1 Copyright © 2017 by ASME

Proceedings of the 2017 Manufacturing Science and Engineering Conference MSEC2017

June 4 - 8, 2017, University of Southern California, Los Angeles, CA

MSEC2017-2939(draft)

NUMERICAL MODELING OF METAL-BASED ADDITIVE MANUFACTURING PROCESS USING LEVEL SET METHODS

Qian Ye and Shikui Chen* Department of Mechanical Engineering, State University of New York at Stony Brook

Stony Brook, New York, 11790, USA E-mail: qian.ye@stonybrook.edu, shikui.chen@stonybrook.edu

ABSTRACT Modern computer technology enables people to

simulate additive manufacturing (AM) process at high

fidelity, which has proven to be an effective way to

analyze, predict, and design the AM processes. In this

paper, a new method is proposed to simulate the

melting process of metal powder based AM. The

physics is described using partial differential

equations for heat transfer and Laminar flow. The

level set methods are employed to track the motion of

free surface between liquid and solid phases. The

issues, including free surface evolution, phase

changes, and velocity field calculation are

investigated. The convergence problem is examined in

order to improve the efficiency of solving this

multiphysics problem.

Keywords: Additive Manufacturing, metal powder,

modeling, simulation, level-set method

1 INTRODUCTION The additive manufacturing technology produces

parts by adding material in layers [1]. In contrast to

conventional manufacturing methods which remove

the excess material, additive manufacturing uses an

energy source to melt the material and deposit it layer

by layer [2]. AM allows fabricating products with a

significant freedom in the shape, composition, and

function [3].

A variety of materials can be involved in the AM

process, such as metals, ceramics or composites. In

this paper, we focus on metal-based AM process since

metal products are most widely used in industry.

Metal-based AM products can be achieved either in a

direct way or in an indirect way [4]. In the direct way,

the metal particles are fully melted in the AM process.

Selective Laser Melting (SLM) and Electron Beam

Melting (EBM) are two standard direct methods. In

SLM or EBM, the powder is spread in a layer and then

melted by a laser or electron beam. When the

temperature decreases, the molten material solidifies

and forms a layer of the product. This procedure is

repeated with each layer adhered to the last until the

product is built. In the indirect way, the layers of the

products are bonded by a binder. The Binder Jetting

based AM is one of the commonly used indirect

methods. The metal particles are firstly sintered, and

after infiltration, the metal particles are bonded

together.

Though AM techniques have been widely

developed these years, the process is not thoroughly

understood, which lays the foundation for the

monitoring and control of AM. As shown in Figure 1,

both the direct and the indirect ways involve complex

multi-physics phenomena, for example, melting and

solidification, fluid flow, heat transfer, vaporization or

radiation. Also, in the SLM or EBM processes, the

wetting mechanism should be studied as well [1].

Those physical phenomena are intimately bound up

with the AM process parameters which directly affect

the quality of the products [2]. However, those critical

parameters, such as laser power, scan speed, scan

spacing, particle size or packing density, are usually

decided empirically. Those methods based on

empirical evidence are inaccurate and hard to operate

when the manufacturing process changes.

Nowadays, predictive science-based simulation

and modeling technologies have emerged as a

powerful and efficient tool to study the impact of the

AM process parameters. Phenomena such as the heat

transfer in solid, heat absorption from the heat source,

2 Copyright © 2017 by ASME

or heat radiation between solid with the surrounding

gas, have been successfully simulated [3, 4]. Fan and

Liou [5] simulated the AM process of Titanium alloy

and compared the simulation results with experiment,

which showed a good agreement. However, some of

the phenomena like the Marangoni convection due to

surface tension gradients, the motion of liquid, the

radiation losses at the fluid/gas interface, are not been

developed well in the current computational models

because of the complexity in physics.

EEEEEEEEEEEEEEEEEEEEEEEE

Figure 1. Simulation of selective electron beam melting

processes[6]

The objective of this paper is to build a numerical

model to simulate the metal-based AM process at the

mesoscale. In this study, the simulation model is based

on the SLM process, which the material is spread in a

layer and then melted by a laser beam. This research is

focused on Titanium Alloy powders with a size

between 10 to150 . The heat transfer

phenomena is modeled by solving the PDEs of physics

transportation. By applying the level set method, the

complex changes of particle geometries and the

liquid/gas interface can be captured at high fidelity.

