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Linear dependence of bivariate Minimal Support and Locally Refined B-splines over LR-meshes [article]

Francesco Patrizi, Tor Dokken
2019 arXiv   pre-print
In this paper we address the minimal number of B-splines that can form a linear dependence relation for Minimal Support B-splines (MS B-splines) and for Locally Refinable B-splines (LR B-splines) on LR-meshes  ...  We show that the minimal number is six for MS B-splines, and eight for LR B-splines. The risk of linear dependency is consequently significantly higher for MS B-splines than for LR B-splines.  ...  There are many other definitions of B-splines over meshes with local refinements, such as PHT-splines [11] , PB-splines [12] and LR B-splines [13] .  ... 
arXiv:1811.04919v2 fatcat:rwkgh5lrw5cnfb7cad6utse4aa

Linear dependence of bivariate Minimal Support and Locally Refined B-splines over LR-meshes

Francesco Patrizi, Tor Dokken
2019 Computer Aided Geometric Design  
In this paper we address the minimal number of Minimal Support B-splines (MS B-splines) and of Locally Refined B-splines (LR B-splines) that can form a linear dependence relation.  ...  We show that such minimal numbers are six for MS B-splines and eight for LR B-splines. Further results are established to help detecting collections of B-splines that are linearly independent.  ...  There are many other definitions of B-splines over meshes with local refinements, such as PHT-splines (Deng et al., 2008) , PB-splines (Engleitner and Jüttler, 2017) and LR B-splines (Dokken et al.  ... 
doi:10.1016/j.cagd.2019.101803 fatcat:rnovrdcmi5d7xii763noe6xp2y

Adaptive refinement with locally linearly independent LR B-splines: Theory and applications

Francesco Patrizi, Carla Manni, Francesca Pelosi, Hendrik Speleers
2020 Computer Methods in Applied Mechanics and Engineering  
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.  ...  refinement may produce linearly dependent sets of LR B-splines.  ... 
doi:10.1016/j.cma.2020.113230 fatcat:hjsajvksqzfb5ebpzjuwcczcye

Adaptive refinement with locally linearly independent LR B-splines: Theory and applications [article]

Francesco Patrizi, Carla Manni, Francesca Pelosi, Hendrik Speleers
2020 arXiv   pre-print
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.  ...  mesh refinement may produce linearly dependent sets of LR B-splines.  ... 
arXiv:2001.11236v1 fatcat:lujzi5j6yzdutnjbujldzmbxc4

Trivariate spline representations for computer aided design and additive manufacturing

Tor Dokken, Vibeke Skytt, Oliver Barrowclough
2018 Computers and Mathematics with Applications  
The ideas of (Truncated) Hierarchical B-splines, T-splines and LR B-splines are outlined and the approaches are compared.  ...  We discuss the feasibility for multi-material AM for B-rep based CAD, STL, sculptured triangles as well as trimmed and block-structured trivariate locally refined spline representations.  ...  Acknowledgements This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 680448. References  ... 
doi:10.1016/j.camwa.2018.08.017 fatcat:wqw53vfajrg77pr4cftlpr6cam

Isogeometric analysis using LR B-splines

Kjetil André Johannessen, Trond Kvamsdal, Tor Dokken
2014 Computer Methods in Applied Mechanics and Engineering  
The recently proposed locally refined B-splines, denoted LR B-splines, by Dokken et al. [6] may have the potential to be a framework for isogeometric analysis to enable future interoperable computer aided  ...  In this paper, we propose local refinement strategies for adaptive isogeometric analysis using LR B-splines and investigate its performance by doing numerical tests on well known benchmark cases.  ...  Acknowledgement The authors acknowledge the financial support from the Norwegian Research Council and the industrial partners of the ICADA-project; Ceetron, Det Norske Veritas and Statoil.  ... 
doi:10.1016/j.cma.2013.09.014 fatcat:j4jmes3p7vgw3ihaazhcya2ieu

SURFACE APPROXIMATION OF COASTAL REGIONS: LR B-SPLINE FOR DETECTION OF DEFORMATION PATTERN

G. Kermarrec, V. Skytt, T. Dokken
2022 ISPRS Annals of the Photogrammetry, Remote Sensing and Spatial Information Sciences  
The result is frequently that noise is approximated and that surfaces have unwanted oscillations.Locally Refined (LR) B-spline surfaces were introduced to face this challenge and provide a tool for approximating  ...  We further highlight the potential of LR B-spline volumes for spatio-temporal visualisation of deformation patterns.  ...  ACKNOWLEDGEMENTS This study is supported by the Deutsche Forschungsgemeinschaft under the project KE2453/2-1 and the Norwegian research council under grant number 270922.  ... 
doi:10.5194/isprs-annals-v-2-2022-119-2022 fatcat:ghdwmboggzambgqpbm2dcwpcou

Locally refined spline surfaces for terrain and sea bed data: tailored approximation, export and analysis tools [article]

Vibeke Skytt, Tor Dokken
2021 arXiv   pre-print
The novel Locally Refined B-spline (LR B-spline) surface format is suited for representing terrain and seabed data in a compact way.  ...  In regions where many data points belong to the same smooth area LR B-splines allow a very lean representation of the shape by locally adapting the resolution of the spline space to the size and local  ...  In Section 3.1 we address a single B-spline, the tensor product of two B-splines, the univariate minimal support B-spline bases and the bivariate minimal support tensor product Bspline basis.  ... 
arXiv:2010.01980v2 fatcat:eofxarmajrey3cxlcpatbetc3e

