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  • Starke, Gerhard  (4)
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  • 1
    Online Resource
    Online Resource
    Comparison of geometrically nonlinear LSFEM formulations based on different hyperelastic models
    Wiley ; 2013
    In:  PAMM Vol. 13, No. 1 ( 2013-12), p. 97-98
    Details
    In: PAMM, Wiley, Vol. 13, No. 1 ( 2013-12), p. 97-98
    Abstract: This contribution deals with the solution of geometrically nonlinear elastic problems solved by the least‐squares mixed finite element method (LSFEM). The degrees of freedom (displacements and stresses) will be approximated using suitable spaces, namely W 1, p with p 〉 4 and H ( div ,Ω). In order to define the stress response of the material, different hyperelastic free energy functions will be presented. The residual forms ℛ I of the balance of momentum and the constitutive equation build a system of differential equations of first order. Choosing suitable weighting operators and applying L 2 ‐norms lead to a least‐squares functional ℱ( P,u ). The interpolation of the unknowns is accomplished using a standard polynomial interpolation for the displacements and vector‐valued Raviart‐Thomas functions for the approximation of the stresses. The formulations presented will be compared considering a uni‐axial spatial tension test. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)
    Type of Medium: Online Resource
    ISSN: 1617-7061 , 1617-7061
    URL: Issue
    URL: Article
    DOI: 10.1002/pamm.v13.1
    DOI: 10.1002/pamm.201310044
    Language: English
    Publisher: Wiley
    Publication Date: 2013
    detail.hit.zdb_id: 2078931-2
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    Online Resource
  • 2
    Online Resource
    Online Resource
    An improved mixed finite element based on a modified least‐squares formulation for hyperelasticity
    Wiley ; 2014
    In:  PAMM Vol. 14, No. 1 ( 2014-12), p. 243-244
    Details
    In: PAMM, Wiley, Vol. 14, No. 1 ( 2014-12), p. 243-244
    Abstract: The present work deals with the solution of geometrically nonlinear elastic problems solved by the least‐squares finite element method (LSFEM). The main goal is to obtain an improved performance and an accurate approximation in particular for lower‐order elements. Basis for the mixed element is a first‐order stress‐displacement formulation resulting from a classical least‐squares method. Similar to the ideas in SCHWARZ ET AL. [1] a modified weak form is derived by the introduction of an additional term controlling the stress symmetry condition. The approximation of the unknowns follows the same procedures as for a conventional least‐squares method, see e.g. CAI & STARKE [2]. The proposed modified formulation is compared to recently developed classical LSFEMs, in order to show the improvement of performance and accuracy. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)
    Type of Medium: Online Resource
    ISSN: 1617-7061 , 1617-7061
    URL: Issue
    URL: Article
    DOI: 10.1002/pamm.v14.1
    DOI: 10.1002/pamm.201410109
    Language: English
    Publisher: Wiley
    Publication Date: 2014
    detail.hit.zdb_id: 2078931-2
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    Online Resource
  • 3
    Online Resource
    Online Resource
    Solving hyperelastic problems using mixed LSFEM
    Wiley ; 2012
    In:  PAMM Vol. 12, No. 1 ( 2012-12), p. 215-216
    Details
    In: PAMM, Wiley, Vol. 12, No. 1 ( 2012-12), p. 215-216
    Abstract: The focus of this contribution is the solution of hyperelastic problems using the least‐squares finite element method (LSFEM). In particular a mixed least‐squares finite element formulation is provided and applied on geometrically nonlinear problems. The basis for the element formulation is a div‐grad first‐order system consisting of the equilibrium condition and the constitutive equation both written in a residual form. An L 2 ‐norm is adopted on the residuals leading to a functional depending on displacements and stresses which has to be minimized. Therefore the first variations with respect to both free variables have to be zero. The solution can then be found by applying Newton's Method. For the continuous approximation of the displacements in W 1, p with p 〉 2, standard polynomials are used. Shape functions belonging to a Raviart‐Thomas space are applied for the stress interpolation. These vector‐valued functions ensure a conforming discretization of the Sobolev space H ( div , Ω). Finally the proposed formulation is tested in a numerical example. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)
    Type of Medium: Online Resource
    ISSN: 1617-7061 , 1617-7061
    URL: Issue
    URL: Article
    DOI: 10.1002/pamm.v12.1
    DOI: 10.1002/pamm.201210098
    Language: English
    Publisher: Wiley
    Publication Date: 2012
    detail.hit.zdb_id: 2078931-2
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    Online Resource
  • 4
    Online Resource
    Online Resource
    On a least‐squares formulation for hyperelastic, transversely isotropic problems
    Wiley ; 2014
    In:  PAMM Vol. 14, No. 1 ( 2014-12), p. 247-248
    Details
    In: PAMM, Wiley, Vol. 14, No. 1 ( 2014-12), p. 247-248
    Abstract: 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)
    Type of Medium: Online Resource
    ISSN: 1617-7061 , 1617-7061
    URL: Issue
    URL: Article
    DOI: 10.1002/pamm.v14.1
    DOI: 10.1002/pamm.201410111
    Language: English
    Publisher: Wiley
    Publication Date: 2014
    detail.hit.zdb_id: 2078931-2
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    Online Resource
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