Implementation and Accuracy Analysis of ETD Method in FDTD Simulation Environment

Bojana Nikolić, Bojan Dimitrijević, Slavoljub Aleksić Faculty of Electronic Engineering

University of Niš Niš, Serbia

Mićo Gaćanović Faculty of Electrical Engineering

University of Banja Luka Banja Luka, Bosnia and Hercegovina

Abstract—In this paper, the influence of two different time approximation techniques on the accuracy of the finite difference time domain (FDTD) method has been analysed. The following approximation techniques have been considered: time-average (TA) and exponential time differencing (ETD). Both algorithms have been implemented in an own developed FDTD simulation environment. In particular, an example of an on-body antenna on the human tissue is considered, as a possible case of interest.

Keywords—exponential time differencing; finite difference time domain method; time-average approximation

I. INTRODUCTION A rapid development of computer technology today has

made the finite difference time domain (FDTD) method become the primary available tool for the design of antenna and microwave circuit components, EMC/EMI analysis, and the prediction of radio propagation [1], [2]. FDTD is a full wave time domain differential equation based technique. It is a versatile method that was proposed by Yee [3] originally for two dimensional problems with metal boundaries. Initially the FDTD method was applied to scattering problems and subsequently has become one of the most popular methods used to simulate and analyse problems in electromagnetics, ranging from antennas, microwave wave circuits, electromagnetic compatibility (EMC) issues, bioelectro-magnetics, electromagnetic scattering to novel materials and nanophotonics [1], [4], [5]. Since it is a time-domain method, solutions can cover a wide frequency range with a single simulation run. Basic advantage of this method is its simplicity and guaranteed convergence with proper choice of parameters. The disadvantages of the FDTD are high memory and computational requirements. However, this drawback becomes less and less significant, since the computational power will continue to grow exponentially in the future.

Mathematical theorems for the FDTD formulation, concerning issues such as accuracy, convergence, dispersion, computational complexity and stability are available in [6].

The perfectly matched layer (PML) absorbing media [7] has proven to be the most robust and efficient technique for the termination of FDTD computational domain [8]. Its implementation, referred to as the convolutional PML (CPML), proofs to be superior over the other implementations of the PML, offering a number of advantages [9]. CPML is based on

the stretched coordinate form [10] and the use of complex frequency shift (CFS) of PML parameters [11] and its application is completely independent of the host medium.

In the available literature, effects of lossy media and conductors have been considered by introducing the conductivity into the original formulation of lossless FDTD update equations [12]. The temporal and spatial discretization is usually done employing second-order accurate two-point central difference method [13]. A time approximation of the components that aren’t available in required moments of time, can be performed using several different methods. One common scheme is the time-average (TA) [1], [13]. This approximation is based on the average in time between the field values at the next and the previous half time step relative to the current position in time, so one step apart. The method is second order accurate and conditionally stabile.

Other variants of time approximation include the time-forward (TF) [14], time-backward (TB) [15] and the exponential time differencing (ETD) [12], [16], [17].

In the time-forward approximation a field component at one time step is approximated using its value at the previous step backward in time. In [14] this FDTD scheme is applied to the propagation analysis through a highly conductive nonlinear magnetic material. By following the pattern similar to the one of TF method, in the time-backward approximation a field component at one time step is approximated using its value at the next step forward in time. In [15] authors apply the method to the solution of the electromagnetic fields within an arbitrary dielectric scatterer of the order of one wavelength in diameter. TF and TB methods are both first order accurate. For this reason they won’t be part of the accuracy analysis presented in this paper, although both algorithms have been implemented in an own developed FDTD simulation environment.

The original ETD algorithm has been applied to the simulation of electric conductive media and isotropic lossy dielectrics [16], [18]. Subsequently, the ETD schemes for the simulation of wave propagation in magnetized plasma are developed [19]. In [19], the original approximations of ETD by Taylor series schemes are presented. The ETD method has a second-order temporal accuracy.

In [16] authors determine the stability condition and analyse the accuracy of the exponential and average time-approximation schemes for FDTD in an isotropic,

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homogeneous lossy dielectric with electric and magnetic conductivities. Paper [17] presents a comparative 3-D dispersion analysis of FDTD schemes for doubly lossy media, where both electric and magnetic conductivities are nonzero. Among the FDTD schemes presented are TA, TF, TB and ETD.

In this paper, the influence of two different time approximation techniques on the accuracy of the FDTD method has been analysed. The following techniques have been considered: time-average (TA) and exponential time differencing (ETD). Both algorithms have been implemented in an own developed FDTD simulation environment. In particular, an example of an on-body antenna on the human tissue is considered, as a possible case of interest.

II. FDTD SCHEME As a general case, we consider a homogeneous isotropic

dielectric medium with electric and magnetic losses. The Maxwell’s curl equations in differential form are written as

HEtE

EHt

H

*

where ε, σ, μ and σ* are the permittivity, electric conductivity, permeability and magnetic conductivity, respectively. It should be mentioned that the magnetic conductivity σ* is, strictly speaking, not physically realistic, but is often used for artificial absorption at simulation space boundaries and it will be included in this analysis, as such.

