Isogeometric collocation on planar multi-patch domains. (English) Zbl 1441.65112
Summary: We present an isogeometric framework based on collocation to construct a \(C^2\)-smooth approximation of the solution of the Poisson’s equation over planar bilinearly parameterized multi-patch domains. The construction of the used globally \(C^2\)-smooth discretization space for the partial differential equation is simple and works uniformly for all possible multi-patch configurations. The basis of the \(C^2\)-smooth space can be described as the span of three different types of locally supported functions corresponding to the single patches, edges and vertices of the multi-patch domain. For the selection of the collocation points, which is important for the stability and convergence of the collocation problem, two different choices are numerically investigated. The first approach employs the tensor-product Greville abscissae as collocation points, and shows for the multi-patch case the same convergence behavior as for the one-patch case [F. Auricchio et al., Math. Models Methods Appl. Sci. 20, No. 11, 2075–2107 (2010; Zbl 1226.65091)], which is suboptimal in particular for odd spline degree. The second approach generalizes the concept of superconvergent points from the one-patch case (cf. [C. Anitescu et al., Comput. Methods Appl. Mech. Eng. 284, 1073–1097 (2015; Zbl 1425.65193); H. Gomez and L. de Lorenzis, “The variational collocation method”, ibid. 309, 152–181 (2016; doi:10.1016/j.cma.2016.06.003); M. Montardini et al., “Optimal-order isogeometric collocation at Galerkin superconvergent points”, ibid. 316, 741–757 (2017; doi:10.1016/j.cma.2016.09.043)]) to the multi-patch case. Again, these points possess better convergence properties than Greville abscissae in case of odd spline degree.
MSC:
| 65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |
| 65D17 | Computer-aided design (modeling of curves and surfaces) |
| 68U07 | Computer science aspects of computer-aided design |
Keywords:
isogeometric analysis; collocation; superconvergent points; second order continuity; multi-patch domain; Poisson’s equationReferences:
| [1] | Auricchio, F.; Beirão da Veiga, L.; Hughes, T. J.R.; Reali, A.; Sangalli, G., Isogeometric collocation methods, Math. Models Methods Appl. Sci., 20, 11, 2075-2107 (2010) · Zbl 1226.65091 |
| [2] | Anitescu, C.; Jia, Y.; Zhang, Y. J.; Rabczuk, T., An isogeometric collocation method using superconvergent points, Comput. Methods Appl. Mech. Engrg., 284, 1073-1097 (2015) · Zbl 1425.65193 |
| [3] | Gomez, H.; De Lorenzis, L., The variational collocation method, Comput. Methods Appl. Mech. Engrg., 309, 152-181 (2016) · Zbl 1439.74489 |
| [4] | Montardini, M.; Sangalli, G.; Tamellini, L., Optimal-order isogeometric collocation at Galerkin superconvergent points, Comput. Methods Appl. Mech. Engrg., 316, 741-757 (2017) · Zbl 1439.65187 |
| [5] | Beirão da Veiga, L.; Buffa, A.; Sangalli, G.; Vázquez, R., Mathematical analysis of variational isogeometric methods, Acta Numer., 23, 157-287 (2014), 5 · Zbl 1398.65287 |
| [6] | Cottrell, J. A.; Hughes, T. J.R.; Bazilevs, Y., Isogeometric Analysis: Toward Integration of CAD and FEA (2009), John Wiley & Sons: John Wiley & Sons Chichester, England · Zbl 1378.65009 |
| [7] | 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., 194, 39-41, 4135-4195 (2005) · Zbl 1151.74419 |
