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Preconditioning a Newton-Krylov solver for all-speed melt pool flow physics. (English) Zbl 1453.76084
Summary: 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 \(3 \times 3\) block system into a sequence of two \(2 \times 2\) block systems: velocity-pressure, \(\mathbf{v} P\), and velocity-temperature, \(\mathbf{v} T\). We compare the performance of the \(\mathbf{v} P\)-\(\mathbf{v} T\) 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-Bénard melt convection, compressible internally heated convection, and 3D laser-induced melt pool flow. Numerical results demonstrate that the \(\mathbf{v} P\)-\(\mathbf{v} T\) 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 4th-order).
MSC:
76M10 Finite element methods applied to problems in fluid mechanics
65F08 Preconditioners for iterative methods
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76N06 Compressible Navier-Stokes equations
80A22 Stefan problems, phase changes, etc.
80A17 Thermodynamics of continua
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References:
[1] (2013), ALE3D Web Page
[2] Anderson, J. D.; Wendt, J., Computational Fluid Dynamics, vol. 206 (1995), Springer
[3] Anisimov, S.; V, K., Instabilities in laser-matter interaction, Laser Part. Beams, 14, 4, 797 (1996)
[4] Balay, S.; Buschelman, K.; Eijkhout, V.; Gropp, W. D.; Kaushik, D.; Knepley, M. G.; McInnes, L. C.; Smith, B. F.; Zhang, H., PETSc Users Manual (2004), Argonne National Laboratory, Technical Report ANL-95/11 - Revision 2.1.5
[5] Batchelor, G. K., An Introduction to Fluid Dynamics (2000), Cambridge university press · Zbl 0152.44402
[6] Beccantini, A.; Studer, E.; Gounand, S.; Magnaud, J.-P.; Kloczko, T.; Corre, C.; Kudriakov, S., Numerical simulations of a transient injection flow at low Mach number regime, Int. J. Numer. Methods Eng., 76, 5, 662-696 (2008) · Zbl 1195.76334
[7] Benzi, M., Preconditioning techniques for large linear systems: a survey, J. Comput. Phys., 182, 2, 418-477 (2002) · Zbl 1015.65018
[8] Briley, W.; Taylor, L.; Whitfield, D., High-resolution viscous flow simulations at arbitrary Mach number, J. Comput. Phys., 184, 1, 79-105 (2003) · Zbl 1118.76338
[9] Brown, P. N.; Woodward, C. S., Preconditioning strategies for fully implicit radiation diffusion with material-energy transfer, SIAM J. Sci. Comput., 23, 2, 499-516 (2001) · Zbl 0992.65102
[10] Chacón, L.; Stanier, A., A scalable, fully implicit algorithm for the reduced two-field low-βextended MHD model, J. Comput. Phys., 326, 763-772 (2016) · Zbl 1373.76340
[11] Choi, Y.-H.; Merkle, C. L., The application of preconditioning in viscous flows, J. Comput. Phys., 105, 2, 207-223 (1993) · Zbl 0768.76032
[12] Chorin, A. J., A numerical method for solving incompressible viscous flow problems, J. Comput. Phys., 135, 2, 118-125 (1997) · Zbl 0899.76283
[13] Cleary, A.; Falgout, R.; Henson, V.; Jones, J.; Manteuffel, T.; McCormick, S.; Miranda, G.; Ruge, J., Robustness and scalability of algebraic multigrid, SIAM J. Sci. Comput., 21, 5, 1886-1908 (2000) · Zbl 0959.65049
[14] Coleman, T. F.; Moré, J. J., Estimation of sparse Jacobian matrices and graph coloring problems, SIAM J. Numer. Anal., 20, 1, 187-209 (1983) · Zbl 0527.65033
[15] Cyr, E. C.; Shadid, J. N.; Tuminaro, R. S., Stabilization and scalable block preconditioning for the Navier-Stokes equations, J. Comput. Phys., 231, 2, 345-363 (2012) · Zbl 1426.76241
