Specific numerical eigenfunctions of Helmholtz equation in 3D for ellipsoids












6












$begingroup$


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?










share|improve this question











$endgroup$

















    6












    $begingroup$


    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?










    share|improve this question











    $endgroup$















      6












      6








      6





      $begingroup$


      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?










      share|improve this question











      $endgroup$




      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






      share|improve this question















      share|improve this question













      share|improve this question




      share|improve this question








      edited 11 hours ago









      user64494

      3,57811022




      3,57811022










      asked Mar 26 at 22:24









      George GiannoulisGeorge Giannoulis

      574




      574






















          2 Answers
          2






          active

          oldest

          votes


















          9












          $begingroup$

          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}, Ellipsoid[{0, 0, 0}, {0.75, 0.6, 0.6}]], 4,
          Method -> {"Eigensystem" -> {"FEAST", "Interval" -> {425, 500}}}]

          {{427.961, 428.783, 430.026, 430.156},...}


          And here are the density plots:



          Table[DensityPlot[funs[[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 -> vals[[i]], ColorFunction -> Hue,
          PlotLegends -> Automatic, PlotRange -> All], {i, 1, 4}]


          enter image description here



          Slice density plots:



          Table[SliceDensityPlot3D[funs[[i]][x, y, z], 
          Element[ {x, y, z}, Ellipsoid[{0, 0, 0}, {0.75, 0.6, 0.6}]],
          PlotRange -> All, PlotLabel -> vals[[i]],
          PlotTheme -> "Minimal"], {i, Length[vals]}]


          enter image description here



          And density plots:



          Table[DensityPlot3D[funs[[i]][x, y, z], 
          Element[ {x, y, z}, Ellipsoid[{0, 0, 0}, {0.75, 0.6, 0.6}]],
          PlotRange -> All, PlotLabel -> vals[[i]],
          PlotTheme -> "Minimal"], {i, Length[vals]}]


          enter image description here






          share|improve this answer











          $endgroup$













          • $begingroup$
            Thank you for your answer but I need to clarify something technical here. Does NDEigensystems compute eigenmodes from start, ie 0 and then narrows its search to the desired interval (425, 500 HZ here) or does it start from 425 Hz and then stops at 500 Hz?
            $endgroup$
            – George Giannoulis
            yesterday










          • $begingroup$
            @GeorgeGiannoulis, I think the latter, but you could have a look at the FEAST algorithm.Thought that version is not the same as the one linked in Mathematica but that shlould not matter. NDEigensystem makes use if Eigensystem (like in your code) which then uses FEAST from a library.
            $endgroup$
            – user21
            17 hours ago










          • $begingroup$
            OK one last thing here. I can't seem to understand what the boubdary is in your code. Is it a cube,a sphere, an ellispoid? Something else?
            $endgroup$
            – George Giannoulis
            12 hours ago










          • $begingroup$
            @GeorgeGiannoulis, it's the ellipsoidI have updated the code.
            $endgroup$
            – user21
            12 hours ago










          • $begingroup$
            Great! I d like to add some density plots though for the eigenvalues. My code looks something like this:
            $endgroup$
            – George Giannoulis
            12 hours ago



















          6












          $begingroup$

          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.






          share|improve this answer











          $endgroup$














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            2 Answers
            2






            active

            oldest

            votes








            2 Answers
            2






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            9












            $begingroup$

            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}, Ellipsoid[{0, 0, 0}, {0.75, 0.6, 0.6}]], 4,
            Method -> {"Eigensystem" -> {"FEAST", "Interval" -> {425, 500}}}]

            {{427.961, 428.783, 430.026, 430.156},...}


            And here are the density plots:



            Table[DensityPlot[funs[[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 -> vals[[i]], ColorFunction -> Hue,
            PlotLegends -> Automatic, PlotRange -> All], {i, 1, 4}]


            enter image description here



            Slice density plots:



            Table[SliceDensityPlot3D[funs[[i]][x, y, z], 
            Element[ {x, y, z}, Ellipsoid[{0, 0, 0}, {0.75, 0.6, 0.6}]],
            PlotRange -> All, PlotLabel -> vals[[i]],
            PlotTheme -> "Minimal"], {i, Length[vals]}]


            enter image description here



            And density plots:



