Stabilized bi-cubic Hermite Bézier finite element method with application to gas-plasma interactions occurring during massive material injection in tokamaks. (English) Zbl 1538.76102
Summary: Development of a numerical tool based upon the high-order, high-resolution Galerkin finite element method (FEM) often encounters two challenges: First, the Galerkin FEMs give central approximations to the differential operators and their use in the simulation of the convection-dominated flows may lead to the dispersion errors yielding entirely wrong numerical solutions. Secondly, high-order, high-resolution numerical methods are known to produce high wave-number oscillations in the vicinity of shocks/discontinuities in the numerical solution adversely affecting the stability of the method. We present the stabilized finite element method for plasma fluid models to address the two challenges. The numerical stabilization is based on two strategies: Variational Multiscale (VMS) and the shock-capturing approach. The former strategy takes into account (the approximation of) the effect of the unresolved scales onto resolved scales to introduce upwinding in the Galerkin FEM. The latter adaptively adds the artificial viscosity only in the vicinity of shocks. These numerical stabilization strategies are applied to stabilize the bi-cubic Hermite Bézier FEM in the computational framework of the nonlinear magnetohydrodynamics (MHD) code JOREK. The application of the stabilized FEM to the challenging simulation of Shattered Pellet Injection (SPI) in JET-like plasma is presented. It is shown that the developed numerical stabilization model improves the stability of the underlying numerical algorithm and the computational cost required to reveal the complex physics is reduced. The physical and numerical models presented can be used to perform expensive simulations of the plasma applications in large computational domains such as JET, and ITER.
MSC:
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 76W05 | Magnetohydrodynamics and electrohydrodynamics |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 35L65 | Hyperbolic conservation laws |
| 76L05 | Shock waves and blast waves in fluid mechanics |
Keywords:
stabilized FEM; variational multi-scale methods; shock-capturing stabilization; bi-cubic Hermite Bézier FEM; magnetohydrodynamics; massive material injection in tokamak plasmaReferences:
| [1] | Sovinec, C.; Glasser, A.; Gianakon, T.; Barnes, D.; Nebel, R.; Kruger, S.; Schnack, D.; Plimpton, S.; Tarditi, A.; Chu, M., Nonlinear magnetohydrodynamics simulation using high-order finite elements, J. Comput. Phys., 195, 1, 355-386 (2004) · Zbl 1087.76070 |
| [2] | Ferraro, N.; Jardin, S., Calculations of two-fluid magnetohydrodynamic axisymmetric steady-states, J. Comput. Phys., 228, 20, 7742-7770 (2009) · Zbl 1391.76324 |
| [3] | Czarny, O.; Huysmans, G., Bézier surfaces and finite elements for MHD simulations, J. Comput. Phys., 227, 16, 7423-7445 (2008) · Zbl 1141.76035 |
| [4] | Huysmans, G., ELMs: MHD instabilities at the transport barrier, Plasma Phys. Control. Fusion, 47, 12B, B165-B178 (2005) |
