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

Overmuchmemory and time of CPU have been taken bymultiresolution 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-BORMRTD) 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-MRTDmethod. 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 MRTDmethods.


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][6][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.

Equations of BOR-MRTD
The electric and magnetic fields are expanded by Fourier series as where e  , e V , h  , h V are Fourier coefficients and e = E  , h = H  . is azimuth angle;  is modulus. is related to cos ; V is related to sin .Substituting (2a) and (2b) to (1a)-(1f), (1a)-(1f) are rewritten as The electric and magnetic fields are expanded by Daubechies' scaling function in space domain and by Haar's scaling function in time domain.
The distribution of field components is shown in Figure 1.
The equations of BOR-MRTD method are presented as follows: The coefficient () is equal to For Daubechies' scaling function with two vanishing moments ( 2 ), the coefficients are shown in Table 1; for  > 2, () are zeros due to the compact support of Daubechies' scaling function; for  < 0, () are given by the symmetry relation (−1 − ) = −().

Convolution Perfect Matched Layer
Based on equations of ADI-BOR-MRTD scheme, we can present equations of CPML with consulting paper [11].
In the matched layer, the coefficients   and   are defined as follows [12,13]: where  is the distance from the spot in the matched layer to the interface between computational domain and matched layer,  is the thickness of matched layer, and  is a polynomial coefficient. max is defined as follows: where  =  max / opt is positive and  is positive too.

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.

The Ball.
The radius of metal ball is 1 meter.The results are shown in Figure 2 and Table 2.
Figure 2 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.

The Cylinder with
Half-Ball-Hat.The metal cylinder with half-ball-hat is designed as Figure 3.
The results are shown as Figure 4 and Table 3.
From Figure 4 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.

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.

Figure 2 :
Figure 2: Single station RCS of metal ball.

Table 2 :
Comparison of the time and memory of CPU.

Table 3 :
Comparison of the time and memory of CPU.