Comptes Rendus
Magnetic field based finite element method for magneto-static problems with discontinuous electric potential distributions
Comptes Rendus. Mécanique, Volume 351 (2023) no. S1, pp. 53-72.

We introduce two finite element formulations to approximate magneto-static problems with discontinuous electric potential based respectively on the electrical scalar potential and the magnetic field. This work is motivated by our interest in Liquid Metal Batteries (LMBs), a promising technology for storing intermittent renewable sources of energy in large scale energy storage devices. LMBs consist of three liquid layers stably stratified and immiscible, with a light liquid metal on top (negative electrode), a molten salt in the middle (electrolyte) and a heavier liquid metal on bottom (positive electrode). Energy is stored in electrical potential differences that can be modeled as jumps at each electrode-electrolyte interface. This paper focuses on introducing new finite element methods for computing current and potential distributions, which account for internal voltage jumps in liquid metal batteries. Two different formulations that use as primary unknowns the electrical potential and magnetic field, respectively, are presented. We validate them using various manufactured test cases, and discuss their applications for simulating the current distribution during the discharge phase in a liquid metal battery.

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DOI: 10.5802/crmeca.184
Keywords: magnetohydrodynamics, finite element methods, interior penalty techniques, discontinuous electric potential, liquid metal batteries

Sabrina Bénard 1; Loic Cappanera 2; Wietze Herreman 3; Caroline Nore 4

1 Université Paris-Saclay, CNRS, LISN, 91400 Orsay, France
2 Department of Mathematics, University of Houston, Houston, Texas 77204, USA
3 Université Paris-Saclay, CNRS, FAST, 91400 Orsay, France
4 Laboratoire Interdisciplinaire des Sciences du Numérique; LISN, Université Paris-Saclay, Bât 507, Campus Universitaire F-91405 Orsay, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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     title = {Magnetic field based finite element method for magneto-static problems with discontinuous electric potential distributions},
     journal = {Comptes Rendus. M\'ecanique},
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Sabrina Bénard; Loic Cappanera; Wietze Herreman; Caroline Nore. Magnetic field based finite element method for magneto-static problems with discontinuous electric potential distributions. Comptes Rendus. Mécanique, Volume 351 (2023) no. S1, pp. 53-72. doi : 10.5802/crmeca.184. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.184/

[1] Marie Odile Bristeau; Roland Glowinski; Jacques Periaux Numerical methods for the Navier–Stokes equations. Applications to the simulation of compressible and incompressible viscous flows, Comput. Phys. Rep., Volume 6 (1987) no. 1-6, pp. 73-187 | DOI

[2] Roland Glowinski; Patrick Le Tallec Augmented Lagrangian and operator-splitting methods in nonlinear mechanics, SIAM Studies in Applied Mathematics, 9, Society for Industrial and Applied Mathematics, 1989 | DOI | Zbl

[3] Roland Glowinski Finite element methods for incompressible viscous flow, Numerical methods for fluids (Part 3) (Handbook of Numerical Analysis), Volume 9, Elsevier, 2003, pp. 3-1176 | DOI | Zbl

[4] Roland Glowinski; Stanley J. Osher; Wotao Yin Splitting methods in communication, imaging, science, and engineering, Scientific Computation, Springer, 2017

[5] Roland Glowinski Lectures on numerical methods for non-linear variational problems, Springer, 2008

[6] Jean-Luc Guermond; Peter Minev; Jie Shen An overview of projection methods for incompressible flows, Comput. Methods Appl. Mech. Eng., Volume 195 (2006) no. 44-47, pp. 6011-6045 | DOI | MR | Zbl

[7] Hojong Kim; Dane A. Boysen; Jocelyn M. Newhouse; Brian L. Spatocco; Brice Chung; Paul J. Burke; David J. Bradwell; Kai Jiang; Alina A. Tomaszowska; Kangli Wang; Weifeng Wei; Luis A. Ortiz; Salvador A. Barriga; Sophie M. Poizeau; Donald R. Sadoway Liquid Metal Batteries: Past, Present, and Future, Chem. Rev., Volume 113 (2013) no. 3, pp. 2075-2099 | DOI

[8] Norbert Weber; Steffen Landgraf; Kashif Mushtaq; Michael Nimtz; Paolo Personnettaz; Tom Weier; Ji Zhao; Donald R. Sadoway Modeling discontinuous potential distributions using the finite volume method, and application to liquid metal batteries, Electrochim. Acta, Volume 318 (2019), pp. 857-864 | DOI

