Concurrent shape and build orientation optimization for FDM additive manufacturing using the principal stress lines (PSL)

Abstract

Additive Manufacturing (AM) with the consisting constantly evolving technologies is a particularly popular research area. Based on the shape forming freedom, size, shape, and topology optimization techniques can be validated by AM produced parts. However, in every manufacturing process, AM also has some adverse inherent properties. One and maybe the most significant optimization problem is the mechanical anisotropy caused by the layered structure. In this paper, a simultaneous build orientation and shape optimization method is presented. Both of the approaches are intended to increase the mechanical performance of the produced parts. Shape optimization was accomplished by varying the cross-section of the beam geometries, based on the angle between a PSL section and the characteristic load direction. To test the efficiency and validate the method 2D structures (with relatively small 3rd dimension) and their tensile properties were tested. Based on the results, we can prove that the PSL method works and help to increase the mechanical performance by 19.2% with only 7.8% size increment.

1. Introduction

Recently, Additive Manufacturing (AM) and field-related optimization processes are gaining prominence due to the rapid development of these techniques, the improvement of manufacturing solutions, and the compulsion to the market demands. The new approach as the shape must follow the function, leads to continuous development and improvement of both AM solutions and geometric optimization of parts. One of the most exploited advantages of AM is the high degree of freedom in the forms that can be produced. It is often utilized to reduce volume and mass, while maintaining adequate stiffness or compliance. Although most related research works with major simplifications and often ignores the limitations of manufacturing technology. All AM processes are identical in that they produce a given component layer by layer by adding material. Sub-categories within AM are affected to varying degrees by this layered structure, but typically the main problems on which optimization procedures focus to the staircase effect, surface and dimensional accuracy, support structure, related costs, thermal conditions, or anisotropy. Therefore, the researchers aim to keep their adverse effects to a minimum. In most cases, the print orientation of the part is changed (relative to the printing plane or print tray), as this can directly suppress the formation of layers within the product. Pandey et al. [1] investigated the staircase effect of the 3D printed part, and found the most suitable print orientation that fulfills the preset requirements, using multicriteria genetic algorithm. They found that there are two limiting situations, one with minimum average part surface roughness with maximum production time and another is minimum production time with maximum average part surface roughness. Cheng et al. [2] also created a muti-objective optimization of the build orientation, where the primary objective was to find the best dimensional accuracy, and a secondary objective was to minimize the build time. Morgan et al. [3] developed a software, which aims to find the optimal orientation with minimal support structure required. As per the optimal build orientation considering a mechanical behavior, Cheng and To Ref. [4] investigated the residual stress inside the printed parts and found the orientation, where this adverse effect is minimal. Furthermore, since print orientation only allows the optimal version to be found from a limited set, this does not necessarily mean that individual objectives cannot be further optimized. Therefore, the next step would be the appropriate geometric optimization considering the effect of print orientation. Leutenecker-Twielsiek et al. [5] presented a framework of design guidelines, where they collected most of the necessary process characteristics, design principles, and design rules. Although, geometric and build orientation optimization are not coordinated in the present state of research, typically only one follows the other.

Geometry optimization is preferred in both discrete and continuous ways, resulting in a component shape that provides maximum stiffness while performing a given function. Since the generated form can be manufactured without modifications, the applied optimization techniques can be directly evaluated so the relationship is clear, between the virtual design and the physical part. One of the many topological optimization procedures is the Principal Stress Lines (PSL) is the latest approach, which is more frequently used, as the correlation of the basis of the shape optimization and the manufacturing process is rather simple. With the PSL technique, the force flow inside the part can be traced and the material distribution can be coordinated according to it. Sales et al. [6] used PLS to create a line-based topology optimization of 2D structures, where the printed lines built up the parts follow the PSLs. Similarly, Xu et al. [7] created a lightweight and low vibration amplitude web design method for gears. Based on their experimental results the optimized gear compared to the solid one is 20.5% lighter and has a 29.5% vibration reduction. With the utilization of a multi-axis AM solution Tam and Mueller [8] were able to create a spatial version of this kind of optimization technique. In their work, the deposited lines of the Fused Deposition Modeling (FDM) AM solution follow the PSL directions, which opens new possibilities for structural optimization.

