arXiv subjects

Alessandra Sestini

Publications and source records attributed to Alessandra Sestini.

At least 19 recordsLinked to original sources

arXiv2026

An algorithmic approach to direct spline products: procedures and computational aspects

We introduce an efficient algorithmic procedure for implementing the direct formula that represents the product of splines in the B-spline basis. We first demonstrate the relevance of this direct approach through numerical evidence showing that implicit methods, such as collocation, may fail in some instances due to severe ill-conditioning of the associated system matrices, whereas the direct formula remains robust. We then recast the direct formula into an algorithmic framework based on the Oslo Algorithm and subsequently enhance it, through a factorization of the terms to be computed, to dramatically improve computational efficiency. Extensive numerical experiments illustrate the substantial reduction in computational cost achieved by the proposed method. Implementation aspects are also discussed to ensure numerical stability and applicability.

math.NA↗

arXiv2025

A Smoothly Varying Quadrature Approach for 3D IgA-BEM Discretizations: Application to Stokes Flow Simulations

We introduce a novel quadrature strategy for Isogeometric Analysis (IgA) boundary element discretizations, specifically tailored to collocation methods. Thanks to the dimensionality reduction and the natural handling of unbounded domains, boundary integral formulations are particularly appealing in the IgA framework. However, they require the evaluation of boundary integrals whose kernels exhibit singular or nearly singular behavior. Even when the kernel is not singular, its numerical evaluation becomes challenging whenever the integration region lies close to a collocation point. These integrals of polar and nearly singular functions represent the main computational difficulty of IgA-BEM and motivate the development of efficient and accurate quadrature rules. Unlike traditional methods that classify integrals as singular, nearly singular, or regular, our approach employs a desingularizing change of variables that smoothly adapts to the physical distance from singularities in the boundary integral kernels. The transformation intensifies near the polar point and progressively weakens when integrating over portions of the domain that are farther from it, ultimately leaving the integrand unchanged in the limit of a diametrically opposed region. This automatic calibration enhances accuracy and robustness by eliminating the traditional classification step, to which the approximation quality is often highly sensitive. Moreover, integration is performed directly over B-spline supports rather than over individual elements, reducing computational cost, particularly for higher-degree splines. The proposed method is validated through boundary element benchmarks for the three dimensional Stokes problem, where we achieve excellent convergence rates.

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arXiv2025

Decoupling solution and quadrature nodes in meshless Nystr\"om methods for second-kind Fredholm integral equations

We introduce a meshless Nystr\"om method for Fredholm integral equations of the second kind with smooth kernels in which the solution and quadrature nodes are chosen independently. Meshless moment-free quadrature formulas discretize the integral operator on scattered nodes, while local reconstruction with polyharmonic spline radial basis functions transfers values from a coarser set of solution nodes to a finer set of quadrature nodes. This construction yields a high-order method applicable to complex domains and irregular node distributions. We provide a well-posedness and convergence analysis of the resulting discretization. Under natural assumptions, we establish unique solvability for sufficiently dense node sets and derive an error bound that separates the contributions of quadrature and reconstruction. The overall convergence order is determined by the lower of the quadrature and reconstruction orders. When these orders coincide, we propose a coarse-grid parameter sweep to identify a nearly optimal ratio between the densities of solution and quadrature nodes. Numerical experiments on planar domains confirm the predicted rates and demonstrate a substantial increase in computational efficiency over the classical Nystr\"om method, especially for narrow kernels.

math.NA↗

arXiv2025

Piecewise rational rotation-minimizing motions via data stream interpolation

When a moving frame defined along a space curve is required to keep an axis aligned with the tangent direction of motion, the use of rotation-minimizing frames (RMF) avoids unnecessary rotations in the normal plane. The construction of rigid body motions using a specific subset of quintic curves with rational RMFs (RRMFs) is here considered. In particular, a novel geometric characterization of such subset enables the design of a local algorithm to interpolate an assigned stream of positions, together with an initial frame orientation. To achieve this, the translational part of the motion is described by a parametric $G^1$ spline curve whose segments are quintic RRMFs, with a globally continuous piecewise rational rotation-minimizing frame. A selection of numerical experiments illustrates the performances of the proposed method on synthetic and arbitrary data streams.

