A Cartesian Grid Method for Modeling Charge Distribution on Interfaces via Augmented Technique

Haifeng Ji,Qian Zhang,Bin Zhang

doi:10.1155/2018/7521273

Abstract

In the electrostatic field computations, second-order elliptic interface problems with nonhomogeneous interface jump conditions need to be solved. In realistic applications, often the total electric quantity on the interface is given. However, the charge distribution on the interface corresponding to the nonhomogeneous interface jump condition is unknown. This paper proposes a Cartesian grid method for solving the interface problem with the given total electric quantity on the interface. The proposed method employs both the immersed finite element with the nonhomogeneous interface jump condition and the augmented technique. Numerical experiments are presented to show the accuracy and efficiency of the proposed method.

Highlights

We consider an isolated conductor that is placed near other charges or in an external electric field
For the problem discussed in this paper, only the total electric quantity on the interface is known, not the charge density distribution on the interface which is related to the nonhomogeneous interface jump condition of this problem
We present a new Cartesian grid method based on the immersed finite element (IFE) method and the augmented technique [12, 13]

Summary

Introduction

We consider an isolated conductor that is placed near other charges or in an external electric field. If the charge distribution q(x, y) on the interface Γ is given, the potential φ(x, y) can be solved efficiently by the immersed finite element (IFE) method for nonhomogeneous interface jump conditions (see [2, 3]). The elliptic interface problem can be solved efficiently by the IFE method with the given nonhomogeneous interface jump q(x, y). For the problem discussed in this paper, only the total electric quantity on the interface is known, not the charge density distribution on the interface which is related to the nonhomogeneous interface jump condition of this problem. The augmented variable is chosen such that the nonhomogeneous interface jump condition and the total electric quantity are satisfied.

Augmented Technique for Given Electric Quantity on Interfaces

Immersed Finite Element for Nonhomogeneous Interface Jump Conditions

Enforce the Total Electric Quantity on the Interface

Numerical Experiments

Conclusion