Numerical resolution of the hyperbolic heat equation using smoothed mathematical functions instead of Heaviside and Dirac delta distributions

Rivera, M. J.; Trujillo, Macarena; Romero‐García, Vicente; Molina, Juan Antonio López; Berjano, Enrique

doi:10.1016/j.icheatmasstransfer.2013.05.017

Cited by 16 publications

(8 citation statements)

References 18 publications

(6 reference statements)

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“…To obviate this difficulty, it has been suggested that the singular function (here, the Diracdelta function) should be regularized in order to achieve a smoother representation of the singular function and to stabilize the unwanted oscillatory behavior of solutions near the singularities. Various regularization techniques have been developed in Waldén (1999), Wei et al (2002), Engquist et al (2005), Burko and Khanna (2007), Sundararajan et al (2007), Towers (2007), and Rivera et al (2013). Waldén (1999) solved some elliptic and parabolic problems using the finite difference method and showed that the right regularization of the singular source terms can improve the convergence rate of solutions.…”

Section: Introductionmentioning

confidence: 99%

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A Differential Quadrature Procedure with Regularization of the Dirac-delta Function for Numerical Solution of Moving Load Problem

Eftekhari

2015

Lat. Am. j. solids struct.

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The differential quadrature method (DQM) is one of the most elegant and efficient methods for the numerical solution of partial differential equations arising in engineering and applied sciences. It is simple to use and also straightforward to implement. However, the DQM is well-known to have some difficulty when applied to partial differential equations involving singular functions like the Dirac-delta function. This is caused by the fact that the Dirac-delta function cannot be directly discretized by the DQM. To overcome this difficulty, this paper presents a simple differential quadrature procedure in which the Dirac-delta function is replaced by regularized smooth functions. By regularizing the Dirac-delta function, such singular function is treated as non-singular functions and can be easily and directly discretized using the DQM. To demonstrate the applicability and reliability of the proposed method, it is applied here to solve some moving load problems of beams and rectangular plates, where the location of the moving load is described by a time-dependent Dirac-delta function. The results generated by the proposed method are compared with analytical and numerical results available in the literature. Numerical results reveal that the proposed method can be used as an efficient tool for dynamic analysis of beam-and plate-type structures traversed by moving dynamic loads.

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Section: Introductionmentioning

confidence: 99%

“…More recently, Towers (2007) proposed two closely related finite difference methods for discretization of the Dirac-delta function. Most recently, Rivera et al (2013) proposed various smooth functions for approximation of the Dirac-delta function.…”

Section: Introductionmentioning

confidence: 99%

A Differential Quadrature Procedure with Regularization of the Dirac-delta Function for Numerical Solution of Moving Load Problem

Eftekhari

2015

Lat. Am. j. solids struct.

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“…In order to simulate the tracer injection in an instant of time, a Gaussian pulse function was employed [33] y t = 1…”

Section: Tracer Simulationmentioning

confidence: 99%

Simulations of a Single-Phase Flow in a Compound Parabolic Concentrator Reactor

Pérez

Nava

2018

International Journal of Photoenergy

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This paper deals with the analysis and interpretation of flow visualization and residence time distribution (RTD) in a compound parabolic concentrator (CPC) reactor using computational fluid dynamics (CFD). CFD was calculated under turbulent flow conditions solving the Reynolds averaged Navier–Stokes (RANS) equation expressed in terms of turbulent viscosity and the standard k−ε turbulent model in 3D. A 3D diffusion-convection model was implemented in the CPC reactor to determine the RTD. The fluid flow visualization and RTD were validated with experimental results. The CFD showed that the magnitude of the velocity field remains almost uniform in most of the bulk reactor, although near and inside the 90° connectors and the union segments, the velocity presented low- and high-speed zones. Comparisons of theoretical and experimental RTD curves showed that the k−ε model is appropriate to simulate the nonideal flow inside the CPC reactor under turbulent flow conditions.

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“…Engquist et al [25] presented two methods to construct consistent approximations for the regularization of the Dirac-delta function. Rivera et al [26] proposed numerical resolution of the hyperbolic heat equation using smoothed mathematical functions for approximation of the Heaviside and Dirac-delta functions.…”

Section: Introductionmentioning

confidence: 99%

A New Numerical Procedure for Vibration Analysis of Beam under Impulse and Multiharmonics Piezoelectric Actuators

Belkourchia

Azrar

2020

Journal of Applied Mathematics

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The dynamic behavior of structures with piezoelectric patches is governed by partial differential equations with strong singularities. To directly deal with these equations, well adapted numerical procedures are required. In this work, the differential quadrature method (DQM) combined with a regularization procedure for space and implicit scheme for time discretization is used. The DQM is a simple method that can be implemented with few grid points and can give results with a good accuracy. However, the DQM presents some difficulties when applied to partial differential equations involving strong singularities. This is due to the fact that the subsidiaries of the singular functions cannot be straightforwardly discretized by the DQM. A methodological approach based on the regularization procedure is used here to overcome this difficulty and the derivatives of the Dirac-delta function are replaced by regularized smooth functions. Thanks to this regularization, the resulting differential equations can be directly discretized using the DQM. The efficiency and applicability of the proposed approach are demonstrated in the computation of the dynamic behavior of beams for various boundary conditions and excited by impulse and Multiharmonics piezoelectric actuators. The obtained numerical results are well compared to the developed analytical solution.

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Numerical resolution of the hyperbolic heat equation using smoothed mathematical functions instead of Heaviside and Dirac delta distributions

Abstract: ElsevierRivera Ortun, MJ.; Trujillo Guillen, M.; Romero García, V.; López Molina, JA.; Berjano Zanón, E. (2013). Numerical resolution of the hyperbolic heat equation using smoothed mathematical functions instead of Heaviside and Dirac delta distributions.

Cited by 16 publications

References 18 publications

A Differential Quadrature Procedure with Regularization of the Dirac-delta Function for Numerical Solution of Moving Load Problem

A Differential Quadrature Procedure with Regularization of the Dirac-delta Function for Numerical Solution of Moving Load Problem

Simulations of a Single-Phase Flow in a Compound Parabolic Concentrator Reactor

A New Numerical Procedure for Vibration Analysis of Beam under Impulse and Multiharmonics Piezoelectric Actuators

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