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

mathematical optimization - Minimizing using indices, error: Part::pkspec1: The expression cannot be used as a part specification

I want to use Minimize where the variables to minimize are indices pointing into an array. Here a MWE that hopefully shows what my problem is. vars = u@# & /@ Range[3]; cons = Flatten@ { Table[(u[j] != #) & /@ vars[[j + 1 ;; -1]], {j, 1, 3 - 1}], 1 vec1 = {1, 2, 3}; vec2 = {1, 2, 3}; Minimize[{Total@((vec1[[#]] - vec2[[u[#]]])^2 & /@ Range[1, 3]), cons}, vars, Integers] The error I get: Part::pkspec1: The expression u[1] cannot be used as a part specification. >> Answer Ok, it seems that one can get around Mathematica trying to evaluate vec2[[u[1]]] too early by using the function Indexed[vec2,u[1]] . The working MWE would then look like the following: vars = u@# & /@ Range[3]; cons = Flatten@{ Table[(u[j] != #) & /@ vars[[j + 1 ;; -1]], {j, 1, 3 - 1}], 1 vec1 = {1, 2, 3}; vec2 = {1, 2, 3}; NMinimize[ {Total@((vec1[[#]] - Indexed[vec2, u[#]])^2 & /@ R...
Post a Comment
Read more

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

What is and isn't a valid variable specification for Manipulate?

I have an expression whose terms have arguments (representing subscripts), like this: myExpr = A[0] + V[1,T] I would like to put it inside a Manipulate to see its value as I move around the parameters. (The goal is eventually to plot it wrt one of the variables inside.) However, Mathematica complains when I set V[1,T] as a manipulated variable: Manipulate[Evaluate[myExpr], {A[0], 0, 1}, {V[1, T], 0, 1}] (*Manipulate::vsform: Manipulate argument {V[1,T],0,1} does not have the correct form for a variable specification. >> *) As a workaround, if I get rid of the symbol T inside the argument, it works fine: Manipulate[ Evaluate[myExpr /. T -> 15], {A[0], 0, 1}, {V[1, 15], 0, 1}] Why this behavior? Can anyone point me to the documentation that says what counts as a valid variable? And is there a way to get Manpiulate to accept an expression with a symbolic argument as a variable? Investigations I've done so far: I tried using variableQ from this answer , but it says V[1...
Post a Comment
Read more