Skip to main content

finite element method - Swinging Beam example from Help. Not Understanding/Not Working


To help me understand finite element analysis with Mathematica I have been reading the Finite Element Method User Guide in Help and am stuck on the example in Coupled PDEs. Here there is a static beam problem and a swinging beam problem.


I have taken the code from the static beam problem and this works nicely


 Needs["NDSolve`FEM`"]

\[CapitalOmega] = ImplicitRegion[True, {x, y}];
mesh = ToElementMesh[\[CapitalOmega], {{0, 5}, {0, 1}}, 

   "MaxCellMeasure" -> 0.1];
vd = NDSolve`VariableData[{"DependentVariable", 
     "Space"} -> {{u, v}, {x, y}}];
sd = NDSolve`SolutionData["Space" -> ToNumericalRegion[mesh]];

bcDu0 = DirichletCondition[u[x, y] == 0, x == 0];
bcDv0 = DirichletCondition[v[x, y] == 0, x == 0];
bcNL = NeumannValue[-1, x == 5];
initBCs = 
  InitializeBoundaryConditions[vd, sd, {{bcDu0}, {bcDv0, bcNL}}];


diffusionCoefficients = 
  "DiffusionCoefficients" -> {{{{-(Y/(1 - \[Nu]^2)), 
        0}, {0, -((Y (1 - \[Nu]))/(2 (1 - \[Nu]^2)))}}, {{0, -((
         Y \[Nu])/(1 - \[Nu]^2))}, {-((Y (1 - \[Nu]))/(
         2 (1 - \[Nu]^2))), 
        0}}}, {{{0, -((Y (1 - \[Nu]))/(2 (1 - \[Nu]^2)))}, {-((
         Y \[Nu])/(1 - \[Nu]^2)), 
        0}}, {{-((Y (1 - \[Nu]))/(2 (1 - \[Nu]^2))), 
        0}, {0, -(Y/(1 - \[Nu]^2))}}}} /. {Y -> 10^3, \[Nu] -> 33/100};

initCoeffs = 
  InitializePDECoefficients[vd, sd, {diffusionCoefficients}];


methodData = InitializePDEMethodData[vd, sd];
discretePDE = DiscretizePDE[initCoeffs, methodData, sd];
split = {#[[1]] + 1 ;; #[[2]], #[[2]] + 1 ;; #[[3]]} &[
   methodData["IncidentOffsets"]];
discreteBCs = DiscretizeBoundaryConditions[initBCs, methodData, sd];
{load, stiffness, damping, mass} = discretePDE["SystemMatrices"];


DeployBoundaryConditions[{load, stiffness}, discreteBCs];

solution = LinearSolve[stiffness, load];
uif = ElementMeshInterpolation[{mesh}, solution[[split[[1]]]]];
vif = ElementMeshInterpolation[{mesh}, solution[[split[[2]]]]];
dmesh = ElementMeshDeformation[mesh, {uif, vif}];

I can plot results as follows using


 mesh["Wireframe"]

ContourPlot[uif[x, y], {x, y} \[Element] mesh, 
 ColorFunction -> "Temperature", AspectRatio -> Automatic]
ContourPlot[vif[x, y], {x, y} \[Element] mesh, 
 ColorFunction -> "Temperature", AspectRatio -> Automatic]
Show[{
  mesh["Wireframe"],
  dmesh["Wireframe"[
    "ElementMeshDirective" -> Directive[EdgeForm[Red], FaceForm[]]]]}]

to give



Mathematica graphics


All this is good but it goes wrong for me when I move onto the swinging beam which modifies the above code. I can't get this to run even if I copy directly from Help. Here is the relevant set-up code from Help


    massCoefficients = "MassCoefficients" -> {{1, 0}, {0, 1}};
initCoeffs = 
  InitializePDECoefficients[vd, 
   sd, {diffusionCoefficients, massCoefficients}];
discretePDE = DiscretizePDE[initCoeffs, methodData, sd];
{load, stiffness, damping, mass} = discretePDE["SystemMatrices"];
rayleighDamping = 0.1*mass + 0.04*stiffness;
DeployBoundaryConditions[{load, stiffness, rayleighDamping, mass}, 

  discreteBCs];
dof = methodData["DegreesOfFreedom"];
init = dinit = ConstantArray[0, {dof, 1}];
sparsity = 
  ArrayFlatten[{{mass["PatternArray"], 
     mass["PatternArray"]}, {rayleighDamping["PatternArray"], 
     rayleighDamping["PatternArray"]}}];

It is the next bit of the solution code that gets stuck. The time monitor stops around 0.00158 so I have tried running the code to 0.0015 rather than 45 as in Help but it still gets stuck.


