Two-dimensional Shape Optimal Design by the Boundary Element Method

In this paper, boundary element method is used to optimize the shape of two-dimensional elastic bodies, and some problems in the application are satisfactorily treated. In addition, features of calculation of design sensitivity vectors in optimization and singularity of integration are analysed in detail. A practical example and photoelastic experiment verify that using boundary element method to analyse the structure is suitable in the shape optimazation. In the optimal shape problem we assume the surface tractions on all boundaries of domain to be known. The equations obtained by the BEM are a set of dependent equations. The displacement boundary condition must be given when the equations are solved by the displacement method. Then the first equations are modified as a set of contradictory equations. We solve the equations by the least-square method and obtain a more accurate solution of the equations. Sensitivity analysis is the most critical part in the optimization process. The fundamental solutions T and U of BEM are singular. This results in a higher order singularity in calculation of sensitivity analysis. These difficulties can be avoided considering the feature of shape optimization. The singular integrals including derivative of T could be deleted. The integrals related to derivative of U can be transformed into general integrals. Here we analyse the case of linear element discretized, the same result may be obtained for the quadratic element. An example is presented. A picture taken in the photoelastic experiment is given. It is verified that boundary of the final design is uniformly stressed. The purpose of this paper is to prove BEM to be an efficient and suitable numerical method of analysis in shape optimal problems.