Dennis Sullivan, Ph. D. Professor of Electrical and Computer Engineering University of Idaho © Dennis Sullivan 2006 1 One-dimensional Simulation in Free Space




НазваниеDennis Sullivan, Ph. D. Professor of Electrical and Computer Engineering University of Idaho © Dennis Sullivan 2006 1 One-dimensional Simulation in Free Space
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A Brief Introduction to

The Finite-Difference Time-Domain

(FDTD) Method



Dennis Sullivan, Ph. D.

Professor of Electrical and Computer Engineering

University of Idaho


© Dennis Sullivan 2006


1.1 One-dimensional Simulation in Free Space

Electromagnetics is governed by the time-dependent Maxwell’s curl equations, which in free space are

(1.1 a)

. (1.1 b)

E and H are vectors in three dimensions, but if we consider only one dimension

(1.2 a)

. (1.2 b)

To put these equations in a computer, we approximate the derivatives with the “finite-difference” approximations:

(1.3 a)

. (1.3 b)

In these two equations, time is specified by the superscripts, i. e., “n” actually means a time , and “k” actually means the distance . (It might seem more sensible to use as the incremental step, since in this case we are going in the z direction. However, is so commonly used for a spatial increment that I will use .)

We rearrange the above equations to :

(1.4 a)

. (1.4 b)

Notice that the calculations are interleaved in both space and time. In Eq. (1.4 a), for example, the new value of is calculated from the previous value of and the most recent values of . This is the fundamental paradigm of the finite-difference time-domain (FDTD) method Fig. 1.1) [1].

Eq. (1.4 a) and (1.4 b) look very similar. However, and differ by several orders of magnitude:

,

.

Therefore, and will differ by several orders of magnitude. This is circumvented by making the following change of variables [2]:

. (1.5)

Substituting this into Eq. (1.4a) and (1.4b) gives

(1.6a)

(1.6b)

Now both and will have the same order of magnitude. We will call this “normalized” units. Physicist call this Gaussian units. Note that



and



This quantity is called the “impedance of free space.”

Once the cell size is chosen, then the time step is determined by

(1.7)

where is the speed of light in free space. Therefore,

(1.8)



Figure 1.1. A diagram of the calculation of E and H fields in FDTD.


Re-writing Eq. (1.6 a) and (1.6 b) in C computer code gives the following:

ex[k] = ex[k] + 0.5*( hy[k-1] - hy[k] ) (1.9 a)

hy[k] = hy[k] + 0.5*( ex[k] - ex[k+1] ) (1.9 b)

Note that the n or n+1/2 or n-1/2 in the superscripts is gone. Time is implicit in the FDTD method. In Eq. (1.9 a), the ex on the right side of the equal sign is the previous value at n - 1/2, and the ex on the left side is the new value, n+1/2, which is being calculated. Position, however, is explicit. The only difference is that k + 1/2 and k - 1/2 are rounded off to k and k-1 in order to specify a position in an array in the program. Figure 1.2 illustrates a simulation in free space. The following things are worth noting:


1. The and values are calculated by separate loops, and they employ the interleaving described above.

2. After the values are calculated, the source is calculated. This is done by simply specifying a value of at the point k = 1, and overriding what was previously calculated. This is referred to as a “hard source,” because a specific value is imposed on the FDTD grid.





Figure 1.2. A one-dimensional FDTD simulation in free space.

Simulation in a Lossless Dielectric Material.


Now consider the case where the medium is not free space but a medium that has a relative dielectric constant other than one. That mean Eq. (1.2 a) must be written

.

{In this class we will not be dealing with magnetic material, so the permeability is always . Therefore, Eq. (1.2 b) does not change.} If we got through the same finite-difference approximation and switch to normalized units, Eq (1.6 a) becomes

ex[k] = ex[k] + cb[k]*( hy[k-1] - hy[k] ),

cb[k]=0.5/epsilon.


1.2 Simulation in a Lossy medium

Once more we will start with the time-dependent Maxwell's curl equations, but we will write them in a more general form, which will allow us to simulate propagation in media which have conductivity:

(1.10 a)

. (1.10 b)

is the current density, which can also be written

,

where is the conductivity. Putting this into Eq. (1.10 a) and dividing through by the dielectric constant we get

.

