pith. machine review for the scientific record. sign in

arxiv: 2605.04573 · v1 · submitted 2026-05-06 · 🧮 math.NA · cs.NA

Recognition: unknown

Mixed Finite Elements for Geometrically Exact Beams using Discontinuous Rotations and Discrete Curvature

Alexander Humer , Ivo Steinbrecher , Astrid Pechstein
Authors on Pith no claims yet

Pith reviewed 2026-05-08 16:15 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords mixed finite elementsgeometrically exact beamsSimo-Reissner beamsdiscontinuous rotationsdiscrete curvaturefinite element formulation
0
0 comments X

The pith

A mixed finite-element formulation for geometrically exact beams treats moments as independent variables to allow discontinuous rotations per element.

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

The paper develops a mixed finite element method for Simo-Reissner beams that introduces the moment vector as an additional independent field. This choice permits rotations to be approximated discontinuously inside each element, which simplifies the overall discretization. The concept of discrete curvature ensures mathematical consistency when rotations are allowed to jump at element interfaces. For linear constitutive laws the formulation is obtained via a Legendre transform of the curvature energy, and the total Lagrangian setting with multiplicative relative-rotation interpolation preserves objectivity and path independence. Numerical benchmarks confirm optimal convergence rates that hold irrespective of beam slenderness or polynomial degree, including the special case of element-wise constant rotations.

Core claim

The central claim is that a mixed variational formulation for geometrically exact beams, obtained by introducing the moment vector as an independent field, enables an element-local discontinuous approximation of rotations while remaining consistent through the notion of discrete curvature. Objectivity is retained by interpolating relative rotations via a multiplicative split, and path independence follows directly from the total Lagrangian description.

What carries the argument

mixed variational principle with moment vector as independent field and discrete curvature to accommodate discontinuous rotations

Load-bearing premise

The Legendre transform of the curvature strain energy produces a stable and consistent discrete system for every slenderness ratio and every element order.

What would settle it

A benchmark computation on a highly slender beam with the lowest-order element that exhibits locking or suboptimal convergence rates would falsify the claims.

read the original abstract

We propose a novel mixed finite-element formulation for geometrically exact (Simo--Reissner) beams that introduces the moment vector as additional independent field. The specific mixed form allows for an element-local, discontinuous approximation of rotations, which is key to a simple and efficient discretization framework. The concept of discrete curvature provides a mathematically consistent treatment of rotation discontinuities. For linear constitutive laws, the mixed form is derived via a Legendre transform of the curvature-related strain energy. Objectivity is retained at the discrete level by interpolating relative rotations through a multiplicative split of the rotation field; path-independence is inherent to the total Lagrangian setting and verified numerically. Several benchmarks demonstrate optimal rates of convergence and accuracy, irrespective of the beam's slenderness and order of approximation. Notably, the lowest-order element entirely avoids rotation interpolation by employing element-constant rotations only.

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 proposes a novel mixed finite-element formulation for geometrically exact Simo-Reissner beams. It introduces the moment vector as an independent field, derived for linear constitutive laws via a Legendre transform of the curvature strain energy. This enables element-local discontinuous rotation approximations, with discrete curvature providing consistent treatment of rotation jumps. Objectivity is retained via multiplicative interpolation of relative rotations, and the total Lagrangian setting ensures path-independence. Numerical benchmarks are presented to demonstrate optimal convergence rates independent of beam slenderness and polynomial order, including a lowest-order element with element-constant rotations only.

Significance. If the stability and convergence properties hold as indicated by the benchmarks, the formulation would offer a practical simplification for discretizing geometrically exact beams by avoiding continuous rotation fields and associated interpolation complexities. The approach could be particularly useful for slender beams and varying element orders. The paper credits the total Lagrangian framework and multiplicative split for preserving key properties, and the numerical evidence for optimal rates across slenderness is a positive aspect, though the lack of supporting analysis limits the strength of the claims.

