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The Taylor-SPH Meshfree Method: Basis and Validation

Received: 1 February 2015     Accepted: 1 February 2015     Published: 2 July 2015
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Abstract

This paper presents the basis and validation of the Taylor-SPH meshless method formulated in terms of stresses and velocities which can be applied to Solid Dynamic problems. The proposed method consists of applying first the time discretization by means of a Taylor series expansion in two steps and a corrected SPH method for the space discretization. In order to avoid numerical instabilities, two different sets of particles are used in the time discretization. To validate the Taylor-SPH method, it has been applied to solve the propagation of shock waves in elastic materials and the results have been compared with those obtained with a corrected SPH discretization combined with a 4th order Runge-Kutta time integration. The Taylor-SPH method is shown to be stable, robust and efficient and it provides more accurate results than those obtained with the standard SPH along with the Runge-Kutta time integration scheme. Numerical dispersion and diffusion are eliminated and only a reduced number of particles is required to obtain accurate results.

Published in Applied and Computational Mathematics (Volume 4, Issue 4)
DOI 10.11648/j.acm.20150404.17
Page(s) 286-295
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2015. Published by Science Publishing Group

Keywords

Taylor-SPH (TSPH), Meshfree, Runge-Kutta, Shock Wave, Stability, Dynamics

References
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[15] M. Mabssout, M. Pastor, M.I. Herreros, M. Quecedo, “A Runge-Kutta, Taylor-Galerkin scheme for hyperbolic systems with source terms. Application to shock wave propagation in viscoplastic geomaterials”. Int. J. Numer. Analytical Meth. Geomechanics, (2006); 30(13): 1337-1355.
[16] M. Mabssout, M. Herreros, “Taylor-SPH vs Taylor–Galerkin for shock waves in viscoplastic continua”, Eur. J. Comput. Mech. 20 (5–6) (2011) 281–308.
[17] M. Mabssout, M. I. Herreros, “Runge-Kutta vs Taylor-SPH. Two time integration schemes for SPH with application to Soil Dynamics”, App. Math. Modelling, 37(5), pp. 3541-3563, 2013.
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Cite This Article
  • APA Style

    H. Idder, M. Mabssout, M. I. Herreros. (2015). The Taylor-SPH Meshfree Method: Basis and Validation. Applied and Computational Mathematics, 4(4), 286-295. https://doi.org/10.11648/j.acm.20150404.17

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    ACS Style

    H. Idder; M. Mabssout; M. I. Herreros. The Taylor-SPH Meshfree Method: Basis and Validation. Appl. Comput. Math. 2015, 4(4), 286-295. doi: 10.11648/j.acm.20150404.17

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    AMA Style

    H. Idder, M. Mabssout, M. I. Herreros. The Taylor-SPH Meshfree Method: Basis and Validation. Appl Comput Math. 2015;4(4):286-295. doi: 10.11648/j.acm.20150404.17

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  • @article{10.11648/j.acm.20150404.17,
      author = {H. Idder and M. Mabssout and M. I. Herreros},
      title = {The Taylor-SPH Meshfree Method: Basis and Validation},
      journal = {Applied and Computational Mathematics},
      volume = {4},
      number = {4},
      pages = {286-295},
      doi = {10.11648/j.acm.20150404.17},
      url = {https://doi.org/10.11648/j.acm.20150404.17},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20150404.17},
      abstract = {This paper presents the basis and validation of the Taylor-SPH meshless method formulated in terms of stresses and velocities which can be applied to Solid Dynamic problems. The proposed method consists of applying first the time discretization by means of a Taylor series expansion in two steps and a corrected SPH method for the space discretization. In order to avoid numerical instabilities, two different sets of particles are used in the time discretization. To validate the Taylor-SPH method, it has been applied to solve the propagation of shock waves in elastic materials and the results have been compared with those obtained with a corrected SPH discretization combined with a 4th order Runge-Kutta time integration. The Taylor-SPH method is shown to be stable, robust and efficient and it provides more accurate results than those obtained with the standard SPH along with the Runge-Kutta time integration scheme. Numerical dispersion and diffusion are eliminated and only a reduced number of particles is required to obtain accurate results.},
     year = {2015}
    }
    

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    T1  - The Taylor-SPH Meshfree Method: Basis and Validation
    AU  - H. Idder
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    T2  - Applied and Computational Mathematics
    JF  - Applied and Computational Mathematics
    JO  - Applied and Computational Mathematics
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    PB  - Science Publishing Group
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    UR  - https://doi.org/10.11648/j.acm.20150404.17
    AB  - This paper presents the basis and validation of the Taylor-SPH meshless method formulated in terms of stresses and velocities which can be applied to Solid Dynamic problems. The proposed method consists of applying first the time discretization by means of a Taylor series expansion in two steps and a corrected SPH method for the space discretization. In order to avoid numerical instabilities, two different sets of particles are used in the time discretization. To validate the Taylor-SPH method, it has been applied to solve the propagation of shock waves in elastic materials and the results have been compared with those obtained with a corrected SPH discretization combined with a 4th order Runge-Kutta time integration. The Taylor-SPH method is shown to be stable, robust and efficient and it provides more accurate results than those obtained with the standard SPH along with the Runge-Kutta time integration scheme. Numerical dispersion and diffusion are eliminated and only a reduced number of particles is required to obtain accurate results.
    VL  - 4
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Author Information
  • Faculty of Science and Technology, Laboratory of Mechanics and Civil Engineering, Tangier, Morocco

  • Faculty of Science and Technology, Laboratory of Mechanics and Civil Engineering, Tangier, Morocco

  • CEDEX, Madrid, Spain

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