| [5] | Hoelzl, M.; Huijsmans, G.; Pamela, S.; Bécoulet, M.; Nardon, E.; Artola, F.; Nkonga, B.; Atanasiu, C.; Bandaru, V.; Bhole, A.; Bonfiglio, D.; Cathey, A.; Czarny, O.; Dvornova, A.; Fehér, T.; Fil, A.; Franck, E.; Futatani, S.; Gruca, M.; Guillard, H.; Haverkort, J.; Holod, I.; Hu, D.; Kim, S.; Korving, S.; Kos, L.; Krebs, I.; Kripner, L.; Latu, G.; Liu, F.; Merkel, P.; Meshcheriakov, D.; Mitterauer, V.; Mochalskyy, S.; Morales, J.; Nies, R.; Nikulsin, N.; Orain, F.; Pratt, J.; Ramasamy, R.; Ramet, P.; Reux, C.; Särkimäki, K.; Schwarz, N.; Verma, P. S.; Smith, S.; Sommariva, C.; Strumberger, E.; van Vugt, D.; Verbeek, M.; Westerhof, E.; Wieschollek, F.; Zielinski, J., The JOREK non-linear extended MHD code and applications to large-scale instabilities and their control in magnetically confined fusion plasmas, Nucl. Fusion, 61, Article 065001 pp. (2021) |
| [6] | Pamela, S.; Huijsmans, G.; Hoelzl, M., A generalised formulation of G-continuous Bezier elements applied to non-linear MHD simulations, J. Comput. Phys., Article 111101 pp. (2022) · Zbl 07540342 |
| [7] | Haverkort, J. W.; Blank, H. J.; Huijsmans, G.; Pratt, J.; Koren, B., Implementation of the full viscoresistive magnetohydrodynamic equations in a nonlinear finite element code, J. Comput. Phys., 281-302 (2016) · Zbl 1349.76216 |
| [8] | Pamela, S. J.P.; Bhole, A.; Huijsmans, G. T.A.; Nkonga, B.; Hoelzl, M.; Krebs, I.; Strumberger, E., Extended full-MHD simulation of non-linear instabilities in tokamak plasmas, Phys. Plasmas, 27, 10, Article 102510 pp. (2020) |
| [9] | Hender, T.; Wesley, J.; Bialek, J.; Bondeson, A.; Boozer, A.; Buttery, R.; Garofalo, A.; Goodman, T.; Granetz, R.; Gribov, Y.; Gruber, O.; Gryaznevich, M.; Giruzzi, G.; Günter, S.; Hayashi, N.; Helander, P.; Hegna, C.; Howell, D.; Humphreys, D.; Huysmans, G.; Hyatt, A.; Isayama, A.; Jardin, S.; Kawano, Y.; Kellman, A.; Kessel, C.; Koslowski, H.; Haye, R. L.; Lazzaro, E.; Liu, Y.; Lukash, V.; Manickam, J.; Medvedev, S.; Mertens, V.; Mirnov, S.; Nakamura, Y.; Navratil, G.; Okabayashi, M.; Ozeki, T.; Paccagnella, R.; Pautasso, G.; Porcelli, F.; Pustovitov, V.; Riccardo, V.; Sato, M.; Sauter, O.; Schaffer, M.; Shimada, M.; Sonato, P.; Strait, E.; Sugihara, M.; Takechi, M.; Turnbull, A.; Westerhof, E.; Whyte, D.; Yoshino, R.; Zohm, H., Chapter 3: MHD stability, operational limits and disruptions, Nucl. Fusion, 47, S128-S202 (2007) |
| [10] | Izzo, V.; Parks, P.; Eidietis, N.; Shiraki, D.; Hollmann, E.; Commaux, N.; Granetz, R.; Humphreys, D.; Lasnier, C.; Moyer, R.; Paz-Soldan, C.; Raman, R.; Strait, E., The role of MHD in 3D aspects of massive gas injection, Nucl. Fusion, 55, Article 073032 pp. (2015) |
| [11] | Combs, S. K.; Meitner, S. J.; Baylor, L. R.; Caughman, J. B.O.; Commaux, N.; Fehling, D. T.; Foust, C. R.; Jernigan, T. C.; McGill, J. M.; Parks, P. B.; Rasmussen, D. A., Alternative techniques for injecting massive quantities of gas for plasma-disruption mitigation, IEEE Trans. Plasma Sci., 38, 3, 400-405 (2010) |
| [12] | Baylor, L.; Meitner, S.; Gebhart, T.; Caughman, J.; Herfindal, J.; Shiraki, D.; Youchison, D., Shattered pellet injection technology design and characterization for disruption mitigation experiments, Nucl. Fusion, 59, Article 066008 pp. (2019) |
