Differentiation of Mass and Stiffness Matrices for High Order Sensitivity Calculations in Finite Element-Based Equilibrium Problems

Abstract

Extension of sensitivity methods to include higher order terms depends on the ability to compute higher order derivatives of the mass and stiffness matrices. This paper presents a method based on the use of cubic polynomials to fit mass and stiffness matrices across a range of interest of the design variable. The method is illustrated through an example which uses Padé approximants to expand the solution to a statics problem. The design variable is the thickness of one part of a plate with fixed boundaries. The solution gives a very good approximation over fivefold change in the value of the design variable.