Dispersion optimized quadratures for isogeometric analysis
Curtin University（科廷大学）；Curtin Institute for Computation, Curtin University（科廷计算研究所）；Mineral Resources, Commonwealth Scientific and Industrial Research Organisation (CSIRO)（联邦科学与工业研究组织矿产资源部）
We develop and analyze quadrature blending schemes that minimize the dispersion error of isogeometric analysis up to polynomial order seven with maximum continuity in the span ($C^{p-1}$). The schemes yield two extra orders of convergence (superconvergence) on the eigenvalue errors, while the eigenfunction errors are of optimal convergence order. Both dispersion and spectrum analysis are unified in the form of a Taylor expansion for eigenvalue errors. As a consequence, the schemes increase the accuracy and robustness of isogeometric analysis for wave propagation as well as the differential eigenvalue problems. We analyze the methods' robustness and efficacy and utilize numerical examples to verify our analysis of the performance of the proposed schemes.
It is well-known that outliers appear in the high-frequency region in the approximate spectrum of isogeometric analysis of the second-order elliptic operator. Recently, the outliers have been eliminated by a boundary penalty technique. The essential idea is to impose extra conditions arising from the differential equation at the domain boundary. In this paper, we extend the idea to remove outliers in the superconvergent approximate spectrum of isogeometric analysis with optimally-blended quadrature rules. We show numerically that the eigenvalue errors are of superconvergence rate $h^{2p+2}$ an…
This paper introduces optimally-blended quadrature rules for isogeometric analysis and analyzes the numerical dispersion of the resulting discretizations. To quantify the approximation errors when we modify the inner products, we generalize the Pythagorean eigenvalue theorem of Strang and Fix. The proposed blended quadrature rules have advantages over alternative integration rules for isogeometric analysis on uniform and non-uniform meshes as well as for different polynomial orders and continuity of the basis. The optimally-blended schemes improve the convergence rate of the method by two orde…
The Influence of Quadrature Errors on Isogeometric Mortar Methods
Istituto Universitario di Studi Superiori Pavia（帕维亚高等大学研究所）；Istituto di Matematica Applicata e Tecnologie Informatiche del CNR（意大利国家研究委员会应用数学与信息技术研究所）；M2 – Zentrum Mathematik, Technische Universität München（慕尼黑工业大学数学中心）
We study the numerical solution of forward and inverse acoustic scattering problems by randomly shaped obstacles in three-dimensional space using a fast isogeometric boundary element method. Within the isogeometric framework, realizations of the random scatterer can efficiently be computed by simply updating the NURBS mappings which represent the scatterer. This way, we end up with a random deformation field. In particular, we show that the knowledge of the deformation field's expectation and covariance at the surface of the scatterer are already sufficient to compute the surface Karhunen-Loèv…
Fast formation of isogeometric Galerkin matrices by weighted quadrature
Università degli Studi di Cassino e del Lazio Meridionale（卡西诺及拉齐奥梅里迪奥纳莱大学）；Università degli Studi di Pavia（帕维亚大学）；Istituto di Matematica Applicata e Tecnologie Informatiche “E. Magenes” del CNR（CNR应用数学与信息技术研究所“E. Magenes”）
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）
This paper presents a novel method for solving partial differential equations on three-dimensional CAD geometries by means of immersed isogeometric discretizations that do not require quadrature schemes. It relies on a new developed technique for the evaluation of polynomial integrals over spline boundary representations that is exclusively based on analytical computations. First, through a consistent polynomial approximation step, the finite element operators of the Galerkin method are transformed into integrals involving only polynomial integrands. Then, by successive applications of the div…
A new methodology in isogeometric analysis (IGA) is presented. This methodology delivers low-cost variable-scale approximations (surrogates) of the matrices which IGA conventionally requires to be computed from element-scale quadrature formulas. To generate surrogate matrices, quadrature must only be performed on certain elements in the computational domain. This, in turn, determines only a subset of the entries in the final matrix. The remaining matrix entries are computed by a simple B-spline interpolation procedure. Poisson's equation, membrane vibration, plate bending, and Stokes' flow pro…
Optimal spectral approximation of $2n$-order differential operators by mixed isogeometric analysis
Curtin University（科廷大学）；Curtin Institute for Computation, Curtin University（科廷计算研究所）；Commonwealth Scientific and Industrial Research Organisation (CSIRO)（联邦科学与工业研究组织）
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））
Dispersion-minimized mass for isogeometric analysis
Curtin University（科廷大学）；Curtin Institute for Computation, Curtin University（科廷计算研究所，科廷大学）；Department of Applied Geology, Western Australian School of Mines, Curtin University（西澳矿业学院应用地质系，科廷大学）；Mineral Resources, Commonwealth Scientific and Industrial Research Organisation (CSIRO)（联邦科学与工业研究组织（CSIRO）矿产资源部）
Isogeometric spectral approximation for elliptic differential operators
Curtin University（科廷大学）；Curtin Institute for Computation, Curtin University（科廷大学计算研究所）；Department of Applied Geology, Curtin University（科廷大学应用地质系）；Mineral Resources, Commonwealth Scientific and Industrial Research Organisation (CSIRO)（联邦科学与工业研究组织矿产资源部）
An isogeometric approach for solving the Laplace-Beltrami equation on a two-dimensional manifold embedded in three-dimensional space using a Galerkin method based on Catmull-Clark subdivision surfaces is presented and assessed. The scalar-valued Laplace-Beltrami equation requires only C0 continuity and is adopted to elucidate key features and properties of the isogeometric method using Catmull-Clark subdivision surfaces. Catmull-Clark subdivision bases are used to discretise both the geometry and the physical field. A fitting method generates control meshes to approximate any given geometry wi…
Stable distributions are an important class of infinitely-divisible probability distributions, of which two special cases are the Cauchy distribution and the normal distribution. Aside from a few special cases, the density function for stable distributions has no known analytic form, and is expressible only through the variate's characteristic function or other integral forms. In this paper we present numerical schemes for evaluating the density function for stable distributions, its gradient, and distribution function in various parameter regimes of interest, some of which had no pre-existing…
This paper is devoted to a new first order Taylor-like formula where the corresponding remainder is strongly reduced in comparison with the usual one which appears in the classical Taylor's formula. To derive this new formula, we introduce a linear combination of the first derivative of the concerned function, which is computed at n+1 equally-spaced points between the two points where the function has to be evaluated. We show that an optimal choice of the weights in the linear combination leads to minimizing the corresponding remainder. Then, we analyze the Lagrange P1- interpolation error est…
