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Efficient computation of heat distribution of processed workpiece with coupling to bottom water vapor in water-jet guided laser machining

Jianxin Zhu and Wencheng Lin

Open Access Open Access

Abstract

In this paper, we present a novel approach for calculating the heat distribution within a processed workpiece subjected to laser irradiation while accounting for the influence of bottom water vapor. A comprehensive mathematical model is introduced and numerical techniques using difference approximation are employed. Initially, the three-dimensional heat equation, originally defined in the rectangular coordinate system, is transformed into a corresponding model within the cylindrical coordinate system, incorporating a nonlinear boundary condition to account for coupling effects. Subsequently, leveraging the axial symmetry of the heat distribution, the three-dimensional model is simplified into a two-dimensional one. This simplified model is solved using the alternating direction implicit scheme coupled with the Crank-Nicolson method. Moreover, we develop a high-precision numerical treatment for the nonlinear boundary condition within the cylindrical coordinate system. To validate our methodology, simulation experiments are conducted on three distinct samples. Our comparative results demonstrate the feasibility and efficiency of the proposed approach in the context of water-jet guided laser processing.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Brittle and hard materials, such as beryllium, fused quartz, diamond, and other difficult-to-process materials, are widely used in the fields of aerospace, new energy, and other cutting-edge equipment manufacturing [1–4]. Traditional machining methods often lead to damage when working with these materials. Laser processing, which utilizes high thermal energy to cut, melt, and modify material surfaces, offers a non-contact alternative [5,6]. However, as industries evolve, demands for processing increasingly complex hard and brittle materials have grown. Conventional laser processing techniques have proven inadequate for addressing these demands, leading to the emergence of water-jet guided laser processing [7,8]. Water-jet guided laser processing, often referred to as the laser and water-jet composite removal method, represents an innovative synergy between laser technology and micro-water jet systems, abbreviated as the water-jet guided laser micro-processing method [9–11]. In this method, a high-energy laser beam is coupled into a fine water-jet. The water-jet serves as an optical conduit, guiding the laser through total internal reflection within the water stream. This enables precise laser propagation along the water-jet, ultimately directing the laser onto the workpiece's surface for precise machining and etching [12–14].

Compared to laser-only removal methods, the composite removal approach offers numerous advantages [15]. Firstly, water-jet guided laser micro-machining represents a novel, specialized, and precision-oriented laser processing method. This method offers the distinct advantage of leaving no heat-affected zone after cutting. Consequently, the resultant slag removal is cleaner, enabling deeper material cuts with exceptional cutting accuracy, high repeatability, and cost-effectiveness [16,17]. Secondly, the composite removal method facilitates the cutting of multi-layer materials with air insulation, effectively extending the working distance [18]. Thirdly, thanks to its ability to process materials with high cooling efficiency, minimal heat-affected areas, and limited material bending or deformation, this method is particularly well-suited for working with heat-sensitive or hard brittle materials, such as beryllium materials used in missile applications [19,20].

Research on the theoretical aspects of water-jet guided laser processing has been limited. In 2002, Yang et al. [21] established a mathematical model for water-guided laser etching of silicon wafers. Their study comprehensively addressed laser energy input, water-jet cooling, and material melting and removal, yielding promising experimental results. In 2003, Wagner et al. [22] processed 150µm thick stainless steel using both water-jet guided laser and traditional laser methods. Their findings revealed that water-guided laser processing produced a smaller heat-affected zone while maintaining equivalent processing efficiency. Moreover, the water-jet guided laser method yielded hole edges free from burrs or burn marks, resulting in noticeably superior micro-hole quality compared to traditional laser processing. In 2007, Wang et al. [23,24] developed a mathematical model for water-jet guided laser machining and conducted simulation analysis using ANSYS finite element software. Their study demonstrated satisfactory alignment between numerical simulations and experimental results. In 2009, Yang et al. [25] employed the water-jet guided laser method to cut and punch silicon wafers, effectively preventing micro-cracks, debris, and burrs that typically occur during conventional processing. Their experiments underscored the water-guided laser method's advantages in terms of high cutting efficiency and cutting shape flexibility. In 2011, Zhan et al. [26] established a heat transfer model for water-jet guided laser processing of silicon materials using the finite volume method. They conducted numerical simulations and analyzed liquid flow and heat conduction within the material molten pool during the drilling process. Furthermore, they introduced the concept of water-jet guided laser double-pulse drilling through numerical simulations. Researchers worldwide have made significant progress in testing water-jet guided laser machining of micro-pores, yielding promising results. In 2011, Wang et al. [18] performed experimental studies on the water-jet guided laser punching process for TC4 thin plates. Their results demonstrated minimal accumulation of molten slag at the inlet and outlet edges of water-guided laser punching. Additionally, the processed material exhibited no noticeable burn marks, and the hole's roundness and permeability were satisfactory. In 2021, Zhu et al. [27] presented an initial model of heat conduction distribution following laser irradiation on a processed workpiece. They addressed the three-dimensional heat conduction equation, converting it from a Cartesian coordinate system to a cylindrical one, leveraging the laser beam's unique geometry. Furthermore, based on the axial symmetry of heat distribution along the laser beam's direction, they simplified the equation into a two-dimensional heat conduction equation, creating an alternating direction implicit (ADI) scheme for rapid temperature field distribution calculations within the workpiece. Notably, their model did not account for the coupling effect of thermal radiation from the bottom water vapor.

Common numerical methods for solving complex problems like this include the finite element method (FEM) and the finite difference method (FDM), both of which represent distinct approaches. The FEM is well-suited for handling irregular and complex geometric shapes. However, it involves the computation of a stiffness matrix (calculated through numerous numerical integrations), and controlling truncation errors can be challenging [28]. On the other hand, the FDM is more convenient for regular geometric objects. Its primary advantages lie in its intuitive difference format and ease of controlling truncation errors [28]. In the context of water-jet guided laser processing, our focus in this paper revolves around regular geometric models. Consequently, we opt for the FDM, benefiting from its intuitive format and manageable truncation error. Leveraging the ADI scheme with the Crank-Nicolson method, as constructed in our prior work under cylindrical coordinates [27], we address the three-dimensional heat conduction problem model with nonlinear boundary conditions.

While some scholars have delved into the physical mechanisms and mathematical models governing the interaction between lasers and materials, as well as the coupling mechanisms between lasers and water-jets, optimization of processing parameters, and the integration and miniaturization of laser equipment, they have not thoroughly explored the temperature field's variation patterns. Consequently, this has led to significant approximation errors in existing models, elevated overall errors in the discrete numerical methods employed, and a substantial increase in computational complexity. In essence, the existing models fall short in terms of accuracy and computational efficiency. This paper addresses these limitations by presenting a high-precision mathematical model and an efficient finite difference method for the precise calculation of the temperature field in water-jet guided laser machining.

As is commonly understood, the inclusion of a third kind of boundary condition involving the coupling of heat transfer and radiation in the mathematical model results in a fourth-order temperature equation, rendering it nonlinear in nature. When employing the FDM, challenges pertaining to solution existence and uniqueness, as well as the effectiveness and error order of the final matrix format, become inevitable. In our prior work [27], we established that the ADI scheme with the Crank-Nicolson method is unconditionally stable and convergent in the context of laser processing. We not only employed the first boundary value condition to confirm the method's second-order convergence but also demonstrated its feasibility and accuracy in models featuring a third boundary value condition (linear boundary condition involving heat transfer exclusively).

In this paper, we introduce a FDM capable of effectively addressing the water-jet guided laser processing model. Our focus lies in the analysis of the three-dimensional heat conduction problem with the coupling of bottom water vapor and the resultant heat distribution in the material following water-jet guided laser irradiation. Our paper is structured as follows. In Part 2, we present the three-dimensional problem model in the original rectangular coordinate system and the corresponding two-dimensional problem model in the cylindrical coordinate system. In Part 3, we describe the ADI scheme with the Crank-Nicolson method and the matrix scheme in the cylindrical coordinate system. In Part 4, we discretize and address all boundary conditions, paying special attention to the nonlinear boundary condition with a coupling effect, encompassing both heat transfer and radiation. In Part 5, we identify and resolve the challenges posed by the nonlinear boundary condition for the matrix format of the ADI scheme with the Crank-Nicolson method, maintaining second-order accuracy. In Part 6, we provide three examples, using stainless steel 316, titanium, and beryllium as processing materials. Our objectives are threefold: first, to validate the feasibility and accuracy of our algorithm by comparing it with MATLAB's PDETOOL software; second, to highlight the distinctions between our method and conventional laser processing; and third, to examine the heat distributions within the materials post-water-jet guided laser irradiation. In Part 7, we discuss the impact of processing time with water-jet guided laser, based on the mathematical model. In Part 8, we conclude this paper.

2. Mathematical modelling

A schematic diagram of a standard waterjet-guided laser processing setup with coupling to the bottom water vapor is presented in Fig. 1. This diagram is obtained by modifying the Fig. 1 in Ref. [29], with the addition of coupling of the bottom water vapor. In this paper, we consider the coupling effect from the bottom water vapor under laser irradiation.

 figure: Fig. 1.

Fig. 1. Schematic of a standard waterjet-guided laser processing setup with coupling to the bottom water vapor.

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The three-dimensional heat conduction problem model for the nonlinear boundary condition (including the heat transfer and the radiation) in the rectangular coordinate system is considered as follows:

$$\frac{{\partial \hat{T}}}{{\partial t}} = \nabla ({\alpha \cdot \nabla \hat{T}} )\textrm{ + }f,({x,y,z} )\in \varOmega ,t \in ({0,{T_e}} ] ;$$
where $\Omega = \{ (x,y,z)|0 \le x \le X,{y^2} + {z^2} \le {R^2}\}$ represents a cylinder centered at the origin, $\hat{T}$ is the temperature, t is the time, f is the internal heat source function, $\alpha$ is the thermal diffusivity coefficient ($\alpha \ge 0$), and ${T_e}$ is the total time of heat conduction. The domain of heat equation Eq. (1) is shown in Fig. 2.

 figure: Fig. 2.

Fig. 2. Sketch of the domain of heat equation.

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The initial condition is:

$$\hat{T}({x,y,z,0} )= \hat{\varphi }({x,y,z} ),\textrm{0} \le x \le X,{y^2} + {z^2} \le {R^2};$$

Boundary value conditions are:

$$\left\{ \begin{array}{l} {\left. { - \kappa \left( {\nabla \hat{T} \cdot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over n} } \right)} \right|_{{S_\textrm{3}}}} = \beta q\left( {\sqrt {{y^2} + {z^2}} } \right)|{_{_{{S_3}}}} ,\textrm{where }{\textrm{S}_3} = \left\{ { {({y,z} )} |x = 0,\sqrt {{y^2} + {z^2}} \le d} \right\};\\ { {\hat{T}} |_{{S_4}}} = \textrm{293K}({\textrm{2}{\textrm{0}^\textrm{o}}\textrm{C}} ),\textrm{where }{S_4} = \left\{ { {({y,z} )} |x = 0,d < \sqrt {{y^2} + {z^2}} \le R} \right\};\\ {\left. { - \kappa \left( {\nabla \hat{T} \cdot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over n} } \right)} \right|_{{S_2}}} = { {H({\hat{T} - {T_a}} )} |_{{S_2}}},\textrm{where }{S_2} = \{{ {({x,y,z} )} |0 < x < X,{y^2} + {z^2} = {R^2}} \};\\ - {\left. {\kappa \left( {\nabla \hat{T} \cdot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over n} } \right)} \right|_{{S_1}}} = { {H({\hat{T} - {T_a}} )|{_{{S_1}}} + \varepsilon \sigma \cdot ({{{\hat{T}}^4} - {T_a}^4} )} |_{{S_1}}}\textrm{,where }{S_1} = \{{ {({y,z} )} |x = X,{y^2} + {z^2} \le {R^2}} \}; \end{array} \right.$$
where $\Gamma $ is the laser irradiation area, d is the laser radius, $\beta$ is the material's absorption rate of laser energy, H is the heat dissipation coefficient, ${T_a}$ is the initial environmental temperature, $\varepsilon \in [{0, 1} ]$, $\sigma$ is the Stefan-Boltzman constant, $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over n}$ is the normal direction outside the boundary, q is the external surface heat source function, $\kappa$ is the heat conductivity, and $\nabla = \left\{ {\frac{\partial }{{\partial x}},\frac{\partial }{{\partial y}},\frac{\partial }{{\partial z}}} \right\}$ is the gradient operator.

