Ribbons and groups: A thin rod theory for catheters and filaments

Abstract

We use the rotation group and its algebra to provide a novel description of deformations of special Cosserat rods or thin rods that have negligible shear. Our treatment was motivated by the problem of the simulation cf catheter navigation in a network of blood vessels, where this description is directly useful. In this context, we derive the Euler differential equations that characterize equilibrium configurations of stretch-free thin rods. We apply perturbation methods, used in time-dependent quantum theory, to the thin rod equations to describe incremental deformations of partially constrained rods. Further, our formalism leads naturally to a new and efficient finite element method valid for arbitrary deformations of thin rods with negligible stretch. Associated computational algorithms are developed and applied to the simulation of catheter motion inside an artery network.