One-point quadrature of higher-order finite and virtual elements in nonlinear analysis

authored by

Tobias Bode

Abstract

In the present article, a stability- and consistency-preserving integration scheme for polynomial Galerkin approaches of arbitrary order is presented. The basis is formed by Taylor series expansions of the stresses with respect to the strains, which in turn are expanded towards the spatial directions. With a split of the material and geometric nonlinearities and the assumption of a material behavior linearly variable within an element, the strain energy in elements of arbitrary shape and polynomial order can be evaluated exactly. Therefore, geometric moments have to be calculated in preprocessing, requiring only evaluations of derivatives at a single integration point during the analysis. The moments can be effectively integrated analytically over the boundary of the elements. As one of the manifold applications, the use in the context of second order virtual elements is elaborated for which the assembly time can be significantly reduced. The combination with the automatic differentiation and expression optimization software AceGen provides performant element routines. In the numerical examples, the integration scheme shows promising accuracy and makes the application in more complex material models up to computational homogenization attractive.

Organisation(s)

Institute of Continuum Mechanics
PhoenixD: Photonics, Optics, and Engineering - Innovation Across Disciplines

Type

Article

Journal

Computational mechanics

Volume

73

Pages

1187-1202

No. of pages

16

ISSN

0178-7675

Publication date

05.2024

Publication status

Published

Peer reviewed

Yes

ASJC Scopus subject areas

Computational Mathematics, Mechanical Engineering, Ocean Engineering, Applied Mathematics, Computational Mechanics, Computational Theory and Mathematics

Electronic version(s)

https://doi.org/10.1007/s00466-023-02406-8 (Access: Open)