However, to take full advantage of this new optimization technique, it should also take into account the harmful characteristics of the specific AM technology. The standard, ISO/ASTM 52900:2015 Additive manufacturing – General principles – Terminology, classifies the different technologies into seven categories. The anisotropic behavior resulting from the layered structure mainly affects Material Extrusion (ME) and Direct Energy Deposition (DED) techniques named by the standard. These are commonly used, since the cost involvement compared to some other techniques the assessment of spatial anisotropy is key to the PSL optimization mentioned above. To highlight this adverse effect many researchers investigated the differences in mechanical resistance of tensile loads in a parallel and perpendicular direction to the layers [[9], [10], [11], [12]], and also the compression loads [[13], [14], [15]]. Based on their results parts with tension loads perpendicular to the layers have less resistance, than with tension parallel to the layers, and parts with compression parallel to the layers have less resistance than with perpendicular to the layers.

In this paper, a complement solution for the existing shape optimization methods based on PSL has been proposed. With that the adverse effect of the layered 3D printed structures can be taken into account, thus the result of the optimization method gives a more reliable part.

2. Methodology

2.1. Assessment of anisotropy

To assess and authentically validate the presented solution, mechanical tests were performed on specimens manufactured with FDM technology. As it was presented in the introduction section the anisotropy affects more AM technologies, but FDM suffers the most of its adverse effects. Therefore, the proposed method is most substantial for this technology. PLA raw material was used to create the test specimens and the investigated parts as well since it is the most commonly used raw material for commercial FDM 3D printers, and the particular properties of the material were already investigated in many studies. Specifically, for the experiments, Prusament PLA Green filament has been used. Table 1 contains the general properties of PLA.

Table 1.

PLA material properties.

Material	PLA
Filament diameter [mm]	1.75
Printing temperature [°C]	190–220
Bed temperature [°C]	50–60
Density [g/cm3]	1,24
Colour	White
Tensile strength [MPa]	∼47
Tensile modulus [MPa]	2100

To produce the specimens Prusa i3 MK3S printer has been used. The printing parameters were 0.2 mm layer thickness, 0.4 mm nozzle diameter, 45° raster angle, 100% infill, 20 mm/s print speed, and 215 °C extruder temperature. The specimens were printed using only concentric “contour lines”, as it was proved to be the most reliable [16]. A standardized tensile test has been carried out based on the EN ISO 527–1:2012 Plastics. Determination of tensile properties. Zwick Roell Z100 universal test machine has been used, and each test have been done the same day in room temperature. In order to compare the effect of spatial anisotropy different specimen groups have been manufactured with different angle positions. It is known that the angle between the build plate is the most important, as this affects the position of the layers relative to the load direction, so the effect of the angle in the XY plane is negligible. Fig. 1 a) shows the tilted print orientation of the test specimens relatively to the build plate. Fig. 1 b) represents the best load case, which is parallel to the layers, and Fig. 1 c) is the worst load case perpendicular to the layers.

Fig. 1.

Test specimens printed in different angle positions to test the spatial anisotropy a) selected angle of rotation, b) most favourable layer orientation, and c) least favourable layer orientation regarding the tensile load case.

Since, it is known that for tension the mechanical resistance gradually decreasing as the angle is increased, in this work the counterbalance this phenomenon was the main goal. For most of the parts the pure linear tension cannot be achieved, therefore, at some locations inside the part the emerged stress points in an adverse direction. However, if the design domain allows it the loaded cross-sections can be modified to the appropriate extent, and with that the proportional resistance can be improved.

2.2. Computation of the PSLs

The computation of the PSLs follows the logic that is already well described in other research. Tam and Mueller [8] presented a work where they stated that with the FDM deposition along the principal stress lines a sort of topology optimization can be performed. The basis for finding these honored “force-flow” lines [17], or in the case of the computation problem the direction of the lines at a given node is to calculate the Principal Stresses (PS). PSs are the components of the stress tensor when the corresponding coordinate system is rotated in such a way, that the shear stress components become zero (Fig. 2).

Fig. 2.

Representation of the determination of principal stress directions.

