Orsdxy

Dear wolfram community,

I hope my problem is clear and easy to solve.

I have already solved the following heat equation over a domain:

pde=1/r D[r*D[T[t, r, z], {r, 1}], {r, 1}] + D[D[T[t, r, z], {z, 1}], {z, 1}] -D[T[t, r, z], {t}]





Δz = 50*10^(-4);(*[m]*)

R = 5*10^(-2);(*[m]*)

τ = 3.5*10^(-2)(*[s]*)

Tw = 200;(*[K]*)

Ω = ImplicitRegion[0 <= r <= R && -2*Δz <= z <= 0, {r, z}]



bc = {

    DirichletCondition[T[t, r, z] == Tw, 

    z == -2*Δz],

    DirichletCondition[T[t, r, z] == Tw, r == R]

    }; 

(*Note that the unspecified boundaries are set to Neuman zero by  default*)



ic = {T[0, r, z] == 300 (*[K]*)};





sol = NDSolve[{pde == 0, bc, T, {t, 0, τ},{r, z} ∈ Ω]

I optained an Interpolationfunction. Now I want to use this function as an initial value (T[0,r,z]) to solve a new heat equation over an enlarged domain (domnew). The new domain is essentially the old domain but enlarged by an amount of z:

 omeganew = ImplicitRegion[0 <= r <= R && -4*Δz <= z <= 0, {r, z}];

For the new solution I want the Temperature distribution from the previous solution to be valid at the lower part of omeganew and a different function fc[r,z] (lets assume for the purpose of simplicity that fc[r,z]= constant) to be valid within the region that is newly added. So I need a shifted version of the previous solution with respect to the z-coordinate.

In a nutshel, I would like to extract the information (Temperaturevalues at an abitrary time t) obtained by NDsolve (e.g. sol) to construct a Unitstep function (is there a better alternative?) that I would then use as the initial condition for the new domain (omeganew).

Can anyone please help me?

Here is a way to do it. Let' set up the model:

Δz = 50*10^(-4);(*[m]*)R = 5*10^(-2);(*[m]*)τ = 

 3.5*10^(-2);(*[s]*)

pde = 1/r D[r*D[T[t, r, z], {r, 1}], {r, 1}] + D[D[T[t, r, z], {z, 1}], {z, 1}] - D[T[t, r, z], {t}];

Ω = ImplicitRegion[0 <= r <= R && -2*Δz <= z <= 0, {r, z}];

bc = {DirichletCondition[T[t, r, z] == Tw, z == -2*Δz], DirichletCondition[T[t, r, z] == Tw, r == R]};

If you now call NDSolveValue you will get a solution (looks like it's zero but that is a different issue)

sol = NDSolveValue[{pde == 0, bc, T[0, r, z] == 0}, T, {t, 0, τ}, {r, z} ∈ Ω];

If you evaluate the inteprolating function out side of the region you will get a warning and an Indetermiante as an answer.

sol[0, -1, 3]

Indeterminate

To change that you can use:

sol = NDSolveValue[{pde == 0, bc, T[0, r, z] == 0}, 

   T, 

   {t, 0, τ}, 

   {r, z} ∈ Ω, 

   {"ExtrapolationHandler" -> {5 &, "WarningMessage" -> False}}

   ];

Now you will get the extrapolation value specified (5) and no warning:

sol[0, -1, 3]

5

With this you can then call NDSolveValue on a different domain with a different initial value like so:

sol2 = NDSolveValue[{pde == 0, bc, T[0, r, z] == sol[τ, r, z]}, 

   T, 

   {t, 0, τ}, 

   {r, z} ∈ ImplicitRegion[ 0 <= r <= R && -10*Δz <= z <= 0, {r, z}]

   ];