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Experimental flapping wing optimization and uncertainty quantification using limited samples

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Abstract

Flapping wing micro air vehicles are capable of hover and forward flight with high maneuverability. However, flapping wing flight is difficult to simulate accurately because it is a more complex phenomenon than fixed wing or rotorcraft flight. Consequently, the optimization of flapping wing behavior based on simulation is limited and, therefore, we have elected to optimize a wing experimentally. Specifically, we use experimental data to optimize the flapping wing structure for maximum thrust production in hover mode. We point out the similarities or otherwise between experimental optimization and the more common simulation-based optimization. Experimental optimization is hampered by noisy data, which is due to manufacturing variability and testing/measurement uncertainty in this study. These uncertainties must be reduced to an acceptable level and this requires their quantification. Therefore, improvements in manufacturing and testing procedures were implemented to reduce the noise. Another challenge is to limit the number of experiments for reducing time and cost. This is realized by using surrogates, or meta-models, to approximate the response (in this case, thrust) of the wing. In order to take into account the uncertainty, or noise, in the response, we use a Gaussian Process surrogate with noise and a 2nd order polynomial response surface. We apply a surrogate-based optimization algorithm called Efficient Global Optimization with different sampling criteria and multiple surrogates. This enables us to select multiple points per optimization cycle, which is especially useful in this case as it is more time efficient to manufacture multiple wings at once and this also serves as insurance against failed designs.

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Notes

  1. Wing numbers were assigned to every wing we tested. Before this stage we had 20 wings from initial DOE, 20 wings from revised design space DOE, and 6 high aspect ratio wings, totaling 46.

  2. These are actual aspect ratio values as defined by \( \frac{b^2}{S} \) and not aspect ratio numbers.

References

  • Aliakbarian B, De Faveri D, Converti A, Perego P (2008) Optimisation of olive oil extraction by means of enzyme processing aids using response surface methodology. Biochem Eng J 42(1):34–40

    Article  Google Scholar 

  • Ansari SA, Zbikowski R, Knowles K (2008a) Insectlike Flapping Wings in the Hover Part 1: Effect of Wing Kinematics. J Aircr 45(6):1945–1954

    Article  Google Scholar 

  • Ansari SA, Knowles K, Zbikowski R (2008b) Insectlike Flapping Wings in the Hover Part 2: Effect of Wing Geometry. J Aircr 45(6):1976–1990

    Article  Google Scholar 

  • Azuma A, Azuma S, Watanabe I, Furuta T (1985) Flight mechanics of a dragonfly. J Exp Biol 116(1):79–107

    Google Scholar 

  • Box GE, Wilson KB (1951) On the experimental attainment of optimum conditions, Journal of the Royal Statistical Society. J R Stat Soc Ser B 13(1):1–45

  • Brignoni LA, Garimella SV (1999) Experimental optimization of confined air jet impingement on a pin fin heat sink. Components and Packaging Technologies IEEE Transactions on 22(3):399–404

  • Chang K, Rue J, Ifju P, Haftka RT, Schmitz T, Tyler C, Chaudhuri A, Ganguly V (2014) Analysis of Thrust Production in Small Synthetic Flapping Wings, Residual Stress, Thermomechanics & Infrared Imaging, Hybrid Techniques and Inverse Problems, Springer International Publishing, 8, 1–8, Conference Proceedings of the Society for Experimental Mechanics Series

  • Chaudhuri A, Haftka RT (2014) Efficient Global Optimization with Adaptive Target Setting. AIAA J 52(7):1573–1578

    Article  Google Scholar 

  • Chimakurthi SK, Tang J, Palacios R, Cesnik CES, Shyy W (2009) Computational aeroelasticity framework for analyzing flapping wing micro air vehicles. AIAA J 47(8):1865–1878

    Article  Google Scholar 

  • Combes SA, Daniel TL (2003a) Into thin air: contributions of aerodynamic and inertial-elastic forces to wing bending in the hawkmoth Manduca sexta. J Exp Biol 206:2999–3006

    Article  Google Scholar 

  • Combes SA, Daniel TL (2003b) Flexural stiffness in insect wings II. Spatial distribution and dynamic wing bending. J Exp Biol 206:2989–2997

