# Integral over a surface, given experimental data

I have a mesh of a 3D surface composed by triangles, and I have the value of a function $$u(x,y,z)$$ in every vertex of the mesh (every vertex of each triangle).

I need to calculate the following integral:

$$\iint_S u_{xx}dA$$

Normally, I calculate this integral as a sum over each triangle supposing that $$u$$ is linear on each triangle (because I know the value in the vertex o each triangle). So I can use the basis polinomial (typical of fem). My problem is that this basis must be polynomials of degree 1, but I need the second partial derivate, so this integral will be zero.

How can I calculate this integral?

• didn't know it, I'll google now to see what it's about Dec 18 '20 at 19:27
• Is function $u(\vec{r})$ known for the whole space? Or only on the surface? If it is known then maybe one can use the Stokes theorem to convert the surface integral to a volume integral. Dec 19 '20 at 6:09
• I have data only on the (not closed) surface. Dec 21 '20 at 23:50

1. Calculate $$u_{xx}$$ on each point or vertex by using vtkGradientFilter.
2. Do the integration over the surface by using vtkIntegrateAttributes.
It should work. You can follow this procedure of taking derivative to calculate $$u_{xx}$$ by using FDM and then calculate the integral in any other programming language or libraries.