Stable mixed finite elements for linear elasticity with thin inclusions

• 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
• DOI: 10.1007/s10596-020-10013-2

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.