# 2D surface integral in MATLAB for Finite Element Calculation

## Main Question or Discussion Point

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!

Thanks in advance!

Cheers
Vivek

## Answers and Replies

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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
Science Advisor
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.

Please resist the urge to necro post to these ancient threads.

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
Science Advisor
Homework Helper
Why? This thread shows up for anyone who google searches "matlab surface integral". Wouldn't it be better if they found a reply?
I'm just repeating the policy of the moderators of this forum. They prefer not to have these old posts dredged up from the past.

However, if you wish to create a new thread of your own, dealing with this subject, you can always refer to previous threads on the same topic.

If you seek another opinion on this point, please feel free to contact one of the moderators, say via PM, and start a discussion about necro posting to these old threads. Policies do change at PF from time to time.

Ah I see. Thank you SteamKing. I'll be happy to review the policy and comply in the future.