Can I complete a Gaussian equation using indeces?

Howdy, all.

I've attached all my code for a project involving the Panel Method for aero/hydrodynamics. Everything is working fine until I get to the last few lines (ignoring graphing code):

Essentially, I started writing with indeces, which was probably a mistake. The Gaussian requires the matrices [A][x]=[b] be rearranged to give [x] = [A]\[b]. In my code, [A] is represented by Aindex(i,j) and [b] is Bindex(i,j). [x] would be Lambda(i,j), lines 42-43.

All values prior to executing Lambda(i,j) give correct values. I'm fairly certain the problem is trying to treat indeces like matrices. I've tried every division operator I know of (./, \, etc.) and I've tried inverting both indeces (shotgunning).

Is there a way I can compute the Gaussian without having to start from scratch with matrices?

Final correct values for lambda are given:

such that the correct value of lambda is the numerical term multiplied times 10. Appreciate your help!

Answer 1

[Shivam Lahoti]

Hi Christopher,

I can understand that you want to calculate Gaussian without having to start from scratch with matrices. There are some issues in your code that need to be addressed:

• The inner loop in your last for-loop should be over ‘j’, not ‘i’ again.
• You are trying to divide Aindex by Bindex elementwise, which is not correct for solving the system of equations.
• Lambda should be a vector, not a matrix since it represents the unknowns [x] in the system.

Here is a corrected version of the last part of your code:

% The system matrix and right-hand side vector 
Aindex = zeros(n, n); 
Bindex = zeros(n, 1); 
% Filling the system matrix Aindex and the vector Bindex 
for i= 1:n 
    for j= 1:n 
        Aindex(i,j)= I(i,j)/(2*pi); 
    end 
    Bindex(i)= (-Vinf)*cos(Beta(i)); 
end 
% Solving for the unknowns Lambda 
Lambda = Aindex\Bindex; 
% Calculating Check, V, and Pressure Coefficient 
Check = Lambda/Vinf; 
V = zeros(n, 1); 
PressCoeff = zeros(n, 1); 
for i= 1:n 
    sumJ = 0; 
    for j= 1:n 
        sumJ = sumJ + (Lambda(j)/(2*pi))*J(i,j); 
    end 
    V(i) = sumJ + (Vinf*sin(Beta(i))); 
    PressCoeff(i) = 1 - ((V(i)/Vinf)^2); 
end 
% Plotting 
plot(PressCoeff,'b') 
set(gca,'XTick',0:pi/2:2*pi)  
set(gca,'XTickLabel',{'0','pi/2','pi','3*pi/2','2*pi'}) 

This corrected code first constructs the system matrix Aindex and the right-hand side vector Bindex. Then, it solves for Lambda using the backslash operator. After that, it calculates the velocity V and pressure coefficient PressCoeff using the computed Lambda.

I hope this was helpful.

Regards,

Shivam Lahoti.