major comments (2)
  1. [Abstract and formulation derivation] The central claim that the method achieves optimal convergence 'irrespective of the beam's slenderness and order of approximation' rests on numerical benchmarks, but the mixed formulation obtained via Legendre transform lacks an explicit discrete inf-sup condition or uniform stability estimate. This is load-bearing for the assertion of robustness in the slender limit and for arbitrary orders, as the discontinuous rotations and discrete curvature are introduced precisely to enable the mixed form.
  2. [Numerical benchmarks] The lowest-order element, which employs element-constant rotations only, is highlighted as avoiding rotation interpolation; however, the paper does not provide a specific analysis or additional tests confirming that this element remains stable and accurate for extreme slenderness ratios without introducing locking or other pathologies.
minor comments (2)
  1. [Abstract] The abstract could more explicitly reference the specific benchmarks (e.g., number of test cases, range of slenderness ratios, and polynomial orders tested) to better support the convergence claims.
  2. Notation for the discrete curvature and the multiplicative split of the rotation field should be introduced with a brief reminder of their relation to the continuous fields to improve readability for readers unfamiliar with the Simo-Reissner model.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review. The comments correctly identify that our convergence claims rest primarily on numerical evidence rather than a full theoretical stability analysis. We address each major comment below, indicate planned revisions, and note where we cannot provide additional analysis within the current scope.

read point-by-point responses

  1. Referee: [Abstract and formulation derivation] The central claim that the method achieves optimal convergence 'irrespective of the beam's slenderness and order of approximation' rests on numerical benchmarks, but the mixed formulation obtained via Legendre transform lacks an explicit discrete inf-sup condition or uniform stability estimate. This is load-bearing for the assertion of robustness in the slender limit and for arbitrary orders, as the discontinuous rotations and discrete curvature are introduced precisely to enable the mixed form.

    Authors: We acknowledge that the manuscript provides no discrete inf-sup condition or uniform stability estimate for the mixed formulation. The claims of optimal convergence independent of slenderness and polynomial order are supported exclusively by the numerical benchmarks in Section 5, which include multiple slenderness ratios (from thick to very slender) and approximation orders up to cubic, with the lowest-order element showing no degradation or locking. The mixed moment field and discrete curvature are introduced precisely to permit stable discontinuous rotations while preserving objectivity via multiplicative interpolation. In the revised manuscript we will (i) rephrase the abstract and introduction to state that robustness is demonstrated numerically rather than proven theoretically, and (ii) add a short discussion paragraph in the formulation section that explicitly notes the absence of an inf-sup analysis and identifies it as future work. This is a partial revision because a complete theoretical proof lies beyond the present scope. revision: partial

  2. Referee: [Numerical benchmarks] The lowest-order element, which employs element-constant rotations only, is highlighted as avoiding rotation interpolation; however, the paper does not provide a specific analysis or additional tests confirming that this element remains stable and accurate for extreme slenderness ratios without introducing locking or other pathologies.

    Authors: The lowest-order element is already included in the convergence studies of Section 5 and exhibits optimal rates across the tested slenderness range. To strengthen the evidence, the revised manuscript will contain additional targeted numerical experiments using this element at extreme slenderness ratios (L/h up to 10^6) under both bending and torsion loads. These tests will explicitly monitor for locking, spurious modes, or loss of accuracy. We therefore accept the referee's suggestion and will incorporate the new results. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation rests on standard Legendre transform and total Lagrangian setting

full rationale

The core mixed weak form is obtained by applying the Legendre transform to the curvature strain energy for linear constitutive laws, then introducing the moment as independent field. This algebraic step is independent of the subsequent discrete choices (discontinuous rotations, discrete curvature) and of the numerical benchmarks. Path-independence follows directly from the total Lagrangian formulation. No self-citations are invoked as load-bearing uniqueness theorems, no fitted parameters are relabeled as predictions, and no ansatz is smuggled via prior work. The convergence rates are reported from numerical experiments rather than being presupposed by the derivation equations themselves.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The central claim rests on standard assumptions from geometrically exact beam theory and the introduction of discrete curvature and the moment field as new discretization devices; no fitted numerical parameters are mentioned.

