Specific Numerical Eigenfunctions of Helmholtz equation in 3D for ellipsoids
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I am trying to compute the eigenfunctions of an oblate spheroid (a=75 cm and b=60 cm) using Mathematica's FEM package and Chris' answer from here. Specifically, I am looking for eigenfrequencies around 433, 893, 913 and 2400 MGHz. Is there any way I could narrow my search besides getting all eigenfrequencies initially and then looking for the desired outcome which is impractical?
Here is my code for the first 4 eigenmodes:
Needs["NDSolve`FEM`"];
helmholzSolve3D[g_, numEigenToCompute_Integer,
opts : OptionsPattern] :=
Module[{u, x, y, z, t, pde, dirichletCondition, mesh, boundaryMesh,
nr, state, femdata, initBCs, methodData, initCoeffs, vd, sd,
discretePDE, discreteBCs, load, stiffness, damping, pos, nDiri,
numEigen, res, eigenValues, eigenVectors,
evIF},
(*Discretize the region*)
If[Head[g] === ImplicitRegion || Head[g] === ParametricRegion,
mesh = ToElementMesh[DiscretizeRegion[g, opts], opts],
mesh = ToElementMesh[DiscretizeGraphics[g, opts], opts]];
boundaryMesh = ToBoundaryMesh[mesh];
(*Set up the PDE and boundary condition*)
pde = D[u[t, x, y, z], t] - Laplacian[u[t, x, y, z], {x, y, z}] +
u[t, x, y, z] == 0;
dirichletCondition = DirichletCondition[u[t, x, y, z] == 0, True];
(*Pre-process the equations to obtain the FiniteElementData in
StateData*)nr = ToNumericalRegion[mesh];
{state} =
NDSolve`ProcessEquations[{pde, dirichletCondition,
u[0, x, y, z] == 0}, u, {t, 0, 1}, Element[{x, y, z}, nr]];
femdata = state["FiniteElementData"];
initBCs = femdata["BoundaryConditionData"];
methodData = femdata["FEMMethodData"];
initCoeffs = femdata["PDECoefficientData"];
(*Set up the solution*)vd = methodData["VariableData"];
sd = NDSolve`SolutionData[{"Space" -> nr, "Time" -> 0.}];
(*Discretize the PDE and boundary conditions*)
discretePDE = DiscretizePDE[initCoeffs, methodData, sd];
discreteBCs = DiscretizeBoundaryConditions[initBCs, methodData, sd];
(*Extract the relevant matrices and deploy the boundary conditions*)
load = discretePDE["LoadVector"];
stiffness = discretePDE["StiffnessMatrix"];
damping = discretePDE["DampingMatrix"];
DeployBoundaryConditions[{load, stiffness, damping}, discreteBCs];
(*Set the number of eigenvalues ignoring the Dirichlet positions*)
pos = discreteBCs["DirichletMatrix"]["NonzeroPositions"][[All, 2]];
nDiri = Length[pos];
numEigen = numEigenToCompute + nDiri;
(*Solve the eigensystem*)
res = Eigensystem[{stiffness, damping}, -numEigen];
res = Reverse /@ res;
eigenValues = res[[1, nDiri + 1 ;; Abs[numEigen]]];
eigenVectors = res[[2, nDiri + 1 ;; Abs[numEigen]]];
evIF = ElementMeshInterpolation[{mesh}, #] & /@ eigenVectors;
(*Return the relevant information*)
{eigenValues, evIF, mesh}]
{ev, if, mesh} =
helmholzSolve3D[Ellipsoid[{0, 0, 0}, {0.75, 0.6, 0.6}], 4,
MaxCellMeasure -> 0.025]
Table[
DensityPlot[
if[[i]][x, y, 0.1], {x, -1, 1}, {y, -1, 1},
RegionFunction -> Function[{x, y}, x^2/0.75^2 + y^2/0.6^2 < 1],
PlotLabel -> ev[i] ,
ColorFunction -> Hue,
PlotLegends -> Automatic
],
{i, 1, 4}
]
Any suggestions?
differential-equations numerics finite-element-method
$endgroup$
add a comment |
$begingroup$
I am trying to compute the eigenfunctions of an oblate spheroid (a=75 cm and b=60 cm) using Mathematica's FEM package and Chris' answer from here. Specifically, I am looking for eigenfrequencies around 433, 893, 913 and 2400 MGHz. Is there any way I could narrow my search besides getting all eigenfrequencies initially and then looking for the desired outcome which is impractical?
