Least-Squares Continuous Sensitivity Equations for an Infinite Plate with a Hole

Abstract

Continuous sensitivity methods compute design or shape parameter gradients from the continuous system of partial differential equations instead of the discretized system as is more commonly done in multidisciplinary optimization. The continuous sensitivity equations avoid the need to calculate the problematic mesh sensitivities of the discrete method, and are thus more computationally efficient for some applications. Although there exists an extensive body of literature of continuous sensitivity applications for fluid dynamics problems, the application to structural elasticity problems is far more limited, in part due to the complications of determining the stress sensitivity boundary conditions. To partially fill this void, this paper develops and solves the least-squares, continuous sensitivity equations for a classic 2D plane-stress problem. The continuous sensitivity system equations and sensitivity boundary conditions are derived and the problem is posed in first-order form. The problem is solved using a least-squares finite element method incorporating higherorder polynomial elements. The least-squares and continuous results are presented and compared to the analytic solution and analytic sensitivities.