Warpage Analysis and Control of Thin-Walled Structures Manufactured by Laser Powder Bed Fusion

Thin-walled structures are of great interest because of their use as lightweight components in aeronautical and aerospace engineering. The fabrication of these components by additive manufacturing (AM) often produces undesired warpage because of the thermal stresses induced by the manufacturing process and the components’ reduced structural stiffness. The objective of this study is to analyze the distortion of several thin-walled components fabricated by Laser Powder Bed Fusion (LPBF). Experiments are performed to investigate the sensitivity of the warpage of thin-walled structures fabricated by LPBF to different design parameters such as the wall thickness and the component height in several open and closed shapes. A 3D-scanner is used to measure the residual distortions in terms of the out-of-plane displacement. Moreover, an in-house finite element software is firstly calibrated and then used to enhance the original design in order to minimize the warpage induced by the LPBF printing process. The outcome of this shows that open geometries are more prone to warping than closed ones, as well as how vertical stiffeners can mitigate component warpage by increasing stiffness.


Introduction
Additive manufacturing (AM) is an industrial process increasingly integrated in the production chain. Nowadays, the aerospace and aeronautical industries are "printing" several high-performance metallic components [1,2]. Laser Powder Bed Fusion (LPBF), also known as Selective Laser Melting (SLM), is the most widely adopted AM technology to fabricate complex structures through a layer-by-layer metal deposition sequence, as it provides remarkable geometrical accuracy and satisfactory mechanical properties from a wide range of metallic powders [3,4]. The LPBF process is carried out in a closed chamber with controlled atmosphere (e.g., argon) to prevent oxidation. The 3D-printing process consists of the following steps: (i) a new powder layer (30~60 µm) is spread, (ii) the highenergy laser beam selectively melts the power bed according to a user defined scanning sequence, and, (iii) the base-plate is lowered to allow for the deposition of a new powder layer. This sequence is repeated until the building process is completed. At the end of the AM process, the loose powder is removed, allowing for the final cooling stage at ambient temperature.
In AM, the material undergoes repeated heating and cooling cycles, high temperature gradients are generated and, consequently, large thermal deformations are induced. The temperature gradient between layers produces residual stresses and plastic deformations because the material cannot freely expand and contract. As a result, the accumulated

Experimental Campaign
The experimental campaign is carried out using a Concept Laser M2 powder bed machine (Concept Laser, Lichtenfels, Germany). This equipment uses a Yb-fibre laser with a maximum input power of 400 W (D4 Sigma Gaussian beam). The building process takes place in a closed chamber (250 × 250 × 280 mm 3 ) with an annealed Ti-6Al-4V base-plate (250 × 250 × 25 mm 3 ) and a controlled atmosphere of pure argon (the oxygen content is restricted to 100 ppm).
In this research, the gas atomized Ti-6Al-4V powder used for building has a spherical shape, with a size of 15~53 µm. The chemical composition of this powder is given in Table 1. Before the printing process, the powder is dried in a vacuum oven (Fengxun, Shanghai, China) (vacuum degree less than 80 kPa) at 125 • C during 2.5 h. The 3D-printing is characterized by a 90 • rotating scan pattern. Table 2 reports the processing parameters adopted for the manufacturing process. In addition, no preheating is used in this work.  Figure 1 shows the thin-walled structures printed on the LPBF base-plate after removing the loose powder. These include: • Single-wall structures of different thicknesses and heights; • Cylindrical structures of different diameters and heights; • Square-section structures of several thicknesses; • Open-section structures (e.g., semi-cylinder, L-shape, etc.).  The 3D-printing is characterized by a 90° rotating scan pattern. Table 2 reports the processing parameters adopted for the manufacturing process. In addition, no preheating is used in this work.  200 30 1000 100 100 Figure 1 shows the thin-walled structures printed on the LPBF base-plate after removing the loose powder. These include: Open-section structures (e.g., semi-cylinder, L-shape, etc.).  Figure 2 shows the Breuckmann SmartSCAN3D scanner (Aniwaa, Central Singapore, Singapore) with the measurement accuracy of 0.015 mm, used to measure the actual distortion of all the components after the 3D-printing process. The relative component warpage is calculated by the Geomagic Qualify software (3D Systems, Rock Hill, Washington, DC, USA) by comparing the experimental 3D-scan images with the nominal CAD geometries. It should be mentioned that all the measurements are done before the removal of the substrate (cutting phase).  Figure 2 shows the Breuckmann SmartSCAN3D scanner (Aniwaa, Central Singapore, Singapore) with the measurement accuracy of 0.015 mm, used to measure the actual distortion of all the components after the 3D-printing process. The relative component warpage is calculated by the Geomagic Qualify software (3D Systems, Rock Hill, Washington, DC, USA) by comparing the experimental 3D-scan images with the nominal CAD geometries. It should be mentioned that all the measurements are done before the removal of the substrate (cutting phase).

