# Determination of acceptable weld bead geometries for wire arc additive manufacturing

Diego Verbist, Frédéric Vignat, Matthieu Museau, Sylvain Lavernhe

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Diego Verbist, Frédéric Vignat, Matthieu Museau, Sylvain Lavernhe. Determination of acceptable weld bead geometries for wire arc additive manufacturing. International Journal of Advanced Manufacturing Technology, 2026, 10.1007/s00170-025-17155-6. http://dx.doi.org/10.1007/s00170-025-17155-6

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# Determination of acceptable weld bead geometries for Wire Arc Additive Manufacturing

Diego Verbist<sup>(1)*</sup>, Frederic Vignat<sup>(1)</sup>, Matthieu Museau<sup>(1)</sup>, Sylvain Lavernhe<sup>(2)</sup>

<sup>(1)</sup> Univ. Grenoble Alpes, CNRS, Grenoble INP, G-SCOP, 38000, Grenoble, France

<sup>(2)</sup> Université Paris-Saclay, ENS Paris-Saclay, LURPA, 91190, Gif-sur-Yvette, France

*diego.verbist-galvez1@grenoble-inp.fr

**Keywords:** WAAM, weld bead overlapping models, tolerance constraints, process planning

**Abstract:** The Wire Arc Additive Manufacturing (WAAM) technology is extensively utilized in various industrial applications, particularly for the repair and remanufacturing of damaged components. However, two prevalent challenges in these applications are the variability of substrate geometries and the poor dimensional accuracy of the deposited material. This paper aims to present a methodology for establishing geometric and manufacturing constraints tailored to the current geometry of the component. These constraints serve to guide the selection of bead geometries and process parameters, thereby enhancing dimensional accuracy and reducing material waste.

## 1 Introduction

The Wire Arc Additive Manufacturing (WAAM) technology enables the creation of metallic components by layering weld beads, facilitating the production of both thin walls and voluminous structures. This technology offers a higher material deposition rate compared to other additive manufacturing (AM) processes, allowing for the faster and more energy-efficient production of large-scale parts. The capabilities of this process, coupled with its relatively low cost, make it particularly suitable for applications in repair and remanufacturing [1], [2]. In these scenarios, WAAM is used to deposit new material onto the degraded surfaces of used or damaged parts, thereby reversing superficial wear and damage. Notably, this application necessitates that the manufacturing process be adaptable to the varying geometries of the worn parts, which entails dealing with variable substrate geometries.

Moreover, a significant limitation of this technology is that the deposited parts often exhibit a rough and undulating surface quality. Consequently, finishing operations are typically required to achieve the desired surface finish. A certain excess material allowance is necessary to ensure adequate cutting conditions during these operations, leading to waste in material, time, and energy. For economic and environmental reasons, this allowance should be minimized. This principle aligns with the concept of
Near-Net-Shape manufacturing, where the initial shaping of parts should closely approximate the intended final geometry (the Net-Shape).

However, producing Near-Net-Shape parts relies on precise control of the deposited geometry. In WAAM, this requires controlling the geometry of individual weld beads and how they fit together. Limousin et al. [3] demonstrated that the geometry of weld beads is closely related to the thermal state of the part during deposition. This state is influenced by various interdependent kinematic and thermal phenomena, which are further affected by the variable substrate geometries, making the achievement of a Near-Net-Shape geometry challenging.

Within this context, various published works have proposed methods to control weld bead geometries [4]. Models based on sine, parabola, ellipse and circular arc functions have been used to approximate the geometries of weld beads [5], [6] and to understand the interaction between adjacent beads [5], [6], [7], [8], [9], [10], [11], [12]. The accuracy of these models has been shown to depend on the feedstock material [7]. Additionally, substantial effort has been put into developing tools to predict weld bead geometries based on process parameters, using machine learning algorithms [13], [14], [15], [16], [17] or physics-based models [18], [19], [20].

Alternatively, in-process monitoring of the melt pool has been studied to detect unplanned geometrical variations and enhance process stability [21], [22], [23]. Path planning strategies have also been created to guide parameter choices and improve the stability of the obtained geometries [24], [25], [26] or to reduce the occurrence of certain defects.

Despite recent advancements in WAAM, two key challenges continue to hinder the reliable production of Near-Net-Shape (NNS) parts:

1. The limited integration of manufacturing and geometric tolerance constraints in the selection of weld bead geometries suited to both the current substrate and the target part geometry.

2. The precarious control of deposition conditions, which depend on complex and dynamic interactions between process parameters and uncontrolled phenomena, such as the evolving geometry and thermal state of the substrate. These interactions complicate precise process control and make it difficult to identify suitable conditions for achieving the desired bead geometries.

