Spline Functions: Basic Theory

Schumaker, Larry L.

doi:10.1017/cbo9780511618994

Cited by 1,044 publications

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“…Let Π 0 be the standard spline quasi-interpolant into the space V 0 , which is built using the dual basis functions as detailed in [36,Theorem 12.6]. Given any u ∈ V , define…”

Section: Initial Results In One Dimensionmentioning

confidence: 99%

“…Due to well-known stability and approximation properties of the quasi-interpolant in [36,Theorem 12.7], the following bound holds:…”

Section: Initial Results In One Dimensionmentioning

confidence: 99%

“…The functions N p i are a partition of unity, as shown in [36]. In Figure 3.1 we present an example consisting of n = 9 cubic basis functions generated from the open knot vector ξ = {0, 0, 0, 0, 1/6, 1/3, 1/2, 2/3, 5/6, 1, 1, 1, 1}.…”

Section: B-splines B-splines Inmentioning

confidence: 99%

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Overlapping Schwarz Methods for Isogeometric Analysis

Veiga¹,

Cho²,

Pavarino³

et al. 2012

SIAM J. Numer. Anal.

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Abstract. We construct and analyze an overlapping Schwarz preconditioner for elliptic problems discretized with isogeometric analysis. The preconditioner is based on partitioning the domain of the problem into overlapping subdomains, solving local isogeometric problems on these subdomains, and solving an additional coarse isogeometric problem associated with the subdomain mesh. We develop an h-analysis of the preconditioner, showing in particular that the resulting algorithm is scalable and its convergence rate depends linearly on the ratio between subdomain and "overlap sizes" for fixed polynomial degree p and regularity k of the basis functions. Numerical results in two-and three-dimensional tests show the good convergence properties of the preconditioner with respect to the isogeometric discretization parameters h, p, k, number of subdomains N , overlap size, and also jumps in the coefficients of the elliptic operator.Key words. domain decomposition methods, overlapping Schwarz, scalable preconditioners, isogeometric analysis, finite elements, NURBS AMS subject classifications. 65N55, 65N30, 65F10 DOI. 10.1137/110833476 1. Introduction. Isogeometric analysis (IGA) based on NURBS (nonuniform rational B-splines) was introduced by Hughes, Cottrell, and Bazilevs in [27] as an innovative numerical methodology for the analysis of PDE problems, allowing for an exact description of CAD-type geometries. NURBS are a standard in the computeraided design (CAD) community mainly because such spline functions allow excellent representations of free-form surfaces, and there are very efficient algorithms for their evaluation, refinement, and derefinement. In the isogeometric framework, the NURBS basis functions representing the CAD geometry are also used as the basis for the discrete solution space of PDEs, following an isoparametric paradigm. In addition to exact representation of CAD geometries, another advantage of using NURBS basis functions is the higher control on the regularity of the discrete space. For instance, spaces of global C k regularity are easily built, thus allowing for fewer degrees of freedom, better performance in case of vibrations, easier approximation of higher order problems, and other advantages. IGA methodologies have been summarized in the recent book [18] and studied in, e.g., [2,4,9,10,19,23,28,29,11,5,8]. IGA methods are having a growing impact on fields as diverse as fluid dynamics [6,7,40,15,26], structural mechanics [3,1,12,20,30,39], and electromagnetics [17,16].The discrete systems produced by isogeometric methods are better conditioned than the systems produced by standard finite elements or finite differences, but their conditioning can still degenerates rapidly for decreasing mesh size h, increasing poly-

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“…Let Π 0 be the standard spline quasi-interpolant into the space V 0 , which is built using the dual basis functions as detailed in [36,Theorem 12.6]. Given any u ∈ V , define…”

Section: Initial Results In One Dimensionmentioning

confidence: 99%

“…Due to well-known stability and approximation properties of the quasi-interpolant in [36,Theorem 12.7], the following bound holds:…”

Section: Initial Results In One Dimensionmentioning

confidence: 99%

See 1 more Smart Citation

Overlapping Schwarz Methods for Isogeometric Analysis

Veiga¹,

Cho²,

Pavarino³

et al. 2012

SIAM J. Numer. Anal.

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“…Parametric NURBS surfaces are based on polynomial B-splines and are defined by a set of control points which allow local shape control [3,[5][6][7][8]. The main reason for using rational NURBS instead of (non-rational) polynomial Bsplines is that NURBS are able to exactly reproduce conic sections [9].…”

Section: Continuous Closed Surfacesmentioning

confidence: 99%

Smooth shapes with spherical topology: Beyond traditional modeling, efficient deformation, and interaction

2017

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Existing shape models with spherical topology are typically designed either in the discrete domain using interpolating polygon meshes or in the continuous domain using smooth but non-interpolating schemes such as subdivision or NURBS. Both polygon models and subdivision methods require a large number of parameters to model smooth surfaces. NURBS need fewer parameters but have a complicated rational expression and non-uniform shifts in their formulation. We present a new method to construct deformable closed surfaces, which includes exact spheres, by combining the best of two worlds: a smooth, interpolating model with a continuously varying tangent plane and well-defined curvature at every point on the surface.Our formulation is considerably simpler than NURBS and requires fewer parameters than polygon meshes. We demonstrate the generality of our method with applications including intuitive user-interactive shape modeling, continuous surface deformation, shape morphing, reconstruction of shapes from parameterized point clouds, and fast iterative shape optimization for image segmentation. Comparisons with discrete methods and non-interpolating approaches highlight the advantages of our framework.

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“…The exposition found in these sections is mostly based on classical material, which can be found in basic textbooks on numerical analysis and on spline interpolation, i.g. [3], [2], [9] and [10]. For a more extensive empirical investigation on interpolation using splines, we refer to the paper by Rolain [8].…”

Section: Outlinementioning

confidence: 99%

Frequency domain identification of continuous-time output error models, Part II: Non-uniformly sampled data and B-spline output approximation

Gillberg¹,

Ljung

2010

Automatica

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This paper concerns the subject of identi cation of continuous-time output error (OE) models based on non-uniformly sampled output data. The exact method for doing this is well known in the time-domain, where the continuous-time system is discretized, simulated and the result is tted in a mean square sense to measured data. The material presented here is based on a method proposed in a sister paper [5] which dealt with the same topic but for the case of uniformly sampled data. In this text it will be shown how that method entails that the output should be reconstructed using a B-spline with uniformly distributed knots. This representation can then be used to directly identify the continuous-time system without proceeding via discretization. Only the relative degree of the model is used to choose the order of the spline. AbstractThis paper concerns the subject of identification of continuous-time output error (OE) models based on non-uniformly sampled output data. The exact method for doing this is well known in the time-domain, where the continuous-time system is discretized, simulated and the result is fitted in a mean square sense to measured data. The material presented here is based on a method proposed in a sister paper [5] which dealt with the same topic but for the case of uniformly sampled data. In this text it will be shown how that method entails that the output should be reconstructed using a B-spline with uniformly distributed knots. This representation can then be used to directly identify the continuous-time system without proceeding via discretization.Only the relative degree of the model is used to choose the order of the spline.Copyright c 2007 IFAC

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Spline Functions: Basic Theory

Cited by 1,044 publications

References 0 publications

Overlapping Schwarz Methods for Isogeometric Analysis

Overlapping Schwarz Methods for Isogeometric Analysis

Smooth shapes with spherical topology: Beyond traditional modeling, efficient deformation, and interaction

Frequency domain identification of continuous-time output error models, Part II: Non-uniformly sampled data and B-spline output approximation

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