Speed-up of scattered field formulation of FDTD method combining with time-domain EFIE by using spherical harmonic expansion of Green’s function

Abstract

This paper proposes a scattered field formulation of the finite-difference time domain method (S-FDTD) combining with a time-domain electric field integral equation (TD-EFIE) based on surface equivalence theorem. The scattered field formulation enables us to reduce computation cost effectively for the case that scatterers are located at a far distance from field sources of electromagnetic wave since we need to calculate the scattered field component only on the region in the vicinity of the scatterers. Then it is required for the use of the scattered field formulation that the field source for the incident field component has to be expressed by any analytical solutions such as a point dipole source, plane wave, and so on. In this work, the scattered field formulation of the FDTD method is applied to cases that there are no analytical expressions of the field sources, in which the field source contains conductors such as antennas, combining with the TD-EFIE on a virtual surface which encloses the field sources. Then, it is known that calculation of the TD-EFIE itself is time consuming. This paper considers speed-up of the S-FDTD simulation based on the TD-EFIE using a spherical harmonic expansion of Green’s function of Helmholtz equation, additionally.

Keywords

Finite-difference time-domain method microwave propagation scattered field formulation surface equivalence theorem time-domain electric field integral equation spherical harmonic expansion Green’s function of Helmholtz equation

1. Introduction

The Finite-Difference Time-Domain (FDTD) method is one of most popular numerical schemes for microwave simulations, and it has been applied to various industry applications such as EMC problems, electromagnetic CAD, and so on. In general, computation cost of the FDTD method is lower than those of other numerical schemes since it is not necessary to store any matrices and to solve the matrix equations in the FDTD simulation. However the FDTD computation for comparably large scale calculations such as 1,000 × 1,000 × 1,000 grid space, which often appear in practical situation in the electromagnetic CAD, require over 100GB memory and takes several tens hours even using high-end PCs. Then, the scattered field formulation of the FDTD (S-FDTD) method [1–3] can be effectively employed for saving memory and computation time in the FDTD simulation [4] for electromagnetic scattering problems if the field source can be described analytically such as a point dipole source, plane-like incident wave, and so on. In the S-FDTD scheme, the electromagnetic field value is split into incident and scattered field components, and only the scattered field component is calculated numerically using given distribution of the incident field component around the scatterers. In the other words, the S-FDTD method cannot be applied when the field source contains materials or conductors such as dipole antenna, horn antenna in which the field source cannot be expressed analytically and any numerical simulation are required for the calculation of the source field distribution itself.

To aim to enlarge usage of the S-FDTD method, this paper discusses the scattered field formulation of the FDTD method combining with a time-domain integral equation (TD-IE) for the case that the field source contains materials or conductors and cannot be expressed in any analytical forms. Combination of the time-domain integral equation with the FDTD method itself was already proposed [5–14] to be used for an absorbing boundary condition (ABC) based on the surface equivalent theorem [15,16], which is a different type from Mur’s and perfect matched layer (PML) ABC. According to the surface equivalence theorem, electromagnetic fields outside a virtual surface S, which encloses the field sources, are calculated by the TD-IE to use the electromagnetic field value on S. If we simulate the electromagnetic fields on S by the FDTD method, the electromagnetic fields outside S can be expressed analytically by the TD-IE. The analytical expression of the field value outside S can be used for the field value on the absorbing boundary, in addition for boundary condition in the S-FDTD. However, it is known that the computation of the TD-IE is extremely time consuming, because we need to carry out the numerical surface integral of the TD-IE for every observation points. In this work, it is also shown that effective speed-up of the S-FDTD simulation based on the TD-IE can be achieved employing a spherical harmonic expansion of Green’s function of the Helmholtz equation. In the spherical harmonic expression, the coordinates of observation and integral points in the TD-IE are split each other, therefore, we need to carry out the numerical surface integral only once and the values of the numerical surface integral can be re-used for calculations of the electromagnetic field on every observation points, and accordingly the computation cost for the TD-IE can be reduced effectively.

