CAM colloquium - Friday, February 24
3:30 p.m.
655 Rhodes Hall

Speaker: Jon Wilkening, U.C. Berkeley

 

Title: Shape Optimization of Swimming Sheets

Abstract: Motivated by the propulsion mechanisms adopted by gastropods, we consider shape optimization of a flexible sheet which moves by propagating deformation waves along its body. The self-propelled sheet is separated from a rigid substrate by a thin film of viscous Newtonian fluid. We use a lubrication approximation to model the dynamics and derive the relevant Euler-Lagrange equations to optimize swimming speed and efficiency. The optimization equations are solved numerically using two different schemes: a limited memory BFGS method that uses cubic splines to represent the wave profile, and a multi-shooting Runge-Kutta approach that uses the Levenberg-Marquardt method to vary the parameters of the equations until the constraints are satisfied. The former approach is less efficient but generalizes nicely to the non-lubrication setting. For each optimization problem, we obtain a one parameter family of solutions that become singular in a self-similar fashion as the parameter approaches a critical value. We explore the validity of the lubrication approximation near this singular limit by monitoring higher order corrections to the zeroth order theory. These corrections are themselves validated by comparison with finite element solutions of the full Stokes equations.

 

Refreshments at 4:30 in 657 Rhodes Hall.