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arxiv: 2512.19408 · v2 · submitted 2025-12-22 · 🧮 math.NA · cs.CE· cs.NA· cs.RO· cs.SY· eess.SY· math.DS

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

Philipp L. Kinon , Simon R. Eugster , Peter Betsch This is my paper

Pith reviewed 2026-05-16 20:33 UTC · model grok-4.3

classification 🧮 math.NA cs.CEcs.NAcs.ROcs.SYeess.SYmath.DS
keywords Cosserat rodsport-Hamiltonian systemsmixed finite element methodenergy-momentum integrationfinite rotationsstructure preservationrod dynamics
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The pith

Mixed port-Hamiltonian formulation for Cosserat rods enables structure-preserving discretization and energy-momentum consistent integration.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops an energy-based model for the nonlinear dynamics of spatial Cosserat rods with large displacements and rotations. It uses a mixed formulation with independent displacement, velocity, and stress variables that remains objective and locking-free. Finite rotations are handled via a director approach yielding a constant mass matrix, resulting in an infinite-dimensional nonlinear port-Hamiltonian system. Time-differentiated compliance relations allow kinematic constraints like inextensibility. A structure-preserving finite element method then produces a finite-dimensional system retaining the port-Hamiltonian structure for consistent integration schemes.

Core claim

The authors establish a mixed formulation of Cosserat rod dynamics as an infinite-dimensional nonlinear port-Hamiltonian system with quadratic energy functional, governed by partial differential-algebraic equations. The formulation incorporates a director representation of finite rotations and uses time-differentiated stress-strain compliance to enforce constraints. Upon structure-preserving finite element discretization, the system retains its port-Hamiltonian structure in finite dimensions, which supports the construction of energy-momentum consistent time integrators. Dissipative effects and actuation mechanisms integrate directly into this framework.

What carries the argument

The mixed formulation with independent displacement, velocity, and stress fields in a port-Hamiltonian setting, combined with director-based rotation representation and time-differentiated compliance for constraints.

If this is right

  • Supports design of energy-momentum consistent integration schemes for the rod dynamics.
  • Accommodates dissipative material models such as the generalized-Maxwell model without additional modifications.
  • Enables non-standard actuation like pneumatic chambers or tendons within the same energy-based structure.
  • Maintains objectivity and avoids locking in simulations of large rotations and displacements.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar port-Hamiltonian discretizations could apply to other flexible body models in computational mechanics.
  • The constant mass matrix from the director formulation may simplify the analysis of constrained mechanical systems.
  • Energy consistency could improve long-term stability in simulations of robotic or biological rod-like structures.

Load-bearing premise

The time-differentiated form of the stress-strain compliance relations preserves the overall port-Hamiltonian structure even when imposing kinematic constraints.

What would settle it

A simulation exhibiting drift in total energy or momentum for an inextensible Cosserat rod under the proposed discretization scheme would indicate failure to preserve the port-Hamiltonian structure.

Figures

Figures reproduced from arXiv: 2512.19408 by Peter Betsch, Philipp L. Kinon, Simon R. Eugster.

Figure 1
Figure 1. Figure 1: Schematic depiction of a Cosserat rod configuration. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Schematic rheological model for the generalized-Maxwell approach. [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Schematic depiction for the actuation of a Cosserat rod, where the [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Flying spaghetti: Snapshots of the kayak-rowing motion with azimuth and elevation perspective angles (55, 15) and the colormap for time tn with ∈ [0, 15]. 0 5 10 15 0 200 400 600 tn 0 5 10 15 0 25 tn 0 5 10 15 −1 0 1 ·10−11 tn [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Flying spaghetti: Total energy Hˆ (xn), input work Wext n , total energy increments Hˆ (xˆn+1) − Hˆ (xˆn), and energy balance violation ∆En. At t = 5, the system is closed as both boundary inputs become zero and the beam continues with a free flight, where total energy, linear and angular momentum should be preserved. The resulting motion fits well to the depictions from [91, 93, 94], see snapshots in [PI… view at source ↗
Figure 6
Figure 6. Figure 6: Flying spaghetti: Total linear momentum components [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Flying spaghetti: Analytical solution ( ) and numerical solution ( ) for 1-component of the center of mass as well as deviation ∆ri with i ∈ {1 , 2 , 3 }. of the beam during motion [90, 92], given by r =    r1(t) 0 4    , r1(t) =    3 + 2 15 t 3 , for t ≤ 2.5, 43 6 − 5t + 2t 2 − 2 15 t 3 , for 2.5 < t ≤ 5, − 19 2 + 5t, for t > 5. (75) This is closely matched by the simulated center of mass r h… view at source ↗
Figure 8
Figure 8. Figure 8: Flying spaghetti: Convergence of the relative error for centerline position [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Nonlinear oscillation of a cantilever: Tip velocities in cross-section frame [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Nonlinear oscillation of a cantilever: Tip positions in inertial frame [PITH_FULL_IMAGE:figures/full_fig_p024_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Nonlinear oscillation of a cantilever: Orthonormality constraints [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Nonlinear oscillation of a cantilever: L 2 -norms of the shear and dilatation strain components computed from displacement quantities, i.e., ||Γi ||Ω with i ∈ {1 , 2 , 3 }. L 2 -norms of the drift between the curvature and torsion strain measures computed from the stress quantities and the ones computed from displacement quantities, denoted as||∆Kj ||Ω for j ∈ { 1 , 2 , 3 } [PITH_FULL_IMAGE:figures/full_… view at source ↗
Figure 13
Figure 13. Figure 13: Nonlinear oscillation of a cantilever: Total energy [PITH_FULL_IMAGE:figures/full_fig_p025_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Quasistatic cantilever problem: Centerline position [PITH_FULL_IMAGE:figures/full_fig_p026_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Dynamic maneuver of a soft robotic arm: Phase angle function, amplitude function and [PITH_FULL_IMAGE:figures/full_fig_p027_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Dynamic maneuver of a soft robotic arm: Snapshots of the centerline with azimuth and [PITH_FULL_IMAGE:figures/full_fig_p027_16.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a mixed port-Hamiltonian formulation for the nonlinear dynamics of spatial Cosserat rods with large displacements and rotations. Independent fields are introduced for displacement, velocity and stress; finite rotations are handled via a director formulation yielding a constant mass matrix and an objective, locking-free model. The infinite-dimensional system is expressed as nonlinear PDAEs with quadratic Hamiltonian. Time-differentiated compliance relations replace algebraic constitutive laws to enforce kinematic constraints such as inextensibility without Lagrange multipliers. A structure-preserving finite-element discretization is shown to produce a finite-dimensional port-Hamiltonian system, enabling energy-momentum consistent time integration. Dissipative (generalized Maxwell) and non-standard actuation terms integrate naturally; numerical examples are provided to illustrate the framework.

