{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T11:14:00Z","timestamp":1777547640341,"version":"3.51.4"},"reference-count":57,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2022,2,17]],"date-time":"2022-02-17T00:00:00Z","timestamp":1645056000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,2,17]],"date-time":"2022-02-17T00:00:00Z","timestamp":1645056000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Found Comput Math"],"published-print":{"date-parts":[[2022,10]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this paper, we address the problem of constructing <jats:inline-formula><jats:alternatives><jats:tex-math>$$C^2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mi>C<\/mml:mi>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> cubic spline functions on a given arbitrary triangulation <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathcal {T}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>T<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. To this end, we endow every triangle of <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathcal {T}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>T<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> with a Wang\u2013Shi macro-structure. The <jats:inline-formula><jats:alternatives><jats:tex-math>$$C^2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mi>C<\/mml:mi>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> cubic space on such a refined triangulation has a stable dimension and optimal approximation power. Moreover, any spline function in this space can be locally built on each of the macro-triangles independently via Hermite interpolation. We provide a simplex spline basis for the space of <jats:inline-formula><jats:alternatives><jats:tex-math>$$C^2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mi>C<\/mml:mi>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> cubics defined on a single macro-triangle which behaves like a Bernstein\/B-spline basis over the triangle. The basis functions inherit recurrence relations and differentiation formulas from the simplex spline construction, they form a nonnegative partition of unity, they admit simple conditions for <jats:inline-formula><jats:alternatives><jats:tex-math>$$C^2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mi>C<\/mml:mi>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> joins across the edges of neighboring triangles, and they enjoy a Marsden-like identity. Also, there is a single control net to facilitate control and early visualization of a spline function over the macro-triangle. Thanks to these properties, the complex geometry of the Wang\u2013Shi macro-structure is transparent to the user. Stable global bases for the full space of <jats:inline-formula><jats:alternatives><jats:tex-math>$$C^2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mi>C<\/mml:mi>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> cubics on the Wang\u2013Shi refined triangulation <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathcal {T}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>T<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> are deduced from the local simplex spline basis by extending the concept of minimal determining\u00a0sets.