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An asymmetric Mixed Finite Element for incompressible Elasticity based on a modified Least‐Squares Formulation
Author(s) -
Schwarz Alexander,
Schröder Jörg
Publication year - 2008
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200810341
Subject(s) - compressibility , finite element method , elasticity (physics) , mathematics , norm (philosophy) , mathematical analysis , least squares function approximation , minification , mixed finite element method , element (criminal law) , physics , mathematical optimization , mechanics , thermodynamics , statistics , estimator , political science , law
Abstract The main goal of this work is to derive a mixed finite element for quasi incompressible elastic problems at small strains. Basis for the element formulation is a weak form resulting from a least–squares method, see e.g. Jiang [1]. Therefore, the L 2 ‐norm minimization of the residuals of the given first order system of differential equations leads to a functional depending on displacements, strains and stresses. In this context an enrichment of the strain space is introduced and the respective weak form is derived. The proposed approach differs from a classical least–squares method in the elimination of the additional strains on element level. As a consequence the resulting matrices become asymmetric. This reduction results in a two field formulation related to displacements and stresses. The approximation of these unknows follows the same procedures as for a conventional least–squares method, see e.g. Cai & Starke [2] or Schwarz & Schröder [6]. At the end a numerical examples for quasi incompressible elasticity is presented. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)