Isogeometric boundary element analysis using unstructured T-splines

Computer Methods in Applied Mechanics and Engineering

Volumn 254, Issue , 2013, Pages 197-221

  Scott, M A ^d   Simpson, R N ^b   Evans, J A ^a   Lipton, S ^a   Bordas, S P A ^b   Hughes, T J R ^a   Sederberg, T W ^c  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

^b CARDIFF UNIVERSITY   (United Kingdom)

^c Brigham Young University   (United States)

^d Department of Electrical and Computer Engineering   (United States)

Author keywords

Boundary elements; Collocation; Design through analysis; Isogeometric analysis; T splines

Indexed keywords

BOUNDARY ELEMENTS; COLLOCATION; DESIGN-THROUGH-ANALYSIS; ISOGEOMETRIC ANALYSIS; T-SPLINES;

COMPUTER AIDED DESIGN; RATIONAL FUNCTIONS; SPLINES; THREE DIMENSIONAL;

BOUNDARY ELEMENT METHOD;

EID: 84870805346     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cma.2012.11.001     Document Type: Article

References (116)

1. Hughes T.J.R., Cottrell J.A., Bazilevs Y. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement. Comput. Methods Appl. Mech. Engrg. 2005, 194:4135-4195.
2. Cottrell J.A., Hughes T.J.R., Bazilevs Y. Isogeometric Analysis: Toward Integration of CAD and FEA 2009, Wiley, Chichester.
3. Cottrell J.A., Reali A., Bazilevs Y., Hughes T.J.R. Isogeometric analysis of structural vibrations. Comput. Methods Appl. Mech. Engrg. 2006, 195:5257-5296.
4. Evans J.A., Bazilevs Y., BabuŠka I., Hughes T.J.R. n-Widths, sup-infs, and optimality ratios for the k-version of the isogeometric finite element method. Comput. Methods Appl. Mech. Engrg. 2009, 198(21-26):1726-1741.
5. Akkerman I., Bazilevs Y., Calo V., Hughes T.J.R., Hulshoff S. The role of continuity in residual-based variational multiscale modeling of turbulence. Comput. Mech. 2008, 41:371-378.
6. Bazilevs Y., Calo V.M., Cottrell J.A., Hughes T.J.R., Reali A., Scovazzi G. Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows. Comput. Methods Appl. Mech. Engrg. 2007, 197:173-201.
7. Bazilevs Y., Akkerman I. Large eddy simulation of turbulent Taylor-Couette flow using isogeometric analysis and residual-based variational multiscale method. J. Comput. Phys. 2010, 229:3402-3414.
8. Bazilevs Y., Michler C., Calo V.M., Hughes T.J.R. Isogeometric variational multiscale modeling of wall-bounded turbulent flows with weakly enforced boundary conditions on unstretched meshes. Comput. Methods Appl. Mech. Engrg. 2010, 199(13-16):780-790.
9. Bazilevs Y., Calo V.M., Zhang Y., Hughes T.J.R. Isogeometric fluid-structure interaction analysis with applications to arterial blood flow. Comput. Mech. 2006, 38:310-322.
10. Bazilevs Y., Calo V.M., Hughes T.J.R., Zhang Y. Isogeometric fluid-structure interaction: theory, algorithms, and computations. Comput. Mech. 2008, 43:3-37.
11. Zhang Y., Bazilevs Y., Goswami S., Bajaj C., Hughes T.J.R. Patient-specific vascular NURBS modeling for isogeometric analysis of blood flow. Comput. Methods Appl. Mech. Engrg. 2007, 196:2943-2959.
12. Bazilevs Y., Gohean J.R., Hughes T.J.R., Moser R.D., Zhang Y. 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. Engrg. 2009, 198(45-46):3534-3550.
13. Auricchio F., da Veiga L.B., Lovadina C., Reali A. The importance of the exact satisfaction of the incompressibility constraint in nonlinear elasticity: mixed FEMs versus NURBS-based approximations. Comput. Methods Appl. Mech. Engrg. 2010, 199:314-323.
