# The Alternating Direction Implicit Body of Revolution Multiresolution Time Domain Method with Convolution Perfect Matched Layer

Authors: Chen Yiwang; Ni Jiazheng; Liu Yawen; Dong Xiaohua; Zhang Pin
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

Overmuch memory and time of CPU have been taken by multiresolution time domain (MRTD) method in three-dimension issues. In order to solve this problem, the alternating direction implicit body of revolution multiresolution time domain (ADI-BOR-MRTD) scheme is presented. Firstly, based on body of revolution finite difference time domain (BOR-FDTD) method, equations of body of revolution multiresolution time domain (BOR-MRTD) method are implemented. Then alternating direction implicit (ADI) is introduced into BOR-MRTD method. Lastly, convolution perfect matched layer (CPML) is applied for ADI-BOR-MRTD method. Numerical results demonstrate that ADI-BOR-MRTD method saves more memory and time of CPU than FDTD and MRTD methods.


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## Body

## 1. Introduction

As an efficient numerical algorithm, the multiresolution time domain (MRTD) method was applied in electromagnetic field computation in 1996 by Krumpholz and Katehi [1] firstly. Compared with the finite difference time domain (FDTD) method, the MRTD method has lower numerical dispersion and saves more memory and time of CPU [1, 2].The time index and the calculating efficiency of the MRTD method are generally limited by the Courant-Friedrich-Levy (CFL) stability condition. However, the alternating direction implicit (ADI) technique can overcome the CFL limitation [3]. Chen and Zhang had published the ADI-MRTD scheme in 2001 [4]. The time step size for the ADI-MRTD is only limited by modeling accuracy of the calculation. Then, the study on the numerical dispersion, absorbing boundary conditions, and the application in the one-dimension photoelectronic band-gap of the ADI-MRTD scheme are developed gradually [5–7].The body of revolution is an important target in electromagnetic field computation. In order to calculate the body of revolution with less memory and time of CPU, the ADI-BOR-MRTD scheme is presented. At the end of the work, the convolution perfect matched layer (CPML) formulations are derived for the ADI-BOR-MRTD scheme.


