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Error "Limits of integration must be double or single scalars." when solve equations with "fsolve" function

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Yuting
Yuting 2023-8-1
评论： Harald 2023-8-2
I want to solve the balance equations like following:
% momentum and force balance
f1 = int(pl*x,x,-L,0)+int(pr*x,x,0,L)+M/d;
f2 = int(pl,x,-L,0)*sin(alph+tilt)+int(pr,x,0,L)*sin(alph-tilt)-G/d-2*L*P_e*sin(alph)*cos(tilt);
f3 = int(pl,x,-L,0)*cos(alph+tilt)-int(pr,x,0,L)*cos(alph-tilt)+2*L*P_e*sin(alph)*sin(tilt);
initial_guess = [0.5, 0.1, 2];
equations = matlabFunction(f1,f2,f3, 'Vars', {x1, y1, Uratio});
result = fsolve(@(vars) equations(vars(1), vars(2), vars(3)), initial_guess);
Then error happend after running:
Error using integral
Limits of integration must be double or single scalars.
Error in symengine>@(x)x.*(Uratio.^3.*(Uratio.*(x.^4.*(-cos(pi./6.0-atan(x1./y1).*2.0).*(x1.^2./4.0+y1.^2./4.0)+cos(pi./6.0-atan(x1./y1).*2.0).*(x1.^2.*(3.0./4.0)+y1.^2.*(3.0./4.0))+x1.^2+y1.^2)+(x.^2.*(x1.^2+y1.^2).^2)./2.0+x.^6./6.0-x.^3.*cos(pi./1.2e+1-ata.........
Error in integralCalc/iterateScalarValued (line 314)
fx = FUN(t);
Error in integralCalc/vadapt (line 132)
[q,errbnd] = iterateScalarValued(u,tinterval,pathlen);
Error in integralCalc (line 75)
[q,errbnd] = vadapt(@AtoBInvTransform,interval);
Error in integral (line 87)
Q = integralCalc(fun,a,b,opstruct);
Error in symengine>@(x1,y1,Uratio)deal(integral(@(x)x.*(Uratio.^3.*(Uratio.*(x.^4.*(-cos(pi./6.0-atan(x1./y1).*2.0).*(x1.^2./4.0+y1.^2./4.0)+cos(pi./6.0-atan(x1./y1).*2.0).*(x1.^2.*(3.0./4.0)+y1.^2.*(3.0./4.0))+x1.^2+y1.^2)+(x.^2.*(x1.^2+y1.^2).^2)./2.0+x.^6./6.0-x.^3.*cos(pi./1.2e+1-atan(x1./y1)).*(x1.^2+y1.^2).^(3.0./2.0).*(4.0./3.0)-x.^5.*cos(pi./1.2e+1-atan(x1./y1)).*sqrt(x1.^2+y1.^2).*(4.0./5.0))+(Uratio.*integral(@(x)(sin(pi./1.2e+1-atan(x1./y1)).^2.*(x1.^2+y1.^2)+(x-cos(pi./1.2e+1-atan(x1./y1)).*sqrt(x1.^2+y1.^2)).^2).^(3.0./2.0),0.0,x).*(cos(pi./6.0-atan(x1./y1).*2.0).*(x1.^2.*4.5265984e-5+y1.^2.*4.5265984e-5)-cos(pi.*(1.1e+1./6.0)-atan(x1./y1).*2.0).*(x1.^2.*4.5265984e-5+y1.^2.*4.5265984e-5)-cos(pi./6.0-atan(x1./y1).*2.0).*(x1.^2.*1.357.........
Error in fsolvetest_3para>@(vars)equations(vars(1),vars(2),vars(3)) (line 74)
result = fsolve(@(vars) equations(vars(1), vars(2), vars(3)), initial_guess);
Error in fsolve (line 264)
fuser = feval(funfcn{3},x,varargin{:});
Error in fsolvetest_3para (line 74)
result = fsolve(@(vars) equations(vars(1), vars(2), vars(3)), initial_guess);
Caused by:
Failure in initial objective function evaluation. FSOLVE cannot continue.
  2 个评论
Yuting
Yuting 2023-8-1
the entire code is:
alph=pi*15/180;
L=0.116;
T_ice=18;
addweight=0.212;
T0=15;
dL=0.02;
%tilt=2.5*pi/180;
tilt=0;
heightofweight=12.5;
forcefromcable=0;
mu_L=0.001;
rho_S=920*0.95;
rho_L=1000;
Cp_s=2049.41;
h_m=334000+Cp_s*T_ice;
K_L=0.57;
P_e=102770;
% 101300+1000*9.8*0.15
g1=addweight*9.8;
g2=0.38*9.8;
heightofmasscenter=0.07;
heightofcable=0.13;
syms x1 y1 Uratio x
d = pi*L*sin(alph)/2;
M = g1*(dL+heightofweight*tan(tilt))*cos(tilt)+g2*heightofmasscenter*sin(tilt)-forcefromcable*heightofcable*sin(tilt);
G = g1+g2;
beta = atan(x1/y1);
xL = sqrt(x1^2+y1^2)*cos(pi-alph-beta);
DL = sqrt(x1^2+y1^2)*sin(pi-alph-beta);
xR = sqrt(x1^2+y1^2)*cos(beta-alph);
DR = sqrt(x1^2+y1^2)*sin(beta-alph);
LL = DL^2+(xL-x)^2;
RR = DR^2+(xR-x)^2;
C = Uratio*(int(x*RR^2,x,0,L)-int(x*LL^2,x,0,-L))/(int(LL^1.5,x,0,-L)-int(RR^1.5,x,0,L));
P0 = 12*mu_L*rho_S^4*h_m^3*Uratio^3*(Uratio*int(LL^2*x,x,0,-L)+C*int(LL^1.5,x,0,-L))/(rho_L*T0^3*K_L^3)+P_e;
pl = -12*mu_L*rho_S^4*h_m^3*Uratio^3*(Uratio*int(LL^2*x,x,0,x)+C*int(LL^1.5,x,0,x))/(rho_L*T0^3*K_L^3)+P0;
pr = -12*mu_L*rho_S^4*h_m^3*Uratio^3*(Uratio*int(RR^2*x,x,0,x)+C*int(RR^1.5,x,0,x))/(rho_L*T0^3*K_L^3)+P0;
f1 = int(pl*x,x,-L,0)+int(pr*x,x,0,L)+M/d;
f2 = int(pl,x,-L,0)*sin(alph+tilt)+int(pr,x,0,L)*sin(alph-tilt)-G/d-2*L*P_e*sin(alph)*cos(tilt);
f3 = int(pl,x,-L,0)*cos(alph+tilt)-int(pr,x,0,L)*cos(alph-tilt)+2*L*P_e*sin(alph)*sin(tilt);
disp('balance equations done')
initial_guess = [0.5, 0.1, 2];
equations = matlabFunction(f1,f2,f3, 'Vars', {x1, y1, Uratio});
result = fsolve(@(vars) equations(vars(1), vars(2), vars(3)), initial_guess);