    Article  Google Scholar 

  • de Croon GCHE, de Clerq KME, Ruijsink R, Remes B, de Wagter C (2009) Design, aerodynamics, and vision-based control of the DelFly. International Journal of Micro Air Vehicles 1(2):71–97

    Article  Google Scholar 

  • de Lima CJ, Coelho LF, Da Silva GP, Alvarez G, Contiero J (2010) L (+) Lactic Acid Production by New Lactobacillus Rhamnosus B 103. Journal of Microbial & Biochemical Technology 2(3):64–69

    Article  Google Scholar 

  • Deelder AM, Snoijink JJ (1975) Ploem J S, “Experimental optimization of the DASS system for immunodiagnosis of some helminth infections. Ann N Y Acad Sci 254(1):119–134

    Article  Google Scholar 

  • Dey AN (1976) Experimental Optimization of Li/SOCl2 Primary Cells with Respect to the Electrolyte and the Cathode Compositions. J Electrochem Soc 123(9):1262–1264

    Article  Google Scholar 

  • Dickinson MH, Lehmann F-O, Sane SP (1999) Wing Rotation and the Aerodynamic Basis of Insect Flight. Science 284(5422):1954–1960

    Article  Google Scholar 

  • Dransfield R.D. and Brightwell R., How to Get On Top of Statistics: Design & Analysis for Biologists, with R, Influential Points LLP, 2012.

  • Ellington CP, Van den Berg C, Willmott AP, Thomas ALR (1996) Leading-edge vortices in insect flight. Nature 384(19/26):626–630

    Article  Google Scholar 

  • Elmen GW (1935) Magnetic Alloys of Iron Nickel, and Cobalt. Trans Am Inst Electr Eng 54(12):1292–1299

    Article  Google Scholar 

  • Fisher RA (1921) Studies in Crop Variation. I. An examination of the yield of dressed grain from Broadbalk, The. J Agric Sci 11(02):107–135

  • Fisher RA (1925) The influence of rainfall on the yield of wheat at Rothamsted. Philosophical Transactions of the Royal Society of London, Series B, Containing Papers of a Biological Character 213:89–142

    Article  Google Scholar 

  • Ginsbourger D., Le Riche R. and Carraro L., Kriging is well-suited to parallelize optimization, Computational Intelligence in Expensive Optimization Problems, 2, Springer Berlin Heidelberg, 131–162, 2010.

  • Glüer C-C, Blake G, Lu Y, Blunt BA, Jergas M, Genant HK (1995) Accurate assessment of precision errors: how to measure the reproducibility of bone densitometry techniques. Osteoporos Int 5(4):262–270

    Article  Google Scholar 

  • Gogulapati A, Friedmann PP (2014) Approximate aerodynamic and aeroelastic modeling of flapping wings in forward flight. AIAA J 52(1):212–218

    Article  Google Scholar 

  • Gogulapati A, Friedmann PP, Kheng E, Shyy W (2013a) Approximate aeroelastic modeling of flapping wings in hover. AIAA J 51(3):567–583

    Article  Google Scholar 

  • Gogulapati A, Friedmann PP, Martins JRRA (2013b) Optimization of the Kinematics of a Flapping wing MAV in Hover for Enhanced Performance, 54th AIAA/ASME/ASCE/AHS/ASC Structures. Structural Dynamics and Materials Conference, Boston, MA, pp 8–11. doi:10.2514/6.2013-1646

    Google Scholar 

  • Haftka RT, Scott EP, Cruz JR (1998) Optimization and experiments: a survey. Appl Mech Rev 51:435

    Article  Google Scholar 

  • Hedrick TL, Tobalske BW, Ros IG, Warrick DR, Biewener AA (2012) Morphological and kinematic basis of the hummingbird flight stroke: scaling of flight muscle transmission ratio, Proceedings of The Royal Society B (Biological sciences), 279 (1735), 1986–1992

    Google Scholar 

  • Hubback JA (1946) Sampling for rice yield in Bihar and Orissa, Sankhyā. Ind J Stat (1933–1960) 7(3):281–294, Originally published by Government of India as Bulletin No. 166, 1927

  • Jones DR (2001) A taxonomy of global optimization methods based on response surfaces. J Glob Optim 21(4):345–383

    Article  MATH  Google Scholar 

  • Jones D, Schonlau M, Welch W (1998) Efficient global optimization of expensive black-box functions. J Glob Optim 13(4):455–492

    Article  MATH  MathSciNet  Google Scholar 

  • Jongerius SR, Lentink D (2010) Structural Analysis of a Dragonfly Wing. Exp Mech 50(9):1323–1334

    Article  Google Scholar 

  • Kawamura Y., Souda S., Nishimoto S. and Ellington C.P., Clapping-wing micro air vehicle of insect size, Bio-mechanisms of swimming and flying, Kato N and Kamimura S (eds), Springer Verlag, 319–330, 2008.