Fig. 1.

Surface charge and current on S and surface equivalent theorem.

2. Scattered field formulation of FDTD method [1–3]

In the S-FDTD method, the electromagnetic fields E, H (total field) are split into two components, the incident E ^inc, H ^inc and the scattered E ^scat, H ^scat field components as follows [1], $\begin{eqnarray}\displaystyle \mathbf{E}=\mathbf{E}^{\text{inc}}+\mathbf{E}^{\text{scat}},\quad \mathbf{H}=\mathbf{H}^{\text{inc}}+\mathbf{H}^{\text{scat}}. & & \displaystyle\end{eqnarray}$ (1) For example, in the simplest cases in which there exist only perfect electric conductors (PEC) in the region, E ^scat, H ^scat satisfy the following Maxwell’s equation, $\begin{eqnarray} \displaystyle \text{} & \text{} & \displaystyle {\displaystyle \frac{\partial \mathbf{E}^{\text{scat}}}{\partial t}}={\displaystyle \frac{1}{{\varepsilon}}}{\nabla}\times \mathbf{H}^{\text{scat}}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle {\displaystyle \frac{\partial \mathbf{H}^{\text{scat}}}{\partial t}}={\displaystyle {-}\frac{1}{{\mu}}}{\nabla}\times \mathbf{E}^{\text{scat}}. \end{eqnarray}$ (2)

Then, the boundary condition for the electric field at the PEC is expressed by $\begin{eqnarray}\displaystyle \mathbf{E}=\mathbf{E}^{\text{inc}}+\mathbf{E}^{\text{scat}}=0. & & \displaystyle\end{eqnarray}$ (3) Accordingly when free space solutions to the incident field components E ^inc, H ^inc are given analytically, the scattered field components E ^scat, H ^scat can be calculated numerically to solve (2) regarding the − E ^inc, − H ^inc being the equivalent field sources or boundary conditions at the PEC. In this S-FDTD formulation, we need to simulate only the scattered field components E ^scat, H ^scat by the FDTD scheme and then the FDTD grid discretization is carried out only for the local region around the scatterers which is much smaller than the whole region including the field source. Therefore the scattered field formulation enables us to reduce the computation cost effectively. Then the incident field components E ^inc, H ^inc must be given by any analytical forms such as in point dipole source, plane incident wave, otherwise the scattered field formulation cannot be used, for example, for the case that the field source consists of conductors such as antennas.

Fig. 2.

Overview of S-FDTD simulation combining with TD-IE for electric field.

Fig. 3.

Surface integral on virtual surface S for every observation point x.