## 2. Equations of BOR-MRTD

In cylindrical coordinates, Maxwell’s equations should be written as(1a)ε∂Eρ∂t+σEρ=1ρ∂Hz∂φ-∂Hφ∂z(1b)ε∂Eφ∂t+σEφ=∂Hρ∂z-∂Hz∂ρ(1c)ε∂Ez∂t+σEz=1ρ∂ρHφ∂ρ-1ρ∂Hρ∂φ(1d)-μ∂Hρ∂t-σmHρ=1ρ∂Ez∂φ-∂Eφ∂z(1e)-μ∂Hφ∂t-σmHφ=∂Eρ∂z-∂Ez∂ρ(1f)-μ∂Hz∂t-σmHz=1ρ∂ρEφ∂ρ-1ρ∂Eρ∂φ.The electric and magnetic fields are expanded by Fourier series as(2a)E=∑m=0∞eucos⁡mϕ+evsin⁡mϕ(2b)H=∑m=0∞hucos⁡mϕ+hvsin⁡mϕ,where eu,ev,hu,hv are Fourier coefficients and e=Em, h=Hm. ϕ is azimuth angle; m is modulus. u is related to cos⁡mϕ; v is related to sin⁡mϕ.Substituting (2a) and (2b) to (1a)–(1f), (1a)–(1f) are rewritten as(3a)ε∂eρ∂t=±mρhz-∂hφ∂z(3b)ε∂eφ∂t=∂hρ∂z-∂hz∂ρ(3c)ε∂ez∂t=1ρ∂ρhφ∂ρ∓mρhρ(3d)μ∂hρ∂t=±mρez+∂eφ∂z(3e)μ∂hφ∂t=-∂eρ∂z+∂ez∂ρ(3f)μ∂hz∂t=-1ρ∂ρeφ∂ρ∓mρeρ.The electric and magnetic fields are expanded by Daubechies’ scaling function in space domain and by Haar’s scaling function in time domain.(4a)eρr→,t=∑i,j,k,n=-∞+∞ei+1/2,j,kαρ,nΦntαi+1/2ραjφαkz(4b)eφr→,t=∑i,j,k,n=-∞+∞ei,j+1/2,kαφ,nΦntαiραj+1/2φαkz(4c)ezr→,t=∑i,j,k,n=-∞+∞ei,j,k+1/2αz,nΦntαiραjφαk+1/2z(4d)hρr→,t=∑i,j,k,n=-∞+∞hi,j+1/2,k+1/2αρ,n+1/2Φn+1/2tαiραj+1/2φαk+1/2z(4e)hφr→,t=∑i,j,k,n=-∞+∞hi+1/2,j,k+1/2αφ,n+1/2Φn+1/2tαi+1/2ραjφαk+1/2z(4f)hzr→,t=∑i,j,k,n=-∞+∞hi+1/2,j+1/2,kαz,n+1/2Φn+1/2tαi+1/2ραj+1/2φαkz.e i , j , k α ζ , n and hi,j,kαζ,n are the field coefficients, with ζ=ρ,φ,z. The indexes i, j, k, and n are the space indices and time indices as ρ=iΔρ, φ=jΔφ, z=kΔz, and t=nΔt, where Δρ, Δφ, Δz, and Δt represent the space and time discretization intervals in ρ-, φ-, z-, and t-direction. The function Φ(t) is Haar’s scaling function [8], and α is Daubechies’ scaling function [9].The distribution of field components is shown in Figure1.Figure 1
Distribution of field components for BOR-MRTD.The equations of BOR-MRTD method are presented as follows:(5a)eρn+1i+12,k=eρni+12,k±mΔti+1/2εΔρhzn+1/2i+12,k-ΔtεΔz∑lalhφn+1/2i+12,k+l+12(5b)eφn+1i,k=eφni,k+ΔtεΔz∑lalhρn+1/2i,k+l+12-ΔtεΔρ∑lalhzn+1/2i+l+12,k.(5c)ezn+1i,k+12=ezni,k+12∓mΔtεiΔρhρn+1/2i,k+12+ΔtiεΔρ∑lali+l+12hφn+1/2i+l+12,k+12(5d)hρn+1/2i,k+12=hρn-1/2i,k+12+Δtμ1Δz∑laleφni,k+l+1±miΔρezni,k+12(5e)hφn+1/2i+12,k+12=hφn-1/2i+12,k+12+ΔtμΔρ∑lalezni,k+l+12-ΔtμΔz∑laleρni+12,k+l+1(5f)hzn+1/2i+12,k=hzn-1/2i+12,k+Δtμi+1/2Δρ∓meρni+12,k-∑lali+l+1eφni+l+1,k.The coefficient al is equal to(6)al≡∂αm′+1/2x∂x,αm-1x=1π∫0∞α^ω2sin⁡ωl+0.5dω.For Daubechies’ scaling function with two vanishing moments (D2), the coefficients are shown in Table 1; for l>2, al are zeros due to the compact support of Daubechies’ scaling function; for l<0, al are given by the symmetry relation a-1-l=-al.Table 1
The coefficientsa(l) of D2.
l a ( l ) 0 1.22916661202745 1 −0.09374997764746 2 0.01041666418309Whenρ=0, namely, i=0, the values of eϕ,ez,hρ are singular. It can be solved via Ampere’s law:(7)∮cH·dl=ε∂∂t∬sE·ds+∬sσE·ds.Taking the integral of (7),(8)ezn+10,k+12=ezn0,k+12+4ΔtεΔρhφn+1/212,k+12,m=0.According to distribution ofeφ,ez,hρ, when m≠1, hρ0,k+1/2=eφ0,k=0, and when m>0, ez0,k+1/2=0. When ρ=0, it is not necessary to calculate eφ, because the coefficient of eφ0,k is 0 in (5f). hρ0,k+1/2 is just useful to calculate eφ0,k, so it is needless too. The conclusion is that ez with m=0 is the only field component to be calculated when ρ=0.


