# Adaptive trajectories generation for additive manufacturing for structural parts with variable thickness

A. Giordano, Cyril Bordreuil, F. Bugarin, S. Segonds

► To cite this version:

A. Giordano, Cyril Bordreuil, F. Bugarin, S. Segonds. Adaptive trajectories generation for additive manufacturing for structural parts with variable thickness. Structures, 2025, 80, pp.109980. 10.1016/j.istruc.2025.109980. hal-05466173

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Submitted on 19 Jan 2026

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# Adaptive trajectories generation for additive manufacturing for structural parts with variable thickness

A. Giordano<sup>a,*</sup> <sup>ID</sup>, C. Bordreuil<sup>b</sup>, F. Bugarin<sup>a</sup>, S. Segonds<sup>a</sup>

<figure><img src="image_2.png" /></figure>

## ARTICLE INFO

### Keywords:

- Wire arc directed energy deposition (WA-DED)
- Wire arc additive manufacturing (WAAM)
- Three-dimensional (3D) toolpath
- Deposit height modulation
- Variable thickness parts
- Curved surface deposition

## ABSTRACT

Wire Arc Directed Energy Deposition (WA-DED) has demonstrated its ability to produce thin metallic structures due to its high material deposition rate and freedom of movement. However, in additive manufacturing, the surface quality of the raw part depends on generating an optimized toolpath to limit defects on the final part. Traditional deposition strategies in manufacturing are often at a constant height and do not allow for the production of a wide range of topologies. WA-DED has the advantage of modulating the deposition height during fabrication, allowing the creation of complex shapes while increasing the final precision. This paper presents a new approach aimed at creating 3D trajectories with variable deposition height on curved layers for complex surfaces with variable thickness. This method is used on thick parts and thick parts with local thickness variation requiring local deposition of more than a single bead. This method is based on isothermal surfaces derived from a thermal conduction problem on the geometry to be manufactured, specifically adapted to overcome the problems of multi-bead deposition and disturbance of the thermal conduction flow induced by the variation in thickness of the part. The curved layers derived from isothermal surfaces serve as a support for creating the trajectories and a map of the inter-layer distance is calculated by ray casting. The trajectories, in the form of a pseudo-spiral, take into account the distance between the layers to adapt the deposition height, which varies across the surface to be fabricated. Finally, the fabrication order is carried out incrementally in the direction of fabrication. Numerical and experimental validation on a thin part with locally variable thickness, topologically optimized, has validated the effectiveness of the method.

## 1. Introduction

Additive Manufacturing (AM) is the technology that opens up new horizons in manufacturing by reducing production costs and offering tremendous geometric freedom in the shape of manufactured parts. It relies on the layer-by-layer assembly of materials to create 3D structures. Wire Arc Additive Manufacturing (WAAM), a type of DED (ISO/ASTM 52900) [1] process, stands out for its ability to produce large parts with high material deposition rates [2,3]. It also enables the fabrication of mechanical components of choice due to its highly attractive buy-to-fly ratio compared to conventional machining technologies [4–6]. WAAM is an additive manufacturing technology based on an arc welding process coupled with a robotic arm or, less commonly, a 3-axis computer numerical control (CNC) machine. The component is built layer by layer by melting a filler wire using an electric arc. However, unlike

Selective Laser Melting (SLM), this process does not allow for high deposition accuracy. The surface quality of the raw parts typically requires machining to achieve the desired precision in “functional” areas with machining technologies. Despite these drawbacks, WAAM is a promising technology for large parts due to its ease of implementation compared to powder-based techniques. WAAM also offers the possibility to add surfaces or elements to existing parts. This technology allows for the modulation of deposition height during the process, an advantage that is currently underexploited but offers a new dynamic for trajectory creation. Current deposition strategies are primarily in 2.5D and do not fully utilize the workspace that WAAM can provide. Indeed recent studies have shown that the substantial performance sacrifices typically associated with 2.5 or 3-axis approaches can be drastically reduced with multi-axis AM-based approach [7]. Also, identifying a build direction that can eliminate the need for support structures in the case of complex

<footnote><sup>a</sup> Institut Clément Ader (ICA), Université de Toulouse, UT3, CNRS, IMT Mines Albi, INSA Toulouse, ISAE-SUPAERO, Toulouse, France</footnote>

<footnote><sup>b</sup> Laboratoire de Mécanique et de Génie Civil, Université de Montpellier / CNRS, Montpellier, France</footnote>

<footnote>* Corresponding author.
Email addresses: arthur.giordano@univ-tse3.fr (A. Giordano), cyril.bordreuil@umontpellier.fr (C. Bordreuil), florian.bugarin@univ-tlse3.fr (F. Bugarin), stephane.segonds@univ-tlse3.fr (S. Segonds).</footnote>
build geometries can be challenging for complex multibranch parts [8]. Moreover, Ding et al. [9] explained that the manufacturing strategy in additive manufacturing is directly related to the component's surface precision. However, manufacturing strategies for parts whose thickness could be more than one single bead are rarely discussed. In this article, a new trajectory generation method for thick parts using a variation in deposited height during manufacturing is presented. The variation in deposition height is interesting because the part to be manufactured is segmented using 3D surfaces that are not separated by a constant height. The deposited material bead is modeled as an elliptical cylinder for which the cylinder's height is not the same as its width due to material sagging during deposition. This strategy is based on a thermal scalar field previously presented by Giordano et al. [10], who developed a method to generate 3D trajectories with deposition modulation on shell parts. Currently, the possibility of modulating the deposition height is very rarely used, and the manufacturing of locally thick parts and parts with variable thickness remains a challenge. In fact, the variation in the thickness of the parts often results from a side effect of height modulation of the single bead used to produce the part. It is therefore imposed on the design and does not correspond to a controlled modulation of thickness that can be intentionally achieved, for example by placing multiple beads side by side within the thickness of the manufactured part. Parts with variable thickness are essential, particularly in the field of structural components. Applied to thick parts, this method constrains the trajectory to conform to the geometric requirements of the component being manufactured, increasing the final precision of the part and allowing for rapid trajectory generation on all types of geometries. In this article, we apply the proposed method to the manufacturing of structural parts with variable thicknesses resulting from a topology optimization. More precisely, a gyroid is studied as it can be used as an elementary reference element that can be combined with other elementary gyroids to manufacture large lattice structures presenting high stiffness over weight ratio [11,12].