[16] Cyr, E. C.; Shadid, J. N.; Tuminaro, R. S.; Pawlowski, R. P.; Chacón, L., A new approximate block factorization preconditioner for two-dimensional incompressible (reduced) resistive MHD, SIAM J. Sci. Comput., 35, 3, B701-B730 (2013) · Zbl 1273.76269
[17] Danaila, I.; Moglan, R.; Hecht, F.; Masson, S. L., A Newton method with adaptive finite elements for solving phase-change problems with natural convection, J. Comput. Phys., 274, 826-840 (2014) · Zbl 1351.76056
[18] Dantzig, J. A., Modelling liquid – solid phase changes with melt convection, Int. J. Numer. Methods Eng., 28, 8, 1769-1785 (1989)
[19] De Sterck, H.; Yang, U.; Heys, J., Reducing complexity in parallel algebraic multigrid preconditioners, SIAM J. Matrix Anal. Appl., 27, 4, 1019-1039 (2006) · Zbl 1102.65034
[20] Ehlen, G.; Ludwig, A.; Sahm, P. R., Simulation of time-dependent pool shape during laser spot welding: transient effects, Metall. Trans. A, 34, 12, 2947-2961 (2003)
[21] Eisenstat, S. C.; Walker, H. F., Choosing the forcing terms in an inexact Newton method, SIAM J. Sci. Comput., 17, 1, 16-32 (1995) · Zbl 0845.65021
[22] Elman, H.; Howle, V. E.; Shadid, J.; Shuttleworth, R.; Tuminaro, R., A taxonomy and comparison of parallel block multi-level preconditioners for the incompressible Navier-Stokes equations, J. Comput. Phys., 227, 3, 1790-1808 (2008) · Zbl 1290.76023
[23] Elman, H. C.; Howle, V. E.; Shadid, J. N.; Tuminaro, R. S., A parallel block multi-level preconditioner for the 3d incompressible Navier-Stokes equations, J. Comput. Phys., 187, 2, 504-523 (2003) · Zbl 1061.76058
[24] Evans, K. J.; Knoll, D. A.; Pernice, M., Development of a 2-D algorithm to simulate convection and phase transition efficiently, J. Comput. Phys., 219, 1, 404-417 (2006) · Zbl 1102.76037
[25] Falgout, R. D.; Yang, U. M., HYPRE: a library of high performance preconditioners, (International Conference on Computational Science (2002), Springer), 632-641 · Zbl 1056.65046
[26] Ghia, U.; Ghia, K.; Shin, C., High-Re solutions for incompressible flow using the Navier-Stokes equations and a Multigrid method, J. Comput. Phys., 48, 347-411 (1982) · Zbl 0511.76031
[27] Guillard, H.; Viozat, C., On the behaviour of upwind schemes in the low Mach number limit, Comput. Fluids, 28, 1, 63-86 (1999) · Zbl 0963.76062
[28] Henson, V.; Yang, U., BoomerAMG: a parallel algebraic multigrid solver and preconditioner, Developments and Trends in Iterative Methods for Large Systems of Equations - In Memorium Rudiger Weiss. Developments and Trends in Iterative Methods for Large Systems of Equations - In Memorium Rudiger Weiss, Appl. Numer. Math., 41, 1, 155-177 (2002) · Zbl 0995.65128
[29] Karypis, G., Metis and parmetis, (Encyclopedia of Parallel Computing (2011), Springer), 1117-1124
[30] Khairallah, S. A.; Anderson, A. T., Mesoscopic simulation model of selective laser melting of stainless steel powder, J. Mater. Process. Technol., 214, 2627-2636 (2014)
[31] Khairallah, S. A.; Anderson, A. T.; Rubenchik, A.; King, W. E., Laser powder-bed fusion additive manufacturing: physics of complex melt flow and formation mechanisms of pores, spatter, and denudation zones, Acta Mater., 108, 36-45 (2016)
[32] Knoll, D.; Keyes, D., Jacobian-free Newton-Krylov methods: a survey of approaches and applications, J. Comput. Phys., 193, 2, 357-397 (2004) · Zbl 1036.65045
[33] Knoll, D.; McHugh, P.; Keyes, D., Newton-Krylov methods for low-Mach-number compressible combustion, AIAA J., 34, 5, 961-967 (1996) · Zbl 0900.76406
