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Adaptive isogeometric methods with hierarchical splines: An overview
1. | Dipartimento di Matematica e Informatica 'U. Dini', Università degli Studi di Firenze, viale Morgagni 67a, 50134 Firenze, Italy |
2. | Institute of Mathematics, Ecole Polytechnique Fédérale de Lausanne, Station 8, 1015 Lausanne, Switzerland |
3. | Istituto di Matematica Applicata e Tecnologie Informatiche "E. Magenes" del CNR, via Ferrata 5, 27100 Pavia, Italy |
We consider an adaptive isogeometric method (AIGM) based on (truncated) hierarchical B-splines and present the study of its numerical properties. By following [
References:
[1] |
M. Actis, P. Morin and M. S. Pauletti, A new perspective on hierarchical spline spaces for adaptivity, arXiv: 1808.02053 |
[2] |
Y. Bazilevs, V. M. Calo, J. A. Cottrell, J. Evans, T. J. R. Hughes, S. Lipton, M. A. Scott and T. W. Sederberg,
Isogeometric analysis using T-Splines, Comput. Methods Appl. Mech. Engrg., 199 (2010), 229-263.
doi: 10.1016/j.cma.2009.02.036. |
[3] |
L. Beirão da Veiga, A. Buffa, D. Cho and G. Sangalli,
Analysis-Suitable T-splines are Dual-Compatible, Comput. Methods Appl. Mech. Engrg., 249/252 (2012), 42-51.
doi: 10.1016/j.cma.2012.02.025. |
[4] |
L. Beirão da Veiga, A. Buffa, G. Sangalli and R. Vázquez,
Analysis-suitable T-splines of arbitrary degree: Definition, linear independence and approximation properties, Math. Models Methods Appl. Sci., 23 (2013), 1979-2003.
doi: 10.1142/S0218202513500231. |
[5] |
A. Bonito and R. H. Nochetto,
Quasi-optimal convergence rate of an adaptive discontinuous Galerkin method, SIAM J. Numer. Anal., 48 (2010), 734-771.
doi: 10.1137/08072838X. |
[6] |
C. Bracco, C. Giannelli and R. Vázquez, Refinement algorithms for adaptive isogeometric methods with hierarchical splines, Axioms, 7 (2018), 43.
doi: 10.3390/axioms7030043. |
[7] |
A. Buffa and E. M. Garau,
Refinable spaces and local approximation estimates for hierarchical splines, IMA J. Numer. Anal., 37 (2017), 1125-1149.
doi: 10.1093/imanum/drw035. |
[8] |
A. Buffa and E. M. Garau,
A posteriori error estimators for hierarchical B-spline discretizations, Math. Models Methods Appl. Sci., 28 (2018), 1453-1480.
doi: 10.1142/S0218202518500392. |
[9] |
A. Buffa, E. M. Garau, C. Giannelli and G. Sangalli, On quasi-interpolation operators in spline spaces, in Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations (eds. G. R. Barrenechea et al.), vol. 114, Lecture Notes in Computational Science and Engineering, 2016, 73-91.
doi: 10.1007/978-3-319-41640-3_3. |
[10] |
A. Buffa and C. Giannelli,
Adaptive isogeometric methods with hierarchical splines: Error estimator and convergence, Math. Models Methods Appl. Sci., 26 (2016), 1-25.
doi: 10.1142/S0218202516500019. |
[11] |
A. Buffa and C. Giannelli,
Adaptive isogeometric methods with hierarchical splines: Optimality and convergence rates, Math. Models Methods Appl. Sci., 27 (2017), 2781-2802.
doi: 10.1142/S0218202517500580. |
[12] |
A. Buffa, C. Giannelli, P. Morgenstern and D. Peterseim,
Complexity of hierarchical refinement for a class of admissible mesh configurations, Comput. Aided Geom. Design, 47 (2016), 83-92.
