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Mixed finite element discretization of intrinsic geometrically exact beams for explicit multibody dynamics

T0 review · 3 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read A mixed finite element discretization of the intrinsic beam formulation assembles flexible multibody systems as port-Hamiltonian ODEs without Lagrange multipliers and preserves total energy exactly.

desk verdict Solid, honest construction of mixed-FE discretizations of the intrinsic pH beam with constraint-free interconnection — but the no-Lagrange-multiplier claim is narrower than the abstract implies. read the letter →

arxiv 2607.20245 v1 pith:3MZBNGPE submitted 2026-07-22 math.NA cs.NAphysics.comp-ph

Andrea Brugnoli , Philipp L. Kinon , Francesco Sanfedino , Peter Betsch , Olivier A. Bauchau This is my paper · ORCID
classification math.NAcs.NAphysics.comp-ph MSC 65M6074K1070E5537M15
keywords GeometricallyexactbeamsMixedfiniteelementsPort-HamiltoniansystemsStructurepreservingdiscretizationMultibodydynamicsEnergy-preservingtimeintegrationKinematicconstraintsIntrinsicbeamformulation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper aims to show that a well-known intrinsic description of geometrically exact beams—written in terms of velocities and stress resultants instead of displacements and rotations—can be discretized with mixed finite elements so that all standard boundary conditions become natural. That linear-operator structure lets the authors assemble flexible multibody systems, including closed loops such as a four-bar mechanism, as port-Hamiltonian ordinary differential equations, without Lagrange multipliers or the stiff differential-algebraic equations they normally introduce. If the argument is right, the payoff is practical: the interconnected systems retain an exact energy-preserving time integration scheme (implicit midpoint), with fewer degrees of freedom and fewer nonlinear iterations per time step than established energy-momentum schemes. This points toward simpler and more stable simulation of highly flexible robots, deployable structures, and other flexible multibody mechanisms.

What carries the argument

The central mechanism is the mixed finite element treatment of the intrinsic beam equations, whose differential operators are linear. By choosing which pairs of equations are integrated by parts, four discrete causality types (free, clamped, pinned, guided) are obtained, each with different natural boundary ports. Pairs with opposite causality are then coupled by a feedback (gyrator) interconnection, which encodes Newton's third law and yields a globally skew-symmetric matrix; nonlinearities remain confined to the rotation-dependent interconnection operator. This keeps the final system port-Hamiltonian and quadratic in energy, so implicit midpoint integration preserves total energy exactly.

What would settle it

Build or simulate a multibody topology in which three beams converge to a single pivot (the case the paper itself flags as conflict-prone). If the proposed feedback-interconnection recipe cannot produce a well-posed ODE there without Lagrange multipliers or artificial inertia, the central 'no algebraic constraints' claim does not hold for that class of topologies; observing the method fall back to constraints or regularization would refute the general claim.

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Extended reading notes

Core claim

The central claim is that the intrinsic beam formulation has a linear differential operator, so the velocity and stress equations can be integrated by parts in different combinations. Doing so generates four discrete port-Hamiltonian variants—free, clamped, pinned, and guided—whose boundary ports accept either forces/moments or velocities, whatever the physical setting requires. Pairing variants with opposite causality through a feedback interconnection—which encodes equal velocity and opposite force at a joint—produces a global skew-symmetric structure with nonlinearities confined to the interconnection and rotation operators. The paper verifies the construction on a cantilever, an L-frame,

Load-bearing premise

The no-constraints construction works only when every joint can be assigned a consistent causality—one subsystem supplying velocity and accepting force, the other the reverse—and these assignments fit together across the whole network; for topologies where three or more beams meet at one pivot this may fail, and the paper concedes that constraints or added inertia then return.

