Two-dimensional time-domain volume integral equations for scattering of inhomogeneous objects

Authors


Abstract

[1] This paper proposes a time-domain volume integral equation based method for analyzing the transient scattering from a two-dimensional inhomogeneous cylinder by invoking the volume equivalence principle for both the transverse magnetic and electric cases. The cylinder is discretized into triangular cells, and the electric flux is chosen as the unknown. For the transverse magnetic case, the electric flux is defined on the surfaces of the triangles. For the transverse electric case, because of the electric charges induced inside and on the surface of the cylinder, the electric flux is defined on the edges of the triangles, and expanded in space in terms of two-dimensional surface roof-top basis functions. The time-domain volume integral equation is solved by using a marching-on-in-time scheme. Numerical results obtained using this method are in excellent agreement with the data obtained using the finite-difference time-domain method.

1. Introduction

[2] Numerical electromagnetic methods are basically classified into two methods, namely, the differential and integral equation based methods. The differential equation based methods need to use the truncation boundary conditions; however, the integral equation based methods can implicitly satisfy the radiation condition. As far as the integral equation based methods are concerned, for the two-dimensional objects, the frequency-domain methods are intensively studied for the perfectly electrically conducting (PEC), homogeneous and inhomogeneous dielectric bodies [Harrington, 1993; Richmond, 1965, 1966]. In the time-domain, the integral equation based methods for PEC and homogeneous objects are often investigated [Bennett, 1968; Mittra, 1976; Vechinski and Rao, 1992], but the methods for the inhomogeneous objects are seldom studied.

[3] This paper proposes a time-domain volume integral equation (TDVIE) based method for analyzing the scattering from inhomogeneous cylinders. Though this TDVIE based method has been used to analyze the scattering from the nonlinear objects under the transverse magnetic (TM) illumination [Wang et al., 2000], it is not trivial extension of the TM case to the transverse electric (TE) case. For the sake of completeness, this paper presents both the TM and TE cases.

[4] Consider an inhomogeneous dielectric cylinder in the free space. It is illuminated by an incident plane wave. The dielectric cylinder is assumed to have the same permeability as that in the free space (μ = μ0). The permittivity is assumed to be ε(r). To facilitate the derivation of the integral equation, the total electric field is decomposed into the incident and scattered components as

equation image

[5] By invoking the volume equivalence principle, the scattered field is characterized by an equivalent volume current density J(r, t) that radiates in the unbounded free space and is given by

equation image

where D(r, t) = ε(r)E(r, t) denotes the electric flux density, and κ(r) = (ε(r) − ε0)/ε(r) is the contrast ratio. For the nonlinear objects [Wang et al., 2000], ε(r) is a function of the total electric field E(r, t). Here, we only consider the linear inhomogeneous objects in which ε(r) is just dependent on the position r.

[6] In the following two sections, the TDVIE based method will be presented for the TM and TE cases, respectively. The marching-on-in-time (MOT) based scheme will be proposed to solve the TDVIE. In section 4, some numerical results are presented for inhomogeneous circular and square cylinders. For the sake of comparison, the results obtained using the finite-difference time-domain (FDTD) method are also presented. The results obtained using the TDVIE and FDTD methods are in excellent agreement with each other.

2. TM Case

[7] First, the TM case is considered. In this case, the incident electric field is expressed as Einc(r, t) = equation imageEzinc(r, t), and equation image is the unit vector of the infinitely long axis direction. Therefore, the equivalent volume current density J(r, t) has only the z component, and so do the scattered field, the electric flux density, and the vector potential.

2.1. Formulation of EFIE

[8] The scattered field may be expressed as

equation image

and the potential A(r, t) can be written as

equation image

where S denotes the transverse surface of the cylinder, the symbol “⊗” is the temporal convolution operator, and g(R, t) is the two-dimensional Green's function and given by

equation image

where u(·) is the Heaviside step function, c denotes the speed of light in the free space, and R = ∣rr′∣ is the distance from the source point (r′) to the field point (r).

