Energy law preserving continuous finite element schemes for a gas metal arc welding system
本站小编 Free考研考试/2022-01-02
Yanhai Lin,1, Yongyue Jiang21Fujian Province University Key Laboratory of Computation Science and School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China 2School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China
Abstract In this paper a modified continuous energy law was explored to investigate transport behavior in a gas metal arc welding (GMAW) system. The energy law equality at a discrete level for the GMAW system was derived by using the finite element scheme. The mass conservation and current density continuous equation with the penalty scheme was applied to improve the stability. According to the phase-field model coupled with the energy law preserving method, the GMAW model was discretized and a metal transfer process with a pulse current was simulated. It was found that the numerical solution agrees well with the data of the metal transfer process obtained by high-speed photography. Compared with the numerical solution of the volume of fluid model, which was widely studied in the GMAW system based on the finite element method Euler scheme, the energy law preserving method can provide better accuracy in predicting the shape evolution of the droplet and with a greater computing efficiency. Keywords：phase field;gas metal arc welding (GMAW);metal transfer;discrete energy law;finite element method;numerical solution
PDF (743KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Yanhai Lin, Yongyue Jiang. Energy law preserving continuous finite element schemes for a gas metal arc welding system. Communications in Theoretical Physics, 2021, 73(2): 025004- doi:10.1088/1572-9494/abd0e3
1. Introduction
An in-depth understanding of the behavior of gas metal arc welding (GMAW) is very important because of the increasing demands of engineering practices, such as controlling the welding process to obtain more weld quality and higher productivity. In a GMAW system, the arc is always chosen as the heating source for melting the work-piece within the joining process. This technology is widely applied in railway, shipbuilding and other engineering processes because of its high flexibility, low cost, easy automation and excellent quality. Metal transfer, which plays an important role in GMAW, is a complex process involving multi-field coupling like electro-magnetic fields, fluid flow and heat fields. In the metal transfer process, most of the heat from the arc, electrodes and heating melts the wire and the work-piece at the same time.
One of the main important purposes in theoretical and experimental studies of the metal transfer process is to make a good prediction of the welding results with high precision. Firstly, based on a new steady mathematical model, a numerical analysis of the magnetic diffusion equation, the energy transport equation and the velocity and temperature of electrodes was created to predict the metal transport within inert gas welding arcs [1]. Chen et al [2] used the volume of fluid (VOF) technique to investigate the flow transport behavior of metal processes with consideration of electro-magnetic forces, surface tension, gravitational force and arc plasmas. Based on the same technique, Hu and Tsai [3, 4] also simulated a GMAW system that included the electrode, the arc and the weld pool by creating a new mathematical-physical model. Hu and Tsai [3] focused on the behavior of arc plasmas, while Hu and Tsai [4] focused on the results of the metal, such as transfer, the melting flow process and temperature, impingement onto the work-piece, and the welding pool formation and so on. Furthermore, Haidar [5] created a more detailed model for predicting thermodynamic behavior in GMAW and considered more factors such as viscous drag, inertia action, gravitational force, arc pressure and so on. Impacts of welding pool evaporation and thermodynamic behavior on the formation of the welding pool were investigated by Zacharia et al [6] which could make a prediction of the weld pool with higher precision.
Recently, Anzehaee and Haeri [7] undertook an investigation of control of the thermal transport of the work-piece within a GMAW system by controlling the melting rate, heating transfer and detaching droplets scale. Later, Rao et al [8] presented a detailed complete mathematical-physical model to investigate the influences of the thermal Marangoni effect on the liquid-solid interface, forces at free metal surfaces and the energy source at the plasma-anode surface on the transfer behavior of a GMAW system. Feng et al [9] studied the effects of weld pool inside or outside flows on the characteristics of droplets and metal transfer within a GMAW system. Cheon et al [10] investigated the finger-shaped evolution in the GMAW process by using a computational fluid dynamics (CFD) based numerical method with the commercial software Flow3D. Wu et al [11, 12] experimentally investigated the effect of an additional electro-thermal-magnetic field on arc metal stream, liquid flow and heat transfer of a welding pool within a high-speed GMAW system. Xiong et al [13] numerically and experimentally presented the heat transfer behavior for a thin-walled part within a GMAW system. The effect of the variable preheating temperature distribution was taken into account. Sachajdak et al [14] presented CFD modeling for flow and metal heat transfer of a GMAW system by using the finite volume method (VFM). Komen et al [15] numerically presented the behavior of molten metal droplet transport and welding pool flow within a GMAW system using the incompressible smoothed particle hydrodynamics method. Also, some scholars have studied the GMAW process to focus on the arc [16, 17], welding current [18, 19], fume formation [20] and plasma [21, 22] and so on.
