Minimizing part deformation in laser deposition by evolutionary optimization of laser beam path

Additive manufacturing (AM) technology provides fabrication of complex, custom-designed components and represents a key technology in the ongoing digital transformation, especially within the framework of Industry 4.0 (Blakey-Milner et al., 2021; Madhavadas et al., 2022). Laser-based additive manufacturing (LBAM) has emerged as a prominent approach within AM, and the key LBAM techniques, such as selective laser melting (SLM) (Lupi et al., 2023) and laser-based directed energy deposition (DED-LB) (Shamsaei et al., 2015; Svetlizky et al., 2022; Thompson et al., 2015), stand out for their ability to melt and deposit material layers with a low heat-affected zone, enabling the fabrication of complex three-dimensional structures.

Delamination and cracking during the AM process, as well as deformations such as warping, bending and overall geometrical inaccuracies of AM parts—which often cannot be eliminated even by heat treatment—remain key problems in LBAM. The rapid heating and cooling cycles inherent to LBAM, caused by the melting and solidification of metal powder, induce large temperature gradients. These, in turn, generate residual stresses that can compromise both the stability of the AM process and the dimensional accuracy and structural integrity of the final part. Therefore, controlling the temperature distribution during the deposition process is critical for producing high-quality and geometrically accurate 3D-AM parts and can be achieved by optimizing the laser beam (LB) path.

Numerous studies have demonstrated that the LB path significantly influences temperature distribution and, consequently, the magnitude and pattern of deformations in printed parts. Early work by Mercelis and Kruth (2006) identified residual stresses as a primary source of deformation in SLM-manufactured components and emphasized the laser scanning strategy as one of the key parameters affecting stress profiles. A comprehensive review by Wang et al. (2022) confirms that the path-dependent residual stress and deformation are two critical defects of the parts fabricated by direct energy deposition and that the optimal design of the scanning strategy is an important controlling method. Experimental studies confirm the strong correlation between scanning strategies and residual stress distribution, showing how these affect resulting part deformations (Bian et al., 2020; Carraturo et al., 2020).

The importance of integrating optimization techniques and predictive simulations to minimize residual stress and distortion has also been emphasized in recent studies (Dar et al., 2025; Potočnik et al., 2024; Twumasi et al., 2025). Liu et al. (2024) designed a parameter optimization algorithm to adjust the local scanning parameters to avoid defects and thus enhance the performance consistency of the formed parts. Twumasi et al. (2025) proposed a physics-guided, machine-learning approach to optimize scan paths for desired microstructure outcomes, using a surrogate machine learning model to reduce computational costs and deep reinforcement learning to generate optimized scan paths for target microstructure. These studies highlight the critical need for advanced numerical frameworks that incorporate deformation minimization into scan path planning—an objective directly addressed in the present work.

Evolutionary algorithms have been introduced for generating and optimizing DED-LB deposition head paths in AM (Nassehi et al., 2015), and various other approaches have been proposed for tool path planning, aiming to optimize temperature distribution in the built part (Ding et al., 2014; Ren et al., 2019; Ren et al., 2019). Evolutionary methods were also applied to enhance the thermo-mechanical properties in wire-arc AM by generating continuous optimized tool paths (Zhou et al., 2022). The impact of different scanning strategies on the thermo-mechanical behavior and geometrical accuracies of fabricated parts using SLM was examined in Jia et al. (2023).

Part distortion and geometrical inaccuracy remain a significant challenge in AM; therefore, novel slicing and path planning algorithms, such as non-planar AM, are promising potential tools for improving part accuracy when integrated into the pre-processing phase of 3D printing (Schaechtl et al., 2022). A novel adaptive toolpath generation algorithm in laser powder bed fusion was proposed by Qin et al. (2023). The method minimizes thermal gradients to reduce part distortion, incorporating collision-free and smoothing constraints combined with an island-based strategy, and experimental results show significantly reduced distortion compared to traditional methods, demonstrating effectiveness in AM of complex 3D parts. A multi-objective optimization framework was proposed by Jabón et al. (2023) in the combination of AM with topology optimization to reduce support structures by 30% on average while maintaining mass and only slightly decreasing part stiffness.

