A Hybrid Shape Optimization Method Based on Implicit Differentiation and Node Removal Techniques

Abstract

Abstract This paper presents a hybrid shape optimal design methodology using an implicit differentiation approach for sensitivity analysis and a node removal technique for shape alteration. The approach presented attempts to overcome the weaknesses inherent in each individual technique. The basic idea is to combine the sensitivity analysis, which forms the analytical basis for the algorithm, and a node removal technique, which grossly modifies the shape without the need for a remeshing after each iteration. The sensitivity analysis is based on the finite element equilibrium equation and the implicit differentiation technique. It examines the effect positional changes of the boundary nodes have on the stress values. Using the sensitivity results, a sequential linear programming algorithm is utilized to determine optimum positions of the boundary nodes. These optimization results are provided as inputs to an algorithm that decides which boundary nodes should be removed. By removing boundary nodes, the boundary elements change to either a triangular or a non-existent type. This shape modification procedure starts from the boundary elements and moves toward the internal elements. Only two iterations of finite element analysis are required to modify one boundary layer. To maintain the structural integrity and the connectivity of the elements in the model, a connectivity check is performed after each iteration. Three design examples are given to illustrate the accuracy and the steps involved in the proposed optimal design methodology.