Yong Cao

Asymptotic boundary conditions with immersed finite elements for interface magnetostatic/electrostatic field problems with open boundary

Abstract Many of the magnetostatic/electrostatic field problems encountered in aerospace engineering, such as plasma sheath simulation and ion neutralization process in space, are not confined to finite domain and non-interface problems, but characterized as open boundary and interface problems. Asymptotic boundary conditions (ABC) and immersed finite elements (IFE) are relatively new tools to handle open boundaries and interface problems respectively. Compared with the traditional truncation approach, asymptotic boundary conditions need a much smaller domain to achieve the same accuracy. When regular finite element methods are applied to an interface problem, it is necessary to use a body-fitting mesh in order to obtain the optimal convergence rate. However, immersed finite elements possess the same optimal convergence rate on a Cartesian mesh, which is critical to many applications. This paper applies immersed finite element methods and asymptotic boundary conditions to solve an interface problem arising from electric field simulation in composite materials with open boundary. Numerical examples are provided to demonstrate the high global accuracy of the IFE method with ABC based on Cartesian meshes, especially around both interface and boundary. This algorithm uses a much smaller domain than the truncation approach in order to achieve the same accuracy.

An iterative immersed finite element method for an electric potential interface problem based on given surface electric quantity

Interface problems involving the non-homogeneous flux jump condition are critical for engineering designs in the magnetostatic/electrostatic field. In applications, such as plasma simulation, we often only know the total electric quantity on the surface of the object, not the charge density distribution on the surface which appears as the non-homogeneous flux jump condition in the usual interface problems considered in the literature for the magnetostatic/electrostatic field. Based on structured meshes independent of the interface, this article proposes an iterative method that employs both the immersed finite element (IFE) method with non-homogeneous flux jump conditions and the regular finite element method with ghost nodes introduced in the object to solve the 2D interface problem for the potential field according to the given total electric quantity on the surface of the object. Numerical experiments are provided to illustrate the accuracy and efficiency of the proposed method.

A splitting extrapolation for solving nonlinear elliptic equations with d-quadratic finite elements

Nonlinear elliptic partial differential equations are important to many large scale engineering and science problems. For this kind of equations, this article discusses a splitting extrapolation which possesses a high order of accuracy, a high degree of parallelism, less computational complexity and more flexibility than Richardson extrapolation. According to the problems, some domain decompositions are constructed and some independent mesh parameters are designed. Multi-parameter asymptotic expansions are proved for the errors of approximations. Based on the expansions, splitting extrapolation formulas are developed to compute approximations with high order of accuracy on a globally fine grid. Because these formulas only require us to solve a set of smaller discrete subproblems on different coarser grids in parallel instead of on the globally fine grid, a large scale multidimensional problem is turned into a set of smaller discrete subproblems. Additionally, this method is efficient for solving interface problems.

Immersed finite element methods for unbounded interface problems with periodic structures

Interface problems arise in many physical and engineering simulations involving multiple materials. Periodic structures often appear in simulations with large or even unbounded domain, such as magnetostatic/electrostatic field simulations. Immersed finite element (IFE) methods are efficient tools to solve interface problems on a Cartesian mesh, which is desirable to many applications like particle-in-cell simulation of plasma physics. In this article, we develop an IFE method for an interface problem with periodic structure on an infinite domain. To cope with the periodic boundary condition, we modify the stiffness matrix of the IFE method. The new matrix is maintained symmetric positive definite, so that the linear system can be solved efficiently. Numerical examples are provided to demonstrate features of this method.

Decoupling the Stationary Navier-Stokes-Darcy System with the Beavers-Joseph-Saffman Interface Condition

This paper proposes a domain decomposition method for the coupled stationary Navier-Stokes and Darcy equations with the Beavers-Joseph-Saffman interface condition in order to improve the efficiency of the finite element method. The physical interface conditions are directly utilized to construct the boundary conditions on the interface and then decouple the Navier-Stokes and Darcy equations. Newton iteration will be used to deal with the nonlinear systems. Numerical results are presented to illustrate the features of the proposed method.

Numerical Simulation Study on Barrel Erosion of Ion Thruster Accelerator Grid

The barrel erosion, one type of accelerator grid erosion in an ion thruster, is important and needs more detailed studies. In this paper, a three-dimensional numerical simulation code is developed using a hybrid immersed-finite-element particle-in-cell method and Monte Carlo collisions method; the dynamics of a neutral atom is modeled using a direct simulation Monte Carlo method; and a numerical simulation study is run to investigate the mechanism of aperture barrel erosion of an ion thruster accelerator grid. Simulation results indicate that the aperture barrel erosion of ion thruster accelerator grid is entirely caused by the charge-exchange ions under nominal operating condition, and the incident charge-exchange ions in four regions, namely the upstream region, the extraction (center) region, the extraction (edge) region, and downstream of the accelerator grid, are the causes of aperture barrel erosion. High-energy charge-exchange ions originated from upstream region are not well focused and play an im...

An algorithm using the finite volume element method and its splitting extrapolation

This paper is to present a new efficient algorithm by using the finite volume element method and its splitting extrapolation. This method combines the local conservation property of the finite volume element method and the advantages of splitting extrapolation, such as a high order of accuracy, a high degree of parallelism, less computational complexity and more flexibility than a Richardson extrapolation. Because the splitting extrapolation formulas only require us to solve a set of smaller discrete subproblems on different coarser grids in parallel instead of on the globally fine grid, a large scale multidimensional problem is turned into a set of smaller discrete subproblems. Additionally, this method is efficient for solving interface problems if we regard the interfaces of the problems as the interfaces of the initial domain decomposition.

Modeling and an immersed finite element method for an interface wave equation

Abstract The electromagnetic field, which is governed by Maxwell’s equation, plays a key role in plasma simulation. In this article, we first derive the interface conditions when we rewrite the interface Maxwell’s equation, whose problem domain involves complex media such as objects of different materials, into a parabolic–hyperbolic type of interface model, which is a modified wave equation by adding a first order time derivative term due to the lossy medium. Based on the interface conditions and the existing bilinear immersed finite element space for the interface Poisson equation, we propose an immersed finite element method for the spatial discretization of this parabolic–hyperbolic interface equation on a Cartesian mesh independent of the interface. Then we use a second order finite difference method for the temporal discretization in order to develop the full discretization scheme. Compared with the unstructured body-fitting mesh which is needed by the traditional finite element method for interface problems, the Cartesian mesh independent of the interface will significantly benefit the plasma simulation in the electromagnetic field, especially the particle models (such as the particle-in-cell method). This method provides a new efficient way to study varying electromagnetic field with objects of different materials on a Cartesian mesh independent of the interface, hence builds a solid foundation for the further study on the motion of plasma in this electromagnetic field. Numerical examples are provided to demonstrate the features of the proposed method.

An improved immersed finite element particle-in-cell method for plasma simulation

Abstract The particle-in-cell (PIC) method has been widely used for plasma simulation, because of its noise-reduction capability and moderate computational cost. The immersed finite element (IFE) method is efficient for solving interface problems on Cartesian meshes, which is desirable for the PIC method. The combination of these two methods provides an effective tool for plasma simulation with complex interface/boundary. This paper introduces an improved IFE–PIC method that enhances the performance in both IFE and PIC aspects. For the electric field solver, we adopt the newly developed partially penalized IFE method with enhanced accuracy. For PIC implementation, we introduce a new interpolation technique to ensure the conservation of the charge. Numerical examples are provided to demonstrate the features of the improved IFE–PIC method.