# An Inhomogeneous Model for Laser Welding of Industrial Interest

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction to the Problem

## 2. Melting–Resolidification Process

## 3. Governing Equation

#### 3.1. Material and Geometries

#### 3.2. Domains Not Belonging to the Laser Welding Domain: Model, Initial and Boundary Conditions

#### 3.3. Laser Welding Domain: The Model

#### 3.4. Full Model in the Domain

#### 3.5. Realistic Formulations for Both Volumetric and Superficial Laser Heat Sources

#### 3.5.1. Classical Gaussian Laser Heat Source

#### 3.5.2. Conical Laser Heat Source

#### 3.5.3. Ellipsoid Laser Heat Source

**Remark**

**1.**

## 4. The Galerkin-FEM Approach

#### 4.1. Some Remarks on Existence, Uniqueness and Regularity of the Solution

**Theorem**

**1**

**.**Let $\mathrm{\Omega}\subset {\mathbb{R}}^{n}$ be a bounded domain, with a ${C}^{2}$ boundary $\partial \mathrm{\Omega}$. For a finite time $\tilde{t}>0$, we consider the following nonlinear parabolic second-order PDE boundary value problem [23]:

- (1)
- ${p}_{1}$, ${p}_{2}$, ${p}_{3}$, and ${p}_{4}$ are non-negative constants;
- (2)
- $\mathrm{\Phi}\left(T\right(\mathbf{x},t\left)\right)$ is a positive and bounded real function of class ${C}^{1}((0,\tilde{t}\phantom{\rule{0.166667em}{0ex}}]\times \mathrm{\Omega})$ with a bounded derivative;
- (3)
- $K\left(T\right(\mathbf{x},t\left)\right)$ assumed to satisfy the following inequality:$$0<{K}_{m}\le K\left(T(\mathbf{x},t)\right)\le {K}_{M},\phantom{\rule{1.em}{0ex}}\forall (\mathbf{x},t)\in (0,\tilde{t}\phantom{\rule{0.166667em}{0ex}}]\times \mathrm{\Omega};$$
- (4)
- $\mathrm{\Psi}\left(T\right(\mathbf{x},t\left)\right)$ is a positive bounded real function;
- (5)
- ${r}^{\left(v\right)}(\mathbf{x},t)\in {L}^{p}((0,\tilde{t}\phantom{\rule{0.166667em}{0ex}}]\times \mathrm{\Omega})$ with $p\ge 2$;
- (6)
- ${r}^{\left(s\right)}(\mathbf{x},t)\in {W}_{p}^{1-\frac{1}{2p},2-\frac{1}{p}}((0,\tilde{t}\phantom{\rule{0.166667em}{0ex}}]\times \partial \mathrm{\Omega})$;
- (7)
- ${T}_{0}\left(\mathbf{x}\right)\in {W}_{\infty}^{2-\frac{2}{p}}\left(\mathrm{\Omega}\right)$, verifying$$\begin{array}{c}\hfill K\left({T}_{0}\left(\mathbf{x}\right)\right){\displaystyle \frac{\partial {T}_{0}\left(\mathbf{x}\right)}{\partial \widehat{\mathbf{n}}}}+{p}_{1}[{T}_{0}\left(\mathbf{x}\right)-{\theta}_{1}]+{p}_{2}[{T}_{0}\left(\mathbf{x}\right)-{\theta}_{2}]+{p}_{3}[{T}_{0}^{4}\left(\mathbf{x}\right)-{\theta}_{3}^{4}]={p}_{4}\phantom{\rule{0.166667em}{0ex}}{r}^{\left(s\right)}(\mathbf{x},0);\end{array}$$

**Remark**

**2.**

#### 4.2. Galerkin-FEM Basics

## 5. Results of Computations

^{2}K), and ${h}^{bench}=20{h}^{air}$ (as experimentally suggested from metallurgical experiments [35,36,37,38,39,40] to guarantee the correct penetration without favoring the thermal degradation of the alloy structure, and it is such that it produces significant effects even at depth).

#### 5.1. Mesh Creation

#### 5.2. Exploiting Classical Gaussian Laser Heat Source

#### 5.3. Exploiting Conical 3D Laser Heat Source

#### 5.4. Exploiting Ellipsoid Laser Heat Source

## 6. Conclusions and Perspectives

- -
- The proposed model is quite realistic since it simulates the welding process of Al-Si 5% alloy plates, used on an industrial level with appreciable results, placed side by side, which takes the form of the creation of a fusion/resolidification bead, in compliance with regulatory standards, without the addition of additional material which, usually, causes unwanted thickening.
- -
- The process implemented, based on a temperature melting range of the alloy, fits well with the needs of modern metallurgy.
- -
- The further boundary condition, taking into account the fact that the plates are placed on a workbench, lays the first foundations for more accurate modeling, which begins to take into account any thermal losses.
- -
- As further highlighted in Figure 12, the laser beam modeled by the classic 3D Gaussian formulation fits well with the proposed analytical approach, leaving the operator the right choice of laser beam power (strictly linked to the laser electrical current) to avoid damage to the material structure.
- -
- The proposed polynomial-shaped substitute thermal capacity makes it possible to obtain maps, highlighting gradual variations in temperature, which agree well with the fact that the analytical solution is unique and regular (also providing the volumetric solid-state fraction, which, as is known, defines the mechanical properties of the weld). Obviously, the topic is far from being fully studied, and several more studies need to be conducted to optimize the proposed process and then evaluate with standardized characterization tests. Particularly, the ongoing research foresees the following future developments:

