Optimization Optimization is the methodology for obtaining the

Optimization • Optimization is the methodology for obtaining the best alternative from all possible ones. • Unconstrained optimization has only an objective function. – What is the route and travel speed that minimize the distance, or travel time, or number of turns from UF to your home? • Design variables are road segments and speeds. • In constrained optimization we add constraints. – No more than 3 stop lights on the way (inequality). – Drive exactly at speed limit (equality constraint).

Standard formulation • The standard formulation of an optimization problem is: • Minimize: What do we do for maximization? • Inequalities have less than zero. What do we do when we have the opposite sense?

Example 1. 4. 1 (but differently) •

Formulation • Normalize and standardize constraints • Optimization problem • Need analysis to calculate constraints in terms of design variable (often comes from a computer code rather than an equation).

Analysis •

Getting acquainted via Matlab • Em=3. 45; Ef=124; r=Em/Ef; Vf=linspace(0, 0. 4, 101); E 1 o. Ef=Vf+r*(1 -Vf); E 1 o. E 2=Vf. ^2+Vf. *(1 -Vf). *(1/r+r)+(1 -Vf). ^2; E 2 o. E 1=1. /E 1 o. E 2; plot(Vf, E 1 o. Ef, Vf, E 2 o. E 1, 'r-') xlabel('Vf'); legend('E 1/Ef', 'E 2/E 1', 'Location', 'North')

Solution • Narrow range and plot constraints Vf=linspace(0. 1, 0. 3, 101); E 1 o. Ef=Vf+r*(1 -Vf); E 1 o. E 2=Vf. ^2+Vf. *(1 -Vf). *(1/r+r)+(1 -Vf). ^2; E 2 o. E 1=1. /E 1 o. E 2; g 1=0. 2 -E 1 o. Ef; g 2=0. 15 -E 2 o. E 1; plot(Vf, g 1, Vf, g 2, 'r-') xlabel('Vf'); ylabel('g 1, g 2') legend('g 1', 'g 2', 'Location', 'North') • Feasible domain when both are negative Solution is 0. 1675. Textbook says 0. 18 (one student found that the denominator should be 1 -r not 1+r, giving 0. 1771

Stacking sequence optimization • Most of this course is concerned with optimization of the angles of the fiber in a laminate composed of unidirectional plies. • The objective function is either the thickness (total number of plies) or the load that can be carried by a laminate of given thickness. • Constraints will be on strains, stresses, buckling loads and natural frequencies.