pith. sign in

arxiv: 2602.14566 · v1 · pith:SHYDTCVInew · submitted 2026-02-16 · 💻 cs.CE

Simultaneous analysis of curved Kirchhoff beams and Kirchhoff--Love shells embedded in bulk domains

Jonas Neumeyer , Michael Wolfgang Kaiser , Thomas-Peter Fries This is my paper

Pith reviewed 2026-05-22 11:55 UTC · model grok-4.3

classification 💻 cs.CE
keywords Bulk Trace FEMlevel setsKirchhoff beamsKirchhoff-Love shellsmixed-hybrid formulationembedded structureshigher-order convergenceC0 elements
0
0 comments X

The pith

A mixed-hybrid Bulk Trace FEM lets standard C0 Lagrange elements from the bulk domain model multiple curved Kirchhoff beams and shells at once via level sets.

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

The paper develops a numerical method to treat an entire collection of curved beams and shells whose shapes are defined by level sets of a single scalar function inside a larger bulk domain. By extending the Kirchhoff-Love equations across all level sets simultaneously, the problem is recast as a bulk problem that can be discretized with ordinary finite elements. A mixed-hybrid formulation is introduced to relax the C1 continuity normally required for displacement-based Kirchhoff models, allowing standard C0-continuous Lagrange elements while retaining higher-order accuracy. Several numerical examples confirm optimal convergence rates and serve as reference cases for future embedded-structure studies.

Core claim

The authors propose a mixed-hybrid and higher-order accurate Bulk Trace FEM that enables the use of standard C0-continuous Lagrange elements with dimensionality of the bulk domain for simultaneous modeling of beams and shells on level sets, with the geometries of the structures implied by level sets of a scalar function over the bulk domain.

What carries the argument

Mixed-hybrid Bulk Trace FEM that approximates displacements and moments on level-set traces using bulk-domain Lagrange elements of arbitrary polynomial degree.

If this is right

  • Standard C0-continuous Lagrange elements from the bulk domain can be used for higher-order accurate solutions of Kirchhoff beam and shell models.
  • Multiple beams and shells lying on different level sets can be analyzed within a single bulk computation without generating separate surface meshes.
  • The approach achieves optimal convergence rates in displacement and stress quantities as verified through numerical tests.
  • Level-set geometry descriptions allow parametric variations of the embedded structures by simply modifying the scalar level-set function.

Where Pith is reading between the lines

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

  • The single-bulk-mesh treatment could simplify coupled problems in which beams or shells exchange forces or heat with surrounding material inside the same domain.
  • The benchmark cases introduced in the paper provide concrete reference solutions that other embedded or trace finite element schemes can be tested against.
  • The mixed-hybrid relaxation of continuity requirements may carry over to time-dependent or moderately nonlinear Kirchhoff-type models while preserving the bulk discretization advantage.

Load-bearing premise

The geometries of the beams and shells are accurately implied by level sets of a scalar function over the bulk domain, and the Kirchhoff-Love assumptions of small displacements without shear deformations hold for all structures simultaneously.

What would settle it

A benchmark test with a known exact solution for a curved Kirchhoff shell in which the measured L2 or energy-norm error fails to decrease at the optimal rate for the chosen polynomial degree would falsify the higher-order accuracy of the mixed-hybrid formulation.

Figures

Figures reproduced from arXiv: 2602.14566 by Jonas Neumeyer, Michael Wolfgang Kaiser, Thomas-Peter Fries.

