A new hybrid adaptive mesh algorithm based on Voronoi tessellations and equi-distribution principle: Algorithms and numerical experiments

• A new hybrid h-adaptive mesh algorithm with the capability to variable mesh topology.
• The algorithm is based on equi-distribution principle and Voronoi tessellations.
• The coarsening and refinement processes can be asynchronous or synchronous.
• A new double thresholds marking strategy and a local smoothing have been introduced.
• The hybrid algorithm can converge to arbitrary mesh resolution.

A new hybrid h-adaptive algorithm is design to optimize unstructured polygonal meshes in 2D based on Voronoi tessellations and its topological dual, Delaunay triangulations. The algorithm consists of several unique features: first, dynamic change of mesh topology (reconnection of nodes) is realized through refinement and coarsening iterations with specific feature (i.e. Voronoi diagram); second, two moving mesh strategies are included, one is the weighted global smoothing for mesh regulation which is based on centroidal Voronoi tessellations (CVTs), the other is a local smoothing for the purpose of speed-up of the adaptive iterations; third, a new marking strategy with double threshold values guarantee the convergence of adaptive iterations. The advantage of the hybrid algorithm is that the goal mesh resolution can converge to an arbitrarily small number; it asymptotically converges to a straight line in 2D (i.e., the volume equi-distribution error approaches zero). The h-adaptation is referred to as asynchronous process, if the processes of coarsening and refinement are connected through several global mesh regulation steps. Otherwise the hybrid adaptive process is synchronous. This will provide options to meet special requirement such as smooth transition of physical fields. The numerical results show that the numerical errors are indeed well distributed in an averaged sense in the domain, where the mesh sizes are optimal, and nodes of the mesh smoothly follow the underlying geometry or physics. The new hybrid algorithm demonstrates its potential for effectiveness, efficiency and robustness to handle generation and optimization of polygonal meshes for extreme scenarios with a wide spectrum of length scales.