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

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.

arXiv comment39 pages, 16 figures

References in corpus (8)

• A Partitioned Finite Element Method for power-preserving discretization of open systems of conservation laws
• Port-Hamiltonian formulation and symplectic discretization of Plate models. Part I : Mindlin model for thick plates
• Port-Hamiltonian formulation and symplectic discretization of plate models. Part II : Kirchhoff model for thin plates
• A port-Hamiltonian approach to modeling the structural dynamics of complex systems
• Port-Hamiltonian flexible multibody dynamics
• A family of total Lagrangian Petrov-Galerkin Cosserat rod finite element formulations
• Discrete nonlinear elastodynamics in a port-Hamiltonian framework
• Discrete gradient methods for port-Hamiltonian differential-algebraic equations