Here is my code for the first 4 eigenmodes:
Needs["NDSolve`FEM`"];
helmholzSolve3D[g_, numEigenToCompute_Integer,
opts : OptionsPattern] :=
Module[{u, x, y, z, t, pde, dirichletCondition, mesh, boundaryMesh,
nr, state, femdata, initBCs, methodData, initCoeffs, vd, sd,
discretePDE, discreteBCs, load, stiffness, damping, pos, nDiri,
numEigen, res, eigenValues, eigenVectors,
evIF},
(*Discretize the region*)
If[Head[g] === ImplicitRegion || Head[g] === ParametricRegion,
mesh = ToElementMesh[DiscretizeRegion[g, opts], opts],
mesh = ToElementMesh[DiscretizeGraphics[g, opts], opts]];
boundaryMesh = ToBoundaryMesh[mesh];
(*Set up the PDE and boundary condition*)
pde = D[u[t, x, y, z], t] - Laplacian[u[t, x, y, z], {x, y, z}] +
u[t, x, y, z] == 0;
dirichletCondition = DirichletCondition[u[t, x, y, z] == 0, True];
(*Pre-process the equations to obtain the FiniteElementData in
StateData*)nr = ToNumericalRegion[mesh];
{state} =
NDSolve`ProcessEquations[{pde, dirichletCondition,
u[0, x, y, z] == 0}, u, {t, 0, 1}, Element[{x, y, z}, nr]];
femdata = state["FiniteElementData"];
initBCs = femdata["BoundaryConditionData"];
methodData = femdata["FEMMethodData"];
initCoeffs = femdata["PDECoefficientData"];
(*Set up the solution*)vd = methodData["VariableData"];
sd = NDSolve`SolutionData[{"Space" -> nr, "Time" -> 0.}];
(*Discretize the PDE and boundary conditions*)
discretePDE = DiscretizePDE[initCoeffs, methodData, sd];
discreteBCs = DiscretizeBoundaryConditions[initBCs, methodData, sd];
(*Extract the relevant matrices and deploy the boundary conditions*)
load = discretePDE["LoadVector"];
stiffness = discretePDE["StiffnessMatrix"];
damping = discretePDE["DampingMatrix"];
DeployBoundaryConditions[{load, stiffness, damping}, discreteBCs];
(*Set the number of eigenvalues ignoring the Dirichlet positions*)
pos = discreteBCs["DirichletMatrix"]["NonzeroPositions"][[All, 2]];
nDiri = Length[pos];
numEigen = numEigenToCompute + nDiri;
(*Solve the eigensystem*)
res = Eigensystem[{stiffness, damping}, -numEigen];
res = Reverse /@ res;
eigenValues = res[[1, nDiri + 1 ;; Abs[numEigen]]];
eigenVectors = res[[2, nDiri + 1 ;; Abs[numEigen]]];
evIF = ElementMeshInterpolation[{mesh}, #] & /@ eigenVectors;
(*Return the relevant information*)
{eigenValues, evIF, mesh}]
{ev, if, mesh} =
helmholzSolve3D[Ellipsoid[{0, 0, 0}, {0.75, 0.6, 0.6}], 4,
MaxCellMeasure -> 0.025]
Table[
DensityPlot[
if[[i]][x, y, 0.1], {x, -1, 1}, {y, -1, 1},
RegionFunction -> Function[{x, y}, x^2/0.75^2 + y^2/0.6^2 < 1],
PlotLabel -> ev[i] ,
ColorFunction -> Hue,
PlotLegends -> Automatic
],
{i, 1, 4}
]
Any suggestions?
differential-equations numerics finite-element-method
$endgroup$
add a comment |
$begingroup$
I am trying to compute the eigenfunctions of an oblate spheroid (a=75 cm and b=60 cm) using Mathematica's FEM package and Chris' answer from here. Specifically, I am looking for eigenfrequencies around 433, 893, 913 and 2400 MGHz. Is there any way I could narrow my search besides getting all eigenfrequencies initially and then looking for the desired outcome which is impractical?
