A Higher‐Order Multiscale Finite Element Method Based on the Ultra‐Weak Mixed Formulation for 3D Solids

Abstract

A new version of the multiscale finite element method (MsFEM) for elastic solid bodies, based on the mixed ultra‐weak formulation is presented. The mixed formulation ensures fast stress convergence, while its ultra‐weak variant facilitates the construction of conforming shape functions. Furthermore, the stability guaranteed by the discontinuous Petrov–Galerkin (DPG) methodology, combined with the high approximability inherent to MsFEM, ensures the convergence of the numerical solution. The key innovation of the proposed MsFEM modification lies in the computation of the prolongation operator between fine and coarse finite element meshes. The proposed approach, which accounts for high‐order approximation, has been successfully verified on several representative numerical examples for 3D elasticity problems and can also be applied to direct FEM computations (without upscaling) for heterogeneous materials, when the multigrid method is used to solve the systems of algebraic equations for the ultra‐weak mixed formulation.