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Spline-based smooth beam-to-beam contact model

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Abstract

The contact between bodies is a complex phenomenon that involves mechanical interaction, frictional sliding and heat transfer, among others. A common (and convenient) approach for the mechanical interaction in a finite element framework is to directly use the geometry of the elements to formulate the contact. The main drawback lies in the sharp corners that occur when straight finite elements are connected leading eventually to contact singularities. To circumvent this issue, particularly in the context of beam-to-beam contact, the present work proposes a pointwise contact formulation based on smooth C1 continuous spline contact elements. The proposed spline-based formulation, which can be directly attached to any quadratic beam finite element formulation, guarantees a smooth description for the whole set of elements, where contact takes place. A specific nonlinear normal contact interaction law and a rheological model for friction, both with elastic and viscous damping contributions, are developed increasing robustness in practical applications. To demonstrate this robustness, specific examples are considered including comparisons with a similar surface-to-surface formulation and an alternative smooth contact scheme, smooth contact with finite elements having sharp corners, modeling of a knot tightening with self-contact, and a simulation involving multiple pointwise contacts.

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References

  1. Jin S, Sohn D, Im S (2016) Node-to-node scheme for three-dimensional contact mechanics using polyhedral type variable-node elements. Comput Methods Appl Mech Eng 304:217–242. https://doi.org/10.1016/j.cma.2016.02.019

    Article  MathSciNet  MATH  Google Scholar 

  2. Xing W, Zhang J, Song C, Tin-Loi F (2019) A node-to-node scheme for three-dimensional contact problems using the scaled boundary finite element method. Comput Methods Appl Mech Eng 347:928–956. https://doi.org/10.1016/j.cma.2019.01.015

    Article  MathSciNet  MATH  Google Scholar 

  3. Wriggers P, Rust WT (2019) A virtual element method for frictional contact including large deformations. Eng Comput (Swansea, Wales) 36:2133–2161. https://doi.org/10.1108/EC-02-2019-0043

    Article  Google Scholar 

  4. Paggi M, Wriggers P (2016) Node-to-segment and node-to-surface interface finite elements for fracture mechanics. Comput Methods Appl Mech Eng 300:540–560. https://doi.org/10.1016/j.cma.2015.11.023

    Article  MathSciNet  MATH  Google Scholar 

  5. Khoei AR, Biabanaki SOR, Parvaneh SM (2013) 3D dynamic modeling of powder forming processes via a simple and efficient node-to-surface contact algorithm. Appl Math Model 37:443–462. https://doi.org/10.1016/j.apm.2012.03.010

    Article  MathSciNet  MATH  Google Scholar 

  6. Crisfield MA (2000) Re-visiting the contact patch test. Int J Numer Methods Eng 48:435–449. https://doi.org/10.1002/(SICI)1097-0207(20000530)48:3%3c435::AID-NME891%3e3.0.CO;2-V

    Article  MATH  Google Scholar 

  7. De Lorenzis L, Wriggers P, Hughes TJR (2014) Isogeometric contact: a review. GAMM Mitteilungen 37:85–123. https://doi.org/10.1002/gamm.201410005

  8. Heinstein MW, Laursen TA (2003) A three-dimensional surface-to-surface projection algorithm for non-coincident domains. Commun Numer Methods Eng 19:421–432. https://doi.org/10.1002/cnm.601

    Article  MathSciNet  MATH  Google Scholar 

  9. Zimmerman BK, Ateshian GA (2018) A surface-to-surface finite element algorithm for large deformation frictional contact in febio. J Biomech Eng 140. https://doi.org/10.1115/1.4040497

  10. Wriggers P, Zavarise G (1997) On contact between three-dimensional beams undergoing large deflections. Commun Numer Methods Eng 13:429–438. https://doi.org/10.1002/(SICI)1099-0887(199706)13:6%3c429::AID-CNM70%3e3.0.CO;2-X

    Article  MathSciNet  MATH  Google Scholar 

  11. Litewka P, Wriggers P (2001) Frictional contact between 3D beams. Comput Mech 28:26–39. https://doi.org/10.1007/s004660100266

    Article  MATH  Google Scholar 

  12. Pietrzak G, Curnier A (1999) Large deformation frictional contact mechanics: continuum formulation and augmented Lagrangian treatment. Comput Methods Appl Mech Eng 177:351–381. https://doi.org/10.1016/S0045-7825(98)00388-0

