Parametric Study of Inverse Heat Source Model Based on Molten Pool Morphology for Selective Laser Melting

Abstract

Selective laser melting is a commonly employed additive manufacturing technique that facilitates the fabrication of intricate geometries through the laser-induced melting of powder materials. The quality of the produced parts is significantly influenced by the molten pool morphology, which is affected by parameters such as laser power, scanning rate, and powder characteristics. However, the selection of unknown parameters within the heat source model significantly impacts the simulation outcomes and must be carefully considered. This study addresses this issue by proposing an inversion method for accurately determining the parameters of the Goldak double ellipsoid heat source model using molten pool morphology as a reference. A pattern search algorithm combined with Bayesian inference was employed to invert and estimate the heat source parameters. The results demonstrated that the inversed parameters significantly improved the prediction accuracy of molten pool geometry. The inverse parameters χ₀, χ₁, and χ₂ were 1.17, 1.00, and 2.08, respectively. The study provides valuable insights into the use of image-based methods for parameter inversion and offers a more reliable tool for improving the precision of simulations. These findings have important implications for optimizing processing conditions and enhancing the overall quality of additively-manufactured components.

1. Introduction

Selective laser melting (SLM) is a prominent additive manufacturing technology capable of producing components with complex geometries. This technique utilizes a laser heat source to rapidly fuse micron-scale powder particles, which then solidify in a layer-by-layer manner along a pre-defined geometric path, ultimately completing the part fabrication process [1]. SLM is widely used in industries such as aerospace, transportation, and biomedical engineering [2,3,4,5]. However, the SLM process is accompanied by challenges such as high cooling rates, melt pool dynamics, and residual stresses, which can lead to deformation and pores in the finished parts, ultimately reducing their service life [6,7]. The formation of defects during additive manufacturing is closely linked to the temperature history during the preparation phase [8,9,10]. The accuracy of these predictions is heavily dependent on the heat source model employed, making it a critical area of research. Developing accurate heat source models and determining their parameters are vital to improving the prediction reliability in SLM.

Numerical methods are essential for modeling and analyzing thermal distribution, molten pool morphology, residual stresses, and the evolution of the solidified structure [10,11,12,13]. High simulation accuracy and efficiency are necessary for these analyses. Among the key factors influencing prediction accuracy, the heat source model and its parameters play a crucial role. Currently, common heat source models used in simulations include the surface heat source [14], Gaussian body heat [15], Goldak double ellipsoid heat source [16], cylindrical heat source [17], and conical heat source models [18]. In laser powder bed fusion (L-PBF), where the laser diameter is typically 100 μm and the powder layer thickness is on the micrometer scale, many studies have used a surface heat source model to represent the heat input [19]. Fu et al. [20] applied a surface heat source model to predict the molten pool size of a laser-prepared Ti6Al4V alloy. However, their results showed that the predicted molten pool width and depth were smaller than those observed experimentally, primarily due to the limited energy penetration of the surface heat source. Similar conclusions were drawn in another study [21]. Moreover, Li et al. [12] found that the experimental molten pool depth exceeded the predicted values based on the surface heat source model. In contrast, the body heat source model, with its superior energy penetration capacity, predicts greater depths of laser energy penetration. Ding et al. [22] proposed a model for predicting the molten pool temperature using Gaussian and double ellipsoidal heat sources. However, the relationship between these inputs and the molten pool geometry has not been experimentally validated. Chukkan et al. [23] improved the predictions for the laser welding process of 316 L stainless steel by using a conical-cylindrical composite heat source model. These findings highlight the significant relationship between molten pool geometry and pore formation, offering an effective method for studying material melting mechanisms [9,24,25]. Furthermore, the depth-to-width ratio of the molten pool is crucial for identifying lock-hole regions, making molten pool morphometry an essential factor for defect prediction in additively manufactured components.

