An efficient MP algorithm for structural shape optimization problems

摘要

Integral methods ― such as the Finite Element Method (FEM) and the Boundary Element Method (BEM) ― are frequently used in structural optimization problems to solve systems of partial differential equations. Therefore, one must take into account the large computational requirements of these sophisticated techniques at the time of choosing a suitable Mathematical Programming (MP) algorithm for this kind of problems. Among the currently available MP algorithms, Sequential Linear Programming (SLP) seems to be one of the most adequate to structural optimization. Basically, SLP consist in constructing succesive linear approximations to the original non linear optimization problem within each step. However, the application of SLP may involve important malfunctions. Thus, the solution to the approximated linear problems can fail to exist, or may lead to a highly unfeasible point of the original non linear problem; also, large oscillations often occur near the optimum, precluding the algorithm to converge. In this paper, we present an improved SLP algorithm with line-search, specially designed for structural optimization problems. In each iteration, an approximated linear problem with aditional side constraints is solved by Linear Programming. The solution to this linear problem defines a search direction. Then, the objective function and the non linear constraints are quadratically approximated in the search direction, and a line-search is performed. The algorithm includes strategies to avoid stalling in the boundary of the feasible region, and to obtain alternate search directions in the case of incompatible linearized constraints. Techniques developed by the authors for efficient high-order shape sensitivity analysis are referenced.