Implicit Definition of a Cube -vs- built in Cuboid function

Questions
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1

Hello,

I am trying to get an FEM solution to the steady state heat equation in a cube. I have been playing around with the different ways to define the cubic region and have found some inconsitencies and was wondering if someone could offer up a reason for why the solutions are not the same? My first method was to use the built in Cuboid function and NDSolveValue spits out a solution just fine but then I try to implicitly define a cube and NDSolveValue fails. It seems like the implict generation does not fully generate a cube while the Cuboid function does. Why is this true?

a = 5;
R = ImplicitRegion[
   0 <= x <= a && 0 <= y <= a && 0 <= z <= a, {x, y, z}];
Rdis = DiscretizeRegion[R]
RegionPlot3D[R, Axes -> True, AxesLabel -> {x, y, z}]


R2 = Cuboid[{0, 0, 0}, {a, a, a}];
R2dis = DiscretizeRegion[R2]
RegionPlot3D[R2, Axes -> True, AxesLabel -> {x, y, z}]
2

My experience with DiscretizeRegion is that it is a promising prototype which still needs a lot of work. Let’s hope we get a better version with Mathematica 11…

3

Can you post the code with the NDSolve that fails? When I use the two regions for integration I get identical results,

Integrate[Exp[-x^2 - y^2 - z^2], {x, y, z} \[Element] R] === 
 Integrate[Exp[-x^2 - y^2 - z^2], {x, y, z} \[Element] R2]
(* True *)
4

<< NDSolveFEM
a = 5;
R = ImplicitRegion[
0 <= x <= a && 0 <= y <= a && 0 <= z <= a, {x, y, z}];
Rdis = DiscretizeRegion[R]
RegionPlot3D[R, Axes → True, AxesLabel → {x, y, z}]

lap = Laplacian[T[x, y, z], {x, y, z}];
\[CapitalGamma] = {DirichletCondition[T[x, y, z] == 100, 
    0 <= x <= a && 0 <= y <= a && z == a], 
   DirichletCondition[
    T[x, y, z] == 
     0, (0 <= x <= a && 0 <= y <= a && z == 0) || (x == 0 && 
       0 <= y <= a && 0 <= z <= a) || (x == a && 0 <= y <= a && 
       0 <= z <= a) || (0 <= x <= a && y == 0 && 
       0 <= z <= a) || (0 <= x <= a && y == a && 0 <= z <= a)]};

sol = NDSolveValue[{lap == 0, \[CapitalGamma]}, 
   T, {x, y, z} \[Element] Rdis];
5

Michael,
There is definitely some kind of bug here in the DiscretizeRegion function, I made a post here trying to get to the bottom of it.

6

Thanks! The answer on that page does not fully make sense to me but it will be enough to figure out how to move forwards!