<\/jats:p>","DOI":"10.1007\/s10208-022-09553-z","type":"journal-article","created":{"date-parts":[[2022,2,17]],"date-time":"2022-02-17T20:06:39Z","timestamp":1645128399000},"page":"1309-1350","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Construction of $$C^2$$ Cubic Splines on Arbitrary Triangulations"],"prefix":"10.1007","volume":"22","author":[{"given":"Tom","family":"Lyche","sequence":"first","affiliation":[]},{"given":"Carla","family":"Manni","sequence":"additional","affiliation":[]},{"given":"Hendrik","family":"Speleers","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,2,17]]},"reference":[{"key":"9553_CR1","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1023\/A:1014299228104","volume":"16","author":"P Alfeld","year":"2002","unstructured":"Alfeld, P., Schumaker, L.L.: Smooth macro-elements based on Powell\u2013Sabin triangle splits. Adv. Comput. Math. 16, 29\u201346 (2002)","journal-title":"Adv. Comput. Math."},{"key":"9553_CR2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-63841-6","volume-title":"Smooth B\u00e9zier Surfaces over Unstructured Quadrilateral Meshes, Lecture Notes of the Unione Matematica Italiana","author":"M Bercovier","year":"2017","unstructured":"Bercovier, M., Matskewich, T.: Smooth B\u00e9zier Surfaces over Unstructured Quadrilateral Meshes. Lecture Notes of the Unione Matematica Italiana, vol.\u00a022. Springer, Berlin (2017)"},{"key":"9553_CR3","doi-asserted-by":"publisher","first-page":"101814","DOI":"10.1016\/j.cagd.2020.101814","volume":"78","author":"A Blidia","year":"2020","unstructured":"Blidia, A., Mourrain, B., Xu, G.: Geometrically smooth spline bases for data fitting and simulation. Comput. Aided Geom. Des. 78, 101814 (2020)","journal-title":"Comput. Aided Geom. Des."},{"key":"9553_CR4","doi-asserted-by":"publisher","first-page":"923","DOI":"10.1007\/s00211-019-01063-5","volume":"143","author":"A Bressan","year":"2019","unstructured":"Bressan, A., Sande, E.: Approximation in FEM, DG and IGA: a theoretical comparison. Numer. Math. 143, 923\u2013942 (2019)","journal-title":"Numer. Math."},{"key":"9553_CR5","doi-asserted-by":"publisher","first-page":"112598","DOI":"10.1016\/j.cma.2019.112598","volume":"357","author":"J Cao","year":"2019","unstructured":"Cao, J., Chen, Z., Wei, X., Zhang, Y.J.: A finite element framework based on bivariate simplex splines on triangle configurations. Comput. Methods Appl. Mech. Eng. 357, 112598 (2019)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"9553_CR6","doi-asserted-by":"publisher","first-page":"294","DOI":"10.1016\/j.cagd.2018.03.025","volume":"62","author":"CL Chan","year":"2018","unstructured":"Chan, C.L., Anitescu, C., Rabczuk, T.: Isogeometric analysis with strong multipatch $$C^1$$-coupling. Comput. Aided Geom. Des. 62, 294\u2013310 (2018)","journal-title":"Comput. Aided Geom. Des."},{"key":"9553_CR7","doi-asserted-by":"publisher","first-page":"197","DOI":"10.1016\/0022-247X(83)90014-8","volume":"94","author":"CK Chui","year":"1983","unstructured":"Chui, C.K., Wang, R.H.: Multivariate spline spaces. J. Math. Anal. Appl. 94, 197\u2013221 (1983)","journal-title":"J. Math. Anal. Appl."},{"key":"9553_CR8","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719208","volume-title":"The Finite Element Method for Elliptic Problems, Classics in Applied Mathematics","author":"PG Ciarlet","year":"2002","unstructured":"Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Classics in Applied Mathematics, vol.\u00a040. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2002)"},{"key":"9553_CR9","unstructured":"Clough, R.W., Tocher, J.L.: Finite element stiffness matrices for analysis of plates in bending. In: Proceedings of the Conference on Matrix Methods in Structural Mechanics, pp. 515\u2013545. Wright-Patterson Air Force Base (1965)"},{"key":"9553_CR10","doi-asserted-by":"publisher","first-page":"1667","DOI":"10.1090\/S0025-5718-2013-02664-6","volume":"82","author":"E Cohen","year":"2013","unstructured":"Cohen, E., Lyche, T., Riesenfeld, R.F.: A B-spline-like basis for the Powell\u2013Sabin 12-split based on simplex splines. Math. Comp. 82, 1667\u20131707 (2013)","journal-title":"Math. Comp."