The VOF does not consider the thermocapillary/Marangoni effect, which is a very important factor in the study of multi-phase fluids with a clear surface. Thermocapillary/Marangoni forces are always caused by surface tension, which is always due to temperature or concentration gradients. Borcia and Bestehorn [23] presented the effect of a deformable surface on Marangoni fluid flow within liquid-gas two-phase flows using the phase-field technique. Using the same method, Anderson et al [24] considered solidification of a pure substance that includes convections in the fluid-phase, and both the solid-phase and fluid-phase were treated as viscous fluids in the model. Furthermore, Borcia and coworkers [25, 26] extended the phase-field technique when describing the Marangoni driven flow within gas-liquid or liquid-liquid systems in drops, bubbles and thin liquid film. Guo et al [27] presented continuous finite element schemes for a phase-field model within a double-layer liquid Benard-Marangoni driven problem, and both the buoyancy forces and the Marangoni forces were taken into account. Furthermore, Guo and Lin [28] also presented the influences of thermo-dynamical factors on binary quasi-incompressible fluid by using the phase-field technique. It was found that the interface condition of the traditional sharp surface model could be recovered from the phase-field model, and the phase-field model treated different phases as a single phase.
Recently, Yang et al [29] presented a welding system to study the cable-type welding wire GMAW process, and a high-speed camera system and other electrical equipment was used to examine droplet formation, arc shape and metal transfer. Zhao and Chung [30] presented the numerical results of magneto-hydrodynamic metal transport in a GMAW system by using the phase-field technique. The transport phenomenon could be shown clearly, and they found that the numerical simulation results closely matched the experimental data. Zhao and Chung [31] presented the effect of heating thermal-dynamics on heat and metal processes within a pulsed GMAW system. Furthermore, Zhao et al [32] presented the influences of pulsing parameters on the metal process within GMAW based on the same phase-field method, and different sets of current waveforms were taken into account and compared. In view of the phase-field technique coupled with the continuous finite method, Jiang and coworkers [33-35] presented a new energy law model for a simple GMAW system. It was found that the simulation process with the new energy law gave good agreement with the experimental result captured by high-speed electronic equipment.
Motivated by the above motional works (Zhao and coworkers [30-32] and Jiang and coworkers [33-35]), we consider in this regard an energy law preserving technique for a GMAW system by using the continuous finite element technique. A continuous energy law and a similar energy law at discrete level were obtained.
2. Governing equations, model and energy law