The double ellipsoid heat source model involves multiple parameters that govern the distribution of energy within the material. However, discrepancies between analytically derived heat source parameters and experimental results have prompted significant interest in accurately identifying these parameters. Bai et al. [26] introduced an equivalent heat source model based on thermal-physical behavior analysis to examine the impact of solid phase changes and other influencing factors on the mechanical properties. They used the response surface method to optimize the heat source parameters for specific welding conditions. Similarly, Chen et al. [27] employed an optimization algorithm to investigate the temperature model for laser-welded plates, using inverse techniques to minimize discrepancies between the simulated and experimental molten pool dimensions. Li et al. [28] optimized their heat source parameters by minimizing an objective function based on welding microstructure images. Zhang et al. [29] used peak temperatures at key points along the molten pool contour to determine absorption coefficients for their heat source model. However, typical SLM settings involve a confined environment, complicating accurate temperature measurements. Moreover, the use of infrared spectroscopy often requires expensive custom equipment. As a result, there is a growing need for cost-effective and convenient methods to determine the heat source model parameters. To standardize inverse parameter problems, it is essential to validate the uncertainty associated with these parameters. Jakkareddy et al. [30] used liquid crystal thermography to obtain steady-state temperatures, applying the Metropolis–Hastings Markov Chain Monte Carlo (MH-MCMC) algorithm based on Bayesian inference to address inverse problems related to boundary conditions, such as convection and heat transfer coefficients, and to analyze associated uncertainties. Parthasarathy and Balaji [31] performed a numerical study on unsteady-state heat conduction problems, utilizing MH-MCMC based on Bayesian inference to invert parameters like emissivity, thermal conductivity, and convective coefficients, while conducting a comprehensive uncertainty analysis. Kumar et al. [32] investigated the sensitivity of temperature estimates to thermophysical parameters and heat generation, validating the accuracy of the estimated parameters. Therefore, uncertainty analysis is critical for inverse parameter studies.

The objective of this study was the development of a high-precision prediction model by means of the fusion of Bayesian inference and pattern search algorithms. It was based on experimental molten pool morphology, with the purpose of inverting the modified parameters (χ₀, χ₁, χ₂) of the Goldak double ellipsoid heat source model. This method has the potential to markedly reduce the experimental cost of molten pool morphology detection and provide a theoretical basis for optimizing process parameters (e.g., laser power, scanning path), thus enhancing the forming quality and reliability of complex components.

2. Simulation and Experiment

2.1. Forward Model

To ensure the accuracy of the resulting calculations, the model was based on the following theoretical assumptions:

(1)

The powder particle layer was treated as a continuous medium, with volume changes during processing being neglected.

(2)

Thermophysical parameters, including density, thermal conductivity, and specific heat capacity, are sensitive to temperature variations and are influenced by the treatment of the temperature field during thermal conduction.

(3)

The convection heat transfer coefficient and surface emissivity were considered constant and independent of temperature.

(4)

The laser was modeled as a heat source.

2.1.1. Thermophysical Parameters

The thermophysical parameters were determined based on previous studies [33]. For SLM, more accurate calculations are required due to the reduction in thermal conductivity caused by high-energy laser reflection phenomena among the powder particles. Additive manufacturing involves a range of complex physical processes, and simplifications are employed to reduce the overall complexity and computational burden. The relationship between powder layer density, thermal conductivity, and porosity is simplified in Equations (1)–(3) [33].

$$\varphi ={\displaystyle \frac{{\rho }_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{l}}-{\rho }_{\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{d}\mathrm{e}\mathrm{r}}}{{\rho }_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{l}}}}$$

(1)

$${\rho }_{\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{d}\mathrm{e}\mathrm{r}}=\left(1-\varphi \right){·\rho }_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{l}}$$

(2)

$$k_{\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{d}\mathrm{e}\mathrm{r}}=\left(1-\varphi \right){·k}_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{l}}$$

(3)

where ρ_material is the solid density, ρ_powder is the density of the material in the powder state, k_material is the solid thermal conductivity, k_powder is the powder thermal conductivity, and the powder layer porosity is set as ϕ = 0.4. The thermophysical parameters of the molten phase can be determined by means of Equations (4)–(8) [33].

$$\rho =\left({\rho }_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{l}}-{\rho }_{\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{d}\mathrm{e}\mathrm{r}}\right)f_L+{\rho }_{\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{d}\mathrm{e}\mathrm{r}}$$

