Stable Mixed Finite Elements for Linear Elasticity with Thin Inclusions
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.
Cite
@article{arxiv.1903.01757,
title = {Stable Mixed Finite Elements for Linear Elasticity with Thin Inclusions},
author = {Wietse M. Boon and Jan M. Nordbotten},
journal= {arXiv preprint arXiv:1903.01757},
year = {2019}
}