},{"key":"9553_CR11","doi-asserted-by":"publisher","DOI":"10.1201\/9781439864203","volume-title":"Geometric Modeling with Splines","author":"E Cohen","year":"2001","unstructured":"Cohen, E., Riesenfeld, R.F., Elber, G.: Geometric Modeling with Splines. A K Peters Ltd., Natick (2001)"},{"key":"9553_CR12","doi-asserted-by":"publisher","DOI":"10.1002\/9780470749081","volume-title":"Isogeometric Analysis: Toward Integration of CAD and FEA","author":"JA Cottrell","year":"2009","unstructured":"Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y.: Isogeometric Analysis: Toward Integration of CAD and FEA. Wiley, New York (2009)"},{"key":"9553_CR13","doi-asserted-by":"publisher","first-page":"543","DOI":"10.1137\/0727034","volume":"27","author":"D Diener","year":"1990","unstructured":"Diener, D.: Instability in the dimension of spaces of bivariate piecewise polynomials of degree $$2r$$ and smoothness order $$r$$. SIAM J. Numer. Anal. 27, 543\u2013551 (1990)","journal-title":"SIAM J. Numer. Anal."},{"key":"9553_CR14","doi-asserted-by":"publisher","first-page":"61","DOI":"10.1016\/S0167-8396(97)81785-2","volume":"15","author":"P Dierckx","year":"1997","unstructured":"Dierckx, P.: On calculating normalized Powell\u2013Sabin B-splines. Comput. Aided Geom. Des. 15, 61\u201378 (1997)","journal-title":"Comput. Aided Geom. Des."},{"key":"9553_CR15","doi-asserted-by":"publisher","first-page":"331","DOI":"10.1016\/j.cagd.2012.12.005","volume":"30","author":"T Dokken","year":"2013","unstructured":"Dokken, T., Lyche, T., Pettersen, K.F.: Polynomial splines over locally refined box-partitions. Comput. Aided Geom. Des. 30, 331\u2013356 (2013)","journal-title":"Comput. Aided Geom. Des."},{"key":"9553_CR16","doi-asserted-by":"publisher","first-page":"485","DOI":"10.1016\/j.cagd.2012.03.025","volume":"29","author":"C Giannelli","year":"2012","unstructured":"Giannelli, C., J\u00fcttler, B., Speleers, H.: THB-splines: the truncated basis for hierarchical splines. Comput. Aided Geom. Des. 29, 485\u2013498 (2012)","journal-title":"Comput. Aided Geom. Des."},{"key":"9553_CR17","doi-asserted-by":"publisher","first-page":"1257","DOI":"10.1007\/s10543-015-0600-y","volume":"56","author":"J Gro\u0161elj","year":"2016","unstructured":"Gro\u0161elj, J.: A normalized representation of super splines of arbitrary degree on Powell\u2013Sabin triangulations. BIT Numer. Math. 56, 1257\u20131280 (2016)","journal-title":"BIT Numer. Math."},{"key":"9553_CR18","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.cagd.2017.05.003","volume":"57","author":"J Gro\u0161elj","year":"2017","unstructured":"Gro\u0161elj, J., Speleers, H.: Construction and analysis of cubic Powell\u2013Sabin B-splines. Comput. Aided Geom. Des. 57, 1\u201322 (2017)","journal-title":"Comput. Aided Geom. Des."},{"key":"9553_CR19","doi-asserted-by":"crossref","unstructured":"Gro\u0161elj, J., Speleers, H.: Super-smooth cubic Powell\u2013Sabin splines on three-directional triangulations: B-spline representation and subdivision. J. Comput. Appl. Math. 386, art. 113245 (2021)","DOI":"10.1016\/j.cam.2020.113245"},{"key":"9553_CR20","doi-asserted-by":"publisher","first-page":"401","DOI":"10.1007\/BF01888166","volume":"7","author":"AK Ibrahim","year":"1991","unstructured":"Ibrahim, A.K., Schumaker, L.L.: Super spline spaces of smoothness $$r$$ and degree $$d\\ge 3r + 2$$. Constr. Approx. 7, 401\u2013423 (1991)","journal-title":"Constr. Approx."