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.

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 Γℓ=0,…,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 Γℓ. 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

We consider fast solvers for large linear systems arising from the Galerkin approximation based on B-splines of classical ddimensional\ud elliptic problems, d ≥ 1, in the context of isogeometric analysis. Our ultimate goal is to design iterative algorithms\ud with the following two properties. First, their computational cost is optimal, that is linear with respect to the number of degrees of\ud freedom, i.e. the resulting matrix size. Second, they are totally robust, i.e., their convergence speed is substantially independent of\ud all the relevant parameters: in our case, these are the matrix size (related to the fineness parameter), the spline degree (associated to\ud the approximation order), and the dimensionality d of the problem.We review several methods like PCG, multigrid, multi-iterative\ud algorithms, and we show how their numerical behavior (in terms of convergence speed) can be understood through the notion of\ud spectral distribution, i.e. through a compact symbol which describes the global eigenvalue behavior of the considered stiffness\ud matrices. As a final step, we show how we can design an optimal and totally robust multi-iterative method, by taking into account\ud the analytic features of the symbol. A wide variety of numerical experiments, few open problems and perspectives are presented\ud and critically discussed.\ud ⃝c 2014 Elsevier B.V. All rights reserved

We develop a multi-degree polar spline framework with applications to both geometric modeling and isogeometric analysis. First, multi-degree splines are introduced as piecewise non-uniform rational B-splines (NURBS) of non-uniform or variable polynomial degree, and a simple algorithm for their construction is presented. Then, an extension to two-dimensional polar configurations is provided by means of a tensor-product construction with a collapsed edge. Suitable combinations of these basis functions yield C k smooth polar splines for any k ≥ 0. We show that it is always possible to construct a set of smooth polar spline basis functions that form a convex partition of unity and possess locality. Explicit constructions for k ∈ {0, 1, 2} are presented. Optimal approximation behavior is observed numerically, and examples of free-form design, smooth hole-filling, and high-order partial differential equations demonstrate the applicability of the developed framework.