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Won Sang Chung - One of the best experts on this subject based on the ideXlab platform.

S A Eftekhari - One of the best experts on this subject based on the ideXlab platform.

  • a differential quadrature procedure with direct projection of the heaviside function for numerical solution of moving load problem
    Latin American Journal of Solids and Structures, 2016
    Co-Authors: S A Eftekhari
    Abstract:

    OWING TO ITS PARTICULAR CHARACTERISTICS, THE DIRECT DISCRETIZATION OF THE Dirac-Delta FUNCTION IS NOT FEASIBLE WHEN POINT DISCRETIZATION METHODS LIKE THE DIFFERENTIAL QUADRATURE METHOD (DQM) ARE AP-PLIED. A WAY FOR OVERCOMING THIS DIFFICULTY IS TO APPROXIMATE (OR REGULARIZE) THE Dirac-Delta FUNCTION WITH SIMPLE MATHEMATICAL FUNC-TIONS. BY REGULARIZING THE Dirac-Delta FUNCTION, SUCH SINGULAR FUNC-TION IS TREATED AS NON-SINGULAR FUNCTIONS AND CAN BE EASILY AND DIRECTLY DISCRETIZED USING THE DQM. ON THE OTHER HAND, IT IS POSSI-BLE TO COMBINE THE DQM WITH THE INTEGRAL QUADRATURE METHOD (IQM) TO HANDLE THE Dirac-Delta FUNCTION. ALTERNATIVELY, ONE MAY USE ANOTHER DEI¬NITION OF THE Dirac-Delta FUNCTION THAT THE DERIVA-TIVE OF THE HEAVISIDE FUNCTION, H(X), IS THE Dirac-Delta FUNCTION, I´(X), IN THE DISTRIBUTION SENSE, NAMELY, DH(X)/DX = I´(X). THIS APPROACH HAS BEEN REFERRED IN THE LITERATURE AS THE DIRECT PROJECTION APPROACH. IT HAS BEEN SHOWN THAT ALTHOUGH THIS APPROACH YIELDS HIGHLY OSCILLATORY APPROXIMATION OF THE Dirac-Delta FUNCTION, IT CAN YIELD A NON-OSCILLATORY APPROXIMATION OF THE SOLUTION. IN THIS PAPER, WE FIRST PRESENT A MODIFIED DIRECT PROJECTION APPROACH THAT ELIMI-NATES SUCH DIFFICULTY (OSCILLATORY APPROXIMATION OF THE Dirac-Delta FUNCTION). WE THEN DEMONSTRATE THE APPLICABILITY AND RELIABILITY OF THE PROPOSED METHOD BY APPLYING IT TO SOME MOVING LOAD PROBLEMS OF BEAMS AND RECTANGULAR PLATES.

  • differential quadrature procedure for in plane vibration analysis of variable thickness circular arches traversed by a moving point load
    Applied Mathematical Modelling, 2016
    Co-Authors: S A Eftekhari
    Abstract:

    Abstract Point discretization methods such as the differential quadrature method (DQM) are well known to have difficulties in solving partial differential equations that involve the Dirac-Delta function because the Dirac-Delta function is a generalized singularity function and it cannot be discretized directly using the DQM. To overcome this difficulty, a simple differential quadrature methodology is proposed in this study, where the Dirac-Delta function is expanded into a Fourier trigonometric series. By expanding the Dirac-Delta function into a Fourier trigonometric series, this singular function is treated as non-singular functions, which can be discretized easily and directly using the DQM. The applicability of the proposed method is demonstrated by the in-plane vibration analysis of variable thickness circular arches traversed by a moving point load. The numerical results show that the proposed method is highly accurate and reliable.

  • a note on mathematical treatment of the Dirac Delta function in the differential quadrature bending and forced vibration analysis of beams and rectangular plates subjected to concentrated loads
    Applied Mathematical Modelling, 2015
    Co-Authors: S A Eftekhari
    Abstract:

    Abstract There are many physical and mechanical phenomena that can be well described by means of the Dirac-Delta function. For instance, the bending and vibration behavior of structures under concentrated loads, impulsive loading, moving loads, and impact loading; and thermoelastic vibration behavior of structures under heat source points can be mathematically modeled by means of the Dirac-Delta function. It is well known that such phenomena or problems can be easily handled by weak form based methods such as the Ritz and finite element methods. However, the strong form based methods such as the finite difference and differential quadrature methods may encounter some difficulties in mathematical modeling and treatment of the Dirac-Delta function. This is mainly caused by the fact that the properties of the Dirac-Delta function are in the form of integrals and not in the form of derivatives. To overcome this difficulty, this paper presents a combined differential quadrature–integral quadrature method in which such type of problems can be easily and accurately modeled. Its accuracy and reliability are demonstrated through the static and dynamic analysis of beams and rectangular plates under concentrated loads. This paper also presents a simple differential quadrature formulation for the analysis of rectangular plates. The proposed formulation first reduces the original plate problem to two simple beam problems. Each beam problem in then discretized using the differential quadrature method (DQM) in a simple manner. Compared with the conventional DQM, the proposed DQM is superior since its implementation and programming are easier and simpler.

  • a differential quadrature procedure with regularization of the Dirac Delta function for numerical solution of moving load problem
    Latin American Journal of Solids and Structures, 2015
    Co-Authors: S A Eftekhari
    Abstract:

    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.

