# Data-based Geometrical Model Of Weld Bead Deposit Applied For Wire And Arc Additive Manufacturing

Chetra Mang, Xavier Lorang, Ramdane Tami, François Rouchon

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Chetra Mang, Xavier Lorang, Ramdane Tami, François Rouchon. Data-based Geometrical Model Of Weld Bead Deposit Applied For Wire And Arc Additive Manufacturing. 2025. hal-05093138

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Distributed under a Creative Commons Attribution 4.0 International License
DATA-BASED GEOMETRICAL MODEL OF WELD BEAD DEPOSIT
APPLIED FOR WIRE AND ARC ADDITIVE MANUFACTURING

Chetra Mang¹, Xavier Lorang¹, Ramdane Tami¹, and François Rouchon¹,²

¹IRT SystemX, 2 Bd Thomas Gobert,
91120 Palaiseau, France

²Université Paris-Saclay, ENS Paris-Saclay,
LURPA, 91190 Gif-sur-Yvette, France

{chetra.mang, xavier.lorang}@irt-systemx.fr

{ramdane.tami, francois.rouchon}@irt-systemx.fr

ABSTRACT

The present work is carried out in the framework of the WAS project [1] which deals with directed
energy deposition (DED) process. The process relies on an automatized welding process in which
a part is built by a successively deposed metal bead. Under a two-dimensional hypothesis, we
propose a meta-model based on geometrical parameters for a weld bead geometrical model using the
equilibrium between the hydrostatic pressure and the capillarity force. The geometrical parameter
models include contact angle and tri-phase contact line position. Different experimental setups are
carried out to generate data-base for training a geometrical parameter model. Strategies for building
geometrical parameter meta-model is proposed in this paper. Then, the data-based geometrical model
is validated with a distributed overlapping distant thick wall.

**Keywords** Wire and arc additive manufacturing · Young-Laplace equation · Bead geometry · Meta-model.

# 1 Introduction

Metal additive manufacturing (MAM) has grown, in recent years, very strong interest in academic research and industrial applications. Among MAM processes, wire and arc additive manufacturing (WAAM) became very popular through its advantages in manufacturing medium and large-scale components [2] thanks to the possibility of large weld bead deposition. However, weld bead deposition is sensitive to process parameters and substrate temperature. With minor changes in these parameters, the geometric defects of the components can be significant. To be able to control these defects, we need to be able to simulate the geometry of the parts according to the process parameters. Modeling a weld bead geometry allows computing the weld bead dimensions and the geometric shape to assemble the weld bead with a given process parameter. To do so, one can either use empirical law by observing the correlation between the process parameters and the molten pool geometric shape and dimension or physical law to model the molten pool geometric shape. Ding et al. [3] proposed a 2D geometric model of a weld bead based on simple geometry such as parabolic, circle arc, and cosine functions. The molten pool’s height, width, and area section from experiments is used to build an empirical law for the function parameters. Cao et al. [4] modeled the 2D geometric shape of a weld bead based on Gaussian, exponential, parabolic and sine functions. But, the authors used only one process parameter for correlating with the function parameters. Xiong et al. [5] employed frequently used functions such as parabolic, circle arc, and cosine function to model a 2D weld bead geometry. The authors compared the simulated profiles and experimental ones. The evolution of the profiles depends on the ratio of wire speed and torch speed. The authors [3, 4, 5] also proposed a weld bead 2D overlapping model using an additional parameter to connect from one weld bead to another to ensure volume conservation. Rios et al. [6] proposed a 2D weld bead geometry model based on an analytical solution of a physical hydrostatic and capillary force equilibrium equation. The authors used the hypothesis of the infinite cylinder for modeling. Different process parameters are used to compare the simulated and experimental profiles. Mohebbi et al. [7] proposed the same methodology using the analytical solution to model a 2D weld bead geometry. The authors demonstrated the simulation of stacking weld beads compared to the experimental profiles. Up to now, we can only use
empirical law to simulate 2D weld bead geometry overlapping or analytical solution from physical law to simulate 2D weld bead geometry stacking. But, simulating 2D weld bead assembly faces many challenges. In this work, we propose a hybrid model of weld bead geometry based on a meta-model of the input parameters for the physical law of a 2D weld bead geometry. This hybrid model allows simulation of the weld bead assembly to obtain the geometry of the component to be manufactured. The paper is divided into two parts. The first part describes the methodologies to build experimental data as well as to construct a physical model based on hydrostatic and capillary force equilibrium. The second part focuses on the results and discussion. In this part, we improve our model based on the outcomes from the methodologies.