| [13] | Lehnen, M.; Aleynikova, K.; Aleynikov, P.; Campbell, D.; Drewelow, P.; Eidietis, N.; Gasparyan, Y.; Granetz, R.; Gribov, Y.; Hartmann, N.; Hollmann, E.; Izzo, V.; Jachmich, S.; Kim, S.-H.; Kooan, M.; Koslowski, H.; Kovalenko, D.; Kruezi, U.; Loarte, A.; Maruyama, S.; Matthews, G.; Parks, P.; Pautasso, G.; Pitts, R.; Reux, C.; Riccardo, V.; Roccella, R.; Snipes, J.; Thornton, A.; de Vries, P., Disruptions in ITER and strategies for their control and mitigation, J. Nucl. Mater., 463, 39-48 (2015), Plasma-surface interactions 21 |
| [14] | Hoelzl, M.; Hu, D.; Nardon, E.; Huijsmans, G. T.A., First predictive simulations for deuterium shattered pellet injection in ASDEX upgrade, Phys. Plasmas, 27, 2, Article 022510 pp. (2020) |
| [15] | von Neumann, J.; Richtmyer, R. D., A method for the numerical calculations of hydrodynamical shocks, J. Appl. Phys., 21, 232-238 (1950) · Zbl 0037.12002 |
| [16] | Wilkins, M. L., Calculation of elastic plastic flow, Methods Comput. Phys., 3 (1964) |
| [17] | Wilkins, M. L., Use of artificial viscosity in multidimensionnal fluid dynamic calculations, J. Comput. Phys., 36, 381-403 (1980) |
| [18] | Godunov, S.; Zabrodine, A.; Ivanov, M.; Kraiko, A.; Prokopov, G., Résolution numérique des problèmes multidimensionnels de la dynamique des gaz (1979), Mir · Zbl 0421.65056 |
| [19] | Donea, J.; Giuliani, S.; Laval, H.; Quartapelle, L., Time-accurate solution of advection-diffusion problems by finite elements, Comput. Methods Appl. Mech. Eng., 45, 1-3, 123-145 (1984) · Zbl 0514.76083 |
| [20] | Ambrosi, D.; Quartapelle, L., A Taylor-Galerkin method for simulating nonlinear dispersive water waves, J. Comput. Phys., 146, 2, 546-569 (1998) · Zbl 0932.76031 |
| [21] | Brezzi, F.; Bristeau, M.-O.; Franca, L. P.; Mallet, M.; Rogé, G., A relationship between stabilized finite element methods and the Galerkin method with bubble functions, Comput. Methods Appl. Mech. Eng., 96, 1, 117-129 (1992) · Zbl 0756.76044 |
| [22] | Brezzi, F.; Franca, L. P.; Russo, A., Further considerations on residual-free bubbles for advective-diffusive equations, Comput. Methods Appl. Mech. Eng., 166, 1-2, 25-33 (1998) · Zbl 0934.65126 |
| [23] | Brooks, A.; Hughes, T., Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., 32, 199-259 (1982) · Zbl 0497.76041 |
| [24] | Hughes, T. J.R.; Scovazzi, G.; Franca, L. P., Multiscale and Stabilized Methods (2004), John Wiley & Sons, Ltd |
| [25] | Hughes, T.; Scovazzi, G.; Tezduyar, T., Stabilized methods for compressible flows, J. Sci. Comput., 43, 343-368 (2010) · Zbl 1203.76130 |
| [26] | Johnson, C.; Szepessy, A.; Hansbo, P., On the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws, Math. Comput., 54, 189, 107-129 (1990) · Zbl 0685.65086 |
| [27] | Tezduyar, T. E.; Senga, M., Stabilization and shock-capturing parameters in SUPG formulation of compressible flows, Comput. Methods Appl. Mech. Eng., 195, 13, 1621-1632 (2006) · Zbl 1122.76061 |
| [28] | Bassi, F.; Cecchi, F.; Franchina, N.; Rebay, S.; Savini, M., High-order discontinuous Galerkin computation of axisymmetric transonic flows in safety relief valves, Comput. Fluids, 49, 1, 203-213 (2011) · Zbl 1271.76163 |
