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. 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.. 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.