5. Miscellaneous models

In order to completely model the AM process it can be necessary to include an energy source, laser or e-beam, and if powders are used it can be necessary to track their position within a powder bed or blown powder.

#### 5.1. Laser modeling

The most common laser distributions used in metal AM process are top hat, Figure 5b, and Gaussian, Figure 5a. Just as the material must be divided into the domain, a laser must be divided into smaller pieces for proper modeling. This is done by dividing the projected area into segments, which can be thought of as rays from the laser. The flux that is generated within each of these divisions of the laser is found by taking the integral of the functions over one division. This was done for the Gaussian laser distribution where Figure 5a was generated by integrating Eq. (41),

$$\phi(\mathbf{x}, y) = \phi\_0 \sqrt{1 - \frac{\mathbf{x}^2}{r\_0^2} - \frac{y^2}{r\_0^2}} \tag{41}$$

5.2. E-beam modeling

intensity in the z-direction can be expressed as Eq. (42).

Q x \_ ð Þ¼� ; <sup>y</sup>; <sup>z</sup> <sup>η</sup>bηePB

5.3. Powder bed and blown powder models

<sup>ϕ</sup><sup>z</sup> <sup>¼</sup> <sup>1</sup>

ϕxyϕ<sup>z</sup>

<sup>0</sup>:<sup>75</sup> �2:<sup>25</sup> <sup>z</sup>

<sup>S</sup> ¼ �ηbηePB

the beam power, S is the absolute penetration depth, and db beam diameter [39].

The other choice of a heat source for metal AM is an electron beam. Typically an electron beam is modeled as a Gaussian heat source, just as the previously mentioned laser. However, an electron beam will also penetrate the surface heating the material for a given depth. The

> S <sup>2</sup>

The xy and z intensities can then be combined to determine the heat flux, as shown in Eq. (43),

where η<sup>b</sup> is the beam control efficiency, η<sup>e</sup> is the energy conservation at the part surface, PB is

The last element of the metal AM process that is typically necessary is the addition of material. In the wire feed DED process this is done by modeling the wire as a solid material, just as the substrate, and treating it in a similar fashion. When using either a powder bed or blown powder, this is not possible. Due to the stochastic nature of the powders, it can be challenging to model their behavior. When modeling the powder bed setup, there are two prevailing methods that have been used: discrete element method (DEM) and geometric methods.

The DEM technique tracks the powder particles on an individual basis to determine their final position in the build volume. This simulation technique typically begins by dropping particles (the blue particles in Figure 6a), or sets of particles, from a random x and y position but a designated height in the domain. From there, the particles position and velocity (the red vectors in Figure 6a) are tracked as the particle is subjected to the major forces of gravity, contact, and friction. In some cases, smaller interaction forces, such as Van der Waal forces [40] or JKR interaction forces [41], are added to increase the accuracy of the simulation. The particles are allowed settle to their resting point (the green particles in Figure 6a) and more particles are added as needed to the simulation. This technique is very appealing to the simulation powder spreading in powder bed AM due to the ease of adding a powder spreader to the simulation without much additional effort. This results in the ability to simulation the entire process from the layering of the powder to laser interaction to melting and solidification [42]. The geometric method is not as realistic for powder bed process but is computationally only a small fraction of the DEM technique. In this technique, the area to be filled is analyzed without regard to how the powder would flow. The first geometric method is referred to as the compression algorithm. In this technique, the particles are randomly spaced within the domain. The particles are then moved in the direction of compression, typically the direction

4ln ð Þ 0:1 πd<sup>2</sup> bS e

<sup>þ</sup> <sup>1</sup>:<sup>5</sup> <sup>z</sup>

<sup>S</sup> <sup>þ</sup> <sup>0</sup>:<sup>75</sup> (42)

<sup>b</sup> �<sup>3</sup> <sup>z</sup>

S <sup>2</sup>

� <sup>2</sup> <sup>z</sup> <sup>S</sup> <sup>þ</sup> <sup>1</sup> (43)

Modeling and Simulation of Metal AM http://dx.doi.org/10.5772/intechopen.78144 105

<sup>4</sup>ln ð Þ <sup>0</sup>:<sup>1</sup> <sup>x</sup>2þy<sup>2</sup> ð Þ d2

where ϕð Þ x; y is the intensity as a specific ð Þ x; y location, ϕ<sup>0</sup> is the initial laser power, and r<sup>0</sup> is the laser radius [19]. The top hat beam, Figure 5b, is much easier to model. In this model, the laser power is simply divided by the number of divisions that can be placed within the laser boundaries. The heat flux that is found is then applied to a specific location of the domain based on the current laser location. These intensities are then multiplied by α, the material absorptivity, to determine the actual amount of energy that is absorbed by the material. This process can be applied to any distribution that can be imagined.