| [8] | Bercovier, M.; Matskewich, T., (Smooth Bézier Surfaces over Unstructured Quadrilateral Meshes. Smooth Bézier Surfaces over Unstructured Quadrilateral Meshes, Lecture Notes of the Unione Matematica Italiana (2017), Springer) · Zbl 1422.65006 |
| [9] | Blidia, A.; Mourrain, B.; Villamizar, N., G \({}^1\)-smooth splines on quad meshes with 4-split macro-patch elements, Comput. Aided Geom. Design, 52-53, 106-125 (2017) · Zbl 1366.65012 |
| [10] | Collin, A.; Sangalli, G.; Takacs, T., Analysis-suitable G \({}^1\) multi-patch parametrizations for \(C{}^1\) isogeometric spaces, Comput. Aided Geom. Design, 47, 93-113 (2016) · Zbl 1418.65017 |
| [11] | Kapl, M.; Buchegger, F.; Bercovier, M.; Jüttler, B., Isogeometric analysis with geometrically continuous functions on planar multi-patch geometries, Comput. Methods Appl. Mech. Engrg., 316, 209-234 (2017) · Zbl 1439.65178 |
| [12] | Kapl, M.; Sangalli, G.; Takacs, T., Dimension and basis construction for analysis-suitable G^1 two-patch parameterizations, Comput. Aided Geom. Design, 52-53, 75-89 (2017) · Zbl 1366.65106 |
| [13] | Kapl, M.; Sangalli, G.; Takacs, T., An isogeometric \(C^1\) subspace on unstructured multi-patch planar domains, Comput. Aided Geom. Design, 69, 55-75 (2019) · Zbl 1470.65016 |
| [14] | M. Kapl, G. Sangalli, T. Takacs, Isogeometric analysis with \(C^1\) functions on unstructured quadrilateral meshes. Technical Report 1812.09088, arXiv.org, 2018. · Zbl 1470.65016 |
| [15] | Kapl, M.; Vitrih, V.; Jüttler, B.; Birner, K., Isogeometric analysis with geometrically continuous functions on two-patch geometries, Comput. Math. Appl., 70, 7, 1518-1538 (2015) · Zbl 1443.65346 |
| [16] | Karčiauskas, K.; Nguyen, T.; Peters, J., Generalizing bicubic splines for modeling and IGA with irregular layout, Comput.-Aided Des., 70, 23-35 (2016) |
| [17] | Mourrain, B.; Vidunas, R.; Villamizar, N., Dimension and bases for geometrically continuous splines on surfaces of arbitrary topology, Comput. Aided Geom. Design, 45, 108-133 (2016) · Zbl 1418.41010 |
| [18] | Nguyen, T.; Peters, J., Refinable \(C^1\) spline elements for irregular quad layout, Comput. Aided Geom. Design, 43, 123-130 (2016) · Zbl 1418.65026 |
| [19] | D. Toshniwal, H. Speleers, T.J.R. Hughes, Analysis-suitable spline spaces of arbitrary degree on unstructured quadrilateral meshes. Technical Report 16, Institute for Computational Engineering and Sciences (ICES), 2017. · Zbl 1439.65017 |
| [20] | Toshniwal, D.; Speleers, H.; Hughes, T. J.R., 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., 327, 411-458 (2017) · Zbl 1439.65017 |
| [21] | Chan, C. L.; Anitescu, C.; Rabczuk, T., Isogeometric analysis with strong multipatch \(C{}^1\)-coupling, Comput. Aided Geom. Design, 62, 294-310 (2018) · Zbl 1505.65050 |
| [22] | Kapl, M.; Vitrih, V., Space of C^2-smooth geometrically continuous isogeometric functions on two-patch geometries, Comput. Math. Appl., 73, 1, 37-59 (2017) · Zbl 1368.65023 |
| [23] | Kapl, M.; Vitrih, V., Space of \(C{}^2\)-smooth geometrically continuous isogeometric functions on planar multi-patch geometries: Dimension and numerical experiments, Comput. Math. Appl., 73, 10, 2319-2338 (2017) · Zbl 1373.65013 |
| [24] | Kapl, M.; Vitrih, V., Dimension and basis construction for \(C^2\)-smooth isogeometric spline spaces over bilinear-like \(G^2\) two-patch parameterizations, J. Comput. Appl. Math., 335, 289-311 (2018) · Zbl 1386.65084 |
