Development of Techniques for Modeling the Static Buckling of Euler Beam and Dynamic Response of Kirchhoff Rods: Application to Surgical Simulation and Training

Abstract

In this dissertation, we present novel schemes for a static simulation of a buckled Euler beam with curve channel constraints in two dimensional space and simulation of the dynamic response of a soft Kirchhoff rod in three dimension space at real time rate. The aim of this model is to provide a robust and fast means for simulating endoscopes and surgical threads for training and surgical simulation purposes. Finding a static configuration of a buckled cantilever elastic beam constrained in a curved solid channel subject to end forces is a simple model of endoscopy and it is posed as the minimization of an energy functional. We solve it by a novel technique, a variant of a dynamic programming approach called the Viterbi algorithm. The core idea of this approach is to discretize the variables describing the potential energy and to construct a set of admissible configurations of the beam. The Viterbi algorithm is then employed to search through the set of possible beam configurations and locate the one with the minimum potential energy in a very computationally efficient way. The new approach does not require any gradient computations and could be considered as a direct search method, and thus can be guaranteed to find the global minimum potential energy. Also the constraints can be automatically satisfied by constructing the proper set of all the possible configurations. The approach can also be used to find feasible starting configurations associated with conventional minimizing algorithms.

We also discuss a novel scheme based on discrete variational integrators to study the dynamics of an inextensible thin Kirchhoff rod which is a model for a surgical thread. The benefits of such approach are that it is a very efficient scheme that guarantees conservation of momentum and energy over very long times so that a real time simulator can be operated over long periods of time. In addition, we report on an innovative technique to capture the inextensibility as well as the internal dissipation of the rod efficiently. Finally, a new collision avoidance scheme based on a continuous penalty force is employed to simulate the interaction of the rod with the surrounding medium. The simulations performed capture the formation of plectoneme, i.e. a loop of helices twisted together. Lastly, the scheme is employed to simulate the tying of a square knot. This model can be used to simulate surgical threads at real time rate.