1 Introduction

Numerical simulation is considered as a practical alternative to real-time observation methods when analyzing the laser keyhole welding process because of the limitations of experimental methods, such as the difficulty in observing the inside of a workpiece during processing [1–3]. Although the JWRI X-ray transmission in situ observation method [4] has been developed to improve the observation, other limitations, such as expense, resolution, and framing speed, still exist.

The laser keyhole welding process involves the absorption and reflection of a laser ray which comes into contact with a workpiece. A certain quantity of laser ray energy is absorbed by the workpiece, and the remaining energy is carried away by the reflected ray to another location in the keyhole. Inside the keyhole, this phenomenon occurs repeatedly and is known as multiple-reflection. It is one of the important physical phenomena in the laser keyhole welding process because it enhances the amount of energy absorbed by the workpiece from the laser [5].

To analyze the multi physics phenomena of the laser keyhole welding, several numerical models were used, such as the heat flux of the Gaussian heat source [5], the recoil pressure with the Clausisus–Clapeyron equation [6], the Marangoni flow considering the temperature gradient [7], the buoyancy force with Boussinesq approximation, the additional shear stress and heat source due to vapor ejected through the keyhole entrance [8, 9], and the bubble formation, assumed to be an adiabatic bubble [9].

Cho et al. performed numerical simulations of the laser keyhole welding considering multiple-reflection [9–11]. The multiple-reflection was calculated using a ‘Ray tracing’ methodology. Through ray tracing, several ray properties, such as the current location of the ray, the location of the reflection, and the reflected direction, were calculated by solving the proper discriminant. In this paper, this ray tracing method is called the Direct Search Method (DSM).

Ahn et al. performed numerical simulations of laser drilling and scribing with another ray tracing method by solving the surface equation and ray direction vector equation [12]. This ray tracing method is referred to in this paper as the Progressive Search Method (PSM).

In the present study, laser keyhole welding simulations were performed using the Computational Fluid Dynamics (CFD) method along with the free surface detection algorithm and the volume of fluid (VOF) technique [13]. The details of DSM and PSM are introduced in Section 2. Simulations using the same were performed to compare these two methods. Transmission and scattering models were applied in PSM.

2 Methodology

2.1 Definition of laser beam and event occurrence

2.1.1 Definition of laser beam

In both the DSM and PSM, a laser beam is defined as a bundle of rays. In DSM, one ray is defined as a single ray; however, in PSM, one ray is defined as a set of sub-rays, as shown in Fig. 1. In DSM, one ray has only one sub-ray. In the bundle of rays, all the rays have different ray starting points. However, in the same set of sub-rays, all sub-rays have the same starting point.

Fig. 1
figure 1

Definition of laser beam; a DSM and b PSM

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The reason for using the set of sub-rays concept is to better depict the multi-path of a ray. A single ray can be transmitted into a workpiece or scattered by the plasma plume. In such cases, the ray will have multiple paths and be divided into absorbed rays and transmitted rays with energy attenuated by absorption or attenuated by scattering. In DSM, this process cannot be depicted because of the single ray definition. However, this process can be depicted in PSM because of the set of sub-rays concept.

2.1.2 Definition of event occurrence

In the laser keyhole welding process, various events occur, such as absorption, reflection, transmission, and scattering. In DSM, energy separation by events such as absorptivity or reflectivity is calculated by the coefficient of the event. However, a stochastic approach is used in PSM for depicting the multi-path.

As shown in Fig. 2, the reflected and absorbed ray energies are calculated as ‘RI’ and ‘(1-R) I’, respectively, in DSM. The term R refers to the reflectivity obtained from Fresnel reflection and I is the ray energy. In PSM, as already mentioned, if the number of sub-rays in one ray is N, the number of reflected sub-rays is almost ‘RN’ and the number of non-reflected sub-rays is almost ‘(1-R)N’. Non-reflected sub-rays will be absorbed at the reflection point or transmitted into the workpiece.

Fig. 2
figure 2

Schematic of event calculation; a DSM and b PSM

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The sub-rays are reflected at the same reflection point and move with the same reflected direction. In this case, event means reflection. In PSM, reflection occurs when ‘R’ is bigger than the generated uniform random number ‘r’. This condition is calculated in all sub-rays. Therefore, the results for reflection in the two methods are almost the same when the number of sub-rays is sufficiently large, because of the uniform random number. Uniform random number means that the probability of a random number being generated within a range is uniform. For example, if 10 uniform random integers are generated in the range 1 ∼ 2, this includes 5 integers with the value 1 and 5 integers with the value 2. However, the order of generation is random, such as [1, 1, 2, 1, 2, 1, 2, …].

