Optimal shape design for minimum Lagrangian

One of the most efficient tools for solving optimal shape design problems offers the minimization of Lagrangian with respect to selected internal parameters. It starts with the formulation of the Lagrangian variational principle with an additional assumption that the volume of domain (in 2D the area) describing the structure in the undeformed state remains constant during the deformation process. It can be shown that the boundary density of the deformation energy at each boundary point in the optimal state is constant. A new position of the boundary is calculated from simple formula using the method of steepest descent. As the difference in energy density at boundary points may be very large, new positions of the boundary points are alternatively determined in a special way. Since the problem is strongly non-linear, iterative procedure has to be proposed in an efficient way. The proposed approach incorporating boundary element method appears to be very promising for solving shape optimization of structures.