Counting the dimension of splines of mixed smoothness A general recipe, and its application to planar meshes of arbitrary topologies
In this paper, we study the dimension of bivariate polynomial splines of mixed smoothness on polygonal meshes. Here, “mixed smoothness” refers to the choice of different orders of smoothness across different edges of the mesh. To study the dimension of spaces of such splines, we use tools from homol...
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| Vydané v: | Advances in computational mathematics Ročník 47; číslo 1 |
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| Abstract | In this paper, we study the dimension of bivariate polynomial splines of mixed smoothness on polygonal meshes. Here, “mixed smoothness” refers to the choice of different orders of smoothness across different edges of the mesh. To study the dimension of spaces of such splines, we use tools from homological algebra. These tools were first applied to the study of splines by Billera (Trans. Am. Math. Soc.
310
(1), 325–340,
1988
). Using them, estimation of the spline space dimension amounts to the study of the Billera-Schenck-Stillman complex for the spline space. In particular, when the homology in positions 1 and 0 of this complex is trivial, the dimension of the spline space can be computed combinatorially. We call such spline spaces “lower-acyclic.” In this paper, starting from a spline space which is lower-acyclic, we present sufficient conditions that ensure that the same will be true for the spline space obtained after relaxing the smoothness requirements across a subset of the mesh edges. This general recipe is applied in a specific setting: meshes of arbitrary topologies. We show how our results can be used to compute the dimensions of spline spaces on triangulations, polygonal meshes, and T-meshes with holes. |
|---|---|
| AbstractList | In this paper, we study the dimension of bivariate polynomial splines of mixed smoothness on polygonal meshes. Here, “mixed smoothness” refers to the choice of different orders of smoothness across different edges of the mesh. To study the dimension of spaces of such splines, we use tools from homological algebra. These tools were first applied to the study of splines by Billera (Trans. Am. Math. Soc.
310
(1), 325–340,
1988
). Using them, estimation of the spline space dimension amounts to the study of the Billera-Schenck-Stillman complex for the spline space. In particular, when the homology in positions 1 and 0 of this complex is trivial, the dimension of the spline space can be computed combinatorially. We call such spline spaces “lower-acyclic.” In this paper, starting from a spline space which is lower-acyclic, we present sufficient conditions that ensure that the same will be true for the spline space obtained after relaxing the smoothness requirements across a subset of the mesh edges. This general recipe is applied in a specific setting: meshes of arbitrary topologies. We show how our results can be used to compute the dimensions of spline spaces on triangulations, polygonal meshes, and T-meshes with holes. In this paper, we study the dimension of bivariate polynomial splines of mixed smoothness on polygonal meshes. Here, “mixed smoothness” refers to the choice of different orders of smoothness across different edges of the mesh. To study the dimension of spaces of such splines, we use tools from homological algebra. These tools were first applied to the study of splines by Billera (Trans. Am. Math. Soc. 310 (1), 325–340, 1988). Using them, estimation of the spline space dimension amounts to the study of the Billera-Schenck-Stillman complex for the spline space. In particular, when the homology in positions 1 and 0 of this complex is trivial, the dimension of the spline space can be computed combinatorially. We call such spline spaces “lower-acyclic.” In this paper, starting from a spline space which is lower-acyclic, we present sufficient conditions that ensure that the same will be true for the spline space obtained after relaxing the smoothness requirements across a subset of the mesh edges. This general recipe is applied in a specific setting: meshes of arbitrary topologies. We show how our results can be used to compute the dimensions of spline spaces on triangulations, polygonal meshes, and T-meshes with holes. |
