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On a least‐squares formulation for hyperelastic, transversely isotropic problems
Author(s) -
Steeger Karl,
Schwarz Alexander,
Schröder Jörg,
Starke Gerhard,
Müller Benjamin
Publication year - 2014
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.201410111
Subject(s) - hyperelastic material , transverse isotropy , interpolation (computer graphics) , mathematics , finite element method , norm (philosophy) , mathematical analysis , isotropy , least squares function approximation , geometry , physics , classical mechanics , motion (physics) , statistics , quantum mechanics , estimator , political science , law , thermodynamics
In the present work a mixed finite element based on a least‐squares formulation is proposed. In detail, the provided constitutive relation is based on a hyperelastic free energy including terms describing a transversely isotropic material behavior. Basis for the element formulation is a weak form resulting from a least‐squares method, see e.g. [1]. The L 2 ‐norm minimization of the residuals of the given first‐order system of differential equations leads to a functional depending on displacements and stresses. The interpolation of the unknowns is executed using different approximation spaces for the stresses ( W q (div, Ω)) and the displacements ( W 1,p (Ω)), under consideration of suitable p and q . For the approximation of the stresses vector‐valued shape functions of Raviart‐Thomas type, related to the edges of the respective triangular element, are applied. Standard interpolation polynomials are used for the continuous approximation of the displacements. The performance of the proposed formulation will be investigated considering a numerical example. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)