| [29] | Guermond, J.-L.; Pasquetti, R.; Popov, B., Entropy viscosity method for nonlinear conservation laws, J. Comput. Phys., 230, 11, 4248-4267 (2011), Special issue High Order Methods for CFD Problems · Zbl 1220.65134 |
| [30] | Dzanic, T.; Trojak, W.; Witherden, F., Utilizing time-reversibility for shock capturing in nonlinear hyperbolic conservation laws, Comput. Fluids, 247, Article 105652 pp. (2022) · Zbl 1521.35116 |
| [31] | Speleers, H.; Manni, C.; Pelosi, F.; Sampoli, M. L., Isogeometric analysis with Powell-Sabin splines for advection-diffusion-reaction problems, Comput. Methods Appl. Mech. Eng., 221-222, 132-148 (2012) · Zbl 1253.65026 |
| [32] | Giorgiani, G.; Guillard, H.; Nkonga, B.; Serre, E., A stabilized Powell-Sabin finite-element method for the 2D Euler equations in supersonic regime, Comput. Methods Appl. Mech. Eng., 340, 216-235 (2018) · Zbl 1440.76065 |
| [33] | Shadid, J.; Pawlowski, R.; Cyr, E.; Tuminaro, R.; Chacón, L.; Weber, P., Scalable implicit incompressible resistive MHD with stabilized FE and fully-coupled Newton-Krylov-AMG, Comput. Methods Appl. Mech. Eng., 304, 1-25 (2016) · Zbl 1423.76275 |
| [34] | Lin, P.; Shadid, J.; Hu, J.; Pawlowski, R.; Cyr, E., Performance of fully-coupled algebraic multigrid preconditioners for large-scale VMS resistive MHD, J. Comput. Appl. Math., 344, 782-793 (2018) · Zbl 1443.76152 |
| [35] | Tang, Q.; Chacón, L.; Kolev, T. V.; Shadid, J. N.; Tang, X.-Z., An adaptive scalable fully implicit algorithm based on stabilized finite element for reduced visco-resistive MHD, J. Comput. Phys., 454, Article 110967 pp. (2022) · Zbl 07518056 |
| [36] | Trelles, J. P.; Modirkhazeni, S. M., Variational multiscale method for nonequilibrium plasma flows, Comput. Methods Appl. Mech. Eng., 282, 87-131 (2014) · Zbl 1423.76511 |
| [37] | Goedbloed, J. P.H.; Poedts, S., Principles of Magnetohydrodynamics: With Applications to Laboratory and Astrophysical Plasmas (2004), Cambridge University Press |
| [38] | S.I. Braginskii, Transport processes in a plasma, 1965. |
| [39] | Hu, D.; Nardon, E.; Hoelzl, M.; Wieschollek, F.; Lehnen, M.; Huijsmans, G.; van Vugt, D. C.; Kim, S.-H., Radiation asymmetry and MHD destabilization during the thermal quench after impurity shattered pellet injection, Nucl. Fusion, 61, Article 026015 pp. (2021) |
| [40] | Nardon, E.; Fil, A.; Hoelzl, M.; Huijsmans, G., Progress in understanding disruptions triggered by massive gas injection via 3D non-linear MHD modelling with JOREK, Plasma Phys. Control. Fusion, 59 (2017) |
| [41] | Nardon, E.; Hu, D.; Hoelzl, M.; Bonfiglio, D., Fast plasma dilution in ITER with pure deuterium shattered pellet injection, Nucl. Fusion, 60, Article 126040 pp. (2020) |
| [42] | Nardon, E.; Hu, D.; Artola, F. J.; Bonfiglio, D.; Hoelzl, M.; Boboc, A.; Carvalho, P.; Gerasimov, S.; Huijsmans, G.; Mitterauer, V.; Schwarz, N.; Sun, H., Thermal quench and current profile relaxation dynamics in massive-material-injection-triggered tokamak disruptions, Plasma Phys. Control. Fusion, 63, Article 115006 pp. (2021) |
| [43] | Hu, D.; Huijsmans, G. T.A.; Nardon, E.; Hoelzl, M.; Lehnen, M.; Bonfiglio, D., Collisional-radiative non-equilibrium impurity treatment for JOREK simulations, Plasma Phys. Control. Fusion, 63, Article 125003 pp. (2021) |
| [44] | http://open.adas.ac.uk |