Given the complexity of the problem, the proposed problem-solving strategy is divided into two stages:

1. Definition of the admissible design space for weld bead geometries. In this stage, geometric constraints are established based on machining allowance requirements and NNS limitations. These constraints are tailored to the final part geometry (defined in Figure 5), and are applied to current substrate, identifying which weld bead geometries can be deposited on the current substrate while respecting the geometric constraints. This enables the definition of a space of acceptable weld bead geometries.
2. Determination of process parameter sets capable of depositing the selected bead geometries, accounting for the substrate's thermal state. In this stage, geometric measurements and thermal simulations are used to construct empirical models that capture the relationships between bead geometry, physical conditions, and input process parameters. These models (first presented in a previous publication [27]) enable the identification of operating points adapted to deposit target weld bead geometries under specific substrate thermal conditions. This effectively transforms the space of acceptable weld bead geometries into a space of acceptable operating points.

The overall methodology (illustrated in Figure 1) integrates these two stages to determine admissible weld bead geometries and corresponding user-defined process parameters for each layer, based on the substrate's geometric and thermal state. This workflow is embedded within an iterative algorithm that adapts to the predicted or measured substrate state, enabling a suitable selection of bead geometry and process settings throughout multi-layer deposition.

<figure><img src="image_2.png" /><figcaption>Fig. 1. Diagram of the global proposed methodology</figcaption></figure>

This article concentrates on the first stage of the proposed methodology, which involves defining a space of acceptable weld bead geometries based solely on geometric criteria. Subsequent stages of the overall framework, notably the selection of parameters and the iterative algorithm, will be addressed in a future publication. The research question guiding the work presented here is as follows:

**How can acceptable weld bead geometries for manufacturing Near-Net-Shape parts in WAAM be defined accounting for varying substrate geometries?**

To address this question, this article is structured into three sections: Section 2 introduces the various models used to represent weld bead geometries and their overlapping behaviours. Section 3 explains how geometric constraints can be derived from manufacturing and Near-Net-Shape requirements. Section 4 presents two methods for identifying acceptable weld bead geometries based on these constraints and discusses the advantages and applications of each method. Finally, Section 5 shows the experimental validation of the models used in this work.
## 2 Weld bead geometry models

To better understand and analyse the geometries of weld beads and their stacking interactions, it is necessary to establish models to represent their geometries and quantify the observed variability.

### 2.1 Individual Weld bead geometry models

To understand the problem, the first step is to define the geometry of single weld beads. Several weld bead geometry approximation models have been considered in the literature. Xiong et al. [5] and Ding et al. [9] studied parabola, circular arc, and sine function models, determining them to be reliable approximations of the weld bead profile. They found the pertinence of each model to be dependent on the tested process parameters. For instance, the parabola was more accurate for lower deposition ratios (ratio between the feed speed and travel speed of the torch), while the circular arc was deemed more accurate for higher ratios (above 12). Zhang et al. [26] observed 3 distinct bead profile shapes in aluminium-based WAAM, depending mainly on the feed rate: Ω-shaped, arc-shaped, and triangular-like. The first two of these are better represented by the circular weld bead model. These variations underscore the importance of selecting a model adapted to the feed-stock material and intended operating window. Limousin et al. [3] investigated the effects of different thermal parameters on the geometry of Al-Mg alloy weld beads in WAAM and found the circular arc model to be particularly well-suited for representing aluminium alloy weld beads. Given that the experimental tests in our work involve an Al-Mg alloy feedstock, this model is chosen to represent the beads. This approach allows for the description of weld bead geometries using three parameters: the bead radius R, the elevation of the centre Zc relative to the substrate, and the cross-sectional area A of the bead above the substrate (an indicator of the amount of deposited material).

<figure><img src="image_3.png" /><figcaption>Fig. 2. Visualisation of different weld beads represented by the circular arc model</figcaption></figure>

In the simplest scenario (represented in Fig 2), where the bead is deposited on a flat substrate, the area can be calculated as a function of the bead's radius and the elevation of its centre relative to the substrate's surface. By using trigonometric relations, based on the circular model and the position of the centre of the weld bead relative to the substrate, we obtain equations (1) and (2).

The formula changes slightly depending on whether the centre of the weld bead is below or above the substrate's surface (thus whether Zc is positive or negative):

$$ if \ Z_c \ge 0: \quad A = \pi R^2 - \cos^{-1}\left(\frac{Z_c}{R}\right) * R^2 + \left(Z_c * \sqrt{R^2 - Z_c^2}\right) \quad (1) $$
$$ if \ Z_c < 0: \quad A = \cos^{-1}\left(\frac{Z_c}{R}\right) * R^2 - \left(Z_c * \sqrt{R^2 - Z_c^2}\right) \quad (2) $$

Knowing two of these parameters, the third can be calculated using the previous formulas. For the remainder of this paper, the studies will primarily focus on the bead radius (R) and cross-sectional area (A) as the main parameters describing the weld bead geometry.

