THB-splines: The truncated basis for hierarchical splines

The construction of classical hierarchical B-splines can be suitably modified in order to define locally supported basis functions that form a partition of unity. We will show that this property can be obtained by reducing the support of basis functions defined on coarse grids, according to finer levels in the hierarchy of splines. This truncation not only decreases the overlapping of supports related to basis functions arising from different hierarchical levels, but it also improves the numerical properties of the corresponding hierarchical basis - which is denoted as truncated hierarchical B-spline (THB-spline) basis. Several computed examples will illustrate the adaptive approximation behavior obtained by using a refinement algorithm based on THB-splines.

[1]  Günther Greiner,et al.  Interpolating and approximating scattered 3D-data with hierarchical tensor product B-splines , 2010 .

[2]  Jiansong Deng,et al.  Surface modeling with polynomial splines over hierarchical T-meshes , 2007, 2007 10th IEEE International Conference on Computer-Aided Design and Computer Graphics.

[3]  David R. Forsey,et al.  Multiresolution Surface Reconstruction for Hierarchical B-splines , 1998, Graphics Interface.

[4]  David R. Forsey,et al.  Hierarchical B-spline refinement , 1988, SIGGRAPH.

[5]  Hendrik Speleers,et al.  Quasi-hierarchical Powell-Sabin B-splines , 2009, Comput. Aided Geom. Des..

[6]  Larry L. Schumaker,et al.  Nonnegativity preserving macro-element interpolation of scattered data , 2010, Comput. Aided Geom. Des..

[7]  Thomas J. R. Hughes,et al.  On linear independence of T-spline blending functions , 2012, Comput. Aided Geom. Des..

[8]  Bert Jüttler,et al.  Enhancing isogeometric analysis by a finite element-based local refinement strategy , 2012 .

[9]  G. Sangalli,et al.  Linear independence of the T-spline blending functions associated with some particular T-meshes , 2010 .

[10]  R ForseyDavid,et al.  Hierarchical B-spline refinement , 1988 .

[11]  Jiansong Deng,et al.  Polynomial splines over hierarchical T-meshes , 2008, Graph. Model..

[12]  B. Simeon,et al.  A hierarchical approach to adaptive local refinement in isogeometric analysis , 2011 .

[13]  B. Simeon,et al.  Adaptive isogeometric analysis by local h-refinement with T-splines , 2010 .

[14]  David R. Forsey,et al.  Surface fitting with hierarchical splines , 1995, TOGS.

[15]  Voon Pang Kong,et al.  Range Restricted Interpolation Using Cubic Bezier Triangles , 2004, WSCG.

[16]  T. Hughes,et al.  Local refinement of analysis-suitable T-splines , 2012 .

[17]  Bert Jüttler,et al.  Bases and dimensions of bivariate hierarchical tensor-product splines , 2013, J. Comput. Appl. Math..

[18]  H. Nguyen-Xuan,et al.  Isogeometric analysis using polynomial splines over hierarchical T-meshes for two-dimensional elastic solids , 2011 .

[19]  Nicholas S. North,et al.  T-spline simplification and local refinement , 2004, SIGGRAPH 2004.

[20]  Ahmad H. Nasri,et al.  T-splines and T-NURCCs , 2003, ACM Trans. Graph..