Shape Optimization of Peristaltic Pumping
Shawn Walker, CIMS


Transport is a fundamental aspect of biology and peristaltic pumping is a  fundamental mechanism to accomplish this; it is also important in many  industrial processes.  We present a variational method for optimizing  peristaltic pumping in a two dimensional periodic channel with moving  walls to pump fluid.  No a prior  assumption is made on the wall motion,  except that the shape is static in a moving wave frame.  Thus, we pose an  infinite dimensional optimization problem and solve it with finite  elements.  Sensitivities of the cost and constraints are computed  variationally via shape differential calculus and $L^2$-type projections  are used to compute quantities such as curvature and boundary stresses.  Our Optimization method falls under the category of sequential quadratic  programming (SQP) methods.  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.