As illustrated in Figure 2, the radius and cross-sectional area can evolve independently, resulting in various bead geometries. The cross-sectional area represents the amount of material within the weld bead and is thus a consequence of the material deposition rate (the ratio between the wire feed speed and the travel speed of the torch). Conversely, the primary parameter determining the radius of the weld beads and the height of their centre is found to be the thermal environment during deposition [3], which is itself a result of the interaction between the heat source and the temperature field of the substrate. For instance, the beads with a constant cross-sectional area, as depicted in Figure 2, are deposited under different thermal conditions. Higher deposition temperatures lead to a greater spread of the bead, resulting in a larger radius and a lower elevation of the centre.

Next, it is important to understand how these weld beads interact with the given substrate shape, which, in most cases, will be constituted by previously deposited weld beads. This would enable the generalisation of the circular weld bead model for different substrate geometries. The principles remain the same:

* The centre of the weld bead is assumed to be located exactly on the deposition trajectory.

* The considered cross-section area of a weld bead does not include the area that intersects with the substrate

To understand these interactions, two common case scenarios were considered:

* Vertical stacking, used to create thin-wall structures.

* Horizontal overlap, used to create cladding layers or fill a more important volume.

## 2.2 Thin wall structures built by vertically stacking individual weld beads

To comprehend the interaction of weld beads when stacked vertically, previous research by Limousin et al. [3] conducted tests depositing aluminium weld beads on both flat and cylindrical substrates with different radii, and at different pre-heating temperatures. Their findings indicate that radius of newly deposited weld beads increases with the rise of temperatures surrounding deposition (calculated through thermal simulation), while that the substrate's shape has a negligible impact. Assuming no material loss during deposition and a constant deposition rate, this hypothesis suggests that both the radius and the cross-sectional area remain unaffected by the substrate geometry.

By adapting the previous equations (1, 2) to a non-planar substrate, it is feasible to select an expected weld bead geometry (radius and area) and calculate its position relative to the substrate, predicting the geometry of the overlapping structure.
<figure><img src="image_4.png" /><figcaption>**Fig. 3.** Representation of vertically overlapping weld beads.</figcaption></figure>

As illustrated in Figure 3, the considered cross-sectional area of the new weld bead ($A_1$) excludes the portion of the circle that overlaps with the previous weld bead. To maintain the cross-sectional area and radius, the position (centre height) of the weld bead adjusts according to the substrate geometry. This vertical bead overlapping model has been shown to accurately represent the geometry of aluminium thin-walled structures in WAAM [7].

## 2.3 Voluminous structures filled with horizontally overlapping weld beads.

When depositing weld beads side by side, a portion of the predicted geometry of each weld bead overlaps with the adjacent bead. The material intended to occupy this overlapped area is displaced and hypothesized to be located in the valley between the two weld beads. Literature has explored the formation of these valleys; Suryakumar et al. [8] demonstrated that the valley shape varies with the lateral centre distance (distance between beads).

To predict the geometry of overlapped structures (beads plus valley), it is crucial to consider both the geometry of individual weld beads and the distance between their centres. Various geometric models approximate the valley profile (e.g., flat, arc) [8]. Notably, Ding et al. [9] developed the Tangent Overlapping Model (TOM), which approximates the valley geometry with a tangent to the newly deposited weld bead. Ding et al. [9] defined a critical lateral centre-distance $d^*$ in millimetres as a function of weld bead size (assuming identical weld beads). This value serves as a centre distance limit below which the resulting deposition geometries become unstable. For this study, the TOM was adapted to the circular weld bead model (as seen in Fig 4), governed by the following principles:

* The profile of the valley is a tangent to the new weld bead

* The centre of the weld bead is assumed to be aligned with the torch (vertical) and the contact point in the substrate (determined by the tool-path).
* The new weld bead geometry (bead plus valley) must conserve the same initially predicted radius and cross-sectional area. Therefore, the valley must compensate the overlapping area between the beads. If this is not impossible, the geometry is deemed unstable and is not represented by the model.

<figure><img src="image_5.png" /><figcaption>**Fig. 4.** Representation of the adapted TOM model for laterally overlapping circular beads</figcaption></figure>

With these principles, given the substrate geometry (including the previous bead), the initially predicted geometry of the new weld bead (radius and area), and the lateral centre distance, it is possible to determine the geometry of the newly deposited structure (beads plus valley).