doi: 10.1016/j.cagd.2016.04.003. |
[13] |
C. de Boor,
A Practical Guide to Splines, Springer, revised ed., 2001. |
[14] |
T. Dokken, T. Lyche and K. F. Pettersen,
Polynomial splines over locally refined box-partitions, Comput. Aided Geom. Design, 30 (2013), 331-356.
doi: 10.1016/j.cagd.2012.12.005. |
[15] |
M. R. Dörfel, B. Jüttler and B. Simeon,
Adaptive isogeometric analysis by local h-refinement with T-splines, Comput. Methods Appl. Mech. Engrg., 199 (2010), 264-275.
doi: 10.1016/j.cma.2008.07.012. |
[16] |
W. Dörfler,
A convergent adaptive algorithm for Poisson's equation, SIAM J. Numer. Anal., 33 (1996), 1106-1124.
doi: 10.1137/0733054. |
[17] |
E. J. Evans, M. A. Scott, X. Li and D. C. Thomas,
Hierarchical T-splines: Analysis-suitability, Bézier extraction, and application as an adaptive basis for isogeometric analysis, Comput. Methods Appl. Mech. Engrg., 284 (2015), 1-20.
doi: 10.1016/j.cma.2014.05.019. |
[18] |
M. Feischl, G. Gantner, A. Haberl and D. Praetorius,
Adaptive 2D IGA boundary element methods, Engineering Analysis with Boundary Elements, 62 (2016), 141-153.
doi: 10.1016/j.enganabound.2015.10.003. |
[19] |
M. Feischl, G. Gantner, A. Haberl and D. Praetorius,
Optimal convergence for adaptive IGA boundary element methods for weakly-singular integral equations, Numer. Math., 136 (2017), 147-182.
doi: 10.1007/s00211-016-0836-8. |
[20] |
G. Gantner, D. Haberlik and D. Praetorius,
Adaptive IGAFEM with optimal convergence rates: Hierarchical B-splines, Math. Models Methods Appl. Sci., 27 (2017), 2631-2674.
doi: 10.1142/S0218202517500543. |
[21] |
E. Garau and R. Vázquez,
Algorithms for the implementation of adaptive isogeometric methods using hierarchical B-splines, Appl. Numer. Math., 123 (2018), 58-87.
doi: 10.1016/j.apnum.2017.08.006. |
[22] |
C. Giannelli and B. Jüttler,
Bases and dimensions of bivariate hierarchical tensor-product splines, J. Comput. Appl. Math., 239 (2013), 162-178.
doi: 10.1016/j.cam.2012.09.031. |
[23] |
C. Giannelli, B. Jüttler, S. K. Kleiss, A. Mantzaflaris, B. Simeon and J. Špeh,
THB-splines: An effective mathematical technology for adaptive refinement in geometric design and isogeometric analysis, Comput. Methods Appl. Mech. Engrg., 299 (2016), 337-365.
doi: 10.1016/j.cma.2015.11.002. |
[24] |
C. Giannelli, B. Jüttler and H. Speleers,
THB-splines: The truncated basis for hierarchical splines, Comput. Aided Geom. Design, 29 (2012), 485-498.
doi: 10.1016/j.cagd.2012.03.025. |
[25] |
C. Giannelli, B. Jüttler and H. Speleers,
Strongly stable bases for adaptively refined multilevel spline spaces, Adv. Comput. Math., 40 (2014), 459-490.
doi: 10.1007/s10444-013-9315-2. |
[26] |
P. Hennig, M. Kästner, P. Morgenstern and D. Peterseim,
Adaptive mesh refinement strategies in isogeometric analysis— A computational comparison, Comput. Methods Appl. Mech. Engrg., 316 (2017), 424-448.
doi: 10.1016/j.cma.2016.07.029. |
[27] |
T. J. R. Hughes, J. A. Cottrell and Y. Bazilevs,
Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Engrg., 194 (2005), 4135-4195.
