Finite element mesh not resolving features

Question

I am trying to make a finite element mesh in 2D. The features are not found by ToElementMesh. This is a shelf bracket with nails. I build the region using RegionDifference and RegionUnion and the result is clear.

   Needs["NDSolve`FEM`"]
L = 0.2; (* Bracket side length *)
d = 0.02;  (* End edge lenght *)
r = L - d;   (* radius of curved edge *)
r1 = d;    (* radius of hole *)
L2 = 0.004; (* nail thickness *)
L3 = 0.04; (* nail length *)
L4 = L/8; (* location of nail from bottom *)
Y = 10^3;  (* modulus of elasticity *)
ν = 33/100;  (* Poisson ratio *)

reg = RegionDifference[
   RegionUnion[
    Rectangle[{0, 0}, {L, L}],
    Rectangle[{-L3, L4 - L2/2}, {0, L4 + L2/2}],
    Rectangle[{-L3, (L - L4) - L2/2}, {0, (L - L4) + L2/2}]
    ],
   RegionUnion[
    Disk[{L, 0}, r],
    Disk[{2 d, L - 2 d}, r1]]
   ];
Show[Region[reg], PlotRange -> All, Frame -> True]

This region looks good. Now I try to mesh.

mesh = ToElementMesh[reg, "BoundaryMeshGenerator" -> "Continuation", 
   "MaxBoundaryCellMeasure" -> 0.001, "MaxCellMeasure" -> 0.0001];
Show[mesh["Wireframe"], PlotRange -> All]

The nails have been lost and the edges are very poorly represented. I have tried different options for ToElementMesh but have not hit on one that works. What is happening? Version 11.1 for Windows. Thanks

Edit

Helpful user21 gave a solution below which works for the resolution in his example. I have tried to increase the resolution and this is what happens.

mesh = ToElementMesh[reg, {{-0.04`, 0.2`}, {0.`, 0.2`}}, 
   "MaxBoundaryCellMeasure" -> 0.001, 
   "BoundaryMeshGenerator" -> {"RegionPlot", "SamplePoints" -> 200} ];
Show[mesh["Wireframe"], PlotRange -> All]

The curved boundaries and the lower part of the vertical straight boundary are badly formed. I would like a high resolution because when doing stress calculations gradients of the solution are needed. Any more thoughts?

Combining the suggestions from the answers below is successful.

Further edit

If you use this mesh you will need to use the workaround from this question. Add the option "ImproveBoundaryPosition" -> False to ToElementMesh. The mesh coordinates are not all contained within the solution interpolation function. So the final ToElementMesh expression was

mesh = ToElementMesh[reg, {{-L3, L}, {0, L}}, 
   "MaxBoundaryCellMeasure" -> 0.001, MaxCellMeasure -> 1.1*^-6, 
   AccuracyGoal -> 8, MeshQualityGoal -> 1, 
   "BoundaryMeshGenerator" -> {"RegionPlot", "SamplePoints" -> 200},
   "ImproveBoundaryPosition" -> False];

and it worked to give this mesh

Answer 1

I've had success using AccuracyGoal -> 8 for meshing regions with small dimensions. (Note: Use fractions for greater accuracy)

Needs["NDSolve`FEM`"]
L = 2/10;  (*Bracket side length*)
d = 2/100; (*End edge lenght*)
r = L - d; (*radius of curved edge*)
r1 = d;    (*radius of hole*)
L2 = 4/1000; (*nail thickness*)
L3 = 4/100;  (*nail length*)
L4 = L/8;    (*location of nail from bottom*)
Y = 10^3;    (*modulus of elasticity*)
ν = 33/100;  (*Poisson ratio*)

reg = 
 RegionDifference[
  RegionUnion[Rectangle[{0, 0}, {L, L}], 
  Rectangle[{-L3, L4 - L2/2}, {0, L4 + L2/2}], 
  Rectangle[{-L3, (L - L4) - L2/2}, {0, (L - L4) + L2/2}]], 
  RegionUnion[Disk[{L, 0}, r], Disk[{2 d, L - 2 d}, r1]]];
Show[Region[reg], PlotRange -> All, Frame -> True]

mesh = ToElementMesh[reg, {{-4/100, 2/10}, {0, 2/10}}, 
   "MaxBoundaryCellMeasure" -> 0.001, MaxCellMeasure -> 1*^-5, 
    AccuracyGoal -> 7, MeshQualityGoal -> 1,
   "BoundaryMeshGenerator" -> {"RegionPlot", "SamplePoints" -> 50}];
Show[mesh["Wireframe"], PlotRange -> All]

