A 3D immersed finite element method with non-homogeneous interface flux jump for applications in particle-in-cell simulations of plasma-lunar surface interactions

Motivated by the need to handle complex boundary conditions efficiently and accurately in particle-in-cell (PIC) simulations, this paper presents a three-dimensional (3D) linear immersed finite element (IFE) method with non-homogeneous flux jump conditions for solving electrostatic field involving complex boundary conditions using structured meshes independent of the interface. This method treats an object boundary as part of the simulation domain and solves the electric field at the boundary as an interface problem. In order to resolve charging on a dielectric surface, a new 3D linear IFE basis function is designed for each interface element to capture the electric field jump on the interface. Numerical experiments are provided to demonstrate the optimal convergence rates in L 2 and H 1 norms of the IFE solution. This new IFE method is integrated into a PIC method for simulations involving charging of a complex dielectric surface in a plasma. A numerical study of plasma-surface interactions at the lunar terminator is presented to demonstrate the applicability of the new method.

[1]  T. Lin,et al.  THE IMMERSED FINITE ELEMENT METHOD FOR PARABOLIC PROBLEMS USING THE LAPLACE TRANSFORMATION IN TIME DISCRETIZATION , 2013 .

[2]  Zhilin Li,et al.  An Immersed Finite Element Method for Elasticity Equations with Interfaces , 2005 .

[3]  Gian Luca Delzanno,et al.  Comparison of dust charging between orbital-motion-limited theory and particle-in-cell simulations , 2015, 1611.08658.

[4]  Zhilin Li,et al.  The Immersed Interface Method: Numerical Solutions of PDEs Involving Interfaces and Irregular Domains (Frontiers in Applied Mathematics) , 2006 .

[5]  Xiaoming He,et al.  An iterative immersed finite element method for an electric potential interface problem based on given surface electric quantity , 2015, J. Comput. Phys..

[6]  Xiaoming He,et al.  Interior penalty bilinear IFE discontinuous Galerkin methods for elliptic equations with discontinuous coefficient , 2010, J. Syst. Sci. Complex..

[7]  Xiaoming He.  Bilinear Immersed Finite Elements for Interface Problems , 2009 .

[8]  C. Birdsall,et al.  Plasma Physics via Computer Simulation , 2018 .

[9]  Xiaoming He,et al.  Immersed finite element methods for elliptic interface problems with non-homogeneous jump conditions , 2011 .

[10]  Hui Xie,et al.  A FINITE ELEMENT METHOD FOR ELASTICITY INTERFACE PROBLEMS WITH LOCALLY MODIFIED TRIANGULATIONS. , 2011, International journal of numerical analysis and modeling.

[11]  Wing Kam Liu,et al.  Immersed finite element method for rigid body motions in the incompressible Navier–Stokes flow , 2008 .

[12]  Mohamed Ben-Romdhane,et al.  Higher-Order Immersed Finite Element Spaces for Second-Order Elliptic Interface Problems with Quadratic Interface , 2014 .

[13]  Y. Shkuratov,et al.  Regolith Layer Thickness Mapping of the Moon by Radar and Optical Data , 2001 .

[14]  Z. Wang,et al.  Immersed finite element methods for 4th order differential equations , 2011, J. Comput. Appl. Math..

[15]  Tao Lin,et al.  Linear and bilinear immersed finite elements for planar elasticity interface problems , 2012, J. Comput. Appl. Math..

[16]  Xiaoming He,et al.  Three-dimensional IFE-PIC numerical simulation of background pressure's effect on accelerator grid impingement current for ion optics , 2015 .

[17]  Tao Lin,et al.  A locking-free immersed finite element method for planar elasticity interface problems , 2013, J. Comput. Phys..

[18]  Ivo Babuska,et al.  The finite element method for elliptic equations with discontinuous coefficients , 1970, Computing.

[19]  Théodore Papadopoulo,et al.  A Trilinear Immersed Finite Element Method for Solving the Electroencephalography Forward Problem , 2010, SIAM J. Sci. Comput..

[20]  Xiaoming He,et al.  A selective immersed discontinuous Galerkin method for elliptic interface problems , 2014 .

[21]  Grant Heiken,et al.  Book-Review - Lunar Sourcebook - a User's Guide to the Moon , 1991 .

[22]  Joseph Wang,et al.  Ion Propulsion Simulations Using Parallel Supercomputer , 2005 .

[23]  Zhilin Li,et al.  Immersed Interface Finite Element Methods for Elasticity Interface Problems with Non-Homogeneous Jump Conditions , 2009 .

[24]  A. Poppe,et al.  Modeling, Theoretical and Observational Studies of the Lunar Photoelectron Sheath , 2011 .

[25]  Gian Luca Delzanno,et al.  Orbital-motion-limited theory of dust charging and plasma response , 2014, 1503.07820.

[26]  B. Heinrich.  Finite Difference Methods on Irregular Networks , 1987 .

[27]  D. Ming.  Lunar sourcebook. A user's guide to the moon , 1992 .

[28]  Raed Kafafy,et al.  Whole Subscale Ion Optics Simulation: Direct Ion Impingement and Electron Backstreaming , 2005 .

[29]  Ronald Fedkiw,et al.  The immersed interface method. Numerical solutions of PDEs involving interfaces and irregular domains , 2007, Math. Comput..

[30]  Zhilin Li.  The immersed interface method using a finite element formulation , 1998 .

[31]  Xiaoming He,et al.  A Bilinear Immersed Finite Volume Element Method For the Diffusion Equation with Discontinuous Coefficient , 2009 .