# 2 Experimental methodologies

In the framework of this paper, we describe the experimental setup, the manufactured specimens, and the measurement methods to build a database for the hybrid model. Then, we detail the construction of the physical model based on the data.

## 2.1 Experimental setup

### 2.1.1 Wire and arc additive manufacturing machine

Wire and Arc Additive Manufacturing (WAAM) is classified as a Directed Energy Deposition (DED) technology by the International Organization of Standardization (ISO) [8] as it uses an electric arc as a thermal source and a wire as feedstock. This process is derived from welding and its standard uses with industrial equipment: welding power generator, torch and wire feeding systems. A 6-axis anthropomorphic robot can provide the relative motion between the torch and the part. WAAM can use several welding techniques such as gas metal arc welding (GMAW), gas tungsten arc welding (GTAW), or plasma arc welding (PAW) (Fig. 1). Derived from GMAW, an improved technology also widely used is Cold Metal Transfer (CMT) as it offers a lower heat input [7] as the short circuit are controlled by moving the wire forward and backward. In GMAW, the electric arc is directly formed between the wire, and the part. Thus, the geometry of the torch presents an axisymmetric tube which makes the control of the tool orientation easier. GTAW and PAW use a non-consumable electrode in tungsten, requiring a separate wire feed process where the tool orientation may become an issue. The experimental tests conducted here were done with a Fanuc Arc Mate 120iC and its controller R-30iB. The welding system used is a Fronius TPS 600i with CMT process. The synergic line used is the a CMT Universal 3248.

### 2.1.2 Manufacturing of specimens

In order to study the weld bead geometry and their assembly, the artifacts built are two thick walls (A and B), a thin wall (C), and four one-layer walls (Di). For every wall, the wire used is the Boehler $X90 - IG$ with a diameter of 1.0 mm. The welding gas used is an $Ar:AlM14(Ar + O21\% + CO23\%)$ with a flow rate of $22L/min$. The wire speed rate is set to $8m/min$, and the torch speed is set to $0.75m/min$. The substrate on which the walls are built is a low alloy steel $S355$. The distance between the contact tip and the part aimed for each layer is $15mm$. However, this parameter is set indirectly during the programming of the toolpath, and due to the instability that the process may encounter, this parameter may vary. With these process parameters, the bead aimed has approximately a width of $7mm$ and a height of $2mm$.

A first $150mm$ length thick wall (A) of 10 layers is manufactured with a raster strategy where the first bead of each layer is built on the first bead of the previous layer. Here, the programmed layer distance is set as 85% of the height of a single weld bead on a plan substrate that is $1.7mm$. A fixed overlapping distance of $4.673mm$ is considered. This distance represents 66.7% of the width of a single bead on a plan substrate. This ratio corresponds to the optimal distance between two juxtaposed beads according to the Flat-top Overlapping Model [4, 5, 10].

As for the thin wall, each layer is built in the opposite direction. Then a second-thick wall (B) of 10 layers is also built with the same process parameters and the same height between each layer as the first thick wall. For this second-thick wall, the overlapping distance between two consecutive beads is variable in order to juxtapose beads in various conditions and have different contact angles. The distribution of the overlapping space between the six weld beads of the ten layers is shown in Table 1.

The thin wall (C) has a $150mm$ length, and in order to equilibrate the start and stop zone, the bead is built in the alternative direction. From our previous experience, the thickness of the 25-layers is programmed at 60% of the height of a single bead on a plan substrate. Thus, the programmed height of each layer of the thin wall is $1.2mm$. For the four one-layer walls ($D_i$), the overlapping distance between two consecutive beads is variable in order to juxtapose beads
<figure><img src="image_3.png" /><figcaption>Figure 1: Schematic diagram of (a) GTAW, (b) PAW, and (c) GMAW from [9].</figcaption></figure>

in various conditions and have different contact angles. The distribution of the overlapping distance between the five beads of each wall is listed in Table 2.

<figure><img src="image_4.png" /><figcaption>Figure 2: Distance between two weld beads.</figcaption></figure>

## 2.2 Measurements

During the manufacturing process, the wire speed and the robot's positions at the tool control point (TCP) are monitored through the robot controller. Between each layer built, manufacturing is stopped to digitalize the geometry of the artifact. The measurement process is explained in this section, and then the cloud-point post-processing is detailed.

