Abstract
Weak scaling performance of a recently developed fully kinetic, 3-D parallel immersed-finite-element particle-in-cell framework, namely PIFE-PIC, was investigated. A nominal 1-D plasma charging problem, the lunar photoelectron sheath at a low Sun elevation angle, was chosen to validate PIFE-PIC against recently derived semi-analytic solutions of a 1-D photoelectron sheath. The weak scaling performance test shows that the overall efficiency of PIFE-PIC is insensitive to the number of macroparticles and, counterintuitively, more domain decomposition iterations in the field-solve of PIC may lead to faster computing due to better convergence of field solutions at early stages of PIC iteration. The PIFE-PIC framework was then applied to simulate plasma charging of a wavy lunar surface with 324,000 cells and 150 million macroparticles demonstrating the capability of PIFE-PIC in resolving local-scale plasma environment near the surface of the Moon.
Similar content being viewed by others
References
McKay DS, Heiken G, Basu A, Blanford G, Simon S, Reedy R, French BM, Papike J (1991) Chapter 7: The lunar regolith. In: Heiken GH, Vaniman DT, French BM (eds) Lunar sourcebook: a user’s guide to the moon, pp 285–356. Cambridge University Press, Cambridge
Shkuratov YG, Bondarenko NV (2001) Regolith layer thickness mapping of the moon by radar and optical data. Icarus 149(2):329–338
Han D, Wang J, He X-M (2018) Immersed-finite-element particle-in-cell simulations of plasma charging at lunar terminator. J. Spacecr Rockets 55(6):1490–1497
Han D, Wang P, He X-M, Lin T, Wang J (2016) A 3D immersed finite element method with non-homogeneous interface flux jump for applications in particle-in-cell simulations of plasma-lunar surface interactions. J Comput Phys 321:965–980
Han D, Wang J, He X-M (2016) A non-homogeneous immersed-finite-element particle-in-cell method for modeling dielectric surface charging in plasmas. IEEE Trans Plasma Sci 44(8):1326–1332
Han D, He X, Wang JJ (2018) PIFE-PIC: a 3-D parallel immersed finite element particle-in-cell framework for plasma simulations. In: AIAA SciTech forum 2018. AIAA 2018-2196, Kissimmee, Florida
Han D, He X, Lund D, Zhang X (2021) PIFE-PIC: parallel immersed finite element particle-in-cell for 3-D kinetic simulations of plasma–material interactions. SIAM J Sci Comput 43(3):235–257
Zhao J, Wei X, Du X, He X, Han D (2021) Photoelectron sheath and plasma charging on the lunar surface: semianalytic solutions and fully-kinetic particle-in-cell simulations. IEEE Trans Plasma Sci 49(10):3036–3050
Burman E, Claus S, Hansbo P, Larson MG, Massing A (2015) CutFEM: discretizing geometry and partial differential equations. Int J Numer Methods Eng 104(7):472–501
LeVeque RJ, Li ZL (1994) The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J Numer Anal 31(4):1019–1044. https://doi.org/10.1137/0731054
Fries T-P, Belytschko T (2010) The extended/generalized finite element method: an overview of the method and its applications. Int J Numer Methods Eng 84(3):253–304. https://doi.org/10.1002/nme.2914
Zhou YC, Zhao S, Feig M, Wei GW (2006) High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources. J Comput Phys 213(1):1–30. https://doi.org/10.1016/j.jcp.2005.07.022
Babuška I, Banerjee U (2012) Stable generalized finite element method (SGFEM). Comput Methods Appl Mech Eng 201(204):91–111. https://doi.org/10.1016/j.cma.2011.09.012
Li Z (1998) The immersed interface method using a finite element formulation. Appl Numer Math 27(3):253–267
Gong Y, Li B, Li Z (2008) Immersed-interface finite-element methods for elliptic interface problems with non-homogeneous jump conditions. SIAM J Numer Anal 46:472–495
