Engineering Question

ECE 452 – Assignment 2
Due at 4 pm, Monday, October 17, 2022
Please submit your assignment as a PDF file on eClass.
1. The 3D diffusion equation ut u
2   is discretized to second order accuracy O(x
, y
, z
) using the Finite Difference method as
೙ ିଶ௨೔,ೕ,ೖ
೙ ା௨೔షభ,ೕ,ೖ

మ +
೙ ିଶ௨೔,ೕ,ೖ
೙ ା௨೔,ೕషభ,ೖ

∆௬మ +
೙ ିଶ௨೔,ೕ,ೖ
೙ ା௨೔,ೕ,ೖషభ


where 𝑢௜,௝,௞
௡ = 𝑢(𝑖∆𝑥,𝑗∆𝑦, 𝑘∆𝑧, 𝑛∆𝑡). Determine the stability condition of the above discrete
2. The propagation of electromagnetic waves in 2D is governed by the wave equation

where n(x, y) is the refractive index of the medium and c = 3 × 108
m/s is the speed of light in
vacuum. We would like to solve the above equation using the Finite Difference Time Domain
(FDTD) method. The computation domain is restricted to a rectangular region of size a × b
and first-order Radiating Boundary Conditions based on the one-way wave equation are
applied to all four boundaries.
(a) Give a FD discretization of the wave equation for an interior node (i, j).
(b) Derive the FD equations for nodes on the left, right, top and bottom boundaries.
(c) Implement the FDTD method in a MATLAB program to solve the above wave equation.
An outline of the program is given at the end of the assignment for your reference.
(d) Use your program to run the following simulations:
Simulation 1:
This is a test run to verify that your program works correctly. Define the computational
domain to be the region 0 ≤ x ≤ 10m, 5m ≤ y ≤ 5m, and set the index of the medium
to be free space everywhere, n(x, y) = 1. Apply a point source located at (2.5m, 0). The
source emits a Gaussian pulse modulating a carrier signal given by:
sin( )
/ 2
( , , ) exp
t T
u x y t s s 

 
  ,
where  = 2c/ is the frequency,  = 1.0m is the wavelength, w = 8.0fs is the pulse
width, and T0 = 4.0fs is the time offset of the pulse centre. Use grid sizes x = y = 0.05m
and set the time step to the maximum allowable by the CFL stability condition. Initialize
the fields at the first two time steps to 0 (i.e., set 0
0 ui, j  ui j  ). Run the simulation for
175 time steps and provide the following results:
(i) A 3D plot or a contour plot of the field distribution u(x, y) at time step n = 175. (Use
mesh or surf commands for 3D plot and contour for contour plot)
(ii) A plot of the field u versus x along the line y = 0 at time step n = 175.
Print out and hand in a copy of your MATLAB program.
Simulation 2:
In this simulation we would like to study the diffraction of a plane wave by a single slit.
We consider a plate with a slit of width a = 5m illuminated by a plane wave with
wavelength  = 1.0m, as shown in the figure below. The plate is 0.5m thick and is
assumed to be made of a material of very high refractive index (n = 10) so that it is strongly
reflective. In the simulation, set the computational domain to be  ≤ x ≤ 25m, 15m ≤
y ≤ 15m, with grid sizes x = y = 0.05m. Use a line source located at xs = 1.0m to
generate a plane wave with unit amplitude and wavelength  = 1.0m. The field at a node
on the line source is given by
u(x , y,t) sin( t)
s  
where 14.5m ≤ y ≤ 14.5m (so that the line source does not touch the top and bottom
boundaries). Set the time step to the maximum allowable by the CFL stability condition,
run the simulation and provide the following results:
(i) A 3D plot or a contour plot of the field distribution u(x, y) at t = 90fs.
(ii) Suppose a “screen” is placed at location xo = 20m to image the diffraction pattern of
the field intensity. The field distribution on the screen can be expressed as
u(x , y,t) E(x , y) cos( t) o  o  ,
where E(xo, y) is the envelope of the field. Give a plot of the intensity diffraction
pattern, I(y) = E
(xo, y). (The field envelope can be obtained by capturing the maximum
field at each point on the screen over a few periods of oscillation of the wave).
(iii)From Fraunhofer’s theory of diffraction, the intensity distribution of light transmitted
through a single slit is given by
sin( )
( ) 

 o
I y I ,
where   (a /)sin . The angle  is given by tan  y / L, where L = 10m is the
distance from the plate to the screen. Choosing an appropriate value for Io, plot the
above expression on the same plot obtained in (ii) and compare the numerical
simulation to the theoretical intensity distribution.
1 10
a = 5m
y (m)
x (m)
source plate screen
(n = 10)
n = 1 n = 1

