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Questions about NDSolve with Dirichlet/Neumann bcs and direct formulations

Posted 5 months ago
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I try to solve this Simple symmetric continuum mechanical problem but I'm getting different solutions A Simple symmetric continuum mechanical Problem

PDE in Inactive operator form

Operator-Form from documentation example: https://reference.wolfram.com/language/FEMDocumentation/tutorial/FiniteElementBestPractice.html

op = {Inactive[
       Div][({{0, 0, -((Y \[Nu])/((1 - 2 \[Nu]) (1 + \[Nu])))}, {0, 0,
           0}, {-(Y/(2 (1 + \[Nu]))), 0, 0}}.Inactive[Grad][
         w[x, y, z], {x, y, z}]), {x, y, z}] + 
     Inactive[
       Div][({{0, -((Y \[Nu])/((1 - 2 \[Nu]) (1 + \[Nu]))), 
          0}, {-(Y/(2 (1 + \[Nu]))), 0, 0}, {0, 0, 0}}.Inactive[Grad][
         v[x, y, z], {x, y, z}]), {x, y, z}] + 
     Inactive[
       Div][({{-((Y (1 - \[Nu]))/((1 - 2 \[Nu]) (1 + \[Nu]))), 0, 
          0}, {0, -(Y/(2 (1 + \[Nu]))), 0}, {0, 
          0, -(Y/(2 (1 + \[Nu])))}}.Inactive[Grad][
         u[x, y, z], {x, y, z}]), {x, y, z}], 
    Inactive[
       Div][({{0, 0, 0}, {0, 
          0, -((Y \[Nu])/((1 - 2 \[Nu]) (1 + \[Nu])))}, {0, -(Y/(
           2 (1 + \[Nu]))), 0}}.Inactive[Grad][
         w[x, y, z], {x, y, z}]), {x, y, z}] + 
     Inactive[
       Div][({{0, -(Y/(2 (1 + \[Nu]))), 
          0}, {-((Y \[Nu])/((1 - 2 \[Nu]) (1 + \[Nu]))), 0, 0}, {0, 0,
           0}}.Inactive[Grad][u[x, y, z], {x, y, z}]), {x, y, z}] + 
     Inactive[
       Div][({{-(Y/(2 (1 + \[Nu]))), 0, 
          0}, {0, -((Y (1 - \[Nu]))/((1 - 2 \[Nu]) (1 + \[Nu]))), 
          0}, {0, 0, -(Y/(2 (1 + \[Nu])))}}.Inactive[Grad][
         v[x, y, z], {x, y, z}]), {x, y, z}], 
    Inactive[
       Div][({{0, 0, 0}, {0, 
          0, -(Y/(2 (1 + \[Nu])))}, {0, -((
           Y \[Nu])/((1 - 2 \[Nu]) (1 + \[Nu]))), 0}}.Inactive[Grad][
         v[x, y, z], {x, y, z}]), {x, y, z}] + 
     Inactive[
       Div][({{0, 0, -(Y/(2 (1 + \[Nu])))}, {0, 0, 
          0}, {-((Y \[Nu])/((1 - 2 \[Nu]) (1 + \[Nu]))), 0, 
          0}}.Inactive[Grad][u[x, y, z], {x, y, z}]), {x, y, z}] + 
     Inactive[
       Div][({{-(Y/(2 (1 + \[Nu]))), 0, 0}, {0, -(Y/(2 (1 + \[Nu]))), 
          0}, {0, 0, -((
           Y (1 - \[Nu]))/((1 - 2 \[Nu]) (1 + \[Nu])))}}.Inactive[
          Grad][w[x, y, z], {x, y, z}]), {x, y, z}]} /. {Y -> 
     210000, \[Nu] -> 3/10};

Domain (a simple block, c.f. Figure):

x0 = 0; x1 = 50; 
y0 = 0; y1 = 50;
z0 = 0; z1 = 70;

Symmetric boundary conditions and given stress value:

\[CapitalGamma]Xsym = DirichletCondition[{u[x, y, z] == 0}, x == x1];
\[CapitalGamma]Ysym = DirichletCondition[{v[x, y, z] == 0}, y == y0];
\[CapitalGamma]Zsym = DirichletCondition[{w[x, y, z] == 0}, z == z1];
(* Stress component in z-direction Subscript[\[Sigma], zz], Neumann \
boundary condition*)
stress = 350;

