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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 le ..."
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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
Algorithms and Data Structures for Truncated Hierarchical B–splines
"... Abstract. Tensor–product B–spline surfaces are commonly used as standard modeling tool in Computer Aided Geometric Design and for numerical simulation in Isogeometric Analysis. However, when considering tensor–product grids, there is no possibility of a localized mesh refinement without propagation ..."
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Abstract. Tensor–product B–spline surfaces are commonly used as standard modeling tool in Computer Aided Geometric Design and for numerical simulation in Isogeometric Analysis. However, when considering tensor–product grids, there is no possibility of a localized mesh refinement without propagation of the refinement outside the region of interest. The recently introduced truncated hierarchical B–splines (THB– splines) [5] provide the possibility of a local and adaptive refinement procedure, while simultaneously preserving the partition of unity property. We present an effective implementation of the fundamental algorithms needed for the manipulation of THB–spline representations based on standard data structures. By combining a quadtree data structure — which is used to represent the nested sequence of subdomains — with a suitable data structure for sparse matrices, we obtain an efficient technique for the construction and evaluation of THB–splines.
Contents lists available at ScienceDirect Journal of Computational and Applied
"... journal homepage: www.elsevier.com/locate/cam Dimensions of biquadratic and bicubic spline spaces over ..."
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journal homepage: www.elsevier.com/locate/cam Dimensions of biquadratic and bicubic spline spaces over
CHARACTERIZATION OF BIVARIATE HIERARCHICAL QUARTIC BOX SPLINES ON A THREEDIRECTIONAL GRID
"... Abstract. We consider the adaptive refinement of bivariate quartic C2smooth boxspline spaces on the threedirectional (typeI) grid G. The polynomial segments of these box splines belong to a certain subspace of the space of quartic polynomials, which will be called the space of special quartics. ..."
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Abstract. We consider the adaptive refinement of bivariate quartic C2smooth boxspline spaces on the threedirectional (typeI) grid G. The polynomial segments of these box splines belong to a certain subspace of the space of quartic polynomials, which will be called the space of special quartics. Given a finite sequence (Gℓ)ℓ=0,...,N of dyadically refined grids, we obtain a hierarchical grid by selecting cells from each level such that their closure covers the entire domain Ω, which is a bounded subset of R2. A suitable selection procedure allows to define a basis spanning a hierarchical box spline space. As our main result, we derive a characterization of this space. More precisely, under certain mild assumptions on hierarchical grid, the hierarchical spline space is shown to contain all C2smooth functions whose restrictions to the cells of the hierarchical grid are special quartic polynomials. 1.
On the completeness of hierarchical tensorproduct Bsplines
"... Given a grid in Rd, consisting of d biinfinite sequences of hyperplanes (possibly with multiplicities) orthogonal to the d axes of the coordinate system, we consider the spaces of tensorproduct spline functions of a given degree on a multicell domain. Such a domain consists of finite set of cell ..."
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Given a grid in Rd, consisting of d biinfinite sequences of hyperplanes (possibly with multiplicities) orthogonal to the d axes of the coordinate system, we consider the spaces of tensorproduct spline functions of a given degree on a multicell domain. Such a domain consists of finite set of cells which are defined by the grid. A piecewise polynomial function belongs to the spline space if its polynomial pieces on adjacent cells have a contact according to the multiplicity of the hyperplanes in the grid. We prove that the connected components of the associated set of tensorproduct Bsplines, whose support intersects the multicell domain, form a basis of this spline space. More precisely, if the intersection of the support of a tensorproduct Bspline with the multicell domain consists of several connected components, then each of these components contributes one basis function. In order to establish the connection to earlier results, we also present further details relating to the threedimensional case with single knots only. A hierarchical Bspline basis is defined by specifying nested hierarchies of spline spaces and multicell domains. We adapt the techniques from [12] to the more general setting and prove the completeness of this basis (in the sense that its span contains all piecewise polynomial functions on the hierarchical grid with the smoothness specified by the grid and the degrees) under certain assumptions on the domain hierarchy. Finally, we introduce a decoupled version of the hierarchical spline basis that allows to relax the assumptions on the domain hierarchy. In certain situations, such as quadratic tensorproduct splines, the decoupled basis provides the completeness property for any choice of the domain hierarchy.