In DSM, the entire non-reflected ray energy is absorbed at the reflection point due to the single ray definition. However, in PSM, due to the set of sub-rays concept, non-reflected sub-rays can be depicted as absorbed at the reflection point or as sub-rays transmitted into the workpiece. If the uniform random number ‘r’ is smaller than the probability of absorption in workpiece ‘PA’, the sub-ray is absorbed at the reflection point. In the other case, where ‘r’ is bigger than ‘PA’, the sub-ray is transmitted into the workpiece with a refracted direction calculated by Snell’s law. The probability of absorption in the workpiece is related to Beer-Lambert’s law. The details of the transmission process will be explained in Section 2.3.1.

The reason for using a stochastic approach is found in the calculation algorithm. The ray tracing of the sub-rays can be calculated one by one, for each sub-ray, respectively. The ray tracing of the next sub-ray is started when the ray tracing of the present sub-ray is over. In PSM, the ray tracing of the present sub-ray is over when absorption occurs or the present sub-ray is located outside of the simulation domain. In DSM, the ray tracing is over when the energy of a multiple-reflected ray is almost zero or the ray is located outside of the simulation domain.

PSM, as shown in Fig. 3, can be easily understood by the following points:

  1. 1.

    The events of one sub-ray are calculated by the stochastic approach.

  2. 2.

    If absorption occurs, all the energy of one sub-ray is absorbed and ray tracing is over.

  3. 3.

    In other events, such as reflection, transmission, and scattering, the direction of one sub-ray is changed with all energy of one sub-ray.

  4. 4.

    If one sub-ray is located outside of the simulation domain, ray tracing is over.

  5. 5.

    Repeat this process for the all sub-rays in one ray.

Fig. 3
figure 3

Schematic of the PSM process

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2.2 Methodology

2.2.1 Before reflection and contact checking in DSM

Before reflection, the methodologies of the DSM and PSM are the same. However, the contact checking methods are different.

Ray tracing before reflection is performed as follows:

  1. 1.

    The ray is moved step by step and checked to see whether the ray is located on the surface cell (free surface) or not (Fig. 4a).

  2. 2.

    If the located cell is not the surface cell, then the ray is moved to the next location and the surface cell checking is performed (Fig. 4b).

  3. 3.

    When the location of the ray is the surface cell, contact checking is performed.

    In DSM, the cell is divided into sub-cells for accurate calculation, as shown in Fig. 5. The ray is moved from b to a. As shown in Fig. 5, the term Δl is the sub-cell size, a n is the center of the nth sub-cell, l n is the distance between the center of the n th sub-cell and the ray starting point b, l th is the radius of the smallest sphere which can include the sub-cell, and l n_dis is the smallest distance between the center of the n th sub-cell and the ray direction vector \( \overrightarrow{r}=\overrightarrow{a-b} \). The terms l th and l n_dis are expressed as Eqs. (1) and (2) [9].

    $$ {l}_{\mathrm{th}}=\frac{\varDelta l}{2}\sqrt{3} $$
    (1)
    $$ {l}_{\mathrm{n}\_\mathrm{dis}}={l}_{\mathrm{n}}\times sin\left[ arcos\left(\frac{\left|\overrightarrow{a_{\mathrm{n}}-b}\right|\cdot \left|\overrightarrow{a-b}\right|}{\left|\overrightarrow{a_{\mathrm{n}}-b}\right|\left|\overrightarrow{a-b}\right|}\right)\right] $$
    (2)

    To check whether the ray contacts the surface, the discriminant expressed in Eq. (3) is used.

    $$ {l}_{\mathrm{n}\_\mathrm{dis}}\le {l}_{\mathrm{th}} $$
    (3)

    The sub-cells which satisfy Eq. (3) exist in the cell, and the contact between the ray and the surface occurs subsequently. In the other case, contact does not occur and the ray is moved to the next location, and checking the surface cell and contact is performed again until contact occurs or the ray is located outside of the simulation domain.

Fig. 4
figure 4

Ray tracing before reflection

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Fig. 5
figure 5

Schematic of contact checking in DSM

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2.2.2 Before reflection and contact checking in PSM

Contact checking in PSM is different than in DSM, which uses the discriminant. In PSM, the real contact point is calculated by using the surface equation and ray direction vector equation [12] as shown in Fig. 6.

