## Estimation of the heat source of laser pulses by a dual-phase-lag model

### Ching-yu Yang

⇑Department of Mold and Die Engineering, National Kaohsiung University of Applied Sciences, Kaohsiung City 807, Taiwan, ROC

### a r t i c l e

### i n f o

Article history:

Received 4 December 2013

Received in revised form 19 January 2014 Accepted 21 January 2014

Available online 12 March 2014 Keywords:

Short-pulse laser heating Dual-phase-lag model Inverse problem

### a b s t r a c t

Because the duration of laser heating is ultrashort, a sequential method is proposed for estimating the input condition in the dual-phase-lag (DPL) conduction model. The estimation is deduced based on a numerical approach combined with a future time concept. Two cases are presented to show the features and validity of the proposed method. The actual and estimated values are compared to conﬁrm the valid-ity and accuracy of the proposed method.

Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Short-pulse laser heating has had a marked effect on microscale manufacturing and material thermal processes such as laser pat-terning, micromachining, the structural monitoring of thin metal ﬁlms, and laser surface treatments [1–9]. The mechanism of short-pulse laser heating on metals involve the deposition of radi-ation energy on electrons and energy increases within electron gases. It conducts the energy transferred to the lattice through electron–lattice interactions, and the energy is propagated through the media. Recent studies have employed theoretical models of short-pulse laser heating to describe microscopic electron–phonon energy transport[1,10,11]. The two-step process describes the en-ergy transported between phonons and electrons in the short tran-sient through a microscopic view. This model applies the parabolic or hyperbolic form based on the energy conservation principle. The parabolic model can capture the temperature of electrons during the heating and thermalization processes; however, it cannot pre-dict the ﬁnite speed of energy propagation[1,11]. Conversely, the hyperbolic model is constructed using the experimental data col-lected from the subpicosecond laser heating process of thin metal ﬁlms[1,12–18].

Tzou[19]developed a dual-phase-lag (DPL) model comprising two relaxation parameters (sqand

### s

T) to describe microscopicelec-tron–phonon interactions. In this model,

### s

qrepresents the phaselag of the heat ﬂux that captures the rapid transient effect of thermal inertia, and

### s

Trepresents the phase lag of the temperaturegradient, which captures the time delay resulting from the

micro-structural interaction effect [11,20]. The DPL model was developed to consider both the temporal and spatial effects of heat transfer in a one-temperature formulation[11,20]. The DPL model can describe the heat diffusion, thermal wave, phonon-electron interaction, and phonon scattering models when the values of

### s

qand

### s

Tare varied[11,20,21]. When a gold thin ﬁlm was irradiatedusing a short laser heat pulse, it appeared that the theoretical pre-diction of the reﬂectivity changed based on the DPL model corre-sponding to the front surface data available for the thin gold ﬁlm [21]. Because of the generality and accuracy of the DPL model, it was adopted in this study to examine ultrashort duration laser heating on thin ﬁlms. In previous studies, the application of the inverse technique for DPL-based ultrashort duration laser heating has been limited. Tang and Araki [22] and Orlande et al. [23] adopted the DPL model to estimate model parameters. Tang and Araki[22] used the Levenberg–Marquardt algorithm to simulta-neously estimate the relaxation parameters and thermal diffusivity of pulse heat ﬂuxes. To estimate the parameters of metals under the conditions of thermal nonequilibrium between electrons and lattices, Orlande et al.[23] employed the Markov Chain Monte Carlo method within the Bayesian framework. Huang and Lin [24]applied the conjugate gradient method to determine the heat generation that was modeled from the laser pulse, irradiated at the boundary. The result showed that their method was applicable for estimating heat sources formulated using a laser pulse, although the exact values of the heat sources could not be measured. This problem might be derived from its high nonlinearity mathematical description, which results in the difﬁculty of inverse estimation [25]. Using the nonlinear least square error formulation would double the degree of nonlinearity in the problem; therefore, a ro-bust and accurate method must be developed that does not in-crease the nonlinearity.

http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.01.054 0017-9310/Ó 2014 Elsevier Ltd. All rights reserved.

⇑Tel.: +886 936439039.

E-mail address:cyyang@cc.kuas.edu.tw

International Journal of Heat and Mass Transfer 73 (2014) 358–364

Contents lists available atScienceDirect

## International Journal of Heat and Mass Transfer

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j h m ttemperature errors were set within the standard deviation of the measurement errors

### r

= 2.576 and 2.576, implying that the mean of the standard deviation of the measurements is equal to### r

at a 99% conﬁdence level. The relative values error are 1% when### r

= 0.00165 and 5% when### r

= 0.00825.Figs. 3 and 4 show the estimated results and corresponding measurement errors. Large errors tend to correspond with the divergence of the estimations from the actual solution. For exam-ple, the results shown inFig. 3exhibit a large deviation from their exact solutions when the standard deviation of the measurement error

### r

= 0.00825, and particularly when the estimated result of### r

= 0.00165 is superior to that of### r

= 0.00000.Fig. 4shows the esti-mated results where r = 10, 15, and 20, and the measurement error### r

= 0.00165. The ﬁgure also shows that the increasing number of future values stabilizes the estimated result. To further test the applicability of the proposed method, the estimated source was substituted into (10), and the temperature response at x = 0 is shown (Fig. 5). The result indicates that the estimation is accept-able when the measurement error is considered.According to the numerical result of the example, it is evident that the estimation matched the actual value when no measure-ment error occurred. However, the difference becomes noticeable when the measurement error is considered. The value of the rela-tive error increases in conjunction with the measured error. The numerical results demonstrate that the proposed method is robust and stable when the measurement error is included in the estimation.

6. Conclusion

This paper presents an efﬁcient algorithm for determining the heat source of an ultrashort duration laser heating process. The DPL model was adopted and the inverse solution was represented as a closed-form expression that was derived using a ﬁnite-ele-ment-difference method. The special feature of this method is that a nonlinear formulation for the inverse problem is unnecessary. No preselected functional form for the unknown strength was re-quired, and no sensitivity analysis was needed in the algorithm. The problems associated with the properties of the measurement were presented to show the applicability of the proposed method. The results indicated that the proposed method can be used to accurately estimate the input strength, even when using an ultra-short laser input that exhibits a high order of nonlinearity. In con-clusion, the numerical results prove that the proposed method is an accurate and stable inverse technique for solving inverse ultra-short duration laser heating problems. Furthermore, the proposed method is applicable to other inverse problems such as source strength estimations in the high dimension domain.

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* = 0.00165 and r = 20*
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