doi: 10.1016/j.cma.2004.10.008. |
[28] |
K. A. Johannessen, T. Kvamsdal and T. Dokken,
Isogeometric analysis using LR B-splines, Comput. Methods Appl. Mech. Engrg., 269 (2014), 471-514.
doi: 10.1016/j.cma.2013.09.014. |
[29] |
K. A. Johannessen, F. Remonato and T. Kvamsdal,
On the similarities and differences between Classical Hierarchical, Truncated Hierarchical and LR B-splines, Comput. Methods Appl. Mech. Engrg., 291 (2015), 64-101.
doi: 10.1016/j.cma.2015.02.031. |
[30] |
G. Kiss, C. Giannelli, U. Zore, B. Jüttler, D. Großmann and J. Barner,
Adaptive CAD model (re-)construction with THB-splines, Graphical models, 76 (2014), 273-288.
doi: 10.1016/j.gmod.2014.03.017. |
[31] |
R. Kraft, Adaptive and linearly independent multilevel B-splines, in Surface Fitting and Multiresolution Methods (eds. A. Le Méhauté, C. Rabut and L. L. Schumaker), Vanderbilt University Press, Nashville, 1997,209-218. |
[32] |
M. Kumar, T. Kvamsdal and K. A. Johannessen,
Simple a posteriori error estimators in adaptive isogeometric analysis, Comput. Math. Appl., 70 (2015), 1555-1582.
doi: 10.1016/j.camwa.2015.05.031. |
[33] |
M. Kumar, T. Kvamsdal and K. A. Johannessen,
Superconvergent patch recovery and a posteriori error estimation technique in adaptive isogeometric analysis, Comput. Methods Appl. Mech. Engrg., 316 (2017), 1086-1156.
doi: 10.1016/j.cma.2016.11.014. |
[34] |
G. Kuru, C. V. Verhoosel, K. G. van der Zeeb and E. H. van Brummelen,
Goal-adaptive isogeometric analysis with hierarchical splines, Comput. Methods Appl. Mech. Engrg., 270 (2014), 270-292.
doi: 10.1016/j.cma.2013.11.026. |
[35] |
X. Li, J. Zheng, T. W. Sederberg, T. J. R. Hughes and M. A. Scott,
On linear independence of T-spline blending functions, Comput. Aided Geom. Design, 29 (2012), 63-76.
doi: 10.1016/j.cagd.2011.08.005. |
[36] |
G. Lorenzo, M. A. Scott, K. Tew, T. J. R. Hughes and H. Gomez,
Hierarchically refined and coarsened splines for moving interface problems, with particular application to phase-field models of prostate tumor growth, Comput. Methods Appl. Mech. Engrg., 319 (2017), 515-548.
doi: 10.1016/j.cma.2017.03.009. |
[37] |
D. Mokriš, B. Jüttler and C. Giannelli,
On the completeness of hierarchical tensor-product B-splines, J. Comput. Appl. Math., 271 (2014), 53-70.
doi: 10.1016/j.cam.2014.04.001. |
[38] |
P. Morgenstern,
Globally structured three-dimensional analysis-suitable T-splines: definition, linear independence and m-graded local refinement, SIAM J. Numer. Anal., 54 (2016), 2163-2186.
doi: 10.1137/15M102229X. |
[39] |
P. Morgenstern,
Mesh Refinement Strategies for the Adaptive Isogeometric Method, PhD thesis, Institut für Numerische Simulation, Rheinische Friedrich-Wilhelms-Universität Bonn, 2017. |
[40] |
P. Morgenstern and D. Peterseim,
Analysis-suitable adaptive T-mesh refinement with linear complexity, Comput. Aided Geom. Design, 34 (2015), 50-66.
