# Finite element heat conduction with coupled 1D and 3D calculation

I would like to perform a 3d FEM heat transfer calculation, which should have also included fluid dynamic simulation. But I want to simplify it into a 3d FEM without fluid dynamic simulation, but with a boundary condition which also requires to solve a 1d differential equation. A simplified example of "the simplified model" can be seen as follows.

Let's say I have an element and it looks like the following graphic region =
RegionDifference[Cuboid[{0, 0, 0}, {0.006536/2, 0.05, 0.0047}],
Cuboid[{0, 0, 0.0007}, {0.0015, 0.05, 0.0017}]];
DiscretizeRegion[region]


The equations is as follows

eq = With[{lambda = 1, c = 4200, rho = 1000,
w = 0.02}, {lambda Laplacian[u[x, y, z], {x, y, z}] ==
NeumannValue[0,
x == 0 || x == 0.006536/2 || z == 0 || z == 0.05] +
NeumannValue[8 (u[x, y, z] - 20), y == 0] +
NeumannValue[0.8 (u[x, y, z] - 20), y == 0.0047] +
NeumannValue[
1000 (u[x, y, z] - t[z]), (y == 0.0007 &&
x < 0.0015) || (y == 1.7/1000 &&
x < 0.0015) || (x == 0.0015 && (0.0007 < y < 0.0017))],
-c rho 0.003 0.001 w D[t[z], z] == 1000 (t[z] - u[x, y, z]),
t == 30}]


Basically, the outer surfaces have normal robin boundary conditions. The inner surface has also robin boundary conditions, but they require solving t[z]. This t[z] is only dependent on z.

Solving this equation with NDSolveValue

sol = NDSolveValue[eq, {u, t}, {x, y, z} \[Element] region, Method -> {"PDEDiscretization" -> {"FiniteElement", "MeshOptions" -> {"MaxCellMeasure" -> 10^(-6)}}}]


gives quite a lot of errors

Transpose::nmtx: The first two levels of {NDSolvexs$20949,t} cannot be transposed. Part::partw: Part 2 of Transpose[{NDSolvexs$20949,t}] does not exist.
Transpose::nmtx: The first two levels of {NDSolvexs$20949,Function[{x,y,z},30]} cannot be transposed. Part::partw: Part 2 of Transpose[{NDSolvexs$20949,Function[{x,y,z},30]}] does not exist.
Set::partw: Part 2 of Transpose[{NDSolvexs\$20949,t}] does not exist.
Rule::argr: Rule called with 1 argument; 2 arguments are expected.
Function::fpct: Too many parameters in {x,y,z} to be filled from Function[{x,y,z},30][z].
NDSolveValue::overdet: There are fewer dependent variables, {u[x,y,z]}, than equations, so the system is overdetermined.


Does anyone know how to solve this problem? A thousand thanks.

• With the code NDSolveProcessEquations[... , DependentVariables-> {u,t}] , you have an unique and clear error message : "The function t[z] does not have the same number of arguments as independent variables (3)" ( NDSolveProcessEquations is the first step of the process of resolution of the pde used by NDSolve ) – andre314 Mar 22 '17 at 20:58
• @andre so it means that NDSolve is not able to solve the couples 1d and 3D equations? – 407Peezy Mar 23 '17 at 0:07
• Probably, it is not able (now) – andre314 Mar 23 '17 at 0:31

The idea is to transform t[z] to t[x,y,z]. Note that I also changed the predicate of the y==0.0047 to z==0.0047.

region = RegionDifference[
Cuboid[{0, 0, 0}, {0.006536/2, 0.05, 0.0047}],
Cuboid[{0, 0, 0.0007}, {0.0015, 0.05, 0.0017}]];
eq = With[{lambda = 1, c = 4200, rho = 1000,
w = 0.02}, {lambda Laplacian[u[x, y, z], {x, y, z}] ==
NeumannValue[0,
x == 0 || x == 0.006536/2 || z == 0 || z == 0.05] +
NeumannValue[8 (u[x, y, z] - 20), y == 0] +
NeumannValue[0.8 (u[x, y, z] - 20), z == 0.0047] +
NeumannValue[
1000 (u[x, y, z] - t[x, y, z]), (y == 0.0007 &&
x < 0.0015) || (y == 1.7/1000 &&
x < 0.0015) || (x ==
0.0015 && (0.0007 < y < 0.0017))], -c rho 0.003 0.001 w D[
t[x, y, z], z] == 1000 (t[x, y, z] - u[x, y, z]),
t[x, y, 0] == 30}];
sol = NDSolveValue[eq, {u, t}, {x, y, z} \[Element] region,
Method -> {"PDEDiscretization" -> {"FiniteElement",
"MeshOptions" -> {"MaxCellMeasure" -> 10^(-6)}}}]


This gives a message:

NDSolveValue::femibcnd: No DirichletCondition or Robin-type NeumannValue was specified for {u}; the result is not unique up to a constant.
`

but returns two interpolating functions. See if this helps you.