In this study, the simple GMAW system is considered (see figure 1, the schematic of the physical model and computational domain). It is assumed that ${\rm{\Omega }}$ is the bounded computational domain of the weld pool. Here, ${\rm{\Gamma }}$ is the boundary of ${\rm{\Omega }}.$ The governing partial differential equations of the GMAW system with the phase-field model are created as [33-35]:$\begin{eqnarray}\displaystyle \frac{\partial \rho }{\partial t}+{\rm{\nabla }}\cdot (\rho {\boldsymbol{v}})=0,\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{\partial (\rho {\boldsymbol{v}})}{\partial t}+{\rm{\nabla }}\cdot (\rho {\boldsymbol{v}}\otimes {\boldsymbol{v}})=-{\rm{\nabla }}p+{\rm{\nabla }}\cdot \left[\eta ({\rm{\nabla }}{\boldsymbol{v}}+{\rm{\nabla }}{{\boldsymbol{v}}}^{{\rm{T}}})\right]\\ \,+\,\mu {\rm{\nabla }}f+\rho {\boldsymbol{G}}+{\boldsymbol{J}}\times {\boldsymbol{B}},\end{array}\end{eqnarray}$$\begin{eqnarray}\displaystyle \frac{\partial f}{\partial t}+{\boldsymbol{v}}\cdot {\rm{\nabla }}f=M{\rm{\nabla }}\cdot ({\rm{\nabla }}\mu ),\end{eqnarray}$$\begin{eqnarray}\mu =\displaystyle \frac{f({f}^{2}-1)}{\varepsilon }-\varepsilon {\rm{\Delta }}f,\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{\partial (\rho {c}_{p}T)}{\partial t}+{\rm{\nabla }}\cdot (\rho {c}_{p}{\boldsymbol{v}}T)={\rm{\nabla }}\cdot (k{\rm{\nabla }}T)-\rho H\displaystyle \frac{\partial f}{\partial t}\\ \,+\,\displaystyle \frac{{{\boldsymbol{J}}}^{2}}{{\sigma }_{e}}+\displaystyle \frac{5{k}_{{\rm{b}}}}{2e}{\boldsymbol{J}}\cdot {\rm{\nabla }}T,\end{array}\end{eqnarray}$$\begin{eqnarray}{{\rm{\nabla }}}^{2}{\boldsymbol{A}}=-{\mu }_{0}{\boldsymbol{J}},{\rm{\nabla }}\cdot {\boldsymbol{J}}=0,{\boldsymbol{J}}=-{\sigma }_{e}{\rm{\nabla }}{\rm{\Phi }},{\boldsymbol{B}}={\rm{\nabla }}\times {\boldsymbol{A}},\end{eqnarray}$where $\rho $ is the density of fluid, $t$ is time, v is velocity. Here, $p$ is pressure, $\eta $ is viscosity, $\mu $ is the chemical potential, and it represents the mixture energy as it has two parts contributing to separation and mixing, $f$ is the order coefficient of different phases of the mixture, i.e. $f=1$ describes the fluid case while $f=-1$ describes the metal case. G is the gravitational acceleration and ${\boldsymbol{G}}=(0,-9.8).$ J is the current density, B is a self-induced electro-magnetic field, $M$ is a phonological mobility parameter and $\varepsilon $ is the thickness of the interface. In equation (5), $T$ is the temperature, ${c}_{p}$ is specific heat, $k$ is the thermal conductivity coefficient, $H$ is the latent heat, ${\sigma }_{e}$ is electrical conductivity, ${k}_{{\rm{b}}}$ is the Stefan-Boltzmann coefficient and $e$ is electronic charge. A is a magnetic vector and ${\rm{\Phi }}$ is the electrical potential. Furthermore, the boundary conditions and computing domain of Zhao and Chung [30], which consider the phase-field model instead of the VOF method, are applied in the process of calculation. In addition, $f$ and $\mu $ satisfy ${\partial }_{{\bf{n}}}f=0$ and ${\partial }_{{\bf{n}}}\mu =0.$
To improve the efficiency and stability of the calculation process, a positive constant $c$ is presented to modified equations (9)-(10) for the stability as$\begin{eqnarray*}\displaystyle \frac{\partial f}{\partial t}+{\boldsymbol{v}}\cdot {\rm{\nabla }}f=\displaystyle \frac{M}{Pe}{\rm{\Delta }}(\omega +cf),\omega +cf=\mu .\end{eqnarray*}$