Xu et al. (2024) proposed stress-based continuous planar tool path planning for AM, which also presents a novel continuous planar path planning algorithm based on the principal stress orientation field to improve the structural performance of AM parts. The tool paths are generated with a genetic algorithm (GA) by minimizing the idle travel distance of the nozzle, and experimental results demonstrate that the objects fabricated with the proposed method have superior structural performance.

In order to avoid extensive experimental parameter studies in the AM design phase, an AM process-specific design based on numerical simulation is favorable. Poulhaon et al. (2024) proposed an incremental inherent stress model for the fast prediction of part distortion, which simulates the AM process bead-by-bead, accounting for thermo-mechanical coupling and inherent stress evolution, and the study demonstrated massive computational reduction with acceptable accuracy in predicting part distortion trends. Thermo-mechanical modeling of the DED-LB process has been discussed in various contexts. Temperature and residual stress prediction was investigated through finite element modeling (FEM) of the process in the case of a multi-material part, and a good correlation was found between the experimental and simulation results (Li et al., 2020). Temperature-induced residual stress and the resulting deformations are most commonly investigated in the DED-LB process, and finite element simulations have proved a good level of agreement between the temperature profiling and the respective distortion tendencies (Yang et al., 2016). A three-dimensional finite element approach was discussed for identifying the best DED-LB scanning strategy. Temperature histories and residual stresses were analyzed and validated with experimental results to identify the optimal DED-LB process parameters (Zhou et al., 2022).

This study builds upon previous research (Potočnik et al., 2024), where a method for GA-based optimization of LB path in AM was proposed. GA-based optimization was used to find the optimal LB path on a metal substrate to minimize a fitness function, which included thermal fitness (average thermal gradient) and process fitness (suitability for LBAM). In this study, the optimization framework is extended to directly optimize the deformation of the manufactured workpiece. The optimization method and the modifications applied in this study are described in Section 2. The estimation of the workpiece deformations is obtained by FEM thermo-mechanical simulations, as described in Section 3, and the results of the simulations are presented in Section 4. Finally, the conclusions of this study are summarized in Section 5.

The results of this research demonstrate that the GA-based LB path optimization can improve the structural response of the considered metal plate, e.g. decrease in the plate deformation. The proposed LB path optimization framework provides a systematic and automated approach to optimizing the LBAM process.

The method of GA-based optimization of LB paths (Potočnik et al., 2024) is used as an approach to enhance the properties of the AM part. The method involves generating LB paths by searching for the optimal sequence of LB irradiation that minimizes a fitness function, comprising thermal and process fitness components. Various path generators are applied alongside standard ones (e.g. raster, zigzag) to define the initial population of path solutions. A GA-based LB path optimization framework consists of the initialization of LB path solutions and the application of crossover and mutation operators through subsequent GA generations. The LB paths are progressively optimized according to the fitness function, and the method provides an automated approach to an efficient LB path selection in LBAM.

In this study, DED-LB-based deposition on a metal plate is considered, and the deposition geometry (on a grid of 30 × 30 elements, i.e. cells) consists of a frame at the borders of the metal plate, as shown in Figure 1. Figure 1a presents the design mask, Figure 1b the matrix formulation of an example of the LB path with numbers (visible in the zoomed area) denoting the subsequent LB locations, and Figure 1c shows the corresponding LB path, composed of two laser deposition segments, denoted in blue and red.

In general, a matrix formulation of LB paths can be used for the description of different initial LB paths, representing the initial population of solutions in GA-based optimization. However, for the manipulation of LB paths by GA operators (crossover, mutation), it is more suitable that the LB paths are represented by strings (Potočnik et al., 2024).

The GA provides a global optimization solver for smooth or non-smooth optimization problems with any type of constraint (Goldberg and Holland, 1989), and GA-based methods have already been proposed in the context of AM (Liu et al., 2021; Moussa and ElMaraghy, 2020; Vaissier et al., 2019). The GA starts by creating an initial population of solutions, and then this population is repeatedly modified by crossover and mutation operators. In each generation, the best solutions are selected according to the fitness function to produce children for the next generation, and the population evolves over successive generations toward an optimal solution.

