Determining of the laser heat flux for three - dimensional conduction model by the sequential method

 Vietnam Journal of Mechanics, VAST, Vol.42, No. 2 (2020), pp. 95 – 103
 DOI: https://doi.org/10.15625/0866-7136/13753
 DETERMINING OF THE LASER HEAT FLUX FOR
 THREE-DIMENSIONAL CONDUCTION MODEL BY
 THE SEQUENTIAL METHOD
 Nguyen Nhut Phi Long1, Nguyen Hoai Son1,∗, Nguyen Quan2
 1Ho Chi Minh City University of Technology and Education, Vietnam
 2Pham Van Dong University, Quang Ngai, Vietnam
 ∗E-mail: sonnh@hcmute.edu.vn
 Received: 19 April 2019 / Published online: 11 May 2020
 Abstract. When performing a laser processing, one of the parameters to consider is the
 laser heat flux. This is a very important parameter of the processing. It is difficult to di-
 rectly and correctly measure this parameter during the processing. Therefore, to estimate
 this parameter, a solution has been implemented. In this study, the Newton–Raphson
 method has been calibrated as an operational algorithm to evaluate the laser heat flux
 value accurately in the 3-D conduction model. To confirm the effectiveness of the pre-
 sented method, the paper has given two specific applications. It is obtained that the se-
 quential method is a reasonable, correct, and powerful method to determine the inversely
 laser heat flux in the three-dimensional conduction model.
 Keywords: laser processing, laser flux, 3-D heat conduction model, modified Newton–
 Raphson (MNR) method, sequential method.
 1. INTRODUCTION
 In recent years, the rapid development of laser processing technology has gradually
replaced the traditional processing techniques. Comparing the conventional machining
processing, laser processing technology is not affected by the wear of the tool and the
friction during processing because of its non-contact property. Moreover, laser processing
technology has many advantages, such as high energy density, quick fabrication, high
precision, and low cost and non-pollution [1,2].
 The absorbed energy of a laser beam known as laser heat flux is an important param-
eter in laser processing such as laser cladding, laser surface hardening, and laser welding.
However, the direct measurement of this parameter during the process is difficult. As a
result, many researchers used the inverse method to determine this parameter. For in-
stance, Wang et al. [3] calculated this parameter on the surface. His work used the conju-
gate gradient method (CGM) in the surface hardening process by the laser to determine
this parameter inversely. Chen and Xu [4] applied Laplace transform to the governing
 
c 2020 Vietnam Academy of Science and Technology
96 Nguyen Nhut Phi Long, Nguyen Hoai Son, Nguyen Quan
differential equation, boundary conditions, and initial condition to evaluate absorption
in the heating process for surface by laser. Yang et al. [5] applied CGM to estimate the
absorbed energy of the laser beam and depth of the melt zone simultaneously. Sun et
al. [6] used the method of direct sensitivity coefficient for the determining of the surface
heat flux and the absorptivity of the coating surface in the surface hardening process by
the laser. Nevertheless, these studies were only performed on a one-dimensional and
two-dimensional model. The obtained results may be less accurate than the truly three-
dimensional model. Nguyen and Yang [7] used the inverse algorithm in the modified
Newton–Raphson method (MNR method) to determine laser power in order to reach the
required width penetration in the laser welding process. Through two examples, with
every 7 iterations in each example, the speed of the method being applied is excellent.
The error between the estimated width penetration and the setup width penetration in
the two examples, respectively, is 0.25% and 0.2%. Nguyen and Yang [8] continued to use
the MNR method in estimating the absorption coefficient, which is one of the very impor-
tant parameters during laser welding. The determination of this parameter depends on
the temperature is the complex non-linear inverse problem. The long rod with the small
diameter model is heated by the Gauss distribution laser source. The results of the two
examples in this study show that the number of future time step increases, the measure-
ment error decreases, the determined value increases, and this result with the constant
relation of the future time step is less exact the linear type of the future time. Nguyen et
al. [9] used the sequential method to inverse evaluation of the absorption coefficient for
the spot laser welding (three-dimensional cylindrical workpiece) with Friedman’s heat
source model. This assessment, through two examples, includes two main processes: di-
rect analysis and inverse analysis. Firstly, this study applied the effective heat capacity
method to find temperature fields by finite element method (FEM) with the boundary
conditions, and the absorption coefficient assumed as specific values. In the inverse anal-
ysis process, the unknown variables are obtained from exploring points systematical ... M STATEMENT
 Considering the cubic-shaped workpiece with top surface is heated by an incident
laser beam with a radius of rb. The remaining surrounding surfaces of the sample are
 Determining of the laser heat flux for three-dimensional conduction model by the sequential method 97
coated with insulation layers to avoid energy loss. The remaining surrounding surfaces
of the sample are coated with insulation layers to avoid energy loss. Thermocouples are
placed inside to record the temperature, as shown in Fig.1.
