A finite element method based on C0‐continuous assumed gradients

Abstract

SUMMARYThis article presents an alternative approach to assumed gradient methods in FEM applied to threedimensional elasticity. Starting from nodal integration (NI), a general C0-continuous assumed interpolation of the deformation gradient is formulated. The assumed gradient is incorporated using the principle of Hu-Washizu. By dual Lagrange multiplier spaces, the functional is reduced to the displacements as the only unknowns. An integration scheme is proposed where the integration points coincide with the support points of the interpolation. Requirements for regular finite element meshes are explained. Using this interpretation of NI, instabilities (appearance of spurious modes) can be explained. The article discusses and classifies available strategies to stabilize NI such as penalty methods, SCNI, -FEM. Related approaches, such as the smoothed finite element method, are presented and discussed. New stabilization techniques for NI are presented being entirely based on the choice of the assumed gradient interpolation, i.e. nodal-bubble support, edge-based support and support using tensor-product interpolations. A strategy is presented on how the interpolation functions can be derived for various element types. Interpolation functions for the first-order hexahedral element, the first-order and the second-order tetrahedral elements are given. Numerous examples illustrate the strengths and limitations of the new schemes.

Asynchronous variational integration using continuous assumed gradient elements

“…Illustration of various 1st order CAG elements [2]. The circles define the positions of the support points of the assumed gradient field.…”

“…Such elements introduce either an assumed strain field (for infinitesimal strains) or an assumed deformation gradient field (for finite strains). The family includes nodal integration [22–24], smoothed finite element methods (SFEM) [25,26] and continuous assumed gradient (CAG) elements [2]. The mentioned methods were developed to improve the accuracy of low-order continuum elements.…”

“…For more details, the reader is referred to [2]. Section 3 and 4 presents a generalized formulation of asynchronous variational scheme following the ideas in [17] and then applies the formulation to structural elasticity using CAG elements.…”

“…Furthermore, long-term stability is ensured by its variational structure. This article presents AVI in the context of finite elements based on a weakened weak form (W2) Liu (2009) [1], exemplified by continuous assumed gradient elements Wolff and Bucher (2011) [2]. The article presents the main ideas of the modified AVI, gives implementation notes and a recipe for estimating the critical time step.…”

Distance fields on unstructured grids: Stable interpolation, assumed gradients, collision detection and gap function

“…Such is given if special element formulations are used, for example isogeometric analysis [33] or continuous assumed gradients [34].…”