The actual problem for calculating a shape of free surface of the melt when analyzing the processes of wire-based electron-beam surfacing on the substrate, being introduced into additive manufacturing, is the development of adequate mathematical models of heat and mass transfer. The paper proposed a formulation of the problem of melt motion in the framework of the Lagrangian description. The mathematical statement includes the balance equations for mass, momentum and energy, and physical equations for describing heat and mass transfer. The smoothed particle hydrodynamics method was used for numerical simulation of the process of wire-based electron-beam surfacing on the substrate made from same materials (titanium or steel). A finite-difference analog of the equations is given and the algorithm for solving the problem is implemented. To integrate the discretized equations Verlet method was utilized. Algorithms are implemented in the open software package LAMMPS. The numerical simulation results allow the estimation of non-stationary volume temperature distributions, melt flow velocities and pressures, and characteristics of process. The possibility of applying the developed mathematical model to describe additive production is shown. The comparison of numerical calculations with experimental studies showed good agreement.

中文翻译：

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光滑粒子流体动力学方法对电子束导线沉积增材制造的数​​学建模

当分析在加成制造中引入的基于线的电子束在基材上的表面沉积过程时，计算熔体自由表面形状的实际问题是开发了适当的传热和传质数学模型。该论文在拉格朗日描述的框架内提出了熔体运动问题的表述。该数学表达式包括质量，动量和能量的平衡方程式，以及描述热量和质量传递的物理方程式。使用平滑粒子流体动力学方法对由相同材料（钛或钢）制成的基板上的线基电子束堆焊过程进行了数值模拟。给出了方程的有限差分模拟，并实现了求解该问题的算法。为了整合离散方程，采用了Verlet方法。算法在开放式软件包LAMMPS中实现。数值模拟结果可以估算非稳态体积温度分布，熔体流动速度和压力以及过程特性。显示了应用开发的数学模型描述添加剂生产的可能性。数值计算与实验研究的比较显示出很好的一致性。显示了应用开发的数学模型描述添加剂生产的可能性。数值计算与实验研究的比较显示出很好的一致性。显示了应用开发的数学模型描述添加剂生产的可能性。数值计算与实验研究的比较显示出很好的一致性。