We present a numerical analysis method and results using the traction method for optimization problems of domains in which linear elastic problems are defined. In this paper we consider the application of the traction method which was proposed as a solution to domain optimization problems in elliptic boundary value problems. The minimization problems of the mean compliance were treated. Using the Lagrange multiplier method, we obtain the shape gradient functions for these domain optimization problems from the optimality criteria. In this drawing process we consider the variation of the surface force acting on the boundary and the variation of the stiffness function and the body force distributed in the domain. We show successful solutions of the problem of an infinite plate with a hole and of the problems of a rectangular plate clamped at both ends.