            Table[DensityPlot3D[funs[[i]][x, y, z], 
            Element[ {x, y, z}, Ellipsoid[{0, 0, 0}, {0.75, 0.6, 0.6}]],
            PlotRange -> All, PlotLabel -> vals[[i]],
            PlotTheme -> "Minimal"], {i, Length[vals]}]


            enter image description here






            share|improve this answer











            $endgroup$













            • $begingroup$
              Thank you for your answer but I need to clarify something technical here. Does NDEigensystems compute eigenmodes from start, ie 0 and then narrows its search to the desired interval (425, 500 HZ here) or does it start from 425 Hz and then stops at 500 Hz?
              $endgroup$
              – George Giannoulis
              yesterday










            • $begingroup$
              @GeorgeGiannoulis, I think the latter, but you could have a look at the FEAST algorithm.Thought that version is not the same as the one linked in Mathematica but that shlould not matter. NDEigensystem makes use if Eigensystem (like in your code) which then uses FEAST from a library.
              $endgroup$
              – user21
              17 hours ago










            • $begingroup$
              OK one last thing here. I can't seem to understand what the boubdary is in your code. Is it a cube,a sphere, an ellispoid? Something else?
              $endgroup$
              – George Giannoulis
              12 hours ago










            • $begingroup$
              @GeorgeGiannoulis, it's the ellipsoidI have updated the code.
              $endgroup$
              – user21
              12 hours ago










            • $begingroup$
              Great! I d like to add some density plots though for the eigenvalues. My code looks something like this:
              $endgroup$
              – George Giannoulis
              12 hours ago
















            9












            $begingroup$

            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}, Ellipsoid[{0, 0, 0}, {0.75, 0.6, 0.6}]], 4,
            Method -> {"Eigensystem" -> {"FEAST", "Interval" -> {425, 500}}}]

            {{427.961, 428.783, 430.026, 430.156},...}


            And here are the density plots:



            Table[DensityPlot[funs[[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 -> vals[[i]], ColorFunction -> Hue,
            PlotLegends -> Automatic, PlotRange -> All], {i, 1, 4}]


            enter image description here



            Slice density plots:



            Table[SliceDensityPlot3D[funs[[i]][x, y, z], 
            Element[ {x, y, z}, Ellipsoid[{0, 0, 0}, {0.75, 0.6, 0.6}]],
            PlotRange -> All, PlotLabel -> vals[[i]],
            PlotTheme -> "Minimal"], {i, Length[vals]}]


            enter image description here



            And density plots:



            Table[DensityPlot3D[funs[[i]][x, y, z], 
            Element[ {x, y, z}, Ellipsoid[{0, 0, 0}, {0.75, 0.6, 0.6}]],
            PlotRange -> All, PlotLabel -> vals[[i]],
            PlotTheme -> "Minimal"], {i, Length[vals]}]


            enter image description here






            share|improve this answer











            $endgroup$













            • $begingroup$
              Thank you for your answer but I need to clarify something technical here. Does NDEigensystems compute eigenmodes from start, ie 0 and then narrows its search to the desired interval (425, 500 HZ here) or does it start from 425 Hz and then stops at 500 Hz?
              $endgroup$
              – George Giannoulis
              yesterday










            • $begingroup$
              @GeorgeGiannoulis, I think the latter, but you could have a look at the FEAST algorithm.Thought that version is not the same as the one linked in Mathematica but that shlould not matter. NDEigensystem makes use if Eigensystem (like in your code) which then uses FEAST from a library.
              $endgroup$
              – user21
              17 hours ago










            • $begingroup$
              OK one last thing here. I can't seem to understand what the boubdary is in your code. Is it a cube,a sphere, an ellispoid? Something else?
              $endgroup$
              – George Giannoulis
              12 hours ago










            • $begingroup$
              @GeorgeGiannoulis, it's the ellipsoidI have updated the code.
              $endgroup$
              – user21
              12 hours ago