doi: 10.1016/j.cagd.2015.02.003. |
[41] |
P. Morin, R. H. Nochetto and M. S. Pauletti,
An adaptive method for hierarchical splines. A posteriori estimation via local problems, convergence and optimality, In preparation. |
[42] |
R. H. Nochetto, K. G. Siebert and A. Veeser, Theory of adaptive finite element methods: An introduction, in Multiscale, Nonlinear and Adaptive Approximation (eds. R. DeVore and A. Kunoth), Springer Berlin Heidelberg, 2009,409-542.
doi: 10.1007/978-3-642-03413-8_12. |
[43] |
R. H. Nochetto and A. Veeser, Primer of adaptive finite element methods, in Multiscale and Adaptivity: Modeling, Numerics and Applications, vol. 2040 of Lecture Notes in Math., Springer, Heidelberg, 2012,125-225.
doi: 10.1007/978-3-642-24079-9. |
[44] |
D. Schillinger, L. Dedé, M. Scott, J. Evans, M. Borden, E. Rank and T. Hughes,
An isogeometric design-through-analysis methodology based on adaptive hierarchical refinement of NURBS, immersed boundary methods, and T-spline CAD surfaces, Comput. Methods Appl. Mech. Engrg., 249/252 (2012), 116-150.
doi: 10.1016/j.cma.2012.03.017. |
[45] |
L. L. Schumaker,
Spline Functions: Basic Theory, 3rd Edition, Cambridge University Press, 2007.
doi: 10.1017/CBO9780511618994. |
[46] |
M. A. Scott, D. C. Thomas and E. J. Evans,
Isogeometric spline forests, Comput. Methods Appl. Mech. Engrg., 269 (2014), 222-264.
doi: 10.1016/j.cma.2013.10.024. |
[47] |
T. W. Sederberg, D. L. Cardon, G. T. Finnigan, N. S. North, J. Zheng and T. Lyche,
T-spline simplification and local refinement, ACM Trans. Graphics, 23 (2004), 276-283.
doi: 10.1145/1015706.1015715. |
[48] |
H. Speleers and C. Manni,
Effortless quasi-interpolation in hierarchical spaces, Numer. Math., 132 (2016), 155-184.
doi: 10.1007/s00211-015-0711-z. |
[49] |
A.-V. Vuong, C. Giannelli, B. Jüttler and B. Simeon,
A hierarchical approach to adaptive local refinement in isogeometric analysis, Comput. Methods Appl. Mech. Engrg., 200 (2011), 3554-3567.
doi: 10.1016/j.cma.2011.09.004. |
show all references
References:
[1] |
M. Actis, P. Morin and M. S. Pauletti, A new perspective on hierarchical spline spaces for adaptivity, arXiv: 1808.02053 |
[2] |
Y. Bazilevs, V. M. Calo, J. A. Cottrell, J. Evans, T. J. R. Hughes, S. Lipton, M. A. Scott and T. W. Sederberg,
Isogeometric analysis using T-Splines, Comput. Methods Appl. Mech. Engrg., 199 (2010), 229-263.
doi: 10.1016/j.cma.2009.02.036. |
[3] |
L. Beirão da Veiga, A. Buffa, D. Cho and G. Sangalli,
Analysis-Suitable T-splines are Dual-Compatible, Comput. Methods Appl. Mech. Engrg., 249/252 (2012), 42-51.
doi: 10.1016/j.cma.2012.02.025. |
[4] |
L. Beirão da Veiga, A. Buffa, G. Sangalli and R. Vázquez,
Analysis-suitable T-splines of arbitrary degree: Definition, linear independence and approximation properties, Math. Models Methods Appl. Sci., 23 (2013), 1979-2003.
doi: 10.1142/S0218202513500231. |
[5] |
A. Bonito and R. H. Nochetto,
Quasi-optimal convergence rate of an adaptive discontinuous Galerkin method, SIAM J. Numer. Anal., 48 (2010), 734-771.