Scale Effect on the Part Warpage
In this section, the sensitivity to the dimensions (e.g., wall-thickness, building height and component size) of the thin-walled components is examined. Therefore, the warpage is measured by 3D-scanning to obtain its relation with the actual dimensions of the 3D-printed structures.
The first assessment study considers a single-wall geometry with a base length of 50 mm and a total height of 50 mm. Three thicknesses of 1 mm, 2 mm and 5 mm, respectively, are investigated. Figure 3 compares the final warpage distributions as measured for the three wall thicknesses used. It can be seen that the thinnest wall presents the more pronounced warpage, up to 0.41 mm in the middle of the final cambered surface. As the wall-thickness increases, the warpage rapidly decreases; the maximum out-of-plane displacement for the 5 mm thickness wall is only of about 0.03 mm. Hence, as the thickness is increased, the wall warpage is restrained by the enhanced structural stiffness of the component. Simultaneously, higher residual stresses are induced by the printing process.

Scale Effect on the Part Warpage
In this section, the sensitivity to the dimensions (e.g., wall-thickness, building height and component size) of the thin-walled components is examined. Therefore, the warpage is measured by 3D-scanning to obtain its relation with the actual dimensions of the 3D-printed structures.
The first assessment study considers a single-wall geometry with a base length of 50 mm and a total height of 50 mm. Three thicknesses of 1 mm, 2 mm and 5 mm, respectively, are investigated. Figure 3 compares the final warpage distributions as measured for the three wall thicknesses used. It can be seen that the thinnest wall presents the more pronounced warpage, up to 0.41 mm in the middle of the final cambered surface. As the wall-thickness increases, the warpage rapidly decreases; the maximum out-of-plane displacement for the 5 mm thickness wall is only of about 0.03 mm. Hence, as the thickness is increased, the wall warpage is restrained by the enhanced structural stiffness of the component. Simultaneously, higher residual stresses are induced by the printing process.