axioms (2)
  • domain assumption The beam kinematics follow the Simo-Reissner geometrically exact model
    Basis for the entire formulation.
  • domain assumption Constitutive response is linear so that a Legendre transform yields the mixed form
    Explicitly required for the derivation of the mixed variational principle.
invented entities (2)
  • Moment vector as independent field no independent evidence
    purpose: To allow element-local discontinuous rotation approximation in the mixed formulation
    Introduced as an additional unknown to decouple rotations from displacements.
  • Discrete curvature no independent evidence
    purpose: To provide a mathematically consistent measure of curvature across rotation discontinuities
    New concept required to handle jumps in the rotation field.

pith-pipeline@v0.9.0 · 5448 in / 1435 out tokens · 92855 ms · 2026-05-08T16:15:57.019938+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

49 extracted references · 41 canonical work pages

  1. [1]

    Marc-Michel Bousquet et Soc., Lausanne (1744)

    Euler, L.: Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes, Sive Solutio Problematis Isoperimetrici Lattissimo Sensu Accepti. Marc-Michel Bousquet et Soc., Lausanne (1744)

  2. [2]

    Journal für die reine und angewandte Mathematik 1859(56), 285–313 (1859) https: //doi.org/10.1515/crll.1859.56.285

    Kirchhoff, G.: Ueber das Gleichgewicht und die Bewegung eines unendlich dünnen elastischen Stabes. Journal für die reine und angewandte Mathematik 1859(56), 285–313 (1859) https: //doi.org/10.1515/crll.1859.56.285

    work page doi:10.1515/crll.1859.56.285

  3. [3]

    On the correction for shear of the differential equation for transverse vibrations of prismatic bars

    Timoshenko, S.P.: LXVI. On the correction for shear of the differential equation for transverse vibrations of prismatic bars. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 41(245), 744–746 (1921) https://doi.org/10.1080/14786442108636264

    work page doi:10.1080/14786442108636264 1921

  4. [4]

    On one-dimensional finite-strain beam theory: The plane problem

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

    work page doi:10.1007/bf01602645 1972

  5. [5]

    Studies in Applied Mathematics 52(2), 87–95 (1973) https://doi.org/10.1002/sapm197352287

    Reissner, E.: On One-Dimensional Large-Displacement Finite-Strain Beam Theory. Studies in Applied Mathematics 52(2), 87–95 (1973) https://doi.org/10.1002/sapm197352287

    work page doi:10.1002/sapm197352287 1973

  6. [6]

    ZAMP Zeitschrift für angewandte Mathematik und Physik 32(6), 734–744 (1981) https://doi.org/10.1007/BF00946983

    Reissner, E.: On finite deformations of space-curved beams. ZAMP Zeitschrift für angewandte Mathematik und Physik 32(6), 734–744 (1981) https://doi.org/10.1007/BF00946983

    work page doi:10.1007/bf00946983 1981

  7. [7]

    In: Truesdell, C

    Antman, S.S.: The Theory of Rods. In: Truesdell, C. (ed.) Linear Theories of Elasticity and Thermoelasticity vol. VIa/2, pp. 641–703. Springer, Berlin Heidelberg (1973). https: //doi.org/10.1007/978-3-662-39776-3_6

    work page doi:10.1007/978-3-662-39776-3_6 1973

  8. [8]

    A finite strain beam formulation. The three-dimensional dynamic problem. Part I

    Simo, J.C.: A finite strain beam formulation. The three-dimensional dynamic problem. Part I. Computer Methods in Applied Mechanics and Engineering 49(1), 55–70 (1985) https: //doi.org/10.1016/0045-7825(85)90050-7 25

    work page doi:10.1016/0045-7825(85)90050-7 1985

  9. [9]

    part II: Computational aspects

    Simo, J.C., Vu-Quoc, L.: A three-dimensional finite-strain rod model. part II: Computational aspects. Computer Methods in Applied Mechanics and Engineering 58(1), 79–116 (1986) https://doi.org/10.1016/0045-7825(86)90079-4

    work page doi:10.1016/0045-7825(86)90079-4 1986

  10. [10]

    Journal of Applied Mechanics 53(4), 849–854 (1986) https://doi

    Simo, J.C., Vu-Quoc, L.: On the Dynamics of Flexible Beams Under Large Overall Motions— The Plane Case: Part I. Journal of Applied Mechanics 53(4), 849–854 (1986) https://doi. org/10.1115/1.3171870

    work page doi:10.1115/1.3171870 1986

  11. [11]