Answer 2

These should work:

Method 1

ToElementMesh[reg,{{-0.04`, 0.2`}, {0.`, 0.2`}},
"MaxBoundaryCellMeasure" -> 0.005, 
  "BoundaryMeshGenerator" -> {"RegionPlot", 
    "SamplePoints" -> 41}]["Wireframe"]

The problem is that "Coninuation" fails (which needs to be investigated) and the "RegionPlot" one does need the "SamplePoints" (think "PlotPloints"). Hope this helps.

If you need a finer mesh you can use:

ToElementMesh[reg, {{-0.04`, 0.2`}, {0.`, 0.2`}}, 
  "MaxBoundaryCellMeasure" -> 0.005, "MaxCellMeasure" -> 0.00001, 
  "BoundaryMeshGenerator" -> {"RegionPlot", 
    "SamplePoints" -> 41}]["Wireframe"]

Method 2

Here is a better way to do it:

mesh = ToElementMesh[reg, RegionBounds[reg], 
   "BoundaryMeshGenerator" -> {"BoundaryDiscretizeRegion"}, 
   "MaxBoundaryCellMeasure" -> 0.001, "MaxCellMeasure" -> 0.0001];
mesh["Wireframe"]

All of these methods are documented in ToBoundaryMesh. There you can find more info.

Method 3

Here is yet another way to do it, but that will requite version 11.2:

 L = 1./5; d = 1./50; r = L - d; r1 = d; L2 = 1./250; L3 = 1./25; L4 = 
 L/8;
reg = RegionDifference[
   RegionUnion[Rectangle[{0, 0}, {L, L}], 
    Rectangle[{-L3, L4 - L2/2}, {0, L4 + L2/2}], 
    Rectangle[{-L3, (L - L4) - L2/2}, {0, (L - L4) + L2/2}]], 
   RegionUnion[Disk[{L, 0}, r], Disk[{2 d, L - 2 d}, r1]]];

Now we make boundary discretizations of the components and construct the region from those:

bdr = BoundaryDiscretizeRegion /@ reg[[2]];
bmr = Fold[RegionDifference, bdr]

Next, we create a NumericalRegion and attach the boundary mesh to it:

Needs["NDSolve`FEM`"]
nr = ToNumericalRegion[reg, RegionBounds[reg]];
bmesh = ToBoundaryMesh[bmr];
SetNumericalRegionElementMesh[nr, bmesh];

When we now mesh that numerical region have both the symbolic and the boundary representation available and a good approximation can be generated:

mesh = ToElementMesh[nr];
mesh["Wireframe"]

To compare the area of the mesh with the area of the symbolic representation we use:

narea = Total[mesh["MeshElementMeasure"], 2];

and

L = 1/5; d = 1/50; r = L - d; r1 = d; L2 = 1/250; L3 = 1/25; L4 = 
 L/8;
reg2 = RegionDifference[
   RegionUnion[Rectangle[{0, 0}, {L, L}], 
    Rectangle[{-L3, L4 - L2/2}, {0, L4 + L2/2}], 
    Rectangle[{-L3, (L - L4) - L2/2}, {0, (L - L4) + L2/2}]], 
   RegionUnion[Disk[{L, 0}, r], Disk[{2 d, L - 2 d}, r1]]];
sarea = Integrate[1, {x, y} \[Element] reg2];

The result:

narea - sarea
-2.4810104379269227`*^-7

I think this is quite good.