Shape Optimization by GJ-Integral: Localization Method for Composite Material

Abstract

GJ-integral $J_{\omega }(u,\mu )=P_{\omega }(u,\mu )+R_{\omega }(u,\mu )$ is the tool for shape sensitivity analysis of singular points in boundary value problem for partial differential equations, that is, GJ-integral takes value 0 if the solution u is regular inside $\omega$ for any vector field $\mu$. The variation of energies with respect to the movement of singular points are expressed by $R_{\omega }(u,\mu )$ having finite value even if u has not regularity inside $\omega$. We can solve shape optimization problems with respect to the set of singular points using GJ-integral and $H^1$ gradient method (Azegami’s method). Here the singular points are the points on the boundary and on the interface of different materials. This paper provides a brief introduction to the history and basic theorems on GJ-integral. We also give extended results for composite material and its application to the shape optimization problem with some numerical examples by finite element analysis.