Nitsche’s method for two and three dimensional NURBS patch coupling
详细信息 查看全文
- 作者：Vinh Phu Nguyen (1)
Pierre Kerfriden (1)
Marco Brino (3)
Stéphane P. A. Bordas (1) (2)
Elvio Bonisoli (3) - 关键词：Nitsche’s method ; Isogeometric analysis (IGA) ; Multi ; patch NURBS IGA ; Finite element method
- 刊名：Computational Mechanics
- 出版年：2014
- 出版时间：June 2014
- 年：2014
- 卷：53
- 期：6
- 页码：1163-1182
- 全文大小：
- 参考文献：1. Piegl LA, Tiller W (1996) The NURBS book. Springer, Berlin
2. Rogers DF (2001) An introduction to NURBS with historical perspective. Academic Press, San Diego
3. Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194(39-1):4135-195 CrossRef
4. Cottrell JA, Hughes TJR, Bazilevs Y (2009) Isogeometric analysis: toward integration of CAD and FEA. Wiley, Chichester CrossRef
5. Kagan P, Fischer A, Bar-Yoseph PZ (1998) New B-spline finite element approach for geometrical design and mechanical analysis. Int J Numer Methods Eng 41(3):435-58 CrossRef
6. Kagan P, Fischer A (2000) Integrated mechanically based CAE system using B-spline finite elements. Comput Aided Des 32(8-):539-52 CrossRef
7. Cirak F, Ortiz M, Schr?der P (2000) Subdivision surfaces: a new paradigm for thin-shell finite-element analysis. Int J Numer Methods Eng 47(12):2039-072 CrossRef
8. Uhm TK, Youn SK (2009) T-spline finite element method for the analysis of shell structures. Int J Numer Methods Eng 80(4):507-36 CrossRef
9. Kiendl J, Bletzinger K-U, Linhard J, Wüchner R (2009) Isogeometric shell analysis with Kirchhoff-Love elements. Comput Methods Appl Mech Eng 198(49-2):3902-914 CrossRef
10. Benson DJ, Bazilevs Y, Hsu MC, Hughes TJR (2010) Isogeometric shell analysis: the Reissner–Mindlin shell. Comput Methods Appl Mech Eng 199(5-):276-89 CrossRef
11. Benson DJ, Bazilevs Y, Hsu M-C, Hughes TJR (2011) A large deformation, rotation-free, isogeometric shell. Comput Methods Appl Mech Eng 200(13-6):1367-378 CrossRef
12. Beir?o da Veiga L, Buffa A, Lovadina C, Martinelli M, Sangalli G (2012) An isogeometric method for the Reissner–Mindlin plate bending problem. Comput Methods Appl Mech Eng 209-12:45-3 CrossRef
13. Echter R, Oesterle B, Bischoff M (2013) A hierarchic family of isogeometric shell finite elements. Comput Methods Appl Mech Eng 254:170-80 CrossRef
14. Benson DJ, Hartmann S, Bazilevs Y, Hsu M-C, Hughes TJR (2013) Blended isogeometric shells. Comput Methods Appl Mech Eng 255:133-46 CrossRef
15. Kiendl J, Bazilevs Y, Hsu M-C, Wüchner R, Bletzinger K-U (2010) The bending strip method for isogeometric analysis of Kirchhoff–Love shell structures comprised of multiple patches. Comput Methods Appl Mech Eng 199(37-0):2403-416 CrossRef
16. Temizer ?, Wriggers P, Hughes TJR (2011) Contact treatment in isogeometric analysis with NURBS. Comput Methods Appl Mech Eng 200(9-2):1100-112 CrossRef
17. Jia L (2011) Isogeometric contact analysis: geometric basis and formulation for frictionless contact. Comput Methods Appl Mech Eng 200(5-):726-41
18. Temizer ?, Wriggers P, Hughes TJR (2012) Three-dimensional Mortar-Based frictional contact treatment in isogeometric analysis with NURBS. Comput Methods Appl Mech Eng 209-12:115-28 CrossRef
19. De Lorenzis L, Temizer ?, Wriggers P, Zavarise G (2011) A large deformation frictional contact formulation using NURBS-bases isogeometric analysis. Int J Numer Methods Eng 87(13):1278-300
20. Matzen ME, Cichosz T, Bischoff M (2013) A point to segment contact formulation for isogeometric, NURBS based finite elements. Comput Methods Appl Mech Eng 255:27-9 CrossRef
21. Wall WA, Frenzel MA, Cyron C (2008) Isogeometric structural shape optimization. Comput Methods Appl Mech Eng 197(33-0):2976-988 CrossRef
