Good Morning! So welcome to the new lecture series on Computational Electromagnetics and

its Applications So this particular topic is quite interesting for people working in

industries or in Academia mostly focusing on modeling and also modeling related to experiments

Computational Electromagnetics has been a topic that is gaining advantage and attention

since the last 50 years owing to various changes in computer architecture availability of fast

computers and also numerical solvers that are able to moral quite accurately quite complex

problems

So in this particular module series we will focus on finite difference methods

So we will see what is going to be our motivation for this particular lecture And we will follow

the order as it is given here Following the motivation we will look into the background

and background of particularly the finite differencing method And we will introduce

certain finite differencing schemesSo let us go directly to the motivation

The motivation for Numerical Methods is we cannot use analytical methods quite straight

forwardly for most of the practical problems because practical problems has always certain

sense of non linearity So analytical methods fail when the partial differential equations

are not linear

We see that linearizing create serious errors In other words inaccuracy in the solution

space So that is the first motivation for going into the non analytical or any kind

of numerical methods

The second thing is what happens when computational domain is complex what I mean by that is for

example here let us say we are talking about a spiral antenna which has various spirals

several arms of spirals metallic with certain properties let us say conductivity permittivity

and permeability And then backed by certain other materials which could be dielectric

medium or whatsoever And then has also metallic surfaces on the side So this is quite a complex

structure for us to analytically model So Analytical method also fails when the domain

becomes complex

And also when we have boundary conditions let us say we have a set of boundary conditions

defined on the gamma 1 which is equal to u of 0 let us say this is a very hard boundary

condition And then other part of the boundary of the domain let us say gamma 2 has certain

flux normal component of the flux define so this is called as the mixed boundary condition

So when we have two three different types of boundaries also in those conditions it

is very difficult to use if not impossible to use analytical methods Likewise for example

when we have boundary conditions which are also time dependent let us say this is in

the previous case we had straight forward as a constant

Whereas in the other case when we have instead of constant if we have a time variation also

on the boundaries So then also the analytical methods become quite difficult For example

here we have an aluminium plate into you know various units and then we see that the four

corners are having different boundary conditions which are dependent on time

The last but not the least motivation is when you are talking about inhomogeneous and also

anisotropic medium For example in this case when you are trying to model a medium which

is anisotropic in a particular direction and then isotropic elsewhere so what do you see

is it's very difficult to model such mediums accurately using analytical methods

So with this we see that what we have in the case of finite difference method or any numerical

method is it gives us quite a bit of flexibility and also face out the disadvantages or the

lack of flexibility of the analytical methods for modeling more accurately complex problems

It had been said I am going to go into one of the most basic methods which is called

as the Finite difference method Without specifying whether frequency domain or a time domain

we will just look at the special discrimination for the time being

So the method itself was introduced in 1920s by Thorn And he actually named the method

the method of squares And it was mainly used for nonlinear hydrodynamics equation Because

they found out that it's very difficult to use classical methods like we talked before

So he invented a new method which he called the method of squares to model nonlinear hydrodynamic

problem that is the background

But in Electromagnetics the scheme itself was broaden by Ken Yee So Ken Yee introduced

the method in Electrodynamics as you can see for Maxwell equation using two staggered partition

grids So what I mean by staggered partition grid will become more clear later on but the

pictorial representation here gives us a little bit understanding There are two grids one

is green grid the other one is the brown grid and they are staggered in space and infact

they are also staggered in time This is something we will see later on but this is the most

important point So what I want to do also give a little bit background on this methods

I mentioned that algorithm itself was introduced in 1966 by E but the method itself did not

gain attention for almost a decade Nobody really bothered about using the method for

almost a decade

So that being said what was the other methods or what was keeping numerical scientist or

computational scientist busy They were more busy with a well established method called

as method of moments Which we will see later on It was much more evolved it had much more

numerical and mathematical tools involved in it so people were basically using method

of moments They did not pay attention to find a different time domain or find a different

method in general for almost a decade There were other problems also related to finite

difference method The method itself works fine but somehow if you wanted to do any practical

problems you need to define the method along with the boundary conditions if not you are

going to re assimilate a very large problem although what you are interested is very small

area of computational domain

So if for example if I wanted to simulate scattering problem let us say I want to understand

what is a scattering of a particular object let us say I have a car and I am talking about

electromagnetic waves scattered by the car So what I would do is I would model the car

surrounded by certain atmosphere maybe I am using a standard air free space but since