},{"key":"9553_CR21","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1016\/j.cad.2013.08.017","volume":"46","author":"N Jaxon","year":"2014","unstructured":"Jaxon, N., Qian, X.: Isogeometric analysis on triangulations. Comput. Aided Des. 46, 45\u201357 (2014)","journal-title":"Comput. Aided Des."},{"key":"9553_CR22","doi-asserted-by":"publisher","first-page":"55","DOI":"10.1016\/j.cagd.2019.01.002","volume":"69","author":"M Kapl","year":"2019","unstructured":"Kapl, M., Sangalli, G., Takacs, T.: An isogeometric $$C^1$$ subspace on unstructured multi-patch planar domains. Comput. Aided Geom. Des. 69, 55\u201375 (2019)","journal-title":"Comput. Aided Geom. Des."},{"key":"9553_CR23","doi-asserted-by":"publisher","first-page":"191","DOI":"10.1016\/S0167-8396(96)00028-3","volume":"14","author":"MJ Lai","year":"1997","unstructured":"Lai, M.J.: Geometric interpretation of smoothness conditions of triangular polynomial patches. Comput. Aided Geom. Des. 14, 191\u2013199 (1997)","journal-title":"Comput. Aided Geom. Des."},{"key":"9553_CR24","doi-asserted-by":"publisher","first-page":"105","DOI":"10.1007\/PL00005435","volume":"88","author":"MJ Lai","year":"2001","unstructured":"Lai, M.J., Schumaker, L.L.: Macro-elements and stable local bases for splines on Clough\u2013Tocher triangulations. Numer. Math. 88, 105\u2013119 (2001)","journal-title":"Numer. Math."},{"key":"9553_CR25","doi-asserted-by":"publisher","first-page":"335","DOI":"10.1090\/S0025-5718-01-01379-5","volume":"72","author":"MJ Lai","year":"2003","unstructured":"Lai, M.J., Schumaker, L.L.: Macro-elements and stable local bases for splines on Powell\u2013Sabin triangulations. Math. Comp. 72, 335\u2013354 (2003)","journal-title":"Math. Comp."},{"key":"9553_CR26","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511721588","volume-title":"Spline Functions on Triangulations, Encyclopedia of Mathematics and Its Applications","author":"MJ Lai","year":"2007","unstructured":"Lai, M.J., Schumaker, L.L.: Spline Functions on Triangulations. Encyclopedia of Mathematics and Its Applications, vol. 110. Cambridge University Press, Cambridge (2007)"},{"key":"9553_CR27","series-title":"Lecture Notes and Mathematics","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/978-3-319-94911-6_1","volume-title":"Splines and PDEs: From Approximation Theory to Numerical Linear Algebra","author":"T Lyche","year":"2018","unstructured":"Lyche, T., Manni, C., Speleers, H.: Foundations of spline theory: B-splines, spline approximation, and hierarchical refinement. In: Lyche, T., et\u00a0al. (eds.) Splines and PDEs: From Approximation Theory to Numerical Linear Algebra. Lecture Notes in Mathematics, vol. 2219, pp. 1\u201376. Springer, Berlin (2018)"},{"key":"9553_CR28","doi-asserted-by":"publisher","first-page":"76","DOI":"10.1016\/j.cagd.2018.07.004","volume":"65","author":"T Lyche","year":"2018","unstructured":"Lyche, T., Merrien, J.L.: Simplex-splines on the Clough\u2013Tocher element. Comput. Aided Geom. Des. 65, 76\u201392 (2018)","journal-title":"Comput. Aided Geom. Des."},{"key":"9553_CR29","unstructured":"Lyche, T., Merrien, J.L., Sauer, T.: Simplex-splines on the Clough\u2013Tocher split with arbitrary smoothness. In: Manni, C., Speleers, H. (eds.) Geometric Challenges in Isogeometric Analysis, Springer INdAM Series. Springer (to appear)"},{"key":"9553_CR30","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00365-016-9332-8","volume":"45","author":"T Lyche","year":"2017","unstructured":"Lyche, T., Muntingh, G.: Stable simplex spline bases for $$C^3$$ quintics on the Powell\u2013Sabin 12-split. Constr. Approx. 45, 1\u201332 (2017)","journal-title":"Constr. Approx."