## 3. Equations of ADI-BOR-MRTD

In alternating direction implicit method, field components have been calculated att=nΔt and t=n+1/2Δt. The time has been discretized in two steps, namely, n→n+1/2 and n+1/2→n+1 [10]. The ADI-BOR-MRTD equations are presented as follows.First Step ( n → n + 1 / 2 ) (9a) e ρ n + 1 / 2 i + 1 2 , k = e ρ n i + 1 2 , k + Δ t 2 ε ± m i + 1 / 2 Δ ρ h z n + 1 / 2 i + 1 2 , k - 1 Δ z ∑ l a l h φ n i + 1 2 , k + l + 1 2 (9b) e φ n + 1 / 2 i , k = e φ n i , k + Δ t 2 ε 1 Δ z ∑ l a l h ρ n + 1 / 2 i , k + l + 1 2 - 1 Δ ρ ∑ l a l h z n i + l + 1 2 , k (9c) e z n + 1 / 2 i , k + 1 2 = e z n i , k + 1 2 + Δ t 2 ε ∑ l a l i + l + 1 / 2 h φ n + 1 / 2 i + l + 1 / 2 , k + 1 / 2 i Δ ρ ∓ m i Δ ρ h ρ n i , k + 1 2 (9d) h ρ n + 1 / 2 i , k + 1 2 = h ρ n i , k + 1 2 + Δ t 2 μ ∑ l a l e φ n + 1 / 2 i , k + l + 1 Δ z ± m i Δ ρ e z n i , k + 1 2 (9e) h φ n + 1 / 2 i + 1 2 , k + 1 2 = h φ n i + 1 2 , k + 1 2 + Δ t 2 μ ∑ l a l e z n + 1 / 2 i + l + 1 , k + 1 / 2 Δ ρ - ∑ l a l e ρ n i + 1 / 2 , k + l + 1 Δ z (9f) h z n + 1 / 2 i + 1 2 , k = h z n i + 1 2 , k + Δ t 2 i + 1 / 2 μ Δ ρ ∓ m e ρ n + 1 / 2 i + 1 2 , k - ∑ l a l i + l + 1 e φ n i + l + 1 , k .Second Step ( n + 1 / 2 → n + 1 ) (10a) e ρ n + 1 i + 1 2 , k = e ρ n + 1 / 2 i + 1 2 , k + Δ t 2 ε ± m i + 1 / 2 Δ ρ h z n + 1 / 2 i + 1 2 , k - 1 Δ z ∑ l a l h φ n + 1 i + 1 2 , k + l + 1 2 (10b) e φ n + 1 i , k = e φ n + 1 / 2 i , k + Δ t 2 ε 1 Δ z ∑ l a l h ρ n + 1 / 2 i , k + l + 1 2 - 1 Δ ρ ∑ l a l h z n + 1 i + l + 1 2 , k (10c) e z n + 1 i , k + 1 2 = e z n + 1 / 2 i , k + 1 2 + Δ t 2 ε ∑ l a l i + l + 1 / 2 h φ n + 1 / 2 i + l + 1 / 2 , k + 1 / 2 i Δ ρ ∓ m i Δ ρ h ρ n + 1 i , k + 1 2 (10d) h ρ n + 1 i , k + 1 2 = h ρ n + 1 / 2 i , k + 1 2 + Δ t 2 μ ∑ l a l e φ n + 1 / 2 i , k + l + 1 Δ z ± m i Δ ρ e z n + 1 i , k + 1 2 (10e) h φ n + 1 i + 1 2 , k + 1 2 = h φ n + 1 / 2 i + 1 2 , k + 1 2 + Δ t 2 μ ∑ l a l e z n + 1 / 2 i + l + 1 , k + 1 / 2 Δ ρ - ∑ l a l e ρ n + 1 i + 1 / 2 , k + l + 1 Δ z (10f) h z n + 1 i + 1 2 , k = h z n + 1 / 2 i + 1 2 , k + Δ t 2 i + 1 / 2 μ Δ ρ ∓ m e ρ n + 1 / 2 i + 1 2 , k - ∑ l a l i + l + 1 e φ n + 1 i + l + 1 , k .Equations (9a)–(9f) and (10a)–(10f) can be solved by the generalized Thomas method [5].Whenρ=0, namely, i=0, the values of ez are singular as BOR-MRTD scheme. So ez is calculated as follows:( n → n + 1 / 2 ) (11) e z n + 1 / 2 0 , k + 1 2 - Δ t 2 μ ε Δ ρ 2 ∑ l a l e z n + 1 / 2 l + 1 , k + 1 2 = e z n 0 , k + 1 2 + 2 Δ t ε Δ ρ h φ n 1 2 , k + 1 2 - Δ t 2 μ ε Δ ρ Δ z ∑ l a l e ρ n 1 2 , k + l + 1 .( n + 1 / 2 → n + 1 ) (12) e z n + 1 0 , k + 1 2 = e z n + 1 / 2 0 , k + 1 2 + 2 Δ t ε Δ ρ h φ n + 1 / 2 1 2 , k + 1 2 .


