Mathematical Problems in Engineering
Volume 2018 (2018), Article ID 7521273, 7 pages
https://doi.org/10.1155/2018/7521273

Research Article

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

Haifeng Ji,¹ Qian Zhang,² and Bin Zhang³

¹School of Science, Nanjing University of Posts and Telecommunications, Nanjing, Jiangsu 210023, China
²Institute of Information Technology, Nanjing University of Chinese Medicine, Nanjing, Jiangsu 210023, China
³School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China

Correspondence should be addressed to Qian Zhang; zq19880125@163.com

Received 16 December 2017; Revised 1 April 2018; Accepted 8 April 2018; Published 14 May 2018

Academic Editor: Ruben Specogna

Copyright © 2018 Haifeng Ji et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

We consider an isolated conductor that is placed near other charges or in an external electric field. For simplicity, we take a rectangular domain as the computational domain. The conductor occupying is included in the domain . The boundary of is denoted by which is also called an interface in this paper. We assume that . Let , let be the unit normal vector of pointing from to , and let be the permittivity of . The modeling of the electrostatic field leads to the interface problem [1],

together with the interface jump conditions on the interface ,

where the interfacial function represents the charge density on the interface and is also unknown; however, the total electric quantity

is given. For simplicity, we consider a homogeneous Dirichlet boundary condition:

Note that other boundary conditions can be treated using standard techniques. The function represents the external charge density of the part outside of the conductor. The solution is the potential. Once the potential is obtained, the electrostatic field can be computed by . If the charge distribution on the interface is given, then the potential can be solved efficiently by the immersed finite element (IFE) method for nonhomogeneous interface jump conditions (see [2, 3]). The significance of the IFE method [4–7] is the use of structured meshes which are independent of the interface, such as Cartesian meshes. The IFE method modifies the basis function on interface elements according to the interface conditions to capture the jumps of the exact solution. The weak form and the degrees of freedom remain the same as if there was no interface. If the coefficient is a constant without jumps, then the stiffness matrix is the same as that obtained by traditional finite element for the problem without interfaces. And only the right-hand side needs to be modified according to the interface conditions. The elliptic interface problem can be solved efficiently by the IFE method with the given nonhomogeneous interface jump .

However, 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. In [1], the authors proposed an iterative IFE method for this interface problem. An IFE method with nonhomogeneous interface jump conditions and a standard finite element method with ghost nodes are combined to get a “Prediction-Correction-Prediction” iteration. Numerical examples in [1] show that the iterative method is convergent and can solve this problem efficiently. Note that for partial differential equations there are many other methods in the literature [8–11]. In this paper, we present a new Cartesian grid method based on the IFE method and the augmented technique [12, 13]. By introducing the jump of the normal derivative of the exact solution as an augmented variable, we can get an efficient discretization in which the fast Fourier transform- (FFT-) based fast Poisson solver can be applied. The augmented variable is chosen such that the nonhomogeneous interface jump condition and the total electric quantity are satisfied. In the numerical method, the augmented variable is solved by using the GMRES iteration. Compared with the iterative IFE method proposed in [1], the advantage of our Cartesian grid method is that the FFT-based fast Poisson solver can be used. Numerical experiments are also provided in this paper to show the performance of the proposed method.

The rest of the paper is organized as follows. In Section 2, we describe the augmented technique for the interface problem with given electric quantity on the interface. We choose an augmented variable and rewrite the interface problem to a new one for which the FFT-based fast Poisson solver can be applied. In Section 3, we briefly recall the IFE method for the nonhomogeneous interface jump conditions, where the augmented variable is assumed to be given. In Section 4, the constraint of the total electric quantity is enforced and some implementation details about the GMRES iteration are described. Finally, some numerical examples are provided in Section 5 to show the accuracy and efficiency of the proposed method.

2. Augmented Technique for Given Electric Quantity on Interfaces

By introducing an augmented variable and using the fact that the coefficient is a constant, the original problem can be written as

with

It is obvious that the solution of (5)-(6) is dependent on the augmented variable . Thus, we denote the solution of (5)-(6) by . Let the solution of the original model problem (1)-(4) be and define

Then satisfies (5)-(6) with . In other words, . From the original model problem, the augmented variable should be chosen such that the solution of (5)-(6) satisfies

This is the constraint for the choice of the augmented variable. When the conductor is in electrical equilibrium, the electrical potential is a constant and electric field inside the conductor; that is, on the interface . Hence, we have and .

In the continuous case, the augmented method is to find the solution of (5)-(6) and the augmented variable is constrained by

To present the numerical method, first we partition the domain into uniform rectangles with mesh size (, ); then we obtain the triangulation with mesh size by cutting the rectangles along one of diagonals in the same direction. We call an element an interface element if intersects ; otherwise, we call a noninterface element. The sets of all interface elements and noninterface elements are denoted by and , respectively. The interface is approximated by , the union of the line segments connecting the intersections of the interface and the edges of elements. That is,

where is the intersection of the interface and edges of elements. Let be the domain with as its boundary, and . There is a small region around the interface ,

whose area is of order . And the distance between and satisfies .

In the discrete case, the augmented variable is piecewise constant and is defined at line segments on interface elements; that is,

In other words, on the interface , the augmented variable is approximated by the piecewise constant function:

In the numerical method, if the vector (or in the function form) is given, then we can use the IFE method which will be described in the next section to get the discrete solution .