(4)

$$f_L=\left\{\begin{array}{c}\begin{array}{cc}0& T<T_S\end{array}\\ \begin{array}{cc}{\displaystyle \frac{T-T_S}{T_L-T_S}}& T_S\le T\le T_L\end{array}\\ \begin{array}{cc}1& T_L<T\end{array}\end{array}\right.$$

(5)

$$k=\left(k_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{l}}-k_{\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{d}\mathrm{e}\mathrm{r}}\right)f_L+k_{\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{d}\mathrm{e}\mathrm{r}}$$

(6)

$${\alpha }_m={\displaystyle \frac{f_L{\rho }_L-\left(1-f_L\right){\rho }_P}{2\left[f_L{\rho }_L+\left(1-f_L\right){\rho }_P\right]}}$$

(7)

$$c_p={\displaystyle \frac{1}{\rho }}\left[\left(1-f_L\right){\rho }_{\mathrm{p}\mathrm{o}\mathrm{w}\mathrm{d}\mathrm{e}\mathrm{r}}{c}_s+f_L{\rho }_L{c}_L\right]+{\displaystyle \frac{L\partial {\alpha }_m}{\partial T}}$$

(8)

where ρ is the density of the powder layer, f_L is the percentage of liquid phase in molten zone, T_L is the temperature attained at full liquidity, and T_S is the temperature at which powder begins to dissolve (see Equation (5)) [33]. k is the molten state thermal conductivity, c_p is the specific heat capacity, L is the latent heat of phase transition, and α_m is the mass fraction (see Equation (7)) [33]. The thermophysical parameters used in the calculations were derived from previous research [33].

2.1.2. Boundary Condition

It can be observed that a nonlinear temperature transfer is a typical characteristic of SLM. The finite element model is calculated in accordance with the energy balance equation [34,35].

$${\displaystyle \frac{\partial \left(\rho c_{P}T\right)}{\partial t}}+\overrightarrow{u}·\nabla \left(\rho c_{P}T\right)=\nabla ·\left(k·\nabla T\right)+q$$

(9)

To define the boundary conditions of the numerical model, the mathematical framework can be expressed as follows [34,35]:

$$q=k{\left.{\displaystyle \frac{\partial T}{\partial z}}\right|}_{z=d}+q_c+q_r+q_{evap}$$

(10)

where d denotes the height from the substrate surface; q_c is the heat loss due to convection; q_r is the surface radiation; q_evap is the heat of evaporation [34,35].

$$q_c=H_c(T-T_0)$$

(11)

$$q_r=\sigma \epsilon (T^4-T_0^4)$$

(12)

where H_c defines a constant natural convection coefficient (10 W/m²), σ is the Stefan–Boltzmann constant (5.67 × 10⁻⁸ W/(m²·K⁴)), and ε is the material emissivity (0.65) [34,35]. T₀ is the ambient temperature (293 K).

Evaporation occurs on metal surfaces at elevated temperatures, resulting in mass transfer between the molten material and the external environment. The evaporation process is dependent upon the material surface temperature within an external inert gas environment. For the purposes of analysis, the mass loss due to evaporation was disregarded, and the heat loss of molten material can be described by Equation (13) [33,34,35].

$$q_{evap}={\displaystyle \frac{0.82HM}{\sqrt{2\pi MRT}}}{p}_0{e}^{\left({\displaystyle \frac{HM(T-T_b)}{RT{T}_b}}\right)}$$

(13)

where H refers to the enthalpy of evaporation [8], M is the molar mass of the metal, R is the universal gas constant (8.314 J·mol⁻¹·K⁻¹), and p₀ is the saturated vapor pressure. The energy absorbed by the metal from laser radiation leads to its melting [33,34,35].