    Article  MathSciNet  MATH  Google Scholar 

  13. Padmanabhan V, Laursen TA (2001) A framework for development of surface smoothing procedures in large deformation frictional contact analysis. Finite Elem Anal Des 37:173–198. https://doi.org/10.1016/S0168-874X(00)00029-9

    Article  MATH  Google Scholar 

  14. Magliulo M, Zilian A, Beex LAA (2020) Contact between shear-deformable beams with elliptical cross sections. Acta Mech 231:273–291. https://doi.org/10.1007/s00707-019-02520-w

    Article  MathSciNet  MATH  Google Scholar 

  15. Tasora A, Benatti S, Mangoni D, Garziera R (2020) A geometrically exact isogeometric beam for large displacements and contacts. Comput Methods Appl Mech Eng 358:112635. https://doi.org/10.1016/j.cma.2019.112635

  16. Wriggers P, Krstulovic-Opara L, Korelc J (2001) Smooth C1-interpolations for two-dimensional frictional contact problems. Int J Numer Methods Eng 51:1469–1495. https://doi.org/10.1002/nme.227

    Article  MATH  Google Scholar 

  17. Krstulović-Opara L, Wriggers P, Korelc J (2002) A C1-continuous formulation for 3D finite deformation frictional contact. Comput Mech 29:27–42. https://doi.org/10.1007/s00466-002-0317-z

    Article  MathSciNet  MATH  Google Scholar 

  18. Al-Dojayli M, Meguid SA (2002) Accurate modeling of contact using cubic splines. Finite Elem Anal Des 38:337–352. https://doi.org/10.1016/S0168-874X(01)00088-9

    Article  MATH  Google Scholar 

  19. Litewka P (2007) Hermite polynomial smoothing in beam-to-beam frictional contact. Comput Mech 40:815–826. https://doi.org/10.1007/s00466-006-0143-9

    Article  MATH  Google Scholar 

  20. Durville D (2010) Simulation of the mechanical behaviour of woven fabrics at the scale of fibers. Int J Mater Form 3:1241–1251. https://doi.org/10.1007/s12289-009-0674-7

    Article  Google Scholar 

  21. Durville D (2012) Contact-friction modeling within elastic beam assemblies: an application to knot tightening. Comput Mech 49:687–707. https://doi.org/10.1007/s00466-012-0683-0

    Article  MathSciNet  MATH  Google Scholar 

  22. Konyukhov A, Schweizerhof K (2010) Geometrically exact covariant approach for contact between curves. Comput Methods Appl Mech Eng 199:2510–2531. https://doi.org/10.1016/j.cma.2010.04.012

    Article  MathSciNet  MATH  Google Scholar 

  23. Lu J (2011) Isogeometric contact analysis: geometric basis and formulation for frictionless contact. Comput Methods Appl Mech Eng 200:726–741. https://doi.org/10.1016/j.cma.2010.10.001

    Article  MathSciNet  MATH  Google Scholar 

  24. Temizer I, Wriggers P, Hughes TJR (2012) Three-dimensional mortar-based frictional contact treatment in isogeometric analysis with NURBS. Comput Methods Appl Mech Eng 209–212:115–128. https://doi.org/10.1016/j.cma.2011.10.014

    Article  MathSciNet  MATH  Google Scholar 

  25. Meier C, Grill MJ, Wall WA, Popp A (2018) Geometrically exact beam elements and smooth contact schemes for the modeling of fiber-based materials and structures. Int J Solids Struct 154:124–146. https://doi.org/10.1016/j.ijsolstr.2017.07.020

    Article  Google Scholar 

  26. Nishi S, Terada K, Temizer I (2019) Isogeometric analysis for numerical plate testing of dry woven fabrics involving frictional contact at meso-scale. Comput Mech 64:211–229. https://doi.org/10.1007/s00466-018-1666-6

    Article  MathSciNet  MATH  Google Scholar 

  27. de Boor C (1972) On calculating with B-splines. J Approx Theory 6:50–62. https://doi.org/10.1016/0021-9045(72)90080-9

    Article  MathSciNet  MATH  Google Scholar 

  28. Cottrell JA, Hughes TJR, Bazilevs Y (2009) Isogeometric Analysis: Toward Integration of CAD and FEA. Wiley, Chichester

  29. Piegl L, Tiller W (1997) The NURBS book. Springer, Berlin

    Book  MATH  Google Scholar 

  30. Belytschko T, Neal MO (1991) Contact-impact by the pinball algorithm with penalty and Lagrangian methods. Int J Numer Methods Eng 31:547–572. https://doi.org/10.1002/nme.1620310309

