Inverse Model for Simultaneously Estimating Material Parameters and Absorption Coefficient in a Laser-Irradiated Sheet

Abstract

In this work, a heuristic inverse method for simultaneous estimation of thermal conductivity, specific heat, density and absorptivity in a laser-irradiated sheet is proposed. A fast forward model, which can predict the temperature evolution during laser heating is built as the foundation of the inverse model. The forward model comprises of a proper analytical modelling considering three-dimensional heat conduction equation with coupled conduction–convection boundary conditions. The proposed inverse method tries to change the unknown parameters in each step till the predicted temperature close to the recorded temperature. Two different examples of a heating process on aluminium alloy (Al 6061-T6) are considered to demonstrate the efficacy of the inverse method. The accuracy of the inverse method is assessed by simulated experimental temperatures considering temperature-dependent properties in the forward model. The results show that the inversely recovered parameters are sufficiently accurate in calculating the surface temperature at different process conditions. The suggested heuristic inverse method has the potential for the fast computation of parameters for a desired laser heating temperature without needing arduous experiments and unproductive finite element method (FEM) analysis.

Appendix

Following expressions are used in solution:

$${X}_{m}\left({\beta }_{m},x\right)={\beta }_{m}\mathrm{cos}\left({\beta }_{m}x\right)+\left(\frac{h}{k}\right)\mathrm{sin}\left({\beta }_{m}x\right)$$

(A.1)

$${Y}_{n}\left({\gamma }_{n},y\right)={\gamma }_{n}\mathrm{cos}\left({\gamma }_{n}y\right)+\left(\frac{h}{k}\right)\mathrm{sin}\left({\gamma }_{n}y\right)$$

(A.2)

$${Z}_{p}\left({\eta }_{p},z\right)={\eta }_{p}\mathrm{cos}\left({\eta }_{p}z\right)+\left(\frac{h}{k}\right)\mathrm{sin}\left({\eta }_{p}z\right)$$

(A.3)

$$N_x\left(\beta_m\right)=\frac12\left\{\left(\beta_m^2+\left(\frac hk\right)^2\right)\left(L+\frac{\displaystyle\frac hk}{\left(\beta_m^2+\left(\frac hk\right)^2\right)}\right)+\frac hk\right\}$$

(A.4)

$$N_y\left(\gamma_n\right)=\frac12\left\{\left(\gamma_m^2+\left(\frac hk\right)^2\right)\left(W+\frac{\displaystyle\frac hk}{\left(\gamma_m^2+\left(\frac hk\right)^2\right)}\right)+\frac hk\right\}$$

(A.5)

$$N_z\left(\eta_p\right)=\frac12\left\{\left(\eta_m^2+\left(\frac hk\right)^2\right)\left(W+\frac{\displaystyle\frac hk}{\left(\eta_m^2+\left(\frac hk\right)^2\right)}\right)+\frac hk\right\}$$

(A.6)

$$\widehat{\overline{\widetilde{F}} }\left({\beta }_{m},{\gamma }_{n},{\eta }_{p}\right)={T}_{0}\left[\begin{array}{l}\left\{\frac{h}{k{\beta }_{m}}\left(1-\mathrm{cos}\left({\beta }_{m}L\right)\right)+\mathrm{sin}\left({\beta }_{m}L\right)\right\}\left\{\frac{h}{k{\gamma }_{n}}\left(1-\mathrm{cos}\left({\gamma }_{n}W\right)\right)+\mathrm{sin}\left({\gamma }_{n}W\right)\right\}\\ \times \left\{\frac{h}{k{\eta }_{p}}\left(1-\mathrm{cos}\left({\eta }_{p}H\right)\right)+\mathrm{sin}\left({\eta }_{p}H\right)\right\}\end{array}\right]$$

(A.7)