## 1.1. Related works

The complexity of parts produced by additive manufacturing may require adapting the trajectories used. Currently, most strategies fall within the domain of 2.5D, where the CAD model is sliced into parallel planes. These CAD parts are typically uniformly sliced by parallel planes with constant spacing, and less frequently, an adaptive slicing approach [13–15] may be employed, with non-constant spacing based on a defined criterion. Mao et al. [16] implemented adaptive slicing using global deviation as a criterion, while limiting computation time. 2.5D strategies generally allow for this type of slicing depending on the technology used, but parts manufactured with these strategies often exhibit poor surface quality and are subject to the stair-step effect. This leads to low geometric precision due to flat slicing using three degrees of freedom and may result in the appearance of stair-step effects at the end of part manufacturing if the initial and final manufacturing surfaces are not flat and parallel. Etienne et al. [17] developed a model that slices the deformed part to be manufactured and performs a reverse deformation of the toolpaths to revert to the initial geometry. This method results in slightly curved layers, thus utilizing an additional feature offered by three-axis 3D printers. However, other manufacturing processes like WAAM offer up to three additional degrees of freedom when coupled with robotic arms, thus totaling six degrees of freedom. These extra degrees of freedom can be used to adapt to new methods of generating continuous trajectories on curved slicing layers, involving non-planar decomposition. Curved layers are more complex but offer numerous advantages and are increasingly studied [18,19]. The generation of these layers proposed by Fang et al. [20] enables the creation of optimized toolpaths based on the directions of principal stresses, resulting in improved mechanical performance of the manufactured part. This strategy keeps the nozzle aligned with the normal vector of the surface to be fabricated, unlike strategies using three axes where the

nozzle is aligned with the normal vector to the substrate. The main advantage of this technique is to enhance the part quality by eliminating the stair-stepping effect [21]. Several studies have been conducted on curved slicing using the iso-value principle [22,23]. Xu et al. [24] developed a technique to generate trajectories using iso-contours derived from geodesic distances. However, the use of geodesic distance fields does not adequately address the geometric peculiarities of the components to be manufactured, especially when the parts have a non-planar upper free edge. The strategy for continuous 3D trajectory generation presented by Diourte et al. [25] is also based on geodesic distance calculation and implements the use of generators to overcome the issues encountered by strategies based on geodesic distances regarding the geometric peculiarities of the parts. This method is computationally expensive, so it is relevant to reduce trajectory calculation time to make the process versatile and competitive. In this regard, Giordano et al. [10] developed a method based on a thermal scalar field to maximize deposition continuity. The final trajectory contour can thus conform to the topology of the upper free edge and potentially enable additional functionality. Arc welding-based additive manufacturing also allows for height modulation, extending beyond flat slicing strategies [26,27]. In-situ control of deposition height, layer by layer, enables faster modulation and compensation, with reduced cooling time [28,29]. Mbojd et al. [30] have developed a deposition height control system based on critical process parameters. A study was conducted to investigate in-situ compensation of bead height [31]. According to Zhang et al. [32], the deposition height is not constant during fabrication, being greater at the beginning of the bead due to higher energy input. This phenomenon leads to deposition issues and affects the surface quality of the part, thus proper fabrication trajectory planning is crucial to achieve the best possible surface quality. Regarding the filling phase for thick parts, various trajectory patterns such as rasters, zigzags, and spirals [9,33–36] enable layer-by-layer fabrication but may result in insufficient density due to inadequate overlap of beads. To address this density issue, Robinson et al. [37] developed a method that adjusts the scanning angle for a zigzag trajectory. This method has also demonstrated a reduction in residual stresses in the manufactured parts [38].

## 1.2. Contributions

This article presents a three-dimensional trajectory planning strategy with a layer height variation for solid, thick, and locally variable thickness parts which is essential for the manufacture of structural parts. This method is based on a thermal model using a heat conduction law [10]. The underlying idea of using a heat conduction law lies in the rapid solution of a thermal transport equation and in exploiting the conservation characteristics of the thermal flux propagating through the surface to be manufactured. This thermal flux propagates while respecting the constraints imposed by the surface it traverses and adapts to all geometric shapes. This thermal scalar field is associated with Dirichlet boundary conditions, which allow for consideration of all forms of geometries that can be manufactured in WAAM. This diversity of producible parts is facilitated notably by the use of a 6-axis robotic arm equipped with a welding torch, which significantly reduces the stair-stepping effect, undesirable for the final surface finish. The robot also enables modulation, which is one of the main advantages of the technology and the central idea of the strategy proposed in this article. This modulation is constrained within an interval $[h_{min}, h_{max}]$, where $h_{min}$ is the minimum deposition height and $h_{max}$ is the maximum deposition height. Various WAAM technology parameters enable this modulation, such as wire feed speed and welding speed, which can be controlled in real time during the fabrication process. The algorithm presented in this article is based on isothermal surfaces to slice a 3D model into curved layers. The schematic representation of the method is detailed in following Fig. 1. To our knowledge, there is currently no trajectory generation method that makes it possible to manufacture parts in WAAM and 5 axes whose thickness is variable and controlled and which uses the possibility of
<figure><img src="image_3.png" /><figcaption>Fig. 1. Schematic representation of the method.</figcaption></figure>

modulating the deposit height to guarantee an adaptive trajectory limiting arc restarts. It is used to showcase the potential of this method for thick parts with spiral-type trajectories before being extended to the fabrication of thick and locally variable thickness parts.

The document is structured as follows: in Section 2, we discuss the use of the thermal scalar field constrained by boundary conditions, detailing the construction of curved layers from this field. This section also describes the algorithm for generating the printing path with deposition modulation, based on the constructed curved layers. The results of tests of our method are then presented in Section 3, followed by the conclusion in the final section.

## 2. Generation of trajectories with height deposition modulation

Trajectory generation method with deposition height modulation of the present article relies on geometry created using CAD software, as is typically the case for such manufacturing processes. This CAD file is then integrated into finite element analysis software, which calculates heat transfer based on pure heat conduction. This temperature field is subsequently used to determine the deposition trajectory, but it has no

physical connection to the temperature of the part during its fabrication. To obtain accurate information on temperature after calculation on the geometry, it is important to have a coherent mesh. This mesh, whether triangular or tetrahedral depending on the type of part being manufactured, should be as uniform as possible. Elongated triangles should be avoided as they can lead to inaccurate information that would not allow for effective segmentation of iso-surfaces. The interest of this strategy lies in its ability to manufacture solid, thick, and locally variable thickness parts, as described in Sections 2.3.1 and 2.3.2. While this strategy has a common foundation, adjustments are necessary depending on the component to be manufactured due to the inherent requirements of these geometries, thus allowing for a versatile strategy.