$$\left\{\begin{array}{c}\begin{array}{cc}{q}_f\left(x,y,z\right)={\displaystyle \frac{6\sqrt{3}{\chi }_0{f}_{f}P}{a_{f}bc\pi \sqrt{\pi }}}{e}^{\left(-{\displaystyle \frac{3{\left(x-x_0\right)}^2}{{{(\chi }_2{a}_f)}^2}}-{\displaystyle \frac{3{\left(y-y_0\right)}^2}{{\left({\chi }_{1}b\right)}^2}}-{\displaystyle \frac{3z^2}{{\left({\chi }_{2}c\right)}^2}}\right)}& x\ge x_0\end{array}\\ \begin{array}{cc}{q}_r\left(x,y,z\right)={\displaystyle \frac{6\sqrt{3}{\chi }_0{f}_{r}P}{a_{r}bc\pi \sqrt{\pi }}}{e}^{\left(-{\displaystyle \frac{3{\left(x-x_0\right)}^2}{{{(\chi }_2{a}_r)}^2}}-{\displaystyle \frac{3{\left(y-y_0\right)}^2}{{\left({\chi }_{1}b\right)}^2}}-{\displaystyle \frac{3z^2}{{\left({\chi }_{2}c\right)}^2}}\right)}& x<x_0\end{array}\end{array}\right.$$

(14)

where the powder layer thickness is 30 µm and P is the laser power. The values of the front melt length (a_f), back melt length (a_r), melt width (b), and melt depth (c) as well as the front and back energy ratios (f_f and f_r) in the double ellipsoid heat source model were determined based on previous research [33]. χ_i (i = 0, 1, 2) is the inversed parameters.

2.1.3. Physical Model

Figure 1 shows the finite element model and model schema. To reduce the calculation effort, the model size was set to 1000 × 600 × 300 μm. The computational domain was divided into two regions: a refined ortho-hexahedral mesh was employed in the laser processing area to accurately capture the complex temperature field, while an ortho-tetrahedral mesh was used in the remaining regions.

2.2. Inversed Model

The study utilized a Bayesian inference method to estimate parameters. The Bayes theorem states that the posterior probability density function of an estimate is proportional to its likelihood density function. The conditional probability of the estimated vector L given the measured data Y is calculated using Bayes’ formula, as shown below [36,37,38]:

$$P\left(LǀY\right)={\displaystyle \frac{P\left(YǀL\right)\times P\left(L\right)}{P\left(L\right)}}={\displaystyle \frac{P\left(YǀL\right)\times P\left(L\right)}{\int P\left(YǀL\right)\times P\left(L\right)dL}}$$

(15)

where P(LǀY) is the posterior probability density function (PPDF), P(L) is the prior density function, and P(YǀL) is the likelihood density function obtained through a comparison of the experimental and simulated molten pool morphology.

$$P\left(YǀL\right)={\displaystyle \frac{1}{{\left(\sqrt{2\pi {{\sigma }_1}^2}\right)}^n}}{e}^{-\left({\displaystyle \frac{{\tau }^2}{2{\sigma }^2}}\right)}$$

(16)

where σ₁ is the uncertainty between the measured and forward model, n is the measured dimension (A_means) [36], P(L) is shown in Equation (17), and the mean and standard deviation of the estimates are μ_p and σ_p.

$$P\left(L\right)={\displaystyle \frac{1}{{\left(\sqrt{2\pi {\sigma }_p^2}\right)}^n}}{e}^{-\left({\displaystyle \frac{{(L-{\mu }_p)}^2}{2{\sigma }_p^2}}\right)}$$

(17)

Combined with Equations (15)–(17), the PPDF can be expressed as [36,37,38]:

$$P\left(LǀY\right)={\displaystyle \frac{e^{-(\frac{{\tau }^2}{2{\sigma }^2}+\frac{{(L-{\mu }_p)}^2}{2{\sigma }_p^2})}}{\int -(\frac{{\tau }^2}{2{\sigma }^2}+\frac{{(L-{\mu }_p)}^2}{2{\sigma }_p^2})dL}}$$

(18)

In this study, we used the pattern search method in conjunction with Bayesian inference to dynamically generate samples of the parameters (L). Combined with the literature [36,37], the specific steps for the sample generation in parameter estimation are as follows:

• Initialize parameters: Initialize the parameter vector L¹ = ($L_1^1$, $L_2^1$,…, $L_{{\mathrm{n}}^{\prime }}^1$), where n′ is the dimension of the parameter.
• Sampling and optimization: For i = 1, 2, … M (maximum number of iterations), the parameter dimensions j = 1, 2, …, n′.