Outline of the 2D Finite Difference Time Domain Program
% Create and initialize Ny x Nx matrices to store fields at time levels
% n – 1, n, and n + 1:
En_1 = zeros(Ny,Nx); % time level n – 1
En0 = zeros(Ny,Nx); % time level n
En1 = zeros(Ny,Nx); % time level n + 1
% March solution from time step n = 0 to n = nstop:
for n = 1 : nstop,
% Compute interior nodes:
for jj = 2 : Ny-1,
for ii = 2 : Nx-1,
compute En1(jj,ii)
Compute value of E at source point (xs, ys) at time step n + 1
Compute boundary nodes at time step n + 1
% Store fields at the two most recent time levels:
Un_1 = Un0;
Un0 = Un1;

Get 20% Discount on This Paper
Pages (550 words)
Approximate price: -

Try it now!

Get 20% Discount on This Paper

We'll send you the first draft for approval by at
Total price:

How it works?

Follow these simple steps to get your paper done

Place your order

Fill in the order form and provide all details of your assignment.

Proceed with the payment

Choose the payment system that suits you most.

Receive the final file

Once your paper is ready, we will email it to you.

Our Services

Paper Helper has assembled a team of highly skilled writers with diverse experience in the online writing circles. Our aim is to become a one stop shop for all your Academic/ online writing. Check out below our amazing service!


Essay Writing Services

At Paper Helper, we prioritize on all aspects that creates a good grade such as impeccable grammar, proper structure, zero-plagiarism, and conformance to guidelines. The principal purpose of essay writing is to present the author's evaluation concerning a singular subject about which they have made. Since Professionalism is the mother of every success, try our team of experienced writers in helping you complete your essays and other assignments.


Admission Papers

You have been trying to join that prestigious institution you long yearned for, but the hurdle of an admission essay has become a stumbling block. We have your back, with our proven team that has gained invaluable experience over time, your chance of joining that institution is now! Just let us work on that essay.How do you write an admission essay? How do you begin the essay? For answers, try Quality Custom Writers Now!


Editing and Proofreading

Regardless of whether you're pleased with your composing abilities, it's never an impractical notion to have a second eye go through your work. The best editing services leaves no mistake untouched. We recognize the stuff needed to polish up a writing; as a component of our editing and proofreading, we'll change and refine your write up to guarantee it's amazing, and blunder free. Our group of expert editors will examine your work, giving an impeccable touch of English while ensuring your punctuation and sentence structures are top-notch.


Technical papers

We pride ourselves in having a team of clinical writers. The stringent and rigorous vetting process ensures that only the best persons for job. We hire qualified PhD and MA writers only. We equally offer our team of writers bonuses and incentives to motivate their working spirit in terms of delivering original, unique, and informative content. They are our resources drawn from diverse fields. Therefore your technical paper is in the right hands. Every paper is assessed and only the writers with the technical know-how in that field get to work on it.


College Essay Writing

If all along you have been looking for a trustworthy college essay service provider that provides superb academic papers at reasonable prices, then be glad that you search has ended with us. We are your best choice! Get high-quality college essay writing from our magnificent team of knowledgeable and dedicated writers right now!


Quality Assignment/Homework Help

We give the students premium quality assignments, without alarming them with plagiarism and referencing issues. We ensure that the assignments stick to the rules given by the tutors. We are specific about the deadlines you give us. We assure you that you will get your papers well in advance, knowing that you will review and return it if there are any changes, which should be incorporated.

togel macau togel terbesar bocoran admin jarwo data macau