            • $begingroup$
              Great! I d like to add some density plots though for the eigenvalues. My code looks something like this:
              $endgroup$
              – George Giannoulis
              12 hours ago














            9












            9








            9





            $begingroup$

            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}, Ellipsoid[{0, 0, 0}, {0.75, 0.6, 0.6}]], 4,
            Method -> {"Eigensystem" -> {"FEAST", "Interval" -> {425, 500}}}]

            {{427.961, 428.783, 430.026, 430.156},...}


            And here are the density plots:



            Table[DensityPlot[funs[[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 -> vals[[i]], ColorFunction -> Hue,
            PlotLegends -> Automatic, PlotRange -> All], {i, 1, 4}]


            enter image description here



            Slice density plots:



            Table[SliceDensityPlot3D[funs[[i]][x, y, z], 
            Element[ {x, y, z}, Ellipsoid[{0, 0, 0}, {0.75, 0.6, 0.6}]],
            PlotRange -> All, PlotLabel -> vals[[i]],
            PlotTheme -> "Minimal"], {i, Length[vals]}]


            enter image description here



            And density plots:



            Table[DensityPlot3D[funs[[i]][x, y, z], 
            Element[ {x, y, z}, Ellipsoid[{0, 0, 0}, {0.75, 0.6, 0.6}]],
            PlotRange -> All, PlotLabel -> vals[[i]],
            PlotTheme -> "Minimal"], {i, Length[vals]}]


            enter image description here






            share|improve this answer











            $endgroup$



            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}, Ellipsoid[{0, 0, 0}, {0.75, 0.6, 0.6}]], 4,
            Method -> {"Eigensystem" -> {"FEAST", "Interval" -> {425, 500}}}]

            {{427.961, 428.783, 430.026, 430.156},...}


            And here are the density plots:



            Table[DensityPlot[funs[[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 -> vals[[i]], ColorFunction -> Hue,
            PlotLegends -> Automatic, PlotRange -> All], {i, 1, 4}]


            enter image description here



            Slice density plots:



            Table[SliceDensityPlot3D[funs[[i]][x, y, z], 
            Element[ {x, y, z}, Ellipsoid[{0, 0, 0}, {0.75, 0.6, 0.6}]],
            PlotRange -> All, PlotLabel -> vals[[i]],
            PlotTheme -> "Minimal"], {i, Length[vals]}]


            enter image description here



            And density plots:



            Table[DensityPlot3D[funs[[i]][x, y, z], 
            Element[ {x, y, z}, Ellipsoid[{0, 0, 0}, {0.75, 0.6, 0.6}]],
            PlotRange -> All, PlotLabel -> vals[[i]],
            PlotTheme -> "Minimal"], {i, Length[vals]}]


            enter image description here







            share|improve this answer














            share|improve this answer



            share|improve this answer








            edited 10 hours ago

























            answered 2 days ago









            user21user21

            20.1k45385




            20.1k45385












            • $begingroup$
              Thank you for your answer but I need to clarify something technical here. Does NDEigensystems compute eigenmodes from start, ie 0 and then narrows its search to the desired interval (425, 500 HZ here) or does it start from 425 Hz and then stops at 500 Hz?
              $endgroup$
              – George Giannoulis
              yesterday










            • $begingroup$
              @GeorgeGiannoulis, I think the latter, but you could have a look at the FEAST algorithm.Thought that version is not the same as the one linked in Mathematica but that shlould not matter. NDEigensystem makes use if Eigensystem (like in your code) which then uses FEAST from a library.
              $endgroup$
              – user21
              17 hours ago










            • $begingroup$
              OK one last thing here. I can't seem to understand what the boubdary is in your code. Is it a cube,a sphere, an ellispoid? Something else?
              $endgroup$
              – George Giannoulis
              12 hours ago










            • $begingroup$
              @GeorgeGiannoulis, it's the ellipsoidI have updated the code.
              $endgroup$
              – user21
              12 hours ago










            • $begingroup$
              Great! I d like to add some density plots though for the eigenvalues. My code looks something like this:
              $endgroup$
              – George Giannoulis
              12 hours ago


















