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Scalable implicit incompressible resistive MHD with stabilized FE and fully-coupled Newton-Krylov-AMG. (English) Zbl 1423.76275

Summary: The computational solution of the governing balance equations for mass, momentum, heat transfer and magnetic induction for resistive magnetohydrodynamics (MHD) systems can be extremely challenging. These difficulties arise from both the strong nonlinear, nonsymmetric coupling of fluid and electromagnetic phenomena, as well as the significant range of time- and length-scales that the interactions of these physical mechanisms produce. This paper explores the development of a scalable, fully-implicit stabilized unstructured finite element (FE) capability for 3D incompressible resistive MHD. The discussion considers the development of a stabilized FE formulation in context of the variational multiscale (VMS) method, and describes the scalable implicit time integration and direct-to-steady-state solution capability. The nonlinear solver strategy employs Newton-Krylov methods, which are preconditioned using fully-coupled algebraic multilevel preconditioners. These preconditioners are shown to enable a robust, scalable and efficient solution approach for the large-scale sparse linear systems generated by the Newton linearization. Verification results demonstrate the expected order-of-accuracy for the stabilized FE discretization. The approach is tested on a variety of prototype problems, that include MHD duct flows, an unstable hydromagnetic Kelvin-Helmholtz shear layer, and a 3D island coalescence problem used to model magnetic reconnection. Initial results that explore the scaling of the solution methods are also presented on up to 128K processors for problems with up to 1.8B unknowns on a CrayXK7.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
76W05 Magnetohydrodynamics and electrohydrodynamics
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[1] Goedbloed, H.; Poedts, S., Principles of magnetohydrodynamics with applications to laboratory and astrophysical plasmas, (2004), Cambridge Univ. Press
[2] Dai, W.; Woodward, P. R., A simple finite difference scheme for multidimensional magnetohydrodynamic equations, J. Comput. Phys., 142, 331, (1998) · Zbl 0932.76048
[3] Ryu, D.; Miniati, F.; Jones, T. W.; Frank, A., A divergence-free upwind code for multi-dimensional magnetohydrodynamics flows, Astrophys. J., 509, 244, (1998)
[4] Balsara, D. S., Divergence-free adaptive mesh refinement for magnetohydrodynamics, J. Comput. Phys., 174, 614-648, (2001) · Zbl 1157.76369
[5] Tóth, G., The \(\nabla \cdot \mathbf{B} = 0\) constraint in shock-capturing magnetohydrodynamics codes, J. Comput. Phys., 161, 605-652, (2000) · Zbl 0980.76051
[6] Tóth, G.; Keppens, R.; Botchev, M. A., Implicit and semi-implicit schemes in the versatile advection code: numerical tests, Astronom. Astrophys., 332, 1159-1170, (1998)
[7] Keppens, R.; Tóth, G.; Botchev, M. A.; Ploeg, A. V.D., Implicit and semi-implicit schemes: algorithms, Internat. J. Numer. Methods Fluids, 30, 335-352, (1999) · Zbl 0951.76059
[8] Aydemir, A. Y.; Barnes, D. C., An implicit algorithm for compressible three-dimensional magnetohydrodynamic calculations, J. Comput. Phys., 59, 1, 108-119, (1985) · Zbl 0568.76112