Figure 1
Figure 1. Figure 1: For an arbitrary bulk domain Ω and level-set function ϕ(x), (a) and (d) show the level￾set functions ϕ(x), (b) and (e) some selected level sets Γ c in Ω. When restricting bulk domains to prescribed level-set intervals, see Eq. (2.2), examples are seen in (c) and (f), including selected levels sets Γ c . 2.2 Geometric quantities and differential operators In order to define the governing equations of a boun… view at source ↗
Figure 2
Figure 2. Figure 2: Vector fields on a selected manifold Γ c and its boundary ∂Γ c . The normal vector m with respect to ∂Ω is shown in gray, the normal vector n with respect to Γ c in blue, the conormal vector q in green, and the tangent vector t in red. bination of coarea formulas formulated over bulk domains and classical integral theorems relevant for individual manifolds. First, the formulation of a family of manifolds m… view at source ↗
Figure 3
Figure 3. Figure 3: Boundary quantities on a selected boundary ∂Γ c . Displacements and force components are split in terms of (a) n and q for beams or (b) n, q, and t for shells. Rotations and moments act around the z-axis for beams or around t for shells. A distinction is made between the Dirichlet boundaries ∂Γ c D,i and Neumann boundaries ∂Γ c N,i [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A three-dimensional illustration of the two-dimensional plane from [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: For some bulk domain Ω in R 2 and R 3 , (a) and (d) shows an example mesh Ω h discretized by cubic elements, (b) and (e) the discretized boundary ∂Ω h , and (c) and (f) the interface mesh Ψh . The lower limit of the level set ϕ min is depicted in blue, the upper limit of the level set ϕ max in red, and some selected level sets of ϕ within these limits in gray. and m must also be evaluated there, i.e., on Ψ… view at source ↗
Figure 6
Figure 6. Figure 6: Neighboring elements T + and T − and the edge or face element F located inbetween. Some selected level sets are indicated in gray. trial functions, are obtained straightforwardly by usual isoparametric mappings of one-, two-, or three-dimensional reference elements to the real, physical elements in the mesh, respectively. Note that equal-order interpolations of the individual mechanical fields (e.g., for u… view at source ↗
Figure 7
Figure 7. Figure 7: Setup of the family of arc-shaped beams in R 2 : (a) Geometry of the discretized bulk domain Ω h and some selected beams Γ c . (b) Scaled deformed configuration with the Euclidean norm of the displacements ∥u∥ as a color plot over the bulk domain. The gray mesh lines and level sets indicate the undeformed configuration. v = ˆv = 0. An analytical solution for an individual arc can be found in [43]. There, n… view at source ↗
Figure 8
Figure 8. Figure 8: Plots of different stress resultants of the family of arc-shaped beams in R 2 over Ω and Γ c . (a) and (d) show the only nonzero principal component of the moment tensor m (1) Γ , (b) and (e) the only nonzero principal component of the physical normal force tensor n real(1) Γ and (c) and (f) the shear force q. 5.2 Family of circular beams over a more general bulk domain in R 2 The geometric setup is reprod… view at source ↗
Figure 9
Figure 9. Figure 9: Convergence studies for the family of arc-shaped beams in R 2 . Convergence rates in the L 2 -error for (a) the displacements u, (b) the only nonzero principal component of the moment tensor m (1) Γ , (c) the only nonzero principal component of the physical normal force tensor n real(1) Γ , and (d) the shear force q and (e) the relative stored energy error εe . the level-set function ψ(x) = p x 2 + y 2 , t… view at source ↗
Figure 10
Figure 10. Figure 10: Setup of the family of circular beams over a more general bulk domain in R 2 : (a) Geometry of the discretized bulk domain Ω h and some selected beams Γ c . (b) Scaled deformed configuration with the Euclidean norm of the displacements ∥u∥ as a color plot over the bulk domain. The gray mesh lines and level sets indicate the undeformed configuration. a pre-asymptotic regime. For the stored energy error ana… view at source ↗
Figure 11
Figure 11. Figure 11: Convergence studies for the family of circular beams over a more general bulk domain in R 2 . Convergence rates in (a) the absolute residual error εres,1, (b) the relative residual error εres,2, and (c) the relative stored energy error εe [PITH_FULL_IMAGE:figures/full_fig_p027_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Setup of the family of simply supported shells in R 3 : (a) Geometry of the discretized bulk domain Ω h and some selected shells Γ c . (b) Scaled deformed configuration with the Euclidean norm of the displacements ∥u∥ as a contour plot over some selected shells. The gray mesh lines imply the undeformed configuration. Regarding the test case parameters, the load vector is set to f = [0, 0, −100]T, Young’s … view at source ↗
Figure 13
Figure 13. Figure 13: Convergence studies for the family of simply supported shells in R 3 . Convergence rates in (a) the absolute residual error εres,1, (b) the relative residual error εres,2, and (c) the relative stored energy error εe . 5.4 Family of clamped cupolas on a curved surface in R 3 Adopted from [35], the last test case represents a family of spherical cupolas, described by the level-set function ϕ(x) = p x 2 + y … view at source ↗
Figure 14
Figure 14. Figure 14: (a) for the geometry of the bulk domain and the level-sets and [PITH_FULL_IMAGE:figures/full_fig_p030_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Convergence studies for the family of clamped cupolas on a curved surface in R 3 . Con￾vergence rates in (a) the absolute residual error εres,1, (b) the relative residual error εres,2, and (c) the relative stored energy error εe . 6 Conclusions and outlook In this work, a mechanical model and a corresponding numerical method are proposed for the simultaneous analysis of geometrically linear Kirchhoff beam… view at source ↗
read the original abstract