Editorial extensions

If this is right

  • If the central claim is right, planar flexible multibody systems—including closed kinematic loops such as the four-bar mechanism—can be simulated as ordinary differential equations rather than differential-algebraic equations, eliminating Lagrange multipliers and the associated stiffness.
  • The discrete total energy is preserved exactly under implicit midpoint integration for unforced systems, and a discrete power balance holds when external inputs are present.
  • The method uses fewer degrees of freedom per beam and fewer nonlinear iterations per time step than the compared energy-momentum schemes on the four-bar benchmark, which could reduce simulation cost in practice.
  • Mixed boundary conditions (e.g., a cantilever obtained by clamping one half and freeing the other) arise naturally by interconnecting two causality-compatible beam discretizations rather than by enforcing constraints.
  • Because all nonlinearities are confined to the interconnection and rotation operators, the discrete system retains a port-Hamiltonian structure that can be exploited for control and estimation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the causality-compatibility condition can be characterized automatically (e.g., as a graph-theoretic matching), the assembly procedure could be turned into a generic library for arbitrary flexible multibody networks; the paper leaves this automation as future work.
  • The four causality variants are not beam-specific: the same integration-by-parts and mixed-element recipe should transfer to other linear-operator port-Hamiltonian models, such as heat or wave propagation, yielding constraint-free network couplings there too.
  • Since angular momentum is not preserved by the proposed time integrator, simulations of long-time rotational motion may need a momentum-correction step or a different time scheme; a natural test is to measure drift in total angular momentum on the flying-spaghetti benchmark over long horizons.
  • The method's efficiency advantage in nonlinear iterations appears on one benchmark; a stronger test would compare iteration counts and wall-clock time for larger spatial problems and three-dimensional extensions.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper develops mixed finite element discretizations of the intrinsic (Hodges) port-Hamiltonian formulation of planar geometrically exact beams. Four causality variants are constructed (free, clamped, pinned, guided) by integrating different pairs of equations by parts, yielding natural weak enforcement of kinematic or dynamic boundary conditions. Feedback interconnections of these variants are used to assemble a cantilever and a four-bar mechanism without Lagrange multipliers. The implicit midpoint scheme is applied to the resulting pH descriptor system, and Proposition 1 proves exact discrete energy conservation for unforced systems, with a power balance for forced systems. Numerical benchmarks (flying spaghetti, flexible pendulum, L-shaped frame, four-bar mechanism) are compared with results from the literature.

Significance. If the advertised generality were established, the contribution would be significant: it would give DAE-free, energy-preserving pH multibody beam models with clear FE construction. The concrete contributions are valuable and largely self-contained: the four mixed FE discretizations are explicit, the energy-conservation proof in Prop. 1 is standard and appears correct, and the benchmarks against independent references ([36], [14], [13]) support the discretization. The main weakness is that the central 'without algebraic constraints' claim is broader than what is proved or demonstrated; the paper's own conclusion concedes the main limitation. This overstatement affects the abstract and introduction, but the underlying construction for the demonstrated examples appears sound.

major comments (3)
  1. [Abstract; Secs. 4 and 7] The unqualified statement that 'multibody systems can be assembled without algebraic constraints' is not established for general network topologies. The paper's own Conclusion (Sec. 7) concedes: 'when multiple beams converge to a single pivot node one should be treated as free and the other as pinned but this may generate conflicts with the rest of the network. To avoid Lagrange multipliers in general regularization approaches ... may be used.' No automated causality-assignment algorithm or existence condition is provided, and the port selection in Sec. 4 is manual and example-specific. This is load-bearing for the headline claim. The abstract and introduction should be qualified, and the topology conditions under which the constraint-free construction works should be stated, or an algorithm/proof should be supplied.
  2. [Sec. 6.3 and Table 2] The 'no algebraic constraints' framing is internally contradicted by two of the paper's own examples. In the pendulum benchmark, Table 2 sets rhoA=rhoI=0, so the assembled matrix E in Eq. (33) is singular and the system is a descriptor system/DAE, not an ODE. In the rigid L-frame (Sec. 6.3), the authors set compliance entries to zero and state that 'the corresponding stress type quantities immediately act as Lagrange multipliers.' These are algebraic constraints. The claim should be restricted to the absence of kinematic joint constraints in the regular (positive-definite E) case, and the regularity assumptions behind Prop. 1 and the 'no algebraic constraints' statements should be stated explicitly.
  3. [Sec. 5, Prop. 1] The energy-conservation proof is correct for the unforced, nonsingular case and for the exact solution of the implicit-midpoint algebraic system. However, the Newton-iteration comparison in Sec. 6.4 uses a tolerance of 1e-5, and the L-frame examples (Sec. 6.3) are forced by q(t), so they do not verify the unforced energy-conservation statement. The paper should distinguish more clearly between exact energy conservation (Prop. 1, unforced) and the empirical energy balance of the forced examples; this also affects the interpretation of Figs. 20 and 21, where the energy increments are not of machine-precision size.
minor comments (5)
  1. [Sec. 4.2] In the Beam 3 equations, 'yP1,nr = B^T_{P3,vl} z_{P3}' should presumably be 'yP3,nl' or 'yP3,nr'; the same expression is written twice and appears to be a typo.
  2. [Title/Abstract] The phrase 'explicit multibody dynamics' is misleading, since the time integration scheme is implicit (Sec. 5). Clarify whether 'explicit' refers to the absence of algebraic constraints or to some other meaning.
  3. [Abstract] The sentence 'Furthermore the scheme appear to require less Newton iterations' should read 'the scheme appears to require fewer Newton iterations.' Also, a quantitative statement of the comparison (problem, tolerance, code) would help.
  4. [Sec. 3.5, Eq. (28)] In the descriptor form, the notation z_q for the constant gravity gradient in the upper block of Eq. (34) is slightly confusing because the state-dependent gradient of the Hamiltonian is not written out. A brief explanation of the discrete gradient convention would improve readability.
  5. [Sec. 6.3] For the L-shaped frame, the paper states that rigidity is realized by zeroing compliances, but it does not explain how the resulting singular system is solved or how the algebraic variables are initialized. Adding this detail would strengthen confidence in the method's applicability to constrained cases.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the FE discretization, interconnections, and energy proof are derived in-paper; the main caveat is a scope limitation, not a circular step.