Scale Effect on the Part Warpage
In this section, the sensitivity to the dimensions (e.g., wall-thickness, building height and component size) of the thin-walled components is examined. Therefore, the warpage is measured by 3D-scanning to obtain its relation with the actual dimensions of the 3D-printed structures.
The first assessment study considers a single-wall geometry with a base length of 50 mm and a total height of 50 mm. Three thicknesses of 1 mm, 2 mm and 5 mm, respectively, are investigated. Figure 3 compares the final warpage distributions as measured for the three wall thicknesses used. It can be seen that the thinnest wall presents the more pronounced warpage, up to 0.41 mm in the middle of the final cambered surface. As the wall-thickness increases, the warpage rapidly decreases; the maximum out-of-plane displacement for the 5 mm thickness wall is only of about 0.03 mm. Hence, as the thickness is increased, the wall warpage is restrained by the enhanced structural stiffness of the component. Simultaneously, higher residual stresses are induced by the printing process.  A second study analyses three different wall heights (37 mm, 43 mm and 50 mm, respectively) for the same single-walled geometry and a fixed thickness of 1 mm. Figure 4 compares the residual warpages of the three thin-walled structures. It is possible to observe that increasing the component height results in larger part warpage. This is not only because of the reduced wall thickness, but also because the higher the component, the less sensitive it is to the clamping conditions at the base and, thus, the more free it is to deform. A second study analyses three different wall heights (37 mm, 43 mm and 50 mm, respectively) for the same single-walled geometry and a fixed thickness of 1 mm. Figure 4 compares the residual warpages of the three thin-walled structures. It is possible to observe that increasing the component height results in larger part warpage. This is not only because of the reduced wall thickness, but also because the higher the component, the less sensitive it is to the clamping conditions at the base and, thus, the more free it is to deform. Next, a closed thin-wall cylindrical structure is studied. The diameter is 50 mm and the wall thickness is to 1 mm. The cylinder heights are of 30 mm, 50 mm and 70 mm, respectively. Figure 5 shows the contour-fills of the warpage for the three cylindrical structures with different heights. It can be found that increasing the height leads to larger warpages as observed for the single-wall. However, the maximum warpage is remarkably lower than in both the single-wall and the square thin-wall structures. In this case, the (cylindrical) shape noticeably improves the structural stiffness of the printed structure for the same reduced wall thickness and component height. A further assessment study is performed assuming a thin-wall cylindrical geometry with a thickness of 1 mm and a height of 50 mm. Different cylinder diameters of 50 mm, 70 mm and 90 mm are studied. Figure 6 depicts the contour-fills of the final warpage of Next, a closed thin-wall cylindrical structure is studied. The diameter is 50 mm and the wall thickness is to 1 mm. The cylinder heights are of 30 mm, 50 mm and 70 mm, respectively. Figure 5 shows the contour-fills of the warpage for the three cylindrical structures with different heights. It can be found that increasing the height leads to larger warpages as observed for the single-wall. However, the maximum warpage is remarkably lower than in both the single-wall and the square thin-wall structures. In this case, the (cylindrical) shape noticeably improves the structural stiffness of the printed structure for the same reduced wall thickness and component height. A second study analyses three different wall heights (37 mm, 43 mm and 50 mm, respectively) for the same single-walled geometry and a fixed thickness of 1 mm. Figure 4 compares the residual warpages of the three thin-walled structures. It is possible to observe that increasing the component height results in larger part warpage. This is not only because of the reduced wall thickness, but also because the higher the component, the less sensitive it is to the clamping conditions at the base and, thus, the more free it is to deform. Next, a closed thin-wall cylindrical structure is studied. The diameter is 50 mm and the wall thickness is to 1 mm. The cylinder heights are of 30 mm, 50 mm and 70 mm, respectively. Figure 5 shows the contour-fills of the warpage for the three cylindrical structures with different heights. It can be found that increasing the height leads to larger warpages as observed for the single-wall. However, the maximum warpage is remarkably lower than in both the single-wall and the square thin-wall structures. In this case, the (cylindrical) shape noticeably improves the structural stiffness of the printed structure for the same reduced wall thickness and component height. A further assessment study is performed assuming a thin-wall cylindrical geometry with a thickness of 1 mm and a height of 50 mm. Different cylinder diameters of 50 mm, 70 mm and 90 mm are studied. Figure 6 depicts the contour-fills of the final warpage of A further assessment study is performed assuming a thin-wall cylindrical geometry with a thickness of 1 mm and a height of 50 mm. Different cylinder diameters of 50 mm, 70 mm and 90 mm are studied. Figure 6 depicts the contour-fills of the final warpage of the three cylinders. As expected, by increasing the diameter, the warpage is also more Finally, the last assessment studies a closed square thin-wall section. The same base length and height of 50 mm are used, while wall thicknesses of 1 mm, 2 mm and 5 mm are investigated. Figure 7 shows the contour-fills of the residual warpage for the three AM structures. Once again, it can be observed that the thicker the wall-thickness, the smaller the observed warpage. Therefore, a pronounced bulging is shown by the component with the thinnest wall. Nevertheless, the overall structural stiffness of the closed section allows for a reduction of the maximum out-of-plane displacements: 0.37 mm compared to 0.41 mm as measured for the open single-wall. This is because the deformation of each one of the walls conforming the square thin-wall section is constrained by the substrate and also laterally. Notably, the cylinder of similar dimensions shows a much better performance than the square thin-wall section of similar dimensions, Figures  5b and 7a, respectively. This is explained as follows. In LPBF, high tensile stresses in the hoop/wall direction exist at the top for two components. Large tensile stresses also happen along the corners of the square part due to the high cooling rates at such locations; this builds up compressive stresses in the central region of each square wall so that any small disturbance can favour the bulging perpendicular to the wall. This is not the case for the cylinder. Finally, the last assessment studies a closed square thin-wall section. The same base length and height of 50 mm are used, while wall thicknesses of 1 mm, 2 mm and 5 mm are investigated. Figure 7 shows the contour-fills of the residual warpage for the three AM structures. Once again, it can be observed that the thicker the wall-thickness, the smaller the observed warpage. Therefore, a pronounced bulging is shown by the component with the thinnest wall. Nevertheless, the overall structural stiffness of the closed section allows for a reduction of the maximum out-of-plane displacements: 0.37 mm compared to 0.41 mm as measured for the open single-wall. This is because the deformation of each one of the walls conforming the square thin-wall section is constrained by the substrate and also laterally. Notably, the cylinder of similar dimensions shows a much better performance than the square thin-wall section of similar dimensions, Figures 5b and 7a, respectively. This is explained as follows. In LPBF, high tensile stresses in the hoop/wall direction exist at the top for two components. Large tensile stresses also happen along the corners of the square part due to the high cooling rates at such locations; this builds up compressive stresses in the central region of each square wall so that any small disturbance can favour the bulging perpendicular to the wall. This is not the case for the cylinder. Finally, the last assessment studies a closed square thin-wall section. The same base length and height of 50 mm are used, while wall thicknesses of 1 mm, 2 mm and 5 mm are investigated. Figure 7 shows the contour-fills of the residual warpage for the three AM structures. Once again, it can be observed that the thicker the wall-thickness, the smaller the observed warpage. Therefore, a pronounced bulging is shown by the component with the thinnest wall. Nevertheless, the overall structural stiffness of the closed section allows for a reduction of the maximum out-of-plane displacements: 0.37 mm compared to 0.41 mm as measured for the open single-wall. This is because the deformation of each one of the walls conforming the square thin-wall section is constrained by the substrate and also laterally. Notably, the cylinder of similar dimensions shows a much better performance than the square thin-wall section of similar dimensions, Figures  5b and 7a, respectively. This is explained as follows. In LPBF, high tensile stresses in the hoop/wall direction exist at the top for two components. Large tensile stresses also happen along the corners of the square part due to the high cooling rates at such locations; this builds up compressive stresses in the central region of each square wall so that any small disturbance can favour the bulging perpendicular to the wall. This is not the case for the cylinder.