By using equations (26)+(27)+(28)-(29)+(30)+ (31)-(32), the continuous energy law can be given as:$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{\rm{d}}E}{{\rm{d}}t}=\displaystyle {\int }_{{\rm{\Omega }}}\left(\displaystyle \frac{\rho {\boldsymbol{g}}}{F{r}^{2}}\cdot {\boldsymbol{v}}+NMe(({\boldsymbol{J}}\times ({\rm{\nabla }}\times {\boldsymbol{A}}))\cdot {\boldsymbol{v}}+{\boldsymbol{J}}\cdot {\boldsymbol{A}})\right.\\ \,\,\,+\,\left.\displaystyle \frac{R}{\overline{{\mu }_{0}}{\Pr }}{\rm{\nabla }}{\boldsymbol{A}}:{\rm{\nabla }}({\boldsymbol{J}}T)\right){\rm{d}}{\boldsymbol{x}}-\displaystyle \frac{1}{{Re}}{\parallel {\rm{\nabla }}{\boldsymbol{v}}\parallel }_{{L}^{2}}^{2}\\ \,\,\,-\,\displaystyle \frac{M}{Pe}{\parallel {\rm{\nabla }}\mu \parallel }_{{L}^{2}}^{2}-\displaystyle \frac{1}{{\Pr }}{\parallel {\rm{\nabla }}T\parallel }_{{L}^{2}}^{2}-\displaystyle \frac{NMe}{\overline{{\mu }_{0}}}{\parallel {\rm{\nabla }}{\boldsymbol{A}}\parallel }_{{L}^{2}}^{2}\end{array}\end{eqnarray}$where$\begin{eqnarray}\begin{array}{c}E=\frac{\rho }{2}{\parallel {\boldsymbol{v}}\parallel }_{{L}^{2}}^{2}+{\parallel {\rm{\nabla }}f\parallel }_{{L}^{2}}^{2}+{\int }_{{\rm{\Omega }}}\left(\psi +\frac{\rho HfT}{{\Pr }}\right){\rm{d}}{\boldsymbol{x}},\\ \psi =\frac{1}{4\varepsilon }{({f}^{2}-1)}^{2}.\end{array}\end{eqnarray}$
3. Discrete format of the finite element method
We rewrite the continuity condition (7) and current density continuity condition (25) with penalty formulation [33-36] to improve the efficiency and stability of the calculation process with ${\rm{\nabla }}\cdot {\boldsymbol{v}}+\delta p=0,$ ${\rm{\nabla }}\cdot {\boldsymbol{J}}+\delta T=0$, where $\delta ={10}^{-6}$ is a small penalty. A finite-difference-scheme in time and a conformal ${C}^{0}$-finite-element technique in space are set to provide expressions of the weak form of the research (see Jiang and coworkers [33-36]). Here, ${\rm{\Delta }}t$ is the time-step size and ${{\boldsymbol{v}}}_{h}^{n},$ ${p}_{h}^{n},$ ${f}_{h}^{n},$ ${\omega }_{h}^{n},$ ${{\boldsymbol{J}}}_{h}^{n},$ ${{\boldsymbol{A}}}_{h}^{n},$ ${T}_{h}^{n}$ are approximations of ${{\boldsymbol{v}}}_{h}^{n}={\boldsymbol{v}}(n{\rm{\Delta }}t),$ ${p}_{h}^{n}=p(n{\rm{\Delta }}t),$ ${f}_{h}^{n}=f(n{\rm{\Delta }}t),$ ${\omega }_{h}^{n}=\omega (n{\rm{\Delta }}t),$ ${{\boldsymbol{J}}}_{h}^{n}={\boldsymbol{J}}(n{\rm{\Delta }}t),$ ${{\boldsymbol{A}}}_{h}^{n}={\boldsymbol{A}}(n{\rm{\Delta }}t),$ ${T}_{h}^{n}=T(n{\rm{\Delta }}t).$ And ${{\boldsymbol{v}}}_{h}^{n+1},$ ${p}_{h}^{n+1},$ ${f}_{h}^{n+1},$ ${\omega }_{h}^{n+1},$ ${{\boldsymbol{J}}}_{h}^{n+1},$ ${{\boldsymbol{A}}}_{h}^{n+1},$ ${T}_{h}^{n+1}$ are the approximations at time ${t}^{n+1}=(n+1){\rm{\Delta }}t.