2. 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).

3. 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?

4. Example 1.4.1 (but differently)

5. 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).

6. Analysis

7. Getting acquainted via Matlab Em=3.45; Ef=124;r=Em/Ef; Vf=linspace(0,0.4,101); E1oEf=Vf+r*(1-Vf); E1oE2=Vf.^2+Vf.*(1-Vf).*(1/r+r)+(1-Vf).^2; E2oE1=1./E1oE2; plot(Vf,E1oEf,Vf,E2oE1,'r-') xlabel('Vf'); legend('E1/Ef','E2/E1','Location','North')

8. Solution Narrow range and plot constraints Feasible domain when both are negative Vf=linspace(0.1,0.3,101); E1oEf=Vf+r*(1-Vf); E1oE2=Vf.^2+Vf.*(1-Vf).*(1/r+r)+(1-Vf).^2; E2oE1=1./E1oE2; g1=0.2-E1oEf;g2=0.15-E2oE1; plot(Vf,g1,Vf,g2,'r-') xlabel('Vf'); ylabel('g1,g2') legend('g1','g2','Location','North') Solution is 0.1675. Textbook says 0.18 (one student found that the denominator should be 1-r not 1+r, giving 0.1771

9. 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.