Papers by Carlotta Giannelli
Isogeometric analysis with C1 hierarchical functions on planar two-patch geometries
Computers & Mathematics with Applications
Fault and gradient fault detection and reconstruction from scattered data
Computer Aided Geometric Design
Computer Aided Geometric Design
We present a configurable trajectory planning strategy on planar paths for offline definition of ... more We present a configurable trajectory planning strategy on planar paths for offline definition of time-dependent C 2 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.
Computer Methods in Applied Mechanics and Engineering
In the present work we introduce a complete set of algorithms to efficiently perform adaptive ref... more In the present work we introduce a complete set of algorithms to efficiently perform adaptive refinement and coarsening by exploiting truncated hierarchical B-splines (THB-splines) defined on suitably graded isogeometric meshes, that are called admissible mesh configurations. We apply the proposed algorithms to two-dimensional linear heat transfer problems with localized moving heat source, as simplified models for additive manufacturing applications. We first verify the accuracy of the admissible adaptive scheme with respect to an overkilled solution, for then comparing our results with similar schemes which consider different refinement and coarsening algorithms, with or without taking into account grading parameters. This study shows that the THB-spline admissible solution delivers an optimal discretization for what concerns not only the accuracy of the approximation, but also the (reduced) number of degrees of freedom per time step. In the last example we investigate the capability of the algorithms to approximate the thermal history of the problem for a more complicated source path. The comparison with uniform and non-admissible hierarchical meshes demonstrates that also in this case our adaptive scheme returns the desired accuracy while strongly improving the computational efficiency.
Adaptive fitting with THB-splines: Error analysis and industrial applications
Computer Aided Geometric Design
Axioms
The construction of suitable mesh configurations for spline models that provide local refinement ... more The construction of suitable mesh configurations for spline models that provide local refinement capabilities is one of the fundamental components for the analysis and development of adaptive isogeometric methods. We investigate the design and implementation of refinement algorithms for hierarchical B-spline spaces that enable the construction of locally graded meshes. The refinement rules properly control the interaction of basis functions at different refinement levels. This guarantees a bounded number of nonvanishing (truncated) hierarchical B-splines on any mesh element. The performances of the algorithms are validated with standard benchmark problems.
Journal of Computational and Applied Mathematics
A minimal twist frame (f 1 (ξ), f 2 (ξ), f 3 (ξ)) on a polynomial space curve r(ξ), ξ ∈ [ 0, 1 ] ... more A minimal twist frame (f 1 (ξ), f 2 (ξ), f 3 (ξ)) on a polynomial space curve r(ξ), ξ ∈ [ 0, 1 ] is an orthonormal frame, where f 1 (ξ) is the tangent and the normal-plane vectors f 2 (ξ), f 3 (ξ) have the least variation between given initial and final instances f 2 (0), f 3 (0) and f 2 (1), f 3 (1). Namely, if ω = ω 1 f 1 +ω 2 f 2 +ω 3 f 3 is the frame angular velocity, the component ω 1 does not change sign, and its arc length integral has the smallest value consistent with the boundary conditions. We consider construction of curves with rational minimal twist frames, based on the Pythagoreanhodograph curves of degree 7 that have rational rotation-minimizing Euler-Rodrigues frames (e 1 (ξ), e 2 (ξ), e 3 (ξ)) -i.e., the normal-plane vectors e 2 (ξ), e 3 (ξ) have no rotation about the tangent e 1 (ξ). A set of equations that govern the construction of such curves with prescribed initial/final points and tangents, and total arc length, is derived. For the resulting curves f 2 (ξ), f 3 (ξ) are then obtained from e 2 (ξ), e 3 (ξ) by a monotone rational normal-plane rotation, subject to the boundary conditions. A selection of computed examples is included to illustrate the construction, and it is shown that the desirable feature of a uniform rotation rate (i.e., ω 1 = constant) can be accurately approximated.
An immersed-isogeometric model: Application to linear elasticity and implementation with THBox-splines
Journal of Computational and Applied Mathematics
International Journal for Numerical Methods in Engineering
The isogeometric formulation of Boundary Element Method (BEM) is investigated within the adaptivi... more 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.
Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering
The Pythagorean-hodograph (PH) curves offer distinct advantages in planning curvilinear paths for... more The Pythagorean-hodograph (PH) curves offer distinct advantages in planning curvilinear paths for unmanned or autonomous air, ground, or underwater vehicles. Although several authors have discussed their use in these contexts, prior studies contain misconceptions about the properties of PH curves or invoke heuristic approximate constructions when exact methods are available. To address these issues, the present study provides a basic introduction to the key properties of PH curves, and describes some exact constructions of particular interest in path planning. These include maintenance of minimum safe separations within vehicle swarms; (2) construction of paths of different shape but identical arc length, ensuring simultaneous arrival of vehicles travelling at a constant speed; (3) determination of the curvature extrema of PH paths, and their modification to satisfy a given curvature bound; and (4) construction of curvature-continuous paths of bounded curvature through fields of polygonal obstacles.
Curvature continuous path planning and path finding based on PH splines with tension
Computer-Aided Design
Computer Aided Geometric Design, 2016
An adaptive isogeometric method based on d-variate hierarchical spline constructions can be deriv... more An adaptive isogeometric method based on d-variate hierarchical spline constructions can be derived by considering a refine module that preserves a certain class of admissibility between two consecutive steps of the adaptive loop .
Computer Aided Geometric Design, 2017
We introduce an adaptive scattered data fitting scheme as extension of local least squares approx... more 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.
On Quasi-Interpolation Operators in Spline Spaces
Lecture Notes in Computational Science and Engineering, 2016
Advances in Computational Mathematics, 2016
A rotation-minimizing frame (f 1 , f 2 , f 3 ) on a space curve r(ξ) defines an orthonormal basis... more A rotation-minimizing frame (f 1 , f 2 , f 3 ) on a space curve r(ξ) defines an orthonormal basis for R 3 in which f 1 = r ′ /|r ′ | is the curve tangent, and the normal-plane vectors f 2 , f 3 exhibit no instantaneous rotation about 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 r ′ (ξ) = A(ξ) i A * (ξ) for some quaternion polynomial A(ξ). By introducing the notion of the rotation indicatrix and the core of the quaternion polynomial A(ξ), 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.
Computer Methods in Applied Mechanics and Engineering, 2016
Isogeometric analysis is a recently developed framework based on finite element analysis, where t... more Isogeometric analysis is a recently developed framework based on finite element analysis, where the simple building blocks in geometry and solution space are replaced by more complex and geometricallyoriented compounds. Box splines are an established tool to model complex geometry, and form an intermediate approach between classical tensor-product B-splines and splines over triangulations. Local refinement can be achieved by considering hierarchically nested sequences of box spline spaces. Since box splines do not offer special elements to impose boundary conditions for the numerical solution of partial differential equations (PDEs), we discuss a weak treatment of such boundary conditions. Along the domain boundary, an appropriate domain strip is introduced to enforce the boundary conditions in a weak sense. The thickness of the strip is adaptively defined in order to avoid unnecessary computations. Numerical examples show the optimal convergence rate of box splines and their hierarchical variants for the solution of PDEs.
Splines over regular triangulations in numerical simulation
Computer-Aided Design, 2016
Path planning with obstacle avoidance by <mml:math altimg="si1.gif" display="inline" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xml...
Computer-Aided Design, 2016
BIT Numerical Mathematics, 2016
Spline quasi-interpolation (QI) is a general and powerful approach for the construction of low co... more 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 quasiinterpolation 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.
Computer Methods in Applied Mechanics and Engineering, 2016
Local refinement with hierarchical B-spline structures is an active topic of research in the cont... more Local refinement with hierarchical B-spline structures is an active topic of research in the context of geometric modeling and isogeometric analysis. By exploiting a multilevel control structure, we show that truncated hierarchical B-spline (THB-spline) representations support interactive modeling tools, while simultaneously providing effective approximation schemes for the manipulation of complex data sets and the solution of partial differential equations via isogeometric analysis. A selection of illustrative 2D and 3D numerical examples demonstrates the potential of the hierarchical framework.
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Papers by Carlotta Giannelli