14. Auricchio F., da Veiga L.B., Buffa A., Lovadina C., Reali A., Sangalli G. A fully locking-free isogeometric approach for plane linear elasticity problems: a stream function formulation. Comput. Methods Appl. Mech. Engrg. 2007, 197:160-172.
15. Elguedj T., Bazilevs Y., Calo V.M., Hughes T.J.R. B̄ and F̄ projection methods for nearly incompressible linear and non-linear elasticity and plasticity using higher-order NURBS elements. Comput. Methods Appl. Mech. Engrg. 2008, 197:2732-2762.
16. Cottrell J.A., Hughes T.J.R., Reali A. Studies of refinement and continuity in isogeometric analysis. Comput. Methods Appl. Mech. Engrg. 2007, 196:4160-4183.
17. Benson D.J., Bazilevs Y., Hsu M.C., Hughes T.J.R. Isogeometric shell analysis: the Reissner-Mindlin shell. Comput. Methods Appl. Mech. Engrg. 2010, 199(5-8):276-289.
18. Benson D.J., Bazilevs Y., Hsu M.C., Hughes T.J.R. A large deformation, rotation-free, isogeometric shell. Int. J. Numer. Methods Engrg. 2011, 200:1367-1378.
19. Benson D.J., Bazilevs Y., De Luycker E., Hsu M.C., Scott M.A., Hughes T.J.R., Belytschko T. A generalized finite element formulation for arbitrary basis functions: from isogeometric analysis to XFEM. Int. J. Numer. Methods Engrg. 2010, 83:765-785.
20. Echter R., Bischoff M. Numerical efficiency, locking and unlocking of NURBS finite elements. Comput. Methods Appl. Mech. Engrg. 2010, 199(5-8):374-382.
21. Kiendl J., Bazilevs Y., Hsu M.C., Wuechner R., Bletzinger K.U. The bending strip method for isogeometric analysis of Kirchhoff-Love shell structures comprised of multiple patches. Comput. Methods Appl. Mech. Engrg. 2010, 199(37-40):2403-2416.
22. Gomez H., Calo V.M., Bazilevs Y., Hughes T.J.R. Isogeometric analysis of the Cahn-Hilliard phase-field model. Comput. Methods Appl. Mech. Engrg. 2008, 197:4333-4352.
23. Gomez H., Hughes T.J.R., Nogueira X., Calo V.M. Isogeometric analysis of the isothermal Navier-Stokes-Korteweg equations. Comput. Methods Appl. Mech. Engrg. 2010, 199(25-28):1828-1840.
24. Borden M.J., Scott M.A., Verhoosel C.V., Landis C.M., Hughes T.J.R. A phase-field description of dynamic brittle fracture. Comput. Methods Appl. Mech. Engrg. 2012, 217:77-95.
25. Lipton S., Evans J.A., Bazilevs Y., Elguedj T., Hughes T.J.R. Robustness of isogeometric structural discretizations under severe mesh distortion. Comput. Methods Appl. Mech. Engrg. 2010, 199(5-8):357-373.
26. Wall W.A., Frenzel M.A., Cyron C. Isogeometric structural shape optimization. Comput. Methods Appl. Mech. Engrg. 2008, 197:2976-2988.
27. Qian X. Full analytical sensitivities in NURBS based isogeometric shape optimization. Comput. Methods Appl. Mech. Engrg. 2010, 199(29-32):2059-2071.
28. Nagy A.P., Abdalla M.M., Gurdal Z. Isogeometric sizing and shape optimization of beam structures. Comput. Methods Appl. Mech. Engrg. 2010, 199(17-20):1216-1230.
29. Nagy A.P., Abdalla M.M., Gurdal Z. On the variational formulation of stress constraints in isogeometric design. Comput. Methods Appl. Mech. Engrg. 2010, 199(41-44):2687-2696.
30. Buffa A., Sangalli G., Vázquez R. Isogeometric analysis in electromagnetics: B-splines approximation. Comput. Methods Appl. Mech. Engrg. 2010, 199(17-20):1143-1152.