Fig. 6
figure 6

Schematic of contact checking in PSM

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The surface equation is expressed as Eqs. (4) and (5), and the ray direction vector equation is expressed as Eq. (6).

$$ \varphi \left({x}_1,{x}_2,{x}_3\right)=0 $$
(4)
$$ \begin{array}{l}\varphi \left({x}_1,{x}_2,{x}_3\right)={\varphi}_{000}\left(\varDelta {x}_1-{x}_1\right)\left(\varDelta {x}_2-{x}_2\right)\left(\varDelta {x}_3-{x}_3\right)+{\varphi}_{100}{x}_1\left(\varDelta {x}_2-{x}_2\right)\left(\varDelta {x}_3-{x}_3\right)\\ {}+{\varphi}_{110}{x}_1{x}_2\left(\varDelta {x}_3-{x}_3\right)+{\varphi}_{010}\left(\varDelta {x}_1-{x}_1\right){x}_2\left(\varDelta {x}_3-{x}_3\right)+{\varphi}_{001}\left(\varDelta {x}_1-{x}_1\right)\left(\varDelta {x}_2-{x}_2\right){x}_3\\ {}+{\varphi}_{101}{x}_1\left(\varDelta {x}_2-{x}_2\right){x}_3+{\varphi}_{111}{x}_1{x}_2{x}_3+{\varphi}_{011}\left(\varDelta {x}_1-{x}_1\right){x}_2{x}_3,\end{array} $$
(5)
$$ \left\{\begin{array}{l}{x}_1={X}_1+t{u}_1\\ {}{x}_2={X}_2+t{u}_2\\ {}{x}_3={X}_3+t{u}_3\end{array}\right. $$
(6)

where (x 1, x 2, x3) and (X 1, X 2, X 3) are the contact point and the ray starting point, respectively. The term t is the distance between (x 1, x 2, x 3) and (X 1, X 2, X 3), and (u 1, u 2, u 3) is the direction vector of the ray. The signed distance of node ϕ ijk and the cell size (Δx 1, Δx 2, Δx 3) are shown in Fig. 7. The signed distance of the node is decided as shown in Fig. 8.

Fig. 7
figure 7

Schematic of signed distance

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Fig. 8
figure 8

Decision of signed distance

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The reasons for using the signed distance concept are to solve the surface equation and to reconstruct the free surface. In Piecewise Linear Interface Construction (PLIC) [14], used in VOF, a discontinuous surface can occur, which can block or cause leakage of the ray. Therefore, the surface in PSM is reconstructed as a continuous surface by the Level-Set method.

In Fig. 8, the signed distance ϕ 11, between point P11 and the surfaces of the cells (a, b, c, d), which surround the point P11, is expressed as Eqs. (7a–d).

$$ {\varphi}_{11,\mathrm{a}}=\left({n}_{1,00}\varDelta {x}_{1,0}+{n}_{2,00}\varDelta {x}_{2,0}\right)/2-{d}_{00} $$
(7a)
$$ {\varphi}_{11,\mathrm{b}}=\left(-{n}_{1,10}\varDelta {x}_{1,1}+{n}_{2,10}\varDelta {x}_{2,0}\right)/2-{d}_{10} $$
(7b)
$$ {\varphi}_{11,\mathrm{c}}=\left(-{n}_{1,11}\varDelta {x}_{1,1}-{n}_{2,11}\varDelta {x}_{2,1}\right)/2-{d}_{11} $$
(7c)
$$ {\varphi}_{11,\mathrm{d}}=\left({n}_{1,01}\varDelta {x}_{1,0}-{n}_{2,01}\varDelta {x}_{2,1}\right)/2-{d}_{01} $$
(7d)

where n a,ij is the x a direction component of the surface normal vector for the cell which has the center C ij, d ij is the smallest distance from the center C ij of the cell to the surface in the cell.

The smallest or largest value of ϕ 11 is chosen, and then the signed distance values of the cross points between the surface and cell edges are assumed to be zero. The other signed distance values of each node are decided by the proportion of the distance between the nodes and cross points. If the node is on the surface, the sign is negative, otherwise, positive. In this paper, the largest value of ϕ 11 was used.

