Geometrically nonlinear finite element implementation based on highly accurate 1st and 2nd numerical derivative scheme using hyper-dual numbers

Abstract

This paper proposes a novel implementation scheme of geometrically nonlinear finite element programs, which automatically compute exact internal force vectors and element stiffness matrices by numerically differentiating a strain energy function at each element. This method can significantly simplify the complex implementation procedure which is often observed in conventional finite element implementations, since it never requires B matrices, stress tensors, and elastic tensors by hand. The proposed method is based on a highly accurate numerical derivatives which use hyper-dual numbers and never suffer from any round-off and truncation errors. Several numerical examples are performed to demonstrate the effectiveness and robustness of the proposed method.