Numerical Modeling
In LPBF the following two operations are repeated for each layer (see Figure 8):

1.
Recoating: the baseplate is lowered to accommodate a new powder layer; 2.
Laser-scanning: the laser heat source generates a molten pool which follows a userdefined trajectory to selectively melt the powder bed consolidating the new layer comprising the AM build.

Numerical Modeling
In LPBF the following two operations are repeated for each layer (see Figure 8): 1. Recoating: the baseplate is lowered to accommodate a new powder layer; 2. Laser-scanning: the laser heat source generates a molten pool which follows a user-defined trajectory to selectively melt the powder bed consolidating the new layer comprising the AM build. The most suitable numerical strategy to approach the simulation of the LPBF process is by the inherent strain method [32]. This method allows for a fast prediction of the distortions and residual stresses of the fabricated component. Therefore, the detailed simulation of the laser movement along the scanning trajectory is replaced by a layer-by-layer (or even multi-layer) analysis. This assumption is generally accepted when simulating LPBF processes because the scanning speed is very high (up to 1 m/s) compared to other AM technologies and the TMAZ is very localized, and it affects only the last few (2-3) layers. As a consequence, thermo-plastic (inherent) strains are only generated within the TMAZ.
This given, the original CAD geometry is firstly sliced according to the thickness of the powder-bed and then the FE discretization is generated accordingly. The birth-death FE activation technique is adopted [19,33] to progressively add the elements belonging to each new layer to the computational domain, Ω. In the initial FE discretization of the nominal CAD geometry, all elements are inactive. Inactive elements are not computed or assembled into the global system of equations. The numerical simulation of the building process consists of a sequence of layer activations. Hence, at each time-step, the elements belonging to the next layer are switched on (activated), becoming part of the current computational domain. For each time-step, the thermo-mechanical analysis is performed for all the active elements, until the whole layer-by-layer deposition sequence is completed.