The calculation of the stress angle with which the PS directions can be found is rather difficult, but for 2D cases, the problem can easily be solved. In this article, only 2D shapes were investigated, since the spatial anisotropy of the AM technology is only influenced by the angle between the parts and the build tray. At the combined solving equation (1), the principal stress orientation, θ_P, can be computed by setting τ′_xy = 0.

$$0=\left({\sigma }_{yy}-{\sigma }_{xx}\right)\mathit{sin}{\theta }_P\mathit{cos}{\theta }_P+{\tau }_{xy}\left({\mathit{cos}}^2{\theta }_P-{\mathit{sin}}^2{\theta }_P\right)$$ (1)

This gives (2)

$$\mathit{tan}\left(2{\theta }_P\right)=\frac{2{\tau }_{xy}}{{\sigma }_{xx}-{\sigma }_{yy}}$$ (2)

and the transformation matrix is (3)

$$\mathbf{Q}=\left[\begin{array}{cc}\mathit{cos}{\theta }_P& \mathit{sin}{\theta }_P\\ -\mathit{sin}{\theta }_P& \mathit{cos}{\theta }_P\end{array}\right]$$ (3)

To calculate the magnitude of the resulted normal stresses θ_P can be written back to the equations (σ₁, σ₂) (4)

$$\left({\sigma }_1,{\sigma }_2\right)=\frac{{\sigma }_{xx}+{\sigma }_{yy}}{2}\pm \sqrt{{\left(\frac{{\sigma }_{xx}-{\sigma }_{yy}}{2}\right)}^2+{\tau }_{xy}^2}$$ (4)

Or they could also be obtained by using ${\sigma }^{\prime }=\mathbf{Q}\bullet \sigma \bullet {\mathbf{Q}}^{\mathbf{T}}$ with Q based on θ_P.

After the calculation of the PS directions for each node inside the investigated domain, tracing the PSL must be performed. The assessment of the PSL is an incremental process. Generally, the aim is to connect the support point(s) or surface(s) with the loaded surface(s)/line(s). Therefore, the starting point of a PSL is located on the support, and the endpoint is located on the loaded surface(s)/line(s). The determination of the trajectory is the following: 1) A location inside the domain is determined by moving from the initial node to a given extent according to the PS direction of the initial node 2) the nearest node to this location will be the next member of the PSL, so it is connected with the initial node 3) the resulting node will be the next initial node 4) repeat until the formed PSL reach a load point or surface. In Fig. 3 the selection of the nodes can be seen. Here, an extracted small area with nine nodes illustrates the incremental trajectory building. As it can be seen if it is assumed that the first initial nodes are the three bottom ones, by stepping one unit in the σ₁ PS direction they can find the next node and connect them. Also, it can be stated from the figure, that the connecting line doesn't perfectly align with the PS direction, but has the smallest possible mismatch. Therefore, it can be pointed out that the density of the mesh and its uniformity play a critical role in PSL generation.

Fig. 3.

Formation of PSL

2.3. Problem formulation of the anisotropy for PSL based geometry

As it can be seen in the related works, the great benefit of the shape optimization based on PSL is to add material along the resulted lines, so the geometry follows the force flow. Generally, with traditional subtractive methods, the machining of such a shape would be unpleasantly difficult and costly, but with AM it can be performed without major manufacturing compliance while the optimized geometry could mean significant cost savings. Typically, the material is formed on the PSL as struts with a uniform cross-section. This kind of geometry formulation could be sufficient if we consider that the part is homogenous and has isotropic mechanical properties. However, since the FDM (and some other AM technologies) have spatial anisotropy, the generated struts that are positioned adversely could drastically reduce the mechanical resistance of the part. To balance this issue two solution has been performed simultaneously.

• 1)Selecting the optimal printing orientation of the part minimizes the unfavourable problem. Since the boundary conditions of the part are known, the direction of all the PSL can be concluded and the orientation where most of the Stress components are pointing in the appropriate direction can be chosen.
• 2)However, with the orientation selection only a reduction of the problem is possible. Since the relationship of the decrease in the mechanical resistance is known an angle-based cross-section optimization can be performed, with which it can be assured that the limitation of the AM technology does not affect the force-flow-based geometry.

It is worth to mention that the PSL-based optimization does not consider the magnitude of the stresses only its direction. Therefore, if there is a local stress maximum point within the domain, which could cause the failure of the part might remain in the optimized geometry as well. With the above-mentioned technique, only the manufacturing resulting issues are corrected.

In Fig. 4 the representation of a single PSL can be seen. The red arrows are showing the σ₁ and the blue arrows the σ₂ stresses. The concluded characteristic stress directions are represented with dashed lines.

Fig. 4.

Characteristic stress directions of a single PSL.

