Layout Optimization : A Mathematical Programming Approach

This paper presents applications of specially tailored methods of the mathematical
programming approach for solving topology design problems formulated as an optimal
material distribution, providing a way to enlarge the scope of potential applications of
topology optimization. The first feature of the methodology is to resort to dual maximization to solve optimization problems with a huge number of design variables, using the concept of Sequential Convex Programming (SCP). Here a central theme is the choice of convex approximations suited to the problems. First order approximations are firstly considered and compared. However, to increase the performances of the procedures (for example, to reduce the number of steps to arrive to a stationary design), a new approach to build un-expensive second approximation schemes, especially relevant for large scale problems, have been developed and validated. To illustrate the efficiency of the mathematical programming approach, the final part of the paper deals with an efficient treatment of perimeter constraints. Such constraints are essential to regularize material distribution with non optimal microstructures, but perimeter constraints are very difficult to handle in a numerical procedure for practical designs. The paper presents a solution to this problem that has been implemented and validated on a wide range of
applications.