On [H1]3×3, [H(curl)]3 and H(sym Curl) finite elements for matrix-valued Curl problems View Full Text


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Article Info

DATE

2022-09-09

AUTHORS

Adam Sky, Ingo Muench, Patrizio Neff

ABSTRACT

In this work we test the numerical behaviour of matrix-valued fields approximated by finite element subspaces of [H1]3×3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[{{\,\mathrm{\textit{H}^1}\,}}]^{3\times 3}$$\end{document}, [H(curl)]3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[ H (\textrm{curl}{})]^3$$\end{document} and H(symCurl)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ H (\textrm{sym}\, \textrm{Curl}{})$$\end{document} for a linear abstract variational problem connected to the relaxed micromorphic model. The formulation of the corresponding finite elements is introduced, followed by numerical benchmarks and our conclusions. The relaxed micromorphic continuum model reduces the continuity assumptions of the classical micromorphic model by replacing the full gradient of the microdistortion in the free energy functional with the Curl. This results in a larger solution space for the microdistortion, namely [H(curl)]3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[ H (\textrm{curl}{})]^3$$\end{document} in place of the classical [H1]3×3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[{{\,\mathrm{\textit{H}^1}\,}}]^{3 \times 3}$$\end{document}. The continuity conditions on the microdistortion can be further weakened by taking only the symmetric part of the Curl. As shown in recent works, the new appropriate space for the microdistortion is then H(symCurl)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ H (\textrm{sym}\, \textrm{Curl}{})$$\end{document}. The newly introduced space gives rise to a new differential complex for the relaxed micromorphic continuum theory. More... »

PAGES

5

References to SciGraph publications

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  • 2019-04-03. A canonical rate-independent model of geometrically linear isotropic gradient plasticity with isotropic hardening and plastic spin accounting for the Burgers vector in CONTINUUM MECHANICS AND THERMODYNAMICS
  • 2013-11-07. A unifying perspective: the relaxed linear micromorphic continuum in CONTINUUM MECHANICS AND THERMODYNAMICS
  • 2021-07-02. Korn inequalities for incompatible tensor fields in three space dimensions with conformally invariant dislocation energy in CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
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    54 matrix-valued field
    55 microdistortions
    56 micromorphic continuum model
    57 micromorphic continuum theory
    58 micromorphic model
    59 model
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    61 numerical benchmarks
    62 part
    63 place
    64 problem
    65 recent work
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    67 solution space
    68 space
    69 subspace
    70 symmetric part
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