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

By Thiago O. Quinelato, Abimael F. D. Loula, Maicon R. Correa and Todd Arbogast


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 elements347120142CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO - CNPQCOORDENAÇÃO DE APERFEIÇOAMENTO DE PESSOAL DE NÍVEL SUPERIOR - CAPESFUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULO - FAPESP141009/2013-6; 312388/2016-0BEX 6993/15-02017/23338-8Quinelato acknowledges financial support from CAPES, the Coordination for the Improvement of Higher Education Personnel, Brazil process BEX 6993/15-0 and CNPq, the National Council for Scientific and Technological Development, Brazil grant 141009/2013-6. Loula acknowledges financial support from CNPq grant 312388/2016-0. Correa acknowledges financial support from FAPESP, the Research Foundation of the State of São Paulo, Brazil (Grant 2017/23338-8). Arbogast acknowledges financial support from U.S. National Science Foundation grant DMS-141875

Topics: Quadrilátero, Quadrilaterals, Linear elasticity, Mixed finite element method, Quadrilateral element, Full H(div)-approximation, Hybrid formulation
Publisher: Países Baixos
Year: 2020
DOI identifier: 10.1016/j.cma.2018.12.013
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