This paper is organized as follows: Section 2

describes the mathematical method and simulation

techniques for modeling the metal-base AM process.

Two numerical examples on power-based SLM

melting process are presented in Section 3. The

computational results and future work are discussed in

Section 4.

2 SIMULATION OF MELTING PROCESS OF POWDER-BASED SLM

Level Set Methods for AM process modeling

As introduced in section 1, there are many

physical phenomena involved in the AM process. The

setting of these physical models is necessary for the

numerical simulation. In this section, two important

physical models are introduced.

One is the modeling of phase change problem.

One way to model this problem is to simulate the

smallest scales of the solid-liquid interface [11]. This

model is easy to understand but needs a powerful

computing tool, so it only fits very small domains

(0.1m to 10mm). For the macroscopic transport

problem, the representative elementary volume (REV)

is introduced. This model selects a zone to include a

representative and uniform sampling of the mushy

region such that the local scale solidification processes

can be described by variables averaged over the REV

[2]. Based on the concept, the governing equations for

the conservation in the process can be developed and

solved.

The melt metal flow in the AM processes is

considered as a free-surface flow, which means the

viscous stresses in the melt metal flow is assumed to

be zero. Both the Lagrangian method and the Eulerian

method have been applied to model the shape of the

free-surface flow [12]. However, the Lagrangian

method becomes more complex in handling fluid

problems with topological changes [11]. The Eulerian

method is more suitable to model moving interface

problems.

In this paper, the level set method is employed for

free surface tracking. The basic idea of the level set

methods is to use an implicit function with one higher

dimension to represent the shape as the zero level set,

as shown in equation 1. Imagine we have an interface

Γ on a bounded region Ω . The level-set 𝜙 can be

defined as equation 1 [7]. So we can easily get the

interface Γ(𝑡) when 𝜙 is equal to 0.

{

𝜙(𝑥, 𝑡) < 0 𝑖𝑛 Ω(t)

𝜙(𝑥, 𝑡) = 0 𝑜𝑛 Γ(𝑡)

𝜙(𝑥, 𝑡) > 0 𝑖𝑛 𝑅𝑛\Ω(𝑡) (1)

Embedding the design in one higher dimension

allows the flexibility in topological changes such as

boundary merging or splitting in the AM process. In

equation 1, after differentiating the level set function

𝜙 = 0 with respect to time t, we get the Hamilton

Jacobi equation, as shown in equation 2.

𝜙𝑡 + 𝐮 ∙ ∇𝜙 = 0, (2)

where 𝐮 is the velocity function of the interface. The

free-surface motion of the melt pools can be captured

by solving the Hamilton-Jacobi partial differential

equation [8, 9]. The subsequent motion is decided by

the velocity field 𝐮 , which is based on the external

physical or geometry information of the interface.

After we get the initial level set 𝜙 (x, y, t00), the

interface at time t is defined by 𝜙(𝑥(𝑡), 𝑦(𝑡), 𝑡) = 0.

The key to the level set method is to calculate a proper

velocity field and use it to update the new level set

function. The application of level set methods to fluid

dynamics problem has been carried out, for instance,

Chung and Das [10] used the level set method to solve

Absorption

Melting

Heat

transfer

Radiation

Vaporizati

Fluid Flow

Wetting

Melting/

Solidification

3 Copyright © 2017 by ASME

the Stefan problem and successfully tracked the

interface between the liquid phase and solid phase.

Physics governing equations In this model, the computational domain for the

SLM process modeling includes the melt pools,

deposited metal powder, and the surrounding air, as

shown in Figure 1. The continuum model is applied to

calculate the properties of the solid-liquid co-exist

area. The melted metal is assumed to be a Newtonian

fluid, and the melt pool is considered to be an

incompressible laminar flow.

For the whole study domain, the conservation

equations are shown as below.