22. Manh ND, Evgrafov A, Gersborg AR, Gravesen J (2011) Isogeometric shape optimization of vibrating membranes. Comput Methods Appl Mech Eng 200(13-6):1343-353 CrossRef
23. Qian X, Sigmund O (2011) Isogeometric shape optimization of photonic crystals via Coons patches. Comput Methods Appl Mech Eng 200(25-8):2237-255 CrossRef
24. Qian X (2010) Full analytical sensitivities in NURBS based isogeometric shape optimization. Comput Methods Appl Mech Eng 199(29-2):2059-071 CrossRef
25. Simpson RN, Bordas SPA, Trevelyan J, Rabczuk T (2012) A two-dimensional isogeometric boundary element method for elastostatic analysis. Comput Methods Appl Mech Eng 209-12:87-00 CrossRef
26. Scott MA, Simpson RN, Evans JA, Lipton S, Bordas SPA, Hughes TJR, Sederberg TW (2013) Isogeometric boundary element analysis using unstructured T-splines. Comput Methods Appl Mech Eng 254:197-21 CrossRef
27. Gomez H, Hughes TJR, Nogueira X, Calo VM (2010) Isogeometric analysis of the isothermal Navier–Stokes–Korteweg equations. Comput Methods Appl Mech Eng 199(25-8):1828-840 CrossRef
28. Bazilevs Y, Akkerman I (2010) Large eddy simulation of turbulent Taylor–Couette flow using isogeometric analysis and the residual-based variational multiscale method. J Comput Phys 229(9):3402-414 CrossRef
29. Nielsen PN, Gersborg AR, Gravesen J, Pedersen NL (2011) Discretizations in isogeometric analysis of Navier-Stokes flow. Comput Methods Appl Mech Eng 200(45-6):3242-253 CrossRef
30. Bazilevs Y, Calo VM, Hughes TJR, Zhang Y (2008) Isogeometric fluid-structure interaction: theory, algorithms, and computations. Comput Mech 43:3-7 CrossRef
31. Bazilevs Y, Gohean JR, Hughes TJR, Moser RD, Zhang Y (2009) Patient-specific isogeometric fluid-structure interaction analysis of thoracic aortic blood flow due to implantation of the Jarvik 2000 left ventricular assist device. Comput Methods Appl Mech Eng 198(45-6):3534-550 CrossRef
32. Gómez H, Calo VM, Bazilevs Y, Hughes TJR (2008) Isogeometric analysis of the Cahn–Hilliard phase-field model. Comput Methods Appl Mech Eng 197(49-0):4333-352 CrossRef
33. Verhoosel CV, Scott MA, Hughes TJR, de Borst R (2011) An isogeometric analysis approach to gradient damage models. Int J Numer Methods Eng 86(1):115-34 CrossRef
34. Fischer P, Klassen M, Mergheim J, Steinmann P, Müller R (2010) Isogeometric analysis of 2D gradient elasticity. Comput Mech 47:325-34 CrossRef
35. Masud A, Kannan R (2012) B-splines and NURBS based finite element methods for Kohn–Sham equations. Comput Methods Appl Mech Eng 241-44:112-27 CrossRef
36. Cottrell JA, Reali A, Bazilevs Y, Hughes TJR (2006) Isogeometric analysis of structural vibrations. Comput Methods Appl Mech Eng 195(41-3):5257-296 CrossRef
37. 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(49-0):4104-124 CrossRef
38. Thai CH, Nguyen-Xuan H, Nguyen-Thanh N, Le T-H, Nguyen-Thoi T, Rabczuk T (2012) Static, free vibration, and buckling analysis of laminated composite Reissner-Mindlin plates using NURBS-based isogeometric approach. Int J Numer Methods Eng 91(6):571-03 CrossRef
39. Wang D, Liu W, Zhang H (2013) Novel higher order mass matrices for isogeometric structural vibration analysis. Comput Methods Appl Mech Eng 260:63-7 CrossRef
40. Evans JA, Bazilevs Y, Babu?ka I, Hughes TJR (2009) n-Widths, sup-infs, and optimality ratios for the k-version of the isogeometric finite element method. Comput Methods Appl Mech Eng 198(21-6):1726-741 CrossRef
41. Verhoosel CV, Scott MA, de Borst R, Hughes TJR (2011) An isogeometric approach to cohesive zone modeling. Int J Numer Methods Eng 87(15):336-60 CrossRef
42. Mo?s N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46(1):131-50 CrossRef