In view of the axial symmetry of the heat distribution, the above problem model can be simplified by using the cylindrical coordinates, and as a result, the three-dimensional problem can be converted into a two-dimensional one. The corresponding two-dimensional problem model is as follows [27]:

$$\frac{{\partial T}}{{\partial t}}\textrm{ = }\alpha \cdot \left( {\frac{{{\partial^2}T}}{{\partial {r^2}}} + \frac{1}{r} \cdot \frac{{\partial T}}{{\partial r}} + \frac{{{\partial^2}T}}{{\partial {x^2}}}} \right) + f,0 < x < X,0 < r < R,0 < t \le {T_e},$$
where $r = \sqrt {{y^2} + {z^2}}$.

The initial condition is:

$$T({x,r,0} )= \varphi ({x,r} ),0 \le x \le X,0 \le r \le R.$$

Boundary value conditions are:

$$\left\{ \begin{array}{l} {\left. { - \kappa \left( {\nabla T \cdot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over n} } \right)} \right|_{x = 0}} = \beta q(r ),0 \le r\, \le d;\\ { T |_{{S_4}}} = \textrm{293K}({\textrm{2}{\textrm{0}^\textrm{o}}\textrm{C}} ),\textrm{where }{S_4} = \left\{ { {({y,z} )} |x = 0,d < \sqrt {{y^2} + {z^2}} \le R} \right\};\\ {\left. { - \kappa \left( {\nabla T \cdot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over n} } \right)} \right|_{x = X}} = H({T - {T_a}} )|{_{x = X}} + \varepsilon \sigma \cdot ({{T^4} - {T_a}^4} )\, |{_{x = X}} ,0 \le r \le R;\\ {\left. { - \kappa \left( {\nabla T \cdot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over n} } \right)} \right|_{r = R}} = H({T - {T_a}} )|{_{r = R}} ,0 < x < X;\\ \kappa {\left. {\frac{{\partial T}}{{\partial r}}} \right|_{r = 0}} = 0,0 < x < X; \end{array} \right.$$
where $\nabla = \left\{ {\frac{\partial }{{\partial x}},\frac{\partial }{{\partial r}}} \right\}$.

The space and time variables are ${x_i} = 0 + i \cdot {h_1},(i = 0,1, \ldots ,M\textrm{ ),}$ ${r_j} = 0 + j \cdot {h_2},$ $(j = 1,2, \ldots ,M)$ and ${t_k} = 0 + k \cdot \tau ,(k = 1,2, \ldots ,N),$ respectively, where ${h_1} = ({X - 0} )/M$, ${h_2} = ({R - 0} )/M$ and $\tau = ({{T_e} - 0} )/N.$

3. Difference scheme for the ADI with the Crank-Nicolson method

In our previous study where $\sigma$ was taken zero and only heat transfer condition was considered (no coupling of bottom water vapor) [27], we discussed and proposed the ADI scheme with Crank-Nicolson method in the cylindrical coordinate system in the context of laser processing, and the convergence, unconditional ${L^2}$ stability, feasibility and accuracy of the proposed method were also verified. Therefore, we will directly give below the difference scheme for the ADI with Crank-Nicolson method with the nonlinear boundary value condition of coupling:

$$\left\{ \begin{array}{c} \left( {I - \frac{{\alpha \tau }}{2}\hat{\delta }_x^2} \right)\left( {I - \frac{{\alpha \tau }}{2}\left( {\hat{\delta }_y^2 + \frac{1}{{{r_j}}}{{\hat{\delta }}_y}} \right)} \right)T_{i,j}^{k + 1} = \left( {I + \frac{{\alpha \tau }}{2}\hat{\delta }_x^2} \right)\left( {I + \frac{{\alpha \tau }}{2}\left( {\hat{\delta }_y^2 + \frac{1}{{{r_j}}}{{\hat{\delta }}_y}} \right)} \right)T_{i,j}^k ,\\ 1 \le i,j \le M - 1 , 1 \le k \le N - 1 ; \end{array} \right.$$
$$T({x,r,0} )= \varphi ({x,r} ),0 \le x \le X,0 \le r \le R;$$
$$\left\{ \begin{array}{l} { { - \kappa ({\nabla T \cdot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over n}} )} |_{x = 0}} = \beta q(r ),0 \le r\, \le d;\\ { { - \kappa ({\nabla T \cdot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over n}} )} |_{x = X}} = H({T - {T_a}} )|{_{x = X}} + \varepsilon \sigma \cdot ({{T^4} - {T_a}^4} )\, |{_{x = X}} ,0 \le r \le R;\\ { { - \kappa ({\nabla T \cdot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over n}} )} |_{r = R}} = H({T - {T_a}} ),0 < x < X;\\ \kappa {\left. {\frac{{\partial T}}{{\partial r}}} \right|_{r = 0}} = 0,0 < x < X; \end{array} \right.$$
where $\hat{\delta }_x^2T_{i,j}^k = \frac{{T_{i - 1,j}^k - 2T_{i,j}^k + T_{i + 1,j}^k}}{{{h_1}^2}},\hat{\delta }_y^2T_{i,j}^k = \frac{{T_{i,j - 1}^k - 2T_{i,j}^k + T_{i,j + 1}^k}}{{{h_2}^2}},{\hat{\delta }_y}T_{i,j}^k = \frac{{T_{i,j\textrm{ + 1}}^k - T_{i,j - 1}^k}}{{\textrm{2}{h_2}}}\, .$

The detailed matrix formats of the interior point discrete results of Eq. (7) are listed in Appendix A.

4. Discrete scheme for the boundary conditions and the error analysis

The second-order discrete format of $T_{0,j}^k,T_{i,M}^k$ and $T_{i,0}^k$ can be obtained in the previous paper [27]:

$$T_{0,j}^k = T_{1,j}^k + \frac{{\beta {h_1}q({{r_j}} )}}{\kappa },0 \le j \le M;$$
$$T_{i,M}^k = \frac{\kappa }{{\kappa + {h_2}H}}T_{i,M - 1}^k + \frac{{{h_2}H}}{{\kappa + {h_2}H}}{T_a},0 \le i \le M;$$
$$T_{i,0}^k = T_{i,1}^k,0 \le i \le M.$$

4.1 Discrete scheme for the nonlinear boundary condition of coupling

The nonlinear boundary condition with coupling of the heat transfer and radiation is discretized as follows.

For ${\left. { - \kappa \left( {\nabla T \cdot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over n} } \right)} \right|_{x = X}} = H({T - {T_a}} )|{_{x = X}} + \varepsilon \sigma \cdot ({{T^4} - {T_a}^4} )\, |{_{x = X}} ,0 \le r \le R\,$, it is discretized as the center at $x = X$,

$$\kappa \frac{{ - \hat{T}_{M + 1,j}^k + T_{M - 1,j}^k}}{{2{h_1}}} = { {[{H({T - {T_a}} )+ \varepsilon \sigma \cdot ({{T^4} - T_a^4} )} ]} |_{x = X}},0 \le j \le M;$$
where $\hat{T}_{M + 1,j}^k$ is the virtual symmetric point of $T_{M - 1,j}^k$ about $x = X$, and is estimated to be:
$$T_{M,j}^k \approx \frac{{\hat{T}_{M + 1,j}^k + T_{M - 1,j}^k}}{2},0 \le j \le M;$$
$${ T |_{x = X}} \approx \frac{{\hat{T}_{M + 1,j}^k + T_{M - 1,j}^k}}{2},0 \le j \le M\, .$$

According to Eqs. (13), (14) and (15), we have:

$$\varepsilon \sigma \cdot {({T_{M,j}^k} )^4} + \left( {H + \frac{\kappa }{{{h_1}}}} \right) \cdot T_{M,j}^k - \frac{\kappa }{{{h_1}}} \cdot T_{M - 1,j}^k - H \cdot {T_a} - \varepsilon \sigma \cdot T_a^4 \approx 0.$$

Equation (16) can find the only solution which meets the conditions through theory, and the judgment process is as follows.

Denote

$$a = \varepsilon \sigma ,b = 0,c = 0,d = H + \frac{\kappa }{{{h_1}}},e ={-} \frac{\kappa }{{{h_1}}} \cdot T_{M - 1,j}^k - H \cdot {T_a} - \varepsilon \sigma \cdot T_a^4;$$
then Eq. (16) can be transformed into:
$$a \cdot {({T_{M,j}^k} )^4} + b \cdot {({T_{M,j}^k} )^3} + c \cdot {({T_{M,j}^k} )^2} + d \cdot T_{M,j}^k + e \approx 0.$$

By Appendix B, we can find an appropriate root of Eq. (18) as follows:

$$T_{M,j}^k \approx \frac{{ - b + {\mathop{\rm sgn}} (\tilde{E}) \cdot \sqrt {\frac{{\tilde{D} + \sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}}}{3}} + \sqrt {\frac{{2\tilde{D} - \left( {\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}} \right) + 2\sqrt z }}{3}} }}{{4a}},$$
where
$$\left\{ \begin{array}{l} {z_1} = \tilde{A}\tilde{D} + 3\left( {\frac{{ - \tilde{B} + \sqrt {{{\tilde{B}}^2} - 4\tilde{A}\tilde{C}} }}{2}} \right),{z_2} = \tilde{A}\tilde{D} + 3\left( {\frac{{ - \tilde{B} - \sqrt {{{\tilde{B}}^2} - 4\tilde{A}\tilde{C}} }}{2}} \right),\\ z = {{\tilde{D}}^2} - \tilde{D} \cdot \left( {\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}} \right) + {\left( {\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}} \right)^2} - 3\tilde{A}, \end{array} \right.$$
and
$$\left\{ {\begin{array}{l} {\tilde{D} = 3{b^2} - 8ac = 0,}\\ {\tilde{E} ={-} {b^3} + 4abc - 8{a^2}d ={-} 8 \cdot {{({\varepsilon \sigma } )}^2} \cdot \left( {H + \frac{\kappa }{{{h_1}}}} \right),}\\ {\tilde{F} = 3{b^4} + 16{a^2}{c^2} - 16a{b^2}c + 16{a^2}bd - 64{a^3}e}\\ { ={-} 64 \cdot {{({\varepsilon \sigma } )}^3} \cdot \left( { - \frac{\kappa }{{{h_1}}} \cdot T_{M - 1,j}^k - H \cdot {T_a} - \varepsilon \sigma \cdot T_a^4} \right), }\\ {\tilde{A} = {{\tilde{D}}^2} - 3\tilde{F} = 3 \cdot 64 \cdot {{({\varepsilon \sigma } )}^3} \cdot \left( { - \frac{\kappa }{{{h_1}}} \cdot T_{M - 1,j}^k - H \cdot {T_a} - \varepsilon \sigma \cdot T_a^4} \right), }\\ {\tilde{B} = \tilde{D}\tilde{F} - 9{{\tilde{E}}^2} ={-} 9 \cdot 64 \cdot {{({\varepsilon \sigma } )}^4} \cdot {{\left( {H + \frac{\kappa }{{{h_1}}}} \right)}^2}, }\\ {\tilde{C} = {{\tilde{F}}^2} - 3\tilde{D}{{\tilde{E}}^2} = {{64}^2} \cdot {{({\varepsilon \sigma } )}^6} \cdot {{\left( { - \frac{\kappa }{{{h_1}}} \cdot T_{M - 1,j}^k - H \cdot {T_a} - \varepsilon \sigma \cdot T_a^4} \right)}^2}.} \end{array}} \right.$$

4.2 Error analysis of discrete scheme

According to Eqs. (20) and (21), the absolute value of the difference between the left and right sides of Eq. (19) is as follows:

$$\left\{ \begin{aligned} |{\hat{R} - \hat{L}} |&= \left|{\frac{1}{{4a}}\left[ \begin{array}{l} \sqrt {\frac{{2\sqrt {{{\left( {\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}} \right)}^2} - \left[ {576 \cdot {{({\varepsilon \sigma } )}^3} \cdot \left( { - \frac{\kappa }{{{h_1}}} \cdot T_{M - 1,j}^k - H \cdot {T_a} - \varepsilon \sigma \cdot T_a^4} \right)} \right]} - \left( {\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}} \right)}}{3}} \\ - \sqrt {\frac{{\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}}}{3}} \end{array} \right] - T_{M,j}^k} \right|\\ &{ = O(h_1^2),}\end{aligned} \right.$$
where $\hat{R}$ and $\hat{L}$ are the right and left of Eq. (19), respectively.