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回答（1 个）

Harald
Harald 2023-8-1
Hi Yuting,
since you are solving the equations numerically anyway, my recommendation would be to perform all the computations numerically from the start, i.e. write a function that computes [f1, f2, f3] from (x1, y1, Uratio). I would only make an exception from this if the symbolic computations are really simple or can be simplified significantly.
This should not only help resolve the error but it will likely also improve performance.
Best wishes,
Harald
  4 个评论
显示 2更早的评论
Yuting
Yuting 2023-8-2
Hi @Harald,
Thanks for your suggestion! And we find a place which might cause the error when we rewrited the formulas.
In the following two lines, the limits in function "int" are 0 and x:
pl = -12*mu_L*rho_S^4*h_m^3*Uratio^3*(Uratio*int(LL^2*x,x,0,x)+C*int(LL^1.5,x,0,x))/(rho_L*T0^3*K_L^3)+P0;
pr = -12*mu_L*rho_S^4*h_m^3*Uratio^3*(Uratio*int(RR^2*x,x,0,x)+C*int(RR^1.5,x,0,x))/(rho_L*T0^3*K_L^3)+P0;
After replaced the int into integral, the upper limit x caused the error as it's not a scalar.
Do you have any possible solution?
Thanks again!
Yuting
Harald
Harald 2023-8-2
Hi @Yuting,
I would avoid using the same variable as the integration variable and the limit as it has potential for confusion.
Since the integral will depend on the upper limit, I would create an anonymous function handle for this:
pl = @(x) -12*mu_L*rho_S^4*h_m^3*Uratio^3*(Uratio*integral(@(xi) LL(xi)^2*xi,0,x)+C*integral(@(xi) LL(xi)^1.5, 0,x))/(rho_L*T0^3*K_L^3)+P0;
This assumes that LL has been created as a function handle, e.g.
LL = @(x) DL^2+(xL-x)^2;
I hope this helps.
Best wishes,
Harald

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