 Fig. 1. Three-dimensional laser heating model
 The equations of the 3-D heat conduction model take the form
 ∂  ∂T  ∂  ∂T  ∂  ∂T  ∂T
 k + k + k = ρC in Ω, 0 ≤ t ≤ t , (1)
 ∂x ∂x ∂y ∂y ∂z ∂z ∂t f
 !
 ∂T −(x2 + y2)
 − =
 k I exp 2 at Γs, (2)
 ∂n rb
 ∂T
 = 0 at otherwise surfaces, (3)
 ∂n
 T(x, y, z, 0) = T0, (4)
  W   kg 
where k is a thermal conductivity ; ρ is a density ; C is a heat capacity
 m◦C m3
 J 
 ; T(x, y, z, t) is a temperature field (◦C), T is the initial temperature (◦C); I is
 kg◦C 0
  W 
a laser heat flux and r is an effective laser beam radius (mm); n is unit normal
 m2 b
vector.
 When other input parameters, the boundary conditions and the laser heat intensity,
are specified prior, the finite element method [10–12] is utilized to find the temperature
field in the Ω domain. When the temperature history is measured inside the workpiece,
the laser heat flux of the surface is estimated by the inverse problem.
 3. THE SEQUENTIAL METHOD
 At each time step, this method consists of four problems to be solved as follows:
98 Nguyen Nhut Phi Long, Nguyen Hoai Son, Nguyen Quan
3.1. Direct/Forward problem
 At the time t = tm, Eqs. (1)–(4) become
 ∂  ∂T  ∂  ∂T  ∂  ∂T  ∂T
 k m + k m + k m = ρC m in Ω, t = t , (5)
 ∂x ∂x ∂y ∂y ∂z ∂z ∂t m
 !
 ∂T −(x2 + y2)
 − m =
 k Im exp 2 at Γs, (6)
 ∂n rb
 ∂T
 m = 0 at otherwise surfaces, (7)
 ∂n
 T(x, y, z, tm−1) = Tm−1, (8)
 In the forward problem, data on temperature changes at one or several points in the
solder are used to calculate the inverse of this coefficient. The concept of the quantity of
time in the future (the future time) is proposed and used to ensure stability and continuity
during the setup process. While setting the value at time step t = tm, the value of the
variable set at the time step t = t1, t = t2,..., t = tm−1 has been set and some values set
in the next time steps are assumed to be constant or linear in relation to the current set
value. Then, the unknown surface laser heat flux is introduced as follows
 Im+r = Im+r−1 = ... = Im+1 = Im, (9)
where r is the quantity of time in the future.
 And then, the solution is performed from t = tm to t = tm+r (r steps) for the direct
problem shown in Eqs. (5)–(8), and the unknown absorbed energy of the laser beam is
set by Eq. (9).
3.2. Sensitivity problem
 The MNR method implements several loops to calculate the value of the variable to
be searched at each time step. Because the MNR method is the gradient-based method,
sensitivity analysis is needed to achieve the search step in each loop. After Eqs. (5)–(8)
 ∂
has been derived for both sides
 ∂Im
 ∂  ∂X  ∂  ∂X  ∂  ∂X  ∂X
 k m + k m + k m = ρC m in Ω, t = t , (10)
 ∂x ∂x ∂y ∂y ∂z ∂z ∂t m
 ∂Xm ∂  2 2 2 

 − k = Im exp(−(x + y )/rb) at Γs, (11)
 ∂n ∂Im
 ∂X
 m = 0 at otherwise surfaces, (12)
 ∂n
 X(x, y, z, tm−1) = Xm−1 = 0, (13)
 Eqs. (10)–(13) are the linear equations and their dependent variable is Xm, concerning
independent variables x, y, z, and t. Hence, the solution of this sensitivity problem can be
solved as the forward problem by the finite element method.
