Isogeometric collocation for solving the biharmonic equation over planar multi-patch domains. (English) Zbl 1539.65190
Summary: We present an isogeometric collocation method for solving the biharmonic equation over planar bilinearly parameterized multi-patch domains. The developed approach is based on the use of the globally \(C^4\)-smooth isogeometric spline space (Kapl and Vitrih, 2021) to approximate the solution of the considered partial differential equation, and proposes as collocation points two different choices, namely on the one hand the Greville points and on the other hand the so-called superconvergent points. Several examples demonstrate the potential of our collocation method for solving the biharmonic equation over planar multi-patch domains, and numerically study the convergence behavior of the two types of collocation points with respect to the \(L^2\)-norm as well as to equivalents of the \(H^s\)-seminorms for \(1 \leq s \leq 4\).
In the studied case of spline degree \(p = 9\), the numerical results indicate in case of the Greville points a convergence of order \(\mathcal{O}(h^{p - 3})\) independent of the considered (semi)norm, and show in case of the superconvergent points an improved convergence of order \(\mathcal{O}(h^{p - 2})\) for all (semi)norms except for the equivalent of the \(H^4\)-seminorm, where the order \(\mathcal{O}(h^{p - 3})\) is anyway optimal.
In the studied case of spline degree \(p = 9\), the numerical results indicate in case of the Greville points a convergence of order \(\mathcal{O}(h^{p - 3})\) independent of the considered (semi)norm, and show in case of the superconvergent points an improved convergence of order \(\mathcal{O}(h^{p - 2})\) for all (semi)norms except for the equivalent of the \(H^4\)-seminorm, where the order \(\mathcal{O}(h^{p - 3})\) is anyway optimal.
MSC:
| 65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |
| 65D17 | Computer-aided design (modeling of curves and surfaces) |
Keywords:
isogeometric analysis; collocation; superconvergent points; fourth order continuity; multi-patch domain; biharmonic equationReferences:
| [1] | 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 |
| [2] | Hughes, T. J.R.; Cottrell, J. A., Isogeometric analysis: CAD, finite elements, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Engrg., 194, 39-41, 4135-4195, 2005 · Zbl 1151.74419 |
| [3] | Calabrò, F.; Sangalli, G.; Tani, M., Fast formation of isogeometric Galerkin matrices by weighted quadrature, Comput. Methods Appl. Mech. Engrg., 316, 606-622, 2017, Special Issue on Isogeometric Analysis: Progress and Challenges · Zbl 1439.65012 |
| [4] | Giust, A.; Jüttler, B., Weighted isogeometric collocation based on spline projectors, Comput. Methods Appl. Mech. Engrg., 391, Article 114554 pp., 2022 · Zbl 1507.65258 |
| [5] | Hughes, T. J.R.; Reali, A.; Sangalli, G., Efficient quadrature for NURBS-based isogeometric analysis, Comput. Methods Appl. Mech. Engrg., 199, 5, 301-313, 2010, Computational Geometry and Analysis · Zbl 1227.65029 |
| [6] | Jüttler, B.; Mantzaflaris, A.; Perl, R.; Rumpf, M., On numerical integration in isogeometric subdivision methods for PDEs on surfaces, Comput. Methods Appl. Mech. Engrg., 302, 131-146, 2016 · Zbl 1425.65033 |
| [7] | Mantzaflaris, A.; Jüttler, B., Integration by interpolation and look-up for Galerkin-based isogeometric analysis, Comput. Methods Appl. Mech. Engrg., 284, 373-400, 2015, Isogeometric Analysis Special Issue · Zbl 1425.65169 |
| [8] | Pan, M.; Jüttler, B.; Giust, A., Fast formation of isogeometric Galerkin matrices via integration by interpolation and look-up, Comput. Methods Appl. Mech. Engrg., 366, Article 113005 pp., 2020 · Zbl 1442.65392 |
| [9] | Peters, J., Geometric continuity, (Handbook of computer aided geometric design, 2002, North-Holland: North-Holland Amsterdam), 193-227 |
| [10] | 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 |
| [11] | 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 |
| [12] | Hughes, T. J.R.; Sangalli, G.; Takacs, T.; Toshniwal, D., Chapter 8 - smooth multi-patch discretizations in isogeometric analysis, (Bonito, Andrea; Nochetto, Ricardo H., Geometric Partial Differential Equations - Part II. Geometric Partial Differential Equations - Part II, volume 22 of Handbook of Numerical Analysis, 2021, Elsevier), 467-543 · Zbl 1473.65302 |
| [13] | Kapl, M.; Sangalli, G.; Takacs, T., Isogeometric analysis with C^1 functions on unstructured quadrilateral meshes, SMAI J. Comput. Math., 5, 67-86, 2019 · Zbl 1437.65194 |
| [14] | 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 |
| [15] | 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 |
| [16] | 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 |
| [17] | 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 |