[9] Combining equations (2)–(4) gives

equation image

[10] Combining equations (1), (2), and (6) yields

equation image

which may represent the 2-D TM transient scattering problem for the inhomogeneous dielectric cylinder. Solving this equation, Esca(r, t) in the whole space can be evaluated by using equation (6).

2.2. Numerical Method

[11] To numerically solve equation (7), the cylinder cross-section S is discretized into Ns triangular cells with the cross sections Sk, k = 1, 2, ⋯, Ns. The time is discretized with the time step Δt, t = (i − 1)Δt, i = 1, 2, ⋯, Nt, here Nt denotes the number of the total time steps. The time step size is chosen as Δt = 0.1fmax−1, where fmax is the maximum frequency of the incident wave.

[12] To apply the method of moments, D(r, t) is expanded in terms of Ns spatial basis functions as

equation image

where Dk(t) may be interpreted as the electric flux flowing past the kth cell at the time t, Pk(r) is a pulse function associated with the kth cell, and defined as

equation image

[13] Putting equation (8) into equation (7), and testing it with δ-function at rm, gives

equation image

where rm and rk denote the central positions of cells m and k, respectively, and Vki(rm, t) is defined as

equation image

where R = ∣rmrk∣ is the distance from the source cell k to the field cell m, and the limits of integration may be expressed as

equation image

If ti2ti1, then Vki(rm, t) = 0.

[14] Equation (10) is a system of linear equations. To solve it, Vki(rm, t) should be evaluated. To do this, Dk(t) is interpolated using a quadratic polynomial, that is

equation image

where

equation image

only aki is used because of the second derivative in equation (11), the other two coefficients are given for the TE case discussed in the next section. Here, the index i is conveniently introduced to represent the present time. Substituting equation (13) into equation (11) gives

equation image

where φ1(t, i, R) is given in Appendix A of this paper. Define

equation image

for the nonself terms(mk), np-point quadrature rule may be used to evaluate this integration,

equation image

where Wg(j) is the weight [Jin, 1993], and Rj is the distance from the sampling point to the field point rm.

[15] For the self terms(m = k), if ti2tR/c, ψ1(t, i, R) can be calculated using equation (17), but if ti2 = tR/c, φ1(t, i, R) becomes singular when R approaches 0. To evaluate ψ1(t, i, R), φ1(t, i, R) is separated into two parts,

equation image

where φ′1(t, i, R) and φ″1(t, i, R) are nonsingular and singular parts, respectively,

equation image

[16] Applying equations (17) and (59) given in Appendix A, ψ1(t, i, R) can be expressed as

equation image

[17] Now, ψ1(t, i, R) has been evaluated for both self and nonself terms, then Vki(rm, t) can be rewritten as

equation image

[18] Using equations (10) and (21), it is easy to formulate a system of Ns linear equations. Solving it at each time step, Dk(t) can be obtained inside the cylinder, and then the electric field everywhere can be calculated at any time.

3. TE Case

[19] This section discusses the TE case. In this case, the direction of the incident electric field lies in the x-y plane, the electric charges are induced inside and on the surface of the cylinder, and hence, the scalar and vector potentials are used to express the scattered fields.

3.1. Formulation of EFIE

[20] The scattered field may be expressed as

equation image

where A(r, t) is the vector potential produced by the equivalent current density J(r, t) given in equation (2),

equation image

and Φ(r, t) is the scalar potential produced by the equivalent charge density σ(r, t)

equation image

[21] The current density J(r, t) and charge density σ(r, t) satisfy the continuity equation

equation image

[22] Combining equations (2) and (23) gives

equation image

[23] Combining equations (2), (24), and (25), we get

equation image

[24] Using equation (22), equation (1) can be written as

equation image

which may represent the EFIE for 2-D TE volume scattering problem.

3.2. Numerical Method

[25] In the TM case, the pulse function is used to expand the electric flux density, but it does not fit for the TE case because of the electric charges induced inside and on the surface of the cylinder. Here the roof-top basis function, developed by Rao et al. [1982], is applied to expand the electric flux density (Figure 1).

Figure 1.

Two-dimensional surface roof-top basis function.