[9] Carolina Duczek; Norbert Weber; Omar E. Godinez-Brizuela; Tom Weier Simulation of potential and species distribution in a Li||Bi liquid metal battery using coupled meshes, Electrochim. Acta (2022), 141413 | DOI

[10] Geoffrey A. Prentice; Charles W. Tobias A Survey of Numerical Methods and Solutions for Current Distribution Problems, J. Electrochem. Soc., Volume 129 (1982) no. 1, pp. 72-78 | DOI

[11] Adam Z. Weber; John Newman Modeling Transport in Polymer-Electrolyte Fuel Cells, Chem. Rev., Volume 104 (2004) no. 10, pp. 4679-4726 | DOI

[12] Norbert Weber; Michael Nimtz; Paolo Personnettaz; Tom Weier; Donald R. Sadoway Numerical simulation of mass transfer enhancement in liquid metal batteries by means of electro-vortex flow, Journal of Power Sources Advances, Volume 1 (2020), 100004 | DOI

[13] Giovanna Guidoboni; Roland Glowinski; Nicola Cavallini; Suncica Canic Stable loosely-coupled-type algorithm for fluid–structure interaction in blood flow, J. Comput. Phys., Volume 228 (2009) no. 18, pp. 6916-6937 | DOI | MR | Zbl

[14] Jean-Luc Guermond; R. Laguerre; Jacques Léorat; Caroline Nore An interior penalty Galerkin method for the MHD equations in heterogeneous domains, J. Comput. Phys., Volume 221 (2007) no. 1, pp. 349-369 | DOI | MR | Zbl

[15] Jean-Luc Guermond; R. Laguerre; Jacques Léorat; Caroline Nore Nonlinear magnetohydrodynamics in axisymmetric heterogeneous domains using a Fourier/Finite Element technique and an Interior Penalty Method, J. Comput. Phys., Volume 228 (2009) no. 8, pp. 2739-2757 | DOI | MR | Zbl

[16] Jean-Luc Guermond; Jacques Léorat; Francky Luddens; Caroline Nore; Adolfo Ribeiro Effects of discontinuous magnetic permeability on magnetodynamic problems, J. Comput. Phys., Volume 230 (2011) no. 16, pp. 6299-6319 | DOI | MR | Zbl

[17] Alexandre Ern; Annette F. Stephansen; Paolo Zunino A discontinuous Galerkin method with weighted averages for advection–diffusion equations with locally small and anisotropic diffusivity, IMA J. Numer. Anal., Volume 29 (2009) no. 2, pp. 235-256 | MR | Zbl

[18] Martin Costabel A coercive bilinear form for Maxwell’s equations, J. Math. Anal. Appl., Volume 157 (1991) no. 2, pp. 527-541 | DOI | MR | Zbl

[19] Andrea Bonito; Jean-Luc Guermond Approximation of the eigenvalue problem for the time harmonic Maxwell system by continuous Lagrange finite elements, Math. Comput., Volume 80 (2011) no. 276, pp. 1887-1910 | DOI | MR | Zbl

[20] Andrea Bonito; Jean-Luc Guermond; Francky Luddens An interior penalty method with C0 finite elements for the approximation of the Maxwell equations in heterogeneous media: convergence analysis with minimal regularity, ESAIM, Math. Model. Numer. Anal., Volume 50 (2016) no. 5, pp. 1457-1489 | DOI | Zbl

[21] Andrea Bonito; Jean-Luc Guermond; Francky Luddens Regularity of the Maxwell equations in heterogeneous media and Lipschitz domains, J. Math. Anal. Appl., Volume 408 (2013) no. 2, pp. 498-512 | DOI | MR | Zbl

[22] W. Herreman; S. Bénard; Caroline Nore; Paolo Personnettaz; L. Cappanera; Jean-Luc Guermond Solutal buoyancy and electrovortex flow in liquid metal batteries, Phys. Rev. Fluids, Volume 5 (2020), 074501 | DOI | Zbl

[23] Paolo Personnettaz; Steffen Landgraf; Michael Nimtz; Norbert Weber; Tom Weier Mass transport induced asymmetry in charge/discharge behavior of liquid metal batteries, Electrochem. commun., Volume 105 (2019), 106496 | DOI

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