| [45] | Wesson, J.; Campbell, D., Tokamaks, International Series of Monographs on Physics (2004), Clarendon Press · Zbl 1111.82054 |
| [46] | Summers, H. P.; McWhirter, R. W.P., Radiative power loss from laboratory and astrophysical plasmas. I. Power loss from plasmas in steady-state ionisation balance, J. Phys. B, At. Mol. Phys., 12, 2387-2412 (1979) |
| [47] | Grad, H.; Rubin, H., Hydromagnetic equilibria and force-free fields, J. Nucl. Energy, 7, 3-4, 284-285 (1954), 1958 |
| [48] | Shafranov, V., On magnetohydrondynamical equilibrium configurations, Sov. Phys. JETP (1958) |
| [49] | Pamela, S.; Huijsmans, G.; Thornton, A.; Kirk, A.; Smith, S.; Hoelzl, M.; Eich, T., A wall-aligned grid generator for non-linear simulations of MHD instabilities in tokamak plasmas, Comput. Phys. Commun., 243, 41-50 (2019) · Zbl 07674814 |
| [50] | Stangeby, P., The Plasma Boundary of Magnetic Fusion Devices, Series in Plasma Physics and Fluid Dynamics (2000), Taylor & Francis |
| [51] | Maire, P.-H.; Nkonga, B., Multi-scale Godunov-type method for cell-centered discrete Lagrangian hydrodynamics, J. Comput. Phys., 228, 3, 799-821 (2009) · Zbl 1156.76039 |
| [52] | Balsara, D. S., Multidimensional HLLE Riemann solver: application to Euler and magnetohydrodynamic flows, J. Comput. Phys., 229, 6, 1970-1993 (2010) · Zbl 1303.76140 |
| [53] | Balsara, D. S.; Vides, J.; Gurski, K.; Nkonga, B.; Dumbser, M.; Garain, S.; Audit, E., A two-dimensional Riemann solver with self-similar sub-structure - alternative formulation based on least squares projection, J. Comput. Phys., 304, 138-161 (2016) · Zbl 1349.76157 |
| [54] | Balsara, D. S.; Nkonga, B., Multidimensional Riemann problem with self-similar internal structure - part III - a multidimensional analogue of the HLLI Riemann solver for conservative hyperbolic systems, J. Comput. Phys., 346, 25-48 (2017) · Zbl 1378.76056 |
| [55] | Dumbser, M.; Balsara, D. S., A new efficient formulation of the HLLEM Riemann solver for general conservative and non-conservative hyperbolic systems, J. Comput. Phys., 304, 275-319 (2016) · Zbl 1349.76603 |
| [56] | Billaud, M.; Gallice, G.; Nkonga, B., A simple stabilized finite element method for solving two phase compressible-incompressible interface flows, Comput. Methods Appl. Mech. Eng., 200, 9-12, 1272-1290 (2011) · Zbl 1225.76191 |
| [57] | Bayona, C.; Baiges, J.; Codina, R., Solution of low Mach number aeroacoustic flows using a variational multi-scale finite element formulation of the compressible Navier-Stokes equations written in primitive variables, Comput. Methods Appl. Mech. Eng. (2018) |
| [58] | Shakib, F.; Hughes, T. J.; Johan, Z., A new finite element formulation for computational fluid dynamics: X. The compressible Euler and Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., 89, 1, 141-219 (1991), Second World Congress on Computational Mechanics |
| [59] | Bazilevs, Y.; Calo, V.; Cottrell, J.; Hughes, T.; Reali, A.; Scovazzi, G., Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows, Comput. Methods Appl. Mech. Eng., 197, 1, 173-201 (2007) · Zbl 1169.76352 |
| [60] | Nkonga, B.; Tarcisio-Costa, J.; Vides, J., VMS Finite Element for MHD and Reduced-MHD in Tokamak Plasmas (2016), Inria Sophia Antipolis; Université de Nice-Sophia Antipolis, Research Report RR-8892 |