# 3 Geometrical acceptability criteria

The previously established geometric models are employed to define possible admissible deposited geometries. To accomplish this, it is crucial to define the final part geometry, which serves as the foundation for establishing allowance constraints. These constraints involve specifying a target workpiece geometry expected at the conclusion of the deposition process. Subsequently, the strategy for depositing the next weld bead must be considered. Typically, two cases are examined:

* Vertical overlap: Weld beads are stacked one on top of another to create thin-walled structures

* Horizontal overlap: weld beads are deposited side by side to fill a larger volume

## 3.1 Vertical overlap

A relevant case study illustrating the application of this step to thin-walled structures is presented in Figure 5: a wall with increasing thickness. The final geometry of the part, post-finishing machining operations, is depicted in green. To ensure optimal conditions for finishing machining operations, it is necessary to include excess material beyond what is required to fill the part's volume. This excess material is referred to as the minimum allowance limit and is represented by the yellow lines in Figure 5, set at a distance $T_{min}$ from the surface of the final part.
To minimize material waste and reduce cutting force variations during machining operations, a maximum allowance limit $V_i$ is implemented. This limit aims to mitigate shape defects.

<figure><img src="image_6.png" /><figcaption>Fig. 5. Wall with increasing thickness case study with allowance limits</figcaption></figure>

Using these limits and the current substrate geometry, it is feasible to evaluate which bead geometries can be used to produce an acceptable part geometry. To be deemed acceptable, the deposited geometry must remain within the geometric allowance limits, represented by the red and yellow lines. A target workpiece geometry is established between these limits (grey contour in Figure 5), and a tolerance for shape variations $(\pm \frac{V_i}{2})$ is defined.

## 3.2 Horizontal overlap

Similar geometric constraints can be established for horizontally overlapping weld beads (Figure 6). The final CAD geometry of the part, post-finishing, serves as the basis for defining tolerance limits. A target workpiece geometry is then established within these limits, and a tolerance value is set to minimize excess material and variations in cutting forces during finishing operations.
<figure><img src="image_7.png" /><figcaption>**Fig. 6.** Representation of planar constraints on laterally overlapping beads.</figcaption></figure>

When constructing a part through successive layers of weld beads, each layer can be considered as an intermediate target workpiece, as seen in Figure 7:

<figure><img src="image_8.png" /><figcaption>**Fig. 7.** Process architecture for the creation of parts in WAAM.</figcaption></figure>

Implementing similar constraints on each of these components can enhance the planarity of each layer, thereby reducing the occurrence and propagation of defects in subsequent layers. Geometric boundaries, as illustrated in Figure 6 by the red and orange lines, restrict the space of admissible geometries. The profile of the deposited part or layer must remain within these boundaries to be deemed acceptable. The established weld bead models and geometric constraints are utilized to define a space of acceptable weld bead geometries.
# 4 Constructing the Space of Acceptable Bead Geometries (SABG)

Summarizing the scenarios discussed above:

<table><caption>**Table 1.** Summary of necessary parameters for the definition of weld bead geometries in different stacking scenarios.</caption><thead><tr><td>Scenario</td><td>Vertical Stacking</td><td>Horizontal overlapping</td></tr></thead><tbody><tr><td>Parameters needed to define weld bead geometries</td><td>radius, cross-section area</td><td>radius, cross-section area, centre distance</td></tr></tbody></table>

Determining the boundaries of the SABG involves solving a set of implicit nonlinear equations that link the bead geometries and the geometric constraints. This can be achieved analytically on a case-by-case basis aided by simplifying hypotheses such as assuming a flat substrate or considering previous beads as perfect circles.

However, the analytical method is limited when dealing with non-standard substrate geometries and quickly grows in complexity with each additional variable. Therefore, to consider more complex cases (such as the horizontal overlapping case) and to account for non-standard workpiece geometries, a numerical solution within a CAD environment is proposed.

In this section, both approaches are presented and illustrated with the case-studies defined previously: the analytical approach is applied to the vertical stacking scenario, while the numerical approach is used for the more complex horizontal overlapping scenario.

## 4.1 Analytical method

The previously established geometric constraints can be expressed as mathematical equations. This method is illustrated in the following case study of a wall with increasing thickness (Figure 8):

Let $R_1$ and $R_2$ be the radii of the previous weld ($C_1$) bead and the new one ($C_2$), respectively. $Z_{c1}$ is the elevation of the centre of the previous weld bead.

