Shape Optimization using the Boundary Element Method and a SAND Interior Point Algorithm for Constrained Optimization

Simultaneous Analysis and Design technique (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, equilibrium is only obtained at the end of the optimization process. Therefore, it is not necessary to solve the equilibrium equation per iteration of the optimization process. In the literature, some advantages in the application of the SAND technique have been recognized [

1

].

The shape optimization problem consists of looking for the geometry that minimizes an objective function, like mass or compliance, subject to mechanical constraints. In the present work a discrete model of the geometry that employs spline interpolation curves and defines the shape as a function of nodal coordinates is used.

The Boundary Element Method (BEM) is a technique for structural analysis based on an integral form of the equilibrium equations [

2

]. For linear elasticity problems, the BEM needs only a mesh on the boundary of the structure and its corresponding state variables, displacements and stresses, therein defined. 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.

In this paper the BEM formulation is used to define the shape optimization problem, the evaluation of the derivatives of the equilibrium equations is shown and the shape optimization problem is dealt with using an interior point algorithm based on the SAND technique.

Numerical results for two-dimensional linear elasticity problems are presented to illustrate the effi- ciency of the proposed technique.