Detailed estimations of Eq. (22) are shown in Appendix C.

5. Difficulties and solutions for the nonlinear boundary condition of coupling in matrix format

From the final discrete solution scheme (19) of the coupled boundary condition, $T_{M,j}^{k + 1/2}$ and $T_{M - 1,j}^{k + 1/2}$ are not linear. If the Eq. (19) is directly substituted into the matrix scheme (7), the matrix scheme (7) cannot be solved.

Hence, we need to ensure the effectiveness of the coupling boundary condition and the solvability of the matrix format of the ADI with Crank-Nicolson. Therefore, we will use the interpolation method for $T_{M,j}^{k + 1/2}$ in the matrix format and ensure that the error of the final result is $O({{\tau^2} + h_1^2 + h_2^2} )$.

The interpolation method for $T_{M,j}^{k + 1/2}$ is as follows:

$$\begin{aligned} T_{M,j}^{k + 1/2} &\approx \frac{{T_{M,j}^k + T_{M,j}^{k + 1}}}{2} \approx \frac{{T_{M,j}^k}}{2} + \frac{1}{2} \cdot ({2 \cdot T_{M - 1,j}^{k + 1} - T_{M - 2,j}^{k + 1}} )\\ & \approx \frac{{T_{M,j}^k}}{2} + \frac{1}{2} \cdot [{2 \cdot ({2 \cdot T_{M - 1,j}^{k + 1/2} - T_{M - 1,j}^k} )- ({2 \cdot T_{M - 2,j}^{k + 1/2} - T_{M - 2,j}^k} )} ]\\ & \approx \frac{{T_{M,j}^k}}{2} - T_{M - 1,j}^k + \frac{{T_{M - 2,j}^k}}{2} + 2 \cdot T_{M - 1,j}^{k + 1/2} - T_{M - 2,j}^{k + 1/2}, \end{aligned}$$
where the total truncation error ${R_c}$ generated by Eq. (23) is:
$$\begin{aligned} {R_c} &={-} \frac{{{\tau ^2}}}{8} \cdot \frac{{{\partial ^2}T({{x_M},{r_j},{t_{k + 1/2}}} )}}{{\partial {t^2}}} + \frac{{h_1^2}}{2} \cdot \frac{{{\partial ^2}T({{x_M},{r_j},{t_{k + 1}}} )}}{{\partial {x^2}}} + \frac{{{\tau ^2}}}{4} \cdot \frac{{{\partial ^2}T({{x_{M - 1}},{r_j},{t_{k + 1}}} )}}{{\partial {t^2}}}\\ & - \frac{{{\tau ^2}}}{8} \cdot \frac{{{\partial ^2}T({{x_{M - 2}},{r_j},{t_{k + 1}}} )}}{{\partial {t^2}}} + o({{\tau^2} + h_1^2} )= O({{\tau^2} + h_1^2} ). \end{aligned}$$

6. Examples

The following examples are suitable for cylindrical coordinates. Taking advantage of the axial symmetry of heat distribution, the three-dimensional problems (1) ∼ (3) can be converted into two-dimensional problems in $({x,r} )$ coordinates.

$$\frac{{\partial T}}{{\partial t}}\textrm{ = }\alpha \cdot \left( {\frac{{{\partial^2}T}}{{\partial {r^2}}} + \frac{1}{r} \cdot \frac{{\partial T}}{{\partial r}} + \frac{{{\partial^2}T}}{{\partial {x^2}}}} \right) + f,0 \le x \le X,0 < r < R,0 < t \le {T_e};$$
$${ {T({x,r,t} )} |_{t = 0}} = {T_a},$$
$$\left\{ \begin{array}{l} {\left. { - \kappa \left( {\nabla T \cdot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over n} } \right)} \right|_{x = 0}} = \beta q(r ),0 \le r \le d;\\ { T |_{{S_4}}} = \textrm{293K}({\textrm{2}{\textrm{0}^\textrm{o}}\textrm{C}} ); \\ {\left. { - \kappa \left( {\nabla T \cdot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over n} } \right)} \right|_{x = X}} = H({T - {T_a}} )|{_{x = X}} + \varepsilon \sigma \cdot ({{T^4} - {T_a}^4} )\, |{_{x = X}} ,0 \le r \le R; \, \\ - {\left. {\kappa \left( {\nabla T \cdot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over n} } \right)} \right|_{r = R}} = H({T - {T_a}} )\, |{_{r = R}} ,0 < x < X;\textrm{ }\, \kappa {\left. {\frac{{\partial T}}{{\partial r}}} \right|_{r = 0}} = 0,0 < x < X;\, \end{array} \right.$$
where $\alpha = \frac{\kappa }{{\rho c}}$ is the thermal diffusivity coefficient, $\sigma = 5.67 \times {10^{ - 8}}\textrm{W/}({{\textrm{m}^\textrm{2}} \cdot {\textrm{K}^\textrm{4}}} )$ and $\varepsilon = 0.5.$

Specifically, the input of laser energy acting on the material surface is in the form of Gaussian heat flux. Then, we have

$$\beta q(r )= \beta \cdot \frac{{2P}}{{\pi \cdot {d^2}}}{\text{e}^{ - \frac{{2{r^2}}}{{{d^2}}}}},$$
where $d = 0.001\textrm{m}$ is the laser radius, and P is the laser power.

Next, three examples will be used to discuss the temperature distribution in the material after the water-jet guided laser irradiates the material surface. We set the initial temperature ${T_0}$ and the ambient temperature ${T_a}$ to be $293K({{{20}^o}C} )$. The absorptivity of the material to the laser is $\beta = 1$.

First, we use Examples 1 and 2 to verify the feasibility and accuracy of our method. Then, simulations of the ordinary laser and the water-jet guided laser processings of the beryllium material for missile manufacturing are carried out in MATLAB in Example 3.

Example 1. Using difference scheme for the nonlinear boundary condition with coupling (heat transfer and radiation) effect to simulate the processing of the 316 stainless steel material (shown in Table 1) by the water-jet guided laser for the verification of the effectiveness of the algorithm.

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Table 1. The thermal property parameters of 316 stainless steel

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Using our method and PDETOOL respectively to solve the problem on MATLAB, we get Figs. 3 and 4.

 figure: Fig. 3.

Fig. 3. Numerical solutions of difference equations $({h = \textrm{0}\textrm{.001/50},\tau = \textrm{0}\textrm{.01/500,}H\textrm{ = 1}\textrm{.5}} )$.

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 figure: Fig. 4.

Fig. 4. Numerical solutions obtained by the PDETOOL of MATLAB.

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The approximate solution in Fig. 3 is compared with the cross sections of the numerical solution in Fig. 4, as shown in Fig. 5.

 figure: Fig. 5.

Fig. 5. (a) Cross section contrast by the ADI with Crank-Nicolson method (left figure). (b) Cross section contrast by the PDETOOL of MATLAB (right figure).

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From Figs. 3 to 5, the numerical solution obtained by the ADI with the Crank-Nicolson method for the nonlinear boundary condition with coupling (heat transfer and radiation) effect is in good agreement with the numerical solution given by the PDETOOL of MATLAB under the same environment within a certain small error. This agreement shows that our method is feasible and effective in the simulation of water-jet guided laser processing.

In addition, the temperature distribution inside the 316 stainless steel material is described as follows. Along the depth x direction, it begins to drop quickly, then slowly moves, and finally tends to the experimental environment temperature of $293K$. However, along the material radius r direction, it follows a Gaussian distribution.

Example 2. Using difference scheme for the nonlinear boundary condition with coupling (heat transfer and radiation) effect to simulate the processing of the titanium material (shown in Table 2) by the water-jet guided laser for the verification of the effectiveness of the algorithm.

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Table 2. The thermal property parameters of titanium

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Using our method and PDETOOL respectively to solve the problem on MATLAB, we get Figs. 6 and 7.

 figure: Fig. 6.

Fig. 6. Numerical solutions of difference equations $({h = \textrm{0}\textrm{.001/50},\tau = \textrm{0}\textrm{.01/500,}H\textrm{ = 1}\textrm{.5}} )$.

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 figure: Fig. 7.

Fig. 7. Numerical solutions obtained by PDETOOL of MATLAB.

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The approximate solution in Fig. 6 is compared with the cross sections of the numerical solution in Fig. 7, as shown in Fig. 8.

 figure: Fig. 8.

Fig. 8. (a) Cross section contrast by the ADI with Crank-Nicolson method (left figure). (b) Cross section contrast by the PDETOOL of MATLAB (right figure).

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Based on the temperature distribution depicted in Figs. 6 to 8, titanium material exhibits superior heat resistance. When subjecting the water-jet guided laser processing materials to a laser power and time respectively of only $200W$ and $0.01\textrm{s}$, the temperature within the material along the depth x direction behaves as follows: it initially experiences a rapid decline, followed by a gradual decrease, eventually stabilizing at the experimental environment temperature of $293K$. Consequently, the laser is unable to penetrate the material. Next, we extend the processing time to $0.1\textrm{s}$ and obtain the results shown in Figs. 9 and 10.

 figure: Fig. 9.

Fig. 9. Numerical solutions of difference equations $({h = \textrm{0}\textrm{.001/50},\tau = \textrm{0}\textrm{.1/500,}H\textrm{ = 1}\textrm{.5}} )$.

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 figure: Fig. 10.

Fig. 10. Numerical solutions obtained by PDETOOL of MATLAB.

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The approximate solution in Fig. 9 is compared with the cross sections of the numerical solution in Fig. 10, as shown in Fig. 11.

 figure: Fig. 11.

Fig. 11. (a) Cross section contrast by the ADI with Crank-Nicolson method (left figure). (b) Cross section contrast by the PDETOOL of MATLAB (right figure).

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Analyzing the temperature distribution in Figs. 9 to 11, we observe that as the processing time extends to $0.1\textrm{s}$, the heat effectively permeates the titanium material. The temperature within the material exhibits an initial rapid reduction in magnitude, followed by a slower decline.