The allowance limits are defined as follows:

* $T_{min}$: the minimum allowance limit

* $V_i + T_{min}$: the maximum allowance limit

The target workpiece geometry is defined by:

* e: the initial thickness

* α: the slope angle

* H: the total height of the wall
<figure><img src="image_9.png" /><figcaption>**Fig. 8.** Case study: wall with increasing thickness.</figcaption></figure>

Finding the acceptable range of geometries involves determining valid combinations of radius (R) and cross-sectional area (A). First, a range of acceptable radius values is proposed:

* The weld beads must cover the area within the minimal allowance limits. Considering the increasing thickness of the wall, the minimum radius for C₂ is set as the radius of C₁: R₂min = R₁ (Fig 9a-9c).

* The beads must also respect the maximal allowance limit. The geometry of C₂ with the largest possible radius will be tangent to the maximal allowance limit and coincide with the intersection of C₁ and the minimal allowance limit (Fig 9e).

* Thus, the profile of the overlapping weld beads must remain within the set boundaries: the interval [Rmin, Rmax].

For each radius value within these limits, the range of acceptable centre positions is different. Each centre position, for any given radius, changes the weld bead cross section area. Hence, by limiting the position of the bead profiles, these geometrical constrains effectively restrict both the radius and cross section area of the weld beads.

All limits can be calculated as functions of the centre height and radius of the previous bead, the target geometry dimensions, and the allowance margins. Notably, the limits depend on the substrate geometry, whether it is the base plate or the geometry inherited from the previous layer. This approach allows for the analytical definition of geometric limits that delineate the Space of Acceptable Bead Geometries (SABG), as represented in Figure 10.
<figure><img src="image_10.png" /><figcaption>Fig. 9. Examples acceptable weld bead centre positions for different radius values. These positions illustrate the limits of the SABG represented in Figure 10.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>**Fig. 10.** Representation of the SABG for a weld bead in the case study: R₀ = 2.4mm, Z₀=0mm, e=2mm, alpha=5°, Tmin=0.5mm, Vi=0.8mm</figcaption></figure>

To illustrate the SABG and the different geometries it encompasses, the boundary points defining the SABG are depicted in Figure 9:

* Initially, a configuration with the minimum radius and smallest area (lowest acceptable centre position) corresponds to an exact superposition of the previous bead, as shown in Fig 9a.

* Increasing the cross-sectional area (by elevating the centre position) results in geometries such as the intermediate position illustrated in Fig 9b. The upper limit is reached in Figure 9c, as further increasing the area without altering the radius would violate the minimal allowance limits.

* To proceed, the bead radius must be increased alongside the cross-sectional area (Fig 9d), up to the maximum radius for which only one acceptable cross-sectional area exists (Fig 9e).

* From this position, reducing the radius necessitates a corresponding reduction in area:

- For larger radius values, the minimum cross-sectional area is constrained by the intersection with the maximum allowance limit (red lines). This results in geometries similar to Fig 9f.

- For smaller radius values, this minimum area is determined by the requirement for C₂ to cover the top of C₁. This produces geometries such as that in Fig 9h.

- The transition point between these two behaviours occurs when C₂ is tangent to both the maximum allowance limits and the top of C₁, as depicted in Fig 9g.

## 4.2 Numerical method

The geometric models, constraints, and resulting SABG presented in this paper can be constructed using mathematical formulas, although with some simplifying hypotheses. However, this method is limited when considering non-standard substrate geometries, as the formulas would need revision for different situations. To address this, the second
approach considered is the numerical resolution of the problem within a CAD
environment.

CAD software enables the importation of a 3D-scanned substrate geometry into the work environment, updating the geometric models to adapt to the current workpiece geometry. The following chain of operations was used to build the horizontal overlapping model based on the current substrate geometry.

<figure><img src="image_12.png" /><figcaption>**Fig. 11.** Algorithm for constructing the horizontal overlapping model using 3D scanning and CAD software</figcaption></figure>

As shown in Figure 11, the substrate geometry is measured using a Handyscan 700 3D
scanner. Rhino 8 is used to import the point cloud representing the substrate. This
geometry is then reoriented, centred, and sliced to obtain a series of cross-section curves describing the substrate's profile.

The average of these curves serves as the substrate's profile. Based on this profile, geometric tolerance constraints can be defined as described in Section 3.2. For instance, in Figure 12, the highest point of the first weld bead ($C_1$) is used as the target mean height for the layer, with the acceptable design space defined by the blue lines above and below this point.

The range of possible weld bead geometries for the new bead ($C_2$) is defined by various combinations of radius and area values. Among these, a series of potential weld bead geometries is preselected based on whether the height of their summit falls within the acceptable boundaries.

