Least-Squares Continuous Sensitivity Shape Optimization for Structural Elasticity Applications

Abstract

A general first-order formulation of the continuous sensitivity equations is introduced that improves the accuracy of the sensitivity boundary conditions. In the continuous sensitivity method, design or shape parameter gradients are computed from a continuous system of partial differential equations instead of the discretized system, which avoids the problematic mesh sensitivity calculations of discrete methods for shape variation problems. The first-order formulation for both the underlying elasticity and sensitivity equations is amenable to solution by a high-order polynomial least-squares finite element model. The continuous sensitivity boundary-value problem, which can be posed in local or total derivative form, is generally simpler in local sensitivity form for shape variation problems. Although the local form is not unique and total material sensitivities are generally desired for structural elasticity problems, the local sensitivity solution is easily transformed to a material derivative. The first-order formulation and the transformation to material derivatives are demonstrated for two elasticity example problems. The least-squares continuous sensitivity solutions are presented and compared with analytic sensitivities and finite difference results and should prove convenient validation benchmarks for other continuous sensitivity applications.