Generalized multiscale finite element treatment of a heterogeneous nonlinear strain-limiting elastic model

In this work, we consider a nonlinear strain-limiting elastic model in heterogeneous domains. We investigate heterogeneous material with soft and stiff inclusions and perforations that are important to understand an elastic solid's behavior and crack-tip fields. Numerical solutions of problems in computational domains with inclusions and perforations require the construction of a sufficiently fine grid that resolves heterogeneity on the grid level. Approximations on such grids lead to a large system of equations with large computational costs. To reduce the size of the system and provide an accurate solution, we present a generalized multiscale finite element approximation on the coarse grid. In this method, we construct multiscale basis functions in each local domain associated with the coarse-grid cell and based on the construction of the snapshot space and solution of the local spectral problem reduce the size of the snapshot space. Two types of multiscale basis function construction are presented. The first type is a general case that can handle any boundary conditions on the global boundary of the heterogeneous domain. The considered problem requires an accurate approximation of the crack-surface boundary. In the second type of multiscale basis functions, we incorporate global boundary conditions in the basis construction process which provide an accurate approximation of the stress and strain on the crack boundary. We present numerical results for three cases of heterogeneity: soft inclusions, stiff inclusions, and perforations. A numerical investigation is presented for the two examples of loading on the domain with and without crack boundary conditions. The presented generalized multiscale finite element solver provides an accurate solution with a large reduction of the discrete system size. Our results illustrate the significant error reduction on the crack surface when we use the second type of basis functions.