| ArticleNumber | 6 |
| Author | Toshniwal, Deepesh DiPasquale, Michael |
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| Keywords | Spline dimension formulas 41A15 Splines Mixed smoothness Polygonal meshes with holes 13D02 |
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| References | ToshniwalDSpeleersHHughesTJRSmooth cubic spline spaces on unstructured quadrilateral meshes with particular emphasis on extraordinary points: Geometric design and isogeometric analysis considerationsComput. Methods Appl. Mech. Eng.2017327411458372577710.1016/j.cma.2017.06.008 AlfeldPSchumakerLLOn the dimension of bivariate spline spaces of smoothness r and degree d = 3r + 1Numer. Math.1990571651661106237210.1007/BF01386434 Chui, C.K.: Multivariate Splines, vol. 54. Siam (1988) MourrainBOn the dimension of spline spaces on planar T-meshesMath. Comput.201483286847871314369510.1090/S0025-5718-2013-02738-X ToshniwalDVillamizarNDimension of polynomial splines of mixed smoothness on T-meshesComput. Aided Geom. Des.202080101880410195910.1016/j.cagd.2020.101880 GeramitaASchenckHKFat points, inverse systems, and piecewise polynomial functionsJ. Algebra19982041116128162394910.1006/jabr.1997.7361 SchenckHStillmanMLocal cohomology of bivariate splinesJ. Pure Appl. Algebra1997117535548145785410.1016/S0022-4049(97)00026-1 ZengCWuMDengFDengJDimensions of spline spaces over non-rectangular T-meshesAdv. Comput. Math.201642612591286357120510.1007/s10444-016-9461-4 DiPasqualeMDimension of mixed splines on polytopal cellsMath. Comput.201887310905939373922310.1090/mcom/3224 CottrellJAHughesTJRBazilevsYIsogeometric Analysis: Toward Integration of CAD and FEA2009HobokenWiley10.1002/9780470749081 BraccoCLycheTManniCRomanFSpeleersHGeneralized spline spaces over T-meshes: Dimension formula and locally refined generalized B-splinesAppl. Math. Comput.201627218719834181231410.65027 SchenckHStillmanMA family of ideals of minimal regularity and the hilbert series of Cr(Δ)Adv. Appl. Math.1997192169182145949610.1006/aama.1997.0533 IbrahimAKSchumakerLLSuper spline spaces of smoothness r and degree d ≥ 3r + 2Constr. Approx.199173401423112041210.1007/BF01888166 SchenckHComputational Algebraic Geometry, vol. 582003CambridgeCambridge University Press10.1017/CBO9780511756320 McDonaldTSchenckHPiecewise polynomials on polyhedral complexesAdv. Appl. Math.20094218293247531510.1016/j.aam.2008.06.001 BilleraLJHomology of smooth splines: Generic triangulations and a conjecture of strangTrans. Am. Math. Soc.1988310132534096575710.1090/S0002-9947-1988-0965757-9 SchumakerLLBounds on the dimension of spaces of multivariate piecewise polynomialsRocky Mt. J. Math.198414125126473617710.1216/RMJ-1984-14-1-251 HughesTJRCottrellJABazilevsYIsogeometric Analysis: CAD, finite elements, NURBS, exact geometry and mesh refinementComput. Methods Appl. Mech. Eng.200519441354195215238210.1016/j.cma.2004.10.008 AlfeldPSchumakerLLThe dimension of bivariate spline spaces of smoothness r for degree d ≥ 4r + 1Constr. Approx.19873118919788955410.1007/BF01890563 HongDSpaces of bivariate spline functions over triangulationJ. Approx. Theory199171567511173080756.41017 Toshniwal, D., Mourrain, B., Hughes, T.J.R.: Advances in Computational Mathematics (accepted). arXiv:https://doi.org/1903.05949 [math] https://doi.org/10.1007/s10444-020-09829-4 (2019) Alfeld, P., Piper, B., Schumaker, L.L.: Spaces of bivariate splines on triangulations with holes. In: Proceedings of China-U.S. Joint Conference on Approximation Theory (Hangzhou, 1985), vol. 3, pp. 1–10 (1987) FarinGEHoschekJKimM-SHandbook of Computer Aided Geometric Design2002AmsterdamElsevier1003.68179 HatcherAAlgebraic Topology2002CambridgeCambridge University Press1044.55001 StrangGPiecewise polynomials and the finite