Stress mixed polyhedral finite elements for two-scale elasticity models with relaxed symmetry

• Published in: Computers & mathematics with applications (1987) 2024-04, Vol.159, p.302-318
• Main Authors: Devloo, Philippe R.B., Fernandes, Jeferson W.D., Gomes, Sônia M., Shauer, Nathan
• Format: Article
• Language: English
• Subjects: Computational linear elasticity; Hybridization; Mixed finite elements; Multiscale method; Polyhedral meshes; Relaxed symmetry
• ISSN: 0898-1221; 1873-7668
• DOI: 10.1016/j.camwa.2024.02.030

We consider two-scale stress mixed finite element elasticity models using H(div)-conforming tensor approximations for the stress variable, whilst displacement and rotation fields are introduced to impose divergence and symmetry constraints. The variables are searched in composite FE spaces based on polyhedral subdomains, formed by the conglomeration of local shape-regular micro partitions. The two-scale characteristic is expressed in terms of refined discretizations inside the subdomains versus coarser normal components of tensors over their boundaries (traction), with respect to mesh size, polynomial degree, or both. General error estimates are derived and stability is proved for five particular cases, associated with stable single-scale local tetrahedral space settings. 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. A static condensation procedure is designed for computational implementation. There is a global problem for primary variables at the coarser level, with a drastic reduction in the number of degrees of freedom, solving the traction variable and piecewise polyhedral rigid body motion components of the displacement. The fine details of the solution (secondary variables) are recovered by local Neumann problems in each polyhedron, the traction variable playing the role of boundary data. In this sense, the proposed formulation can be interpreted as an equivalent Multiscale Hybrid Mixed method, derived from a global-local characterization of the exact solution. A numerical example with known smooth solution is simulated to attest convergence properties of the method based on local BDFM divergence-compatible finite element pairs. Application to a problem with highly heterogeneous material is analyzed for robustness verification. •Multiscale method for 3D mixed weak symmetry elasticity approximations.•Theoretical analysis for the convergence of the stress and displacement.•Numerical results confirm theoretical convergence rates.•Method applied to highly heterogeneous 3D model.•Method results in elementwise equilibrated stress field.