2.4. Investigated geometries

To test the efficiency of the proposed theorem two geometries were investigated. The first one is a single beam, similar to the tensile test specimen. With this shape, the verification of the cross-section modification can be performed. The second shape is a beam with a cut-out in the middle, which helps to check the efficiency of curved PSLs. Instead of using a standardized tensile test specimen, in this geometry the load paths will be curved, the relevance of the proposed optimization solution can be seen. The initial geometries are just the definition of the envelope of the design domain, as with the PLS technique the optimized shape takes a form of a structural member based shape. The initial geometries can be seen in Fig. 5 a) simple beam and Fig. 5 b) beam with a cut-out.

Fig. 5.

Investigated initial geometries a) simple beam, and b) beam with a cut-out.

2.5. Algorithm overview

To perform the build orientation and the shape optimization MATLAB R2022a software has been used. The FEA simulation, to reduce the repetitive manual import-export issue was also executed in MATLAB using the Partial Differential Equation Toolbox. Since both of the investigated geometries have a small thickness, the plane-stress method has been used for the numerical simulations. As was mentioned earlier the computation of the PSLs is highly dependent on the mesh quality, therefore, closely uniform linear tetrahedral elements have been used with small maximal element sizes. The preference of the linear elements over the quadratic elements also serves the minimization of the rounding error at PSL computation. After the numerical simulation, the algorithm computes the PS directions for each node. With this information first by the mean of the local directions, the characteristic direction is defined and interpreted for the whole part. This direction which is calculated by the means of each tension stress direction must point parallel with the printer's X-axis (Fig. 1.), in order to minimize the adverse layering effect. At the same time the tracing of all the PSLs which connect the supported and loaded edges (in 2D) is performed. This generates a large number of trajectories depending on the resolution of the mesh, so the PSLs must be reduced to a minimum in order to obtain an optimized strut structure (explained on the section below). Basically, the determination of the minimum number of PSL is affected by the number of boundary edges, eg.: fix supports and loaded edges, and whether the geometry is convex or non-convex. Once this is done, each PSL must be assigned an initial cross-sectional value which is swept through the PSL to obtain the strut geometry. Meanwhile, if the PSL is not straight or not parallel to the X-axis, the cross-sectional value should be varied locally according to the available anisotropy measurement results. Based on the simple correlation $\sigma =F/A$, if a cross-section fracture at lower force due to the adverse effect of the layering, with the increase of the cross-sectional area the same resistance can be achieved. The logic of the proposed algorithm can be seen in Fig. 6.

Fig. 6.

Proposed algorithm.

3. Results and discussion

3.1. Measured data regarding the FDM anisotropy

The results of the tensile tests can be seen in Fig. 7, the numbers in bracket means the serial number of the pieces in the sample group. It can be stated that in each three investigated angle orientations the curves are closely identical, and the deviation is minimal. Also, the material is experiencing plastic deformation throughout the test, but the initial slope of the curves, which determines Young's modulus according to the standard are equal for each nine tested specimens. Therefore, it can be concluded, that the angle position only affects the maximum tensile strength.

Fig. 7.

Tensile test results.

3.2. Results of PSL generation

The results of the PSL generation for both shapes can be seen in Fig. 8 a) and b). As it was assumed, inside the simple beam with only tension stresses the lines are pointing from the supported face to the loaded face parallel to each other. For making the generation faster and reducing the computation errors, some simplification can be made. If a shape has at least one plane or axis of symmetry, it is sufficient to plot the resulting force-flow inside only one of the reflective parts, which can be applied to the other side as well. However, by keeping both sides the inaccuracy caused by the mesh can be reduced, as the mean of each close-symmetric PSL can be traced. Therefore, in the case of the beam with cut-out, only the bottom half of the shape sliced with the horizontal symmetry axis was kept. The neglect of the bisection with the vertical symmetry axis made it possible to more accurately represent the PLSs formed on the two sides.

Fig. 8.

Resulted PSL in a) beam and b) beam with cut-out.

Only one typical PSL is required to perform the optimization, to which the initial cross-section can be assigned. As was mentioned, the number of necessary PSLs affects the number of supported and loaded edges and the shape of the part (convex/concave). In the case of the beam, the determination is simple, only one support and one loaded edge are connected with a single line in a convex part, so the necessary number of PSL is one. Likewise, the beam with cut-out has only one support and loaded edge, but is a non-convex part. Even though all the PSLs are starting and ending on the same edges, the trajectories go differently. This difference is more evident in the simplified version of the half-cut test form of the beam with a cut-out shape. In that form, one support and two loaded edges are associated, which leads to the conclusion that the number of necessary PSLs is two.

