Stable Mixed Finite Elements for Linear Elasticity with Thin Inclusions
This work provides a rigorous numerical framework for engineers modeling composites with thin inclusions, though it is an incremental extension of mixed finite element theory to mixed-dimensional geometries.
The authors develop and analyze stable mixed finite element methods for linear elasticity in composite materials with thin inclusions modeled as lower-dimensional manifolds, proving well-posedness and convergence with local momentum conservation.
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.