1 Introduction

Electron beam power bed fusion (PBF-EB) is an additive manufacturing process that utilizes a high energy electron beam for the local consolidation of metal powder[1]. The layerwise fabrication from the power bed enables the tool free fabrication of complex geometries. High vacuum conditions in combination with high process temperatures and the ability to tailor the spatio-temporal energy input enables the processing of a large variety of metal alloys [2,3,4,5,6,7,8] and even the local tailoring of microstructure and properties [9,10,11].

The selection of appropriate processing parameters and hatching strategies in PBF-EB is critical for the fabrication of defect-free parts with predetermined homogeneous properties. Complex geometries are commonly fabricated using line-based hatching strategies, e.g., the cross-snake strategy. Melt pool formation in line-based hatching strategies is governed by the cumulative heating effect, where the residual heat of previous hatch lines affects the melt pool formation of subsequent hatch lines [12,13,14]. This enables the energy efficient melting of large areas compared to alternative point melting strategies [10].

In cuboid geometries with constant scan length fabricated using constant processing parameters, the build-up of the cumulative heating effect ultimately results in quasi-stationary (QS) thermal conditions and a recurring spatio-temporal melt pool evolution along the hatch direction [12,13,14]. Therefore, the identification of suitable process parameter combinations from the center of standardized cuboid specimen is by default based on the QS melt pool evolution.

Scan length-induced differences of the cumulative heating effect prevent the formation of QS thermal conditions in complex geometries, fabricated with constant processing parameters. Recent research, however, showed that scan length-induced differences in the melt pool formation can be eliminated by an adaptation of the scan strategy. The return time compensation [15] enables the fabrication of complex cross-sections with predetermined melt pool geometries, independent of the local scan length. This is achieved by tailoring the local beam return time and the resulting local magnitude of the cumulative heating effect according to the QS thermal conditions of their reference bounding rectangle [15]. This scan strategy-based compensation approach enables the fabrication of complex cross-sections with constant processing parameters, which prevents the influence of second-order effects on the local melt pool formation, e.g., variations of the beam diameter as a function of beam power [16].

Independent of the geometry, the build-up of the cumulative heating effect on the hatch scale until QS thermal conditions are reached results in distinct differences of the spatio-temporal melt pool evolution between the start of the hatch, where the cumulative heating effect is not yet completely developed, and the QS region. This behavior is not only limited to first leading edge of the geometry, but all edges without prior energy input lack cumulative heating and differ from the desired QS melt pool geometry. Local deviations of the melt pool geometry from the desired QS geometry in regions with underdeveloped cumulative heating can result in local deviation of the desired properties. These include differences in the consolidation behavior, including defect formation and surface modifications [17, 18], solidification conditions, and microstructure development [19, 20].

With the goal to ensure the defect-free fabrication of complex geometries, process control approaches were developed for different scales and aspects of the powder bed fusion process, e.g., turning point and overhang effects on the hatch scale [21,22,23] and heat accumulation effects on the part scale [24]. In laser powder bed fusion (PBF-LB), control systems were established either based on feedback control during the process [22, 25], utilizing real-time process data, or based on feed forward control, which compensates the process based on empirical relationships or an underlying process model [24]. In both cases, compensation strategies in PBF-LB are based on process parameter-based control through the adaptation of beam power and velocity to compensate for local heat accumulation on the hatch scale [26,27,28,29,30]. Different optimal control methods were developed based on simplified thermal models to control the melt pool geometry and the maximum surface temperature by piece wise adaptation of the beam power [29, 31]. Ogoke et al. [32] employed a reinforcement learning approach to determine optimal control policies that adapt beam power or beam velocity with the objective to reach a desired melt pool geometry. On the part scale, Riensche et al. [24] applied a similar methodology, combining a graph-based part scale thermal model with a heuristic-based feed forward control approach to reduce heat accumulation over the build process in complex geometries.

In PBF-EB, the limited response time of the beam current control with respect to the beam velocity limits the possibilities for a fast adaptation of the beam power over the course of single hatch lines. Therefore, the implementation of process parameter-based control systems in PBF-EB is challenging and only limited research on control approaches for complex geometries was performed. Forslund et al. [33] successfully implemented a greedy optimization algorithm, that adapts the beam velocity and beam diameter based on a simplified thermal model to achieve constant maximum surface temperatures, but also stated that the transfer to complex geometries is challenging. Although different proprietary compensation strategies are already implemented in commercial machines of the manufacturer ARCAM (Arcam AB, Mölndal, Sweden), only limited information is available about their design [34,35,36,37].

The objective of this work is to derive a complementary scan-strategy-based approach to reduce regions with underdeveloped cumulative heating in line-based PBF-EB with the goal to achieve homogeneous thermal conditions for melt pool formation over complex cross-sections. For this purpose, an optimization problem based on a simplified thermal model is solved for a cuboid model geometry with constant scan length. Based on the solution of the optimization problem, a generalized scan-strategy-based approach to reduce regions with underdeveloped cumulative heating is derived for arbitrary cross-sections and is numerically validated for different process parameter and geometry combinations.

The remainder of this article is structured as follows. First, Sect. 2 discusses the formulation of the optimization problem for the model geometry with constant scan length, including the derivation of the objective function, definition of the search space, and the description of the simplified thermal model including its underlying assumptions and limitations. Sect. 3.1 presents the results of the optimization problem and Sect. 3.2 derives and verifies a reduced problem formulation. The generalization of these results into a scanning strategy-based approach for complex geometries is detailed in Sect. 3.3 and the limitations and further extension possibilities are discussed.

2 Methodology

2.1 Hatch scale residual heat accumulation in electron beam powder bed fusion

Prior to the formulation of the optimization problem, the optimization goal and the solution approach have to be defined. The spatio-temporal temperature and melt pool evolution in line-based hatching strategies are governed by a superposition of the residual thermal energy from previous layers and the preheating step \(E_{\text {p}}\), the contribution of thermal energy from the cumulative heating effect \(E_{\text {c}}\) and the thermal energy input of each individual hatch line \(E_{\text {l}}\). Therefore, the melt pool at any position of the hatch emerges based on a defined thermal state \(\Psi\), established by the superposition of \(E_{\text {p}}\) and \(E_{\text {c}}\), and the energy input of each line \(E_{\text {l}}\).

