Conservative multi-material remap for staggered multi-material Arbitrary Lagrangian-Eulerian methods

Remapping is one of the essential parts of most multi-material Arbitrary Lagrangian-Eulerian (ALE) methods. In this paper, we present a new remapping approach in the framework of 2D staggered multi-material ALE on logically rectangular meshes. It is based on the computation of the second-order material mass fluxes (using intersections/overlays) to all neighboring cells, including the corner neighbors. Fluid mass is then remapped in a flux form as well as all other fluid quantities (internal energy, pressure). We pay a special attention to the remap of nodal quantities, performed also in a flux form. An optimization-based approach is used for the construction of the nodal mass fluxes. The flux-corrected remap (FCR) approach for flux limiting is employed for the nodal velocity remap, which enforces bound preservation of the remapped constructed velocity field. Several examples of numerical calculations are presented, which demonstrate properties of our remapping method in the context of a full ALE algorithm.

[1]  R. B. Pember,et al.  A Comparison of Staggered-Mesh Lagrange Plus Remap and Cell-Centered Direct Eulerian Godunov Schemes for Rulerian Shock Hydrodynamics , 2000 .

[2]  Mikhail Shashkov,et al.  Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/fld.1574 Closure models for multimaterial cells in arbitrary Lagrangian–Eulerian hydrocodes ‡ , 2022 .

[3]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

[4]  J. R. Kamm,et al.  A comparative study of various pressure relaxation closure models for one‐dimensional two‐material Lagrangian hydrodynamics , 2011 .

[5]  J. S. Peery,et al.  Multi-Material ALE methods in unstructured grids , 2000 .

[6]  P Anninos.  Kull ALE: II. Grid Motion on Unstructured Arbitrary Polyhedral Meshes , 2002 .

[7]  R. N. Hill,et al.  Constrained optimization framework for interface-aware sub-scale dynamics closure model for multimaterial cells in Lagrangian and arbitrary Lagrangian-Eulerian hydrodynamics , 2014, J. Comput. Phys..

[8]  Pierre-Henri Maire,et al.  Two-Step Hybrid Remapping (Conservative Interpolation) for Multimaterial Arbitrary Lagrangian-Eulerian Methods LA-UR-10-05438 , 2010 .

[9]  P. Anninos,et al.  Multiphase Advection and Radiation Diffusion with Material Interfaces on Unstructured Meshes , 2002 .

[10]  Jérôme Breil,et al.  Hybrid remap for multi-material ALE , 2011 .

[11]  D. Benson.  Computational methods in Lagrangian and Eulerian hydrocodes , 1992 .

[12]  J. Hyvärinen,et al.  An Arbitrary Lagrangian-Eulerian finite element method , 1998 .

[13]  Jérôme Breil,et al.  A two-dimensional unstructured cell-centered multi-material ALE scheme using VOF interface reconstruction , 2010, J. Comput. Phys..

[14]  R. Anderson,et al.  An arbitrary Lagrangian-Eulerian method with adaptive mesh refinement for the solution of the Euler equations , 2004 .

[15]  Joseph Falcovitz,et al.  VIP (Vector Image Polygon) multi-dimensional slope limiters for scalar variables , 2013 .

[16]  P. Knupp,et al.  Reference Jacobian optimization-based rezone strategies for arbitrary Lagrangian Eulerian methods , 2002 .

[17]  John K. Dukowicz,et al.  Incremental Remapping as a Transport/Advection Algorithm , 2000 .

[18]  J. Michael Owen,et al.  Arbitrary Lagrangian Eulerian remap treatments consistent with staggered compatible total energy conserving Lagrangian methods , 2013, J. Comput. Phys..

[19]  J. R. Kamm,et al.  Development of a sub-scale dynamics model for pressure relaxation of multi-material cells in Lagrangian hydrodynamics , 2010 .

[20]  Patrick M. Knupp,et al.  Winslow Smoothing on Two-Dimensional Unstructured Meshes , 1999, Engineering with Computers.

