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/weights, parsing from the first/last subinterval to the middle of the interval. These closed formulae are based on the semi-classical Jacobi type orthogonal polynomials.
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…
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）
In this paper we study the dimension of bivariate polynomial splines of mixed smoothness on polygonal meshes. Here, "mixed smoothness" refers to the choice of different orders of smoothness across different edges of the mesh. To study the dimension of spaces of such splines, we use tools from Homological Algebra. These tools were first applied to the study of splines by Billera (1988). Using them, estimation of the spline space dimension amounts to the study of the generalized Billera-Schenck-Stillman complex for the spline space. In particular, when the homology in positions one and zero of t…
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…
The implementation of finite element methods (FEMs) for nonlocal models with a finite range of interaction poses challenges not faced in the partial differential equations (PDEs) setting. For example, one has to deal with weak forms involving double integrals which lead to discrete systems having higher assembly and solving costs due to possibly much lower sparsity compared to that of FEMs for PDEs. In addition, one may encounter non-smooth integrands. In many nonlocal models, nonlocal interactions are limited to bounded neighborhoods that are ubiquitously chosen to be Euclidean balls, resulti…
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））
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）