Approximate magnitudes of each individual energy contribution and the resulting thermal history at different positions are depicted schematically for a cuboid geometry in Fig. 1. Although the temperature distribution after the preheating step of a complex geometry is not completely uniform in practice, \(E_{\text {p}}\) is assumed to be constant within the scope of this study, to focus on the influence of the cumulative heating effect. The magnitude of the cumulative heating effect \(E_{\text {c}}\) at any position of the hatch is determined by a complex relationship between part geometry, local material parameters, scanning strategy, as well as the utilizes processing parameters, and is governed by the energy input of previous hatch lines and their respective energy loss until the current hatch line. Therefore, at leading edges of a geometry, the cumulative heating effect does not contribute to the melt pool formation for the first hatch line and the melt pool is solely determined by the superposition of \(E_{\text {p}}\), described by the preheating temperature \(T_{\text {p}}\), and \(E_{\text {l}}\), as shown in Fig. 1c.

Fig. 1
figure 1

Build-up of the cumulative heating effect over subsequent hatch lines in the center of a cuboid geometry with constant processing parameters: a schematic of the cross-section and b superposition of thermal energies for subsequent hatch lines: \(E_{\text {p}}\) residual energy from the previous layer and the preheating step (dashed black), and \(E_{\text {c}}\) cumulative heating on the hatch scale (red area). Target thermal states with and without cumulative heating and compensation approaches (A),(B) are indicated by the grey and blue arrows. Exemplary temperature evolution at c the first and d the 15th hatch lines

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Over the course of the hatch, the magnitude of the cumulative heating effect increases continuously until a QS state, when a constant cumulative heating contribution is reached, as shown in Fig. 1b. The increase of \(E_{\text {c}}\) and the corresponding temperature contribution \(T_c\) over multiple hatch lines, until a QS state is reached as indicated in Fig. 1d at line 15, results in the characteristic regions with an underdeveloped melt pool at the beginning of any geometry.

Based on the continuous build-up of the cumulative heating effect over the course of the hatch, two thermal states qualify as potential optimization goal. The first state is equivalent to the thermal conditions of an independent melt line, where the melt pool forms without any contribution of cumulative heating, as indicated in Fig. 1b (A). The second state is characterized by a completely developed cumulative heating contribution and is equivalent to the state in the QS region of the hatch, as indicated in Fig. 1b (B).

To pursue these target states, there are two different solution approaches based on distinct modifications of the processing strategy. In order for the first approach to achieve the thermal state of an independent melt line, the naturally occurring cumulative heating effect between neighboring melt tracks has to be compensated for by reducing the energy input of every subsequent melt line, as indicated by Fig. 1b (A).

The second approach, in contrast, leverages the natural build-up of residual heat on the hatch scale and aims to establish the QS conditions \(E_{\text {qs}}\), indicated by the blue dotted line in Fig. 1b (B), from the beginning of the geometry. Since there is lack of cumulative heating at locations with only a limited amount of preceding hatch lines, additional energy has to be deposited at these locations, as indicated in Fig. 1b (B). In contrast to the first approach, this approach limits the number of melt lines that require adaptation, which significantly reduces the computational cost and enables the solution of the problem even for larger geometries.

2.2 Optimization formulation

The goal of the optimization is to control the melt pool formation at leading edges of the geometry to reduce the extent of regions with underdeveloped cumulative heating using the second approach [Fig. 1b (B)] through a line-wise process modification. To converge towards an optimal set of parameters and minimize a defined objective function, optimization algorithms iteratively update candidate solutions within a defined search space according to either gradient information or according to a heuristic search. Both the definition of the objective function, that is able to evaluate the quality of a candidate solution and guide the optimization algorithm towards the global minimum, and the definition of an appropriate search space are critical for the convergence and the quality of the final solution.

Objective function To define a suitable objective function, that is able to quantify the difference between the thermal conditions of a candidate solution and the thermal conditions in the QS region, the characteristics of the target thermal conditions have to be identified first. The QS thermal conditions arise from a sequence of energy input instances from a Gaussian shaped beam, which are separated in space, by subsequent melt tracks, and time, by defined temporal intervals, and lead to the characteristic temperature curves in Fig. 1d). The time between subsequent energy input instances depends on the local return time \(t_{\text {ret}}\), which is a function of the location of the point of interest in the geometry.

Hence, an objective function can be defined, for each position x, y, z within the outline of the geometry, based on the temperature difference between the thermal history of the candidate solution \(T_{x,y,z}\) (Fig. 1c) and the thermal history \(T_{x',y,z}\) (Fig. 1d) of the corresponding position with the same local return time in the QS region of the hatch \(x',y,z\). Since this temperature-based objective function considers the complete spatio-temporal evolution of the temperature field over the whole melting time \(t_{max}\), it would both require the solution of an expensive multi-objective optimization problem as well as the non-trivial temporal alignment of the thermal histories between the candidate solution \(T_{x,y,z}\) and the QS region \(T_{x',y,z}\).

Therefore, we propose a simplified objective function based on the maximum melt pool depth \(\Lambda _{x,y}\) at each position of the hatch, which is closely related to the spatio-temporal temperature evolution, but only requires the spatial alignment of candidate solution and target state. The maximum melt pool depth at any position x, y can be calculated based on the melt pool envelope which is defined as the maximum melt pool depth that is present at any time step of the hatch \(\Lambda _{x,y}= max(\{\Lambda _{x,y}(t): t = 0,...,t_{\text {max}}\})\). A weighted sum scalarization of the objective values at each position x, y with higher weights \(\omega _{x,y}\) towards the leading edge of the geometry enables the unification of all objectives into a single objective function \(\xi\). For a candidate solution \(\Xi\), the simplified objective function can be defined as the sum squared error between the maximum melt pool depth of the candidate solution \(\Lambda _{x,y}(\Xi )\) and the corresponding melt pool depth in the QS state \(\Lambda _{x',y}\) for each position x, y

$$\begin{aligned} {\begin{matrix} \xi (\Xi ) = & \sum _{x,y} \omega _{x,y}\cdot (\Lambda _{x',y}-\Lambda _{x,y}(\Xi ))^2 \\ & x,y \in \mathbb {R}^2. \\ \end{matrix}} \end{aligned}$$
(1)

To ensure the spatial alignment, the first melt line of the candidate solution is aligned with a melt line in the QS region. In addition, only positions where a melt pool was present are considered in the calculation of the final objective value, emphasizing a sharp edge at the boundary of the geometry.

Search space In general, line-based hatching strategies in PBF-EB offer the potential for a line-wise modification of the beam power P, velocity v, line offset \(l_o\), and beam diameter \(d_{\text {b}}\). However, since the melt pool depth-based objective function (Eq. (1)) does not explicitly encode the required dynamics of the energy input in the QS region, they have to be enforced by constraining the search space.

Since the temperature evolution in the QS region is governed by a sequence of beam interactions that lead to an incremental increase of the temperature, as seen in Fig. 1d, it is not possible to deposit large amounts of energy during the first hatch lines to achieve the required melt pool depth. Instead of depositing excessive energy within a single melt line, that can lead to deviations from the desired QS behavior, energy has to be deposited repeatedly until the required melt pool depth is reached.

