# 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.

## Notes

### Acknowledgments

I would like to express my gratitude to Prof. Kimura and to Prof. O. Pironneau and Prof. F. Hecht for FreeFem++. I am deeply grateful to Prof. Azegami who provided the knowledge on shape optimization. This work was supported by JSPS KAKENHI Grant Number 16K05285.

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