To find the contact point (x 1, x 2, x 3), Eq. (8) is solved with variables by using Eqs. (9a and b).

$$ {a}_3{t}^3+{a}_2{t}^2+{a}_1t+{a}_0=0 $$
(8)
$$ \begin{array}{l}{a}_3={u}_1{u}_2{u}_3\varDelta {\varphi}_0,\\ {}{a}_2={u}_1{u}_2\left(\varDelta {\varphi}_0{X}_3-\varDelta {\varphi}_3\varDelta {x}_3\right)+{u}_2{u}_3\left(\varDelta {\varphi}_0{X}_1-\varDelta {\varphi}_1\varDelta {x}_1\right)+{u}_3{u}_1\left(\varDelta {\varphi}_0{X}_2-\varDelta {\varphi}_2\varDelta {x}_2\right),\\ {}{a}_1={u}_1\left(\varDelta {\varphi}_0{X}_2{X}_3-\varDelta {\varphi}_2{X}_3\varDelta {x}_2-\varDelta {\varphi}_3{X}_2\varDelta {x}_3+\varDelta {\varphi}_{23}\varDelta {x}_2\varDelta {x}_3\right)\\ {}\kern1em +{u}_2\left(\varDelta {\varphi}_0{X}_3{X}_1-\varDelta {\varphi}_3{X}_1\varDelta {x}_3-\varDelta {\varphi}_1{X}_3\varDelta {x}_1+\varDelta {\varphi}_{31}\varDelta {x}_3\varDelta {x}_1\right)\\ {}\kern1em +{u}_3\left(\varDelta {\varphi}_0{X}_1{X}_2-\varDelta {\varphi}_1{X}_2\varDelta {x}_1-\varDelta {\varphi}_2{X}_1\varDelta {x}_2+\varDelta {\varphi}_{12}\varDelta {x}_1\varDelta {x}_2\right),\\ {}{a}_0=-{\varphi}_{000}\varDelta {x}_1\varDelta {x}_2\varDelta {x}_3+\varDelta {\varphi}_0{X}_1{X}_2{X}_3+\varDelta {\varphi}_{12}{X}_3\varDelta {x}_1{x}_2\\ {}+\varDelta {\varphi}_{23}{X}_1\varDelta {x}_2{x}_3+\varDelta {\varphi}_{31}{X}_2\varDelta {x}_3{x}_1-\left(\varDelta {\varphi}_1{X}_2{X}_3\varDelta {x}_1+\varDelta {\varphi}_2{X}_3{X}_1\varDelta {x}_2+\varDelta {\varphi}_3{X}_1{X}_2\varDelta {x}_3\right),\end{array} $$
(9a)
$$ \begin{array}{l}\varDelta {\varphi}_0={\varphi}_{000}+{\varphi}_{110}+{\varphi}_{101}+{\varphi}_{011}-\left({\varphi}_{100}+{\varphi}_{010}+{\varphi}_{001}+{\varphi}_{111}\right),\\ {}\varDelta {\varphi}_1={\varphi}_{000}+{\varphi}_{011}-\left({\varphi}_{010}+{\varphi}_{001}\right),\\ {}\varDelta {\varphi}_2={\varphi}_{000}+{\varphi}_{101}-\left({\varphi}_{100}+{\varphi}_{001}\right),\\ {}\varDelta {\varphi}_3={\varphi}_{000}+{\varphi}_{110}-\left({\varphi}_{100}+{\varphi}_{010}\right),\\ {}\varDelta {\varphi}_{12}={\varphi}_{000}-{\varphi}_{001},\\ {}\varDelta {\varphi}_{23}={\varphi}_{000}-{\varphi}_{100},\\ {}\varDelta {\varphi}_{31}={\varphi}_{000}-{\varphi}_{010}.\end{array} $$
(9b)

The sub-ray contacts the surface in the located cell when the following conditions are satisfied:

  1. 1.

    The contact point (x 1, x 2, x 3) should exist in the located cell

  2. 2.

    t should be positive.

2.2.3 Next contact point of reflected ray in DSM after reflection

When contact between the ray and surface occurs, the ray is separated into a reflected ray and non-reflected ray. As already mentioned, in DSM, all the energy of the non-reflected ray is absorbed at the reflection point. In PSM, the non-reflected sub-rays are separated into absorbed sub-rays and transmitted sub-rays. In this section, the method for finding the next contact point of the reflected ray or sub-rays is explained.