There are two methods of applying the laser. The first is to define the surface and apply the flux directly to this region of the material and not attempting to determine if there is anything blocking the laser projection. The other, and more realistic method, is to perform ray tracing and apply the heat flux to the first location that is struck by the ray. Both method work but ray tracing is preferred for the sake of reality but can be costly to compute.

Figure 5. Example of heat flux from example distributions.

#### 5.2. E-beam modeling

5. Miscellaneous models

powder bed or blown powder.

5.1. Laser modeling

104 3D Printing

integrating Eq. (41),

In order to completely model the AM process it can be necessary to include an energy source, laser or e-beam, and if powders are used it can be necessary to track their position within a

The most common laser distributions used in metal AM process are top hat, Figure 5b, and Gaussian, Figure 5a. Just as the material must be divided into the domain, a laser must be divided into smaller pieces for proper modeling. This is done by dividing the projected area into segments, which can be thought of as rays from the laser. The flux that is generated within each of these divisions of the laser is found by taking the integral of the functions over one division. This was done for the Gaussian laser distribution where Figure 5a was generated by

s

where ϕð Þ x; y is the intensity as a specific ð Þ x; y location, ϕ<sup>0</sup> is the initial laser power, and r<sup>0</sup> is the laser radius [19]. The top hat beam, Figure 5b, is much easier to model. In this model, the laser power is simply divided by the number of divisions that can be placed within the laser boundaries. The heat flux that is found is then applied to a specific location of the domain based on the current laser location. These intensities are then multiplied by α, the material absorptivity, to determine the actual amount of energy that is absorbed by the material. This

There are two methods of applying the laser. The first is to define the surface and apply the flux directly to this region of the material and not attempting to determine if there is anything blocking the laser projection. The other, and more realistic method, is to perform ray tracing and apply the heat flux to the first location that is struck by the ray. Both method work but ray

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>x</sup><sup>2</sup> r2 0 � y2 r2 0

(41)

ϕð Þ¼ x; y ϕ<sup>0</sup>

process can be applied to any distribution that can be imagined.

tracing is preferred for the sake of reality but can be costly to compute.

Figure 5. Example of heat flux from example distributions.

The other choice of a heat source for metal AM is an electron beam. Typically an electron beam is modeled as a Gaussian heat source, just as the previously mentioned laser. However, an electron beam will also penetrate the surface heating the material for a given depth. The intensity in the z-direction can be expressed as Eq. (42).

$$\phi\_z = \frac{1}{0.75} \left( -2.25 \left( \frac{z}{S} \right)^2 + 1.5 \frac{z}{S} + 0.75 \right) \tag{42}$$

The xy and z intensities can then be combined to determine the heat flux, as shown in Eq. (43),

$$\dot{Q}(\mathbf{x},y,z) = -\eta\_b \eta\_\epsilon P\_B \frac{\phi\_{xy}\phi\_z}{S} = -\eta\_b \eta\_\epsilon P\_B \frac{4\ln\left(0.1\right)}{\pi d\_b^2 S} e^{\frac{4\ln\left(0.1\right)\left(\frac{\pi^2}{4} + \frac{y^2}{2}\right)}{\left(S\right)^2}} \left(-3\left(\frac{z}{S}\right)^2 - 2\frac{z}{S} + 1\right) \tag{43}$$

where η<sup>b</sup> is the beam control efficiency, η<sup>e</sup> is the energy conservation at the part surface, PB is the beam power, S is the absolute penetration depth, and db beam diameter [39].

#### 5.3. Powder bed and blown powder models

The last element of the metal AM process that is typically necessary is the addition of material. In the wire feed DED process this is done by modeling the wire as a solid material, just as the substrate, and treating it in a similar fashion. When using either a powder bed or blown powder, this is not possible. Due to the stochastic nature of the powders, it can be challenging to model their behavior. When modeling the powder bed setup, there are two prevailing methods that have been used: discrete element method (DEM) and geometric methods.

The DEM technique tracks the powder particles on an individual basis to determine their final position in the build volume. This simulation technique typically begins by dropping particles (the blue particles in Figure 6a), or sets of particles, from a random x and y position but a designated height in the domain. From there, the particles position and velocity (the red vectors in Figure 6a) are tracked as the particle is subjected to the major forces of gravity, contact, and friction. In some cases, smaller interaction forces, such as Van der Waal forces [40] or JKR interaction forces [41], are added to increase the accuracy of the simulation. The particles are allowed settle to their resting point (the green particles in Figure 6a) and more particles are added as needed to the simulation. This technique is very appealing to the simulation powder spreading in powder bed AM due to the ease of adding a powder spreader to the simulation without much additional effort. This results in the ability to simulation the entire process from the layering of the powder to laser interaction to melting and solidification [42].

