Solve constrained nonlinear optimization problems using exterior penalty function and ALM. When solving the following problems, there are a few best practices that you should remember:
• Convert side bounds into a pair of inequality constraints
• All problems must be posed in standard form
• Normalize all constraints
1. The two-bar truss shown is symmetric about the y axis. The design variables x1 and x2, respectively, represent the nondimensional area of cross section of the members A1/Aref and the nondimensional position of joints 1 and 2, x/h. Here, Aref is the reference value of the area (A) and h is the height of the truss. The coordinates of joint 3 are held constant. The weight of the truss F(X) is to be minimized without exceeding the permissible stress, σ0.
The weight of the truss is given by:
F(X) = 2ρ h x2 Aref √(1 + x1^2)
Where ρ is the weight density, P is the applied load, and E the Young's modulus. The stresses induced in members 1 and 2 (σ1 and σ2) are given by:
σ1(X) = P(1 + x1) √(1 + x1^2) / (2√2 x1 x2 Aref)
σ2(X) = P(x1 − 1) √(1 + x1^2) / (2√2 x1 x2 Aref)
In addition, upper and lower bounds are placed on the design variables as:
0.1 ≤ x1 ≤ 2.0
0.1 ≤ x2 ≤ 2.3
Use a starting location of [3; 3] and find the solution to the problem using the following data: E = 30 × 10^6 psi, ρ = 0.283 lb/in^3, P = 12,000 lb, σ0 = 20,000 psi, h = 90 in, Aref = 1 in^2.
Optimize the problem using Exterior penalty function and Augmented Lagrangian Method. Document your current design variable location and the constraint values for each iteration.