Although changes of the beam velocity v can easily tailor the local energy input of each hatch line resulting in an increased melt pool depth, the amount of beam interactions is still limited towards the edge of the geometry and does not correspond to the desired sequence of beam interactions in the QS region. Variations of the v also directly influence the local return time in between interactions and can lead to local differences of \(E_{\text {c}}\). Changes of the beam power P, in contrast, can tailor the local energy input without affecting the local return times. However, current electron gun hardware is limited by the rate of change of the beam power, so that rapid changes of beam power between adjacent hatch lines are not possible.

Based on these constraints, the search space \(\Gamma\) that preserves the energy input dynamics and respects current hardware limitations is limited to the line-wise adaptation of the line offset \(\Gamma = \left\{ i \in \mathbb {N}: lo_i \right\}\) for each hatch line i. Further constraints regards the search space, that bound the possible line offset values between zero, corresponding to a repetition of a melt line, and the base line offset \(0 \le lo_i \le lo_{\text {b}}\), were implemented.

Optimization algorithm We consider the minimization of the melt pool depth-based objective function (Eq. (1)) as a black-box problem, since there is neither a closed form nor direct derivative information available. The optimization is performed using the Covariance-Matrix-Adaptation Evolution-Strategy (CMA) [38], which is a stochastic optimization algorithm, which iteratively updates a population of candidate solutions based on their objective values until a desired optimal solution is obtained. To explore the search space efficiently, the algorithm only samples candidate solutions from a fraction of the total search space, represented by a multivariate normal distribution \(\mathcal {N}\), in each iteration and then evolves the location and shape of the distribution to minimize the underlying objective function. The multivariate normal distribution \(\mathcal {N}(\mu ,\Sigma )\ \in \ \mathbb {R}^i\) is defined by its mean vector \(\mu\), which determines the location of the distribution within the search space, and its covariance \(\Sigma\), which determines the overall shape and the scale of the distribution in each dimension. In our case, the search space \(\Gamma \in \ \mathbb {R}^i\) comprises in each i dimensions, all possible values that the corresponding line offsets \(lo_i\) can take. The optimization procedure is detailed in Algorithm 1.

Algorithm 1
figure a

CMA-ES with bound constraints

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For each iteration j of the optimization, the CMA-ES samples a total of \(\lambda\) candidate solutions from the distribution \(\mathcal {N}(\mu ,\Sigma )\) to yield the corresponding line offsets \(lo_i\) and the corresponding objective values \(\xi (\Xi _k)\) for each candidate solution k are calculated based on the thermal model in Sect. 2.3 using Eq. (1). To progress towards an optimal solution, the CMA-ES updates the parameters of the underlying distribution (\(\mu ,\Sigma )\) for the next iteration, by moving the mean of the distribution towards candidate solutions with lower objective values and updating the shape and variance of the distribution to extend in the direction of lower objective values. The evolution of the distribution follows the natural gradient of the current generation and the mean vector and covariance matrix are updated to decrease the expected objective value and ultimately result in an optimal solution [39]. Constraints of the search space were handled by bound constraints, which implement a repair method which projects infeasible values back onto the value of the violated boundary [40]. Further details about the implementation and the update of the underlying distribution can be found elsewhere [38, 40].

2.3 Heat transfer model

The thermal model underlying the calculation of the spatio-temporal temperature evaluation for each candidate solution is based on an explicit finite difference method, that solves the heat equation in three dimensions

$$\begin{aligned} \frac{\partial T}{\partial t} = \alpha \nabla ^{2}T + \frac{Q}{c\rho }; \end{aligned}$$
(2)

T describes the temperature, \(\alpha\) is the thermal diffusivity, Q defines the source term of the electron beam, \(\rho\) is density, and c the specific heat. The heat source term \({Q} = \eta \cdot I(x,y) \cdot P\) is described as a surface heat flux with a Gaussian distribution of beam intensity I and beam diameter of \(d_{\text {b}}\) at the location (x, y) which is calculated according to

$$\begin{aligned} I(x,y)=\frac{1}{2 \pi {d_{\text {b}}}^2}\exp ({\frac{(x-x_{\text {b}})^2+(y-y_{\text {b}})^2}{2 \pi {d_{\text {b}}}^2}}), \end{aligned}$$

where \(x_{\text {b}}\) and \(y_{\text {b}}\) define the location of the center of the beam.

To focus on the determination of a suitable approach to reduce regions with underdeveloped thermal conditions and to achieve the computational speed required for an efficient optimization the thermal model only considers heat conditions and effects of fluid convection, radiation, vaporization, and latent heat release are neglected.

The model operates on a cubic lattice with constant lattice spacing dx in all three dimensions with a constant time step dt and is implemented on GPU using NUMPY [41] and JAX [42] for time-efficient calculation. The beam path and its corresponding local beam parameters are defined segment-wise for each hatch vector i. The transition of the beam between subsequent hatch lines is not explicitly modeled and is assumed to be instantaneous.

For the calculation of the spatio-temporal melt pool evolution of each candidate solution, the simulation domain is initialized as a continuum with constant material parameters and preheating temperature \(T_{\text {p}}\). Directional dependent thermal conductivity and the interface between powder and already consolidated material at the edge of the geometry are not considered by the thermal model. Dirichlet boundary conditions with a value of Tp are imposed on the bottom and side surfaces of the simulation domain, while a Neumann boundary condition, imposing a no flux condition, is applied to the top surface. The size of the simulation domain is large enough, that the boundary conditions do not interfere with the spatio-temporal melt pool evolution of the hatch.

2.3.1 Model geometry

For the optimization of a cuboid geometry with a line length \(l_{\text {m}}\) of 15 mm, we utilize Ti-6Al-4V as model material and a parameter combination with a high cumulative heating contribution, that results in the formation of a permanent persistent melt pool is selected [43]. As base processing parameters, a base line offset \(lo_{\text {b}}\) of 100 \(\upmu\)m with a beam power P of 950 W and a beam velocity v of 5.72 ms\(^{-1}\) at a preheating temperature \(T_{\text {p}}\) of 1023 K are chosen.

The search space for this specific process parameter combination includes \(i \in 1,...,9\) melt lines and the optimization to obtain an optimal distribution of line offsets is carried out for a total of \(\epsilon =75\) generations with \(\lambda =10\) candidate solutions each. The underlying distribution is initialized with a mean line offset of \(\mu =[75,...,75]\) \(\upmu\)m for each of the ten melt lines and the initial standard deviation of the covariance matrix \(\Sigma\), which defines the distribution width in each dimension for each line i, was set to 20 \(\upmu\)m. Prior to the optimization, the target thermal state and melt pool depth distribution is calculated using the thermal model. For the calculation of the maximum melt pool depth distribution of a candidate solution, the simulation domain (Fig. 2) is limited in size, to include only 50 hatch lines, since the region of interest is located at the leading edge of the geometry.

