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））
Tomas Teijeiro, Jamie M. Taylor, Ali Hashemian, David Pardo
We propose the use of machine learning techniques to find optimal quadrature rules for the construction of stiffness and mass matrices in isogeometric analysis (IGA). We initially consider 1D spline spaces of arbitrary degree spanned over uniform and non-uniform knot sequences, and then the generated optimal rules are used for integration over higher-dimensional spaces using tensor product sense. The quadrature rule search is posed as an optimization problem and solved by a machine learning strategy based on gradient-descent. However, since the optimization space is highly non-convex, the success of the search strongly depends on the number of quadrature points and the parameter initialization. Thus, we use a dynamic programming strategy that initializes the parameters from the optimal solution over the spline space with a lower number of knots. With this method, we found optimal quadrature rules for spline spaces when using IGA discretizations with up to 50 uniform elements and polynomial degrees up to 8, showing the generality of the approach in this scenario. For non-uniform partitions, the method also finds an optimal rule in a reasonable number of test cases. We also assess the generated optimal rules in two practical case studies, namely, the eigenvalue problem of the Laplace operator and the eigenfrequency analysis of freeform curved beams, where the latter problem shows the applicability of the method to curved geometries. In particular, the proposed method results in savings with respect to traditional Gaussian integration of up to 44% in 1D, 68% in 2D, and 82% in 3D spaces.
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）
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）
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）
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）
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）
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）
