Evolution of solidification texture during additive manufacturing

Striking differences in the solidification textures of a nickel based alloy owing to changes in laser scanning pattern during additive manufacturing are examined based on theory and experimental data. Understanding and controlling texture are important because it affects mechanical and chemical properties. Solidification texture depends on the local heat flow directions and competitive grain growth in one of the six <100> preferred growth directions in face centered cubic alloys. Therefore, the heat flow directions are examined for various laser beam scanning patterns based on numerical modeling of heat transfer and fluid flow in three dimensions. Here we show that numerical modeling can not only provide a deeper understanding of the solidification growth patterns during the additive manufacturing, it also serves as a basis for customizing solidification textures which are important for properties and performance of components.


A. Model assumptions
Several simplifying assumptions are made to make the complex, threedimensional, transient calculations tractable. The densities of the solid and liquid metals are assumed to be constant. The surface of the growing layer is assumed to be flat. The loss of alloying elements due to vaporization and its effects on both the heat loss and composition change are not considered in the calculations.

B. Governing equations
The model solves the conservation equations for mass, momentum, and energy in transient three-dimensional form. These equations are available in standard text books [3] and in many of our previous publications [4,5] . The specific discretization scheme and the solution methodology for transient three dimensional form are also discussed in details in the literature [3,4] . Spatially non-uniform grids, with finer grid spacing near the axis of the laser beam were used for efficient calculation of variables.
The governing equations were discretized by following a control volume method [3] .
The velocity components and the scalar variables were stored at different locations to enhance the convergence and stability of the computational scheme. At each time step, the three components of velocities and the enthalpy were iterated following a sequence known as the SIMPLE algorithm [3] . The implicit computational scheme adapted is unconditionally stable. The discretized linear equations were solved using a Gaussian elimination technique known as the tri-diagonal matrix algorithm [3] .

C. Computational domain and calculation procedure
The

D. Boundary conditions and convergence criteria
At the beginning of the simulation, all the cells above the substrate are assigned properties of an inert gas and the initial temperature of the domain is taken as the room temperature (298 K). The variation of all variables across the mid-section longitudinal symmetry plane is set to zero. In the remaining surfaces, heat loss by radiation and convection is applied as boundary conditions for the solution of the enthalpy equation. At the top surface of the melt pool, the velocities arising from the surface tension variation due to temperature gradient are applied for the solution of momentum equations [5,6] . Velocities are set to be zero at other surfaces since they are solid and the melt pool does not extend there.
At any given time step, the iterations were terminated when two convergence criteria were satisfied. The magnitudes of the residuals of enthalpy and the three components of velocities, and the overall heat balance were checked after every iteration. The largest imbalance of any variable on the two sides of a discretization equation for all interior grid points had to be less than 0.1%. In addition, the overall heat balance criterion required that the sum of the total heat loss from the domain and the heat accumulation had to be almost equal to the heat input into the calculation domain. Their difference had to be less than 0.5% of the heat input for this convergence criterion to be satisfied. The criteria were selected so that the final results were not adversely affected while maintaining computational speed.