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Constructing analysis-suitable parameterization of computational domain from CAD boundary by variational harmonic method. (English) Zbl 1349.65079
J. Comput. Phys. 252, 275-289 (2013); corrigendum ibid. 274, 865 (2014).
Summary: In isogeometric analysis, parameterization of computational domain has great effects as mesh generation in finite element analysis. In this paper, based on the concept of harmonic mapping from the computational domain to parametric domain, a variational harmonic approach is proposed to construct analysis-suitable parameterization of computational domain from CAD boundary for 2D and 3D isogeometric applications. Different from the previous elliptic mesh generation method in finite element analysis, the proposed method focuses on isogeometric version, and converts the elliptic PDE into a nonlinear optimization problem, in which a regular term is integrated into the optimization formulation to achieve more uniform and orthogonal iso-parametric structure near convex (concave) parts of the boundary. Several examples are presented to show the efficiency of the proposed method in 2D and 3D isogeometric analysis.

##### MSC:
 65D17 Computer-aided design (modeling of curves and surfaces) 65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
ISOGAT
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##### References:
 [1] Aigner, M.; Heinrich, C.; Jüttler, B.; Pilgerstorfer, E.; Simeon, B.; Vuong, A.-V., Swept volume parametrization for isogeometric analysis, (Hancock, E.; Martin, R., The Mathematics of Surfaces, MoS XIII 2009, Lecture Notes in Computer Science, vol. 5654, (2009), Springer), 19-44 · Zbl 1253.65182 [2] Akkermana, I.; Bazilevsa, Y.; Keesb, C. E.; Farthing, M. W., Isogeometric analysis of free-surface flow, Journal of Computational Physics, 230, 4137-4152, (2011) · Zbl 1343.76040 [3] Auricchio, F.; da Veiga, L. B.; Buffa, A.; Lovadina, C.; Reali, A.; Sangalli, G., A fully “locking-free” isogeometric approach for plane linear elasticity problems: A stream function formulation, Computer Methods in Applied Mechanics and Engineering, 197, 160-172, (2007) · Zbl 1169.74643 [4] Bazilevs, Y.; Beirao de Veiga, L.; Cottrell, J. A.; Hughes, T. J.R.; Sangalli, G., Isogeometric analysis: approximation, stability and error estimates for refined meshes, Mathematical Models and Methods in Applied Sciences, 6, 1031-1090, (2006) · Zbl 1103.65113 [5] Bazilevs, Y.; Calo, V. M.; Hughes, T. J.R.; Zhang, Y., Isogeometric fluid structure interaction: theory, algorithms, and computations, Computational Mechanics, 43, 3-37, (2008) · Zbl 1169.74015 [6] Bazilevs, Y.; Calo, V. M.; Cottrell, J. A.; Evans, J.; Hughes, T. J.R.; Lipton, S.; Scott, M. A.; Sederberg, T. W., Isogeometric analysis using T-splines, Computer Methods in Applied Mechanics and Engineering, 199, 229-263, (2010) · Zbl 1227.74123 [7] Burkhart, D.; Hamann, B.; Umlauf, G., Iso-geometric analysis based on catmull-Clark subdivision solids, Computer Graphics Forum, 29, 1575-1584, (2010) [8] Cohen, E.; Martin, T.; Kirby, R. M.; Lyche, T.; Riesenfeld, R. F., Analysis-aware modeling: understanding quality considerations in modeling for isogeometric analysis, Computer Methods in Applied Mechanics and Engineering, 199, 334-356, (2010) · Zbl 1227.74109 [9] Cottrell, J. A.; Hughes, T. J.R.; Bazilevs, Y., Isogeometric analysis: toward integration of CAD and FEA, (2009), Wiley · Zbl 1378.65009 [10] Cottrell, J. A.; Hughes, T. J.R.; Reali, A., Studies of refinement and continuity in isogeometric analysis, Computer Methods in Applied Mechanics and Engineering, 196, 4160-4183, (2007) · Zbl 1173.74407 [11] Cottrell, J. A.; Reali, A.; Bazilevs, Y.; Hughes, T. J.R., Isogeometric analysis of structural vibrations, Computer Methods in Applied Mechanics and Engineering, 195, 5257-5296, (2006) · Zbl 1119.74024 [12] Crouseilles, N.; Ratnani, A.; Sonnendrücker, E., An isogeometric analysis approach for the study of the gyrokinetic quasi-neutrality equation, Journal of Computational Physics, 231, 373-393, (2012) · Zbl 1246.82101 [13] Dörfel, M.; Jüttler, B.; Simeon, B., Adaptive isogeometric analysis by local h-refinement with T-splines, Computer Methods in Applied Mechanics and Engineering, 199, 264-275, (2010) · Zbl 1227.74125 [14] Duvigneau, R., An introduction to isogeometric analysis with application to thermal conduction, (June 2009), INRIA Research Report RR-6957 [15] Escobar, J. M.; Cascón, J. M.; Rodríguez, E.; Montenegro, R., A new approach to solid modeling with trivariate T-spline based on mesh optimization, Computer Methods in Applied Mechanics and Engineering, 200, 3210-3222, (2011) · Zbl 1230.74223 [16] Farin, G.; Hansford, D., Discrete coons patches, Computer Aided Geometric Design, 16, 691-700, (1999) · Zbl 0997.65033 [17] Gill, P.; Murray, W.; Wright, M., Practical optimization, (1981), Springer [18] Gomez, H.; Calo, V. M.; Bazilevs, Y.; Hughes, T. J.R., Isogeometric analysis of the Cahn-Hilliard