3.2.1. Basis Functions and Testing Procedure

[26] To evaluate the scattered field by MoM, we should first expand D(r, t) in space and time,

equation image

where Ne is the number of edges defining the triangles, fn(r) is the basis function associated with the nth edge, and expressed as [Rao et al., 1982]

equation image

where ln is the length of the nth edge, An± is the area of the triangle Tn±. Points in the triangle Tn+ may be designated either by the position vector r defined with respect to the origin of the coordinate system, or by the position vector ρn+ defined with respect to the free vertex of Tn+. The vector ρn is similarly defined, but it is directed toward the free vertex of Tn. Plus and minus signs are used to determine the positive flux reference direction which is assumed to be from Tn+ to Tn.

[27] Performing the divergence operator on the basis function, gives

equation image

[28] To test equation (28), the inner product is defined as 〈f, g〉 ≡ ∫Sf · gds, and the testing function is chosen as the same as the expansion basis function. Testing each term in equation (28) with fm, we get the following equation,

equation image

where ρmc± is the vector of the centroid of Tm±, and rmc± is the vector of the centroid of Tm± with respect to the origin O. Equation (32) implies that the permittivity ε(r) (and so the contrast ratio κ(r) discussed in next section) takes its value at the centroid of its corresponding triangle.

[29] The integration equation image is given in Appendix A. Next, we discuss how to evaluate the scalar and vector potentials.

3.2.2. Evaluation of Scalar Potential

[30] Following the method developed by Schaubert et al. [1984], and substituting equation (29) into equation (27), yields

equation image

[31] Similar to the TM case, Dk(t) is expanded in terms of quadratic polynomial, then

equation image

where

equation image

and φ1, φ2 and φ3 are given in Appendix A. Define the surface and line integrations as

equation image
equation image
equation image

[32] The field points are at the centroids of triangles, and the integral path for the line integration is along the edges of triangles, so there are no singular terms in the line integration. Therefore, Φl can be easily evaluated numerically. Next, we discuss how to evaluate the surface integrations. Here, we only need to demonstrate how to evaluate equation image, and equation image can be similarly calculated. Define

equation image

[33] When two triangles associated with edges m and k are not the same one, these three integrations can be numerically integrated. Here, np-point quadrature rule is used again

equation image

[34] When the source and field cells coincide, integrands in equation (39) are singular if ti2 = tR/c. For nonsingular integrations, they can be evaluated numerically using equation (40), but for singular integrations, the integrands are separated into two parts as

equation image

where

equation image

[35] The terms with single prime are nonsingular and can be evaluated numerically, and the singular integrations can be analytically integrated in Appendix A (Figure 2), so equation (39) may be written as

equation image
Figure 2.

Demonstration of parameters used in the triangle integrations.

[36] After evaluating ψ1, ψ2 and ψ3 for nonself and self terms, we can rewrite equation image as

equation image

where

equation image

We can similarly evaluate equation image, and then calculate the scalar potential using equation (34).

3.2.3. Evaluation of Time Derivative of Vector Potential

[37] Substituting equation (29) into equation (26), and using equation (13), we have

equation image

where

equation image

Next, we discuss how to evaluate the surface integrations. For nonself terms and the nonsingular terms of the self patches, we may still use the np-point quadrature rule to perform the integration,

equation image

where equation image is the free vertex of triangle Tk+. For the self terms, we can deal with the singular integration similarly to those for scalar potentials. Doing so, we have

equation image

where u is given in Appendix A. We can similarly evaluate equation image, and then calculate the time derivative of the vector potential using equation (46).

4. Numerical Results

[38] To validate the method proposed in this paper, we present some numerical results for two two-dimensional inhomogeneous scatterers, namely, circular and square cylinders. The radius of the circular cylinder is a = 0.5 m, a total of 138 triangular cells are used to discretize this cylinder, and its relative permittivity is given by

equation image

[39] The side length of the square cylinder is l = 1.5 m, it is divided into 450 triangular cells, and its relative permittivity is given by

equation image

In equations (50) and (51), r is the distance from the center of the cylinder.