Fig. 2 depicts the three geometries that will be studied in this article. Geometries 2(a) and (b), derived from our previous work, will be used to validate the method numerically. The part 2(c), based on one of Schwarz's minimal surfaces, is used for the experimental validation of this new trajectory strategy, with its local thickness variations.

### 2.1. Isovalues slicing

In layer-based additive manufacturing, each layer is considered at an iso-height, composed of points with the same z coordinate, creating flat slicing surfaces. The idea of our method stems from that proposed in our previous article [10]. This strategy utilizes a virtual heat transfer, replacing the slicing of geometries created using CAD software, based on iso-height surfaces, with isothermal surfaces. The heat conduction process propagates through the 3D part, from the substrate in contact with the bottom free edge of the part to the top free edge. The temperature field obtained allows slicing into isothermal surfaces, where all points of the same layer have the same temperature, which can be flat or curved. This thermal problem is considered as pure conduction, without convection or radiation, calculated using a finite element code. This virtual heat transfer is modeled by the following equation:

$$ \frac{\rho C_p \partial T}{\partial t} = \Delta(k\Delta T) + Q \quad (1) $$

where T is the temperature field, Q is a volumetric heat source, and ρ, C_p, k are respectively the density, thermal capacity, and thermal

<figure><img src="image_4.png" /><figcaption>(a) Solid part based on the part from our previous article.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>(c) Thick part with ribs for local thickness variation.</figcaption></figure>

<caption>Fig. 2. Solid part with non-planar boundary and thick part with ribs for local thickness variation.</caption>
<figure><img src="image_6.png" /><figcaption>Fig. 3. Method of generating the thermal isosurfaces by interpolating the information at the nodes of the mesh.</figcaption></figure>

conductivity of the material. Our assumption of pure conduction transforms Eq. (1) into:

$$ \Delta u = 0 \qquad (2) $$

This thermal field is subject to Dirichlet conditions on the boundaries of the domain. The first boundary condition is a constant temperature called $T_{lower}$ and the second condition is called $T_{upper}$. These two boundary conditions are applied to points on the free surfaces with $T_{lower} < T_{upper}$ and are chosen arbitrarily. Their values have no influence on the method of partitioning the geometry into isothermal surfaces. Once the thermal calculation is performed, each point of the 3D geometry has temperature information resulting from the resolution of Eq. (2). With this information, it is possible to create the $n_{layers}$ isothermal surfaces denoted $S = S_1, \dots, S_{n_{layers}}$.

Fig. 3 illustrates the creation of isothermal surfaces using the previously calculated thermal scalar field. This figure clearly illustrates that the surfaces are not parallel. The local variation in distance between the surfaces induces local variation in both bead width and thickness making it necessary to calculate the local radial displacement of the torch for deposition of the following beads which will manufacture part's thickness. Each mesh node, having its own temperature, allows the creation of isotherms by interpolation between two mesh nodes. In Fig. 3, the isothermal surface $S_i$ is the isotherm with a value of 33, extracted based on the number of layers. It is important to note that the isosurfaces are not equidistant. The number of layers is calculated based on the minimum height $h_{min}$ of the deposition modulation range [$h_{min}, h_{max}$]. The calculation of the number of layers to be manufactured depends on the chosen average deposition height and is determined using the following equation:

$$ n_{layers} = \left\lfloor \frac{\min(G_{layers})}{h_{min}} \right\rfloor \qquad (3) $$

Where G represents the set of geodesic distances calculated between the two free edges, and $h_{min}$ is the lower limit of the deposition modulation interval. The geodesic distance [39] is defined as the distance, denoted $\varphi(Pi_{lower}, Pi_{upper})$, representing the shortest distance between two points $Pi_{lower}$ and $Pi_{upper}$ projected onto the surface, in our case, the outer skin of the 3D geometry. This geodesic distance is expressed as follows:

$$ G = \varphi(Pi_{lower}, Pi_{upper}) \qquad (4) $$

where $Pi_{lower}$ is a point belonging to the lower free edge, and $Pi_{upper}$ is a point belonging to the upper free edge (Fig. 4).

The thermal conduction problem, in steady state, conserves its heat flux. Indeed, the heat flux moves through the geometry while conserving its flux. In other words, when the section it traverses decreases, the heat flux increases, resulting in variable spacings during the division into isothermal surfaces. In some cases, these variations can be too significant and may not correspond to the modulation range allowed by the technology used. Therefore, it is crucial to smooth out these deviations between the isothermal surfaces using the previously calculated

<figure><img src="image_7.png" /><figcaption>Fig. 4. Calculation of geodesic distances to calculate the number of layers.</figcaption></figure>

geodesic distance with the scalar thermal field. To establish this empirical relationship, there is no direct link between these quantities. To create a link, the extraction of a slice of constant width of the geometry to be manufactured is necessary. This sample will undergo the same treatment as the part to be manufactured, starting with the application of the pure thermal conduction problem as well as a division into isosurfaces. The size of the sample is chosen so that its geodesic length will determine the number of isosurfaces to be extracted and will allow the creation of an isosurface every millimeter. After this step, a realignment between the isothermal surfaces of the sample and those of the global geometry is performed. This involves determining, for each of the isosurfaces of the sample, the temperature value of the scalar field of the complete part. This operation allows obtaining a division of the part using the isothermal surfaces, with relatively constant distances between them so that the variation in distance remains within the achievable range of deposition height variation.

## 2.2. Height map and welding bead geometry

Before proceeding to create the trajectory on each of the curved layers, it is important to know at every point on these layers their distance from the previous one. Indeed, each surface is isothermal but not equidistant, so generating a height map on each of the curved surfaces is necessary to determine the amount of material to be deposited. To calculate this height map, the principle of ray casting [40] is used as it allows for easily determining the distance between two curved surfaces relative to their normal. The equation for ray casting is as follows:

$$ R_{origin} = R_0 = [X_0, Y_0, Z_0] $$

$$ R_{direction} = R_d = [X_d, Y_d, Z_d] $$

$$ \text{where } X_d^2 + Y_d^2 + Z_d^2 = 1 $$

We obtain the ray equation:

$$ R(t) = R_0 + t * R_d, \text{ where } t > 0 \qquad (5) $$

Now, the plane used for the intersection calculation must be defined using a plane equation, from which its normal vector $P_{normal}$ can be derived (Eq. 6).

$$ P = A * x + B * y + C * z + D = 0 $$

$$ \text{where } A^2 + B^2 + C^2 = 1 \qquad (6) $$

$$ P_{normal} = P_n = [A, B, C] $$

To finalize and obtain the distance between the origin of the ray $R_{origin}$ and the intersection point with the plane $P$, we need to substitute Eq. (5) into the above plane equation and solve for $t$.