From the comprehensive analysis of Figs. 6 to 11, it becomes evident that, regardless of the duration of the material’s exposure to water-jet guided laser processing, the results obtained using our algorithm closely align with those achieved through MATLAB’s PDETOOL under equivalent conditions. This substantiates the feasibility and effectiveness of our algorithm as discussed above.

Example 3. Utilizing a difference scheme to simulate the processing of beryllium material (shown in Table 3) for missiles through water-jet guided laser, particularly considering the nonlinear boundary condition involving the coupling of heat transfer and radiation effect, proves essential.

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Table 3. The thermal property parameters of beryllium

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Given that beryllium is characterized by its high hardness and brittleness, conventional and standard laser methods fail to achieve fast, complete, and clean processing. However, the water-jet guided laser method effectively meets these requirements. In this context, we will employ our algorithm to validate this assertion. We will conduct a comprehensive comparison between ordinary laser processing and water-jet guided laser processing, followed by an in-depth analysis of the temperature distribution within the material.

The results of the ordinary laser processing (the ADI with Crank-Nicolson method) of the beryllium materials are as follows (shown in Fig. 12) [27]:

 figure: Fig. 12.

Fig. 12. Numerical solutions of difference equations $({h = \textrm{0}\textrm{.001/50},\tau = \textrm{0}\textrm{.01/500,}H\textrm{ = 1}\textrm{.5}} )$.

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Using difference scheme for the nonlinear boundary condition with coupling (heat transfer and radiation) effect to solve the problem on MATLAB, we get Fig. 13.

 figure: Fig. 13.

Fig. 13. Numerical solutions of difference equations $({h = \textrm{0}\textrm{.001/50},\tau = \textrm{0}\textrm{.01/500,}H\textrm{ = 1}\textrm{.5}} )$.

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The approximate solution in Fig. 12 is compared with the cross sections of the numerical solution in Fig. 13, as shown in Figs. 14.

 figure: Fig. 14.

Fig. 14. (a) Cross section contrast by the ordinary laser (left figure). (b) Cross section contrast by the water-jet guided laser (right figure).

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The comparison between ordinary laser processing and water-jet guided laser processing is presented as follows. Figures 12 and 13 illustrate that both processing methods lead to a decrease in material temperature, transitioning from a rapid to a gradual decline.

Examining Fig. 14, we observe that along the material's depth (direction), the results of water-jet guided laser processing consistently remain slightly lower than those of ordinary laser processing. This subtle difference effectively reduces the extent of thermal impact on the material and can even achieve a rapid cooling effect, thereby mitigating the thermal deformation of the beryllium material.

7. Analysis of material processing time by the water-jet guided laser

As demonstrated in Examples 2 and 3, the water-jet guided laser is capable of rapidly penetrating the material in a short duration to accomplish the processing. Nevertheless, in practical applications, as the water-jet guided laser method is frequently employed for processing hard or brittle materials, the overall processing time may not be very brief. Consequently, when the water-jet guided laser traverses the material multiple times to complete the processing, the water located at the material's base may experience heating and even boiling, potentially impacting the surrounding environmental temperature ${T_\textrm{a}}$.

In Examples 1, 2 and 3 of this paper, we assume that water-jet guided laser only penetrates the material once, so it is not necessary to consider the change of ${T_\textrm{a}}$. However, when the water-jet guided laser penetrates the material multiple times to complete the processing, that is, the processing time is not very short, the nonlinear boundary condition with coupling (heat transfer and radiation) effect is recast as follows:

$${ {\kappa ({\nabla T \cdot n} )} |_{x = X}} = H({T - {T_a}} )|{_{x = X}} + \varepsilon \sigma \cdot ({{T^4} - {T_a}^4} )\, |{_{x = X}} ,0 \le r \le R\, .$$

Equation (29) shows that, as the processing time increases, the temperature of the water at the bottom of the material increases, thereby affecting ${T_\textrm{a}}$. However, an increase of ${T_\textrm{a}}$ will affect both heat transfer and radiation. When ${T_\textrm{a}}$ increases, the heat required to penetrate the bottom of the material decreases. That is, as the processing time increases, the water-jet guided laser will penetrate the material more and more easily.

8. Conclusions

In this paper, we present an efficient FDM that effectively transforms the three-dimensional problem into two dimensions. This method is designed for calculating the temperature distribution within materials processed using water-jet guided laser technology. Specifically, our approach combines the ADI scheme with the Crank-Nicolson method, addressing the nonlinear boundary condition encompassing heat transfer and radiation effects. We offer an alternative solution that is both feasible and accurate. Our method not only streamlines the three-dimensional problem, enhancing overall work efficiency, but also conserves valuable working time and storage space. Through three different examples, we have effectively explored the temperature distribution within materials subjected to water-jet guided laser processing. First, given that most researchers tend to work with problem models in the rectangular coordinate system, we establish the initial problem model in this coordinate system to ensure its wider applicability. Leveraging the axial symmetry of heat distribution, we convert the three-dimensional equation from the rectangular coordinate system into a two-dimensional equation in the cylindrical coordinate system. Second, in the context of water-jet guided laser processing, we concentrate on the analysis of the nonlinear boundary condition involving the coupling of heat transfer and radiation effects within the cylindrical coordinate system. We provide theoretical confirmation of the existence and uniqueness of the solution for the coupled boundary value discrete equation. Third, we address the issue arising in the matrix format of the ADI with the Crank-Nicolson method due to $T_{M,j}^{k + 1/2}$ by implementing interpolation techniques, ensuring that the discrete format of the boundary condition (nonlinear) maintains at least second-order accuracy. Fourth, using our algorithm, we conduct simulation experiments on MATLAB through various examples, comparing the results with those from MATLAB's PDETOOL. This allows us to draw the following conclusions.

1. From Examples 1 and 2, it is evident that the ADI with Crank-Nicolson algorithm in the cylindrical coordinate system for water-jet guided laser processing is effective.

2. Example 3 highlights that the cooling rate within the material subjected to water-jet guided laser processing is faster. This enhanced cooling effectively prevents material deformation during processing and maintains thermal conductivity, thus reducing adverse temperature effects. Compared to ordinary laser processing, water-jet guided laser processing ensures improved material toughness and integrity.

3. Analysis of the mathematical model, in conjunction with the assurance of material toughness and integrity, demonstrates that as processing time increases, water-jet guided laser processing can efficiently and easily penetrate the material for complete processing.

To enhance work efficiency and broaden the range of solution methods, it is essential to analyze various problems and apply suitable methods to address challenges. Thus, we employ the FDM to tackle the problem model in this paper, offering the advantages of an intuitive format and precise control over truncation errors. The three-dimensional heat conduction problem model presented in this article is universally applicable. Regardless of whether the material irradiated by the water-jet guided laser is cylindrical or not, as long as a circular pattern is maintained at the center of the water-jet guided laser spot, the problem model in this paper remains relevant and effective.

Appendix

A. Matrix formats

The matrix formats of the interior point discrete results of Eq. (7) are listed as follows:

$$\begin{array}{l} S \cdot \bar{T}_{i - 1}^{k + 1/2} + E \cdot \bar{T}_i^{k + 1/2} + S \cdot \bar{T}_{i + 1}^{k + 1/2} = V \cdot \bar{T}_i^k,\\ G \cdot \bar{T}_i^{k + 1} = L \cdot \bar{T}_{i - 1}^{k + 1/2} + W \cdot \bar{T}_i^{k + 1/2} + L \cdot \bar{T}_{i + 1}^{k + 1/2}, \end{array}$$
where ${\bar{T}_i} = {[{{T_{i,1}},{T_{i,2}}, \cdots ,{T_{i,M - 2}},{T_{i,M - 1}}} ]^T},0 \le i \le M\, .$

Further, Eq. (30) can be written as the following form:

$$\left[ {\begin{array}{@{}ccccc@{}} E&S&{}&{}&{}\\ S&E&S&{}&{}\\ {}& \ddots & \ddots & \ddots &{}\\ {}&{}&S&E&S\\ {}&{}&{}&S&E \end{array}} \right]{\left[ {\begin{array}{@{}c@{}} {{{\bar{T}}_1}}\\ {{{\bar{T}}_2}}\\ \vdots \\ {{{\bar{T}}_{M - 2}}}\\ {{{\bar{T}}_{M - 1}}} \end{array}} \right]^{k + \frac{1}{2}}} = \left[ {\begin{array}{@{}ccccc@{}} V&{}&{}&{}&{}\\ {}&V&{}&{}&{}\\ {}&{}& \ddots &{}&{}\\ {}&{}&{}&V&{}\\ {}&{}&{}&{}&V \end{array}} \right]{\left[ {\begin{array}{@{}c@{}} {{{\bar{T}}_1}}\\ {{{\bar{T}}_2}}\\ \vdots \\ {{{\bar{T}}_{M - 2}}}\\ {{{\bar{T}}_{M - 1}}} \end{array}} \right]^k} + \left[ {\begin{array}{@{}c@{}} { - S{{\bar{T}}_0}^{k + 1/2} + {{\bar{\omega }}_1}}\\ {{{\bar{\omega }}_2}}\\ \vdots \\ {{{\bar{\omega }}_{M - 2}}}\\ { - S{{\bar{T}}_M}^{k + 1/2} + {{\bar{\omega }}_{M - 1}}} \end{array}} \right],$$
$$\left[ {\begin{array}{@{}ccccc@{}} G&{}&{}&{}&{}\\ {}&G&{}&{}&{}\\ {}&{}& \ddots &{}&{}\\ {}&{}&{}&G&{}\\ {}&{}&{}&{}&G \end{array}} \right]{\left[ {\begin{array}{@{}c@{}} {{{\bar{T}}_1}}\\ {{{\bar{T}}_2}}\\ \vdots \\ {{{\bar{T}}_{M - 2}}}\\ {{{\bar{T}}_{M - 1}}} \end{array}} \right]^{k + 1}} = \left[ {\begin{array}{@{}ccccc@{}} W&L&{}&{}&{}\\ L&W&L&{}&{}\\ {}& \ddots & \ddots & \ddots &{}\\ {}&{}&L&W&L\\ {}&{}&{}&L&W \end{array}} \right]{\left[ {\begin{array}{@{}c@{}} {{{\bar{T}}_1}}\\ {{{\bar{T}}_2}}\\ \vdots \\ {{{\bar{T}}_{M - 2}}}\\ {{{\bar{T}}_{M - 1}}} \end{array}} \right]^{k + \frac{1}{2}}} + \left[ {\begin{array}{@{}c@{}} {L{{\bar{T}}_0}^{k + 1/2} - {{\bar{e}}_1}}\\ { - {{\bar{e}}_2}}\\ \vdots \\ { - {{\bar{e}}_{M - 2}}}\\ {L{{\bar{T}}_M}^{k + 1/2} - {{\bar{e}}_{M - 1}}} \end{array}} \right],$$
where,
$${\hat{r}_1} = \frac{{\alpha \cdot \tau }}{{{h_1}^2}},{\hat{r}_2} = \frac{{\alpha \cdot \tau }}{{{h_2}^2}},{b_1}(j )= \left( {1 + \frac{1}{{2j}}} \right) \cdot {\hat{r}_2},{b_2}(j )= \left( {1 - \frac{1}{{2j}}} \right) \cdot {\hat{r}_2};$$
$$S\textrm{ = diag}{( - \frac{{{{\hat{r}}_1}}}{2}, \cdots , - \frac{{{{\hat{r}}_1}}}{2})_{({M - 1} )\times ({M - 1} )}},L\textrm{ = diag}{(\frac{{{{\hat{r}}_1}}}{2}, \cdots , \frac{{{{\hat{r}}_1}}}{2})_{({M - 1} )\times ({M - 1} )}};$$
$$E = \textrm{diag}{((1 + {\hat{r}_1}), \cdots , (1 + {\hat{r}_1}))_{({M - 1} )\times ({M - 1} )}},W = \textrm{diag}{((1 - {\hat{r}_1}), \cdots , (1 - {\hat{r}_1}))_{({M - 1} )\times ({M - 1} )}};$$
$$V\textrm{ = }{\left[ {\begin{array}{ccccc} {1 - {{\hat{r}}_2}}&{\frac{{{b_1}(1 )}}{2}}&{}&{}&{}\\ {\frac{{{b_2}(2 )}}{2}}&{1 - {{\hat{r}}_2}}&{\frac{{{b_1}(2 )}}{2}}&{}&{}\\ {}& \ddots & \ddots & \ddots &{}\\ {}&{}&{\frac{{{b_2}({M - 2} )}}{2}}&{1 - {{\hat{r}}_2}}&{\frac{{{b_1}({M - 2} )}}{2}}\\ {}&{}&{}&{\frac{{{b_2}({M - 1} )}}{2}}&{1 - {{\hat{r}}_2}} \end{array}} \right]_{({M - 1} )\times ({M - 1} )}},$$
$$G\textrm{ = }{\left[ {\begin{array}{ccccc} {1 + {{\hat{r}}_\textrm{2}}}&{ - \frac{{{b_1}(1 )}}{2}}&{}&{}&{}\\ { - \frac{{{b_2}(2 )}}{2}}&{1 + {{\hat{r}}_\textrm{2}}}&{ - \frac{{{b_1}(2 )}}{2}}&{}&{}\\ {}& \ddots & \ddots & \ddots &{}\\ {}&{}&{ - \frac{{{b_2}({M - 2} )}}{2}}&{1 + {{\hat{r}}_\textrm{2}}}&{ - \frac{{{b_1}({M - 2} )}}{2}}\\ {}&{}&{}&{ - \frac{{{b_2}({M - 1} )}}{2}}&{1 + {{\hat{r}}_\textrm{2}}} \end{array}} \right]_{({M - 1} )\times ({M - 1} )}},$$
$$\scalebox{0.95}{$\begin{array}{@{}c@{}}{\bar{\omega }_i} = {\left[ {\frac{{{b_2}(1 )}}{2}T_{i,0}^k,0, \cdots ,0,\frac{{{b_1}({M - 1} )}}{2}T_{i,M}^k} \right]^T}_{({M - 1} )\times 1},{\bar{e}_i} = {\left[ { - \frac{{{b_2}(1 )}}{2}T_{i,0}^{k + 1},0, \cdots ,0, - \frac{{{b_1}({M - 1} )}}{2}T_{i,M}^{k + 1}} \right]^T}_{({M - 1} )\times 1};\\ 1 \le i \le M - 1,1 \le k \le N - 1 .\end{array}$}$$

