Mathematical modeling and stability analysis for laser cutting via asymptotic expansion

Abstract

A new model for laser cutting is presented. It is derived by an asymptotic expansion of a free boundary problem for the arising melt and describes the coupled movements of the melt boundaries by a system of two nonlinear partial differential equations. The paper extends former models by presenting a complete first-order expansion allowing to include arbitrary higher-order terms. A stationary solution is derived and investigated for stability since damped perturbations do not disturb the solidification process whereas growing perturbations lead to unwanted ripple structures at the cutting front. New higher order terms in the model are identified with physical phenomena which are shown to provide stabilization of the process. The stability analysis is evaluated in a parameter study with varying cutting velocities for 1µm fiber laser cuttings and validated with published experimental and simulated data for the same study. Increasing the cutting speed provides larger stability areas leading to smoother cutting fronts. As a novelty, published ripple depth tendencies are shown to be met qualitatively by our model and typical ripple wavelengths are predicted in the order of magnitude. Hence, our model provides a purely analytical method to investigate the cutting quality.