3. Scattered field formulation of FDTD method with time-domain integral equation

We here consider to apply the S-FDTD method for the case that the field source contains conductors in which the FDTD numerical simulation is required to solve the electromagnetic fields around the field source. In Fig. 1(a), it is depicted that the field source produces electromagnetic fields, and the field source is surrounded by a virtual surface S. Then, the surface equivalence theorem (Fig. 1(b)) tells us that the field source including antenna inside S can be replaced equivalently by the surface current density J = n × H, surface charge density 𝜌 = ϵ n ⋅ E, magnetic surface current density M = E × n and magnetic surface charge density 𝜌_M = μ n ⋅ H on S (n is a unit normal to the surface S) [15,16]. In other word, electromagnetic fields E, H at any time t and position x outside S can be expressed analytically by the following time-domain integral equation, $\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}\mathbf{E}(t,\mathbf{x})=-{\displaystyle \frac{1}{{\varepsilon}}}{\displaystyle \frac{{\varepsilon}}{4{\pi}}}{\nabla}\times \displaystyle \int _{S}{\displaystyle \frac{\mathbf{M}\left(t-{\displaystyle \frac{\mathbf{x}-\mathbf{x}^{\prime }}{c}},\mathbf{x}^{\prime }\right)}{|\mathbf{x}-\mathbf{x}^{\prime }|}}dS^{\prime }\\ \hspace{60.0pt}-∼{\displaystyle \frac{{\mu}}{4{\pi}}}{\displaystyle \frac{\partial }{\partial t}}\displaystyle \int _{S}{\displaystyle \frac{\mathbf{J}\left(t-{\displaystyle \frac{\mathbf{x}-\mathbf{x}^{\prime }}{c}},\mathbf{x}^{\prime }\right)}{|\mathbf{x}-\mathbf{x}^{\prime }|}}dS^{\prime }-{\displaystyle \frac{1}{4{\pi}{\epsilon}}}{\nabla}\displaystyle \int _{S}{\displaystyle \frac{{\rho}\left(t-{\displaystyle \frac{\mathbf{x}-\mathbf{x}^{\prime }}{c}},\mathbf{x}^{\prime }\right)}{|\mathbf{x}-\mathbf{x}^{\prime }|}}dS^{\prime }\\ \mathbf{H}(t,\mathbf{x})=-{\displaystyle \frac{1}{{\mu}}}{\displaystyle \frac{{\mu}}{4{\pi}}}{\nabla}\times \displaystyle \int _{S}{\displaystyle \frac{\mathbf{J}\left(t-{\displaystyle \frac{\mathbf{x}-\mathbf{x}^{\prime }}{c}},\mathbf{x}^{\prime }\right)}{|\mathbf{x}-\mathbf{x}^{\prime }|}}dS^{\prime }\\ \hspace{60.0pt}-∼{\displaystyle \frac{{\varepsilon}}{4{\pi}}}{\displaystyle \frac{\partial }{\partial t}}\displaystyle \int _{S}{\displaystyle \frac{\mathbf{M}\left(t-{\displaystyle \frac{\mathbf{x}-\mathbf{x}^{\prime }}{c}},\mathbf{x}^{\prime }\right)}{|\mathbf{x}-\mathbf{x}^{\prime }|}}dS^{\prime }-{\displaystyle \frac{1}{4{\pi}{\mu}}}{\nabla}\displaystyle \int _{S}{\displaystyle \frac{{\rho}_{M}\left(t-{\displaystyle \frac{\mathbf{x}-\mathbf{x}^{\prime }}{c}},\mathbf{x}^{\prime }\right)}{|\mathbf{x}-\mathbf{x}^{\prime }|}}dS^{\prime }\end{array} & & \displaystyle\end{eqnarray}$ (4) to use the time-domain signals of electromagnetic fields of J, M, 𝜌 and 𝜌_M on the virtual surface S [17 ]. This concept can be introduced effectively into the S-FDTD formulation for perfect electric conductor (PEC) scatterers as in Fig. 2. Firstly, we perform the standard FDTD simulation in the vicinity of the field source (denoted by V ₀ in Fig. 2) which contains all antenna conductors and the virtual surface S, and store the time-domain signals of M, J, 𝜌 calculated by the upper equation of ((4)) for the electric field (TD-EFIE) on the virtual surface S. Then if we substitute the time-domain signals of M, J, 𝜌 into ((4)), we can obtain the analytical expression for the electromagnetic fields everywhere outside S in the surface integral form, which can be applied to the incident field component E ^inc of the S-FDTD method for scattering phenomena around any scatterers outside S. In this simulation of Fig. 2, two FDTD calculations, the standard FDTD scheme at V ₀ and the S-FDTD scheme at V _s are carried out at the same time. In particular, the S-FDTD scheme at V _s is completely same as those of the standard FDTD method because E ^scat and H ^scat satisfy the same equation ((2)) as those of the standard FDTD method. In principle, there are no explicit field excitation in the S-FDTD simulation at V _s, but the field excitation for E ^scat and H ^scat are done by the boundary condition ((3)) at the scatterer (Fig. 2). Then it is easily understood that memory requirement of the S-FDTD method is smaller than that of the FDTD method for the full region V. It should be notified that this scheme can be applied to any types of field sources if the virtual surface S encloses all materials of the field source.

