Non-linear mathematical model containing double integral fit using DEA

Questions
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1

Hello guys!
I need your help to find the parameters by fitting the data using a differential evolution algorithm on a non-linear mathematical model containing double integral which can not be determined analytically but numerically. I used multiple functions to reduce the expression of integration,

F = {{\[Lambda]1, 0, 0}, {0, \[Lambda]1, 0}, {0, 0, 1/\[Lambda]1^2}};
c = Transpose[F] . F;
EGreen = 1/2 (c - IdentityMatrix[3]);
bleft = F . Transpose[F];
Nref = {Sin[\[CapitalTheta]] Cos[\[CapitalPhi]], 
   Sin[\[CapitalTheta]] Sin[\[CapitalPhi]], Cos[\[CapitalTheta]]};
ncur = F . Nref;
w1 = D[c1/(2 c2) (Exp[c2 (Imat)^3] - 1), Imat]; 
Imat = Simplify[
   Tr[EGreen. (TensorProduct[Nref, Nref])] + Sqrt[
    Tr[EGreen^2 . (TensorProduct[Nref, Nref])]]];
Hmod = 1/
    2 (TensorProduct[Nref, Nref] + 
      1/(2 Sqrt[
        trEEH]) (EGreen . (TensorProduct[Nref, 
            Nref]) + (TensorProduct[Nref, Nref]) . EGreen)) /. 
   trEEH -> Tr[EGreen . EGreen . (TensorProduct[Nref, Nref])];
hmod = Simplify[F . Hmod . Transpose[F]];

Sigma11BiaxialMatchedInvariant[n_, b_, c1_, c2_] := 
  Table[0.3 (\[Lambda]1^2 - 1/\[Lambda]1^4) + 
    NIntegrate[
     4 n Sqrt[b/(2 Pi)]
        Exp[2 b (Cos[\[Alpha] - \[CapitalPhi]] \
Sin[\[CapitalTheta]])^2]/
       Erfi[Sqrt[2 b]]*2 w1*(-hmod[[3, 3]] + hmod[[1, 1]])*
       Sin[\[CapitalTheta]] /. {\[Alpha] -> 0}, {\[CapitalPhi], 0, 
      2 \[Pi]}, {\[CapitalTheta], 0, \[Pi]}], {\[Lambda]1, 0.5, 1.5, 
    0.04}];

fit = NonlinearModelFit[data, 
  Sigma11BiaxialMatchedInvariantSigma11[n, b, c1, c2], {n, b, c1, c2}, \[Lambda]1, 
  Method -> {NMinimize, 
    Method -> {"DifferentialEvolution", "ScalingFactor" -> 0.9, 
      "CrossProbability" -> 0.1, 
      "PostProcess" -> {FindMinimum, Method -> "QuasiNewton"}}}]

I have attached the data. Please help me!

GST33.mat (15.7 KB)

2

I don’t think you can get much help without giving the data, trEEH, and AI11 not to mention exactly where you need help. Do you get an error message? Warning message? If so, what did it say? Does NonlinearModelFit keep running forever?

3

I need help in NonlinearModelFit to fit the function Sigma11BiaxialMatchedInvariant which has parameters n, b, c1 and c2. The function Sigma11BiaxialMatchedInvariant is a numerical integration which has a variable \[Lambda]1. The function Sigma11BiaxialMatchedInvariant contains the expression w1 and hmod which has been written. Yes! it is running very late and then it is returning the same expression every time. I need to find parameters n, b, c1 and c2.

4

I have already attached data in GST33.mat file. As far as trEEH concerned, I have already substituted the expression trEEH -> Tr[EGreen . EGreen . (TensorProduct[Nref, Nref])] . You can check the editted code for AI11. Sorry for the error.