# Full domain of Dirichlet boundary not displaying in 3DPlot

I have code to solve the 2D wave equation on a given region with an initial condition that is sinusoidal in one part of the region and 0 elsewhere. Here is an example on a rectangular region that works just fine, with plot shown at t=0:

weq = Laplacian[u[x, y, t], {x, y}] == D[u[x, y, t], {t, 2}];

bc = {u[x, 2 Pi, t] == 0, u[x, 0, t] == 0, u[0, y, t] == 0,
u[Pi, y, t] == 0};

ic = {u[x, y, 0] ==
If[0 <= x <= Pi && 0 <= y <= Pi, Sin[m x] Sin[n y], 0],
Derivative[0, 0, 1][u][x, y, 0] == 0};

nsol2 = Quiet[NDSolve[{weq, bc, ic} /. {m -> 2, n -> 2},
u, {x, 0, Pi}, {y, 0, 2 Pi}, {t, 0, 5}]];

ListAnimate[
Table[Plot3D[u[x, y, t] /. nsol2, {x, 0, \[Pi]}, {y, 0, 2 \[Pi]},
AxesLabel -> Automatic,
PlotRange -> {{0, \[Pi]}, {0, 2 \[Pi]}, {-1, 1}},
BoxRatios -> {1, 2, 0.5}], {t, 0, 5, 0.5}]] I want to keep the same initial condition ic that is sinusoidal in the [{0,0},{Pi,Pi}] rectangle and zero elsewhere, but with a curved guide as a Dirichlet boundary instead of the rectangular boundary used above.

bc2 = DirichletCondition[u[x, y, t] == 0, True]
tube = Region[
RegionUnion[Rectangle[{0, 0}, {Pi, Pi}],
Rectangle[{2 Pi, -2 Pi}, {3 Pi, -Pi}],
Region[Annulus[{Pi, -Pi}, {Pi, 2 Pi}, {0, Pi/2}]]]]

benttubesol =
Quiet[NDSolve[{weq, bc2, ic} /. {m -> 2, n -> 2},
u, {x, y} \[Element] tube, {t, 0, 15}]]

ListAnimate[
Table[Plot3D[u[x, y, t] /. benttubesol, {x, y} \[Element] tube,
AxesLabel -> Automatic,
PlotRange -> {{0, 3 \[Pi]}, {-2 Pi, \[Pi]}, {-1, 1}},
BoxRatios -> {1, 1, 0.5}], {t, 0, 15, 1}]] The [{0,0},{Pi,Pi}] rectangle is mysteriously absent from the 3DPlot, although the part of the tube that does appear evolves correctly in time (the above is after some time has passed). Why could this be? I think it's related to the if function in ic, but I don't know what I could change to fix it.

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• It seems like you are missing a bit of code in the first part of your answer that describes your nsolve2 ? Are you wanting to plot both solutions together? – Dunlop May 23 at 5:26
• @Dunlop Thank you; I've added it. I don't want to plot them both together, I just find it strange that the [{0,0},{Pi,Pi}] rectangle appears in that first case and not the second, where I also want it to be. – user2188518 May 23 at 5:39
• I see what you mean. You can also plot the solution for the second one if you restrict the plot range to the [{0,0},{Pi,Pi}] rectangle only. So it is solving there, it seems like there is something strange going on with the 3D plot – Dunlop May 23 at 5:59
• Try this: ListAnimate[ Table[Plot3D[ u[x, y, t] /. benttubesol, {x, -1, 3 \[Pi]}, {y, -2 \[Pi], \[Pi]}, AxesLabel -> Automatic, PlotRange -> {{-1, 3 \[Pi]}, {-2 Pi, \[Pi]}, {-1, 1}}, BoxRatios -> {1, 1, 1}], {t, 0, 15, 1}]] – Dunlop May 23 at 6:04

Here's something to show the issue in a more minimal example

tube=
Region[
RegionUnion[
Rectangle[{0, 0}, {Pi, Pi}],
Rectangle[{2 Pi, -2 Pi}, {3 Pi, -Pi}], Region[Annulus[{Pi, -Pi}, {Pi, 2 Pi}, {0, Pi/2}]]]
];
Show[
Plot3D[x^2 + y^2,
{x, y} \[Element] Rectangle[{0, 0}, {Pi, Pi}],
PlotStyle -> Red
],
Plot3D[x^2 + y^2,
{x, y} \[Element] tube,
PlotStyle -> Blue
]
] If you instead use DiscretizeRegion on tube it seems to work

Show[
Plot3D[x^2 + y^2,
{x, y} \[Element] Rectangle[{0, 0}, {Pi, Pi}],
PlotStyle -> Red
],
Plot3D[x^2 + y^2,
{x, y} \[Element] DiscretizeRegion@tube,
PlotStyle -> Blue
]
] Suggests that it's a bug in Plot3D. I'd email Wolfram about it.

• Thank you so much; I'll reach out to them! – user2188518 May 23 at 12:08