Shape optimisation by the boundary element method: a comparison between mathematical programming and normal movement approaches

Abstract

The aim of this work is to find the best boundary shape of a structural component under certain loading, to have minimum weight, or uniformly distributed equivalent stresses. Two shape optimisation algorithms are developed. One of them is a mathematical programming method, and considers nodal coordinates on the design boundary directly as the design variables, while the other one is rather an optimality criterion approach, based on normal movement of the design boundary. Solving the optimisation problem of stress concentration for a perforated plate which has an analytical solution, shows that the presented mathematical programming method results in almost the same as the analytical solution. Nevertheless, increasing the number of design variables to find more smooth shapes in mathematical methods can cause severe programming problems. Comparing the result of this method with that of the optimality criterion indicates that the latter is much easier to apply without any limit on the number of design variables. To calculate stresses at every iteration, the boundary element method (BEM) is used. Therefore both algorithms benefit from a simple mesh generation based on equal length elements, which provides the possibility of solving multiply-connected domains or geometrically complicated mechanical components. Both methods are used to find the optimum shape of a circular plate under radial loading with four design holes. Finally, the problem of the best topology and shape of circular disks is solved by the optimality criterion approach. Also it is proposed that ‘a fully stressed design algorithm which starts from the best topology design, has the best shape for weight optimisation.