代写ME2623 Engineering Mathematics and Programming代做R编程

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Module Code: ME2623

Module Title: Engineering Mathematics and Programming

Assessment Title: Solving the 2D Wave Equation Using the Finite Difference Method

Weighting: 40%

Main objectives of the assessment:

To develop and document a MATLAB program that solves the 2D wave equation using the finite difference method, demonstrating an understanding of numerical methods, code implementation, and validation techniques.

Brief Description of the assessment:

The student is required to develop a MATLAB program to solve the 2D wave equation using the finite difference method. Additionally, the student must provide a user guide explaining the functionality of the code, validate its results, and propose potential enhancements for future applications.

Learning outcomes for the assessment:

1.    Understanding of analytical methods that are

useful for Mechanical and Aerospace Engineering

2.    Understanding how to employ programming to solve basic engineering computational problems.

3.    Applying best-practice programming techniques to solve Mathematical models of Engineering

problems.

4.    Understanding the usefulness of programming techniques in the process of solving Engineering problems.

5.    Presenting computational results in a clear and concise manner including validation and

verification.

Assessment and marking criteria

This assignment will be assessed based on several criteria, each contributing to the final grade as

indicated by the percentage in brackets:

● Explaining the interface (20%)

● Providing a quickstart example (20%)

● Code annotation (20%)

● Verification and validation (20%)

● Suggesting future extensions (10%)

● Including an appendix with the code (10%)

Your submission will be assessed based on:

•     Accuracy   and    stability of the    numerical implementation.

•     Clarity    and    comprehensiveness    of your documentation.

•     Creativity     and     feasibility     of     proposed extensions.

Assessment method by which a student can demonstrate learning outcomes:

Completion of the assignment defined above, which must be submitted via WISEflow for marking.

Format for the assessment/coursework (Guidelines on the expected format and length of submission):

The university takes academic misconduct very seriously. Further information about this subject can be found here:

• Senate regulation 6: https://www.brunel.ac.uk/about/documents/pdf/Senate-Regulation-6-2020-07-01.pdf

• Library: https://www.brunel.ac.uk/life/library/SubjectSupport/Plagiarism

The university’s policy on using artificial intelligence in your studies can be found here:

https://students.brunel.ac.uk/study/using-artificial-intelligence-in-your-studies

Distribution date to students: 6th December 2024

Submission Deadline: 6th January 2025

Indicative Reading List:

Further information:

Assignments must be submitted to WISEflow in PDF format (unless stated otherwise), using Arial size

11 font and limited to no more than three pages. Students are strongly advised to submit their assignments well ahead of the deadline.

Late submissions are subject to penalties.

Assignment Brief

Solve the following free-vibration 2D hyperbolic equation using the finite difference method:


Initial condition:


Boundary condition:

u(0,y, t)=u(1,y, t)=u(x,0, t)=u(x,1, t)=0,    t>0

Stability condition:

It is suggested that Δx=ΔY. Ensure your numerical scheme satisfies the explicit stability condition:

Note: the stability condition of the explicit time scheme of the 2D wave equation

Using afunction with the following interface,

function u = solve_wave_eq(uinitial,c, Lx, Ly, Mx, My,maxIter,showplot)

Where:

Inputs

u_initial

Initial displacement field (size MX XMY) representing u(x,y,0) .

c

Wave speed (c= 1.5).

Lx,Ly

Length of the domain in the x andy directions

Mx,My

Number of cells in the andy directions (e.g., MX=20, MY=20).

dt

Time step size (must satisfy the stability condition)

maxIter

Maximum number of time steps to solve

showplot

Boolean flag to enable/disable plotting of the wave

Output

u

Final displacement field after solving the 2D wave equation for maxIter steps

Task

1. Explain the Interface

1.1  Describe what an interface is in computing and explain the scope of the function in terms of its inputs and outputs.

1.2  Detail the shapes of uinitial and the returned u.

2. Quickstart Example

2.1 Write a quickstart guide explaining how to use the function. Include at least one example test case

(e.g.,   u initial=s in(2mx).s in(4my),     L X=LY=1,    MX=MY= 20)

2.2  Include plots showing the wave propagation at various time steps, and explain the physical significance of these results.

2.3  If the code doesn’t work as expected, explain what behaviour you would anticipate.

3. Code Annotation

3.1 Choose two critical snippets of code and describe how they work:

3.1.1 Calculating u at the next time step using the finite difference method. 3.1.2 Implementing the boundary conditions.

4. Verification and Validation

4.1  Discuss how you verified and validated the code.

Verification: Ensuring the code is solving the equations correctly (e.g., stability tests).

Validation: Confirming the code produces realistic and accurate results (e.g., comparing to analytical solutions or expected behaviours).