Various standard path generators (raster, zigzag, spiral, etc.) are used to define the initial population of LB path solutions. Additionally, a stochastic-based search path generator was introduced (Potočnik et al., 2024), which is designed to search for random and preferably long continuous paths within the available path design topology (determined by the design mask).

In contrast to our previous study (based on thermal and process fitness), the fitness J in this research is based on the evaluation of the structural fitness of LB path solutions, which is expressed as the maximum deformation of the AM part in the vertical direction (δ_z):

Low values of the fitness J correspond to overall low deformation of the workpiece, and therefore the optimization procedure aims at minimizing the fitness value J.

The crossover operator combines genetic information (i.e. segments of LB paths) from two parent solutions and generates one or more offspring. Figure 2 illustrates the crossover operation by combining two parent solutions (Figure 2a and b) into a child LB path (Figure 2c).

The mutation operator introduces additional diversity into the population by randomly altering the genetic material of the selected parent LB path to generate a child LB path. Figure 3 shows an example of a mutation operation that introduces small random changes to explore the search space and potentially find better solutions. To show the progression of the LB path, the path segments in Figures 2 and 3 are denoted with blue for the first deposited segment, red for the last segment and shades of purple for the intermediate segments.

In this study, the following modifications (with respect to our previous study (Potočnik et al., 2024)) were introduced to implement the optimization framework based on the FEM thermo-mechanical simulations.

1. Fitness in this study (as explained above) consists of structural evaluation only (e.g. maximum workpiece deformation).

2. Elimination of singleton deposition points, namely path generators, crossover and mutation operators, can (due to the stochastic nature of operations) introduce LB paths with several disconnected points (singletons). This is unfavorable both from the point of view of LBAM process implementation, as well as for the FEM simulations, where a single-point deposition cannot be implemented, and laser-based depositions are only available as lines. Previously, the process fitness was applied to penalize unfavorable LB paths, and in this study, the process fitness is directly integrated into the path-generating procedure, which is upgraded as follows:

   1. Identify singletons in an LB path.

   2. For each singleton, search for adjacent path segments and try to connect the singleton to the beginning or end of one of the adjacent path segments.

   3. Repeat the procedure until all singletons are fused to adjacent path segments.

3. Adding random path generators (in combination with the previous point (b)) provides additional expressiveness of the GA optimization in the exploration of the available space of potential solutions.

The principal equations that are a basis for the thermo-mechanical FEM in this study are presented below. The equation for heat conduction through a solid is given as follows (Li et al., 2020; Megahed et al., 2016):

where $k$ is the thermal conductivity (⁠$W/K.m$⁠), $T$ is the temperature (K), $q$ is the rate of heat flux/convection/radiation/internal heat generation inside the volume (W), $ρ$ is the density of the material (⁠$kg/m3$⁠), c is the specific heat of the material (⁠$J/kg.K$⁠), $t$ is the time (s) and $▽$ is the divergence. $k▽2T$ represents the rate of heat conduction and $ρc∂T∂t$ is the rate of energy storage inside the volume, which represents the transient term in the simulation.

Temperature and heat flow analysis from the transient thermal model are required to compute the thermal expansion in the entire domain and thus define the thermo-mechanical properties. The mesh type and geometry remain the same throughout the simulation. The Lagrangian framework has been applied, as it has been widely adopted for modeling deformation in such applications. The governing mechanical equilibrium equation for stress is given as:

where $σ$ is the stress tensor associated with the material behavior law and $fint$ is the internal forces. Strain and stress tensors are linked by the equation:

where $C$ is the fourth-order material stiffness tensor and $ϵe$ is the second-order elastic strain tensor. Total strain tensor $ϵ$ is composed of three components namely elastic strain tensor (⁠$ϵe)$⁠, plastic strain tensor (⁠$ϵp)$ and thermal strain tensor (⁠$ϵth)$ as follows:

where $E$ is the Young’s modulus, $v$ is the Poisson’s coefficient, $g(σy)$ is a function associated with the material behavior, $σy$ is the yield stress, $α$ is the thermal expansion coefficient, $θ$ is the nodal temperature and $θo$ is the initial temperature.

