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Energetically Optimal Flapping Wing Motions via Adjoint-Based Optimization and High-Order Discretizations

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Frontiers in PDE-Constrained Optimization

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 163))

Abstract

A globally high-order numerical discretization of time-dependent conservation laws on deforming domains, and the corresponding fully discrete adjoint method, is reviewed and applied to determine energetically optimal flapping wing motions subject to aerodynamic constraints using a reduced space PDE-constrained optimization framework. The conservation law on a deforming domain is transformed to one on a fixed domain and discretized in space using a high-order discontinuous Galerkin method. An efficient, high-order temporal discretization is achieved using diagonally implicit Runge-Kutta schemes. Quantities of interest, such as the total energy required to complete a flapping cycle and the integrated forces produced on the wing, are discretized in a solver-consistent way, that is, via the same spatiotemporal discretization used for the conservation law. The fully discrete adjoint method is used to compute discretely consistent gradients of the quantities of interest and passed to a black-box, gradient-based nonlinear optimization solver. This framework successfully determines an energetically optimal flapping trajectory such that the net thrust of the wing is zero to within 9 digits after only 12 optimization iterations.

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References

  1. Roger Alexander. Diagonally implicit Runge-Kutta methods for stiff ODEs. SIAM J. Numer. Anal., 14(6):1006–1021, 1977.

    Article  MathSciNet  MATH  Google Scholar 

  2. Yves Allaneau, Matthew Culbreth, and Antony Jameson. A computational framework for low Reynolds number 3d flapping wings simulations. In 20th AIAA Computational Fluid Dynamics Conference, Honolulu, Hawaii, June 27–30 2011.

    Google Scholar 

  3. Douglas N Arnold, Franco Brezzi, Bernardo Cockburn, and L Donatella Marini. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM Journal on Numerical Analysis, 39(5):1749–1779, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  4. Sai K Banala and Sunil K Agrawal. Design and optimization of a mechanism for out-of-plane insect winglike motion with twist. Journal of Mechanical Design, 127(4):841–844, 2005.

    Article  Google Scholar 

  5. James M Birch, William B Dickson, and Michael H Dickinson. Force production and flow structure of the leading edge vortex on flapping wings at high and low Reynolds numbers. Journal of Experimental Biology, 207(7):1063–1072, 2004.

    Article  Google Scholar 

  6. Frank M Bos, Bas W van Oudheusden, and Hester Bijl. Wing performance and 3-d vortical structure formation in flapping flight. Journal of Fluids and Structures, 42:130–151, 2013.

    Article  Google Scholar 

  7. Chris Chabalko, Richard D Snyder, Philip S Beran, and Gregory Parker. The physics of an optimized flapping wing micro air vehicle. In Proceedings of the 47th AIAA Aerospace Science Meeting Including The New Horizons Forum and Aerospace Exposition, AIAA Paper, number 2009–801, Orlando, Florida, January 5–8 2009.

    Google Scholar 

  8. Tony F Chan and Wing Lok Wan. Analysis of projection methods for solving linear systems with multiple right-hand sides. SIAM Journal on Scientific Computing, 18(6):1698–1721, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  9. Jung-Sun Choi, Liangyu Zhao, Gyung-Jin Park, Sunil K Agrawal, and Raymond K Kolonay. Enhancement of a flapping wing using path and dynamic topology optimization. AIAA Journal, 49(12):2616–2626, December 2011.

    Article  Google Scholar 

  10. Bernardo Cockburn and Chi-Wang Shu. Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput., 16(3):173–261, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  11. Matthew Culbreth, Yves Allaneau, and Antony Jameson. High-fidelity optimization of flapping airfoils and wings. In 29th AIAA Applied Aerodynamics Conference, Honolulu, Hawaii, June 27–30 2011.

    Google Scholar 

  12. Benoît Desjardins, Emmanuel Grenier, P-L Lions, and Nader Masmoudi. Incompressible limit for solutions of the isentropic Navier–Stokes equations with Dirichlet boundary conditions. Journal de Mathématiques Pures et Appliquées, 78(5):461–471, 1999.

    Article  MathSciNet  MATH  Google Scholar 

  13. Michael H Dickinson, Fritz-Olaf Lehmann, and Sanjay P Sane. Wing rotation and the aerodynamic basis of insect flight. Science, 284(5422):1954–1960, 1999.

    Article  Google Scholar 

  14. Charbel Farhat, Christoph Degand, Bruno Koobus, and Michel Lesoinne. Torsional springs for two-dimensional dynamic unstructured fluid meshes. Computer methods in applied mechanics and engineering, 163(1):231–245, 1998.