Mechanical Problem
The stress analysis enforces the conservation of the balance of momentum and the continuity equations within the current active domain: where b are the body forces per unit of volume and K(T) is the temperature-dependent bulk modulus and p and s, are the hydrostatic (pressure) and the deviatoric components of the Cauchy stress tensor, σ. Thus: The most suitable numerical strategy to approach the simulation of the LPBF process is by the inherent strain method [32]. This method allows for a fast prediction of the distortions and residual stresses of the fabricated component. Therefore, the detailed simulation of the laser movement along the scanning trajectory is replaced by a layer-bylayer (or even multi-layer) analysis. This assumption is generally accepted when simulating LPBF processes because the scanning speed is very high (up to 1 m/s) compared to other AM technologies and the TMAZ is very localized, and it affects only the last few (2-3) layers. As a consequence, thermo-plastic (inherent) strains are only generated within the TMAZ. This given, the original CAD geometry is firstly sliced according to the thickness of the powder-bed and then the FE discretization is generated accordingly. The birth-death FE activation technique is adopted [19,33] to progressively add the elements belonging to each new layer to the computational domain, Ω. In the initial FE discretization of the nominal CAD geometry, all elements are inactive. Inactive elements are not computed or assembled into the global system of equations. The numerical simulation of the building process consists of a sequence of layer activations. Hence, at each time-step, the elements belonging to the next layer are switched on (activated), becoming part of the current computational domain. For each time-step, the thermo-mechanical analysis is performed for all the active elements, until the whole layer-by-layer deposition sequence is completed.

Mechanical Problem
The stress analysis enforces the conservation of the balance of momentum and the continuity equations within the current active domain: where b are the body forces per unit of volume and K(T) is the temperature-dependent bulk modulus and p and s, are the hydrostatic (pressure) and the deviatoric components of the Cauchy stress tensor, σ. Thus: The strain tensor, ε(u) = s u, can also be split in its volumetric and deviatoric parts, e vol = ·u and e, respectively, so that: The deviatoric stress tensor is computed as: where G(T) is the temperature-dependent shear modulus and e inh is the inherent strain tensor, as: e inh = e T I + e vp (6) which typically includes both the thermal deformation, e T (T), and the visco-plastic strains, e vp . Commonly, the inherent strain method does not contemplate the solution of the thermal problem. Therefore, the inherent strain tensor is assumed as a (constant) user-defined material parameter depending on the material and process characterization. Its value can be obtained either experimentally or numerically, through the detailed simulation of the molten pool solidification.
In this work, a staggered thermo-mechanical solution is adopted to solve the AM problem [34,35]. Therefore, both the thermal and the mechanical analyses are carried out in a layer-by-layer manner. On the one hand, the thermal deformations can be computed from the actual (non-uniform) temperature field (inherent shrinkage method). On the other hand, the (inherent) visco-plastic deformations are defined as: e vp = diag(e l , e t , e z ) = I(e l , e t , e z ) T where e l , e t and e z are the longitudinal (aligned with the scanning direction), transversal and vertical (building direction) visco-plastic components, respectively [33][34][35]. In LPBF, the thickness of the deposited layer is very small, so e z can be neglected. Additionally, thin-walled parts can be assumed to be in a quasi-plane-stress condition, so that the only inherent strain component to be considered is the one along the mid-line of the cross-section, regardless of the scanning pattern.

Thermal Problem
H being the hatch spacing, the total travelled scanning length for each layer is: where A is the area of the current layer. The volume of the Heat Affected Zone (HAZ) is given by: where d is the laser beam diameter and s is the penetration (depth) of the melt-pool, while V = A·t is the volume of the layer, being t its thickness. In this work, d = H (no overlapping) and s = t (the melt-pool depth is the layer thickness). Thus, the volume of the HAZ corresponds to the volume of the layer being activated. Hence, the heat source density accounting for the total energy input during the scanning sequence of the whole layer is computed as: where η is the laser absorptivity and P is the laser power. This heat source is delivered during the whole scanning phase, which is characterized by a time interval, ∆t scan computed as: where V scan is the scanning velocity. Observe that the LPBF process also includes the recoating operation, that is a cooling phase of duration ∆t cool .
Often, the thermo-mechanical analysis is accelerated by lumping several layers into a single time-step. Therefore, the equivalent time-step of the analysis results in: ∆t eq = n·(∆t scan + ∆t cool ) (12) where n is the number of lumped layers (in this work n = 10) and the power input is uniformly spread in the lumped volume, V lump , so that:

Geometrical Models and FE Meshes
The coupled thermo-mechanical analysis is performed using the in-house FE software, COMET, developed at the International Centre for Numerical Methods in Engineering (CIMNE) [36,37]. The definition of the CAD geometries, and the corresponding the FE mesh generation as well as the pre/post-processing operations are carried out using the inhouse pre/post-processor GiD [38]. The hardware used to run the simulation incorporates an Intel(R) Core (TM) i7-9700, 3.0 GHz processor and 16.0 GB of RAM. Figure 9 shows the FE meshes of three different thin-walled geometries including both open and closed structures with a fixed thickness of 1 mm. The dimensions of these structures and the corresponding numbers of the hexahedral elements and nodes are listed in Table 3.
where Vscan is the scanning velocity. Observe that the LPBF process also includes the recoating operation, that is a cooling phase of duration Δtcool. Often, the thermo-mechanical analysis is accelerated by lumping several layers into a single time-step. Therefore, the equivalent time-step of the analysis results in: Δteq = n·(Δtscan + Δtcool) (12) where n is the number of lumped layers (in this work n = 10) and the power input is uniformly spread in the lumped volume, Vlump, so that:

Geometrical Models and FE Meshes
The coupled thermo-mechanical analysis is performed using the in-house FE software, COMET, developed at the International Centre for Numerical Methods in Engineering (CIMNE) [36,37]. The definition of the CAD geometries, and the corresponding the FE mesh generation as well as the pre/post-processing operations are carried out using the in-house pre/post-processor GiD [38]. The hardware used to run the simulation incorporates an Intel(R) Core (TM) i7-9700, 3.0 GHz processor and 16.0 GB of RAM. Figure 9 shows the FE meshes of three different thin-walled geometries including both open and closed structures with a fixed thickness of 1 mm. The dimensions of these structures and the corresponding numbers of the hexahedral elements and nodes are listed in Table 3.  The average mesh size in all cases is of about 1 × 1 mm 2 within the horizontal XY plane, and 0.3 mm in the building direction (a lumping layer-height). All these thin-walled work-pieces have the same height of 50 mm. The LPBF process is simulated with a sequence of 167 time-steps through a layer-by-layer activation strategy. Each layer used for the simulation has a thickness of 0.3 mm, thus including 10 physical layers (30 µm of loose powder is spread at each recoating step).
The temperature-dependent thermal and mechanical properties used to feed the constitutive model of Ti-6Al-4V alloy have been calibrated in a previous work [5], as shown in Table 4.  The average mesh size in all cases is of about 1 × 1 mm 2 within the horizontal XY plane, and 0.3 mm in the building direction (a lumping layer-height). All these thin-walled workpieces have the same height of 50 mm. The LPBF process is simulated with a sequence of 167 time-steps through a layer-by-layer activation strategy. Each layer used for the simulation has a thickness of 0.3 mm, thus including 10 physical layers (30 µm of loose powder is spread at each recoating step).
The temperature-dependent thermal and mechanical properties used to feed the constitutive model of Ti-6Al-4V alloy have been calibrated in a previous work [5], as shown in Table 4.
The thermal boundary conditions adopted in all the proposed analyses are defined through a heat convection coefficient of 12.7 W/( • C·m 2 ) and an emissivity of 0.35 [39]. The ambient temperature is set to 26 • C during the whole LPBF process. The efficiency of the LPBF process, when Ti-6Al-4V loose powder is used, is defined in terms of the heat power absorption coefficient fixed to η = 0.4 [39]. Table 4. Temperature-dependent material properties of Ti-6Al-4V alloy [5]. All the thin-walled structures are printed at the same time (see Figure 1). Nevertheless, the simulation strategy assumes that each component is printed separately. This is because the heat conductivity of the loose powder is extremely low, so that most of the heat is evacuated by radiation through the upper surface of the layer being printed as well as by heat conduction through the baseplate. Therefore, the in-plane (XY) heat flux through the loose powder is negligible compared to the heat flow in the vertical (building) direction.