element methodBull. Am. Math. Soc.19737961128113732706010.1090/S0002-9904-1973-13351-8 Grayson, D.R., Stillman, M.E.: Macaulay2, a software system for research in algebraic geometry. Available at https://faculty.math.illinois.edu/Macaulay2 Strang, G.: The dimension of piecewise polynomial spaces, and one-sided approximation. In: Conference on the Numerical Solution of Differential Equations, pp 144–152. Springer (1974) ToshniwalDHughesTJRPolynomial splines of non-uniform degree on triangulations: Combinatorial bounds on the dimensionComput. Aided Geom. Des.201975101763401609910.1016/j.cagd.2019.07.002 SchenckHSorokinaTSubdivision and spline spacesConstr. Approx.2018472237247376927710.1007/s00365-017-9367-5 Jia, R.Q.: Lower bounds on the dimension of spaces of bivariate splines. In: Multivariate Approximation and Interpolation (Duisburg, 1989), vol. 94 of International Series of Numerical Mathematics, pp. 155–165. Birkhäuser, Basel (1990) MourrainBVillamizarNHomological techniques for the analysis of the dimension of triangular spline spacesJ. Symb. Comput.201350564577299689610.1016/j.jsc.2012.10.002 M DiPasquale (9830_CR11) 2018; 87 D Toshniwal (9830_CR19) 2017; 327 GE Farin (9830_CR1) 2002 B Mourrain (9830_CR29) 2013; 50 9830_CR22 H Schenck (9830_CR8) 1997; 117 T McDonald (9830_CR10) 2009; 42 B Mourrain (9830_CR13) 2014; 83 9830_CR21 9830_CR20 JA Cottrell (9830_CR2) 2009 LL Schumaker (9830_CR5) 1984; 14 LJ Billera (9830_CR7) 1988; 310 P Alfeld (9830_CR6) 1987; 3 H Schenck (9830_CR9) 1997; 19 H Schenck (9830_CR25) 2003 G Strang (9830_CR3) 1973; 79 9830_CR14 C Zeng (9830_CR16) 2016; 42 P Alfeld (9830_CR24) 1990; 57 TJR Hughes (9830_CR31) 2005; 194 A Geramita (9830_CR28) 1998; 204 AK Ibrahim (9830_CR27) 1991; 7 H Schenck (9830_CR17) 2018; 47 A Hatcher (9830_CR23) 2002 9830_CR30 D Hong (9830_CR26) 1991; 7 9830_CR4 D Toshniwal (9830_CR18) 2020; 80 C Bracco (9830_CR15) 2016; 272 D Toshniwal (9830_CR12) 2019; 75 |
| References_xml | – reference: BilleraLJHomology of smooth splines: Generic triangulations and a conjecture of strangTrans. Am. Math. Soc.1988310132534096575710.1090/S0002-9947-1988-0965757-9 – reference: Strang, G.: The dimension of piecewise polynomial spaces, and one-sided approximation. In: Conference on the Numerical Solution of Differential Equations, pp 144–152. Springer (1974) – reference: SchenckHStillmanMA family of ideals of minimal regularity and the hilbert series of Cr(Δ)Adv. Appl. Math.1997192169182145949610.1006/aama.1997.0533 – reference: ToshniwalDHughesTJRPolynomial splines of non-uniform degree on triangulations: Combinatorial bounds on the dimensionComput. Aided Geom. Des.201975101763401609910.1016/j.cagd.2019.07.002 – reference: Chui, C.K.: Multivariate Splines, vol. 54. Siam (1988) – reference: FarinGEHoschekJKimM-SHandbook of Computer Aided Geometric Design2002AmsterdamElsevier1003.68179 – reference: HongDSpaces of bivariate spline functions over triangulationJ. Approx. Theory199171567511173080756.41017 – reference: SchenckHSorokinaTSubdivision and spline spacesConstr. Approx.2018472237247376927710.1007/s00365-017-9367-5 – reference: Grayson, D.R., Stillman, M.E.: Macaulay2, a software system for research in algebraic geometry. Available at https://faculty.math.illinois.edu/Macaulay2/ – reference: StrangGPiecewise polynomials and the finite element methodBull. Am. Math. Soc.19737961128113732706010.1090/S0002-9904-1973-13351-8 – reference: SchenckHComputational Algebraic Geometry, vol. 582003CambridgeCambridge University Press10.1017/CBO9780511756320 – reference: Jia, R.Q.: Lower bounds on the dimension of spaces of bivariate splines. In: Multivariate Approximation and Interpolation (Duisburg, 1989), vol. 94 of International Series of Numerical Mathematics, pp. 155–165. Birkhäuser, Basel (1990) – reference: Toshniwal, D., Mourrain, B., Hughes, T.J.R.: Advances in Computational Mathematics (accepted). arXiv:https://doi.org/1903.05949 [math] https://doi.org/10.1007/s10444-020-09829-4 (2019) – reference: CottrellJAHughesTJRBazilevsYIsogeometric Analysis: Toward