## 4. Convolution Perfect Matched Layer

Based on equations of ADI-BOR-MRTD scheme, we can present equations of CPML with consulting paper [11].( n → n + 1 / 2 ) 13 a e ρ n + 1 / 2 i + 1 2 , k = C A ρ i + 1 2 , k e ρ n i + 1 2 , k + C B ρ i + 1 2 , k ± m i + 1 / 2 Δ ρ h z n + 1 / 2 i + 1 2 , k - 1 κ k Δ z ∑ l a l h φ n i + 1 2 , k + l + 1 2 - ψ e ρ z n i + 1 2 , k 13 a i ψ e ρ z n + 1 / 2 i + 1 2 , k = Q k ψ e ρ z n i + 1 2 , k + P k Δ z ∑ l a l h φ n i + 1 2 , k + l + 1 2 13 b e ϕ n + 1 / 2 i , k = C A φ i , k e φ n i , k + C B φ i , k 1 κ k Δ z ∑ l a l h ρ n + 1 / 2 i , k + l + 1 2 - 1 κ i Δ ρ ∑ l a l h z n i + l + 1 2 , k + ψ e φ z n + 1 / 2 i , k - ψ e φ ρ n i , k 13 b i ψ e φ z n + 1 / 2 i , k = Q k ψ e φ z n i , k + P k Δ z ∑ l a l h ρ n + 1 / 2 i , k + l + 1 2 13 b i i ψ e φ ρ n + 1 / 2 i , k = Q i ψ e φ ρ n i , k + P i Δ ρ ∑ l a l h z n i + l + 1 2 , k 13 c e z n + 1 / 2 i , k + 1 2 = C A z i , k + 1 2 e z n i , k + 1 2 + C B z i , k + 1 2 ∑ l a l i + l + 1 / 2 h φ n + 1 / 2 i + l + 1 / 2 , k + 1 / 2 i κ i Δ ρ ∓ m i Δ ρ h ρ n i , k + 1 2 + ψ e z ρ n + 1 / 2 i , k + 1 2 13 c i ψ e z ρ n + 1 / 2 i , k + 1 2 = Q i ψ e z ρ n i , k + 1 2 + P i Δ ρ ∑ l a l h φ n + 1 / 2 i + l + 1 2 , k + 1 2 13 d h ρ n + 1 / 2 i , k + 1 2 = C P x i , k + 1 2 h ρ n i , k + 1 2 + C Q x i , k + 1 2 ∑ l a l e φ n + 1 / 2 i , k + l + 1 κ k + 1 / 2 Δ z ± m i Δ ρ e z n i , k + 1 2 + ψ h ρ z n + 1 / 2 i , k + 1 2 13 d i ψ h ρ z n + 1 / 2 i , k + 1 2 = Q k + 1 / 2 ψ h ρ z n i , k + 1 2 + P k + 1 / 2 Δ z ∑ l a l e φ n + 1 / 2 i , k + l + 1 13 e h φ n + 1 / 2 i + 1 2 , k + 1 2 = C P φ i + 1 2 , k + 1 2 h φ n i + 1 2 , k + 1 2 + C Q φ i + 1 2 , k + 1 2 ∑ l a l e z n + 1 / 2 i + l + 1 , k + 1 / 2 κ i + 1 / 2 Δ ρ - ∑ l a l e ρ n i + 1 / 2 , k + l + 1 κ i + 1 / 2 Δ z + ψ h φ ρ n + 1 / 2 i + 1 2 , k + 1 2 - ψ h φ z n i + 1 2 , k + 1 2 13 e i ψ h φ z n + 1 / 2 i + 1 2 , k + 1 2 = Q k + 1 / 2 ψ h φ z n i + 1 2 , k + 1 2 + P k + 1 / 2 Δ z ∑ l a l e ρ n i + 1 2 , k + l + 1 13 e i i ψ h φ ρ n + 1 / 2 i + 1 2 , k + 1 2 = Q i + 1 / 2 ψ h φ ρ n i + 1 2 , k + 1 2 + P i + 1 / 2 Δ ρ ∑ l a l e z n + 1 / 2 i + l + 1 , k + 1 2 13 f h z n + 1 / 2 i + 1 2 , k = C P z i + 1 2 , k h z n i + 1 2 , k + C Q z i + 1 2 , k ∓ m e ρ n + 1 / 2 i + 1 / 2 , k i + 1 / 2 Δ ρ - ∑ l a l i + l + 1 e φ n i + l + 1 , k κ i + 1 / 2 i + 1 / 2 Δ ρ - ψ h z ρ n i + 1 2 , k 13 f i ψ h z ρ n + 1 / 2 i + 1 2 , k = Q i + 1 / 2 ψ h z ρ n i + 1 2 , k + P i + 1 / 2 Δ ρ ∑ l a l E y n i + l + 1 , k .