Polynomial splines over locally refined box-partitions
TL;DR: This work addresses progressive local refinement of splines defined on axes parallel box-partitions and corresponding box-meshes in any space dimension and obtains a collection of hierarchically scaled 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.
About: This article is published in Computer Aided Geometric Design.The article was published on 2013-03-01. It has received 358 citations till now. The article focuses on the topics: Box spline & Spline (mathematics).
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TL;DR: This work reviews the recent status of methodologies and techniques related to the construction of digital twins mostly from a modeling perspective to provide a detailed coverage of the current challenges and enabling technologies along with recommendations and reflections for various stakeholders.
Abstract: Digital twin can be defined as a virtual representation of a physical asset enabled through data and simulators for real-time prediction, optimization, monitoring, controlling, and improved decision making. Recent advances in computational pipelines, multiphysics solvers, artificial intelligence, big data cybernetics, data processing and management tools bring the promise of digital twins and their impact on society closer to reality. Digital twinning is now an important and emerging trend in many applications. Also referred to as a computational megamodel, device shadow, mirrored system, avatar or a synchronized virtual prototype, there can be no doubt that a digital twin plays a transformative role not only in how we design and operate cyber-physical intelligent systems, but also in how we advance the modularity of multi-disciplinary systems to tackle fundamental barriers not addressed by the current, evolutionary modeling practices. In this work, we review the recent status of methodologies and techniques related to the construction of digital twins mostly from a modeling perspective. Our aim is to provide a detailed coverage of the current challenges and enabling technologies along with recommendations and reflections for various stakeholders.
660 citations
Cites methods from "Polynomial splines over locally ref..."
...One popular method for local refinement is the Locally Refined B-splines (LR B-Splines), see [92] and [93]....
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01 Jan 2010
TL;DR: A construction of rational isotropic curves with a prescribed tangent field which leads to the description of all rational minimal surfaces is presented.
Abstract: We will deal with the translation surfaces which are the shapes generated by translating one curve along another one. We focus on the geometry of translation surfaces generated by two algebraic curves in space and study their properties, especially those useful for geometric modelling purposes. It is a classical result that each minimal surface may be obtained as a translation surface generated by an isotropic curve and its complex conjugate. Thus, we can study the minimal surfaces as special instances of translation surfaces. All the results about translation surfaces will be directly applied also to minimal surfaces. Finally, we present a construction of rational isotropic curves with a prescribed tangent field which leads to the description of all rational minimal surfaces. A close relation to surfaces with Pythagorean normals will be also discussed.
319 citations
Cites background or methods from "Polynomial splines over locally ref..."
...The formation of these tool paths is usually computed with the aid of Computer Aided Manufacturing (CAM) software [3]....
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...) and Ω0 = [0, 4], Ω1 = [1, 3] and Ω2 = [2, 3]....
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...The small remaining scattering at large radii is attributed to the effect of boundary conditions [3,37] and not to the inaccuracy of the proposed method....
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...Examples that consider an initial reference spline in the univariate case may be found in [3,17,29]....
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...3 Local Basis While Bézier extraction was only described for spaces of piecewise tensor-product polynomials with Bernstein polynomials as local generators, there is no practical issue to extend it to other local bases such as enriched splines spaces as those from [3]....
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TL;DR: This review paper collects several results that form part of the theoretical foundation of isogeometric methods, and analyses variational techniques for the numerical resolution of PDEs based on splines or NURBS and provides optimal approximation and error estimates in several cases of interest.
Abstract: This review paper collects several results that form part of the theoretical foundation of isogeometric methods. We analyse variational techniques for the numerical resolution of PDEs based on splines or NURBS and we provide optimal approximation and error estimates in several cases of interest. The theory presented also includes estimates for T-splines, which are an extension of splines allowing for local refinement. In particular, we focus our attention on elliptic and saddle point problems, and we define spline edge and face elements. Our theoretical results are demonstrated by a rich set of numerical examples. Finally, we discuss implementation and efficiency together with preconditioning issues for the final linear system. © Cambridge University Press 2014.
298 citations
Cites background from "Polynomial splines over locally ref..."
...LRsplines (Dokken et al. 2013) and hierarchical splines (Vuong et al. 2011) were proposed more recently in the isogeometric literature, and represent a valid alternative to T-splines....
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TL;DR: This review article provides a concise introduction to the basics of the finite cell method, and summarizes recent developments of the technology, with particular emphasis on the research topics in which the author has been actively involved.
Abstract: The finite cell method is an embedded domain method, which combines the fictitious domain approach with higher-order finite elements, adaptive integration, and weak enforcement of unfitted essential boundary conditions. Its core idea is to use a simple unfitted structured mesh of higher-order basis functions for the approximation of the solution fields, while the geometry is captured by means of adaptive quadrature points. This eliminates the need for boundary conforming meshes that require time-consuming and error-prone mesh generation procedures, and opens the door for a seamless integration of very complex geometric models into finite element analysis. At the same time, the finite cell method achieves full accuracy, i.e. optimal rates of convergence, when the mesh is refined, and exponential rates of convergence, when the polynomial degree is increased. Due to the flexibility of the quadrature based geometry approximation, the finite cell method can operate with almost any geometric model, ranging from boundary representations in computer aided geometric design to voxel representations obtained from medical imaging technologies. In this review article, we first provide a concise introduction to the basics of the finite cell method. We then summarize recent developments of the technology, with particular emphasis on the research topics in which we have been actively involved. These include the finite cell method with B-spline and NURBS basis functions, the treatment of geometric nonlinearities for large deformation analysis, the weak enforcement of boundary and coupling conditions, and local refinement schemes. We illustrate the capabilities and advantages of the finite cell method with several challenging examples, e.g. the image-based analysis of foam-like structures, the patient-specific analysis of a human femur bone, the analysis of volumetric structures based on CAD boundary representations, and the isogeometric treatment of trimmed NURBS surfaces. We conclude our review by briefly discussing some key aspects for the efficient implementation of the finite cell method.
