LR B-splines
LR B-splines to approximate bathymetry datasets: An improved statistical criterion to judge the goodness of fit
Vibeke Skytt , Gaël Kermarrec , Tor Dokken

The task of representing remotely sensed scattered point clouds with mathematical surfaces is ubiquitous to reduce a high number of observations to a compact description with as few coefficients as possible. To reach that goal, locally refined B-splines provide a simple framework to perform surface approximation by allowing an iterative local refinement. Different setups exist (bidegree of the splines, tolerance, refinement strategies) and the choice is often made heuristically, depending on the applications and observations at hand. In this article, we introduce a statistical information criterion based on the t-distribution to judge the goodness of fit of the surface approximation for remote sensing data with outliers. We use a real bathymetry dataset and illustrate how concepts from model selection can be used to select the most adequate refinement strategy of the LR B-splines.

International Journal of Applied Earth Observation and Geoinformation, 2022
Effective grading refinement for locally linearly independent LR B-splines
Francesco Patrizi

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.

BIT Numerical Mathematics, 2022
Linear dependence of bivariate Minimal Support and Locally Refined B-splines over LR-meshes
Francesco Patrizi , Tor Dokken

The focus on locally refined spline spaces has grown rapidly in recent years due to the need in Isogeoemtric analysis (IgA) of spline spaces with local adaptivity: a property not offered by the strict regular structure of tensor product B-spline spaces. However, this flexibility sometimes results in collections of B-splines spanning the space that are not linearly independent. 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. Further results are established to help detecting collections of B-splines that are linearly independent.

Computer Aided Geometric Design, 2020
Error-controlled adaptive LR B-splines XIGA for assessment of fracture parameters in through-cracked Mindlin-Reissner plates
Tiantang Yu , Hongting Yuan , Jiming Gu , Satoyuki Tanaka , Tinh Quoc Bui

This paper presents an error-controlled adaptive extended isogeometric analysis (XIGA) for assessment of fracture behavior for through-cracked Mindlin-Reissner plates. The locally refined (LR) B-splines that facilitate the local refinement are used as the basis functions for the approximations, while the kinematics of plates are described in terms of the Mindlin-Reissner plate theory. Appropriate enrichment functions for cracked plates are incorporated into the formulation to represent geometrically displacement discontinuities across the crack faces and singularity of the stress in the vicinity at the crack tips, thus cracks are modeled topologically independent of the computational mesh. Here we extend the Zienkiewicz-Zhu method-based error estimation to the XIGA formulation for cracks in Mindlin-Reissner plates such that non-smoothnesses and stress singularity are reflected accurately. We conduct the error-controlled adaptivity for local mesh refinement to enhance the accuracy and performance. Fracture parameters (i.e., mixed-mode intensity factors) are calculated through the contour interaction integral method in the context of Mindlin-Reissner plate theory. Several numerical examples for through-cracked plates are studied to show the accuracy and effectiveness of the proposed formulation in modeling fracture of cracked moderately thick plates.

Engineering Fracture Mechanics, 2020
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Adaptive refinement with locally linearly independent LR B-splines: Theory and applications
Francesco Patrizi , Carla Manni , Francesca Pelosi , Hendrik Speleers

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.

Computer Methods in Applied Mechanics and Engineering, 2020
LR B-Splines implementation in the Altair RadiossTM solver for explicit dynamics IsoGeometric Analysis
Matthieu Occelli , Thomas Elguedj , Salim Bouabdallah , Lionel Morançay

IsoGeometric Analysis has shown to be a very promising tool for an integrated design and analysis process. A challenging task is still to move IGA from a proof of concept to a convenient design tool for industry and this work contributes to this endeavor. This paper deals with the implementation of the IGA concept into Altair Radioss finite element solver in order to address crash and stamping simulation applications. To this end, the necessary ingredients to a smooth integration of IGA in a traditional finite element code have been identified and adapted to the existing code architecture. A solid B-Spline element has been developed in Altair Radioss and then, an existing contact interface has been extended in order to work seamlessly with both NURBS and Lagrange finite elements. As local refinement is needed for solution approximation, an analysis is made in terms of analysis suitability and implementation aspects for several Spline technologies. The Locally Refined B-Spline (LRBS) technology is implemented and is validated on industrial benchmarks, for validation cases conventionally used for industrial codes like stamping and drop test.

