On mesh refinement procedures for polygonal virtual elements

This work concerns the application of adaptive refinement procedures to meshes of unstructured polygonal virtual elements. Adaptive refinement indicators previously proposed by the authors, and investigated for meshes of structured quadrilateral elements, are studied in more general applications. Specifically, the performance of the indicators is studied on unstructured polygonal meshes, and for cases of compressible and nearly-incompressible materials. Localized refinement of unstructured meshes is a non-trivial task as the algorithm must be robust, and must accommodate a wide variety of geometric possibilities. To this end, an element refinement algorithm is presented based on strategic seeding of Voronoi tessellations and is suitable for both structured and unstructured meshes. Furthermore, it is not known apriori whether the previously proposed refinement indicators will be reliable or effective in the presence of unstructured mesh geometries and nearly-incompressible materials. Thus, the performance of the refinement procedures is studied through a broad numerical campaign. The results demonstrate that the high degree of efficacy and efficiency previously exhibited by the adaptive procedures is also achieved in the cases of irregular unstructured/Voronoi meshes and near-incompressibility.