Shape optimization of peristaltic pumping

Shawn W. Walker, Michael J. Shelley

Research output: Contribution to journalArticlepeer-review


Transport is a fundamental aspect of biology and peristaltic pumping is a fundamental mechanism to accomplish this; it is also important to many industrial processes. We present a variational method for optimizing the wave shape of a peristaltic pump. Specifically, we optimize the wave profile of a two dimensional channel containing a Navier-Stokes fluid with no assumption on the wave profile other than it is a traveling wave (e.g. we do not assume it is the graph of a function). Hence, this is an infinite-dimensional optimization problem. The optimization criteria consists of minimizing the input fluid power (due to the peristaltic wave) subject to constraints on the average flux of fluid and area of the channel. Sensitivities of the cost and constraints are computed variationally via shape differential calculus and we use a sequential quadratic programming (SQP) method to find a solution of the first order KKT conditions. We also use a merit-function based line search in order to balance between decreasing the cost and keeping the constraints satisfied when updating the channel shape. Our numerical implementation uses a finite element method for computing a solution of the Navier-Stokes equations, adjoint equations, as well as for the SQP method when computing perturbations of the channel shape. The walls of the channel are deformed by an explicit front-tracking approach. In computing functional sensitivities with respect to shape, we use L2-type projections for computing boundary stresses and for geometric quantities such as the tangent field on the channel walls and the curvature; we show error estimates for the boundary stress and tangent field approximations. As a result, we find optimized shapes that are not obvious and have not been previously reported in the peristaltic pumping literature. Specifically, we see highly asymmetric wave shapes that are far from being sine waves. Many examples are shown for a range of fluxes and Reynolds numbers up to Re = 500 which illustrate the capabilities of our method.

Original languageEnglish (US)
Pages (from-to)1260-1291
Number of pages32
JournalJournal of Computational Physics
Issue number4
StatePublished - Feb 20 2010


  • Finite element method
  • PDE-constrained optimization
  • Peristalsis
  • Pumping
  • Shape optimization
  • Variational method

ASJC Scopus subject areas

  • Numerical Analysis
  • Modeling and Simulation
  • Physics and Astronomy (miscellaneous)
  • General Physics and Astronomy
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics


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