### 2.2.1 Wire speed and robot speed measurement

The wire speed and TCP position are measured directly through the controller, which obtains the wire speed information from the power generator and uses the robot's internal sensors to get the flange's position relative to the world frame. Thus, every 12 milliseconds, the controller gives the six coordinates of the robot's flange relative to its world flange and the wire speed. In order to obtain the position of the TCP relative to the part frame in which the tool path is programmed, post-processing is required.

Considering the position and orientation of the tool frame relative to the flange and the part frame relative to the world frame, these transformations are known and necessary for the programming of the toolpath by the controller.
<table><caption>Table 1: Overlapping distribution for the thick wall B (mm).</caption><thead><tr><th colspan="6">Distance fraction between beads</th></tr><tr><th colspan="6">f<sub>i</sub> = <sup>d<sub>i</sub></sup>&frasl;<sub>L</sub>, L = 7mm</th></tr><tr><th colspan="6">Layer (10<sup>-1</sup>)</th></tr><tr><th></th><th>1-2</th><th>2-3</th><th>3-4</th><th>4-5</th><th>5-6</th></tr></thead><tbody><tr><th>1</th><td>7.38</td><td>6.19</td><td>6.85</td><td>6.67</td><td>6.98</td></tr><tr><th>2</th><td>5.81</td><td>7.34</td><td>7.13</td><td>7.44</td><td>5.87</td></tr><tr><th>3</th><td>7.08</td><td>5.98</td><td>6.32</td><td>6.06</td><td>5.61</td></tr><tr><th>4</th><td>5.91</td><td>5.52</td><td>6.09</td><td>6.16</td><td>7.48</td></tr><tr><th>5</th><td>6.92</td><td>5.72</td><td>6.64</td><td>5.76</td><td>7.28</td></tr><tr><th>6</th><td>6.53</td><td>6.40</td><td>7.14</td><td>6.89</td><td>6.74</td></tr><tr><th>7</th><td>7.19</td><td>7.22</td><td>6.23</td><td>6.48</td><td>7.04</td></tr><tr><th>8</th><td>7.41</td><td>5.67</td><td>5.95</td><td>5.63</td><td>5.58</td></tr><tr><th>9</th><td>6.78</td><td>6.44</td><td>6.55</td><td>6.26</td><td>6.81</td></tr><tr><th>10</th><td>7.00</td><td>5.83</td><td>6.61</td><td>6.37</td><td>6.12</td></tr></tbody></table>

<table><caption>Table 2: : Overlapping distribution for the walls $D_i$ (mm).</caption><thead><tr><th colspan="5">Distance fraction between beads</th></tr><tr><th colspan="5">f<sub>i</sub> = <sup>d<sub>i</sub></sup>&frasl;<sub>L</sub>, L = 7mm</th></tr><tr><th colspan="5">Wall (10<sup>-1</sup>)</th></tr><tr><th></th><th>1-2</th><th>2-3</th><th>3-4</th><th>4-5</th></tr></thead><tbody><tr><th>D<sub>1</sub></th><td>6.6</td><td>6.6</td><td>6.6</td><td>6.6</td></tr><tr><th>D<sub>2</sub></th><td>7.113</td><td>6.332</td><td>5.992</td><td>4.817</td></tr><tr><th>D<sub>3</sub></th><td>6.172</td><td>5.536</td><td>5.152</td><td>6.685</td></tr><tr><th>D<sub>4</sub></th><td>7.487</td><td>6.838</td><td>6.9311</td><td>4.996</td></tr></tbody></table>

## 2.2.2 Metal deposition scannings of the specimens

To digitalize each layer during the manufacturing process, the scanning system CREAFORM HANDY SCAN BLACK Elite is used. This scanner relies on stereo-vision and must be operated manually; thus, the manufacturing had to be interrupted between each layer in order to have access to the part for scanning. In order to digitalize and express each layer in the same coordinate system, the scanner uses calibration targets directly stuck on the substrate and the substrate's support. The scanner enables the capture of numerous cross beams made with two sets of eleven blue lines. The scanner detects the intersections of those lines up to 1.3 million measurements in seconds. In order to digitalize the whole part, the operator needs to sweep the laser crosses across all the concerned surfaces. The cloud-point generated has a resolution of 0.1 mm resulting cloud point of approximately 300 000 points for the first layers to 750 000 for the last ones. A significant amount of points of the substrate in each layer are captured, and processed. The digitalization process is illustrated Fig. 3.