    Article  MATH  Google Scholar 

  15. Charbel Farhat, Philippe Geuzaine, and Céline Grandmont. The discrete geometric conservation law and the nonlinear stability of ale schemes for the solution of flow problems on moving grids. Journal of Computational Physics, 174(2):669–694, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  16. Bradley Froehle and Per-Olof Persson. Nonlinear elasticity for mesh deformation with high-order discontinuous Galerkin methods for the Navier-Stokes equations on deforming domains. In Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2014, pages 73–85. Springer, 2015.

    Google Scholar 

  17. Bradley Michael Froehle. High-order discontinuous Galerkin fluid-structure interaction methods. University of California, Berkeley, 2013.

    Google Scholar 

  18. Ryan B George, Mark B Colton, Christopher A Mattson, and Scott L Thomson. A differentially driven flapping wing mechanism for force analysis and trajectory optimization. International Journal of Micro Air Vehicles, 4(1):31–49, 2012.

    Article  Google Scholar 

  19. Mehdi Ghommem, Nathan Collier, Antti H Niemi, and Victor M Calo. On the shape optimization of flapping wings and their performance analysis. Aerospace Science and Technology, 32(1):274–292, 2014.

    Article  Google Scholar 

  20. Mehdi Ghommem, Muhammad R Hajj, Dean T Mook, Bret K Stanford, Philip S Beran, Richard D Snyder, and Layne T Watson. Global optimization of actively morphing flapping wings. Journal of Fluids and Structures, 33:210–228, 2012.

    Article  Google Scholar 

  21. Peter A Gnoffo et al. CFD validation studies for hypersonic flow prediction. AIAA paper, 1025:2001, 2001.

    Google Scholar 

  22. Peter G Ifju, David A Jenkins, Scott Ettinger, Yongsheng Lian, Wei Shyy, and Martin R Waszak. Flexible-wing-based micro air vehicles. In 40th AIAA Aerospace Sciences Meeting & Exhibit, number 2002–0705, pages 1–5, Reno, Nevada, January 14–17 2002.

    Google Scholar 

  23. Antony Jameson, Wolfgang Schmidt, Eli Turkel, et al. Numerical solutions of the Euler equations by finite volume methods using Runge-Kutta time-stepping schemes. AIAA paper, 1259:1981, 1981.

    Google Scholar 

  24. KD Jones and MF Platzer. Numerical computation of flapping-wing propulsion and power extraction. In 35th Aerospace Sciences Meeting and Exhibit, volume 97, page 0826, Reno, Nevada, January 6–9 1997. AIAA.

    Google Scholar 

  25. Martin Jones and Nail K Yamaleev. Adjoint based shape and kinematics optimization of flapping wing propulsive efficiency. 43rd AIAA Fluid Dynamics Conference. San Diego, CA, 2013. AIAA 2013–2472, 2013.

    Google Scholar 

  26. Bruno Koobus and Charbel Farhat. Second-order time-accurate and geometrically conservative implicit schemes for flow computations on unstructured dynamic meshes. Computer Methods in Applied Mechanics and Engineering, 170(1):103–129, 1999.

    Article  MathSciNet  MATH  Google Scholar 

  27. Fritz-Olaf Lehmann. The mechanisms of lift enhancement in insect flight. Naturwissenschaften, 91(3):101–122, 2004.

    Article  Google Scholar 

  28. Michel Lesoinne and Charbel Farhat. Geometric conservation laws for flow problems with moving boundaries and deformable meshes, and their impact on aeroelastic computations. Computer methods in applied mechanics and engineering, 134(1):71–90, 1996.

    Article  MATH  Google Scholar 

  29. Chi-Kun Lin. On the incompressible limit of the compressible Navier-Stokes equations. Communications in partial differential equations, 20(3–4):677–707, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  30. Karthik Mani and Dimitri J Mavriplis. Unsteady discrete adjoint formulation for two-dimensional flow problems with deforming meshes. AIAA Journal, 46(6):1351–1364, 2015/06/22 2008.

    Article  Google Scholar 

  31. V Maple. Waterloo maple software. University of Waterloo, Version, 5, 1994.

    Google Scholar 

  32. James M. McMichael and Michael S. Francis. Micro air vehicles - towards a new dimension in flight. Technical report, DARPA, August 7 1997.

    Google Scholar 

  33. Siva K. Nadarajah and Antony Jameson. Optimum shape design for unsteady flows with time-accurate continuous and discrete adjoint method. AIAA Journal, 45(7):1478–1491, 2007.

    Article  Google Scholar 

  34. Eric J Nielsen, Boris Diskin, and Nail K Yamaleev. Discrete adjoint-based design optimization of unsteady turbulent flows on dynamic unstructured grids. AIAA Journal, 48(6):1195–1206, 2010.

    Article  Google Scholar 

  35. Kui Ou, Patrice Castonguay, and Antony Jameson. 3d flapping wing simulation with high order spectral difference method on deformable mesh. In 49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, volume 1316, page 2011, Orlando, Florida, January 4–7 2011.