    Journal of Applied Mechanics 53(4), 855 (1986) https://doi.org/ 10.1115/1.3171871

    Simo, J.C., Vu-Quoc, L.: On the Dynamics of Flexible Beams Under Large Overall Motions— The Plane Case: Part II. Journal of Applied Mechanics 53(4), 855 (1986) https://doi.org/ 10.1115/1.3171871

    work page doi:10.1115/1.3171871 1986

  12. [12]

    Acta Mechanica 224(7), 1493–1525 (2013) https://doi.org/10.1007/s00707-013-0818-1

    Humer, A.: Exact solutions for the buckling and postbuckling of shear-deformable beams. Acta Mechanica 224(7), 1493–1525 (2013) https://doi.org/10.1007/s00707-013-0818-1

    work page doi:10.1007/s00707-013-0818-1 2013

  13. [13]

    Acta Mechanica (2019) https://doi.org/ 10.1007/s00707-019-02472-1

    Humer, A., Pechstein, A.S.: Exact solutions for the buckling and postbuckling of a shear- deformable cantilever subjected to a follower force. Acta Mechanica (2019) https://doi.org/ 10.1007/s00707-019-02472-1

    work page doi:10.1007/s00707-019-02472-1 2019

  14. [14]

    International Journal for Numerical Methods in Engineering 14(7), 961–986 (1979) https: //doi.org/10.1002/nme.1620140703

    Bathe, K.-J., Bolourchi, S.: Large displacement analysis of three‐dimensional beam structures. International Journal for Numerical Methods in Engineering 14(7), 961–986 (1979) https: //doi.org/10.1002/nme.1620140703

    work page doi:10.1002/nme.1620140703 1979

  15. [15]

    International Journal for Numerical Methods in Engineering 26(11), 2403–2438 (1988) https: //doi.org/10.1002/nme.1620261105

    Cardona, A., Geradin, M.: A beam finite element non‐linear theory with finite rotations. International Journal for Numerical Methods in Engineering 26(11), 2403–2438 (1988) https: //doi.org/10.1002/nme.1620261105

    work page doi:10.1002/nme.1620261105 1988

  16. [16]

    International Journal for Numerical Methods in Engineering 38(21), 3653–3673 (1995) https://doi.org/10.1002/nme.1620382107

    Ibrahimbegović, A., Frey, F., Kožar, I.: Computational aspects of vector‐like parametriza- tion of three‐dimensional finite rotations. International Journal for Numerical Methods in Engineering 38(21), 3653–3673 (1995) https://doi.org/10.1002/nme.1620382107

    work page doi:10.1002/nme.1620382107 1995

  17. [17]

    Computer Methods in Applied Mechanics and Engineering 149(1-4), 49–71 (1997) https://doi.org/10.1016/ S0045-7825(97)00059-5

    Ibrahimbegovic, A.: On the choice of finite rotation parameters. Computer Methods in Applied Mechanics and Engineering 149(1-4), 49–71 (1997) https://doi.org/10.1016/ S0045-7825(97)00059-5

    1997

  18. [18]

    Archives of Computational Methods in Engineering 26(1), 163–243 (2019) https://doi.org/10.1007/s11831-017-9232-5

    Meier, C., Popp, A., Wall, W.A.: Geometrically Exact Finite Element Formulations for Slen- der Beams: Kirchhoff–Love Theory Versus Simo–Reissner Theory. Archives of Computational Methods in Engineering 26(1), 163–243 (2019) https://doi.org/10.1007/s11831-017-9232-5

    work page doi:10.1007/s11831-017-9232-5 2019

  19. [19]

    Proceedings of the Royal Society of London

    Crisfield, M.A., Jelenić, G.: Objectivity of strain measures in the geometrically exact three- dimensional beam theory and its finite-element implementation. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 455(1983), 1125–1147 (1999) https://doi.org/10.1098/rspa.1999.0352

    work page doi:10.1098/rspa.1999.0352 1983

  20. [20]

    Computer Methods in Applied Mechanics and Engineering 171(1-2), 141–171 (1999) https://doi.org/10.1016/S0045-7825(98)00249-7