            • $begingroup$
              Thank you for your answer but I need to clarify something technical here. Does NDEigensystems compute eigenmodes from start, ie 0 and then narrows its search to the desired interval (425, 500 HZ here) or does it start from 425 Hz and then stops at 500 Hz?
              $endgroup$
              – George Giannoulis
              yesterday










            • $begingroup$
              @GeorgeGiannoulis, I think the latter, but you could have a look at the FEAST algorithm.Thought that version is not the same as the one linked in Mathematica but that shlould not matter. NDEigensystem makes use if Eigensystem (like in your code) which then uses FEAST from a library.
              $endgroup$
              – user21
              17 hours ago










            • $begingroup$
              OK one last thing here. I can't seem to understand what the boubdary is in your code. Is it a cube,a sphere, an ellispoid? Something else?
              $endgroup$
              – George Giannoulis
              12 hours ago










            • $begingroup$
              @GeorgeGiannoulis, it's the ellipsoidI have updated the code.
              $endgroup$
              – user21
              12 hours ago










            • $begingroup$
              Great! I d like to add some density plots though for the eigenvalues. My code looks something like this:
              $endgroup$
              – George Giannoulis
              12 hours ago
















            $begingroup$
            Thank you for your answer but I need to clarify something technical here. Does NDEigensystems compute eigenmodes from start, ie 0 and then narrows its search to the desired interval (425, 500 HZ here) or does it start from 425 Hz and then stops at 500 Hz?
            $endgroup$
            – George Giannoulis
            yesterday




            $begingroup$
            Thank you for your answer but I need to clarify something technical here. Does NDEigensystems compute eigenmodes from start, ie 0 and then narrows its search to the desired interval (425, 500 HZ here) or does it start from 425 Hz and then stops at 500 Hz?
            $endgroup$
            – George Giannoulis
            yesterday












            $begingroup$
            @GeorgeGiannoulis, I think the latter, but you could have a look at the FEAST algorithm.Thought that version is not the same as the one linked in Mathematica but that shlould not matter. NDEigensystem makes use if Eigensystem (like in your code) which then uses FEAST from a library.
            $endgroup$
            – user21
            17 hours ago




            $begingroup$
            @GeorgeGiannoulis, I think the latter, but you could have a look at the FEAST algorithm.Thought that version is not the same as the one linked in Mathematica but that shlould not matter. NDEigensystem makes use if Eigensystem (like in your code) which then uses FEAST from a library.
            $endgroup$
            – user21
            17 hours ago












            $begingroup$
            OK one last thing here. I can't seem to understand what the boubdary is in your code. Is it a cube,a sphere, an ellispoid? Something else?
            $endgroup$
            – George Giannoulis
            12 hours ago




            $begingroup$
            OK one last thing here. I can't seem to understand what the boubdary is in your code. Is it a cube,a sphere, an ellispoid? Something else?
            $endgroup$
            – George Giannoulis
            12 hours ago












            $begingroup$
            @GeorgeGiannoulis, it's the ellipsoidI have updated the code.
            $endgroup$
            – user21
            12 hours ago




            $begingroup$
            @GeorgeGiannoulis, it's the ellipsoidI have updated the code.
            $endgroup$
            – user21
            12 hours ago












            $begingroup$
            Great! I d like to add some density plots though for the eigenvalues. My code looks something like this:
            $endgroup$
            – George Giannoulis
            12 hours ago




            $begingroup$
            Great! I d like to add some density plots though for the eigenvalues. My code looks something like this:
            $endgroup$
            – George Giannoulis
            12 hours ago











            6












            $begingroup$

            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.






            share|improve this answer











            $endgroup$


















              6












              $begingroup$

              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.






              share|improve this answer











              $endgroup$
















                6












                6








                6





                $begingroup$

                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.






                share|improve this answer











                $endgroup$



                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.







                share|improve this answer














                share|improve this answer



                share|improve this answer








                edited 2 days ago

























                answered Mar 26 at 22:32









                Henrik SchumacherHenrik Schumacher

                58.4k581161




                58.4k581161






























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