( n + 1 / 2 → n + 1 ) 14 a e ρ n + 1 i + 1 2 , k = C A x i + 1 2 , k e ρ n + 1 / 2 i + 1 2 , k + C B x i + 1 2 , k ± m i + 1 / 2 Δ ρ h z n + 1 / 2 i + 1 2 , k - 1 κ k Δ z ∑ l a l h φ n + 1 i + 1 2 , k + l + 1 2 - ψ e ρ z n + 1 i + 1 2 , k 14 a i ψ e ρ z n + 1 i + 1 2 , k = Q k ψ e ρ z n + 1 / 2 i + 1 2 , k + P k Δ z ∑ l a l h φ n + 1 i + 1 2 , k + l + 1 2 14 b e φ n + 1 i , k = C A φ i , k e φ n + 1 / 2 i , k + C B φ i , k 1 κ k Δ z ∑ l a l h ρ n + 1 / 2 i , k + l + 1 2 - 1 κ i Δ ρ ∑ l a l h z n + 1 i + l + 1 2 , k + ψ e φ z n + 1 / 2 i , k - ψ e φ ρ n + 1 i , k 14 b i ψ e φ z n + 1 i , k = Q k ψ e φ z n + 1 / 2 i , k + P k Δ z ∑ l a l h ρ n + 1 / 2 i , k + l + 1 2 14 b i i ψ e φ ρ n + 1 i , k = Q i ψ e φ ρ n + 1 / 2 i , k + P i Δ ρ ∑ l a l h z n + 1 i + l + 1 2 , k 14 c e z n + 1 i , k + 1 2 = C A z i , k + 1 2 e z n + 1 / 2 i , k + 1 2 + C B z i , k + 1 2 ∑ l a l i + l + 1 / 2 h φ n + 1 / 2 i + l + 1 / 2 , k + 1 / 2 i κ i Δ ρ ∓ m i Δ ρ h ρ n + 1 i , k + 1 2 + ψ e z ρ n + 1 / 2 i , k + 1 2 14 c i ψ e z ρ n + 1 i , k + 1 2 = Q i ψ e z ρ n + 1 / 2 i , k + 1 2 + P i Δ ρ ∑ l a l h φ n + 1 / 2 i + l + 1 2 , k + 1 2 14 d h ρ n + 1 i , k + 1 2 = C P ρ i , k + 1 2 h ρ n + 1 / 2 i , k + 1 2 + C Q ρ i , k + 1 2 ∑ l a l e φ n + 1 / 2 i , k + l + 1 κ k + 1 / 2 Δ z ± m i Δ ρ e z n + 1 i , k + 1 2 + ψ h ρ z n + 1 / 2 i , k + 1 2 14 d i ψ h ρ z n + 1 i , k + 1 2 = Q k + 1 / 2 ψ h ρ z n + 1 / 2 i , k + 1 2 + P k + 1 / 2 Δ z ∑ l a l e φ n + 1 / 2 i , k + l + 1 14 e h φ n + 1 i + 1 2 , k + 1 2 = C P φ i + 1 2 , k + 1 2 h φ n + 1 / 2 i + 1 2 , k + 1 2 + C Q φ i + 1 2 , k + 1 2 ∑ l a l e z n + 1 / 2 i + l + 1 , k + 1 / 2 Δ ρ - ∑ l a l e ρ n + 1 i + 1 / 2 , k + l + 1 Δ z + ψ h φ ρ n + 1 / 2 i + 1 2 , k + 1 2 - ψ h φ z n + 1 i + 1 2 , k + 1 2 14 e i ψ h φ z n + 1 i + 1 2 , k + 1 2 = Q k + 1 / 2 ψ h φ z n + 1 / 2 i + 1 2 , k + 1 2 + P k + 1 / 2 Δ z ∑ l a l e ρ n + 1 i + 1 2 , k + l + 1 14 e i i ψ h φ ρ n + 1 i + 1 2 , k + 1 2 = Q i + 1 / 2 ψ h φ ρ n + 1 / 2 i + 1 2 , k + 1 2 + P i + 1 / 2 Δ ρ ∑ l a l e z n + 1 / 2 i + l + 1 , k + 1 2 14 f h z n + 1 i + 1 2 , k = C P z i + 1 2 , k h z n + 1 / 2 i + 1 2 , k + C Q z i + 1 2 , k ∓ m e ρ n + 1 / 2 i + 1 / 2 , k i + 1 / 2 Δ ρ - ∑ l a l i + l + 1 e φ n + 1 i + l + 1 , k κ i + 1 / 2 i + 1 / 2 Δ ρ - ψ h z ρ n + 1 i + 1 2 , k 14 f i ψ h z ρ n + 1 i + 1 2 , k = Q i + 1 / 2 ψ h z ρ n + 1 / 2 i + 1 2 , k + P i + 1 / 2 Δ ρ ∑ l a l e φ n + 1 i + l + 1 , k ,where15aCAρi+12,k=4ε-σΔt4ε+σΔtCPρi,k+12=4μ-σmΔt4μ+σmΔtCBρi+12,k=2Δt4ε+σΔtCQρi,k+12=2Δt4μ+σmΔt15bCAφi,k=4ε-σΔt4ε+σΔtCPφi+12,k+12=4μ-σmΔt4μ+σmΔtCBφi,k=2Δt4ε+σΔtCQφi+12,k+12=2Δt4μ+σmΔt15cCAzi,k+12=4ε-σΔt4ε+σΔtCPzi+12,k=4μ-σmΔt4μ+σmΔtCBzi,k+12=2Δt4ε+σΔtCQzi+12,k=2Δt4μ+σmΔt15dPξ=σξσξκξ+κξ2αξe-σξ/κξ+αξΔt/2ε0-1,Qξ=e-σξ/κξ+αξΔt/2ε0,ξ=i,k.Equations13a–13fi and 14a–14fi can be also solved by the generalized Thomas method [5]. The value of ez is calculated the same as (11) and (12).In the matched layer, the coefficientsσi and κi are defined as follows [12, 13]:16σρ=σmaxρdmκρ=1+κmax-1ρdm,where ρ is the distance from the spot in the matched layer to the interface between computational domain and matched layer, d is the thickness of matched layer, and m is a polynomial coefficient. σmax is defined as follows:17σmax=kσoptσopt=m+1150πεrΔ,where k=σmax/σopt is positive and α is positive too.