phase-field model, Computer Methods in Applied Mechanics and Engineering, 197, 4333-4352, (2008) · Zbl 1194.74524 [19] Hughes, T. J.R.; Cottrell, J. A.; Bazilevs, Y., Isogeometric analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement, Computer Methods in Applied Mechanics and Engineering, 194, 4135-4195, (2005) · Zbl 1151.74419 [20] Li, X.; Guo, X.; Wang, H.; He, Y.; Gu, X.; Qin, H., Harmonic volumetric mapping for solid modeling applications, (Proceedings of the 2007 ACM symposium on Solid and Physical Modeling, (2007)), 109-120 [21] Martin, T.; Cohen, E.; Kirby, R. M., Volumetric parameterization and trivariate B-spline Fitting using harmonic functions, Computer Aided Geometric Design, 26, 648-664, (2009) · Zbl 1205.65094 [22] T. Nguyen, B. Jüttler, Using approximate implicitization for domain parameterization in isogeometric analysis, in: International Conference on Curves and Surfaces, Avignon, France, 2010. [23] Nguyen-Thanh, N.; Nguyen-Xuan, H.; Bordasd, S. P.A.; Rabczuk, T., Isogeometric analysis using polynomial splines over hierarchical T-meshes for two-dimensional elastic solids, Computer Methods in Applied Mechanics and Engineering, 200, 1892-1908, (2011) · Zbl 1228.74091 [24] Nguyen-Thanh, N.; Kiendl, J.; Nguyen-Xuan, H.; Wüchner, R.; Bletzinger, K. U.; Bazilevs, Y.; Rabczuk, T., Rotation free isogeometric thin shell analysis using PHT-splines, Computer Methods in Applied Mechanics and Engineering, 200, 3410-3424, (2011) · Zbl 1230.74230 [25] Pettersen, K. F.; Skytt, V., Spline volume fairing, (7th International Conference on Curves and Surfaces, Avignon, France, June 24-30, 2010, Lecture Notes in Computer Science, vol. 6920, (2012)), 553-561 · Zbl 1352.65064 [26] Ratnani, A.; Sonnendrücker, E., An arbitrary high-order spline finite element solver for the time domain Maxwell equations, Journal of Scientific Computing, 51, 87-106, (2012) · Zbl 1247.78042 [27] Spekreijse, S. P., Elliptic grid generation based on Laplace equations and algebraic transformations, Journal of Computational Physics, 118, 38-61, (1995) · Zbl 0823.65120 [28] Wang, C. L.; Tang, K., Non-self-overlapping Hermite interpolation mapping: a practical solution for structured quadrilateral meshing, Computer-Aided Design, 37, 271-283, (2005) [29] Wang, C. L.; Tang, K., Non-self-overlapping structured grid generation on an n-sided surface, International Journal for Numerical Methods in Fluids, 46, 961-982, (2004) · Zbl 1062.65128 [30] Wang, C. L.; Tang, K., Algebraic grid generation on trimmed surface using non-self-overlapping coons patch mapping, International Journal for Numerical Methods in Engineering, 60, 1259-1286, (2004) · Zbl 1059.65113 [31] Wang, W.; Zhang, Y.; Liu, L.; Hughes, T. J.R., Trivariate solid T-spline construction from boundary triangulations with arbitrary genus topology, Computer-Aided Design, 45, 351-360, (2013) [32] Wu, N. J.; Tsay, T. K.; Yang, T. C.; Chang, H. Y., Orthogonal grid generation of an irregular region using a local polynomial collocation method, Journal of Computational Physics, 243, 58-73, (2013) · Zbl 1349.65658 [33] Xia, J.; He, Y.; Yin, X.; Han, S.; Gu, X., Direct product volume parameterization using harmonic fields, (Proceedings of IEEE International Conference on Shape Modeling and Applications, (2010)), 3-12 [34] Xia, J.; He, Y.; Han, S.; Fu, C. W.; Luo, F.; Gu, X., Parameterization of star shaped volumes using greenʼs functions, (Proceedings of Geometric Modeling and Processing 2010, vol. 6130, (2010)), 219-235 [35] Xu, G.; Mourrain, B.; Duvigneau, R.; Galligo, A., Optimal analysis-aware parameterization of computational domain in 3D isogeometric analysis, Computer-Aided Design, 45, 812-821, (2013) [36] Xu, G.; Mourrain, B.; Duvigneau, R.; Galligo, A., Parameterization of computational domain in isogeometric analysis: methods and comparison, Computer Methods in Applied Mechanics and Engineering, 200, 2021-2031, (2011) · Zbl 1228.65232 [37] Xu, G.; Mourrain, B.; Duvigneau, R.; Galligo, A., Analysis-suitable volume parameterization of multi-block computational domain in isogeometric applications, Computer-Aided Design, 45, 395-404, (2013) [38] Zhang, Y.; Wang, W.; Hughes, T. J.R., Solid T-spline construction from boundary representations for genus-zero geometry, Computer Methods in Applied Mechanics and Engineering, 201, 185-197, (2012) · Zbl 1348.65057 [39] Zhang, Y.; Wang, W.; Hughes, T. J.R., Conformal solid T-spline construction from boundary T-spline representations, Computational Mechanics, 51, 1051-1059, (2013) · Zbl 1367.65024 [40] Zhang, Y.; Jia, Y.; Wang, S. Y., An improved nearly-orthogonal structured mesh generation system with smoothness control functions, Journal of Computational Physics, 231, 5289-5305, (2012) · Zbl 1251.65166 [41] Zhang, Y.; Jia, Y.; Wang, S. Y., 2D nearly orthogonal mesh generation with controls on distortion function, Journal of Computational Physics, 218, 549-571, (2006) · Zbl 1107.65110
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