| [18] | 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., 316, 1005-1061, 2017 · Zbl 1439.65016 |
| [19] | Kapl, M.; Vitrih, V., \( C^s\)-Smooth isogeometric spline spaces over planar multi-patch parameterizations, Adv. Comput. Math., 47, 47, 2021 · Zbl 1541.65009 |
| [20] | Lai, M.-J.; Schumaker, L. L., Spline functions on triangulations, (Volume 110 of Encyclopedia of Mathematics and Its Applications, 2007, Cambridge University Press: Cambridge University Press Cambridge) · Zbl 1185.41001 |
| [21] | Speleers, H., Construction of normalized B-splines for a family of smooth spline spaces over Powell-Sabin triangulations, Constr. Approx., 37, 1, 41-72, 2013 · Zbl 1264.41014 |
| [22] | Grošelj, J., A normalized representation of super splines of arbitrary degree on Powell-Sabin triangulations, BIT Numer. Math., 56, 4, 1257-1280, 2016 · Zbl 1365.41007 |
| [23] | 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 |
| [24] | 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 |
| [25] | Gomez, H.; De Lorenzis, L., The variational collocation method, Comput. Methods Appl. Mech. Engrg., 309, 152-181, 2016 · Zbl 1439.74489 |
| [26] | 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 |
| [27] | 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 |
| [28] | 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 |
| [29] | 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 |
| [30] | 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 |
| [31] | Fahrendorf, F.; Morganti, S.; Reali, A.; Hughes, T. J.R.; De Lorenzis, L., Mixed stress-displacement isogeometric collocation for nearly incompressible elasticity and elastoplasticity, Comput. Methods Appl. Mech. Engrg., 369, 113112, 48, 2020 · Zbl 1506.74060 |
| [32] | 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 |
| [33] | Reali, A.; Hughes, T. J.R., An introduction to isogeometric collocation methods, (Isogeometric Methods for Numerical Simulation, 2015, Springer), 173-204 · Zbl 1327.74144 |
| [34] | Morganti, S.; Fahrendorf, F.; De Lorenzis, L.; Evans, J. A.; Hughes, T. J.R.; Reali, A., Isogeometric collocation: A mixed displacement-pressure method for nearly incompressible elasticity, CMES Comput. Model. Eng. Sci., 129, 3, 1125-1150, 2021 |
| [35] | Ayala, T.; Videla, J.; Anitescu, C.; Atroshchenko, E., Enriched isogeometric collocation for two-dimensional time-harmonic acoustics, Comput. Methods Appl. Mech. Engrg., 365, 113033, 32, 2020 · Zbl 1442.74248 |
| [36] | Atroshchenko, E.; Calderon Hurtado, A.; Anitescu, C.; Khajah, T., Isogeometric collocation for acoustic problems with higher-order boundary conditions, Wave Motion, 110, 102861, 24, 2022 · Zbl 1524.76276 |
| [37] | Zampieri, E.; Pavarino, L. F., Isogeometric collocation discretizations for acoustic wave problems, Comput. Methods Appl. Mech. Engrg., 385, Article 114047 pp., 2021 · Zbl 1502.65080 |
| [38] | 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 |
| [39] | Reali, A.; Gomez, H., An isogeometric collocation approach for Bernoulli-Euler beams and Kirchhoff plates, Comput. Methods Appl. Mech. Engrg., 284, 623-636, 2015 · Zbl 1423.74553 |
| [40] | 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 |
| [41] | 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 |
| [42] | Casquero, H.; Liu, L.; Zhang, Y.; Reali, A.; Gomez, H., Isogeometric collocation using analysis-suitable T-splines of arbitrary degree, Comput. Methods Appl. Mech. Engrg., 301, 164-186, 2016 · Zbl 1425.65195 |
| [43] | Maurin, F.; Greco, F.; Coox, L.; Vandepitte, D.; Desmet, W., Isogeometric collocation for Kirchhoff-Love plates and shells, Comput. Methods Appl. Mech. Engrg., 329, 396-420, 2018 · Zbl 1439.74449 |
| [44] | Gomez, H.; Reali, A.; Sangalli. Accurate, G., Efficient, and (iso)geometrically flexible collocation methods for phase-field models, J. Comput. Phys., 262, 153-171, 2014 · Zbl 1349.82084 |
| [45] | 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 |
| [46] | Kapl, M.; Vitrih, V., Isogeometric collocation on planar multi-patch domains, Comput. Methods Appl. Mech. Engrg., 360, Article 112684 pp., 2020 · Zbl 1441.65112 |
| [47] | Bartezzaghi, A.; Dedè, L.; Quarteroni, A., Isogeometric analysis of high order partial differential equations on surfaces, Comput. Methods Appl. Mech. Engrg., 295, 446-469, 2015 · Zbl 1425.65145 |
| [48] | 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 |
| [49] | Kapl, M.; Sangalli, G.; Takacs, T., Construction of analysis-suitable G \({}^1\) planar multi-patch parameterizations, Comput.-Aided Des., 97, 41-55, 2018 |
| [50] | 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 |
| [51] | Bruaset, A. M., A survey of preconditioned iterative methods, (volume 328 of Pitman Research Notes in Mathematics Series, 1995, Longman Scientific & Technical: Longman Scientific & Technical Harlow) · Zbl 0834.65014 |
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