    Jelenić, G., Crisfield, M.A.: Geometrically exact 3D beam theory: implementation of a strain- invariant finite element for statics and dynamics. Computer Methods in Applied Mechanics and Engineering 171(1-2), 141–171 (1999) https://doi.org/10.1016/S0045-7825(98)00249-7

    work page doi:10.1016/s0045-7825(98)00249-7 1999

  21. [21]

    In: Haug, E.J

    Wehage, R.A.: Quaternions and Euler Parameters — A Brief Exposition. In: Haug, E.J. (ed.) Computer Aided Analysis and Optimization of Mechanical System Dynamics, pp. 147–180. Springer, Berlin, Heidelberg (1984). https://doi.org/10.1007/978-3-642-52465-3_5

    work page doi:10.1007/978-3-642-52465-3_5 1984

  22. [22]

    Nikravesh, P.E., Wehage, R.A., Kwon, O.K.: Euler Parameters in Computational Kinematics and Dynamics. Part 1. Journal of Mechanisms, Transmissions, and Automation in Design 107(3), 358–365 (1985) https://doi.org/10.1115/1.3260722 26

    work page doi:10.1115/1.3260722 1985

  23. [23]

    Computer Methods in Applied Mechanics and Engineering 198(49-52), 3944–3956 (2009) https://doi.org/10.1016/j.cma.2009.09.002

    Zupan, E., Saje, M., Zupan, D.: The quaternion-based three-dimensional beam theory. Computer Methods in Applied Mechanics and Engineering 198(49-52), 3944–3956 (2009) https://doi.org/10.1016/j.cma.2009.09.002

    work page doi:10.1016/j.cma.2009.09.002 2009

  24. [24]

    Acta Mechanica 224(8), 1709–1729 (2013) https://doi.org/10.1007/s00707-013-0824-3

    Zupan, E., Saje, M., Zupan, D.: On a virtual work consistent three-dimensional Reissner– Simo beam formulation using the quaternion algebra. Acta Mechanica 224(8), 1709–1729 (2013) https://doi.org/10.1007/s00707-013-0824-3

    work page doi:10.1007/s00707-013-0824-3 2013

  25. [25]

    A projection-based quaternion discretization of the geometrically exact beam model

    Wasmer, P., Betsch, P.: A projection‐based quaternion discretization of the geometrically exact beam model. International Journal for Numerical Methods in Engineering 125(20), 7538 (2024) https://doi.org/10.1002/nme.7538

    work page doi:10.1002/nme.7538 2024

  26. [26]

    Computer Methods in Applied Mechanics and Engineering 268, 451–474 (2014) https://doi.org/10.1016/j.cma.2013.10.008

    Sonneville, V., Cardona, A., Brüls, O.: Geometrically exact beam finite element formulated on the special Euclidean group. Computer Methods in Applied Mechanics and Engineering 268, 451–474 (2014) https://doi.org/10.1016/j.cma.2013.10.008

    work page doi:10.1016/j.cma.2013.10.008 2014

  27. [27]

    International Journal for Numerical Methods in Engineering 112(9), 1129–1153 (2017) https://doi.org/10.1002/nme.5548

    Sonneville, V., Brüls, O., Bauchau, O.A.: Interpolation schemes for geometrically exact beams: A motion approach. International Journal for Numerical Methods in Engineering 112(9), 1129–1153 (2017) https://doi.org/10.1002/nme.5548

    work page doi:10.1002/nme.5548 2017

  28. [28]

    International Journal for Numerical Methods in Engineering 124(13), 2965–2994 (2023) https://doi.org/10.1002/ nme.7236

    Harsch, J., Sailer, S., Eugster, S.R.: A total Lagrangian, objective and intrinsically locking- free Petrov–Galerkin SE (3) Cosserat rod finite element formulation. International Journal for Numerical Methods in Engineering 124(13), 2965–2994 (2023) https://doi.org/10.1002/ nme.7236

    2023

  29. [29]

    International Journal for Numerical Methods in Engineering 17(4), 615–631 (1981) https://doi.org/10.1002/nme.1620170409