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arXiv2025

A positive meshless finite difference scheme for scalar conservation laws with adaptive artificial viscosity driven by fault detection

We present a meshless finite difference method for multivariate scalar conservation laws that generates positive schemes satisfying a local maximum principle on irregular nodes and relies on artificial viscosity for shock capturing. Coupling two different numerical differentiation formulas and the adaptive selection of the sets of influence allows to meet a local CFL condition without any {\it a priori}\ time step restriction. The artificial viscosity term is chosen in an adaptive way by applying it only in the vicinity of the sharp features of the solution identified by an algorithm for fault detection on scattered data. Numerical tests demonstrate a robust performance of the method on irregular nodes and advantages of adaptive artificial viscosity. The accuracy of the obtained solutions is comparable to that for standard monotone methods available only on Cartesian grids.

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arXiv2024

Robust path following for autonomous vehicles with spatial PH quintic splines

The distinctive feature of a polynomial parametric speed let polynomial Pythagorean-hodograph (PH) curves be attractive for the design of accurate and efficient application algorithms. We propose a robust path following scheme for the construction of smooth spatial motions by exploiting PH spline curves. In order to cover a general configuration setting, we present a guidance law which is suitable both for fully-actuated and (more common) under-actuated vehicles, which cannot control all the degrees of freedom. The robustness of the guidance law is enhanced by also taking into account the influence of wind or currents into the equations of motion. A selection of numerical experiments validates the effectiveness of the control strategy when $C^1$ spatial PH quintic interpolants are suitably considered for both kinematic and dynamic simulations.

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arXiv2024

Local spline refinement driven by fault jump estimates for scattered data approximation

We present new fault jump estimates to guide local refinement in surface approximation schemes with adaptive spline constructions. The proposed approach is based on the idea that, since discontinuities in the data should naturally correspond to sharp variations in the reconstructed surface, the location and size of jumps detected in the input point cloud should drive the mesh refinement algorithm. To exploit the possibility of inserting local meshlines in one or the other coordinate direction, as suggested by the jump estimates, we propose a quasi-interpolation (QI) scheme based on locally refined B-splines (LR B-splines). Particular attention is devoted to the construction of the local operator of the LR B-spline QI scheme, which properly adapts the spline approximation according to the nature and density of the scattered data configuration. A selection of numerical examples outlines the performance of the method on synthetic and real datasets characterized by different geographical features.

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arXiv2022

IgA-BEM for 3D Helmholtz problems on multi-patch domains using B-spline tailored numerical integration

An Isogeometric Boundary Element Method (IgA-BEM) is considered for the numerical solution of Helmholtz problems on 3D bounded or unbounded domains, admitting a smooth conformal multi-patch representation of their finite boundary surface. The discretization space is formed by $C^0$ inter-patch continuous basis functions whose restriction to a patch simplifies to the span of tensor product B-splines composed with the given patch parameterization. For both regular and singular integration, the proposed model utilizes a numerical procedure defined on the support of each trial B-spline function, which makes possible a function--by--function implementation of the matrix assembly phase. Spline quasi-interpolation is the common ingredient of all the considered quadrature rules; in the singular case it is combined with a B-spline recursion over the spline degree and with a singularity extraction technique, extended to the multi-patch setting for the first time. A threshold selection strategy is proposed to automatically distinguish between nearly singular and regular integrals. Numerical examples on relevant benchmarks show that the expected convergence orders are achieved with uniform discretization and a small number of uniformly spaced quadrature nodes.