B. Process of the finding root $T_{M,j}^k$

Based on Tianheng formula, the judgment and the multiple root discriminant of Eq. (18) are listed as follows.

The total discriminant $\Delta $ is:

$$\Delta = {\tilde{B}^2} - 4\tilde{A}\tilde{C} > 0.$$

According to Eq. (39) and Tianheng formula, Eq. (18) only has two unequal real roots and a pair of conjugate imaginary roots. Therefore, we obtain roots as follows:

$$\begin{aligned} {x_{1,2}} &= \frac{{ - b + {\mathop{\rm sgn}} (\tilde{E}) \cdot \sqrt {\frac{{\tilde{D} + \sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}}}{3}} \pm \sqrt {\frac{{2\tilde{D} - \left( {\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}} \right) + 2\sqrt z }}{3}} }}{{4a}},\\ {x_{3,4}} &= \frac{{ - b - {\mathop{\rm sgn}} (\tilde{E}) \cdot \sqrt {\frac{{\tilde{D} + \sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}}}{3}} }}{{4a}} \pm \frac{{\sqrt {\frac{{ - 2\tilde{D} + \left( {\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}} \right) + 2\sqrt z }}{3}} }}{{4a}} \cdot \textrm{i}, \end{aligned}$$
where ${\mathop{\rm sgn}}$ is the symbolic function, $\textrm{i} = \sqrt { - 1}$, and
$${z_{1,2}} = \tilde{A}\tilde{D} + 3\left( {\frac{{ - \tilde{B} \pm \sqrt {{{\tilde{B}}^2} - 4\tilde{A}\tilde{C}} }}{2}} \right),z = {\tilde{D}^2} - \tilde{D} \cdot \left( {\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}} \right) + {\left( {\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}} \right)^2} - 3\tilde{A}.$$

Since the solution of our univariate quartic equation is the temperature $T_{M,j}^k$, where $T_{M,j}^k$ is a positive real number, we can round off the imaginary root ${x_{3,4}}$. Then, denote:

$$\begin{aligned} {x_1} &= \frac{{ - b + {\mathop{\rm sgn}} (\tilde{E}) \cdot \sqrt {\frac{{\tilde{D} + \sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}}}{3}} + \sqrt {\frac{{2\tilde{D} - \left( {\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}} \right) + 2\sqrt z }}{3}} }}{{4a}},\\ {x_2} &= \frac{{ - b + {\mathop{\rm sgn}} (\tilde{E}) \cdot \sqrt {\frac{{\tilde{D} + \sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}}}{3}} - \sqrt {\frac{{2\tilde{D} - \left( {\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}} \right) + 2\sqrt z }}{3}} }}{{4a}}. \end{aligned}$$

From Eq. (21), we have

$$\tilde{E} ={-} 8{({\varepsilon \sigma } )^2} \cdot \left( {H + \frac{\kappa }{{{h_1}}}} \right) < 0. $$

Therefore,

$${\mathop{\rm sgn}} (\tilde{E}) < 0.$$

According to Eqs. (43) and (44), we have

$${x_2} < 0.$$

For ${x_1}$, we conclude that ${x_1} > 0$. The detail analysis is given as follows.

Assuming ${x_1} > 0,$ and we have $b = 0$. Therefore, the following equations hold:

$$\sqrt {\frac{{ - \left( {\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}} \right) + 2\sqrt z }}{3}} > \sqrt {\frac{{\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}}}{3}} ,$$
$$\begin{array}{l} \Rightarrow - \left( {\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}} \right) + 2\sqrt z > \sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}},\\ \Rightarrow \sqrt z > \sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}},\\ \Rightarrow \sqrt {{{\left( {\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}} \right)}^2} - 3\tilde{A}} > \sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}. \end{array}$$

Consider the fact:

$$\tilde{A} = {\tilde{D}^2} - 3\tilde{F} = 3 \cdot 64 \cdot {({\varepsilon \sigma } )^3} \cdot \left( { - \frac{\kappa }{{{h_1}}} \cdot T_{M - 1,j}^k - H \cdot {T_a} - \varepsilon \sigma \cdot T_a^4} \right) < 0.$$

Combining Eqs. (47) and (48), we conclude that Eq. (46) is always valid. Hence, the assumption (${x_1} > 0$) is valid.

In summary, the unique solution of Eq. (18) is:

$$T_{M,j}^k \approx {x_1} = \frac{{ - b + {\mathop{\rm sgn}} (\tilde{E}) \cdot \sqrt {\frac{{\tilde{D} + \sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}}}{3}} + \sqrt {\frac{{2\tilde{D} - \left( {\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}} \right) + 2\sqrt z }}{3}} }}{{4a}}.$$

C. Process of the error estimation for $\hat{R} - \hat{L}$

Taylor expansion is applied to $\hat{R} - \hat{L}$, and only the first two terms are preserved (the latter high-order infinitesimal is omitted). The results are as follows.

Since $\sqrt {x - y} \approx \sqrt x - y \cdot \frac{1}{{2\sqrt x }},$ we have

$$\begin{aligned} |{\hat{R} - \hat{L}} |&\approx \left|{\frac{1}{{4a}}\left[ \begin{array}{l} \sqrt {\frac{{\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}} - \frac{{576 \cdot {{({\varepsilon \sigma } )}^3} \cdot \left( { - \frac{\kappa }{{{h_1}}} \cdot T_{M - 1,j}^k - H \cdot {T_a} - \varepsilon \sigma \cdot T_a^4} \right)}}{{\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}}}}}{3}} \\ - \sqrt {\frac{{\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}}}{3}} \end{array} \right] - T_{M,j}^k} \right|\\ & \approx \left|{\frac{1}{{4a}}\left[ {\frac{{ - 288 \cdot {{({\varepsilon \sigma } )}^3} \cdot \left( { - \frac{\kappa }{{{h_1}}} \cdot T_{M - 1,j}^k - H \cdot {T_a} - \varepsilon \sigma \cdot T_a^4} \right)}}{{\sqrt 3 \cdot {{\left( {\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}} \right)}^{3/2}}}}} \right] - T_{M,j}^k} \right|. \end{aligned}$$

According to the Eqs. (16) and (50), we have

$$|{\hat{R} - \hat{L}} |\approx \left|{\frac{{72 \cdot {{({\varepsilon \sigma } )}^3} \cdot {{({T_{M,j}^k} )}^4} + 72 \cdot {{({\varepsilon \sigma } )}^2} \cdot \left( {H + \frac{\kappa }{{{h_1}}}} \right) \cdot T_{M,j}^k - \sqrt 3 \cdot {{\left( {\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}} \right)}^{3/2}} \cdot T_{M,j}^k}}{{\sqrt 3 \cdot {{\left( {\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}} \right)}^{3/2}}}}} \right|,$$
when ${h_1} \to 0$, $\sqrt[3]{{{z_1}}}$ and $\sqrt[3]{{{z_2}}}$ are all greater than 0, so the following inequality holds
$$\sqrt[3]{{{z_1} + {z_2}}} \le \sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}.$$

Based on Eqs. (51) and (52), we have

$$\begin{aligned} |{\hat{R} - \hat{L}} |&\le \left|{\frac{{72 \cdot {{({\varepsilon \sigma } )}^3} \cdot {{({T_{M,j}^k} )}^4} + 72 \cdot {{({\varepsilon \sigma } )}^2} \cdot \left( {H + \frac{\kappa }{{{h_1}}}} \right) \cdot T_{M,j}^k - \sqrt 3 \cdot {{\left( {\sqrt[3]{{{z_1} + {z_2}}}} \right)}^{3/2}} \cdot T_{M,j}^k}}{{\sqrt 3 \cdot {{\left( {\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}} \right)}^{3/2}}}}} \right|\\& \le \left|{\frac{{72 \cdot {{({\varepsilon \sigma } )}^3} \cdot {{({T_{M,j}^k} )}^4}}}{{\sqrt 3 \cdot {{\left( {\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}} \right)}^{3/2}}}}} \right|. \end{aligned}$$

By Taylor expansion, we get:

$$\sqrt[3]{{{z_1} + {z_2}}} \approx \sqrt[3]{{{z_1}}} + {z_2} \cdot \frac{1}{3}z_1^{ - \frac{2}{3}},\sqrt[3]{{{z_1} + {z_2}}} \approx \sqrt[3]{{{z_2}}} + {z_1} \cdot \frac{1}{3}z_2^{ - \frac{2}{3}}.$$