A set of curved beams and shells is geometrically implied by level sets of a scalar function over some bulk domain. The mechanical model for each structure is based on the Kirchhoff--Love theory, that is, small displacements without shear deformations are considered. These models for individual geometries are extended to bulk models, simultaneously modeling the whole set of beams/shells on all level sets. A major focus is on the numerical analysis of such models. A mixed-hybrid and higher-order accurate Bulk Trace FEM is proposed that enables the use of standard $C^0$-continuous Lagrange elements with dimensionality of the bulk domain. That is, the higher-order continuity requirements of displacement-based formulations in context of the Kirchhoff--Love theory are successfully alleviated. Several numerical tests confirm the accuracy and higher-order convergence of the proposed methodology, also qualifying as benchmark test cases in future studies.

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

1 major / 1 minor

Summary. The paper proposes a mixed-hybrid Bulk Trace FEM for simultaneous modeling of curved Kirchhoff beams and Kirchhoff-Love shells geometrically implied by level sets of a scalar function in a bulk domain. Individual Kirchhoff-Love models (small displacements, no shear) are extended to bulk models, and the method uses standard C0-continuous Lagrange elements of bulk dimensionality to alleviate higher-order continuity requirements. Numerical tests are claimed to confirm accuracy and higher-order convergence, positioning the work as providing benchmark cases.

Significance. If the stability and convergence claims hold, the approach would enable efficient implicit embedding of mixed 1D/2D thin structures in bulk meshes without specialized high-continuity elements, offering a practical tool for structural analysis with complex level-set geometries.

major comments (1)
  1. [Numerical analysis / stability discussion] The central claim of higher-order convergence with standard C0 Lagrange bulk elements requires uniform stability of the mixed-hybrid spaces (displacements plus hybrid multipliers for moments/rotations). No explicit inf-sup analysis or numerical check of the inf-sup constant is reported under mesh refinement or varying level-set curvatures/gradients, leaving the stability (and thus the convergence) conditional on unverified assumptions about the trace operators and hybridization.
minor comments (1)
  1. [Abstract] The abstract states that 'several numerical tests confirm the accuracy and higher-order convergence' but provides no details on the specific test geometries, curvature ranges, or error measures used; this should be expanded for reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address the major comment point by point below, providing clarifications and indicating planned revisions where appropriate.

read point-by-point responses

  1. Referee: The central claim of higher-order convergence with standard C0 Lagrange bulk elements requires uniform stability of the mixed-hybrid spaces (displacements plus hybrid multipliers for moments/rotations). No explicit inf-sup analysis or numerical check of the inf-sup constant is reported under mesh refinement or varying level-set curvatures/gradients, leaving the stability (and thus the convergence) conditional on unverified assumptions about the trace operators and hybridization.

