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$\begingroup$ This is close to how i pelting uv maps was done in 3D in the past. Triangulate the mesh then make every edge a spring with rest length at starting length, then force the boundaries to the shape you want (circle) then let the spring dynamics handle the interiors $\endgroup$
– joojaa
Commented Jul 15, 2018 at 14:31
1

$\begingroup$ @joojaa I just wanted to brag around with my classical education ;) Indeed, these things are standard tasks in the domain of texture mapping -- which does not mean that they were easy. The art is to guarantee that there are no flipped triangles. There is much more to it than it would fit into a single post. $\endgroup$
– Henrik Schumacher
Commented Jul 15, 2018 at 14:34
2

$\begingroup$ Nice use of the low level FEM functions. You might be interested in bndvertices === dbc["DirichletRows"] and normalprojections === MapThreadDot[ R["BoundaryNormals"][[1]], (gradu @@@ (0.5 (p[[i]] + p[[j]]))).(-J)]. The first part can be done in NDSolve directly, I'd need think about a way to compute the normal for the second part to be done in NDSolve. $\endgroup$
– user21
Commented Jul 16, 2018 at 6:33
2

$\begingroup$ You can also use something like: g = GraphicsRow[{Graphics[{Texture[tex], ElementMeshToGraphicsComplex[R, VertexTextureCoordinates -> texcoords]}], Graphics[{Texture[tex], ElementMeshToGraphicsComplex[R, VertexTextureCoordinates -> texcoords, "CoordinateConversion" -> (f[Sequence @@ ##] & /@ # &)]}]}, ImageSize -> Full] $\endgroup$
– user21
Commented Jul 16, 2018 at 6:44
2

$\begingroup$ @user21 Thank you so much! These tiny give-aways are actually really rewarding. The thing with the normal in the second equation was indeed bugging me for quite some time. Suggestions to simplify that are welcome. $\endgroup$
– Henrik Schumacher
Commented Jul 16, 2018 at 7:00

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