A B-spline basis for C1 quadratic splines on triangulations with a 10-split

Abstract The paper considers the macro-element splitting technique that refines every triangle of the initial triangulation into ten smaller triangles. The resulting refinement is an extension of the well-known Powell–Sabin 6-split and enables a construction of polynomial C 1 splines of degree two interpolating first order Hermite data at the vertices of the initial triangulation. A particular construction, called a balanced 10-split, is presented that allows a numerically stable B-spline representation of such splines. This amounts to, firstly, defining locally supported basis functions for the macro-element space that form a convex partition of unity, and, secondly, expressing the coefficients of the spline represented in this basis by the means of spline values and derivatives at the vertices of the initial triangulation.

[1]  Hendrik Speleers,et al.  Numerical solution of partial differential equations with Powell-Sabin splines , 2006 .

[2]  Larry L. Schumaker,et al.  Bounds on the dimension of spaces of multivariate piecewise polynomials , 1984 .

[3]  A. Bultheel,et al.  Stable multiresolution analysis on triangles for surface compression , 2006 .

[4]  Hendrik Speleers,et al.  Isogeometric analysis with Powell–Sabin splines for advection–diffusion–reaction problems , 2012 .

[5]  Paul Dierckx,et al.  On calculating normalized Powell-Sabin B-splines , 1997, Comput. Aided Geom. Des..

[6]  A. Serghini,et al.  Polar forms and quadratic spline quasi-interpolants on Powell--Sabin partitions , 2009 .

[7]  D. Sbibih,et al.  C1 Quadratic and C2 Quartic Macro-Elements on a Modified Morgan-Scott Triangulation , 2013 .

[8]  Hendrik Speleers,et al.  Construction of Normalized B-Splines for a Family of Smooth Spline Spaces Over Powell–Sabin Triangulations , 2013 .

[9]  Ahmed Tijini,et al.  A normalized basis for C1 cubic super spline space on Powell-Sabin triangulation , 2014, Math. Comput. Simul..

[10]  Abdelleh Lamnii,et al.  A normalized basis for condensed C1 Powell-Sabin-12 splines , 2015, Comput. Aided Geom. Des..

[11]  Victor Klee,et al.  Finding the Smallest Triangles Containing a Given Convex Polygon , 1985, J. Algorithms.

[12]  Tianjun Wang,et al.  A C2-quintic spline interpolation scheme on triangulation , 1992, Comput. Aided Geom. Des..

[13]  Hendrik Speleers.  A new B-spline representation for cubic splines over Powell-Sabin triangulations , 2015, Comput. Aided Geom. Des..

[14]  Tamal K. Dey,et al.  Delaunay Mesh Generation , 2012, Chapman and Hall / CRC computer and information science series.

[15]  Gerald Farin,et al.  Triangular Bernstein-Bézier patches , 1986, Comput. Aided Geom. Des..

[16]  Jan Groselj,et al.  C1 cubic splines on Powell-Sabin triangulations , 2016, Appl. Math. Comput..

[17]  Alok Aggarwal,et al.  An Optimal Algorithm for Finding Minimal Enclosing Triangles , 1986, J. Algorithms.

[18]  Malcolm A. Sabin,et al.  Piecewise Quadratic Approximations on Triangles , 1977, TOMS.

[19]  Tom Lyche,et al.  A B-spline-like basis for the Powell-Sabin 12-split based on simplex splines , 2013, Math. Comput..

[20]  Hendrik Speleers,et al.  Multivariate normalized Powell-Sabin B-splines and quasi-interpolants , 2013, Comput. Aided Geom. Des..

[21]  Carla Manni,et al.  Quadratic spline quasi-interpolants on Powell-Sabin partitions , 2007, Adv. Comput. Math..

[22]  Paul Dierckx,et al.  Subdivision of uniform Powell-Sabin splines , 1999, Comput. Aided Geom. Des..

[23]  Hendrik Speleers,et al.  A normalized basis for reduced Clough-Tocher splines , 2010, Comput. Aided Geom. Des..

[24]  Adhemar Bultheel,et al.  Automatic construction of control triangles for subdivided Powell-Sabin splines , 2004, Comput. Aided Geom. Des..

[25]  Larry L. Schumaker,et al.  Spline functions on triangulations , 2007, Encyclopedia of mathematics and its applications.