# 2D surface integral in MATLAB for Finite Element Calculation

Hi everyone,

As part of a project, I am required to numerically compute the expression

$$b_{i}^{e} &=& \frac{E_{0}^{i}k_0^2(\epsilon_r-1/\mu_r)}{2\Delta^e}\left[\iint\limits_{\Omega^e}(a_i^e + b_i^e x + c_i^e y)e^{-jk_0 x} dx dy\right] \nonumber\\&&- \frac{jk_0 E_0^i r'}{2\Delta^e \mu_r}\left[\int_{\phi_{1}^{s_2}}^{\phi_{2}^{s_2}}(a_i^e \cos\phi + b_i^e r'\cos^2\phi + c_i^e r'\sin\phi\cos\phi)e^{-jk_0r'\cos\phi}d\phi\right]$$

specifically, compute the integrals numerically. The problem is that $\Omega^e$, the domain of integration of the first integral is a triangle (whose vertex coordinates are well known).

I am unable to figure out a way to do this integral computationally in MATLAB. That is, how does one compute an area integral in MATLAB when the x and y coordinates are coupled (and bounded to lie in a spatial region).

If there is a documented way of doing this, or a preexisting function, I would prefer to use it and go ahead with my work, rather than reinvent the wheel. Any inputs would be greatly appreciated!

Cheers
Vivek

Hi Vivek,

One thing you could do is define a function of x and y that will return one in the domain and zero outside.

For example, the following function returns one only in the 45-45-90 triangle near the origin given by x<3, y>0, x>y:
domain = @(x,y) heaviside(x-y)*heaviside(3-x)*heaviside(y).

Then you can integrate over a square including your triangular domain and multiply the integral argument by this domain function.

I'm sure you've solved this problem by now, but I stumbled across it while searching for surface integration techniques and figured I might as well reply.
Cheers,
David

SteamKing
Staff Emeritus
Homework Helper
Hi Vivek,

One thing you could do is define a function of x and y that will return one in the domain and zero outside.

For example, the following function returns one only in the 45-45-90 triangle near the origin given by x<3, y>0, x>y:
domain = @(x,y) heaviside(x-y)*heaviside(3-x)*heaviside(y).

Then you can integrate over a square including your triangular domain and multiply the integral argument by this domain function.

I'm sure you've solved this problem by now, but I stumbled across it while searching for surface integration techniques and figured I might as well reply.
Cheers,
David
Vivek hasn't posted in a year.

Why? This thread shows up for anyone who google searches "matlab surface integral". Wouldn't it be better if they found a reply?

SteamKing
Staff Emeritus