[9] Park, W.; Breslau, J.; Chen, J.; Fu, G. Y.; Jardin, S. C.; Klasky, S.; Menard, J.; Pletzer, A.; Stratton, B. C.; Stutman, D.; Strauss, H. R.; Sugiyama, L. E., Nonlinear simulation studies of tokamaks and STS, Nucl. fusion, 43, 6, 483-489, (2003)
[10] Jardin, S. C.; Breslau, J. A., Implicit solution of the four-field extended-magnetohydrodynamic equations using high-order high-continuity finite elements, Phys. Plasmas, 12, 5, (2005)
[11] Harned, D. S.; Kerner, W., Semi-implicit method for three-dimensional compressible magnetohydrodynamic simulation, J. Comput. Phys., 60, 62-75, (1985) · Zbl 0581.76057
[12] Schnack, D. D.; Barnes, D. C.; Harned, D. S.; Caramana, E. J., Semi-implicit magnetohydrodynamic calculations, J. Comput. Phys., 70, 330-354, (1987) · Zbl 0615.76109
[13] Harned, D. S.; Mikic, Z., Accurate semi-implicit treatment of the Hall effect in magnetohydrodynamic computations, J. Comput. Phys., 83, 1-15, (1989) · Zbl 0672.76051
[14] Sovinec, C. R.; Glasser, A. H.; Gianakon, T. A.; Barnes, D. C.; Nebel, R. A.; Kruger, S. E.; Schnack, D. D.; Plimpton, S. J.; Tarditi, A.; Chu, M. S., Nonlinear magnetohydrodynamics simulation using high-order finite elements, J. Comput. Phys., 195, 1, 355-386, (2004) · Zbl 1087.76070
[15] Hujeirat, A., IRMHD: an implicit radiative and magnetohydrodynamical solver for self-gravitating systems, Mon. Not. R. Astron. Soc., 298, 310-320, (1998)
[16] A.C. Robinson, et al. ALEGRA: An arbitrary Lagrangian-Eulerian multimaterial, multiphysics code, in: AIAA 2008-1235 46th AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, 2008.
[17] G. Toth, R. Keppens, Versatile advection code, http://www.phys.uu.nl/ toth/.
[18] Marinak, M. M.; Kerbel, G. D.; Gentile, N. A.; Jones, O.; Munro, S. P.D.; Dittrich, T. R.; Haan, S. W., Three-dimensional hydra simulations of national ignition facility targets, Phys. Plasmas, 8, 2275-2280, (2001)
[19] Chacón, L.; Knoll, D. A.; Finn, J. M., Implicit, nonlinear reduced resistive MHD nonlinear solver, J. Comput. Phys., 178, 1, 15-36, (2002) · Zbl 1139.76328
[20] Chacón, L.; Knoll, D. A.; Finn, J. M., An implicit nonlinear reduced resistive MHD solver, J. Comput. Phys., 178, 15-36, (2002) · Zbl 1139.76328
[21] Shumlak, U.; Loverich, J., Approximate Riemann solver for the two-fluid plasma model, J. Comput. Phys., 187, 620-638, (2003) · Zbl 1061.76526
[22] Chacón, L., A non-staggered, conservative, \(\nabla \cdot \mathbf{B} = \mathbf{0}\), finite-volume scheme for 3D implicit extended magnetohydrodynamics in curvilinear geometries, Comput. Phys. Comm., 163, 143-171, (2004) · Zbl 1196.76040
[23] Knoll, D. A.; Chacón, L., Coalescence of magnetic islands in the low-resistivity, Hall-MHD regime, Phys. Rev. Lett., 96, 13, 135001-135004, (2006)
[24] Chacón, L., An optimal, parallel, fully implicit Newton-Krylov solver for three-dimensional visco-resistive magnetohydrodynamics, Phys. Plasmas, 15, (2008)
[25] Shadid, J. N.; Pawlowski, R. P.; Banks, J. W.; Chacón, L.; Lin, P. T.; Tuminaro, R. S., Towards a scalable fully-implicit fully-coupled resistive MHD formulation with stabilized FE methods, J. Comput. Phys., 229, 20, 7649-7671, (2010) · Zbl 1425.76312
[26] Shumlak, U.; Lilly, R.; Reddell, N.; Sousa, E.; Srinivasan, B., Advanced physics calculations using a multi-fluid plasma model, Comput. Phys. Comm., 182, 1767-1770, (2011)
[27] S.C. Jardin, Review of implicit methods for the magnetohydrodynamic description of magnetically confined plasmas, J. Comput. Phys. 231 822. · Zbl 1452.76002
