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Optimization

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?

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Optimization

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  1. 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).

  2. 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?

  3. Example 1.4.1 (but differently) • Choose the fiber fraction for Kevlar epoxy, so as to minimize it, and satisfy the following constraints • Objective function is equal to the single design variable. • To formulate constraints we need the moduli of fiber and matrix

  4. 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).

  5. Analysis • It pays to invest in a bit of algebra • This reveals that the solution will not depend on the individual moduli but only on their ratio. • For =0 get

  6. Getting acquainted via Matlab • Plot ratios as function of volume fraction • Solution near =0.2 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')

  7. 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

  8. 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.

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