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HectorRedal September 9, 2010 16:46

How can be calculate a surface integral in the finite element method?

I would like to post the following question about how to calculate the integral of a surface in the finite element method.

As far as I know, when calculating, for example the stiff matrix, in the finite element formulation, the integral for such calculation is extended over the whole domain, for example a cube.

I am developing a software for cfd that uses the finite element formulation, and there are some integrals that should be performed over the surface of the domain (These integrals are surface integrals).

For integrals that are applied over the domain, the jacobi is used for differential transformation.

My doubt arises when trying to calculate the surface integrals. As far as I know, the jacobi could not be used.
Which kind of differential transformation should be used?

I would appreciate any kind of help / suggestion.

Thanks in advance,


Rami September 13, 2010 05:25

Hello Hector,
I assume (for simplicity) you use brick elements of any order. You integrate as usual (see below for details), except the following: Suppose you use the isoparametric coordinates (r,s,t) with the appropriate transformation to the (x,y,z) coordinates, and suppose your surface is defined by, say, r=1. The you simply use the shape functions (and/or their derivatives) as appearing in the integrand with the substitution of r=1, and instead of the Jacobian relating dV (the volume differential) to dr*ds*dt you will had a relation between dA (the area differential) and ds*dt. That's it. Now integrate either exactly (if possible) or numerically (e.g., Gauss quadrature on the square -1<s<1, -1<t<1).

Good luck,

HectorRedal September 13, 2010 15:58

Hello Rami,

Thank you for your answer.
I will try to follow your instructions.



samkam786 March 13, 2015 09:25

hey !
I have been trying to formulate the FEM equation to calculate the temperature across the 2D cross section.. but the extended volume is not in a form of cube, but in the form of a quadrant. I can't use axisymmetric elements... Any hints ?

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