Significance. If the exact preservation of the port-Hamiltonian structure after differentiation and discretization is rigorously established, the work would supply a valuable systematic route to energy-consistent integrators for geometrically exact rod models. The natural accommodation of dissipation, actuation and constraints within the same structure is particularly useful for flexible multibody and soft-robotics applications.

major comments (2)
  1. [§3.2] §3.2: The time-differentiated compliance form is introduced to enforce inextensibility and shear-rigidity. This step replaces an algebraic constitutive relation with a differential one and thereby modifies the underlying Dirac structure; an explicit verification that the resulting infinite-dimensional system remains port-Hamiltonian (i.e., that the interconnection operator stays skew-symmetric) is required before the discretization step can be claimed to inherit the structure exactly.
  2. [§4] §4: The finite-element discretization is asserted to yield a finite-dimensional PH system whose interconnection matrix remains skew-symmetric and whose Hamiltonian remains quadratic. No explicit algebraic check or numerical test is supplied confirming that the projected differentiated compliance terms preserve skew-symmetry once the nonlinear director-frame kinematics are taken into account.
minor comments (2)
  1. Notation for the director triad and its time derivatives could be introduced more explicitly at the beginning of §2 to aid readability.
  2. The abstract states that numerical examples illustrate the framework; a short quantitative statement (e.g., observed convergence rates or energy drift) would strengthen the abstract.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the need for explicit structure-preservation proofs. We address each major point below and will revise the manuscript to incorporate the requested verifications.

read point-by-point responses

  1. Referee: [§3.2] §3.2: The time-differentiated compliance form is introduced to enforce inextensibility and shear-rigidity. This step replaces an algebraic constitutive relation with a differential one and thereby modifies the underlying Dirac structure; an explicit verification that the resulting infinite-dimensional system remains port-Hamiltonian (i.e., that the interconnection operator stays skew-symmetric) is required before the discretization step can be claimed to inherit the structure exactly.

    Authors: We agree that an explicit verification of skew-symmetry after time differentiation is necessary. In the revised manuscript we will add a dedicated derivation in §3.2 that computes the modified interconnection operator explicitly and proves its skew-symmetry with respect to the natural duality pairing, confirming that the infinite-dimensional system retains port-Hamiltonian structure. revision: yes

  2. Referee: [§4] §4: The finite-element discretization is asserted to yield a finite-dimensional PH system whose interconnection matrix remains skew-symmetric and whose Hamiltonian remains quadratic. No explicit algebraic check or numerical test is supplied confirming that the projected differentiated compliance terms preserve skew-symmetry once the nonlinear director-frame kinematics are taken into account.

    Authors: We acknowledge the value of an explicit algebraic check. The revision will include in §4 the projected discrete interconnection matrix and a direct algebraic demonstration that skew-symmetry is preserved under the nonlinear director-frame kinematics. We will also augment one numerical example with a quantitative energy-momentum conservation test to corroborate the discrete structure. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper presents a mixed formulation derived from standard port-Hamiltonian modeling principles applied to Cosserat rod kinematics, followed by a time-differentiated compliance relation to handle constraints and a finite-element discretization explicitly constructed to retain the PH structure. No step reduces the central claim (preservation of finite-dimensional PH structure) to a fitted parameter, self-referential definition, or unverified self-citation chain; the discretization is shown to produce a skew-symmetric interconnection matrix and quadratic Hamiltonian by direct construction from the weak form, with the time-differentiation justified as a modeling choice rather than an assumption that presupposes the result. External benchmarks such as energy-momentum consistency in numerical examples are independent of the derivation itself. This is the expected self-contained mathematical development with no load-bearing circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on standard port-Hamiltonian theory for infinite-dimensional systems and classical Cosserat rod kinematics; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)
  • domain assumption Port-Hamiltonian structure for systems governed by partial differential-algebraic equations
    Invoked to obtain the infinite-dimensional nonlinear PH system with quadratic energy.
  • domain assumption Director formulation for finite rotations yields constant mass matrix and avoids singularities
    Standard assumption in rod theories to represent SO(3) rotations.

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