43. De Luycker E, Benson DJ, Belytschko T, Bazilevs Y, Hsu MC (2011) X-FEM in isogeometric analysis for linear fracture mechanics. Int J Numer Methods Eng 87(6):541-65 CrossRef
44. Ghorashi SS, Valizadeh N, Mohammadi S (2012) Extended isogeometric analysis for simulation of stationary and propagating cracks. Int J Numer Methods Eng 89:1069-101 CrossRef
45. Tambat A, Subbarayan G (2012) Isogeometric enriched field approximations. Comput Methods Appl Mech Eng 245-46:1-1 CrossRef
46. Borden MJ, Verhoosel CV, Scott MA, Hughes TJR, Landis CM (2012) A phase-field description of dynamic brittle fracture. Comput Methods Appl Mech Eng 217-20:77-5 CrossRef
47. Nguyen VP, Nguyen-Xuan H (2013) High-order B-splines based finite elements for delamination analysis of laminated composites. Compos Struct 102:261-75 CrossRef
48. Nguyen VP, Kerfriden P, Bordas S (2013) Isogeometric cohesive elements for two and three dimensional composite delamination analysis. Compos Part B, 2013. http://arxiv.org/abs/1305.2738
49. Takacs T, Jüttler B (2011) Existence of stiffness matrix integrals for singularly parameterized domains in isogeometric analysis. Comput Methods Appl Mech Eng 200(49-2):3568-582 CrossRef
50. Xu G, Mourrain B, Duvigneau R, Galligo A (2011) Parameterization of computational domain in isogeometric analysis: methods and comparison. Comput Methods Appl Mech Eng 200(23-4):2021-031 CrossRef
51. Cohen E, Martin T, Kirby RM, Lyche T, Riesenfeld RF (2010) Analysis-aware modeling: understanding quality considerations in modeling for isogeometric analysis. Comput Methods Appl Mech Eng 199(5-):334-56 CrossRef
52. Schmidt R, Kiendl J, Bletzinger K-U, Wüchner R (2010) Realization of an integrated structural design process: analysis-suitable geometric modelling and isogeometric analysis. Comput Vis Sci 13(7):315-30 CrossRef
53. Zhou X and Lu J (2005) Nurbs-based galerkin method and application to skeletal muscle modeling. In Proceedings of the 2005 ACM symposium on solid and physical modeling, SPM -5. ACM, New York, pp 71-8
54. Xu G, Mourrain B, Duvigneau R, Galligo A (2013) Optimal analysis-aware parameterization of computational domain in 3D isogeometric analysis. Comput Aided Des 45(4):812-21 CrossRef
55. Xu G, Mourrain B, Duvigneau R, Galligo A (2012) Analysis-suitable volume parameterization of multi-block computational domain in isogeometric applications. Comput Aided Des 45(2):395-04 CrossRef
56. Sederberg TW, Zheng J, Bakenov A, Nasri A (2003) T-splines and T-NURCCs. ACM Trans Gr 22:477-84 CrossRef
57. 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(5-):229-63 CrossRef
58. D?rfel MR, Jüttler B, Simeon B (2010) Adaptive isogeometric analysis by local h-refinement with T-splines. Comput Methods Appl Mech Eng 199(5-):264-75 CrossRef
59. Scott MA, Borden MJ, Verhoosel CV, Sederberg TW, Hughes TJR (2011) Isogeometric finite element data structures based on Bézier extraction of T-splines. Int J Numer Methods Eng 88(2):126-56 CrossRef
60. Nguyen-Thanh N, Nguyen-Xuan H, Bordas SPA, Rabczuk T (2011) Isogeometric analysis using polynomial splines over hierarchical t-meshes for two-dimensional elastic solids. Comput Methods Appl Mech Eng 200(21):1892-908 CrossRef
61. Nguyen-Thanh N, Kiendl J, Nguyen-Xuan H, Wüchner R, Bletzinger KU, Bazilevs Y, Rabczuk T (2011) Rotation free isogeometric thin shell analysis using PHT-splines. Comput Methods Appl Mech Eng 200(47-8):3410-424 CrossRef
62. Rosolen A, Arroyo M (2013) Blending isogeometric analysis and local maximum entropy meshfree approximants. Comput Methods Appl Mech Eng 264:95-07 CrossRef
63. Nguyen VP, Kerfriden P, Claus S, Bordas SPA (2013) Nitsche’s method for mixed dimensional analysis: conforming and non-conforming continuum-beam and continuum-plate coupling. Comput Methods Appl Mech Eng 2013. http://arxiv.org/abs/1308.2910