I do not have a proper termination so I have to simulate the car surrounded by a very very

big volume and although I am only interested is what is going on around the car I need

to simulate a very big problem because I did not have very accurate and stable boundary

conditions So that was one of the biggest problem of this method for a long long time

that changed over a period So what happened was in 1975 it was both Taflove and Broadwyn

they brought certain improvement to the stability of this method they basically computed the

stability criteria and they also improved the methods functioning for a steady state

solution for a sinusoidal input and also in 1977 Holand kuns and Lee applied this method

for a broadband application They basically send in a pulse and simulated it for a broadband

But all these things are only still working on the method itself And later on someone

called as Murr came and did something called as absorbing boundary condition

Let me explain this in a slide Let us say this is a scatterer and then the scattering

is happening so there was a possibility to truncate the entire problem using a certain

absorbing boundary conditions So we will not focus too much on the absorbing boundary condition

right now we will talk about it later on but its important to know that the computational

domain let us say we call it as omega it is being truncated using certain conditions here

at the boundary and this is what we call it as ABC : Absorbing Boundary Conditions And

some called as Beroje a French Engineer working for the electricity corporation of France

He broadens an idea which revolutionized and even popularized this method further called

as perfectly matched layer

We will talk about this also during this course what is the meaning of perfectly matched layer

and how does it work But right now for the motivation its enough to know instead of putting

one single boundary what Beroje did was quite revolutionary We will see why it is revolutionary

later on So he put a truncation layer and this is the perfectly matched layer

So we will discuss all these terms while we go forward But I wanted to give you a little

bit on the historic front of the development of this method So it is quite clear there

were quite a lot of other things that were required for a method to be popularized or

widely applied This is not just the method itself but also the tools that are required

around the method It could be on the stability conditions it could be on the requirement

of certain truncation techniques or perfectly match layer so on and so forth

So with this we will start looking at basically what is the meaning of this finite difference

method We will look at it what are the structure of it in the next slides

So the finite difference method itself is an algebraic form Let us explain this a little

bit further let us say I am interested in finding out the value of certain differential

on a particular grid space

Let us say I have a grid space and I have to compute the value of my function on the

grid space So let me say I am interested in finding out what will be the functions first

derivative on each of these points At Point number 1 this is the derivative Point number

2 this will be the derivative and so on and so on and so forth So what is happening here

is if this is x and this is f of x what I am doing is I am computing the value of f

of x and each of these points and then I am differentiating

What I am interested is I am interested in the different by dx this is what I am interested

in So if these are the two axis and umm the X axis is the independent variable and the

Y axis is the dependent variable I am computing the df by dx in each point Algebraically this

is equivalent to say I am writing down this as a function of certain weights

This will become clear later on but what will essentially happen is if I have a function

df by dx let us say I am interested in first differential second differential so on and

so forth And this is basically given as a sum of certain weights which I call it here

C I F K times the function itself define in those points So I goes from 0 to n if I equal

to 0 we are talking about only one thing and so on and so forth

So in this case what is happening is there are certain weights I am multiplying on this

so you can basically write this as value which is containing f of x i and then there is certain

weight functions c of k will be the value of d to the power of k of f by dx to the power

of k So this essentially algebraic equation where the weight functions are multiplied

for those value so that is what we mean by saying the finite difference approximations

are algebraic in nature

Right now if you are not able to understand this it is totally fine we will explain this

step by step later on Right now what you need to know is the finite difference approximations

are basically algebraic They are in the form an algebraic equation

The second thing is as you can see here the value at a point depends on values at some

neighboring points What I mean that by this is I know the value of these points given

in black I am interested in knowing a value let us say at this point so the value at this

point can be found out the using the values at these neighboring points So we are talking

only in terms of special derivative here special aspect here but if you also take the time

axis into play so you can compute the value of certain points in space and time using

the values of certain neighboring points in space and time as well So the entire process

of finite difference goes through 3 simple steps So when I say three simple steps what

