Shape design of elastostatic structures based on local perturbation analysis

Abstract

This paper describes a non-gradient formulation for solving shape optimal design problems involving structures in plane stress or having an axially symmetric geometry. The minimization of the maximum von Mises stress value at a traction free boundary poses a non-linear optimization problem in which the design variables do not appear explicitly in the formulation. The most commonly used approach is to apply a standard non-linear programming technique. There exists in this field no universally accepted solution method. The major difficulty of shape optimization in connection with FEM is to perform an accurate and efficient sensitivity analysis. The perturbation analysis introduced here takes advantage of the character of the problem. It is based on methods from the theory of notches. The results are applied to an FE-model of the structural component. The iterative method with such a direction of search works efficiently even for a large number of design variables as shown by Schnack (1977b, 1978, 1979, 1980, 1983 and 1985). Using a dynamic programming formulation (see also Schnack and Sporl 1986), the existence of a solution for the shape optimal problem will be discussed. Examples of applications to structural components from mechanical engineering are presented to demonstrate the power of this approach.