The reflected direction of ray \( \overrightarrow{r_1} \) is expressed as Eq. (10) by Fresnel-reflection, where the ray direction vector is before reflection \( \overrightarrow{r}=\overrightarrow{a-b} \) and \( \overrightarrow{N} \) is the surface normal vector. As shown in Fig. 9a, the reflected ray starts from the center of the sub-cell where contact occurs because the exact contact point cannot be calculated in DSM.

Fig. 9
figure 9

Starting point comparison; a DSM and b PSM

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$$ \overrightarrow{r_1}=\overrightarrow{r}-2\left(\overrightarrow{r}\cdot \overrightarrow{N}\right)\overrightarrow{N} $$
(10)

Finding the next contact point is performed as follows:

  1. 1.

    Select the primary candidate cells which satisfy Eq. (3) with \( \overrightarrow{r_1} \) and the cell size as Δl, instead of \( \overrightarrow{r} \) and sub-cell size, respectively (Fig. 10a).

  2. 2.

    Select the secondary candidate cells which satisfy Eq. (3) with the sub-cell size as Δl among primary candidate cells. Then, those cells which do not satisfy Eq. (3), such as the cell ② in Fig. 8, are excluded from the secondary candidate cells (Fig. 10b, c).

  3. 3.

    Finally, select the cell which has the shortest l n, distance between the center of the cell and the reflection point, as the next contact point of the reflected ray (Fig. 10d).

Fig. 10
figure 10

Schematic of finding the next contact point of the reflected ray in DSM

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The reasons for double checking cell size and sub-cell size are to save computational time and to increase the simulation accuracy. Selecting primary candidate cells with cell size can reduce the computational time because the number of calculations with sub-cell size is reduced. If there is no process with cell size, all surface cells should be calculated by Eq. (3) with the sub-cell size. Using the sub-cell size, as already mentioned, can increase the simulation accuracy.

2.2.4 Next contact point of reflected ray in PSM after reflection

In PSM, the reflected direction of the sub-ray is also defined by Fresnel-reflection. The real contact point is obtained Eq. (8). Therefore, the reflected ray starts from the real contact point as shown in Fig. 9b.

The ray tracing after reflection in PSM is almost the same as with the ray tracing before reflection and is performed as follows and as shown in Fig. 11:

  1. 1.

    The sub-ray is moved step by step in the reflected direction, and then checked to see whether this sub-ray is located on the surface cell (free surface) or not.

  2. 2.

    If the located cell is not the surface cell, then the ray is moved to the next location and surface cell checking is performed.

  3. 3.

    When the location of the ray is the surface cell, contact checking is performed.

Fig. 11
figure 11

Schematic of finding the next contact point of the reflected ray in PSM

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2.3 New models in PSM

In PSM, there are two new models, the transmission and scattering models.

2.3.1 Transmission model

When contact and reflection occur, non-reflected sub-rays are divided into absorbed sub-rays and transmitted sub-rays. A sub-ray is not reflected and becomes a non-reflected sub-ray when reflectivity ‘R’ is less than the uniform random number (r) in the range 0 ∼ 1. R is calculated by Fresnel-reflection. Among non-reflected sub-rays, some sub-rays which have ‘r’ greater than the probability of absorption ‘PA’ are transmitted into the workpiece, and the other sub-rays are absorbed in the reflection point. As shown in Fig. 12, the number of non-reflected sub-rays is almost ‘(1-R)N’, the number of absorbed sub-rays is almost ‘PA(1-R)N’, and the number of transmitted sub-rays is almost ‘(1-PA)(1-R)N’, where ‘R’ is reflectivity, if the total number of sub-rays is ‘N’. The next step is performed as follows:

  1. 1.

    The transmitted sub-ray is moved with θ t and the probability of absorption is calculated again.

  2. 2.

    If ‘r’ is greater than ‘PA’, this sub-ray is transmitted again.

  3. 3.

    In the other case, this sub-ray is absorbed at the located cell, and ray tracing is over.