are those steps That what we are going to say Let us say we take simple one dimensional

example to illustrate the logic and we can expand it to multi dimensions

Let us say we have a domain which is a one dimensional domain given by this line what

we are essentially doing here is we need to divide this solution domain into grids of

nodes Say for example 3 nodal points Of course when you do numerical methods your domain

will be much larger and your number of nodes will be also much larger here we are illustrating

it only for the sake of simplicity with three points

And then the second step is you are trying to approximate the differentials what I mean

by differentials is the differential equations by certain difference equation For example

in this case this f prime at x equal to x not is nothing but first derivative of f with

respect to x at x not is equal to certain value that we are computing for f at x not

plus delta x minus f x not divided by the step size in x So the step size in x is the

value here In other words we can write this in a much more simple way as follows:

So let us say this is the grid and what we are having here is this is the size of delta

x So what we are doing next is we are trying to go forward with certain difference equation

in order to approximate the value of the differential at certain point Likewise as you can see we

have to have certain boundary conditions and initial conditions BCs are the boundary conditions

and ICs are initial conditions

So when we say boundary conditions we are interested in the x coordinate of the boundary

Since it is a one time problem what we are talking about is let us say x equal to 0 here

and x equal to 1 So we are skipping certain values of u at x equal to 0 here and x equal

to 1 as 0 for all time variables Similarly we are talking about certain initial conditions

if it is a static problem then we are not interested in the time variables but in the

case of the time varying problem we set the initial conditions as time equal to 0 certain

value So this value is a t equal to 0

As you can see the number of initial condition also changes depending on the order of the

problem itself that we will see in the next slides so what you need to know is when we

have a problem the solution process for finite difference method is you create a step 1 and

then you do the following thing you divide the domain into certain nodal points and then

in the step 2 you approximate the differentials using certain difference equations and then

in the step 3 you use certain given boundary conditions and initial conditions so that

you can compute the value of the problem space or the solution space serially in time

So let us look now at certain fundamental form of finite differencing So let us say

we have a problem where the solution is f of x and the value of the x coordinate is

given we have a step size of delta x this is x not a step before is x not minus delta

x and the step one forward is x not plus delta x

So let us say we are interested in knowing what is the value of certain differentials

based on this f of x So let us say we are interested in knowing the value of the first

differential of f of x at the point x not So we can do this in different ways

The first way to do that is we take the forward differencing What we mean by forward differencing

is we say at x not the value of f dash that is the first differential at x equal to x

not is going to only depend on the value that is one step in the forward direction from

x not and x not itself

So it does not really matter what the value is here And the value is given by this equation

where we are taking the value at d minus the value at p divided by the step size itself

So this is a very simple forward differencing scheme

Likewise we can do the same thing with backward differencing scheme So here we say that the

value at x not the first differential of f at x not with respect to x is going to depend

only on the value at x not minus delta x and x not itself It does not matter what the value

is here it is going to depend only on this So as you can see both these methods whether

you are doing a forward differencing or backward differencing there is a kind of a bias The

bias is it is saying what ever going to be the value of differential at one point is

going to be only dependent on what is in the forward direction or in the backward direction

So a safer bet would be is to take the value at both x not minus delta x and x not plus

delta x and that is exactly what we do in the scheme of central differencing as you

can see hereSo what we are saying is instead of going and doing the differencing either

only focusing on x not plus delta x or only focusing on x not minus delta x let us take

both of them That is what we are exactly doing here We are saying the value of the first

differential at x not is going to depend on both these values at x not minus delta x and

x not plus delta x and of course here would be twice the delta x delta x here plus delta

x here And this is the central differencing method

So let us summarize all of them in one slide so we have the forward differencing scheme

which is focusing only on forward value the backward differencing scheme which is focusing

mainly on the backward value and the central differencing which is taking both to delta

x

So with that being said we can stop here and we can summarize what we have seen So we have

seen now the motivation for going into the finite differencing method why we are approaching

finite difference scheme as compared to analytical method and we also gave you the historical

background on the method itself And we have introduced some very basic notions of differencing

Of course we have to develop them further if we wanted to model any meaningful problems

for applications in electromagnetics So with that being said we will come back again and

we will focus on the further techniques in finite differencing Thank you