| [61] | Beam, R. M.; Warming, R. F., Alternating direction implicit methods for parabolic equations with a mixed derivative, SIAM J. Sci. Stat. Comput., 1, 1, 131-159 (1980) · Zbl 0462.65060 |
| [62] | Guillard, H.; Lakhlili, J.; Loseille, A.; Loyer, A.; Nkonga, B.; Ratnani, A.; Elarif, A., Tokamesh: a software for mesh generation in Tokamaks (2018), CASTOR, Research Report RR-9230 |
| [63] | Ashish, B.; Boniface, N.; Stanislas, P.; Guido, H.; Matthias, H., Treatment of polar grid singularities in the bi-cubic Hermite-Bézier approximations: isoparametric finite element framework, J. Comput. Phys., 471, Article 111611 pp. (2022) · Zbl 07605582 |
| [64] | Kurganov, A.; Petrova, G.; Popov, B., Adaptive semidiscrete central-upwind schemes for nonconvex hyperbolic conservation laws, SIAM J. Sci. Comput., 29, 6, 2381-2401 (2007) · Zbl 1154.65356 |
| [65] | Orszag, S. A.; Tang, C.-M., Small-scale structure of two-dimensional magnetohydrodynamic turbulence, J. Fluid Mech., 90, 1, 129-143 (1979) |
| [66] | Price, D. J., Smoothed particle magnetohydrodynamics - IV. Using the vector potential, Mon. Not. R. Astron. Soc., 401, 1475-1499 (2010) |
| [67] | Vides, J.; Nkonga, B.; Audit, E., A simple two-dimensional extension of the HLL Riemann solver for hyperbolic systems of conservation laws, J. Comput. Phys., 280, 643-675 (2015) · Zbl 1349.76403 |
| [68] | Fil, A., Modeling of massive gas injection triggered disruptions in tokamak plasmas (2015), PhD thesis, Thèse de doctorat dirigée par Peter Beyer, Physique et sciences de la matière, Aix-Marseille |
| [69] | Ferraro, N.; Lyons, B.; Kim, C.; Liu, Y.; Jardin, S., 3D two-temperature magnetohydrodynamic modeling of fast thermal quenches due to injected impurities in tokamaks, Nucl. Fusion, 59, Article 016001 pp. (2018) |
| [70] | Kim, C. C.; Liu, Y.; Parks, P. B.; Lao, L. L.; Lehnen, M.; Loarte, A., Shattered pellet injection simulations with NIMROD, Phys. Plasmas, 26, 4, Article 042510 pp. (2019) |
| [71] | Izzo, V. A., Dispersive shell pellet injection modeling and validation for DIII-D disruption mitigation, Phys. Plasmas, 28, 8, Article 082502 pp. (2021) |
| [72] | Zafar, A.; Zhu, P.; Ali, A.; Zeng, S.; Li, H., Effects of helium massive gas injection level on disruption mitigation on EAST, Plasma Sci. Technol., 23, Article 075103 pp. (2021) |
| [73] | Zeng, S.; Zhu, P.; Izzo, V.; Li, H.; Jiang, Z., MHD simulations of cold bubble formation from 2/1 tearing mode during massive gas injection in a tokamak, Nucl. Fusion, 62, Article 026015 pp. (2021) |
| [74] | Parks, P.; Turnbull, R. J., Effect of transonic flow in the ablation cloud on the lifetime of a solid hydrogen pellet in a plasma, Phys. Fluids, 21, 10, 1735-1741 (1978) · Zbl 0389.76082 |
| [75] | Sergeev, V.; Bakhareva, O.; Kuteev, B., Studies of the impurity pellet ablation in the high-temperature plasma of magnetic confinement devices, Plasma Phys. Rep., 32, 363-377 (2006) |
| [76] | P. Parks, The ablation rate of light-element pellets with a kinetic treatment for penetration of plasma electrons through the ablation cloud, 2020, unpublished. |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
