Stable mixed finite elements for linear elasticity with thin inclusions

We consider mechanics of composite materials in which thin inclusions are modeled by lower-dimensional manifolds. By successively applying the dimensional reduction to junctions and intersections within the material, a geometry of hierarchically connected manifolds is formed which we refer to as mix...

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Veröffentlicht in:Computational geosciences Jg. 25; H. 2; S. 603 - 620
Hauptverfasser: Boon, W. M., Nordbotten, J. M.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Cham Springer International Publishing 01.04.2021
Springer Nature B.V
Schlagworte:
A priori analysis
Composite materials
Differential equations
Differential geometry
Dimensions
Earth and Environmental Science
Earth Sciences
Elasticity
Geotechnical Engineering & Applied Earth Sciences
Hydrogeology
Inclusions
Intersections
Linear elasticity
Manifolds
Manifolds (mathematics)
Mathematical analysis
Mathematical Modeling and Industrial Mathematics
Mechanics
Mixed finite element
Mixed-dimensional
Momentum
Norms
Operators (mathematics)
Original Paper
Partial differential equations
Soil Science & Conservation
Stability
Tensors
Weak symmetry
65N12
74K20
74S05
Linear elasticity
A priori analysis
Weak symmetry
Mixed-dimensional
Mixed finite element
65N30
ISSN:1420-0597, 1573-1499, 1573-1499
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Beschreibung
Zusammenfassung:We consider mechanics of composite materials in which thin inclusions are modeled by lower-dimensional manifolds. By successively applying the dimensional reduction to junctions and intersections within the material, a geometry of hierarchically connected manifolds is formed which we refer to as mixed-dimensional. The governing equations with respect to linear elasticity are then defined on this mixed-dimensional geometry. The resulting system of partial differential equations is also referred to as mixed-dimensional, since functions defined on domains of multiple dimensionalities are considered in a fully coupled manner. With the use of a semi-discrete differential operator, we obtain the variational formulation of this system in terms of both displacements and stresses. The system is then analyzed and shown to be well-posed with respect to appropriately weighted norms. Numerical discretization schemes are proposed using well-known mixed finite elements in all dimensions. The schemes conserve linear momentum locally while relaxing the symmetry condition on the stress tensor. Stability and convergence are shown using a priori error estimates and confirmed numerically.
Bibliographie:ObjectType-Article-1
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ISSN:1420-0597
1573-1499
1573-1499
DOI:10.1007/s10596-020-10013-2