<figure><img src="image_5.png" /><figcaption>Figure 3: Digitalization with the CREAFORM HANDY SCAN Black Elite.</figcaption></figure>

## 2.2.3 Specimen cloudpoint post-processing

The cloud point of each layer generated is expressed in the same coordinate system, which is not the programming coordinate system (which is also the part coordinate system). In order to fit the programmed toolpath with the digitalized
geometry, the top plan of the substrate is used as a reference. RANdom SAmple Consensus (RANSAC) algorithm is used to identify the substrate’s plan. The points of the wall, considered as outliers by the RANSAC algorithm, are then filtered such that the (OXY) plan of the coordinate system match with the identified substrate plan. In order to determine the average section of the wall after each layer, the point of the wall is arbitrarily selected with a region of interest of 90 mm approximately placed in the center of the wall, excluding 30 mm of each ends to get rid of the starting and extinction zone of the electric arcs. The selected points are then projected in an orthogonal plan relative to the toolpath. The average profile of the point cloud projection is computed with the LOcally Estimated Scatterplot Smoothing (LOWESS) method [13]. The whole method has been implemented under Python 3.7 and sklearn library for the RANSAC algorithm and stats models for the LOWESS. The computed average profiles of the overlapping one-layer walls ($D_i$), thin wall C, thick walls (A and B) are illustrated Figures 4, 5 and 6, respectively.

<figure><img src="image_6.png" /><figcaption>Figure 4: Average profiles of the overlapping walls $D_i$.</figcaption></figure>

# 3 Modeling methodologies

Firstly, this part describes the hypothesis used in physical law to model a 2D weld bead geometry. Secondly, it addresses the use of data to enrich the physical model.

## 3.1 Geometry model of welding bead

### 3.1.1 Physical nature of weld bead geometry

Molten pool under the electrical arc can be acted by several kinds of forces such as Lorentz force, buoyance force, arc shear stress and surface tension force [7]. By taking into account these forces, the molten pool geometry can be a
<figure><img src="image_7.png" /><figcaption>Figure 5: Stacking profiles of the thin wall C.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Figure 6: Average profiles of the thick wall with (left) constant overlapping distance A and (right) distributed overlapping distance B.</figcaption></figure>

very complicated shape. Under 2D hypothesis (axisymmetric or infinite cylindrical), as in [11], it is stated that the approximated geometry of the molten pool can be handled by a static liquid state which is the equilibrium between the hydrostatic pressure and the capillarity force. This equilibrium can be expressed as Laplace equation:

$$ \Delta P(z) = \gamma C(z). \qquad (1) $$

$$ P(z) - P(0) = \gamma(C(z) - C(0)) = -\rho_m g(z - 0); \qquad (2) $$

where

* $P(z), P(0)$ are the pressure at $z, z = 0$ respectively;

* $\gamma$ is the surface tension force which has the relation with the capillarity length as following:

$$ \kappa = \sqrt{\frac{\gamma}{\rho_m g}}; \qquad (3) $$

* $g$ is the gravity;

* $\rho_m$ is the metal density at temperature $T_m$ (melting temperature of the metal);
• and $C(z), C(0) = C_0$ are the curvatures at $z, z = 0$ respectively.

<figure><img src="image_9.png" /><figcaption>Figure 7: Topology of welding beads.</figcaption></figure>

The curvature $C$ is expressed as the inverse sum of the principal curvature radii $R_1$ and $R_2$:

$$C = \frac{1}{R_1} + \frac{1}{R_2}; \qquad (4)$$

With two-dimensional hypothesis, two scenarios can be displayed:

• Axisymmetrical hypothesis: $R_1 = R_2 = R \Rightarrow C = \frac{2}{R}$;

• Cylindrical hypothesis: $R_1 = R, R_2 \to \infty \Rightarrow C = \frac{1}{R}$.

Therefore, the equation (2) can be written as:

$$0.5\kappa^{-2}z^2 - C_0z + \lambda\left(\frac{\frac{dx}{dz}}{\sqrt{1 + \left(\frac{dx}{dz}\right)^2}}\right) = 0; \qquad (5)$$

where

$$\lambda = \begin{cases} 1 & \text{if cylindrical hypothesis} \\ 2 & \text{if axisymmetrical hypothesis} \end{cases} \qquad (6)$$

The shape of water drop has been studied in [12]. The drop is under the hypothesis of the spherical cap form. As the molten pool is the accumulation of the metal liquid drops while the wire arc is melting, a bead deposition can be considered as a successive merge of the spherical caps. With this consideration, we can suppose the axisymmetrical hypothesis for the bead geometry.

