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MECSOL 2022
8th International Symposium on Solid Mechanics
A Multiscale Hybrid-Mixed Method for Three-Dimensional Linear Elasticity
Submission Author:
Nathan Shauer , SP , Brazil
Co-Authors:
Nathan Shauer, Philippe Devloo, Sonia Gomes, Jeferson Wilian Dossa Fernandes
Presenter: Nathan Shauer
doi://10.26678/ABCM.MECSOL2022.MSL22-0217
Abstract
We consider two-scale hybrid-mixed finite element elasticity models using H(div)-conforming tensor approximations for the stress variable, whilst displacement and rotation fields represent multipliers to impose divergence and symmetry constraints. The discretization is based on general polyhedral meshes with flat faces (polytopes) in a global-local context. There is a global problem for primary variables at the coarser level, solving normal stress trace (traction), piecewise-defined over a partition of the mesh skeleton (facets), and piecewise polytope rigid motions. The fine details of the solution (secondary variables) are obtained by completely independent local Neumann problems in each polytopal subdomain, the traction variable playing the role of boundary data. As compared to the traction accuracy, internal variables may be enriched with respect to internal mesh size, internal polynomial degree, or both. The polytopal grid does not have to match across the facets, but a mild compatibility constraint is required. Stability and error estimates are proved for the method using a variety of two-scale space configurations (associated with known stable single-scale space settings for tetrahedral local sub-partitions). Enhanced accuracy rates for displacement and super-convergent divergence of the stress can be obtained. Stress, rotation, and stress symmetry errors keep the same accuracy order determined by the traction discretization. This analysis expands to three-dimensional elasticity problems the one presented in [1] for polygonal elastic domains. Numerical examples are analyzed to attest convergence properties of the method. Additionally, an application problem where the material is highly heterogeneous is analyzed to verify robustness. [1] P. R. B.Devloo, A. M. Farias, S. M. Gomes, W. Pereira, A. J. B dos Santos, F. Valentin, New H(div)-conforming multiscale hybrid-mixed methods for the elasticity problem on polygonal meshes. ESAIM-M2AN 55 (2021) 1005-1037.
Keywords
multiscale, Mixed Finite Elements, Linear Elasticity, Hybridization