<figure><img src="image_13.png" /><figcaption>Fig. 12. Extreme example of an acceptable geometry of the overlap model built in Rhino 3D Grasshopper. All configurations in which both point B and the summit of $C_2$ are within the blue margins are considered acceptable.</figcaption></figure>

Each potential geometry of $C_2$ can be placed at different centre distance values relative to the centre of $C_1$. A range of centre-distance values allowing for intersection between the beads is tested. For each iteration, the position of point B is adjusted to ensure that the area of the model (bead + valley, depicted in red in Figure 14) is equal to the expected cross-sectional area for that bead geometry.

To be deemed acceptable, the geometry must fulfil the following conditions:

* The area of the model (bead + valley) is equal to the expected $C_2$ cross-sectional area.

* Both point B and the summit of $C_2$ remain within the tolerance margins.

Otherwise, that geometry is deemed unacceptable and discarded, and the next set of parameters is tested.

After testing all bead geometry – centre-distance pairings, the acceptable combinations can then be represented in a graph with points of {radius, area, centre distance} coordinates. An SABG is then defined as a convex hull volume containing these points, as illustrated in Figure 13.
<figure><img src="image_14.png" /><figcaption>**Fig. 13.** SABG in radius, cross-section area and centre distance axes. The data points used to build the volume are represented in black. The black contours represent sections of the volume at different D levels.</figcaption></figure>

# 5 Experimental validation and discussion

The experiments presented in this part are all conducted in a WAAM platform, consisting of a YASKAWA MA1440 six-axis robot and a TPS CMT 4000 Fronius welding station. The feedstock material is Al5356, and the welding synergy is set to CMT 875.

## 5.1 Validation of the vertical overlapping model

This geometric model for thin-walled structures was initially presented by Manokruang et al. [7] who demonstrated its accuracy in representing the geometry of vertically stacked aluminium weld beads in a wall with constant thickness in WAAM. The case study in this article is employed to validate the vertical overlapping circular model. For this purpose, weld beads with increasing material deposition rates are stacked vertically. After the deposition of each layer, the workpiece geometry is 3D scanned, and the resulting mesh is sliced into cross-sections. The profile of each layer is defined as an average curve of these cross-sectional curves.
A circular overlapping model is then constructed, consisting of circles sized according to the expected radius and cross-sectional area values for each layer. Given the significant variability inherent in the manufacturing process, the radius and cross-sectional area values used correspond to measurements of the actual deposited geometries.

Finally, the obtained circles are compared to the scanned profiles by calculating the
distance between all points in the profile and the closest point on the circles. This
comparison assesses whether the model can accurately represent the profile of the
scanned geometry.

<figure><img src="image_15.png" /><figcaption>**Fig. 14.** Representation of two different deposited walls. In black the average curves of each layer. In red the circles corresponding to the model.</figcaption></figure>

Figure 14b illustrates an initial test involving two vertically deposited beads. The maximum error observed is located at the base of the upper bead, amounting to 12% of the bead's radius. Conversely, the top of the bead exhibits an error of 4%.

Figure 14b also depicts a wall constructed with larger weld beads and additional layers, with the final three layers examined in this example. Consistent with previous observations, the highest error in this model is found at the base of the weld beads, where the measured error ranges between 2% and 5% of the respective bead's radius. Near the top of each weld bead, the error ranges between 1% and 2% of the radius.

These observations suggest that the model more accurately represents the geometry of
larger, vertically stacked weld beads. The highest deviations are found at the base of
the weld beads, whereas the top of the beads is predicted with greater accuracy.

**5.2 Validation of the lateral overlapping model**

Regarding the validation of the lateral overlapping model, the aim of the experiments is to compare the fitted circular bead profiles and valley geometries to that of weld beads deposited side by side. To reduce the effects of certain perturbations unique to
CMT-based WAAM, process parameters were maintained as constant as possible for the deposition of all weld beads:

* Wire Feed Speed (WFS): 5 m/min

* Travel Speed (TS): 60 cm/min

* Substrate preheating temperature: 200°C.

These parameters were used to deposit a first series of weld beads (C₁), sufficiently spaced to avoid interaction (40mm) and with pauses to allow for temperature stabilisation. The geometry of the deposited weld beads was scanned and fitted with circular arcs. Radius and area were measured and an SABG was calculated using the model (which is displayed later in Fig 19).

After re-stabilisation of substrate temperature, a second series of weld beads (C₂) was deposited (with identical parameters) adjacent to the first series at various centre distance values within the range that allows for overlap.