  • Keenmon M, Klingebiel K, Won H, Andriukov A (2012) Development of the Nano Hummingbird: A Tailless Flapping Wing Micro Air Vehicle, 50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition. Nashville, Tennessee, pp 9–12. doi:10.2514/6.2012-588

    Google Scholar 

  • Kesel AB, Philippi U, Nachtigall W (1998) Biomechanical aspects of the insect wing: an analysis using the finite element method. Comput Biol Med 28(4):423–437

    Article  Google Scholar 

  • Kim H, Papila M, Mason WH, Haftka RT, Watson LT, Grossman B (2001) Detection and Repair of Poorly Converged Optimization Runs. AIAA J 39(12):2242–2249

    Article  Google Scholar 

  • King DL, Buck ME (1991) Experimental optimization of an anisotropic etching process for random texturization of silicon solar cells, Photovoltaic specialists conference, conference record of the twenty second IEEE (pp 303–308), October

  • Lentink D, Jongerius SR, Bradshaw NL (2009) The scalable design of flapping micro air vehicles inspired by insect flight. In: Floreano D, Zufferey JC, Srinivasan MV, Ellington CP (eds) Flying insects and robots. Springer, Berlin, Heidelberg, pp 185–205, The scalable design of flapping micro air vehicles inspired by insect flight

  • Mahalanobis PC (1946) Sample surveys of crop yields in India, Sankhyā: The Indian Journal of Statistics (1933–1960) 7(3):269–280

  • Mahalanobis PC (1967) The Sample Census of the Area Under Jute in Bengal in 1940, Sankhyā: The Indian Journal of Statistics, Series B 29(1/2):81–182

  • Mckay MD, Beckman RJ, Conover WJ (1979) A comparison of three methods for selecting values of input variables from a computer code. Technometrics 21:239–245

    MATH  MathSciNet  Google Scholar 

  • O'Hara RP, Palazotto AN (2012) The morphological characterization of the forewing of the Manduca sexta species for the application of biomimetic flapping wing micro air vehicles. Bioinspir Biomim 7(4):046011

  • Otto JC, Landman D, Patera AT (1996) A surrogate approach to the experimental optimization of multielement airfoils, Proceedings of the Forum on CFD for Design and Optimization, ASME International Mechanical Engineering Conference and Exposition, San Francisco, CA (pp. 4–6), September, 1996

  • Parajó JC, Santos V, Domínguez H, Vázquez M (1995) NH4OH-Based pretreatment for improving the nutritional quality of single-cell protein (SCP). Appl Biochem Biotechnol 55(2):133–149

    Article  Google Scholar 

  • Pornsin-Sirirak TN, Tai YC, Ho CH, Keennon M (2001) Microbat-A palm-sized electrically powered omithopter, NASA/JPL Workshop on Biomorphic Robotics. Pasadena, USA

    Google Scholar 

  • Pourtakdoust SH, Aliabadi SK (2012) Evaluation of flapping wing propulsion based on a new experimentally validated aeroelastic model. Scientia Iranica 19(3):472–482

    Article  Google Scholar 

  • Price KV, Storn RM, Lampinen JA (2005) Differential Evolution: A practical Approach to Global Optimization. Springer, New York

    Google Scholar 

  • Ren H, Wang X, Li X, Chen Y (2013) Effects of Dragonfly Wing Structure on the Dynamic Performances. Journal of Bionic Engineering 10(1):28–38

    Article  Google Scholar 

  • Rue J, Chang K, Ifju P, Haftka RT, Schmitz T, McIntire J, Tyler C, Ganguly V, Chaudhuri A (2013) Fabrication and analysis of small flapping wings. In: Imaging methods for novel materials and challenging applications, 3rd edn. Springer, New York, pp 337–344

