Skip to main content
SpringerLink
Log in

Vanquishing volumetric locking in quadratic NURBS-based discretizations of nearly-incompressible linear elasticity: CAS elements

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

Quadratic NURBS-based discretizations of the Galerkin method suffer from volumetric locking when applied to nearly-incompressible linear elasticity. Volumetric locking causes not only smaller displacements than expected, but also large-amplitude spurious oscillations of normal stresses. Continuous-assumed-strain (CAS) elements have been recently introduced to remove membrane locking in quadratic NURBS-based discretizations of linear plane curved Kirchhoff rods (Casquero et al. in Comput Methods Appl Mech Eng 360:112765, 2022). In this work, we propose two generalizations of CAS elements (named CAS1 and CAS2 elements) to overcome volumetric locking in quadratic NURBS-based discretizations of nearly-incompressible linear elasticity. CAS1 elements linearly interpolate the strains at the knots in each direction for the term in the variational form involving the first Lamé parameter while CAS2 elements linearly interpolate the dilatational strains at the knots in each direction. For both element types, a displacement vector with \(C^1\) continuity across element boundaries results in assumed strains with \(C^0\) continuity across element boundaries. In addition, the implementation of the two locking treatments proposed in this work does not require any additional global or element matrix operations such as matrix inversions or matrix multiplications. The locking treatments are applied at the element level and the nonzero pattern of the global stiffness matrix is preserved. The numerical examples solved in this work show that CAS1 and CAS2 elements, using either two or three Gauss–Legrendre quadrature points per direction, are effective locking treatments since they not only result in more accurate displacements for coarse meshes, but also remove the spurious oscillations of normal stresses.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Cottrell JA, Reali A, Bazilevs Y, Hughes TJR (2006) Isogeometric analysis of structural vibrations. Comput Methods Appl Mech Eng 195:5257–5296

    Article  MathSciNet  MATH  Google Scholar 

  2. Hughes TJR, Reali A, Sangalli G (2008) Duality and unified analysis of discrete approximations in structural dynamics and wave propagation: comparison of \( p\)-method finite elements with \(k\)-method NURBS. Comput Methods Appl Mech Eng 197:4104–4124

    Article  MathSciNet  MATH  Google Scholar 

  3. Hughes TJR, Evans JA, Reali A (2014) Finite element and NURBS approximations of eigenvalue, boundary-value, and initial-value problems. Comput Methods Appl Mech Eng 272:290–320

    Article  MathSciNet  MATH  Google Scholar 

  4. Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194:4135–4195

    Article  MathSciNet  MATH  Google Scholar 

  5. Cottrell JA, Hughes TJR, Bazilevs Y (2009) Isogeometric analysis: toward integration of CAD and FEA. Wiley, Hoboken

    Book  MATH  Google Scholar 

  6. Lipton S, Evans J, Bazilevs Y, Elguedj T, Hughes TJR (2010) Robustness of isogeometric structural discretizations under severe mesh distortion. Comput Methods Appl Mech Eng 199:357–373

    Article  MATH  Google Scholar 

  7. Elguedj T, Bazilevs Y, Calo VM, Hughes TJR (2008) \(\overline{B}\) and \(\overline{F}\) projection methods for nearly incompressible linear and non-linear elasticity and plasticity using higher-order NURBS elements. Comput Methods Appl Mech Eng 197:2732–2762

    Article  MATH  Google Scholar 

  8. Flanagan D, Belytschko T (1981) A uniform strain hexahedron and quadrilateral with orthogonal hourglass control. Int J Numer Methods Eng 17(5):679–706

    Article  MATH  Google Scholar 

  9. Belytschko T, Ong JSJ, Liu WK, Kennedy JM (1984) Hourglass control in linear and nonlinear problems. Comput Methods Appl Mech Eng 43(3):251–276

    Article  MATH  Google Scholar 

  10. Reese S, Küssner M, Reddy BD (1999) A new stabilization technique for finite elements in non-linear elasticity. Int J Numer Methods Eng 44(11):1617–1652

    Article  MathSciNet  MATH  Google Scholar 

  11. Reese S, Wriggers P, Reddy B (2000) A new locking-free brick element technique for large deformation problems in elasticity. Comput Struct 75(3):291–304

