3D Plot in Matlab - not correct

Question

I am trying to calculate the potential function in a given coplanar rectangular structure. Here is the equation i deduced by using mathematics, here is the sumary: Assumption

Now I try to plot it in MATLAB, here is my code:

function [x,y,v] = potentialFM(a1,a2,a3,b)
syms n;

%range in normalized values
x=0:1/20:1;
y=0:(b/a3)/20:(b/a3);
[X,Y] = meshgrid(x,y);

%normalized values
an1=a1/a3;
an2=a2/a3;
bn=b/a3;

%symbol k 
k=(n.*2+1)*(pi/2);

C1=(1/(k.^2)).*(2/(an1-an2));
C2=cos(k.*an2)-cos(k.*an1);
C3=1/(sinh(k.*bn));
Vx=cos(X*k);
Vy=sinh(Y*k);
v=symsum(C1.*C2.*C3.*Vx.*Vy,n,1,20);
end

So, Could someone have a try to my code and check where is an error. The result should look like the assumed graph in the link but i cannot achieve it.

Best Regards,

fRz

Answer 1

You don't have a problem with plotting, but with the evaluation of your function. Why do you use symbolics if you want a numeric result ready for plotting?

Here's a simple fix – n is now simply a loop variable, k another normal variable, and the sum is computed cumulatively using the variable v :

function [x,y,v] = potentialFM(a1,a2,a3,b)

%range in normalized values
x=0:1/20:1;
y=0:(b/a3)/20:(b/a3);
[X,Y] = meshgrid(x,y);

%normalized values
an1=a1/a3;
an2=a2/a3;
bn=b/a3;

v = 0;
for n = 1 : 20
    k=(n.*2+1)*(pi/2);

    C1=(1/(k.^2)).*(2/(an1-an2));
    C2=cos(k.*an2)-cos(k.*an1);
    C3=1/(sinh(k.*bn));
    Vx=cos(X*k);
    Vy=sinh(Y*k);
    v = v + C1.*C2.*C3.*Vx.*Vy;
end

This is not necessarily the most efficient implementation, maybe some of the computations could be vectorized – but it works. Choosing arbitrarily the parameters,

[x,y,v] = potentialFM(1, 2, 3, 4);
surf(x, y, v)

gives:

Answer 2

Another attempt: If you need the value of the infinite sum and hope that symsum can find a closed form expression, there is no need to mix it with numerics. Instead, declare all the variables as symbolic:

syms n an1 an2 bn X Y
k=(n.*2+1)*(pi/2);
C1=(1/(k.^2)).*(2/(an1-an2));
C2=cos(k.*an2)-cos(k.*an1);
C3=1/(sinh(k.*bn));
Vx=cos(X*k);
Vy=sinh(Y*k);
v=symsum(C1.*C2.*C3.*Vx.*Vy,n,1,inf);

symsum returns a result, but it looks like this:

-(8*sum(((exp(-(pi*X*(2*n + 1)*i)/2)/2 + exp((pi*X*(2*n + 1)*i)/2)/2)*(exp(-(pi*Y*(2*n + 1))/2)/2 - exp((pi*Y*(2*n + 1))/2)/2)*(exp(-(pi*an1*(2*n + 1)*i)/2)/2 + exp((pi*an1*(2*n + 1)*i)/2)/2 - exp(-(pi*an2*(2*n + 1)*i)/2)/2 - exp((pi*an2*(2*n + 1)*i)/2)/2))/((2*n + 1)^2*(exp(-(pi*bn*(2*n + 1))/2)/2 - exp((pi*bn*(2*n + 1))/2)/2)), n == 1..Inf))/(pi^2*(an1 - an2))

As you can see, the Symbolic Math Toolbox was able to perform some simplifications (?) but could not solve the core of the problem: There is still an infinite sum in there. As far as I understand, this indicates that there is no closed form expression (or at least that finding it is beyond the capabilities of the toolbox).