Virtual element method with adaptive refinement for problems of two-dimensional complex topology models from an engineering perspective

Virtual element methods (VEM) provide great flexibility in solving numerical problems defined on arbitrarily shaped polygonal or polyhedral discretizations. In this paper, we develop a framework for two dimensional elastic problems defined on complex topology models using high-order virtual element methods from an engineering perspective. The VEM discrete formulations are detailedly derived following the rule used in standard FEM. An arbitrarily complex model is first embedded into a rectangular domain which is then discretized into a structured grid. The elements intersecting with the boundaries are further adaptively refined through a quad-tree refinement strategy controlled by a subdivision level or an approximation error. An optimization method is proposed to avoid the generation of tiny elements and two averaged schemes for stress recovery in post-processing are discussed. The behavior of the proposed VEM is thoroughly studied and the results are compared with analytical solutions and that obtained from FEM. The heavy burden placed on meshing complex CAD geometries is greatly alleviated and the convergence studies confirm the accuracy and convergence of the method.