31. Cohen E., Martin T., Kirby R.M., Lyche T., Riesenfeld R.F. Analysis-aware modeling: understanding quality considerations in modeling for isogeometric analysis. Comput. Methods Appl. Mech. Engrg. 2010, 199(5-8):334-356.
32. Lu J. Circular element: isogeometric elements of smooth boundary. Comput. Methods Appl. Mech. Engrg. 2009, 198(30-32):2391-2402.
33. Kim H., Seo Y., Youn S. Isogeometric analysis for trimmed CAD surfaces. Comput. Methods Appl. Mech. Engrg. 2009, 198(37-40):2982-2995.
34. Costantini P., Manni C., Pelosi F., Sampoli M.L. Quasi-interpolation in isogeometric analysis based on generalized B-splines. Comput. Aided Geom. Des. 2010, 27(8):656-668.
35. M. Aigner, C. Heinrich, B. Juettler, E. Pilgerstorfer, B. Simeon, A.V. Vuong, Swept volume parameterization for isogeometric analysis, in: E.R. Hancock, R.R. Martin, M.A. Sabin (Eds.), Mathematics of Surfaces XIII, Lecture Notes in Computer Science, vol. 5654, 2009, pp. 19-44.
36. Martin T., Cohen E., Kirby R.M. Volumetric parameterization and trivariate B-spline fitting using harmonic functions. Comput. Aided Geom. Des. 2009, 26(6):648-664.
37. Wang W., Zhang Y. Wavelets-based NURBS simplification and fairing. Comput. Methods Appl. Mech. Engrg. 2010, 199(5-8):290-300. 10.1016/j.cma.2009.04.003.
38. Wang W., Zhang Y., Scott M.A., Hughes T.J.R. Converting an unstructured quadrilateral mesh to a standard T-spline surface. Comput. Mech. 2011, 48:477-498.
39. Sederberg T.W., Zheng J., Bakenov A., Nasri A. T-splines and T-NURCCs. ACM Trans. Graph. 2003, 22:477-484.
40. Sederberg T.W., Finnigan G.T., Li X., Lin H. Watertight trimmed NURBS. ACM Trans. Graph. 2008, 27:1-79.
41. H. Ipson, T-spline merging, Master's Thesis, Brigham Young University, 2005.
42. Sederberg T.W., Cardon D.L., Finnigan G.T., North N.S., Zheng J., Lyche T. T-spline simplification and local refinement. ACM Trans. Graph. 2004, 23:276-283.
43. Scott M.A., Li X., Sederberg T.W., Hughes T.J.R. Local refinement of analysis-suitable T-splines. Comput. Methods Appl. Mech. Engrg. 2012, 213:206-222.
44. Bazilevs Y., Calo V.M., Cottrell J.A., Evans J.A., Hughes T.J.R., Lipton S., Scott M.A., Sederberg T.W. Isogeometric analysis using T-splines. Comput. Methods Appl. Mech. Engrg. 2010, 199(5-8):229-263.
45. Dörfel M., Jüttler B., Simeon B. Adaptive isogeometric analysis by local h-refinement with T-splines. Comput. Methods Appl. Mech. Engrg. 2009, 199(5-8):264-275.
46. M.A. Scott, T-splines as a Design-Through-Analysis technology, Ph.D. Thesis, The University of Texas at Austin, 2011.
47. Li X., Zheng J., Sederberg T.W., Hughes T.J.R., Scott M.A. On linear independence of T-spline blending functions. Comput. Aided Geom. Des. 2012, 29:63-76.
48. Verhoosel C.V., Scott M.A., Hughes T.J.R., de Borst R. An isogeometric analysis approach to gradient damage models. Int. J. Numer. Methods Engrg. 2011, 86:115-134.
49. Verhoosel C.V., Scott M.A., de Borst R., Hughes T.J.R. An isogeometric approach to cohesive zone modeling. Int. J. Numer. Methods Engrg. 2011, 87:336-360.
50. Bazilevs Y., Hsu M.-C., Scott M.A. Isogeometric fluid-structure interaction analysis with emphasis on non-matching discretizations, and with application to wind turbines. Comput. Methods Appl. Mech. Engrg. 2012, 249-252:28-41.