[28] Nedelec, J., Mixed finite elements in \(\mathbf{R}^3\), Numer. Math., 35, 315-341, (1980) · Zbl 0419.65069
[29] Evans, C.; Hawley, J., Simulation of magnetohydrodynamic flows: a constrained transport method, Astrophys. J., 332, 659-677, (1988)
[30] Hyman, J.; Shashkov, M., Adjoint operators for the natural discretizations of the divergence, gradient and curl on logically rectangular grids, Appl. Numer. Math., 25, 413-442, (1997) · Zbl 1005.65024
[31] Bochev, P. B.; Robinson, A. C., Matching algorithms with physics: exact sequences of finite element spaces, (Estep, D.; Tavener, S., Preservation of Stability under Discretization, (2001), Colorado State University, SIAM Philadelphia), 145-165
[32] P.B. Bochev, J.J. Hu, A.C. Robinson, R.S. Tuminaro, Towards robust 3D Z-pinch simulations: discretization and fast solvers for magnetic diffusion in heterogeneous conductors, Electron. Trans. Numer. Anal. 15, http://etna.msc.kent.edu. Special issue for the Tenth Copper Mountain Conference on Multigrid Methods. · Zbl 1201.76041
[33] Schotzau, D., Mixed finite element methods for stationary incompressible magneto-hydrodynamics, Numer. Math., 96, 771-800, (2004) · Zbl 1098.76043
[34] Chacón, L.; Knoll, D. A.; Finn, J. M., Hall MHD effects in the 2-d Kelvin-Helmholtz/tearing instability, Phys. Lett. A, 308, 187-197, (2003) · Zbl 1086.81559
[35] Lankalapalli, S.; Flaherty, J.; Shephard, M.; Strauss, H., An adaptive finite element method for magnetohydrodynamics, J. Comput. Phys., 225, 363-381, (2007) · Zbl 1118.76039
[36] Gunzburger, M.; Meir, A.; Peterson, J., On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary, incompressible magnetohydrodynamics, Math. Comp., 56, 3, 523-563, (1991) · Zbl 0731.76094
[37] Gerbeau, J.-F., A stabilized finite element method for the incompressible magnetohydrodynamic equations, Numer. Math., 87, 83-111, (2000) · Zbl 0988.76050
[38] Costabel, M.; Dauge, M., Singularities of electromagnetic fields in polyhedral domains, Arch. Ration. Mech. Anal., 151, 3, 221-276, (2000) · Zbl 0968.35113
[39] Costabel, M.; Dauge, M., Weighted regularization of Maxwell equations in polyhedral domains, Numer. Math., 93, 2, 239-277, (2002) · Zbl 1019.78009
[40] Salah, N. B.; Soulaimani, A.; Habashi, W. G.; Fortin, M., A conservative stabilized finite element method for the magento-hydrodyanamics equations, Internat. J. Numer. Methods Fluids, 29, 535-554, (1999) · Zbl 0938.76049
[41] Codina, R.; Hernández-Silva, N., Stabilized finite element approximation of the stationary magento-hydrodyanamics equations, Comput. Mech., 38, 344-355, (2006) · Zbl 1160.76025
[42] Codina, R.; Hernández-Silva, N., Approximation of the thermally coupled MHD problem using a stabilized finite element method, J. Comput. Phys., 230, 1281-1303, (2011) · Zbl 1391.76317
[43] Brezzi, F., On existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers, RAIRO Model. Math. Anal. Numer., 21, 129-151, (1974) · Zbl 0338.90047
[44] Gunzburger, M., Finite element methods for viscous incompressible flows, (1989), Academic Press Boston · Zbl 0697.76031
[45] Badia, S.; Codina, R.; Planas, R., On an unconditionally convergent stabilized finite element approximation of resistive magnetohydrodynamics, J. Comput. Phys., 234, 399-416, (2013) · Zbl 1284.76248