When applying FDTD algorithm, the temporal and spatial discretization of (1)-(2) is done by second-order accurate two-point central difference method. The computational space is divided into cuboidal elementary cells called Yee cells. The elementary Yee cell together with the E and H field components displacement is shown in Fig. 1. A staggered spatial mesh is used for interleaved placement of the electric and magnetic fields: the temporal location of the E and H field components differs by a half time step. Every E component is surrounded by four circulating H components, and every H component is surrounded by four circulating E components. A time approximation, which is necessary for the components in moments of time that aren’t provided by the algorithm, is performed using several different methods, namely: time-average (TA), the time-forward (TF), time-backward (TB) and the exponential time differencing (ETD). The TA method, as most commonly used, was already part of an own so far developed FDTD simulation environment and its implementation, along with undertaken code optimizations, is described in detail in [20]. The additional three algorithms (TF, TB and ETD) were implemented in an equivalent manner, so that it doesn’t affect the simulation time, when simulations are run under the same conditions. However, TF and TB methods won’t be part of the accuracy analysis presented in this paper, since they are both only first order accurate.

Fig. 1. Electric and magnetic field vector components in the elementary cell.

The generalized update equation of FDTD schemes for H and E field components can be written as

21

1 n

En

En HbEaE

nH

nH

nEbHaH

21

21

where Ea , Eb , Ha and Hb are the electric and magnetic field update coefficients.

The exact update equations for H and E field components can be presented as (for the brevity only equations for Hx and Ex field components are presented)

y

EEb

z

EEb

HaH

nkjiz

nkjiz

Hx

nkjiy

nkjiy

Hx

nkjixHx

nkjix

21,,21,1,,

,21,1,21,,

2121,21,,

2121,21,

z

HHb

y

HHb

EaE

nkjiy

nkjiy

Ex

nkjiz

nkjiz

Ex

nkjixEx

nkjix

2121,,21

2121,,21

,

21,21,21

21,21,21

,

,,21,1

,,21

respectively. Similar update equations can be written for y and z components of E and H field. The H field components are

first to be executed. Update coefficients Eva , , Evb , , Hva , and

Hvb , ( zyxv ,, ) are given in Table 1, for both time approximation approaches.

TABLE I. UPDATE COEFFICIENTS FOR ETD AND TA TIME APPROXIMATION APPLIED IN FDTD

Update Coefficients

Time approximation methods ETD TA

Hva ,

v

v t*exp

vv

vvtt

2*12*1

Hvb , v

vv t**exp1

vv

vt

t

2*1

Eva ,

v

v texp

vv

vvtt

2121

Evb , v

vv t

exp1 vv

vt

t

21

It should be noted that by applying the second-order Pade approximation to the ETD update coefficients

32121 t

tte t

the TA update coefficients can be obtained. Thus, TA can be considered as an approximation of ETD scheme and both of them are with the second order accuracy. In computer implementation, these update coefficients are prepared by the algorithm at the beginning of the simulation run and stored in

indexing tables. This provides the same simulation time regardless of the applied time approximation method.

III. SIMULATION RESULTS AND DISCUSION Both TA and ETD algorithm have been implemented in an

own developed FDTD simulation environment. The simulations and the accuracy analysis are intentionally conducted on the example of the simplest and well known type of antenna, so that the influence of using different time approximation methods on the obtained results can be traced and properly interpreted. The analysed scenario is close to the case of on-body dipole antenna placed on the surface of a human tissue. The substrate that corresponds to the human tissue is characterized by the appropriate values of εr and σ parameters (εr=30 and σ=2S/m). Total length of the antenna is L=18mm.

In Fig. 2 a display of parameters 11s and 11z versus frequency is presented for the examined test case. Blue curve refers to the modulus of s11 parameter, while the red one refers to the modulus of z11 parameter. Frequency axis is scaled so that one division corresponds to 0.542535GHz. Parameters

11s and 11z are given in dB and normalized by 5dB and 15dB, respectively.

In Fig. 3 a deviation of |s11| parameter from its exact values versus frequency is presented for the case of ETD and TA method applied. The results are given in dB. As referent exact values the simulation results obtained for very small time step are used. It can be noticed that the use of ETD method for time

Fig. 2. Parameters |s11| and |z11| versus frequency as obtained in simulation interface.

approximation yields slightly better results. Having in mind other advantages of ETD method (especially regarding stability) and no additional complexity of its implementation, there are no doubts that this method should be included in simulation algorithm. Although the achieved improvement in this particular examined case is only to some extent, the implemented ETD method can be a significant tool in simulation and analysis of some other cases of interest.

5 6 7 8 9 10 11 12 13 14 15-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

|s

11| (

dB)

f (GHz)

TA ETD

Fig. 3. Deviation from the exact values (in dB) of |s11| in the case of ETD and TA method applied.

IV. CONCLUSION In this paper, the influence of two different time

approximation techniques on the accuracy of the FDTD method has been analysed on the example of an on-body antenna on the human tissue. In particular, time-average and exponential time differencing techniques have been implemented in an own developed FDTD simulation environment. These methods are chosen as the second-order accurate and the most frequently proposed in the literature. The performed implementation of both approximation techniques and later analysis confirmed the same complexity of their algorithms, equal simulation time and a slightly better accuracy of ETD technique. Having that in mind and some other advantages of ETD method (especially regarding stability) there is no doubt that this method should be included in simulation algorithm. This can be potentially a significant tool in simulation and analysis of some other cases of interest.

ACKNOWLEDGMENT This work is supported in part by the Ministry of

Education, Science and Technological Development of Serbia within the Project TR-32051 and TR-33008.

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