over(B, -) and over(F, -) projection methods for nearly incompressible linear and non-linear elasticity and plasticity using higher-order NURBS elements

Computer Methods in Applied Mechanics and Engineering

Volumn 197, Issue 33-40, 2008, Pages 2732-2762

  Elguedj, T ^a   Bazilevs, Y ^a   Calo, V M ^a   Hughes, T J R ^a  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

Author keywords

Incompressibility; Isogeometric analysis; Non linear elasticity; NURBS; over(B, ) method; over(F, ) method; Plasticity; Volumetric locking

Indexed keywords

DEFORMATION; ELASTICITY; ITERATIVE METHODS; NEWTON-RAPHSON METHOD; PLASTICITY; POTENTIAL ENERGY;

ISOGEOMETRIC ANALYSIS; NON-UNIFORM RATIONAL B-SPLINES; VOLUMETRIC LOCKING;

COMPUTATIONAL GEOMETRY;

COMPUTATIONAL GEOMETRY; DEFORMATION; ELASTICITY; ITERATIVE METHODS; NEWTON-RAPHSON METHOD; PLASTICITY; POTENTIAL ENERGY;

EID: 44749085922     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cma.2008.01.012     Document Type: Article

References (71)

1. Akkerman I., Bazilevs Y., Calo V.M., Hughes T.J.R., and Hulshoff S. The role of continuity in residual-based variational multiscale modeling of turbulence. Comput. Mech. 41 (2008) 371-378
2. Armero F., and Glaser S. On the formulation of enhanced strain finite elements in finite deformation. Engrg. Comput. 14 (1997) 759-791
3. Auricchio F., Beirão da Veiga L., Lovadina C., and Reali A. An analysis od some mixed-enhanced finite elements for plane linear elasticity. Comput. Methods Appl. Mech. Engrg. 194 (2005) 1075-1092
4. Auricchio F., Beirão da Veiga L., Buffa A., Lovadina C., Reali A., and Sangalli G. A fully locking-free isogeometric approach for plane linear elasticity problems: a stream function formulation. Comput. Methods Appl. Mech. Engrg. 197 (2007) 160-172
5. F. Auricchio, L. Beirão da Veiga, C. Lovadina, A. Reali, The importance of the exact satisfaction of the incompressibility constraint in nonlinear elasticity: mixed FEMs versus NURBS-based approximations, Comput. Methods Appl. Mech. Engrg. (2008), in press.
6. Barth T.J. Numerical methods for gasdynamic systems on unstructured meshs. In: Kröner, Ohlberger, and Rohde (Eds). An Introduction to Recent Developments in Theory and Numerics for Conservation Laws. Lecture Notes in Computational Science and Engineering vol. 5 (1998), Springer-Verlag, Heidelberg 195-285
7. Y. Bazilevs. Isogeometric Analysis of turbulence and fluid-structure interaction. PhD thesis, University of Texas at Austin, 2007.
8. Bazilevs Y., Calo V.M., Cottrell J.A., Hughes T.J.R., Reali A., and Scovazzi G. Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows. Comput. Methods Appl. Mech. Engrg. 197 (2007) 173-201
9. Bazilevs Y., Calo V.M., Zhang Y., and Hughes T.J.R. Isogeometric fluid-structure interaction analysis with applications to arterial blood flow. Comput. Mech. 38 (2006) 310-322
10. Bazilevs Y., Beirao da Veiga L., Cottrell J.A., Hughes T.J.R., and Sangalli G. Isogeometric Analysis: approximation, stability and error estimates for h-refined meshes. Math. Models Methods Appl. Sci. 16 7 (2006) 1031-1090
11. Bazilevs Y., Michler C., Calo V.M., and Hughes T.J.R. Weak dirichlet boundary conditions for wall-bounded turbulent flows. Comput. Methods Appl. Mech. Engrg. 196 (2007) 4853-4862
12. Belytschko T., Ong J.S.J., Liu W.K., and Kennedy J.M. Hourglass control in linear and nonlinear problems. Comput. Methods Appl. Mech. Engrg. 43 (1984) 251-276
13. Brezzi F., and Fortin M. Mixed and Hybrid Finite Element Methods (1991), Springer, Berlin, Heidelberg, New York
14. Brink U., and Stein E. On some mixed finite element methods for incompressible and nearly incompressible finite elasticity. Comput. Mech. 19 (1996) 105-119
15. Brooks A.N., and Hughes T.J.R. Streamline-Upwind-Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 32 (1982) 199-259
16. Büchter N., Ramm E., and Roehl D. Three-dimensionnal extension of non-linear shell formulation based on the enhanced assumed strain concept. Int. J. Numer. Methods Engrg. 37 (1994) 2551-2568
17. Chavan K.S., Lamichhane B.P., and Wohlmuth B.I. Locking-free finite element methods for linear and nonlinear elasticity in 2D and 3D. Comput. Methods Appl. Mech. Engrg. 196 (2007) 4075-4086
18. Cohen E., Riesenfeld R., and Elber G. Geometric Modeling with Splines: An Introduction (2001), A.K. Peters Ltd., Wellesley, MA
19. Cook R.D., Malkus D.S., and Plesha M.E. Concepts and Applications of Finite Element Analysis (1989), Wiley, New York