We now revert to our simple one-dimensional equation:


,

and make the change of variable in Eq. (1.5) which gives

(1.11 a)

. (1.11 b)

Next take the finite difference approximations for both the temporal and spatial derivatives similar to Eq. (1.3 a):


(1.12)

Notice that the last term in Eq. (1.11 a) is approximated as the average across two time steps in Eq. (1.12). From the previous section

,

so Eq. (1.12) becomes



or

.

From these we can get the computer equations

ex[k] = ca[k]*ex[k] + cb[k] *( hy[k-1] - hy[k] ) (1.13 a)

hy[k] = hy[k] + 0.5*( ex[k] - ex[k+1] ), (1.13 b)

where

eaf = dt*sigma/(2*epsz*epsilon) (1.14 a)

ca[k] = (1. - eaf)/(1. + eaf) (1.14 b)

cb[k] =0.5/(epsilon*(1. + eaf)). (1.14 c)


Figure 1.3 shows a simulation of a pulse hitting a lossy medium that has a dielectric constant of 5 and a conductivity of 0.05. The pulse is generated at the far left side and propagates to the right. Notice that the waveform in the medium is absorbed as it propagates.









Figure 1.3. FDTD simulation of a pulse hitting a lossy material with ,


2.1 Reformulation Using the Flux Density

Up until now, we have been using the form of Maxwell’s equations given in Eq. (1.1), which uses only the E and the H fields. However, a more general form is

(2.1 a)

(2.1 b)

, (2.1 c)

where D is the electric flux density. Notice that Eq. (2.1 b) is written in the frequency domain. The reason for this will be explained later. We will begin by normalizing these equations, using

(2.2 a)

, (2.2 b)

which leads to

(2.3 a)

(2.3 b)

. (2.3 c)

We saw in section one that this form of Eq. (2.3 a) and (2.3 c) will lead to the very simple finite difference equations, Eq. (1.4 a) and (1.4 b). The only change is the use of D instead of E. However, we still have to get Eq. (2.3 b) into a time domain difference equation for implementation into FDTD. The first task is to get it from the frequency domain to the time domain. We will assume we are dealing with a lossy dielectric medium, of the form

(2.4)

The dielectric constant and conductivity of most media vary at different frequencies. The pulses we have been using as a source in chapters one and two contain a spectrum of frequencies. In order to simulate frequency dependent material, we will need a way to account for this. One of the most significant developments in the FDTD method was a means to simulate frequency dependent materials [2].

We will start with a very simple example to illustrate the ideas. Suppose we have a medium whose dielectric constant and conductivity vary over the frequency range of 10 to 400 MHz as shown in Table 2.1.


Frequency (MHz)

40 97 0.69

100 72 0.89

200 56 1.28

300 54 1.37

433 43 1.43


Table 2.1. Muscle tissue properties at various frequncies


A material like this can be adequately represented by the following formulation:

. (2.5)

This is referred to as the DeBye formulation. In this formulation, there is a dielectric constant and a conductivity , but there is also a frequency dependent term. In order to simulate this medium in FDTD, Eq. (2.5) must be put into the sampled time domain. Let us define the last term times the E field as

(2.6)

The inverse Fourier Transform of the DeBye term is where u(t) is the Heavyside, or rectangular function, which is 0 for t<0 and 1 thereafter. Equation 2.6 in the frequency domain becomes the convolution



in the time domain. We now have to approximate this as a summation in the sampled time domain:

(2.7)

Notice that



Substituting this value into Eq. (2.7) above gives

. (2.8)

The conductivity is calculated in a similar way. Instead of a convolution, it is a simple integration



Now we calculate the flux density from

(2.9)

and solving for

(2.10 a)

(2.10 b)

(2.10 c)

Once again, note that everything concerning the medium is contained in Eq. (2.10 a) through Eq. (2.10 c); the calculation of the flux density and magnetic field are unchanged.

Other methods have been developed to simulate frequency-dependent tissues [3, 4, 5].


2.2 Frequency Domain Calculation

In biological applications, the quantity of interest is the specific absorption rate, or SAR

. (2.11)

To determine this, we need the magnitude of the E field when the simulation has reached steady state. In this section we describe one method to determine this which also allows us to get results for several frequencies with one simulation.

We start by using the Gaussian pulse as the source. If it is narrow enough, is a good approximation to an impulse. We then iterate the FDTD program until the pulse has died out, and take the Fourier transform of the E fields in the slab. If we have the Fourier transform of the E field at a point, then we know the amplitude and phase of the E field which would result from illumination by any sinusoidal source. This, too, has a very serious drawback: the E field for all the time domain data at every point of interest would have to be stored until the FDTD program is through iterating so the Fourier transform of the data could be taken, presumably using a fast Fourier transform algorithm. This presents a logistical nightmare.