{uifWL, vifWL, wifWL} = 
 NDSolveValue[{op == {0, 
     0, -NeumannValue[stress, 
       z == z0]}, \[CapitalGamma]Xsym, \[CapitalGamma]Ysym, \
\[CapitalGamma]Zsym}, {u, v, w}, {x, x0, x1}, {y, y0, y1}, {z, z0, 
   z1}]

Show the result

Define stress tensor depending on the displacements u,v,w:

\[Nu] = 3/10; g = 210000/(2 (1 + \[Nu])); (* material constants*)
e = Subscript[\[Epsilon], xx] + Subscript[\[Epsilon], yy] + 
   Subscript[\[Epsilon], zz];
Subscript[\[Sigma], xx] = 
 2 g (Subscript[\[Epsilon], 
    xx] + \[Nu]/(1 - 2 \[Nu]) e); Subscript[\[Tau], xy] = 
 g Subscript[\[Gamma], xy];
Subscript[\[Sigma], yy] = 
 2 g (Subscript[\[Epsilon], 
    yy] + \[Nu]/(1 - 2 \[Nu]) e); Subscript[\[Tau], xz] = 
 g Subscript[\[Gamma], xz];
Subscript[\[Sigma], zz] = 
 2 g (Subscript[\[Epsilon], 
    zz] + \[Nu]/(1 - 2 \[Nu]) e); Subscript[\[Tau], yz] = 
 g Subscript[\[Gamma], yz];

stressTensor = ( {
    {Subscript[\[Sigma], xx], Subscript[\[Tau], xy], Subscript[\[Tau],
      xz]},
    {Subscript[\[Tau], xy], Subscript[\[Sigma], yy], Subscript[\[Tau],
      yz]},
    {Subscript[\[Tau], xz], Subscript[\[Tau], yz], Subscript[\[Sigma],
      zz]}
   } ) /. {Subscript[\[Epsilon], xx] -> 
\!\(\*SuperscriptBox[\(u\), 
TagBox[
RowBox[{"(", 
RowBox[{"1", ",", "0", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, y, z], Subscript[\[Epsilon], yy] -> 
\!\(\*SuperscriptBox[\(v\), 
TagBox[
RowBox[{"(", 
RowBox[{"0", ",", "1", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, y, z], Subscript[\[Epsilon], zz] -> 
\!\(\*SuperscriptBox[\(w\), 
TagBox[
RowBox[{"(", 
RowBox[{"0", ",", "0", ",", "1"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, y, z], Subscript[\[Gamma], xy] -> 
\!\(\*SuperscriptBox[\(u\), 
TagBox[
RowBox[{"(", 
RowBox[{"0", ",", "1", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, y, z] + 
\!\(\*SuperscriptBox[\(v\), 
TagBox[
RowBox[{"(", 
RowBox[{"1", ",", "0", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, y, z], Subscript[\[Gamma], xz] -> 
\!\(\*SuperscriptBox[\(u\), 
TagBox[
RowBox[{"(", 
RowBox[{"0", ",", "0", ",", "1"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, y, z] + 
\!\(\*SuperscriptBox[\(w\), 
TagBox[
RowBox[{"(", 
RowBox[{"1", ",", "0", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, y, z], Subscript[\[Gamma], yz] -> 
\!\(\*SuperscriptBox[\(v\), 
TagBox[
RowBox[{"(", 
RowBox[{"0", ",", "0", ",", "1"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, y, z] + 
\!\(\*SuperscriptBox[\(w\), 
TagBox[
RowBox[{"(", 
RowBox[{"0", ",", "1", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, y, z]}

