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CONVERT THE FOLLOWING MATLAB CODE FROM SOURCE PANEL METHOD TO
VORTEX PANEL METHOD:
clc;clear all;close all;
Vinf=100; % freestream velocity
R=1; % cylinder radius
n=4; % number of panels
alpha=2; % angle of attack
dtheta=2*pi/n;
theta=pi+pi/n:-dtheta:-pi+pi/n;
X=R*cos(theta);
Y=R*sin(theta);
for i=1:n
% angle of flow with tangent of panel
phi(i)=-alpha+atan2((Y(i+1)-Y(i)),(X(i+1)-X(i)));
% angle of flow with normal of panel
beta(i)=phi(i)+pi/2;
x_mid(i)=(X(i+1)+X(i))/2;
y_mid(i)=(Y(i+1)+Y(i))/2;
S(i)=sqrt((Y(i+1)-Y(i))^2+(X(i+1)-X(i))^2);
end
% Source Panel Method
for j=1:n
neighbors(:,j)=[1:j-1 j+1:n];
xi=x_mid(j);
yi=y_mid(j);
for i=1:n-1
m=neighbors(i,j);
Xj=X(m);
Yj=Y(m);
Xj1=X(m+1);
Yj1=Y(m+1);
A=-(xi-Xj)*cos(phi(m))-(yi-Yj)*sin(phi(m));...
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Modify the given code below which uses Simpson's 3/8 method with
6 subintervals to answer the question above.
if a==b
I=0
else
N=3;
h=(b-a)/N;
x=a:h:b;
y=Fun(x);
I=3*h/8*(y(1)+2*sum(y(4:3:(N-2)))+y(N+1));
N=2*N;
check=0;
% Calculating subsequent valus of I
while check==0;
h=(b-a)/N;
x=a:h:b;
y=Fun(x);
% Composite Simpson's 3/8 method, Eq. (9.22)
I_new=3*h/8*(y(1)+2*sum(y(4:3:(N-2)))+y(N+1));
I_new=I_new+3*h/8*3*(sum(y(2:3:(N-1)))+sum(y(3:3:N)));
% Compare solution with that calculated in the previous
iteration
error=abs((I-I_new)/I)*100; % (*)
if error>0.1 % continue iteration
check=0;
N=N*2;
I=I_new;
elseif error<=0.1 % end iteration
check=1;
I=I_new;
end
end
end...
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(c) (Optional) Calculate the fo/2 Padé approximant to e, where θ is rea I. Show that the leading-order error is θ3/6 Consider Eq. (1), with U = 0 at x = 0 and x 1, approximated at the mesh points xm-Tnh, mに1 N-1, Nh = 1, along the time level tn nk by the set of difference equations and λ = k/h2, and TN-1 is the matrix of order N-1 repre- senting the second order operator 62 Investigate the stability...
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4. (20 pts) Suppose the boundary-value problem y" – y=x, 0 < x < 1; y(0) = y'(1) = 0 Let h = 1/n, X; = jh, where j = 0,1,..., n and u; y(x;). Consider two "exterior" mesh points 2-1 = -h and 2n+1 = 1+h. Write out an 0(ha) approximate linear tridiagonal system for {u}. Hint: Let u-1 = y(x-1) = y(-h) and Un+1 = y(2n+1) = y(1 + h). Then using f(a+h) – f(a – h). f'(a)...
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Question 1 [Total 20 marks] (a) [5 marks] In a steady-state two-dimensional heat flow problem, the temperature, u, at any point in the domain (t, ) satisfies the differential equation u y(2-y) u= U0F With the given temperature boundary condition as follows: u(x, 0) = 0, u(x, 2) = x(4-x), 0 < x < 4 Calculate the temperature at the interior points a, b, and c using a mesh size h-1.
Question 1 [Total 20 marks] (a) [5 marks] In...
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Solve Laplace's equation on \(-\pi \leq x \leq \pi\) and \(0 \leq y \leq 1\),$$ \frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0 $$subject to periodic boundary conditions in \(x\),$$ \begin{aligned} u(-\pi, y) &=u(\pi, y) \\ \frac{\partial u}{\partial x}(-\pi, y) &=\frac{\partial u}{\partial x}(\pi, y) \end{aligned} $$and the Dirichlet conditions in \(y\),$$ u(x, 0)=h(x), \quad u(x, 1)=0 $$
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Use the following pseudocode for the Newton-Raphson method to
write MATLAB code to approximate the cube root (a)1/3 of
a given number a with accuracy roughly within 10-8 using
x0 = a/2. Use at most 100 iterations. Explain steps by commenting
on them.
