Duality and unified analysis of discrete approximations in structural dynamics and wave propagation: Comparison of p-method finite elements with k-method NURBS

Hughes, Thomas J.R.; Reali, Alessandro; Sangalli, Giancarlo

doi:10.1016/j.cma.2008.04.006

Cited by 373 publications

(267 citation statements)

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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].…”

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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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mentioning

confidence: 99%

Overlapping Schwarz Methods for Isogeometric Analysis

Veiga¹,

Cho²,

Pavarino³

et al. 2012

SIAM J. Numer. Anal.

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“…Here, N N N and @N N N @x x x are the shape matrix and its derivative with respect to the physical coordinate x x x; see [9,10,6]. Due to the space discretization, the exact solution to Eq.…”

Section: Introductionmentioning

confidence: 99%

“…(2) contains the numerical dispersion error. Usually the analysis of the numerical dispersion error and its improvement for many space-discretization techniques such as the finite elements, spectral elements, isogeometric elements and others starts with the analysis and modifications of the elemental mass and stiffness matrices; see [10,1,2,7,8,15,16,17,18,22,11,14,20,21]. For example, one simple and effective finite-element technique for acoustic and elastic wave propagation problems is based on the calculation of the mass matrix M M M in Eq.…”

Section: Introductionmentioning

confidence: 99%

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Optimal Reduction of Numerical Dispersion for Wave Propagation Problems. Application to Isogeometric Elements.

Idesman¹

2017

Proceedings of the 6th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COM

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Abstract. Based on the optimal coefficients of the stencil equation, a numerical technique for the reduction of the numerical dispersion error has been suggested. New isogeometric elements with the reduced numerical dispersion error for wave propagation problems have been developed with the suggested approach. By the minimization of the order of the dispersion error of the stencil equation, the order of the dispersion error is improved from order 2p (the conventional isogeometric elements) to order 4p (the isogeometric elements with reduced dispersion) where p is the order of the polynomial approximations. Because all coefficients of the stencil equation are obtained from the minimization procedure, the obtained accuracy is maximum possible. The corresponding elemental mass and stiffness matrices of the isogeometric elements with reduced dispersion are calculated with help of the optimal coefficients of the stencil equation. The analysis of the dispersion error of the isogeometric elements with the lumped mass matrix has also shown that independent of the procedures for the calculation of the lumped mass matrix, the second order of the dispersion error cannot be improved with the conventional stiffness matrix. However, the dispersion error with the lumped mass matrix can be improved from the second order to order 2p by the modification of the stiffness matrix. The numerical examples confirm the computational efficiency of the isogeometric elements with reduced dispersion that significantly reduce the computation time at a given accuracy. The numerical results obtained by the new and conventional isogeometric elements may include spurious oscillations due to the dispersion error. These oscillations can be quantified and filtered by the two-stage timeintegration technique developed recently. The approach developed can be directly applied to other space-discretization techniques with similar stencil equations. 946Available online at www.eccomasproceedia.org Eccomas Proceedia COMPDYN (2017) 946-954

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“…The recent exponential growth of the isogeometric analysis [24][25][26][27][28][29] has raised attention on collocation methods based on NURBS and/or T-Splines trial functions. 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.…”

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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Duality and unified analysis of discrete approximations in structural dynamics and wave propagation: Comparison of p-method finite elements with k-method NURBS

Cited by 373 publications

References 16 publications

Overlapping Schwarz Methods for Isogeometric Analysis

Overlapping Schwarz Methods for Isogeometric Analysis

Optimal Reduction of Numerical Dispersion for Wave Propagation Problems. Application to Isogeometric Elements.

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

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