Preconditioning a Newton-Krylov solver for all-speed melt pool flow physics

Abstract In this paper, we introduce a multigrid block-based preconditioner for solving linear systems arising from a Discontinuous Galerkin discretization of the all-speed Navier-Stokes equations with phase change. The equations are discretized in conservative form with a reconstructed Discontinuous Galerkin (rDG) method and integrated with fully-implicit time discretization schemes. To robustly converge the numerically stiff systems, we use the Newton-Krylov framework with a primitive-variable formulation (pressure, velocity, and temperature), which is better conditioned than the conservative-variable form at low-Mach number. In the limit of large acoustic CFL number and viscous Fourier number, there is a strong coupling between the velocity-pressure system and the linear systems become non-diagonally dominant. To effectively solve these ill-conditioned systems, an approximate block factorization preconditioner is developed, which uses the Schur complement to reduce a 3 × 3 block system into a sequence of two 2 × 2 block systems: velocity-pressure, v P , and velocity-temperature, v T . We compare the performance of the v P - v T Schur complement preconditioner to classic preconditioning strategies: monolithic algebraic multigrid (AMG), element-block SOR, and primitive variable block Gauss-Seidel. The performance of the preconditioned solver is investigated in the limit of large CFL and Fourier numbers for low-Mach lid-driven cavity flow, Rayleigh-Benard melt convection, compressible internally heated convection, and 3D laser-induced melt pool flow. Numerical results demonstrate that the v P - v T Schur complement preconditioned solver scales well both algorithmically and in parallel, and is robust for highly ill-conditioned systems, for all tested rDG discretization schemes (up to 4 th -order).

[1]  J. Delplanque,et al.  High-order fully implicit solver for all-speed fluid dynamics , 2018, Shock Waves.

[2]  E. Turkel,et al.  Preconditioned methods for solving the incompressible low speed compressible equations , 1987 .

[3]  A. Chorin A Numerical Method for Solving Incompressible Viscous Flow Problems , 1997 .

[4]  Homer F. Walker,et al.  Choosing the Forcing Terms in an Inexact Newton Method , 1996, SIAM J. Sci. Comput..

[5]  Robert D. Falgout,et al.  hypre: A Library of High Performance Preconditioners , 2002, International Conference on Computational Science.

[6]  Hong Luo,et al.  Fully-Implicit Orthogonal Reconstructed Discontinuous Petrov-Galerkin Method for Multiphysics Problems , 2015 .

[7]  D. Keyes,et al.  Jacobian-free Newton-Krylov methods: a survey of approaches and applications , 2004 .

[8]  Zhanhua Ma,et al.  Solid velocity correction schemes for a temperature transforming model for convection phase change , 2006 .

[9]  John N. Shadid,et al.  A taxonomy and comparison of parallel block multi-level preconditioners for the incompressible Navier-Stokes equations , 2008, J. Comput. Phys..

[10]  M. Benzi Preconditioning techniques for large linear systems: a survey , 2002 .

[11]  C. Munz,et al.  The extension of incompressible flow solvers to the weakly compressible regime , 2003 .

[12]  John N. Shadid,et al.  A New Approximate Block Factorization Preconditioner for Two-Dimensional Incompressible (Reduced) Resistive MHD , 2013, SIAM J. Sci. Comput..

[13]  D. Spalding,et al.  A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows , 1972 .

[14]  Sergei I. Anisimov,et al.  Instabilities in Laser-Matter Interaction , 1995 .

[15]  Hong Luo,et al.  Fully-implicit orthogonal reconstructed Discontinuous Galerkin method for fluid dynamics with phase change , 2016, J. Comput. Phys..

[16]  A. Rubenchik,et al.  Laser powder-bed fusion additive manufacturing: Physics of complex melt flow and formation mechanisms of pores, spatter, and denudation zones , 2015, 1512.02593.

[17]  D. Korzekwa,et al.  Truchas – a multi-physics tool for casting simulation , 2009 .

[18]  George Karypis,et al.  METIS and ParMETIS , 2011, Encyclopedia of Parallel Computing.

[19]  Ionut Danaila,et al.  A Newton method with adaptive finite elements for solving phase-change problems with natural convection , 2014, J. Comput. Phys..

[20]  J. J. Moré,et al.  Estimation of sparse jacobian matrices and graph coloring problems , 1983 .

[21]  R. Gilgenbach S.I. Anisimov and V.A. Khokhlov, Instabilities in Laser-Matter Interaction , CRC Press, Boca Raton, FL (1995). 141 pages, $99.95 (U.S.)/120.00 (Foreign). ISBN 0–8493-8660–8. , 1996 .

[22]  Jan Vierendeels,et al.  Modelling of natural convection flows with large temperature differences : a benchmark problem for low Mach number solvers. Part 1, Reference solutions , 2005 .

[23]  Dana A. Knoll,et al.  On Preconditioning Newton-Krylov Methods in Solidifying Flow Applications , 2001, SIAM J. Sci. Comput..

[24]  Dana A. Knoll,et al.  Physics-Based Preconditioners for Ocean Simulation , 2013, SIAM J. Sci. Comput..

[25]  Hong Luo,et al.  A Reconstructed Discontinuous Galerkin Method Based on a Hierarchical Hermite WENO Reconstruction for Compressible Flows on Tetrahedral Grids , 2012 .

[26]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[27]  G. Batchelor,et al.  An Introduction to Fluid Dynamics , 1968 .