On the other hand, it is known that the computation of the surface integral (4) is extremely time consuming since we need to carry out the surface integrals numerically for every observation point x and time t (Fig. 3). Indeed, in the S-FDTD combining with the time-domain electric field equation (TD-EFIE) of (4), the time-domain surface integrals are performed for all grids of PEC scatterers.

Fig. 4.

Numerical model of five conductor Yagi-antenna and scatterer.

4. Spherical harmonic expression of Green function

We here consider to employ a spherical harmonic expansion of Green’s function of the Helmholtz equation for speed-up of the computation of the TD-EFIE. Straightforward calculation of the surface integral of the TD-EFIE of ((4)) for all observation points results in extremely heavy computation since the computation of the TD-EFIE for all FDTD grids of the scatterers contains the large number of multiplications, I × N, where I and N are the numbers of scatterer grid x outside S and integral points x’ on S. In many cases of simulations of scattering phenomena, we interested only in the time harmonic behavior e ^iωt of the fields, and then, the computation cost in the calculation of the TD-EFIE ((4)) can be reduced effectively by a use of the following spherical harmonic expansion [18] of the Green’s function of the Helmholtz equation, $\begin{eqnarray} \displaystyle {\displaystyle \frac{e^{-i{\frac{{\omega}}{c}}|\mathbf{x}-\mathbf{x}^{\prime }|}}{|\mathbf{x}-\mathbf{x}^{\prime }|}}=-i{\displaystyle \frac{{\omega}}{c}}\mathop{\sum }_{l=0}^{\infty }\mathop{\sum }_{m=-l}^{l}(2l+1)(-1)^{m}h_{l}^{(2)}\left({\displaystyle \frac{{\omega}}{c}}r\right)j_{l}\left({\displaystyle \frac{{\omega}}{c}}r^{\prime }\right)P_{l}^{m}(\cos {\theta})P_{l}^{-m}(\cos {\theta}^{\prime })e^{im({\varphi}-{\varphi}^{\prime })} & & \displaystyle \end{eqnarray}$ (5) where x = (r, θ, φ), x ^′ = (r, ^′θ^′, φ^′) are spherical coordinates, j _l and $h_{l}^{(2)}$ are the spherical Bessel function and the spherical Hankel function of the second kind respectively, $P_{l}^{m}$ (cosθ) is the associated Legendre polynomial. For example, the surface integral of the first term of ((4)) is expressed in the following spherical harmonic expansion form to use ((5)), $\begin{eqnarray} \displaystyle \text{} & \text{} & \displaystyle \int _{S}{\displaystyle \frac{\mathbf{M}({\omega},\mathbf{x}^{\prime })e^{-i{\frac{{\omega}}{c}}|\mathbf{x}-\mathbf{x}^{\prime }|}}{|\mathbf{x}-\mathbf{x}^{\prime }|}}dS^{\prime }=-i{\displaystyle \frac{{\omega}}{c}}\mathop{\sum }_{l=0}^{\infty }\mathop{\sum }_{m=0}^{l}2C_{m}h_{l}^{(2)}\left({\displaystyle \frac{{\omega}}{c}}r\right)\sqrt{{\displaystyle \frac{(2l+1)(l-m)!}{2(l+m)!