[46] Dedner, A.; Kemm, F.; Kroner, D.; Munz, C.-D.; Schnitzer, T.; Wesenberg, M., Hyperbolic divergence cleaning for the MHD equations, J. Comput. Phys., 175, 645-673, (2002) · Zbl 1059.76040
[47] Moreau, R., Magneto-hydrodynamics, (1990), Kluwer Dordrecht
[48] Davidson, P. A., An introduction to magnetohydrodynamics, (2001), Cambridge Univ. Press · Zbl 0974.76002
[49] Strauss, H. R., Nonlinear, 3-dimensional magnetohydrodynamics of noncircular tokamaks, Phys. Fluids, 19, 1, 134-140, (1976)
[50] Hazeltine, R. D.; Kotschenreuther, M.; Morrison, P. J., A four-field model for tokamak plasma dynamics, Phys. Fluids, 28, 8, 2466-2477, (1985) · Zbl 0584.76124
[51] Drake, J. F.; Antonsen, T. M., Nonlinear reduced fluid equations for toroidal plasmas, Phys. Fluids, 27, 4, 898-908, (1984) · Zbl 0555.76097
[52] Brooks, A. N.; Hughes, T., Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 32, 199-259, (1982) · Zbl 0497.76041
[53] Hughes, T.; Franca, L.; Balestra, M., A new finite element formulation for computational fluid dynamics: V. circumventing the babuska-Brezzi condition: A stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations, Comput. Methods Appl. Mech. Engrg., 59, 85-99, (1986) · Zbl 0622.76077
[54] Hughes, T., Multiscale phenomena: green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods, Comput. Methods Appl. Mech. Engrg., 127, 387-401, (1995) · Zbl 0866.76044
[55] Hughes, T.; Feijoo, G.; Mazzei, L.; Quincy, J., The variational multiscale method: A paradigm for computational mechanics, Comput. Methods Appl. Mech. Engrg., 166, 3-24, (1998) · Zbl 1017.65525
[56] Hughes, T. J.R.; Scovazzi, G.; Franca, l. P., (Stein, Erwin; de Borst, Rene; Hughes., Thomas J. R., Chapter 2: Multiscale and Stabilized Methods in Encyclopedia of Computational Mechanics, Fluids, vol. 3, (2007), John Wiley)
[57] Cyr, E. C.; Shadid, J. N.; Tuminaro, R. S.; Pawlowski, R. P.; Chacón, L., A new approximate block factorization preconditioner for 2D incompressible (reduced) resistive MHD, SISC, 35, B701-B730, (2013) · Zbl 1273.76269
[58] Codina, R., On stabilized finite element methods for linear systems of convection-diffusion-reaction equations, Comput. Methods Appl. Mech. Engrg., 188, 61-88, (2000) · Zbl 0973.76041
[59] Codina, R., Stabilized finite element approximation of transient incompressible flows using orthogonal subscales, Comput. Methods Appl. Mech. Engrg., 191, 4295-4321, (2002) · Zbl 1015.76045
[60] Collis, S.; Heinkenschloss, M., Analysis of the streamline-upwind/Petrov-Galerkin method applied to the solution of optimal control problems, technical report TR02-01, (2002), Department of Computational and Applied Mathematics, Rice University Houston, TX 77005-1892
[61] Sondak, D.; Shadid, J. N.; Oberai, A. A.; Pawlowski, R. P.; Cyr, E. C.; Smith, T. M., A new class of finite element variational multiscale turbulence models for incompressible magnetohydrodynamics, J. Comput. Phys., 295, 596-616, (2015) · Zbl 1349.76140
[62] Hughes, T.; Mallet, M., A new finite element formulation for computational fluid dynamics: III. the generalized streamline operator for multidimentional advective-diffusive systems, Comput. Methods Appl. Mech. Engrg., 58, 305-328, (1986) · Zbl 0622.76075
[63] Shakib, F., Finite element analysis of the compressible Euler and Navier-Stokes equations, (1989), Division of Applied Mathematics, Stanford University, (Ph.D. thesis)
