Carlotta Giannelli

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

Adaptive isogeometric methods with hierarchical splines: Error estimator and convergence

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

On the completeness of hierarchical tensor-product B-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

Bases and dimensions of bivariate hierarchical tensor-product splines

\Gamma ^{\ell }

Design of rational rotation–minimizing rigid body motions by Hermite interpolation

. 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.

Adaptive CAD model (re-)construction with THB-splines

The problem of developing an adaptive isogeometric method (AIGM) for solving elliptic second-order partial differential equations with truncated hierarchical B-splines of arbitrary degree and different order of continuity is addressed. The adaptivity analysis holds in any space dimensions. We consider a simple residual-type error estimator for which we provide a posteriori upper and lower bound in terms of local error indicators, taking also into account the critical role of oscillations as in a standard adaptive finite element setting. The error estimates are properly combined with a simple marking strategy to define a sequence of admissible locally refined meshes and corresponding approximate solutions. The design of a refine module that preserves the admissibility of the hierarchical mesh configuration between two consectutive steps of the adaptive loop is presented. The contraction property of the quasi-error, given by the sum of the energy error and the scaled error estimator, leads to the convergence proof of the AIGM.

Algorithms and Data Structures for Truncated Hierarchical B–splines

Abstract Given a grid in R d , consisting of d bi-infinite sequences of hyperplanes (possibly with multiplicities) orthogonal to the d axes of the coordinate system, we consider the spaces of tensor-product spline functions of a given degree on a multi-cell 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 tensor-product B -splines, whose support intersects the multi-cell domain, form a basis of this spline space. More precisely, if the intersection of the support of a tensor-product B -spline with the multi-cell 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 three-dimensional case with single knots only. A hierarchical B -spline basis is defined by specifying nested hierarchies of spline spaces and multi-cell domains. We adapt the techniques from Giannelli and Juttler (2013) 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 tensor-product splines, the decoupled basis provides the completeness property for any choice of the domain hierarchy.

Adaptive isogeometric methods with hierarchical splines: Optimality and convergence rates

We prove that the dimension of bivariate tensor-product spline spaces of bi-degree (d,d) with maximum order of smoothness on a multi-cell domain (more precisely, on a set of cells from a tensor-product grid) is equal to the number of tensor-product B-spline basis functions, defined by only single knots in both directions, acting on the considered domain. A certain reasonable assumption on the configuration of the cells is required. This result is then generalized to the case of piecewise polynomial spaces, with the same smoothness properties mentioned above, defined on a multi-grid multi-cell domain (more precisely, on a set of cells from a hierarchy of tensor-product grids). Again, a certain reasonable assumption regarding the configuration of cells is needed. Finally, it is observed that this construction corresponds to the classical definition of hierarchical B-spline bases. This allows to conclude that this basis spans the full space of spline functions on multi-grid multi-cell domains under reasonable assumptions.

Path planning with obstacle avoidance by G 1 PH quintic splines

The construction of space curves with rational rotation-minimizing frames (RRMF curves) by the interpolation of G1 Hermite data, i.e., initial/final points pi and pf and frames (ti, ui, vi) and (tf , uf , vf ), is addressed. Noting that the RRMF quintics form a proper subset of the spatial Pythagorean–hodograph (PH) quintics, characterized by a vector constraint on their quaternion coefficients, and that C1 spatial PH quintic Hermite interpolants possess two free scalar parameters, sufficient degrees of freedom for satisfying the RRMF condition and interpolating the end points and frames can be obtained by relaxing the Hermite data from C1 to G1. It is shown that, after satisfaction of the RRMF condition, interpolation of the end frames can always be achieved by solving a quadratic equation with a positive discriminant. Three scalar freedoms then remain for interpolation of the end–point displacement pf −pi, and this can be reduced to computing the real roots of a degree 6 univariate polynomial. The nonlinear dependence of the polynomial coefficients on the prescribed data precludes simple a priori guarantees for the existence of solutions in all cases, although existence is demonstrated for the asymptotic case of densely–sampled data from a smooth curve. Modulation of the hodograph by a scalar polynomial is proposed as a means of introducing additional degrees of freedom, in cases where solutions to the end–point interpolation problem are not found. The methods proposed herein are expected to find important applications in exactly specifying rigid–body motions along curved paths, with minimized rotation, for animation, robotics, spatial path planning, and geometric sweeping operations.