full rationale

The derivation is self-contained. The paper exhibits the weak forms (15)-(24), the finite element matrices (18), (20), (22), (24), the feedback interconnections for the cantilever and four-bar examples in Sec. 4, and proves discrete energy preservation in Proposition 1 using skew-symmetry and Eq. (31). The benchmarks include the independent flying-spaghetti reference [36] and the pendulum reference [14]; the four-bar comparison with PH-EM/PH-EMi from [13] is a comparison with a published alternative method, not an input used to derive the new scheme. The self-citations ([13], [18], [19], [20], [30]) are used for background, for pointers to prior methodology, or as comparison references, but the load-bearing mathematical steps are re-derived in the paper itself; no uniqueness theorem or fitted parameter is imported. The Sec. 7 concession that a consistent global causality assignment is required for constraint-free assembly narrows the abstract's unconditional claim, and the abstract's wording therefore overstates generality, but this is a scope limitation rather than a circular derivation. No prediction reduces by construction to a fitted quantity, and no step is equivalent to its own input by definition, so no circular step is identified.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The method's contribution is genuinely a discretization; it introduces no new physics, no fitted constitutive constants, and no invented entities. The free parameters are numerical settings (split point, solver tolerance, rigid-case compliances). The main external weights are: continuous equivalence (thesis [9]), the pH structure of the intrinsic equations (prior literature), and the mixed-FE pH discretization framework (mostly the authors' own prior work). The limiting premise is the global causality-assignment condition, explicitly conceded in Sec. 7.

free parameters (3)
  • Cantilever split point L1 = L2 = L/2 = L/2
    Sec. 4.1 splits the cantilever at an arbitrary point 'for simplicity'; the dynamics should be split-invariant, but no split-independence test is shown, so it functions as a hand-chosen numerical setting.
  • Newton solver tolerance for method comparison = 10^-5
    Sec. 6.4 sets the Newton tolerance to 10^-5 for all compared methods; the 'fewer Newton iterations' claim is measured at this hand-chosen operating point and could shift with tolerance or Jacobian strategy.
  • Rigid L-frame: zeroed compliances = 0
    Sec. 6.3 realizes rigidity by setting compliance parameters to zero, turning stress variables into Lagrange multipliers; a modeling/benchmark choice rather than a fit, listed for exhaustiveness.
assumptions (5)
  • domain assumption Continuous equivalence of the Hodges intrinsic formulation and the Simo-Reissner displacement formulation.
    Invoked in Sec. 2 ('in the continuous setting... those formulations are equivalent') and attributed to the PhD thesis [9]; the entire strategy of simulating the intrinsic model and reconstructing positions a posteriori rests on this equivalence.
  • domain assumption Formal skew-adjointness of the intrinsic beam interconnection operator J_m.
    Sec. 2.3 asserts J_m is formally skew-adjoint, citing the pH structure established in [11,17,12]; the discrete energy balance and the skew-symmetry of the assembled matrices inherit this property.
  • domain assumption The mixed-FE pH discretization framework (dual-field/partitioned FE, natural boundary conditions) transfers to the intrinsic beam and preserves exact structural relations.
    Sec. 3 builds on [29,30,19]; the locking-free claim and the strong relations (d_s CG1 subset DG0, Ct d_t n_h = d_s v_h - b2 Pi_DG0 w_h) are asserted from that framework rather than re-derived here.
  • domain assumption Global causality-assignment consistency for the interconnection graph.
    Sec. 4 requires each joint to pair dual causal ports; the Conclusion concedes that multiple beams at one pivot node can conflict with the rest of the network, so the constraint-free construction only provably works for causally orientable topologies (cantilever, four-bar).
  • standard math Discrete compatibility M_ry 1 = a (Eq. 31).
    Used in Prop. 1 to convert the discrete gravity potential into an exact energy difference; holds because CG1/DG0 basis functions form a partition of unity, so it is a structural property of the chosen spaces rather than an extra postulate.

