Efficient Method for Derivatives of Nonlinear Stiffness Matrix

Abstract

Structural design often includes geometrically nonlinear analysis to reduce structural weight and increase energy efficiency. The full-order finite element model can perform the geometrically nonlinear analysis, but its computational cost is expensive. Therefore, nonlinear reduced-order models (NLROMs) have been developed to reduce costs. The non-intrusive NLROM has a lower cost than the other due to the approximation of the nonlinear internal force by a polynomial of reduced coordinates based on the Taylor expansion. The constants in the polynomial, named reduced stiffnesses, are derived from the derivative of the structure’s tangential stiffness matrix with respect to the reduced coordinates. The precision of the derivative of the tangential stiffness affects the reduced stiffness, which in turn significantly influences the accuracy of the NLROM. Therefore, this study evaluates the accuracy of the derivative of the tangential stiffness calculated by the methods: finite difference, complex step, and hyper-dual step. Analytical derivatives of the nonlinear stiffness are developed to provide references for evaluating the accuracy of the numerical methods. We propose using the central difference method to calculate the stiffness coefficients of NLROM due to its advantages, such as accuracy, low computational cost, and compatibility with commercial finite element software.