Depending on Eqs. (53) and (54), we have

$$|{\hat{R} - \hat{L}} |\le \left|{\frac{{72 \cdot {{({\varepsilon \sigma } )}^3} \cdot {{({T_{M,j}^k} )}^4}}}{{\sqrt 3 \cdot {{\left( {2 \cdot \sqrt[3]{{{z_1} + {z_2}}} - \frac{1}{3}\left( {{z_1} \cdot \frac{1}{3}z_2^{ - \frac{2}{3}} + {z_2} \cdot \frac{1}{3}z_1^{ - \frac{2}{3}}} \right)} \right)}^{3/2}}}}} \right|,$$
when ${h_1} \to 0$, $\sqrt 3 \cdot {\left( {2 \cdot \sqrt[3]{{{z_1} + {z_2}}} - \frac{1}{3}\left( {{z_1} \cdot \frac{1}{3}z_2^{ - \frac{2}{3}} + {z_2} \cdot \frac{1}{3}z_1^{ - \frac{2}{3}}} \right)} \right)^{3/2}}$ includes $\frac{1}{{h_1^4}}$, and should be the largest item. Since the comparison of other items is negligible, we obtain the limits for Eq. (55) as follows:
$$\begin{aligned} \mathop {\lim }\limits_{{h_1} \to 0} \left|{\frac{{\hat{R} - \hat{L}}}{{h_1^2}}} \right|&\le \mathop {\lim }\limits_{{h_1} \to 0} \left|{\frac{{\frac{{72 \cdot {{({\varepsilon \sigma } )}^3} \cdot {{({T_{M,j}^k} )}^4}}}{{\sqrt 3 \cdot {{\left( {2 \cdot \sqrt[3]{{{z_1} + {z_2}}} - \frac{1}{3}\left( {{z_1} \cdot \frac{1}{3}z_2^{ - \frac{2}{3}} + {z_2} \cdot \frac{1}{3}z_1^{ - \frac{2}{3}}} \right)} \right)}^{3/2}}}}}}{{h_1^2}}} \right|\\ & \le \mathop {\lim }\limits_{{h_1} \to 0} \left|{\frac{{Q \cdot h_1^4}}{{h_1^2}}} \right|= 0, \end{aligned}$$
where Q is the coefficient of item $h_1^4$.

Therefore, from Eq. (56), the error order of Eq. (19) is at least 2.

Funding

Ministry of Science and Technology of the People's Republic of China (2018YFB1107600).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (14)
Fig. 1.
Fig. 1. Schematic of a standard waterjet-guided laser processing setup with coupling to the bottom water vapor.

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Fig. 2.
Fig. 2. Sketch of the domain of heat equation.

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Fig. 3.
Fig. 3. Numerical solutions of difference equations $({h = \textrm{0}\textrm{.001/50},\tau = \textrm{0}\textrm{.01/500,}H\textrm{ = 1}\textrm{.5}} )$ .

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Fig. 4.
Fig. 4. Numerical solutions obtained by the PDETOOL of MATLAB.

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Fig. 5.
Fig. 5. (a) Cross section contrast by the ADI with Crank-Nicolson method (left figure). (b) Cross section contrast by the PDETOOL of MATLAB (right figure).

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Fig. 6.
Fig. 6. Numerical solutions of difference equations $({h = \textrm{0}\textrm{.001/50},\tau = \textrm{0}\textrm{.01/500,}H\textrm{ = 1}\textrm{.5}} )$ .

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Fig. 7.
Fig. 7. Numerical solutions obtained by PDETOOL of MATLAB.

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Fig. 8.
Fig. 8. (a) Cross section contrast by the ADI with Crank-Nicolson method (left figure). (b) Cross section contrast by the PDETOOL of MATLAB (right figure).

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Fig. 9.
Fig. 9. Numerical solutions of difference equations $({h = \textrm{0}\textrm{.001/50},\tau = \textrm{0}\textrm{.1/500,}H\textrm{ = 1}\textrm{.5}} )$ .

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Fig. 10.
Fig. 10. Numerical solutions obtained by PDETOOL of MATLAB.

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Fig. 11.
Fig. 11. (a) Cross section contrast by the ADI with Crank-Nicolson method (left figure). (b) Cross section contrast by the PDETOOL of MATLAB (right figure).

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Fig. 12.
Fig. 12. Numerical solutions of difference equations $({h = \textrm{0}\textrm{.001/50},\tau = \textrm{0}\textrm{.01/500,}H\textrm{ = 1}\textrm{.5}} )$ .

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Fig. 13.
Fig. 13. Numerical solutions of difference equations $({h = \textrm{0}\textrm{.001/50},\tau = \textrm{0}\textrm{.01/500,}H\textrm{ = 1}\textrm{.5}} )$ .

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Fig. 14.
Fig. 14. (a) Cross section contrast by the ordinary laser (left figure). (b) Cross section contrast by the water-jet guided laser (right figure).

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Tables (3)

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Table 1. The thermal property parameters of 316 stainless steel

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Table 2. The thermal property parameters of titanium

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Table 3. The thermal property parameters of beryllium

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Equations (56)