## 5. Numerical Results

ADI-BOR-MRTD method has been tested by a metal ball and a metal cylinder with half-ball-hat. For comparison, they have been also calculated by FDTD and MRTD methods.CPU is Intel(R) Core(TM) i3 2.93 GHz; the memory bank is 1.93 GB; the Mac OS is Microsoft Windows XP Professional; the operating system is Fortran 90 Complier.


### 5.1. The Ball

The radius of metal ball is 1 meter. The results are shown in Figure2 and Table 2.(1)
FDTD:Δx×Δy×Δz=2cm×2cm×2cm, Δt=3.33×10-11s, and the cell lattice is 138×138×138 with eight-cell-thick matched layer.(2)
MRTD:Δx×Δy×Δz=10cm×10cm×10cm, Δt=11.11×10-11s, and the cell lattice is 56×56×56 with eight-cell-thick matched layer.(3)
ADI-BOR-MRTD:Δρ×Δz=10cm×10cm, Δt=22.22×10-11s, the cell lattice is 28 × 56 with eight-cell-thick matched layer, and the modulus range is m=0~16.Table 2
Comparison of the time and memory of CPU.
FDTD MRTD ADI-BOR-MRTD CPU time/s 1652 5 0.4 Memory/MB 172.3 13.4 2.6Figure 2
Single station RCS of metal ball.Figure2 shows that when the frequency is less than 500 MHz, the differences among three numerical results are less than 2 dB, which validate the feasibility of the ADI-BOR-MRTD method. Moreover, Table 2 demonstrates that the ADI-BOR-MRTD method has taken less time and memory of CPU than the other two methods.


