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Primal and mixed finite element formulations for the relaxed micromorphic model.

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Title: Primal and mixed finite element formulations for the relaxed micromorphic model.
Authors: Sky, Adam1 (AUTHOR) adam.sky@tu-dortmund.de, Neunteufel, Michael2 (AUTHOR) michael.neunteufel@tuwien.ac.at, Muench, Ingo1 (AUTHOR) ingo.muench@tu-dortmund.de, Schöberl, Joachim2 (AUTHOR) joachim.schoeberl@tuwien.ac.at, Neff, Patrizio3 (AUTHOR) patrizio.neff@uni-due.de
Source: Computer Methods in Applied Mechanics & Engineering. Sep2022, Vol. 399, pN.PAG-N.PAG. 1p.
Subjects: Intrinsic motivation, Porous materials, Degrees of freedom, Micropolar elasticity, Metamaterials, Mathematical continuum
Abstract: The classical Cauchy continuum theory is suitable to model highly homogeneous materials. However, many materials, such as porous media or metamaterials, exhibit a pronounced microstructure. As a result, the classical continuum theory cannot capture their mechanical behaviour without fully resolving the underlying microstructure. In terms of finite element computations, this can be done by modelling the entire body, including every interior cell. The relaxed micromorphic continuum offers an alternative method by instead enriching the kinematics of the mathematical model. The theory introduces a microdistortion field, encompassing nine extra degrees of freedom for each material point. The corresponding elastic energy functional contains the gradient of the displacement field, the microdistortion field and its Curl (the micro-dislocation). Therefore, the natural spaces of the fields are [ H 1 ] 3 for the displacement and [ H (curl) ] 3 for the microdistortion, leading to unusual finite element formulations. In this work we describe the construction of appropriate finite elements using Nédélec and Raviart–Thomas subspaces, encompassing solutions to the orientation problem and the discrete consistent coupling condition. Further, we explore the numerical behaviour of the relaxed micromorphic model for both a primal and a mixed formulation. The focus of our benchmarks lies in the influence of the characteristic length L c and the correlation to the classical Cauchy continuum theory. • Existence and uniqueness results for primal and mixed formulations of the relaxed micromorphic model. • A priori convergence estimates. • A detailed description of the construction of the appropriate finite elements on tetrahedra. • An intrinsic solution to the orientation problem of vectorial elements. • A specialized scheme for an exact discrete consistent coupling condition. [ABSTRACT FROM AUTHOR]
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Database: Engineering Source
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Abstract:The classical Cauchy continuum theory is suitable to model highly homogeneous materials. However, many materials, such as porous media or metamaterials, exhibit a pronounced microstructure. As a result, the classical continuum theory cannot capture their mechanical behaviour without fully resolving the underlying microstructure. In terms of finite element computations, this can be done by modelling the entire body, including every interior cell. The relaxed micromorphic continuum offers an alternative method by instead enriching the kinematics of the mathematical model. The theory introduces a microdistortion field, encompassing nine extra degrees of freedom for each material point. The corresponding elastic energy functional contains the gradient of the displacement field, the microdistortion field and its Curl (the micro-dislocation). Therefore, the natural spaces of the fields are [ H 1 ] 3 for the displacement and [ H (curl) ] 3 for the microdistortion, leading to unusual finite element formulations. In this work we describe the construction of appropriate finite elements using Nédélec and Raviart–Thomas subspaces, encompassing solutions to the orientation problem and the discrete consistent coupling condition. Further, we explore the numerical behaviour of the relaxed micromorphic model for both a primal and a mixed formulation. The focus of our benchmarks lies in the influence of the characteristic length L c and the correlation to the classical Cauchy continuum theory. • Existence and uniqueness results for primal and mixed formulations of the relaxed micromorphic model. • A priori convergence estimates. • A detailed description of the construction of the appropriate finite elements on tetrahedra. • An intrinsic solution to the orientation problem of vectorial elements. • A specialized scheme for an exact discrete consistent coupling condition. [ABSTRACT FROM AUTHOR]
ISSN:00457825
DOI:10.1016/j.cma.2022.115298