We provide explicit expressions for quadrature rules on the space of $C^1$ quintic splines with uniform knot sequences over finite domains. The quadrature nodes and weights are derived via an explicit recursion that avoids an intervention of any numerical solver and the rule is optimal, that is, it requires the minimal number of nodes. For each of $n$ subintervals, generically, only two nodes are required which reduces the evaluation cost by $2/3$ when compared to the classical Gaussian quadrature for polynomials. Numerical experiments show fast convergence, as $n$ grows, to the "two-third" quadrature rule of Hughes et al. for infinite domains.
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…
A theory of spline quadrature rules for arbitrary continuity class in a closed interval $[a, b]$ with arbitrary nonuniform subintervals based on semi-classical orthogonal Jacobi polynomials is proposed. For continuity class $c \ge 2$ this theory depends on a conjecture.
We present weighted quadrature for hierarchical B-splines to address the fast formation of system matrices arising from adaptive isogeometric Galerkin methods with suitably graded hierarchical meshes. By exploiting a local tensor-product structure, we extend the construction of weighted rules from the tensor-product to the hierarchical spline setting. The proposed algorithm has a computational cost proportional to the number of degrees of freedom and advantageous properties with increasing spline degree. To illustrate the performance of the method and confirm the theoretical estimates, a selec…
Closed formulae for all Gaussian or optimal, 1-parameter quadrature rules in a compact interval [a, b] with non uniform, asymmetric subintervals, arbitrary number of nodes per subinterval for the spline classes $S_{2N, 0}$ and $S_{2N+1, 1}$, i.e. even and odd degree are presented. Also rules for the 2 missing spline classes $S_{2N-1, 0}$ and $S_{2N, 1}$ (the so called 1/2-rules), i.e. odd and even degree are presented. These quadrature rules are explicit in the sense, that they compute the nodes and their weights in the first/last boundary subinterval and, via a recursion the other nodes/weigh…
Bézier projection: a unified approach for local projection and quadrature-free refinement and coarsening of NURBS and T-splines with particular application to isogeometric design and analysis
Brigham Young University（杨百翰大学）；University of Colorado, Boulder（科罗拉多大学博尔德分校）
On the error bounds of the Gauss-type quadrature formulae associated with spaces of parabolic and cubic spline functions with double equidistant knots
Faculty of Mathematics and Informatics, Sofia University ”St. Kliment Ohridski”（索非亚大学圣克利门特奥赫里斯基数学与信息学院）；Faculty of Pharmacy, Medical University of Sofia（索非亚医科大学药学院）
Numerical Solution of two dimensional coupled viscous Burgers Equation using the Modified Cubic B Spline Differential Quadrature Method
DDU Gorakhpur University（德杜戈拉克普尔大学）；ISRO Telemetry, Tracking and Command Network (ISTRAC)（印度空间研究组织遥测、跟踪和指挥网络）；ISRO Satellite Center (ISAC)（印度空间研究组织卫星中心）
Efficient assembly based on B-spline tailored quadrature rules for the IgA-SGBEM
University of Parma（帕尔马大学）；University of Cassino and southern Lazio（卡西诺及南拉齐奥大学）；University of Siena（锡耶纳大学）；University of Pavia（帕维亚大学）；University of Florence（佛罗伦萨大学）
Efficient mass and stiffness matrix assembly via weighted Gaussian quadrature rules for B-splines
BCAM – Basque Center for Applied Mathematics（巴斯克应用数学中心）；Curtin University（科廷大学）；Commonwealth Scientific and Industrial Research Organization (CSIRO)（联邦科学与工业研究组织）
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…
The spatial impulse response (SIR) method is a well-known approach to calculate transient acoustic fields of arbitrary-shape transducers. It involves the evaluation of a time-dependent surface integral. Although analytic expressions of the SIR exist for some geometries, numerical methods based on the discretization of transducer surfaces have become the standard. The proposed method consists of representing the transducer as a non-uniform rational B-spline (NURBS) surface, and decomposing it into smooth Bézier patches onto which quadrature rules can be deployed. The evaluation of the SIR can t…
Machine Learning Discovery of Optimal Quadrature Rules for Isogeometric Analysis
BCAM – Basque Center for Applied Mathematics（巴斯克应用数学中心（BCAM））；CUNEF University（CUNEF大学）；University of the Basque Country (UPV/EHU)（巴斯克大学（UPV/EHU））；Ikerbasque – Basque Foundation for Sciences（巴斯克科学基金会（Ikerbasque））
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）
University of Science and Technology of China（中国科学技术大学）；The Hong Kong Polytechnic University（香港理工大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
The polynomial spline collocation method is proposed for solution of Volterra integral equations of the first kind with special piecewise continuous kernels. The Gauss-type quadrature formula is used to approximate integrals during the discretisation of the proposed projection method. The estimate of accuracy of approximate solution is obtained. Stochastic arithmetics is also used based on the Contrôle et Estimation Stochastique des Arrondis de Calculs (CESTAC) method and the Control of Accuracy and Debugging for Numerical Applications (CADNA) library. Applying this approach it is possible to …
Refined Gauss--Lobatto bounds for odd-order convexity
Czech Academy of Sciences（捷克科学院）；Institute of Mathematics and Computer Science Jagiellonian University（雅盖隆大学数学与计算机科学研究所）；University of Bielsko–Biała（别尔斯科比亚拉大学）