$$ t = \frac{-(P_n \cdot R_0 + D)}{P_n \cdot R_d}, \text{ where } t > 0 \qquad (7) $$

The dot product $P_n \cdot R_d$ must be strictly negative to satisfy the intersection condition between the ray and the plane. Thus, the coordinates
<figure><img src="image_8.png" /><figcaption>Fig. 5. Calculation of the height map on the support surface for material deposition using a radius throw.</figcaption></figure>

of the intersection point are given by the following equation:

$$r_i = [x_i, y_i, z_i] = [X_0 + X_d * t, Y_0 + Y_d * t, Z_0 + Z_d * t]$$

This leads us to the final distance D between the ray origin $R_0$ and the intersection point $r_i$ calculated using Eq. (8).

$$D = \sqrt{(x_i - X_0)^2 + (y_i - Y_0)^2 + (z_i - Z_0)^2} \quad (8)$$

This ray casting method is robust and precise, offering great flexibility for calculating distances between two points. It is used here to calculate, at each point on surface n, the distance to the intersection point of the ray on surface n + 1, with a ray having the normal at the origin point of the ray, as illustrated in Fig. 5.

These height maps are crucial for adapting the path based on the height of the bead to be deposited, but especially based on its width. Indeed, the bead is considered as an elliptical cylinder, where the width is conditioned by the height according to a factor governed by the following equation: $w_{bead} = k * H$. The choice of geometry for the weld bead is validated with experimental tests that have determined its shape. This assumption allows setting a factor k relating the deposition height h to the width $w_{bead}$ based on process parameters. A second parameter is also studied, especially when manufacturing thick parts where multiple weld beads are placed next to each other. In this specific case, it is important to calculate the inter-bead distance, i.e., the distance between the two neutral fibers of these elliptical beads, and to anticipate a distance smaller than this distance. This distance is a combination of the inter-bead distance, coupled with a reduction coefficient to determine a distance ensuring the overlap of the last deposited bead on the already solidified one [41,42]. These two parameters have been studied and are presented in more detail in Section 3.1. The size of the bead greatly influences the geometry of the part to be produced, especially for so-called thick parts. Indeed, during material deposition, the width of the weld bead is imposed based on the desired height for each curved layer. This constraint requires that geometries created using CAD software take this into account. The number of beads can only be an integer, so it is important to consider this parameter from the design stage onwards, in order to create geometries that closely resemble the expected final state after fabrication.

## 2.3. Trajectory creation strategy on an isosurface

The last step of this strategy is the creation of tool paths on each of the curved surfaces that serve as support for the future trajectory. Filling each layer can be done using different patterns [35,43]. Pseudo-spiral filling is preferred in our study as it allows filling the layer from the outside to the inside without transition, thus creating quasi-continuous trajectories while preserving the external appearance of the part. This choice helps limit material sagging on the exterior of the geometry and contains it more towards the interior. The filling strategy is the same for all three types of geometries, but it requires adjustments based on the constraints of the part to be manufactured. The first step is to start with the fabrication of the outer contour, which will result in a good surface finish. The construction of this contour, as well as all other segments of the trajectory, uses the isothermal surfaces as support and is linked to the height map. Extracting the free edge serves as the basis for

calculating the trajectory of each layer. To create the first path of the deposition trajectory, each point of the outer free edge is translated along the normal to the outer skin of the 3D geometry, by $k * h/2$ with k being the sag coefficient of the bead (see Section 2.2) and h the height at the point to be moved. The displacement is done according to the following equation:

$$p_j = p_i - n * k * h/2 \quad (9)$$

where $p_i$ is a point from the outer contour and $p_j$ is the point that has been displaced to create the start of the tool path.

The first contour being created, the trajectory generation is similar for solid parts and thick parts, except that thick parts will require the removal of unnecessary tool path segments for part creation.

### 2.3.1. Solid parts

Creating the trajectory for solid parts is simpler than for thick parts but relies on the same strategy. The trajectory for each curved layer is still created from the outer contour (Fig. 6). The points of this contour are all translated along the normal to the geometry surface, creating a second segment of the final trajectory. To generate the other trajectory segments, all points of the previous contour are also translated by a distance equal to the given height at each point (Eq. 9), but this time with respect to a tangent vector. This creation mechanism is illustrated in Fig. 7.

The complete trajectory is created from the previous segments that are connected to each other. To avoid creating a step effect and to distribute the heat input into the part during fabrication, the strategy chosen to connect these contours involves starting each layer from points that are not exactly superimposed. This point is defined using four fictitious passage points located at the corners of an imaginary

<figure><img src="image_9.png" /><figcaption>Fig. 6. Example of an extraction of the outer contour and translation of its points on a simple surface.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 7. Extraction of the outer contour and translation of its points on a solid part.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>Fig. 8. Example of spiral trajectory on a circular surface.</figcaption></figure>

plane, with the geometry centered with respect to the center of the coordinate system, which has coordinates (0.0, 0.0, 0.0), with the following coordinates:

* $p_1 = (-100, 100, 0),$

* $p_2 = (100, 100, 0),$

* $p_3 = (100, -100, 0),$

* $p_4 = (-100, -100, 0).$

The starting point of each layer is the point closest to these passage points, and each layer sees this passage point change in the following order: $p_1 \to p_2 \to p_3 \to p_4 \to p_1$ etc. This aims to avoid concentrating the heat provided by the process always in the same location, while maintaining production efficiency by avoiding wasting time on robot traversal time to change starting points. The robot's path is represented in Fig. 8 by the orange segments, allowing for a complete visualization of the tool path and differentiation of fabrication phases. An additional constraint added to the trajectory is that the direction of traversal is reversed from one contour to the next. Even contours have a clockwise traversal direction, while odd contours have a counterclockwise traversal direction. This peculiarity is also used to limit the creation of hot zones in the part during its fabrication and enable as optimal a fabrication as possible in terms of residual stress distribution.

### 2.3.2. Thick structural parts

Locally thick parts are of most interest for structural design. For thick parts, we will use the same strategy as for manufacturing solid parts, only an additional step is required. Indeed, some trajectory segments

generated for a solid part are not necessary for this type of part. The trajectory is first calculated on the solid geometry. The segments not to be kept on this final trajectory are removed using a voiding of the solid geometry in order to obtain the geometry to be manufactured and at the same time the final trajectory of the thick part. Indeed, if the points of the trajectory are contained on the solid surface, then they are kept. The remaining points are excluded, allowing the creation of the desired tool path.

