Mixed formulation and structure-preserving discretization of Cosserat rod dynamics in a port-Hamiltonian framework

TL;DR

This paper introduces a novel energy-based modeling and structure-preserving discretization framework for nonlinear Cosserat rod dynamics, enabling accurate simulations with large rotations and complex actuation.

Contribution

It develops a mixed formulation with independent variables and a finite element discretization that preserves the port-Hamiltonian structure, accommodating large displacements, rotations, and dissipative effects.

Findings

Finite element discretization yields a finite-dimensional PH system.

The framework naturally incorporates dissipative behaviors and actuation methods.

Numerical examples demonstrate the effectiveness of the proposed approach.

Abstract

An energy-based modeling framework for the nonlinear dynamics of spatial Cosserat rods undergoing large displacements and rotations is proposed. The mixed formulation features independent displacement, velocity and stress variables and is further objective and locking-free. Finite rotations are represented using a director formulation that avoids singularities and yields a constant mass matrix. This results in an infinite-dimensional nonlinear port-Hamiltonian (PH) system governed by partial differential-algebraic equations with a quadratic energy functional. Using a time-differentiated compliance form of the stress-strain relations allows for the imposition of kinematic constraints, such as inextensibility or shear-rigidity. A structure-preserving finite element discretization leads to a finite-dimensional system with PH structure, thus facilitating the design of an energy-momentum consistent integration scheme. Dissipative material behavior (via the generalized-Maxwell model) and non-standard actuation approaches (via pneumatic chambers or tendons) integrate naturally into the framework. As illustrated by selected numerical examples, the present framework establishes a new approach to energy-momentum consistent formulations in computational mechanics involving finite rotations.