  (a)

  Generate a random number v~U (0,1).

  (b)

  Create a candidate sample L_j^∗~N ($L_{\mathrm{i}}^{\mathrm{j}}$, ${\sigma }_{L_{\mathrm{j}}^{\mathrm{i}}}^2$) based on the current sample and uncertainty ${\sigma }_{L_{\mathrm{j}}^{\mathrm{i}}}^2$.

  (c)

  P(LǀY) evaluates the objective function on a random sample of points L^∗ = ($L_1^{\mathbf{*}}$, $L_2^{\mathbf{*}}$,…, $L_{{\mathrm{n}}^{\prime }}^{\mathbf{*}}$).

  (d)

  Based on Balaji’s study [37,38], the acceptance rates can be expressed as:

$$\alpha =\min (1,{\displaystyle \frac{P\left(L^{\mathbf{*}}ǀY\right)}{P\left(L_{\mathrm{i}}ǀY\right)}})$$

(19)

If v > α, accept L_i+1 = L^∗; otherwise (v ≥ α), go to step 2 with L_i+1 = L_i

3.

Termination: Terminates the computation when the number of iterations reaches M or the objective function value converges.

2.3. Experiment

The samples were produced using SLM equipment (BLT-S210, manufactured by Bright Laser Technologies, with the source of the equipment being Xi’an, China) in an Ar environment to prevent oxidation of the material. The material for the experiment was Ti6Al4V (~53 μm), 50 wt.% β-Ti alloy powder (~53 μm), and 0.25 wt.% Fe₂O₃ nanoparticles (~30 nm), which were mechanically mixed by a ball mill [33]. Figure 2b,c exhibits the microscopic morphology of the powder particles. Figure 2d illustrates the samples prepared by SLM. As illustrated in

Table 1. Process parameters.

Number	Laser Power (W)	Scanning Rate (mm/s)	Layer Thickness (μm)
Sample 1	150	800	30
Sample 2	150	1000	30
Sample 3	150	1200	30
Sample 4	150	1400	30
Sample 5	175	1200	30
Sample 6	180	1200	30

, the process parameters for single-pass specimen preparation were as follows: three specimens were prepared for each process parameter, and three measurements were performed for each specimen. This approach was adopted to enhance the reliability of the experiment.

Table 2. Compositions of the Ti6Al4V and β-Ti alloys.

Alloying Elements	O	N	H	Fe	Al	V	Ti
Composition of Ti6Al4V (wt.%)	0.0961	0.0072	0.003	0.1626	6.19	4.26	Bal.
Composition of β-Ti (wt.%)	0.04	0.015	0.002	0.2	-	-	Bal.

shows the mass fraction of each element for the raw materials (Ti6Al4V and β-Ti). In conclusion, the molten pool of specimens was examined using scanning electron microscopy (SEM, SU5000) manufactured by Hitachi High-Tech Corporation, with the source of the equipment being Dalian, China.

3. Analysis of Inversed Parameters

3.1. Process of Inversed Parameters

It is essential that the inversed parameters are accurately defined. Currently, the primary techniques for determining the uncertain parameters of the double ellipsoid heat source model include trial-and-error, fitted formulae, and inversion methods. This paper presents a mathematical model based on the inversion of molten pool images that can be extended to study other materials and models. Accordingly, the problem of inversed parameters under investigation is formulated in Equation (20), with a mathematical model involving design variables, an objective function, and constraints.

$$\begin{gathered} \mathrm{Min:} f\left({\chi }_0,{\chi }_1,{\chi }_2\right)=\tau \\ \begin{array}{cc}\mathrm{Subject\; to:}& 0\le {\chi }_0\le 5\hfill \\ & 0\le {\chi }_1\le 10\\ & 0\le {\chi }_2\le 10\end{array} \end{gathered}$$

(20)

The inversion problem involves the parameters χ₀, χ₁, and χ₂. The inversion process is illustrated in Figure 3, and the steps are as follows:

(1)

The parameters are initially set to values (χ₀′, χ₁′, χ₂′), and then substituted into the heat source model to calculate the temperature distribution.