Here is an alternative [6, 7]. Suppose we want to calculate the Fourier transform of the E field E(t) at a frequency . This can be done by the equation



Notice that the lower limit of the integral is 0 because the FDTD program assumes all causal functions. The upper limit is , the time at which the FDTD iteration is halted. Re-writing the above in a finite difference form

,

(2.12)

where T is the number of iterations and is the time step, so . Equation 2.12 may be divided into its real and imaginary parts

, (2.13)



which may be implemented in computer code by


real_pt[m,k] = real_pt[m,k] + ex[k]*cos(2*pi*freq(m)*dt*n) (2.14 a)

imag_pt[m,k] = imag_pt[m,k] + ex[k]*sin(2*pi*freq(m)*dt*n) (2.14 b)


For every point k, in the region of interest, we require only two buffers for every frequency of interest. At any point k, from the real part of , real_pt[m,k], and the imaginary part imag_pt[m,k], we can determine the amplitude and phase at the frequency:




amp[m,k] = sqrt(pow(real_pt[m,k],2.)+pow(imag_pt[m,k],2.))

phase[m,k] = atan2(imag_pt[m,k],real_pt[m,k]).


Once the magnitude of at a certain E field is calculated, the corresponding SAR can be calculated. Figure 2.1 illustrates an example of the calculation of SAR at two different frequencies.









Figure 2.1. Calculation of SAR in a tissue for two different frequencies.


3.1 Three-dimensional Simulation

The original FDTD paradigm was described by the "Yee Cell," (Fig. 3.1), named, of course, after Kane Yee [1]. Note that the E and H fields are assumed interleaved around a cell whose origin is at the location I, J, K. Every E field is located 1/2 cell width from the origin in the direction of its orientation; every H field is offset 1/2 cell in each direction except that of its orientation.



Figure 3.1. The Yee cell.


Not surprisingly, we will start with Maxwell's equations

(3.1 a)

(3.1 b)

. (3.1 c)

Once again, we will drop the ~ notation, but it will always be assumed that we are referring to the normalized values.

Eqs. (3.1.a) and (3.1.c) produce six scalar equations, two of which are:

(3.2 a)

. (3.2 a)

The first step is to take the finite difference approximations.

(3.3 a)


. (3.3 b)

The relationship between E and D, corresponding to Eq. (3.1 b) is exactly the same as the one-dimensional or two-dimensional cases, except now there will be three equations. Different materials can be specified at each cell within the FDTD program (Fig. 3.2).




Figure 3.2. Different properties can be specified for each cell in an FDTD program.


Many simulations model the applicator as well as the body being radiated. A simple dipole antenna is illustrated in Fig. 3.3. It consists of two metal arms. A dipole antenna functions by having current run through the arms, which results in radiation. FDTD simulates a dipole in the following way: The metal of the arms is specified by setting the appropriate parameters to zero in the cells corresponding to metal. This insures that the corresponding field at this point remains zero, as it would if that point were inside metal. The source is specified by setting the field in the gap to a certain value. (In a real dipole antenna, the field in the gap would be the result of the current running through the metal arms.) Notice that we could have specified a current in the following manner: Ampere’s circuital law says

,

ie., the current through a surface is equal to the line integral of the H field around the surface.



Figure 3.3. A dipole antenna is simulated by specifying the cells of the antenna arms with values that insure the E field will remain at zero. The input stimulation is accomplished by specifying an E field at the gap of the dipole. The current in the dipole arms is simulated by the surrounding H fields. The E fields will radiate outward.


3.2 The Perfectly Matched Layer (PML)

Without the proper truncation of the problem space by appropriate boundary conditions, unwanted reflections would return to cause errors in the simulation (Fig. 3.4). These outgoing waves must be eliminated by an absorbing boundary condition (ABC).



Figure 3.4. Without an absorbing boundary condition, outgoing waves would be reflected back into the problem space (left). The perfectly matched layer (PML) is one of the best means of truncating outgoing waves (right).


One of the most flexible and efficient ABCs is the perfectly matched layer (PML) developed by Berenger [8]. The basic idea is this: if a wave is propagating in medium A and it impinges upon medium B, the amount of reflection is dictated by the intrinsic impedances of the two media (Fig. 3.5)

, (3.4)

which are determined by the dielectric constants and permeabilities of the two media

. (3.5)



Figure 3.5 The reflection at the interface of two media is dependent on their respective impedances.