4.2 Suggest  at  least  one  validation  case   (e.g., u(x,y,0)=s in(2mx).s in(4my)) and  comparing  with analytical solutions for specific times).

5. Future Extensions

5.1 Suggest  possible  extensions  to  the  code   (e.g.,  adding  absorbing   boundary  conditions,   modelling nonlinear waves, or introducing energy loss). Briefly explain how these could be implemented.

6. Appendix

6.1  Include the full MATLAB code as an appendix. Ensure the code is thoroughly commented to describe each step.

Example Test Case:

Run the following scenario to validate your code:

● u initial(x, y)=s in(2nx).s in(4ny)

● L X=LY=1

● MX=MY=20

●    dt= 0.002 (ensuring stability condition is met)

● max lter= 100

Visualize the results in several time steps (e.g., t=0,0.02,0.04) and describe the behaviour of the wave.

Code Requirements:

i.        Use the explicit finite difference method to solve the wave equation.

ii.        Implement fixed boundary conditions as specified.

iii.        Include input validation to check:

•     dt satisfies the stability condition.

• u_initial matches the size Mx × My .

•     All inputs are numeric and have appropriate dimensions.

Suggestions:

i.        Format your submission to include the required sections: Interface explanation, quickstart example, annotated code, verification/validation, and future extensions.

ii.        Use plots and error analysis from the code to support your explanations.

iii.        Ensure the MATLAB code is included as an appendix with additional comments if necessary.

Notes:

i.        Refer to lecture slides, textbooks, or MATLAB documentation if needed.

ii.        Unit tests are encouraged to ensure correct implementation.

MATLAB Code

clc; clear all; close all;

%2D WAVE EQUATION utt = c^2(uxx+uyy)

%with initial condition  u(x,y,0) = sin(p*pi*x)*sin(q*pi*y), 0 0 % and boundary conditions u(0,y,t) = u(1,y,t)= u(x,0,t)= u(x,1,t) = 0 t>0 c = 1.5;

dx = 0.01; dy = dx;

sigma = 1/sqrt(2); gamma = 1/sqrt(2); %Courant-Friedrich Stability Condition dt = sigma*(dx/c);

t = 0:dt:1; x = 0:dx:1; y = 0:dy:1;

u = zeros(length(x),length(y),length(t)); p = 2; q = 4;

u(:,:,1) = transpose(sin(p.*pi.*x))*sin(q.*pi.*y); %u(x,y,0) = sin(p*pi*x)*sin(q*pi*y) %u(x,y,dt)

for i=2:length(x)-1

for j=2:length(y)-1

u(i,j,2)= (sigma^2)*(u(i+1,j,1)-2*u(i,j,1)+u(i-1,j,1))...

+(gamma^2)*(u(i,j+1,1)-2*u(i,j,1)+u(i,j-1,1))+2*u(i,j,1) - u(i,j,1);

end end

for n=2:length(t)-1

for i=2:length(x)-1

for j=2:length(y)-1

u(i,j,n+1)= (sigma^2)*(u(i+1,j,n)-2*u(i,j,n)+u(i-1,j,n))...

+(gamma^2)*(u(i,j+1,n)-2*u(i,j,n)+u(i,j-1,n)) + 2*u(i,j,n) - u(i,j,n-1);

end end

end

%Analytic solution

SOL = zeros(length(x),length(y),length(t)); S = transpose(sin(p.*pi.*x))*sin(q.*pi.*y); for i=1:length(t)

SOL(:,:,i) = S*cos(c.*pi.*(sqrt(p^2+q^2).*t(i)));

end

%Absolute error E = abs(SOL-u);

[X,Y] = meshgrid(x,y);

max_e = max(max(max(E))); for j=1:length(t)

subplot(2,1,1) colormap(cool);

p1 = surf(X,Y,u(:,:,j));

title(sprintf('2D wave equation at t = %1.2f, con sigma = %1.2fy gamma = %1.2f',t(j),sigma, gamma),'Fontsize',11); xlabel('x','Fontsize',11); ylabel('y','Fontsize',11);

zlabel(sprintf('u(x,y,t = %1.2f)',t(j)),'Fontsize',11); axis ([0 1 0 1 -1 1]);

subplot(2,1,2)

p2 = surf(X,Y,E(:,:,j));

title(sprintf('Absolute error at t = %1.2f',t(j)),'Fontsize',11); xlabel('x','Fontsize',11); ylabel('y','Fontsize',11);

zlabel(sprintf('E(x,y,t = %1.2f)',t(j)),'Fontsize',11); axis ([0 1 0 1 0 max_e]);

pause(0.001); hold on;

if(j~=length(t)) delete(p1);

delete(p2);

end end



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