In this paper we describe an adaptive refinement strategy for LR B-splines. The presented strategy ensures, at each iteration, local linear independence of the obtained set of LR B-splines. This property is then exploited in two applications: the construction of efficient quasi-interpolation schemes and the numerical solution of elliptic problems using the isogeometric Galerkin method.
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…
We present a new refinement strategy for locally refined B-splines which ensures the local linear independence of the basis functions. The strategy also guarantees the spanning of the full spline space on the underlying locally refined mesh. The resulting mesh has nice grading properties which grant the preservation of shape regularity and local quasi uniformity of the elements in the refining process.
We present an adaptive refinement algorithm for T-splines on unstructured 2D meshes. While for structured 2D meshes, one can refine elements alternatingly in horizontal and vertical direction, such an approach cannot be generalized directly to unstructured meshes, where no two unique global mesh directions can be assigned. To resolve this issue, we introduce the concept of direction indices, i.e., integers associated to each edge, which are inspired by theory on higher-dimensional structured T-splines. Together with refinement levels of edges, these indices essentially drive the refinement sch…
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 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…
Locally refined spline surfaces (LRB) is a representation well suited for scattered data approximation. When a data set has local details in some areas and is largely smooth in other, LR B-splines allow the spatial distribution of degrees of freedom to follow the variations of the data set. An LRB surface approximating a data set is refined in areas where the accuracy does not meet a required tolerance. In this paper we address, in a systematic study, different LRB refinement strategies and polynomial degrees for surface approximation. We study their influence on data volume and accuracy when …
Department of Mathematics and Informatics, Vilnius University（维尔纽斯大学数学与信息学系）；Institute of Mathematics and Informatics（维尔纽斯大学数学与信息学研究所）；Institute of Computational Math. and Sci. and Eng. Computing, Academy of Mathematics and System Sciences, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所）
Suitably graded THB-spline refinement and coarsening: Towards an adaptive isogeometric analysis of additive manufacturing processes
University of Pavia（帕维亚大学）；Technische Universität München（慕尼黑工业大学）；Università degli Studi di Firenze（佛罗伦萨大学）；Ecole Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；Istituto di Matematica Applicata e Tecnologie Informatiche “E. Magenes” del CNR（意大利国家研究委员会应用数学与信息技术研究所）
Local h-, p-, and k-Refinement Strategies for the Isogeometric Shifted Boundary Method Using THB-Splines
Technische Universität München（慕尼黑工业大学）；OTH Regensburg（雷根斯堡应用科学大学）；Universidad Politécnica de Cataluña (UPC)（加泰罗尼亚理工大学）；Centre Internacional de Mètodes Numèrics en Enginyeria (CIMNE)（国际工程数值方法中心）
Bases of T-meshes and the refinement of hierarchical B-splines
The University of Auckland（奥克兰大学）；Seoul National University（首尔大学）；Dongguk University（东国大学）；University of Turin（都灵大学）；Asian Institute of Technology（亚洲理工学院）
Adaptive isogeometric phase-field modeling of the Cahn-Hilliard equation: Suitably graded hierarchical refinement and coarsening on multi-patch geometries
We consider two-stage scattered data fitting with truncated hierarchical B-splines (THB-splines) for the adaptive reconstruction of industrial models. The first stage of the scheme is devoted to the computation of local least squares variational spline approximations, exploiting a simple fairness functional to handle data distributions with a locally varying density of points. Hierarchical spline quasi-interpolation based on THB-splines is considered in the second stage of the method to construct the adaptive spline surface approximating the whole scattered data set and a suitable strategy to …
To exploit the advantageous properties of isogeometric analysis (IGA) in a scan-based setting, it is important to extract a smooth geometric domain from the scan data (e.g., voxel data). IGA-suitable domains can be constructed by convoluting the grayscale data using B-splines. A negative side-effect of this convolution technique is, however, that it can induce topological changes in the process of smoothing when features with a size similar to that of the voxels are encountered. This manuscript presents an enhanced B-spline-based segmentation procedure using a refinement strategy based on trun…