    Article  MathSciNet  Google Scholar 

  12. Nagtegaal JC, Parks DM, Rice J (1974) On numerically accurate finite element solutions in the fully plastic range. Comput Methods Appl Mech Eng 4(2):153–177

    Article  MathSciNet  MATH  Google Scholar 

  13. Hughes TJR (1980) Generalization of selective integration procedures to anisotropic and nonlinear media. Int J Numer Methods Eng 15(9):1413–1418

    Article  MathSciNet  MATH  Google Scholar 

  14. Hughes TJR (1977) Equivalence of finite elements for nearly incompressible elasticity. J Appl Mech 44(1):181–183

    Article  Google Scholar 

  15. Malkus DS, Hughes TJR (1978) Mixed finite element methods-reduced and selective integration techniques: a unification of concepts. Comput Methods Appl Mech Eng 15(1):63–81

    Article  MATH  Google Scholar 

  16. Hughes TJR, Malkus DS (1981) A general penalty/mixed equivalence theorem for anisotropic, incompressible finite elements. In: Atluri SN (ed) Hybrid and mixed finite element methods. Wiley, Hoboken, pp 487–496

    Google Scholar 

  17. Wilson E, Taylor R, Doherty W, Ghaboussi J (1973) Incompatible displacement models. In: Fenves SJ (ed) Numerical and computer methods in structural mechanics. Academic Press, New York, pp 43–57

    Google Scholar 

  18. Simo JC, Rifai M (1990) A class of mixed assumed strain methods and the method of incompatible modes. Int J Numer Methods Eng 29(8):1595–1638

    Article  MathSciNet  MATH  Google Scholar 

  19. Simo JC, Armero F (1992) Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes. Int J Numer Methods Eng 33(7):1413–1449

    Article  MATH  Google Scholar 

  20. Glaser S, Armero F (1997) On the formulation of enhanced strain finite elements in finite deformations. Eng Comput 14:759–791

    Article  MATH  Google Scholar 

  21. Kasper EP, Taylor RL (2000) A mixed-enhanced strain method: part I: geometrically linear problems. Comput Struct 75(3):237–250

    Article  Google Scholar 

  22. Kasper EP, Taylor RL (2000) A mixed-enhanced strain method: part II: geometrically nonlinear problems. Comput Struct 75(3):251–260

    Article  Google Scholar 

  23. Auricchio F, da Veiga LB, Lovadina C, Reali A, Sangalli G (2007) A fully “locking-free’’ isogeometric approach for plane linear elasticity problems: a stream function formulation. Comput Methods Appl Mech Eng 197(1–4):160–172

    Article  MathSciNet  MATH  Google Scholar 

  24. Elguedj T, Hughes TJR (2014) Isogeometric analysis of nearly incompressible large strain plasticity. Comput Methods Appl Mech Eng 268:388–416

    Article  MathSciNet  MATH  Google Scholar 

  25. Antolin P, Bressan A, Buffa A, Sangalli G (2017) An isogeometric method for linear nearly-incompressible elasticity with local stress projection. Comput Methods Appl Mech Eng 316:694–719

    Article  MathSciNet  MATH  Google Scholar 

  26. Bressan A (2011) Isogeometric regular discretization for the Stokes problem. IMA J Numer Anal 31(4):1334–1356

    Article  MathSciNet  MATH  Google Scholar 

  27. Bressan A, Sangalli G (2013) Isogeometric discretizations of the Stokes problem: stability analysis by the macroelement technique. IMA J Numer Anal 33(2):629–651

    Article  MathSciNet  MATH  Google Scholar 

  28. Taylor R (2011) Isogeometric analysis of nearly incompressible solids. Int J Numer Methods Eng 87(1–5):273–288

    Article  MathSciNet  MATH  Google Scholar 

  29. Cardoso RP, CesardeSa JMA (2012) The enhanced assumed strain method for the isogeometric analysis of nearly incompressible deformation of solids. Int J Numer Methods Eng 92(1):56–78

    Article  MathSciNet  MATH  Google Scholar 

  30. Moutsanidis G, Li W, Bazilevs Y (2021) Reduced quadrature for FEM, IGA and meshfree methods. Comput Methods Appl Mech Eng 373:113521