$ The revised midpoint schemes are used in the weak form and the discretized formulation as follows$\begin{eqnarray}\displaystyle {\int }_{{\rm{\Omega }}}\left({\rm{\nabla }}\cdot {{\boldsymbol{v}}}_{h}^{n+\tfrac{1}{2}}+\delta {p}_{h}^{n+\tfrac{1}{2}}\right)q{\rm{d}}{\boldsymbol{x}}=0,\end{eqnarray}$$\begin{eqnarray}\displaystyle {\int }_{{\rm{\Omega }}}\left(\begin{array}{l}\rho {{\boldsymbol{v}}}_{\overline{t}}^{n+1}\cdot {\boldsymbol{u}}+\left(\rho {{\boldsymbol{v}}}_{h}^{n+\tfrac{1}{2}}\cdot {\rm{\nabla }}{{\boldsymbol{v}}}_{h}^{n+\tfrac{1}{2}}\right)\cdot {\boldsymbol{u}}+\displaystyle \frac{\rho }{2}\left(\left({\rm{\nabla }}\cdot {{\boldsymbol{v}}}_{h}^{n+\tfrac{1}{2}}\right){{\boldsymbol{v}}}_{h}^{n+\tfrac{1}{2}}\right)\cdot {\boldsymbol{u}}+\displaystyle \frac{1}{{Re}}{\rm{\nabla }}{{\boldsymbol{v}}}_{h}^{n+\tfrac{1}{2}}:{\rm{\nabla }}{\boldsymbol{u}}\\ -{p}_{h}^{n+\tfrac{1}{2}}\left({\rm{\nabla }}\cdot {\boldsymbol{u}}\right)-Te\left({\omega }_{h}^{n+\tfrac{1}{2}}+c{f}_{h}^{n+\tfrac{1}{2}}\right){\rm{\nabla }}{f}_{h}^{n+\tfrac{1}{2}}\cdot {\boldsymbol{u}}-\displaystyle \frac{\rho g}{F{r}^{2}}\cdot {\boldsymbol{u}}+NMe\left({{\boldsymbol{J}}}_{h}^{n+\tfrac{1}{2}}\times \left({\rm{\nabla }}\times {{\boldsymbol{A}}}_{h}^{n+\tfrac{1}{2}}\right)\right)\cdot {\boldsymbol{u}}\end{array}\right){\rm{d}}{\boldsymbol{x}}=0,\end{eqnarray}$$\begin{eqnarray}\begin{array}{c}{\int }_{{\rm{\Omega }}}\left({f}_{\bar{t}}^{n+1}\gamma +\left({{\boldsymbol{v}}}_{h}^{n+\tfrac{1}{2}}\cdot {\rm{\nabla }}{f}_{h}^{n+\tfrac{1}{2}}\right)\gamma +\frac{M}{Pe}{\rm{\nabla }}\right.\\ \,\,\left.\left({\omega }_{h}^{n+\tfrac{1}{2}}+c{f}_{h}^{n+\tfrac{1}{2}}\right)\cdot {\rm{\nabla }}\gamma \right){\rm{d}}{\boldsymbol{x}}=0,\end{array}\end{eqnarray}$$\begin{eqnarray}\displaystyle {\int }_{{\rm{\Omega }}}\left(\left({\omega }_{h}^{n+\tfrac{1}{2}}+c{f}_{h}^{n+\tfrac{1}{2}}\right)\chi -\varepsilon {\rm{\nabla }}{f}_{h}^{n+\tfrac{1}{2}}\cdot {\rm{\nabla }}\chi -P\left({f}_{h}^{n+1},{f}_{h}^{n}\right)\chi \right){\rm{d}}{\boldsymbol{x}}=0,\end{eqnarray}$$\begin{eqnarray}\displaystyle {\int }_{{\rm{\Omega }}}\left(\begin{array}{l}\rho \left({T}_{\overline{t}}^{n+1}+{{\boldsymbol{v}}}_{h}^{n+\tfrac{1}{2}}\cdot {\rm{\nabla }}{T}_{h}^{n+\tfrac{1}{2}}\right)\theta +\displaystyle \frac{1}{{\Pr }}{\rm{\nabla }}{T}_{h}^{n+\tfrac{1}{2}}\cdot {\rm{\nabla }}\theta +\displaystyle \frac{1}{{\Pr }}\rho {H}_{L}{f}_{\overline{t}}^{n+1}\theta \\ -\displaystyle \frac{R}{4{\Pr }}\left({\left|{{\boldsymbol{J}}}_{h}^{n+1}+{{\boldsymbol{J}}}_{h}^{n}\right|}^{2}\right)\theta -\displaystyle \frac{{K}_{{\rm{b}}}}{{\Pr }}{{\boldsymbol{J}}}_{h}^{n+\tfrac{1}{2}}\cdot {\rm{\nabla }}{T}_{h}^{n+\tfrac{1}{2}}\theta \end{array}\right){\rm{d}}{\boldsymbol{x}}=0,\end{eqnarray}$
From the above, the discrete energy law (43) is similar to the continuous energy law (33). A linearization and an iterative method at each time step should be done to (35)-(41) because the discrete scheme is nonlinear implicit. The fixed point theory is applied in the linearization. The following iterative scheme (for $s=1,2,\cdots $) is used at every time ${t}^{n+1},$ i.e. finding ${\overline{{\boldsymbol{v}}}}_{s},$ ${\overline{p}}_{s},$ ${\overline{f}}_{s},$ ${\overline{\omega }}_{s},$ ${\overline{T}}_{s},$ ${{\boldsymbol{A}}}_{s}$ and ${\overline{{\boldsymbol{J}}}}_{s}$ (as the approximation of ${{\boldsymbol{v}}}_{h}^{n+1},$ ${p}_{h}^{n+1},$ ${f}_{h}^{n+1},$ ${\omega }_{h}^{n+1},$ ${T}_{h}^{n+1},$ ${{\boldsymbol{A}}}_{h}^{n+1}$ and ${{\boldsymbol{J}}}_{h}^{n+1},$ respectively) to satisfy$\begin{eqnarray}\displaystyle {\int }_{{\rm{\Omega }}}\left({\rm{\nabla }}\cdot \displaystyle \frac{{\overline{{\boldsymbol{v}}}}_{s}+{{\boldsymbol{v}}}_{h}^{n}}{2}+\displaystyle \frac{\delta }{2}\left({\overline{p}}_{s}+{p}_{h}^{n}\right)\right)q{\rm{d}}{\boldsymbol{x}}=0,\end{eqnarray}$$\begin{eqnarray}\displaystyle {\int }_{{\rm{\Omega }}}\left(\begin{array}{l}\rho \displaystyle \frac{{\overline{{\boldsymbol{v}}}}_{s}-{{\boldsymbol{v}}}_{h}^{n}}{{\rm{\Delta }}t}\cdot {\boldsymbol{u}}+\left(\rho \displaystyle \frac{{\overline{{\boldsymbol{v}}}}_{s-1}+{{\boldsymbol{v}}}_{h}^{n}}{2}\cdot {\rm{\nabla }}\displaystyle \frac{{\overline{{\boldsymbol{v}}}}_{s-1}+{{\boldsymbol{v}}}_{h}^{n}}{2}\right)\cdot {\boldsymbol{u}}+\displaystyle \frac{\rho }{2}\left(\left({\rm{\nabla }}\cdot \displaystyle \frac{{\overline{{\boldsymbol{v}}}}_{s-1}+{{\boldsymbol{v}}}_{h}^{n}}{2}\right)\displaystyle \frac{{\overline{{\boldsymbol{v}}}}_{s-1}+{{\boldsymbol{v}}}_{h}^{n}}{2}\right)\cdot {\boldsymbol{u}}\\ -\displaystyle \frac{{\overline{p}}_{s}+{p}_{h}^{n}}{2}\left({\rm{\nabla }}\cdot {\boldsymbol{u}}\right)+\displaystyle \frac{1}{{Re}}{\rm{\nabla }}\displaystyle \frac{{\overline{{\boldsymbol{v}}}}_{s}+{{\boldsymbol{v}}}_{h}^{n}}{2}:{\rm{\nabla }}{\boldsymbol{u}}-Te\left(\displaystyle \frac{{\overline{\omega }}_{s}+{\omega }_{h}^{n}}{2}+c\displaystyle \frac{{\overline{f}}_{s}+{f}_{h}^{n}}{2}\right){\rm{\nabla }}\displaystyle \frac{{\overline{f}}_{s}+{f}_{h}^{n}}{2}\cdot {\boldsymbol{u}}\\ -\displaystyle \frac{\rho g}{F{r}^{2}}\cdot {\boldsymbol{u}}+NMe\left(\displaystyle \frac{{\overline{{\boldsymbol{J}}}}_{s}+{{\boldsymbol{J}}}_{h}^{n}}{2}\times \left({\rm{\nabla }}\times \displaystyle \frac{{\overline{{\boldsymbol{A}}}}_{s}+{{\boldsymbol{A}}}_{h}^{n}}{2}\right)\right)\cdot {\boldsymbol{u}}\end{array}\right){\rm{d}}{\boldsymbol{x}}=0,\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}\displaystyle {\int }_{{\rm{\Omega }}}\left(\displaystyle \frac{{\overline{f}}_{s}-{f}_{h}^{n}}{{\rm{\Delta }}t}\gamma +\left(\displaystyle \frac{{\overline{{\boldsymbol{v}}}}_{s-1}+{{\boldsymbol{v}}}_{h}^{n}}{2}\cdot {\rm{\nabla }}\displaystyle \frac{{\overline{f}}_{s-1}+{f}_{h}^{n}}{2}\right)\gamma \right.\\ \,+\,\left.\displaystyle \frac{M}{Pe}{\rm{\nabla }}\left(\displaystyle \frac{{\overline{\omega }}_{s}+{\omega }_{h}^{n}}{2}+c\displaystyle \frac{{\overline{f}}_{s}+{f}_{h}^{n}}{2}\right)\cdot {\rm{\nabla }}\gamma \right){\rm{d}}{\boldsymbol{x}}=0,\end{array}\end{eqnarray}$$\begin{eqnarray}\begin{array}{l}\displaystyle {\int }_{{\rm{\Omega }}}\left(\left(\displaystyle \frac{{\overline{\omega }}_{s}+{\omega }_{h}^{n}}{2}+c\displaystyle \frac{{\overline{f}}_{s}+{f}_{h}^{n}}{2}\right)\chi -\varepsilon {\rm{\nabla }}\displaystyle \frac{{\overline{f}}_{s}+{f}_{h}^{n}}{2}\right.\\ \,\cdot \,\left.{\rm{\nabla }}\chi -P\left({\overline{f}}_{s-1},{f}_{h}^{n}\right)\right){\rm{d}}{\boldsymbol{x}}=0,\end{array}\end{eqnarray}$$\begin{eqnarray}{\int }_{{\rm{\Omega }}}\left(\begin{array}{c}\rho \left(\frac{{\bar{T}}_{s}-{T}_{h}^{n}}{{\rm{\Delta }}t}+\frac{{\bar{{\boldsymbol{v}}}}_{s}+{{\boldsymbol{v}}}_{h}^{n}}{2}\cdot {\rm{\nabla }}\frac{{\bar{T}}_{s-1}+{T}_{h}^{n}}{2}\right)\theta +\frac{1}{{\Pr }}{\rm{\nabla }}\frac{{\bar{T}}_{s}+{T}_{h}^{n}}{2}\cdot {\rm{\nabla }}\theta \\ +\frac{1}{{\Pr }}\rho {H}_{L}\frac{{\bar{f}}_{s}-{f}_{h}^{n}}{{\rm{\Delta }}t}\theta -\frac{R}{4{\Pr }}\left({\left|{\bar{{\boldsymbol{J}}}}_{s}+{{\boldsymbol{J}}}_{h}^{n}\right|}^{2}\right)\theta -\frac{{K}_{{\rm{b}}}}{{\Pr }}\frac{{\bar{{\boldsymbol{J}}}}_{s}+{{\boldsymbol{J}}}_{h}^{n}}{2}\cdot {\rm{\nabla }}\frac{{\bar{T}}_{s}+{T}_{h}^{n}}{2}\theta \end{array}\right){\rm{d}}{\boldsymbol{x}}=0,\end{eqnarray}$