[32]  Slimane Adjerid,et al.  A p-th degree immersed finite element for boundary value problems with discontinuous coefficients , 2009 .

[33]  Xiaoming He,et al.  Asymptotic boundary conditions with immersed finite elements for interface magnetostatic/electrostatic field problems with open boundary , 2011, Comput. Phys. Commun..

[34]  Xiaoming He,et al.  Modeling Electrostatic Levitation of Dust Particles on Lunar Surface , 2008, IEEE Transactions on Plasma Science.

[35]  Zhilin Li,et al.  The immersed finite volume element methods for the elliptic interface problems , 1999 .

[36]  Xiaoming He,et al.  Approximation capability of a bilinear immersed finite element space , 2008 .

[37]  Bo Li,et al.  Immersed-Interface Finite-Element Methods for Elliptic Interface Problems with Nonhomogeneous Jump Conditions , 2007, SIAM J. Numer. Anal..

[38]  T. Lin,et al.  HIGHER DEGREE IMMERSED FINITE ELEMENT METHODS FOR SECOND-ORDER ELLIPTIC INTERFACE PROBLEMS , 2014 .

[39]  Xiaoming He,et al.  Immersed Finite Element Particle-in-cell Simulations of Lunar Surface Charging , 2014 .

[40]  Zhilin Li,et al.  An immersed finite element space and its approximation capability , 2004 .

[41]  Xiaoming He,et al.  The convergence of the bilinear and linear immersed finite element solutions to interface problems , 2012 .

[42]  Joseph Wang.  The Immersed Finite Element Method for Plasma Particle Simulations , 2003 .

[43]  Andrew R. Poppe,et al.  The effect of surface topography on the lunar photoelectron sheath and electrostatic dust transport , 2012 .

[44]  Pu Wang,et al.  Immersed Finite Element Particle-In-Cell Modeling of Surface Charging in Rarefied Plasmas , 2010 .

[45]  Raed Kafafy,et al.  Whole Ion Optics Gridlet Simulations Using a Hybrid-Grid Immersed-Finite-Element Particle-in-Cell Code , 2007 .

[46]  Xiaoming He,et al.  Immersed finite element methods for parabolic equations with moving interface , 2013 .

[47]  Weiwei Sun,et al.  Quadratic immersed finite element spaces and their approximation capabilities , 2006, Adv. Comput. Math..

[48]  Xu Zhang,et al.  Nonconforming Immersed Finite Element Methods for Interface Problems , 2013 .

[49]  T. Lin,et al.  An Immersed Finite Element Electric Field Solver for Ion Optics Modeling , 2002 .

[50]  D. M. Cook,et al.  The Theory of the Electromagnetic Field , 1975 .

[51]  James H. Bramble,et al.  A finite element method for interface problems in domains with smooth boundaries and interfaces , 1996, Adv. Comput. Math..

[52]  Kye T. Wee,et al.  An Analysis of a Broken P1-Nonconforming Finite Element Method for Interface Problems , 2009, SIAM J. Numer. Anal..

[53]  Yanping Lin,et al.  Immersed finite element method of lines for moving interface problems with nonhomogeneous flux jump , 2013 .

[54]  Daniel E. Hastings,et al.  Ionospheric plasma flow over large high-voltage space platforms. II: The formation and structure of plasma wake , 1992 .

[55]  S. Noble.  The Lunar Regolith , 2009 .

[56]  V. Vahedi,et al.  Simultaneous Potential and Circuit Solution for Two-Dimensional Bounded Plasma Simulation Codes , 1997 .

[57]  William M. Farrell,et al.  Anticipated electrical environment within permanently shadowed lunar craters , 2010 .

[58]  R. Kafafy.  Immersed Finite Element Particle-In-Cell Simulations of Ion Propulsion , 2005 .

[59]  Do Y. Kwak,et al.  Optimal convergence analysis of an immersed interface finite element method , 2010, Adv. Comput. Math..

[60]  Stefan A. Sauter,et al.  Composite Finite Elements for Elliptic Boundary Value Problems with Discontinuous Coefficients , 2006, Computing.

[61]  Tao Lin,et al.  New Cartesian grid methods for interface problems using the finite element formulation , 2003, Numerische Mathematik.

[62]  D. A. Dunnett.  Classical Electrodynamics , 2020, Nature.

[63]  Tao Lin,et al.  Partially Penalized Immersed Finite Element Methods For Elliptic Interface Problems , 2015, SIAM J. Numer. Anal..

[64]  So-Hsiang Chou,et al.  An Immersed Linear Finite Element Method with Interface Flux Capturing Recovery , 2012 .

[65]  Zhilin Li,et al.  A Symmetric and Consistent Immersed Finite Element Method for Interface Problems , 2014, J. Sci. Comput..

[66]  M. Omizo,et al.  Modeling , 1983, Encyclopedic Dictionary of Archaeology.

[67]  R. Reedy,et al.  THE LUNAR REGOLITH , 2012 .

[68]  R. Kafafy,et al.  Three‐dimensional immersed finite element methods for electric field simulation in composite materials , 2005 .

[69]  Weiwei Sun,et al.  Error estimation of a class of quadratic immersed finite element methods for elliptic interface problems , 2007 .

[70]  Andrew R. Poppe,et al.  Simulations of the photoelectron sheath and dust levitation on the lunar surface , 2010 .

[71]  J. Zou,et al.  Finite element methods and their convergence for elliptic and parabolic interface problems , 1998 .