doi: 10.1137/08072838X. |
[6] |
C. Bracco, C. Giannelli and R. Vázquez, Refinement algorithms for adaptive isogeometric methods with hierarchical splines, Axioms, 7 (2018), 43.
doi: 10.3390/axioms7030043. |
[7] |
A. Buffa and E. M. Garau,
Refinable spaces and local approximation estimates for hierarchical splines, IMA J. Numer. Anal., 37 (2017), 1125-1149.
doi: 10.1093/imanum/drw035. |
[8] |
A. Buffa and E. M. Garau,
A posteriori error estimators for hierarchical B-spline discretizations, Math. Models Methods Appl. Sci., 28 (2018), 1453-1480.
doi: 10.1142/S0218202518500392. |
[9] |
A. Buffa, E. M. Garau, C. Giannelli and G. Sangalli, On quasi-interpolation operators in spline spaces, in Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations (eds. G. R. Barrenechea et al.), vol. 114, Lecture Notes in Computational Science and Engineering, 2016, 73-91.
doi: 10.1007/978-3-319-41640-3_3. |
[10] |
A. Buffa and C. Giannelli,
Adaptive isogeometric methods with hierarchical splines: Error estimator and convergence, Math. Models Methods Appl. Sci., 26 (2016), 1-25.
doi: 10.1142/S0218202516500019. |
[11] |
A. Buffa and C. Giannelli,
Adaptive isogeometric methods with hierarchical splines: Optimality and convergence rates, Math. Models Methods Appl. Sci., 27 (2017), 2781-2802.
doi: 10.1142/S0218202517500580. |
[12] |
A. Buffa, C. Giannelli, P. Morgenstern and D. Peterseim,
Complexity of hierarchical refinement for a class of admissible mesh configurations, Comput. Aided Geom. Design, 47 (2016), 83-92.
doi: 10.1016/j.cagd.2016.04.003. |
[13] |
C. de Boor,
A Practical Guide to Splines, Springer, revised ed., 2001. |
[14] |
T. Dokken, T. Lyche and K. F. Pettersen,
Polynomial splines over locally refined box-partitions, Comput. Aided Geom. Design, 30 (2013), 331-356.
doi: 10.1016/j.cagd.2012.12.005. |
[15] |
M. R. Dörfel, B. Jüttler and B. Simeon,
Adaptive isogeometric analysis by local h-refinement with T-splines, Comput. Methods Appl. Mech. Engrg., 199 (2010), 264-275.
doi: 10.1016/j.cma.2008.07.012. |
[16] |
W. Dörfler,
A convergent adaptive algorithm for Poisson's equation, SIAM J. Numer. Anal., 33 (1996), 1106-1124.
doi: 10.1137/0733054. |
[17] |
E. J. Evans, M. A. Scott, X. Li and D. C. Thomas,
Hierarchical T-splines: Analysis-suitability, Bézier extraction, and application as an adaptive basis for isogeometric analysis, Comput. Methods Appl. Mech. Engrg., 284 (2015), 1-20.
doi: 10.1016/j.cma.2014.05.019. |
[18] |
M. Feischl, G. Gantner, A. Haberl and D. Praetorius,
Adaptive 2D IGA boundary element methods, Engineering Analysis with Boundary Elements, 62 (2016), 141-153.
doi: 10.1016/j.enganabound.2015.10.003. |
[19] |
M. Feischl, G. Gantner, A. Haberl and D. Praetorius,
Optimal convergence for adaptive IGA boundary element methods for weakly-singular integral equations, Numer. Math., 136 (2017), 147-182.
doi: 10.1007/s00211-016-0836-8. |
[20] |
G. Gantner, D. Haberlik and D. Praetorius,
Adaptive IGAFEM with optimal convergence rates: Hierarchical B-splines, Math. Models Methods Appl. Sci., 27 (2017), 2631-2674.