Fig. 12
figure 12

Schematic of transmission model

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‘r’ is updated in every step. The probability of absorption ‘PA’ is expressed as Eq. (11) by Beer-Lambert’s law. θ t is expressed as Eq. (12) by Snell’s law

$$ {P}_{\mathrm{A}}=1- exp\left(- al\right) $$
(11a)
$$ \alpha =4\pi {\kappa}_{2,\mathrm{eff}}/{\lambda}_0 $$
(11b)
$$ {\kappa}_{2,\mathrm{eff}}=q $$
(11c)
$$ {q}^2=\frac{1}{2}\left[\sqrt{{\left({\eta}_2^2-{\kappa}_2^2-{\eta}_1^2{ \sin}^2\theta \right)}^2+4{\eta}_2^2{\kappa}_2^2}-\left({\eta}_2^2-{\kappa}_2^2-{\eta}_1^2{ \sin}^2\theta \right)\right] $$
(11d)
$$ {\eta}_1 \sin \theta ={\eta}_{2,\mathrm{eff}} \sin {\theta}_t=p \tan {\theta}_t $$
(12a)
$$ {\eta}_{2,\mathrm{eff}}=\frac{1}{\sqrt{2}}{\left[\sqrt{{\left({\eta}_2^2-{\kappa}_2^2-{\eta}_1^2{ \sin}^2\theta \right)}^2+4{\eta}_2^2{\kappa}_2^2}+\left({\eta}_2^2-{\kappa}_2^2+{\eta}_1^2{ \sin}^2\theta \right)\right]}^{1/2}=p/ \cos {\theta}_t $$
(12b)
$$ {p}^2=\frac{1}{2}\left[\sqrt{{\left({\eta}_2^2-{\kappa}_2^2-{\eta}_1^2{ \sin}^2\theta \right)}^2+4{\eta}_2^2{\kappa}_2^2}+\left({\eta}_2^2-{\kappa}_2^2-{\eta}_1^2{ \sin}^2\theta \right)\right] $$
(12c)

where η 1 is the refractive index of medium 1 (gas), η 2 and κ 2 are the real and imaginary parts of the complex refractive index of medium 2 (workpiece) [15], λ is the wavelength, and l is the propagation length of the transmitted sub-ray. Because the probability of absorption is almost 100 %, transmission hardly occurs when the workpiece is metal.

Since SS400 is opaque and transmission hardly occurs into a metal workpiece, the workpiece is assumed to be polyimide, a partially transparent opaque material, to validate the transmission effect, and the artificial keyhole shape is assumed to have the different surface gradient through depth direction. α in Eq. (11a) is also assumed to be 14, 1.4, and 0.14 μm−1 to compare the results when α is changed. The purpose of these simulations is the validation of transmission model, and the comparison of the movement and absorption of transmitted rays when α is changed. The simulation conditions except α, therefore, should be the same. However, accurate comparison may be difficult when keyhole shape is changed by melting or vaporization. The workpiece is assumed to be in a solid state and has no phase change to prevent the change of keyhole shapes.

Color contour indicates temperature, and 808 K is the melting temperature of polyimide. As shown in Fig. 13, more transmission occurs when α is smaller. When α becomes smaller, the probability of absorption also becomes smaller by Eq. (11a). Therefore, according to this logic, more transmission and wider temperature distribution occurs with smaller α, and this is a reasonable result.

Fig. 13
figure 13

Results of transmission model with various α; a α = 14 μm−1, b α = 1.4 μm−1, and c α = 0.14 μm−1

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2.3.2 Scattering model

The scattering process is almost the same as the transmission model. The only difference is that the sub-rays are divided by scattering into non-direction changed sub-rays and direction changed sub-rays. As shown in Fig. 14, if the number of sub-rays is ‘N’, the number of non-direction changed sub-rays is almost ‘(1-S)N’ [16] and the number of direction changed sub-rays produced by scattering is almost ‘SN’ when the probability of scattering is ‘S’.

Fig. 14
figure 14

Schematic of scattering model

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The probability of scattering ‘S’ is expressed as Eq. (13) [17] by Beer-Lambert’s law, and scattered angle θ is expressed as Eq. (14) using the Henyey-Greenstein phase function [18, 19].

$$ S=1- exp\left(-{a}_{\mathrm{sca}}l\right) $$
(13a)
$$ {\alpha}_{\mathrm{sca}}=\frac{2{N}_{\mathrm{p}}{\pi}^5{D}^6}{3{\lambda}^4}\left(\frac{{\overline{\eta}}^2-1}{{\overline{\eta}}^2+2}\right) $$
(13b)
$$ \cos \theta =\frac{1}{2e}\left(1+{e}^2-{\left(\frac{1-{e}^2}{1-e+2e\xi}\right)}^2\right) $$
(14)

where N p is the number density of particles, D is the particle diameter, \( \overline{\eta} \) is the complex refractive index, e is the asymmetric parameter (−1 ≤ e ≤ 1), and ξ is the uniform random number in the range 0 ∼ 1.