    Noor, A.K., Peters, J.M.: Mixed models and reduced/selective integration displacement mod- els for nonlinear analysis of curved beams. International Journal for Numerical Methods in Engineering 17(4), 615–631 (1981) https://doi.org/10.1002/nme.1620170409

    work page doi:10.1002/nme.1620170409 1981

  30. [30]

    Computer Methods in Applied Mechanics and Engineering 193(23-26), 2507–2545 (2004) https://doi.org/10.1016/j.cma.2004.01.029

    Nukala, P.K.V.V., White, D.W.: A mixed finite element for three-dimensional nonlinear anal- ysis of steel frames. Computer Methods in Applied Mechanics and Engineering 193(23-26), 2507–2545 (2004) https://doi.org/10.1016/j.cma.2004.01.029

    work page doi:10.1016/j.cma.2004.01.029 2004

  31. [31]

    Herrmann, M., Castello, D., Breuling, J., Garcia, I.C., Greco, L., Eugster, S.R.: A mixed Petrov-Galerkin Cosserat rod finite element formulation. arXiv. arXiv:2507.01552 [math] (2026). https://doi.org/10.48550/arXiv.2507.01552

    work page doi:10.48550/arxiv.2507.01552 2026

  32. [32]

    International Journal of Solids and Structures 26(8), 887–900 (1990) https://doi.org/10

    Saje, M.: A variational principle for finite planar deformation of straight slender elastic beams. International Journal of Solids and Structures 26(8), 887–900 (1990) https://doi.org/10. 1016/0020-7683(90)90075-7

    1990

  33. [33]

    Computers & Structures 39(3-4), 327–337 (1991) https://doi.org/10.1016/0045-7949(91) 90030-P

    Saje, M.: Finite element formulation of finite planar deformation of curved elastic beams. Computers & Structures 39(3-4), 327–337 (1991) https://doi.org/10.1016/0045-7949(91) 90030-P

    work page doi:10.1016/0045-7949(91 1991

  34. [34]

    Computers & Structures 225, 106109 (2019) https://doi.org/10.1016/j.compstruc.2019

    Neunteufel, M., Schöberl, J.: The Hellan–Herrmann–Johnson method for nonlinear shells. Computers & Structures 225, 106109 (2019) https://doi.org/10.1016/j.compstruc.2019. 106109

    work page doi:10.1016/j.compstruc.2019 2019

  35. [35]

    Computer Methods in Applied Mechanics and Engineering 373, 113524 (2021) https://doi.org/10.1016/ j.cma.2020.113524

    Neunteufel, M., Schöberl, J.: A voiding membrane locking with Regge interpolation. Computer Methods in Applied Mechanics and Engineering 373, 113524 (2021) https://doi.org/10.1016/ j.cma.2020.113524

    work page arXiv 2021

  36. [36]

    Acta Mechanica 27 (2025) https://doi.org/10.1007/s00707-025-04255-3

    Platzer, S., Pechstein, A., Humer, A., Krommer, M.: Viscoelastic Kirchhoff–Love shells at finite strains: constitutive modeling and mixed low-regularity finite elements. Acta Mechanica 27 (2025) https://doi.org/10.1007/s00707-025-04255-3

    work page doi:10.1007/s00707-025-04255-3 2025

  37. [37]

    ISSS Journal of Micro and Smart Systems (2025) https://doi.org/10.1007/s41683-025-00138-w

    Platzer, S., Pechstein, A., Humer, A., Krommer, M.: A low-regularity finite element approach for dielectric viscoelastic Kirchhoff–Love shells. ISSS Journal of Micro and Smart Systems (2025) https://doi.org/10.1007/s41683-025-00138-w

    work page doi:10.1007/s41683-025-00138-w 2025

  38. [38]

    Computational Mechanics 66(6), 1377–1398 (2020) https://doi.org/10.1007/s00466-020-01907-0

    Steinbrecher, I., Mayr, M., Grill, M.J., Kremheller, J., Meier, C., Popp, A.: A mortar-type finite element approach for embedding 1D beams into 3D solid volumes. Computational Mechanics 66(6), 1377–1398 (2020) https://doi.org/10.1007/s00466-020-01907-0

    work page doi:10.1007/s00466-020-01907-0 2020

  39. [39]