doi: 10.1142/S0218202517500543. |
[21] |
E. Garau and R. Vázquez,
Algorithms for the implementation of adaptive isogeometric methods using hierarchical B-splines, Appl. Numer. Math., 123 (2018), 58-87.
doi: 10.1016/j.apnum.2017.08.006. |
[22] |
C. Giannelli and B. Jüttler,
Bases and dimensions of bivariate hierarchical tensor-product splines, J. Comput. Appl. Math., 239 (2013), 162-178.
doi: 10.1016/j.cam.2012.09.031. |
[23] |
C. Giannelli, B. Jüttler, S. K. Kleiss, A. Mantzaflaris, B. Simeon and J. Špeh,
THB-splines: An effective mathematical technology for adaptive refinement in geometric design and isogeometric analysis, Comput. Methods Appl. Mech. Engrg., 299 (2016), 337-365.
doi: 10.1016/j.cma.2015.11.002. |
[24] |
C. Giannelli, B. Jüttler and H. Speleers,
THB-splines: The truncated basis for hierarchical splines, Comput. Aided Geom. Design, 29 (2012), 485-498.
doi: 10.1016/j.cagd.2012.03.025. |
[25] |
C. Giannelli, B. Jüttler and H. Speleers,
Strongly stable bases for adaptively refined multilevel spline spaces, Adv. Comput. Math., 40 (2014), 459-490.
doi: 10.1007/s10444-013-9315-2. |
[26] |
P. Hennig, M. Kästner, P. Morgenstern and D. Peterseim,
Adaptive mesh refinement strategies in isogeometric analysis— A computational comparison, Comput. Methods Appl. Mech. Engrg., 316 (2017), 424-448.
doi: 10.1016/j.cma.2016.07.029. |
[27] |
T. J. R. Hughes, J. A. Cottrell and Y. Bazilevs,
Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Engrg., 194 (2005), 4135-4195.
doi: 10.1016/j.cma.2004.10.008. |
[28] |
K. A. Johannessen, T. Kvamsdal and T. Dokken,
Isogeometric analysis using LR B-splines, Comput. Methods Appl. Mech. Engrg., 269 (2014), 471-514.
doi: 10.1016/j.cma.2013.09.014. |
[29] |
K. A. Johannessen, F. Remonato and T. Kvamsdal,
On the similarities and differences between Classical Hierarchical, Truncated Hierarchical and LR B-splines, Comput. Methods Appl. Mech. Engrg., 291 (2015), 64-101.
doi: 10.1016/j.cma.2015.02.031. |
[30] |
G. Kiss, C. Giannelli, U. Zore, B. Jüttler, D. Großmann and J. Barner,
Adaptive CAD model (re-)construction with THB-splines, Graphical models, 76 (2014), 273-288.
doi: 10.1016/j.gmod.2014.03.017. |
[31] |
R. Kraft, Adaptive and linearly independent multilevel B-splines, in Surface Fitting and Multiresolution Methods (eds. A. Le Méhauté, C. Rabut and L. L. Schumaker), Vanderbilt University Press, Nashville, 1997,209-218. |
[32] |
M. Kumar, T. Kvamsdal and K. A. Johannessen,
Simple a posteriori error estimators in adaptive isogeometric analysis, Comput. Math. Appl., 70 (2015), 1555-1582.
doi: 10.1016/j.camwa.2015.05.031. |
[33] |
M. Kumar, T. Kvamsdal and K. A. Johannessen,
Superconvergent patch recovery and a posteriori error estimation technique in adaptive isogeometric analysis, Comput. Methods Appl. Mech. Engrg., 316 (2017), 1086-1156.
doi: 10.1016/j.cma.2016.11.014. |
[34] |
G. Kuru, C. V. Verhoosel, K. G. van der Zeeb and E. H. van Brummelen,
Goal-adaptive isogeometric analysis with hierarchical splines, Comput. Methods Appl. Mech. Engrg., 270 (2014), 270-292.
