Re: Help to solve a large system of Nonlinear equations

Try TK Solver from UTS.

http://www.uts.com/

Fred


"Nicolas Charest" <[EMAIL PROTECTED]> wrote in message
news:[EMAIL PROTECTED]
> I tried to solve a system of 41 nonlinear equation - 41 variables.
> Physically,This system describe a very hyperstatically concrete girder
> behavior under large loads.
>
> I have try to solve it with a software "Systems of Nonlinear
> Equations" found on www.numericalmathematics.com. This software use
> generalized Newton method, but this software can't take a system of 41
> equations.
>
> I have try to solve it with mathematics library PETSc but this
> algortihm need that I build the Jacobian matrix of 41 by 41 element.
> It's very hard because I have about 400 element different of zero.
> Probably, some software build Jacobian matrix.
>
> Finally, the difficulty of this system is to found initial value of
> the 41 independant variables for that it converge.
>
> The system is:
>
> 41 variables:
>
P,M0,M1,M2,M3,M4,Vba,Vca,Vda,Vea,f1p,f2p,f3p,f4p,K1,K3,K4,K5,K6,K7,K8,Lam1,L
am2,
> Lam3,Lam4,f1c,f2c,f3c,f4c,c1,c2,c3,c4,T1x,T2x,T3x,T4x,Tde,Tcd,Tbc,Tab
>
> 41 equations:
>
"-M0+L*P/4-L/8*tan(f4c)*T4x-L/8*tan(f3c)*(T3x+T4x)-L/8*tan(f2c)*(T2x+T3x+T4x
)-L/8*tan(f1c)*(T1x+T2x+T3x+T4x)+T4x*Vea+T3x*(Vea-Vba)+T2x*(Vea-Vca)+T1x*(Ve
a-Vda)-T4x*e-T3x*e-T2x*e-T1x*e=0"
>
"-M1+3*L*P/16-L/8*tan(f4c)*T4x-L/8*tan(f3c)*(T3x+T4x)-L/8*tan(f2c)*(T2x+T3x+
T4x)+T4x*Vda+T3x*(Vda-Vba)+T2x*(Vda-Vca)-T4x*e-T3x*e-T2x*e-T1x*e/2=0"
>
"-M2+L*P/8-L/8*tan(f4c)*T4x-L/8*tan(f3c)*(T3x+T4x)+T4x*Vca+T3x*(Vca-Vba)-T4x
*e-T3x*e-T2x*e/2=0"
> "-M3+L*P/16-L/8*tan(f4c)*T4x+T4x*Vba-T4x*e-T3x*e/2=0"
> "-M4-T4x*e=0"
> "Vba-K1*sin(Lam4*L/8)+e*cos(Lam4*L/8)-L/8*tan(f4c)-e+P*L/(16*T4x)=0"
>
"Vca-K3*sin(Lam3*L/4)-K4*cos(Lam3*L/4)-L/8*tan(f3c)-e-1/(T3x+T4x)*(-P*L/8+L/
8*tan(f4c)*T4x+T3x*Vba)=0"
>
"Vda-K5*sin(3*Lam2*L/8)-K6*cos(3*Lam2*L/8)-L/8*tan(f2c)-e-1/(T2x+T3x+T4x)*(-
3*P*L/16+L/8*tan(f4c)*T4x+L/8*tan(f3c)*T3x+L/8*tan(f3c)*T4x+T3x*Vba+T2x*Vca)
=0"
>
"Vea-K7*sin(Lam1*L/2)-K8*cos(Lam1*L/2)-L/8*tan(f1c)-e-1/(T1x+T2x+T3x+T4x)*(-
P*L/4+L/8*tan(f4c)*T4x+L/8*tan(f3c)*(T3x+T4x)+L/8*tan(f2c)*(T2x+T3x+T4x)+T3x
*Vba+T2x*Vca+T1x*Vda)=0"
>
"f1p-K5*Lam2*cos(3*Lam2*L/8)+K6*Lam2*sin(3*Lam2*L/8)-tan(f2c)+P/(2*(T2x+T3x+
T4x))=0"
>
"f2p-K3*Lam3*cos(Lam3*L/4)+K4*Lam3*sin(Lam3*L/4)-tan(f3c)+P/(2*(T3x+T4x))=0"
> "f3p-K1*Lam4*cos(Lam4*L/8)-e*Lam4*sin(Lam4*L/8)-tan(f4c)+P/(2*T4x)=0"