Fig. 9(a) illustrates the technique of overlaying the trajectory of the fictitiously solid layer onto the actual surface to be manufactured. The curves that will not be retained are shown in blue, while the final trajectory, shown in black in the figure, constructs the curved layer. Next is the step, similar to that of solid parts, of connecting each of the trajectory parts that were created by removing unnecessary points. It is necessary to choose a starting point and an ending point on each of the contours. In this example, the tool path travels clockwise, and the starting point is the point closest to the coordinate (100, 100, 0), see Fig. 9(b).

## 2.4. Strategy for creating the final trajectory

The final step to complete the trajectory for the part to be manufactured involves connecting each previously determined point, particularly for solid and thick parts, to create the toolpath. This final trajectory also incorporates information about the orientation of the welding torch using a Frenet frame. This frame defines the orientation of the torch throughout the toolpath along three axes:

* The first axis $\vec{n}$ corresponds to the orientation axis of the torch and is defined at each point by the normal to the curved surface to be manufactured.

* The second axis $\vec{t}$ is determined by the tangent vector to the surface, created from two consecutive points on the trajectory.

* The third axis $\vec{b}$ is defined by the cross product of the two previous axes, $\vec{n}$ and $\vec{t}$.

This frame is of particular importance as it guides the robot in depositing material under optimal conditions, especially for the part validating our method (see Section 3.3).

## 2.5. Application of the trajectory generation strategy

Prior to this method, the use of CAD software is necessary to create the 3D part to be manufactured. Then, finite element analysis software takes this 3D geometry and solves a pure conduction problem in three dimensions to retrieve the temperature map for creating the isosurfaces. The trajectory generation method is implemented in Python and is associated with "The Visualization Toolkit" (VTK) library [44], which is often used for modeling. This library allows visualization and manipulation of scientific data and provides access to geometric algorithms. In this section, the application of our toolpath generation methodology is

<figure><img src="image_12.png" /><figcaption>Fig. 9. (a). Strategy of point superposition with the surface to be manufactured. (b). Final trajectory on a curved surface derived from a thick part.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>**Fig. 10.** Creation of isothermal surfaces using the thermal scalar field from the pure conduction problem.</figcaption></figure>

<table><caption>**Table 1**
Parameters used for the solid part 2(a).</caption><thead><tr><th colspan="2">Parameters used to generate the solid part path</th></tr></thead><tbody><tr><td>Average deposit <i>h</i></td><td>2.0 mm</td></tr><tr><td>Minimum deposit <i>h</i><sub>min</sub></td><td>1.6 mm</td></tr><tr><td>Maximum deposit <i>h</i><sub>max</sub></td><td>2.3 mm</td></tr><tr><td>Number of layers</td><td>84</td></tr></tbody></table>

presented in two parts: the first deals with solid parts and the latter with thick parts.

### 2.5.1. Solid parts

The solid part is based on the shell part that was manufactured in the article [10], but it is modified to be converted into a solid part. This geometry is chosen in this article due to its geometric peculiarities that make its manufacturing non-trivial, such as its upper free edge and its different curvatures. These constraints imposed on the geometry require the use of the WAAM capabilities and also demonstrate its versatility on parts that are not shells. The use of the pure thermal conduction problem with Dirichlet conditions is applied to this geometry, ensuring that the upper free surface will be manufactured in a single bead (Fig. 10).

The parameters referenced in Table 1 were calculated using our method to validate the feasibility of manufacturing the part. The range of deposition height falls within the variation range achievable with the technology used.

Solving the pure conduction problem using finite element software yields a thermal scalar field across the entire volume. This problem utilizes Dirichlet conditions chosen such that the temperature assigned to points on the surface in contact with the substrate, $T_{lower}$, is set to 0 °C, while the temperature assigned to points on the upper free surface, $T_{upper}$, is 100°C. The number of isothermal surfaces for this geometry is 84, calculated using Eq. (3), which will serve as a foundation for trajectory creation.

Fig. 11 illustrates the trajectory on the solid part created according to our strategy. This trajectory follows the characteristics outlined in Section 2.3.1. Indeed, the generated trajectory exhibits an alternating

<figure><img src="image_14.png" /><figcaption>**Fig. 11.** Trajectory creation is facilitated using the isothermal surfaces and their height map.</figcaption></figure>

direction of travel from one layer to another, while the start of the different beads composing the trajectory of a curved surface occurs with a 90° phase shift clockwise.

### 2.5.2. Thick parts

The geometry chosen to illustrate the section on thick parts is the same as that used for solid parts, but it has been hollowed out from its internal structure. This choice was made to demonstrate that the method can be applied to a geometry, whether solid or hollow. This shape is subjected to a thermal scalar field resulting from pure thermal conduction, as outlined in the previous Section 2.5.1 (Fig. 12).

Once the scalar field is obtained, the creation of the curved surfaces can be carried out. To achieve this, the number of these surfaces must be calculated using Eq. (3). For the fabrication of this part, 84 isothermal surfaces need to be generated, where the height will be calculated for each of them at all points. These height maps, created using ray tracing, are indispensable for generating the trajectory of the part. The maximum modulation range is between [1.8 mm, 2.3 mm], which requires the ability to vary the height of the deposited bead during the part creation process. After generating the height maps, the trajectory is created while adhering to the various criteria that were previously outlined.

Fig. 13 illustrates the path followed by the welding torch to create the part. This path adheres to the same constraints imposed on the solid part. Regarding the trajectory, the geometry was chosen such that the minimum thickness, 8.0 mm, is large enough to allow the thick part of the part to be fabricated with two material beads.

## 3. Application of the method and experimental validation

The experimental validation constitutes an important step in our study, as it allows bridging the gap between trajectory simulation and the fabricated workpiece. This section outlines the equipment used and the experimental campaign conducted on bead deposition, aimed at determining the various process parameters necessary for the successful fabrication of our part. A section details the manufacturing process before comparing the results with their numerical counterparts, thus
<figure><img src="image_15.png" /><figcaption>**Fig. 12.** Creation of the isothermal surfaces using the thermal scalar field derived from the pure thermal conduction problem.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>**Fig. 13.** Trajectory creation for the thick part using the isothermal surfaces and their height maps.</figcaption></figure>

validating the method while ensuring compliance with the geometric constraints of the part.

