Adaptive reconnection-based arbitrary Lagrangian Eulerian method

eW present a new adaptive Arbitrary Lagrangian Eulerian (ALE) method. This method is based on the reconnection-based ALE (ReALE) methodology of Refs. 35,34,6. The main elements in a standard ReALE method are: an explicit Lagrangian phase on an arbitrary polygonal (in 2D) mesh in which the solution and positions of grid nodes are updated; a rezoning phase in which a new grid is defined by changing the connectivity (using Voronoi tessellation) but not the number of cells; and a remapping phase in which the Lagrangian solution is transferred onto the new grid. In the standard ReALE method, the rezoned mesh is smoothed by using one or several steps toward centroidal Voronoi tessellation, but it is not adapted to the solution in any way.In the current paper we present a new adaptive ReALE method, A-ReALE, that is based on the following design principles. First, a monitor function (or error indicator) based on the Hessian of some flow parameter(s) is utilized. Second, an equi-distribution principle for the monitor function is used as a criterion for adapting the mesh. Third, a centroidal Voronoi tessellation is used to adapt the mesh. Fourth, we scale the monitor function to avoid very small and large cells and then smooth it to permit the use of theoretical results related to weighted centroidal Voronoi tessellation.In the A-ReALE method, both number of cells and their locations are allowed to change at the rezone stage on each time step. The number of generators at each time step is chosen to guarantee the required spatial resolution in regions where monitor function reaches its maximum value.We present all details required for implementation of new adaptive A-ReALE method and demonstrate its performance in comparison with standard ReALE method on series of numerical examples.

[1]  Desheng Wang,et al.  Tetrahedral mesh generation and optimization based on centroidal Voronoi tessellations , 2003 .

[2]  Allen Gersho,et al.  Asymptotically optimal block quantization , 1979, IEEE Trans. Inf. Theory.

[3]  G. Sod.  Numerical methods in fluid dynamics , 1985 .

[4]  Pierre-Henri Maire,et al.  A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes , 2009, J. Comput. Phys..

[5]  Lili Ju,et al.  Adaptive Anisotropic Meshing For Steady Convection-Dominated Problems , 2009 .

[6]  Dinshaw S. Balsara,et al.  Maintaining Pressure Positivity in Magnetohydrodynamic Simulations , 1999 .

[7]  C. W. Hirt,et al.  An Arbitrary Lagrangian-Eulerian Computing Method for All Flow Speeds , 1997 .

[8]  Gunzburger,et al.  Advances in Studies and Applications of Centroidal Voronoi Tessellations , 2010 .

[9]  M. Shashkov,et al.  An efficient linearity-and-bound-preserving remapping method , 2003 .

[10]  John K. Dukowicz,et al.  Accurate conservative remapping (rezoning) for arbitrary Lagrangian-Eulerian computations , 1987 .

[11]  W. Heilig,et al.  Experimental and numerical study of the interaction between a planar shock wave and a square cavity , 1996 .

[12]  Frédéric Alauzet,et al.  Time accurate anisotropic goal-oriented mesh adaptation for unsteady flows , 2012, J. Comput. Phys..

[13]  Timothy J. Barth,et al.  Numerical Methods for Gasdynamic Systems on Unstructured Meshes , 1997, Theory and Numerics for Conservation Laws.

[14]  Stéphane Del Pino,et al.  Metric-based mesh adaptation for 2D Lagrangian compressible flows , 2011, J. Comput. Phys..

[15]  Frédéric Alauzet.  High‐order methods and mesh adaptation for Euler equations , 2008 .

[16]  Q. Du,et al.  Recent progress in robust and quality Delaunay mesh generation , 2006 .

[17]  Raphaël Loubère,et al.  ReALE: A Reconnection Arbitrary-Lagrangian―Eulerian method in cylindrical geometry , 2011 .

[18]  Pascal J. Frey,et al.  Fast Adaptive Quadtree Mesh Generation , 1998, IMR.