### 3.1.2 Analytical resolution of Young-Laplace equation

Equation (5) can be solved semi-analytically by using the elliptical integral of the first and second kind $E$ and $F$ respectively. The solution can be written as:

$$x = \pm A(E(\psi, B) - E(\frac{\pi}{2}, B) + F(\psi, B) - F(\frac{\pi}{2}, B)); \qquad (7)$$

where
$$A = \frac{1 + 0.5C_0^2\kappa^2}{\kappa^{-2} + 0.25C_0^2}; \qquad (8)$$

$$B = \frac{\kappa^{-2}}{\kappa^{-2} + 0.25C_0^2}; \qquad (9)$$

and

$$\psi = \cos^{-1}\left(\sqrt{\frac{0.5\kappa^{-2}z^2 - C_0z}{2}}\right). \qquad (10)$$

The analytical expression can be used to model a bead geometry as in [?]. Nevertheless, by using semi-analytical solution, the computation time can be heavy. In the next section, we proposed the discretization of the equation (5).

### 3.1.3 Proposed discretised resolution of Young-Laplace equation

As $\theta(z)$ is the tangent of the bead geometry, we can decompose the equation (5) into a system of two equations as following:

$$\left\{ \begin{array}{l} \frac{dx}{dz} = \frac{1}{\tan(\theta(z))} \\ 0.5k^{-2}z^2 - C_0z = 2(\cos(\theta(z)) - 1) \end{array} \right. \qquad (11)$$

Equation (11) can be discretized using the iterative scheme as following:

$$\left\{ \begin{array}{l} x_{n+1} = x_n + \frac{\Delta z_n}{\tan(\theta_n)} \\ 0.5k^{-2}z_n^2 - C_0z_n = 2(\cos(\theta_n) - 1) \end{array} \right. \qquad (12)$$

where $\alpha$ and $\beta$ are regularized parameters.

With this regularized parameters, it is not sufficient to assure the convergence of the sequence toward the solution. The solution is also depending on the discretized step $\Delta z_n$. The following expression is proposed to assure the best choice of the discretized step $\Delta z_n$ for that the system of sequences converge toward the solution.

$$\Delta z_n = \delta(\kappa, C_0)\Delta z; \qquad (13)$$

In order to obtain the discretized step, the function $\delta(\kappa, C_0)$ is to be determined.

## 3.2 Validation of discretized resolution

To numerically solve the equation 12, $\delta(k, C_0)$ is required to be calibrated with an analytical solution. In our case, we consider steel material. We determine $\delta$ for $C_0 \in [250, 1250]$ to calibrate the numerical model with an analytical model. Figure 8 illustrates the different geometrical shapes of the weld bead according to different values of $C_0$ with the calibrated value of $\delta = 0.8$. We note that the width $W_0$ and height $H_0$ of the weld bead are smaller when the curvature is more significant. We determine the range of the width and height of the weld bead according to the $C_0$ value. These ranges are shown in the Tab.3.

<table><caption>Table 3: Ranges of width and height of the weld bead with respect to curvature.</caption><thead><tr><th></th><th>min</th><th>max</th></tr></thead><tbody><tr><td>$C_0$</td><td>260</td><td>1240</td></tr><tr><td>$H_0$</td><td>1.5</td><td>5.</td></tr><tr><td>$W_0$</td><td>2.5</td><td>10.</td></tr></tbody></table>

Figure 9 illustrates the comparison between the numerical and analytical solutions. The numerical solution is shown to be quantitatively validated according to the analytical one. Furthermore, as displayed in Tab. 4, the computational cost of the numerical solution is 15 times less than that of the analytical one. It is crucial to have a faster computational time to simulate large-scale weld bead deposition.
<figure><img src="image_10.png" /><figcaption>Figure 8: Sensitivity analysis of different $C_0$ for $\theta_0 = \frac{\pi}{2}$.</figcaption></figure>

<table><caption>Table 4: CPU time for weld bead geometrical simulation with analytical and numerical solutions.</caption><thead><tr><th></th><th>CPU time (s)</th></tr></thead><tbody><tr><td>Analytical</td><td>0.33</td></tr><tr><td>Numerical</td><td>0.02</td></tr></tbody></table>

<figure><img src="image_11.png" /><figcaption>Figure 9: Validation of discretized model comparing to the analytical resolution.</figcaption></figure>

### 3.3 Weld bead modeling methodology

This section describes the application of the numerical solution to model a weld bead geometry deposited on a general support via meta-model.

