Optimization of a Flapping Wing

Irina Patrikeeva

HARP REU Program

Mentor: Dr. Kobayashi

August 3, 2011

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ProblemObjectives● 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

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

beamsmembrane

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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

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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]BB → A

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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

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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●

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Program Flow

● Black-box interface

[From DAKOTA User's Manual 5.1]

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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

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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

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Using HOSC

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

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Results

● Pareto set of optimal solutions for drag CD, power CP, and lift coefficients CL

● 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

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Drag-Power-Lift Pareto front

Evaluations: 1000Initial population: 50Generations: 5

Final set of Pareto optimal solutions (red)

Non-optimal solutions from all evaluations (black)

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

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

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Optimal drag coefficient design

● Lowest drag coefficient

– CD = 0.1672

– CP = 0.3493

– CL = 0.3418Cellular representation

Wing topology

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Optimal power coefficient design

● Lowest power coefficient

– CD = 0.1780

– CP = 0.3436– CL = 0.3157

Cellular representation

Wing topology

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Optimal lift coefficient design

● Highest lift coefficient

– CD = 0.1945

– CP = 0.4738– CL = 0.5491

Cellular representation

Wing topology

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Problems

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

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Conclusions

● Multiobjective optimization drag-power, lift trade-offs

● Pareto front optimal solutions

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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.

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Questions?