[19]  W. Bo and M. Shashkov.  R-Adaptive Reconnection-based Arbitrary Lagrangian Eulerian Method-R-ReALE , 2015 .

[20]  P. Gruber,et al.  Optimum Quantization and Its Applications , 2004 .

[21]  J. MacQueen.  Some methods for classification and analysis of multivariate observations , 1967 .

[22]  Jérôme Breil,et al.  A multi-material ReALE method with MOF interface reconstruction , 2013 .

[23]  Bruno Després,et al.  A cell-centered Lagrangian hydrodynamics scheme on general unstructured meshes in arbitrary dimension , 2009, J. Comput. Phys..

[24]  Robert D. Russell,et al.  Adaptivity with moving grids , 2009, Acta Numerica.

[25]  E. Dorfi,et al.  Simple adaptive grids for 1-d initial value problems , 1987 .

[26]  Ralph R. Martin,et al.  Improved initialisation for centroidal Voronoi tessellation and optimal Delaunay triangulation , 2012, Comput. Aided Des..

[27]  Raphaël Loubère,et al.  ReALE: A reconnection-based arbitrary-Lagrangian-Eulerian method , 2010, J. Comput. Phys..

[28]  François Vilar,et al.  A high-order Discontinuous Galerkin discretization for solving two-dimensional Lagrangian hydrodynamics , 2012 .

[29]  C Thompson,et al.  Applied CFD techniques: An introduction based on finite element methods , 2002 .

[30]  Pierre Alliez,et al.  2D Centroidal Voronoi Tessellations with Constraints , 2010 .

[31]  Qiang Du,et al.  Probabilistic methods for centroidal Voronoi tessellations and their parallel implementations , 2002, Parallel Comput..

[32]  Qiang Du,et al.  Centroidal Voronoi Tessellations: Applications and Algorithms , 1999, SIAM Rev..

[33]  Max Gunzburger,et al.  Constrained CVT meshes and a comparison of triangular mesh generators , 2009, Comput. Geom..

[34]  Chenglei Yang,et al.  On centroidal voronoi tessellation—energy smoothness and fast computation , 2009, TOGS.

[35]  J. N. Johnson,et al.  A new parallel algorithm for constructing Voronoi tessellations from distributed input data , 2014, Comput. Phys. Commun..

[36]  Mikhail Shashkov,et al.  The Error-Minimization-Based Strategy for Moving Mesh Methods , 2006 .

[37]  Rémi Abgrall,et al.  A Cell-Centered Lagrangian Scheme for Two-Dimensional Compressible Flow Problems , 2007, SIAM J. Sci. Comput..

[38]  Jeffrey Grandy.  Conservative Remapping and Region Overlays by Intersecting Arbitrary Polyhedra , 1999 .

[39]  Weidong Zhao,et al.  Adaptive Finite Element Methods for Elliptic PDEs Based on Conforming Centroidal Voronoi-Delaunay Triangulations , 2006, SIAM J. Sci. Comput..

[40]  R. Löhner.  An adaptive finite element scheme for transient problems in CFD , 1987 .

[41]  Jérôme Breil,et al.  A swept‐intersection‐based remapping method in a ReALE framework , 2013 .

[42]  Franz Aurenhammer,et al.  Voronoi diagrams—a survey of a fundamental geometric data structure , 1991, CSUR.

[43]  Tamal K. Dey,et al.  Delaunay Mesh Generation , 2012, Chapman and Hall / CRC computer and information science series.

[44]  Tao Tang,et al.  Adaptive Mesh Methods for One- and Two-Dimensional Hyperbolic Conservation Laws , 2003, SIAM J. Numer. Anal..

[45]  Robert D. Russell,et al.  Adaptive Moving Mesh Methods , 2010 .

[46]  Atsuyuki Okabe,et al.  Spatial Tessellations: Concepts and Applications of Voronoi Diagrams , 1992, Wiley Series in Probability and Mathematical Statistics.