doi: 10.1016/j.cma.2013.11.026. |
[35] |
X. Li, J. Zheng, T. W. Sederberg, T. J. R. Hughes and M. A. Scott,
On linear independence of T-spline blending functions, Comput. Aided Geom. Design, 29 (2012), 63-76.
doi: 10.1016/j.cagd.2011.08.005. |
[36] |
G. Lorenzo, M. A. Scott, K. Tew, T. J. R. Hughes and H. Gomez,
Hierarchically refined and coarsened splines for moving interface problems, with particular application to phase-field models of prostate tumor growth, Comput. Methods Appl. Mech. Engrg., 319 (2017), 515-548.
doi: 10.1016/j.cma.2017.03.009. |
[37] |
D. Mokriš, B. Jüttler and C. Giannelli,
On the completeness of hierarchical tensor-product B-splines, J. Comput. Appl. Math., 271 (2014), 53-70.
doi: 10.1016/j.cam.2014.04.001. |
[38] |
P. Morgenstern,
Globally structured three-dimensional analysis-suitable T-splines: definition, linear independence and m-graded local refinement, SIAM J. Numer. Anal., 54 (2016), 2163-2186.
doi: 10.1137/15M102229X. |
[39] |
P. Morgenstern,
Mesh Refinement Strategies for the Adaptive Isogeometric Method, PhD thesis, Institut für Numerische Simulation, Rheinische Friedrich-Wilhelms-Universität Bonn, 2017. |
[40] |
P. Morgenstern and D. Peterseim,
Analysis-suitable adaptive T-mesh refinement with linear complexity, Comput. Aided Geom. Design, 34 (2015), 50-66.
doi: 10.1016/j.cagd.2015.02.003. |
[41] |
P. Morin, R. H. Nochetto and M. S. Pauletti,
An adaptive method for hierarchical splines. A posteriori estimation via local problems, convergence and optimality, In preparation. |
[42] |
R. H. Nochetto, K. G. Siebert and A. Veeser, Theory of adaptive finite element methods: An introduction, in Multiscale, Nonlinear and Adaptive Approximation (eds. R. DeVore and A. Kunoth), Springer Berlin Heidelberg, 2009,409-542.
doi: 10.1007/978-3-642-03413-8_12. |
[43] |
R. H. Nochetto and A. Veeser, Primer of adaptive finite element methods, in Multiscale and Adaptivity: Modeling, Numerics and Applications, vol. 2040 of Lecture Notes in Math., Springer, Heidelberg, 2012,125-225.
doi: 10.1007/978-3-642-24079-9. |
[44] |
D. Schillinger, L. Dedé, M. Scott, J. Evans, M. Borden, E. Rank and T. Hughes,
An isogeometric design-through-analysis methodology based on adaptive hierarchical refinement of NURBS, immersed boundary methods, and T-spline CAD surfaces, Comput. Methods Appl. Mech. Engrg., 249/252 (2012), 116-150.
doi: 10.1016/j.cma.2012.03.017. |
[45] |
L. L. Schumaker,
Spline Functions: Basic Theory, 3rd Edition, Cambridge University Press, 2007.
doi: 10.1017/CBO9780511618994. |
[46] |
M. A. Scott, D. C. Thomas and E. J. Evans,
Isogeometric spline forests, Comput. Methods Appl. Mech. Engrg., 269 (2014), 222-264.
doi: 10.1016/j.cma.2013.10.024. |
[47] |
T. W. Sederberg, D. L. Cardon, G. T. Finnigan, N. S. North, J. Zheng and T. Lyche,
T-spline simplification and local refinement, ACM Trans. Graphics, 23 (2004), 276-283.
doi: 10.1145/1015706.1015715. |
[48] |
H. Speleers and C. Manni,
Effortless quasi-interpolation in hierarchical spaces, Numer. Math., 132 (2016), 155-184.
doi: 10.1007/s00211-015-0711-z. |
[49] |
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