The geometric method is not as realistic for powder bed process but is computationally only a small fraction of the DEM technique. In this technique, the area to be filled is analyzed without regard to how the powder would flow. The first geometric method is referred to as the compression algorithm. In this technique, the particles are randomly spaced within the domain. The particles are then moved in the direction of compression, typically the direction

modeling. Other models that can be included, based on the user's desires, are stress, micro-

Modeling and Simulation of Metal AM http://dx.doi.org/10.5772/intechopen.78144 107

1. Fan and Liou [45] model heat transfer and fluid flow dynamics (VOF) for a blown powder

2. Kumar et al. [46] model heat transfer and fluid flow dynamics (SPH) for a wire feed DED

3. Lee and Zhang [47] model powder bed generation (DEM), heat transfer, fluid flow (VOF)

The support from Department of Energy Grant DE-SC0015207, and NSF grants CMMI 1625736 and EEC 1004839, and Product Innovation and Engineering, LLC., are appreciated. We also appreciate the financial support provided by the Center for Aerospace Manufacturing Tech-

Department of Mechanical and Aerospace Engineering, Missouri University of Science and

[2] Bourell DLD, Beaman JJ, Leu MC, Rosen DWA. Brief history of additive manufacturing and the 2009 roadmap for additive manufacturing: Looking back and looking ahead. In: Workshop on Rapid Technologies. 2009. pp. 5-11. Available from: http://iweb.tntech.edu/

[3] ASTM. Standard Guidelines for Design for Additive Manufacturing. West Conshohocken,

[4] Bhavar V, Kattire P, Patil V, Khot S, Gujar K, Singh R. A review on powder bed fusion technology of metal additive manufacturing. In: 4th International Conference and Exhibi-

PA, USA: American Society for Testing Materials; 2017. p. 52910

tion on Additive Manufacturing Technologies-AM-2014, September; 2014

and microstructure for powder bed AM with Nickel-based super-alloys

nology (CAMT) at the Missouri University of Science and Technology.

structure, surface finish, and more attributes. Some examples where this have been done are:

DED AM method with a laser for Ti-64

AM method with a laser for Ti-64

Acknowledgements

Author details

Aaron Flood\* and Frank Liou

Technology, Rolla, USA

References

\*Address all correspondence to: ajfrk6@mst.edu

[1] Munz OJ. Photo-Glyph Recording; 1951

rrpl/rapidtech2009/bourell.pdf

Figure 6. Powder bed modeling techniques.

of gravity, by the shortest distance that results in a collision of particles. This compressing is repeated until the potential energy is below a user-defined tolerance. The particles are then "shaken", or moved laterally, and the process is repeated. The build volume is then refilled with more particles and the process is repeated until the volume is full [43]. Another geometric method, displayed in Figure 6b, focuses on a tetrahedral mesh. In this method, the build volume is meshed and particles are placed on the nodes and edges based on a set of rules. This allows for an efficient filling of the space that has a packing density approximately matching reality [44].

Each of these methods can be used to simulate the powder bed process, however, the selection can be based on a couple of main factors. If speed is required then a geometric approach should be used. This approach takes on the order of seconds for a full simulation whereas the DEM approach will take hours to days to simulation an identical setup. If physical accuracy is needed then it necessary to use the DEM approach. This is because it tracks the powder particles through time and space. They are subjected to the forces of nature that result in a realistic simulation, whereas the geometric approach is simply a packing problem where the particles are placed where they can fit. This can result in packing densities that are not representatives of natural occurrences.

Modeling of the blown powder DED has only been done with the DEM approach previously discussed. To apply the DEM to blown powder, the nozzle must be modeled as a boundary condition and the particles should be fed through the feed system and tracked to determine when and where they strike the melt pool.

### 6. Conclusion

In order to mathematically model the AM process, it is necessary to couple several distinct mathematical models. The necessary models are thermal, fluids modeling and energy input modeling. Other models that can be included, based on the user's desires, are stress, microstructure, surface finish, and more attributes.

Some examples where this have been done are:


### Acknowledgements

The support from Department of Energy Grant DE-SC0015207, and NSF grants CMMI 1625736 and EEC 1004839, and Product Innovation and Engineering, LLC., are appreciated. We also appreciate the financial support provided by the Center for Aerospace Manufacturing Technology (CAMT) at the Missouri University of Science and Technology.
