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A fractional model for estimating the hole geometry in the laser drilling process of thin metal sheets

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  • Zahra, Waheed K.
  • Abdel-Aty, Mahmoud
  • Abidou, Diaa

Abstract

Fractional calculus has been increasingly attracting interest in various fields of science and engineering where the problems are governed by differential and integral equations. This shift towards adopting such an approach approves its validity since it has shown that different engineering problems could be better represented by fractional than integer order calculus. Therefore, in this work, fractional calculus is employed in order to simulate a previously addressed problem of metal laser drilling process using meshless local Petrov–Galerkin (MLPG). Both approximations of shifted and weighted shifted Grünwald–Letnikov are used and compared with each other in terms of the expected hole geometry and its closeness to the experimental data. Moreover, the fractional order derivative is considered to be both constant and variable in order to show its impact on the expected outcome of the hole profile. Specifically speaking, for this problem of fixed laser absorptivity, it is shown that the fractional derivative order needs to be variable in order to make the numerical results best match the experimental data in both stages of transient and steady-state.

Suggested Citation

  • Zahra, Waheed K. & Abdel-Aty, Mahmoud & Abidou, Diaa, 2020. "A fractional model for estimating the hole geometry in the laser drilling process of thin metal sheets," Chaos, Solitons & Fractals, Elsevier, vol. 136(C).
    • Handle: RePEc:eee:chsofr:v:136:y:2020:i:c:s0960077920302435
    DOI: 10.1016/j.chaos.2020.109843
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    References listed on IDEAS

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    1. Abdon Atangana & Adem Kılıçman, 2013. "The Use of Sumudu Transform for Solving Certain Nonlinear Fractional Heat-Like Equations," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-12, May.
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    3. Zahra, W.K. & Van Daele, M., 2017. "Discrete spline methods for solving two point fractional Bagley–Torvik equation," Applied Mathematics and Computation, Elsevier, vol. 296(C), pages 42-56.
    4. Collins, Jeb & Gremaud, Pierre, 2011. "A simple model for laser drilling," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 81(8), pages 1541-1552.
    5. Alkahtani, Badr Saad T. & Atangana, Abdon, 2016. "Analysis of non-homogeneous heat model with new trend of derivative with fractional order," Chaos, Solitons & Fractals, Elsevier, vol. 89(C), pages 566-571.
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