A fully Sinc‐Galerkin method for Euler–Bernoulli beam models

Smith, Ralph C.; Bowers, Kenneth L.; Lund, John

doi:10.1002/num.1690080207

Cited by 27 publications

(8 citation statements)

References 15 publications

(10 reference statements)

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“…In [91] and [152], numerical methods are developed for cantilever beam problems consisting of (1.9) subject to the initial conditions (1.10) (in [152], go = gl = 0) and the boundary conditions…”

Section: One Dimensional Problemsmentioning

confidence: 99%

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Numerical Methods for Schrödinger-Type Problems

Fairweather¹,

Khebchareon²

2002

Applied Optimization

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Several physical phenomena are modeled by initial-boundary value problems which can be formulated as SchrOdinger -type systems of partial differential equations. In this paper, two classes of problems of this kind, Schrodinger equations, which arise in various areas of physics, and certain vibration problems from civil and mechanical engineering, are considered. A survey of numerical methods for solving linear and nonlinear problems in one and several space variables is presented, with special attention being devoted to the parabolic wave equation, the cubic Schrodinger equation, and to fourth order parabolic equations arising in vibrating beam and plate problems. Recently developed finite element methods for solving SchrOdinger-type systems are also outlined.Keywords: Schrodinger systems, linear and nonlinear Schrodinger equations, parabolic wave equation, vibrating beams and plates, fourth order parabolic equations, finite difference methods, finite element Galerkin methods, orthogonal spline collocation methods, spectral methods IntroductionWe consider the numerical solution of problems that can be written as initialboundary value problems (IBVPs) for real systems of partial differential equations of the form

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Section: One Dimensional Problemsmentioning

confidence: 99%

“…In [91], a finite element Galerkin method is proposed in which C 1 piecewise cubic polynomials (the space of piecewise Hermite cubics) are used in space, and the time stepping is done using the approach developed in [167]. A space-time Sinc-Galerkin method is presented in [152].…”

Section: One Dimensional Problemsmentioning

confidence: 99%

Numerical Methods for Schrödinger-Type Problems

Fairweather¹,

Khebchareon²

2002

Applied Optimization

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“…The main disadvantage of this approach lies in the illconditioning of the solution matrix stemming from Eq. (13). In fact, the condition number of the resulting matrix becomes enormous.…”

Section: Special Treatment At the Boundariesmentioning

confidence: 99%

“…Even at this high order, the consistency of the method in exhibiting an exponential convergence rate could be shown. In a subsequent study, Smith, Bowers and Lund 13 have applied the Sinc-Galerkin method to several examples involving fourth-order time-dependent partial differential equations in both space and time.…”

Section: Introductionmentioning

confidence: 99%

Navier-Stokes Solution of the Driven-Cavity Problem Using Sinc-Collocation Elements

Narasimhan

Majdalani

2002

32nd AIAA Fluid Dynamics Conference and Exhibit

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Different numerical approaches have been proposed in the past to solve the Navier-Stokes equations. Conventional methods have often relied on finite-difference, finite-element, and boundary-element techniques. Multi-grid methods have been recently introduced because they help promote a faster convergence rate of the error residual. A difficulty plaguing numerical methods today is the inability to treat singularities at or near boundaries. Such difficulties become even more pronounced when coupled with the need to handle semiinfinite and infinite domains. Sinc-based numerical algorithms have the advantage of handling singularities, boundary layers, and semi-infinite domains very effectively. In addition, they typically require fewer nodal points while providing an exponential convergence rate in solving linear differential equations. This study involves a first step in applying the Sinc-based algorithm to solve a nonlinear set of partial differential equations. The example we consider arises in the context of a driven-cavity flow in two space dimensions. As such, the steady and incompressible Navier-Stokes equations are solved by means of two-dimensional Sinc collocation in conjunction with the primitive variable method and a pressure correction algorithm based on artificial compressibility. Simulations are also carried out using forward differences, central differences, and a commercial code. Results are compared with one another and with the Sinc-collocation approximation. It is found that the error in the Sinc-collocation approximation outperforms other solutions, especially near the singular corners of the cavity.

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“…In particular, they have become very popular in solving initial and boundary value problems of ordinary and partial differential equations including those with Dirichlet, Neuman and other boundary conditions [13,17,26,27]. There are several advantages to using approximations based on Sinc numerical methods.…”

Section: Introductionmentioning

confidence: 99%

Application of the Sinc method to a dynamic elasto-plastic problem

Abdella

Küçük

2009

Journal of Computational and Applied Mathematics

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This paper presents the application of Sinc bases to simulate numerically the dynamic behavior of a one-dimensional elastoplastic problem. The numerical methods that are traditionally employed to solve elastoplastic problems include finite difference, finite element and spectral methods. However, more recently, biorthogonal wavelet bases have been used to study the dynamic response of a uniaxial elasto-plastic rod [Giovanni F. Naldi, Karsten Urban, Paolo Venini, A wavelet-Galerkin method for elastoplasticity problems, Report 181, RWTH Aachen IGPM, and Math. Modelling and Scient. Computing, vol. 10, 2000]. In this paper the Sinc-Galerkin method is used to solve the straight elasto-plastic rod problem. Due to their exponential convergence rates and their need for a relatively fewer nodal points, Sinc based methods can significantly outperform traditional numerical methods [J. Lund, K.L. Bowers, Sinc Methods for Quadrature and Differential Equations, SIAM, Philadelphia, 1992]. However, the potential of Sinc-based methods for solving elastoplasticity problems has not yet been explored. The aim of this paper is to demonstrate the possible application of Sinc methods through the numerical investigation of the unsteady one dimensional elastic-plastic rod problem.

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A fully Sinc‐Galerkin method for Euler–Bernoulli beam models

Cited by 27 publications

References 15 publications

Numerical Methods for Schrödinger-Type Problems

Numerical Methods for Schrödinger-Type Problems

Navier-Stokes Solution of the Driven-Cavity Problem Using Sinc-Collocation Elements

Application of the Sinc method to a dynamic elasto-plastic problem

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