Quasi-optimal mesh generation for heterogeneous materials: A fully adaptive virtual element method

The mesh flexibility of the virtual element method (VEM) has made it increasingly popular in the context of adaptive remeshing. Recent work by the authors presented a method for generating quasi-optimal meshes of VEs for homogeneous elastic problems. The prevalence of heterogeneous materials in engineering applications necessitates an efficient, accurate, and robust tool for their analysis. Herein, the VEM mesh flexibility provides significant further advantages as complex matrix and inclusion geometries can be trivially realized. Nevertheless, to date an adaptive VEM has not been applied to heterogeneous materials. In this work, a fully adaptive non-hierarchical remeshing procedure is presented for the generation of quasi-optimal meshes of VEs for 2D heterogeneous elastic problems, thus, representing the novel presentation of adaptive refinement and coarsening procedures for heterogeneous bodies in the VEM context. The procedure novelly permits independently prescribed accuracy targets for the matrix and inclusion regions of heterogeneous bodies and generates quasi-optimal meshes for each region. Numerical results demonstrate the accuracy, efficacy, and robustness of the proposed procedure over a broad range of heterogeneous characteristics including material properties, inclusion shape and location regularity, and inclusion quantities. Furthermore, since the adaptive procedure is hierarchy-free, quasi-optimal meshes can be generated from any initial mesh.