×

NURBS-SEM: a hybrid spectral element method on NURBS maps for the solution of elliptic PDEs on surfaces. (English) Zbl 1440.65253

Summary: Non Uniform Rational B-spline (NURBS) patches are a standard way to describe complex geometries in Computer Aided Design tools, and have gained a lot of popularity in recent years also for the approximation of partial differential equations, via the Isogeometric Analysis (IGA) paradigm. However, spectral accuracy in IGA is limited to relatively small NURBS patch degrees (roughly \(p \leq 8)\), since local condition numbers grow very rapidly for higher degrees. On the other hand, traditional Spectral Element Methods (SEM) guarantee spectral accuracy but often require complex and expensive meshing techniques, like transfinite mapping, that result anyway in inexact geometries. In this work we propose a hybrid NURBS-SEM approximation method that achieves spectral accuracy and maintains exact geometry representation by combining the advantages of IGA and SEM. As a prototypical problem on non trivial geometries, we consider the Laplace-Beltrami and Allen-Cahn equations on a surface. On these problems, we present a comparison of several instances of NURBS-SEM with the standard Galerkin and Collocation Isogeometric Analysis (IGA).

MSC:

65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
65D07 Numerical computation using splines

Software:

NumPy; PetIGA; SciPy
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Hughes, T.; Cottrell, J.; 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
[2] Cottrell, J.; Hughes, T.; Bazilevs, Y., Isogeometric Analysis: Toward Integration of CAD and FEA (2009), Wiley · Zbl 1378.65009
[3] Kiendl, J.; Bletzinger, K.-U.; Linhard, J.; Wüchner, R., Isogeometric shell analysis with Kirchhoff-Love elements, Comput. Methods Appl. Mech. Engrg., 198, 3902-3914 (2009) · Zbl 1231.74422
[4] Kiendl, J.; Hsu, M.-C.; Wu, M.; Reali, A., Isogeometric Kirchhoff-Love shell formulations for general hyperelastic materials, Comput. Methods Appl. Mech. Engrg., 291, 280-303 (2015) · Zbl 1423.74177
[5] Benson, D. J.; Bazilevs, Y.; Hsu, M. C.; Hughes, T. J.R., Isogeometric shell analysis: The Reissner-Mindlin shell, Comput. Methods Appl. Mech. Engrg., 199, 276-289 (2010) · Zbl 1227.74107
[6] Dornisch, W.; Klinkel, S.; Simeon, B., Isogeometric Reissner-Mindlin shell analysis with exactly calculated director vectors, Comput. Methods Appl. Mech. Engrg., 253, 491-504 (2013) · Zbl 1297.74070
[7] Dornisch, W.; Klinkel, S., Treatment of Reissner-Mindlin shells with kinks without the need for drilling rotation stabilization in an isogeometric framework, Comput. Methods Appl. Mech. Engrg., 276, 35-66 (2014) · Zbl 1423.74571
[8] Uhm, T.-K.; Youn, S.-K., T-spline finite element method for the analysis of shell structures, Internat. J. Numer. Methods Engrg., 80, 507-536 (2009) · Zbl 1176.74198
[9] Heltai, L.; Kiendl, J.; DeSimone, A.; Reali, A., A natural framework for isogeometric fluid-structure interaction based on BEM-shell coupling, Comput. Methods Appl. Mech. Engrg., 316, 522-546 (2017) · Zbl 1439.74108
[10] Heltai, L.; Arroyo, M.; DeSimone, A., Nonsingular isogeometric boundary element method for Stokes flows in 3D, Comput. Methods Appl. Mech. Engrg., 268, 514-539 (2014) · Zbl 1295.76022
[11] 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
[12] Bartezzaghi, A.; Dedè, L.; Quarteroni, A., Isogeometric analysis of geometric partial differential equations, Comput. Methods Appl. Mech. Engrg., 311, 625-647 (2016) · Zbl 1439.65145
[13] Dedè, L.; Quarteroni, A., Isogeometric analysis for second order partial differential equations on surfaces, Comput. Methods Appl. Mech. Engrg., 284, 807-834 (2015) · Zbl 1425.65156
[14] Piegl, L.; Tiller, W., The NURBS Book (1995), Springer-Verlag · Zbl 0828.68118
[15] Cottrell, J. A.; Reali, A.; Bazilevs, Y.; Hughes, T. J.R., Isogeometric analysis of structural vibrations, Comput. Methods Appl. Mech. Engrg., 195, 41-43, 5257-5296 (2006) · Zbl 1119.74024
[16] Beirão da Veiga, L.; Buffa, A.; Rivas, J.; Sangalli, G., Some estimates for h-p-k-refinement in isogeometric analysis, Numer. Math., 118, 2, 271-305 (2011) · Zbl 1222.41010