64. Nitsche J (1971) Uber ein variationsprinzip zur losung von dirichlet-problemen bei verwendung von teilraumen, die keinen randbedingungen unterworfen sind. Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg 36:9-5 CrossRef
65. Hansbo A, Hansbo P (2002) An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems. Comput Methods Appl Mech Eng 191(4748):5537-552 CrossRef
66. Dolbow J, Harari I (2009) An efficient finite element method for embedded interface problems. Int J Numer Methods Eng 78:229-52 CrossRef
67. Becker R, Hansbo P, Stenberg R (2003) A finite element method for domain decomposition with non-matching grids. ESAIM Math Modell Numer Anal 37:209-25 CrossRef
68. Hansbo A, Hansbo P, Larson MG (2003) A finite element method on composite grids based on Nitsche’s method. ESAIM Math Modell Numer Anal 37:495-14 CrossRef
69. Sanders J, Puso MA (2012) An embedded mesh method for treating overlapping finite element meshes. Int J Numer Methods Eng 91:289-05 CrossRef
70. Sanders JD, Laursen T, Puso MA (2011) A Nitsche embedded mesh method. Comput Mech 49(2):243-57 CrossRef
71. Burman E, Hansbo P (2012) Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method. Appl Numer Math 62(4):328-41 CrossRef
72. Fernández-Méndez S, Huerta A (2004) Imposing essential boundary conditions in mesh-free methods. Comput Methods Appl Mech Eng 193(1214):1257-275 CrossRef
73. Ruess M, Schillinger D, Bazilevs Y, Varduhn V, Rank E (2013) Weakly enforced essential boundary conditions for NURBS-embedded and trimmed NURBS geometries on the basis of the finite cell method. Accepted for publication, Int J Numer Methods Eng
74. Baiges J, Codina R, Henke F, Shahmiri S, Wall WA (2012) A symmetric method for weakly imposing Dirichlet boundary conditions in embedded finite element meshes. Int J Numer Methods Eng 90(5):636-58 CrossRef
75. Embar A, Dolbow J, Harari I (2010) Imposing Dirichlet boundary conditions with Nitsche’s method and spline-based finite elements. Int J Numer Methods Eng 83(7):877-98
76. Burman E, Fernández MA (2009) Stabilization of explicit coupling in fluidstructure interaction involving fluid incompressibility. Comput Methods Appl Mech Eng 198(5):766-84 CrossRef
77. Bazilevs Y, Hughes TJR (2007) Weak imposition of Dirichlet boundary conditions in fluid mechanics. Comput Fluids 36(1):12-6 CrossRef
78. Ruess M, Schillinger D, Bazilevs Y, Rank E (2013) Weakly enforced essential boundary conditions for NURBS-embedded and trimmed NURBS geometries on the basis of the finite cell method. Int J Numer Methods Eng 95(10):811-46 CrossRef
79. Wriggers P, Zavarise G (2008) A formulation for frictionless contact problems using a weak form introduced by Nitsche. Comput Mech 41(3):407-20 CrossRef
80. Sanders JD, Dolbow JE, Laursen TA (2009) On methods for stabilizing constraints over enriched interfaces in elasticity. Int J Numer Methods Eng 78:1009-036 CrossRef
81. Griebel M, Schweitzer MA (2002) A particle-partition of unity method—Part V: boundary conditions. In: Hildebrandt S, Karcher H (eds) Geometric analysis and nonlinear partial differential equations. Springer, Berlin, pp 519-42
82. Rhino. CAD modeling and design toolkit. www.rhino3d.com
83. Hughes TJR (2000) The finite element method: linear static and dynamic finite element analysis. Dover Publications, Mineola
84. Nayroles B, Touzot G, Villon P (1992) Generalizing the finite element method: diffuse approximation and diffuse elements. Comput Mech 10(5):307-18 CrossRef
85. Belytschko T, Lu YY, Gu L (1994) Element-free galerkin methods. Int J Numer Methods Eng 37(2):229-56