### 3.1. Materials

The chosen geometry is fabricated using a KUKA 6-axis robot and a welding torch, similar to the one presented in [10]. The torch

<figure><img src="image_17.png" /><figcaption>**Fig. 14.** The system used to manufacture the parts consists in a KUKA robotic arm equipped with a CMT welding torch.</figcaption></figure>

employs the CMT technology developed by Fronius [45], which reduces energy input during the welding phase. This reduces the size of the melt pool and the formation of droplets, thereby minimizing spattering. The multi-axis system comprising the robotic arm provides a large working area and the freedom of movement offered by WAAM. Parts are manufactured to avoid kinematic singularities by placing their reference frame at the center of the workspace.

The position of the welding torch (see Fig. 14) is tracked throughout the fabrication process using the robot's supervisory software. Coupled with an Ethernet XML module, this allows communication with the robot during fabrication to manage process parameters. Fabrication instructions received by the robot are verified using a library specific to the welding system. The material used for the Schwarz-P fabrication is G3Si steel with a shielding gas composed of 92% Argon and 8% CO₂. A 6 mm substrate plate, made of the same steel, is used to support the part and hold it in place during production. The process and material properties used for fabricating our part are listed in Table 2.

It's crucial to be able to accurately modulate the deposition height, which is one of the key advantages of WAAM and central to our method. This modulation is achieved through mastery of the process parameters. It's possible to vary, during fabrication, the wire feed speed (WFS), torch speed (TS), deposition height (BH), and deposition width (BW) of the weld bead. The voltage (U) and current (I) parameters, which influence the electric arc, regulate the heat input during fabrication. Considering the deposited material bead with an elliptical section, we can relate the aforementioned process parameters to the following relationship,
<caption>**Table 2**</caption>

Different properties of the G3Si steel used to manufacture the part utilized to validate our method.

<table><caption>**Table 3**</caption><thead><tr><th colspan="2">G3Si steel properties</th></tr></thead><tbody><tr><td>Volumetric mass</td><td>7800 kg.m<sup>-3</sup></td></tr><tr><td>Melt temperature</td><td>1700 K</td></tr><tr><td>CMT torch characteristics</td><td></td></tr><tr><td>Diameter torch</td><td>24 mm</td></tr><tr><td>Diameter wire</td><td>1 mm</td></tr></tbody></table>

Experimental campaign for height deposition modulation with constant wire speed.

<table><thead><tr><th></th><th>TS<br>mm.s<sup>-1</sup></th><th>WFS<br>m.min<sup>-1</sup></th><th>BH<br>mm</th><th>BW<br>mm</th></tr></thead><tbody><tr><td rowspan="3">Test 1</td><td rowspan="3">0.005</td><td rowspan="3">2.4</td><td>2.5</td><td>3.6</td></tr><tr><td>2.4</td><td>3.5</td></tr><tr><td>2.5</td><td>3.5</td></tr><tr><td><strong>Mean</strong></td><td><strong>0.003</strong></td><td><strong>2.4</strong></td><td><strong>2.53</strong></td><td><strong>3.53</strong></td></tr><tr><td rowspan="3">Test 2</td><td rowspan="3">0.005</td><td rowspan="3">2.4</td><td>2.1</td><td>3.1</td></tr><tr><td>2.1</td><td>3.0</td></tr><tr><td>2.2</td><td>3.2</td></tr><tr><td><strong>Mean</strong></td><td><strong>0.005</strong></td><td><strong>2.4</strong></td><td><strong>2.13</strong></td><td><strong>3.1</strong></td></tr><tr><td rowspan="3">Test 3</td><td>0.005</td><td>2.4</td><td>2.1</td><td>2.3</td></tr><tr><td>0.007</td><td>2.4</td><td>2.1</td><td>2.3</td></tr><tr><td>0.007</td><td>2.4</td><td>2.0</td><td>2.2</td></tr><tr><td><strong>Mean</strong></td><td><strong>0.007</strong></td><td><strong>2.4</strong></td><td><strong>2.03</strong></td><td><strong>2.26</strong></td></tr></tbody></table>

derived from Ref. [21], with *k* being the bead collapse coefficient.

$$BH = \frac{\pi * d_{it}^2 * WFS}{4 * BW * k * TS} \quad (10)$$

An experimental campaign was conducted to test deposition modulation on the simple geometry of a cylinder. This aimed to validate two aspects: firstly, the modulation of deposition height during the fabrication process, and secondly, to determine the optimal overlap coefficient between two beads. The first series of experiments aimed to associate sets of process parameters that could produce a given deposition height and width, based on the following two parameters:

* Torch linear speed in mm.s⁻¹,

* Wire feed speed in m.min⁻¹.

The voltage, current, and wire feed speed are kept constant to facilitate the study at values of 10.3 V, 68 A, and 2.4 m.min⁻¹ respectively. The tests are repeated three times to obtain an average of the results and are conducted over ten layers to reach a steady state and smooth out any potential imperfections related to the start-up of the process. Heights are calculated over the total height obtained from the 10 layers, then divided by the same number, and are presented in the following Table 3.

A second campaign is conducted to test the evolution of both the height and width of the deposition during fabrication. A cylinder composed of ten beads is fabricated to reach a steady state, as in the first experimental campaign. This cylinder is divided into 4 quarters where different process parameters are tested. The first and third quarters are fabricated with a torch linear speed of 0.005 mm.s⁻¹, while the second quarter has a torch speed of 0.007 mm.s⁻¹. The fourth quarter experiences a slowdown in torch speed compared to quarters 1 and 2, with a speed of 0.003 mm.s⁻¹. The results obtained are presented in the Table 4.

## 3.2. Topological optimization of the Schwarz-P shape

We have decided not to manufacture the solid part because this type of component is of limited interest in additive manufacturing. Indeed, it is often more advantageous to produce such a part through subtractive

<caption>**Table 4**</caption>

Experimental campaign to verify the evolution of height deposition modulation during fabrication with constant wire speed.