[21]  Raphaël Loubère,et al.  A subcell remapping method on staggered polygonal grids for arbitrary-Lagrangian-Eulerian methods , 2005 .

[22]  Mack Kenamond,et al.  Compatible, total energy conserving and symmetry preserving arbitrary Lagrangian-Eulerian hydrodynamics in 2D rz - Cylindrical coordinates , 2014, J. Comput. Phys..

[23]  Hervé Guillard,et al.  A five equation reduced model for compressible two phase flow problems , 2005 .

[24]  L. Margolin.  Introduction to “An Arbitrary Lagrangian-Eulerian Computing Method for All Flow Speeds” , 1997 .

[25]  Mikhail J. Shashkov,et al.  One-step hybrid remapping algorithm for multi-material arbitrary Lagrangian-Eulerian methods , 2012, J. Comput. Phys..

[26]  Gabi Luttwak.  Second order discrete rezoning , 2008 .

[27]  Bruno Després,et al.  Numerical resolution of a two-component compressible fluid model with interfaces , 2007 .

[28]  P Anninos.  Kull ALE: I. Unstructured Mesh Advection, Interface Capturing, and Multiphase 2T RHD with Material Interfaces , 2002 .

[29]  Mikhail Shashkov,et al.  Interface-aware sub-scale dynamics closure model , 2012 .

[30]  Mikhail J. Shashkov,et al.  Reconstruction of multi-material interfaces from moment data , 2008, J. Comput. Phys..

[31]  Jérôme Breil,et al.  Two-step hybrid conservative remapping for multimaterial arbitrary Lagrangian-Eulerian methods , 2011, J. Comput. Phys..

[32]  A. Huerta,et al.  Arbitrary Lagrangian–Eulerian Methods , 2004 .

[33]  Hyung Taek Ahn,et al.  Multi-material interface reconstruction on generalized polyhedral meshes , 2007, J. Comput. Phys..

[34]  G. B. Zimmerman,et al.  An algorithm for time evolving volume fractions in mixed zones in Lagrangian hydrodynamics calculations , 2009 .

[35]  James J. Quirk,et al.  On the dynamics of a shock–bubble interaction , 1994, Journal of Fluid Mechanics.

[36]  Richard Liska,et al.  Symmetry-preserving momentum remap for ALE hydrodynamics , 2013 .

[37]  Pierre-Henri Maire,et al.  A high-order one-step sub-cell force-based discretization for cell-centered Lagrangian hydrodynamics on polygonal grids , 2011 .

[38]  Mark de Berg,et al.  Computational geometry: algorithms and applications , 1997 .

[39]  Joseph O'Rourke,et al.  Computational Geometry in C. , 1995 .

[40]  M. Shashkov,et al.  The Construction of Compatible Hydrodynamics Algorithms Utilizing Conservation of Total Energy , 1998 .

[41]  S. Zalesak.  Fully multidimensional flux-corrected transport algorithms for fluids , 1979 .

[42]  H. Bruce Stewart,et al.  Two-phase flow: Models and methods , 1984 .

[43]  Raphaël Loubère,et al.  Staggered Lagrangian Discretization Based on Cell-Centered Riemann Solver and Associated Hydrodynamics Scheme , 2011 .

[44]  Rao V. Garimella,et al.  A comparative study of interface reconstruction methods for multi-material ALE simulations , 2010 .

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

[46]  Patrick M. Knupp,et al.  Fundamentals of Grid Generation , 2020 .

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

[48]  Pavel Váchal,et al.  Symmetry- and essentially-bound-preserving flux-corrected remapping of momentum in staggered ALE hydrodynamics , 2013, J. Comput. Phys..

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

[50]  Dimitri J. Mavriplis,et al.  Revisiting the Least-squares Procedure for Gradient Reconstruction on Unstructured Meshes , 2003 .

[51]  P Anninos.  New VOF interface capturing and reconstruction algorithms , 1999 .

[52]  Joseph Falcovitz,et al.  Slope limiting for vectors: A novel vector limiting algorithm , 2011 .