions become eigenvectors. Using the eigenvectors as the orthonormal bases in KLE, we define transformations from the coordinates of KLE to the physical coordinates and construct small problems for the variables of KLE. Our proposal is to solve the small problems instead of the original ones. The feasibility of the proposed method is illustrated by testing the numerical schemes using an end mean compliance minimization problem of a threedimensional hyper elastic body.