When the value of e is positive, the probability of forward scattering is higher. On the other hand, the probability of backward scattering is higher when the value of e is negative. If the value of e is zero, scattering occurs with the same probability in all directions (isotropic scattering). When the absolute value of e is almost 0.3, the Henyey-Greenstein phase function can be the approximate expression of the Rayleigh scattering phase function [20]. For clear comparison, α sca in Eq. (13a) is assumed to be 2000 m−1 and for the two e in Eq. (14), the values 0.3 and 0.6 are used. The workpiece is SS400. Due to the same reason of Section 2.3.1, the workpiece for the validation of scattering model is also assumed to be in a solid state and have no phase change to prevent the change of free surface shapes, because the simulation conditions, except e, should be the same.

The color contour indicates absorbed power. As shown in Fig. 15, stronger power absorption is observed near the center of the laser beam when the scattering model is not applied. If the scattering model is applied, the results have a wider distribution of power absorption and weaker power absorption in observed near the center of the laser beam. If a smaller e value is applied, more scattering is observed, as shown in Fig. 15b, c. The scattered angles, cosθ in Eq. (14), with various e values, are shown in Fig. 16. A large e value has a higher possibility of large cosθ value. A large cosθ value indicates a small value of θ. Therefore, the scattered angle θ becomes larger when a smaller e value is applied. According to this logic, the difference between result (b) and (c) is reasonable.

Fig. 15
figure 15

Results of scattering model: a without scattering; b e = 0.3, αsca = 2000 m−1; and c e = 0.6, αsca = 2000 m−1

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Fig. 16
figure 16

Scattered angles with various e

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3 Results and discussion

3.1 Laser keyhole welding simulations with DSM and PSM

The experimental and simulation results are shown in Fig. 17. For experiments, a disk laser with 1.03-μm wavelength was used. Laser power was 3 kW and the focused laser diameter was 200 μm. Welding speed was 0.5 m/min. The workpiece was SS400 with 10 mm thickness. Figure 17a is the result of the experiment, Fig. 17b is the result of the simulation using DSM, Fig. 17c is the result of the simulation using PSM without scattering model, and Fig. 17d is the result of the simulation using PSM with scattering model.

Fig. 17
figure 17

Experimental and simulation results: a experimental result; b simulation result using DSM; c simulation result using PSM without scattering; and d simulation result using PSM with scattering

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As shown in Fig. 17, the simulation results with DSM and PSM have almost the same bead shape, and there is only a small difference between the DSM and PSM in the bottom part of the bead shape. In these simulations, a 200 μm cell size was used due to the limitations of computational memory and time. If a sufficiently smaller cell size is used, the difference in simulation accuracy may become larger because the real contact point is calculated in PSM, it is a more physical method of numerical ray tracing than DSM as already mentioned in Section 2. As shown between Fig. 17c, d, the scattering effect seems to be minor because two results are almost the same. The number of scattered sub-ray is small. PSM has a faster speed of calculation than DSM. DSM needs almost double the calculation time of PSM (DSM = 55 h, PSM without scattering = 24 h, and PSM with scattering = 38 h). When multiple-reflection is calculated in DSM, Eq. (3) is calculated for all surface cells, and this process causes longer calculation time. And scattering model also causes longer calculation time.

4 Conclusion

The conclusions obtained in this paper are summarized as follows:

  1. 1.

    The procedures for DSM and PSM were introduced.

  2. 2.

    The differences between the two methods are as follows:

    • PSM uses a real equation of the surface and ray direction vector, while DSM uses a discriminant.

    • The reflected ray is moved step by step in PSM, while only once by the discriminant in DSM.

    • The reflected ray starts from the real contact point in PSM, but from the center of the contacted sub-cell in DSM.

    • PSM can depict the multi-path of the ray due to the set of sub-rays concept. On the other hand, DSM cannot depict the multi-path of the ray due to the single ray definition

  3. 3.

    PSM is a more physical numerical method and depicts various phenomena, such as transmission and scattering.

  4. 4.

    The PSM was validated to include the models of the transmission and scattering for numerical simulations of the laser keyhole welding

  5. 5.

    The simulation results with DSM and PSM have almost the same bead shape in the laser welding of mild steels, while PSM has a faster speed of calculation.

  6. 6.

    Scattering model seems to have minor effect in laser keyhole welding simulations.