# Finding eigenvalues for Laplacian operator for 3D shape with Neumann boundary conditions

I've just begun to use the Mathematica so my question may seem to be naive. To get a solution for my problem I looked at the example provided in help.

{vals, funs} =
NDEigensystem[{-Laplacian[u[x, y], {x, y}],
DirichletCondition[u[x, y] == 0, True]},
u[x, y], {x, y} ∈ Disk[], 6];


And I tried to apply it for 3D shape like this.

Ω =
Pyramid[{{aa, aa, 0}, {aa, -aa, 0}, {-aa, -aa, 0}, {-aa, aa, 0}, {0,
0, hh}}];
Γ =
ImplicitRegion[((Abs[x] >= Abs[y] &&
z == hh*(1 - Abs[x]/aa)) || (Abs[y] >= Abs[x] &&
z == hh*(1 - Abs[y]/aa)) || z == 0), {{x, -aa, aa}, {y, -aa,
aa}, {z, 0, hh}}];
aa := 1;
hh := 1;
NDEigensystem[{-Laplacian[f[x, y, z], {x, y, z}] ==
NeumannValue[0, Element[{x, y, z}, Γ]]}, f[x, y, z],
Element[{x, y, z}, Ω], 6]


But unfortunately it returns the same thing as I have at input. Can anybody explain to me what I'm doing wrong?
I edited this post to make things more clear. I'm looking for natural frequencies of pyramid bounded area. So, my end-goal is to find eigenvalues and eigenfunctions for the Helmholtz equation with Neumann boundary condition u'[x,y,z]=0 on the surface of a pyramid. This task seemed to me to be difficult to start with so I decided to find first eigenvalues for the Laplacian operator. Here is the Helmholtz equation:

 Laplacian[u(x,y,z),{x,y,z}] + k^2u(x,y,z) =0

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As shown in the docs for NDEigensystem, the proper way to specify a homogeneous boundary condition with NeumannValue is to add it to the linear operator (only homogeneous boundary conditions are supported at this time, V10.4.1).

Block[{aa = 1, hh = 1},
Ω =
Pyramid[{{aa, aa, 0}, {aa, -aa, 0}, {-aa, -aa, 0}, {-aa, aa,
0}, {0, 0, hh}}];
eigs = NDEigensystem[{-Laplacian[f[x, y, z], {x, y, z}] +
NeumannValue[0, True]}, f[x, y, z],
Element[{x, y, z}, Ω], 6]

];

Grid[Partition[
SliceContourPlot3D[#2,
"CenterPlanes", {x, y, z} ∈ Ω,
PlotRange -> Full, PlotLabel -> #1, ColorFunction -> "Rainbow",
Contours -> 19] &, eigs],
3]]


• Thank you Michael, your remark on how to specify correctly Neumann boundary conditions is very helpful. Fortunately, in my case the condition is homogeneous. It did work for my Mathematica 10.3 as well. Are you sure that NeumannValue(0, True) is the same thing as NeumannValue[0, Element[{x, y, z}, Γ]] ?
– Alex
Jul 19, 2016 at 2:17
• @Alex Isn't Gamma the boundary of the pyramid? If not, could you describe what it is? (The True in NeumannValue[..., True] defines the condition to apply to the whole boundary of the pyramid.) Jul 19, 2016 at 3:48
• Yes, Gamma is supposed to be the boundary of the pyramid. I was just surprised that Mathematica can calculate the derivative on a surface without explicitly given surface.
– Alex
Jul 19, 2016 at 11:53
• @Alex There's some discussion about how NeumannValue[val, pred] works here and a little more here. The way FEM works seems to add val to the components of the appropriate system matrices corresponding to boundary elements for which pred is true. (M certainly computes the boundary elements when discretizing a region; try emesh["BoundaryElements"] for any ElementMesh.) Jul 19, 2016 at 14:43