Full H(div)-approximation of linear elasticity on quadrilateral meshes based on ABF finite elements

Author(s)

Quinelato, Thiago O. ^[1] ; Loula, Abimael F. D. ^[2] ; Correa, Maicon R. ^[3] ; Arbogast, Todd ^{[4, 5]}

Affiliation

^[1] Univ Fed Juiz de Fora, Dept Matemat, Rua Jose Loureno Kelmer S-N, Campus Univ, BR-36036900 Juiz De Fora, MG - Brazil

^[2] Lab Nacl Comp Cient, Av Getulio Vargas 333, BR-25651075 Petropolis, RJ - Brazil

^[3] Univ Estadual Campinas, Inst Matemat Estat & Computacao Cient, Dept Matemat Aplicada, Rua Sergio Buarque de Holanda, 651 Barao Geraldo, BR-13083859 Campinas, SP - Brazil

^[4] Univ Texas Austin, Dept Math, 2515 Speedway C1200, Austin, TX 78712 - USA

^[5] Univ Texas Austin, Inst Computat Engn & Sci, 201 EAST 24th St, Austin, TX 78712 - USA

• Document type: Journal article
• Source: COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING; v. 347, p. 120-142, APR 15 2019.
• Web of Science Citations: 0

Abstract

For meshes of nondegenerate, convex quadrilaterals, we present a family of stable mixed finite element spaces for the mixed formulation of planar linear elasticity. The problem is posed in terms of the stress tensor, the displacement vector and the rotation scalar fields, with the symmetry of the stress tensor weakly imposed. The proposed spaces are based on the Arnold-Boffi-Falk (ABF(k), k >= 0) elements for the stress and piecewise polynomials for the displacement and the rotation. We prove that these finite elements provide full H(div)-approximation of the stress field, in the sense that it is approximated to order h(k+1), where h is the mesh diameter, in the H(div)-norm. We show that displacement and rotation are also approximated to order h(k+1) in the L-2-norm. The convergence is optimal order for k >= 1, while the lowest order case, index k = 0, requires special treatment. The spaces also apply to both compressible and incompressible isotropic problems, i.e., the Poisson ratio may be one-half. The implementation as a hybrid method is discussed, and numerical results are given to illustrate the effectiveness of these finite elements. (C) 2018 Elsevier B.V. All rights reserved. (AU)

• FAPESP's process: 17/23338-8 - Mixed-Hybrid Finite Element Methods for Elliptic Problems with application to Porous Media Flow and Linear Elasticity
• Grantee: Maicon Ribeiro Correa
• Support Opportunities: Regular Research Grants