    Authors: We acknowledge that an explicit inf-sup analysis would provide a stronger theoretical basis. The current work emphasizes the formulation of the mixed-hybrid Bulk Trace FEM and its practical numerical performance rather than a complete stability theory for arbitrary level sets. Section 5 reports multiple benchmark tests demonstrating optimal higher-order convergence for beams and shells under mesh refinement, including examples with curved geometries and different level-set gradients; these results show no degradation indicative of instability. To directly address the concern, the revised manuscript will add a subsection with numerical computations of the inf-sup constant on successively refined meshes for representative level-set configurations. revision: partial

Circularity Check

0 steps flagged

No circularity: Bulk Trace FEM derivation relies on standard FEM theory and numerical verification

full rationale

The paper extends Kirchhoff-Love beam/shell models to bulk domains via level-set geometry and proposes a mixed-hybrid discretization using standard C0 Lagrange elements. Central claims of higher-order accuracy and simultaneous modeling are supported by explicit numerical convergence tests on benchmark cases rather than by any self-referential definitions, fitted parameters renamed as predictions, or load-bearing self-citations. The derivation chain begins from classical thin-structure kinematics and trace operators, then applies hybridization to relax continuity requirements; these steps are independent of the target results and do not reduce to the inputs by construction. External benchmarks (convergence rates) are used to qualify the method, confirming self-contained analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only abstract available; ledger populated from stated assumptions in abstract. No free parameters or invented entities are mentioned.

axioms (2)
  • domain assumption Kirchhoff-Love theory applies: small displacements without shear deformations for all structures on level sets.
    Stated directly in abstract as the mechanical model basis.
  • domain assumption Geometries of beams and shells are exactly implied by level sets of a scalar function over the bulk domain.
    Core modeling choice described in abstract.

pith-pipeline@v0.9.0 · 5679 in / 1215 out tokens · 31731 ms · 2026-05-22T11:55:22.726050+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

60 extracted references · 60 canonical work pages

  1. [1]

    Arnold, D.N.; Brezzi, F.: Mixed and nonconforming finite element methods: imple- mentation, postprocessing and error estimates.ESAIM Math. Model. Numer. Anal., 19, 7–32, 1985

    work page 1985

  2. [2]

    Vieweg+Teubner Verlag, Braunschweig, 1985

    Başar, Y.; Krätzig, W.B.:Mechanik der Flächentragwerke. Vieweg+Teubner Verlag, Braunschweig, 1985. REFERENCES33

    work page 1985

  3. [3]

    InEncyclopedia of Computational Mechanics

    Bischoff, M.; Ramm, E.; Irslinger, J.: Models and Finite Elements for Thin-Walled Structures. InEncyclopedia of Computational Mechanics. (Stein, E.; Borst, R.; Hughes, T.J.; Hughes, T.J., Eds.), John Wiley & Sons, Chichester, 2 edition, 2017

    work page 2017

  4. [4]

    44,Springer Series in Computational Mathematics

    Boffi, D.; Brezzi, F.; Fortin, M.:Mixed Finite Element Methods and Applications, Vol. 44,Springer Series in Computational Mathematics. Springer, Berlin, 2013

    work page 2013

  5. [5]

    Numér.,8, 129–151, 1974

    Brezzi, F.: On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers.RAIRO Anal. Numér.,8, 129–151, 1974

    work page 1974

  6. [6]

    Burger, M.: Finite element approximation of elliptic partial differential equations on implicit surfaces.Comput. Vis. Sci.,12, 87–100, 2009

    work page 2009

  7. [7]

    Methods Appl

    Burman, E.; Hansbo, P.; Larson, M.G.: A stabilized cut finite element method for partial differential equations on surfaces: The Laplace-Beltrami operator.Comput. Methods Appl. Mech. Eng.,285, 188–207, 2015

    work page 2015

  8. [8]

    ESAIM Math

    Burman, E.; Hansbo, P.; Larson, M.G.; Massing, A.: Cut finite element methods for partial differential equations on embedded manifolds of arbitrary codimensions. ESAIM Math. Model. Numer. Anal.,52, 2247–2282, 2018

    work page 2018

  9. [9]

    Methods Appl

    Cenanovic, M.; Hansbo, P.; Larson, M.G.: Cut finite element modeling of linear membranes.Comput. Methods Appl. Mech. Eng.,310, 98–111, 2016

    work page 2016

  10. [10]

    Struct.,66, 19–36, 1998

    Chapelle, D.; Bathe, K.J.: Fundamental considerations for the finite element analysis of shell structures.Comput. Struct.,66, 19–36, 1998

    work page 1998

  11. [11]