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Cite this review

Pith. "Pith review of Mixed finite element discretization of intrinsic geometrically exact beams for explicit multibody dynamics." pith.science (2026). https://pith.science/paper/3MZBNGPE

@misc{pith2026260720245,
  author       = {Pith},
  title        = {Pith review of: Mixed finite element discretization of intrinsic geometrically exact beams for explicit multibody dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3MZBNGPE}},
  note         = {Machine review of arXiv:2607.20245}
}
read the original abstract

The Reissner-Simo and Hodges models are two equivalent continuous descriptions of finite-strain beam dynamics. The Reissner-Simo formulation uses displacements and rotations, while the Hodges formulation is intrinsic and avoids both variables. Although equivalent in theory, the two approaches behave differently after discretization and offer distinct numerical advantages. In this work, we develop a structure-preserving discretization of the intrinsic formulation. Because the intrinsic equations involve linear differential operators, both kinematic and dynamic boundary conditions can be imposed naturally using mixed finite elements. The resulting formulation also enables multibody systems to be assembled without algebraic constraints, avoiding the stiff differential-algebraic equations typically introduced by kinematic constraints. We demonstrate the approach on different examples, also showing that closed kinematic loops can be modeled without algebraic constraints. The resulting interconnected systems retain a port-Hamiltonian structure,with all nonlinearities confined to the interconnection operator. This structure allows exact energy preservation when combined with implicit midpoint time integration. Furthermore the scheme appear to require less Newton iterations compared to existing energy preserving scheme.

Figures

Figures reproduced from arXiv: 2607.20245 by the authors.

