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Blending moving least squares techniques with NURBS basis functions for nonlinear isogeometric analysis

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Abstract

IsoGeometric Analysis (IGA) is increasing its popularity as a new numerical tool for the analysis of structures. IGA provides: (i) the possibility of using higher order polynomials for the basis functions; (ii) the smoothness for contact analysis; (iii) the possibility to operate directly on CAD geometry. The major drawback of IGA is the non-interpolatory characteristic of the basis functions, which adds a difficulty in handling essential boundary conditions. Nevertheless, IGA suffers from the same problems depicted by other methods when it comes to reproduce isochoric and transverse shear strain deformations, especially for low order basis functions. In this work, projection techniques based on the moving least square (MLS) approximations are used to alleviate both the volumetric and the transverse shear lockings in IGA. The main objective is to project the isochoric and transverse shear deformations from lower order subspaces by using the MLS, alleviating in this way the volumetric and the transverse shear locking on the fully-integrated space. Because in IGA different degrees in the approximation functions can be used, different Gauss integration rules can also be employed, making the procedures for locking treatment in IGA very dependent on the degree of the approximation functions used. The blending of MLS with Non-Uniform Rational B-Splines (NURBS) basis functions is a methodology to overcome different locking pathologies in IGA which can be also used for enrichment procedures. Numerical examples for three-dimensional NURBS with only translational degrees of freedom are presented for both shell-type and plane strain structures.

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References

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

    Article  MATH  Google Scholar 

  2. Cottrell JA, Hughes TJR, Reali A (2007) Studies of refinement and continuity in isogeometric structural analysis. Comput Methods Appl Mech Eng 196:4160–4183

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  4. 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  MATH  MathSciNet  Google Scholar 

  5. Hughes TJR, Reali A, Sangalli G (2010) Efficient quadrature for NURBS-based isogeometric analysis. Comput Methods Appl Mech Eng 199:301–313

    Article  MATH  MathSciNet  Google Scholar 

  6. Benson DJ, Bazilevs Y, Hsu MC, Hughes TJR (2010) Isogeometric shell analysis: the Reissner–Mindlin shell. Comput Methods Appl Mech Eng 199:276–289

    Article  MATH  MathSciNet  Google Scholar 

  7. Echter R, Bischoff M (2010) Numerical efficiency, locking and unlocking of NURBS finite elements. Comput Methods Appl Mech Eng 199:374–382

    Article  MATH  Google Scholar 

  8. Hughes TJR, Cohen M, Haroun M (1978) Reduced and selective integration techniques in finite element analysis of plates. Nucl Eng Des 46:203–222

    Article  Google Scholar 

  9. Hughes TJR (2000) The finite element method: linear static and dynamic finite element analysis, 2nd edn. Dover Editions, New Jersey

    Google Scholar 

  10. Liu WK, Guo Y, Tang S, Belytschko T (1998) A multiple-quadrature eight-node hexahedral finite element for large deformation elastoplastic analysis. Comput Methods Appl Mech Eng 154:69–132

    Article  MATH  MathSciNet  Google Scholar 

  11. Cesar de Sa JMA, Owen DRJ (1986) The imposition of the incompressible constraints in finite elements - a review of approaches with a new insight to the locking phenomenon. In: Taylor C, Hinton E, Owen DRJ (eds) Numerical methods for non-linear problems, Pineridge Press, Yugoslavia

  12. Bathe KJ (1996) Finite element procedures, 2nd edn. Prentice-Hall, New Jersey

    Google Scholar 

  13. Taylor RL (2010) Isogeometric analysis of nearly incompressible solids, Int J Numer Methods Eng. doi:10.1002/nme.3048

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

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  16. Andelfinger U, Ramm E, Roehl D (1992) 2d and 3d-enhanced assumed strain elements and their application in plasticity. In: Proceedings of the 4th International Conference on Computational Plasticity, Swansea, pp 1997–2007

  17. Simo JC, Armero F, Taylor RL (1993) Improved versions of assumed enhanced strain tri-linear elements for 3d finite deformation problems. Comput Methods Appl Mech Eng 110:359–386

    Article  MATH  MathSciNet  Google Scholar 

  18. Roehl D, Ramm E (1996) Large elasto-plastic finite element analysis of solids and shells with the enhanced assumed strain concept. Int J Solids Struct 33:3215–3237

    Article  MATH  MathSciNet  Google Scholar 

  19. de Souza Neto EA, Peric D, Dutko M, Owen DRJ (1996) Design of simple low order finite elements for large strain analysis of nearly incompressible solids. Int J Solids Struct 33:3277–3296

    Article  MATH  Google Scholar 

  20. Korelc J, Wriggers P (1996) Consistent gradient formulation for a stable enhanced strain method for large deformation. Eng Comput 13:103–123

    Article  Google Scholar 

  21. Glaser S, Armero F (1996) On the formulation of enhanced strain finite elements in finite deformation. Eng Comput 14:759–791

    Article  Google Scholar 

  22. Bischoff M, Ramm E, Braess D (1999) A class of equivalent enhanced assumed strain and hybrid stress finite elements. Comput Mech 22:443–449