271 citations
TL;DR: This paper proposes local refinement strategies for adaptive isogeometric analysis using LR B-splines and investigates its performance by doing numerical tests on well known benchmark cases.
241 citations
Cites background or methods from "Polynomial splines over locally ref..."
...While it is possible to define LR splines by using non-weighted B-splines, as d one in [6], we will here only consider weighted ones as to maintain the partition of unity which is impor tant in finite element methods....
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...[6] may have the potential to form an alternative framework for future interoperable CA D and FEA systems....
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...This result is generalized in [6] to also address ge neral multiplicities and any dimension....
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...[6] may have the potential to be a framework for isogeometric analys is to enable future interoperable computer aided design and finite element analysis....
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...We now must decide on whether to useΞ1 and insert[0, 4] or to useΞ3 and insert[2, 6]....
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References
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TL;DR: In this article, the concept of isogeometric analysis is proposed and the basis functions generated from NURBS (Non-Uniform Rational B-Splines) are employed to construct an exact geometric model.
5,137 citations
Book•
15 Sep 2009TL;DR: Providing a systematic approach to the topic, the authors begin with a tutorial introducing the foundations of Isogeometric Analysis, before advancing to a comprehensive coverage of the most recent developments in the technique.
Abstract: The authors are the originators of isogeometric analysis, are excellent scientists and good educators. It is very original. There is no other book on this topic. Ren de Borst, Eindhoven University of Technology Written by leading experts in the field and featuring fully integrated colour throughout, Isogeometric Analysis provides a groundbreaking solution for the integration of CAD and FEA technologies. Tom Hughes and his researchers, Austin Cottrell and Yuri Bazilevs, present their pioneering isogeometric approach, which aims to integrate the two techniques of CAD and FEA using precise NURBS geometry in the FEA application. This technology offers the potential to revolutionise automobile, ship and airplane design and analysis by allowing models to be designed, tested and adjusted in one integrative stage. Providing a systematic approach to the topic, the authors begin with a tutorial introducing the foundations of Isogeometric Analysis, before advancing to a comprehensive coverage of the most recent developments in the technique. The authors offer a clear explanation as to how to add isogeometric capabilities to existing finite element computer programs, demonstrating how to implement and use the technology. Detailed programming examples and datasets are included to impart a thorough knowledge and understanding of the material. Provides examples of different applications, showing the reader how to implement isogeometric models Addresses readers on both sides of the CAD/FEA divide Describes Non-Uniform Rational B-Splines (NURBS) basis functions
2,302 citations
TL;DR: T-splines, a generalization of NURBS enabling local refinement, have been explored as a basis for isogeometric analysis in this paper, and they have shown good results on some elementary two-dimensional and three-dimensional fluid and structural analysis problems and attain good results in all cases.
975 citations
01 Jul 2003
TL;DR: A generalization of non-uniform B-spline surfaces called T-splines, which are C2 (in the absence of multiple knots), and T-NURCCs (Non-Uniform Rational Catmull-Clark Surfaces with T-junctions) are a superset of both T- Splines and Catm Mull-Clark surfaces, which can handle any NURBS or CatmULL-Clark model as special cases.
Abstract: This paper presents a generalization of non-uniform B-spline surfaces called T-splines. T-spline control grids permit T-junctions, so lines of control points need not traverse the entire control grid. T-splines support many valuable operations within a consistent framework, such as local refinement, and the merging of several B-spline surfaces that have different knot vectors into a single gap-free model. The paper focuses on T-splines of degree three, which are C2 (in the absence of multiple knots). T-NURCCs (Non-Uniform Rational Catmull-Clark Surfaces with T-junctions) are a superset of both T-splines and Catmull-Clark surfaces. Thus, a modeling program for T-NURCCs can handle any NURBS or Catmull-Clark model as special cases. T-NURCCs enable true local refinement of a Catmull-Clark-type control grid: individual control points can be inserted only where they are needed to provide additional control, or to create a smoother tessellation, and such insertions do not alter the limit surface. T-NURCCs use stationary refinement rules and are C2 except at extraordinary points and features.
849 citations
01 Jun 1988
TL;DR: This work presents a method of localizing the effect of refinement through the use of overlays, which are hierarchically controlled subdivisions, and introduces two editing techniques that are effective when using overlays.
Abstract: Refinement is usually advocated as a means of gaining finer control over a spline curve or surface during editing. For curves, refinement is a local process. It permits the change of control vertices and subsequent editing of the detail in one region of the curve while leaving control vertices in other regions unaffected. For tensor-product surfaces, however, refinement is not local in the sense that it causes control vertices far from a region of interest to change as well as changing the control vertices that influence the region. However, with some care and understanding it is possible to restrict the influence of refinement to the locality at which an editing effect is desired. We present a method of localizing the effect of refinement through the use of overlays, which are hierarchically controlled subdivisions. We also introduce two editing techniques that are effective when using overlays: one is direct surface manipulation through the use of edit points and the other is offset referencing of control vertices.
774 citations