Fig. 2
figure 2

Schematic illustrating the dimensions of the the simulation domain and position of the hatch (blue) for the calculation of a candidate solution

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To obtain the maximum melt pool depth distribution \(\Lambda _{x,y}\), the melt pool envelope is determined in three dimensions, encompassing each cell of the simulation domain above the liquidus temperature \(T_{\text {l}}\), for each time step t of the simulation. The estimated material parameters for the model material for a temperature of \(\sim 1000\ K\) adapted from Rausch et al. [44] and an overview of the simulation parameters are given in Table 1.

Table 1 Material properties (Ti–6Al–4V) and simulation parameters
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3 Results and discussion

3.1 Model geometry

Figure 3 details the results of the optimization procedure described in Sect. 2 for a rectangular cross-section. The convergence of the objective value over 75 . generations in Fig. 3a is complemented by the evolution of the corresponding line offsets of each candidate solution in Fig. 3f and selected maximum melt pool depth distributions in Fig. 3b–e. Figure 3b shows the the target maximum melt pool depth distribution in the QS region of the hatch, which is characterized by three different regions with constant maximum melt pool depth over the course of the hatch in x-direction. The region in the center of the hatch exhibits a maximum melt pool depth of 200 \(\upmu\)m. Towards the turning points of the hatch in positive and negative y-direction, two regions with higher maximum melt pool depth emerge due to the effects of reduced local return times.

Fig. 3
figure 3

Convergence plot a for the optimization of the cuboid geometry, processed with a beam power of \(P =\) 950 W a velocity of \(v =\) 5.72 ms\(^{-1}\), a line length of \(l_{\text {m}} =\) 15 mm, a line offset of \(l_o =\) 100 \(\upmu\)m at a preheating temperature of \(T_{\text {p}} =\) 1023 K. The objective value \(\xi (\Xi )\) is determined based on Eq. (1) according to the target melt pool depth distribution (b). Selected maximum melt pool depth distributions: initial (c), intermediate (d), and optimal (e). Line offset evolution over the course of the optimization (f). Line offset values of the optimal solution are indicated below their corresponding melt line

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Figure 3c shows the maximum melt pool depth distribution of the initial candidate solution \(\Xi _0\) with a constant line offset of \(lo_i: \{i \in 1,...,10\}\) = 100 \(\upmu\)m at the leading edge of the geometry. The maximum melt pool depth at each position of the first melt line at the leading edge is significantly lower compared to the target QS region. At the center of the hatch, the melt pool depth reaches a maximum of 100 \(\upmu\)m and towards the turning points the melt pool depth reaches a maximum of 125 \(\upmu\)m.

Over the course of the hatch, the maximum melt pool depth continuously increases and regions with higher maximum melt pool depth extend further towards the center until the QS state is reached. This is the case, when the extent of both regions at the turning points matches the extent of the corresponding regions in Fig. 3b, at about 3.75 mm into the geometry. In contrast to the QS state with a constant extent of the region at the turning points over the course of the hatch in x-direction, the underdeveloped cumulative heating leads to a curvature of these regions towards the turning points of the hatch.

In Fig. 3a, the evolution of the objective values for each candidate solution \(\xi (\Xi )\) over the course of the optimization is depicted. To quantify the improvement of each candidate solution with respect to the initial state, the objective value \(\xi (\Xi )\) is presented as a fraction of the objective value of the initial configuration \(\xi _0 = \xi (\Xi _{000})\). The subscript indicates the generation j, denoted by the first two digits, and the candidate solution k within the generation. Therefore, the initial candidate solution has a value of \(\xi /\xi _0 =1\). Over the course of the first 20–25, there is a large decrease of the objective value, reaching minimum objective values of \(\xi /\xi _0 \in\) [0.15–0.2]. The variance of the objective values for candidate solution within a generation, however, is still as high as for the first generation of the optimization.

Figure 3d shows the maximum melt pool depth distribution of a candidate solution in the second generation \(\Xi _{023}\). Already in this case, regions with the desired maximum melt pool depth extend further towards the leading edge of the geometry and exhibit a reduced curvature towards the turning points. Compared to the initial configuration, the QS state is reached at a shorter distance from then edge of the geometry.

Over the course of the remaining 50 . generations, the variance of the objective values within a generation decreases continuously and the objective value converges towards a minimum value of \(\xi /\xi _0 = 0.08\), which is already reached in generation 67 .. Figure 3e shows the corresponding maximum melt pool depth distribution of the optimal candidate solution \(\Xi _{671}\). The regions with the desired maximum melt pool depth in the center and towards the turning points of the hatch extend almost completely towards the leading edge of the geometry and the QS melt pool depth distribution is reached starting from the beginning of the geometry. The extent of the regions with higher maximum melt pool depth towards the turning points also matches closely with the extent of the corresponding regions in the QS region and shows no curvature towards the turning points. Compared to the desired QS state, these regions extend even further towards the center of the hatch at the location of the first 2–5 melt lines.

The evolution of the candidate solutions and the corresponding line offsets \(lo_i: \{i \in 1,...,10\}\) over the course of the optimization are shown in Fig. 3f. The line offsets corresponding to the candidate solutions highlighted in Fig. 3c–e are highlighted in the same color. The evolution of the line offsets shows a clear trend. Starting from the initial configuration \(\Xi _0\), where all line offsets are equal to the base line offset of \(lo_{\text {b}}=\)100 \(\upmu\)m, the line offset continuously decreases for the first 6 . melt lines and converges towards the minimum bound of 0 \(\upmu\)m. The remaining line offsets converge towards the original base line offset of 100 \(\upmu\)m, which results in a hatch pattern, where 7 . melt lines are located approximately at the position of the first melt line before the position of the beam changes. This divides the hatch into two distinct phases, a build-up phase, where the first 6 . melt lines deposit additional energy at the location of the first hatch line, and the propagation phase, where the actual hatch pattern starts with the first melt line at the edge of the geometry.

The corresponding surface temperature distributions at different transient time steps during the build-up phase and the propagation phase of the optimal solution are depicted in Fig. 4a. The energy input of the first hatch line in the build-up phase leads to the formation of a trailing melt pool. With each additional line of the build-up phase, additional energy is deposited and the melt pool incrementally elongates in beam movement direction until a persistent melt pool forms with the final line of the build-up phase. The first melt line of the propagation phase, that is also located at the leading edge of the geometry, leads to the formation of a persistent melt pool that covers the complete line length of the geometry. The comparison between melt pool envelopes at the start of the propagation phase and the QS region shows a good agreement.

