Calculating mixed partial derivatives on a 3D mesh

[KareemErgawy]

I am working on implementing a PDE model that simulates a certain physical phenomenon on the surface of a 3D mesh.

The model involves calculating mixed partial derivatives of a scalar function defined on the vertices of the mesh.

What I tried so far (which is not giving good results), is this:

1. First, I calculate the gradient of the scalar function over the mesh. This gives me the first derivative in the x, y, and z directions (the gradient vector field over the mesh).
2. Since the gradient field is defined per face (as opposed to per vertex), I interpolate the gradient field to the vertices by performing a weighted averaging over the triangles that share a vertex. I am using the areas of the triangles as the weights.
3. After this interpolation, the gradient (first order derivative is now define "interpolated" over the vertices). I use this interpolated vector field as input to the gradient operator again and get the gradient of each component in the vector.

I know that this is incorrect (or at least inaccurate) since the scalar function originally defined over the mesh goes out the range of values it should be restricted to if the model was implemented correctly.

So, how can I calculate an accurate descritized approximation of mixed partial derivatives over a 3D mesh?

 

[Nidum]

Sophisticated methods for dealing with this sort of problem can be found in the theory of finite element analysis .

 

[Geofleur]

Perhaps you can compare your numerical result with analytically derived results at each stage and find out where your method is going wrong.

 

[KareemErgawy]

Sophisticated methods for dealing with this sort of problem can be found in the theory of finite element analysis .

Thanks for the reply. Can you refer me to some resources that might be helpful in that regard?

 

[Nidum]

Can you describe what problem you are actually trying to solve - it is quite possible that there is a ready made FE analysis method available .

If no ready made method is available I will give you references to the underlying principles of FE and to programs which can be tailored to suit your needs .

 

[KareemErgawy]

Can you describe what problem you are actually trying to solve - it is quite possible that there is a ready made FE analysis method available .

If no ready made method is available I will give you references to the underlying principles of FE and to programs which can be tailored to suit your needs .

Yup, what I am trying to implement specifically is the model proposed in this paper: http://dl.acm.org/citation.cfm?id=156977. If you don't have access to it, I can write down the equations here. But at least the abstract should give a good idea about the phenomenon it is simulating.

One thing to note is that my goal is to simulate crystal growth but on the surface on a 3D mesh instead of a 2D grid. Please note that I am treating the surface of the 3D mesh as a texture (not explicitly parametrized; however). So I have a scalar function defined on the vertices of a 3D mesh, say theta. This function ranges from 0 to 1 over the vertices of the mesh. Over time, there is a PDE that simulates the change of values (phases) over the mesh surface.

Hope that clarifies thing a little bit more and thanks for help.

 

[Chestermiller]

Is this a rectangular mesh, or are you using a triangular mesh?

Chet