  • Schutte JF, Haftka RT, Fregly BJ (2007) Improved global convergence probability using multiple independent optimizations. Int J Numer Methods Eng 76(6):678–702

    Article  Google Scholar 

  • Shkarayev S, Silin D (2010) Applications of Actuator Disk Theory to Membrane Flapping Wings. AIAA J 48:2227–2234

    Article  Google Scholar 

  • Shyy W, Aono H, Chimakurthi SK, Trizila P, Kang CK, Cesnik CES, Liu H (2010) Recent progress in flapping wing aerodynamics and aeroelasticity. Prog Aerosp Sci 46(7):284–327

    Article  Google Scholar 

  • Tamai M, Wang Z, Rajagopalan G, Hui H, He G (2007) Aerodynamic performance of a corrugated dragonfly airfoil compared with smooth airfoils at low Reynolds numbers, Proceedings of the 45th AIAA Aerospace Sciences Meeting and Exhibit. Reno, Nevada, pp 8–11. doi:10.2514/6.2007-483

    Google Scholar 

  • Tanyildizi MS, Özer D, Elibol M (2005) Optimization of α-amylase production by Bacillus sp. using response surface methodology. Process Biochem 40(7):2291–2296

    Article  Google Scholar 

  • Trietsch D (1999) Statistical quality control: a loss minimization approach, World Sci, pp 170–173

  • Viana FAC, Haftka RT, Steffen V Jr (2009) Multiple surrogates: how cross-validation errors can help us to obtain the best predictor. Struct Multidiscip Optim 39(4):439–457

    Article  Google Scholar 

  • Viana FAC, Haftka RT (2010) Surrogate-based optimization with parallel simulations using the probability of improvement, 13th AIAA/ ISSMO Multidisciplinary analysis and optimization conference, Fort Worth, USA, September 13–15

  • Viana FAC, Haftka RT, Watson LT (2012) Efficient global optimization assisted by multiple surrogate techniques. J Glob Optim 56(2):669–689. doi:10.1007/s10898-012-9892-5

    Article  Google Scholar 

  • Viana FAC, Haftka RT, Hamman R, Venter G (2011) Efficient global optimization with experimental data: revisiting the paper helicopter design, 52nd AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics and materials conference, Denver, Colorado, 4–7 April

  • Willmott AP, Ellington CP (1997) The mechanics of flight in the hawkmoth Manduca sexta I Kinematics of hovering and forward flight. J Exp Biol 200:2705–2722

    Google Scholar 

  • Wootton RJ, Herbert RC, Young PG, Evans KE (2003) Approaches to the structural modeling of insect wings. Phil Trans R Soc B 358:1577–1587

    Article  Google Scholar 

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Acknowledgments

This work was supported by Air Force Office of Scientific Research (AFOSR) grant FA9550-11-1-0066 from Dr. David Stargel, Grant Monitor. The authors would also like to thank Dr. Diane Villanueva for valuable comments and discussions on the research.

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Authors and Affiliations

  1. Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL, USA

    Anirban Chaudhuri, Raphael T. Haftka, Peter Ifju & Kelvin Chang

  2. Department of Mechanical Engineering and Engineering Sciences, University of North Carolina at Charlotte, Charlotte, NC, USA

    Christopher Tyler & Tony Schmitz

Authors
  1. Anirban Chaudhuri
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  2. Raphael T. Haftka
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  3. Peter Ifju
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  4. Kelvin Chang
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  5. Christopher Tyler
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Correspondence to Anirban Chaudhuri.

Appendix: Length of leading edge and root for different aspect ratios of 2nd EGO cycle

Appendix: Length of leading edge and root for different aspect ratios of 2nd EGO cycle

Keeping the area of the wing constant at 2,945.2 mm2, the lengths of leading edge and root were found by calculating all the possible errors incurred in the area due to rounding off the lengths to a single decimal place and selecting the option that was closest in realizing this area constraint. The rounding off of the lengths to one decimal place was required in order to comply with the precision of the manufacturing process. The values of these lengths are given in Table 15.

Table 15 Lengths of leading edge and root for the aspect ratios predicted by 2nd EGO cycle
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Chaudhuri, A., Haftka, R.T., Ifju, P. et al. Experimental flapping wing optimization and uncertainty quantification using limited samples. Struct Multidisc Optim 51, 957–970 (2015). https://doi.org/10.1007/s00158-014-1184-x

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