 Determining of the laser heat flux for three-dimensional conduction model by the sequential method 99
3.3. Modified Newton–Raphson (MNR) method
 In this research, the MNR method is seen in the operational algorithm [7]. After
having the solution of two problems: direct and sensitivity, the inverse analysis process
is satisfied when using this method. In the MNR method, the variable to be set is shown
in a non-linear equation. This equation is built directly from the comparison between the
calculated temperature in the forward problem and the temperature obtained from the
 j
sensor. Therefore, the response value of the sensor value Φmeas and the calculated value
 j
Φcal are predefined. The solution of this equation: the estimation of the unknown heat
flux on the surface (Im) can be changed at each time step.
 j j
 Φ = Φcal − Φmeas = 0, (14)
where j = m, m + 1, . . . , m + r is the quantity of the equations about the comparison
between the measured and the calculated temperature.
 It is assumed that the undetermined surface laser heat at the time step, t = tm, is set
as χ. The derivative of Φ with respect to χ can be illustrated as follows
 ∂Φ
 Ψ = , (15)
 ∂χ
where Ψ is the sensitivity matrix n summary, through the MNR method, the undeter-
mined surface laser heat equation is as follows
 χk+1 = χk + ∆k, (16)
where ∆k is a linear least-squares solution for a set of over-determined linear equations,
and it can be derived as follows
 −1
 h T i T
 ∆k = − Ψ (χk)Ψ(χk) Ψ (χk)Φ(χk), (17)
 −1
 h T i T
 ⇒ χk+1 = χk − Ψ (χk)Ψ(χk) Ψ (χk)Φ(χk). (18)
3.4. The stopping criteria
 To end the loop process, the stopping criterion has been applied. In the modified
Newton-Raphson method, Eqs. (14)–(17) are used to establish the unaware value χ. The
∆k in Eq. (17) is called a step size that runs from χk to χk+1. When the stopping criterion
is contented, the loop of Eq. (18) to estimate the value of χ at each time step will stop. To
stop iteration, two criteria which were proposed by Frank and Wolfe [13] are selected as
follows
 kχk+1 − χκk / kχk+1k ≤ δ, (19)
 kJ (χk+1) − J (χκ)k / kJ (χk+1)k ≤ ε, (20)
where δ and ε are stop values. Those are very small positive numbers.
 r 2
 h i i i
 kJ (χk+1)k = ∑ Φc − Φmeas (21)
 i=1
100 Nguyen Nhut Phi Long, Nguyen Hoai Son, Nguyen Quan
 4. COMPUTATIONAL ALGORITHM [8]
 First, specified the quantity of time in the future (r), the 3D mesh model of the do-
main, the time step (∆t) in the direct problem; given a stop condition value (δ, ε) and the
initial value (χ0). The solution for χk at time step t = tm is as follows:
 Step 1: Let j = m and T(x, y, z, tj−1) is known;
 j
 Step 2: Collect the measured temperature Φmeas;
 Step 3: Assume the initial guess χ0;
 Step 4: Calculate the sensitivity matrix, Ψ, by Eqs. (10–(13));
 Step 5: Solve the direct problem by Eqs. (5)–(8), and then compute the calculated
 j
temperature Φcal;
 j j
 Step 6: Construct Φ by Φmeas and Φcal;
 Step 7: Knowing Ψ and Φ, determine the step size ∆k by Eq. (17);
 Step 8: Knowing ∆k and χk , calculate χk+1 through Eq. (18);
 Step 9: Terminate the iteration if the stopping criterion (Eqs. (19)–(20)) is satisfied.
Otherwise, return to Step 5;
 Step 10: Stop the process if the final time step is attached. Otherwise, let j = m + 1
return to Step 2.
 5. RESULTS AND DISCUSSIONS
 To demonstrate the proposed method, two applications of the different heat flux are
performed and the cubic-shaped substrate with the dimension of 20 × 20 × 20 (mm) is
considered. The workpiece is made by Al1100 which thermal properties is as following:
  J   W   kg 
C = 904 , k = 222 and ρ = 2710 . The thermocouple is located at
 kg◦C m◦C m3
TK1(0, 0, −0.001).
