On solving constrained shape optimization problems for finding the optimum shape of a bar cross-section

The problems of optimization of cylindrical bar cross-sections are formulated in variational forms. The functional considered characterizes torsional and bending rigidities, and the area of cross-section of the bar. The shape of the boundary of the cross-section is taken as a design variable. The problem is first expressed as an optimal control problem. Then by using an embedding method, the class of admissible shapes is replaced by a class of positive Borel measures. The optimization problem in measure space is then approximated by a linear programming problem. The optimal measure representing optimal shape is approximated by the solution of this finite dimensional linear programming problem. Numerical examples are also given.