According to this, the determination was done by extracting the mean of the PSLs. Since all of the necessary PSL has been traced by forming an average PSL based on them a good representation of the simplified load path can be highlighted. Also, the local maximum point (at the inner corners in this case) have a collecting effect, and the distance between the PSLs at those points is denser. By this, later on using the optimization solution, these sharp corners could be smoothed, without knowing the actual local stress magnitudes. For the beam with cut-out, two distinct characteristic PSL lines that have the same start and end edge were used. The resulting trajectories are shown in Fig. 9 a) and b).

Fig. 9.

Characteristic PSL of a) beam and b) beam with cut-out.

3.3. Modified cross-section of beam shape

As it can be seen from the results of the beam shape all the PSLs are going vertically with the initially proposed orientation. Therefore, the optimal print orientation would be laid down on the build tray parallel with the X or Y axis (presented in Fig. 1.) of the longitudinal axis of the shape. Furthermore, since the characteristic PSL does not have any curvature, this print orientation could be the best for the mechanical resistance improvement. Conversely, the lack of curvature makes it a good tool to verify the cross-section optimization theory. Faced with the choice of the most favourable printing orientation, the pieces were examined in the angular positions used in the measurement of the anisotropy for the sake of the test. Theoretically, if a more complex shape has a section that has some area that has a sub-optimal PSL section, its cross-section should be increased to the necessary extent.

Fundamentally, the optimized shape must be able to bear as much load as possible. Here, the maximum means the load or amount of force that a φ = 0 deg PSL section can resist. Assuming that the initial cross-section (10 × 4 mm) is sufficient, it can be assumed that the φ = 15 deg and φ = 30 deg PLS sections should be optimized to have the same resistance as the laid piece. To achieve this the following calculation method can be used (5, 6):

$$A_{15\mathit{deg}}=\frac{F_{0\mathit{deg}}}{{\sigma }_{15\mathit{deg}}}$$ (5)

$$A_{30\mathit{deg}}=\frac{F_{0\mathit{deg}}}{{\sigma }_{30\mathit{deg}}}$$ (6)

The data used for the calculation can be seen in Table 2.

Table 2.

Calculated modified cross-section for beam shape.

	0 deg	15 deg	30 deg
Force [N]	2437	2209	2026
Aoriginal [mm2]	40	40	40
σ [MPa]	60.9	55.225	50.65
Amodified [mm2]	40	∼44	∼48

Even though the optimization technique was performed on 2D shapes, for the verification a 3D geometry had to be used. The thickness of the pieces remains the same (4 mm) to make it possible to check the efficiency of the cross-section change with a renewed tensile test. Consequently, the change in cross-section remains proportional to the “thickness” of the planar PSL. In the case of the φ = 15 deg piece, the increase is 10% and for the φ = 30 deg the increase is 20%. The results of the modified tested pieces can be seen in Fig. 10 a) and b) including ±2σ.

Fig. 10.

Results of the beam shape with different cross sections a) original pieces, b) modified pieces.

It can be said that the increased percentage demanded of the loaded surface area is equal to the percentage decrease in the measured force. If linearity is assumed, the cross-section increase demand of the positions between the tested angles can easily be obtained, also the necessary increase for higher angles.

3.4. Modified cross-section of beam with cut-out shape

The obtained proportional cross-sectional increase was used to optimize the beam with a cut-out shape. Here, proportionally with the angle between the PSL section and the characteristic PSL many cross-section segments have been prepared. Therefore, for each segment of the PSL a unique cross-section was created. For the smooth transition the thickness change is proportional, taking into account that the resulting contours should not be less than the required thickness for the given segment. For comparison, apart from the optimized shape, a simple version with a uniform cross-section was created. The representation of the resulted shapes can be seen in Fig. 11 a) and b).

Fig. 11.

Generated cross sections for the beam with cut-out shape a) with uniform cross-section, and b) optimized cross section.

As for the initial cross-section 10 mm has been chosen. This is advantageous because the geometry formed by the two PSLs present is connected to each other in the constricted parts (at the top and lower ends). Given that the traced PLSs are not perfectly smooth, due to the resolution, each surface has been rounded to avoid the resulting undesired stress peaks. The results of the tensile test can be seen in Fig. 12.

Fig. 12.

Results of the beam with cut-out shape with uniform and with optimized cross-sections.

