In the context of isogeometric analysis, globally isogeometric spaces over unstructured quadrilateral meshes allow the direct solution of fourth order partial differential equations on complex geometries via their Galerkin discretization. The design of such smooth spaces has been intensively studied in the last five years, in particular for the case of planar domains, and is still task of current research. In this paper, we first give a short survey of the developed methods and especially focus on the approach [28]. There, the construction of a specific isogeometric spline space for the class of so-called analysis-suitable multi-patch parametrizations is presented. This particular class of parameterizations comprises exactly those multi-patch geometries, which ensure the design of spaces with optimal approximation properties, and allows the representation of complex planar multi-patch domains. We present known results in a coherent framework, and also extend the construction to parametrizations that are not analysis-suitable by allowing higher-degree splines in the neighborhood of the extraordinary vertices and edges. Finally, we present numerical tests that illustrate the behavior of the proposed method on representative examples.
Mots-clés : Isogeometric Analysis, $C^{1}$ isogeometric functions, geometric continuity, extraordinary vertices, planar multi-patch domain
@article{SMAI-JCM_2019__S5__67_0, author = {Kapl, Mario and Sangalli, Giancarlo and Takacs, Thomas}, title = {Isogeometric analysis with $C^1$ functions on planar, unstructured quadrilateral meshes}, journal = {The SMAI Journal of computational mathematics}, pages = {67--86}, publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles}, volume = {S5}, year = {2019}, doi = {10.5802/smai-jcm.52}, language = {en}, url = {http://www.numdam.org/articles/10.5802/smai-jcm.52/} }
TY - JOUR AU - Kapl, Mario AU - Sangalli, Giancarlo AU - Takacs, Thomas TI - Isogeometric analysis with $C^1$ functions on planar, unstructured quadrilateral meshes JO - The SMAI Journal of computational mathematics PY - 2019 SP - 67 EP - 86 VL - S5 PB - Société de Mathématiques Appliquées et Industrielles UR - http://www.numdam.org/articles/10.5802/smai-jcm.52/ DO - 10.5802/smai-jcm.52 LA - en ID - SMAI-JCM_2019__S5__67_0 ER -
%0 Journal Article %A Kapl, Mario %A Sangalli, Giancarlo %A Takacs, Thomas %T Isogeometric analysis with $C^1$ functions on planar, unstructured quadrilateral meshes %J The SMAI Journal of computational mathematics %D 2019 %P 67-86 %V S5 %I Société de Mathématiques Appliquées et Industrielles %U http://www.numdam.org/articles/10.5802/smai-jcm.52/ %R 10.5802/smai-jcm.52 %G en %F SMAI-JCM_2019__S5__67_0
Kapl, Mario; Sangalli, Giancarlo; Takacs, Thomas. Isogeometric analysis with $C^1$ functions on planar, unstructured quadrilateral meshes. The SMAI Journal of computational mathematics, Volume S5 (2019), pp. 67-86. doi : 10.5802/smai-jcm.52. http://www.numdam.org/articles/10.5802/smai-jcm.52/
[1] Domain Decomposition Methods and Kirchhoff-Love Shell Multipatch Coupling in Isogeometric Analysis, Isogeometric Analysis and Applications 2014 (Jüttler, B.; Simeon, B., eds.), Springer, 2015, pp. 73-101 | DOI | Zbl
[2] The TUBA family of plate elements for the matrix displacement method, The Aeronautical Journal, Volume 72 (1968) no. 692, pp. 701-709 | DOI
[3] A fully “locking-free” isogeometric approach for plane linear elasticity problems: a stream function formulation, Comput. Methods Appl. Mech. Engrg., Volume 197 (2007) no. 1, pp. 160-172 | DOI | MR | Zbl
[4] Isogeometric Analysis of high order Partial Differential Equations on surfaces, Comput. Methods Appl. Mech. Engrg., Volume 295 (2015), pp. 446-469 | DOI | MR | Zbl
[5] Mathematical analysis of variational isogeometric methods, Acta Numer., Volume 23 (2014), pp. 157-287 | DOI | MR
[6] A large deformation, rotation-free, isogeometric shell, Comput. Methods Appl. Mech. Engrg., Volume 200 (2011) no. 13, pp. 1367-1378 | DOI | MR | Zbl
