An adaptive multiscale finite element method for strain localization analysis with the Cosserat continuum theory

Strain localization analysis based on finite element method (FEM) usually requires an intensive computation to capture an accurate shear band and the limit stress, especially in the heterogeneous problems, and thus costs much computational resource and time. This paper proposes an adaptive multiscale FEM (AMsFEM) to improve the computational efficiency of strain localization analysis in heterogeneous solids. In the multiscale analysis, h- and p-adaptive strategies are proposed to update the fine and coarse meshes, respectively. The problem of mesh dependence is handled by the Cosserat continuum theory. In the fine-scale adaptive procedure, triangular elements are taken to discretize the fine-scale domain, and the newest vertex bisection is utilized for refinement based on the gradient of displacement. In the coarse-scale adaptive procedure, a multi-node coarse element technique is considered. By introducing a probability density function for each side of a coarse element, the optimal positions of the newly added coarse nodes can be determined. With the proposed adaptive multiscale procedure, the computational DOFs are reduced smartly and massively. Three representative heterogeneous examples demonstrate that the proposed method can accurately capture shear bands, with an improved computational efficiency and robust convergence.