H Hassanabadi - One of the best experts on this subject based on the ideXlab platform.

Jose Alexandre Nogueira - One of the best experts on this subject based on the ideXlab platform.

  • Dirac Delta function potential in quasiposition representation of a minimal length scenario
    European Physical Journal C, 2018
    Co-Authors: M F Gusson, Oakes A O Goncalves, R O Francisco, R G Furtado, Julio C Fabris, Jose Alexandre Nogueira
    Abstract:

    A minimal-length scenario can be considered as an effective description of quantum gravity effects. In quantum mechanics the introduction of a minimal length can be accomplished through a generalization of Heisenberg’s uncertainty principle. In this scenario, state eigenvectors of the position operator are no longer physical states and the representation in momentum space or a representation in a quasiposition space must be used. In this work, we solve the Schroedinger equation with a Dirac $$\Delta $$ -function potential in quasiposition space. We calculate the bound state energy and the coefficients of reflection and transmission for the scattering states. We show that leading corrections are of order of the minimal length $$({ O}(\sqrt{\beta }))$$ and the coefficients of reflection and transmission are no longer the same for the Dirac Delta well and barrier as in ordinary quantum mechanics. Furthermore, assuming that the equivalence of the 1s state energy of the hydrogen atom and the bound state energy of the Dirac $${{\Delta }}$$ -function potential in the one-dimensional case is kept in a minimal-length scenario, we also find that the leading correction term for the ground state energy of the hydrogen atom is of the order of the minimal length and $$\varDelta x_{\min } \le 10^{-25}$$ m.

  • Dirac Delta function potential in quasiposition representation of a minimal length scenario
    arXiv: High Energy Physics - Theory, 2017
    Co-Authors: M F Gusson, Oakes A O Goncalves, R O Francisco, R G Furtado, Julio C Fabris, Jose Alexandre Nogueira
    Abstract:

    A minimal-length scenario can be considered as an effective description of quantum gravity effects. In quantum mechanics the introduction of a minimal length can be accomplished through a generalization of Heisenberg's uncertainty principle. In this scenario, state eigenvectors of the position operator are no longer physical states and the representation in momentum space or a representation in a quasiposition space must be used. In this work, we solve the Schroedinger equation with Dirac $\Delta$-function potential in quasiposition space. We calculate the bound state energy and the coefficients of reflection and transmission for scattering states. We show that leading corrections are of order of the minimal length $({\sl O}(\sqrt{\beta}))$ and the coefficients of reflection and transmission are no longer the same for the Dirac Delta well and barrier as in ordinary quantum mechanics. Furthermore, assuming that the equivalence of the 1s state energy of the hydrogen atom and the bound state energy of the Dirac $\Delta$-function potential in 1-dim is kept in a minimal-length scenario, we also find that the leading correction term for the ground state energy of the hydrogen atom is of order of the minimal length and $\Delta x_{min} \le 10^{-25}$ m.

Boris N. Khoromskij - One of the best experts on this subject based on the ideXlab platform.

  • range separated tensor decomposition of the discretized Dirac Delta and elliptic operator inverse
    Journal of Computational Physics, 2020
    Co-Authors: Boris N. Khoromskij
    Abstract:

    Abstract In this paper, we introduce the operator dependent range-separated (RS) tensor approximation of the discretized Dirac Delta function (distribution) in R d . It is constructed by application of the elliptic operator to the RS tensor representation of the associated Green kernel discretized on the d-dimensional Cartesian grid. The proposed local-global decomposition of the Dirac Delta can be applied for solving the potential equations in a non-homogeneous medium when the density in the right-hand side is given by a large sum of pointwise singular charges. As an example of applications, we describe the regularization scheme for solving the Poisson-Boltzmann equation that models the electrostatics in bio-molecules. We show how the idea of the operator dependent RS tensor decomposition of the Dirac Delta can be generalized to the closely related problem on range-separated tensor representation of the elliptic resolvent. This approach paves the way for application of tensor numerical methods to elliptic problems with non-regular data. Numerical tests confirm the expected localization properties of the RS tensor approximation of the Dirac Delta represented on a tensor grid in 3D.

  • range separated tensor representation of the discretized multidimensional Dirac Delta and elliptic operator inverse
    arXiv: Numerical Analysis, 2018
    Co-Authors: Boris N. Khoromskij
    Abstract:

    In this paper, we introduce the operator dependent range-separated tensor approximation of the discretized Dirac Delta in $\mathbb{R}^d$. It is constructed by application of the discrete elliptic operator to the range-separated decomposition of the associated Green kernel discretized on the Cartesian grid in $\mathbb{R}^d$. The presented operator dependent local-global splitting of the Dirac Delta can be applied for solving the potential equations in non-homogeneous media when the density in the right-hand side is given by the large sum of pointwise singular charges. We show how the idea of the operator dependent RS splitting of the Dirac Delta can be extended to the closely related problem on the range separated tensor representation of the elliptic resolvent. The numerical tests confirm the expected localization properties of the obtained operator dependent approximation of the Dirac Delta represented on a tensor grid. As an example of application, we consider the regularization scheme for solving the Poisson-Boltzmann equation for modeling the electrostatics in bio-molecules.