Fig. 4
figure 4

Surface temperature fields at different transient time steps in the build-up and in the propagation phase at the leading edge of the geometry and in the QS region of the hatch (a). Comparison of the thermal histories (b) of a point located at the center of the leading edge fabricated with the optimal solution (red) and the corresponding point in the QS region (blue). The highest peak of the thermal history is magnified in the inset plot. Build-up and propagation phase are highlighted in red and grey

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To evaluate the optimal solution quantitatively, the thermal history of the point located at the center of the leading edge, Fig. 4a (\(p_{0}\)), is compared with the thermal history of the corresponding point with the same return time in the QS region, Fig. 4a (\(p_{\text {qs}}\)), in Fig. 4b. To ensure the comparability between temperature curves, the highest peak temperatures of the first line in the propagation phase and the corresponding line in the QS region, which are present when the beam is located directly above the respective point of interest, are aligned. The step-wise increase of the temperature in multiple discrete intervals with energy input and heat dissipation matches between the optimal solution and the QS state. Although there are differences in the magnitude of the peak temperature between the build-up phase and the QS region, each beam interaction in the build-up phase only results in a moderate, incremental increase of the temperature before the next beam interaction. This ultimately leads to the alignment of the temperature curves at the start of the propagation phase with the thermal history in the QS region, as shown in Fig. 4b.

In the QS region of the hatch, at a point \(p_{\text {qs}}\) located on the hatch line l, the staggered energy input from preceding hatch lines leads to the incremental build-up of a characteristic thermal state \(\Psi (p_{\text {qs}})\), defined by its temperature distribution \(T(p_{\text {qs}})\) and its local temperature gradients \(\nabla T(p_{\text {qs}})\). The melt pool, which governs the final material properties at \(p_{\text {qs}}\), forms based on the superposition of this characteristic thermal state \(\Psi (p_{\text {qs}})\) and the energy input from the hatch line l at \(p_{\text {qs}}\). In case of the uncompensated strategy, the thermal state at the leading edge \(\Psi (p_{0})\) is determined only by the residual heat from previous layers and the preheating step. During melt pool formation, the superposition of \(\Psi (p_{0})\) with the energy input from the first hatch line at \(p_0\) leads to high-temperature gradients \(\nabla T(p_0)\) and a high heat flux into the base material, which results only on the formation of a small trailing melt pool, as seen in Figs. 1c and 4a for the first line. At the location of subsequent hatch lines, respectively, only one additional line contributes to the build-up of the thermal state through the cumulative heating effect. For the build-up of the characteristic QS thermal state, however, a defined amount of hatch lines are necessary. Therefore, the QS region emerges only at a defined distance from the edge of the geometry, when \(\Psi (p_{\text {l}}) = \Psi (p_{\text {qs}})\).

The step-wise energy input during the build-up phase of the optimal solution mimics the staggered energy input in the QS region through multiple consecutive beam interactions at the leading edge of the geometry. With each beam interaction, additional energy is deposited and the local thermal gradient \(\nabla T(p_0)\) at the leading edge decreases. Consequently, the underlying thermal state is incremented towards the desired thermal state of the QS region \(\Psi (p_0) \rightarrow \Psi (p_{\text {qs}})\), as demonstrated in Fig. 4. This results in the step-wise increase of the melt pool size at \(p_0\) until the target melt pool shape and thermal history are achieved by the superposition of \(\Psi (p_0)\) after the build-up phase with the energy input of the first line in the propagation phase. The deposition of additional energy at the leading edge to build-up \(\Psi (p_0)\) simultaneously leads to the build-up of the thermal state at subsequent hatch lines \(\Psi (p_{\text {l}}): l \ge 1\). Therefore, only the adaptation of the energy input at the leading edge is necessary to counteract the underdeveloped cumulative heating.

Opposed to the staggered energy input in the QS region, the beam is always located directly above the leading edge during the build-up phase of the optimal solution. Therefore, a large fraction of the total energy input is deposited at \(p_0\), which is necessary to counteract the high heat flux into the base material during the build-up phase, but also results in higher peak temperatures compared to the QS region (Fig. 4). Elevated peak temperatures can lead to excessive evaporation and induce local changes in the alloy composition [16, 45, 46]. However, in the uncompensated case, the lack of cumulative heating at the edge of the geometry already leads to lower peak temperatures and reduced evaporation compared to the QS region, which can lead to unwanted changes of the microstructure [20]. Higher peak temperatures during the build-up phase increase the evaporation at the leading edge of the geometry and homogenize the evaporation behavior.

As mentioned in Sect. 2.3, the interface between powder and already consolidation material is not considered in the scope of this study. However, these effects have a large influence on the build-up of the underlying thermal state at the leading edge \(\Psi (p_0)\), since they limit the amount of heat flux into the base material and reduce the amount of required energy input to achieve thermal condition equivalent to the QS region. A suitable optimal build-up phase to accommodate variations of the local heat flux into the material, however, can be identified using the same methodology with an adapted thermal model that includes these effects.

3.2 Reduced problem formulation

Based on the structure of the optimal solution in Sect. 3.1, a reduced optimization problem can be formulated. Instead of solving the multivariate continuous optimization problem, the problem can be reduced into a one-dimensional, discrete problem. In this case, the optimization space \(\Gamma _{\text {reduced}} = \left\{ n \in \mathbb {N} \right\}\) only consists of the number of melt lines n located at the edge of the geometry in the build-up phase. This significant reduction of the search space enables the computation of a large fraction of the search space and direct access to the optimal solution without extensive search. In addition, the solution provides a simplified hatching strategy, which enables the transfer to complex geometries, which is further detailed in Sect. 3.3.

Figure 5 shows the results of the reduced problem formulation analog to the cuboid geometry in Sect. 3.1 with the same processing parameters. The maximum melt pool depth distributions and the corresponding objective values, determined according to Eq. (1), are depicted for a subset of the search space with different build-up sizes \(n \in {0,...,11}\). The melt pool depth distribution without additional energy input \(n=0\) shows a considerable region with underdeveloped cumulative heating at the leading edge. With increasing build-up size, the maximum melt pool depth at the leading edge increases until the melt pool depth distribution is in good agreement with the QS region at a build-up size of 6 .. A larger build-up size leads to a further increase of the maximum melt pool depth at the leading edge and results in significant overmelting. The corresponding objective values show a step-wise decrease of the objective value until a minimum value is reached at a build-up size of 6 .. With a larger build-up size, the objective value increases again.

