Download A Quasi-Static FDTD Approximation Reduces Computation Time

A Quasi-Static FDTD Approximation Reduces Computation Time
G. McKinnon1
1
GE Medical Systems, Waukesha, WI, United States
Synopsis
When using the finite difference time domain (FDTD) algorithm to calculate gradient coil induced electric fields in tissue,
convergence is very slow. This is due to the relative permittivity values of tissue being in the order of 105 at these low frequencies.
Method of moment calculations, on the other hand, typically invoke a quasi-static approximation whereby, among other things, the
displacement current is ignored.
Here it is shown that a similar quasi-static approximation can be introduced into the FDTD approach, by setting the relative
permittivity to unity, resulting in a significant improvement in the convergence speed.
Introduction
Much of the work in modeling electric fields induced in tissue via changing magnetic fields [1,2] has used the method of moments
technique. Here a quasi-static approximation to Maxwell’s equations is used, whereby it is assumed that the conduction currents in the
tissue are much greater than the displacement currents. In effect this ignores the permittivity. The same idea has been introduced into
an analytical solution for the electric field in the case of a conducting circular cylinder placed within a cylindrical gradient coil [3].
Recently there has been interest in using the FDTD algorithm to study these gradient induced electric field [4].
The FDTD technique solves (or simulates) Maxwell’s equations in the time domain, which implicitly includes the effect of tissue
permittivity. Basically electromagnetic disturbances propagate within the FDTD space at the speed of light, interact with the scattering
objects, and eventually reach a steady state (which may be periodic in the case of a periodic electromagnetic source).
However tissue, at the low frequencies associated with gradient coils, has very high relative permittivity (Er) values, in the order of
105 [5]. Hence the propagation speed of electromagnetic disturbances in tissue is about 300 times slower than the free space speed of
light. It is postulated here that this causes slow convergence of the FDTD simulations.
If this is the case, then it is worth investigating whether reducing the relative permittivity to unity can accelerate convergence,
provided of course that the errors introduced by the quasi-static approximation are negligible for the problem at hand.
Methods
A FDTD whole body gradient coil simulation was developed, with the gradient coil being driven by a constantly ramping current.
This should produce a constant electric field once a steady state is reached. Within the FDTD space was placed a 40cm diameter
circular cylinder, 160cm long, with electrical properties of muscle at 3 kHz (Er = 105, σ = 0.33 siemens/m [5]).
To speed the calculations the FDTD space consisted of only 60,000 cells, with a rather coarse 32mm resolution. The simulation was
run for 400,000 time steps (corresponding only to about 20us of gradient ramping time), and at every 40th time step the twodimensional electric field map over a coronal plane (at y=0) was recorded. This was repeated with the relative permittivity set to unity.
The electric field at the last time step, of the high permittivity case, was assumed to be the reference (correct) field, and all other
field maps were compared with this. An error measurement was defined as the L2 norm of the difference field, normalized by the L2
norm of the reference field. This error measurement, then, was used to track the convergence of the FDTD simulation
Results
Figure 1 is a graph showing the residual error verse FDTD time step, for the two different relative permittivity values. This
demonstrates how the iterations converge much faster with a relative permittivity of unity, compared with that when the relative
permittivity is 105. The graphs also demonstrate that the solutions converge to
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the same value irrespective of the permittivity.
Figure 1
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Proc. Intl. Soc. Mag. Reson. Med. 11 (2003)
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Er=1
Er=100000
20.00
15.00
10.00
5.00
iterations (*1000)
400
370
340
310
280
250
220
190
160
130
70
100
40
0.00
10
References
[1] Wang & Eisenberg, IEEE Trans on Magn. 30:5015-5023 (1994)
[2] Brand & Heid, Magn Reson Med. 48:731-734 (2002)
[3] Bowtell & Bowley, Magn Reson Med. 44:782-790 (2000)
[4] Liu, Crozier, Zhao & Lawrence, Concepts in Magn Reson. 15:26-36 (2002)
[5] Gabriel, http://www.brooks.af.mil/AFRL/HED/hedr/reports/dielectric/
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rms error (%)
Discussion
Introducing this quasi-static approximation into the FDTD algorithm can
reduce the computation by an order of magnitude without drastically affecting
the accuracy in certain low frequency electromagnetic simulations.
However these results were obtained from a simple uniform conducting
object. Future work is needed to investigate the validity of this approximation
with more human-like electromagnetic models.