(2)

The molten pool morphology obtained by SEM (Figure 4(a1)) is used to create a binary image in MATLAB 2021 (Figure 4(a2)). Similarly, the temperature field obtained from the model calculations (Figure 4(b1)) is processed in the same manner, resulting in a binarized image (Figure 4(b2)).

(3)

The matrices of the two binary maps are extracted and compared to analyze their differences.

(4)

The temperature field is recalculated by applying the pattern search method based on Bayesian inference (as detailed in Section 2.2) to update and iterate the parameter values (χ₀, χ₁, χ₂). Once the required conditions are met, such as the error satisfying the necessary criteria or the maximum number of iterations being reached, the iteration process is terminated, and the final parameter values are output.

It is crucial to emphasize that the experimental images are aligned with the temperature map in terms of dimensions, spatial coordinates, and resolution, ensuring the accuracy and consistency of the results.

To analyze the error between the experimental and simulation results, it is essential to define the error within the mathematical model. The image was then subjected to binary processing, where the pixel value at any given point within the molten pool region was set to 255, and the pixel value outside the molten pool was set to 0 (see Equation (21)). For laser power P = 200 W and scanning rate V = 1200 mm/s, the molten pool image (Figure 4(a1,b1)) were binarized to obtain the morphology, as illustrated by the white region in Figure 4(a2,b2). Next, the pixel values of the molten pool (Figure 4(a2)) were extracted to form a matrix A₁ (see Equation (22)), and the same operation was performed on the simulated image (Figure 4(b2)) to obtain a matrix A₂ (Equation (23)). The laser processing area can be classified into the molten, heat-affected, and matrix zones, with the molten pool boundary being critical in determining the accuracy of the inversion. The pixel values of the molten pool differed from those of other areas in the image. In this paper, using Avizo 2019 software, the threshold value for the pixel points at the molten pool boundary was determined to be 118. The image was then filtered with a threshold value of 45 to ultimately define the molten pool boundary. (Figure 4d). During matrix operations, any negative values were replaced with 0, ensuring that no negative value remained in the matrix obtained during the phase subtraction operation (Equations (24) and (25)). A value of f_ij = 0 indicates no difference, whereas a non-zero value of f_ij signifies an error between the simulated and experimental results. The number of pixel points in the molten pool image was experimentally obtained by counting the pixel points, resulting in I₀ = 3630. Therefore, the error between the experimental and simulated results can be calculated using Equation (26). The white areas in Figure 4c represent the errors between the experimental and simulated results.

$$A_l=a_{ij}=\left\{\begin{array}{c}\begin{array}{cc}0& outside\end{array}\\ \begin{array}{cc}255& inside\end{array}\end{array}\right.$$

(21)

$$A_1=\left[\begin{array}{cc}255& 255\\ 255& 0\end{array}\right]$$

(22)

$$A_2=\left[\begin{array}{cc}255& 0\\ 0& 255\end{array}\right]$$

(23)

$$f_1=A_1-A_2=\left[\begin{array}{cc}0& 255\\ 255& 0\end{array}\right]$$

(24)

$$f_2=A_2-A_1=\left[\begin{array}{cc}0& 0\\ 0& 255\end{array}\right]$$

(25)

$$f=f_1+f_2=\left[\begin{array}{cc}0& 255\\ 255& 255\end{array}\right]$$

(26)

$$\tau =\sum _{i=1}^{512}\sum _{j=1}^{512}{\displaystyle \frac{\left[f_{ij}\right]}{255\times I_0}}$$

(27)

3.2. Estimation of Inversed Parameters (χ₀, χ₁, χ₂)

Figure 5 illustrates the relationship between the number of iterations and the error, which remained around 15% after 100 iterations. The inversed parameters for Equation (13) were as follows: χ₀ = 1.17, χ₁ = 1, and χ₂ = 2.08. Sensitivity analysis (SA), based on probability and mathematical statistics, aimed to provide insights into the influence of the heat source model parameters on the discrepancies between the simulation and experimental results. The parameters for the SA were as follows: (1) Factor: Inversed parameters (χ₀, χ₁, χ₂); (2) Levels: Each parameter was varied by ±15% of its final value; (3) Response: The objective function f (χ₀, χ₁, χ₂) was chosen as the response function; (4) Calculation framework: The number of calculations was set to 7, with the selected parameters listed in

Table 3. Sensitivity analysis.