},{"key":"9553_CR31","doi-asserted-by":"publisher","first-page":"141","DOI":"10.1016\/0021-9045(92)90138-E","volume":"69","author":"C Manni","year":"1992","unstructured":"Manni, C.: On the dimension of bivariate spline spaces on generalized quasi-cross-cut partitions. J. Approx. Theory 69, 141\u2013155 (1992)","journal-title":"J. Approx. Theory"},{"key":"9553_CR32","doi-asserted-by":"publisher","first-page":"211","DOI":"10.1007\/978-3-0348-6289-9_14","volume-title":"Multivariate Approximation Theory, International Series of Numerical Mathematics","author":"CA Micchelli","year":"1979","unstructured":"Micchelli, C.A.: On a numerically efficient method for computing multivariate $$B$$-splines. In: Schempp, W., Zeller, K. (eds.) Multivariate Approximation Theory, International Series of Numerical Mathematics, vol.\u00a051, pp. 211\u2013248. Birkh\u00e4user, Basel (1979)"},{"key":"9553_CR33","doi-asserted-by":"publisher","first-page":"2993","DOI":"10.1090\/S0002-9947-07-03976-1","volume":"359","author":"M Neamtu","year":"2007","unstructured":"Neamtu, M.: Delaunay configurations and multivariate splines: a generalization of a result of B. N. Delaunay. Trans. Am. Math. Soc. 359, 2993\u20133004 (2007)","journal-title":"N. Delaunay. Trans. Am. Math. Soc."},{"key":"9553_CR34","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-76406-9","volume-title":"Subdivision Surfaces","author":"J Peters","year":"2008","unstructured":"Peters, J., Reif, U.: Subdivision Surfaces. Springer, Berlin (2008)"},{"key":"9553_CR35","doi-asserted-by":"publisher","first-page":"163","DOI":"10.1016\/0167-8396(84)90028-1","volume":"1","author":"T Poeschl","year":"1984","unstructured":"Poeschl, T.: Detecting surface irregularities using isophotes. Comput. Aided Geom. Des. 1, 163\u2013168 (1984)","journal-title":"Comput. Aided Geom. Des."},{"key":"9553_CR36","doi-asserted-by":"publisher","first-page":"316","DOI":"10.1145\/355759.355761","volume":"3","author":"MJD Powell","year":"1977","unstructured":"Powell, M.J.D., Sabin, M.A.: Piecewise quadratic approximations on triangles. ACM Trans. Math. Softw. 3, 316\u2013325 (1977)","journal-title":"ACM Trans. Math. Softw."},{"key":"9553_CR37","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-04919-8","volume-title":"B\u00e9zier and B-Spline Techniques. Mathematics and Visualization","author":"H Prautzsch","year":"2002","unstructured":"Prautzsch, H., Boehm, W., Paluszny, M.: B\u00e9zier and B-Spline Techniques. Mathematics and Visualization. Springer, Berlin (2002)"},{"key":"9553_CR38","doi-asserted-by":"publisher","first-page":"323","DOI":"10.1016\/0167-8396(89)90032-0","volume":"6","author":"L Ramshaw","year":"1989","unstructured":"Ramshaw, L.: Blossoms are polar forms. Comput. Aided Geom. Des. 6, 323\u2013358 (1989)","journal-title":"Comput. Aided Geom. Des."},{"key":"9553_CR39","doi-asserted-by":"publisher","first-page":"174","DOI":"10.1006\/jath.1996.3079","volume":"90","author":"U Reif","year":"1997","unstructured":"Reif, U.: A refineable space of smooth spline surfaces of arbitrary topological genus. J. Approx. Theory 90, 174\u2013199 (1997)","journal-title":"J. Approx. Theory"},{"key":"9553_CR40","doi-asserted-by":"publisher","first-page":"541","DOI":"10.1016\/0377-0427(85)90047-0","volume":"12\u201313","author":"P Sablonni\u00e8re","year":"1985","unstructured":"Sablonni\u00e8re, P.: Composite finite elements of class $$C^k$$. J. Comput. Appl. Math. 12\u201313, 541\u2013550 (1985)","journal-title":"J. Comput. Appl. Math."