Calibration of the Numerical Model
In order to calibrate the numerical model used for LPBF, the warpage profiles of the thin-wall semi-cylindrical, cylindrical and L-shaped structures are computationally predicted and compared with the experimental 3D-scan results. Figure 10a,b shows the out-of-plane displacements of the semi-cylindrical and cylindrical parts, respectively. The displacements at several control points are specifically used as a reference to assess the accuracy of the simulation. Similarly, Figure 11 compares the experimental and computed displacements of the L-shaped part. Once calibrated, the numerical model is able to faithfully reproduce the experimental results obtained by 3D scanning. It can be seen in Figure 10a that a visible bulge (of approximately 0.4 mm) occurs at opposite sides of the semi-cylindrical part, while the cylindrical geometry just shows very small radial displacements (less than 0.08 mm), these being more pronounced in the mid-section. The open L-shaped thin-wall structure also shows large deformations at the edges of each wall, as shown in Figure 11 and successfully predicted by the numeri- very small radial displacements (less than 0.08 mm), these being more pronounced in the mid-section. The open L-shaped thin-wall structure also shows large deformations at the edges of each wall, as shown in Figure 11 and successfully predicted by the numerical model.
The slight discrepancies between the experimental and computed results are attributed to the mutual thermal influence among all the components as they are printed together (see Figure 1); this influence is disregarded in the FE simulation. Figure 11. Comparison of the out-of-plane displacement fields obtained through (a) 3D scanning and (b) numerical simulation for the L-shaped structure. It can be seen in Figure 10a that a visible bulge (of approximately 0.4 mm) occurs at opposite sides of the semi-cylindrical part, while the cylindrical geometry just shows very small radial displacements (less than 0.08 mm), these being more pronounced in the midsection. The open L-shaped thin-wall structure also shows large deformations at the edges of each wall, as shown in Figure 11 and successfully predicted by the numerical model.
The slight discrepancies between the experimental and computed results are attributed to the mutual thermal influence among all the components as they are printed together (see Figure 1); this influence is disregarded in the FE simulation.

Results and Discussion on the Warpage Mechanism of the Thin-Walled Structures
Understanding the mechanical response of the 3D-printed geometries is necessary to establish a procedure to alleviate the warpage phenomenon in LPBF manufacture. As an example, the above results clearly illustrate that open thin-wall structures are more prone to deformation than closed ones.
Focusing on the open thin-walled semi-cylindrical geometry (Figure 10a), it is observed that the formation of the bulging progresses as the LPBF printing process moves away from the substrate as shown in Figure 12a. The bulging gets more pronounced when printing the upper half of the component because the mechanical constraining from the substrate restrains the development of the distortion in the bottom half. Figure 12b,c shows the evolution of the longitudinal and vertical stresses, σ xx and σ zz , respectively, at two stages of the manufacturing. It can be noted in Figure 12b that the internal surface sustains compressive stresses while tensile stresses are induced in the external surface. As a consequence, the wall deforms outwards. The vertical tensile stresses induced by the high cooling rate and the small bending stiffness of these structures also promote the bulging as shown in Figure 12c.
Contrarily, the closed thin-walled cylindrical part presents a much higher structural stiffness, and thus the AM process only produces a small deformation in the radial direction.
Dunbar et al. [40] studied the warpage of two thin-walled cylinders built by SLM. They also found that the warpage is relatively small. Throughout the printing process, the maximum warpage occurs several layers below the most recently deposited layer. This confers a final concave shape to the component.
Next, the warpage mechanism of the thin-wall L-shape structure is analyzed. Figure 13 shows the evolution of the longitudinal stresses (σ xx ) and the vertical stresses (σ zz ). During the building process, large tensile stresses develop at the top and two side edges while compressive stresses are produced in the central area of the built. As a result, warpage is visible along the lateral edges (see Figure 11). Note that a similar stress distribution and bulging phenomenon also take place when fabricating the single-wall and the square thin-walled section (see Figures 4 and 5).

Structural Optimization for Warpage Control
As above mentioned, the main difficulty for fabricating thin-walled AM components is their low structural stiffness and the high thermal stresses, especially at the sharp corners. In order to minimize the part warpage, a structural optimization strategy is proposed to enhance the overall stiffness of the component and particularly when

Structural Optimization for Warpage Control
As above mentioned, the main difficulty for fabricating thin-walled AM components is their low structural stiffness and the high thermal stresses, especially at the sharp corners. In order to minimize the part warpage, a structural optimization strategy is proposed to enhance the overall stiffness of the component and particularly when Yang et al. [41] also carried out a distortion analysis of thin-walled L-shaped and cross structures; their results showed that the side with longitudinal scan suffers larger distortion, whereas the side with transverse scan is distortion-free. Obviously, the mechanical response of thin-walled structures is closely related with the scan strategy.