Test	χ0	χ1	χ2
T1	1.17	1	2.08
T2	1.17	1	1.77
T3	1.17	1	2.39
T4	1.17	0.85	2.08
T5	1.17	1.15	2.08
T6	0.99	1	2.08
T7	1.52	1	2.08

.

As illustrated in Figure 6, when the inversed parameters (χ₀ = 1.17, χ₁ = 1, and χ₂ = 2.08) were used as the baseline, adjusting any one of these variables resulted in a corresponding increase in the errors of the others. A variation in χ₁ and χ₂ led to an error of approximately 35%. In contrast, a 15% reduction in χ₀ resulted in an error of 71.87%, demonstrating that χ₀ had the greatest influence on the error.

Similar to the mesh independence study in the finite element method, it is essential to investigate the impact of samples on the parameters.

Table 4. The effect of the number of samples on the parameters (χ₀, χ₁, χ₂).

Number of Samples	Parameters	Actual	Mean	MAP	SD
100	χ0	1.17	1.193	1.386	0.313
	χ1	1	1.025	0.956	0.223
	χ2	2.08	2.214	2.481	0.567
150	χ0	1.17	1.282	1.234	0.116
	χ1	1	1.113	1.165	0.131
	χ2	2.08	2.270	2.423	0.256
200	χ0	1.17	1.208	1.210	0.026
	χ1	1	0.857	0.821	0.024
	χ2	2.08	2.153	2.262	0.043

presents the effect of various samples on the mean, maximum a posteriori (MAP), and standard deviation (SD) of the estimated parameters. The results indicate that the SD decreased as the samples increased, which aligns with findings reported in the literature [32]. The posterior probability density functions of the parameters (χ₀, χ₁, χ₂) are shown in Figure 7.

3.3. Estimation of Parameters (P, V)

The purpose of the uncertainty analysis is to quantify the uncertainties in the measured results and the derived quantities [30,39]. In the SLM process, the calculation of output melt pool morphology is represented by the following equation:

$$A=f(P,V)$$

(28)

Once the processing parameters are determined, it is essential to ensure a stable heat input during SLM. Meanwhile, the limitations of the image noise and processing algorithms must also be considered. The pixel matrix A_measure extracted from the experimental results may exhibit an error ΔA when compared with the true matrix A_true [30,39], as follows:

$${\Delta A=A}_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{e}}-A_{\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{e}}$$

(29)

The input parameter uncertainty can be determined by using the error propagation formula as follows:

$${\sigma }_A=\pm \sqrt{{\sum }_{i=1}^N{\left({\displaystyle \frac{\partial A}{\partial p}}{\sigma }_p\right)}^2}$$

(30)

where p represents the measured value, σ_A is the uncertainty in the derived quantity, and σ_p is the error in the measured quantity. Combined with the literature [39], the uncertainties in the measured and derived quantities are presented in

Table 5. Uncertainty in the measured primary quantities and derived quantities.

Quantities	Uncertainty
Power	±10 W
Velocity	±60 mm/s
Threshold value	±5.9

.

The process parameters (P and V) were further estimated. The molten pool morphology at P = 200 W and V = 1200 mm/s was determined through simulation.

Table 6. The effect of the number of samples on power.

Number of Samples	Parameters	The Actual Value of P (W)	The Mean Value of P (W)	The MAP Value of P (W)	The SD Value of P (W)
100	P	200	207.036	217.489	4.120
150	P	200	206.771	205.602	3.951
200	P	200	206.730	208.858	3.884

and

Table 7. The effect of the number of samples on velocity.

Number of Samples	Parameters	The Actual Value of V (mm/s)	The Mean Value of V (mm/s)	The MAP Value of V (mm/s)	The SD Value of V (mm/s)
100	V	1200	1256.667	1243.333	32.510
150	V	1200	1255.833	1173.343	31.667
200	V	1200	1259.166	1246.667	30.141

show the impact of samples on the parameters P and V. It can be observed that the SD decreased as the samples increased. The posterior probability density functions of P and V are shown in Figure 8. The molten pool dimensions, specifically depth and width, were used to visually characterize the morphology. By employing the estimated parameters (χ₀, χ₁, χ₂), P, and V (as listed in Table 6 and Table 7), the molten pool morphology was predicted using the forward model.

