4.2. Boundary condition

The system of equations resulting from Eq. (10) can be solved once the prescribed boundary conditions (BCs) have been substituted. In FE thermal analysis, the possible BCs are:


Boundary conditions depend on how the system interacts with the external environment:

i. Top surface:

The working chamber is filled with Argon to reduce the alloy powder oxidation. Natural convection applies overall on the top surface, apart from the localized area where the

Figure 12. A flow chart representing the main algorithm.

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Figure 12. A flow chart representing the main algorithm.

(1 mm x 0.2 mm x 0.3 mm) is discretized with a coarser mesh. The height must ensure that the bottom border will not interfere with the surface temperature. Mapped mesh guarantees nodal consistency at the interface between powder and base. Convergence analysis has been done to validate the mesh size. Powder elements affected by the heat source, are assigned

A mapped mesh employing hexahedral 8-node elements is adopted to reduce the computational cost while maintaining high thermal field resolution. Specifically, thermal brick elements

Different thermal properties can be associated with the same element type, hence it makes it possible to distinguish the different behavior of powder, base plate, and molten pool. Referring to Figure 11, the base plate elements (cyan) are associated with constant thermal properties, as it is supposed that the base plate is not affected by the thermal field. This assumption helps to reduce the non-linearity. Different element properties are also associated with the layer in order to distinguish between the inert powder bed (blue elements) and the grains that undergo the melting process. These elements are depicted in red and represent the dimension of the

As mentioned before, the algorithm is iterative and the system must be solved at each laser spot application. The diagram in Figure 12 shows how the solving algorithm is carried out.

The system of equations resulting from Eq. (10) can be solved once the prescribed boundary

conditions (BCs) have been substituted. In FE thermal analysis, the possible BCs are:

4. Flux due to radiation ruled by the fourth power of the absolute temperature

Boundary conditions depend on how the system interacts with the external environment:

The working chamber is filled with Argon to reduce the alloy powder oxidation. Natural convection applies overall on the top surface, apart from the localized area where the

3. Flux due to convection ruled by the temperature difference

molten pool. Different thermal properties have just been explained in Section 2.1.

with different material properties, as it is shown in Figure 11.

140 Finite Element Method - Simulation, Numerical Analysis and Solution Techniques

called SOLID70 with the following characteristics are used:

• Conduction and enthalpy capabilities

• Mapped mesh

4.2. Boundary condition

1. Imposed temperature

2. Imposed heat flux

i. Top surface:

• Eight nodes (no mid-edge node capability)

• Applicable to a 3-D transient thermal analysis

laser heat flux is imposed. Since the emitting radiation flux makes the analysis highly nonlinear, its effect is not considered here. To solve this problem, an empirical relationship has been proposed [9, 18], which combines the effect of radiation and convection into a lumped heat transfer coefficient.

ii. Lateral surfaces:

Since the powder conductivity is very low, lateral surfaces can be considered as adiabatic, hence the heat flux imposed is equal to zero (q xð Þ¼ ; t 0).

iii. Bottom surface:

In SLM machines, the base plate is heated between 80�C and 130� C, depending on the machine model. Bottom nodes are constrained with imposed temperature or with convection conditions. In this work, the bottom surface is constrained with convection boundary condition. As a consequence, a convection coefficient must be chosen in order to reproduce the convective exchange conditions into the base plate.

BC applied to the numerical model is summarized in Figure 11.

Not only boundary conditions, but also initial conditions (ICs) are requested to solve the numerical model. Initial conditions can be imposed setting up a starting temperature for all the nodes. These temperatures are used in transient solutions as the first step temperatures, hence at a time equal to zero:

$$\mathbf{T(x,t=0..n')} = \mathbf{T\_0(x)}\tag{14}$$

OLD\_MESH and NEW\_MESH refer to listed mesh entities. Each row of the list contains elements and nodes tracking number, their spatial coordinates, and the related properties.

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Figure 13. Comparison between OLD\_MESH and NEW\_MESH.

The procedure able to assign correctly the temperature and material properties between two different mesh environments is called mapping procedure and is carried out in sequence by MatLab® and ANSYS®. A mapping algorithm is a useful tool that is able to save nodal temperatures from the previous load step, evaluate and assign material type with respect to the element average temperature, and finally, restore the data in the subsequent iteration as initial conditions. To avoid misunderstanding, it is worth noticing the difference between nodal and element properties: temperatures are the values assigned to nodes, while the material number is assigned to the elements. Due to ANSYS® programming language, different thermal behaviors can be assigned to the same element type using material numbers. This is the reason why in this work the expression material properties has been used with the same meaning as thermal properties.

The flowchart presented in Figure 14 helps to understand the mapping algorithm.

be simply assigned as the initial condition with respect to the entities number.

mon and uncommon mesh.

At the beginning, the elements and nodes are listed by ANSYS® in a file with the related material number and temperature. This occurs in the post-procedure step related to the n cycle (NEW\_MESH). The file is imported into MatLab® and compared with the previous mesh file, just saved before from the n-1 cycle (OLD\_MESH). Referring to Figure 13, elements and nodes are compared with respect to their spatial location and divided into two groups: common and uncommon entities. The dashed squares in Figure 13 highlight the difference between com-

Data coming from the previous analysis (step n-1) are assigned to the next one (step n) regardless of the grouped entities. Since the common elements and nodes share the same spatial location, the properties are simply transferred from the OLD\_MESH to the NEW\_MESH. The mapping algorithm takes the (element and node) spatial coordinates from OLD\_MESH and searches the corresponding location in NEW\_MESH (notice that the reference point for the element localization is the centroid). The mapping based on spatial coordinates is needed as the common entities share the same location but not the same tracking number because of the different meshes. Consequently, the temperatures and material numbers are transferred to NEW\_MESH and can

Moreover, since transient solution occurs at each cycle, initial condition must also be set at the beginning of each load step. It follows that initial condition applied to load step n are the nodal temperature obtained from the solution at step n-1.
