Shape optimization using the boundary element method and a SAND interior point algorithm for constrained optimization

The shape optimization problem consists in looking for the geometry that minimizes an objective function, like mass or compliance, subject to mechanical constraints. The boundary element method (BEM) is used for the structural analysis. For linear elasticity problems, it needs only a mesh on the boundary of the structure. This characteristic makes the BEM a natural method for shape optimization, since only the boundary is needed to define the optimization problem and to carry out the structural analysis. The simultaneous analysis and design formulation (SAND) for structural optimization considers the state variables as unknowns of the optimization problem and includes the equilibrium equations as equality constraints. In this way, it is not necessary to solve the equilibrium equation per iteration; the equilibrium is only obtained at the end of the optimization process. In this paper, the shape optimization problem is dealt with using the BEM formulation to define a SAND optimization problem that is solved using an interior point algorithm. Numerical results for two-dimensional linear elasticity problems are presented to illustrate the efficacy of the proposed technique.