20. Cottrell J.A., Hughes T.J.R., and Reali A. Studies of refinement and continuity in isogeometric structural analysis. Comput. Methods Appl. Mech. Engrg. 197 (2007) 4160-4183
21. Cottrell J.A., Reali A., Bazilevs Y., and Hughes T.J.R. Isogeometric analysis of structural vibrations. Comput. Methods Appl. Mech. Engrg. 195 (2006) 5257-5296
22. Cesar de Sa J.M.A., and Natal Jorge R.M. New enhanced-strain elements for incompressible problems. Int. J. Numer. Methods Engrg. 44 (1999) 229-248
23. de Souza Neto E.A., Peric D., Dutko M., and Owen D.R.J. Design of simple low order finite elements for large strain analysis of nearly incompressible solids. Int. J. Solids Struct. 33 (1996) 3277-3296
24. de Souza Neto E.A., Peric D., Huang G.C., and Owen D.R.J. Remarks on the stability of enhanced strain elements in finite elasticity and elastocplasticity. In: Owen D.R.J., Onate E., and Hinton E. (Eds). Computational Plasticity - Fundamentals and Applications (1995), Pineridge Press
25. de Souza Neto E.A., Andrade Pires F.M., and Owen D.R.J. F-bar based linear triangles and tetrahedra for finite strain analysis of nearly incompressible solids. Part I: formulation and benchmarking. Int. J. Numer. Methods Engrg. 62 (2005) 353-383
26. Dolbow J., and Belytschko T. Volumetric locking in the element free Galerkin method. Int. J. Numer. Methods Engrg. 46 (1999) 925-942
27. Düster A., Hartmann S., and Rank E. p-FEM applied to finite isotropic hyperelastic bodies. Comput. Methods Appl. Mech. Engrg. 192 (2003) 5147-5166
28. Farin G.E. NURBS Curves and Surfaces: from Projective Geometry to Practical Use (1995), A.K. Peters Ltd., Natick, MA
29. Flanagan D., and Belytschko T. A uniform strain hexahedron and quadrilateral with orthogonal hourglass control. Int. J. Numer. Methods Engrg. 17 (1981) 679-706
30. Flory R.J. Thermodynamic relations for highly elastic materials. Trans. Farday Soc. 57 (1969) 829-838
31. U. Heisserer, S. Hartmann, A. Düster, Z. Yosibash, On volumetric locking-free behaviour of p-version finite elements under finite deformations, Commun. Numer. Methods Engrg., in press, doi:10.1002/cnm.1008.
32. Huerta A., and Fernandez-Mendez S. Locking in the incompressible limit for the Element-Free Galerkin method. Int. J. Numer. Methods Engrg. 51 (2001) 1362-1383
33. Hughes T.J.R. Equivalence of finite elements for nearly incompressible elasticity. J. Appl. Mech., Trans. ASME 44 1 (1977) 181-183
34. Hughes T.J.R. Generalization of selective integration procedure to anisotropic and nonlinear media. Int. J. Numer. Methods Engrg. 15 (1980) 1413-1418
35. Hughes T.J.R. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis (2000), Dover Publications, Mineola, NY
36. T.J.R. Hughes, H. Allik, Finite elements for compressible and incompressible continua, in: Proceedings of the Symposium on Civil Engineering, Vanderbilt University, Nashville, TN, 1969, pp. 27-62.
37. Hughes T.J.R., Cottrell J.A., and Bazilevs Y. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Engrg. 194 (2005) 4135-4195
38. Hughes T.J.R., Franca L.P., and Balestra M. A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuska-Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accomodating equal order interpolations. Comput. Methods Appl. Mech. Engrg. 59 (1986) 85-99
39. Hughes T.J.R., Franca L.P., and Hulbert G.M. A new finite element formulation for computational fluid dynamics: VIII. The Galerkin Least Squares method for advective-diffusive equations. Comput. Methods Appl. Mech. Engrg. 73 (1999) 173-189
40. Hughes T.J.R., and Malkus D.S. A general penalty/mixed equivalence theorem for anisotropic incompressible finite elements. In: Atluri S.N., Gallagher R.H., and Zienkiewicz O.C. (Eds). Hybrid and Mixed Finite Element Methods (1983), John Wiley, London 487-496
41. Hughes T.J.R., and Pister K.S. Consistent linearization in mechanics of solid and structures. Comput. Struct. 9 (1978) 391-397
42. T.J.R. Hughes, A. Reali, G. Sangalli, 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. Engrg., submitted for publication. Available as ICES report 07-29, http://www.ices.utexas.edu/research/reports/2007/0729.pdf.
43. T.J.R. Hughes, R.L. Taylor, J.L. Sackman, Finite element formulation and solution of contact-impact problems in continuum mechanics-III, SESM Report 75-3, Department of Civil Engineering, The University of California, Berkeley, 1975.
