10
$\begingroup$

There is a code for solving a wave equation over an arbitrarily shaped region.

Ω = 
 RegionDifference[RegionUnion[Disk[], Rectangle[{0, 0}, {2, 2}]], 
 Disk[{1/4, 1/4}, 1/5]];

uifWave = NDSolveValue[{D[u[t, x, y], t, t] - Inactive[Laplacian][u[t, x, y], {x, y}] == 0, 
   u[0, x, y] == E^(-5*((x - 3/2)^2 + (y - 3/2)^2)), 
   Derivative[1, 0, 0][u][0, x, y] == 0, 
 DirichletCondition[u[t, x, y] == 0, 
     True]}, 
        u, {t, 0, 2 π}, {x, y} ∈ Ω] // 
    Quiet;

framesWEQ = 
 Table[Plot3D[
 uifWave[t, x, y], {x, y} ∈ uifWave["ElementMesh"], 
 PlotRange -> {-1, 1}, Boxed -> False, Axes -> False, 
 Mesh -> None], {t, 0, 2 π, 2 π/50}];




  Manipulate[framesWEQ[[i]], {{i, 16, "time"}, 1, Length[framesWEQ], 1},
  SaveDefinitions -> True]

enter image description here

How to solve this equation over the Black Sea shape region?

 mask = GeoGraphics[{GeoStyling[RGBColor["Aqua"]], 
 Polygon[Entity["Ocean", "BlackSea"]]}, GeoBackground -> None]

enter image description here

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  • 3
    $\begingroup$ Ω = DiscretizeRegion[ Entity["Ocean", "BlackSea"]["Polygon"] /. GeoPosition -> Identity] $\endgroup$ – Edmund Nov 30 '16 at 11:21
  • $\begingroup$ Somehow the answer seems trivially obvious. Could you say more about the problem you're facing? $\endgroup$ – Michael E2 Nov 30 '16 at 11:23
  • 1
    $\begingroup$ @Edmund That, like my Ω = Cases[mask, _Polygon, Infinity][[1]], doesn't work. I mean, it produces a region, but does not produce a solution to the equations. $\endgroup$ – corey979 Nov 30 '16 at 11:29
  • $\begingroup$ Might need {x, y} ∈ Ω in Plot3D instead of "ElementMesh". $\endgroup$ – Edmund Nov 30 '16 at 11:32
  • 1
    $\begingroup$ @corey979 I get a solution for uifWave that plots but I think the wave is very small over the area. May need to increase the amplitude in u. $\endgroup$ – Edmund Nov 30 '16 at 11:38
6
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If you use something like this:

Needs["NDSolve`FEM`"];
dg = DiscretizeGraphics[
   Entity["Ocean", "BlackSea"]["Polygon"] /. 
    GeoPosition -> Identity];
mesh = ToElementMesh[dg, MaxCellMeasure -> 0.005 (*, "MeshOrder"->1 *)]

You will be able to refine the mesh to your needs. This will then do the time integration and monitor it's progress:

Monitor[uifWave = 
  NDSolveValue[{D[u[t, x, y], t, t] - 
      Inactive[Laplacian][u[t, x, y], {x, y}] == 0, 
    u[0, x, y] == Exp[-10 ((x - 44)^2 + (y - 31)^2)], 
    Derivative[1, 0, 0][u][0, x, y] == 0, 
    DirichletCondition[u[t, x, y] == 0, True]}, 
   u, {t, 0, 2}, {x, y} \[Element] mesh, 
   EvaluationMonitor :> (monitor = Row[{"t = ", CForm[t]}])], monitor]

For completeness:

framesWEQ = 
  Table[Plot3D[uifWave[t, x, y], {x, y} \[Element] dg, 
    PlotRange -> {-1, 1}, Boxed -> False, Axes -> False, 
    Mesh -> None], {t, 0, 2, 2/50}];
Manipulate[framesWEQ[[i]], {{i, 16, "time"}, 1, Length[framesWEQ], 1},
  SaveDefinitions -> True]
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8
$\begingroup$

It works with Edmund's suggestion, only the initial state is changed to

u[0, x, y] == Exp[-10 ((x - 44)^2 + (y - 31)^2)].

Coordinates {44, 31} are inside the Black Sea.

For the sake of simplicity I've restricted the region to the western third of the sea.

Ω =  DiscretizeRegion[Entity["Ocean", "BlackSea"]["Polygon"] /. GeoPosition -> Identity,
     {{40, 50}, {27, 33}}]

enter image description here

Here is the solution for {t,0,4}:

enter image description here

EDIT: Another gif with 100 frames and PlotPoints -> 50

enter image description here

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  • 1
    $\begingroup$ Looks like the wavelength is too short for the granularity of the discretization ... $\endgroup$ – Szabolcs Nov 30 '16 at 20:20
  • $\begingroup$ @Szabolcs Yes, an accurate solution needs some tweaking :) $\endgroup$ – Gyebro Nov 30 '16 at 20:22
  • 5
    $\begingroup$ Just being picky; these are membrane waves. We need to solve for water surface waves which we don't know how to do yet see here. $\endgroup$ – Hugh Nov 30 '16 at 21:29

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