51. Schillinger D., Dedè L., Scott M.A., Evans J.A., Borden M.J., Rank E., Hughes T.J.R. An isogeometric design-through-analysis methodology based on adaptive hierarchical refinement of NURBS, immersed boundary methods, and T-spline CAD surfaces. Comput. Methods Appl. Mech. Engrg. 2012, 249-252:116-150.
52. Y. Zhang, W. Wang, T.J.R. Hughes, Solid T-spline construction from boundary representations for genus-zero geometry, ICES, Report 11-40.
53. Borden M.J., Scott M.A., Evans J.A., Hughes T.J.R. Isogeometric finite element data structures based on Bézier extraction of NURBS. Int. J. Numer. Methods Engrg. 2011, 87:15-47.
54. Scott M.A., Borden M.J., Verhoosel C.V., Sederberg T.W., Hughes T.J.R. Isogeometric finite element data structures based on Bézier extraction of T-splines. Int. J. Numer. Methods Engrg. 2011, 88:126-156.
55. Beirão da Veiga L., Buffa A., Cho D., Sangalli G. Isogeometric analysis using T-splines on two-patch geometries. Comput. Methods Appl. Mech. Engrg. 2011, 200(21-22):1787-1803.
56. Jaswon M. Integral equation methods in potential theory - I. Proc. R. Soc. Lond. 1963, A275:23-32.
57. Symm G. Integral equation methods in potential theory - II. Proc. R. Soc. Lond. 1963, A275:33-46.
58. Cruse T. Numerical solutions in three dimensional elastostatics. Int. J. Solids Struct. 1969, 5:1259-1274.
59. Lachat J., Watson J. Effective numerical treatment of boundary integral equations: a formulation for three dimensional elastostatics. Int. J. Numer. Methods Engrg. 1976, 10:991-1005.
60. Cervera E., Trevelyan J. Evolutionary structural optimisation based on boundary representation of NURBS: Part I: 2D algorithms. Comput. Struct. 2005, 23-24(83):1902-1916.
61. Cervera E., Trevelyan J. Evolutionary structural optimisation based on boundary representation of NURBS: Part II: 3D algorithms. Comput. Struct. 2005, 23-24(83):1917-1929.
62. C. Politis, A.I. Ginnis, P.D. Kaklis, K. Belibassakis, C. Feurer, An isogeometric BEM for exterior potential-flow problems in the plane, in: 2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling, SPM '09, 2009, pp. 349-354.
63. Simpson R., Bordas S., Trevelyan J., Rabczuk T. A two-dimensional isogeometric boundary element method for elastostatic analysis. Comput. Methods Appl. Mech. Engrg. 2012, 209-212:87-100.
64. Li K., Qian X. Isogeometric analysis and shape optimization via boundary integral. Comput. Aided Des. 2011, 43(11):1427-1437.
65. Arnold D., Wendland W.L. On the asymptotic convergence of collocation methods. Math. Comput. 1983, 41(164):349-381.
66. Saranen J., Wendland W.L. On the asymptotic convergence of collocation methods with spline functions of even degree. Math. Comput. 1985, 45(171):91-108.
67. Hsiao G., Prössdorf S. A generalization of the Arnold-Wendland lemma to a modified collocation method for boundary integral equations in R3. Math. Nachr. 1993, 163:133-144.
68. Costabel M., McLean W. Spline collocation for strongly elliptic equations on the torus. Numer. Math. 1992, 62(4):511-538.
69. Juhl P. A note on the convergence of the direct collocation boundary element method. J. Sound Vib. 1998, 212(4):703-719.
70. Hughes T.J.R. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis 2000, Dover Publications, Mineola, NY.
