PDE-Driven Level Sets and Shape Sensitivity for Structural Topology Optimization

Abstract

This paper addresses the problem of structural shape and topology optimization. A level set method is adopted as an alternative approach to the popular homogenization based methods. The paper focuses on four areas of discussion: (1) The level-set model of the structure’s shape is characterized as a region and global representation; the shape boundary is embedded in a higher-dimensional scalar function as its “iso-surface.” Changes of the shape and topology are governed by a partial differential equation (PDE). (2) The velocity vector of the Hamilton-Jacobi PDE is shown to be naturally related to the shape derivative from the classical shape variational analysis. Thus, the level set method provides a natural setting to combine the rigorous shape variations into the optimization process. Finally, the benefit and the advantages of the developed method are illustrated with several 2D examples that have been extensively used in the recent literature of topology optimization, especially in the homogenization based methods.