Structural Optimization for Warpage Control
As above mentioned, the main difficulty for fabricating thin-walled AM components is their low structural stiffness and the high thermal stresses, especially at the sharp corners. In order to minimize the part warpage, a structural optimization strategy is proposed to enhance the overall stiffness of the component and particularly when open sections must be fabricated. The idea consists in adding vertical stiffeners able to resist the thermal stresses by locally increasing the structural stiffness and, at the same time, reducing the cooling rate at the edges of AM components as shown in Figure 14. open sections must be fabricated. The idea consists in adding vertical stiffeners able to resist the thermal stresses by locally increasing the structural stiffness and, at the same time, reducing the cooling rate at the edges of AM components as shown in Figure 14. Figure 15 shows the final warpage of both the open semi-cylindrical and L-shaped thin-walled parts with and without the vertical stiffeners. The warpage minimization is noticeably achieved. Figure 16 shows the residual von Mises stresses of the open semi-cylindrical and L-shaped thin-walled structures. These residual stresses are mitigated by lowering the cooling rates at the edges of the two structures, thus increasing the local heat accumulation. Note that there still exist large residual stresses in the bottom area of both components. However, substrate preheating just before LPBF can successfully mitigate the residual stresses in this area to achieve high-quality fabrication of thin-walled components [5,7].    open sections must be fabricated. The idea consists in adding vertical stiffeners able to resist the thermal stresses by locally increasing the structural stiffness and, at the same time, reducing the cooling rate at the edges of AM components as shown in Figure 14. Figure 15 shows the final warpage of both the open semi-cylindrical and L-shaped thin-walled parts with and without the vertical stiffeners. The warpage minimization is noticeably achieved. Figure 16 shows the residual von Mises stresses of the open semi-cylindrical and L-shaped thin-walled structures. These residual stresses are mitigated by lowering the cooling rates at the edges of the two structures, thus increasing the local heat accumulation. Note that there still exist large residual stresses in the bottom area of both components. However, substrate preheating just before LPBF can successfully mitigate the residual stresses in this area to achieve high-quality fabrication of thin-walled components [5,7].    Figure 16 shows the residual von Mises stresses of the open semi-cylindrical and L-shaped thin-walled structures. These residual stresses are mitigated by lowering the cooling rates at the edges of the two structures, thus increasing the local heat accumulation. Note that there still exist large residual stresses in the bottom area of both components. However, substrate preheating just before LPBF can successfully mitigate the residual stresses in this area to achieve high-quality fabrication of thin-walled components [5,7].

Conclusions
In this work, the warpage of several thin-walled structures fabricated by LPBF is investigated. The major conclusions of this study are:

•
The wall thickness plays a significant role on the warpage of the final part. Thicker walled structures present reduced warpage.

•
Increasing the build height as well as reducing the wall curvature causes larger warpage, as shown for the cylindrical structures.

•
Open sections (e.g., semi-cylinder, L-shape, etc.) are more prone to warpage than closed ones (e.g., cylinder and square section) because of their reduced structural stiffness.

•
The use of vertical stiffeners enables locally enhancing the structural stiffness of thin-walled structures, minimizing the residual warpage induced by the LPBF process. • FE analysis of LPBF processes is a useful tool to analyse different thin-walled structures in order to predict the actual warpage. The developed FE model has been calibrated with experimental 3D-scanning images.

Conclusions
In this work, the warpage of several thin-walled structures fabricated by LPBF is investigated. The major conclusions of this study are: 1.
The wall thickness plays a significant role on the warpage of the final part. Thicker walled structures present reduced warpage.

2.
Increasing the build height as well as reducing the wall curvature causes larger warpage, as shown for the cylindrical structures.

3.
Open sections (e.g., semi-cylinder, L-shape, etc.) are more prone to warpage than closed ones (e.g., cylinder and square section) because of their reduced structural stiffness.

4.
The use of vertical stiffeners enables locally enhancing the structural stiffness of thinwalled structures, minimizing the residual warpage induced by the LPBF process. 5.
FE analysis of LPBF processes is a useful tool to analyse different thin-walled structures in order to predict the actual warpage. The developed FE model has been calibrated with experimental 3D-scanning images.