71. Buffa A., Cho D., Sangalli G. Linear independence of the T-spline blending functions associated with some particular T-meshes. Comput. Methods Appl. Mech. Engrg. 2010, 199(23-24):1437-1445.
72. Sederberg T.W., Zheng J., Song X. Knot intervals and multi-degree splines. Comput. Aided Geom. Des. 2003, 20:455-468.
73. Catmull E., Clark J. Recursively generated B-spline surfaces on arbitrary topological meshes. Comput. Aided Des. 1978, 10:350-355.
74. Doo D., Sabin M. Behaviour of recursive division surfaces near extraordinary points. Comput. Aided Des. 1978, 10(6):356-360.
75. C. Loop, Smooth subdivision surfaces based on triangles, Master's Thesis, University of Utah, 1987.
76. Sabin M. Recent progress in subdivision: a survey. Adv. Multiresolut. Geom. Model. 2005, 203-230.
77. Reif U. A unified approach to subdivision algorithms near extraordinary vertices. Comput. Aided Geom. Des. 1995, 12(2):153-174.
78. Reif U., Schröder P. Curvature integrability of subdivision surfaces. Adv. Comput. Math. 2001, 14(2):157-174.
79. Stam J. Exact evaluation of Catmull-Clark subdivision surfaces at arbitrary parameter values. Proceedings of the 25th Annual Conference on Computer Graphics and Interactive Techniques 1998, 395-404. ACM.
80. Cashman T.J., Augsdörfer U.H., Dodgson N.A., Sabin M.A. NURBS with extraordinary points: high-degree, non-uniform, rational subdivision schemes. ACM Trans. Graph. 2009, 28:46:1-46:9.
81. T.W. Sederberg, J. Zheng, D. Sewell, M. Sabin, Non-uniform recursive subdivision surfaces, in: Proceedings of the 25th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH '98, 1998, pp. 387-394.
82. Cirak F., Ortiz M., Schröder P. Subdivision surfaces: a new paradigm for thin shell analysis. Int. J. Numer. Methods Engrg. 2000, 47:2039-2072.
83. C1-conforming subdivision elements for finite deformation thin shell analysis. Int. J. Numer. Methods Engrg. 2001, 51:813-833.
84. Cirak F., Scott M.J., Antonsson E.K., Ortiz M., Schröder P. Integrated modeling, finite-element analysis, and engineering design for thin-shell structures using subdivision. Comput. Aided Des. 2002, 34:137-148.
85. Cirak F., Ortiz M., Pandolfi A. A cohesive approach to thin-shell fracture and fragmentation. Comput. Methods Appl. Mech. Engrg. 2005, 194(21-24):2604-2618.
86. Burkhart D., Hamann B., Umlauf G. Isogeometric finite element analysis based on Catmull-Clark subdivision solids. Computer Graphics Forum 2010, vol. 29:1575-1584. Wiley Online Library.
87. Boehm W. Visual continuity. Comput. Aided Des. 1988, 20(6):307-311.
88. Degen W.L.F. Explicit continuity conditions for adjacent Bézier surface patches. Comput. Aided Geom. Des. 1990, 7(1-4):181-189.
89. GC1 continuity conditions between two adjacent rational Bézier surface patches. Comput. Aided Geom. Des. 1990, 7(1-4):151-163.
90. Zheng J., Wang G., Liang Y. Curvature continuity between adjacent rational Bézier patches. Comput. Aided Geom. Des. 1992, 9(5):321-335.
91. Peters J. Patching Catmull-Clark meshes. Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques 2000, 255-258. ACM Press/Addison-Wesley Publishing Co.
92. G2 tensor product splines over extraordinary vertices. Computer Graphics Forum 2008, vol. 27:1373-1382. Blackwell Science Ltd.
93. G1 smooth B-spline surfaces. Comput. Aided Geom. Des. 2004, 21(9):893-913.
