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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Published in:	Computational geosciences Vol. 25; no. 2; pp. 603 - 620
Main Authors:	Boon, W. M., Nordbotten, J. M.
Format:	Journal Article
Language:	English
Published:	Cham Springer International Publishing 01.04.2021 Springer Nature B.V
Subjects:	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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Abstract	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.
AbstractList	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.
Author	Boon, W. M. Nordbotten, J. M.
Author_xml	– sequence: 1 givenname: W. M. orcidid: 0000-0003-4080-2369 surname: Boon fullname: Boon, W. M. email: wietse@kth.se organization: Institute for Modelling Hydraulic and Environmental Systems, University of Stuttgart, Department of Mathematics, KTH Royal Institute of Technology – sequence: 2 givenname: J. M. orcidid: 0000-0003-1455-5704 surname: Nordbotten fullname: Nordbotten, J. M. organization: Department of Mathematics, University of Bergen
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Cites_doi	10.1016/j.advwatres.2017.10.036 10.1137/S1064827503429363 10.1007/978-3-642-36519-5 10.1002/9781118137086 10.1007/s002110100348 10.1002/nme.2579 10.1007/BF01396415 10.1007/s10596-020-10002-5 10.1007/s10444-011-9240-1 10.1007/BF01389710 10.1007/s10231-020-01013-1 10.1137/17M1139102 10.1007/s10208-016-9315-y 10.1002/2015WR017290 10.1002/mma.1670020302 10.1007/978-3-319-93873-8_7 10.1002/2017WR020943 10.1017/S0962492906210018 10.1093/oso/9780198501787.001.0001
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References_xml	– reference: AwanouGRectangular mixed elements for elasticity with weakly imposed symmetry conditionAdv. Comput. Math.201338235136710.1007/s10444-011-9240-1 – reference: CaillerieDNedelecJThe effect of a thin inclusion of high rigidity in an elastic bodyMath. Meth. Appl. Sci.19802325127010.1002/mma.1670020302 – reference: Evans, L.: Partial Differential Equations. Orient Longman (1998) – reference: Nordbotten, J.M., Boon, W.M.: Modeling, structure and discretization of mixed-dimensional partial differential equations. In: Domain Decomposition Methods in Science and Engineering XXIV, Lecture Notes in Computational Science and Engineering (2017) – reference: BoffiDFortinMBrezziFMixed Finite Element Methods and Applications. Springer Series in Computational Mathematics2013BerlinSpringer10.1007/978-3-642-36519-5 – reference: FlemischBBerreIBoonWFumagalliASchwenckNScottiAStefanssonITatomirABenchmarks for single-phase flow in fractured porous mediaAdv. Water Resour.201811123925810.1016/j.advwatres.2017.10.036 – reference: Keilegavlen, E., Berge, R., Fumagalli, A., Starnoni, M., Stefansson, I., Varela, J., Berre, I.: Porepy: an open-source software for simulation of multiphysics processes in fractured porous media. arXiv:1908.09869 (2019) – reference: QuarteroniAValliADomain Decomposition Methods for Partial Differential Equations1999New YorkOxford University Press – reference: SævikPNixonCInclusion of topological measurements into analytic estimates of effective permeability in fractured mediaWater Resour. Res.201753119424944310.1002/2017WR020943 – reference: ArnoldDNWintherRMixed finite elements for elasticityNumer. Math.200292340141910.1007/s002110100348 – reference: Boon, W.M., Nordbotten, J.M., Vatne, J.E.: Functional analysis and exterior calculus on mixed-dimensional geometries. Annali di Matematica Pura ed Applicata (1923-) (2020) – reference: Licht, M.W.: Complexes of discrete distributional differential forms and their homology theory. Foundations of Computational Mathematics (2016) – reference: BoonWMNordbottenJMYotovIRobust discretization of flow in fractured porous mediaSIAM J. Numer. Anal.20185642203223310.1137/17M1139102 – reference: GeuzaineCRemacleJFGmsh: a 3-d finite element mesh generator with built-in pre-and post-processing facilitiesInt. J. Numer. Meth. Eng.200979111309133110.1002/nme.2579 – reference: ArnoldDNFalkRSWintherRFinite element exterior calculus, homological techniques, and applicationsActa Numerica200615115510.1017/S0962492906210018 – reference: MartinVJaffréJRobertsJEModeling fractures and barriers as interfaces for flow in porous mediaSIAM J. Sci. Comput.20052651667169110.1137/S1064827503429363 – reference: BjørnaråTINordbottenJMParkJVertically integrated models for coupled two-phase flow and geomechanics in porous mediaWater Resour. Res.20165221398141710.1002/2015WR017290 – reference: BrezziFDouglasJMariniLDTwo families of mixed finite elements for second order elliptic problemsNumer. Math.198547221723510.1007/BF01389710 – reference: NordbottenJMCeliaMAGeological Storage of CO2: Modeling Approaches for Large-Scale Simulation2011New YorkWiley10.1002/9781118137086 – reference: Alnæs, M., Blechta, J., Hake, J., Johansson, A., Kehlet, B., Logg, A., Richardson, C., Ring, J., Rognes, M.E., Wells, G.N.: The fenics project version 1.5. Archive of Numerical Software 3(100) (2015) – reference: ArnoldDNFalkRSWintherRDifferential complexes and stability of finite element methods II: the elasticity complexIMA Volumes Math. Appl.200614247 – reference: Ciarlet, P.: Mathematical Elasticity, Vol III, Theory of Shells. Mathematical Elasticity. Elsevier Science (2000) – reference: NedelecJMixed finite elements in $\mathbb {R}^{3}$ℝ3Numer. Math.198035331534110.1007/BF01396415 – reference: Bjørnarå, T.I.: Model development for efficient simulation of co2 storage. Ph.D. thesis, The University of Bergen (2018) – volume: 111 start-page: 239 year: 2018 ident: 10013_CR15 publication-title: Adv. Water Resour. doi: 10.1016/j.advwatres.2017.10.036 – volume: 26 start-page: 1667 issue: 5 year: 2005 ident: 10013_CR19 publication-title: SIAM J. Sci. Comput. doi: 10.1137/S1064827503429363 – volume-title: Mixed Finite Element Methods and Applications. Springer Series in Computational Mathematics year: 2013 ident: 10013_CR8 doi: 10.1007/978-3-642-36519-5 – volume-title: Geological Storage of CO2: Modeling Approaches for Large-Scale Simulation year: 2011 ident: 10013_CR22 doi: 10.1002/9781118137086 – volume: 142 start-page: 47 year: 2006 ident: 10013_CR2 publication-title: IMA Volumes Math. 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SubjectTerms	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
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Title	Stable mixed finite elements for linear elasticity with thin inclusions
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