86. Liu WK, Jun S, Zhang YF (1995) Reproducing kernel particle methods. Int J Numer Methods Fluids 20:1081-106 CrossRef
87. Chen JS, Wu CT, Yoon S, You Y (2001) A stabilized conforming nodal integration for Galerkin meshfree methods. Int J Numer Methods Eng 50:435-66 CrossRef
88. Nguyen VP, Rabczuk T, Bordas S, Duflot M (2008) Meshless methods: a review and computer implementation aspects. Math Comput Simul 79(3):763-13 CrossRef
89. Borden MJ, Scott MA, Evans JA, Hughes TJR (2011) Isogeometric finite element data structures based on Bézier extraction of NURBS. Int J Numer Methods Eng 87(15):15-7 CrossRef
90. Henderson A (2007) ParaView guide, a parallel visualization application. Kitware Inc
91. Ugural AC, Fenster SK (1995) Advanced strength and applied elasticity, 3rd edn. Prentice-Hall, Englewood Cliffs
92. Nguyen VP, Bordas SPA, Rabczuk T (2013) Isogeometric analysis: an overview and computer implementation aspects. Comput Aided Geom Des. http://arxiv.org/abs/1205.2129
93. Hughes TJR, Reali A, Sangalli G (2010) Efficient quadrature for NURBS-based isogeometric analysis. Comput Methods Appl Mech Eng 199(5-):301-13 CrossRef
94. Auricchio F, Calabro F, Hughes TJR, Reali A, Sangalli G (2012) A simple algorithm for obtaining nearly optimal quadrature rules for NURBS-based isogeometric analysis. Comput Methods Appl Mech Eng 249-52:15-7 CrossRef
95. Fritz A, Hüeber S, Wohlmuth BI (2004) A comparison of mortar and Nitsche techniques for linear elasticity. CALCOLO 41(3):115-37 CrossRef
96. Sukumar N, Chopp DL, Mo?s N, Belytschko T (2000) Modelling holes and inclusions by level sets in the extended finite element method. Comput Methods Appl Mech Eng 190:6183-200 CrossRef
97. Sevilla R, Fernández-Méndez S, Huerta A (2008) NURBS-enhanced finite element method (NEFEM). Int J Numer Methods Eng 76(1):56-3 CrossRef
98. Tornincasa S, Bonisoli E, Kerfriden P, Brino M (2014) Investigation of crossing and veering phenomena in an isogeometric analysis framework. In Proceedings of IMAC XXXII, Orlando pp 3- Feb 2014
99. Ruess M, Schillinger D, Bazilevs Y, Ozcan A, Rank E (2013) Weakly enforced boundary and coupling conditions in isogeometric analysis. In Proceedings of 12th US national congress on computational mechanics. Raleigh, North Carolina, July 22-5
100. Noels L, Radovitzky R (2006) A general discontinuous Galerkin method for finite hyperelasticity. Formulation and numerical applications. Int J Numer Methods Eng 68(1):64-7 CrossRef
101. Radovitzky R, Seagraves A, Tupek M, Noels L (2011) A scalable 3D fracture and fragmentation algorithm based on a hybrid, discontinuous Galerkin, cohesive element method. Comput Methods Appl Mech Eng 200(14):326-44 CrossRef - 作者单位：Vinh Phu Nguyen (1)
Pierre Kerfriden (1)
Marco Brino (3)
Stéphane P. A. Bordas (1) (2)
Elvio Bonisoli (3)
1. Cardiff University, Queen’s Buildings, The Parade, Cardiff, Wales, CF24 3AA, UK
3. Politecnico di Torino - DIGEP, corso Duca degli Abruzzi 24, 10129?, Turin, Italy
2. Faculté des Sciences, de la Technologie et de la Communication, University of Luxembourg, 6, rue Richard Coudenhove-Kalergi, 1359?, Luxembourg City, Luxembourg - ISSN：1432-0924
文摘
We present a Nitche’s method to couple non-conforming two and three-dimensional non uniform rational b-splines (NURBS) patches in the context of isogeometric analysis. We present results for linear elastostatics in two and and three-dimensions. The method can deal with surface-surface or volume-volume coupling, and we show how it can be used to handle heterogeneities such as inclusions. We also present preliminary results on modal analysis. This simple coupling method has the potential to increase the applicability of NURBS-based isogeometric analysis for practical applications.