Free boundary conditions in MATHEMATICA - is this right? Second opinion

Question

I'm trying to prescribe free boundary conditions for a non-linear evolution equation in mathematica and I wanted as second opinion on whether or not what I am doing is right.

The boundary conditions have been marked with a comment, viz., (FREE BOUNDARY CONDITIONS)

I'd also like to run this for pinned boundary conditions.

Needs["VectorAnalysis`"]
Needs["DifferentialEquations`InterpolatingFunctionAnatomy`"];
Clear[Eq5, Complete, h, S, G, E1, K1, D1, VR, M]
Eq5[h_, {S_, G_, E1_, K1_, D1_, VR_, M_}] := \!\(
\*SubscriptBox[\(\[PartialD]\), \(t\)]h\) + 
    Div[-h^3 G Grad[h] + 
      h^3 S Grad[Laplacian[h]] + (VR E1^2 h^3)/(D1 (h + K1)^3)
        Grad[h] + M (h/(1 + h))^2 Grad[h]] + E1/(h + K1) == 0;
SetCoordinates[Cartesian[x, y, z]];
Complete[S_, G_, E1_, K1_, D1_, VR_, M_] := 
  Eq5[h[x, y, t], {S, G, E1, K1, D1, VR, M}];
TraditionalForm[Complete[S, G, E1, K1, D1, VR, M]]




  L = 185.62; TMax = 100; km = 0.0381; 
Off[NDSolve::mxsst]; 
Off[NDSolve::ibcinc]; 
hSol = h /. NDSolve[{Complete[100, 0, 0, 0, 0.001, 0, 5], 



(*FREE BOUNDARY CONDITIONS*)

     Derivative[2, 0, 0][h][0, y, t] == 0, 
     Derivative[2, 0, 0][h][L, y, t] == 0, 
     Derivative[0, 2, 0][h][x, 0, t] == 0, 
     Derivative[0, 2, 0][h][x, L, t] == 0, 
     Derivative[3, 0, 0][h][0, y, t] == 0, 
     Derivative[3, 0, 0][h][L, y, t] == 0, 
     Derivative[0, 3, 0][h][x, 0, t] == 0, 
     Derivative[0, 3, 0][h][x, L, t] == 0, 

    (*FREE BOUNDARY CONDITIONS*)

     h[x, y, 0] == 1 + (-0.05*Cos[2*Pi*(x/L)] - 0.05*Sin[2*Pi*(x/L)])*
                 Cos[2*Pi*(y/L)]}, 
    h, {x, 0, L}, {y, 0, L}, {t, 0, TMax}][[1]]
hGrid = InterpolatingFunction[hSol]; 
{TMin, TRup} = InterpolatingFunctionDomain[hSol][[3]]

I'm guessing that `hsol` on the penultimate line of your code should have been `hSol` (with a capital S). Since `hSol` is already an `InterplatingFunction` the definition of `hGrid` doesn't really make sense. — Heike, Jan 11 '12 at 15:36
@Heike Thanks for pointing that out. Other than that, what do you think of the implementation of free boundary conditions (The second and third derivatives of the 'edges' are set to zero)? — dearN, Jan 11 '12 at 15:40
@Heike I was thinking so too from my bending moment days. But I just had to check anyway! Thanks! :) — dearN, Jan 11 '12 at 19:08

score 1 · Accepted Answer · answered Jan 12 '12 at 19:08

The consensus achieved from reading the comments is that the implementation of free boundary conditions in the code above is correct.

More detail should be available in books dealing with mechanics of materials or strength of materials in chapters referring to Bending moments and shear stress diagrams where very often free-free or free-fixed or fixed-fixed boundary conditions are used.

Free boundary conditions in MATHEMATICA - is this right? Second opinion

1 Answers1