Isogeometric Collocation Methods

Auricchio, Ferdinando; Veiga, L. Beirão da; Hughes, Thomas J.R.; Reali, Alessandro; Sangalli, Giancarlo

doi:10.1142/s0218202510004878

Cited by 333 publications

(328 citation statements)

References 21 publications

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“…[26,25,32,33]), where integrals in the weak form are eliminated by selecting the test function as the Dirac delta, formally constructed as the limit of a sequence of smooth functions with compact support that converge to a distribution, satisfying the so-called sifting property; or by enforcing directly the strong form of the governing equations to be satisfied in a discrete number of (collocation) points (e.g. [23,27,28,24]). We adopt the latter approach.…”

Section: Collocation and Discretization Of The Governing Equationsmentioning

confidence: 99%

“…The linearized governing equations are collocated at the standard Greville abscissa [23]. The Greville abscissa related to a knot vector U = [u 0 , .…”

Section: Collocation and Discretization Of The Governing Equationsmentioning

confidence: 99%

“…IGA-C is a new promising method introduced in [23] based on the discretization of the strong form of the governing equations. The method takes advantage of the higher-continuity properties of the approximating functions adopted in IGA and permits reducing significantly the computational costs by eliminating the need for numerical quadrature [24].…”

Section: Introductionmentioning

confidence: 99%

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Isogeometric collocation for three-dimensional geometrically exact shear-deformable beams

Marino

2016

Computer Methods in Applied Mechanics and Engineering

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Highlights• Isogeometric collocation is extended to geometrically exact beams.• Consistent linearization of the strong form of the governing equations is derived.• Incremental rotations are parametrized through Eulerian rotation vectors.• Numerical tests show efficiency and high accuracy. AbstractWe extend the isogeometric collocation method to the geometrically nonlinear beams. An exact kinematic formulation, able to represent three-dimensional displacements and rotations without any restriction in magnitude, is presented without the introduction of the moving frame concept. A displacement-based formulation is adopted. Full linearization of the strong form of the governing equations is derived consistently with the underlying geometric structure of the configuration manifold. Incremental rotations are parametrized through Eulerian rotation vectors and configuration updates are performed by means of the exponential map. Numerical tests demonstrate that the proposed combination of isogeometric collocation method with the chosen rotations parametrization results in an efficient computational scheme able to model complex problems with high accuracy.

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Section: Collocation and Discretization Of The Governing Equationsmentioning

confidence: 99%

“…The linearized governing equations are collocated at the standard Greville abscissa [23]. The Greville abscissa related to a knot vector U = [u 0 , .…”

Section: Collocation and Discretization Of The Governing Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Isogeometric collocation for three-dimensional geometrically exact shear-deformable beams

Marino

2016

Computer Methods in Applied Mechanics and Engineering

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“…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].…”

mentioning

confidence: 99%

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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“…Given their higher-order smoothness and their favorable approximation properties, NURBS basis functions constitute an efficient interpolation tool for collocating differential operators, i.e., for the approximation of the strong formulation of PDEs; furthermore they apply even to irregular or complex geometric domains. The newly developed IGA Collocation methods [8,[30][31][32][33], set in the realm of weighted residuals [35], have been applied to a variety of solid, structural mechanics and engineering problems, with excellent performances in terms of efficiency, accuracy and robustness if compared to standard Galerkin methods.…”

Section: Introductionmentioning

confidence: 99%

NURBS-Based Collocation Methods for the Structural Analysis of Shells of Revolution

Bellis

Artioli

2016

Metals

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Abstract:In this work we present a collocation method for the structural analysis of shells of revolution based on Non-Uniform Rational B-Spline (NURBS) interpolation. The method is based on the strong formulation of the equilibrium equations according to Reissner-Mindlin theory, with Fourier series expansion of dependent variables, which makes the problem 1D. Several numerical tests validate convergence, accuracy, and robustness of the proposed methodology, and its feasibility as a tool for the analysis and design of complex shell structures.

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Isogeometric Collocation Methods

Cited by 333 publications

References 21 publications

Isogeometric collocation for three-dimensional geometrically exact shear-deformable beams

Isogeometric collocation for three-dimensional geometrically exact shear-deformable beams

Overlapping Schwarz Methods for Isogeometric Analysis

NURBS-Based Collocation Methods for the Structural Analysis of Shells of Revolution

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