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

Abstract

• Fractional calculus is employed in the simulation of the laser drilling process. • Temperature is locally approximated by meshless local Petrov–Galerkin method. • Constant and variable fractional orders are investigated. • The fractional model yields better results than the integer order model. 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.