PDE-driven level sets, shape sensitivity and curvature flow for structural topology optimization

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. (3) Perimeter regularization is incorporated in the method to make the optimization problem well-posed. It also produces an effect of the geometric heat equation, regularizing and smoothing the geometric boundaries as an anisotropic filter. (4)We further describe numerical techniques for efficient and robust implementation of the method, by embedding a rectilinear grid in a fixed finite element mesh defined on a reference design domain. This would separate the issues of accuracy in numerical calculations of the physical equation and in the level-set model propagation. 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.