[64] Brackbill, J. U.; Barnes, D. C., The effect of nonzero \(\nabla \cdot \mathbf{B}\) on the numerical solution of the magnetohydrodynamic equations, J. Comput. Phys., 35, 3, 426-430, (1980) · Zbl 0429.76079
[65] Marder, B., A method for incorporating Gauss law into electromagnetic pic codes, J. Comput. Phys., 68, 48-55, (1987) · Zbl 0603.65079
[66] Ascher, U. M.; Petzold, L. R., Computer methods for ordinary differential equations and differential-algebraic equations, (1998), SIAM · Zbl 0908.65055
[67] Dennis, J. E.; Schnabel, R. B., Numerical methods for unconstrained optimization and nonlinear equations, series in automatic computation, (1983), Prentice-Hall Englewood Cliffs, NJ
[68] Brown, P. N.; Saad, Y., Convergence theory of nonlinear Newton-Krylov algorithms, SIAM J. Optim., 4, 297-330, (1994) · Zbl 0814.65048
[69] Dembo, R. S.; Eisenstat, S. C.; Steihaug, T., Inexact Newton methods, SIAM J. Numer. Anal., 19, 400-408, (1982) · Zbl 0478.65030
[70] Eisenstat, S. C.; Walker, H. F., Choosing the forcing terms in an inexact Newton method, SIAM J. Sci. Comput., 17, 16-32, (1996) · Zbl 0845.65021
[71] Eisenstat, S. C.; Walker, H. F., Globally convergent inexact Newton methods, SIAM J. Optim., 4, 393-422, (1994) · Zbl 0814.65049
[72] Shadid, J. N.; Tuminaro, R. S.; Walker, H. F., An inexact Newton method for fully-coupled solution of the Navier-Stokes equations with heat and mass transport, J. Comput. Phys., 137, 155-185, (1997) · Zbl 0898.76066
[73] Pawlowski, R. P.; Shadid, J. N.; Simonis, J. P.; Walker, H. F., Globalaization techniques for Newton-Krylov methods and applications to the fully-coupled solution of the Navier-Stokes equations, SIAM Rev., 48, 700-721, (2006) · Zbl 1110.65039
[74] Phipps, E. T.; Pawlowski, R. P., (Efficient Expression Templates for Operator Overloading-based Automatic Differentiation, Lecture Notes in Computational Science and Engineering, vol. 87, (2012), Springer), 309-320 · Zbl 1252.65057
[75] Heroux, M.; Bartlett, R.; Howle, V.; Hoekstra, R.; Hu, J.; Kolda, T.; Lehoucq, R.; Long, K.; Pawlowski, R.; Phipps, E.; Salinger, A.; Thornquist, H.; Tuminaro, R.; Willenbring, J.; Williams, A., An overview of trilinos project, ACM Trans. Math. Software, 31, 3, 397-423, (2005) · Zbl 1136.65354
[76] Shadid, J. N.; Salinger, A. G.; Pawlowski, R. P.; Lin, P. T.; Hennigan, G. L.; Tuminaro, R. S.; Lehoucq, R. B., Stabilized FE computational analysis of nonlinear steady state transport/reaction systems, Comput. Methods Appl. Mech. Engrg., 195, 1846-1871, (2006) · Zbl 1178.76240
[77] Saad, Y., Iterative methods for sparse linear systems, (2003), SIAM · Zbl 1002.65042
[78] Quarteroni, A.; Valli, A., Domain decomposition methods for partial differential equations, (1999), Oxford University Press Oxford · Zbl 0931.65118
[79] Tuminaro, R.; Tong, C.; Shadid, J.; Devine, K. D.; Day, D., On a multilevel preconditioning module for unstructured mesh Krylov solvers: two-level Schwarz, Commun. Numer. Methods Eng., 18, 383-389, (2002) · Zbl 0999.65101
[80] Shadid, J.; Tuminaro, R.; Devine, K.; Henningan, 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
[81] Trottenberg, U.; Oosterlee, C.; Schüller, A., Multigrid, (2001), Academic Press London
[82] Hackbusch, W., (Multigrid Methods and Applications, Computational Mathematics, vol. 4, (1985), Springer-Verlag Berlin) · Zbl 0577.65118