[28]  David E. Keyes,et al.  Newton-Krylov Methods for Low-Mach-Number Compressible Combustion , 1996 .

[29]  V. Voller,et al.  A fixed grid numerical modelling methodology for convection-diffusion mushy region phase-change problems , 1987 .

[30]  Christophe Eric Corre,et al.  Numerical simulations of a transient injection flow at low Mach number regime , 2008 .

[31]  R. Tuminaro,et al.  A parallel block multi-level preconditioner for the 3D incompressible Navier--Stokes equations , 2003 .

[32]  A. Ludwig,et al.  Simulation of time-dependent pool shape during laser spot welding: Transient effects , 2003 .

[33]  Luis Chacón,et al.  Jacobian–Free Newton–Krylov Methods for the Accurate Time Integration of Stiff Wave Systems , 2005, J. Sci. Comput..

[34]  Barry F. Smith,et al.  Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations , 1996 .

[35]  A. Stanier,et al.  A scalable, fully implicit algorithm for the reduced two-field low-β extended MHD model , 2016, J. Comput. Phys..

[36]  Katherine J. Evans,et al.  Development of a 2-D algorithm to simulate convection and phase transition efficiently , 2006, J. Comput. Phys..

[37]  C. L. Merkle,et al.  The application of preconditioning in viscous flows , 1993 .

[38]  V. E. Henson,et al.  BoomerAMG: a parallel algebraic multigrid solver and preconditioner , 2002 .

[39]  Lafayette K. Taylor,et al.  High-resolution viscous flow simulations at arbitrary Mach number , 2003 .

[40]  Philip L. Roe,et al.  Characteristic time-stepping or local preconditioning of the Euler equations , 1991 .

[41]  Paul Lin,et al.  Performance of fully coupled algebraic multilevel domain decomposition preconditioners for incompressible flow and transport , 2006 .

[42]  F. White Viscous Fluid Flow , 1974 .

[43]  E. Turkel,et al.  PRECONDITIONING TECHNIQUES IN COMPUTATIONAL FLUID DYNAMICS , 1999 .

[44]  Wayne A. Smith,et al.  Preconditioning Applied to Variable and Constant Density Flows , 1995 .

[45]  Hong Luo,et al.  A set of parallel, implicit methods for a reconstructed discontinuous Galerkin method for compressible flows on 3D hybrid grids , 2014 .

[46]  Hong Luo,et al.  A Hermite WENO reconstruction-based discontinuous Galerkin method for the Euler equations on tetrahedral grids , 2012, J. Comput. Phys..

[47]  Jonathan A. Dantzig,et al.  MODELLING LIQUID-SOLID PHASE CHANGES WITH MELT CONVECTION , 1989 .

[48]  M. Liou A Sequel to AUSM , 1996 .

[49]  Michael Pernice,et al.  A Multigrid-Preconditioned Newton-Krylov Method for the Incompressible Navier-Stokes Equations , 2001, SIAM J. Sci. Comput..

[50]  Van Emden Henson,et al.  Robustness and Scalability of Algebraic Multigrid , 1999, SIAM J. Sci. Comput..

[51]  Carol S. Woodward,et al.  Preconditioning Strategies for Fully Implicit Radiation Diffusion with Material-Energy Transfer , 2001, SIAM J. Sci. Comput..

[52]  William J. Rider,et al.  Physics-Based Preconditioning and the Newton-Krylov Method for Non-equilibrium Radiation Diffusion , 2000 .

[53]  John N. Shadid,et al.  Stabilization and scalable block preconditioning for the Navier-Stokes equations , 2012, J. Comput. Phys..

[54]  Paul Lin,et al.  Performance of fully coupled domain decomposition preconditioners for finite element transport/reaction simulations , 2005 .

[55]  Richard C. Martineau,et al.  On physics-based preconditioning of the Navier-Stokes equations , 2009, J. Comput. Phys..

[56]  Marcello Lappa,et al.  A mathematical and numerical framework for the analysis of compressible thermal convection in gases at very high temperatures , 2016, J. Comput. Phys..

[57]  S. Khairallah,et al.  Mesoscopic Simulation Model of Selective Laser Melting of Stainless Steel Powder , 2014 .

[58]  U. Ghia,et al.  High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method , 1982 .

[59]  Robert Nourgaliev,et al.  Recovery Discontinuous Galerkin Jacobian-Free Newton-Krylov Method for Multiphysics Problems , 2010 .

[60]  Hans De Sterck,et al.  Reducing Complexity in Parallel Algebraic Multigrid Preconditioners , 2004, SIAM J. Matrix Anal. Appl..

[61]  John N. Shadid,et al.  On a multilevel preconditioning module for unstructured mesh Krylov solvers: two-level Schwarz , 2002 .

[62]  H. Guillard,et al.  On the behaviour of upwind schemes in the low Mach number limit , 1999 .

[63]  David K. Gartling,et al.  A finite element method for low-speed compressible flows☆ , 2003 .

[64]  M. Liou,et al.  A New Flux Splitting Scheme , 1993 .

[65]  Per-Olof Persson,et al.  Newton-GMRES Preconditioning for Discontinuous Galerkin Discretizations of the Navier--Stokes Equations , 2008, SIAM J. Sci. Comput..

[66]  Meng-Sing Liou,et al.  A sequel to AUSM, Part II: AUSM+-up for all speeds , 2006, J. Comput. Phys..