[34] Knoll, D.; Vanderheyden, W.; Mousseau, V.; Kothe, D., On preconditioning Newton-Krylov methods in solidifying flow applications, SIAM J. Sci. Comput., 23, 2, 381-397 (2001) · Zbl 1125.65315
[35] Knoll, D. A.; Mousseau, V.; Chacón, L.; Reisner, J., Jacobian-Free Newton-Krylov methods for the accurate time integration of Stiff wave systems, J. Sci. Comput., 25, 1, 213-230 (2005) · Zbl 1203.65071
[36] Korzekwa, D., Truchas – a multi-physics tool for casting simulation, Int. J. Cast Met. Res., 22, 1-4, 187-191 (2009)
[37] Lappa, M., A mathematical and numerical framework for the analysis of compressible thermal convection in gases at very high temperatures, J. Comput. Phys., 313, 687-712 (2016) · Zbl 1349.76773
[38] Le Quéré, P.; Weisman, C.; Paillère, H.; Vierendeels, J.; Dick, E.; Becker, R.; Braack, M.; Locke, J., Modelling of natural convection flows with large temperature differences: a benchmark problem for low Mach number solvers, part 1: reference solutions, Modél. Math. Anal. Numér., 39, 03, 609-616 (2005) · Zbl 1130.76047
[39] Lin, P.; Sala, M.; Shadid, J.; Tuminaro, R. S., Performance of fully coupled algebraic multilevel domain decomposition preconditioners for incompressible flow and transport, Int. J. Numer. Methods Eng., 67, 208-225 (2006) · Zbl 1110.76315
[40] Liou, M.-S., A sequel to AUSM: AUSM+, J. Comput. Phys., 129, 2, 364-382 (1996) · Zbl 0870.76049
[41] Liou, M.-S., A sequel to AUSM, part II: AUSM+-up for all speeds, J. Comput. Phys., 214, 1, 137-170 (2006) · Zbl 1137.76344
[42] Liou, M.-S.; Steffen, C. J., A new flux splitting scheme, J. Comput. Phys., 107, 1, 23-39 (1993) · Zbl 0779.76056
[43] Luo, H.; Xia, Y.; Li, S.; Nourgaliev, R.; Cai, C., A Hermite WENO reconstruction-based discontinuous Galerkin method for the Euler equations on tetrahedral grids, J. Comput. Phys., 231, 5489-5502 (2012) · Zbl 1426.76288
[44] Luo, H.; Xia, Y.; Spiegel, S.; Nourgaliev, R.; Jiang, Z., A reconstructed discontinuous Galerkin method based on a hierarchical WENO reconstruction for compressible flows on tetrahedral grids, J. Comput. Phys., 236, 477-492 (2013) · Zbl 1286.65125
[45] Ma, Z.; Zhang, Y., Solid velocity correction schemes for a temperature transforming model for convection phase change, Int. J. Numer. Methods Heat Fluid Flow, 16, 2, 204-225 (2006)
[46] Martinez, M. J.; Gartling, D. K., A finite element method for low-speed compressible flows, Comput. Methods Appl. Mech. Eng., 193, 21, 1959-1979 (2004) · Zbl 1067.76062
[47] Mousseau, V.; Knoll, D.; Rider, W., Physics-based preconditioning and the Newton-Krylov method for non-equilibrium radiation diffusion, J. Comput. Phys., 160, 2, 743-765 (2000) · Zbl 0949.65092
[48] Munz, C.-D.; Roller, S.; Klein, R.; Geratz, K. J., The extension of incompressible flow solvers to the weakly compressible regime, Comput. Fluids, 32, 2, 173-196 (2003) · Zbl 1042.76045
[49] Newman, C.; Knoll, D. A., Physics-based preconditioners for ocean simulation, SIAM J. Sci. Comput., 35, 5, S445-S464 (2013) · Zbl 1406.86005
[50] Nourgaliev, R.; Greene, P.; Weston, B.; Barney, R.; Anderson, A.; Khairallah, S.; Delplanque, J.-P., High-order fully-implicit solver for all-speed fluid dynamics: AUSM ride from nearly-incompressible variable-density flows to shock dynamics, Int. J. Shock Waves Deton. Explos. (2019)
[51] Nourgaliev, R.; Luo, H.; Schofield, S.; Dunn, T.; Anderson, A.; Weston, B.; Delplanque, J.-P., Fully-Implicit Orthogonal Reconstructed Discontinuous Petrov-Galerkin Method for Multiphysics Problems (2015), Lawrence Livermore National Laboratory: Lawrence Livermore National Laboratory Livermore, USA, Technical Report LLNL-TR-664250 · Zbl 1349.76248