Here is my code for the first 4 eigenmodes:
Needs["NDSolve`FEM`"];
helmholzSolve3D[g_, numEigenToCompute_Integer,
opts : OptionsPattern] :=
Module[{u, x, y, z, t, pde, dirichletCondition, mesh, boundaryMesh,
nr, state, femdata, initBCs, methodData, initCoeffs, vd, sd,
discretePDE, discreteBCs, load, stiffness, damping, pos, nDiri,
numEigen, res, eigenValues, eigenVectors,
evIF},
(*Discretize the region*)
If[Head[g] === ImplicitRegion || Head[g] === ParametricRegion,
mesh = ToElementMesh[DiscretizeRegion[g, opts], opts],
mesh = ToElementMesh[DiscretizeGraphics[g, opts], opts]];
boundaryMesh = ToBoundaryMesh[mesh];
(*Set up the PDE and boundary condition*)
pde = D[u[t, x, y, z], t] - Laplacian[u[t, x, y, z], {x, y, z}] +
u[t, x, y, z] == 0;
dirichletCondition = DirichletCondition[u[t, x, y, z] == 0, True];
(*Pre-process the equations to obtain the FiniteElementData in
StateData*)nr = ToNumericalRegion[mesh];
{state} =
NDSolve`ProcessEquations[{pde, dirichletCondition,
u[0, x, y, z] == 0}, u, {t, 0, 1}, Element[{x, y, z}, nr]];
femdata = state["FiniteElementData"];
initBCs = femdata["BoundaryConditionData"];
methodData = femdata["FEMMethodData"];
initCoeffs = femdata["PDECoefficientData"];
(*Set up the solution*)vd = methodData["VariableData"];
sd = NDSolve`SolutionData[{"Space" -> nr, "Time" -> 0.}];
(*Discretize the PDE and boundary conditions*)
discretePDE = DiscretizePDE[initCoeffs, methodData, sd];
discreteBCs = DiscretizeBoundaryConditions[initBCs, methodData, sd];
(*Extract the relevant matrices and deploy the boundary conditions*)
load = discretePDE["LoadVector"];
stiffness = discretePDE["StiffnessMatrix"];
damping = discretePDE["DampingMatrix"];
DeployBoundaryConditions[{load, stiffness, damping}, discreteBCs];
(*Set the number of eigenvalues ignoring the Dirichlet positions*)
pos = discreteBCs["DirichletMatrix"]["NonzeroPositions"][[All, 2]];
nDiri = Length[pos];
numEigen = numEigenToCompute + nDiri;
(*Solve the eigensystem*)
res = Eigensystem[{stiffness, damping}, -numEigen];
res = Reverse /@ res;
eigenValues = res[[1, nDiri + 1 ;; Abs[numEigen]]];
eigenVectors = res[[2, nDiri + 1 ;; Abs[numEigen]]];
evIF = ElementMeshInterpolation[{mesh}, #] & /@ eigenVectors;
(*Return the relevant information*)
{eigenValues, evIF, mesh}]
{ev, if, mesh} =
helmholzSolve3D[Ellipsoid[{0, 0, 0}, {0.75, 0.6, 0.6}], 4,
MaxCellMeasure -> 0.025]
Table[
DensityPlot[
if[[i]][x, y, 0.1], {x, -1, 1}, {y, -1, 1},
RegionFunction -> Function[{x, y}, x^2/0.75^2 + y^2/0.6^2 < 1],
PlotLabel -> ev[i] ,
ColorFunction -> Hue,
PlotLegends -> Automatic
],
{i, 1, 4}
]
Any suggestions?
differential-equations numerics finite-element-method
$endgroup$
I am trying to compute the eigenfunctions of an oblate spheroid (a=75 cm and b=60 cm) using Mathematica's FEM package and Chris' answer from here. Specifically, I am looking for eigenfrequencies around 433, 893, 913 and 2400 MGHz. Is there any way I could narrow my search besides getting all eigenfrequencies initially and then looking for the desired outcome which is impractical?
