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I'm trying to minimize a function in order to find a slope critical heigth on a solope stability analisys. The functions taken from (Chen 1975) are the following:

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I added all the equations to mma but NMinimize doesn't work. Here is the mma code:

Lr0[thetah_, thetao_] := 
  Sin[thetah - thetao]/(Sin[thetah + alpha]) - 
   Sin[thetah + 
      beta]/(Sin[thetah + alpha] Sin[
       beta - alpha]) (Exp[(thetah - thetao) Tan[phi] ] Sin[
        thetah + alpha] - Sin[thetao + alpha]);
f1[thetah_, 
   thetao_] := ((3 Tan[phi] Cos[thetah] + Sin[thetah]) Exp[
     3 (thetah - thetao) Tan[phi]] - (3 Tan[phi] Cos[thetao] + 
     Sin[thetao]))/(3 (1 + 9 Tan[phi]^2)) ;
f2[thetah_, thetao_] := 
  Lr0[thetah, thetao]/
   6 (2 Cos[thetao] - Lr0[thetah, thetao] Cos[alpha]) Sin[
    thetao + alpha];
f3[thetah_, thetao_] := 
 1/6 Exp[(thetah - thetao) Tan[phi]] (Sin[(thetah - thetao)] - 
    Lr0[thetah, thetao] Sin[(thetah + alpha)]); (Cos[thetao] - 
  Lr0[thetah, thetao] Cos[alpha] + 
  Cos[thetah] Exp[(thetah - thetao) Tan[phi]])
ff[thetah_, thetao_] := (
  Sin[beta] (Exp[2 (thetah - thetao) Tan[phi] ] - 
     1) (Sin[thetah + alpha] Exp[(thetah - thetao) Tan[phi]] - 
     Sin[thetao + alpha]))/(
  2 Sin[beta - alpha] Tan[
    phi] (f1[thetah, thetao] - f2[thetah, thetao] - 
     f3[thetah, thetao]));
fff = ff[thetah, thetao] /. beta -> 45. Pi/180 /. 
    phi -> 20. Pi/180 /. alpha -> 0. // N
sol = NMinimize[fff, {thetah, thetao}, Method -> "RandomSearch"]

Function fff

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Book's solution:

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Reference

Chen, Wai-Fah (1975). Limit analysis and soil plasticity. Elsevier, New York.

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  • $\begingroup$ ...; fff = ff[thetah, thetao]/.{beta->45 Pi/180, phi->20 Pi/180, alpha->0}; NMinimize[fff, {thetah,thetao}, Method->"RandomSearch", WorkingPrecision->32] quickly returns "NMinimize failed to converge to a solution. The function may be unbounded" {-8.413500149*^22, {thetah->-0.252703647, thetao->-0.986343945}} $\endgroup$ – Bill Jan 29 at 18:57
  • $\begingroup$ This is a very ugly function (see edited question). I'm wondering how the book's author solved this in 1975 without mma. He even build tables with the results. $\endgroup$ – Diogo Jan 29 at 19:16
  • $\begingroup$ Making use of your current code and Maple, I obtained a confirmation of the Bill's result. The code on demand. Do you correctly reproduce formulas from the cited book in Mathematica? $\endgroup$ – user64494 Jan 29 at 20:17
  • $\begingroup$ @user64494 I've checked more then 5 times.. $\endgroup$ – Diogo Jan 29 at 20:19
  • $\begingroup$ The book's name is: Limit Analisys in Soil Plasticity 1975 $\endgroup$ – Diogo Jan 29 at 20:20

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