Fourier Series in Orthogonal Polynomials

Осиленкер, Б. П.

doi:10.1142/4039

Cited by 37 publications

(24 citation statements)

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“…A particular choice of the LLS approximation is a truncated expansion in a system of orthogonal polynomials, also known as Fourier series in orthogonal polynomials [28,30]. Let p j (θ), j = 0, 1, 2, .…”

Section: Approximation With Orthogonal Polynomialsmentioning

confidence: 99%

A two-step model reduction approach for mechanical systems with moving loads

Stykel

Vasilyev

2016

Journal of Computational and Applied Mathematics

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Abstract. We consider model order reduction of mechanical systems with moving loads. Such systems have a time-varying input matrix that makes it difficult the direct application of standard model reduction methods. In this paper, we present a two-step model reduction approach for systems with moving loads which is based on a low-rank approximation of the input matrix and applying Krylov subspace methods to the resulting linear time-invariant system with a modified input. Numerical results demonstrate the properties of the proposed model reduction method.Key words. mechanical systems, moving loads, model reduction 1. Introduction. In structural dynamics, the moving load problem has received a lot of attention because of its importance in many practical applications. Mechanical systems with moving loads arise, for example, in modelling of bridges with moving vehicles [17,32], cableways [17], cranes [42] or working gears [39]. One of the most popular methods for simulation of the dynamic behaviour of such systems is a finite element method (FEM) based on a variational formulation of the structural mechanics problem, e.g., [41]. In the engineering literature, the principle of virtual work is used to derive the FEM approximations, see [5]. The FEM discretization of a system subjected to moving loads yields a linear time-varying (LTV) second-order system

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Section: Approximation With Orthogonal Polynomialsmentioning

confidence: 99%

A two-step model reduction approach for mechanical systems with moving loads

Stykel

Vasilyev

2016

Journal of Computational and Applied Mathematics

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“…In definition (14), P , p are the Jacobi polynomials of order p. We use ultraspheric polynomials corresponding to the choice = = 1. Multidimensional modal expansions are constructed by taking tensor product of 1-D modal expansions.…”

Section: Finite Element Modelmentioning

confidence: 99%

“…We will first state the following relationships which will be useful later. For a proof of these equalities, see [14].…”

Section: Orthogonality Of Modal Basesmentioning

confidence: 99%

Orthogonality of modal bases in hp finite element models

Prabhakar

Reddy

2007

Numerical Methods in Fluids

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SUMMARYIn this paper, we exploit orthogonality of modal bases (SIAM J. Sci. Comput. 1999; 20:1671-1695) used in hp finite element models. We calculate entries of coefficient matrix analytically without using any numerical integration, which can be computationally very expensive. We use properties of Jacobi polynomials and recast the entries of the coefficient matrix so that they can be evaluated analytically. We implement this in the context of the least-squares finite element model although this procedure can be used in other finite element formulations. In this paper, we only develop analytical expressions for rectangular elements. Spectral convergence of the L 2 least-squares functional is verified using exact solution of Kovasznay flow. Numerical results for transient flow over a backward-facing step are also presented. We also solve steady flow past a circular cylinder and show the reduction in computational cost using expressions developed herein.

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“…Let μ be a finite positive Borel measure supported on an infinite subset of R. It is well-known that the polynomial kernels (also called reproducing, Christoffel-Darboux or Dirichlet kernels) associated with the sequences of orthogonal polynomials corresponding to μ are frequently used as a basic tool in spectral analysis, convergence of orthogonal expansions [2,23,27], and other aspects of mathematical analysis (see [26] and the references therein). In the setting of orthogonal polynomial theory these kernels have been especially used by Freud and Nevai [4,21,22] and, more recently, the remarkable Lubinsky's works [9,10] have caused heightened interest in this topic.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic behavior of the partial derivatives of Laguerre kernels and some applications

Marcellán

Pérez-Valero

Quintana

2015

Journal of Mathematical Analysis and Applications

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Orthogonal polynomials Laguerre polynomials Asymptotics Diagonal Christoffel-Darboux kernels Sobolev-type orthogonal polynomialsThe aim of this paper is to present some new results about the asymptotic behavior of the partial derivatives of the kernel polynomials associated with the Gamma distribution. We also show how these results can be used in order to obtain the inner relative asymptotics for certain Laguerre-Sobolev type polynomials.

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Fourier Series in Orthogonal Polynomials

Cited by 37 publications

References 0 publications

A two-step model reduction approach for mechanical systems with moving loads

A two-step model reduction approach for mechanical systems with moving loads

Orthogonality of modal bases in hp finite element models

Asymptotic behavior of the partial derivatives of Laguerre kernels and some applications

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