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**T-spline finite element method for the analysis of shell structures.**
*(English)*
Zbl 1176.74198

Summary: A T-spline surface is a nonuniform rational B-spline (NURBS) surface with T-junctions, and is defined by a control grid called T-mesh. The T-mesh is similar to a NURBS control mesh except that in a T-mesh, a row or column of control points is allowed to terminate in the inner parametric space. This property of T-splines makes local refinement possible. In the present study, shell formulation based on the T-spline finite element method (FEM) is presented. Shell formulation based on NURBS or T-splines has fundamental limitations because rotational DOFs, which are necessary in the shell formulation, cannot be defined on control points. In this study, the simple mapping scheme, in which every control point is mapped into one geometric point on the surface, is employed to eliminate the limitations. Using this mapping scheme, T-spline FEM can be easily extended to the analysis of shells. The proposed shell formulation is verified through various benchmarking problems. This study is a part of the efforts by the authors for the integration of CAD-CAE processes.

### Keywords:

nonuniform rational B-splines (NURBS); T-splines; finite element method; isogeometric analysis; CAD; CAE### Software:

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\textit{T.-K. Uhm} and \textit{S.-K. Youn}, Int. J. Numer. Methods Eng. 80, No. 4, 507--536 (2009; Zbl 1176.74198)

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### References:

[1] | Moore, New 48 D.O.F. quadrilateral shell element with variable-order polynomial and rational B-spline geometries with rigid body modes, International Journal for Numerical Methods in Engineering 20 pp 2121– (1984) · Zbl 0565.73064 |

[2] | Shen, Multivariate spline element analysis for plate bending problems, Computers and Structures 40 pp 1343– (1991) |

[3] | Pengcheng, Bending analysis of plates and spherical shells by multivariable spline element method based on generalized variational principle, Computers and Structures 55 pp 151– (1995) · Zbl 0900.73750 |

[4] | Fan, Nine node spline element for free vibration analysis of general plates, Journal of Sound and Vibration 165 pp 85– (1993) · Zbl 0925.73794 |

[5] | Düster, The p-version of the finite element method for three-dimensional curved thin walled structures, International Journal for Numerical Methods in Engineering 52 pp 673– (2001) · Zbl 1058.74079 |

[6] | Hughes, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Computer Methods in Applied Mechanics and Engineering 194 pp 4135– (2005) · Zbl 1151.74419 |

[7] | Reali A. An isogeometric analysis approach for the study of structural vibrations. M.S. Thesis, Università degli Studi di Pavia, 2004. |

[8] | Reali, An isogeometric analysis approach for the study of structural vibrations, Journal of Earthquake Engineering 10 pp 1– (2006) |

[9] | Bazilevs, Isogeometric fluid-structure interaction analysis with applications to arterial blood flow, Computational Mechanics 38 pp 310– (2006) · Zbl 1161.74020 |

[10] | Cottrell, Isogeometric analysis of structural vibrations, Computer Methods in Applied Mechanics and Engineering 195 pp 5257– (2006) · Zbl 1119.74024 |

[11] | Cottrell, Studies of refinement and continuity in isogeometric structural analysis, Computer Methods in Applied Mechanics and Engineering 196 pp 4160– (2007) · Zbl 1173.74407 |

[12] | Zhang, Patient-specific vascular NURBS modeling for isogeometric analysis of blood flow, Computer Methods in Applied Mechanics and Engineering 196 pp 2943– (2007) · Zbl 1121.76076 |

[13] | Sederberg, T-splines and T-NURCCs, ACM Transactions on Graphics 22 pp 477– (2003) |

[14] | Sederberg, T-spline simplification and local refinement, ACM Transactions on Graphics 23 pp 276– (2004) |

[15] | Kim K-S. A study on T-spline finite element analysis with local refinement. M.S. Thesis, Kaist, 2007. |

[16] | Kim K-S, Seo Y-D, Youn S-K. Spline-based finite element analysis with T-spline local refinement. Proceedings of the International Conference on Computational Methods (ICCM 2007), Hiroshima, 2007; 188. |

[17] | Uhm T-K, Seo Y-D, Kim H-J, Youn S-K. T-spline finite element method with local refinement. Proceedings of the Ninth U.S. National Congress on Computational Mechanics (USNCCM IX), San Francisco, July 2007. |

[18] | Uhm, A locally refinable T-spline finite element method for CAD/CAE integration, Structural Engineering and Mechanics 30 pp 225– (2008) |

[19] | Uhm T-K, Youn S-K. The analysis of shell structures using T-spline finite element method. Proceedings of the Eighth World Congress on Computational Mechanics (WCCM 8), Venice, June 2008. |

[20] | Seo Y-D, Youn S-K. Integrated structural optimization with T-Spline finite element method (TSFEM). Proceedings of the Eighth World Congress on Computational Mechanics (WCCM 8), Venice, June 2008. |

[21] | Kim H-J, Seo Y-D, Uhm T-K, Youn S-K. T-spline finite element analysis of the trimmed surface. Proceedings of the Eighth World Congress on Computational Mechanics (WCCM 8), Venice, June 2008. |

[22] | Dörfel, Adaptive isogeometric analysis by local h-refinement with T-splines, Computer Methods in Applied Mechanics and Engineering (2008) · Zbl 1227.74125 |

[23] | Bazilevs Y, Calo VM, Cottrell JA, Evans JA, Hughes TJR, Lipton S, Scott MA, Sederberg TW. Isogeometric analysis using T-splines. ICES Report 08-26, The University of Texas at Austin, 2008. |

[24] | Cho, Development of geometrically exact new shell elements based on general curvilinear co-ordinates, International Journal for Numerical Methods in Engineering 56 pp 81– (2003) |

[25] | Roh, The application of geometrically exact shell elements to B-spline surfaces, Computer Methods in Applied Mechanics and Engineering 193 pp 2261– (2004) · Zbl 1067.74585 |

[26] | Roh, Integration of geometric design and mechanical analysis using B-spline functions on surface, International Journal for Numerical Methods in Engineering 62 pp 1927– (2005) · Zbl 1118.74354 |

[27] | Roh HY. Integrated approach for geometric modeling and non-linear finite element analysis of shells using B-spline surfaces. Ph.D. Thesis, Seoul National University, 2005. |

[28] | Inoue, A NURBS finite element method for product shape design, Journal of Engineering Design 16 pp 157– (2005) |

[29] | Cirak, Subdivision surfaces: a new paradigm for thin-shell finite-element analysis, International Journal for Numerical Methods in Engineering 47 pp 2039– (2000) · Zbl 0983.74063 |

[30] | Cirak, Integrated modeling, finite-element analysis, and engineering design for thin-shell structures using subdivision, Computer-Aided Design 34 pp 137– (2002) |

[31] | Cirak, Fully C1-conforming subdivision elements for finite deformation thin-shell analysis, International Journal for Numerical Methods in Engineering 51 pp 813– (2001) · Zbl 1039.74045 |

[32] | Piegl, The NURBS Book (Monographs in Visual Communication) (1997) |

[33] | Macneal, A proposed standard set of problems to test finite element accuracy, Finite Elements in Analysis and Design 1 pp 3– (1985) |

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