Abstract
The problem of constructing a normalized hierarchical basis for adaptively refined spline spaces is addressed. Multilevel representations are defined in terms of a hierarchy of basis functions, reflecting different levels of refinement. When the hierarchical model is constructed by considering an underlying sequence of bases \(\{\Gamma ^{\ell }\}_{\ell =0,\ldots ,N-1}\) with properties analogous to classical tensor-product B-splines, we can define a set of locally supported basis functions that form a partition of unity and possess the property of coefficient preservation, i.e., they preserve the coefficients of functions represented with respect to one of the bases \(\Gamma ^{\ell }\). Our construction relies on a certain truncation procedure, which eliminates the contributions of functions from finer levels in the hierarchy to coarser level ones. Consequently, the support of the original basis functions defined on coarse grids is possibly reduced according to finer levels in the hierarchy. This truncation mechanism not only decreases the overlapping of basis supports, but it also guarantees strong stability of the construction. In addition to presenting the theory for the general framework, we apply it to hierarchically refined tensor-product spline spaces, under certain reasonable assumptions on the given knot configuration.
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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. Comput. Methods Appl. Mech. Eng. 199, 229–263 (2010)
de Boor, C.: A Practical Guide to Splines. (revised ed.) Springer (2001)
de Boor, C., Höllig, K., Riemenschneider, S.: Box Splines. Springer-Verlag, New York (1993)
Costantini, P., Lyche, T., Manni, C.: On a class of weak Tchebycheff systems. Numer. Math. 101, 333–354 (2005)
Dæhlen, M., Lyche, T., Mørken, K., Schneider, R., Seidel, H.P.: Multiresolution analysis over triangles, based on quadratic Hermite interpolation. J. Comput. Appl. Math. 119, 97–114 (2000)
Dokken, T., Lyche, T., Pettersen, K.F.: Polynomial splines over locally refined box-partitions. Comput. Aided Geom. Des. 30, 331–356 (2013)
Dörfel, M.R., Jüttler, B., Simeon, B.: Adaptive isogeometric analysis by local h-refinement with T-splines. Comput. Methods Appl. Mech. Eng. 199, 264–275 (2010)
Farouki, R.T., Goodman, T.N.T.: On the optimal stability of the Bernstein basis. Math. Comput. 65, 1553–1566 (1996)
Forsey, D.R., Bartels, R.H.: Hierarchical B-spline refinement. Comput. Graph. 22, 205–212 (1988)
Giannelli, C., Jüttler, B., Speleers, H.: THB-splines: the truncated basis for hierarchical splines. Comput. Aided Geom. Des. 29, 485–498 (2012)
Greville, T.N.E.: On the normalisation of the B-splines and the location of the nodes for the case of unequally spaced knots. In: Shisha, O. (ed.) Inequalities, pp. 286–290. Academic, New York (1967)
Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Eng. 194, 4135–4195 (2005)
Kleiss, S.K., Jüttler, B., Zulehner, W.: Enhancing isogeometric analysis by a finite element-based local refinement strategy. Comput. Methods Appl. Mech. Eng. 213–216, 168–182 (2012)
Kraft, R.: Adaptive and linearly independent multilevel B-splines. In: Le Méhauté, A., Rabut, C., Schumaker, L.L. (eds.) Surface Fitting and Multiresolution Methods, pp. 209–218. Vanderbilt University Press, Nashville (1997)
Kraft, R.: Adaptive und linear unabhängige Multilevel B-Splines und ihre Anwendungen. Ph.D. thesis, Universität Stuttgart (1998)
Lai, M.J., Schumaker, L.L.: Spline Functions on Triangulations. Cambridge University Press, Cambridge (2007)
Lyche, T., Scherer, K.: On the sup-norm condition number of the multivariate triangular Bernstein basis. In: Nürnberger, G., Schmidt, J.W., Walz, G. (eds.) Multivariate Approximation and Splines, pp. 141–151. Birkhäuser Verlag, Basel (1997)
Nguyen-Thanh, N., Nguyen-Xuan, H., Bordas, S.P.A., Rabczuk, T.: Isogeometric analysis using polynomial splines over hierarchical T-meshes for two-dimensional elastic solids. Comput. Methods Appl. Mech. Eng. 200, 1892–1908 (2011)
Piegl, L., Tiller, W.: The NURBS Book. Springer, Berlin (1997)
Prautzsch, H.: The location of the control points in the case of box splines. IMA J. Numer. Anal. 6, 43–49 (1986)
Prautzsch, H., Boehm, W., Paluszny, M.: Bézier and B-spline Techniques. Springer, Berlin (2002)
Schumaker, L.L.: Spline Functions: Basic Theory, 3rd edn. Cambridge University Press, Cambridge (2007)
Speleers, H., Dierckx, P., Vandewalle, S.: Quasi-hierarchical Powell–Sabin B-splines. Comput. Aided Geom. Des. 26, 174–191 (2009)
Speleers, H., Dierckx, P., Vandewalle, S.: On the local approximation power of quasi-hierarchical Powell–Sabin splines. In: Dæhlen, M., Floater, M., Lyche, T., Merrien, J., Mørken, K., Schumaker, L. (eds.) Mathematical Methods for Curves and Surfaces, pp. 419–433. Lecture Notes in Computer Science 5862, (2010)
Vuong, A.V., Giannelli, C., Jüttler, B., Simeon, B.: A hierarchical approach to adaptive local refinement in isogeometric analysis. Comput. Methods Appl. Mech. Eng. 200, 3554–3567 (2011)
Wu, M., Xu, J., Wang, R., Yang, Z.: Hierarchical bases of spline spaces with highest order smoothness over hierarchical T-subdivisions. Comput. Aided Geom. Des. 29, 499–509 (2012)
Yvart, A., Hahmann, S.: Hierarchical triangular splines. ACM Trans. Graph. 24, 1374–1391 (2005)
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Giannelli, C., Jüttler, B. & Speleers, H. Strongly stable bases for adaptively refined multilevel spline spaces. Adv Comput Math 40, 459–490 (2014). https://doi.org/10.1007/s10444-013-9315-2
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DOI: https://doi.org/10.1007/s10444-013-9315-2