 Dynamic["time: " <> ToString[CForm[currentTime]]]

AbsoluteTiming[
 tif = NDSolveValue[{
    mass.u''[ t] + rayleighDamping.u'[ t] + stiffness.u[ t] == load
    , u[ 0] == init, u'[ 0] == dinit}, u, {t, 0, 0.0015}
   , Method -> {"EquationSimplification" -> "Residual"}
   , Jacobian -> {Automatic, Sparse -> sparsity}
   , EvaluationMonitor :> (currentTime = t;)
   ]]

Are there any suggestions for how to make this work?




Answer



It seems that my version of Mathematica was corrupted. I did a Shift+Control while starting Mathematica (see here for good instructions) and the code now works perfectly. Thanks to user21 and MarcoB for confirming that the code works. Wolfram Support also confirmed that the code works and suggested a restart.


Out of interest I compared a benchmark from before the Shift+Control to after the restart. The benchmark is found by running


    Needs["Benchmarking`"]
    BenchmarkReport[]

Before restart: Mathematica graphics


After restart: Mathematica graphics


As you can see I have gone from anti-penultimate to top-of-the-heap!


Thanks for the help



Comments

Post a Comment

Popular posts from this blog

functions - Get leading series expansion term?

Given a function f[x] , I would like to have a function leadingSeries that returns just the leading term in the series around x=0 . For example: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x)] x and leadingSeries[(1/x + 2 + (1 - 1/x^3)/4)/(4 + x)] -(1/(16 x^3)) Is there such a function in Mathematica? Or maybe one can implement it efficiently? EDIT I finally went with the following implementation, based on Carl Woll 's answer: lds[ex_,x_]:=( (ex/.x->(x+O[x]^2))/.SeriesData[U_,Z_,L_List,Mi_,Ma_,De_]:>SeriesData[U,Z,{L[[1]]},Mi,Mi+1,De]//Quiet//Normal) The advantage is, that this one also properly works with functions whose leading term is a constant: lds[Exp[x],x] 1 Answer Update 1 Updated to eliminate SeriesData and to not return additional terms Perhaps you could use: leadingSeries[expr_, x_] := Normal[expr /. x->(x+O[x]^2) /. a_List :> Take[a, 1]] Then for your examples: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x), x] leadingSeries[Exp[x], x] leadingSeries[(1/x + 2 + (1 - 1/x...
Post a Comment
Read more

How to thread a list

I have data in format data = {{a1, a2}, {b1, b2}, {c1, c2}, {d1, d2}} Tableform: I want to thread it to : tdata = {{{a1, b1}, {a2, b2}}, {{a1, c1}, {a2, c2}}, {{a1, d1}, {a2, d2}}} Tableform: And I would like to do better then pseudofunction[n_] := Transpose[{data2[[1]], data2[[n]]}]; SetAttributes[pseudofunction, Listable]; Range[2, 4] // pseudofunction Here is my benchmark data, where data3 is normal sample of real data. data3 = Drop[ExcelWorkBook[[Column1 ;; Column4]], None, 1]; data2 = {a #, b #, c #, d #} & /@ Range[1, 10^5]; data = RandomReal[{0, 1}, {10^6, 4}]; Here is my benchmark code kptnw[list_] := Transpose[{Table[First@#, {Length@# - 1}], Rest@#}, {3, 1, 2}] &@list kptnw2[list_] := Transpose[{ConstantArray[First@#, Length@# - 1], Rest@#}, {3, 1, 2}] &@list OleksandrR[list_] := Flatten[Outer[List, List@First[list], Rest[list], 1], {{2}, {1, 4}}] paradox2[list_] := Partition[Riffle[list[[1]], #], 2] & /@ Drop[list, 1] RM[list_] := FoldList[Transpose[{First@li...
Post a Comment
Read more

front end - keyboard shortcut to invoke Insert new matrix

I frequently need to type in some matrices, and the menu command Insert > Table/Matrix > New... allows matrices with lines drawn between columns and rows, which is very helpful. I would like to make a keyboard shortcut for it, but cannot find the relevant frontend token command (4209405) for it. Since the FullForm[] and InputForm[] of matrices with lines drawn between rows and columns is the same as those without lines, it's hard to do this via 3rd party system-wide text expanders (e.g. autohotkey or atext on mac). How does one assign a keyboard shortcut for the menu item Insert > Table/Matrix > New... , preferably using only mathematica? Thanks! Answer In the MenuSetup.tr (for linux located in the $InstallationDirectory/SystemFiles/FrontEnd/TextResources/X/ directory), I changed the line MenuItem["&New...", "CreateGridBoxDialog"] to read MenuItem["&New...", "CreateGridBoxDialog", MenuKey["m", Modifiers-...
Post a Comment
Read more