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Intersecting boundary problems when solving PDEs

Hi everyone,

the Wolfram Language both in the cloud and in Mathematica 10 offers the possibility to solve PDEs on quite general surfaces. In the documentation it is shown that the Laplace equation can be solved on a 3D model of the space shuttle like this:

mr = DiscretizeGraphics[ExampleData[{"Geometry3D", "SpaceShuttle"}]];

uif = NDSolveValue[{Inactive[Laplacian][u[x, y, z], {x, y, z}] == 0, 
    DirichletCondition[u[x, y, z] == 1, z <= -1.3]}, 
   u, {x, y, z} \[Element] mr];

Needs["NDSolve`FEM`"]
ElementMeshSurfacePlot3D[uif, Boxed -> False, ViewPoint -> {0, -4, 2}]

This is really nice and gives

enter image description here

The problem shows up when we take more irregular objects like 

mr = DiscretizeGraphics[ExampleData[{"Geometry3D", "Cow"}]]

enter image description here

or 

mr = DiscretizeGraphics[ExampleData[{"Geometry3D", "Horse"}]]

When you execute the integration you get:

NDSolveValue::fememib: The input has or generated an intersecting boundary and cannot be processed. >>

Similarly, 

BoundaryDiscretizeGraphics[ExampleData[{"Geometry3D", "Cow"}]]

gives

BoundaryMeshRegion::binsect: "The boundary curves self-intersect or cross each other in BoundaryMeshRegion[{{0.10799399763345718`,0.08466669917106628`,-0.21893799304962158`},{0.1157900020480156`,0.08869930356740952`,-0.22847199440002441`},{0.10719799995422363`,0.09556479752063751`,-0.23278899490833282`},<<46>>,{-0.30769699811935425`,0.093573197722435`,-0.22068199515342712`},<<2853>>},{{},{},{Polygon[{<<1>>}]}}]. \!\(\*ButtonBox[\">>\",
Appearance->{Automatic, None},
BaseStyle->\"Link\",
ButtonData:>\"paclet:ref/BoundaryMeshRegion\",
ButtonNote->\"BoundaryMeshRegion::binsect\"]\)"

So the problem is that there are self-intersections of the surface. Is there any elegant approach to fixing that?

Cheers, Marco
POSTED BY: Marco Thiel
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Hi everyone, I have a similar problem that Mathematica fails to solve. Would any options help? I want to solve 3D Laplace equation where one of the boundaries of the region is dynamic, i.e. I can update it and make a next cycle. I have experimented with different shapes of the surface. For rather smooth ones it works. For the ones with many edges it fails by generating self-intersections.
The code below works.

xmin = -1; xmax = +1; ymin = -1; ymax = 1;
shape[x_, y_] := 0.05*Sin[15 x + 0.8]*Sin[15 y + 0.8];
realshape[x_, y_] = Piecewise[{{shape[x, y], shape[x, y] > 0}}, 0];
bottomregion = 
  ImplicitRegion[-0.1 < z < 
    shape[x, y], {{x, xmin, xmax}, {y, ymin, ymax}, {z, -0.1, 
     xcut}}];
ToElementMesh[meshbottom, "MaxCellMeasure" -> 0.003]

but if I replace shape with realshape, then it fails.

POSTED BY: Sergey Slizovskiy
Attachments:
POSTED BY: J Gregory Moxness

Hi Gregory,

thanks a lot. Indeed, now it works:

mr = Import["~/Desktop/cowfix4.stl", "BoundaryMeshRegion"];
uif = NDSolveValue[{Inactive[Laplacian][
      u[x, y, z], {x, y, z}] == -0.01, 
    DirichletCondition[u[x, y, z] == 1, z <= 0.08]}, 
   u, {x, y, z} \[Element] mr];

Needs["NDSolve`FEM`"]
ElementMeshSurfacePlot3D[uif, Boxed -> False, ViewPoint -> {0, -4, 2}]

enter image description here

I noticed that you had to trim the cow's tail.

Thanks,

Marco
POSTED BY: Marco Thiel
POSTED BY: Marco Thiel

That is generally a very hard problem and WL is unlikely in the near future to be able to solve 'healing' of intersections in 3D. Supposedly, there are "safe" and very expensive algorithms for doing this.

TetGenDetectIntersectingFacets returns the intersecting facets, then one can try to repair them manually.

POSTED BY: Bruce Miller
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