To simulation the weld bead profile, we, first, need to solve equation (12). To do so,

* The geometrical parameters such as the tri-phase contact line position $x_0$, the contact angle $\theta$, and the curvature $C_0$ shown in Figure 10 are to be determined;
* For the tri-phase contact line position $x_0$, we propose a method described in detail in section 3.4.2 and its meta-model in section 4.1.3;

* For the contact angle, numerical measurement is performed on experimental profiles of specimen (A). The numerical measurement technique is described in section 3.4.2 and the corresponding contact angle meta-model in section 4.1.2;

* The curvature $C_0$ may be calculated according to the area $A(C_0)$ limiting by the cross-section of the weld bead and the support such as an optimization problem as follows:

$$C_0^* = \arg\min_{C_0} |A(C_0) - A_0| \qquad (14)$$

Where $A_0$ is the area computed by the flow rate of the metal wire as $\frac{\pi r_f^2 v_f}{v_r}$ ($1 - a$) where $r_f$ is the wire radius, $v_f$ is the wire speed, $v_r$ is the torch speed

* $a$ is the solidification shrinkage determined by the method employed in section 4.1.1.

<figure><img src="image_12.png" /><figcaption>Figure 10: Initial conditions for simulating a weld bead depositing on a support.</figcaption></figure>

## 3.4 Meta-model for geometrical parameters

### 3.4.1 Tri-phase contact line position

When a weld bead is deposited on plan support as in Figure 11a, the weld bead forms a circle arc. The semi-width $r_i$ of the weld bead can be computed from the area and contact angle of the weld bead. Based on this length and weld bead overlapping distance $d_i$, the tri-phase contact line position can be computed by the intersection of a circle of radius $R_i(r_i, d_i)$ and its center which is the projection of the torch position on the support as shown in Figure 11b. $R_i(r_i, d_i)$ is determined based on experimental tri-phase contact line position which the left intersection of the experimental profile and the support.

<figure><img src="image_13.png" /><figcaption>Figure 11: Definition of tri-phase contact line position according.</figcaption></figure>
### 3.4.2 Contact angle

The contact angle $\theta$ is defined as an angle tangential to the weld bead geometry at the support, as shown in Fig. 12. This angle is formed by the equilibrium at the tri-phase line, which is an intersection of liquid, gas, and solid. While a drop is in an equilibrium state, the drop geometry satisfies Young Laplace equation. For 2D hypothesis, the drop’s shape is much closer to a circular arc. Consequently, using a circle to better fit the form of the drop is the best alternative. However, the circle fit can generate an error in configuration where the contact angle is larger than $\frac{\pi}{2}$. Therefore, a polynomial correction is necessary in this case. This method is proposed in [13].

<figure><img src="image_14.png" /><figcaption>Figure 12: Angle which is positive at the front (contact angle $\theta$) and negative at the rear (support angle $\alpha$).</figcaption></figure>

The following steps describe the algorithm that we use in the mentioned paper:

* Create a closed three-point set at the support; Fit a circle arc passing by the three points, noted y (Figure 13a);

* Calculate the tangent to the circle at the contact point $P_0$:

$$ \frac{dy}{dx}|_{P_0} = \frac{x_0 - x_{center}}{\sqrt{R^2 - (x_0 - x_{center})^2}} \quad (15) $$

* Compute the gap between the point set of the profile and the circle;

* Fit a polynomial, noted $\tilde{y}$;

* Compute the tangent of the polynomial at the contact point $P_0$: $\frac{d\tilde{y}}{dx}|_{P_0}$;

* Compute the angle at the contact point with the polynomial correction:

$$ \theta|_{P_0} = atan(\frac{dy}{dx}|_{P_0} + \frac{d\tilde{y}}{dx}|_{P_0}) \quad (16) $$