The solution is correct:

wifWL[0, 0, z0] (* displacement in z-direction *)
(stressTensor /. {u -> uifWL, v -> vifWL, w -> wifWL} /. {x -> 25, 
     y -> 25, z -> z0}) // Chop[#, 10^-9] & // MatrixForm

correct Solution

Same PDE and boundaries just with Active operator form gives a different (incorrect ) solution

opActive = Activate[op]
NDSolveValue[{opActive == {0, 
     0, -NeumannValue[stress, 
       z == z0]}, \[CapitalGamma]Xsym, \[CapitalGamma]Ysym, \
\[CapitalGamma]Zsym}, {u, v, w}, {x, x0, x1}, {y, y0, y1}, {z, z0, 
   z1}];

%[[3]][0, 0, z0]
(stressTensor /. {u -> %%[[1]], v -> %%[[2]], w -> %%[[3]]} /. {x -> 
      25, y -> 25, z -> z0}) // Chop[#, 10^-9] & // MatrixForm

incorrect result

Comment: According to Wolfram Language documentation:

This is due to the different interpretation of the "Coefficient Form of a PDE" in Inactive and Active operator form in conjunction with the Neumann[] boundary condition; see explanations in

https://reference.wolfram.com/language/FEMDocumentation/tutorial/FiniteElementBestPractice.html https://ipfs-sec.stackexchange.cloudflare-ipfs.com/mathematica/A/question/100455.html

This means that such formulations, which contain Neumann[] boundary conditions, must always be formulated with the Inactivated operator representation.


1. Question: Is there a possibility to solve a PDE in the Active operator form with Neumann boundary conditions in such a way that the boundary conditions are correctly considered (I would like to avoid the Inactive form)?

First approach avoidance of NeumannValue[] and DirichletCondition[]-Function instead direct input of all boundary conditions

Neumann boundaries in direct Form:

boundaryNeumannInDirectForm = {
    (stressTensor.{0, 1, 0} == {0, 0, 0} /. y -> y1) // Thread,
    (stressTensor.{-1, 0, 0} == {0, 0, 0} /. x -> x0) // Thread,
    (stressTensor.{0, 0, -1} == {0, 0, -stress} /. z -> z0) // Thread
    } // Simplify // Flatten

Dirichlet boundaries in direct Form:

boundaryDirichletInDirectForm = {
  u[x, y, z] == 0 /. x -> x1, v[x, y, z] == 0 /. y -> y0, 
  w[x, y, z] == 0 /. z -> z1
  }

NDSolveValue gives an error:

NDSolveValue[{(opActive == {0, 0, 0}) // Thread, 
  Flatten@{boundaryDirichletInDirectForm, 
    boundaryNeumannInDirectForm}}, {u, v, w}, {x, x0, x1}, {y, y0, 
  y1}, {z, z0, z1}]

enter image description here


2. Question: What's the problem here, with the direct formulation? Why does WL interpret the Neumann boundary conditions as Dirichlet boundary conditions and how can this be avoided?

A similar example from the WL-Documentation seems to work:

pde = {\!\(
\*SubscriptBox[\(\[PartialD]\), \(t\)]\(u[t, x]\)\) == \!\(
\*SubscriptBox[\(\[PartialD]\), \(x\)]\((\((v[t, x] - 1)\)\ 
\*SubscriptBox[\(\[PartialD]\), \(x\)]u[t, x])\)\) + (16 x t - 2 t - 
        16 (v[t, x] - 1)) (u[t, x] - 1) + 10 x E^(-4 x), \!\(
\*SubscriptBox[\(\[PartialD]\), \(t\)]\(v[t, x]\)\) == \!\(
\*SubscriptBox[\(\[PartialD]\), \({x, 2}\)]\(v[t, x]\)\) + \!\(
\*SubscriptBox[\(\[PartialD]\), \(x\)]\(u[t, x]\)\) + 4 u[t, x] - 4 + 
     x^2 - 2 t - 10 t E^(-4 x)};
bc = {u[0, x] == 1, v[0, x] == 1, u[t, 0] == 1, v[t, 0] == 1, 
   3 u[t, 1] + 
\!\(\*SuperscriptBox[\(u\), 
TagBox[
RowBox[{"(", 
RowBox[{"0", ",", "1"}], ")"}],
Derivative],
MultilineFunction->None]\)[t, 1] == 3, 5 
\!\(\*SuperscriptBox[\(v\), 
TagBox[
RowBox[{"(", 
RowBox[{"0", ",", "1"}], ")"}],
Derivative],
MultilineFunction->None]\)[t, 1] == E^4 (u[t, 1] - 1)};
NDSolve[{pde, bc}, {u, v}, {x, 0, 1}, {t, 0, 2}]

For easier use I have attached the wl file.