<figure><img src="image_16.png" /><figcaption>**Fig. 15.** Comparison between the measured centre-distances and the programmed step-over distances. The obtained data-points are represented in blue, while the grey dotted line serves as a reference for when both coordinates are equal.</figcaption></figure>

As seen in Figure 15, most test results revealed a significant discrepancy between the apparent lateral centre distance of the obtained beads and the expected centre-distance (corresponding to the programmed step-over distance in the machine trajectory settings). An example of this occurrence is shown in Fig 16.
<figure><img src="image_17.png" /></figure>

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

<figure><img src="image_19.png" /></figure>

<figure><img src="image_20.png" /></figure>
<figure><img src="image_21.png" /><figcaption>**Fig. 16.** Comparison between the expected overlapping geometries and the scanned profile of the overlapping beads. a) Scanned profiles of $C_1$ and extracted average profile. b) Circular fit of $C_2$ on the average profile. c) Expected $C_2$ geometry based on the geometry of $C_1$ and the programmed step-over distance. d) Scanned profiles of the overlapping geometry and circular fits. e) Overlapping model adjusted to fit the scanned geometry and comparison with the expected position of the bead's centre.</figcaption></figure>

The observed discrepancy between the expected and actual centre positions of $C_2$ likely results from unplanned interactions between the deposition process and the pre-existing profile of bead $C_1$. Two mechanisms are hypothesised:

1. Arc deflection: The welding arc may be diverted toward the adjacent bead due to geometric or electromagnetic influences, causing the molten droplets to be preferentially deposited closer to the neighbouring weld bead.

2. Thermocapillary forces: Temperature gradients within the melt pool can induce surface tension-driven flow, directing material toward cooler regions with lower surface tension, typically near the previously deposited bead.

This phenomenon has been previously documented in the literature by Li et al. [10], who developed a neural network-based algorithm to predict the actual centre distance between beads. Their algorithm utilizes the bead's dimensions (approximated by a parabola) and the programmed centre distance as input data. However, the algorithm is calibrated for a specific feedstock and welding mode, limiting its general applicability. Alternatively, Zhang et al. [26] proposed an alternative path-planning strategy for cladding layers, in which parallel weld beads are first deposited, sufficiently spaced from each-other to avoid overlap and melt pool interactions. A second series of beads is then deposited effectively "filling the gaps" between the previous weld beads. This approach ensures that each bead is deposited in a more symmetric thermal and geometric environment, reducing thermal gradients geometric interactions, mitigating centre-distance deviation.

These approaches, however, are more relevant for path-planning and process parameter determination, which will be addressed in future work.

Additionally, it is observed that when the intersection between $C_2$ and $C_1$ exceeds the compensating capacity of the valley, the centre of $C_2$ shifts upward to accommodate
displaced material. This shift alters the valley geometry, producing a visible bulge at the summit of C₂ that is misaligned with the bead's geometric centre. Meanwhile, the lower region of the valley, near the intersection with C₁, exhibits reduced material accumulation. Similar behaviour was consistently observed in other samples with excessive overlap. The trend indicates that reducing the step-over distance increases bead-to-bead attraction, as evidenced by the growing discrepancy between the programmed step-over and the measured centre distance (Figure 15), and leads to more pronounced deformation of the valley profile.

These observations raise questions about the accuracy of the proposed overlapping model. To evaluate its validity, we focus on two critical locations for assessing the flatness of the top surface: the summit of bead C₂ and the bottom of the valley formed between C₁ and C₂. These correspond to points A and B in the model shown in Figure 16e.

To enable this comparison, the overlapping model is first adjusted to fit the scanned geometries (as seen in Figure 16e).

* Circles are fitted to the free side of each bead (not accounting) to determine their radius the position of their centres.

* The centre distance is measured between the centres of the fitted circular arcs.

* The cross-sectional area of the scanned C₂ profile is measured by comparing it with the scanned profile of C₁ (scanned before C₂ was deposited). This area is then used as a target for the area of C₂ in the model (circular bead plus tangent valley), allowing the program to adjust the position of the valley to minimize the difference between the two cross-sectional areas.

The final result is a geometric construction composed of two intersecting circular arcs and a line representing the valley between the beads (Figure 16c).

The model's geometry is subsequently compared to the 3D-scanned profile of the beads, with particular attention given to the height at the bottom of the valley (compared to point B in the model) and the summit of C₂ (compared to point A in the model) as illustrated in Figure 16b. These points are measured on both the scanned geometry and the fitted model.
<figure><img src="image_22.png" /><figcaption>**Fig. 17.** Height comparison between scanned profiles and the fitted model. Green and blue curves: tendencies of the measured heights at the top of C₂ and the bottom of the valley respectively. Blue shaded region: range of centre-distance values considered by the acceptability algorithm. Yellow rectangle: potential range of acceptable configurations.</figcaption></figure>

The graph in Figure 17 demonstrates that within its range of validity (blue shaded region), the adapted tangent overlapping model accurately assesses the elevation of both the bottom of the valley and the summit of C₂, with the highest relative error measured being 5.1%.