A hierarchical construction of LR meshes in 2D

Andrea Bressan, Bert Jüttler
2015 Computer Aided Geometric Design  
We describe a construction of LR-spaces whose bases are composed of locally linearly independent B-splines which also form a partition of unity.  ...  ., 2013) and similarly to the hierarchical B-spline framework (Forsey and Bartels, 1988 ) the construction of the mesh is based on a priori choice of a sequence of nested tensor B-spline spaces.  ...  Acknowledgments The financial support of this research by the Austrian Science Fund (FWF) through the NFN "Geometry + Simulation" (S117) is gratefully acknowledged. Special thanks go to Dr.  ... 
doi:10.1016/j.cagd.2015.06.002 fatcat:gofjwhcsjfg5xko3oyqmtfqw7i

A Versatile Strategy for the Implementation of Adaptive Splines [chapter]

Andrea Bressan, Dominik Mokriš
2017 Lecture Notes in Computer Science  
nested refinement and hierarchical LR-splines.  ...  This paper presents an implementation framework for spline spaces over T-meshes (and their d-dimensional analogs). The aim is to share code between the implementation of several spline spaces.  ...  Acknowledgments The authors have been supported by the Austrian Science Fund (FWF, NFN S117 "Geometry + Simulation") and by the Seventh Framework Programme of the EU (project EXAMPLE, GA No. 324340).  ... 
doi:10.1007/978-3-319-67885-6_3 fatcat:4cb4ai2ttbaqffcmg7c6adkqd4

Adaptive isogeometric finite element analysis of steady-state groundwater flow

Yared W. Bekele, Trond Kvamsdal, Arne M. Kvarving, Steinar Nordal
2015 International journal for numerical and analytical methods in geomechanics (Print)  
A posteriori error estimates are calculated to identify which B-splines should be locally refined. The error estimates are calculated based on recovery of the L 2 -projected solution.  ...  In this paper, we apply adaptive isogeometric finite element analysis using locally refined (LR) Bsplines to address these types of problems.  ...  The authors gratefully acknowledge the support. Special thanks to all in the IFEM software team at SINTEF Applied Mathematics and Dr. Kjetil A. Johannessen for his work on LR B-splines.  ... 
doi:10.1002/nag.2425 fatcat:ov2njnhakvds5hl7o3eycfsiym

Post-processing and visualization techniques for isogeometric analysis results

Annette Stahl, Trond Kvamsdal, Christian Schellewald
2017 Computer Methods in Applied Mechanics and Engineering  
The presented post-processing and visualization techniques thereby strengthen the relation between geometry, analysis and visualization.  ...  This is achieved by facilitating the same mathematical function space used for geometry and analysis also for post-processing and visualization purposes.  ...  Acknowledgments The authors acknowledge the financial support from the Norwegian Research Council and the industrial partners of the ICADA-project (NFR Project 187993); Ceetron, Det Norske Veritas and  ... 
doi:10.1016/j.cma.2016.10.040 fatcat:if2uyoutlzhuxislqse2ujzvsy

Simple a posteriori error estimators in adaptive isogeometric analysis

Mukesh Kumar, Trond Kvamsdal, Kjetil André Johannessen
2015 Computers and Mathematics with Applications  
The proposed a posteriori error based adaptive h-refinement methodology using LR B-splines are tested on classical elliptic benchmark problems.  ...  The idea is based on a Serendipity 1 pairing of discrete approximation spaces S p,k is considered as an enrichment of the original basis of S p,k h (M) by means of the k-refinement, a typical unique feature  ...  Acknowledgements The authors acknowledge the the financial support from the Norwegian Research Council and the industrial partners of the ICADA project (RCN grant no: 187993) (Ceetron, DNV GL and Statoil  ... 
doi:10.1016/j.camwa.2015.05.031 fatcat:l5rwayfulfe5xksuhr7mfwlioe

THB-spline approximations for turbine blade design with local B-spline approximations [article]

Cesare Bracco, Carlotta Giannelli, David Grossmann, Sofia Imperatore, Dominik Mokris, Alessandra Sestini
2020 arXiv   pre-print
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  ...  The INdAM support through GNCS and Finanziamenti Premiali SUNRISE is also gratefully acknowledged.  ... 
arXiv:2003.08706v2 fatcat:vjn4crdudfcktnduuiegv232ia

A framework for efficient isogeometric computations of phase-field brittle fracture in multipatch shell structures

Davide Proserpio, Marreddy Ambati, Laura De Lorenzis, Josef Kiendl
2020 Computer Methods in Applied Mechanics and Engineering  
Furthermore, we present a multistep predictor-corrector algorithm for adaptive local refinement with LR NURBS, which can accurately predict and refine the region around the crack even in cases where fracture  ...  Finally, we investigate and compare the numerical efficiency of loosely vs. strongly staggered solution schemes and of the second-vs. fourth-order phase-field models.  ...  Acknowledgments The authors gratefully acknowledge the support from the Research Council of Norway, NFR, and the German Academic Exchange Service, DAAD, through the project FATIKA -Isogeometric phase-field  ... 
doi:10.1016/j.cma.2020.113363 fatcat:sab4wmmspncinkcj7hobjjuvom
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