| [25] | Kapl, M.; Vitrih, V., Solving the triharmonic equation over multi-patch planar domains using isogeometric analysis, J. Comput. Appl. Math., 358, 385-404 (2019) · Zbl 1415.65042 |
| [26] | Toshniwal, D.; Speleers, H.; Hiemstra, R.; Hughes, T. J.R., Multi-degree smooth polar splines: A framework for geometric modeling and isogeometric analysis, Comput. Methods Appl. Mech. Engrg. (2016) |
| [27] | Hoschek, J.; Lasser, D., Fundamentals of Computer Aided Geometric Design (1993), A K Peters Ltd.: A K Peters Ltd. Wellesley, MA · Zbl 0788.68002 |
| [28] | Peters, J., Geometric continuity, (Handbook of Computer Aided Geometric Design (2002), North-Holland: North-Holland Amsterdam), 193-227 |
| [29] | Groisser, D.; Peters, J., Matched G \({}^k\)-constructions always yield \(C{}^k\)-continuous isogeometric elements, Comput. Aided Geom. Design, 34, 67-72 (2015) · Zbl 1375.65026 |
| [30] | Beirão da Veiga, L.; Lovadina, C.; Reali, A., Avoiding shear locking for the Timoshenko beam problem via isogeometric collocation methods, Comput. Methods Appl. Mech. Engrg., 241-244, 38-51 (2012) · Zbl 1353.74045 |
| [31] | Schillinger, D.; Evans, J. A.; Reali, A.; Scott, M. A.; Hughes, T. J.R., Isogeometric collocation: Cost comparison with Galerkin methods and extension to adaptive hierarchical NURBS discretizations, Comput. Methods Appl. Mech. Engrg., 267, 170-232 (2013) · Zbl 1286.65174 |
| [32] | Reali, A.; Hughes, T. J.R., An introduction to isogeometric collocation methods, (Isogeometric Methods for Numerical Simulation (2015), Springer), 173-204 · Zbl 1327.74144 |
| [33] | Marino, E.; Kiendl, J.; De Lorenzis, L., Explicit isogeometric collocation for the dynamics of three-dimensional beams undergoing finite motions, Comput. Methods Appl. Mech. Engrg., 343, 530-549 (2019) · Zbl 1440.74185 |
| [34] | Kiendl, J.; Marino, E.; De Lorenzis, L., Isogeometric collocation for the Reissner-Mindlin shell problem, Comput. Methods Appl. Mech. Engrg., 325, 645-665 (2017) · Zbl 1439.74433 |
| [35] | Evans, J. A.; Hiemstra, R. R.; Hughes, T. J.R.; Reali, A., Explicit higher-order accurate isogeometric collocation methods for structural dynamics, Comput. Methods Appl. Mech. Engrg., 338, 208-240 (2018) · Zbl 1440.74464 |
| [36] | Fahrendorf, F.; De Lorenzis, L.; Gomez, H., Reduced integration at superconvergent points in isogeometric analysis, Comput. Methods Appl. Mech. Engrg., 328, 390-410 (2018) · Zbl 1439.65163 |
| [37] | Auricchio, F.; Beirão da Veiga, L.; Hughes, T. J.R.; Reali, A.; Sangalli, G., Isogeometric collocation for elastostatics and explicit dynamics, Comput. Methods Appl. Mech. Engrg., 249-252, 2-14 (2012) · Zbl 1348.74305 |
| [38] | Jia, Y.; Anitescu, C.; Zhang, Y. J.; Rabczuk, T., An adaptive isogeometric analysis collocation method with a recovery-based error estimator, Comput. Methods Appl. Mech. Engrg., 345, 52-74 (2019) · Zbl 1440.65248 |
| [39] | Deng, J.; Chen, F.; Xin Li, X.; Hu, C.; Tong, W.; Yang, Z.; Feng, Y., Polynomial splines over hierarchical T-meshes, Graph. Models, 70, 4, 76-86 (2008) |
| [40] | De Lorenzis, L.; Evans, J. A.; Hughes, T. J.R.; Reali, A., Isogeometric collocation: Neumann boundary conditions and contact, Comput. Methods Appl. Mech. Engrg., 284, 21-54 (2015) · Zbl 1423.74947 |
| [41] | Schillinger, D.; Borden, M. J.; Stolarski, H. K., Isogeometric collocation for phase-field fracture models, Comput. Methods Appl. Mech. Engrg., 284, 583-610 (2015) · Zbl 1423.74848 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
