1. Optimization of a Flapping Wing Irina Patrikeeva HARP REU Program Mentor: Dr. Kobayashi August 3, 2011

2. 2 Problem Objectives ● Optimize design of a flapping wing and flight kinematics ● Best design = maximum lift, minimum drag, and minimum power ● Motivation ● Artificial flapping wings for air vehicles ● Exploration of feasible wing topologies ● Better understanding of flight kinematics

3. 3 Structural Model ● Wing made of thin membrane and beams ● Topology obtained by cellular division ● Uniform beam thickness beams membrane

4. 4 Kinematics ● Wing is divided into a series of span stations ● Up-down flapping motion through angle β ● Plunging motion in z-direction ● Small elastic deformations z

5. 5 Wing Topology Generation ● Propagating cellular division process ● Each edge assigned a letter ● Each letter assigned a production rule, e.g. A → B[+A]x[-A]B B → A

6. 6 Genetic Algorithms ● Wing configuration encoded as a genome ● Fitness function ● Next generation formed from most fit individuals ● Crossover ● Random mutations ● Population evolves towards an optimal solution

7. 7 Methods ● DAKOTA: Design Analysis Kit for Optimization Terascale Applications ● Extensible problem-solving environment ● Multi-objective genetic algorithm ● Interface between user supplied code and iterative system analysis method ●

8. 8 Program Flow ● Black-box interface [From DAKOTA User's Manual 5.1]

9. 9 Problem Formulation ● Optimize three functions: drag, lift and power coefficients ● Input design variables: – 1445 topology variables: wing mesh – 153 kinematics variables: flight motions ● Given lower and upper bounds ● No constraints

10. 10 Flight Representation ● Fixed frequency ω = 40 rad/s ● External flow velocity U ∞ = 10 m/s ● Angle of attack α = 4° ● 3 motions = 3 Fourier series – Plunging motion – Flapping motion – Pitching motion

11. 11 Using HOSC ● Concurrent execution of function evaluation ● DAKOTA automatically exploits parallelism ● Evaluation of 1 individual < 10 sec

12. 12 Results ● Pareto set of optimal solutions for drag C D, power C P, and lift coefficients C L ● Pareto set is a set of solutions such that it's impossible to improve one coefficient without making either of the other two worse off

13. 13 Drag-Power-Lift Pareto front Evaluations: 1000 Initial population: 50 Generations: 5 Final set of Pareto optimal solutions (red) Non-optimal solutions from all evaluations (black)

14. 14 Pareto Optimal Front ● Initial population: 50 individuals ● Set of Pareto optimal values: 159 designs

15. 15 Extremes of Pareto Front ● Lowest drag coefficient ● Lowest power coefficient ● Highest lift coefficient

16. 16 Optimal drag coefficient design ● Lowest drag coefficient – C D = 0.1672 – C P = 0.3493 – C L = 0.3418 Cellular representation Wing topology

17. 17 Optimal power coefficient design ● Lowest power coefficient – C D = 0.1780 – CP = 0.3436 – CL = 0.3157 Cellular representation Wing topology

18. 18 Optimal lift coefficient design ● Highest lift coefficient – C D = 0.1945 – CP = 0.4738 – CL = 0.5491 Cellular representation Wing topology

19. 19 Problems ● Optimization is time-consuming ● Pre-processing and post-processing ● Convergence of GA's

20. 20 Conclusions ● Multiobjective optimization drag-power, lift trade-offs ● Pareto front optimal solutions

21. 21 Acknowledgments Thank you Dr. Kobayashi, Dr. Brown, and students of HARP REU Program, and everyone else who helped make this summer great! This material is based upon work supported by the National Science Foundation under Grant No. 0852082. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

22. 22 Questions?