Approximation Properties for Blended B-Splines on Unstructured Quadrilateral Mesh

Abstract

In this paper, the approximation properties for blended B-splines on unstructured quadrilateral mesh are analyzed. It is demonstrated that the blended B-splines can generate bi-cubic polynomials when being applied on regular and irregular elements. Moreover, the construction of dual basis for standard B-splines is extended to blended B-splines and the boundedness of the corresponding linear functional is established. Finally, the blended B-splines exhibit the optimal convergence in both the parametric and physical domains.

References

[1] Bazilevs Y, Veiga L, Cottrell JA, Hughes TJ, Sangalli G. Isogeometric analysis: approximation, stability and error estimates for h-refined meshes. Math. Models Methods Appl. Sci., 2006, 16(07): 1031-1090

[2] Brenner SC. The Mathematical Theory of Finite Element Methods, 2008 New York Springer

[3] Burkhart, D., Hamann, B., Umlauf, G.: Iso-geometric finite element analysis based on Catmull–Clark subdivision solids. In: Computer Graphics Forum, vol. 29, pp. 1575–1584. Wiley Online Library (2010)

[4] Collin A, Sangalli G, Takacs T. Analysis-suitable $G^1$ multi-patch parametrizations for $C^1$ isogeometric spaces. Comput. Aided Geomet. Design, 2016, 47: 93-113

[5] Collin A, Sangalli G, Takacs T. Approximation properties of multi-patch $G^1$ isogeometric spaces. Comput. Aided Geom. Des, 2016, 47: 93-113

[6] Da Veiga LB, Cho D, Sangalli G. Anisotropic NURBS approximation in isogeometric analysis. Comput. Methods Appl. Mech. Eng., 2012, 209: 1-11

[7] Da Veiga LB, Buffa A, Sangalli G, Vázquez R. Mathematical analysis of variational isogeometric methods. Acta Numer., 2014, 23: 157-287

[8] Deng, F., Yang, T., Liu, J., Deng, J.: Isogeometric analysis on implicit domains: approximation, stability and error estimates. Commun. Math. Stat. 1–26 (2023)

[9] Deng J, Chen F, Li X, Hu C, Tong W, Yang Z, Feng Y. Polynomial splines over hierarchical T-meshes. Graph. Models, 2008, 74: 76-86

[10] Dokken T, Lyche T, Pettersen KF. Polynomial splines over locally refined box-partitions. Comput. Aided Geomet. Design, 2013, 30: 331-356

[11] Evans JA, Bazilevs Y, Babuška I, Hughes TJR. n-Widths, sup-infs, and optimality ratios for the k-version of the isogeometric finite element method. Comput. Methods Appl. Mech. Eng., 2009, 198(21–26): 1726-1741

[12] Giannelli C, Jüttler B, Speleers H. THB-splines: the truncated basis for hierarchical splines. Comput. Aided Geomet. Design, 2012, 29(7): 485-498

[13] Hughes TJ, Cottrell JA, Bazilevs Y. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Eng., 2005, 194(39–41): 4135-4195

[14] Kang H, Li X, Deng J, Chen F. Truncated hierarchical Loop subdivision surfaces and application in isogeometric analysis. Comput. Math. Appl., 2016, 72: 2041-2055

[15] Kapl M, Buchegger F, Bercovier M, Jüttler B. Isogeometric analysis with geometrically continuous functions on planar multi-patch geometries. Comput. Methods Appl. Mech. Eng., 2017, 316: 209-234

[16] Kapl M, Sangalli G, Takacs T. Construction of analysis-suitable $G^1$ planar multi-patch parameterizations. Comput. Aided Des., 2018, 97: 41-55

[17] Li X, Li X. AS++ T-splines: arbitrary degree, nestedness and approximation. Numer. Math., 2021, 148: 795-816

[18] Li X, Zheng J, Sederberg TW, Hughes TJR, Scott MA. On the linear independence of T-splines blending functions. Comput. Aided Geomet. Design, 2012, 29: 63-76

