Steady State Temperature

The steady state temperature (T1 and T2 ) at two points (mid-point and the free end) of the one dimensional fin correspond to the minimum of the function given below. Determine the temperature T1 and T2.

Figure 1: Steady State Temperature Problem Description

Mathematical formulation:

Minimize:

clc; 
// Objective function
function f=ObjectiveFunction(t)
    f = 0.6382*t(1)^2 + 0.3191*t(2)^2 - 0.2809*t(1)*t(2) - 67.906*t(1) - 14.29*t(2);
endfunction
// Initial guess
x0 = [100 200];
disp(x0, "Initial guess given to the solver is ")
input("Press enter to proceed: ")
[xopt,fopt,exitflag,output,gradient,hessian] = fminunc(ObjectiveFunction,x0)
// Result representation
clc
if exitflag == 0 then  
    disp("Optimal Solution Found")
    disp(xopt', "The optimum solution obtained is")
    disp(fopt, "The optimum objective function value is")
elseif exitflag == 1 then 
    disp("Maximum Number of Iterations Exceeded. Output may not be optimal.")
    disp(xopt', "The solution obtained is")
    disp(fopt, "The objective function value is")
else
    disp("Error encountered")
end
disp(output)

Expected Output:

Optimal Solution Found   
 
 The optimum solution obtained is   
 
    64.36    50.72  
 
 The optimum objective function is   
 
  - 2547.72  
 
   Iterations: 2
   Cpu_Time: 0.011
   Objective_Evaluation: 3
   Dual_Infeasibility: 1.167D-09
   Message: "Optimal Solution Found"

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Steady State Temperature

Steady State Temperature