Advances in Engineering Software, 2019
Multi-inclusions modeling by adaptive XIGA based on LR B-splines and multiple level sets
Jiming Gu , Tiantang Yu , Thanh-Tung Nguyen , Satoyuki Tanaka , Tinh Quoc Bui

In this paper, we present an effective computational approach that combines an adaptive extended isogeometric analysis (XIGA) method with locally refined (LR) B-splines and level set methods for modeling multiple inclusions in two-dimensional (2D) elasticity problems. The advantage of XIGA is to model inclusions without considering internal inclusion interfaces by additional functions. Multiple level set functions are used to represent the location of inclusion interfaces and to define enrichment functions. Local refinement for adaptive XIGA using LR B-splines is based on the posterior error estimator. We use the strategy of structured mesh refinement to implement local refinement in adaptive XIGA. Numerical experiments for multiple inclusions with complicated geometries are presented to demonstrate the accuracy and performance of the proposed approach. In addition, numerical results indicate that the adaptive XIGA with local refinement achieves faster convergence rate than that of the XIGA with uniform global refinement.

Finite Elements in Analysis and Design, 2018
Adaptive local surface refinement based on LR NURBS and its application to contact
Christopher Zimmermann , Roger A. Sauer

A novel adaptive local surface refinement technique based on Locally Refined Non-Uniform Rational B-Splines (LR NURBS) is presented. LR NURBS can model complex geometries exactly and are the rational extension of LR B-splines. The local representation of the parameter space overcomes the drawback of non-existent local refinement in standard NURBS-based isogeometric analysis. For a convenient embedding into general finite element code, the Bézier extraction operator for LR NURBS is formulated. An automatic remeshing technique is presented that allows adaptive local refinement and coarsening of LR NURBS. In this work, LR NURBS are applied to contact computations of 3D solids and membranes. For solids, LR NURBS-enriched finite elements are used to discretize the contact surfaces with LR NURBS finite elements, while the rest of the body is discretized by linear Lagrange finite elements. For membranes, the entire surface is discretized by LR NURBS. Various numerical examples are shown, and they demonstrate the benefit of using LR NURBS: Compared to uniform refinement, LR NURBS can achieve high accuracy at lower computational cost.

Computational Mechanics, 2017
Superconvergent patch recovery and a posteriori error estimation technique in adaptive isogeometric analysis
Mukesh Kumar , Trond Kvamsdal , Kjetil André Johannessen

In this article, we address adaptive methods for isogeometric analysis based on local refinement guided by recovery based a posteriori error estimates. Isogeometric analysis was introduced a decade ago and an impressive progress has been made related to many aspects of numerical methods and advanced applications. However, related to adaptive mesh refinement guided by a posteriori error estimators, rather few attempts are pursued besides the use of classical residual based error estimators. In this article, we explore a feature common for Isogeometric analysis (IGA), namely the use of structured tensorial meshes that facilitates superconvergence behavior of the gradient in the Galerkin discretization. By utilizing the concept of structured mesh refinement using LR B-splines, our aim is to facilitate superconvergence behavior for locally refined meshes as well. Superconvergence behavior matches well with the use of recovery based a posteriori estimator in the Superconvergent Patch Recovery (SPR) procedure. However, to our knowledge so far, the SPR procedure has not been exploited in the IGA community.

Computer Methods in Applied Mechanics and Engineering, 2017
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Divergence-conforming discretization for Stokes problem on locally refined meshes using LR B-splines
Kjetil André Johannessen , Mukesh Kumar , Trond Kvamsdal

To solve the incompressible flow problems using isogeometric analysis, the div-compatible spline spaces were originally introduced by Buffa et al. (2011), and later developed by Evans (2011). In this paper, we extend the div-compatible spline spaces with local refinement capability using Locally Refined (LR) B-splines over rectangular domains. We argue that the spline spaces generated on locally refined meshes will satisfy compatibility provided they span the entire function spaces as governed by Mourrain (2014) dimension formula. We will in this work use the structured refined LR B-splines as introduced by Johannessen et al. (2014). Further, we consider these div-compatible LR B-spline spaces to approximate the velocity and pressure fields in mixed discretization for Stokes problem and a set of standard benchmark tests are performed to show the stability, efficiency and the conservation properties of the discrete velocity fields in adaptive isogeometric analysis.