Here is my code for the first 4 eigenmodes:
Needs["NDSolve`FEM`"];
helmholzSolve3D[g_, numEigenToCompute_Integer,
opts : OptionsPattern] :=
Module[{u, x, y, z, t, pde, dirichletCondition, mesh, boundaryMesh,
nr, state, femdata, initBCs, methodData, initCoeffs, vd, sd,
discretePDE, discreteBCs, load, stiffness, damping, pos, nDiri,
numEigen, res, eigenValues, eigenVectors,
evIF},
(*Discretize the region*)
If[Head[g] === ImplicitRegion || Head[g] === ParametricRegion,
mesh = ToElementMesh[DiscretizeRegion[g, opts], opts],
mesh = ToElementMesh[DiscretizeGraphics[g, opts], opts]];
boundaryMesh = ToBoundaryMesh[mesh];
(*Set up the PDE and boundary condition*)
pde = D[u[t, x, y, z], t] - Laplacian[u[t, x, y, z], {x, y, z}] +
u[t, x, y, z] == 0;
dirichletCondition = DirichletCondition[u[t, x, y, z] == 0, True];
(*Pre-process the equations to obtain the FiniteElementData in
StateData*)nr = ToNumericalRegion[mesh];
{state} =
NDSolve`ProcessEquations[{pde, dirichletCondition,
u[0, x, y, z] == 0}, u, {t, 0, 1}, Element[{x, y, z}, nr]];
femdata = state["FiniteElementData"];
initBCs = femdata["BoundaryConditionData"];
methodData = femdata["FEMMethodData"];
initCoeffs = femdata["PDECoefficientData"];
(*Set up the solution*)vd = methodData["VariableData"];
sd = NDSolve`SolutionData[{"Space" -> nr, "Time" -> 0.}];
(*Discretize the PDE and boundary conditions*)
discretePDE = DiscretizePDE[initCoeffs, methodData, sd];
discreteBCs = DiscretizeBoundaryConditions[initBCs, methodData, sd];
(*Extract the relevant matrices and deploy the boundary conditions*)
load = discretePDE["LoadVector"];
stiffness = discretePDE["StiffnessMatrix"];
damping = discretePDE["DampingMatrix"];
DeployBoundaryConditions[{load, stiffness, damping}, discreteBCs];
(*Set the number of eigenvalues ignoring the Dirichlet positions*)
pos = discreteBCs["DirichletMatrix"]["NonzeroPositions"][[All, 2]];
nDiri = Length[pos];
numEigen = numEigenToCompute + nDiri;
(*Solve the eigensystem*)
res = Eigensystem[{stiffness, damping}, -numEigen];
res = Reverse /@ res;
eigenValues = res[[1, nDiri + 1 ;; Abs[numEigen]]];
eigenVectors = res[[2, nDiri + 1 ;; Abs[numEigen]]];
evIF = ElementMeshInterpolation[{mesh}, #] & /@ eigenVectors;
(*Return the relevant information*)
{eigenValues, evIF, mesh}]
{ev, if, mesh} =
helmholzSolve3D[Ellipsoid[{0, 0, 0}, {0.75, 0.6, 0.6}], 4,
MaxCellMeasure -> 0.025]
Table[
DensityPlot[
if[[i]][x, y, 0.1], {x, -1, 1}, {y, -1, 1},
RegionFunction -> Function[{x, y}, x^2/0.75^2 + y^2/0.6^2 < 1],
PlotLabel -> ev[i] ,
ColorFunction -> Hue,
PlotLegends -> Automatic
],
{i, 1, 4}
]
Any suggestions?