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$$\frac{{\partial \hat{T}}}{{\partial t}} = \nabla ({\alpha \cdot \nabla \hat{T}} )\textrm{ + }f,({x,y,z} )\in \varOmega ,t \in ({0,{T_e}} ] ;$$
$$\hat{T}({x,y,z,0} )= \hat{\varphi }({x,y,z} ),\textrm{0} \le x \le X,{y^2} + {z^2} \le {R^2};$$
$$\left\{ \begin{array}{l} {\left. { - \kappa \left( {\nabla \hat{T} \cdot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over n} } \right)} \right|_{{S_\textrm{3}}}} = \beta q\left( {\sqrt {{y^2} + {z^2}} } \right)|{_{_{{S_3}}}} ,\textrm{where }{\textrm{S}_3} = \left\{ { {({y,z} )} |x = 0,\sqrt {{y^2} + {z^2}} \le d} \right\};\\ { {\hat{T}} |_{{S_4}}} = \textrm{293K}({\textrm{2}{\textrm{0}^\textrm{o}}\textrm{C}} ),\textrm{where }{S_4} = \left\{ { {({y,z} )} |x = 0,d < \sqrt {{y^2} + {z^2}} \le R} \right\};\\ {\left. { - \kappa \left( {\nabla \hat{T} \cdot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over n} } \right)} \right|_{{S_2}}} = { {H({\hat{T} - {T_a}} )} |_{{S_2}}},\textrm{where }{S_2} = \{{ {({x,y,z} )} |0 < x < X,{y^2} + {z^2} = {R^2}} \};\\ - {\left. {\kappa \left( {\nabla \hat{T} \cdot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over n} } \right)} \right|_{{S_1}}} = { {H({\hat{T} - {T_a}} )|{_{{S_1}}} + \varepsilon \sigma \cdot ({{{\hat{T}}^4} - {T_a}^4} )} |_{{S_1}}}\textrm{,where }{S_1} = \{{ {({y,z} )} |x = X,{y^2} + {z^2} \le {R^2}} \}; \end{array} \right.$$
$$\frac{{\partial T}}{{\partial t}}\textrm{ = }\alpha \cdot \left( {\frac{{{\partial^2}T}}{{\partial {r^2}}} + \frac{1}{r} \cdot \frac{{\partial T}}{{\partial r}} + \frac{{{\partial^2}T}}{{\partial {x^2}}}} \right) + f,0 < x < X,0 < r < R,0 < t \le {T_e},$$
$$T({x,r,0} )= \varphi ({x,r} ),0 \le x \le X,0 \le r \le R.$$
$$\left\{ \begin{array}{l} {\left. { - \kappa \left( {\nabla T \cdot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over n} } \right)} \right|_{x = 0}} = \beta q(r ),0 \le r\, \le d;\\ { T |_{{S_4}}} = \textrm{293K}({\textrm{2}{\textrm{0}^\textrm{o}}\textrm{C}} ),\textrm{where }{S_4} = \left\{ { {({y,z} )} |x = 0,d < \sqrt {{y^2} + {z^2}} \le R} \right\};\\ {\left. { - \kappa \left( {\nabla T \cdot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over n} } \right)} \right|_{x = X}} = H({T - {T_a}} )|{_{x = X}} + \varepsilon \sigma \cdot ({{T^4} - {T_a}^4} )\, |{_{x = X}} ,0 \le r \le R;\\ {\left. { - \kappa \left( {\nabla T \cdot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over n} } \right)} \right|_{r = R}} = H({T - {T_a}} )|{_{r = R}} ,0 < x < X;\\ \kappa {\left. {\frac{{\partial T}}{{\partial r}}} \right|_{r = 0}} = 0,0 < x < X; \end{array} \right.$$
$$\left\{ \begin{array}{c} \left( {I - \frac{{\alpha \tau }}{2}\hat{\delta }_x^2} \right)\left( {I - \frac{{\alpha \tau }}{2}\left( {\hat{\delta }_y^2 + \frac{1}{{{r_j}}}{{\hat{\delta }}_y}} \right)} \right)T_{i,j}^{k + 1} = \left( {I + \frac{{\alpha \tau }}{2}\hat{\delta }_x^2} \right)\left( {I + \frac{{\alpha \tau }}{2}\left( {\hat{\delta }_y^2 + \frac{1}{{{r_j}}}{{\hat{\delta }}_y}} \right)} \right)T_{i,j}^k ,\\ 1 \le i,j \le M - 1 , 1 \le k \le N - 1 ; \end{array} \right.$$
$$T({x,r,0} )= \varphi ({x,r} ),0 \le x \le X,0 \le r \le R;$$
$$\left\{ \begin{array}{l} { { - \kappa ({\nabla T \cdot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over n}} )} |_{x = 0}} = \beta q(r ),0 \le r\, \le d;\\ { { - \kappa ({\nabla T \cdot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over n}} )} |_{x = X}} = H({T - {T_a}} )|{_{x = X}} + \varepsilon \sigma \cdot ({{T^4} - {T_a}^4} )\, |{_{x = X}} ,0 \le r \le R;\\ { { - \kappa ({\nabla T \cdot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over n}} )} |_{r = R}} = H({T - {T_a}} ),0 < x < X;\\ \kappa {\left. {\frac{{\partial T}}{{\partial r}}} \right|_{r = 0}} = 0,0 < x < X; \end{array} \right.$$
$$T_{0,j}^k = T_{1,j}^k + \frac{{\beta {h_1}q({{r_j}} )}}{\kappa },0 \le j \le M;$$
$$T_{i,M}^k = \frac{\kappa }{{\kappa + {h_2}H}}T_{i,M - 1}^k + \frac{{{h_2}H}}{{\kappa + {h_2}H}}{T_a},0 \le i \le M;$$
$$T_{i,0}^k = T_{i,1}^k,0 \le i \le M.$$
$$\kappa \frac{{ - \hat{T}_{M + 1,j}^k + T_{M - 1,j}^k}}{{2{h_1}}} = { {[{H({T - {T_a}} )+ \varepsilon \sigma \cdot ({{T^4} - T_a^4} )} ]} |_{x = X}},0 \le j \le M;$$
$$T_{M,j}^k \approx \frac{{\hat{T}_{M + 1,j}^k + T_{M - 1,j}^k}}{2},0 \le j \le M;$$
$${ T |_{x = X}} \approx \frac{{\hat{T}_{M + 1,j}^k + T_{M - 1,j}^k}}{2},0 \le j \le M\, .$$
$$\varepsilon \sigma \cdot {({T_{M,j}^k} )^4} + \left( {H + \frac{\kappa }{{{h_1}}}} \right) \cdot T_{M,j}^k - \frac{\kappa }{{{h_1}}} \cdot T_{M - 1,j}^k - H \cdot {T_a} - \varepsilon \sigma \cdot T_a^4 \approx 0.$$
$$a = \varepsilon \sigma ,b = 0,c = 0,d = H + \frac{\kappa }{{{h_1}}},e ={-} \frac{\kappa }{{{h_1}}} \cdot T_{M - 1,j}^k - H \cdot {T_a} - \varepsilon \sigma \cdot T_a^4;$$
$$a \cdot {({T_{M,j}^k} )^4} + b \cdot {({T_{M,j}^k} )^3} + c \cdot {({T_{M,j}^k} )^2} + d \cdot T_{M,j}^k + e \approx 0.$$
$$T_{M,j}^k \approx \frac{{ - b + {\mathop{\rm sgn}} (\tilde{E}) \cdot \sqrt {\frac{{\tilde{D} + \sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}}}{3}} + \sqrt {\frac{{2\tilde{D} - \left( {\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}} \right) + 2\sqrt z }}{3}} }}{{4a}},$$
$$\left\{ \begin{array}{l} {z_1} = \tilde{A}\tilde{D} + 3\left( {\frac{{ - \tilde{B} + \sqrt {{{\tilde{B}}^2} - 4\tilde{A}\tilde{C}} }}{2}} \right),{z_2} = \tilde{A}\tilde{D} + 3\left( {\frac{{ - \tilde{B} - \sqrt {{{\tilde{B}}^2} - 4\tilde{A}\tilde{C}} }}{2}} \right),\\ z = {{\tilde{D}}^2} - \tilde{D} \cdot \left( {\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}} \right) + {\left( {\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}} \right)^2} - 3\tilde{A}, \end{array} \right.$$
$$\left\{ {\begin{array}{l} {\tilde{D} = 3{b^2} - 8ac = 0,}\\ {\tilde{E} ={-} {b^3} + 4abc - 8{a^2}d ={-} 8 \cdot {{({\varepsilon \sigma } )}^2} \cdot \left( {H + \frac{\kappa }{{{h_1}}}} \right),}\\ {\tilde{F} = 3{b^4} + 16{a^2}{c^2} - 16a{b^2}c + 16{a^2}bd - 64{a^3}e}\\ { ={-} 64 \cdot {{({\varepsilon \sigma } )}^3} \cdot \left( { - \frac{\kappa }{{{h_1}}} \cdot T_{M - 1,j}^k - H \cdot {T_a} - \varepsilon \sigma \cdot T_a^4} \right), }\\ {\tilde{A} = {{\tilde{D}}^2} - 3\tilde{F} = 3 \cdot 64 \cdot {{({\varepsilon \sigma } )}^3} \cdot \left( { - \frac{\kappa }{{{h_1}}} \cdot T_{M - 1,j}^k - H \cdot {T_a} - \varepsilon \sigma \cdot T_a^4} \right), }\\ {\tilde{B} = \tilde{D}\tilde{F} - 9{{\tilde{E}}^2} ={-} 9 \cdot 64 \cdot {{({\varepsilon \sigma } )}^4} \cdot {{\left( {H + \frac{\kappa }{{{h_1}}}} \right)}^2}, }\\ {\tilde{C} = {{\tilde{F}}^2} - 3\tilde{D}{{\tilde{E}}^2} = {{64}^2} \cdot {{({\varepsilon \sigma } )}^6} \cdot {{\left( { - \frac{\kappa }{{{h_1}}} \cdot T_{M - 1,j}^k - H \cdot {T_a} - \varepsilon \sigma \cdot T_a^4} \right)}^2}.} \end{array}} \right.$$
$$\left\{ \begin{aligned} |{\hat{R} - \hat{L}} |&= \left|{\frac{1}{{4a}}\left[ \begin{array}{l} \sqrt {\frac{{2\sqrt {{{\left( {\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}} \right)}^2} - \left[ {576 \cdot {{({\varepsilon \sigma } )}^3} \cdot \left( { - \frac{\kappa }{{{h_1}}} \cdot T_{M - 1,j}^k - H \cdot {T_a} - \varepsilon \sigma \cdot T_a^4} \right)} \right]} - \left( {\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}} \right)}}{3}} \\ - \sqrt {\frac{{\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}}}{3}} \end{array} \right] - T_{M,j}^k} \right|\\ &{ = O(h_1^2),}\end{aligned} \right.$$
$$\begin{aligned} T_{M,j}^{k + 1/2} &\approx \frac{{T_{M,j}^k + T_{M,j}^{k + 1}}}{2} \approx \frac{{T_{M,j}^k}}{2} + \frac{1}{2} \cdot ({2 \cdot T_{M - 1,j}^{k + 1} - T_{M - 2,j}^{k + 1}} )\\ & \approx \frac{{T_{M,j}^k}}{2} + \frac{1}{2} \cdot [{2 \cdot ({2 \cdot T_{M - 1,j}^{k + 1/2} - T_{M - 1,j}^k} )- ({2 \cdot T_{M - 2,j}^{k + 1/2} - T_{M - 2,j}^k} )} ]\\ & \approx \frac{{T_{M,j}^k}}{2} - T_{M - 1,j}^k + \frac{{T_{M - 2,j}^k}}{2} + 2 \cdot T_{M - 1,j}^{k + 1/2} - T_{M - 2,j}^{k + 1/2}, \end{aligned}$$
$$\begin{aligned} {R_c} &={-} \frac{{{\tau ^2}}}{8} \cdot \frac{{{\partial ^2}T({{x_M},{r_j},{t_{k + 1/2}}} )}}{{\partial {t^2}}} + \frac{{h_1^2}}{2} \cdot \frac{{{\partial ^2}T({{x_M},{r_j},{t_{k + 1}}} )}}{{\partial {x^2}}} + \frac{{{\tau ^2}}}{4} \cdot \frac{{{\partial ^2}T({{x_{M - 1}},{r_j},{t_{k + 1}}} )}}{{\partial {t^2}}}\\ & - \frac{{{\tau ^2}}}{8} \cdot \frac{{{\partial ^2}T({{x_{M - 2}},{r_j},{t_{k + 1}}} )}}{{\partial {t^2}}} + o({{\tau^2} + h_1^2} )= O({{\tau^2} + h_1^2} ). \end{aligned}$$
$$\frac{{\partial T}}{{\partial t}}\textrm{ = }\alpha \cdot \left( {\frac{{{\partial^2}T}}{{\partial {r^2}}} + \frac{1}{r} \cdot \frac{{\partial T}}{{\partial r}} + \frac{{{\partial^2}T}}{{\partial {x^2}}}} \right) + f,0 \le x \le X,0 < r < R,0 < t \le {T_e};$$
$${ {T({x,r,t} )} |_{t = 0}} = {T_a},$$
$$\left\{ \begin{array}{l} {\left. { - \kappa \left( {\nabla T \cdot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over n} } \right)} \right|_{x = 0}} = \beta q(r ),0 \le r \le d;\\ { T |_{{S_4}}} = \textrm{293K}({\textrm{2}{\textrm{0}^\textrm{o}}\textrm{C}} ); \\ {\left. { - \kappa \left( {\nabla T \cdot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over n} } \right)} \right|_{x = X}} = H({T - {T_a}} )|{_{x = X}} + \varepsilon \sigma \cdot ({{T^4} - {T_a}^4} )\, |{_{x = X}} ,0 \le r \le R; \, \\ - {\left. {\kappa \left( {\nabla T \cdot \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over n} } \right)} \right|_{r = R}} = H({T - {T_a}} )\, |{_{r = R}} ,0 < x < X;\textrm{ }\, \kappa {\left. {\frac{{\partial T}}{{\partial r}}} \right|_{r = 0}} = 0,0 < x < X;\, \end{array} \right.$$
$$\beta q(r )= \beta \cdot \frac{{2P}}{{\pi \cdot {d^2}}}{\text{e}^{ - \frac{{2{r^2}}}{{{d^2}}}}},$$
$${ {\kappa ({\nabla T \cdot n} )} |_{x = X}} = H({T - {T_a}} )|{_{x = X}} + \varepsilon \sigma \cdot ({{T^4} - {T_a}^4} )\, |{_{x = X}} ,0 \le r \le R\, .$$