[40] The cylinders are illuminated by the incident plane waves, which have the waveforms as

equation image

where E0 = 4.0 V/m, η0 denotes the wave impedance of the free space, equation image is the wave vector, and the time signature of the incident wave is chosen as the Neuman function (the derivative of a Gaussian pulse),

equation image

where t0 = 1.12 × 10−6 s, σ = 6.22 × 1013 s−2. Figures 3 and 4 are numerical results for the TM case, and Figures 5 and 6 are for the TE case.

Figure 3.

Comparison of the electric fields at the center of circular cylinder obtained using TDVIE and FDTD methods for TM case.

Figure 4.

Comparison of the electric fields at the center of square cylinder obtained using TDVIE and FDTD methods for TM case.

Figure 5.

Comparison of the electric fields at the center of circular cylinder obtained using TDVIE and FDTD methods for TE case.

Figure 6.

Comparison of the electric fields at the center of square cylinder obtained using TDVIE and FDTD methods for TE case.

[41] In the TM case, equation image = equation image is chosen. Figures 3 and 4 are the electric fields at the centers of circular and square cylinders, respectively. For the sake of comparisons, the numerical results obtained using the FDTD method are also presented in these two figures. These TDVIE and FDTD results agree very well with each other.

[42] In the TE case, we choose equation image. Figures 5 and 6 present the electric fields at the centers of circular and square cylinders, respectively. It can be seen that the results obtained by using the TDVIE method agree very well with those solved by using the FDTD method.

5. Conclusion

[43] This paper proposes an MOT scheme based TDVIE method for analysis of scattering from inhomogeneous cylinders for both TM and TE cases. For the TM case, the electric flux is defined on the surfaces of triangles, and expanded using the pulse basis functions; for the TE case, it is defined on the edges of triangles, and expanded using the surface roof-top basis functions. Numerical results obtained using the TDVIE method agree very well with those obtained using the FDTD method. Unlike the surface integral equation for the homogeneous scatterers, no resonances and instabilities occur in the time-domain volume integral equations presented in this paper.

[44] Like most schemes based on the time-domain integral equations, the proposed TDVIE solver is computationally expensive. Indeed, while we believe that our scheme has certain advantages over the FDTD methods for modelling fine features and structures that are not easily discretized on the rectangular grids, the above numerical experiments reveal that our TDVIE code is slow compared to our FDTD solver. The TDVIE solver is, however, amenable to acceleration by the two-dimensional plane-wave time-domain (PWTD) method [Lu et al., 2000], a recently developed scheme for rapidly evaluating 2-D wave fields due to the known sources, which can be used to considerably reduce the computational complexity to O(NtlogNt · NslogNs) from its conventional O(Nt2Ns2). Therefore, we anticipate that a PWTD enhanced TDVIE solver will be computationally competitive with its FDTD counterparts.

Appendix A

[45] This appendix presents some integrations used in this paper.

[46] Three integrations with respect to the time are given as

equation image
equation image
equation image

[47] In equation (32), we want to evaluate the integration ∫equation imagedsfm · fk. Using the definition of surface coordinate [Schaubert et al., 1984], it can be shown that

equation image

where the plus sign is taken if triangles associated with edges m and k have the same signs; otherwise, it is minus sign. In this equation, r1, r2, and r3 are vectors from the origin O to the vertices of the triangle associated with the kth edge.

[48] In this paper, we need to evaluate some singular integrations. Following the method developed by Wilton et al. [1984], we can convert the surface integrations into line integrations by applying the Gauss integral theorem. To this end, we define some parameters. As shown in Figure 2, suppose that P1P2 is any side of the field triangle Tm+, and its unit direction is denoted by equation image, and its outward normal unit vector is n. P(rmc+) is the field point. The vectors for points P2 and P1 are ρ+ and ρ, respectively. Assume that PP0 is perpendicular to P1P2 and P0 is on the P1P2 or its extension, and the length of PP0 is h. Any point on the P1P2 can be denoted by the variable l′, which is measured from P0 to P2 and P1 are denoted as l+ and l, respectively. From Figure 2, we have

equation image

then we obtain the following three equations

equation image
equation image
equation image

Ancillary