},{"key":"9553_CR41","doi-asserted-by":"publisher","first-page":"889","DOI":"10.1007\/s00211-019-01097-9","volume":"144","author":"E Sande","year":"2020","unstructured":"Sande, E., Manni, C., Speleers, H.: Explicit error estimates for spline approximation of arbitrary smoothness in isogeometric analysis. Numer. Math. 144, 889\u2013929 (2020)","journal-title":"Numer. Math."},{"key":"9553_CR42","doi-asserted-by":"publisher","first-page":"61","DOI":"10.1016\/j.cagd.2016.05.004","volume":"47","author":"G Sangalli","year":"2016","unstructured":"Sangalli, G., Takacs, T., V\u00e1zquez, R.: Unstructured spline spaces for isogeometric analysis based on spline manifolds. Comput. Aided Geom. Des. 47, 61\u201382 (2016)","journal-title":"Comput. Aided Geom. Des."},{"key":"9553_CR43","doi-asserted-by":"publisher","first-page":"711","DOI":"10.1090\/S0025-5718-05-01813-2","volume":"75","author":"LL Schumaker","year":"2006","unstructured":"Schumaker, L.L., Sorokina, T.: Smooth macro-elements on Powell\u2013Sabin-12 splits. Math. Comp. 75, 711\u2013726 (2006)","journal-title":"Math. Comp."},{"key":"9553_CR44","doi-asserted-by":"publisher","first-page":"477","DOI":"10.1145\/882262.882295","volume":"22","author":"TW Sederberg","year":"2003","unstructured":"Sederberg, T.W., Zheng, J., Bakenov, A., Nasri, A.: T-splines and T-NURCCs. ACM Trans. Graph. 22, 477\u2013484 (2003)","journal-title":"ACM Trans. Graph."},{"key":"9553_CR45","doi-asserted-by":"publisher","first-page":"438","DOI":"10.1016\/j.cagd.2010.05.001","volume":"27","author":"H Speleers","year":"2010","unstructured":"Speleers, H.: A normalized basis for quintic Powell\u2013Sabin splines. Comput. Aided Geom. Des. 27, 438\u2013457 (2010)","journal-title":"Comput. Aided Geom. Des."},{"key":"9553_CR46","doi-asserted-by":"publisher","first-page":"700","DOI":"10.1016\/j.cagd.2010.09.003","volume":"27","author":"H Speleers","year":"2010","unstructured":"Speleers, H.: A normalized basis for reduced Clough\u2013Tocher splines. Comput. Aided Geom. Des. 27, 700\u2013712 (2010)","journal-title":"Comput. Aided Geom. Des."},{"key":"9553_CR47","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1007\/s00365-011-9151-x","volume":"37","author":"H Speleers","year":"2013","unstructured":"Speleers, H.: Construction of normalized B-splines for a family of smooth spline spaces over Powell\u2013Sabin triangulations. Constr. Approx. 37, 41\u201372 (2013)","journal-title":"Constr. Approx."},{"key":"9553_CR48","doi-asserted-by":"publisher","first-page":"297","DOI":"10.1007\/s00365-014-9248-0","volume":"41","author":"H Speleers","year":"2015","unstructured":"Speleers, H.: A family of smooth quasi-interpolants defined over Powell\u2013Sabin triangulations. Constr. Approx. 41, 297\u2013324 (2015)","journal-title":"Constr. Approx."},{"key":"9553_CR49","doi-asserted-by":"publisher","first-page":"42","DOI":"10.1016\/j.cagd.2015.05.002","volume":"37","author":"H Speleers","year":"2015","unstructured":"Speleers, H.: A new B-spline representation for cubic splines over Powell\u2013Sabin triangulations. Comput. Aided Geom. Des. 37, 42\u201356 (2015)","journal-title":"Comput. Aided Geom. Des."},{"key":"9553_CR50","doi-asserted-by":"publisher","first-page":"68","DOI":"10.1016\/j.cam.2015.03.024","volume":"289","author":"H Speleers","year":"2015","unstructured":"Speleers, H., Manni, C.: Optimizing domain parameterization in isogeometric analysis based on Powell\u2013Sabin splines. J. Comput. Appl. Math. 289, 68\u201386 (2015)","journal-title":"J. Comput. Appl. Math."