[52] Nourgaliev, R.; Luo, H.; Weston, B.; Anderson, A.; Schofield, S.; Dunn, T.; Delplanque, J.-P., Fully-implicit orthogonal reconstructed Discontinuous Galerkin method for fluid dynamics with phase change, J. Comput. Phys., 305, 964-996 (2016) · Zbl 1349.76248
[53] Nourgaliev, R.; Park, H.; Mousseau, V., Recovery discontinuous Galerkin Jacobian-Free Newton-Krylov method for multiphysics problems, (Computational Fluid Dynamics Review (2010))
[54] Park, H.; Nourgaliev, R.; Martineau, R. C.; Knoll, D. A., On physics-based preconditioning of the Navier-Stokes equations, J. Comput. Phys., 228, 24, 9131-9146 (2009) · Zbl 1395.65028
[55] Patankar, S. V.; Spalding, D. B., A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows, Int. J. Heat Mass Transf., 15, 10, 1787-1806 (1972) · Zbl 0246.76080
[56] Pernice, M.; Tocci, M., A multigrid-preconditioned Newton-Krylov method for the incompressible Navier-Stokes equations, SIAM J. Sci. Comput., 23, 2, 398-418 (2001) · Zbl 0995.76061
[57] Persson, P.-O.; Peraire, J., Newton-GMRES preconditioning for discontinuous Galerkin discretizations of the Navier-Stokes equations, SIAM J. Sci. Comput., 30, 6, 2709-2733 (2008) · Zbl 1362.76052
[58] Saad, Y., Iterative Methods for Sparse Linear Systems (2003), SIAM: SIAM Philadelphia · Zbl 1002.65042
[59] Saad, Y.; Schultz, M., GMRES: A Generalized Minimal Residual algorithm for solving linear systems, SIAM J. Sci. Stat. Comput., 7, 856-869 (1986) · Zbl 0599.65018
[60] Shadid, J.; Tuminaro, R.; Devine, K.; Hennigan, G.; Lin, P., Performance of fully coupled domain decomposition preconditioners for finite element transport/reaction simulations, J. Comput. Phys., 205, 1, 24-47 (2005) · Zbl 1087.76069
[61] Smith, B.; Bjorstad, P.; Gropp, W., Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations (2004), Cambridge university press
[62] Tidriri, M. D., Hybrid Newton-Krylov/domain decomposition methods for compressible flows, (Proceedings of the Ninth International Conference on Domain Decomposition Methods in Sciences and Engineering (1998)), 532-539
[63] Trefethen, L. N.; Bau, D., Numerical Linear Algebra, vol. 50 (1997), SIAM
[64] Tuminaro, R.; Tong, C.; Shadid, J.; Devine, K.; Day, D., On a multilevel preconditioning module for unstructured mesh Krylov solvers: two-level Schwarz, Commun. Numer. Methods Eng., 18, 6, 383-389 (2002) · Zbl 0999.65101
[65] Turkel, E., Preconditioned methods for solving the incompressible and low speed compressible equations, J. Comput. Phys., 72, 2, 277-298 (1987) · Zbl 0633.76069
[66] Turkel, E., Preconditioning techniques in computational fluid dynamics, Annu. Rev. Fluid Mech., 31, 1, 385-416 (1999)
[67] van Leer, B.; Roe, P.; Lee, W.-T., Characteristic time-stepping or local preconditioning of the Euler equations, AIAA J. (1991)
[68] Voller, V.; Prakash, C., A fixed grid numerical modelling methodology for convection-diffusion mushy region phase-change problems, Int. J. Heat Mass Transf., 30, 8, 1709-1719 (1987)
[69] Weiss, J. M.; Smith, W. A., Preconditioning applied to variable and constant density flows, AIAA J., 33, 11, 2050-2057 (1995) · Zbl 0849.76072
[70] White, F., Viscous Fluid Flow, McGraw-Hill Series in Mechanical Engineering (1991), McGraw-Hill
[71] Xia, Y.; Luo, H.; Nourgaliev, R., An implicit hermite WENO reconstruction-based discontinuous Galerkin on tetrahedral grids, Comput. Fluids, 98, 134-151 (2014)
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