<table><thead><tr><th>Quarter</th><th>TS<br>mm.s<sup>-1</sup></th><th>WFS<br>m.min<sup>-1</sup></th><th>BH<br>mm</th><th>BW<br>mm</th></tr></thead><tbody><tr><td rowspan="3">1</td><td rowspan="3">0.005</td><td rowspan="3">2.4</td><td>2.1</td><td>3.0</td></tr><tr><td>2.1</td><td>3.1</td></tr><tr><td>2.2</td><td>3.2</td></tr><tr><td><strong>Mean</strong></td><td><strong>0.005</strong></td><td><strong>2.4</strong></td><td><strong>2.13</strong></td><td><strong>3.1</strong></td></tr><tr><td rowspan="3">2</td><td rowspan="3">0.007</td><td rowspan="3">2.4</td><td>2.2</td><td>2.3</td></tr><tr><td>2.2</td><td>2.4</td></tr><tr><td>2.1</td><td>2.4</td></tr><tr><td><strong>Mean</strong></td><td><strong>0.007</strong></td><td><strong>2.4</strong></td><td><strong>2.17</strong></td><td><strong>2.37</strong></td></tr><tr><td rowspan="3">3</td><td rowspan="3">0.005</td><td rowspan="3">2.4</td><td>2.1</td><td>3.1</td></tr><tr><td>2.1</td><td>3.0</td></tr><tr><td>2.2</td><td>3.2</td></tr><tr><td><strong>Mean</strong></td><td><strong>0.005</strong></td><td><strong>2.4</strong></td><td><strong>2.13</strong></td><td><strong>3.1</strong></td></tr><tr><td rowspan="3">4</td><td rowspan="3">0.003</td><td rowspan="3">2.4</td><td>2.5</td><td>3.6</td></tr><tr><td>2.6</td><td>3.7</td></tr><tr><td>2.5</td><td>3.7</td></tr><tr><td><strong>Mean</strong></td><td><strong>0.003</strong></td><td><strong>2.4</strong></td><td><strong>2.53</strong></td><td><strong>3.67</strong></td></tr></tbody></table>

<figure><img src="image_18.png" /></figure>

<table><caption>**Fig. 15.** Structural characteristics comparison - Gyroid with constant and variable thickness.</caption><thead><tr><th colspan="2">Configuration initiale</th><th colspan="2">Configuration retenue</th></tr></thead><tbody><tr><td></td><td>Vol. (m³)</td><td>7.11c-4</td><td>8.24c-4</td></tr><tr><td></td><td>Vol. sup.</td><td>0%</td><td>6.61%</td></tr><tr><td></td><td>Vol. conf.</td><td>÷ Vol. ini.</td><td>÷ Vol. ini.</td></tr><tr><th colspan="2">Compression</th><th colspan="2">Compression</th></tr><tr><td>σ<sub>VM</sub> (MPa)</td><td>2.7c2</td><td>σ<sub>VM</sub> (MPa)</td><td>2.20c2</td></tr><tr><td>U<sub>Magni.</sub> (mm)</td><td>3.92c-1</td><td>U<sub>Magni.</sub> (mm)</td><td>1.52c-1</td></tr><tr><td>U<sub>3</sub> (mm)</td><td>-3.92c-1</td><td></td><td>-1.52c-1</td></tr><tr><td></td><td></td><td></td><td></td></tr></tbody></table>

manufacturing. Moreover, manufacturing a solid part would require a very long fabrication time and a significant material consumption. Many research efforts focus on finding repeatable elementary geometric patterns that allow filling entire volumes by overlaying and juxtaposing these elementary patterns. In this context, it is interesting to choose part geometries that maximize stiffness while not being entirely solid to limit the total mass of the system. Topological optimization techniques can be used to determine a geometry that exhibits a good stiffness-to-weight ratio. The geometry chosen to apply our strategy is the one presented in Fig. 2(c), referred to as “Schwarz-P”. This choice was made because this geometry can be used as an elementary structure for filling a given volume and due to its overhangs [11]. First, in the following section, the relevance of the thickness variation of the manufactured part on its structural resistance is discussed by comparing the volumes of material required for the manufacture of gyroids of constant and variable thicknesses. Then, in the next section, the trajectory determination method presented in this article is implemented.
<figure><img src="image_19.png" /><figcaption>**Fig. 16.** (a). Geometry serving as a base for topological optimization based on the parameters in Table 5. (b). Final part to be manufactured resulting from topological optimization on the shape (a). (c). Part section of part resulting from topological optimization highlighting the locally variable thickness nature.</figcaption></figure>

<table><caption>**Table 5**

Parameters used for topological optimization of the thick part 16(b).</caption><thead><tr><th colspan="2">Material used in finite element software</th></tr></thead><tbody><tr><td>Density</td><td>8000 kg.m<sup>-3</sup></td></tr><tr><td>Young's Modulus</td><td>193e3 MPa</td></tr><tr><td>Poisson's ratio</td><td>0.25</td></tr><tr><td>Young's Modulus</td><td>193e3 MPa</td></tr></tbody><thead><tr><th colspan="2">Mechanical parameters used for topological optimization</th></tr></thead><tbody><tr><td>Compressive stress</td><td>50 kN</td></tr><tr><td>yield stress</td><td>485 MPa</td></tr><tr><td>Minimum thickness e<sub>min</sub></td><td>8 mm</td></tr></tbody></table>

### 3.2.1. Variable thickness interest in structural gyroid

The gyroid of constant thickness is named as Gc and the one of variable thickness is named as Gv. The external shapes and dimensions of the gyroids are identical; however, the internal shape of the gyroid Gv

<table><caption>**Table 6**

Parameters used for the thick part, “Schwarz-P”, equipped with ribs 2(c).</caption><thead><tr><th colspan="2">Parameters used to generate the thick part path</th></tr></thead><tbody><tr><td>Average deposit h</td><td>2.0 mm</td></tr><tr><td>Minimum deposit h<sub>min</sub></td><td>1.7 mm</td></tr><tr><td>Maximum deposit h<sub>max</sub></td><td>2.6 mm</td></tr><tr><td>Number of layers</td><td>110</td></tr></tbody></table>

<figure><img src="image_20.png" /><figcaption>**Fig. 18.** Creating the trajectory for the thick part measuring 200 mm × 200 mm × 200 mm using the isothermal surfaces and their height map.</figcaption></figure>

is topologically optimized by locally adding material inside the part to increase its stiffness (Fig. 15).

Analysis of the simulation results shows that an increase of only 6.6% in the mass of the part, with an intelligent arrangement of the material in the form of ribs, makes it possible to reduce the maximum deflexion absorbed by the part by 63%. This clearly illustrates that the possibility of manufacturing locally thick parts allows a significant advance in the application of wire deposition technology to the manufacture of structural parts.

However the resulting geometry is complex due to overhanging areas, which promote material sagging, as well as numerous areas where collisions are possible. While this geometry offers mechanical advantages, it is made more complex to enhance its rigidity. One of the benefits of this type of geometry is the ability to juxtapose and superimpose them to fill large volumes. Therefore, it is essential that the top and bottom surfaces of each of these elementary parts are sufficiently flat to allow the overlapped deposition of the next elementary part. Achieving this

<figure><img src="image_21.png" /><figcaption>**Fig. 17.** Creating isothermal surfaces using the thermal scalar field resulting from the pure conduction problem on the “Schwarz-P” geometry.</figcaption></figure>
<figure><img src="image_22.png" /><figcaption>**Fig. 19.** Scheme showing the difference between two beads deposited with a change in nozzle orientation on the second bead deposited.</figcaption></figure>

geometry thus requires the ability to modulate the deposition height to enable layer-by-layer fabrication. The local variations in thickness that increase stiffness highlight the effectiveness of the method presented.