Guzmán J, Sánchez MA, Sarkis M (2016) Higher-order finite element methods for elliptic problems with interfaces. ESAIM Math Model Numer Anal 50(5):1561–1583
He X-M (2009) Bilinear immersed finite elements for interface problems. Ph.D. dissertation, Virginia Polytechnic Institute and State University
He X-M, Lin T, Lin Y (2008) Approximation capability of a bilinear immersed finite element space. Numer Methods Partial Differ Equ 24(5):1265–1300
Li Z, Lin T, Wu X (2003) New Cartesian grid methods for interface problems using the finite element formulation. Numer Math 96(1):61–98
Preusser T, Rumpf M, Sauter S, Schwen LO (2011) 3D composite finite elements for elliptic boundary value problems with discontinuous coefficients. SIAM J Sci Comput 33(5):2115–2143
Vallaghè S, Papadopoulo T (2010) A trilinear immersed finite element method for solving the electroencephalography forward problem. SIAM J Sci Comput 32(4):2379–2394
Guo R (2019) Design, analysis, and application of immersed finite element methods. Ph.D. dissertation, Virginia Polytechnic Institute and State University
Zhang X (2013) Nonconforming immersed finite element methods for interface problems. Ph.D. dissertation, Virginia Polytechnic Institute and State University
Lin T, Zhang X (2012) Linear and bilinear immersed finite elements for planar elasticity interface problems. J Comput Appl Math 236(18):4681–4699
Lin T, Sheen D, Zhang X (2013) A locking-free immersed finite element method for planar elasticity interface problems. J Comput Phys 247:228–247
Guo R, Lin T, Lin Y (2020) Error estimates for a partially penalized immersed finite element method for elasticity interface problems. ESAIM Math Model Numer Anal 54(1):1–24
Feng W, He X-M, Lin Y, Zhang X (2014) Immersed finite element method for interface problems with algebraic multigrid solver. Commun Comput Phys 15(4):1045–1067
He X-M, Lin T, Lin Y, Zhang X (2013) Immersed finite element methods for parabolic equations with moving interface. Numer Methods Partial Differ Equ 29(2):619–646
Adjerid S, Moon K (2019) An immersed discontinuous Galerkin method for acoustic wave propagation in inhomogeneous media. SIAM J Sci Comput 41(1):139–162
Adjerid S, Lin T, Zhuang Q (2020) Error estimates for an immersed finite element method for second order hyperbolic equations in inhomogeneous media. J Sci Comput 84(2):35
Bai J, Cao Y, He X-M, Liu H, Yang X (2018) Modeling and an immersed finite element method for an interface wave equation. Comput Math Appl 76(7):1625–1638
Adjerid S, Chaabane N, Lin T, Yue P (2019) An immersed discontinuous finite element method for the Stokes problem with a moving interface. J Comput Appl Math 362:540–559
Jones D, Zhang X (2021) A class of nonconforming immersed finite element methods for Stokes interface problems. J Comput Appl Math 392:113493
Guo R, Lin T (2019) A higher degree immersed finite element method based on a Cauchy extension for elliptic interface problems. SIAM J Numer Anal 57(4):1545–1573
He X-M, Lin T, Lin Y (2012) The convergence of the bilinear and linear immersed finite element solutions to interface problems. Numer Methods Partial Differ Equ 28(1):312–330
Lin T, Lin Y, Zhang X (2015) Partially penalized immersed finite element methods for elliptic interface problems. SIAM J Numer Anal 53(2):1121–1144
He X-M, Lin T, Lin Y (2010) Interior penalty bilinear IFE discontinuous Galerkin methods for elliptic equations with discontinuous coefficient, dedicated to David Russell’s 70th birthday. J Syst Sci Complex 23(3):467–483