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arXiv2022

Construction of $G^2$ planar Hermite interpolants with prescribed arc lengths

In this paper we address the problem of constructing $G^2$ planar Pythagorean--hodograph (PH) spline curves, that interpolate points, tangent directions and curvatures, and have prescribed arc-length. The interpolation scheme is completely local. Each spline segment is defined as a PH biarc curve of degree $7$, which results in having a closed form solution of the $G^2$ interpolation equations depending on four free parameters. By fixing two of them to zero, it is proven that the length constraint can be satisfied for any data and any chosen ratio between the two boundary tangents. Length interpolation equation reduces to one algebraic equation with four solutions in general. To select the best one, the value of the bending energy is observed. Several numerical examples are provided to illustrate the obtained theoretical results and to numerically confirm that the approximation order is $5$.

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arXiv2021

A local $C^2$ Hermite interpolation scheme with PH quintic splines for 3D data streams

The construction of smooth spatial paths with Pythagorean-hodograph (PH) quintic spline biarcs is proposed. To facilitate real-time computations of $C^2$ PH quintic splines, an efficient local data stream interpolation algorithm is introduced. Each spline segment interpolates second and first order Hermite data at the initial and final end-point, respectively. In the spline extension of the scheme a $C^2$ smooth connection between successive spline segments is obtained by taking the locally required second-order derivative information from the previous segment. Consequently, the data stream spline interpolant is globally $C^2$ continuous and can be constructed for arbitrary $C^1$ Hermite data configurations. A simple and effective selection of the free parameters that arise in the interpolation problem is proposed. The developed theoretical analysis proves the fourth approximation order of the local scheme while a selection of numerical examples confirms the same accuracy of its spline extension. In addition, the performances of the algorithm are also validated by considering its application to point stream interpolation with automatically generated first-order derivative information.

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arXiv2020

THB-spline approximations for turbine blade design with local B-spline approximations

We consider two-stage scattered data fitting with truncated hierarchical B-splines (THB-splines) for the adaptive reconstruction of industrial models. The first stage of the scheme is devoted to the computation of local least squares variational spline approximations, exploiting a simple fairness functional to handle data distributions with a locally varying density of points. Hierarchical spline quasi-interpolation based on THB-splines is considered in the second stage of the method to construct the adaptive spline surface approximating the whole scattered data set and a suitable strategy to guide the adaptive refinement is introduced. A selection of examples on geometric models representing components of aircraft turbine blades highlights the performances of the scheme. The tests include a scattered data set with voids and the adaptive reconstruction of a cylinder-like surface.

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arXiv2018

An application of numerical differentiation formulas to discontinuity curve detection from irregularly sampled data

We present a method to detect discontinuity curves, usually called faults, from a set of scattered data. The scheme first extracts from the data set a subset of points close to the faults. This selection is based on an indicator obtained by using numerical differentiation formulas with irregular centers for gradient approximation, since they can be directly applied to the scattered point cloud without intermediate approximations on a grid. The shape of the faults is reconstructed through local computations of regression lines and quadratic least squares approximations. In the final reconstruction stage, a suitable curve interpolation algorithm is applied to the selected set of ordered points previously associated with each fault.

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arXiv2018

An adaptive IGA-BEM with hierarchical B-splines based on quasi-interpolation quadrature schemes

The isogeometric formulation of Boundary Element Method (BEM) is investigated within the adaptivity framework. Suitable weighted quadrature rules to evaluate integrals appearing in the Galerkin BEM formulation of 2D Laplace model problems are introduced. The new quadrature schemes are based on a spline quasi-interpolant (QI) operator and properly framed in the hierarchical setting. The local nature of the QI perfectly fits with hierarchical spline constructions and leads to an efficient and accurate numerical scheme. An automatic adaptive refinement strategy is driven by a residual based error estimator. Numerical examples show that the optimal convergence rate of the BEM solution is recovered by the proposed adaptive method.

