Multi-Track Melt Pool Width Modeling in Powder Bed Fusion Additive Manufacturing

Additive manufacturing (AM) differs from conventional subtractive machining as it creates a part by adding material layer by layer, directly from a digital model. Powder bed fusion (PBF) is a specific AM technique that uses high-precision lasers or electron beams as the energy source to fuse polymeric or metallic powder materials together. While PBF has revolutionized the fabrication of complex parts, there are still challenges to its wider adoption. These challenges include issues with reliability and in-process variations caused by uncertain laser-material interactions, environmental vibrations, powder recycling, imperfect interactions of mechanical components, and the recursive thermal histories of materials [1–5].

In PBF, a typical part is built from thousands of thin layers, as shown in Fig. 1. Each layer is created by regulating the energy beam to follow trajectories predetermined in a slicing process based on the part geometry. Once a layer is finished printing, a new thin layer of powder is spread on top, and the process repeats. Modeling this complex dynamic system (Fig. 1) is crucial for understanding and controlling PBF and related techniques. Researchers use finite element modeling to explore energy deposition mechanisms, and control-oriented modeling to build mathematical models that can regulate in-process variations. For instance, Refs. [1,6–8] adopt finite element modeling to investigate the effects of various scan configurations on the thermal fields of powder bed, the geometries of melt pool, and the mechanical properties of printed parts. In control-oriented modeling, Refs. [9–12] employ the low-order system models and further build the nonlinear submodels to cover more process dynamics. Based on these models, subsequent control algorithms such as PID control [13], sliding mode control [11], predictive control [9], repetitive control [2,14], iterative learning-based control [15], and iterative simulation-based control [4,16] have proven effective in improving the dimensional accuracy of printed parts.

This paper presents a novel approach to modeling and examining PBF by combining finite element modeling and control-oriented modeling. First, we develop a finite element model (FEM) to look into the intricate thermal interactions that occur during the PBF process. The FEM then serves as a simulation platform for gathering data and identifying parameters for the proposed modeling schemes. In contrast to the typically used low-order system models, we develop a physics-based analytical model for control-oriented modeling that accounts for the complex dynamic behavior of melt pool width during multi-track PBF process. The proposed control-oriented multi-track model is formulated by applying superposition to a single-track model derived from the Rosenthal equation, with melt pool width as the output. We validate the accuracy of the multi-track melt pool width model using FEM and demonstrate that the developed model can effectively represent the key characteristics of the convoluted multi-track PBF process.

The remainder of this paper is structured as follows. In Sec. 2, we build the FEM of the thermal fields in PBF. Section 3 explores the preliminary physics related to melt pool width. Section 4 develops and analyzes the primary multi-track melt pool width model. Finally, Sec. 5 concludes the paper.

We employ temperature-dependent thermal properties k, c_p, and ρ for both solid and liquid materials. Then, we calculate the thermal properties of the powder material based on the porosity of the solid material. To account for the latent heat of fusion, we introduce the effective heat capacity. The left plot of Fig. 2 displays the bidirectional scan strategy and the built geometry blocks that consist of a substrate and a thin layer of powder bed. The right plot of Fig. 2 illustrates the simulated surface temperature profile at 0.14 s, where the isotherm of T = T_m depicts the melt pool geometry and T_m is the melting point. For more information regarding the thermal properties, process parameters, and meshing scheme utilized in this FEM, please refer to Ref. [4], where it has been experimentally validated.

The developed FEM functions as a simulation platform for predicting the thermal fields of the powder bed throughout the multi-track PBF process. The finite element results are shown in Fig. 3, as well as in the top plots of Figs. 6 and 8. We observe that the start of each track has larger melt pool widths than the rest of the track. This is because in bidirectional scanning, when the energy beam approaches the end of one track, the large latent heat does not have enough time to dissipate out before the next track starts. Later on we will use the data (e.g., melt pool width) generated from the FEM to identify and verify the proposed analytical model. Specifically, we obtain the melt pool width from the FEM-predicted temperature distribution by searching the isotherm of T = T_m for the maximum width.

The Rosenthal equation is commonly used in the context of additive manufacturing to estimate the temperature distribution during the manufacturing process. This equation relates the temperature at a specific point in a material to the process parameters such as laser power, scan speed, and material properties.

The derivation of the Rosenthal equation involves making certain assumptions and simplifications. First, the material’s physical coefficients such as k, ρ, and c_p are assumed to be independent of temperature. The use of average values of these coefficients provides a reasonable approximation and enables a closed-form solution to be obtained. Second, the internal heat generation is neglected, i.e., q_s = 0. Third, the workpiece material is assumed to be homogeneous and isotropic. Additionally, when the powder bed is processed long enough, a quasi-stationary state is presumed to be reached, i.e., the temperature undergoes no change with time in the moving coordinate system (ξ, y, z). Moreover, a point heat source is used instead of a Gaussian distribution. Finally, the effect of latent heat of fusion is considered negligible since the absorbed latent heat evolves later on.