He X-M, Lin T, Lin Y (2014) A selective immersed discontinuous Galerkin method for elliptic interface problems. Math Methods Appl Sci 37(7):983–1002
Lin T, Yang Q, Zhang X (2015) A priori error estimates for some discontinuous Galerkin immersed finite element methods. J Sci Comput 65(3):875–894
Ewing RE, Li Z, Lin T, Lin Y (1999) The immersed finite volume element methods for the elliptic interface problems. Modelling ’98 (Prague). Math Comput Simul 50(1–4):63–76
Cao W, Zhang X, Zhang Z, Zou Q (2017) Superconvergence of immersed finite volume methods for one-dimensional interface problems. J Sci Comput 73(2–3):543–565
He X-M, Lin T, Lin Y (2009) A bilinear immersed finite volume element method for the diffusion equation with discontinuous coefficients, dedicated to Richard E. Ewing on the occasion of his 60th birthday. Commun Comput Phys 6(1):185–202
Lin T, Sheen D, Zhang X (2019) A nonconforming immersed finite element method for elliptic interface problems. J Sci Comput 79(1):442–463
Bai J, Cao Y, Chu Y, Zhang X (2018) An improved immersed finite element particle-in-cell method for plasma simulation. Comput Math Appl 75(6):1887–1899
Bai J, Cao Y, He X-M, Peng E (2021) An implicit particle-in-cell model based on anisotropic immersed-finite-element method. Comput Phys Commun 261:107655
Cao Y, Chu Y, Zhang X, Zhang X (2016) Immersed finite element methods for unbounded interface problems with periodic structures. J Comput Appl Math 307:72–81
Chu Y, Cao Y, He X-M, Luo M (2011) Asymptotic boundary conditions with immersed finite elements for interface magnetostatic/electrostatic field problems with open boundary. Comput Phys Commun 182(11):2331–2338
Chu Y, Han D, Cao Y, He X-M, Wang J (2017) An immersed-finite-element particle-in-cell simulation tool for plasma surface interaction. Int J Numer Anal Model 14(2):175–200
Wang J, He X-M, Cao Y (2007) Modeling spacecraft charging and charged dust particle interactions on lunar surface. In: Proceedings of the 10th spacecraft charging technology conference, Biarritz, France
Cao Y, Chu Y, He X-M, Lin T (2015) An iterative immersed finite element method for an electric potential interface problem based on given surface electric quantity. J Comput Phys 281:82–95
He X-M, Lin T, Lin Y (2011) Immersed finite element methods for elliptic interface problems with non-homogeneous jump conditions. Int J Numer Anal Model 8(2):284–301
Lu C, Wan J, Cao Y, He X-M (2020) A fully decoupled iterative method with three-dimensional anisotropic immersed finite elements for Kaufman-type discharge problems. Comput Methods Appl Mech Eng 372:113345
Lu C, Yang Z, Bai J, Cao Y, He X-M (2020) Three-dimensional immersed finite element method for anisotropic magnetostatic/electrostatic interface problems with non-homogeneous flux jump. Int J Numer Methods Eng 121(10):2107–2127
Cao H, Cao Y, Chu Y, He X-M, Lin T (2018) A Huygens immersed-finite-element particle-in-cell method for modeling plasma-surface interactions with moving interface. Commun Nonlinear Sci Numer Simul 59:132–148
Jian H, Chu Y, Cao H, Cao Y, He X-M, Xia G (2015) Three-dimensional IFE-PIC numerical simulation of background pressure’s effect on accelerator grid impingement current for ion optics. Vacuum 116:130–138
Kafafy R, Wang J (2005) Whole subscale ion optics simulation: direct ion impingement and electron backstreaming. In: 41st AIAA/ASME/SAE/ASEE joint propulsion conference and exhibit. AIAA 2005-3691, Tucson, Arizona
Kafafy R, Wang J (2007) Whole ion optics gridlet simulations using a hybrid-grid immersed-finite-element particle-in-cell code. J Propuls Power 23(1):59–68