}}}P_{l}^{m}(\cos {\theta})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \hspace{36.0pt}\times \{\cos (m{\phi})\mathbf{M}_{CR}^{lm}({\omega})+i\cos (m{\phi})\mathbf{M}_{CI}^{lm}({\omega})+\sin (m{\phi})\mathbf{M}_{SR}^{lm}({\omega})+i\sin (m{\phi})\mathbf{M}_{SI}^{lm}({\omega})\} \end{eqnarray}$ (6) where $\begin{eqnarray} \begin{array}{@{} c@{}}\displaystyle \mathbf{M}_{CR}^{lm}({\omega})=\int _{S}{\displaystyle \frac{\mathbf{M}^{c}({\omega},\mathbf{x}^{\prime })}{2}}j_{l}\left({\displaystyle \frac{{\omega}}{c}}r^{\prime }\right)\tilde{P}_{l}^{m}(\cos {\theta}^{\prime })\cos (m{\phi}^{\prime })dS^{\prime },\\[1.2pc] \displaystyle \mathbf{M}_{CI}^{lm}({\omega})=\int _{S}{\displaystyle \frac{-\mathbf{M}^{s}({\omega},\mathbf{x}^{\prime })}{2}}j_{l}\left({\displaystyle \frac{{\omega}}{c}}r^{\prime }\right)\tilde{P}_{l}^{m}(\cos {\theta}^{\prime })\cos (m{\phi}^{\prime })dS^{\prime },\\[1.2pc] \displaystyle \mathbf{M}_{SR}^{lm}({\omega})=\int _{S}{\displaystyle \frac{\mathbf{M}^{c}({\omega},\mathbf{x}^{\prime })}{2}}j_{l}\left({\displaystyle \frac{{\omega}}{c}}r^{\prime }\right)\tilde{P}_{l}^{m}(\cos {\theta}^{\prime })\sin (m{\phi}^{\prime })dS^{\prime },\\[1.2pc] \displaystyle \mathbf{M}_{SI}^{lm}({\omega})=\int _{S}{\displaystyle \frac{-\mathbf{M}^{s}({\omega},\mathbf{x}^{\prime })}{2}}j_{l}\left({\displaystyle \frac{{\omega}}{c}}r^{\prime }\right)\tilde{P}_{l}^{m}(\cos {\theta}^{\prime })\sin (m{\phi}^{\prime })dS^{\prime }. \end{array} \end{eqnarray}$ (7) Indexes “lm”, “C or S” and “R or I” of the coefficients M _{C/S R/I} ^lm in ((7)) denote multipole components corresponding to the $P_{l}^{m}$ (cosθ), cos(mφ) or sin(mφ) functions in the integrands of ((7)) and real or imaginary parts inside the bracket in ((6)), respectively. The M ^c and M ^s in ((7)) are Fourier coefficients of the time-domain signal of M(t, x) = E(t, x)×n at the position x, $\begin{eqnarray}\displaystyle \mathbf{M}(t,\mathbf{x})={\displaystyle \frac{\mathbf{M}^{c}(0,\mathbf{x})}{2}}+\mathop{\sum }_{n=1}^{\infty }(\mathbf{M}^{c}(n{\omega},\mathbf{x})\cos (n{\omega}t)+\mathbf{M}^{s}(n{\omega},\mathbf{x})\sin (n{\omega}t)). & & \displaystyle\end{eqnarray}$ (8) The second and third terms of ((4)) are expressed in the spherical expansion form by the similar way as in ((6)). When the spherical harmonic expansion as in ((6)) is employed for the TD-IE, we need to carry out the surface integrals for M, J, 𝜌 in ((4)) only once for the calculation of the multi-pole coefficients as in ((7)) since the observation point x and integral point x ^′ are separated each other in ((6)), and ((7)) contains only the integral point x ^′. This means that the above mentioned computation cost of the number of multiplications I × N will be reduced to be I × L ², where L is the maximum number of multi-poles in ((6)). In general, the summation for l in ((6)) can be truncated by about L =10 at most for sufficient accuracy. Accordingly, the computation cost of ((6)) is much smaller than that of the straightforward calculation of numerical integrals in ((4)).

Fig. 5.

Electric field distribution simulated by standard FDTD method (full domain simulation).

Fig. 6.

Electric field around scatterer in vertical cross-section simulated by (a) standard FDTD method, (b) with TD-EFIE and (c) with spherical expansion.

Fig. 7.

Distribution of surface current (a) and charge (b) densities on S.

