A hierarchical design concept for shape optimization based on the interaction of CAGD and FEM

Abstract

The numerical treatment of shape optimization problems requires sophisticated software tools such as Computer Aided Design (CAD), the Finite Element Method (FEM) and a suitable Mathematical Programming (MP) algorithm. Efficiency of the overall procedure is guaranteed if these tools interact optimally. The theoretical and numerical effort for sensitivity analysis reflect the complexity of this engineering problem. In this paper we outline a general modelling concept for shape optimization problems. Hierarchical design models within Computer Aided Geometrical Design (CAGD) and the interaction of geometry and FEM lead to an efficient overall optimization procedure. Our concept has been derived, implemented and tested for shell structures but it is seen to be generally applicable. After a short introduction containing the state of the art we give an overview of the numerical tools used and outline the interaction of CAGD and FEM within the overall optimization procedure. The paper is mainly devoted to the hierarchical design space based on a hierarchical geometrical modelling. The major part of computational effort is consumed by sensitivity analysis related to the number of design variables. Therefore, this number should be limited and only few powerful design variables corresponding to the special interests of the considered problem should be defined. This procedure may lead to a considerable limitation of the design space. Based on a hierarchy in the geometrical model different types of design variables are introduced: design variables with global, regional and local influence. The new method is based on successive activation of these types of design variables. This procedure leads to a considerable reduction of computational time for the sensitivity analysis without loss of geometrical flexibility. A new method of geometrical refinement and a successive adaptively driven expansion and reduction of the design space is described. It is based on the degree elevation or degree reduction of parametric curves and surfaces, respectively. A numerical example illustrates the new method and the efficiency of the overall optimization procedure.