Shape Optimization of Chiral Propellers in 3-D Stokes Flow
Shawn Walker

Abstract: Locomotion at the micro-scale is important in biology and in industrial applications such as targeted drug delivery and micro-fluidics. We present results on the optimal shape of a rigid body locomoting in 3-D Stokes flow. The actuation consists of applying a fixed moment and constraining the body to only move along the moment axis; this models the effect of an external magnetic torque on an object made of magnetically susceptible material. The shape of the object is parametrized by a 3-D centerline with a given cross-sectional shape. No a priori assumption is made on the centerline. We show there exists a minimizer to the infinite dimensional optimization problem in a suitable infinite class of admissible shapes. We develop a variational (constrained) descent method which is well-posed for the continuous and discrete versions of the problem. Sensitivities of the cost and constraints are computed variationally via shape differential calculus. Computations are accomplished by a boundary integral method to solve the Stokes equations, and a finite element method to obtain descent directions for the optimization algorithm. We show examples of locomotor shapes with and without different fixed payload/cargo shapes.