R-Adaptive Reconnection-Based Arbitrary Lagrangian Eulerian Method-R-ReALE (Journal of Mathematical Study, 2015)
We present a new R-adaptive Arbitrary Lagrangian Eulerian (ALE) method,based on the reconnection-based ALE - ReALE methodology [5, 41, 42]. The main elementsin a standard ReALE method are: an explicit Lagrangian phase on an arbitrarypolygonal (in 2D) mesh, followed by a rezoning phase in which a new grid is defined,and a remapping phase in which the Lagrangian solution is transferred onto the newgrid. The rezoned mesh is smoothed by using one or several steps toward centroidalVoronoi tessellation, but it is not adapted to the solution in any way. We present anew R-adaptive ReALE method (R-ReALE, where R stands for Relocation). The newmethod is based on the following design principles. First, a monitor function (or errorindicator) based on Hessian of some flow parameter(s), is utilized. Second, the newalgorithm uses the equidistribution principle with respect to the monitor function ascriterion for defining an adaptive mesh. Third, centroidal Voronoi tessellation is usedfor the construction of the adaptive mesh. Fourth, we modify the raw monitor function(scale it to avoid extremely small and large cells and smooth it to create a smoothmesh), in order to utilize theoretical results related to centroidal Voronoi tessellation.In the R-ReALE method, the number of mesh cells is chosen at the beginning of the calculationand does not change with time, but the mesh is adapted according to the modifiedmonitor function during the rezone stage at each time step. We present all detailsrequired for implementation of the new adaptive R-ReALE method and demonstrateits performance relative to standard ReALE method on a series of numerical examples.
Keywords: Gas Dynamics; R-adaptation; Reconnection; ALE.