# 4 Results and discussions

## 4.1 Geometrical parameter meta-model and experimental results

### 4.1.1 Solidification shrinkage

This section demonstrates how to select a method to quantify the flow loss using the specimen manufacturing with different values of Contact Tip to Work Distance (CTWD) shown in Figure 14. Firstly, part 1, 2 and 3 are chosen to be the post-processing zones. With a quantitative value of these three cross section depositions, we establish a method to compute the flow rate of the wire and the deposition weld bead as follows:

$$ d_{bead} = Av_r \quad (17) $$

and

$$ d_{wire} = \frac{\int_{t_0}^{t_0+\Delta t} ws(t)dt}{\Delta t} \quad (18) $$

As shown in Tab. 5, we compute the wire flow, deposition, and corresponding correction. Table 5 as well shows shrinkage rate of each part of the specimen and their minimum and maximum value. The maximum of the shrinkage rate ($a = \frac{|d_{bead}-d_{wire}|}{d_{wire}}$) is close to one find in the literature [14].
<figure><img src="image_15.png" /><figcaption>Figure 13: Computation of contact angles with the proposed algorithm (a) best circle fitting and (b) large errors due to circle fitting.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Figure 14: Specimen manufactured with different CT in which CT = 10, 15, 20 mm at part 3, 2 and 1 respectively.</figcaption></figure>

### 4.1.2 Correlation between contact angles and support angles

The method mentioned in section 3.4.2 is used to compute the contact angles of each weld bead of the juxtaposition specimens and the thick wall (A) characterized by a constant weld bead overlapping distance. The contact angles are plotted according to the support angle at the contact point see Figure 15. The figure demonstrates the group of the three different scenarios, as shown in Figure 16. The three scenarios correspond to the juxtaposition, the stacking, and the
<table><caption>Table 5: Values of flow of each part computation.</caption><thead><tr><th>Flow m<sup>3</sup>/s</th><th>Part 1</th><th>Part 2</th><th>Part 3</th><th>mean</th><th>max</th></tr></thead><tbody><tr><td>d<sub>bead</sub>(10<sup>-7</sup>)</td><td>1.246</td><td>1.241</td><td>1.116</td><td></td><td></td></tr><tr><td>d<sub>wire</sub>(10<sup>-7</sup>)</td><td>1.356</td><td>1.320</td><td>1.193</td><td></td><td></td></tr><tr><td>a</td><td>0.081</td><td>0.059</td><td>0.064</td><td>0.068</td><td>0.081</td></tr></tbody></table>

stacking on the edges. We can find the empirical laws corresponding to each scenario using linear regression. The empirical contact law is then used for the simulation.

<figure><img src="image_17.png" /><figcaption>Figure 15: Linear regression of each group of contact angles according to each scenario.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>Figure 16: Three scenarios of contact angle configuration according to a support (a) juxtaposition, (b) symmetrical stacking and (c) extremity stacking.</figcaption></figure>

### 4.1.3 Correlation among initial position parameters

As described in section 3.4.1, the correlation between $R_i$, $r_i$ and $d_i$ of the weld bead i is required to establish the tri-phase contact line position meta-model. The experimental value of $R_i$ and the computing value of $r_i$ are plotted in Fig. 17 (left). We can cluster the plotting data into five zones, as in Fig. 17 (right), where the first three zones are the first column of deposited weld beads and the remains of the weld beads is in the last two zones. Because of zero value overlapping distance of the first column, we approximate experimental data $R_i = R_i(r_i)$, otherwise, $R_i = R_i(r_i, d_i)$. The red, yellow and purple points correspond to the deposition of the first column of the weld beads. The red and yellow points correspond to the first and second weld bead deposition, respectively. For the first weld bead, $R_i = r_i$. For the second weld bead, $R_i = 1.4r_i$. The zone 3 (purple points) represent to the column of weld beads depositing at the left-hand side except the first and the second weld beads, and $R_i \approx 1.3r_i$. The zone 2 (cyan points) corresponds to the
last two columns of weld beads deposit on the right-hand side except the first row of the weld beads, and $R_i = r_i + \frac{d_i}{25}$.
The zone 1 (blue points) corresponds to the remaining weld beads, and $R_i = r_i + \frac{d_i}{10.5}$.

Finally, the empirical law of deposition position is established resulting from the experimental data points. As we equally have the empirical law of the contact angle and flow correction for the deposit area, we have complete information about the geometrical parameters of the geometrical model. Therefore, we employ the complete methodology for determining the initial position and contact angle by their meta-models to study the impact of different scenario as well used for validating the model using the different specimen in the following sections.

<figure><img src="image_19.png" /><figcaption>Figure 17: Tri-phase contact line position data of different weld bead scenario.</figcaption></figure>

## 4.2 Impact of each geometrical parameter meta-model on geometrical simulations

The impact of the geometrical parameter meta-models on geometrical simulations is carried out in this section. We investigate four different scenarios as in the Tab. 6.