3. Immersed Finite Element for Nonhomogeneous Interface Jump Conditions

In the IFE method, the interface jump conditions and are used to construct the discrete trial function space . For any , the finite dimensional function is piecewise linear on each element and is broken along to satisfy and on . For the test function space, we use the standard conforming finite element space . It will be shown later that the IFE function can be decomposed as

where and is a piecewise linear function that is nonzero only on noninterface elements. Note that the function depends on the augmented variable .

Given the augmented variable , the IFE method for (5)-(6) is to find such that

The discrete solution is . From (16), it is obvious that the stiffness matrix is the same as that obtained by traditional finite element for the problem without interfaces; only the right-hand side needs to be modified. Thus, we can take advantage of the fast Poisson solver to solve the system of equations efficiently when the augmented variable is known.

3.1. Construction of the Function

First, we describe the function in the space in detail. On a non-interface element, is the standard linear function and the degrees of freedom are functional values on the vertices of the element. On interface elements, for example, on (see Figure 1 for an illustration), the function is constructed as the following piecewise linear function:

The coefficients are chosen such that

where the vector represents the degrees of freedom. Obviously, the function can be decomposed as

where and satisfy

with coefficients chosen such that

Figure 1: An example of the interface element.

4. Enforce the Total Electric Quantity on the Interface

In the discrete case, constraints (9) and (10) are replaced by

In matrix-vector form, the discretization (16) and (22) can be written as

For (23), we have the vector form , where and are vectors. To enforce this constraint, we augmented the linear system with the equation and a Lagrangian multiplier to get

We use the GMRES iteration method to solve the augmented variable and the Lagrangian multiplier first and then to solve by using one more fast Poisson solver. We refer the readers to Section in [14] for the details about the GMRES iteration.

5. Numerical Experiments

In this section, we present some examples to show the accuracy and the efficiency of the proposed numerical method. First, we consider the following example in which the exact solution is given. This example is taken from [1].

Example 1. Consider a conductor that is placed in an externally applied field along the direction. The external charge density . It is easy to verify that the exact solution is

and on the interface

We choose , , and in this example. Not only the potential but also the electric field are computed by using the proposed method. We choose the square domain as the computational domain. We first partition the domain into congruent squares, and then we obtain the triangulation by cutting the squares along one of diagonals in the same direction. We compute errors in for the numerical potential and the numerical electric field and estimate the convergence rate by using

Numerical results reported in Table 1 show that the proposed method achieves first-order convergence in the norm for both the potential and the electric field. The numerical charge distribution on the interface, the numerical potential , and the numerical electric field obtained by the proposed method with are plotted in Figures 2, 3, and 4, respectively.

Table 1: A grid refinement analysis for Example 1 for the proposed method.

Rate

Rate

16

32

2.11712

1.19087

64

0.30758

0.83689

128

1.30749

1.02640

256

0.75350

0.94776

512

1.27473

1.06372

Figure 2: Exact charge distribution (blue) and the numerical charge distribution on the interface (red) obtained by the proposed method with for Example 1.

Figure 3: Numerical potential obtained by the proposed method with for Example 1.

Figure 4: Numerical electric field obtained by the proposed method with for Example 1.

Example 2. Consider with , , and . The computational domain is . The external charge density is set to be inside the rectangular and everywhere else. Dirichlet boundary condition is applied to the left boundary and Neumann boundary condition is applied on the right, bottom, and top boundaries. We choose and the total electric quantity in this example. Note that there is no explicit analytical solution for this example. Numerical results are plotted in Figures 5, 6, and 7. Note that the external charge is negative in a small area and the total electric quantity is positive. In order to maintain the electric field inside the conductor, the positive surface charges should move towards the external negative charges. Figure 5 shows that the positive charges gather on the right side of the circle.

Figure 5: Numerical charge distribution on the interface (red) obtained by the proposed method with for Example 2.

Figure 6: Numerical potential obtained by the proposed method with for Example 2.

Figure 7: Numerical electric field obtained by the proposed method with for Example 2.

Example 3. Consider a conductor with complicated boundary

where

We choose , , and . The external charge density is set to be . Dirichlet boundary condition is applied to the left boundary and Neumann boundary condition is applied on the rest of boundaries. We choose , , and in this example. Similar numerical results are plotted in Figures 8, 9, and 10. From these figures, we can see that the surface charge is distributed according to the potential on the left boundary to maintain the electric field inside the conductor.

Figure 8: Numerical charge distribution on the interface (red) obtained by the proposed method with for Example 3.

Figure 9: Numerical potential obtained by the proposed method with for Example 3.

Figure 10: Numerical electric field obtained by the proposed method with for Example 3.

6. Conclusion

We presented a Cartesian grid method for solving the surface charge distribution problem in electromagnetism. The advantage of the method is that the used mesh does not need to be aligned with the interface. This is quite convenient for the problem with complex interfaces. The proposed method employs both the immersed finite element and the augmented technique. The GMRES iteration and FFT-based fast Poisson solver are used to solve the discrete systems. Numerical examples are provided to show the accuracy and efficiency of the proposed method.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported in part by the Natural Science Foundation of the Jiangsu Higher Education Institution of China (Grant no. 17KJB110014), the National Natural Science Foundation of China (Grant no. 11701291), the Natural Science Foundation of Jiangsu Province (Grant no. BK20160880), and NUPTSF (Grant no. NY216030).

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