$$\begin{array}{l} S \cdot \bar{T}_{i - 1}^{k + 1/2} + E \cdot \bar{T}_i^{k + 1/2} + S \cdot \bar{T}_{i + 1}^{k + 1/2} = V \cdot \bar{T}_i^k,\\ G \cdot \bar{T}_i^{k + 1} = L \cdot \bar{T}_{i - 1}^{k + 1/2} + W \cdot \bar{T}_i^{k + 1/2} + L \cdot \bar{T}_{i + 1}^{k + 1/2}, \end{array}$$
$$\left[ {\begin{array}{@{}ccccc@{}} E&S&{}&{}&{}\\ S&E&S&{}&{}\\ {}& \ddots & \ddots & \ddots &{}\\ {}&{}&S&E&S\\ {}&{}&{}&S&E \end{array}} \right]{\left[ {\begin{array}{@{}c@{}} {{{\bar{T}}_1}}\\ {{{\bar{T}}_2}}\\ \vdots \\ {{{\bar{T}}_{M - 2}}}\\ {{{\bar{T}}_{M - 1}}} \end{array}} \right]^{k + \frac{1}{2}}} = \left[ {\begin{array}{@{}ccccc@{}} V&{}&{}&{}&{}\\ {}&V&{}&{}&{}\\ {}&{}& \ddots &{}&{}\\ {}&{}&{}&V&{}\\ {}&{}&{}&{}&V \end{array}} \right]{\left[ {\begin{array}{@{}c@{}} {{{\bar{T}}_1}}\\ {{{\bar{T}}_2}}\\ \vdots \\ {{{\bar{T}}_{M - 2}}}\\ {{{\bar{T}}_{M - 1}}} \end{array}} \right]^k} + \left[ {\begin{array}{@{}c@{}} { - S{{\bar{T}}_0}^{k + 1/2} + {{\bar{\omega }}_1}}\\ {{{\bar{\omega }}_2}}\\ \vdots \\ {{{\bar{\omega }}_{M - 2}}}\\ { - S{{\bar{T}}_M}^{k + 1/2} + {{\bar{\omega }}_{M - 1}}} \end{array}} \right],$$
$$\left[ {\begin{array}{@{}ccccc@{}} G&{}&{}&{}&{}\\ {}&G&{}&{}&{}\\ {}&{}& \ddots &{}&{}\\ {}&{}&{}&G&{}\\ {}&{}&{}&{}&G \end{array}} \right]{\left[ {\begin{array}{@{}c@{}} {{{\bar{T}}_1}}\\ {{{\bar{T}}_2}}\\ \vdots \\ {{{\bar{T}}_{M - 2}}}\\ {{{\bar{T}}_{M - 1}}} \end{array}} \right]^{k + 1}} = \left[ {\begin{array}{@{}ccccc@{}} W&L&{}&{}&{}\\ L&W&L&{}&{}\\ {}& \ddots & \ddots & \ddots &{}\\ {}&{}&L&W&L\\ {}&{}&{}&L&W \end{array}} \right]{\left[ {\begin{array}{@{}c@{}} {{{\bar{T}}_1}}\\ {{{\bar{T}}_2}}\\ \vdots \\ {{{\bar{T}}_{M - 2}}}\\ {{{\bar{T}}_{M - 1}}} \end{array}} \right]^{k + \frac{1}{2}}} + \left[ {\begin{array}{@{}c@{}} {L{{\bar{T}}_0}^{k + 1/2} - {{\bar{e}}_1}}\\ { - {{\bar{e}}_2}}\\ \vdots \\ { - {{\bar{e}}_{M - 2}}}\\ {L{{\bar{T}}_M}^{k + 1/2} - {{\bar{e}}_{M - 1}}} \end{array}} \right],$$
$${\hat{r}_1} = \frac{{\alpha \cdot \tau }}{{{h_1}^2}},{\hat{r}_2} = \frac{{\alpha \cdot \tau }}{{{h_2}^2}},{b_1}(j )= \left( {1 + \frac{1}{{2j}}} \right) \cdot {\hat{r}_2},{b_2}(j )= \left( {1 - \frac{1}{{2j}}} \right) \cdot {\hat{r}_2};$$
$$S\textrm{ = diag}{( - \frac{{{{\hat{r}}_1}}}{2}, \cdots , - \frac{{{{\hat{r}}_1}}}{2})_{({M - 1} )\times ({M - 1} )}},L\textrm{ = diag}{(\frac{{{{\hat{r}}_1}}}{2}, \cdots , \frac{{{{\hat{r}}_1}}}{2})_{({M - 1} )\times ({M - 1} )}};$$
$$E = \textrm{diag}{((1 + {\hat{r}_1}), \cdots , (1 + {\hat{r}_1}))_{({M - 1} )\times ({M - 1} )}},W = \textrm{diag}{((1 - {\hat{r}_1}), \cdots , (1 - {\hat{r}_1}))_{({M - 1} )\times ({M - 1} )}};$$
$$V\textrm{ = }{\left[ {\begin{array}{ccccc} {1 - {{\hat{r}}_2}}&{\frac{{{b_1}(1 )}}{2}}&{}&{}&{}\\ {\frac{{{b_2}(2 )}}{2}}&{1 - {{\hat{r}}_2}}&{\frac{{{b_1}(2 )}}{2}}&{}&{}\\ {}& \ddots & \ddots & \ddots &{}\\ {}&{}&{\frac{{{b_2}({M - 2} )}}{2}}&{1 - {{\hat{r}}_2}}&{\frac{{{b_1}({M - 2} )}}{2}}\\ {}&{}&{}&{\frac{{{b_2}({M - 1} )}}{2}}&{1 - {{\hat{r}}_2}} \end{array}} \right]_{({M - 1} )\times ({M - 1} )}},$$
$$G\textrm{ = }{\left[ {\begin{array}{ccccc} {1 + {{\hat{r}}_\textrm{2}}}&{ - \frac{{{b_1}(1 )}}{2}}&{}&{}&{}\\ { - \frac{{{b_2}(2 )}}{2}}&{1 + {{\hat{r}}_\textrm{2}}}&{ - \frac{{{b_1}(2 )}}{2}}&{}&{}\\ {}& \ddots & \ddots & \ddots &{}\\ {}&{}&{ - \frac{{{b_2}({M - 2} )}}{2}}&{1 + {{\hat{r}}_\textrm{2}}}&{ - \frac{{{b_1}({M - 2} )}}{2}}\\ {}&{}&{}&{ - \frac{{{b_2}({M - 1} )}}{2}}&{1 + {{\hat{r}}_\textrm{2}}} \end{array}} \right]_{({M - 1} )\times ({M - 1} )}},$$
$$\scalebox{0.95}{$\begin{array}{@{}c@{}}{\bar{\omega }_i} = {\left[ {\frac{{{b_2}(1 )}}{2}T_{i,0}^k,0, \cdots ,0,\frac{{{b_1}({M - 1} )}}{2}T_{i,M}^k} \right]^T}_{({M - 1} )\times 1},{\bar{e}_i} = {\left[ { - \frac{{{b_2}(1 )}}{2}T_{i,0}^{k + 1},0, \cdots ,0, - \frac{{{b_1}({M - 1} )}}{2}T_{i,M}^{k + 1}} \right]^T}_{({M - 1} )\times 1};\\ 1 \le i \le M - 1,1 \le k \le N - 1 .\end{array}$}$$
$$\Delta = {\tilde{B}^2} - 4\tilde{A}\tilde{C} > 0.$$
$$\begin{aligned} {x_{1,2}} &= \frac{{ - b + {\mathop{\rm sgn}} (\tilde{E}) \cdot \sqrt {\frac{{\tilde{D} + \sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}}}{3}} \pm \sqrt {\frac{{2\tilde{D} - \left( {\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}} \right) + 2\sqrt z }}{3}} }}{{4a}},\\ {x_{3,4}} &= \frac{{ - b - {\mathop{\rm sgn}} (\tilde{E}) \cdot \sqrt {\frac{{\tilde{D} + \sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}}}{3}} }}{{4a}} \pm \frac{{\sqrt {\frac{{ - 2\tilde{D} + \left( {\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}} \right) + 2\sqrt z }}{3}} }}{{4a}} \cdot \textrm{i}, \end{aligned}$$
$${z_{1,2}} = \tilde{A}\tilde{D} + 3\left( {\frac{{ - \tilde{B} \pm \sqrt {{{\tilde{B}}^2} - 4\tilde{A}\tilde{C}} }}{2}} \right),z = {\tilde{D}^2} - \tilde{D} \cdot \left( {\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}} \right) + {\left( {\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}} \right)^2} - 3\tilde{A}.$$
$$\begin{aligned} {x_1} &= \frac{{ - b + {\mathop{\rm sgn}} (\tilde{E}) \cdot \sqrt {\frac{{\tilde{D} + \sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}}}{3}} + \sqrt {\frac{{2\tilde{D} - \left( {\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}} \right) + 2\sqrt z }}{3}} }}{{4a}},\\ {x_2} &= \frac{{ - b + {\mathop{\rm sgn}} (\tilde{E}) \cdot \sqrt {\frac{{\tilde{D} + \sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}}}{3}} - \sqrt {\frac{{2\tilde{D} - \left( {\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}} \right) + 2\sqrt z }}{3}} }}{{4a}}. \end{aligned}$$
$$\tilde{E} ={-} 8{({\varepsilon \sigma } )^2} \cdot \left( {H + \frac{\kappa }{{{h_1}}}} \right) < 0. $$
$${\mathop{\rm sgn}} (\tilde{E}) < 0.$$
$${x_2} < 0.$$
$$\sqrt {\frac{{ - \left( {\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}} \right) + 2\sqrt z }}{3}} > \sqrt {\frac{{\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}}}{3}} ,$$
$$\begin{array}{l} \Rightarrow - \left( {\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}} \right) + 2\sqrt z > \sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}},\\ \Rightarrow \sqrt z > \sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}},\\ \Rightarrow \sqrt {{{\left( {\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}} \right)}^2} - 3\tilde{A}} > \sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}. \end{array}$$
$$\tilde{A} = {\tilde{D}^2} - 3\tilde{F} = 3 \cdot 64 \cdot {({\varepsilon \sigma } )^3} \cdot \left( { - \frac{\kappa }{{{h_1}}} \cdot T_{M - 1,j}^k - H \cdot {T_a} - \varepsilon \sigma \cdot T_a^4} \right) < 0.$$
$$T_{M,j}^k \approx {x_1} = \frac{{ - b + {\mathop{\rm sgn}} (\tilde{E}) \cdot \sqrt {\frac{{\tilde{D} + \sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}}}{3}} + \sqrt {\frac{{2\tilde{D} - \left( {\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}} \right) + 2\sqrt z }}{3}} }}{{4a}}.$$
$$\begin{aligned} |{\hat{R} - \hat{L}} |&\approx \left|{\frac{1}{{4a}}\left[ \begin{array}{l} \sqrt {\frac{{\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}} - \frac{{576 \cdot {{({\varepsilon \sigma } )}^3} \cdot \left( { - \frac{\kappa }{{{h_1}}} \cdot T_{M - 1,j}^k - H \cdot {T_a} - \varepsilon \sigma \cdot T_a^4} \right)}}{{\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}}}}}{3}} \\ - \sqrt {\frac{{\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}}}{3}} \end{array} \right] - T_{M,j}^k} \right|\\ & \approx \left|{\frac{1}{{4a}}\left[ {\frac{{ - 288 \cdot {{({\varepsilon \sigma } )}^3} \cdot \left( { - \frac{\kappa }{{{h_1}}} \cdot T_{M - 1,j}^k - H \cdot {T_a} - \varepsilon \sigma \cdot T_a^4} \right)}}{{\sqrt 3 \cdot {{\left( {\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}} \right)}^{3/2}}}}} \right] - T_{M,j}^k} \right|. \end{aligned}$$
$$|{\hat{R} - \hat{L}} |\approx \left|{\frac{{72 \cdot {{({\varepsilon \sigma } )}^3} \cdot {{({T_{M,j}^k} )}^4} + 72 \cdot {{({\varepsilon \sigma } )}^2} \cdot \left( {H + \frac{\kappa }{{{h_1}}}} \right) \cdot T_{M,j}^k - \sqrt 3 \cdot {{\left( {\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}} \right)}^{3/2}} \cdot T_{M,j}^k}}{{\sqrt 3 \cdot {{\left( {\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}} \right)}^{3/2}}}}} \right|,$$
$$\sqrt[3]{{{z_1} + {z_2}}} \le \sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}.$$
$$\begin{aligned} |{\hat{R} - \hat{L}} |&\le \left|{\frac{{72 \cdot {{({\varepsilon \sigma } )}^3} \cdot {{({T_{M,j}^k} )}^4} + 72 \cdot {{({\varepsilon \sigma } )}^2} \cdot \left( {H + \frac{\kappa }{{{h_1}}}} \right) \cdot T_{M,j}^k - \sqrt 3 \cdot {{\left( {\sqrt[3]{{{z_1} + {z_2}}}} \right)}^{3/2}} \cdot T_{M,j}^k}}{{\sqrt 3 \cdot {{\left( {\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}} \right)}^{3/2}}}}} \right|\\& \le \left|{\frac{{72 \cdot {{({\varepsilon \sigma } )}^3} \cdot {{({T_{M,j}^k} )}^4}}}{{\sqrt 3 \cdot {{\left( {\sqrt[3]{{{z_1}}} + \sqrt[3]{{{z_2}}}} \right)}^{3/2}}}}} \right|. \end{aligned}$$
$$\sqrt[3]{{{z_1} + {z_2}}} \approx \sqrt[3]{{{z_1}}} + {z_2} \cdot \frac{1}{3}z_1^{ - \frac{2}{3}},\sqrt[3]{{{z_1} + {z_2}}} \approx \sqrt[3]{{{z_2}}} + {z_1} \cdot \frac{1}{3}z_2^{ - \frac{2}{3}}.$$
$$|{\hat{R} - \hat{L}} |\le \left|{\frac{{72 \cdot {{({\varepsilon \sigma } )}^3} \cdot {{({T_{M,j}^k} )}^4}}}{{\sqrt 3 \cdot {{\left( {2 \cdot \sqrt[3]{{{z_1} + {z_2}}} - \frac{1}{3}\left( {{z_1} \cdot \frac{1}{3}z_2^{ - \frac{2}{3}} + {z_2} \cdot \frac{1}{3}z_1^{ - \frac{2}{3}}} \right)} \right)}^{3/2}}}}} \right|,$$
$$\begin{aligned} \mathop {\lim }\limits_{{h_1} \to 0} \left|{\frac{{\hat{R} - \hat{L}}}{{h_1^2}}} \right|&\le \mathop {\lim }\limits_{{h_1} \to 0} \left|{\frac{{\frac{{72 \cdot {{({\varepsilon \sigma } )}^3} \cdot {{({T_{M,j}^k} )}^4}}}{{\sqrt 3 \cdot {{\left( {2 \cdot \sqrt[3]{{{z_1} + {z_2}}} - \frac{1}{3}\left( {{z_1} \cdot \frac{1}{3}z_2^{ - \frac{2}{3}} + {z_2} \cdot \frac{1}{3}z_1^{ - \frac{2}{3}}} \right)} \right)}^{3/2}}}}}}{{h_1^2}}} \right|\\ & \le \mathop {\lim }\limits_{{h_1} \to 0} \left|{\frac{{Q \cdot h_1^4}}{{h_1^2}}} \right|= 0, \end{aligned}$$
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