    Cockburn, B.; Gopalakrishnan, J.: A Characterization of Hybridized Mixed Methods for Second Order Elliptic Problems.SIAM J. Numer. Anal.,42, 283–301, 2004

    work page 2004

  12. [12]

    Cockburn, B.; Gopalakrishnan, J.; Lazarov, R.: Unified Hybridization of Discontin- uous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order Elliptic Problems.SIAM J. Numer. Anal.,47, 1319–1365, 2009

    work page 2009

  13. [13]

    Comput.,52, 17–29, 1989

    Comodi, M.I.: The Hellan-Herrmann-Johnson Method: some new error estimates and postprocessing.Math. Comput.,52, 17–29, 1989

    work page 1989

  14. [14]

    Delfour, M.C.; Zolésio, J.P.: A Boundary Differential Equation for Thin Shells.J. Differ. Equ.,119, 426–449, 1995. 34 REFERENCES

    work page 1995

  15. [15]

    Delfour, M.C.; Zolésio, J.P.: Tangential Differential Equations for Dynamical Thin/Shallow Shells.J. Differ. Equ.,128, 125–167, 1996

    work page 1996

  16. [16]

    Advances in Design and Control

    Delfour, M.C.; Zolésio, J.P.:Shapes and geometries: Metrics, Analysis, Differential Calculus, and Optimization. Advances in Design and Control. SIAM, Philadelphia, PA, 2 edition, 2011

    work page 2011

  17. [17]

    Dziuk, G.; Elliott, C.M.: An Eulerian approach to transport and diffusion on evolving implicit surfaces.Comput. Vis. Sci.,13, 17–28, 2008

    work page 2008

  18. [18]

    Dziuk, G.; Elliott, C.M.: Eulerian finite element method for parabolic PDEs on im- plicit surfaces.Interfaces Free Bound.,10, 119–138, 2008

    work page 2008

  19. [19]

    Dziuk, G.; Elliott, C.M.: Finite element methods for surface PDEs.Acta Numer.,22, 289–396, 2013

    work page 2013

  20. [20]

    Methods Appl

    Echter, R.; Oesterle, B.; Bischoff, M.: A hierarchic family of isogeometric shell finite elements.Comput. Methods Appl. Mech. Eng.,254, 170–180, 2013

    work page 2013

  21. [21]

    Springer, New York, 1969

    Federer, H.:Geometric measure theory. Springer, New York, 1969

    work page 1969

  22. [22]

    Fries, T.P.: Higher-order surface FEM for incompressible Navier-Stokes flows on man- ifolds.Int. J. Numer. Methods Fluids,88, 55–78, 2018

    work page 2018

  23. [23]

    Methods Appl

    Fries, T.P.; Kaiser, M.W.: On the Simultaneous Solution of Structural Membranes on all Level Sets within a Bulk Domain.Comput. Methods Appl. Mech. Eng.,415, 116223, 2023

    work page 2023

  24. [24]

    Fries, T.P.; Kaiser, M.W.: Simultaneous, Dynamical Analysis of Structural Ropes and Membranes on all Level-sets.Proceedings of the 9th European Congress on Com- putational Methods in Applied Sciences and Engineering (ECCOMAS 2024), Lisbon, Portugal, 2024

    work page 2024

  25. [25]

    Fries, T.P.; Neumeyer, J.; Kaiser, M.W.: A new concept for embedding sub-structures via level-sets.Proceedings of the 16th World Congress on Computational Mechanics (WCCM 2024), Vancouver, Canada, 2024

    work page 2024

  26. [26]

    Fries, T.P.; Schöllhammer, D.: A unified finite strain theory for membranes and ropes. Comput. Methods Appl. Mech. Eng.,365, 113031, 2020. REFERENCES35

    work page 2020

  27. [27]

    Mech.,67, 679–697, 2021

    Gfrerer, M.H.: AC1-continuousTrace-Finite-Cell-Methodforlinearthinshellanalysis on implicitly defined surfaces.Comput. Mech.,67, 679–697, 2021

    work page 2021

  28. [28]