In this section, we use equations (46)-(52) to simulate the dynamic behavior of metal transfer within the GMAW system. In the numerical calculation process, the FreeFem++ platform [33-38] is used to obtain the numerical solutions, and the Tecplot drawing software is applied to show the figure results. The numerical results of v, J, A, $f,$ $\omega ,$ $T$ are given by the continuous finite element technique. A 1.2 millimetre diameter stainless steel is set as the electrode, and 5 millimetres is set as the initial arc length. Pure argon is chosen as the shielding gas. Different values of the pulse current in figure 2 are chosen as the welding current (see more details in table 1). Figure 2 shows different waves of the pulse current. The large figure of figure 2 is the first period of the pulse current and the small figure is three periods. In the first period, $a$ is the peak moment as from t=0 to =2.429, $b$ is the project moment as from t=2.429 to t=4.428, and $c$ is the base moment as from t=4.428 to t=15.00. Table 1 shows specific values of time and current in different moments of the first period of the pulse current in the calculation process.
Figure 2.
New window|Download| PPT slide Figure 2.Different waves of the pulse current: $a$ is the peak moment, $b$ is the project moment and $c$ is the base moment.
Table 1. Table 1.Values of time and current in different moments of the first period of the pulse current.
As the interface has variations in thickness, $f$ varies from −1 to 1, we choose $f=0$ as the position of the interface and show the evolution of the drop in figure 3 at nine different time points: t=0, 1.374, 2.538, 3.068, 4.077, 5.719, 6.012, 6.314 and 6.602 (see more details in figure 2). From figure 3, we can see that as the pulse current during the peak moment rises quickly, the droplet growing up mainly happens during the peak time (t=1.374). From t=2.538 to t=4.077, the necking effect is taken on at the root of the attaching droplet during the project moment, and the interaction between the wire and the droplet becomes thinner and thinner. These changes are due to the electro-magnetic field generated by the changing current. Furthermore, an increase in mass molten metal is another cause of these phenomena. The structure of the droplet changes from circle to pear shape and then to flat ellipse in the vertical direction due to the arc pressure. During the base moment from t=5.719 to t=6.602, as the total effect of the arc pressure, the magnetic force along the radial direction and the gravity of the droplet overcomes the resistance of the surface tension, the droplet breaks up in a short time. Figure 3 also shows that these numerical simulation results match the theory of the metal transfer process better compared with the reported results [33-35].