- Considering non-linearity in thermal conductivity, where the thermal conductivity depends on temperature, would enhance the model’s accuracy. This improvement would better reflect real-world conditions and contribute to more precise predictions;
- Upgrading the model to account for the effects of significant thermal expansion in the alloy would be beneficial. Considering the resulting solidification shrinkage and the associated risk of cracks will further improve the model’s predictive capabilities.
- To better simulate real laser welding behavior, future research should consider phenomena such as the production of vapor bubbles (keyholes) and surface tension imbalances caused by temperature gradients (Marangoni effect). Incorporating these factors will contribute to a more comprehensive understanding of the welding process.
- The model should be extended to account for the non-instantaneous temperature distribution during welding. Additionally, considering the microstructural disturbances caused by the laser beam’s interaction with restricted areas of material will enhance the model’s accuracy.
- Exploring the versatility of laser sources, particularly in terms of power and power density management, would allow for localized heating to facilitate specific metallurgical processes. Integrating this capability into the model would increase its practicality and applicability.
- Future research should investigate the impact of the laser beam’s power density, interaction time with the material, and total energy on the localized treatment phase. Understanding these factors will enable optimization of the welding process.
- Although the power of the laser is high, it must be taken into account that it takes some time for the electrical power, converted into the thermal equivalent, to penetrate the material, causing it to melt. Then it would be appropriate to consider the response time of the material before the heat input from the laser penetrates deeply, and introduce one/two relaxation times, considering a Maxwell–Cattaneo–Vernotte/dual-phase lag-type model.
- The dynamic problem studied, if framed in the context of non-homogeneous materials in contact with each other, requires highly non-uniform meshes (to be appropriately refined and re-refined) with preconditions that are difficult to formulate. Then, we find it interesting to reformulate the problem in this framework to obtain optimized numerical solutions using two-level approaches.
- Finally, it would be desirable for the proposed model to explain the presence of the electric current generating the laser beam so as to clearly highlight the cause–effect link on which to implement specific control actions.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Al-Si 5% specimen: $\mathrm{\Omega}$ divided into ${\mathrm{\Omega}}_{1}$, ${\mathrm{\Omega}}_{2}$ and ${\mathrm{\Omega}}_{3}$.

**Figure 7.**(

**a**) Mesh creation: 3500 finite element (4522 nodes, 3924 edges); (

**b**) fusion zone and welding direction.

**Figure 8.**Welding process. The red point represents the impact area of the laser beam. Initial zone: (

**a**,

**b**); central zone: (

**c**–

**e**) and final zone: (

**f**).

**Figure 9.**Distribution of $T(\mathbf{x},t)$ in $\mathrm{\Omega}$ when the laser beam, modeled using: (

**a**) Gaussian formulation, (

**b**) conical formulation and (

**c**) ellisoidal formulation has reached the final point of the welding path.

**Figure 10.**Distribution of $T(\mathbf{x},t)$ in $\mathrm{\Omega}$ when the laser beam, modeled using: (

**a**) Gaussian formulation, (

**b**) conical formulation and (

**c**) ellipsoidal formulation.

**Figure 11.**Distribution of $T(\mathbf{x},t)$ at five different points on the welding wire (${P}_{0}=(0,40.5,4)$, ${P}_{1}=(25,40.5,4),{P}_{2}=(50,40.5,4)$, ${P}_{3}=(75,40.5,4)$, ${P}_{4}=(100,40.5,4)$) (

**a**–

**c**) and in the middle point ${P}_{1}$ of the welding wire (

**b**,

**d**,

**f**), with Gaussian 3D laser heat source in (

**a**,

**b**), conical 3D laser heat source in (

**c**,

**d**) and ellipsoid 3D laser heat source in (

**e**,

**f**).

**Figure 12.**Thermal cycles in (

**a**) transverse direction, (

**b**) longitudinal direction with Gaussian 3D laser heat source.

**Table 1.**Dimensions of ${\mathrm{\Omega}}_{1}$, ${\mathrm{\Omega}}_{2}$ and ${\mathrm{\Omega}}_{3}$.

${\mathbf{\Omega}}_{\mathit{i}}$ | Length (mm) | Width (mm) | Thickness (mm) |
---|---|---|---|