Figure 1
Figure 1. Derivative of a Lagrange space CG1, leading to a piecewise constant function. Inserting the above finite element approximations into weak form (16), and accounting for the arbitrariness of the test functions, the following set of ordinary differential equations (ODE) is obtained Diag     ρAM⊗2 ρIM hCt ⊗ INe CrhINe     d dt   v w n m   =     0 0 −D⊤ ⊗2 0 0 0 F ⊤ −D⊤ D⊗2 −F 0 0 0 D 0 0      … view at source ↗
Figure 2
Figure 2. Block diagram for the free-free case. Let us now proceed analogously with a different set of boundary conditions: The case where both extremities of the beam are clamped. 3.2. Clamped-Clamped boundary conditions If the last two lines of (15) are integrated by parts then one obtains (ψv, ρA∂tvb)Ω = +(ψv, ∂snb)Ω, (ψω, ρI∂tω)Ω = +(ψω, b ⊤ 2 nb)Ω + (ψω, ∂sm)Ω, (ψn, Ct∂tnb)Ω = −(ψn, b2ω)Ω − (∂sψn, vb)Ω + ⟨ψn, vb⟩∂Ω, (ψm,… view at source ↗
Figure 3
Figure 3. Block diagram for the clamped-clamped case. [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (22 more)
Figure 4
Figure 4. Figure 4: Block diagram for the pinned case. 11 [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: Block diagram for the guided case. 3.5. Extension to the nonlinear case and to gravity effects As we have seen in the previous sections, the discretization of the linear part leads to a system of the form Me □e˙ = J e □e□ + B e □u□, y = (B e □) ⊤e□, (25) where □ indica…
Figure 7
Figure 7. Figure 7: A cantilever beam as interconnected system [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: Four bar mechanism Beam 1    EP 1x˙ P 1 = JP 1(xP 1)zP 1 + BP 1,vluP 1,vl + BP 1,vruP 1,vr, yP 1,nl = B⊤ P 1,vlzP 1, yP 1,nr = B⊤ P 1,vlzP 1, Beam 2    EF 2x˙ F 2 = JF 2(xF 2)zF 2 + BF 2,nluF 2,nl + BF 2,nruF 2,nr, yF 2,vl = B⊤ F 2,nlzF 2, yF 2,vr = B⊤ F 2,nr…
Figure 9
Figure 9. Figure 9: Block diagram for the four bar mechanism [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]
Figure 10
Figure 10. Figure 10: Initial configuration for the flying spaghetti [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]
Figure 11
Figure 11. Figure 11: Configuration of the beam at different instants for [PITH_FULL_IMAGE:figures/full_fig_p018_11.png]
Figure 12
Figure 12. Figure 12: Evolution of the energy and its increments [PITH_FULL_IMAGE:figures/full_fig_p019_12.png]
Figure 13
Figure 13. Figure 13: Pendulum with lumped mass as interconnected system [PITH_FULL_IMAGE:figures/full_fig_p019_13.png]
Figure 14
Figure 14. Figure 14: Pendulum 6.3. L-shaped frame We now consider the motion of a large frame. An analogous setting using geometrically exact beams has been considered in [14], whereas in [37] the same problem is treated using Q4 elements (bilinear four node quadrilateral finite element).…
Figure 15
Figure 15. Figure 15: Evolution of rotation and of the angular velocity of the pendulum. [PITH_FULL_IMAGE:figures/full_fig_p020_15.png]
Figure 23
Figure 23. Figure 23: The results coincide with the findings in [13]. Fig. 26 verifies the discrete-time energy-preservation of [PITH_FULL_IMAGE:figures/full_fig_p020_23.png]
Figure 16
Figure 16. Figure 16: Evolution of energy and energy increments for the pendulum. [PITH_FULL_IMAGE:figures/full_fig_p020_16.png]
Figure 17
Figure 17. Figure 17: L-shaped frame obtained as interconnected of a free beam and a cantilever one [PITH_FULL_IMAGE:figures/full_fig_p021_17.png]
Figure 18
Figure 18. Figure 18: Geometry and loading definition for the L-shaped frame [PITH_FULL_IMAGE:figures/full_fig_p021_18.png]
Figure 19
Figure 19. Figure 19: Output configurations for the L-shaped frames at time instants [PITH_FULL_IMAGE:figures/full_fig_p022_19.png]
Figure 20
Figure 20. Figure 20: Rigid L-frame: evolution of energy and energy increments. [PITH_FULL_IMAGE:figures/full_fig_p022_20.png]
Figure 21
Figure 21. Figure 21: Soft L-frame: evolution of energy and energy increments [PITH_FULL_IMAGE:figures/full_fig_p023_21.png]
Figure 22
Figure 22. Figure 22: Position of point P over time 23 [PITH_FULL_IMAGE:figures/full_fig_p023_22.png]
Figure 23
Figure 23. Figure 23: Configurations for the closed loop multibody mech [PITH_FULL_IMAGE:figures/full_fig_p024_23.png]
Figure 25
Figure 25. Figure 25: Comparison of Newton iterations for different integrators [PITH_FULL_IMAGE:figures/full_fig_p024_25.png]
Figure 26
Figure 26. Figure 26: The intrinsic energy, i.e. the sum of kinetic and deformation energies [PITH_FULL_IMAGE:figures/full_fig_p025_26.png]

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Works this paper leans on

46 extracted references · 15 canonical work pages

  1. [36]

    J. C. Simo, L. Vu-Quoc, On the dynamics of flexible beams under large overall motions—the plane case: Part I, Journal of Applied Mechanics 53 (4) (1986) 849–854.doi:10.1115/1.3171870

    doi:10.1115/1.3171870 1986

  2. [14]

    Ljukovac, A

    S. Ljukovac, A. Ibrahimbegovic, M. C. Husic, Nonlinear dynamics and control of Reissner’s 2D geomet- rically exact beam by distributed port-Hamiltonian system, International Journal for Numerical Methods in Engineering 126 (16) (2025) e70103.doi:https://doi.org/10.1002/nme.70103. 25

    work page doi:10.1002/nme.70103 2025

  3. [13]

    P. L. Kinon, P. Betsch, S. R. Eugster, Energy-momentum-consistent simulation of planar geometrically exact beams in a port-Hamiltonian framework, Multibody System Dynamics (06 2025).doi:10.1007/ s11044-025-10087-9

    2025

  4. [1]

    Albu-Schäffer, O

    A. Albu-Schäffer, O. Eiberger, M. Grebenstein, S. Haddadin, C. Ott, T. Wimbock, S. Wolf, G. Hirzinger, Soft robotics, IEEE Robotics & Automation Magazine 15 (3) (2008) 20–30.doi:10.1109/MRA.2008. 927979. cl-EM PH-EM PH-EMi PH intrinsic #Variables/beam O(4Nel)O(6N el)O(4N el)O(5N el) Fields (r, θ, bv, ω) (r, θ, bv, ω, bn, m) (r, θ, bv, ω) (θ, bv, ω, bn, m)...

