In this paper, we formally investigate two mathematical aspects of Hermite splines which translate to features that are relevant to their practical applications. We first demonstrate that Hermite splines are maximally localized in the sense that their support sizes are minimal among pairs of functions with identical reproduction properties. Then, we precisely quantify the approximation power of Hermite splines for reconstructing functions and their derivatives, and show that they are asymptotically identical to cubic B-splines for these tasks. Hermite splines therefore combine optimal localization and excellent approximation power, while retaining interpolation properties and closed-form expression, in contrast to existing similar approaches. These findings shed a new light on the convenience of Hermite splines for use in computer graphics and geometrical design.
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 sp…
The method of constructing trigonometric Hermite splines, which interpolate the values of some periodic function and its derivatives in the nodes of a uniform grid, is considered. The proposed method is based on the periodicity properties of trigonometric functions and is reduced to solving only systems of linear algebraic equations of the second order; solutions of these systems can be obtained in advance. When implementing this method, it is necessary to calculate the coefficients of interpolation trigonometric polynomials that interpolate the values of the function itself and the values of …
The library QIBSH++ is a C++ object oriented library for the solution of Quasi Interpolation problems. The library is based on a Hermite Quasi Interpolating operator, which was derived as continuous extensions of linear multistep methods applied for the numerical solution of Boundary Value Problems for Ordinary Differential Equations. The library includes the possibility to use Hermite data or to apply a finite difference scheme for derivative approximations, when derivative values are not directly available. The generalization of the quasi interpolation procedure to surfaces and volumes appro…
Hermite parametric surface interpolation based on Argyris element
University of Ljubljana（卢布尔雅那大学）；University of Primorska（普里莫斯卡大学）；INdAM, Unità di Ricerca di Firenze c/o DiMaI “U. Dini”, University of Florence（意大利国家数学研究所佛罗伦萨研究单元迪马伊“乌·迪尼”系，佛罗伦萨大学）
Element-based B-spline basis function spaces: construction and application in isogeometric analysis
University of Electronic Science and Technology of China（电子科技大学）；University of Science and Technology of China（中国科学技术大学）；Wayne State University（韦恩州立大学）；Nanjing University of Aeronautics and Astronautics（南京航空航天大学）
Spline interpolation has been used in several applications due to its favorable properties regarding smoothness and accuracy of the interpolant. However, when there exists a discontinuity or a steep gradient in the data, some artifacts can appear due to the Gibbs phenomenon. Also, preservation of data monotonicity is a requirement in some applications, and that property is not automatically verified by the interpolator. In this paper, we study sufficient conditions to obtain monotone cubic splines based on Hermite cubic interpolators and propose different ways to construct them using non-linea…
Generalized sampling consists in the recovery of a function $f$, from the samples of the responses of a collection of linear shift-invariant systems to the input $f$. The reconstructed function is typically a member of a finitely generated integer-shift-invariant space that can reproduce polynomials up to a given degree $M$. While this property allows for an approximation power of order $(M+1)$, it comes with a tradeoff on the length of the support of the basis functions. Specifically, we prove that the sum of the length of the support of the generators is at least $(M+1)$. Following this resu…
We present a novel method named truncated hierarchical unstructured splines (THU-splines) that supports both local $h$-refinement and unstructured quadrilateral meshes. In a THU-spline construction, an unstructured quadrilateral mesh is taken as the input control mesh, where the degenerated-patch method [18] is adopted in irregular regions to define $C^1$-continuous bicubic splines, whereas regular regions only involve $C^2$ B-splines. Irregular regions are then smoothly joined with regular regions through the truncation mechanism [29], leading to a globally smooth spline construction. Subsequ…
This paper presents a PDE-based planar parameterization framework with support for Truncated Hierarchical B-Splines (THB-splines). For this, we adopt the a posteriori refinement strategy of Dual Weighted Residual and present several adaptive numerical schemes for the purpose of approximating an inversely harmonic geometry parameterization. Hereby, the combination of goal-oriented a posteriori strategies and THB-enabled local refinement avoids over-refinement, in particular in geometries with complex boundaries. To control the parametric properties of the outcome, we introduce the concept of do…
In this paper we present a new class of cubature rules with the aim of accurately integrating weakly singular double integrals. In particular we focus on those integrals coming from the discretization of Boundary Integral Equations for 3D Laplace boundary value problems, using a collocation method within the Isogeometric Analysis paradigm. In such setting the regular part of the integrand can be defined as the product of a tensor product B-spline and a general function. The rules are derived by using first the spline quasi-interpolation approach to approximate such function and then the extens…
In this paper, we describe a general class of $C^1$ smooth rational splines that enables, in particular, exact descriptions of ellipses and ellipsoids - some of the most important primitives for CAD and CAE. The univariate rational splines are assembled by transforming multiple sets of NURBS basis functions via so-called design-through-analysis compatible extraction matrices; different sets of NURBS are allowed to have different polynomial degrees and weight functions. Tensor products of the univariate splines yield multivariate splines. In the bivariate setting, we describe how similar design…
The accuracy and effectiveness of Hermite spectral methods for the numerical discretization of partial differential equations on unbounded domains, are strongly affected by the amplitude of the Gaussian weight function employed to describe the approximation space. This is particularly true if the problem is under-resolved, i.e., there are no enough degrees of freedom. The issue becomes even more crucial when the equation under study is time-dependent, forcing in this way the choice of Hermite functions where the corresponding weight depends on time. In order to adapt dynamically the approximat…
The design of globally $C^s$-smooth ($s \geq 1$) isogeometric spline spaces over multi-patch geometries is a current and challenging topic of research in the framework of isogeometric analysis. In this work, we extend the recent methods [25,28] and [31-33] for the construction of $C^1$-smooth and $C^2$-smooth isogeometric spline spaces over particular planar multi-patch geometries to the case of $C^s$-smooth isogeometric multi-patch spline spaces of an arbitrary selected smoothness $s \geq 1$. More precisely, for any $s \geq 1$, we study the space of $C^s$-smooth isogeometric spline functions …
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, w…
This paper discusses the generation of multivariate $C^\infty$ functions with compact small supports by subdivision schemes. Following the construction of such a univariate function, called \emph{Up-function}, by a non-stationary scheme based on masks of {spline subdivision schemes} of growing degrees, we term the multivariate functions we generate Up-like functions. We generate them by non-stationary schemes based on masks of box-splines of growing supports. To analyze the convergence and smoothness of these non-stationary schemes, we develop new tools which apply to a wider class of schemes …
This paper is devoted to the question of constructing a higher order Faber spline basis for the sampling discretization of functions with higher regularity than Lipschitz. The basis constructed in this paper has similar properties as the piecewise linear classical Faber-Schauder basis except for the compactness of the support. Although the new basis functions are supported on the real line they are very well localized (exponentially decaying) and the main parts are concentrated on a segment. This construction gives a complete answer to Problem 3.13 in Triebel's monograph (see References [47]) …
A spline chaos expansion, referred to as SCE, is introduced for uncertainty quantification analysis. The expansion provides a means for representing an output random variable of interest with respect to multivariate orthonormal basis splines (B-splines) in input random variables. The multivariate B-splines are built from a whitening transformation to generate univariate orthonormal B-splines in each coordinate direction, followed by a tensor-product structure to produce the multivariate version. SCE, as it stems from compactly supported B-splines, tackles locally prominent responses more effec…