},{"key":"9553_CR51","doi-asserted-by":"publisher","first-page":"132","DOI":"10.1016\/j.cma.2012.02.009","volume":"221\u2013222","author":"H Speleers","year":"2012","unstructured":"Speleers, H., Manni, C., Pelosi, F., Sampoli, M.L.: Isogeometric analysis with Powell\u2013Sabin splines for advection\u2013diffusion\u2013reaction problems. Comput. Methods Appl. Mech. Eng. 221\u2013222, 132\u2013148 (2012)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"9553_CR52","doi-asserted-by":"publisher","first-page":"101763","DOI":"10.1016\/j.cagd.2019.07.002","volume":"75","author":"D Toshniwal","year":"2019","unstructured":"Toshniwal, D., Hughes, T.J.R.: Polynomial splines of non-uniform degree on triangulations: combinatorial bounds on the dimension. Comput. Aided Geom. Des. 75, 101763 (2019)","journal-title":"Comput. Aided Geom. Des."},{"key":"9553_CR53","doi-asserted-by":"publisher","first-page":"411","DOI":"10.1016\/j.cma.2017.06.008","volume":"327","author":"D Toshniwal","year":"2017","unstructured":"Toshniwal, D., Speleers, H., Hughes, T.J.R.: Smooth cubic spline spaces on unstructured quadrilateral meshes with particular emphasis on extraordinary points: geometric design and isogeometric analysis considerations. Comput. Methods Appl. Mech. Eng. 327, 411\u2013458 (2017)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"9553_CR54","doi-asserted-by":"publisher","first-page":"585","DOI":"10.1016\/j.cma.2017.11.032","volume":"331","author":"C Wang","year":"2018","unstructured":"Wang, C., Xia, S., Wang, X., Qian, X.: Isogeometric shape optimization on triangulations. Comput. Methods Appl. Mech. Eng. 331, 585\u2013622 (2018)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"9553_CR55","first-page":"205","volume-title":"Approximation, Optimization and Computing: Theory and Applications","author":"RH Wang","year":"1990","unstructured":"Wang, R.H., Shi, X.Q.: $$S_{\\mu +1}^{\\mu }$$ surface interpolations over triangulations. In: Law, A.G., Wang, C.L. (eds.) Approximation, Optimization and Computing: Theory and Applications, pp. 205\u2013208. Elsevier Science Publishers B.V., Amsterdam (1990)"},{"key":"9553_CR56","doi-asserted-by":"crossref","unstructured":"\u017den\u00ed\u0161ek, A.: A general theorem on triangular finite $$C^{(m)}$$-elements. Rev. Fran\u00e7aise Automat. Inf. Recherche Op\u00e9rationnelle S\u00e9r. Rouge 8, 119\u2013127 (1974)","DOI":"10.1051\/m2an\/197408R201191"},{"key":"9553_CR57","doi-asserted-by":"publisher","first-page":"14","DOI":"10.1016\/j.cagd.2017.01.001","volume":"50","author":"Y Zhang","year":"2017","unstructured":"Zhang, Y., Cao, J., Chen, Z.G., Zeng, X.M.: Surface reconstruction using simplex splines on feature-sensitive configurations. Comput. Aided Geom. Des. 50, 14\u201328 (2017)","journal-title":"Comput. Aided Geom. Des."}],"container-title":["Foundations of Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10208-022-09553-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10208-022-09553-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10208-022-09553-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,10,21]],"date-time":"2022-10-21T02:19:22Z","timestamp":1666318762000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10208-022-09553-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,2,17]]},"references-count":57,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2022,10]]}},"alternative-id":["9553"],"URL":"https:\/\/doi.org\/10.1007\/s10208-022-09553-z","relation":{},"ISSN":["1615-3375","1615-3383"],"issn-type":[{"value":"1615-3375","type":"print"},{"value":"1615-3383","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,2,17]]},"assertion":[{"value":"22 July 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 July 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 September 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 February 2022","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}