[7] Isogeometric Analysis for -continuous Mortar Method, Corso di Dottorato in Matematica e Statistica, Università degli Studi di Pavia (2017) (Ph. D. Thesis)
[8] Smooth Bézier Surfaces over Unstructured Quadrilateral Meshes, Lecture Notes of the Unione Matematica Italiana, Springer, 2017, xx+192 pages | DOI | Zbl
[9] G-smooth splines on quad meshes with 4-split macro-patch elements, Comput. Aided Geom. Des., Volume 52–-53 (2017), pp. 106-125 | DOI | Zbl
[10] Adaptively refined multi-patch B-splines with enhanced smoothness, Appl. Math. Comput., Volume 272 (2016), pp. 159-172 | MR | Zbl
[11] Iso-geometric analysis based on Catmull-Clark solid subdivision, Computer Graphics Forum, Volume 29 (2010) no. 5, pp. 1575-1784 | DOI
[12] Isogeometric analysis with strong multipatch C-coupling, Comput. Aided Geom. Des., Volume 62 (2018), pp. 294-310 | DOI | MR | Zbl
[13] Subdivision surfaces: a new paradigm for thin-shell finite-element analysis, Int. J. Numer. Meth. Engng., Volume 47 (2000) no. 12, pp. 2039-2072 | DOI | Zbl
[14] Integrated modeling, finite-element analysis, and engineering design for thin-shell structures using subdivision, Computer-Aided Design, Volume 34 (2002) no. 2, pp. 137-148 | DOI
[15] Analysis-suitable G multi-patch parametrizations for C isogeometric spaces, Comput. Aided Geom. Des., Volume 47 (2016), pp. 93-113 | DOI | MR | Zbl
[16] Isogeometric Analysis: Toward Integration of CAD and FEA, John Wiley & Sons, 2009 | Zbl
[17] Isogeometric analysis of 2D gradient elasticity, Comput. Mech., Volume 47 (2011) no. 3, pp. 325-334 | DOI | MR | Zbl
[18] Isogeometric analysis of the Cahn–Hilliard phase-field model, Comput. Methods Appl. Mech. Engrg., Volume 197 (2008) no. 49, pp. 4333-4352 | DOI | MR
[19] Isogeometric Analysis of Phase–Field Models: Application to the Cahn–Hilliard Equation, ECCOMAS Multidisciplinary Jubilee Symposium: New Computational Challenges in Materials, Structures, and Fluids, Springer, 2009, pp. 1-16
[20] Geometric continuity and convex combination patches, Comput. Aided Geom. Des., Volume 4 (1987) no. 1-2, pp. 79-89 | DOI | MR | Zbl
[21] Matched G-constructions always yield C-continuous isogeometric elements, Comput. Aided Geom. Des., Volume 34 (2015), pp. 67-72 | DOI | MR | Zbl
[22] Nitsche’s method for a coupling of isogeometric thin shells and blended shell structures, Comput. Methods Appl. Mech. Engrg., Volume 284 (2015), pp. 881-905 | DOI | MR
[23] Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Engrg., Volume 194 (2005) no. 39-41, pp. 4135-4195 | DOI | MR | Zbl
[24] On numerical integration in isogeometric subdivision methods for PDEs on surfaces, Comput. Methods Appl. Mech. Engrg., Volume 302 (2016), pp. 131-146 | DOI | MR | Zbl
[25] Isogeometric analysis with geometrically continuous functions on planar multi-patch geometries, Comput. Methods Appl. Mech. Engrg., Volume 316 (2017), pp. 209-234 | DOI | MR
[26] Dimension and basis construction for analysis-suitable G two-patch parameterizations, Comput. Aided Geom. Des., Volume 52–53 (2017), pp. 75-89 | DOI | Zbl
[27] Construction of analysis-suitable G planar multi-patch parameterizations, Computer-Aided Design, Volume 97 (2018), pp. 41-55 | DOI
[28] An isogeometric subspace on unstructured multi-patch planar domains, Comput. Aided Geom. Des., Volume 69 (2019), pp. 55-75 | DOI | MR | Zbl
[29] Isogeometric Analysis with Geometrically Continuous Functions on Two-Patch Geometries, Comput. Math. Appl., Volume 70 (2015) no. 7, pp. 1518-1538 | DOI | MR
[30] Generalizing bicubic splines for modeling and IGA with irregular layout, Computer-Aided Design, Volume 70 (2016), pp. 23-35 | DOI
[31] Smooth multi-sided blending of biquadratic splines, Computers & Graphics, Volume 46 (2015), pp. 172-185
[32] Refinable functions on free-form surfaces, Comput. Aided Geom. Des., Volume 54 (2017), pp. 61-73 | DOI | MR | Zbl
[33] Refinable bi-quartics for design and analysis, Computer-Aided Design (2018), pp. 204-214 | DOI