Use f(x) = x3 − a. Choose a = 2 + w, where w = 3
Algorithm : Newton-Raphson Iteration
Input: f(x)=x3−a, x0 =a/2, tolerance
10-8, maximum number of iterations100 Output: an
approximation (a)1/3 within 10-8 or...
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Solve Laplace's equation, \(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0,0<x<a, 0<y<b\), (see (1) in Section 12.5) for a rectangular plate subject to the given boundary conditions.$$ \begin{gathered} \left.\frac{\partial u}{\partial x}\right|_{x=0}=u(0, y), \quad u(\pi, y)=1 \\ u(x, 0)=0, \quad u(x, \pi)=0 \\ u(x, y)=\square+\sum_{n=1}^{\infty}(\square \end{gathered} $$
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Please complete using matlab
In this problem you will utilize a for loop to compute the an approximation of the value of using the Leibniz's formula for Pi. The formula uses a summation. Follow the instructions correctly and read the questions thoroughly. 1. Set up the initial number of iteration i to obtain six iterations. 2. The approximate value of it can be given by the following formula: Approximated = Enzo 2n+1 Where n is an integer starting at zero...
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PLEASE USE MATLAB. The code is basically given to you with the
algorithm above.
Write a program for solving the system Ar-b by using Gauss-Seidel iterative method discussed in class that uses a computer memory for one iterate only. This could be summarized as follows: Input n -- the size of the system, matrix A-(a(i,j), the vectoir b (b (1), . . . , b(n)), the intitial guess x (x(1), ..., x(n)) (you can use x=b) maximum number of iterations...
Modify the given code below which uses Simpson's 3/8 method with
6 subintervals to answer the question above.
if a==b
I=0
else
N=3;
h=(b-a)/N;
x=a:h:b;
y=Fun(x);
I=3*h/8*(y(1)+2*sum(y(4:3:(N-2)))+y(N+1));
N=2*N;
check=0;
% Calculating subsequent valus of I
while check==0;
h=(b-a)/N;
x=a:h:b;
y=Fun(x);
% Composite Simpson's 3/8 method, Eq. (9.22)
I_new=3*h/8*(y(1)+2*sum(y(4:3:(N-2)))+y(N+1));
I_new=I_new+3*h/8*3*(sum(y(2:3:(N-1)))+sum(y(3:3:N)));
% Compare solution with that calculated in the previous
iteration
error=abs((I-I_new)/I)*100; % (*)
if error>0.1 % continue iteration
check=0;
N=N*2;
I=I_new;
elseif error<=0.1 % end iteration
check=1;
I=I_new;
end
end
end...
(c) (Optional) Calculate the fo/2 Padé approximant to e, where θ is rea I. Show that the leading-order error is θ3/6 Consider Eq. (1), with U = 0 at x = 0 and x 1, approximated at the mesh points xm-Tnh, mに1 N-1, Nh = 1, along the time level tn nk by the set of difference equations and λ = k/h2, and TN-1 is the matrix of order N-1 repre- senting the second order operator 62 Investigate the stability...
4. (20 pts) Suppose the boundary-value problem y" – y=x, 0 < x < 1; y(0) = y'(1) = 0 Let h = 1/n, X; = jh, where j = 0,1,..., n and u; y(x;). Consider two "exterior" mesh points 2-1 = -h and 2n+1 = 1+h. Write out an 0(ha) approximate linear tridiagonal system for {u}. Hint: Let u-1 = y(x-1) = y(-h) and Un+1 = y(2n+1) = y(1 + h). Then using f(a+h) – f(a – h). f'(a)...
Question 1 [Total 20 marks] (a) [5 marks] In a steady-state two-dimensional heat flow problem, the temperature, u, at any point in the domain (t, ) satisfies the differential equation u y(2-y) u= U0F With the given temperature boundary condition as follows: u(x, 0) = 0, u(x, 2) = x(4-x), 0 < x < 4 Calculate the temperature at the interior points a, b, and c using a mesh size h-1.
Question 1 [Total 20 marks] (a) [5 marks] In...
Please complete using matlab
In this problem you will utilize a for loop to compute the an approximation of the value of using the Leibniz's formula for Pi. The formula uses a summation. Follow the instructions correctly and read the questions thoroughly. 1. Set up the initial number of iteration i to obtain six iterations. 2. The approximate value of it can be given by the following formula: Approximated = Enzo 2n+1 Where n is an integer starting at zero...
PLEASE USE MATLAB. The code is basically given to you with the
algorithm above.
Write a program for solving the system Ar-b by using Gauss-Seidel iterative method discussed in class that uses a computer memory for one iterate only. This could be summarized as follows: Input n -- the size of the system, matrix A-(a(i,j), the vectoir b (b (1), . . . , b(n)), the intitial guess x (x(1), ..., x(n)) (you can use x=b) maximum number of iterations...