Describing the influence of laser welding parameters for AZ91D alloy using hybrid quadratic–radial basis function

Abstract

AZ91D magnesium alloy plates of thickness 4 mm were successfully welded by CO₂ laser welding. The heat input and hence the microstructural evolution were characterized by varying the laser power and welding speed. Moreover, microhardness, tensile strength, and elongation of the weldment were characterized. Hybrid models integrating the quadratic function and radial basis function were developed to correlate the laser welding process parameters with the properties of the laser-welded specimens. A comprehensive analysis of the influence of laser welding process parameters on properties is presented. The laser welding process parameters for maximizing the tensile strength, elongation, and microhardness of the weldment were determined using multi-objective optimization. The optimum laser welding parameters for AZ91D alloy were as follows: laser power of 2 kW and welding speed of 4.5 m/min. Inclusive analysis of the microstructural evolution and fracture mechanism of the laser-welded specimens is presented.

Appendix

1.1 The development of hybrid quadratic function and radial basis function

Generally, statistical models are developed to relate the predictor variables (process parameters) and the response variable (properties—wear resistance, intergranular corrosion susceptibility) using regression equations. Based on the non-linear relation between the response variable with the predictor variables, the regression equations can be linear or quadratic or cubic or polynomial-based functions. As the response variables exhibited complex non-linearity characteristics with the predictor variable, hybrid model was used for associating the predictor and response variable. The hybrid model for predicting the responses is a combination of linear function and radial basis function.

Radial Basis Function (RBF) network is a single-layered artificial neural network, in which the RBF activates the function of the network. RBF is a real valued function, whose values are dependent on the center point c or origin. So any function that satisfies Eq. (6) is known as RBF function.

$$\mathrm{\varnothing }\left(x,c\right)=\mathrm{\varnothing }\left(|\left|x-c\right||\right)$$

(6)

Some of the commonly used RBFs include Gaussian, multi quadratic, inverse quadratic, and thin plate spline type. In this study, multi quadratic type RBF was chosen to create the model, as the input parameters to the model were of discrete type. The mathematical equation of the multi quadratic type RBF, each associated with different centers, is given by Eq. (7).

$$\mathrm{\varnothing }\left(|\left|x-{x}_{i}\right||\right)=\sqrt{1+{\left[\varepsilon \left(|\left|x-{x}_{i}\right||\right)\right]}^{2}}$$

(7)

The typical equation for building up of function approximation using RBF network is of the form given in Eq. (8). Each RBF is associated with a different center x (may or may not be unique).

$$RBF=y\left(x\right)=\sum_{i=1}^{N}{w}_{i}*\mathrm{\varnothing }\left(|\left|x-{x}_{i}\right||\right)$$

(8)

where y(x) is the approximation function, N is the number of radial basis functions, w_i is the weights. A typical linear–RBF mathematical model is given in Eq. (9)

$$z=\sum_{i=1}^{n}{a}_{i}*{s}_{i}+RBF$$

(9)

where z is the response variable, and a_i is the coefficient of the predictor variable s_i.

1.2 Plot of experimental value versus predicted value (Fig. 14)

Fig. 14

Plot of experimental value versus predicted value: a microhardness; b tensile strength; c elongation (%)

1.2.1 Heat input–sample calculation

The quantum of heat input in the course of laser welding AZ91 alloy plate was calculated using Eq. (10).

$$\mathrm{Heat\;Input }\left({~}^{\mathrm{J}}\!\left/ \!{~}_{\mathrm{m}}\right.\right)=\frac{\mathrm{Power }\;(\mathrm{kW})\times 1000\times 60}{\mathrm{Welding\;Speed }\;\left({~}^{\mathrm{m}}\!\left/ \!{~}_{\mathrm{min}}\right.\right)}$$

(10)

Considering the specimen LW10 that was laser welded with a laser power of 2 kW and welding speed of 4.5 m/min, the heat input is given by

$$\mathrm{Heat\;Input }\;\left({~}^{\mathrm{J}}\!\left/ \!{~}_{\mathrm{m}}\right.\right)=\frac{2 (\mathrm{kW})\times 1000\times 60}{4.5 \left({~}^{\mathrm{m}}\!\left/ \!{~}_{\mathrm{min}}\right.\right)}$$

$$\mathrm{Heat\;Input }\;\left({~}^{\mathrm{J}}\!\left/ \!{~}_{\mathrm{m}}\right.\right)=26666.67{~}^{\mathrm{J}}\!\left/ \!{~}_{\mathrm{m}}\right.$$