[34] The bending strip method for isogeometric analysis of Kirchhoff-Love shell structures comprised of multiple patches, Comput. Methods Appl. Mech. Engrg., Volume 199 (2010) no. 35, pp. 2403-2416 | DOI | MR
[35] Isogeometric shell analysis with Kirchhoff-Love elements, Comput. Methods Appl. Mech. Engrg., Volume 198 (2009) no. 49, pp. 3902-3914 | DOI | MR | Zbl
[36] Isogeometric Kirchhoff–-Love shell formulations for general hyperelastic materials, Comput. Methods Appl. Mech. Engrg., Volume 291 (2015), pp. 280-303 | DOI | MR | Zbl
[37] Spline functions on triangulations, Encyclopedia of Mathematics and Its Applications, 110, Cambridge University Press, 2007, xvi+592 pages | MR | Zbl
[38] Isogeometric analysis of the advective Cahn-–Hilliard equation: Spinodal decomposition under shear flow, J. Comput. Phys., Volume 242 (2013), pp. 321-350 | DOI | MR | Zbl
[39] Construction of surfaces by assembly of quadrilateral patches under arbitrary mesh topology, Hebrew University of Jerusalem (2001) (Ph. D. Thesis)
[40] Dimension and bases for geometrically continuous splines on surfaces of arbitrary topology, Comput. Aided Geom. Des., Volume 45 (2016), pp. 108-133 | DOI | MR | Zbl
[41] Pairs of bi-cubic surface constructions supporting polar connectivity, Comput. Aided Geom. Des., Volume 25 (2008) no. 8, pp. 621-630 | DOI | MR | Zbl
[42] splines covering polar configurations, Computer-Aided Design, Volume 43 (2011) no. 11, pp. 1322-1329 | DOI
[43] A Comparative Study of Several Classical, Discrete Differential and Isogeometric Methods for Solving Poisson’s Equation on the Disk, Axioms, Volume 3 (2014) no. 2, pp. 280-299 | DOI | Zbl
[44] finite elements on non-tensor-product 2d and 3d manifolds, Appl. Math. Comput., Volume 272 (2016), pp. 148-158 | MR | Zbl
[45] Refinable spline elements for irregular quad layout, Comput. Aided Geom. Des., Volume 43 (2016), pp. 123-130 | DOI | MR | Zbl
[46] Variational formulation and isogeometric analysis for fourth-order boundary value problems of gradient-elastic bar and plane strain/stress problems, Comput. Methods Appl. Mech. Engrg., Volume 308 (2016), pp. 182-211 | DOI | MR
[47] Smooth mesh interpolation with cubic patches, Computer-Aided Design, Volume 22 (1990) no. 2, pp. 109-120 | DOI | MR | Zbl
[48] Parametrizing singularly to enclose data points by a smooth parametric surface, Proceedings of graphics interface, Volume 91 (1991)
[49] Geometric continuity, Handbook of computer aided geometric design, North-Holland, 2002, pp. 193-227 | DOI
[50] Subdivision surfaces, Geometry and Computing, 3, Springer, 2008, xvi+204 pages | MR | Zbl
[51] Biquadratic G-spline surfaces, Comput. Aided Geom. Des., Volume 12 (1995) no. 2, pp. 193-205 | DOI | MR | Zbl
[52] A refinable space of smooth spline surfaces of arbitrary topological genus, J. Approx. Th., Volume 90 (1997) no. 2, pp. 174-199 | DOI | MR | Zbl
[53] Isogeometric shell analysis with NURBS compatible subdivision surfaces, Appl. Math. Comput., Volume 272 (2016), pp. 139-147 | MR | Zbl
[54] Isogeometric Analysis and error estimates for high order partial differential equations in fluid dynamics, Comput. Fluids, Volume 102 (2014), pp. 277-303 | DOI | MR | Zbl
[55] Multi-degree smooth polar splines: A framework for geometric modeling and isogeometric analysis, Comput. Methods Appl. Mech. Engrg., Volume 316 (2017), pp. 1005-1061 | DOI | MR
[56] Analysis-suitable spline spaces of arbitrary degree on unstructured quadrilateral meshes (2017) no. 16 (Technical report)
[57] Smooth cubic spline spaces on unstructured quadrilateral meshes with particular emphasis on extraordinary points: Geometric design and isogeometric analysis considerations, Comput. Methods Appl. Mech. Engrg., Volume 327 (2017), pp. 411-458 | DOI | MR
[58] Subdivision surfaces with isogeometric analysis adapted refinement weights, Computer-Aided Design, Volume 102 (2018), pp. 104-114 | DOI | MR
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