Stable mixed finite elements for linear elasticity with thin inclusions
- Resource Type
- Authors
- Wietse M. Boon; Jan Martin Nordbotten
- Source
- Computational Geosciences
- Subject
- Partial differential equation
Discretization
Cauchy stress tensor
0208 environmental biotechnology
Linear elasticity
Mathematical analysis
010103 numerical & computational mathematics
02 engineering and technology
Differential operator
01 natural sciences
Finite element method
Symmetry (physics)
020801 environmental engineering
Computer Science Applications
Computational Mathematics
Computational Theory and Mathematics
Dimensional reduction
0101 mathematics
Computers in Earth Sciences
Mathematics
- Language
- English
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.