Invalid region for small angles. To Element Mesh. Finite element method

Question

I would like to mesh the region below in order to use it for a calculation by means of NDSolve. Could anyone help me to discretize this region for NDSolve using. Thank you very much.

Needs["NDSolve`FEM`"];

Rwg=7.0; RLiner=8.0; RRib=8.5; RExt=9.0; Theta1=0.0227; Theta2=0.00916; 

Ω=ImplicitRegion[
    !(x^2 + y^2>RLiner^2 && x^2 + y^2< RRib^2 && 0<y<x*Tan[Theta2])&&
 (y>=0 && x^2 + y^2>=Rwg^2 && x^2 + y^2<=RExt^2 && y<=x Tan[Theta1]),{x, y}];


Show[ RegionPlot[Ω], ImageSize -> 300]

ToElementMesh[Ω,MaxCellMeasure -> 0.0001]["Wireframe"]

Answer 1

Look at the boundary mesh generated for the region:

Needs["NDSolve`FEM`"];
Rwg=7.0;RLiner=8.0;RRib=8.5;RExt=9.0;Theta1=0.0227;Theta2=0.00916;
\[CapitalOmega]=ImplicitRegion[!(x^2+y^2>RLiner^2&&x^2+y^2<RRib^2&&0<y<x*Tan[Theta2])&&(y>=0&&x^2+y^2>=Rwg^2&&x^2+y^2<=RExt^2&&y<=x Tan[Theta1]),{x,y}];
ToBoundaryMesh[\[CapitalOmega]]["Wireframe"]

You see there is a line at y==0 which should not be there.

\[CapitalOmega][[1]] /. {x -> 8.4, y -> 0}
True

If you reformulate the region to:

LOWER = -1;
\[CapitalOmega] = 
 ImplicitRegion[! (x^2 + y^2 > RLiner^2 && x^2 + y^2 < RRib^2 && 
      LOWER <= y < x*Tan[Theta2]) && y >= 0 && x^2 + y^2 >= 49.` && 
   x^2 + y^2 <= 81.` && y <= 0.022703899831486306` x, {x, y}]

This example works as expected:

ToElementMesh[\[CapitalOmega]]["Wireframe"]

Now, various version of Mathematica use different (default) algorithms to to discretize regions. Even more so DiscretizeRegion and ToElementMesh may use different algorithms in the same version. What I have shown here is version 10.4

Answer 2

I believe this is a bug: for some reason changing the conditional y<x Tan[] to y/x<Tan[] makes it work:

Rwg = 7.0; RLiner = 8.0; RRib = 8.5; RExt = 9.0;
Theta1 = 0.0227
Theta2 = 0.0091
\[CapitalOmega] = 
  ImplicitRegion[! (x^2 + y^2 > RLiner^2 && x^2 + y^2 < RRib^2 && 
       0 < y/x < Tan[Theta2]) &&
    (y >= 0 && x^2 + y^2 >= Rwg^2 && x^2 + y^2 <= RExt^2 && 
      y/x <= Tan[Theta1]), {x, y}];
Show[RegionPlot[\[CapitalOmega]], ImageSize -> 300]
<< NDSolve`FEM`
ToElementMesh[\[CapitalOmega], MaxCellMeasure -> 0.0001]["Wireframe"]

note the mesh is so dense the wireframe image is just black.

here it is with MaxCellMeasure->0.0025

v10.1/Windows. Minimal example of the issue:

 GraphicsRow@Table[
  RLiner = 8.; RRib = 8.5; 
  o2 = ImplicitRegion[
    RLiner^2 <= x^2 + y^2 <= RRib^2 && 0 <= y <= x Tan[Theta2], {x, 
     y}];
  Show[{RegionPlot[o2, PlotStyle -> Red] , 
    Plot[{x Tan[Theta2]}, {x, 0, 9}]},
   PlotRange -> {0, .8}] , {Theta2, {.09, .075}}]

Answer 3

It can sometimes help out if you Reduce your implicit description of the region first.

ToElementMesh[
  ImplicitRegion[! (x^2 + y^2 > RLiner^2 && x^2 + y^2 < RRib^2 && 
         0 < y < x*Tan[Theta2]) && (y >= 0 && x^2 + y^2 >= Rwg^2 && 
        x^2 + y^2 <= RExt^2 && y <= x Tan[Theta1]) // Reduce // Evaluate,
   {x, y}], MaxCellMeasure -> 0.0001]["Wireframe"]

As has been pointed out, it's a very fine mesh. Here it is with MaxCellMeasure commented out:

Answer 4

This is all very confusing, different versions of Mathematica seem to react differently to the different region functions involved here. What I'm going to write here is only really applicable to version 10.2, I may put a note at the bottom for other versions.

Here is the region,

With[{Rwg = 7.0, RLiner = 8.0, RRib = 8.5, RExt = 9.0, 
  Theta1 = 0.0227, Theta2 = 0.00916},
 region = ! (x^2 + y^2 > RLiner^2 && x^2 + y^2 < RRib^2 && 
       0 < y < x*Tan[Theta2]) && (y >= 0 && x^2 + y^2 >= Rwg^2 && 
      x^2 + y^2 <= RExt^2 && y <= x Tan[Theta1]);
 ]

We try to create an ElementMesh from this region,

Needs["NDSolve`FEM`"];
ToElementMesh[ImplicitRegion[
   region, {x, y}],
  MaxCellMeasure -> 0.0001]["Wireframe"]

During evaluation of In[22]:= ToElementMesh::fememins: The mesh elements are not valid. A set of valid mesh element incidents needs to be positive integers and be able to form a complete sequence starting from 1 to the largest incident present. There are missing incidents; a complete sequence cannot be formed. >>

During evaluation of In[22]:= ToElementMesh::femtemnm: A mesh could not be generated. >>

(* $Failed["Wireframe"] *)

We try MichaelE2's trick of simplifying the conditionals,

ToElementMesh[ImplicitRegion[
   Evaluate@Reduce@region, {x, y}],
  MaxCellMeasure -> 0.0001]["Wireframe"]

During evaluation of In[24]:= ToElementMesh::fememib: The input has or generated an intersecting boundary and cannot be processed. >>

During evaluation of In[24]:= ToElementMesh::femtemnm: A mesh could not be generated. >>

(* $Failed["Wireframe"] *)

Still no luck. But if we first create a MeshRegion and then convert it to an ElementMesh, then it does work, even though it still gives some errors.

dregion = DiscretizeRegion[
   ImplicitRegion[
    Evaluate@Reduce@region, {x, y}],
   Method -> "Continuation", PerformanceGoal -> "Quality"];
ToElementMesh[dregion, MaxCellMeasure -> 0.0001]["Wireframe"]

DiscretizeRegion::drcm: Method->Continuation not able to resolve all components of dimension less than the embedding dimension 2; these may be omitted from the result. >>

ToElementMesh::femimq: The element mesh has insufficient quality of -20.1137. A quality estimate below 0. may be caused by a wrong ordering of element incidents or self-intersecting elements. >>

Edit

As shown in this screenshot, I only get this error in version 10.2, not in 10.1 or 10.3