    Article  MathSciNet  MATH  Google Scholar 

  31. Li W, Moutsanidis G, Behzadinasab M, Hillman M, Bazilevs Y (2022) Reduced quadrature for finite element and isogeometric methods in nonlinear solids. Comput Methods Appl Mech Eng 399:115389

    Article  MathSciNet  MATH  Google Scholar 

  32. Adam C, Hughes TJR, Bouabdallah S, Zarroug M, Maitournam H (2015) Selective and reduced numerical integrations for NURBS-based isogeometric analysis. Comput Methods Appl Mech Eng 284:732–761

    Article  MathSciNet  MATH  Google Scholar 

  33. Nagy AP, Benson DJ (2015) On the numerical integration of trimmed isogeometric elements. Comput Methods Appl Mech Eng 284:165–185

    Article  MathSciNet  MATH  Google Scholar 

  34. Breitenberger M, Apostolatos A, Philipp B, Wüchner R, Bletzinger KU (2015) Analysis in computer aided design: nonlinear isogeometric B-Rep analysis of shell structures. Comput Methods Appl Mech Eng 284:401–457

    Article  MathSciNet  MATH  Google Scholar 

  35. Leidinger L, Breitenberger M, Bauer A et al (2019) Explicit dynamic isogeometric B-Rep analysis of penalty-coupled trimmed NURBS shells. Comput Methods Appl Mech Eng 351:891–927

    Article  MathSciNet  MATH  Google Scholar 

  36. Buffa A, Puppi R, Vázquez R (2020) A minimal stabilization procedure for isogeometric methods on trimmed geometries. SIAM J Numer Anal 58(5):2711–2735

    Article  MathSciNet  MATH  Google Scholar 

  37. Wei X, Marussig B, Antolin P, Buffa A (2021) Immersed boundary-conformal isogeometric method for linear elliptic problems. Comput Mech 68(6):1385–1405

    Article  MathSciNet  MATH  Google Scholar 

  38. Antolin P, Wei X, Buffa A (2022) Robust numerical integration on curved polyhedra based on folded decompositions. Comput Methods Appl Mech Eng 395:114948

    Article  MathSciNet  MATH  Google Scholar 

  39. Toshniwal D, Speleers H, Hughes TJR (2017) Smooth cubic spline spaces on unstructured quadrilateral meshes with particular emphasis on extraordinary points: geometric design and isogeometric analysis considerations. Comput Methods Appl Mech Eng 327:411–458

    Article  MathSciNet  MATH  Google Scholar 

  40. Casquero H, Wei X, Toshniwal D et al (2020) Seamless integration of design and Kirchhoff–Love shell analysis using analysis-suitable unstructured T-splines. Comput Methods Appl Mech Eng 360:112765

    Article  MathSciNet  MATH  Google Scholar 

  41. Hiemstra RR, Shepherd KM, Johnson MJ, Quan L, Hughes TJR (2020) Towards untrimmed NURBS: CAD embedded reparameterization of trimmed B-rep geometry using frame-field guided global parameterization. Comput Methods Appl Mech Eng 369:113227

    Article  MathSciNet  MATH  Google Scholar 

  42. Wei X, Li X, Qian K, Hughes TJR, Zhang YJ, Casquero H (2022) Analysis-suitable unstructured T-splines: multiple extraordinary points per face. Comput Methods Appl Mech Eng 391:114494

    Article  MathSciNet  MATH  Google Scholar 

  43. Shepherd KM, Gu XD, Hughes TJR (2022) Isogeometric model reconstruction of open shells via Ricci flow and quadrilateral layout-inducing energies. Eng Struct 252:113602

    Article  Google Scholar 

  44. Shepherd KM, Gu XD, Hughes TJR (2022) Feature-aware reconstruction of trimmed splines using Ricci flow with metric optimization. Comput Methods Appl Mech Eng 402:115555

    Article  MathSciNet  MATH  Google Scholar 

  45. Toshniwal D (2022) Quadratic splines on quad-tri meshes: construction and an application to simulations on watertight reconstructions of trimmed surfaces. Comput Methods Appl Mech Eng 388:114174

    Article  MathSciNet  MATH  Google Scholar 

  46. Wen Z, Faruque MS, Li X, Wei X, Casquero H (2023) Isogeometric analysis using G-spline surfaces with arbitrary unstructured quadrilateral layout. Comput Methods Appl Mech Eng 408:115965