    work page doi:10.1109/mra.2008 2008

  5. [2]

    O. M. Omisore, S. Han, J. Xiong, H. Li, Z. Li, L. Wang, A review on flexible robotic systems for minimally invasive surgery, IEEE Transactions on Systems, Man, and Cybernetics: Systems 52 (1) (2022) 631–644. doi:10.1109/TSMC.2020.3026174

    arXiv 2022

  6. [3]

    Chang, D

    C.-S. Chang, D. H. Hodges, M. J. Patil, Flight dynamics of highly flexible aircraft, Journal of Aircraft 45 (2) (2008) 538–545.doi:10.2514/1.30890

    work page doi:10.2514/1.30890 2008

  7. [4]

    2020.118835

    Y.-J.Chu, H.-F.Lam, Comparativestudyoftheperformancesofabio-inspiredflexible-bladedwindturbine and a rigid-bladed wind turbine in centimeter-scale, Energy 213 (2020) 118835.doi:10.1016/j.energy. 2020.118835

    arXiv 2020

  8. [5]

    R. Chen, L. Zhou, K. Wu, L. Liu, Y. Liu, X. Li, G. Chen, G. Zheng, J. Luo, A general framework for designing compliant deployable mechanisms via geometrically exact beam theory, Mechanism and Machine Theory 202 (2024) 105778.doi:10.1016/j.mechmachtheory.2024.105778

    arXiv 2024

Show all 46 references
  1. [6]

    Reissner, On one-dimensional finite-strain beam theory: The plane problem, Zeitschrift für angewandte Mathematik und Physik ZAMP 23 (5) (1972) 795–804.doi:10.1007/BF01602645

    E. Reissner, On one-dimensional finite-strain beam theory: The plane problem, Zeitschrift für angewandte Mathematik und Physik ZAMP 23 (5) (1972) 795–804.doi:10.1007/BF01602645

    1972 doi

  2. [7]

    Simo, A finite strain beam formulation

    J. Simo, A finite strain beam formulation. the three-dimensional dynamic problem. part I, Computer Methods in Applied Mechanics and Engineering 49 (1) (1985) 55–70.doi:https://doi.org/10.1016/ 0045-7825(85)90050-7

    1985

  3. [8]

    D. H. Hodges, Geometrically exact, intrinsic theory for dynamics of curved and twisted anisotropic beams, AIAA Journal 41 (6) (2003) 1131–1137.doi:10.2514/2.2054

    2003 doi

  4. [9]

    C.M.Rodriguez, Controlandstabilizationofgeometricallyexactbeams, Ph.D.thesis, Friedrich-Alexander- Universitaet Erlangen-Nuernberg (Germany) (2021)

    2021

  5. [10]

    Palacios, Nonlinear normal modes in an intrinsic theory of anisotropic beams, Journal of Sound and Vibration 330 (8) (2011) 1772–1792.doi:10.1016/j.jsv.2010.10.023

    R. Palacios, Nonlinear normal modes in an intrinsic theory of anisotropic beams, Journal of Sound and Vibration 330 (8) (2011) 1772–1792.doi:10.1016/j.jsv.2010.10.023

    2011 doi

  6. [11]

    Artola, A

    M. Artola, A. Wynn, R. Palacios, Modal-based nonlinear model predictive control for 3-d very flexible structures, IEEE Transactions on Automatic Control 67 (5) (2021) 2145–2160.doi:10.1109/TAC.2021. 3071326

    2021 doi

  7. [12]

    E. P. Ayala, Y. Wu, K. Rabenorosoa, Y. Le Gorrec, Energy-Based Modeling and Control of a Piezotube Actuated Optical Fiber, IEEE/ASME Transactions on Mechatronics 28 (1) (2023) 385–395.doi:10.1109/ TMECH.2022.3199566

    2023

  8. [15]

    Khaneh Masjedi, S

    P. Khaneh Masjedi, S. Jangravi, A. Reali, Isogeometric collocation for locking-free large deflection analysis of geometrically exact beams with intrinsic formulation, Computer Methods in Applied Mechanics and Engineering 446 (2025) 118282.doi:10.1016/j.cma.2025.118282

    2025

  9. [16]