The described thermo-mechanical model is used in an FEM simulation, where a substrate of 100 × 100 × 2 mm is chosen and the DED-LB cladding is performed in this area. The clad is in the form of a square strip of dimensions 90 mm × 90 mm, as shown in Figure 1a. The height of the deposited clad is 1 mm. G-codes are created to define the various user-defined LB paths and are given as inputs to the model. Both the substrate and the deposited material in the form of a single clad were meshed with hexahedral elements of 1 mm thickness. Multiple element sizes were tested, and it was observed that the current size had a good balance between the accuracy and the computational load required for simulations. This meshing resulted in an element count of 21,992. In the coordinate system, deposition is defined in both x and y directions, while the z-axis defines the clad height. The material deposition rate for the simulation is set as 7.5 mm³/sec, and the initial cladding started at room temperature (23 °C). One of the sides of the substrate is constrained to restrict its deformation. The substrate and the deposited material are SS316L, which has a melting point of 1,370 °C. The simulation parameters were intentionally selected to amplify deformation effects (e.g. thin substrate, high deposition rate, minimal constraints) in order to provide a clear proof of concept for evaluating the impact of LB path optimization. Figure 4 shows the AM part, which consists of a thin substrate and a deposited layer, with an example of the deposited clad pattern indicated in green.

FEM simulations were performed in Ansys Mechanical. The simulation process involves transient thermal simulations followed by static structural simulations. Temperature distribution, heat flow and heat flux are computed during the transient thermal simulations. Mechanical behavior aspects such as deformation, stress, strain, etc. are the output of static structural simulations. The element birth and death technique is used to simulate the DED-LB progression. The LB diameter of 1.5 mm is represented by 3 elements, and 1 mm thickness of the clad layer is represented by 1 element. Elements of the substrate are all active prior to the simulation, whereas the elements of the clad layer are activated according to the provided LB path.

The above-described FEM simulations were combined with the GA-based optimization framework presented in Section 2. The initial population of solutions was defined by 9 different LB paths, which consisted of standard paths (raster, zigzag, spiral-in, spiral-out) and several randomly generated paths. Starting from the initial population of solutions, the GA-based evolution was performed. For each LB path solution, the FEM thermo-mechanical simulation was conducted to estimate the maximum substrate deformation (after the finalized deposition), which provides the fitness value J (Eq. (1)) of the solution. After the evaluation of the fitness function J of the initial population, i.e. the first generation of 9 LB path solutions, the next generation was generated by using the crossover and mutation operators. GA-based evolution was followed up to 15 generations. In each generation, novel LB path solutions were generated, and successful solutions were propagated into the subsequent generations by the elitist strategy.

Figure 5 shows an example of an undeformed part (grey color) before deposition and a deformed AM part (in colored scale) showing deformation δ_z of the part after the DED-LB process. The largest deformation δ_z occurs on the right side of the workpiece due to the clamping of the substrate on the left side to the fixed support.

Figure 6 shows GA-based evolution results and the best fitness values for 15 GA generations. For each generation, a boxplot shows the minimum, 1st quartile, median, 3rd quartile and maximum fitness values (outliers are marked by red crosses). In the initial population (1st generation), the spiral-in LB path resulted in the highest deformation (δ_z = 6.7 mm), and the best result (15th generation) showed reduced deformation (δ_z = 4.9 mm), which is a 33.7% improvement with respect to the initial LB path with the highest deformation and an 8.5% improvement with respect to the best LB path (δ_z = 5.4 mm) of the 1st generation. The green line in Figure 6 shows the minimum fitness (deformation) of each generation. Although the generation mean value of J does not decrease monotonically with subsequent generations (due to exploration), we can conclude from the monotonic decrease of the minimum fitness J that the GA-based optimization of the LB path provides a useful tool for minimizing the deformation of the AM part in the laser-based AM process.