    Google Scholar 

  36. Will Pazner and Per-Olof Persson. Stage-parallel fully implicit Runge-Kutta solvers for discontinuous Galerkin fluid simulations. Journal of Computational Physics, 2016.

    Google Scholar 

  37. Jaime Peraire and Per-Olof Persson. The Compact Discontinuous Galerkin (CDG) method for elliptic problems. SIAM Journal on Scientific Computing, 30(4):1806–1824, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  38. Per-Olof Persson, Javier Bonet, and Jaime Peraire. Discontinuous Galerkin solution of the Navier–Stokes equations on deformable domains. Computer Methods in Applied Mechanics and Engineering, 198(17):1585–1595, 2009.

    Article  MATH  Google Scholar 

  39. Per-Olof Persson and Jaime Peraire. Curved mesh generation and mesh refinement using Lagrangian solid mechanics. In Proceedings of the 47th AIAA Aerospace Sciences Meeting and Exhibit, volume 204, 2009.

    Google Scholar 

  40. Ravi Ramamurti and William Sandberg. Simulation of flow about flapping airfoils using finite element incompressible flow solver. AIAA Journal, 39(2):253–260, 2001.

    Article  Google Scholar 

  41. Philip L Roe. Approximate Riemann solvers, parameter vectors, and difference schemes. Journal of Computational Physics, 43(2):357–372, 1981.

    Article  MathSciNet  MATH  Google Scholar 

  42. Wei Shyy, Hikaru Aono, Satish Kumar Chimakurthi, P Trizila, C-K Kang, Carlos ES Cesnik, and Hao Liu. Recent progress in flapping wing aerodynamics and aeroelasticity. Progress in Aerospace Sciences, 46(7):284–327, 2010.

    Article  Google Scholar 

  43. Wei Shyy, Mats Berg, and Daniel Ljungqvist. Flapping and flexible wings for biological and micro air vehicles. Progress in aerospace sciences, 35(5):455–505, 1999.

    Article  Google Scholar 

  44. Wei Shyy, Y Lian, J Tang, H Liu, P Trizila, B Stanford, L Bernal, C Cesnik, P Friedmann, and P Ifju. Computational aerodynamics of low Reynolds number plunging, pitching and flexible wings for MAV applications. Acta Mechanica Sinica, 24(4):351–373, 2008.

    Article  MATH  Google Scholar 

  45. Valeria Simoncini and Efstratios Gallopoulos. An iterative method for nonsymmetric systems with multiple right-hand sides. SIAM Journal on Scientific Computing, 16(4):917–933, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  46. Bret K Stanford and Philip S Beran. Analytical sensitivity analysis of an unsteady vortex-lattice method for flapping-wing optimization. Journal of Aircraft, 47(2):647–662, 2010.

    Article  Google Scholar 

  47. WB Tay, BW Van Oudheusden, and H Bijl. Numerical simulation of X-wing type biplane flapping wings in 3D using the immersed boundary method. Bioinspiration & Biomimetics, 9(3), March 14 2014.

    Article  Google Scholar 

  48. Alexandra H Techet. Propulsive performance of biologically inspired flapping foils at high Reynolds numbers. Journal of Experimental Biology, 211(2):274–279, 2008.

    Article  MathSciNet  Google Scholar 

  49. Ismail H Tuncer and Mustafa Kaya. Optimization of flapping airfoils for maximum thrust and propulsive efficiency. AIAA Journal, 43(11):2329–2336, 2005.

    Article  Google Scholar 

  50. James R Usherwood and Charles P Ellington. The aerodynamics of revolving wings ii. propeller force coefficients from mayfly to quail. Journal of Experimental Biology, 205(11):1565–1576, 2002.

    Google Scholar 

  51. Marnix P van Schrojenstein Lantman and K Fidkowski. Adjoint-based optimization of flapping kinematics in viscous flows. In 21st AIAA Computational Fluid Dynamics Conference, 2013.

    Google Scholar 

  52. Andreas Wächter and Lorenz T Biegler. On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Mathematical programming, 106(1):25–57, 2006.

    Google Scholar 

  53. Jingyi Wang, Matthew J Zahr, and Per-Olof Persson. Energetically optimal flapping flight via a fully discrete adjoint method with explicit treatment of flapping frequency. In 23rd AIAA Computational Fluid Dynamics Conference, page 4412, 2017.

    Google Scholar 

  54. ZJ Wang, Krzysztof Fidkowski, Rémi Abgrall, Francesco Bassi, Doru Caraeni, Andrew Cary, Herman Deconinck, Ralf Hartmann, Koen Hillewaert, HT Huynh, et al. High-order CFD methods: current status and perspective. International Journal for Numerical Methods in Fluids, 72(8):811–845, 2013.