Mass conservation:

𝜕𝜌

𝜕𝑡+ ∇ ⋅ 𝜌𝐕 = 𝟎, (3)

Energy conservation:

∂(𝜌ℎ)

∂𝑡+ ∇ ⋅ (𝜌𝐕ℎ)

= ∇ ⋅ (𝑘∇𝑇) − ∇⋅ 𝜌(ℎ𝑙 − ℎ)𝐕,

(4)

Momentum conservation:

𝜕𝜌𝐕

𝜕𝑡+ ∇ ⋅ (𝜌𝐕𝐕) = 𝛁 ⋅ (𝜇𝑙

𝜌

𝜌𝑙

∇𝐕)

− ∇𝑝 + 𝐴𝐕 + 𝐅𝐬𝐭

+ 𝐅𝐜 + 𝑆,

(5)

where 𝜌, 𝑡, 𝐕, 𝜇, 𝑘, 𝑇, 𝑝, 𝐴, ℎ are density, time, velocity,

molten fluid dynamic viscosity, heat conductivity,

temperature, pressure, permeability, enthalpy,

respectively. 𝐅𝐬𝐭 represents the capillary forces and 𝑆

represents the source term which related to the gravity.

𝐅𝐜 refers to the thermo-capillary force. The

subscription “s” represents solid, “l” represents liquid.

During the process of melting and solidification,

the phase changes in a certain temperature zone, which

lead to a solid-liquid coexist zone called mush zone.

To simulate the momentum of this area accurately, a

damping term represented by the permeability A is

applied [11]:

𝐴 = −𝐾0

(1 − 𝑔𝑙)2

𝑔𝑙3 + 𝜖0

, (6)

where 𝐾0 is the permeability coefficient, 𝑔𝑙 is the

mass fraction of liquid, and 𝜖0 is an small constant.

In this study, the interface between phases is

tracked by the level set method on a fixed grid. The

source terms in the momentum equations are applied

as interfacial forces in the fluid flow, such as capillary

force, buoyancy force, or thermocapillary force [12].

The capillary force and surface tension are used to

describe the liquid contractive tendency when the fluid

flows in a narrow space. In addition to the surface

tension, the forces allow the fluid part to resist the

external forces like gravity. For the liquid and gas

interface, the capillary force is represented by the

following equation:

𝐅𝐬𝐭 = 𝝈𝜅𝐧, (7)

where 𝝈 is the surface tension stress, 𝒏 and 𝜅 are

normal vector and curvature of the liquid and gas

interface, which can be calculated by level set

function. From the equation, it is obvious that the

capillary force works on the normal direction on the

liquid-gas interface.

𝐧 =∇𝜙

|∇𝜙|, (8)

𝜅 = ∇ ⋅ 𝐧, (9)

The body source term S is calculated based on the

Boussinesq approximation [13] which represents the

natural convection of the liquid under non-isothermal

condition. The force is caused by the material

temperature dependent density gradient [14].

𝑆 = 𝜌𝑔[1 − 𝛼(𝑇 − 𝑇𝑟𝑒𝑓)], (10)

where 𝛼, 𝑔 and 𝑇𝑟𝑒𝑓 are the thermal expansion

coefficient, the acceleration of gravity and the

reference temperature, respectively.

For a simple case that the density of metal in the

liquid and solid is the same, this source term can be

simplified as the gravity of the flow [15].

𝑆 = 𝜌𝑔 (11)

The surface tension gradient causes the

thermocapillary force in the tangential direction. By

the thermos-capillary force, the surface of the fluid

flow from the lower surface tension to the higher

surface tension coefficient [16].

𝐅𝐜 = ∇𝑠𝑇𝑑𝜎

𝑑𝑇 , (12)

where 𝜎 is the surface tension, ∇𝑠 𝑇 refers the surface

gradient of temperature field. By multiplying the delta

function 𝛿 , the source terms are only applied on the

interface of liquid and gas. The delta function and can

be defined as a smoothed Dirac delta function as

shown in equation (13)

𝛿(𝜙) =1

ℎ√𝜋e

(𝜙2

ℎ2 )

(13)

where h is a parameter of the smooth Dirac delta.