From the literature (Ren et al., 2019), it is known that temperature distribution in the built part is closely connected to mechanical deformation. With respect to this, temperature distribution of the plates at the end of the AM process at time 271.4 s is shown for the best GA-based LB path with the lowest deformation (δ_z = 4.9 mm) and spiral-in LB path with the highest deformation (δ_z = 6.7 mm) in Figure 7a and b, respectively. Color bands in the figure indicate that the plate with the least deformation (Figure 7a) exhibits higher temperatures at the end of the AM process when compared to the plate with maximum deformation (Figure 7b). A maximum temperature of 877.1 °C was observed for the plate with the least deformation and 806.5 °C for the plate with the highest deformation.

Deformation phenomena appear throughout the printing process, but large gradients during the cooling phase considerably affect the deformation behavior (Kim et al., 2014). Temperature conditions are further characterized by an evaluation of the maximum temperature of both plates throughout the cooling phase (starting at time 271.4 s), as presented in green and red color lines in Figure 7 for the best LB path and the worst LB path. Temperature plots for both LB paths indicate that a higher and rapid heat dissipation in the workpiece (red line) might contribute towards a higher deformation, whereas the workpiece with the low deformation dissipates heat at a lower rate and tends to stay at higher temperatures longer (green line). Higher cooling rates lead to a higher thermal gradient, thereby increasing the chances of deformation leading to warping. The best (GA-generated) LB path enables slower cooling of the workpiece, thus ensuring uniform heat dissipation throughout the surface, and the results obtained are in agreement with the mechanism discussed in the literature (Bai et al., 2023).

Figure 8 presents the temporal evolution of the maximum temperature over the free edge of the workpiece which is furthest away from the fixed support. The temperature plots are shown for the LB paths with a) the best fitness (final GA result) and b) the worst fitness (spiral-in). The peaks indicate the melting point (1,370 °C) at which the element cluster is generated. The best LB path reveals more dispersed temperature peaks compared to the worst LB path which results in only a few condensed high-temperature areas. The plots also display the moving minimum values of the temperature in blue color. The continuous path (spiral-in) does not enable the distribution of heat uniformly throughout the plate as it follows a fixed path with no scope for flexibility to alter its characteristics. Whereas the optimal path (GA result) enables uniform and balanced distribution of heat throughout the plate. The optimal path is not continuous as it is comprised of a multiple-segmented path, and these segments enable a higher level of flexibility with respect to location and direction, such that material deposition is arranged in a more distributed way throughout the plate. It can be presumed that the more distributed temperature peaks at the furthest edge from the fixed support result in overall smaller workpiece deformation.

This study presents an application of a GA-based evolutionary approach for optimizing LB paths in LBAM. The method aims to minimize the deformation in the AM part by automating the selection of optimal LB paths. Numerical FEM thermo-mechanical simulations validated the effectiveness of this approach, showing a reduction in deformation of a substrate compared to the initial (standard and random) LB paths. The best result of the 15th generation showed reduced deformation (δ_z = 4.9 mm) which is a 33.7% improvement with respect to the initial LB path with the highest deformation, and an 8.5% improvement with respect to the best LB path of the 1st generation.

The proposed approach offers a systematic and automated method for LB path optimization, which represents an advancement over existing techniques. The study highlights the important role of optimizing LB paths in enhancing the overall quality and efficiency of the LBAM process. By optimizing the sequence of DED-LB, based on defined fitness criteria, the study provides evidence of improved structural outcomes in LBAM. The current study provides (via thermo-mechanical analysis) evidence of improvement in terms of deformation minimization for the considered AM part (cladded metal plate). Although the research is currently limited to FEM thermo-mechanical simulation results, it lays a foundation for future experimental validation to further verify and refine the proposed GA-based optimization strategy.

Future work will focus on experimental validation of the proposed optimization framework, including the fabrication of physical samples using the optimized LB paths and the assessment of resulting deformation and microstructure. This will allow for verification of simulation results and exploration of how the LB path influences grain morphology, residual stresses and mechanical properties of the part.

Overall, this study contributes valuable insights into the automated optimization of LB paths in AM by presenting a framework that has the potential to enhance LBAM processes and improve part quality by minimizing deformation.