    Article  MathSciNet  Google Scholar 

  55. David J Willis, Emily R Israeli, Per-Olof Persson, Mark Drela, Jaime Peraire, SM Swartz, and Kenneth S Breuer. A computational framework for fluid structure interaction in biologically inspired flapping flight. In 25th AIAA Applied Aerodynamics Conference, volume 1, pages 38–59, Miami, Florida, June 25–28 2007.

    Google Scholar 

  56. David J Willis, Per-Olof Persson, Emily R Israeli, Jaime Peraire, Sharon M Swartz, and Kenneth S Breuer. Multifidelity approaches for the computational analysis and design of effective flapping wing vehicles. In 46th AIAA Aerospace Sciences Meeting and Exhibit, page 2008, Reno, Nevada, January 7–10 2008.

    Google Scholar 

  57. Nail Yamaleev, Boris Diskin, and Eric Nielsen. Adjoint-based methodology for time-dependent optimization. In 12th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference. American Institute of Aeronautics and Astronautics, 2008.

    Google Scholar 

  58. Zhi Yang and Dimitri J Mavriplis. Unstructured dynamic meshes with higher-order time integration schemes for the unsteady Navier-Stokes equations. AIAA paper, 1222(2005):1, 2005.

    Google Scholar 

  59. Matthew J. Zahr. Adaptive model reduction to accelerate optimization problems governed by partial differential equations. PhD thesis, Stanford University, August 2016.

    Google Scholar 

  60. Matthew J Zahr and Charbel Farhat. Progressive construction of a parametric reduced-order model for PDE-constrained optimization. International Journal for Numerical Methods in Engineering, 102(5):1111–1135, 2015.

    Google Scholar 

  61. Matthew J. Zahr and Per-Olof Persson. Performance tuning of Newton-GMRES methods for discontinuous Galerkin discretizations of the Navier-Stokes equations. In 21st AIAA Computational Fluid Dynamics Conference. American Institute of Aeronautics and Astronautics, 2013.

    Google Scholar 

  62. Matthew J Zahr and Per-Olof Persson. High-order, time-dependent aerodynamic optimization using a discontinuous Galerkin discretization of the Navier-Stokes equations. In AIAA Science and Technology Forum and Exposition, San Diego, CA, 2016.

    Google Scholar 

  63. Matthew J. Zahr and Per-Olof Persson. An adjoint method for a high-order discretization of deforming domain conservation laws for optimization of flow problems. Journal of Computational Physics, 326, 516–543, 2016.

    Article  MathSciNet  MATH  Google Scholar 

  64. Matthew J Zahr, Per-Olof Persson, and Jon Wilkening. A fully discrete adjoint method for optimization of flow problems on deforming domains with time-periodicity constraints. Computers & Fluids, Special Issue on USNCCM13 International Symposium on Spectral and High-Order Methods, 2016.

    Google Scholar 

  65. Qiang Zhu. Numerical simulation of a flapping foil with chordwise or spanwise flexibility. AIAA Journal, 45(10):2448–2457, October 2007.

    Article  Google Scholar 

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Acknowledgements

This work was supported in part by the Luis Alvarez Postdoctoral Fellowship by the Director, Office of Science, Office of Advanced Scientific Computing Research, of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231 (MZ), and by the Director, Office of Science, Computational and Technology Research, U.S. Department of Energy under contract number DE-AC02-05CH11231 (PP). The content of this publication does not necessarily reflect the position or policy of any of these supporters, and no official endorsement should be inferred.

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Authors and Affiliations

  1. Mathematics Group, Lawrence Berkeley National Laboratory, Berkeley, CA, 94720, USA

    Matthew J. Zahr & Per-Olof Persson

  2. Department of Mathematics, University of California, Berkeley, Berkeley, CA, 94720, USA

    Matthew J. Zahr & Per-Olof Persson

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  1. Matthew J. Zahr
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  2. Per-Olof Persson
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Correspondence to Matthew J. Zahr .

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Editors and Affiliations

  1. Department of Mathematical Sciences, George Mason University, Fairfax, VA, USA

    Harbir Antil

  2. Center for Computing Research, Sandia National Laboratories, Albuquerque, NM, USA

    Drew P. Kouri

  3. Corporate Strategic Research, ExxonMobil Research and Engineering Company, Annandale, NJ, USA

    Martin-D. Lacasse

  4. Center for Computing Research, Sandia National Laboratories, Albuquerque, NM, USA

    Denis Ridzal

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Zahr, M.J., Persson, PO. (2018). Energetically Optimal Flapping Wing Motions via Adjoint-Based Optimization and High-Order Discretizations. In: Antil, H., Kouri, D.P., Lacasse, MD., Ridzal, D. (eds) Frontiers in PDE-Constrained Optimization. The IMA Volumes in Mathematics and its Applications, vol 163. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-8636-1_7

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