Level-set based shape optimization for plane elastic structures using radial basis functions and Hilbertian descent direction

Abstract

Structural optimization problems are often associated with the so-called shape functionals depending on a shape through its geometry and the state being a solution of given partial differential equation. In such a framework it is convenient to work with the gradient-like method based on a concept of a shape derivative and level set method. The key idea of level set method is to represent the structural boundary with zero level set of given function (level set function—LSF). Now, changing the shape of a structure under optimization is equivalent to transport the LSF in such a direction that ensures decreasing the value of the objective functional. To this end, we make use of coercive bilinear form taken from the weak formulation of elasticity problem to obtain descent direction at each iteration. This descent direction is a solution of an additional variational problem, involving the bilinear form mentioned above and the volumetric expression of the shape derivative plays the role of a linear form. In this paper, we combine level set method with radial basis functions (RBFs) used to approximate LSF. We focus on the so-called multiquadric RBFs, but other classes of RBFs are also briefly considered. This eventually leads to transformation of partial differential equation (linear transport equation governing the evolution of shapes) to a system of linear ordinary differential equations which admits analytical formula for the solution. We apply our method to compliance minimization of a cantilever problem as well as to total potential energy minimization of a structure with kinematic loading. To run all the numerical experiments, we wrote our own code in Wolfram Mathematica environment.