The dark blue dotted lines in Figure 17 represent the geometrical allowance limits used to assess whether a weld bead geometry is acceptable. Finally, the yellow rectangle delineates the true range of acceptable configurations (deduced from the tendencies of the green and blue curves). It is evident that three of the measured configurations meet the acceptability criteria, as both the bottom of the valley and the summit of C₂ remain between the tolerance margins (dark blue dotted lines Fig17). The scanned profiles of these acceptable configurations are displayed in Figure 18.
<figure><img src="image_23.png" /></figure>

<figure><img src="image_24.png" /></figure>

<figure><img src="image_25.png" /><figcaption>**Fig. 18.** Scanned profiles of the deposited acceptable weld bead geometries. These representations show the adjustment of the model (as shown in Figure 16e) to the three acceptable cases.</figcaption></figure>
However, as seen in Figure 19a, when compared to the SABG (calculated based on the geometry of $C_1$), only one of these points (corresponding to Fig 18c) is inside the volume. This indicates it is possible to find acceptable geometries outside the SABG defined by the model. This discrepancy is likely due to lower centre distance values being discarded from the model's dataset, based on the assumption that the position of the centre of $C_2$ is unaffected by the adjacent bead. This assumption leads the model to only consider weld beads which are sufficiently spaced to retain a geometry similar to that observed on a single bead deposited on a flat substrate. However, in certain configurations where the beads are closer together, the overlap area between them is too large to be compensated by the tangent valley; the centre of the bead is then elevated to accommodate the displaced material.

<figure><img src="image_26.png" /><figcaption>**Fig. 19.** Comparison between the SABG and the experimental results. In green, the acceptable deposited beads, and in red the rest of the experimental data.</figcaption></figure>

The acceptability model is thus currently limited regarding the range of geometries it can consider. An interesting phenomenon observed during testing is that certain centre distance values seem unattainable by merely varying the step-over distance of the deposition trajectories. Specifically, attempting to achieve centre distance values between 4 and 4.4 mm resulted in inconsistent $C_2$ geometries that were either completely separated from $C_1$ (with no overlap or valley formed) or were attracted to $C_1$ (resulting in an inferior centre distance).
# 6 Conclusion and perspectives

New weld bead overlapping models based on the circular arc model are proposed to represent the geometry of both horizontally and vertically stacked beads. The geometric models are demonstrated to accurately approximate the geometry of overlapping aluminium weld beads deposited with CMT-based WAAM.

The proposed algorithm involves defining both manufacturing and geometric constraints based on the current substrate geometry and the final part geometry. These constraints are then used in conjunction with the proposed geometric models to establish a Space of Acceptable weld Bead Geometries (SABG) in two common case studies.

The obtained SABG successfully determines a range of geometries that adhere to the established constraints. However, experimental results suggest that a certain range of acceptable geometries is not yet considered by the current rules of the model.

The tests also reveal that certain geometries, notably specific centre distance values, appear unattainable using standard trajectory settings and may require a more in-depth study of potential user handles to enable such results. Alternatively, a different deposition strategy such as the gap-filling overlapping model (GOM) [26] could be considered for use with the established geometrical constraints, avoiding the instability generated by an asymmetrical deposition environment.

Several avenues for future improvement have been identified. First, the overlapping geometry models, both vertical and horizontal, could be enhanced to more accurately capture bead-to-bead interactions and accommodate a broader range of geometries. This would yield a more comprehensive Space of Acceptable Bead Geometries (SABG).

In addition, the methodology used to define the SABG boundaries should be generalized to support more complex three-dimensional filling scenarios, such as combined vertical-horizontal stacking or spiral deposition patterns. The numerical approach presented in this work is well suited for such extensions, as it does not depend on predefined standard substrate geometries and leverages the flexibility of a CAD environment to define geometric constraints.

Finally, a key next step involves developing a decision-support tool to assist users in selecting process parameters that enable the deposition of geometries deemed acceptable by the algorithm. The proposed algorithm also holds strong potential for integration into a CAM environment, where it could serve to improve part-dimensional accuracy when coupled with suitable path-planning strategies.

# Acknowledgements

The work presented in this article was supported by the Association Nationale de la Recherche (ANR) as a part of the AWESOME project (ANR-21-CE10-0013).

The authors would like to thank the S.mart Grenoble Alpes Technological Centre for all the technical assistance and provided equipment.
**Data availability**

All data availability inquiries should be addressed to the corresponding authors.

**Conflicts of interest**

All authors declare that they have no conflicts of interest.

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