The construction of smooth spatial paths with Pythagorean-hodograph (PH) quintic splines is proposed. To facilitate real-time computations, an efficient
local
data stream interpolation algorithm is ...introduced to successively construct each spline segment as a quintic PH biarc interpolating second- and first-order Hermite data at the initial and final end-point, respectively. 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 local interpolation problem is proposed. The developed theoretical analysis proves its fourth approximation order while a selection of numerical examples confirms the same accuracy for the spline extension of the scheme. In addition, the performances of the method are also validated by considering its application to point stream interpolation with automatically generated first-order derivative information.
The class of A-stable symmetric one-step Hermite–Obreshkov (HO) methods introduced by F. Loscalzo in 1968 for dealing with initial value problems is analyzed. Such schemes have the peculiarity of ...admitting a multiple knot spline extension collocating the differential equation at the mesh points. As a new result, it is shown that these maximal order schemes are conjugate symplectic, which is a benefit when the methods have to be applied to Hamiltonian problems. Furthermore, a new efficient approach for the computation of the spline extension is introduced, adopting the same strategy developed for the BS linear multistep methods. The performances of the schemes are tested in particular on some Hamiltonian benchmarks and compared with those of the Gauss–Runge–Kutta schemes and Euler–Maclaurin formulas of the same order.
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.
Spline Quasi-Interpolation (QI) of even degree 2R on general partitions is introduced, where derivative information up to order R≥1 at the spline breakpoints is required and maximal convergence order ...can be proved. Relying on the B-spline basis with possible multiple inner knots, a family of quasi-interpolating splines with smoothness of order R is associated with each R≥1, since there is the possibility of using different local sequences of breakpoints to define each QI spline coefficient. By using suitable finite differences approximations of the necessary discrete derivative information, each QI spline in this family can be associated also with a twin approximant belonging to the same spline space, but requiring just function information at the breakpoints.
Among possible different applications of the introduced QI scheme, a smooth continuous extension of the numerical solution of Gauss-Lobatto and Gauss-Legendre Runge-Kutta methods is here considered. When R>1, such extension is based on the use of the variant of the QI scheme with derivative approximation which preserves the approximation power of the original Runge-Kutta scheme.
Two new classes of quadrature formulas associated to the BS Boundary Value Methods are discussed. The first is of Lagrange type and is obtained by directly applying the BS methods to the integration ...problem formulated as a (special) Cauchy problem. The second descends from the related BS Hermite quasi-interpolation approach which produces a spline approximant from Hermite data assigned on meshes with general distributions. The second class formulas is also combined with suitable finite difference approximations of the necessary derivative values in order to define corresponding Lagrange type formulas with the same accuracy.
A minimal twist frame(f1(ξ),f2(ξ),f3(ξ)) on a polynomial space curve r(ξ), ξ∈0,1 is an orthonormal frame, where f1(ξ) is the tangent and the normal-plane vectors f2(ξ),f3(ξ) have the least variation ...between given initial and final instances f2(0),f3(0) and f2(1),f3(1). Namely, if ω=ω1f1+ω2f2+ω3f3 is the frame angular velocity, the component ω1 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 Pythagorean-hodograph curves of degree 7 that have rational rotation-minimizing Euler–Rodrigues frames(e1(ξ),e2(ξ),e3(ξ)) — i.e., the normal-plane vectors e2(ξ),e3(ξ) have no rotation about the tangent e1(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 f2(ξ),f3(ξ) are then obtained from e2(ξ),e3(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., ω1=constant) can be accurately approximated.
Summary
The isogeometric formulation of the boundary element method (IgA‐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 proposed quadrature schemes are based on a spline quasi‐interpolation (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 Galerkin solution is recovered by the proposed adaptive method.
In this paper, we study the construction of quadrature rules for the approximation of hypersingular integrals that occur when 2D Neumann or mixed Laplace problems are numerically solved using ...Boundary Element Methods. In particular the Galerkin discretization is considered within the Isogeometric Analysis setting and spline quasi-interpolation is applied to approximate integrand factors, then integrals are evaluated via recurrence relations. Convergence results of the proposed quadrature rules are given, with respect to both smooth and non smooth integrands. Numerical tests confirm the behavior predicted by the analysis. Finally, several numerical experiments related to the application of the quadrature rules to both exterior and interior differential problems are presented.
•We introduce new quadrature rules based on quasi interpolation for hypersingular integrals.•We give proofs of the convergence properties of the quadrature with different smoothness hypotheses on the functions.•We perform numerical tests for quadrature itself and for the quadrature applied to IgA-BEM discretization of differential problems.•All results reveal that the new quadratures are accurate and reliable.
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.
•We introduce an adaptive scattered data fitting scheme based on THB-splines.•The scheme employs local polynomial least squares approximations of variable degree.•The local approximations are combined with hierarchical quasi-interpolation.•Examples with real scattered data sets show the effectivity of our approach.
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