[17] Bazilevs, Y.; Beirão Da Veiga, L.; Cottrell, J. A.; Hughes, T. J.R.; Sangalli, G., Isogeometric analysis: approximation, stability and error estimates for h-refined meshes, Math. Models Methods Appl. Sci., 16, 07, 1031-1090 (2006) · Zbl 1103.65113
[18] Sevilla, R.; Fernández-Méndez, S.; Huerta, A., Nurbs-enhanced finite element method (nefem), Internat. J. Numer. Methods Engrg., 76, 1, 56-83 (2008) · Zbl 1162.65389
[19] Lee, J. M., Riemannian Manifolds (1997), Springer-Verlag · Zbl 0905.53001
[20] Klingenberg, W., A Course in Differential Geometry (1978), Springer-Verlag · Zbl 0366.53001
[21] Lablée, O., Spectral Theory in Riemannian Geometry (2015), European Mathematical Society · Zbl 1328.53001
[22] Rosenberg, S., The Laplacian on a Riemannian Manifold (1997), London Mathematical Society · Zbl 0868.58074
[23] Cirak, F.; Ortiz, M.; Schröder, P., Subdivision surfaces: a new paradigm for thin-shell finite-element analysis, Internat. J. Numer. Methods Engrg., 47, 12, 2039-2072 (2000) · Zbl 0983.74063
[24] Manni, C.; Sablonnière, P., Quadratic spline quasi-interpolants on powell-sabin partitions, Adv. Comput. Math., 26, 1, 283-304 (2007) · Zbl 1116.65008
[25] Speleers, H.; Manni, C.; Pelosi, F.; Sampoli, M. L., Isogeometric analysis with powell-sabin splines for advection-diffusion-reaction problems, Comput. Methods Appl. Mech. Engrg., 221-222, 132-148 (2012) · Zbl 1253.65026
[26] Gottlieb, D.; Orszag, S. A., Numerical Analysis of Spectral Methods (1977), SIAM · Zbl 0412.65058
[27] Schillinger, D.; Evans, J. A.; Reali, A.; Scott, M. A.; Hughes, T. J., 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
[28] Auricchio, F.; ao da Veiga, L. B.; Hughes, T.; Reali, A.; Sangalli, G., Isogeometric collocation for elastostatics and explicit dynamics, Comput. Methods Appl. Mech. Engrg., 249-252, 2-14 (2012) · Zbl 1348.74305
[29] Canuto, C.; Hussaini, M.; Quarteroni, A.; Zang, T., Spectral Methods, Fundamentals in Single Domains (2006), Springer-Verlag · Zbl 1093.76002
[30] Canuto, C.; Hussaini, M.; Quarteroni, A.; Zang, T., Spectral Methods, Evolution to Complex Geometries and Applications to Fluid Dynamics (2007), Springer-Verlag · Zbl 1121.76001
[31] Deville, M.; Fischer, P.; Mund, E., High-Order Methods for Incompressible Fluid Flow (2002), Cambridge · Zbl 1007.76001
[32] Auricchio, F.; Da Veiga, L. B.; Hughes, T. J.R.; Reali, A.; Sangalli, G., Isogeometric Collocation Methods, Math. Models Methods Appl. Sci., 20, 11, 2075-2107 (2010) · Zbl 1226.65091
[33] Lorenzis, L. D.; Evans, J.; Hughes, T.; Reali, A., Isogeometric collocation: neumann boundary conditions and contact, Comput. Methods Appl. Mech. Engrg., 284, 21-54 (2015), Isogeometric Analysis Special Issue · Zbl 1423.74947
[34] Boyd, J. P. and, Chebyshev and Fourier Spectral Methods (2001), Dover · Zbl 0994.65128
[35] Rotundo, N.; Kim, T.-Y.; Jiang, W.; Heltai, L.; Fried, E., Error analysis of a b-spline based finite-element method for modeling wind-driven ocean circulation, J. Sci. Comput., 69, 1, 430-459 (2016) · Zbl 1398.76124
[36] Hughes, T.; 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
[37] Auricchio, F.; Calabrò, F.; Hughes, T.; Reali, A.; Sangalli, G., A simple algorithm for obtaining nearly optimal quadrature rules for NURBS-based isogeometric analysis, Comput. Methods Appl. Mech. Engrg., 249-252, 15-27 (2012) · Zbl 1348.65059
[38] 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
[39] Calabró, 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
[40] van der Walt, S.; Colbert, S. C.; Varoquaux, G., The numpy array: A structure for efficient numerical computation, Comput. Sci. Eng., 13, 2, 22-30 (2011)
[41] Dalcin, L.; Collier, N.; Vignal, P.; Cortes, A.; Calo, V., Petiga: A framework for high-performance isogeometric analysis, Comput. Methods Appl. Mech. Engrg., 308, 151-181 (2016) · Zbl 1439.65003
[42] Demko, S., On the existence of interpolating projections onto spline spaces, J. Approx. Theory, 43, 2, 151-156 (1985) · Zbl 0561.41029
[43] E. Jones, T. Oliphant, P. Peterson, et al. 2001-. SciPy: Open source scientific tools for Python. [Online].
[44] 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
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.