thermal stress optimization - fmincon

Hi,

I'm using fmincon for stress optimization but there are errors.

The script looks messy due to complicated parameters but the functions are simple.

% file name: thermalstressobj.m
function f = thermalstressobj(x)
l = linspace(0,x,100);

E_1=8.5E10; nu_1=0.34; G_1=3.17e10;CTE_1=1.7e-5;
E_2=2.2e9; nu_2=0.26; G_2=900e8; CTE_2=3.2e-5;
h_1 = 1e-3; h_2 = 1e-4;
D_1 = E_1*h_1^3/(12*(1-nu_1^2)); D_2 = E_2*h_2^3/(12*(1-nu_2^2));
D=D_1+D_2;
lambda_1=(1-nu_1^2)/(E_1*h_1); lambda_2=(1-nu_2^2)/(E_2*h_2);
lambda = lambda_1+lambda_2+(h_1+h_2)^2/(4*D);
kappa_1 = 2*(1+nu_1)*h_1/(3*E_1); kappa_2 = 2*(1+nu_2)*h_2/(3*E_2);
kappa = kappa_1+kappa_2;
large_K = 1/(((1-nu_1^2)*h_1/E_1 + (1-nu_2^2)*h_2/E_2));
small_K = sqrt(lambda/kappa);
mu = (h_1*D_2-h_2*D_1)/(2*D);
alpha = (large_K*D/(4*D_1*D_2))^(1/4);
epsilon = D/(D_1*D_2)*mu^2/lambda;
q_0 = -2/3*(2-epsilon)*alpha^4*small_K^2-((1+epsilon)/3*small_K^2)^3;
q_1 = 4/3*alpha^4+((1+epsilon)/3*small_K^2)^2;
small_d = q_0^2+q_1^3;
small_a = (-q_0+sqrt(small_d))^(1/3); small_b = -(abs(-q_0-sqrt(small_d)))^(1/3);
mu_1 = -(small_a+small_b)/2 + (1+epsilon)/3*small_K^2; mu_2 = sqrt(3)*(small_a-small_b)/2;
zeta_1 = small_a+small_b;
gamma_1 = sqrt(1/2*(sqrt(mu_1^2+mu_2^2)+mu_1)); gamma_2 = sqrt(1/2*(sqrt(mu_1^2+mu_2^2)-mu_1));
beta_1 = sqrt(zeta_1 + (1+epsilon)/3*small_K^2);
u = beta_1*x; v_1 = gamma_1*x; v_2 = gamma_2*x;
D_0 = (gamma_1 + 3/2*zeta_1/mu_2*gamma_2 + (gamma_2-3/2*zeta_1/mu_2*gamma_1)*sin(2*v_2)/sinh(2*v_1))*tanh(u)-beta_1*(coth(2*v_1)-cos(2*v_2)/sinh(2*v_1));
C_1 = -(CTE_2-CTE_1)*150/(kappa*D_0)*(coth(2*v_1)-cos(2*v_2)/sinh(2*v_1))/cosh(u);
C_3 = (CTE_2-CTE_1)*150/(kappa*D_0)*(sin(v_2)/sinh(v_1) + 3/2*zeta_1/mu_2*cos(v_2)/cosh(v_1))*tanh(u);
C_5 = (CTE_2-CTE_1)*150/(kappa*D_0)*(cos(v_2)/cosh(v_1) - 3/2*zeta_1/mu_2*sin(v_2)/cosh(v_1))*tanh(u);

tau = abs(C_1*sinh(beta_1*l) + C_3*cosh(gamma_1*l).*sin(gamma_2*l)+C_5*sinh(gamma_1*l).*cos(gamma_2*l));
f = max(tau);

and

% file name: thermalstressconstr.m

function [c, ceq] = thermalstressconstr(x)
R_th1 = 1e-3/(401*10e-3^2)+1e-4/(58.7*10e-3^2);
R_th2 = (1/((1e-3)/(401*(2*x).^2) + 1/(1e-3)/(3*(10e-3^2-(2*x).^2))))^-1 + (1/((1e-4)/(58.7*(2*x).^2) + 1/(1e-4)/(3*(10e-3^2-(2*x).^2))))^-1;
c = R_th2/R_th1-1.2;
ceq = [];
end

A maximum stress does not occur in specific points like edge or vertex, so linspace and max(tau) are used.

The constraint is ratio of initial thermal resistance to optimized one.

Actually I'm not familiar with MATLAB so I used optimization tool but I cannot find optimized value.

How can I fix the erroe?

[Walter Roberson]

f = max(tau);

max() is not continuously differentiable, and so violates the assumptions of fmincon()

[Matt J]

Is x supposed to be a scalar? If it is not a scalar, then this line will cause an error

l = linspace(0,x,100);

If x is a scalar, I wonder why you are using fmincon to minimize a 1D function. You could just plot the function and observe where the minimum lies on the graph.

[Jaejin Jeon]

Thank you for your answer!.

Actually, it is a term project of optimization class. So I'm trying to use fmincon, which I learned in the class.

Do you have any idea to modifiy the script or other recommended optimization algorithms?

l is length of a structure

[Matt J]

What about my question? Is x supposed to be a scalar or not?

And if l is supposed to be a length, why have you derived it from linspace?

[Jaejin Jeon]

x is a scalar which meas length of the structure. And linspace is used to find the point that maximum stress occurs wihin the x. Actually it would be better to change the position of x and l in a given script. I set it to write f as a function for x.

Sorry for my bad explanation.

[Walter Roberson]

You cannot expect fmincon to be able to converge on any function that uses max(). The mathematics that fmincon uses to find the minima is not valid on functions that use max() or min()

[Matt J]

That's right. You would need to use fminimax. But even then, it seems way overkill for a 1D function minimization.