Isogeometric analysis is a powerful paradigm which exploits the high smoothness of splines for the numerical solution of high order partial differential equations. However, the tensor-product structure of standard multivariate B-spline models is not well suited for the representation of complex geometries, and to maintain high continuity on general domains special constructions on multi-patch geometries must be used. In this paper we focus on adaptive isogeometric methods with hierarchical splines, and extend the construction of $C^1$ isogeometric spline spaces on multi-patch planar domains to…
We consider an adaptive algorithm for finite element methods for the isogeometric analysis (IGAFEM) of elliptic (possibly non-symmetric) second-order partial differential equations. We employ analysis-suitable T-splines of arbitrary odd degree on T-meshes generated by the refinement strategy of [Morgenstern, Peterseim, Comput. Aided Geom. Design 34 (2015)] in 2D and [Morgenstern, SIAM J. Numer. Anal. 54 (2016)] in 3D. Adaptivity is driven by some weighted residual a posteriori error estimator. We prove linear convergence of the error estimator (which is equivalent to the sum of energy error pl…
Subdivision surfaces provide an elegant isogeometric analysis framework for geometric design and analysis of partial differential equations defined on surfaces. They are already a standard in high-end computer animation and graphics and are becoming available in a number of geometric modelling systems for engineering design. The subdivision refinement rules are usually adapted from knot insertion rules for splines. The quadrilateral Catmull-Clark scheme considered in this work is equivalent to cubic B-splines away from extraordinary, or irregular, vertices with other than four adjacent element…
Adaptive isogeometric analysis with hierarchical box splines
Istituto Nazionale di Alta Matematica, Unità di Ricerca di Firenze c/o DiMaI “U. Dini”, Università di Firenze（意大利高等数学研究所佛罗伦萨研究单元）；Dipartimento di Matematica e Informatica “U. Dini”, Università degli Studi di Firenze（佛罗伦萨大学数学与信息学系）；Dipartimento di Matematica, Università degli Studi di Roma “Tor Vergata”（罗马托维加塔大学数学系）
We propose an adaptive mesh refinement strategy for immersed isogeometric analysis, with application to steady heat conduction and viscous flow problems. The proposed strategy is based on residual-based error estimation, which has been tailored to the immersed setting by the incorporation of appropriately scaled stabilization and boundary terms. Element-wise error indicators are elaborated for the Laplace and Stokes problems, and a THB-spline-based local mesh refinement strategy is proposed. The error estimation .and adaptivity procedure is applied to a series of benchmark problems, demonstrat…
A new perspective on hierarchical spline spaces for adaptivity
Facultad de Ingenieria Quimica, Universidad Nacional del Litoral（国立利托拉尔大学化学工程学院）；Consejo Nacional de Investigaciones científicas y técnicas (CONICET)（国家科学研究委员会）
Complexity of hierarchical refinement for a class of admissible mesh configurations
Rheinische Friedrich-Wilhelms-Universität Bonn（波恩大学）；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.马格内斯’）；Istituto Nazionale di Alta Matematica, Unità di Ricerca di Firenze c/o DiMaI ‘U.˜Dini’, Università di Firenze（佛罗伦萨大学迪马伊‘乌·迪尼’单位，意大利高等数学研究所）
Weakening the tight coupling between geometry and simulation in isogeometric analysis: from sub- and super- geometric analysis to Geometry Independent Field approximaTion (GIFT)
University of Chile（智利大学）；Hangzhou Dianzi University（杭州电子科技大学）；Institute of Computational Engineering, University of Luxembourg（卢森堡大学计算工程研究所）；School of Engineering, Cardiff University（卡迪夫大学工程学院）
Given a domain $Ω\subset \mathbb{R}^n$, the de Rham complex of differential forms arises naturally in the study of problems in electromagnetism and fluid mechanics defined on $Ω$, and its discretization helps build stable numerical methods for such problems. For constructing such stable methods, one critical requirement is ensuring that the discrete subcomplex is cohomologically equivalent to the continuous complex. When $Ω$ is a hypercube, we thus require that the discrete subcomplex be exact. Focusing on such $Ω$, we theoretically analyze the discrete de Rham complex built from hierarchical …
