Abstract
Analysis-suitable T-splines are a topological-restricted subset of T-splines, which are optimized to meet the needs both for design and analysis (Li and Scott Models Methods Appl Sci 24:1141–1164, 2014; Li et al. Comput Aided Geom Design 29:63–76, 2012; Scott et al. Comput Methods Appl Mech Eng 213–216, 2012). The paper independently derives a class of bi-degree \((d_{1}, d_{2})\) T-splines for which no perpendicular T-junction extensions intersect, and provides a new proof for the linearly independence of the blending functions. We also prove that the sum of the basis functions is one for an analysis-suitable T-spline if the T-mesh is admissible based on a recursive relation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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. Eng. 199(5–8), 229–263 (2010)
Buffa, A., Cho, D., Sangalli, G.: Linear independence of the T-spline blending functions associated with some particular T-meshes. Comput. Methods Appl. Mech. Eng. 199(23–24), 1437–1445 (2010)
Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y.: Isogeometric Analysis: Toward Integration of CAD and FEA. Wiley, Chichester (2009)
Dokken, T., Lyche, T., Pettersen, K.F.: Polynomial splines over locally refined box-partitions. Comput. Aided Geom. Design 30, 331–356 (2013)
Finnigan, G.T.: Arbitrary degree T-splines. Master’s thesis, Brigham Young University (2008)
Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement. Comput. Methods Appl. Mech. Eng. 194, 4135–4195 (2005)
Ipson, H.: T-spline merging. Master’s thesis, Brigham Young University (2005)
Li, X., Scott, M.A.: Analysis-suitable T-splines: characterization, refinablility and approximation. Math. Models Methods Appl. Sci. 24, 1141–1164 (2014). doi:10.1142/s0218202513500796
Li, X., Zheng, J., Sederberg, T.W., Hughes, T.J.R., Scott, M.A.: On the linear independence of T-splines blending functions. Comput. Aided Geom. Design 29, 63–76 (2012)
Nguyen-Thanh, N., Nguyen-Xuan, H., Bordas, S.P.A., Rabczuk, T.: Isogeometric analysis using polynomial splines over hierarchical T-meshes for two-dimensional elastic solids. Comput. Methods Appl. Mech. Eng. 200, 1892–1908 (2011)
Scott, M.A., Li, X., Sederberg, T.W., Hughes, T.J.R.: Local refinement of analysis-suitable T-splines. Comput. Methods Appl. Mech. Eng. 213–216, 206–222 (2012)
Sederberg, T.W., Cardon, D.L., Finnigan, G.T., North, N.S., Zheng, J., Lyche, T.: T-spline simplification and local refinement. ACM Trans. Gr. 23(3), 276–283 (2004)
Sederberg, T.W., Finnigan, G.T., Li, X., Lin, H., Ipson, H.: Watertight trimmed NURBS. ACM Trans. Gr. 27(3), Article no. 79 (2008)
Sederberg, T.W., Zheng, J., Bakenov, A., Nasri, A.: T-splines and T-NURCCSs. ACM Trans. Gr. 22(3), 477–484 (2003)
Veiga, L.B., Buffa, A., Sangalli, D.C.G.: Analysis-suitable T-splines are dual-compatible. Comput. Methods Appl. Mech. Eng. 249–252, 42–51 (2012)
Veiga, L.B., Buffa, A., Sangalli, G., Vazquez, R.: Analysis-suitable T-splines of arbitrary degree: definition and properties. Math. Models Methods Appl. Sci. 23, 1979–2003 (2013)
Vuong, A.V., Giannelli, C., Jüttler, B., Simeon, B.: A hierarchical approach to adaptive local refinement in isogeometric analysis. Comput. Methods Appl. Mech. Eng. 200, 3554–3567 (2011)
Wang, A., Zhao, G.: The analysis of t-spline blending functions linear independence. Comput. Aided Design Appl. 8(5), 735–745 (2011)
Wang, A., Zhao, G.: An algorithm of determining t-spline classification. Expert Syst. Appl. 40(18), 7280–7284 (2013)
Wang, A., Zhao, G., Li, Y.D.: An influence-knot set based new local refinement algorithm for T-spline surfaces. Expert Syst. Appl. 41(8), 3915–3921 (2014)
Wang, A., Zhao, G., Li, Y.D.: Linear independence of the blending functions of T-splines without multiple knots. Expert Syst. Appl. 41(8), 3634–3639 (2014)
Wang, P., Xu, J., Deng, J., Chen, F.: Adaptiveisogeometricanalysis using rational pht-splines. Comput. Aided Design 43, 1438–1448 (2011)
Acknowledgments
This work was supported by the NSF of China (No.11031007, No.60903148), the Chinese Universities Scientific Fund, SRF for ROCS SE, the CAS Startup Scientific Research Foundation and NBRPC 2011CB302400.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhang, J., Li, X. On the Linear Independence and Partition of Unity of Arbitrary Degree Analysis-Suitable T-splines. Commun. Math. Stat. 3, 353–364 (2015). https://doi.org/10.1007/s40304-015-0064-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40304-015-0064-z