Fig. 5
figure 5

Maximum melt pool depth distributions for a cuboid geometry with a \(l_{\text {m}}\) of 15 mm, beam power P of 950 W a velocity v of 5.72 ms\(^{-1}\) a base line offset \(lo_{\text {b}}\) of 100 \(\upmu\)m at a preheating temperature \(T_{\text {p}}\) of 1023 K at different build-up sizes based on the reduced problem formulation and the corresponding objective values calculated according to Eq. (1)

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Both, the optimal build-up size with \(n_{\text {opt}}=\) 6 . melt lines, and the minimum objective value \(\xi _{\text {min}}/\xi _{0}(red.)=0.09\) of the reduced problem formulation, match well with the build-up size and the minimum objective value of the complete optimization procedure \(\xi _{\text {min}}(CMA-ES)=0.081\). However, the objective values do not match exactly, since the reduction of the search space to integer values of build-up sizes simultaneously leads to a discretization of the achievable thermal states \(\Psi (p_0)\) and objective values \(\xi _{\text {min}}(red.)\). Despite this limitation, the optimal solution of the reduced problem is able to achieve results with the same quality as the complete optimization problem at a fraction of the computational cost.

This enables the identification of optimal process strategies for a wide range of processing parameters, similar to the identification of processing maps [43]. Since each process parameter combination, defined by its area energy \(E_{\text {a}} = P/vl_o\) and lateral velocity \(v_{\text {lat}}= v l_o/lm\), has a unique cumulative heating contribution and extent of the region with underdeveloped melt pool, each parameter combination requires a dedicated optimal solution. Analog to Fig. 5, Fig. 6a shows the objective values as a function of the build-up size n for different process parameter combinations, with constant area energy \(E_{\text {A}}={1.5}\,\textrm{J}\,{\textrm{mm}}^{-2}\) and different lateral velocities. Analog to the trend of the objective value in Fig. 5, the objective values follow the same trend for each parameter combination and the objective value reaches a minimum at a defined build-up size, before increasing again. Depending on the lateral velocity of the parameter combination, the optimal build-up size changes. While at low lateral velocities, a larger build-up sizes is required to reach the minimum, the build-up size decreases with increasing lateral velocity.

Fig. 6
figure 6

Objective values as a function build-up size n for different process parameter combinations with constant area energy \(E_{\text {A}}={1.5}\,\textrm{J}\,{\textrm{mm}}^{-2}\) and different lateral velocity for cuboid geometries (a). The corresponding minimum objective value is highlighted. Optimal build-up size \(n_opt\) as a function of lateral velocity for process parameters with constant area energy (b)

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Figure 6b depicts the optimal build-up size \(n_{\text {opt}}\) as a function of the lateral velocity with constant area energy for a total of 18 . parameter combinations. With constant energy input, process parameters with a higher lateral velocity and lower return time require a smaller optimal build-up size. Like the build-up of the cumulative heating effect, the build-up of the thermal state \(\Psi\) at the leading edge of the geometry is governed by the local energy input and heat dissipation, which in turn depends on the local heat flux into the material and the time available for heat dissipation. The high return time at low lateral velocities results in a large amount of heat dissipation and consequently only a small amount of thermal energy is retained from each line in the build-up phase. With decreasing return time, time available for heat dissipation decreases and a shorter build-up phase is sufficient to reach the desired thermal state.

Ultimately, independent of the process parameter combination, the respective optimal solutions show a significant improvement of their corresponding objective value with respect to the standard cross-snake solution. However, depending on the specific parameter combination, the discretization of the achievable thermal states leads to variations in the quality of the optimal solution, as indicated by the different ratios \(\xi /\xi _0\) in Fig. 6. Especially, process parameter combinations with low return times and small build-up phases are critical, since the target thermal state can be located between two achievable states. Possible approaches to access intermediate thermal states in between the current discretization are the additional adaptation of the beam power in the build-up phase or the adaptation of the underlying scan strategy to achieve a finer discretization of achievable thermal states, by reducing the base line offset, while keeping a constant area energy and lateral velocity.

3.3 Complex geometries

Regions with underdeveloped cumulative heating are not limited to the first leading edge of the contour, but arise at any edge without prior energy input in complex geometries. Therefore, it is necessary to deposit additional energy at each of these locations by integrating multiple build-up phases. The build-up of the respective thermal states, however, cannot take place prior to the hatch, since the majority of the deposited energy would already be dissipated at the required time. Hence, each individual build-up phase has to be integrated into the hatch sequence at its dedicated place and time, to deliver the required amount of energy without interfering with the original hatch sequence.

For this purpose, we propose an scan strategy-based adaptation of the line-based hatching strategy, that includes the compensation of the residual heat build-up on the hatch scale for complex geometries in PBF-EB The compensation strategy is illustrated by the means of an L-shaped geometry in Fig. 7 and is based on the previously developed return time compensation, which introduced idle segments into the hatching sequence [15]. During idle segments, no energy is deposited, but their integration into the hatch sequence at specific locations enables the extension of the local return time. To establish the same return time characteristic as a reference geometry, typically a rectangle, the original hatch sequence of the reference geometry is maintained, but the hatch lines are split along the outline of the complex geometry into melt segments, within the outline, and idle segments, outside the outline, as shown in Fig. 7a. The first step is to identify the line segments that require compensation. If a part of a melt segment or a complete melt segment is located at the edge of the geometry in direction of the hatch process (positive y-direction) and is not preceded by a melt segment in the preceding line, it requires compensation.

Fig. 7
figure 7

Schematic of the return time compensation (a) and the build-up compensation (b) of regions with underdeveloped cumulative heating for a L-shaped geometry

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As indicated in Fig. 7a for the L-shaped geometry, this yields one subsegment at the beginning of the hatch [A, B] and one subsegment within the geometry [C, D], that require compensation. Based on the definition of the return time compensation strategy, it follows that any melt subsegment, which is located within a complex geometry and is not preceded by another melt subsegment in the previous line, is instead preceded by an idle segment. Due to the limited response time of the beam current control, the beam is deflected around the build platform during idle segments for a defined time \(t_{\text {idle}}\), according to the time the beam would need to cover the length of the idle segment \(t_{\text {idle}}=l_{\text {idle}}/v\), to avoid any energy input while extending the local return time. Instead of deflecting the beam around the build platform, the compensation utilizes parts of the idle time for the targeted deposition of energy and the integration of multiple build-up phases during the hatch sequence.

