Refinement strategies for polygonal meshes applied to adaptive VEM discretization

A new quality preserving polygonal mesh refinement algorithm for Virtual Element Methods

“…This refining criterion is also applied when a cell is recognised as a triangle in spite of the presence of more than three vertices when aligned edges are present. For other polygons the first refinement algorithm presented is based on the idea introduced in the article [15] of using the maximum inertia moment as cutting direction of the element (Algorithm 1). The second one, is a natural extension of the longest edge refinement applied to polygons (Algorithm 2) considering the longest diagonal of the cell.…”

“…In the Algorithm 4 we describe the procedure to refine the polygonal cells with a cut along the inertia axis with the maximum moment improved with respect to [15]. In order to prevent the creation of small edges in the refined cells a small change of the refining direction was proposed in [15] such that new edges shorter than a fraction of the cut edge were collapsed. In this new algorithm we define an adaptive collapsing tolerance that depends on the shape and the edges of the refining polygon M and its neighbouring cells N M .…”

“…A cell with three edges or three sets of aligned edges (a geometric triangle, regardless of the number of vertices) is cut with the Longest Diagonal rule. The main differences between the proposed algorithm and the one presented in [15] is that in deciding if an edge can be cut we involve the effects of the cut on the neighbouring element sharing the edge. If the cut of an edge is suitable for the quality of the element that is performing the cut, but negative for the quality of the other facing element the cut is not performed and the intersection between the cutting line and the edge is collapsed to the closest vertex.…”

“…The presence in the mesh of different type of elements that can be refined independently largely increase the difficulties of obtaining a good quality mesh [12,13,14]. A preliminary work for general convex polygons is recently appeared [15].…”

“…Here we show an evolution of the algorithms presented in [15] that takes into account the geometric properties of the cells such that the refinement results to preserve or improve the quality of the cell. The resulting algorithms is capable to ensure some relevant issue for an optimal refinement strategy: a refinement dependent only on the marked cells at each refinement step; a common improvement of the mesh quality during the refinement iterations; a bound of the number of unknowns of the discrete problem with the number of the cells in the mesh.…”