4. Results and Discussion

4.1. Model Analysis

Figure 9 illustrates the temperature distribution for the process parameters P = 200 W and V = 1200 mm/s. The dark red region represents areas where the temperature exceeded the melting point, indicating that the metal is in a liquid phase. The molten pool morphology is typically characterized by its length (a), width (b), and depth (c). These dimensions are critical metrics for assessing the reliability and accuracy of the model predictions [12,15,19].

The revised Goldak double ellipsoid heat source model (E-model) was employed for the temperature predictions, and the corresponding results are presented. Figure 10 depicts the temperature distribution predicted by the E-model. A characteristic ‘comet tail’ phenomenon was observed at the rear of the molten pool (Figure 10a). A similar phenomenon has also been documented in the literature [12,15].

To further analyze the heat source models, three paths were designed based on Figure 1a: the x-path along the x-axis, the y-path along the y-axis, and the z-path along the z-axis. Figure 11 illustrates the evolution of the temperature and temperature gradient along the x-, y-, and z-paths. The criterion for temperature gradient was formulated as 10⁷ K/m [15].

4.2. Experimental Validation

Figure 12 presents a comparison of the experimental and simulated molten pool dimensions. The experimental results indicate that the molten pool dimensions tended to increase with an increase in energy input, and the model results were consistent with this observation. The comparison of molten pool size results from the experiments and simulations showed that the error between them did not exceed ±5 μm, indicating that the E-model demonstrated satisfactory prediction accuracy.

We visualized the molten pool morphological error between the heat source model and experiments. Figure 13 illustrates the morphological discrepancies between the experimental and the heat source model. At P = 200 W and V = 1200 mm/s, the morphological error was τ_E-model = 14.94%. Combining the experimental and simulation results revealed that the E-model exhibited significantly predictive accuracy. This demonstrates that the E-model, when reversed using experimental molten pool images, can accurately predict the molten pool morphology, dimensions, and temperature.

5. Conclusions

This study addressed the challenge of molten pool morphology prediction in SLM by developing a mathematical model for the inversed heat source parameters. The inversed parameters for the E-model were estimated. These inversed parameters were then applied to the forward model to predict the melt pool morphology, with the results validated experimentally. The findings can be summarized as follows:

(1)

An accurate inversion of the modified parameters of the E-model (χ₀ = 1.17, χ₁ = 1.00, χ₂ = 2.08) was achieved through a combination of Bayesian inference and pattern search algorithms. The result was a reduction in the error between the experimental and simulated molten pool morphology to 14.94%, and the error of the key dimensions (the molten width and depth) was stabilized to within ±5 μm.

(2)

The results of the sensitivity analysis demonstrate that the error in the molten pool morphology increased to 71.87% for a 15% change in the value of χ₀ = 1.17, which was much higher than for χ₁ = 1.00 and χ₂ = 2.08.

(3)

The posterior probability density function for 200 samples was obtained as χ₀ = 1.208 ± 0.026, χ₁ = 0.857 ± 0.024, χ₂ = 2.153 ± 0.043. This proves the convergence of the parameters, provides strong support for the robustness of the model, and accelerates the rapid calibration of the parameters in industrial applications.

(4)

The geometries predicted by the E-model at P = 200 W and V = 1200 mm/s demonstrated a high degree of agreement with the experimental results. In comparison with the earlier findings on the prediction of melt pool geometry, the present model demonstrated an error margin of less than 5% in the estimation of the dimensions of the molten pool.

The research method can be extended to other materials (Ti, Al, Mg alloys, etc.) and multi-pass process molten pool morphology research. Furthermore, the combination of machine learning with this method has the potential to optimize the parameters. By quantifying the uncertainty, this method can establish the foundation for process design, effectively reduce the cost of process trial and error, and accelerate the industrial application of additive manufacturing technology.