    O. A. Bauchau, Flexible multibody dynamics, Vol. 176, Springer, 2011

    2011

  10. [17]

    A. Wynn, Y. Wang, R. Palacios, P. J. Goulart, An energy-preserving description of nonlinear beam vibrations in modal coordinates, Journal of Sound and Vibration 332 (21) (2013) 5543–5558.doi: 10.1016/j.jsv.2013.05.021

    2013 doi

  11. [18]

    Brugnoli, D

    A. Brugnoli, D. Matignon, J. Morlier, A linearly-implicit energy-momentum preserving scheme for ge- ometrically nonlinear mechanics based on non-canonical Hamiltonian formulations, Nonlinear Dynamics 113 (20) (2025) 27539–27566.doi:10.1007/s11071-025-11601-6

    2025 doi

  12. [19]

    De Jong, A

    S. De Jong, A. Brugnoli, R. Rashad, Y. Zhang, S. Stramigioli, A domain decomposition strategy for natural imposition of mixed boundary conditions in port-Hamiltonian systems, Applied Mathematical Modelling (2026) 116775doi:10.1016/j.apm.2026.116775

    2026

  13. [20]

    P. L. Kinon, S. R. Eugster, P. Betsch, Mixed formulation and structure-preserving discretization of cosserat rod dynamics in a port-hamiltonian framework, Computer Methods in Applied Mechanics and Engineering 458 (2026) 118966.doi:10.1016/j.cma.2026.118966

    2026

  14. [21]

    Herrmann, D

    M. Herrmann, D. Castello, J. Breuling, I. C. Garcia, L. Greco, S. R. Eugster, A mixed Petrov-Galerkin Cosserat rod finite element formulation, arXiv preprint arXiv:2507.01552 (2026)

    2026

  15. [22]

    Greco, D

    L. Greco, D. Castello, M. Cuomo, An objective and accurate G1 -conforming mixed Bézier FE-formulation for Kirchhoff–Love rods, Mathematics and Mechanics of Solids 29 (4) (2024) 645–685.doi:10.1177/ 10812865231204972

    2024

  16. [23]

    J. G. Kim, Y. Y. Kim, A new higher-order hybrid-mixed curved beam element, International Journal for Numerical Methods in Engineering 43 (5) (1998) 925–940.doi:10.1002/(SICI)1097-0207(19981115)43: 5<925::AID-NME457>3.0.CO;2-M

    1998 doi

  17. [24]

    Santos, P

    H. Santos, P. Pimenta, J. Moitinho De Almeida, Hybrid and multi-field variational principles for geo- metrically exact three-dimensional beams, International Journal of Non-Linear Mechanics 45 (8) (2010) 809–820.doi:10.1016/j.ijnonlinmec.2010.06.003

    2010 doi

  18. [25]

    Thoma, P

    T. Thoma, P. Kotyczka, H. Egger, On the velocity-stress formulation for geometrically nonlinear elastody- namics and its structure-preserving discretization, Mathematical and Computer Modelling of Dynamical Systems 30 (1) (2024) 701–720.doi:10.1080/13873954.2024.2397486

    2024

  19. [26]

    Mehrmann, R

    V. Mehrmann, R. Morandin, Structure-preserving discretization for port-Hamiltonian descriptor systems, in: 2019 IEEE 58th Conference on Decision and Control (CDC), 2019, pp. 6863–6868.doi:10.1109/ CDC40024.2019.9030180

    2019

  20. [27]

    Meier, A

    C. Meier, A. Popp, W. A. Wall, Geometrically exact finite element formulations for slender beams: Kirchhoff–love theory versus simo–reissner theory, Archives of Computational Methods in Engineering 26 (1) (2019) 163–243.doi:10.1007/s11831-017-9232-5

    2019 doi

  21. [28]

    Macchelli, C

    A. Macchelli, C. Melchiorri, Modeling and control of the Timoshenko beam. the distributed port Hamil- tonian approach, SIAM Journal on Control and Optimization 43 (2) (2004) 743–767.doi:10.1137/ S0363012903429530

    2004

  22. [29]

    F. L. Cardoso-Ribeiro, D. Matignon, L. Lefèvre, A partitioned finite element method for power-preserving discretization of open systems of conservation laws, IMA Journal of Mathematical Control and Information 38 (2) (2020) 493–533.doi:10.1093/imamci/dnaa038

    2020 doi

  23. [30]