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arXiv2017

C2 continuous time dependent feedrate scheduling with configurable kinematic constraints

We present a configurable trajectory planning strategy on planar paths for offline definition of time-dependent C2 piecewise quintic feedrates. The more conservative formulation ensures chord tolerance, as well as prescribed bounds on velocity, acceleration and jerk Cartesian components. Since the less restrictive formulations of our strategy can usually still ensure all the desired bounds while simultaneously producing faster motions, the configurability feature is useful not only when reduced motion control is desired but also when full kinematic control has to be guaranteed. Our approach can be applied to any planar path with a piecewise sufficiently smooth parametric representation. When Pythagoreanhodograph spline curves are considered, the corresponding accurate and efficient CNC interpolator algorithms can be exploited.

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arXiv2017

Adaptive scattered data fitting by extension of local approximations to hierarchical splines

We introduce an adaptive scattered data fitting scheme as extension of local least squares approximations to hierarchical spline spaces. To efficiently deal with non-trivial data configurations, the local solutions are described in terms of (variable degree) polynomial approximations according not only to the number of data points locally available, but also to the smallest singular value of the local collocation matrices. These local approximations are subsequently combined without the need of additional computations with the construction of hierarchical quasi-interpolants described in terms of truncated hierarchical B-splines. A selection of numerical experiments shows the effectivity of our approach for the approximation of real scattered data sets describing different terrain configurations.

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arXiv2016

A comprehensive characterization of the set of polynomial curves with rational rotation-minimizing frames

A rotation-minimizing frame $({\bf f}_1,{\bf f}_2,{\bf f}_3)$ on a space curve ${\bf r}(\xi)$ defines an orthonormal basis for $\mathbb{R}^3$ in which ${\bf f}_1={\bf r}'/|{\bf r}'|$ is the curve tangent, and the normal-plane vectors ${\bf f}_2$, ${\bf f}_3$ exhibit no instantaneous rotation about ${\bf f}_1$. Polynomial curves that admit rational rotation-minimizing frames (or RRMF curves) form a subset of the Pythagorean-hodograph (PH) curves, specified by integrating the form ${\bf r}'(\xi)={\cal A}(\xi)\,{\bf i}\,{\cal A}^*(\xi)$ for some quaternion polynomial ${\cal A}(\xi)$. By introducing the notion of rotation indicatrix and of core of the quaternion polynomial ${\cal A}(\xi)$, a comprehensive characterization of the complete space of RRMF curves is developed, that subsumes all previously known special cases. This novel characterization helps clarify the structure of the complete space of RRMF curves, distinguishes the spatial RRMF curves from trivial (planar) cases, and paves the way toward new construction algorithms.

math.NA↗

arXiv2016

Bivariate hierarchical Hermite spline quasi--interpolation

Spline quasi-interpolation (QI) is a general and powerful approach for the construction of low cost and accurate approximations of a given function. In order to provide an efficient adaptive approximation scheme in the bivariate setting, we consider quasi-interpolation in hierarchical spline spaces. In particular, we study and experiment the features of the hierarchical extension of the tensor-product formulation of the Hermite BS quasi-interpolation scheme. The convergence properties of this hierarchical operator, suitably defined in terms of truncated hierarchical B-spline bases, are analyzed. A selection of numerical examples is presented to compare the performances of the hierarchical and tensor-product versions of the scheme.

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arXiv2015

Solution of a quadratic quaternion equation with mixed coefficients

A comprehensive analysis of the morphology of the solution space for a special type of quadratic quaternion equation is presented. This equation, which arises in a surface construction problem, incorporates linear terms in a quaternion variable and its conjugate with right and left quaternion coefficients, while the quadratic term has a quaternion coefficient placed between the variable and its conjugate. It is proved that, for generic coefficients, the equation has two, one, or no solutions, but in certain special instances the solution set may comprise a circle or a 3-sphere in the quaternion space $\mathbb{H}$. The analysis yields solutions for each case, and intuitive interpretations of them in terms of the four-dimensional geometry of the quaternion space $\mathbb{H}$.

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