Integration of CAD and FEA2009HobokenWiley10.1002/9780470749081 – reference: SchenckHStillmanMLocal cohomology of bivariate splinesJ. Pure Appl. Algebra1997117535548145785410.1016/S0022-4049(97)00026-1 – reference: IbrahimAKSchumakerLLSuper spline spaces of smoothness r and degree d ≥ 3r + 2Constr. Approx.199173401423112041210.1007/BF01888166 – reference: MourrainBVillamizarNHomological techniques for the analysis of the dimension of triangular spline spacesJ. Symb. Comput.201350564577299689610.1016/j.jsc.2012.10.002 – reference: BraccoCLycheTManniCRomanFSpeleersHGeneralized spline spaces over T-meshes: Dimension formula and locally refined generalized B-splinesAppl. Math. Comput.201627218719834181231410.65027 – reference: SchumakerLLBounds on the dimension of spaces of multivariate piecewise polynomialsRocky Mt. J. Math.198414125126473617710.1216/RMJ-1984-14-1-251 – reference: HatcherAAlgebraic Topology2002CambridgeCambridge University Press1044.55001 – reference: ToshniwalDSpeleersHHughesTJRSmooth cubic spline spaces on unstructured quadrilateral meshes with particular emphasis on extraordinary points: Geometric design and isogeometric analysis considerationsComput. Methods Appl. Mech. Eng.2017327411458372577710.1016/j.cma.2017.06.008 – reference: Alfeld, P., Piper, B., Schumaker, L.L.: Spaces of bivariate splines on triangulations with holes. In: Proceedings of China-U.S. Joint Conference on Approximation Theory (Hangzhou, 1985), vol. 3, pp. 1–10 (1987) – reference: AlfeldPSchumakerLLOn the dimension of bivariate spline spaces of smoothness r and degree d = 3r + 1Numer. Math.1990571651661106237210.1007/BF01386434 – reference: HughesTJRCottrellJABazilevsYIsogeometric Analysis: CAD, finite elements, NURBS, exact geometry and mesh refinementComput. Methods Appl. Mech. Eng.200519441354195215238210.1016/j.cma.2004.10.008 – reference: DiPasqualeMDimension of mixed splines on polytopal cellsMath. Comput.201887310905939373922310.1090/mcom/3224 – reference: GeramitaASchenckHKFat points, inverse systems, and piecewise polynomial functionsJ. Algebra19982041116128162394910.1006/jabr.1997.7361 – reference: McDonaldTSchenckHPiecewise polynomials on polyhedral complexesAdv. Appl. Math.20094218293247531510.1016/j.aam.2008.06.001 – reference: ZengCWuMDengFDengJDimensions of spline spaces over non-rectangular T-meshesAdv. Comput. Math.201642612591286357120510.1007/s10444-016-9461-4 – reference: ToshniwalDVillamizarNDimension of polynomial splines of mixed smoothness on T-meshesComput. Aided Geom. Des.202080101880410195910.1016/j.cagd.2020.101880 – reference: AlfeldPSchumakerLLThe dimension of bivariate spline spaces of smoothness r for degree d ≥ 4r + 1Constr. Approx.19873118919788955410.1007/BF01890563 – reference: MourrainBOn the dimension of spline spaces on planar T-meshesMath. Comput.201483286847871314369510.1090/S0025-5718-2013-02738-X – ident: 9830_CR20 – volume-title: Algebraic Topology year: 2002 ident: 9830_CR23 – volume: 83 start-page: 847 issue: 286 year: 2014 ident: 9830_CR13 publication-title: Math. Comput. doi: 10.1090/S0025-5718-2013-02738-X – volume: 42 start-page: 1259 issue: 6 year: 2016 ident: 9830_CR16 publication-title: Adv. Comput. Math. doi: 10.1007/s10444-016-9461-4 – volume: 3 start-page: 189 issue: 1 year: 1987 ident: 9830_CR6 publication-title: Constr. 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| Snippet | In this paper, we study the dimension of bivariate polynomial splines of mixed smoothness on polygonal meshes. Here, “mixed smoothness” refers to the choice of... |
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| SubjectTerms | Computational Mathematics and Numerical Analysis Computational Science and Engineering Mathematical and Computational Biology Mathematical Modeling and Industrial Mathematics Mathematics Mathematics and Statistics Visualization |
| Subtitle | A general recipe, and its application to planar meshes of arbitrary topologies |
| Title | Counting the dimension of splines of mixed smoothness |
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