differential-equations numerics finite-element-method
differential-equations numerics finite-element-method
asked 14 hours ago
George GiannoulisGeorge Giannoulis
573
573
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
You could use something like this:
{vals, funs} =
NDEigensystem[{-Laplacian[u[x, y, z], {x, y, z}] + u[x, y, z],
DirichletCondition[u[x, y, z] == 0, True]}, u,
Element[{x, y, z}, mesh], 4,
Method -> {"Eigensystem" -> {"FEAST", "Interval" -> {425, 500}}}]
{{427.961, 428.783, 430.026, 430.156},...}
$endgroup$
add a comment |
$begingroup$
You may try Eigensystem
with
Method -> {"FEAST", "Interval" -> {a, b}}
to search eigenvalue pairs within an interval. See the documentation of Eigensystem
, Section "Methods", Subsection "FEAST" for more details.
$endgroup$
add a comment |
Your Answer
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
You could use something like this:
{vals, funs} =
NDEigensystem[{-Laplacian[u[x, y, z], {x, y, z}] + u[x, y, z],
DirichletCondition[u[x, y, z] == 0, True]}, u,
Element[{x, y, z}, mesh], 4,
Method -> {"Eigensystem" -> {"FEAST", "Interval" -> {425, 500}}}]
{{427.961, 428.783, 430.026, 430.156},...}
$endgroup$
add a comment |
$begingroup$
You could use something like this:
{vals, funs} =
NDEigensystem[{-Laplacian[u[x, y, z], {x, y, z}] + u[x, y, z],
DirichletCondition[u[x, y, z] == 0, True]}, u,
Element[{x, y, z}, mesh], 4,
Method -> {"Eigensystem" -> {"FEAST", "Interval" -> {425, 500}}}]
{{427.961, 428.783, 430.026, 430.156},...}
$endgroup$
add a comment |
$begingroup$
You could use something like this:
{vals, funs} =
NDEigensystem[{-Laplacian[u[x, y, z], {x, y, z}] + u[x, y, z],
DirichletCondition[u[x, y, z] == 0, True]}, u,
Element[{x, y, z}, mesh], 4,
Method -> {"Eigensystem" -> {"FEAST", "Interval" -> {425, 500}}}]
{{427.961, 428.783, 430.026, 430.156},...}
$endgroup$
You could use something like this:
{vals, funs} =
NDEigensystem[{-Laplacian[u[x, y, z], {x, y, z}] + u[x, y, z],
DirichletCondition[u[x, y, z] == 0, True]}, u,
Element[{x, y, z}, mesh], 4,
Method -> {"Eigensystem" -> {"FEAST", "Interval" -> {425, 500}}}]
{{427.961, 428.783, 430.026, 430.156},...}
answered 6 hours ago
user21user21
20k45285
20k45285
add a comment |
add a comment |
$begingroup$
You may try Eigensystem
with
Method -> {"FEAST", "Interval" -> {a, b}}
to search eigenvalue pairs within an interval. See the documentation of Eigensystem
, Section "Methods", Subsection "FEAST" for more details.
$endgroup$
add a comment |
$begingroup$
You may try Eigensystem
with
Method -> {"FEAST", "Interval" -> {a, b}}
to search eigenvalue pairs within an interval. See the documentation of Eigensystem
, Section "Methods", Subsection "FEAST" for more details.
$endgroup$
add a comment |
$begingroup$
You may try Eigensystem
with
Method -> {"FEAST", "Interval" -> {a, b}}
to search eigenvalue pairs within an interval. See the documentation of Eigensystem
, Section "Methods", Subsection "FEAST" for more details.
$endgroup$
You may try Eigensystem
with
Method -> {"FEAST", "Interval" -> {a, b}}
to search eigenvalue pairs within an interval. See the documentation of Eigensystem
, Section "Methods", Subsection "FEAST" for more details.
edited 5 hours ago
answered 14 hours ago
Henrik SchumacherHenrik Schumacher
58.1k579160
58.1k579160
add a comment |
add a comment |
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