    Article  MATH  Google Scholar 

  31. Gay Neto A, Wriggers P (2020) Numerical method for solution of pointwise contact between surfaces. Comput Methods Appl Mech Eng 365:112971. https://doi.org/10.1016/j.cma.2020.112971

  32. Gay Neto A, Wriggers P (2019) Computing pointwise contact between bodies: a class of formulations based on master–master approach. Comput Mech 64:585–609. https://doi.org/10.1007/s00466-019-01680-9

    Article  MathSciNet  MATH  Google Scholar 

  33. Gay Neto A, Pimenta PM, Wriggers P (2016) A master-surface to master-surface formulation for beam to beam contact. Part I: frictionless interaction. Comput Methods Appl Mech Eng 303:400–429. https://doi.org/10.1016/j.cma.2016.02.005

    Article  MathSciNet  MATH  Google Scholar 

  34. Gay Neto A, Pimenta PM, Wriggers P (2017) A master-surface to master-surface formulation for beam to beam contact. Part II: frictional interaction. Comput Methods Appl Mech Eng 319:146–174. https://doi.org/10.1016/j.cma.2017.01.038

    Article  MathSciNet  MATH  Google Scholar 

  35. Neto AG, Wriggers P (2022) Discrete element model for general polyhedra. Comput Part Mech 9:353–380. https://doi.org/10.1007/s40571-021-00415-z

    Article  Google Scholar 

  36. Gay Neto A, Wriggers P (2020) Master-master frictional contact and applications for beam-shell interaction. Comput Mech 66:1213–1235. https://doi.org/10.1007/s00466-020-01890-6

    Article  MathSciNet  MATH  Google Scholar 

  37. Hertz H (1882) Ueber die Berührung fester elastischer Körper. J fur die Reine und Angew Math 1882:156–171. https://doi.org/10.1515/crll.1882.92.156

    Article  MATH  Google Scholar 

  38. Boisse P, Gasser A, Hivet G (2001) Analyses of fabric tensile behaviour: determination of the biaxial tension-strain surfaces and their use in forming simulations. Compos - Part A Appl Sci Manuf 32:1395–1414. https://doi.org/10.1016/S1359-835X(01)00039-2

    Article  Google Scholar 

  39. Wriggers P (2006) Computational contact mechanics. Springer, Berlin

    Book  MATH  Google Scholar 

  40. Simo JC (1985) A finite strain beam formulation. The three-dimensional dynamic problem. Part I. Comput Methods Appl Mech Eng 49:55–70. https://doi.org/10.1016/0045-7825(85)90050-7

    Article  MATH  Google Scholar 

  41. Simo JC, Vu-Quoc L (1986) A three-dimensional finite-strain rod model. part II: computational aspects. Comput Methods Appl Mech Eng 58:79–116. https://doi.org/10.1016/0045-7825(86)90079-4

    Article  MATH  Google Scholar 

  42. Simo JC, Vu-Quoc L (1991) A geometrically-exact rod model incorporating shear and torsion-warping deformation. Int J Solids Struct 27:371–393. https://doi.org/10.1016/0020-7683(91)90089-X

    Article  MathSciNet  MATH  Google Scholar 

  43. Gay Neto A (2020) Giraffe User’s Manual v. 2.0.0

  44. Gay Neto A, Pimenta PM, Wriggers P (2015) Self-contact modeling on beams experiencing loop formation. Comput Mech 55:193–208. https://doi.org/10.1007/s00466-014-1092-3

    Article  MathSciNet  MATH  Google Scholar 

  45. Ota NSN, Wilson L, Gay Neto A et al (2016) Nonlinear dynamic analysis of creased shells. Finite Elem Anal Des 121:64–74. https://doi.org/10.1016/j.finel.2016.07.008

    Article  MathSciNet  Google Scholar 

  46. Faccio Júnior CJ, Cardozo ACP, Monteiro Júnior V, Gay Neto A (2019) Modeling wind turbine blades by geometrically-exact beam and shell elements: a comparative approach. Eng Struct 180:357–378. https://doi.org/10.1016/j.engstruct.2018.09.032

    Article  Google Scholar 

  47. Craveiro MV, Gay Neto A (2018) Upheaval buckling of pipelines due to internal pressure: a geometrically nonlinear finite element analysis. Eng Struct 158:136–154. https://doi.org/10.1016/j.engstruct.2017.12.010

    Article  Google Scholar 

  48. Craveiro MV, Gay Neto A (2019) Lateral buckling of pipelines due to internal pressure: a geometrically nonlinear finite element analysis. Eng Struct 200:109505. https://doi.org/10.1016/j.engstruct.2019.109505

