**4. Conclusion**

Finally, the ANOVA results (**Table 5**) reveal that the model is significant and the lack-of-fit is not (*p*-value close to 0.05). All coefficients shown in the ANOVA table are significant; the others are not, hence the term of "reduced" cubic model. The multilinear regression coefficients are

thus significant and the final mathematical equation (Eq. (16)) in terms of coded factors is given

The model is determined to be statistically adequate. The study of the residuals did not reveal major inconstancies. It is necessary to check if the model is scientifically adequate and if it can

From the response equation (Eq. (16)), it is clear that the pulse energy (*A*) and the scanning speed (*D*) are the most influent parameters on the scribing depth. Heating a target via laser irradiation depends on the amount of energy brought to the target and the interaction duration. This is why they are the most dominant factors and why the interaction term *AD* appears in

the depth increases with increasing energy input until saturation. It is the classical logarithm trend observed in the literature [37]. The scribing depth also decreases with increasing speed until saturation. Interaction terms such as *AC* (evaluation of the fluence, Eqs. (18) and (19)), *AD* (effect of fluence over time), and *CD* (laser/material interaction over time) appear naturally

( )² ( )²

 pw

() ( )

0 0 2 2

æ öæ ö æö æ ö ç ÷ç ÷ =+ =+ ç÷ ç ÷

² ² () 1 <sup>1</sup> *M z M C <sup>z</sup>* l

 w

0 0

w

èø è ø è øè ø

where *z* is the defocus (m), *O* is the overlap ratio, *f* is the repetition rate (Hz), *v* is the scanning speed (mm/s), λ is the wavelength (nm) and ω<sup>0</sup> (cm) is the spot radius at the focal point. The effect of the number of passes *B* varied depending on the delay between two passes and thus on the scanning speed. The effect saturated around three passes for pulse energies below 85

2 2

*E A <sup>F</sup>* pw

1 1

w

pw

*v D <sup>O</sup>*

Y = 156.89 + 52A + 22.63B – 3.87C -– 39.2D + 8.78AB – 5.75AC – 13AD – 3.6BD + 3.68CD

+ 6.75D2

adjusted = 0.997, *R*<sup>2</sup> predicted = 0.9805). The reduced cubic model is

and *D*<sup>2</sup> are conformant with theory as well. Indeed,

*z C* = = ´ ´ (17)

*zf Cf* =- =- ´ ´ (18)

 l

pw

D + 9.05AB2 (16)

(19)

– 3.73ABD + 9.17A2

close to 1 (*R*<sup>2</sup>

= 0.9986, *R*<sup>2</sup>

182 Modeling and Simulation in Engineering Sciences

– 9.15B2

the equation. Square terms coefficient *A*<sup>2</sup>

w

 w

predict response values with additional experiments.

– 5.85C2

in the response equation because they are linked by Eqs. (17)–(19):

using the least-squares method.

– 18.05A2

*3.3.2. Experimental validation*

Whether it is Finite Element Methods, Monte Carlo or Molecular Dynamics, numerical simulation gives the opportunity to better comprehend the physics of laser-material interaction and in many cases to predict results and optimize the parameters for laser processing. In this chapter, numerical simulation was proven to be valuable to both fundamental science and laser applications in the industry. Computer calculation improvement makes it possible to model complex problems when multiphysical phenomena are coupled and allowed important achievements in the field of laser processing. Simplification assumptions are still required but with time and research efforts, models will become more and more realistic giving the possibility of getting a snapshot inside complex physical mechanisms that cannot be observed through experimental means.
