×

Weighted quadrature for hierarchical B-splines. (English) Zbl 1507.74471

Summary: We present weighted quadrature for hierarchical B-splines to address the fast formation of system matrices arising from adaptive isogeometric Galerkin methods with suitably graded hierarchical meshes. By exploiting a local tensor product structure, we extend the construction of weighted rules from the tensor product to the hierarchical spline setting. The proposed algorithm has a computational cost proportional to the number of degrees of freedom and advantageous properties with increasing spline degree. To illustrate the performance of the method and confirm the theoretical estimates, a selection of 2D and 3D numerical tests is provided.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
65D07 Numerical computation using splines
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74S22 Isogeometric methods applied to problems in solid mechanics

Software:

ISOGAT; Igatools
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Buffa, A.; Gantner, G.; Giannelli, C.; Praetorius, D.; Vázquez, R., Mathematical foundations of adaptive isogeometric analysis, Arch. Comput. Methods. Engrg. (2022), in press, arXiv:2107.02023
[2] Vuong, A.-V.; Giannelli, C.; Jüttler, B.; Simeon, B., A hierarchical approach to adaptive local refinement in isogeometric analysis, Comput. Methods Appl. Mech. Engrg., 200, 3554-3567 (2011) · Zbl 1239.65013
[3] Giannelli, C.; Jüttler, B.; Speleers, H., THB-splines: The truncated basis for hierarchical splines, Comput. Aided Geom. Des., 29, 485-498 (2012) · Zbl 1252.65030
[4] Giannelli, C.; Jüttler, B.; Kleiss, S.; Mantzaflaris, A.; Simeon, B.; Špeh, J., THB-splines: An effective mathematical technology for adaptive refinement in geometric design and isogeometric analysis, Comput. Methods Appl. Mech. Engrg., 299, 337-365 (2016) · Zbl 1425.65026
[5] Buffa, A.; Giannelli, C., Adaptive isogeometric methods with hierarchical splines: Error estimator and convergence, Math. Models Methods Appl. Sci., 26, 1-25 (2016) · Zbl 1336.65181
[6] Buffa, A.; Giannelli, C., Adaptive isogeometric methods with hierarchical splines: Optimality and convergence rates, Math. Models Methods Appl. Sci., 27, 2781-2802 (2017) · Zbl 1376.41004
[7] Gantner, G.; Haberlik, D.; Praetorius, D., Adaptive IGAFEM with optimal convergence rates: Hierarchical B-splines, Math. Models Methods Appl. Sci., 27, 2631-2674 (2017) · Zbl 1376.41006
[8] Hennig, P.; Ambati, M.; De Lorenzis, L.; Kästner, M., Projection and transfer operators in adaptive isogeometric analysis with hierarchical B-splines, Comput. Methods Appl. Mech. Engrg., 334, 313-336 (2018) · Zbl 1440.65209
[9] Carraturo, M.; Giannelli, C.; Reali, A.; Vázquez, R., Suitably graded THB-spline refinement and coarsening: towards an adaptive isogeometric analysis of additive manufacturing processes, Comput. Methods Appl. Mech. Engrg., 348, 660-679 (2019) · Zbl 1440.74379
[10] Kuru, G.; Verhoosel, C.; van der Zee, K.; van Brummelen, E., Goal-adaptive isogeometric analysis with hierarchical splines, Comput. Methods Appl. Mech. and Engrg., 270, 270-292 (2014) · Zbl 1296.65162
[11] Calabrò, F.; Sangalli, G.; Tani, M., Fast formation of isogeometric Galerkin matrices by weighted quadrature, Comput. Methods Appl. Mech. Engrg., 316, 606-622 (2017) · Zbl 1439.65012
[12] 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
[13] Pan, M.; Jüttler, B.; Mantzaflaris, A., Efficient matrix assembly in isogeometric analysis with hierarchical B-splines, J. Comput. Appl. Math., 390, Article 113278 pp. (2021) · Zbl 1458.65153
[14] Hirschler, T.; Antolin, P.; Buffa, A., Fast and multiscale formation of isogeometric matrices of microstructured geometric models, Comput. Mech., 69, 439-466 (2022) · Zbl 07492679
[15] Antolin, P.; Buffa, A.; Calabro, F.; Martinelli, M.; Sangalli, G., Efficient matrix computation for tensor-product isogeometric analysis: The use of sum factorization, Comput. Methods Appl. Mech. Engrg., 285, 817-828 (2015) · Zbl 1425.65143
