Dual methods for discrete structural optimization problems

The purpose of this paper is to present a mathematical programming method developed to solve structural optimization problems involving discrete variables. We work in the following context: the structural responses are computed by the nite elements method and convex and separable approximation schemes are used to generate a sequence of explicit approximate subproblems. Each of them is solved in the dual space with a subgradient-based algorithm (or with a variant of it) specially developed to maximize the not everywhere di erentiable dual function. To show that the application eld is large, the presented applications are issued from di erent domains of structural design, such as sizing of thin-walled structures, geometrical con guration of trusses, topology optimization of membrane or 3-D structures and welding points numbering in car bodies. The main drawback of using the dual approach is that the obtained solution is generally not the global optimum. This is linked to the presence of a duality gap, due to the non-convexity of the primal discrete subproblems. Fortunately, this gap can be quanti ed: a maximum bound on its value can be computed. Moreover, it turns out that the duality gap is decreasing for higher number of variables; the maximum bound on the duality gap is generally negligible in the treated applications. The developed algorithms are very e cient for 2-D and 3-D topology optimization, where applications involving thousands of binary design variables are solved in a very short time