To convert idle segments into build-up segments for targeted energy input during the hatch sequence, different conditions have to be satisfied. Build-up segments have to deposit the same amount of energy in the same time over the same length as the corresponding melt subsegment, that requires compensation. In addition, the energy input of build-up segments is required to take place within the same time interval as the corresponding melt subsegment, without interfering with the original return time relationships of the hatch. With constant processing parameters \(P,v=const.\), the same amount of energy can be deposited in the same time over the length of the melt subsegment, when an idle segment of the same or larger length is available in the preceding line \(l_{\text {idle}} \ge l_{\text {melt}}\). Irrespective of their type, line segments located at the same position at subsequent lines, e.g., \([E,F] \wedge [C,D]\), have the same return time characteristic and are able to deposit their energy within the same time interval of the line. Therefore, it is possible to convert idle segments into build-up segments,while preserving all return time relationships, by shifting the idle segment by a translation vector \(\textbf{v}\) from their original location to the position of the melt subsegment, that required compensation, e.g., \([E,F] \mapsto [C,D]\).

Since, depending on the size of the optimal build-up phase \(n_{\text {opt}}\), multiple targeted energy input instances are required to reach the desired thermal state \(\Psi\), idle segments larger then the melt subsegment, that requires compensation, have to be available at the same position in \(n_{\text {opt}}\) preceding lines. For the construction of the final hatch sequence, each idle segment has to be shifted by its respective translation vector \(\mathbf {v_n} = (0, n\times l_o)\), as shown in Fig. 7b. In case of this example, under the assumption of uniform thermal conductivity at each position of the geometry, each build-up phase requires the same optimal build-up size. However, depending on the local material conditions, e.g., different local thermal conductivity, different locations in the geometry can require a different optimal build-up size. Because of that, the implementation of the compensation enables the local adaptation of the optimal build-up size \(n_{\text {opt}}\), which can be determined for each individual case according to Sect. 3.2 for a given process parameter combination and the local material conditions. In case of this example, however, the optimal build-up size was determined as \(n_{\text {opt}}=4\). To compensate the first melt subsegment at the beginning of the hatch [A, B], the reference geometry has to be extended according to the required build-up size n, to prepend the required amount of idle lines at the start of the hatch, as detailed in Fig. 7b.

The effect of the build-up compensation on the spatio-temporal melt pool evolution is depicted in Fig. 8 for the L-shaped geometry with an edge length of \(l_{\text {m}} =\) 15 mm, processed with an area energy of \(E_{\text {a}}={1.5}\,\textrm{J}\,{\textrm{mm}}^{-2}\) and a lateral velocity of \(v_{\text {lat}}=\) 53 mms\(^{-1}\). The optimal build-up size was determined for uniform material properties according to Fig. 6 as \(n_{\text {opt}}=4\). Figure 8a showcases the evolution of the surface temperature distribution at different transient time steps at the end of each respective segment in the hatch sequence. The type and location of the active segment are indicated in black for each time step. The position of the previous segment is highlighted in grey and the contour of the geometry is indicated by the white dotted line, with the energy input of the first build-up segment in 2), an active melt pool emerges, that is located before the actual melt pool, at the corresponding build-up segment [C, D] within the geometry. The melt pool size increases with each additional line of the build-up phase 2), 5), 6), 9), analog to the results of the cuboid geometry in Fig. 4. Since the build-up segments are not located at the center of the geometry, the energy input intervals and return times are not constant and, consequently, there are short return times \((5) \rightarrow (6)\) and long return time \((6)\rightarrow (9)\) between consecutive build-up segments. After the third build-up segment, when the actual melt pool front reaches the location of the build-up segment, both independent melt pools establish a connection in the center, as seen in Fig. 8 7). Ultimately, after the build-up phase, the final melt pool envelope matches the desired melt pool shape and only minor differences between the region with preceding melt lines and the compensated edge is visible in 12).

Fig. 8
figure 8

Spatio-temporal melt pool evolution over the course of the L-shaped geometry processed with \(E_a={1.5}\,\textrm{J}\,{\textrm{mm}}^{-2}\), \(v_{\text {lat}}=\) 53 mms\(^{-1}\), \(lo=\) 100 \(\upmu\)m, \(T_{\text {p}}=\) 1023 K (a): current location and type of the segment are indicated in black, and location of the previous segment is indicated in grey; comparison of the thermal history of a point located at the center of the edge within the geometry with its corresponding point in the QS region (b): time steps corresponding to the surface temperature images are indicated by their respective titles; maximum melt pool depth distribution of the geometry fabricated with the return time compensation (c) and the build-up compensation of regions with underdeveloped cumulative heating (d)

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Figure 8b details the thermal history at the center of the build-up segment [C, D] within the geometry, highlighted by x in (a). The corresponding transient time steps from (a) are indicated in addition. Similar to the thermal history at the leading edge of the cuboid geometry, the intervals of energy input and heat dissipation match between the build-up compensation and the corresponding point in the QS region with the return time characteristic. The localized energy input in the build-up phase also leads to higher peak temperatures at [C, D]. Ultimately, the build-up compensation is able to achieve the same underlying thermal state for melt pool formation \(\Psi (p) = \Psi (p_{\text {qs}})\) as in the QS region and the temperature curves of the first propagation segment match well.

The effect of the build-up compensation on the maximum melt pool distribution in comparison with the return time compensation is depicted in Fig. 8c and d. The maximum melt pool depth distribution of the geometry processed with the return time compensation in (c) shows regions with underdeveloped melt pool at every leading edge of the geometry. Especially at the center of the geometry, where turning point effects are not present, a melt pool with lower maximum melt pool depth emerges, that reaches up to 2 mm into the geometry. The maximum melt pool depth distribution of the geometry processed with the build-up compensation shows a significant reduction of regions with underdeveloped melt pool at the first leading edge [A, B] and at the edge within the geometry [C, D] and the desired melt pool depth is reached almost from the beginning of each respective edge.

3.3.1 Curved geometries

In case of the L-shaped geometry, the sizes of the available idle segments in previous lines match exactly with the length of the melt segment requiring compensation \(l_{\text {idle}} = l_{\text {melt}}\), and no subdivision of idle segments is necessary. In more complex geometries with curved contours, the length of the melt subsegment, that required compensation, can be smaller than the available idle segments in preceding lines. However, in these cases, the same approach can be used to derive the appropriate build-up segments and hatching sequence. This is demonstrated in Fig. 9 for a circle segment with an optimal build-up size of \(n_{\text {opt}}=3\).