  49. Gay Neto A, Martins C de A (2013) Structural stability of flexible lines in catenary configuration under torsion. Mar Struct 34:16–40. https://doi.org/10.1016/j.marstruc.2013.07.002

  50. Neto AG, Martins CA, Pimenta PM (2014) Static analysis of offshore risers with a geometrically-exact 3D beam model subjected to unilateral contact. Comput Mech 53:125–145. https://doi.org/10.1007/s00466-013-0897-9

    Article  MathSciNet  Google Scholar 

  51. Gay Neto A, Ribeiro Malta E, de Mattos PP (2015) Catenary riser sliding and rolling on seabed during induced lateral movement. Mar Struct 41:223–243. https://doi.org/10.1016/j.marstruc.2015.02.001

    Article  Google Scholar 

  52. Gay Neto A (2016) Dynamics of offshore risers using a geometrically-exact beam model with hydrodynamic loads and contact with the seabed. Eng Struct 125:438–454. https://doi.org/10.1016/j.engstruct.2016.07.005

    Article  Google Scholar 

  53. Faccio Júnior CJ, Gay Neto A (2021) Challenges in representing the biaxial mechanical behavior of woven fabrics modeled by beam finite elements with contact. Compos Struct 257:113330. https://doi.org/10.1016/j.compstruct.2020.113330

  54. Pimenta PM, Yojo T (1993) Geometrically exact analysis of spatial frames. Appl Mech Rev 46:S118–S128. https://doi.org/10.1115/1.3122626

    Article  Google Scholar 

  55. Pimenta PM, Campello EMB (2001) Geometrically nonlinear analysis of thin-walled space frames. In: Proceedings of the Second European conference on computational mechanics, II ECCM

  56. da Costa e Silva C, Maassen SF, Pimenta PM, Schröder J (2020) A simple finite element for the geometrically exact analysis of Bernoulli–Euler rods. Comput Mech 65:905–923. https://doi.org/10.1007/s00466-019-01800-5

  57. Zavarise G, Wriggers P (2000) Contact with friction between beams in 3-D space. Int J Numer Methods Eng 49:977–1006. https://doi.org/10.1002/1097-0207(20001120)49:8%3c977::AID-NME986%3e3.0.CO;2-C

    Article  MATH  Google Scholar 

  58. Ibrahimbegović A, Mamouri S (2000) On rigid components and joint constraints in nonlinear dynamics of flexible multibody systems employing 3D geometrically exact beam model. Comput Methods Appl Mech Eng 188:805–831. https://doi.org/10.1016/S0045-7825(99)00363-1

    Article  MATH  Google Scholar 

  59. Gay Neto A (2017) Simulation of mechanisms modeled by geometrically-exact beams using Rodrigues rotation parameters. Comput Mech 59:459–481. https://doi.org/10.1007/s00466-016-1355-2

    Article  MathSciNet  Google Scholar 

  60. Cao J, Akkerman R, Boisse P et al (2008) Characterization of mechanical behavior of woven fabrics: experimental methods and benchmark results. Compos Part A Appl Sci Manuf 39:1037–1053. https://doi.org/10.1016/j.compositesa.2008.02.016

    Article  Google Scholar 

  61. Refachinho de Campos PR, Gay Neto A (2018) Rigid body formulation in a finite element context with contact interaction. Comput Mech 62:1369–1398. https://doi.org/10.1007/s00466-018-1569-6

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors acknowledge the National Council for Scientific and Technological Development (CNPq) under the grants 168927/2018-7 and 304321/2021-4, and the São Paulo Research Foundation (FAPESP) under the grant 2020/13362-1.

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Authors and Affiliations

  1. Polytechnic School at University of São Paulo, São Paulo, Brazil

    Celso Jaco Faccio Júnior & Alfredo Gay Neto

  2. Leibniz Universität Hannover, Hannover, Germany

    Peter Wriggers

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  1. Celso Jaco Faccio Júnior
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  2. Alfredo Gay Neto
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Correspondence to Celso Jaco Faccio Júnior.

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Appendix: Objectivity test displacements

Appendix: Objectivity test displacements

See Table 9.

Table 9 Objectivity test simulation displacements
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Faccio Júnior, C.J., Gay Neto, A. & Wriggers, P. Spline-based smooth beam-to-beam contact model. Comput Mech 72, 663–692 (2023). https://doi.org/10.1007/s00466-023-02283-1

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