[16] Bressan, A.; Takacs, S., Sum factorization techniques in isogeometric analysis, Comput. Methods Appl. Mech. Engrg., 352, 437-460 (2019) · Zbl 1441.65093
[17] Drzisga, D.; Keith, B.; Wohlmuth, B., The surrogate matrix methodology: Accelerating isogeometric analysis of waves, Comput. Methods Appl. Mech. Engrg., 372, Article 113322 pp. (2020) · Zbl 1506.65206
[18] Moutsanidis, G.; Li, W.; Bazilevs, Y., Reduced quadrature for FEM, IGA and meshfree methods, Comput. Methods Appl. Mech. Engrg., 373, Article 113521 pp. (2021) · Zbl 1506.65050
[19] 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
[20] Zou, Z.; Hughes, T.; Scott, M.; Sauer, R.; Savitha, E., Galerkin formulations of isogeometric shell analysis: Alleviating locking with Greville quadratures and higher-order elements, Comput. Methods Appl. Mech. Engrg., 380, Article 113757 pp. (2021) · Zbl 1506.74463
[21] Mantzaflaris, A.; Jüttler, B.; Khoromskij, B. N.; Langer, U., Low rank tensor methods in Galerkin-based isogeometric analysis, Comput. Methods Appl. Mech. Engrg., 316, 1062-1085 (2017) · Zbl 1439.65185
[22] Karatarakis, A.; Karakitsios, P.; Papadrakakis, M., GPU accelerated computation of the isogeometric analysis stiffness matrix, Comput. Methods Appl. Mech. Engrg., 269, 334-355 (2014) · Zbl 1296.65160
[23] Hughes, T. J.; Reali, A.; Sangalli, G., Efficient quadrature for NURBS-based isogeometric analysis, Comput. Methods Appl. Mech. Engrg., 199, 5-8, 301-313 (2010) · Zbl 1227.65029
[24] Auricchio, F.; Calabro, F.; Hughes, T. J.; Reali, A.; Sangalli, G., A simple algorithm for obtaining nearly optimal quadrature rules for NURBS-based isogeometric analysis, Comput. Methods Appl. Mech. Engrg., 249, 15-27 (2012) · Zbl 1348.65059
[25] Bartoň, M.; Calo, V. M., Gauss-Galerkin quadrature rules for quadratic and cubic spline spaces and their application to isogeometric analysis, Comput.-Aided Des., 82, 57-67 (2017)
[26] Bartoň, M.; Calo, V. M., Optimal quadrature rules for odd-degree spline spaces and their application to tensor-product-based isogeometric analysis, Comput. Methods Appl. Mech. Engrg., 305, 217-240 (2016) · Zbl 1425.65039
[27] Hiemstra, R. R.; Calabro, F.; Schillinger, D.; Hughes, T. J., Optimal and reduced quadrature rules for tensor product and hierarchically refined splines in isogeometric analysis, Comput. Methods Appl. Mech. Engrg., 316, 966-1004 (2017) · Zbl 1439.65170
[28] Schillinger, D.; Hossain, S. J.; Hughes, T. J., Reduced Bézier element quadrature rules for quadratic and cubic splines in isogeometric analysis, Comput. Methods Appl. Mech. Engrg., 277, 1-45 (2014) · Zbl 1425.65177
[29] Adam, C.; Hughes, T. J.; Bouabdallah, S.; Zarroug, M.; Maitournam, H., Selective and reduced numerical integrations for NURBS-based isogeometric analysis, Comput. Methods Appl. Mech. Engrg., 284, 732-761 (2015) · Zbl 1425.65138
[30] Hiemstra, R. R.; Sangalli, G.; Tani, M.; Calabro, F.; Hughes, T. J., Fast formation and assembly of finite element matrices with application to isogeometric linear elasticity, Comput. Methods Appl. Mech. Engrg., 355, 234-260 (2017) · Zbl 1441.74244
[31] Bracco, C.; Giannelli, C.; Vázquez, R., Refinement algorithms for adaptive isogeometric methods with hierarchical splines, Axioms, 7, 3, 43 (2018) · Zbl 1432.65015
[32] Marussig, B., Fast formation and assembly of isogeometric Galerkin matrices for trimmed patches, (Manni, C.; Speleers, H., Geometric Challenges in Isogeometric Analysis. Geometric Challenges in Isogeometric Analysis, Springer INdAM Series, Vol. 49 (2022)) · Zbl 1497.65230
[33] Dörfler, W., A convergent algorithm for Poisson’s equation, SIAM J. Numer. Anal., 33, 1106-1124 (1996) · Zbl 0854.65090
[34] Pauletti, M. S.; Martinelli, M.; Cavallini, N.; Antolin Sanchez, P., Igatools: an isogeometric analysis library, SIAM J. Sci. Comput., 37, 4, 465-496 (2015) · Zbl 1332.65196
[35] Sangalli, G.; Tani, M., Matrix-free weighted quadrature for a computationally efficient isogeometric k-method, Comput. Methods Appl. Mech. Engrg., 338, 117-133 (2018) · Zbl 1440.65230
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.