    Brugnoli, R

    A. Brugnoli, R. Rashad, S. Stramigioli, Dual field structure-preserving discretization of port-Hamiltonian systems using finite element exterior calculus, Journal of Computational Physics 471 (2022) 111601.doi: 10.1016/j.jcp.2022.111601

    2022

  24. [31]

    van der Schaft, D

    A. van der Schaft, D. Jeltsema, Port-hamiltonian systems theory: An introductory overview, Foundations and Trends in Systems and Control 1 (2-3) (2014) 173–378.doi:10.1561/2600000002

    2014 doi

  25. [32]

    Finozzi, F

    A. Finozzi, F. Sanfedino, D. Alazard, Parametric sub-structuring models of large space truss structures for structure/control co-design, Mechanical Systems and Signal Processing 180 (2022) 109427.doi:10.1016/ j.ymssp.2022.109427. 26

    2022

  26. [33]

    Sanfedino, G

    F. Sanfedino, G. Thiébaud, D. Alazard, N. Guercio, N. Deslaef, Advances in fine line-of-sight control for large space flexible structures, Aerospace Science and Technology 130 (2022) 107961.doi:10.1016/j. ast.2022.107961

    2022

  27. [34]

    J. S. Hesthaven, T. Warburton, Nodal Discontinuous Galerkin Methods, 1st Edition, Texts in Applied Mathematics, Springer, New York, NY, 2008.doi:10.1007/978-0-387-72067-8

    2008 doi

  28. [35]

    Brugnoli, D

    A. Brugnoli, D. Alazard, V. Pommier-Budinger, D. Matignon, Port-Hamiltonian flexible multibody dy- namics, Multibody System Dynamics 51 (3) (2021).doi:10.1007/s11044-020-09758-6

    2021 doi

  29. [37]

    Betsch, P

    P. Betsch, P. Steinmann, Conservation properties of a time fe method—part II: Time-stepping schemes for non-linear elastodynamics, International Journal for Numerical Methods in Engineering 50 (8) (2001) 1931–1955.doi:https://doi.org/10.1002/nme.103

    2001 doi

  30. [38]

    Bradbury, R

    J. Bradbury, R. Frostig, P. Hawkins, M. J. Johnson, C. Leary, D. Maclaurin, G. Necula, A. Paszke, J. VanderPlas, S. Wanderman-Milne, Q. Zhang, JAX: composable transformations of Python+NumPy programs, version 0.2.5 (2018)

    2018

  31. [39]

    A. A. Shabana, Dynamics of Multibody Systems, Cambridge University Press, 2020

    2020

  32. [40]

    Stander, E

    N. Stander, E. Stein, An energy-conserving planar finite beam element for dynamics of flexible mechanisms, Engineering Computations 13 (6) (1996) 60–85.doi:10.1108/02644409610128418

    1996 doi

  33. [41]

    O.Brüls, A.Cardona, M.Arnold, Liegroupgeneralized-αtimeintegrationofconstrainedflexiblemultibody systems, Mechanism and Machine Theory 48 (2012) 121–137.doi:10.1016/j.mechmachtheory.2011.07. 017

    2012 doi

  34. [42]

    Y. Yu, X. Ding, J. J. Zhu, Attitude tracking control of a quadrotor uav in the exponential coordinates, Journal of the Franklin Institute 350 (8) (2013) 2044–2068.doi:10.1016/j.jfranklin.2013.05.022

    2013 doi

  35. [43]

    Müller, Screw and lie group theory in multibody kinematics, Multibody System Dynamics 43 (1) (2018) 37–70.doi:10.1007/s11044-017-9582-7

    A. Müller, Screw and lie group theory in multibody kinematics, Multibody System Dynamics 43 (1) (2018) 37–70.doi:10.1007/s11044-017-9582-7

    2018 doi

  36. [44]

    Todesco, O

    J. Todesco, O. Brüls, Highly accurate differentiation of the exponential map and its tangent operator, Mechanism and Machine Theory 190 (2023) 105451.doi:10.1016/j.mechmachtheory.2023.105451

    2023

  37. [45]

    Dopico, Y

    D. Dopico, Y. Zhu, A. Sandu, C. Sandu, Direct and adjoint sensitivity analysis of ordinary differential equation multibody formulations, Journal of Computational and Nonlinear Dynamics 10 (1) (2014) 011012. doi:10.1115/1.4026492

    2014 doi

  38. [46]

    T. D. Barfoot, State Estimation for Robotics: Second Edition, 2nd Edition, Cambridge University Press, 2024. 27

    2024

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.