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Hello, everyone. Welcome to ECE 5340/6340, Numerical Techniques and Electromagnetics. This is the first lecture.
I'm Professor Jamesina Simpson with the Department of Electrical Computer Engineering at the University of Utah.
In this first part we're going to be considering the finite difference time domain method.
And to begin with we're going to look at using FDTD to solve the scalar wave equation.
We'll be working in one dimension to keep things simple at first.
And what we learn here in the beginning applying FDTD, finite difference time domain
method to the scalar wave equation, we'll be able to apply later to Maxwell's equations.
So let's get rid of this. To begin with, we're going to be working in 1D.
We'll be working with solving the scalar wave equation in the X direction.
So this means that our changes in BY and Z direction will be zero.
We're going to be considering at first, to keep things simple, we'll have free space.
And we're also going to say we're not going to have any current sources.
And so we have our scalar wave equation, which you have undoubtedly seen in other courses, physics courses.
You can use this equation to describe the propagation of any kind of waves like sound waves,
water waves, in this case, for this course we'll be considering electromagnetic waves.
So for here, for scalar wave equation, we have our unknown U, partial second derivative with respect to space.
So X will be our space variable minus 1 over V squared, which is our speed.
And our second partial derivative of U with respect to time T, our time variable, and this is equal to zero.
So this is our 1-D scalar wave equation.
So we're going to be using -- we're going to be solving this equation to solve for
the propagation of our electromagnetic waves in one dimension, our unknown U.
We want to solve for. Solve for U. And so we want to solve this equation.
We want to see how U evolves over time and in space. So what are our possible solutions?
Well, we know that U -- U is a function of X and T in this case. Space and time.
And the solutions are of the form, they're two functions.
X -- I don't know why it keeps writing on the wrong side of the screen. All right.
X plus VT and another function G. These are arbitrary F and G that I'm using here.
But it's two different functions. X minus VT. Now, you can plug these in and see that they're solutions.
We can also dissect these a little bit and see that these are waves.
These arguments, function of space and time, and one of them you might recall moves to
the left and to the right, if we're going to say that X here, goes left to right.
This is a positive X direction minus X direction.
If we have, say, a Gaussian, then we could figure out easily, for example, if you can't recall which one of
these is the solution moving to the left or to the right, let's look at the peak of this Gaussian curve.
The peak here, this function is equal to let's say this is the unit of or value of one, unity amplitude
Gaussian and then this function is equal to one at a particular, at a combination of X and T.
So for a particular time let's say the time is evolving.
Time is increasing, as time increases for this argument to stay the same, denoting this
particular point on this curve, then in order for this to stay the same then X has to go down.
This argument to be a constant value, this X has to go down. So as time evolves, X gets smaller.
This is a solution of a wave that is moving in the minus X direction. It's moving left. Minus X.
And on this side, as T goes up, X also has to go up to get a constant argument of the function.
And then this one is a solution of the wave traveling to the right.
So what we'll be doing here is let's see what this looks like on our grid.
We have our unknown U, particular or is a function of X and T. I'm going to label this axis as X.
This is our space. Space axis. And we're going to be solving this scalar wave equation in our computer.
We're going to be creating algorithms, finite difference time domain algorithms to solve the scalar wave
equation and propagation of electromagnetic waves but in our computer we don't have continuous space.
We have to discretize our numbers or we have to discretize our space grid and also in time.
So what I'm going to do here is I'm going to divide this grid up into intervals, and this isn't drawn to scale.
Should be equally spaced if we're going to have an equally homogenous grid setup, grid arrangement, I should say.
And so let's say that the interval between these solutions is delta X. So every delta X will be solving for U.
Then as time progresses, I'll add another dimension here denoting time.
We can also solve for our unknown at different time steps, we can say.
And the interval between time steps is delta T time interval.
So we discretize in both in space and in time, and let's say, for example, if we had like a Gaussian,
and this is not drawn to scale again since we need more grid points to get a nice smooth curve,
that our Gaussian, let's take our solution where the wave is propagating to the right, then for
each of these time solutions that we give, then we would see this wave propagating to the right.
So this would be our solution of U at, say, time step T sub N. (oops) Times step N.
And this would be T sub N plus 1, the previous time step would be TN minus 1. This here, the same kind of thing.
We would have X sub i. I'm going to use subscript i to denote this space location number.
And this would be X sub i plus 1, X sub i minus 1.
I'm using the same notation as in the book that we'll be using in this class, which is the
Taflove Hagness Computational Electromagnetics Finite Difference Time Domain Method Book,
Third Edition, 2005 Edition, from Markethouse. So here X is defined as i times delta X. And T -- whoops.
This is X sub i. X sub i is i times delta X.
Assuming a constant interval between all the different space locations that were unknown.
And T sub N is defined as N delta T.
So each of these solutions here that I've drawn, these are showing snapshots of the fields at time T sub N.
So we can see certain snapshot in time.
We can see over time how this wave propagates in this case the Gaussian propagating to the right.
And our unknown going along with this notation we're using we can write this in shorthand.
U sub i and superscript N. This is U, X sub iT sub N. Trying to make this as readable as possible.
So I'm rewriting this several different times. These all say the same thing. And U i delta X and delta T.
This is four different ways of saying the exact same thing.
This is our unknown U at a particular grid cell space location and a particular time step.