Problem with finding the global minimum with fmincon

I am currently trying to find the global minimum for a strain-energy-function ("Holzapfel-Model") and I am running into multiple problems:
The SEF has the form
With
We can calculate , where
We want to determine the minimum of the least-square function

My solution was to put all these equations into one long one:
fun = @(x) sum((sigma_11 – (lambda_1.^2 – lambda_2.^2 .* lambda_1.^2).*x(1) + 2 .*lambda_1.^2 .*cos(x(4))^2 .* (2.*x(2).*((lambda_1.^2 .* cos(x(4))^2 + lambda_2.^2 .* sin(x(4))^2) – 1) .* exp(x(2).*((lambda_1.^2 .* cos(x(4))^2 + lambda_2.^2 .* sin(x(4))^2)-1).^2))).^2 + (sigma_22 – (lambda_2.^2 – lambda_2.^2 .* lambda_1.^2).*x(1) + 2 .*lambda_2.^2 .*sin(x(4))^2 .* (2.*x(2).*((lambda_1.^2 .* cos(x(4))^2 + lambda_2.^2 .* sin(x(4))^2) – 1) .* exp(x(2).*((lambda_1.^2 .* cos(x(4))^2 + lambda_2.^2 .* sin(x(4))^2)-1).^2))).^2)

and then use the following parameters
x0 = [15,500,12,0.75*pi];
A = [];
b = [];
Aeq = [];
beq = [];
lb = [0,0,0,0];
ub = [inf, inf, inf, pi];
chi = fmincon(fun, x0,A,b,Aeq,beq,lb,ub,nonlcon)
function [c,ceq] = nonlcon(x,lambda_1,lambda_2)
c =1-(lambda_1.^2 .* cos(x(4)).^2 + lambda_2.^2 * sin(x(4)).^2) ;
ceq = [];
end
With these parameters, I can somewhat get close to my data points.
Now my questions:
I don’t think I understood c,ceq correctly. I used c to account for the constraint on I4, but I’m not sure if this was the right way to do it.
With the initial guess for x0, I can get close but it never seems to approach my curve nearly enough. How do I know if I have a good starting guess, and is fmincon even the right approach for this problem.
I have multiple data sets, for different stretch ratios (lambda_1:lambda_2: 1-1, 1-0.75, 0.75-1, 1-0.5,0.5-1) and since they are the same sample, I would like to use those datas to get one set of parameters for all of them. I tried to put all my data into a single vector, (1:30 would be the first data set, 31:^60 the second,…). This does not seem to work well. Should I find the solution for just one curve and than try to average over the parameters? As you guys can see, I am doing this parameter evaluation thing the first time ever and I would greatly appreciate help.I am currently trying to find the global minimum for a strain-energy-function ("Holzapfel-Model") and I am running into multiple problems:
The SEF has the form
With
We can calculate , where
We want to determine the minimum of the least-square function

and then use the following parameters
x0 = [15,500,12,0.75*pi];
A = [];
b = [];
Aeq = [];
beq = [];
lb = [0,0,0,0];
ub = [inf, inf, inf, pi];
chi = fmincon(fun, x0,A,b,Aeq,beq,lb,ub,nonlcon)
function [c,ceq] = nonlcon(x,lambda_1,lambda_2)
c =1-(lambda_1.^2 .* cos(x(4)).^2 + lambda_2.^2 * sin(x(4)).^2) ;
ceq = [];
end
With these parameters, I can somewhat get close to my data points.
Now my questions:
I don’t think I understood c,ceq correctly. I used c to account for the constraint on I4, but I’m not sure if this was the right way to do it.
With the initial guess for x0, I can get close but it never seems to approach my curve nearly enough. How do I know if I have a good starting guess, and is fmincon even the right approach for this problem.
I have multiple data sets, for different stretch ratios (lambda_1:lambda_2: 1-1, 1-0.75, 0.75-1, 1-0.5,0.5-1) and since they are the same sample, I would like to use those datas to get one set of parameters for all of them. I tried to put all my data into a single vector, (1:30 would be the first data set, 31:^60 the second,…). This does not seem to work well. Should I find the solution for just one curve and than try to average over the parameters? As you guys can see, I am doing this parameter evaluation thing the first time ever and I would greatly appreciate help. I am currently trying to find the global minimum for a strain-energy-function ("Holzapfel-Model") and I am running into multiple problems:
The SEF has the form
With
We can calculate , where
We want to determine the minimum of the least-square function

and then use the following parameters
x0 = [15,500,12,0.75*pi];
A = [];
b = [];
Aeq = [];
beq = [];
lb = [0,0,0,0];
ub = [inf, inf, inf, pi];
chi = fmincon(fun, x0,A,b,Aeq,beq,lb,ub,nonlcon)
function [c,ceq] = nonlcon(x,lambda_1,lambda_2)
c =1-(lambda_1.^2 .* cos(x(4)).^2 + lambda_2.^2 * sin(x(4)).^2) ;
ceq = [];
end
With these parameters, I can somewhat get close to my data points.
Now my questions:
I don’t think I understood c,ceq correctly. I used c to account for the constraint on I4, but I’m not sure if this was the right way to do it.
With the initial guess for x0, I can get close but it never seems to approach my curve nearly enough. How do I know if I have a good starting guess, and is fmincon even the right approach for this problem.
I have multiple data sets, for different stretch ratios (lambda_1:lambda_2: 1-1, 1-0.75, 0.75-1, 1-0.5,0.5-1) and since they are the same sample, I would like to use those datas to get one set of parameters for all of them. I tried to put all my data into a single vector, (1:30 would be the first data set, 31:^60 the second,…). This does not seem to work well. Should I find the solution for just one curve and than try to average over the parameters? As you guys can see, I am doing this parameter evaluation thing the first time ever and I would greatly appreciate help. fmincon, nonlinear, curve fitting, optimization MATLAB Answers — New Questions

Cart

Cart