It’s worth pointing out that the Rosenthal equation is not directly derived from Green’s functions or probability density function (PDF) kernels. The Rosenthal equation is a simplified heat conduction equation that takes into account factors like heat generation, heat conduction, and convective cooling, but it’s derived through simplifications and assumptions specific to the additive manufacturing process. Green’s functions or PDF kernels are used in more general mathematical and physical contexts to solve differential equations, including heat conduction equations, but their direct connection to the Rosenthal equation in additive manufacturing is not common.

The assumptions hold reasonably well for all alloys except AlSi10Mg under typical PBF configurations [18].

Melt pool width is a crucial parameter for monitoring part properties during PBF manufacturing. Maintaining a user-defined reference value for melt pool width is essential to achieving uniform part quality [19]. To fulfill this requirement, we present a novel analytical model that emulates the dynamic behavior of melt pool width during the multi-track PBF process. The application of this multi-track melt pool width model can aid in developing control algorithms that mitigate process variations and ensure consistent part quality. In this section, we implement the superposition principle to model the evolution of the multi-track melt pool width, based on the single track expression in Eq. (5). The key idea is that the cumulative thermal effect of previous tracks on the current track is reflected on the increasing initial temperature T₀.

To explain the proposed analytical model in detail, we provide a step-by-step procedure below. The melt pool width of the first track w₁ can be directly calculated by Eq. (5) with T₀ = T₀₁ = T_amb, where T₀₁ indicates the initial temperature of every sample on the first track and equals the ambient temperature T_amb. Parameters q, u_x, k, ρ, and c_p are set to be constant. When the laser point reaches the end of the first track, for every sample on the second track as shown in Fig. 4, ξ = (n − 1)u_xt_s, y = h, and z = 0, where n is the sample number, t_s the sampling time, and h the hatch spacing. Furthermore, we have T₁(ξ, h, 0) = T₀₁(N) + T_r(n) from Eq. (3), where T₁ is the temperature distribution of the laser point at the track end and N is the total number of samples per track. Here, T₀₁(N) indicates the initial temperature of the last sample at the first track. The residual thermal effect of the first track on the second track are reflected on the initial temperature of every sample on the second track: T₀₂(n) = T₁(ξ, h, 0) = T₀₁(N) + T_r(n). Then with T₀ = T₀₂, we can calculate the melt pool width of the second track w₂(n) from Eq. (5). Similarly, for the i th track, we have T_0i(n) = T_0(i−1)(N) + T_r(n), and the melt pool width w_i(n) is the solution of Eq. (5) with T₀ = T_0i.

For an individual track, we notice that the melt pool width reaches its peak value a few samples after the track start, specifically at n = n_p. For example, n_p = 3 in Fig. 3. On the other hand, from the analytical multi-track model, the melt pool width reaches its peak value at the track start since n = 1, ξ = (n − 1)u_xt_s = 0, T_r in Eq. (4) peaks, and then T_0i peaks. To address the mismatch, we make an adaptation to the proposed model by shifting the virtual laser spots out by n_p − 1 samples (see Fig. 5) and introducing a tuning parameter β for the first n_p − 1 samples (as in Algorithm 1). Furthermore, to add more design flexibility, we reformulate T_0i as T_0i(n) = T_0(i−1)(N) + T_r(n)/α by introducing another tuning parameter α. Algorithm 1 outlines the fundamental steps of the proposed analytical model for predicting the melt pool width during the multi-track PBF process.

We employ the FEM built in Sec. 2 to simulate the evolution of the melt pool width among multiple tracks. Using the ten-track FEM data in Fig. 6, we identify the parameters in the proposed multi-rack melt pool width model as n_p = 3, α = 20.8, and β = 0.5. The other parameter values can be found in Table 1. Figure 7 shows the resulted T_r, T₀₂, and w₂, where the analytical melt pool width peaks at the third sample after the shift. Furthermore, we compare in Fig. 8 the twenty-track melt pool results from the identified analytical model and the FEM. From the top plots of Figs. 6 and 8, we can tell that the proposed multi-track model can effectively capture the spikes at the start of each track. Moreover, the model can catch the increasing trend of the melt pool width as the track number increases. This is due to the fact that the initial temperature profile T_0i increases with the track number, as shown in the bottom plots of Figs. 6 and 8. Overall, the proposed model’s melt pool width results closely match those of the FEM, with a difference of 5 μm. In addition, compared to FEM, the proposed model reduces the computational burden to a bare minimum. When modeling the ten-track PBF process as in Fig. 6, it takes 4.5 h using FEM [4] and only seconds using the proposed multi-track melt pool width model. Although the FEM has been experimentally validated in Ref. [4], our future endeavors will involve further verification of the proposed model through the PBF experiments directly.

In this paper, we present a comprehensive approach to analyze the melt pool width during the multi-track PBF process. First, we construct a FEM to simulate the thermal fields of PBF. Next, we develop a multi-track analytical model by applying the superposition principle to a single-track melt pool width model derived from the Rosenthal equation. Based on the FEM data, we identify the parameters and validate the effectiveness of the proposed model. The results demonstrate that the proposed analytical model can effectively be catching the complex dynamics of melt pool width that occur during the multi-track PBF process.

This work is supported by the National Science Foundation under Award No. 1953155.

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.