    Struct.,30, 3355–3364, 2008

    Gimena, L.; Gimena, F.N.; Gonzaga, P.: Structural analysis of a curved beam element defined in global coordinates.Eng. Struct.,30, 3355–3364, 2008

    work page 2008

  29. [29]

    Methods Appl

    Hansbo, P.; Larson, M.G.: Finiteelementmodelingofalinearmembraneshellproblem using tangential differential calculus.Comput. Methods Appl. Mech. Eng.,270, 1–14, 2014

    work page 2014

  30. [30]

    Mech.,53, 611–623, 2014

    Hansbo, P.; Larson, M.G.; Larsson, K.: Variational formulation of curved beams in global coordinates.Comput. Mech.,53, 611–623, 2014

    work page 2014

  31. [31]

    Jankuhn, T.; Olshanskii, M.A.; Reusken, A.: Incompressible fluid problems on em- bedded surfaces: Modeling and variational formulations.Interfaces Free Bound.,20, 353–377, 2018

    work page 2018

  32. [32]

    Math.,21, 43–62, 1973

    Johnson, C.: On the convergence of a mixed finite-element method for plate bending problems.Numer. Math.,21, 43–62, 1973

    work page 1973

  33. [33]

    Methods Appl

    Kabaria, H.; Lew, A.J.; Cockburn, B.: A hybridizable discontinuous Galerkin formu- lation for non-linear elasticity.Comput. Methods Appl. Mech. Eng.,283, 303–329, 2015

    work page 2015

  34. [34]

    Kaiser, M.W.; Fries, T.P.: Curved, linear Kirchhoff beams formulated using tan- gential differential calculus and Lagrange multipliers.Proc. Appl. Math. Mech.,22, e202200042, 2023

    work page 2023

  35. [35]

    Kaiser, M.W.; Fries, T.P.: Simultaneous analysis of continuously embedded Reissner- Mindlin shells in 3D bulk domains.Int. J. Numer. Methods Eng.,125, e7495, 2024

    work page 2024

  36. [36]

    Kaiser, M.W.; Fries, T.P.: Simultaneous solution of implicitly defined curved, linear Timoshenko beams in two-dimensional bulk domains.Proc. Appl. Math. Mech.,24, e202400019, 2024

    work page 2024

  37. [37]

    Kaiser, M.W.; Fries, T.P.: SimultaneoussolutionofincompressibleNavier-Stokesflows on multiple surfaces.Arch. Appl. Mech.,95, 2025. 36 REFERENCES

    work page 2025

  38. [38]

    Methods Appl

    Kiendl, J.; Bletzinger, K.U.; Linhard, J.; Wüchner, R.: Isogeometric shell analysis with Kirchhoff-Love elements.Comput. Methods Appl. Mech. Eng.,198, 3902–3914, 2009

    work page 2009

  39. [39]

    Kirchhoff, G.: Über das Gleichgewicht und die Bewegung einer elastischen Scheibe. J. reine angew. Math.,40, 51–88, 1850

    work page

  40. [40]

    The small free vibrations and deformation of a thin elastic shell

    Love, A.E.H.: XVI. The small free vibrations and deformation of a thin elastic shell. Philos. Trans. R. Soc. A,179, 491–546, 1888

    work page

  41. [41]

    Mindlin, R.D.: Influence of Rotatory Inertia and Shear on Flexural Motions of Isotropic, Elastic Plates.ASME J. Appl. Mech.,18, 31–38, 1951

    work page 1951

  42. [42]

    Academic press, San Diego, 1988

    Morgan, F.:Geometric measure theory: a beginner’s guide. Academic press, San Diego, 1988

    work page 1988

  43. [43]

    Neumeyer, J.; Kaiser, M.W.; Fries, T.P.: Higher-Order, Mixed-HybridFiniteElements for Kirchhoff–Love Shells.Int. J. Numer. Methods Eng.,126, e70175, 2025

    work page 2025

  44. [44]

    Struct.,225, 106109, 2019

    Neunteufel, M.; Schöberl, J.: The Hellan–Herrmann–Johnson method for nonlinear shells.Comput. Struct.,225, 106109, 2019

    work page 2019

  45. [45]

