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Making a thermal plot from values on a triangle mesh

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We have a mesh made up of triangular elements in 2d and a set of values defined for each element. We want to create a thermal plot of these values over the surface of the mesh. For this we need to obtain the value at an arbitrary position on the surface using some kind of interpolation. I've been experimenting with nearest neighbour interpolation but the results have a visual "spottiness" that management does not like.

 

I'm wondering if anyone has worked on a problem like this or knows any alternative methods I can try?

 

 

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