    Article  MATH  Google Scholar 

  23. Cesar de Sa JMA, Natal Jorge RM (1999) New enhanced strain elements for incompressible problems. Int J Numer Methods Eng 44:229–248

    Article  MATH  Google Scholar 

  24. Kasper EP, Taylor RL (2000) A mixed enhanced strain method: part 2. Geometrically non-linear problems. Comput Struct 75:251–260

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  27. Areias PMA, Cesar de Sa JMA, Conceicao Antonio CA, Fernandes AA (2003) Analysis of 3D problems using a new enhanced strain hexahedral element. Int J Numer Methods Eng 58:1637–1682

    Article  MATH  Google Scholar 

  28. Belytschko T, Tsay CS (1983) A stabilization procedure for the quadrilateral plate element with one-point quadrature. Int J Numer Methods Eng 19:405–419

    Article  MATH  Google Scholar 

  29. Belytschko T, Lin JI, Tsay C (1984) Explicit algorithms for the nonlinear dynamics of shells. Comput Methods Appl Mech Eng 42:225–251

    Article  Google Scholar 

  30. Belytschko T, Bachrach W (1986) Efficient implementation of quadrilaterals with high coarse-mesh accuracy. Comput Methods Appl Mech Eng 54:279–301

    Google Scholar 

  31. Cardoso RPR, Yoon JW (2005) One point quadrature shell elements for sheet metal forming analysis. Arch Comput Methods Eng 12:3–66

    Article  MATH  Google Scholar 

  32. Dvorkin EN, Bathe KJ (1984) A continuum mechanics based four-node shell element for general nonlinear analysis. Eng Comput 1:77–88

    Article  Google Scholar 

  33. Cardoso RPR, Yoon JW, Gracio JJ, Barlat F, Cesar de Sa JMA (2002) Development of a one point quadrature shell element for nonlinear applications with contact and anisotropy. Comput Methods Appl Mech Eng 191:5177–5206

    Article  MATH  Google Scholar 

  34. Cardoso RPR, Yoon JW (2005) One point quadrature shell element with through-thickness stretch. Comput Methods Appl Mech Eng 194:1161–1199

    Article  MATH  Google Scholar 

  35. Piegl L, Tiller W (1996) The NURBS book. Springer, Berlin

    Google Scholar 

  36. Austin Cottrell J, Hughes TJR, Bazilevs Y (2009) Isogeometric analysis, toward integration of CAD and FEA. Wiley, United Kingdom

  37. Bazilevs Y, Calo VM, Cottrell JA, Evans JA, Hughes TJR, Lipton S, Scott MA, Sederberg TW (2010) Isogeometric analysis using T-Splines. Comput Methods Appl Mech Eng 199:229–263

    Article  MATH  MathSciNet  Google Scholar 

  38. Liu GR (2003) Mesh free methods, moving beyond the finite element method. CRC Press, New York

    MATH  Google Scholar 

  39. Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P (1996) Meshless methods: an overview and recent developments. Comput Methods Appl Mech Eng 139:3–47

    Article  MATH  Google Scholar 

  40. Cardoso RPR, Yoon JW (2007) One point quadrature shell elements: a study on convergence and patch tests. Comput Mech 40:871–883

    Article  MATH  Google Scholar 

  41. Cardoso RPR, Yoon JW, Mahardika M, Choudhry S, Alves de Sousa RJ, Fontes Valente RA (2008) Enhanced assumed strain (EAS) and assumed natural strain (ANS) methods for one-point quadrature solid-shell elements. Int J Numer Methods Eng 75:156–187

    Article  MATH  MathSciNet  Google Scholar 

  42. Cardoso RPR, Cesar de Sa JMA (2012) The enhanced assumed strain method for the isogeometric analysis of nearly incompressible deformation of solids. Int J Numer Methods Eng 92:56–78

    Article  MathSciNet  Google Scholar 

  43. Cook RD, Malkus DS, Plesha ME (1989) Concepts and applications of finite element analysis. Wiley, New York

    MATH  Google Scholar 

  44. Morley LSD (1963) Skew plates and structures. Pergamon Press, London

    MATH  Google Scholar 

  45. Hughes TJR, Tezduyar TE (1981) Finite elements based upon mindlin plate theory with particular reference to the four-node bilinear isoparametric element. J Appl Mech 48:587–596

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Engineering Design and Mathematics, University of the West of England, Bristol , BS16 1QY, UK

    Rui P. R. Cardoso

  2. Institute for Mechanical Engineering (IDMEC), University of Porto, 4200-465 , Porto, Portugal

    J. M. A. Cesar de Sa

Authors
  1. Rui P. R. Cardoso
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  2. J. M. A. Cesar de Sa
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Correspondence to Rui P. R. Cardoso.

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Cardoso, R.P.R., Cesar de Sa, J.M.A. Blending moving least squares techniques with NURBS basis functions for nonlinear isogeometric analysis. Comput Mech 53, 1327–1340 (2014). https://doi.org/10.1007/s00466-014-0977-5

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  • DOI: https://doi.org/10.1007/s00466-014-0977-5

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