This more complex geometry was topologically optimized using a workspace based on the “Schwarz-P” shape, which was filled while retaining the various through-holes of this part. Another constraint for the optimization process on the minimum thickness was chosen, which is 8.0 mm, as illustrated in Fig. 16(a). Topological optimization is carried out on a compression problem to increase the stiffness of the part. Indeed, this type of part can be used as a pattern to fill cavities while limiting the weight of the structure and increasing its mechanical resistance to compression. In our study, a compression force, along the $\bar{z}$ axis, of 50 kN is applied to a master node connected as a rigid body on the upper surface, and the translations of the nodes on the lower face are blocked. All the parameters used for the topological optimization of the “Schwarz-P” shape are presented in the following sections of the article.

### 3.2.2. Deposition path calculation

In this section, the Gv gyroid is now studied for determining deposition trajectory. Thus, topological optimization results in a thick

geometry with the possibility of local thickness variation due to the four ribs (see Fig. 2(c)). To apply the trajectory generation strategy to this geometry, similar to that used for solid parts, it is necessary to solve the thermal conduction problem using finite elements. The boundary conditions of the problem are the same as before, as described in Section 2.5.1 (Fig. 17).

Once the scalar field is obtained, the creation of curved surfaces can be performed. For this, the number of these surfaces must be calculated using Eq. (3). To manufacture the “Schwarz-P” part, 110 isothermal surfaces need to be generated, and the height will be calculated for each of them at every point. These height maps, created using ray tracing, are essential for generating the trajectory of the part. The following table summarizes the manufacturing parameters determined with the method presented.

The characteristics presented in Table 6 will be used for the fabrication of the “Schwarz-P” part, detailed in Section 3.3. They have been selected to achieve a compromise between fabrication time and the final quality of the part.

Fig. 18 illustrates the path followed by the welding torch to create the part. The geometry of the manufactured part was chosen so that the minimum thickness (see Table 5) is large enough to allow the thick part of the part to be manufactured with two material cords.

### 3.3. Manufacturing of the “Schwarz-P” part

The manufactured part, named “Schwarz-P”, made of G3Si steel, is depicted in Fig. 20. This part was produced using 110 curved layers of non-constant height. The deposition modulation ranged from 1.7 mm to 2.6 mm, demonstrating the high adaptability of the method and technology. Indeed, the greater the range of deposition modulation, the more capable the technology and method are of manufacturing complex parts using continuous overlapping layers. When fabricating thick parts like this, it's necessary to apply at least two beads to form its walls. Previous experiments revealed that the best results, in terms of bead shape and desired final dimensions, were achieved when the second bead and

<figure><img src="image_23.png" /><figcaption>**Fig. 20.** Topologically optimized thick part manufactured using our method.</figcaption></figure>
subsequent ones were deposited with a different torch orientation from the first one. Fig. 19 illustrates how the welding torch orientation is altered when depositing the second and subsequent beads compared to the first one. By adjusting the material deposition angle, it is possible to achieve a smaller bead height compared to the other configuration, limiting the size of the valley between these two beads.

The torch orientation is determined by the vector $\vec{n}$, perpendicular to the surface being fabricated. Additionally, an extra angle of 30° is added using the Frenet frame described earlier (see Section 2.4). This change in orientation aims to enhance the deposition geometry by altering the wetting behavior of the liquid pool, in order to achieve the desired dimensions of the beads. During fabrication, it's essential to ensure that the robot's kinematics aren't strained to maintain the most stable material deposition precision possible. To guarantee optimal deposition, the range of nozzle orientation has been restricted to a maximum deviation from the $\vec{z}$ axis between [-45.0°, 45.0°]. This decision was made based on the constraints of the manufacturing system in use, but it can be adjusted to allow for a broader range of robot motion if necessary.

As seen in Fig. 20, using a wire with a diameter of 1 mm has provided a good balance between surface quality and productivity. The wire diameter directly influences both factors. It helps mitigate the staircase effect, a common issue in additive manufacturing, especially with 2.5D strategies, while also affecting manufacturing time—larger diameters reduce production time but may compromise surface quality. The continuous production of this 200x200x100 mm part was estimated to take approximately 16 h and 50 min, but in reality, only half of the part was produced in about 12 h. It took longer than expected due to pauses made to ensure the proper progress of the fabrication, and because it was not feasible to manufacture this part for 8 h without interruption.

## 4. Conclusion

Many trajectory generation strategies applied to WAAM technology focus on freedom of movement. However, one of its major advantages lies in its ability to modify the deposition height during the fabrication phase by adjusting the process parameters. This paper presents an innovative approach to 3D trajectory planning for additive manufacturing by modulating the deposition height, based on slicing a geometry into curved layers. This strategy is used in this paper on thick parts with local thickness variations. This method begins with solving a pure thermal conduction problem, which accounts for all the topological peculiarities of the geometries to be printed. From this resulting thermal scalar field, isothermal surfaces are generated, serving as a support for creating the printing trajectory in a pseudo-spiral manner. The adaptation of the deposition height and width on these surfaces is determined by inter-layer distance maps. This modulation of the deposition height allows for the fabrication of non-planar layers with variable heights. The implementation of this method is flexible and enables the realization of various part topologies, whether they are solid, thick, or have variable thickness.

We applied this strategy to a topologically optimized part based on the Schwarz-P shape, enhancing its rigidity by introducing ribs to vary its thickness locally. Each curved layer trajectory in the pseudo-spiral is unique, adjusted according to the distance between successive layers. The range of deposition height modulation is constrained based on the results of the experimental campaign and the capabilities allowed by the technology used. This approach opens promising perspectives for improving the precision, performance, and variety of topologies of parts manufactured by additive methods.

### CRediT authorship contribution statement

A. Giordano: Writing – review & editing, Writing – original draft, Validation, Software, Methodology, Conceptualization. C. Bordreuil: Writing – review & editing, Validation, Methodology, Conceptualization. F. Bugarin: Writing – review & editing,

Methodology, Conceptualization. S. Segonds: Writing – review & editing, Validation, Methodology, Conceptualization.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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