${\mathrm{\Omega}}_{1}$ | 39 | 100 | 4 |

${\mathrm{\Omega}}_{2}$ | 39 | 100 | 4 |

${\mathrm{\Omega}}_{3}$ | 2 | 100 | 4 |

**Table 2.**Geometric characterizations of ${\mathrm{\Omega}}_{1}$, ${\mathrm{\Omega}}_{2}$ and ${\mathrm{\Omega}}_{3}$.

${\mathbf{\Omega}}_{\mathit{i}}$ | Length (mm) | $\stackrel{\circ}{{\mathbf{\Omega}}_{\mathit{i}}}$ | Length (mm) |
---|---|---|---|

${\mathrm{\Omega}}_{1}$ | $[0,100]\times [0,39]\times [0,4]$ | ${\stackrel{\circ}{\mathrm{\Omega}}}_{1}$ | $(0,100)\times (0,39)\times (0,4)$ |

${\mathrm{\Omega}}_{2}$ | $[0,100]\times [41,80]\times [0,4]$ | $\stackrel{\circ}{{\mathrm{\Omega}}_{2}}$ | $(0,100)\times (41,80)\times (0,4)$ |

${\mathrm{\Omega}}_{3}$ | $[0,100]\times [39,41]\times [0,4]$ | $\stackrel{\circ}{{\mathrm{\Omega}}_{3}}$ | $(0,100)\times (39,41)\times (0,4)$ |

**Table 3.**Geometric characterizations of $\partial {\mathrm{\Omega}}_{1}$, $\partial {\mathrm{\Omega}}_{2}$ and $\partial {\mathrm{\Omega}}_{3}$.

Label | Dimensions | Label | Dimensions |
---|---|---|---|

${F}_{12}$ | $[0,100]\times \left\{0\right\}\times [0,4]$ | ${F}_{2}$ | $[0,100]\times \left\{41\right\}\times [0,4]$ |

${F}_{7}$ | $[0,100]\times \left\{39\right\}\times [0,4]$ | ${F}_{1}$ | $[0,100]\times \left\{80\right\}\times [0,4]$ |

${F}_{5}$ | $\left\{0\right\}\times [41,80]\times [0,4]$ | ${F}_{4}$ | $[0,100]\times [41,80]\times \left\{4\right\}$ |

${F}_{3}$ | $\left\{100\right\}\times [41,80]\times [0,4]$ | ${F}_{6}$ | $[0,100]\times [41,80]\times \left\{0\right\}$ |

${F}_{10}$ | $\left\{0\right\}\times [39,41]\times [0,4]$ | ${F}_{9}$ | $[0,100]\times [39,41]\times \left\{4\right\}$ |

${F}_{8}$ | $\left\{100\right\}\times [39,41]\times [0,4]$ | ${F}_{11}$ | $[0,100]\times [39,41]\times \left\{0\right\}$ |

${F}_{15}$ | $\left\{0\right\}\times [0,39]\times [0,4]$ | ${F}_{14}$ | $[0,100]\times [0,39]\times \left\{4\right\}$ |

${F}_{13}$ | $\left\{100\right\}\times [0,39]\times [0,4]$ | ${F}_{16}$ | $[0,100]\times [0,39]\times \left\{0\right\}$ |

Parameter | Value | Unit |
---|---|---|

${\mathcal{C}}_{S}$ | $2.943\times {10}^{6}$ | J/(m^{3} K) |

${\mathcal{C}}_{L}$ | $3.07\times {10}^{6}$ | J/(m^{3} K) |

$\lambda $ | 290 | W/(m K) |

${L}_{v}$ | $990.6\times {10}^{6}$ | J/(m^{3}) |

${T}_{S}$ | $850.15$ | K |

${T}_{L}$ | $923.15$ | K |

$\u03f5$ | $0.8$ |

Parameter | Significance | Value | Unit |
---|---|---|---|

v | velocity | 40 | mm/s |

P | power | 2400 | W |

${r}_{U}$ | radius | $1.5$ | mm |

${R}_{f}$ | reflexivity | $0.9$ | |

${I}_{0}$ | intensity | 340 | W/mm^{2} |

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**MDPI and ACS Style**

Munafò, C.F.; Palumbo, A.; Versaci, M.
An Inhomogeneous Model for Laser Welding of Industrial Interest. *Mathematics* **2023**, *11*, 3357.
https://doi.org/10.3390/math11153357

**AMA Style**

Munafò CF, Palumbo A, Versaci M.
An Inhomogeneous Model for Laser Welding of Industrial Interest. *Mathematics*. 2023; 11(15):3357.
https://doi.org/10.3390/math11153357

**Chicago/Turabian Style**

Munafò, Carmelo Filippo, Annunziata Palumbo, and Mario Versaci.
2023. "An Inhomogeneous Model for Laser Welding of Industrial Interest" *Mathematics* 11, no. 15: 3357.
https://doi.org/10.3390/math11153357