94. Loop C. Second order smoothness over extraordinary vertices. Proceedings of the 2004 Eurographics/ACM SIGGRAPH Symposium on Geometry Processing 2004, 165-174. ACM.
95. Loop C., Schaefer S. Approximating Catmull-Clark subdivision surfaces with bicubic patches. ACM Trans. Graph. 2008, 27:1-11.
96. Å. Björck, Numerical Methods for Least Squares Problems, Society for Industrial Mathematics, 1996.
97. Auricchio F., Da Veiga L.B., Hughes T.J.R., Reali A., Sangalli G. Isogeometric collocation methods. Math. Models Methods Appl. Sci. 2010, 20(11):2075-2107.
98. F. Auricchio, L. Beirao de Veiga, T.J.R. Hughes, A. Reali, G. Sangalli, Isogeometric collocation for elastostatics and explicit dynamics, ICES, Report 12-07.
99. Telles J. A self-adaptive co-ordinate transformation for efficient numerical evaluation of general boundary element integrals. Int. J. Numer. Methods Engrg. 1987, 24(5):959-973.
100. Johnston P., Elliott D. A sinh transformation for evaluating nearly singular boundary element integrals. Int. J. Numer. Methods Engrg. 2005, 62(4):564-578.
101. Huang Q., Cruse T. Some notes on singular integral techniques in boundary element analysis. Int. J. Numer. Methods Engrg. 1993, 36(15):2643-2659.
102. Cruse T., Aithal R. Non-singular boundary integral equation implementation. Int. J. Numer. Methods Engrg. 1993, 36(2):237-254.
103. Cruse T. An improved boundary-integral equation method for three dimensional elastic stress analysis. Comput. Struct. 1974, 4:741-754.
104. Liu Y. On the simple-solution method and non-singular nature of the BIE/BEM - a review and some new results. Engrg. Anal. Bound. Elem. 2000, 24:789-795.
105. Becker A. The Boundary Element Method in Engineering 1992, McGraw-Hill Book Company.
106. Aliabadi M. The Boundary Element Method: Applications in Solids and Structures 2002, John Wiley & Sons.
107. T-Splines, Inc., 2011. http://www.tsplines.com/rhino/.
108. G.P. Bazeley, Y.K. Cheung, B.M. Irons, O.C. Zienkiewicz, Triangular elements in plate bendingconforming and nonconforming solutions, in: Proceedings of the First Conference on Matrix Methods in Structural Mechanics, 1966, pp. 547-576.
109. Piegl L., Tiller W. The NURBS Book 1997, Springer-Verlag, New York.
110. Takacs T., Jüttler B. Existence of stiffness matrix integrals for singularly parameterized domains in isogeometric analysis. Comput. Methods Appl. Mech. Engrg. 2011, 200:3568-3582.
111. Hughes T.J.R., Reali A., Sangalli G. Efficient quadrature for NURBS-based isogeometric analysis. Comput. Methods Appl. Mech. Engrg. 2010, 199(5-8):301-313.
112. Mi Y., Aliabadi M. Dual boundary element method for three-dimensional fracture mechanics analysis. Engrg. Anal. Bound. Elem. 1992, 10(2):161-171.
113. Rigby R., Aliabadi M. Out-of-core solver for large, multi-zone boundary element matrices. Int. J. Numer. Methods Engrg. 1995, 38:1507-1533.
114. Subia S., Ingber M., Mitra A. A comparison of the semidiscontinuous element and multiple node with auxiliary boundary collocation approaches for the boundary element method. Engrg. Anal. Bound. Elem. 1995, 15(1):19-27.
115. Mantic V. A new formula for the C-matrix in the Somigliana identity. J. Elast. 1993, 33(3):191-201.
116. Hartmann F. The Somigliana identity on piecewise smooth surfaces. J. Elast. 1981, 11(4):403-423.