[83] Trottenberg, U.; Oosterlee, C.; Schüller, A., Multigrid, (2001), Academic Press London
[84] Briggs, W. L.; Henson, V. E.; McCormick, S., A multigrid tutorial, (2000), SIAM Philadelphia · Zbl 0958.65128
[85] Davis, T., Direct methods for sparse linear systems, (2006), SIAM Philadelphia, PA · Zbl 1119.65021
[86] Sala, M., Amesos 2.0 reference guide, tech. rep. SAND2004-4820, (2004), Sandia National Laboratories, September
[87] Ruge, J.; Stüben, K., Algebraic multigrid (AMG), (McCormick, S. F., Multigrid Methods, Frontiers in Applied Mathematics, vol. 3, (1987), SIAM Philadelphia, PA), 73-130
[88] Gee, M.; Siefert, C.; Hu, J.; Tuminaro, R.; Sala, M., ML 5.0 smoothed aggregation user’s guide, tech. rep. SAND2006-2649, 87185, (2006), Sandia National Laboratories Albuquerque, NM
[89] Lin, P. T.; Sala, M.; Shadid, J. N.; Tuminaro, R. S., Performance of a geometric and an algebraic multilevel preconditioner for incompressible flow and transport, (Computational Mechanics: WCCM VI in conjunction with APCOM’04, (2004), Tsinghus Univ. Press, Springer-Verlag)
[90] Lin, P. T.; Sala, M.; Shadid, J. N.; Tuminaro, R. S., Performance of fully-coupled algebraic multilevel domain decomposition preconditioners for incompressible flow and transport, Int. J. Numer. Methods Eng., 67, 9, 208-225, (2006) · Zbl 1110.76315
[91] Lin, P. T.; Shadid, J. N.; Tuminaro, R. S.; Sala, M.; Hennigan, G. L.; Pawlowski, R. P., A parallel fully-coupled algebraic multilevel preconditioner applied to multiphysics PDE applications: drift-diffusion, flow/transport/reaction, resistive MHD, Internat. J. Numer. Methods Fluids, 64, 1148-1179, (2010) · Zbl 1427.65036
[92] Lin, P. T.; Shadid, J. N.; Tuminaro, R. S.; Sala, M., Performance of a Petrov-Galerkin algebraic multilevel preconditioner for finite element modeling of the semiconductor device drift-diffusion equations, Int. J. Numer. Methods Eng., 84, 448-469, (2010) · Zbl 1202.82003
[93] Tuminaro, R. S.; Heroux, M.; Hutchinson, S. A.; Shadid, J. N., Aztec user’s guide-version 2.1, tech. rep. sand99-8801J, 87185, (1999), Sandia National Laboratories Albuquerque NM, Nov
[94] Batchelor, G. K., An introduction to fluid mechanics, (1967), Cambridge Univ. Press · Zbl 0152.44402
[95] Fadeev, V. M.; Kvartskhava, I. F.; Komarov, N. N., Self-focusing of local plasma currents, Nucl. fusion, 5, 3, 202-209, (1965)
[96] Biskamp, D., Magnetic reconnection in plasmas, (2000), Cambridge University Press Cambridge, UK · Zbl 0891.76094
[97] Knoll, D. A.; Chacón, L., Coalescence of magnetic islands, sloshing, and the pressure problem, Phys. Plasmas, 13, 3, 32307-32311, (2006)
[98] Chacón, L.; Knoll, D. A.; Finn, J. M., Hall MHD effects in the 2-D Kelvin-Helmholtz/tearing instability, Phys. Lett.: A, 308, 2-3, 187-197, (2003) · Zbl 1086.81559
[99] Donea, J.; Huerta, A., Finite element methods for flow problems, (2002), John Wiley
[100] E.G. Phillips, E.C. Cyr, J.N. Shadid, R.P. Pawlowski, Approximate block preconditioners and effective Schur-complement approximations for the dual saddle-point problem of incompressible resistive MHD. 2016, in preperation.
[101] Shadid, J., A fully-coupled Newton-Krylov solution method for parallel unstructured finite element fluid flow, heat and mass transfer simulations, Int. J. CFD, 12, 199-211, (1999) · Zbl 0969.76049
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.