Adding these terms into the momentum equations

in the x and y directions, the final formula of

momentum equations are as below:

𝜕𝜌𝑢

𝜕𝑡+ ∇ ⋅ (𝜌𝐮𝑢) = 𝛁 ⋅ (𝜇𝑙

𝜌

𝜌𝑙

∇𝑢) − ∇𝑝 + 𝐴𝑢

+ 𝐞𝐱 (𝜎𝜅𝒏

+ ∇𝑠𝑇𝑑𝜎

𝑑𝑇) 𝛿(𝜙),

(14)

4 Copyright © 2017 by ASME

𝜕𝜌𝒗

𝜕𝑡+ ∇ ⋅ (𝜌𝐯𝑣) = 𝛁 ⋅ (𝜇𝑙

𝜌

𝜌𝑙

∇𝑣) − ∇𝑝

+ 𝐴𝑣

+ 𝐞𝐲 (𝜎𝜅𝒏

+ ∇𝑠𝑇𝑑𝜎

𝑑𝑇) 𝛿(𝜙)

+ 𝜌𝑔[1

− 𝛼(𝑇 − 𝑇𝑟𝑒𝑓)]𝛿(𝜙),

(15)

Methods for simulating the phase change during the melting-solidification process

The phase change during the melting or

solidification process is very complex. It is a non-

linear problem due to the absorption or releases of

latent energy at the melting point.

In this model, the phase change is considered to

be a temperature-dependent process. The enthalpy-

porosity method [17] is used to simulate this

phenomenon. It is assumed that the phase change

occurs when the temperature reaches the melting

temperature 𝑇𝑚 . According to the temperature field,

the calculation domain can be divided into three

regions: the solid, liquid and mushy zone, as shown in

equation 10 below. We use 1 and 0 to indicate the

liquid and solid region respectively. Region with

fraction between 0 and 1 is the mushy area. This area

is considered to be a porous medium, where both the

porosity and the fluid velocity decreases to 0 in the

solid area.

The fraction of the liquid phase can be calculated

by equation (16):

where ϵ is half of the transition zone of liquid and solid

phase. The solid and liquid phase temperature and

enthalpy are represented by following equations:

The temperature of the solid phase:

𝑇𝑠 = 𝑇𝑚 − 𝜖, (17)

The temperature of the liquid phase:

𝑇𝑙 = 𝑇𝑚 + 𝜖, (18)

The enthalpy in the solid phase:

ℎ𝑠 = ∫ 0

𝑇

𝑐𝑝𝑠𝑑𝑇, (19)

The enthalpy in the liquid phase:

ℎ𝑙 = ∫ 𝑐𝑝𝑠𝑑𝑇 + 𝐿𝑚 + ∫ 𝑐𝑝𝑙𝑑𝑇𝑇

𝑇𝑠

𝑇𝑠

0

= 𝑐𝑝𝑙𝑇

+ (𝑐𝑝𝑠 − 𝑐𝑝𝑙)𝑇𝑠

+ 𝐿𝑚 ,

(20)

where 𝜌, 𝑡, 𝑽, 𝜇, 𝑘, 𝑇, 𝑝, 𝐾, ℎ, 𝑐𝑝 are density, time,

velocity, molten fluid dynamic viscosity, heat

conductivity, temperature, pressure, permeability,

enthalpy, heat capacity, respectively. The subscription

“s” represents solid, “l” represents liquid. 𝐿𝑚

represents the latent heat for melting.

Continuum model for temperature-dependent material properties

There are three phases in the whole domain: the

solid metal, the liquid metal, and gas. A continuum

model is used to represent the physical character of the

entire domain [14]. When calculating the material

properties, the liquid- solid coexist area is considered

as a mush zone. The mass fractions and volume

fractions are used to calculate the mixed thermal

physical properties as below[11, 18]:

𝜌 = 𝑔𝑠𝜌𝑠 + 𝑔𝑙𝜌𝑙 , (21)

𝑐𝑝 = 𝑔𝑠𝑐𝑝𝑠 + 𝑔𝑙𝑐𝑝𝑙 , (22)

𝑘 = 𝑔𝑠𝑘𝑠 + 𝑔𝑙𝜌𝑙 , (23)

ℎ = 𝑓𝑠ℎ𝑠 + 𝑓𝑙ℎ𝑙 , (24)

where 𝑓𝑠 and 𝑓𝑙, 𝑔𝑠 and 𝑔𝑙 refer to mass fractions and

volume fractions for solid and liquid phases,

respectively. 𝑐𝑝 is the heat capacity, 𝑐𝑝𝑠, 𝑐𝑝𝑙 are the

specific heat for solid metal and liquid metal.