Computer Methods in Applied Mechanics and Engineering, 2015
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Simple a posteriori error estimators in adaptive isogeometric analysis
Mukesh Kumar , Trond Kvamsdal , Kjetil André Johannessen

In this article we propose two simple a posteriori error estimators for solving second order elliptic problems using adaptive isogeometric analysis. The idea is based on a Serendipity1 pairing of discrete approximation spaces Shp,k(M)–Shp+1,k+1(M), where the space Shp+1,k+1(M) is considered as an enrichment of the original basis of Shp,k(M) by means of the k-refinement, a typical unique feature available in isogeometric analysis. The space Shp+1,k+1(M) is used to obtain a higher order accurate isogeometric finite element approximation and using this approximation we propose two simple a posteriori error estimators. The proposed a posteriori error based adaptive h-refinement methodology using LR B-splines is tested on classical elliptic benchmark problems. The numerical tests illustrate the optimal convergence rates obtained for the unknown, as well as the effectiveness of the proposed error estimators.

Computers \& Mathematics with Applications, 2015
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On the similarities and differences between Classical Hierarchical, Truncated Hierarchical and LR B-splines
Kjetil André Johannessen , Filippo Remonato , Trond Kvamsdal

Smooth spline functions such as B-splines and NURBS are already an established technology in the field of computer-aided design (CAD) and have in recent years been given a lot of attention from the computer-aided engineering (CAE) community. The advantages of local refinement are obvious for anyone working in either field, and as such, several approaches have been proposed. Among others, we find the three strategies Classical Hierarchical B-splines, Truncated Hierarchical B-splines and Locally Refined B-splines. We will in this paper present these three frameworks and highlight similarities and differences between them. In particular, we will look at the function space they span and the support of the basis functions. We will then analyse the corresponding stiffness and mass matrices in terms of sparsity patterns and conditioning numbers. We show that the basis functions in general do not span the same space, and that conditioning numbers are comparable. Moreover we show that the weighting needed by the Classical Hierarchical basis to maintain partition of unity has significant implications on the conditioning numbers.

Computer Methods in Applied Mechanics and Engineering, 2015
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Isogeometric analysis using LR B-splines
Kjetil André Johannessen , Trond Kvamsdal , Tor Dokken

The recently proposed locally refined B-splines, denoted LR B-splines, by Dokken et al. (2013) [6] may have the potential to be a framework for isogeometric analysis to enable future interoperable computer aided design and finite element analysis. 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. The theory behind LR B-spline is not presented in full details, but the main conceptual ingredients are explained and illustrated by a number of examples.

Computer Methods in Applied Mechanics and Engineering, 2014
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Some properties of LR-splines
Andrea Bressan

Recently a new approach to piecewise polynomial spaces generated by B-spline has been presented by T. Dokken, T. Lyche and H.F. Pettersen, namely Locally Refined splines. In their recent work (Dokken et al., 2013) they define the LR B-spline collection and provide tools to compute the space dimension. Here different properties of the LR-splines are analyzed: in particular the coefficients for polynomial representations and their relation with other properties such as linear independence and the number of B-splines covering each element.

Computer Aided Geometric Design, 2013
Polynomial splines over locally refined box-partitions
Tor Dokken , Tom Lyche , Kjell Fredrik Pettersen

We address progressive local refinement of splines defined on axes parallel box-partitions and corresponding box-meshes in any space dimension. The refinement is specified by a sequence of mesh-rectangles (axes parallel hyperrectangles) in the mesh defining the spline spaces. In the 2-variate case a mesh-rectangle is a knotline segment. When starting from a tensor-mesh this refinement process builds what we denote an LR-mesh, a special instance of a box-mesh. On the LR-mesh we obtain a collection of hierarchically scaled Bsplines, denoted LR B-splines, that forms a nonnegative partition of unity and spans the complete piecewise polynomial space on the mesh when the mesh construction follows certain simple rules. The dimensionality of the spline space can be determined using some recent dimension formulas.

Computer Aided Geometric Design, 2013
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LR B-spline introduction
Google play

While strictly not a paper, this is a great introduction for people interested into learning the mechanics behind LR B-splines. It is an interactive demo featuring line insertions to visualize the refinement algorithm. It is only available on android systems, which means you cannot find this on Apples iOS App Store.

App on Google Play