### 5.2. The Cylinder with Half-Ball-Hat

The metal cylinder with half-ball-hat is designed as Figure3.Figure 3
Structure of cylinder with half-ball-hat.The results are shown as Figure4 and Table 3.(1)
FDTD:Δx×Δy×Δz=1cm×1cm×1cm, Δt=1.67×10-11s, and the cell lattice is 96×96×266 with eight-cell-thick matched layer.(2)
MRTD:Δx×Δy×Δz=5cm×5cm×5cm, Δt=5.56×10-11s, and the cell lattice is 48×48×82 with eight-cell-thick matched layer.(3)
ADI-BOR-MRTD:Δρ×Δz=5cm×5cm, Δt=11.11×10-11s, the cell lattice is 24×82 with eight-cell-thick matched layer, and the modulus range is m=0~16.Table 3
Comparison of the time and memory of CPU.
FDTD MRTD ADI-BOR-MRTD CPU time/s 1550 37 5 Memory/MB 161 13 0.23Figure 4
Single station RCS of metal cylinder with half-ball-hat.From Figure4 we can see that when the frequency is less than 1.5 GHz, the differences among three numerical results are less than 3 dB and the curves are similar. The results in Table 3 have also supported that ADI-BOR-MRTD method has taken less time and memory of CPU than the other two methods.


## 5.1. The Ball

The radius of metal ball is 1 meter. The results are shown in Figure2 and Table 2.(1)
FDTD:Δx×Δy×Δz=2cm×2cm×2cm, Δt=3.33×10-11s, and the cell lattice is 138×138×138 with eight-cell-thick matched layer.(2)
MRTD:Δx×Δy×Δz=10cm×10cm×10cm, Δt=11.11×10-11s, and the cell lattice is 56×56×56 with eight-cell-thick matched layer.(3)
ADI-BOR-MRTD:Δρ×Δz=10cm×10cm, Δt=22.22×10-11s, the cell lattice is 28 × 56 with eight-cell-thick matched layer, and the modulus range is m=0~16.Table 2
Comparison of the time and memory of CPU.
FDTD MRTD ADI-BOR-MRTD CPU time/s 1652 5 0.4 Memory/MB 172.3 13.4 2.6Figure 2
Single station RCS of metal ball.Figure2 shows that when the frequency is less than 500 MHz, the differences among three numerical results are less than 2 dB, which validate the feasibility of the ADI-BOR-MRTD method. Moreover, Table 2 demonstrates that the ADI-BOR-MRTD method has taken less time and memory of CPU than the other two methods.


## 5.2. The Cylinder with Half-Ball-Hat

The metal cylinder with half-ball-hat is designed as Figure3.Figure 3
Structure of cylinder with half-ball-hat.The results are shown as Figure4 and Table 3.(1)
FDTD:Δx×Δy×Δz=1cm×1cm×1cm, Δt=1.67×10-11s, and the cell lattice is 96×96×266 with eight-cell-thick matched layer.(2)
MRTD:Δx×Δy×Δz=5cm×5cm×5cm, Δt=5.56×10-11s, and the cell lattice is 48×48×82 with eight-cell-thick matched layer.(3)
ADI-BOR-MRTD:Δρ×Δz=5cm×5cm, Δt=11.11×10-11s, the cell lattice is 24×82 with eight-cell-thick matched layer, and the modulus range is m=0~16.Table 3
Comparison of the time and memory of CPU.
FDTD MRTD ADI-BOR-MRTD CPU time/s 1550 37 5 Memory/MB 161 13 0.23Figure 4
Single station RCS of metal cylinder with half-ball-hat.From Figure4 we can see that when the frequency is less than 1.5 GHz, the differences among three numerical results are less than 3 dB and the curves are similar. The results in Table 3 have also supported that ADI-BOR-MRTD method has taken less time and memory of CPU than the other two methods.


## 6. Conclusion

This paper has developed an ADI-BOR-MRTD algorithm. Furthermore, the CPML absorbing boundary condition is derived for ADI-BOR-MRTD algorithm. The simulated results suggest that the ADI-BOR-MRTD scheme can save more CPU time and memory than the FDTD and MRTD methods, which proves that the ADI-BOR-MRTD scheme is practicable, especially in the body of revolution case. In the next work, the method would be improved feasible for the frequency more than 500 MHz.



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*Source: 2073563-2017-02-13.xml*