 In addition, the measured temperature is generated from Eqs. (1–(4)) when the input
parameters are preselected and it is presumed to have measurement errors. In other
work, the random errors of measurement are added to the exact temperature. It can be
achieved in the following equation.
 Tmeas = Texact + λσ, (22)
where Tmeas known as the temperature data is measured, Texact known as the exact tem-
perature, λ known as the random number (within −2.576 ≤ λ ≤ 2.756 ) are used from
the IMSL subroutine DRNNOR [14] which presents the 99% confidence, and σ known as
the standard deviation.
 The fine mesh is in the area near the laser with a size of about ∆x ≈ 2.10−5 (mm) and
the coarse mesh is in the other area with a size of about ∆x ≈ 2.10−3 (mm) (as shown in
Fig.2). As well, the time increment is 0.02 (s).
 The relative average errors µ (the RA errors µ) to inspect the deviation of the esti-
mated results from the exact solution obtained from the direct problem shown in Eq. (23)
 1 Nt  f − fˆ
 µ =  , (23)
 ∑  ˆ 
 Nt i=1  f 
 Determining of the laser heat flux for three-dimensional conduction model by the sequential method 101
 Fig. 2. The 3D mesh model of the Ω domain Fig. 3. The results is estimated in Application 1
where f is the inverse determining results with measurement errors, fˆ is the result of
exact solution and, Nt is the number of the temporal step. This equation shows that the
value of µ is small, the setting result is still accurate.
 Two applications are performed following as:
5.1. Application 1
 ◦
 The sample is initially at a uniform temperature T0 = 27 ( C), and then is suddenly
heated by a laser beam with its effective radius of rb = 0.0014 (mm) and constant heat flux
  W 
of q = 80 × 106 at the center of the substrate top surface. Fig.3 shows the results
 m2
of the laser heat flux is estimated in Application 1. Thus, with r = 1 and σ = 0, the heat
flux values are determined to be approximate with the correct solution.
 For the case of the measurement errors,
 Table 1
the estimated results with large error diverge . The RA errors in Application 1 with
 the quantity of time in the future r = 3
from the accurate solution. Tab.1 shows the
RA errors µ of the inverse determined results
when the errors of measurement are involved. Cases Relative average error
 As shown in Tab.1, the RA errors decrease σ = 1 0.0031
with the reduction of the errors of measure- σ = 2 0.0061
ment. Particularly, the RA errors in Tab.1 de-
crease from 0.0031 to 0.0061, about 49% when the errors of measurement reduce from σ
= 1 to σ = 2. In the case of σ = 2, the RA error is 0.0061 and the value is small.
5.2. Application 2
 In this application, the time variation of laser heat flux is assumed as follows
 q(t) = 1.5 × 106 exp(t/2)(sin(t) + cos(t) + t). (24)
 Fig.4 shows the results of the absorbed energy of the laser beam is estimated in
Application 2.
102 Nguyen Nhut Phi Long, Nguyen Hoai Son, Nguyen Quan
 Fig. 4. The results is estimated in Application 2
 Once again, Fig.4 shows that the heat flux Table 2. The RA errors in Application 2 with
values are determined still to reach approxi- the quantity of time in the future r = 3
mate values in the case of measurement free-
error and, even if σ = 1, σ = 2, the estimated Cases Relative average error
values still approximate with the exact solu-
tion. The RA errors of the estimated results in σ = 1 0.0034
Application 2 are shown in Tab.2. σ = 2 0.0063
 In the case of σ = 2, the RA error is 0.0063.
These values are still small. In all cases, the average number of iterations is about 3 at each
time step. The total number of iterations is 750. In other words, the speed of convergence
of the proposed method is fast.
 6. CONCLUSIONS
 In this paper, the laser heat flux for the 3D model is estimated through the use of the
sequential method. With 10 steps in the algorithm, corresponding to each time step, this
method has solved four problems, including the direct/forward problem, the sensitivity
problem, the Modified Newton–Raphson method, the stopping criteria. Two applications
have been raised to confirm the proposed method. The results of the applications show
that in the case of the measurement free-errors, the evaluated values of the absorbed
energy of the laser beam has approximated with the exact solution. And in the remaining
case stated, the relative average errors are also very small.
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