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

  • Matthew J. ZahrEmail author
  • Per-Olof Persson
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 163)


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.



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. 1.Mathematics GroupLawrence Berkeley National LaboratoryBerkeleyUSA
  2. 2.Department of MathematicsUniversity of California, BerkeleyBerkeleyUSA

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