[19] Lipton S, Evans JA, Bazilevs Y, Elguedj T, Hughes TJR. Robustness of isogeometric structural discretizations under severe mesh distortion. Comput. Methods Appl. Mech. Eng., 2010, 199(5–8): 357-373

[20] Liu, J., Deng, F., Ma, H., Deng, J.: Isogeometric analysis using basis functions of splines spaces over hierarchical t-meshes with high level differences. Commun. Math. Stat. 1–28 (2023)

[21] Ma, S., Xu, J., Xu, G.: Fast isogeometric method for fluid–structure interaction simulation of heart valves with gift framework. Commun. Math. Stat. 1–25 (2023)

[22] Ma Y, Ma W. A subdivision scheme for unstructured quadrilateral meshes with improved convergence rate for isogeometric analysis. Graph. Models, 2019, 106 101043

[23] Majeed M, Cirak F. Isogeometric analysis using manifold-based smooth basis functions. Comput. Methods Appl. Mech. Eng., 2017, 316: 547-567

[24] Morganti S, Auricchio F, Benson D, Gambarin F, Hartmann S, Hughes T, Reali A. Patient-specific isogeometric structural analysis of aortic valve closure. Comput. Methods Appl. Mech. Eng., 2015, 284: 508-520

[25] Riffnaller-Schiefer A, Augsdorfer UH, Fellner DW. Isogeometric shell analysis with NURBS compatible subdivision surfaces. Appl. Math. Comput., 2016, 272: 139-147

[26] Schillinger D, Dede L, Scott MA, Evans JA, Borden MJ, Rank E, Hughes TJ. 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. Eng., 2012, 249: 116-150

[27] Schumaker L. Spline Functions: Basic Theory, 2007 Cambridge Cambridge University Press

[28] Scott MA, Simpson RN, Evans JA, Lipton S, Bordas SPA, Hughes TJR, Sederberg TW. Isogeometric boundary element analysis using unstructured T-splines. Comput. Methods Appl. Mech. Eng., 2013, 254: 197-221

[29] Sederberg TW, Cardon DL, Finnigan GT, North NS, Zheng J, Lyche T. T-spline simplification and local refinement. ACM Trans. Graphics, 2004, 23(3): 276-283

[30] Toshniwal D, Speleers H, Hughes TJ. Smooth cubic spline spaces on unstructured quadrilateral meshes with particular emphasis on extraordinary points: Geometric design and isogeometric analysis considerations. Comput. Methods Appl. Mech. Eng., 2017, 327: 411-458

[31] Warren J, Weimer H. Subdivision Methods for Geometric Design, 2002 Netherlands Morgan Kaufmann

[32] Wei X, Zhang YJ, Toshniwal D, Speleers H, Li X, Manni C, Evans JA, Hughes TJ. Blended B-spline construction on unstructured quadrilateral and hexahedral meshes with optimal convergence rates in isogeometric analysis. Comput. Methods Appl. Mech. Eng., 2018, 341: 609-639

[33] Wei X, Li X, Zhang YJ, Hughes TJ. Tuned hybrid nonuniform subdivision surfaces with optimal convergence rates. Int. J. Numer. Meth. Eng., 2021, 122(9): 2117-2144

[34] Zhang J, Li X. On the linear independence and partition of unity of arbitrary degree analysis-suitable T-splines. Commun. Math. Stat., 2015, 3(3): 353-364

[35] Zheng T, Li X. Isogeometric topology optimization based on deep learning. Commun. Math. Stat., 2022, 10(3): 543-564

RIGHTS & PERMISSIONS

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature

Just Accepted

This article has successfully passed peer review and final editorial review, and will soon enter typesetting, proofreading and other publishing processes. The currently displayed version is the accepted final manuscript. The officially published version will be updated with format, DOI and citation information upon launch. We recommend that you pay attention to subsequent journal notifications and preferentially cite the officially published version. Thank you for your support and cooperation.