Hendrik Speleers

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.

Strongly stable bases for adaptively refined multilevel spline spaces

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}ℓ=0,…,N−1

Robust and optimal multi-iterative techniques for IgA Galerkin linear systems

\{\Gamma ^{\ell }\}_{\ell =0,\ldots ,N-1}

A normalized basis for quintic Powell--Sabin splines

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

Effortless quasi-interpolation in hierarchical spaces

\Gamma ^{\ell }

On the spectrum of stiffness matrices arising from isogeometric analysis

. 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.

Quasi-hierarchical Powell--Sabin B-splines

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

Symbol-Based Multigrid Methods for Galerkin B-Spline Isogeometric Analysis

We construct a suitable normalized B-spline representation for C^2-continuous quintic Powell-Sabin splines. The basis functions have a local support, they are nonnegative, and they form a partition of unity. The construction is based on the determination of a set of triangles that must contain a specific set of points. We are able to define control points and cubic control polynomials which are tangent to the spline surface. We also show how to compute the Bezier control net of such a spline in a stable way.

Multivariate normalized Powell-Sabin B-splines and quasi-interpolants

We present a general and simple procedure to construct quasi-interpolants in hierarchical spaces. Such spaces are composed of a hierarchy of nested spaces and provide a flexible framework for local refinement. The proposed hierarchical quasi-interpolants are described in terms of the so-called truncated hierarchical basis. Assuming a quasi-interpolant is selected for each space associated with a particular level in the hierarchy, the hierarchical quasi-interpolants are obtained without any additional manipulation. The main properties (like polynomial reproduction) of the quasi-interpolants selected at each level are locally preserved in the hierarchical construction. We show how to construct hierarchical local projectors, and the local approximation order of the underling hierarchical space is also investigated. The presentation is detailed for the truncated hierarchical B-spline basis, and we discuss its extension to a more general framework.

A normalized basis for reduced Clough-Tocher splines

We study the spectral properties of stiffness matrices that arise in the context of isogeometric analysis for the numerical solution of classical second order elliptic problems. Motivated by the applicative interest in the fast solution of the related linear systems, we are looking for a spectral characterization of the involved matrices. In particular, we investigate non-singularity, conditioning (extremal behavior), spectral distribution in the Weyl sense, as well as clustering of the eigenvalues to a certain (compact) subset of