Adaptive mesh refinement in polygonal finite elements using clustering technique

In the adaptive finite element method, the goal is to achieve the desired accuracy with the minimum degrees of freedom. Thus, mesh refinement is required to adjust the mesh size to distribute the error uniformly over the domain. In recent years, research has paid attention to the polygonal elements due to their flexibility for adapting to complex geometries. The most common approach for creating a polygonal mesh is the Voronoi diagram. The main drawback of this approach is the inconsistency of the generated polygonal mesh and the desired mesh density of the adaptive method. This inconsistency is more noticeable in problems where the mesh density changes rapidly in a small region. The adjustment of the size of the elements in polygonal elements is more difficult than triangular or quadrilateral elements because of the greater number of adjacent elements. In the present study, a target error function is defined as the difference of the generated mesh and the desired mesh density in each Gauss point that has been optimized in an iterative algorithm. In the problems where the mesh density changes rapidly, a clustering technique is used to classify different regions of the domain and the Voronoi nuclei of the mesh are located separately. Some additional modification techniques are employed to improve the mesh quality. In problems with cracks, where the Voronoi diagram method has more challenges and some interlocking elements are generated, the problem is modeled without crack initially and then the crack is added to the model and finally the mesh is adapted to the desired mesh density.