<table><caption>Table 6: Different cases of the study of the geometrical meta-model impact on geometrical simulations via tri-phase contact line.</caption><thead><tr><th>Cases</th><th>Flow rate correction</th><th>Experimental position</th><th>Position with R<sub>i</sub> = r<sub>i</sub></th><th>Position with R<sub>i</sub> = R<sub>i</sub>(r<sub>i</sub>, d<sub>i</sub>)</th></tr></thead><tbody><tr><td>1</td><td>No</td><td>Yes</td><td>No</td><td>No</td></tr><tr><td>2</td><td>Yes</td><td>Yes</td><td>No</td><td>No</td></tr><tr><td>3</td><td>Yes</td><td>No</td><td>Yes</td><td>No</td></tr><tr><td>4</td><td>Yes</td><td>No</td><td>No</td><td>Yes</td></tr></tbody></table>

We use the measured wire speed, the programmed trajectory, the torch speed, and the contact angle to simulate the thick wall (A). We observe the impact of the flow rate correction and the tri-phase contact line position parameters on the geometrical simulations.

For the first case, we use an experimental tri-phase contact line position and the deposit area is computed from the measured wire speed and the torch speed. In this case, we suppose there is no solidification shrinkage. Figure 18a shows that the cross section of the global simulation profile is larger than that of the experiment's one. It indicates that the impact of the solidification shrinkage is non-neglected. We use the same conditions as before for the second case, but with considering the flow correction. We observe that the simulation profiles are well fit with the experimental one as shown in Fig. 18b.

For both third and fourth cases, two tri-phase contact line position models are studied considering flow rate correction for the geometrical simulation. At first, we suppose that $R_i = r_i$, then we compute the tri-phase contact line position. Figure 18c illustrates the comparison between the simulation and experimental profiles. The computing tri-phase contact line positions of the first column's weld bead are largely deviated from the experimental tri-phase contact line positions. When, we use the proposed tri-phase contact line position meta-model, we observe that the geometrical simulation profiles have a good fit to the experimental one for each weld bead as illustrated in Fig. 18d. Therefore,
it is crucial to consider the flow rate correction, contact angle and tri-phase contact line position meta-models in the
geometrical simulation.

<figure><img src="image_20.png" /><figcaption>Figure 18: Geometrical simulation with different scenarios to observe the impact of the geometrical parameters on the geometrical simulation (a) case 1, (b) case 2, (c) case 3 and (d) case 4 respectively.</figcaption></figure>

**4.3 Validation of geometry simulation considering geometrical meta-model**

To validate the geometrical model, the thick wall (C) of variational overlapping distance is used to compare with the geometrical simulation. The measured wire speed, the programmed trajectory, and the torch speed corresponding to this wall are employed. The contact angle and tri-phase contact line position meta-model, and flow correction are used as inputs for geometrical model.

Figure 19 illustrates a complete simulation of this thick wall and the comparison of the experimental and simulated profiles. This is a very challenging case because on the edges, a part of the weld beads is deposed in the empty space. However, the simulation result agrees with the experimental one. This demonstrates the strength of the weld bead geometry model.

**5 Conclusion**

In this article, a methodology has been proposed to simulate the geometry of a part (in a 2D context) for WAAM process.
The simulation strategy is built on first the resolution of Young Laplace equation and secondly on the consideration of a
data based geometrical model. This geometrical model is generated from 3D scan experimental data coming from each
<figure><img src="image_21.png" /><figcaption>Figure 19: Simulation of weld bead assembly with a complete meta-model of input geometry parameters of the thick wall with distributed overlapping distance (a) simulated and experimental profiles and (b) 3D scan cloud point of the middle of the specimen.</figcaption></figure>

weld bead of various specimens. The necessity to consider the wholeness of the data model is demonstrated with the verification of a good agreement between geometrical simulation results and geometrical measurements. To conclude, a first validation result is obtained through a comparison between geometrical simulation results and geometrical measurements obtained on a new part which has not been considered in the experimental database definition. The geometrical model can be extended to the deposition of the weld bead on the complex geometrical and trajectory configuration such as the overhang deposition and random tri-phase contact line position deposition. In the case of complex trajectory, the 2D geometrical model can be extended to 3D geometrical model to capture the manifold of the support. Moreover, the geo-metrical model can also be enriched by enlarging the process parameter ranges and different materials.

## Acknowledgement

The authors would like to thank to the industrial partners of the AFH (Additive Factory Hub) consortium for their supports in the framework of WAS project. This project is as well supported by the French government's aid in the framework of PIA (Programme d'Investissement d'Avenir) for Institut de Recherche Technologique SystemX.
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