problem with DelaunayMesh 3D coordinates

Question

I found DelaunayMesh works fine for 2D. For example,

coordinateList = Tuples[{Range[3], Range[3]}];
DelaunayMesh[coordinateList, PlotTheme -> "Lines"]

gives

Oddly enough, it does not work for simple regular point array like below

coordinateList = Tuples[{Range[3], Range[3], Range[3]}];
DelaunayMesh[coordinateList, PlotTheme -> "Lines"]

which just prints out the original data like

DelaunayMesh[{{1, 1, 1}, {1, 1, 2}, {1, 1, 3}, {1, 2, 1}, {1, 2, 
   2}, {1, 2, 3}, {1, 3, 1}, {1, 3, 2}, {1, 3, 3}, {2, 1, 1}, {2, 1, 
   2}, {2, 1, 3}, {2, 2, 1}, {2, 2, 2}, {2, 2, 3}, {2, 3, 1}, {2, 3, 
   2}, {2, 3, 3}, {3, 1, 1}, {3, 1, 2}, {3, 1, 3}, {3, 2, 1}, {3, 2, 
   2}, {3, 2, 3}, {3, 3, 1}, {3, 3, 2}, {3, 3, 3}}, 
 PlotTheme -> "Lines"]

I have to jiggle each point a little to make DelaunayMesh work. Define

ClearAll[jiggleCoordinateList];
jiggleCoordinateList[coordinateList_, eta_] := Module[{},
  RandomReal[eta*{-1, 1}, Dimensions@coordinateList] + coordinateList
  ]

then

coordinateList = Tuples[{Range[3], Range[3], Range[3]}];
DelaunayMesh[jiggleCoordinateList[coordinateList, 0.00001], 
 PlotTheme -> "Lines"]

gives

But why is that? I can not think of a reason why a regular point array in 3D can not be DelaunayMeshed.

Though a jiggled mesh is fine for display, for calculations, a jiggled mesh is not the same as the original mesh and will introduce error (even though it is small, it is not perfect).

Is there any workaround other than jiggling coordinates?

Answer 1

As a workaround you can use the finite element mesh generator:

Needs["NDSolve`FEM`"]
coordinateList = Tuples[{Range[3], Range[3], Range[3]}];
MeshRegion[ToElementMesh[coordinateList], PlotTheme -> "Lines"]