5. Numerical examples

A model of a numerical example for the proposed method is depicted in Fig. 4. Electromagnetic wave with 2.45 GHz radiated from five conductors Yagi-antenna is scattered by a disk-like PEC scatterer with 2.2 cm radius and 2 mm thickness at about seven wavelengths (90 cm) distance from the Yagi-antenna. The grid size of the FDTD simulation is 2 mm which is about 60 divisions of the wavelength. The virtual surface S (surface of 122 × 152 × 152 grid space) encloses the power input point and all conductors of the Yagi-antenna, and the region V _s encloses only the disk scatterer. Firstly, the standard FDTD simulation for the field source is performed in the region V ₀ which contains the Yagi-antenna and the virtual surface S. Synchronizing with this FDTD simulation, the S-FDTD simulation for scattered fields E ^scat, H ^scat is carried out in the region V _s (232 × 232 × 232 grid size) to use the incident field component E ^inc, which are calculated by the TD-IE on S, as the boundary condition (3).

In Fig. 5, a distribution of electric field in y–z cross-section, which are simulated by the standard FDTD method for a whole region (232 × 232 × 632 grid size) including both the field source and scatterer, is depicted. Two peaks at the scatterer indicate concentration of the field at the disk edge. Electric field distributions in sub-region V _s, which are simulated by the standard FDTD for the full region, the S-FDTD based on the TD-EFIE ((4)) and the TD-EFIE with the spherical harmonic expansion ((6)), are depicted in Fig. 6(a), (b) and (c), respectively. In Fig. 7, examples of distributions of surface current and charge densities on the virtual surface S for the calculations of Fig. 6(b) and (c) are depicted. The maximum number L of the summation for l in ((6)) is truncated by L = 5 in Fig. 6(c). We can confirm that all results in Fig. 6 show a good agreement and the S-FDTD based on the TD-EFIE (Fig. 6(b) and (c)) works well for the case that the field source includes conductors and has no analytical expressions. In general, the maximum number L should be determined depending on the distribution of electromagnetic radiation fields. It is checked that the maximum number L for the spherical expansion ((6)) is converged at L = 5 with less than 1% error in the incident field calculation even if there exist strong directivity of the Yagi antenna in the radiation field. Then, computation time for Fig. 6(a), (b) and (c) are 4 hours, 7.6 days and 3.5 hours by PC with intel core i7, respectively. This means that the use of the spherical harmonic expansion reduces the computation cost effectively in the S-FDTD scheme combining with the TD-EFIE. It can be understood considering the scheme of the S-FDTD with TD-EFIE depicted in Fig. 2 that the computation time and required memory depend on the distance between the field source and scatterer for Fig. 6(a), on the other hand, do not depend for Fig.(b) and (c). In addition, difference between Fig.(b) and (c) is only in the calculation of the incident field component E ^inc in V _s. Accordingly, in the case that the scatterer is located at much more far distance from the Yagi antenna than that of the numerical model of Fig. 4, computation time and memory in the standard FDTD simulation (Fig. 6(a)) increase further, but those of the proposed method (Fig. 6(c)) does not increase anymore.

6. Summary

The S-FDTD method combining with the TD-EFIE has been proposed to aim to apply the S-FDTD method to cases that the field source cannot be expressed in any analytical form. In particular, speed-up of the computation of the S-FDTD based on the TD-EFIE using the spherical harmonic expansion for Green’s function of Helmholtz equation has been considered. It is confirmed by the numerical examples of scattering problem with the Yagi-antenna that the proposed method works well and required memory is effectively reduced. It is also confirmed that computation cost of the TD-EFIE can be reduced effectively employing the spherical harmonic expansion.

In this work, the numerical model which contains only PEC is treated, but the scheme of the S-FDTD method with the TD-EFIE and harmonic expansion itself is common for cases which contain other materials. In that case, only the equations for scattered fields (2) and boundary condition (3) will be replaced by appropriate equations and conditions.

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