Numerical approximation of consistent tangent moduli using complex-step derivative and its application to finite deformation problems

Abstract

In this paper, a numerical computation of consistent tangent moduli using complex-step derivative approximation (CSDA) is presented, and its applications to finite deformation problems are demonstrated. The consistent tangent stiffness is needed to achieve quadratic convergence in integration for boundary value problems. However, some material models lead to complex formulations of the consistent tangent stiffness that can be difficult to implement. This study shows a simple, robust and efficient numerical approximation of the tangent moduli that can be easily implemented within commercial FE software. Especially, the CSDA is focused on for numerical derivatives. The CSDA is proved to be of second order for suitably small perturbation and does not suffer from inherent subtractive cancellations that limits the accuracy of finite difference approximations, such as the forward Euler method and the central difference method, in floating point arithmetic. The implementation and the accuracy of this approach is illustrated through several numerical examples. © 2011 The Japan Society of Mechanical Engineers.