44. G.E. Karniadakis, E.T. Bullister, A.T. Patera, A spectral element method for solution of two- and three-dimensional time dependent Navier-Stokes equations, in: Europe-US Conference on Finite Element Methods for Nonlinear Problems, Springer-Verlag, 1985, p. 803.
45. Kasper E.P., and Taylor R.L. A mixed-enhanced strain method. Part I: Geometrically linear problems. Comput. Struct. 75 (2000) 237-250
46. Kasper E.P., and Taylor R.L. A mixed-enhanced strain method. Part II: Geometrically nonlinear problems. Comput. Struct. 75 (2000) 252-260
47. Klass O., Maniatty A., and Shephard M.S. A stabilized mixed finite element method for finite elasticity. Formulation for linear displacement and pressure interpolation. Comput. Methods Appl. Mech. Engrg. 180 (1999) 65-78
48. Livermore Software Technology Corporation. LS-DYNA Keyword User's Manual, May 2007.
49. B.N. Maker, NIKE3D a non-linear, implicit, three-dimensional finite element code for solid and structural mechanics. Technical Report UCRL-MA-105268 Rev. 1, Lawrence Livermore National Laboratory, University of California, Livermore, 1995.
50. Malkus D.S., and Hughes T.J.R. Mixed finite element methods - reduced and selective integration techniques: a unification of concepts. Comput. Methods Appl. Mech. Engrg. 15 1 (1978) 63-81
51. Maniatty A.M., Liu Y., Klass O., and Shephard M.S. Higher order stabilized finite element method for hyperelastic finite deformation. Comput. Methods Appl. Mech. Engrg. 191 (2002) 1491-1503
52. Nagtegaal J.C., Park D.M., and Rice J.R. On numerically accurate finite element solutions in the fully plastic range. Comput. Methods Appl. Mech. Engrg. 4 (1974) 53-177
53. Piegl L., and Tiller W. The NURBS Book (Monographs in Visual Communication). second ed. (1997), Springer-Verlag, New York
54. Ramesh B., and Maniatty A. Stabilized finite element formulation for elastic-plastic finite deformations. Comput. Methods Appl. Mech. Engrg. 194 (2005) 775-800
55. Reese S., Kussner M., and Reddy B.D. A new stabilization technique for finite elements in non-linear elasticity. Int. J. Numer. Methods Engrg. 44 (1999) 1617-1652
56. Reese S., Wriggers P., and Reddy B.D. A new locking-free brick element technique for large deformation problems in elasticity. Comput. Struct. 75 (2000) 291-304
57. Rogers D.F. An Introduction to NURBS With Historical Perspective (2001), Academic Press, San Diego, CA
58. Simo J.C., and Hughes T.J.R. On the variational foundations of assumed strain methods. J. Appl. Mech. 53 (1986) 51-54
59. Simo J.C., and Taylor R.L. Quasi-incompressible finite elasticity in principal stretches. Continuum basis and numerical algorithm. Comput. Methods Appl. Mech. Engrg. 85 (1991) 273-310
60. Simo J.C. A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition. Part I: continuum formulation. Comput. Methods Appl. Mech. Engrg. 66 (1988) 199-219
61. Simo J.C. A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition. Part II: computational aspects. Comput. Methods Appl. Mech. Engrg. 68 (1988) 1-31
62. Simo J.C., and Armero F. Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes. Int. J. Numer. Methods Engrg. 33 (1992) 1413-1449
63. Simo J.C., and Hughes T.J.R. Computational Inelasticity (1998), Springer-Verlag, New York
64. Simo J.C., and Rifai M.S. A class of mixed assumed strain methods and the method of incompatible modes. Int. J. Numer. Methods Engrg. 29 (1990) 1595-1638
65. Simo J.C., Taylor R.L., and Pister K.S. Variational and projection methods for the volume contraint in finite deformation elasto-plasticity. Comput. Methods Appl. Mech. Engrg. 51 (1985) 177-208
66. Suri M. Analytical and computational assessment of locking in the hp finite element method. Comput. Methods Appl. Mech. Engrg. 133 (1996) 347-371
67. Szabó B., and Babuška I. Finite Element Analysis (1991), Wiley, New York
68. Szabó B., Düster A., and Rank E. The p-version of the finite element method. In: Stein E., de Borst R., and Hughes T.J.R. (Eds). Encyclopedia of Computational Mechanics. Fundamentals vol. 1 (2004), Wiley, New York (chapter 5)
69. Timoshenko S.P., and Goodier J.N. Theory of Elasticity. third ed. (1970), McGraw-Hill, Singapore
70. Wilson E.L., Taylor R.L., Doherty W.P., and Ghaboussi J. Incompatible displacement models. In: Fenves S.J., Perrone N., Robinson A.R., and Schnobrich W.C. (Eds). Numerical and Computer Models in Structural Mechanics (1973), Academic Press, New York
71. Zhang Y., Bazilevs Y., Goswami S., Bajaj C.L., and Hughes T.J.R. Patient-specific vascular NURBS modeling for isogeometric analysis of blood flow. Comput. Methods Appl. Mech. Engrg. 196 29-30 (2007) 2943-2959