Figure 3.
New window|Download| PPT slide Figure 3.The evolution of the droplet in metal transfer with the pulse current at different times.
The energy changes and errors of the discrete energy law are depicted in figure 4, where $E$ (equation (44)) stands for the total free energy and $E+Q$ (equation (45)) is the total energy, as shown in section 3. The result in figure 4 shows the same behavior as that in Jiang et al [33], where $E$ increases over time whereas $E+Q$ decreases. The accuracy of the energy law is validated with the error in the energy law reduced to an order of magnitudes $O({10}^{-10})$, which depends on the tolerance chosen in the fixed point iterative method and seems to be good enough for the computation.
Figure 4.
New window|Download| PPT slide Figure 4.(a) The evolution of the total free energy $E$ and the total energy $E+Q.$ (b) Errors of the discrete energy law.
We use the same method to examine the accuracy of the numerical solution in Jiang et al [33], $\parallel {\rm{\nabla }}\cdot ({\boldsymbol{J}}-{{\boldsymbol{J}}}_{h}^{n})\parallel \,=\parallel {\rm{\nabla }}\cdot {{\boldsymbol{J}}}_{h}^{n}\parallel ={\left(\displaystyle {\int }_{\Omega }{\left|{\rm{\nabla }}\cdot {{\boldsymbol{J}}}_{h}^{n}\right|}^{2}{\rm{d}}{\boldsymbol{x}}\right)}^{\tfrac{1}{2}}$ is computed at a time step in this paper. We show the estimated order $\gamma $ of the continuous finite element method for ${\rm{\nabla }}\cdot {\boldsymbol{J}}$ with$\begin{eqnarray*}\gamma =\displaystyle \frac{\mathrm{lg}\left(\parallel {\rm{\nabla }}\cdot ({\boldsymbol{J}}-{{\boldsymbol{J}}}_{h})\parallel /\parallel {\rm{\nabla }}\cdot ({\boldsymbol{J}}-{{\boldsymbol{J}}}_{0.5h})\parallel \right)}{\mathrm{lg}\,2}\end{eqnarray*}$in table 2 and the value of $\gamma $ is close to 2.
Table 2. Table 2.The value and estimated order of ${\rm{\nabla }}\cdot {\boldsymbol{J}}$ for the numerical solution of the metal transfer.
The numerical simulation results of the droplet metal transfer are compared to the images obtained by the high-speed photography in figure 5 to show the validity of the numerical solution. We focus on two diameters (horizontal and vertical) at the moment that the separated metal droplets touch the welding pool. In addition, the average value of three thousand periods is set as the high-speed photography. The comparison in table 3 also contains the data of two diameters and relative errors given by the VOF model, the phase-field model with the Euler scheme (PFME) and the phase-field model with the energy law preserving method (PFMELP). From the results of table 3, it is obvious that the validity of the size and the geometry of the separated metal droplets is PFMELP>PFME>VOF.
In this paper, a new continuous energy law was created to model the transport behavior and phenomena occurring within a GMAW system. The energy law equality at a discrete level was derived by using the finite element method. The mass conservation and current density continuous equation with the penalty scheme was applied to improve the stability in the computing process. To the best of our knowledge, this continuous energy law and the discrete energy law for GMAW have not been derived before. According to the phase-field model coupled with the energy law preserving method, the GMAW model was discretized and the metal transfer process with a pulse current was simulated. Some findings are as follows:(a)The numerical simulation results of this new energy model match the theory of the metal transfer process better compared with the reported results. (b)The new energy model is suitable for the GMAW system and the validity of the shape or geometry of the separated metal droplets as PFMELP>PFME>VOF.
Acknowledgments
Yanhai Lin was supported by the National Natural Science Foundation of China (Grant No. 11702101), the Fundamental Research Funds for the Central Universities and the Promotion Program for Young and Middle-aged Teacher in Science and Technology Research of Huaqiao University (Grant No. ZQN-PY502), the Natural Science Foundation of Fujian Province (Grant No. 2019J05093), and Quanzhou High-Level Talents Support Plan.
,1, Yongyue Jiang21Fujian Province University Key Laboratory of Computation Science and School of Mathematical Sciences, 
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