Fig. 9
figure 9

Schematic of the build-up compensation to construct the build-up segments in the required hatch sequence for a curved surface

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With the construction of the reference geometry, melt subsegments, that required compensation, are identified along the contour of the geometry on both ends of each melt segment for increasing scan lengths. To achieve the proper hatch sequence and positioning of build-up segments, idle segments of preceding line are split into subsegments according to the boundary of the melt subsegments, that require compensation, as shown in Fig. 9. In case of the melt subsegment [A, B] in line 4, idle segments in \(n=3\) preceding lines (3, 2, 1) are split at the boundaries [A, B], as indicated in Fig. 9. The same approach is used for all remaining melt subsegments [C, D], [E, F] in lines 5, 6 for \(n=3\) preceding lines. While maintaining the original hatch sequence, all idle subsegments are converted into build-up segments and are shifted to the position of their corresponding melt subsegment according to their respective translation vector \(\mathbf {v_n}\), as shown in Fig. 9. The resulting hatch sequence satisfies the necessary conditions for the successful build-up compensation and enables the deposition of the right amount of energy during the required time interval of the hatch for complex outer contours and even holes inside the geometry.

Figure 10 details the spatio-temporal melt pool evolution of a circle segment processed using the build-up compensation. Analog to Fig. 8a, the build-up of the required thermal state takes place simultaneously at different locations at the edge of the geometry.

Fig. 10
figure 10

Spatio-temporal melt pool evolution for a circle segment processed with \(E_{\text {a}}={1.5}\,\textrm{J}\,{\textrm{mm}}^{-2}\), \(v_{\text {lat}}=\) 53 mms\(^{-1}\), \(lo=\) 100 \(\upmu\)m, \(T_{\text {p}}=\) 1023 K: current location and type of the segment are indicated in black; location of the previous segment is indicated in grey

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The effect of the build-up compensation on the maximum melt pool depth distribution of three complex model geometries processed with identical processing parameters (\(E_{\text {a}}={1.5}\,\textrm{J}\,{\textrm{mm}}^{-2}\), \(v_{\text {lat}}=\) 53 mms\(^{-1}\)) is demonstrated in Fig. 11 with respect to the standard cross-snake strategy and the return time compensation. The calculated melt pool depth distribution of the geometries processed with the cross-snake strategy shows a dependency of the maximum melt pool depth on the local return time of the beam. Pronounced regions with underdeveloped cumulative heating can only be identified in case of the square annulus geometry at the first hatch line and in the square hole in Fig. 11d, since the return time effects at the start of both circular geometries with short scan lengths in Fig. 11a, d overlay and conceal the build-up effect. The return time compensations strategy removes the influence of the local scan length on the maximum melt pool depth distribution and establishes a melt pool depth independent of the local scan length, All geometries exhibit pronounced regions with underdeveloped cumulative heating at leading edges, as detailed in Fig. 11b, e, h. With a calculated optimal build-up size of \(n_{\text {opt}}=4\), the build-up compensation leads to a significant reduction of all regions with underdeveloped cumulative heating at outer and inner contours of all three geometries, as shown in Fig. 11 (right). The desired maximum melt pool depth, according to the QS region, is reached almost from the edge of the contour and a defined spatio-temporal melt pool evolution, that produces the same maximum melt pool depth distribution, independent of the geometry, is achieved.

Fig. 11
figure 11

Maximum melt pool depth distributions calculated for a circle (top), a square (center), and circular annulus (bottom) with different hatching strategies, cross-snake (left), return time compensation (center), and build-up compensation (right) with identical processing parameters \(E_{\text {a}}={1.5}\,\textrm{J}\,{\textrm{mm}}^{-2}\), \(v_{\text {lat}}=\) 53 mms\(^{-1}\), \(lo=\) 100 \(\upmu\)m

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Figure 11 demonstrates the effectiveness of the build-up compensation to regions with underdeveloped cumulative heating in different model geometries. To achieve the optimal compensation ofregions with underdeveloped cumulative heating, n build-up segments are required. Therefore, n idle segments have to be located in the corresponding lines preceding the edge of the geometry, to provide the required amount of build-up repetitions. Hence, the minimum feature size for an optimal compensation is determined by \(f_{\text {min}} = n\times l_o\). Geometries, that do not fulfill these requirements, e.g., concave geometries with small intrusion widths, cannot be compensated optimally. However, in these cases, the reduced build-up size still leads to significant improvements of the regions with underdeveloped cumulative heating over the uncompensated case. The construction of the build-up segments and the global hatching sequence of the build-up compensation can also take the underlying local material conditions into account and offers the possibility for the local adaptation of the build-up size. If the numerical determination of the optimal build-up size is not possible, even the experimental determination of the optimal build-up size is possible. In case of the selected process parameter combination in Fig. 11 (right), there are still turning point effects visible in all geometries. To remove the turning point effects, the feed forward compensation can further be combined with the extension of the return times to achieve a completely uniform melt pool, according to a predetermined shape [15]. Remaining idle segments in the hatching sequence, which are not used for the compensation of regions with underdeveloped cumulative heating, can be further utilized for the preheating of different geometries.

4 Summary and conclusion

A model-based optimization problem was formulated based on a simplified thermal model with the goal to reduce the extent of regions with underdeveloped cumulative heating for line-based hatching strategy in PBF-EB through line-wise modification of the process strategy. With the goal to achieve a thermal state equivalent to the quasi-stationary state of the base parameter set, the obtained optimal solution takes the form a scanning strategy-based compensation. While maintaining all process parameters, the compensation strategy introduces a build-up phase, which reduces the line offset of a specific number of hatch lines at the beginning of the hatch to build-up the target thermal conditions at the leading edge of the geometry. Under the assumptions of the thermal model, the introduction of a build-up phase significantly reduces the extent of regions with underdeveloped cumulative heating and is able to achieve the target thermal conditions, that are present in the quasi-stationary region, at the leading edge of the geometry. While the exact number of line offset adaptations in the build-up phase depends heavily on the base processing parameter, geometry, material properties, and utilized thermal model, the introduced problem formulation enables the fast numerical calculation of the optimal build-up size under different circumstances.

A generalized scan strategy-based approach was developed, which integrates multiple build-up phases into the hatching sequence, to reduce regions with underdeveloped cumulative heating contribution in complex cross-sections. For this purpose, subsections of idle segments, which are integrated into the hatching sequence to locally extend return times, are converted into build-up segments. Build-up segments enable the deposition of additional energy at the required place and time within the hatch sequence, to achieve the desired thermal state without disrupting the remaining return time relationships of the hatch. The ability of this scan strategy-based approach to significantly reduce the extent of regions with underdeveloped cumulative heating in complex cross-sections was numerically demonstrated using four different complex model geometries.

Overall, the developed scan strategy-based approach further improves the ability to achieve predetermined homogeneous melt pool geometries and material properties, independent of the underlying geometry. In addition, the scan strategy-based nature of the compensation has the advantage of being independent of the underlying material and process parameters and is applicable to a large variety of complex geometries. The application of this approach, therefore, can enable the reduction of various edge effects, e.g., the reduction of material composition differences due to evaporation in TiAl or the reduction of the polycrystalline shell region in single crystal formation.