    Article  MathSciNet  MATH  Google Scholar 

  47. Fahrendorf F, Morganti S, Reali A, Hughes TJR, De Lorenzis L (2020) Mixed stress-displacement isogeometric collocation for nearly incompressible elasticity and elastoplasticity. Comput Methods Appl Mech Eng 369:113112

  48. Morganti S, Fahrendorf F, De Lorenzis L, Evans JA, Hughes TJR, Reali A (2021) Isogeometric collocation: a mixed displacement-pressure method for nearly incompressible elasticity. CMES Comput Model Eng Sci 129(3):1125–1150

    Google Scholar 

  49. Casquero H, Golestanian M (2022) Removing membrane locking in quadratic NURBS-based discretizations of linear plane Kirchhoff rods: CAS elements. Comput Methods Appl Mech Eng 399:115354

    Article  MathSciNet  MATH  Google Scholar 

  50. Golestanian M, Casquero H (2023) Extending CAS elements to remove shear and membrane locking from quadratic NURBS-based discretizations of linear plane Timoshenko rods. Int J Numer Methods Eng 124(18):3997–4021

    Article  MathSciNet  Google Scholar 

  51. Casquero H, Mathews KD (2023) Overcoming membrane locking in quadratic NURBS–based discretizations of linear Kirchhoff–Love shells: CAS elements. Comput Methods Appl Mech Eng 417:116523

  52. Hughes TJR (2012) The finite element method: linear static and dynamic finite element analysis. Courier Corporation, Chelmsford

    Google Scholar 

  53. Dalcin L, Collier N, Vignal P, Côrtes A, Calo VM (2016) PetIGA: a framework for high-performance isogeometric analysis. Comput Methods Appl Mech Eng 308:151–181

    Article  MathSciNet  MATH  Google Scholar 

  54. Balay S, Adams MF, Brown J, et al (2014) PETSc web page. http://www.mcs.anl.gov/petsc

  55. Cook R, Malkus DS, Plesha M, Witt RJ (2007) Concepts and applications of finite element analysis. Wiley, Hoboken

    Google Scholar 

  56. CesardeSa JMA, Natal Jorge RM (1999) New enhanced strain elements for incompressible problems. Int J Numer Methods Eng 44(2):229–248

    Article  MATH  Google Scholar 

  57. Chavan KS, Lamichhane BP, Wohlmuth BI (2007) Locking-free finite element methods for linear and nonlinear elasticity in 2D and 3D. Comput Methods Appl Mech Eng 196(41–44):4075–4086

    Article  MATH  Google Scholar 

  58. Schröder J, Wick T, Reese S et al (2021) A selection of benchmark problems in solid mechanics and applied mathematics. Arch Comput Methods Eng 28:713–751

    Article  MathSciNet  Google Scholar 

  59. Dolbow J, Belytschko T (1999) Volumetric locking in the element free Galerkin method. Int J Numer Methods Eng 46(6):925–942

    Article  MathSciNet  MATH  Google Scholar 

  60. Huerta A, Fernández-Méndez S (2001) Locking in the incompressible limit for the element-free Galerkin method. Int J Numer Methods Eng 51(11):1361–1383

    Article  MATH  Google Scholar 

  61. Nguyen TH, Hiemstra RR, Schillinger D (2022) Leveraging spectral analysis to elucidate membrane locking and unlocking in isogeometric finite element formulations of the curved Euler–Bernoulli beam. Comput Methods Appl Mech Eng 388:114240

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

H. Casquero and M. Golestanian were partially supported by Ansys Inc. and the NSF Grant CMMI-2138187.

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, University of Michigan - Dearborn, 4901 Evergreen Road, Dearborn, 48128-1491, MI, USA

    Hugo Casquero & Mahmoud Golestanian

Authors
  1. Hugo Casquero
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mahmoud Golestanian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hugo Casquero.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Casquero, H., Golestanian, M. Vanquishing volumetric locking in quadratic NURBS-based discretizations of nearly-incompressible linear elasticity: CAS elements. Comput Mech (2023). https://doi.org/10.1007/s00466-023-02409-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00466-023-02409-5

Keywords

Search

Navigation