    Struct.,305, 107543, 2024

    Neunteufel, M.; Schöberl, J.: The Hellan–Herrmann–Johnson and TDNNS methods for linear and nonlinear shells.Comput. Struct.,305, 107543, 2024

    work page 2024

  46. [46]

    InGeometrically Unfitted Finite Element Methods and Applications

    Olshanskii, M.A.; Reusken, A.: Trace finite element methods for PDEs on surfaces. InGeometrically Unfitted Finite Element Methods and Applications. (Bordas, S.P.A.; Burman, E.; Larson, M.G.; Olshanskii, M.A.; Olshanskii, M.A., Eds.), Vol. 121,Lec- ture notes in computational science and engineering, Springer Nature, Cham, 211–258, 2017

    work page 2017

  47. [47]

    Olshanskii, M.A.; Reusken, A.; Grande, J.: A finite element method for elliptic equa- tions on surfaces.SIAM J. Numer. Anal.,47, 3339–3358, 2009

    work page 2009

  48. [48]

    Olshanskii, M.A.; Reusken, A.; Xu, X.: A stabilized finite element method for advection-diffusion equations on surfaces.IMA J. Numer. Anal.,34, 732–758, 2014

    work page 2014

  49. [49]

    Osher, S.; Fedkiw, R.P.: Level set methods: an overview and some recent results.J. Comput. Phys.,169, 463–502, 2001. REFERENCES37

    work page 2001

  50. [50]

    Springer, Berlin, 2003

    Osher, S.; Fedkiw, R.P.:Level Set Methods and Dynamic Implicit Surfaces. Springer, Berlin, 2003

    work page 2003

  51. [51]

    Rafetseder, K.; Zulehner, W.: A decomposition result for Kirchhoff plate bending problems and a new discretization approach.SIAM J. Numer. Anal.,56, 1961–1986, 2018

    work page 1961

  52. [52]

    Methods Appl

    Rafetseder, K.; Zulehner, W.: A new mixed approach to Kirchhoff–Love shells.Com- put. Methods Appl. Mech. Eng.,346, 440–455, 2019

    work page 2019

  53. [53]

    Reissner, E.: The Effect of Transverse Shear Deformation on the Bending of Elastic Plates.ASME J. Appl. Mech.,12, A69–A77, 1945

    work page 1945

  54. [54]

    Mech.,64, 113–131, 2019

    Schöllhammer, D.; Fries, T.P.: Kirchhoff-Love shell theory based on tangential differ- ential calculus.Comput. Mech.,64, 113–131, 2019

    work page 2019

  55. [55]

    Methods Appl

    Schöllhammer, D.; Fries, T.P.: Reissner-Mindlin shell theory based on tangential differential calculus.Comput. Methods Appl. Mech. Eng.,352, 172–188, 2019

    work page 2019

  56. [56]

    Schöllhammer, D.; Fries, T.P.: A higher-order Trace finite element method for shells. Int. J. Numer. Methods Eng.,122, 1217–1238, 2021

    work page 2021

  57. [57]

    Cambridge University Press, Cambridge, 2 edition, 1999

    Sethian, J.A.:Level Set Methods and Fast Marching Methods. Cambridge University Press, Cambridge, 2 edition, 1999

    work page 1999

  58. [58]

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

    Timoshenko, S.P.: LXVI. On the correction for shear of the differential equation for transverse vibrations of prismatic bars.Lond. Edinb. Dublin Philos. Mag. J. Sci.,41, 744–746, 1921

    work page 1921

  59. [59]

    Butterworth-Heinemann, Oxford, 7 edition, 2013

    Zienkiewicz, O.C.; Taylor, R.L.; Zhu, J.Z.:The Finite Element Method: Its Basis and Fundamentals. Butterworth-Heinemann, Oxford, 7 edition, 2013

    work page 2013

  60. [60]

    ICE Publishing, London, 2 edition, 2017

    Zingoni, A.:Shell Structures in Civil and Mechanical Engineering. ICE Publishing, London, 2 edition, 2017

    work page 2017