    Cambridge university press, Cambridge, GB (1998)

    Shabana, A.A.: Dynamics of Multibody Systems, 2nd ed edn. Cambridge university press, Cambridge, GB (1998)

    1998

  40. [40]

    Wiley, Chichester Weinheim (2001)

    Géradin, M., Cardona, A.: Flexible Multibody Dynamics: a Finite Element Approach. Wiley, Chichester Weinheim (2001)

    2001

  41. [41]

    2: Advanced Topics, Repr

    Crisfield, M.A.: Non-linear Finite Element Analysis of Solids and Structures. 2: Advanced Topics, Repr. with corr edn. Wiley, Chichester (2003). Num Pages: 494

    2003

  42. [42]

    Computational Mechanics 57(5), 817–841 (2016) https://doi.org/10

    Sander, O., Neff, P., Bîrsan, M.: Numerical treatment of a geometrically nonlinear planar Cosserat shell model. Computational Mechanics 57(5), 817–841 (2016) https://doi.org/10. 1007/s00466-016-1263-5

    2016

  43. [43]

    Computer Graphics Forum 25(3), 547–556 (2006) https://doi.org/10.1111/ j.1467-8659.2006.00974.x

    Grinspun, E., Gingold, Y., Reisman, J., Zorin, D.: Computing discrete shape operators on general meshes. Computer Graphics Forum 25(3), 547–556 (2006) https://doi.org/10.1111/ j.1467-8659.2006.00974.x

    work page arXiv 2006

  44. [44]

    Computer Methods in Applied Mechanics and Engineering 194(21- 24), 2444–2463 (2005) https://doi.org/10.1016/j.cma.2004.07.040

    Koschnick, F., Bischoff, M., Camprubí, N., Bletzinger, K.-U.: The discrete strain gap method and membrane locking. Computer Methods in Applied Mechanics and Engineering 194(21- 24), 2444–2463 (2005) https://doi.org/10.1016/j.cma.2004.07.040

    work page doi:10.1016/j.cma.2004.07.040 2005

  45. [45]

    Journal of Structural Mechanics 11(2), 153–176 (1983) https://doi.org/ 10.1080/03601218308907439

    Stolarskl, H., Belytschko, T., Carpenter, N.: Bending and Shear Mode Decomposition in C ◦ Structural Elements. Journal of Structural Mechanics 11(2), 153–176 (1983) https://doi.org/ 10.1080/03601218308907439

    work page doi:10.1080/03601218308907439 1983

  46. [46]

    Computer Methods in Applied Mechanics and Engineering 290, 314–341 (2015) https://doi.org/10.1016/j.cma.2015.02.029

    Meier, C., Popp, A., Wall, W.A.: A locking-free finite element formulation and reduced mod- els for geometrically exact Kirchhoff rods. Computer Methods in Applied Mechanics and Engineering 290, 314–341 (2015) https://doi.org/10.1016/j.cma.2015.02.029

    work page doi:10.1016/j.cma.2015.02.029 2015

  47. [47]

    Finite element formulations for constrained spatial nonlinear beam theories

    Harsch, J., Capobianco, G., Eugster, S.R.: Finite element formulations for constrained spatial nonlinear beam theories. Mathematics and Mechanics of Solids 26(12), 1838–1863 (2021) https://doi.org/10.1177/10812865211000790

    work page doi:10.1177/10812865211000790 2021

  48. [48]

    Multibody System Dynamics 20(1), 51–68 (2008) https://doi.org/10.1007/s11044-008-9105-7

    Romero, I.: A comparison of finite elements for nonlinear beams: The absolute nodal coordi- nate and geometrically exact formulations. Multibody System Dynamics 20(1), 51–68 (2008) https://doi.org/10.1007/s11044-008-9105-7

    work page doi:10.1007/s11044-008-9105-7 2008

  49. [49]

    Rezaian, M

    Eugster, S.R., Hesch, C., Betsch, P., Glocker, Ch.: Director-based beam finite elements relying on the geometrically exact beam theory formulated in skew coordinates. International Journal for Numerical Methods in Engineering 97(2), 111–129 (2014) https://doi.org/10.1002/nme. 4586 28

    work page doi:10.1002/nme 2014