I'm trying to evaluate the following integral:
$$I = \frac{1}{AA'} \iint_{s'} \iint_s |\underline{r}'-\underline{r}| \:ds\:ds' $$
which is the distance between two surfaces, $s$ and $s'$. I'm mainly interested in evaluating this for triangles in 3D space, though quadrilaterals would also be useful. However when I use the equation for a triangle:
$$\underline{r} = \underline{a}s + \underline{b}t + \underline{c}$$ $$\underline{r}' = \underline{d}u + \underline{e}v + \underline{f}$$
I get an integral of the form:
$$I = \frac{1}{AA'} \int_0^1\int_0^1\int_0^1\int_0^1 \left(as^2 + bs + csu + dsv + et^2 + ft + ..... \right) \:ds\:dt\:du\:dv $$
then after the first two integrals Mathematica gives up and hands me a mass of terms and conditionals and can't evaluate any further.
I've had some people suggest reducing the surface integrals down to line integrals using something like Stoke's theorem but I can't find a new integrand that works. I can also calculate the above numerically but even with triangular quadrature it's computationally expensive and I'd like to be able to calculate it for a large array of triangles in a tetrahedral mesh.
Any help on this would be great. TIA