Also, the volume fraction and mass fraction

content the relationship:

𝑓𝑙 =𝑔𝑙𝜌𝑙

𝜌, (25)

𝑓𝑙 + 𝑓𝑠 = 1, (26)

𝑔𝑙 + 𝑔𝑠 = 1, (27)

Modeling of the laser beam The laser beam is molded as a Gaussian beam

with intensity I. The highest intensity located in the

center of the laser beam and the laser intensity keeps

decreasing while away from the center point. The

intensity distribution of laser power along the distance

of the laser beam center are as shown in Figure 2.

(28)

where is the absorptivity coefficient. R and r are the

beam radius and the distance from the calculated point

𝑓𝑙(𝑇)

= {

1, 𝑇 ≥ (𝑇𝑚 + 𝜖),𝑇𝑚 + 𝜖 − 𝑇

2𝜖, (𝑇𝑚 + 𝜖) > 𝑇 ≥ (𝑇𝑚 − 𝜖),

0, T < (𝑇𝑚 − 𝜖),

(16)

5 Copyright © 2017 by ASME

to the beam center. stands for the power of laser

beam.

Figure 2: Power intensity distribution of laser beam

Mapping of material properties with the geometric level set model

In this model, the gas and metal interface is

tracked by the level set methods. The material

properties of metal and gas are mapped from the level

set function by the Heaviside function.

Initially, a domain including metallic particles is

defined and meshed. The gaps with particles are filled

with gas. A level set function is introduced to present

the surface between metal and gas at each time step as

shown below.

{

𝜙(𝒙, 𝑡) > 0 𝑖𝑛 Ω𝑔,

𝜙(𝒙, 𝑡) = 0 𝑜𝑛 Γ(𝑡),

𝜙(𝒙, 𝑡) < 0 𝑖𝑛 Ω𝑚 ,

(29)

where substrate 𝑔 means gas, m represents metal. The

negative values of level set function identify the

metallic phases. The gas domain is represented by the

positive values of the level set function.

When solving the multiphase flow problem, the

level set equations need to be reinitialized after several

calculation iterations to keep mass conservation of the

phases [19]. However, in this research, the model is

built in the software COMSOL 5.0. When applying the

phase re-initialization, we have to stop the calculation

to reinitialize the level set function after several time

steps, then substitute the new into COMSOL again for

the afterward calculation. It is of course

computationally expensive. So in this model, a new

method proposed by Deshpande et.al [20] named as

phase-injection is used to keep the mass conservation.

This method is less computational consuming and

much easier applied in COMSOL. The basic idea is to

add a constraint into the level set function to match

with the calculated amount of mass loss.

The interface moves with the fluid velocity, the

motion of the interface can be described by the

following equation [12]:

𝜕𝜙

𝜕𝑡+ 𝐮 ⋅ ∇𝜙 = 𝜑𝛿𝜙, (30)

where𝜑represents the mass difference before and after

the level set function is updated. The constraint is only

applied to the interface. By applying this method, the

level set function does not need to be a signed distance

function.

To avoid numerical instability arising from the

physical property jump at the liquid/gas interface, the

Heaviside function is introduced to define a transition

region where the physical properties are smoothed.

The Heaviside function is shown in equation (31):

𝐻(𝜙)

= {

0, 𝑖𝑓 𝜙 < −휀,1

2[1 +

𝜙

휀+

1

𝜋sin (

𝜋𝜙

휀)] , 𝑖𝑓|𝜙| ≤ 휀,

1 , 𝑖𝑓 𝜙 > 휀,

(31)

where 휀 represents half of the transition zone around

the interface, which means the smoothed Heaviside

function evolves from 0 to 1 in the thickness of the

level set 𝜙 transition layer [21].

By applying the Heaviside function, the thermal

properties of the whole domain are represented by

𝜌 = 𝜌𝑚 ⋅ 𝐻(𝜙) + 𝜌𝑔 ⋅ (1 −

𝐻(𝜙)), (32)

𝑐𝑝 = 𝑐𝑝𝑚 ⋅ 𝐻(𝜙) + 𝑐𝑝𝑔 ⋅ (1

− 𝐻(𝜙)), (33)

𝑘 = 𝑘𝑚 ⋅ 𝐻(𝜙) + 𝑘𝑔 ⋅ (1 − 𝐻(𝜙)), (34)

where 𝜌, 𝑐𝑝 and k represent the density and heat

capacity, heat conductivity, respectively. The

subscribe m and g represent metal and gas.