It can be seen that the resistance of the specimen with an increased cross-section correspondingly to the unfavourable PSL increased. In contrast, the most striking change can be seen on the broken pieces (Fig. 13 a) and b)). In case of the pieces with uniform cross-section, the fracture always started at the same location closer to the grips. Based on the initial finite element analysis it was assumed since this was the location with the highest stress. However, as it can be seen in the case of the pieces with optimized cross-section, some fracture started at the other side of the curved struts, which leads to the conclusion that the optimization was successful and the two ends has the same mechanical resistance. Therefore, only the manufacturing inaccuracy can be the reason of the different fracture locations.

Fig. 13.

Broken test pieces of the beam with cut-out shape a) uniform cross-section, and b) optimal cross-section.

The three pieces that have the uniform cross section along the entire PSL have been broken at the point where the PSL has angle between the characteristic load direction different than 0 deg. According to the theory, this is an expected result, and the location of the first breakpoint is also influenced by the local stress maximum. Two out of three pieces of the optimized cross-section were broken at the unaltered segment of the cross-section and the third was broken at the segment with the angular position similarly to the control pieces. This means that the optimization has resulted in the fact that the impact of the production technology no longer affects the failure. Compared to the additional surface and force increase can be seen in Table 3.

Table 3.

Area and force increment.

	Uniform	Optimized
Area (mm2)	3390.2	3657.2
Force (N)	1534.4	1829

As a summary, it can be stated, that with a 7.8% 2D area increment the mechanical resistance was improved by 19.2%.

3.5. Limitations of the proposed method

Based on the mechanical test, it can be concluded that the algorithm and the used methods can lead to improved tensile properties. However, there are some concerns and limitations that must be taken into account.

• 1)In this paper, only the tensile load cases were investigated. Even at the beam with a cut-out shape due to the non-uniaxial load, some compression stresses emerged, but because of the geometry they were negligible, since only a few nodes had compression dominancy and in general the magnitude of the compression stresses were small. Therefore, the assessment of the adverse direction for compression stresses was not taken into account.
• 2)The used PSL method only considers the load path, regardless of the magnitude of the stress at the nodes. Some nodes with the local maximum point could be a part of more generated PSL, but overall we don't have information about the critical points of the domain while computing the characteristic PSL. Thus, the local maximum point will remain, which could lead to fracture.
• 3)The traced PSLs are related to the initial geometry, so if the material is too flexible and the part is affected by too much deformation the computed optimum cross-sections wouldn't be valid anymore. In this research the used PLA material is considered a brittle one, therefore, it is assumed that it doesn't show any major deformation prior to failure.
• 4)In the cross-section generation and optimization part of the algorithm, it doesn't take into account the predefined design domain, thus it may form material in areas outside of it. In this case, moving the characteristic PSL or weakening the rigidity of the area may be the proposed solution.

4. Conclusion

In this paper, a concurrent build and shape orientation optimization algorithm was presented. By tracing the PSLs the load paths of the parts can be detected, and based on that the optimal printing orientation, where the adverse effect of the layered structure is minimal can be selected. With this, the mechanical resistance (tensile) can be improved to some extent, but in order to further increase this, a shape optimization technique was presented. The recently trending optimization techniques, where the parts are constructed as beam geometries the maximal stiffness can be achieved by minimizing the mass. With the presented extension of this study, these methods can be supplemented, to match with the manufacturing technological peculiarities. To get better mechanical performance, higher measured force varied cross-sections were used for the beam structures. The size of the cross-sections corresponds to the angle between the section of a PLS and the characteristic load direction. Future work: In this paper, only 2D geometries were investigated and only the adverse effect of the tension loads was considered. In the subsequent works, the proposed method must be extended to 3D geometries and also make an assessment of the compression effects. The proposed method can be used for any deposition based AM technology (DED, LOM, CFF), where the remarkable anisotropy exists. Furthermore, the already existing geometry optimization methods, such as lattice meso-structure generation or generative design can be extended by our method for the anisotropic cases.

Author contribution statement

Márton Tamás Birosz: Conceived and designed the experiments; Performed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data; Wrote the paper.

Ferenc Safranyik: Performed the experiments.

Mátyás Andó: Analyzed and interpreted the data; Wrote the paper.

Funding statement

Project no. TKP2021-NVA-29 has been implemented with the support provided by the Ministry of Innovation and Technology of Hungary from the National Research, Development and Innovation Fund, financed under the TKP2021-NVA funding scheme.

Data availability statement

Data will be made available on request.