Normally, in free space, and , but in our normalized units, in free space. When we added the flux density formulation, we switched to







We have added “fictitious” dielectric constant and permeability that we will use to implement the PML.

Sacks, et al. [9], shows that there are two conditions to form a PML:


1. The impedance going from the background medium to the PML must be constant,

=1. (3.6)

The impedance is one because of our normalized units.

2. In the direction perpendicular to the boundary (the x direction, for instance), the relative dielectric constant and relative permeablitiy must be the inverse of those in the other directions, i.e.,

(3.7 a)

(3.7 b)

We will assume that each of these is a complex quantity of the form

for m = x or y (3.8 a)

for m = x or y (3.8 b)

The following selection of parameters satisfies Eqs. (3.7 a) and (3.7 b) [3]:

(3.9 a)

. (3.9 b)

Substituting Eq. (3.9) into (3.3), the value in Eq. (3.9) becomes

.

This fulfills the first requirement above. If increases gradually as it goes into the PML, causing and to be attenuated. The details of the implementation are mathematically intense [10], and will not be presented here.

Figure 3.6 illustrates the use of the PML. The PML is eight cells thick around the outer layer. The source is a pulse offset from the center of the space. As the pulse radiates outward, when it goes into the PML it is absorbed. Note that the out-going wave remains circular and is not distorted by unwanted reflections.





Figure 3.6. Illustration of the effectiveness of the PML.


The source pulse is generated from an off-center position. The nearest part of the wave hits the PML and is absorbed without distorting the circular shape of the rest of the wave. The PML was 8 cells wide.

References

1. K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. on Antennas and Propagation, vol. AP-17, pp. 585-589, 1996.

2. R. Luebbers, F. Hunsberger, K. Kunz, R. Standler, and M. Schneider, “A frequency-dependent finite-difference time-domain formulation for dispersive materials,” IEEE Trans. Electromag. Compat., vol. EMC-32, pp. 222-227, Aug. 1990.

3. R. M. Joseph, S. C. Hagness, and A. Taflove, “Direct time integration of Maxwell’s equations in linear dispersive media with absorption for scattering and popagationof femtosecond electromagnetic pulses,” Optics Letters, vol. 16, pp. 1412-1414, Sept. 1991.

4. O. P. Gandhi, B. Q. Gao, and Y. Y. Chen, “A frequency-dependent finite-difference time-domain formulation for general dispersive media,” IEEE Trans. Microwave Theory Tech., vol. 41, pp. 658-665, April 1993.

5. D. M. Sullivan, “Frequency-dependent FDTD methods using Z transforms,” IEEE Trans. Antenna Prop., vol. AP-40, pp. 1223-1230, Oct. 1992.

6. C. M. Furse, S. P. Mathur, and O. P. Gandhi, “Improvements ot the finite-difference time-domain method for calculating the radar cross section of a perfectly conducting target,” IEEE Trans. Microwave Theory and Tech., vol. MTT-38, pp. 919-927, July 1990.

7. D. M. Sullivan., “Mathematical methods for treatment planning in deep regional hyperthermia,” IEEE Trans. Microwave Theory and Tech., vol. MTT-39, pp. 864-872, May 1991.

8. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves”, J. Comput. Phys., vol. 114, pp. 185-200, 1994.

9. Z. S. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Anten. and Prop., vol. 43, pp. 1460-1463, Dec. 1995.

10. D. M. Sullivan, “An unsplit step 3-D PML for use with the FDTD method,” IEEE Microwave and Guided Wave Letters, vol. 7, pp. 184-186, July 1997.


Books on FDTD


1. K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics, Boca Raton, FL, CRC Press, 1993.

2. A. Taflove, Computation Electrodynamics: The Finite-Difference Time-Domain Method. Boston, MA: Artech House, 1995.

3. A. Taflove, Advances in Computation Electrodynamics: The Finite-Difference Time-Domain Method. Boston, MA: Artech House, 1998.

4. A. Taflove, S. C. Hagness, Computation Electrodynamics: The Finite-Difference Time-Domain Method, 2nd Edition. Boston, MA: Artech House, 2000.

5. D. M. Sullivan, Electromagnetic Simulation Using the FDTD Method. N.Y.: IEEE Press, 2000.





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