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I've been given a problem to solve and I was hoping for some pointers.

To start with we have a surface which is originally represented as a Bezier patch, a peicewise set of bi-cubic bezier surfaces. It might be important to mention that the surface is "fairly two dimensional", the surface is single-valued for each X-Z coordinate, there have been a few problems where it was convenient just to work in the X-Z plane and then just project to a height value.

To perform different types of analysis on the surface we have created two meshes. The first mesh has points that form quadrangles, as if a grid had been laid over the surface. One of the analysis results is a set of load values on the surface, each node in the mesh has 3d vector for the force at that point. We also have another mesh in which the elements are triangles, this is made in a more sophisticated way and there is no necessary connection between points in the two meshes.

So the problem, what is the best way to get load values for the points in the trimesh? Obviously some interpolation scheme is needed. I've been googling around on the subject but I'm not sure how to proceed. For each point on the trimesh the starting point would be to find the quadrangle containing it in the quadmesh. The interpolation methods I've looked at for 2d or 3d problems generally seem to require a regular grid or lattice as input, whereas here we have a set of 3d points but on a 2d surface. Also another question, when doing vector valued interpolation in this sort of context is it usual to just treat each component independently or are there problems with this. (I've seen some contexts mentioned in which it is problematic, normal vectors in graphics calculations particularly.)

(I'm not sure if this is the best forum to post this in, perhaps this counts as a mechanical engineering problem? I don't have a engineering background myself.)