Kafafy RI, Wang J (2006) A hybrid grid immersed finite element particle-in-cell algorithm for modeling spacecraft–plasma interactions. IEEE Trans Plasma Sci 34(5):2114–2124
Wang J, Cao Y, Kafafy R, Pierru J, Decyk VK (2006) Simulations of ion thruster plume–spacecraft interactions on parallel supercomputer. IEEE Trans Plasma Sci 34(5):2148–2158
Han D, Wang JJ (2013) Simulations of ion thruster plume contamination with a whole grid sputtered Mo source model. In: 49th AIAA/ASME/SAE/ASEE joint propulsion conference and exhibit. AIAA 2013-3888, San Jose, California
Depew D, Han D, Wang J, He X-M, Lin T (2014) Immersed-finite-element particle-in-cell simulations of lunar surface charging. In: Proceedings of the 13th spacecraft charging technology conference, Pasadena, California, June 23–27
Han D (2015) Particle-in-cell simulations of plasma interactions with asteroidal and lunar surfaces. Ph.D. thesis, University of Southern California
Han D, Wang J (2019) 3-D fully-kinetic particle-in-cell simulations of small asteroid charging in the solar wind. IEEE Trans Plasma Sci 47(8):3682–3688
Yu W, Han D, Wang J (2019) Numerical simulations of dust dynamics around small asteroids. IEEE Trans Plasma Sci 47(8):3724–3730
Yu W, Wang JJ, Han D (2016) Numerical modeling of dust dynamics around small asteroids. In: AIAA SPACE forum 2016. AIAA 2016-5447, Long Beach, California
Quarteroni A, Valli A (1999) Domain decomposition methods for partial differential equations. Numerical mathematics and scientific computation. Oxford Science Publications, New York
Smith B, Bjørstad P, Gropp W (1996) Domain decomposition: parallel multilevel methods for elliptic partial differential equations. Cambridge University Press, Cambridge
Guo R, Lin T (2020) An immersed finite element method for elliptic interface problems in three dimensions. J Comput Phys 414(1):109478
Guo R, Zhang X (2021) Solving three-dimensional interface problems with immersed finite elements: a-priori error analysis. J Comput Phys 441:110445
Kafafy R, Lin T, Lin Y, Wang J (2005) Three-dimensional immersed finite element methods for electric field simulation in composite materials. Int J Numer Methods Eng 64(7):940–972
Fu JHM (1971) Surface potential of a photoemitting plate. J Geophys Res (1896–1977) 76(10):2506–2509
Nitter T, Havnes O, Melandsø F (1998) Levitation and dynamics of charged dust in the photoelectron sheath above surfaces in space. J Geophys Res Sp Phys 103(A4):6605–6620
Poppe AR (2011) Modeling, theoretical and observational studies of the lunar photoelectron sheath. Ph.D. thesis, University of Colorado, Boulder
Heiken GH, Vaniman DT, French BM (1991) Lunar sourcebook: a user’s guide to the moon. Cambridge University Press, Cambridge
Acknowledgements
This work was partially supported by a NASA Space Technology Graduate Research Opportunity (D.L.), NASA-Missouri Space Grant Consortium through NASA-EPSCoR-Missouri (X.H. and D.H.) and graduate scholarships (D.L.), as well as NSF through grants DMS-2005272 and DMS-2110833 (X.Z.), DMS-2111039 (X.H. and D.H.), and CBET-2132655 (D.H.). The simulations presented here were performed with computing resources provided by the Center for High Performance Computing Research at Missouri University of Science and Technology through an NSF Grant OAC-1919789.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Lund, D., He, X., Zhang, X. et al. Weak scaling of the parallel immersed-finite-element particle-in-cell (PIFE-PIC) framework with lunar plasma charging simulations. Comp. Part. Mech. 9, 1279–1291 (2022). https://doi.org/10.1007/s40571-022-00470-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40571-022-00470-0