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.
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Open access funding provided by Royal Institute of Technology. The research of the authors was funded in part by the Norwegian Research Council grants 233736, 250223. The first author received from the German Research Foundation (DFG) supporting for this work through funding SFB 1313, Project Number 327154368.
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Boon, W.M., Nordbotten, J.M. Stable mixed finite elements for linear elasticity with thin inclusions. Comput Geosci 25, 603–620 (2021). https://doi.org/10.1007/s10596-020-10013-2
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DOI: https://doi.org/10.1007/s10596-020-10013-2