2. Combining materials for composite-material cars Ford initiated research at a time when they took a look at making cars from composite materials. Graphite-epoxy is too expensive, glass- epoxy is not stiff enough. Grosset, L., Venkataraman, S., and Haftka, R.T., “Genetic optimization of two-material composite laminates,” Proceedings, 16th ASC Technical Meeting, Blacksburg, VA, September 2001

3. Multi-material laminate “Materials”: one material = 1 ply (  matrix or fiber materials) E.g.: glass-epoxy, graphite-epoxy, Kevlar-epoxy… Use two materials in order to combine high efficiency (stiffness) and low cost Graphite-epoxy: very stiff but expensive; glass-epoxy: less stiff, less expensive Objective: use graphite-epoxy only where most efficient, use glass-epoxy for the remaining plies

4. Multi-criterion optimization Two competing objective functions: WEIGHT and COST Design variables: –number of plies –ply orientations –ply materials No single design minimizes weight and cost simultaneously: A design is Pareto-optimal (non-dominated) if there is no design for which both Weight and Cost are lower Goal: construct the trade-off curve between weight and cost (set of Pareto-optimal designs, also called Pareto front)

5. Material properties Graphite-epoxy –Longitudinal modulus, E 1 : 20.01 10 6 psi –Transverse modulus, E 2 : 1.30 10 6 psi –Shear modulus, G 12 : 1.03 10 6 psi –Poisson ’ s ratio, 12 : 0.3 –Ply thickness, t: 0.005 in –Density,  : 5.8 10 -2 lb/in 3 –Ultimate shear strain,  ult : 1.5 10 -2 –Cost index: $8/lb Glass-epoxy –Longitudinal modulus, E 1 : 6.30 10 6 psi –Transverse modulus, E 2 : 1.29 10 6 psi –Shear modulus, G 12 : 6.60 10 5 psi –Poisson ’ s ratio, 12 : 0.27 –Ply thickness, t: 0.005 in –Density,  : 7.2 10 -2 lb/in 3 –Ultimate shear strain,  ult : 2.5 10 -2 –Cost index: $1/lb

6. Optimization problem Minimize weight and cost of a 30”x36” plate By changing ply orientations and material m i subject to a frequency constraint:

7. Constructing the Pareto trade- off curve Simple method: weighting method. A composite function is constructed by combining the 2 objectives: W: weight C: cost  : weighting parameter (0  1) A succession of optimizations with  varying from 0 to 1 is solved. The set of optimum designs builds up the Pareto trade-off curve This is not a fool-proof approach (Why?), and it is time consuming. Genetic algorithms provide a shortcut.

8. Multi-material Genetic Algorithm Two variables for each ply: –Fiber orientation –Material Each laminate is represented by 2 strings: –Orientation string –Material string Example: [45/0/30/0/90] is represented by: –Orientation:45-0-30-0-90 –Material: 2-2-1-2-1 GA maximizes fitness: Fitness = -F 1: graphite- epoxy 2: glass-epoxy

9. Simple vibrating plate problem Minimize the weight (W) and cost (C) of a 36”x30” rectangular laminated plate 19 possible ply angles from 0 to 90 in 5- degree step Constraints: –Balanced laminate (for each +  ply, there must be a -  ply in the laminate) –first natural frequency > 25 Hz Frequency calculated using Classical Lamination Theory

10. How constraints are enforced Balance constraint hard coded in the strings: stacks of ±  are used Example: (45-0-30-25-90) represents [±45/0/±30/±25/90] s Frequency constraint is incorporated into the objective function by a penalty, which is proportional to the constraint violation

11. Genetic operators Roulette wheel selection based on rank Two-point crossover Mutation and permutation Ply deletion and addition Operators apply to each chromosome individually. Best individual passed to the next generation (elitist)

12. Pareto Trade-off curve ($) (lb) A (16.3,16.3) B (6.9,55.1) C point C –64% lighter than A; 17% more expensive –53% cheaper than B; 25% heavier

13. Optimum laminates Cost minimization: [±50 10 /0] s, cost = 16.33, weight = 16.33 Weight minimization: [±50 5 /0] s, cost = 55.12, weight = 6.89 Intermediate design: [±50 2 /±50 5 ] s, cost = 27.82, weight = 10.28 Glass-epoxy in the core layers to increase the thickness Graphite-epoxy as outer plies for a high frequency Midplane Intermediate optimum laminates: sandwich-type laminates

14. Problems two-material laminate Check the reasonableness of the weight ratio of the two extreme laminates (16.3/6.9) with simple calculation. Illustrate the difference between crossover of two laminates when (a) the materials and angles are segregated into two chromosomes and (b) there is one string with the two numbers for each ply being together. Why does the figure of the Pareto front not show values of alpha between 0 and 0.7?