> "f4p-K1*Lam4-tan(f4c)+P/(2*T4x)=0"
>
"K1*sin(Lam4*L/8)-e*cos(Lam4*L/8)-(T3x+T4x)/T4x*(K3*sin(Lam3*L/8)+K4*cos(Lam
3*L/8))-T3x/T4x*e=0"
>
"K1*Lam4*cos(Lam4*L/8)+e*Lam4*sin(Lam4*L/8)+tan(f4c)-P/(2*T4x)-K3*Lam3*cos(L
am3*L/8)+K4*Lam3*sin(Lam3*L/8)-tan(f3c)+P/(2*(T3x+T4x))=0"
>
"K3*sin(Lam3*L/4)+K4*cos(Lam3*L/4)-(T2x+T3x+T4x)/(T3x+T4x)*(K5*sin(Lam2*L/4)
+K6*cos(Lam2*L/4))-T2x/(T3x+T4x)*e=0"
>
"K3*Lam3*cos(Lam3*L/4)-K4*Lam3*sin(Lam3*L/4)+tan(f3c)-P/(2*(T3x+T4x))-K5*Lam
2*cos(Lam2*L/4)+K6*Lam2*sin(Lam2*L/4)-tan(f2c)+P/(2*(T2x+T3x+T4x))=0"
>
"K5*sin(3*Lam2*L/8)+K6*cos(3*Lam2*L/8)-(T1x+T2x+T3x+T4x)/(T2x+T3x+T4x)*(K7*s
in(3*Lam1*L/8)+K8*cos(3*Lam1*L/8))-T1x/(T2x+T3x+T4x)*e=0"
>
"K5*Lam2*cos(3*Lam2*L/8)-K6*Lam2*sin(3*Lam2*L/8)+tan(f2c)-P/(2*(T2x+T3x+T4x)
)-K7*Lam1*cos(3*Lam1*L/8)+K8*Lam1*sin(3*Lam1*L/8)-tan(f1c)+P/(2*(T1x+T2x+T3x
+T4x))=0"
>
"K7*Lam1*cos(Lam1*L/2)-K8*Lam1*sin(Lam1*L/2)-P/(2*(T1x+T2x+T3x+T4x))+tan(f1c
)=0"
> "Lam1-sqr((T1x+T2x+T3x+T4x)/(Ec*I))=0"
> "Lam2-sqr((T2x+T3x+T4x)/(Ec*I))=0"
> "Lam3-sqr((T3x+T4x)/(Ec*I))=0"
> "Lam4-sqr(T4x/(Ec*I))=0"
> "tan(f1c)-(Vea-Vda+(dt-c1)*(1-cos(f1p)))/(L/8+(dt-c1)*sin(f1p))=0"
>
"tan(f2c)-(Vda-Vca-(dt-c1)*(1-cos(f1p))+(dt-c2)*(1-cos(f2p)))/(L/8-(dt-c1)*s
in(f1p)+(dt-c2)*sin(f2p))=0"
>
"tan(f3c)-(Vca-Vba-(dt-c2)*(1-cos(f2p))+(dt-c3)*(1-cos(f3p)))/(L/8-(dt-c2)*s
in(f2p)+(dt-c3)*sin(f3p))=0"
>
"tan(f4c)-(Vba-(dt-c3)*(1-cos(f3p))+(dt-c4)*(1-cos(f4p)))/(L/8-(dt-c3)*sin(f
3p)+(dt-c4)*sin(f4p))=0"
> "c1-Yg-(T1x+T2x+T3x+T4x)*Ec*I/(M1*(Aac*Eac+Ac*Ec))=0"
> "c2-Yg-(T2x+T3x+T4x)*Ec*I/(M2*(Aac*Eac+Ac*Ec))=0"
> "c3-Yg-(T3x+T4x)*Ec*I/(M3*(Aac*Eac+Ac*Ec))=0"
> "c4-Yg-T4x*Ec*I/(M4*(Aac*Eac+Ac*Ec))=0"
> "T1x-cos(f1c)*Et*At1*Tde=0"
> "T2x-cos(f2c)*Et*At2*(Tde+Tcd)/2=0"
> "T3x-cos(f3c)*Et*At3*(Tde+Tcd+Tbc)/3=0"
> "T4x-cos(f4c)*Et*At4*(Tde+Tcd+Tbc+Tab)/4=0"
> "Tde-(L/8+(dt-c1)*sin(f1p))/(L/8*cos(f1c))+1=0"
> "Tcd-(L/8-(dt-c1)*sin(f1p)+(dt-c2)*sin(f2p))/(L/8*cos(f2c))+1=0"
> "Tbc-(L/8-(dt-c2)*sin(f2p)+(dt-c3)*sin(f3p))/(L/8*cos(f3c))+1=0"
> "Tab-(L/8-(dt-c3)*sin(f3p)+(dt-c4)*sin(f4p))/(L/8*cos(f4c))+1=0"
> "Def0+M0*Yg/(I*Ec)+1/(Aac*Eac+Ac*Ec)*(T1x+T2x+T3x+T4x)=0"
>
> 15 Constants: dt=650 L=20000 Yg=288.53 e=361.47 I=5546495774 Ac=105000
>               Aac=10000 At1=2000 At2=2000 At3=2000 At4=2000 Ec=43866
>               Eac=200000 Et=200000 Def0=-0.0030
>
>
> I need help about the method to use, a convenient software, and a
> method to find initial value of the variables so the system converge ?
>
> Sincerely