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I have a PDE describing a bending beam which I want to solve numerically.

a=1;

b=0.1;

Approx[x_, ϵ_] := Approx[x, ϵ] = ϵ/(Pi*(x^2 + ϵ^2)) 

Initial = {p[x, 0] == 0.5, q[x, 0] == 0.5};

CisTrans = 
  {Derivative[0, 1][p][x, t] == 10*(1 - p[x, t]) - p[x, t], 
   Derivative[0, 1][q][x, t] == -q[x, t]};

Deflection = 
  {Derivative[0, 2][w][x, t] + a*Derivative[0, 1][w][x, t] + 
   Derivative[2, 0][u][x, t] == -b*(p[x, t] - q[x, t]) *
   Approx[x - 1, 10^(-1000)]};

s = 
   NDSolveValue[
    {Initial, CisTrans, Deflection,  Derivative[2, 0][w][x, t] == u[x, t], 
     u[1, t] == 0, u[x, 0] == 1, w[x, 0] == 0, w[0, t] == 0, 
     WhenEvent[t == 0.5, Print[t]]}, 
    w, {x, 0, 1}, {t, 0, 1}]

This NDSolveValue solves the PDE, but ignores my WhenEvent, so 0.5 is not printed, while in other PDE's the WhenEvent is not ignored. Does anyone know how to solve this?

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  • $\begingroup$ aside to your actual question, your system produces vastly different results if you adjust the limit of the time domain ( see what you get if you just solve over t,0,1/2 ) . This tells me the default solution method is a poor choice, although I didn't have any luck doing better. $\endgroup$
    – george2079
    Commented May 2, 2018 at 16:25
  • $\begingroup$ another aside I cant make any sense of what Approx is supposed to do. It is essentially zero everywhere and then blows up exactly on the x=1 boundary. (off hand I think it never gets evaluated exactly at the boundary ) $\endgroup$
    – george2079
    Commented May 2, 2018 at 16:32
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    $\begingroup$ IIFC, you have to have explicit time integration (e.g., with the method of lines) for WhenEvent to work, and then it only applies to the time-stepping. (While you may call your variable t, it is being treated as one of the spatial variables in NDSolve above.) $\endgroup$
    – Michael E2
    Commented May 2, 2018 at 16:40
  • $\begingroup$ @george2079 Approx seems to be an approximate Dirac delta. $\endgroup$
    – AccidentalFourierTransform
    Commented May 2, 2018 at 19:32
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    $\begingroup$ @AccidentalFourierTransform good call, so it is. NDSolve will not integrate it properly however. $\endgroup$
    – george2079
    Commented May 2, 2018 at 20:22

1 Answer 1

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This is not a solution to your problem, but a possible source of the problem. Sometimes, when the method to obtain the InterpolatingFunction type solution is Unstructured, it seems that WhenEvent[] call doesn't trigger any response. This is most likely the case for complicated PDE or system of PDE's.

Couple of examples from the docs(ref/NDSolve):

One where WhenEvent[] doesn't work:

\[CapitalOmega] = 
  RegionDifference[RegionUnion[Disk[], Rectangle[{0, -1}, {2, 1}]], 
   Disk[{2, 0}]];
ufun = NDSolveValue[{-Laplacian[u[x, y], {x, y}] == 1., 
       PeriodicBoundaryCondition[u[x, y], (x - 2)^2 + y^2 == 1, 
         Function[x, x - {2, 0}]], DirichletCondition[u[x, y] == 0, 
         0 < x < 2 - 10^(-6) && (y <= -1 || y >= 1)], 
   WhenEvent[x == 0, Print[x]]}, u, 
     Element[{x, y}, \[CapitalOmega]]]

However, in this one it works just fine(even if the method utilised was Unstructured):

\[CapitalOmega] = 
  RegionDifference[
   RegionDifference[Rectangle[{0, 0}, {2, 1}], 
    Rectangle[{9/10, 0}, {11/10, 4/10}]], 
   Rectangle[{9/10, 6/10}, {11/10, 1}]];
sol = First[
  NDSolve[{D[u[t, x, y], {t, 2}] - Laplacian[u[t, x, y], {x, y}] == 0,
     DirichletCondition[u[t, x, y] == 0, True], 
    u[0, x, y] == 2*Exp[-125 ((x - 0.25)^2 + (y - 0.5)^2)], 
    Derivative[1, 0, 0][u][0, x, y] == 0, 
    WhenEvent[t == 0.1, Print@t]}, 
   u, {t, 0, 2}, {x, y} \[Element] \[CapitalOmega]]]
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