Many thanks for the help

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Anonymous User
Anonymous User
Posted 5 months ago

I did not do the whole problem due to time constraint. However I've had similar experiences with NDSolve and my input.

You state you have a correct solution "without using Active operator". I ask you to read the Mathematica's book about how expressions are evaluated (ie, in what order).

When NDSolve begins, some your input is "analyized" and it's parts labeled. But NDSolve must analyze your input to set up the solution. Therefore un-evaluated expressions that happen latently will not be detected by the algorithm which tries to "parse" the input expressions to know what is asked to be solved. It will end up not knowing the full problem or having false lablels inside the problem.

Example - if there are rules -> to evaluate, evaluate them before hand and assign them to variable "pde" so that NDSolve has "the actual problem being solved" laid out for it.

You'll find some "built-in" functions react well to being handed unsolved symbolic expressions, but that other "built-in" use "Hold" on the input and do not evaluate the input before attempting to decipher it. The behavior of analyzing function input is not "exactly the same across all functions".

Write a function of your own that analyzes input for the purpose of solving, and you will quickly see what a problem it would be to analyze input that is not yet itself evaluated. Yet for NDSolve, there is some input that should not be "evaluated by kernel" before it begins it's analysis.

POSTED BY: Anonymous User
Answer

Thank you very much for the answer, but what does this mean in this code example:

pde = {-(1050000/13) D[u[x, y, z], z, z] - 
    1050000/13 D[u[x, y, z], y, y] - 2625000/13 D[w[x, y, z], x, z] - 
    2625000/13 D[ v[x, y, z], x, y] - 
    3675000/13 D[u[x, y, z], x, x] == 
   0, -(1050000/13) D[v[x, y, z], z, z] - 
    2625000/13 D[w[x, y, z], y, z] - 3675000/13 D[v[x, y, z], y, y] - 
    2625000/13 D[u[x, y, z], x, y] - 1050000/13 D[v[x, y, z], x, x] ==
    0,
  -(3675000/13) D[ w[x, y, z], z, z] - 
    2625000/13 D[v[x, y, z], y, z] - 1050000/13 D[w[x, y, z], y, y] - 
    2625000/13 D[u[x, y, z], x, z] - 1050000/13 D[w[x, y, z], x, x] ==
    0}

bcs =
 (* 3 Dirichtlet bcs *)
 {u[50, y, z] == 0,
  v[x, 0, z] == 0,
  w[x, y, 70] == 0,
  (* 9 Neumann bcs*)
  (D[u[x, y, z], y] + D[v[x, y, z], x] == 0) /. {y -> 50},
  (3 D[w[x, y, z], z] + 7 D[v[x, y, z], y] + 3 D[u[x, y, z], x] == 
     0) /. {y -> 50},
  (D[v[x, y, z], z] + D[w[x, y, z], y] == 0) /. {y -> 50},
  (3 D[w[x, y, z], z] + 3 D[v[x, y, z], y] + 7 D[u[x, y, z], x] == 
     0) /. {x -> 0},
  (D[u[x, y, z], y] + D[v[x, y, z], x] == 0) /. {x -> 0},
  (D[u[x, y, z], z] + D[w[x, y, z], x] == 0) /. {x -> 0},
  (D[u[x, y, z], z] + D[w[x, y, z], x] == 0) /. {z -> 0},
  (D[v[x, y, z], z] + D[w[x, y, z], y] == 0) /. {z -> 0},
  (10500 D[w[x, y, z], z] + 4500 D[v[x, y, z], y] + 
      4500 D[u[x, y, z], x] == 13) /. z -> 0}

NDSolveValue[{pde, bcs}, {u, v, w}, {x, 0, 50}, {y, 0, 50}, {z, 0, 
  70}]

I don't understand the error message here (it is not a Dirichlet boundary condition): Error Message

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