Boundary conditions of the free surface The energy balance at the free surface is

considered in this model. The boundary condition can

be described as the sum of the heat loss and heat gain

from the system. In this model, the absorbed heat

simulates the gained heat from the laser beam. Also,

two kinds of heat loss are considered: the convective

heat loss and radiation heat loss , which can be

calculated as equations below [22].

(35)

(36)

where are the heat transfer coefficient,

Stefan-Boltzmann’s constant, surface radiation

emissivity and the atmosphere temperature

respectively.

Finally, the energy balance in the L/G interface

can be presented as the following equation:

(37)

where refers to the heat power distribution at the

deposited particles, because the laser beam is the only

6 Copyright © 2017 by ASME

heat source in this model, by neglecting the heat

absorbing difference in different angles, the heat

power is considered to be equal to laser intensity, that

is,

(38)

Also, the thermal contact between melt pool and

the walls is neglected.

3 SIMULATION RESULTS AND NUMERICAL VERIFICATION

Numerical verification of melting and spreading of a single particle

In this experiment, the algorithm for free surface

tracking and surface tension are tested. In the

verification experiments, the material properties of Ti-

6Al-4V are considered to be constant in both solid and

liquid phases. In this part, the model is built to simulate

the process of melting and spreading of a single

particle onto a substrate. The particle is assumed to be

spherical, which in this 2D case is considered to be a

circle with radius 0.3mm. The size of the study domain

is 1.5 mm in length and 0.7mm in width. The particle

is represented by a level set function . The interface

of particle and gas is considered as the 0 level of .The

objective of this model is to study the case involving

both free surface flow and melting of a metal particle.

However, the effects of convection are neglected. The

method can be extended to a more complex situation

later.

The initial interface of solid and gas is presented

by a level set function. Initially, the interface is a circle

which can be represented by equation (39):

(39)

where refer to the x, y coordinates and the

radius of metal particle.

The problem is solved for the total simulation

time 10ms with time step as 0.01ms. Experimental

results are shown in Figures 3. At each time point, the

figures in the first row represent the changes of

Heaviside function, which can clearly show the

geometry changes of the particle. Figures in the second

row refer the changes of temperature field as time

increasing.

From the observation of computational results,

the process can be divided into three parts:

1. Heating (0ms-0.39ms): during the heating

procedure, the temperature in the domain keeps

increasing. Because the heat conductivity is higher in

metal, so the heat transfers faster in the metal particle.

2. Phase change (0.40ms-0.42ms): during this

period, the particle begins melting. The latent heat

absorption is observed within the range of transition

temperatures as shown in Figure 4. The color contours

represent temperature, and the other legend represents

the heat capacity caused by latent heat.

3. Spreading (0.43ms-10ms): The temperature of

the whole calculation domain is higher than the

melting temperature. The phase change has completed

in the particle. There are only two phases in the

domain: liquid and gas. The liquid begins spreading

under the effects of surface tension and gravity.

t00ms t00.39ms t01ms t010ms

Figure 3: Simulating results

7 Copyright © 2017 by ASME

Figure 4: Heat capacity of latent heat at t= 0.5ms

The level set function at 0 level can clearly show the deformation of the surface as illustrated in figure 5.

Figure 5: Temperature contours and Interface changes (a. t=0.10ms; b. t=1.1ms)

Modeling and simulation of the SLM process In this section, the model for simulating the

melting process of metal-based AM is built. The

governing equations for mass, energy and momentum

conservation are applied to the whole computational

domain. Three phases are considered in the model: the

gas, the liquid metal, and solid metal. Instead of

tracking the interface between the solid and liquid

phases explicitly, the boundary of these two phases is

presented by the fraction of the liquid phase, which

depends on the melting temperature and phase change

transition zone. In this case, each particle is assumed

to be a sphere with the same radius, and the radius of

powers ranges from 10-150 . The initial interface of

solid and gas is presented by a level set

function𝜙(𝑥, 𝑦) = 0 . Initially, the geometry of one

particle is represented as following equation:

(40)

where represent the number of the particle,

the x and y coordinates and, radius, respectively.

Boolean constraints are set between each particle in

order to combine them together in the computational

domain.

(41)

The intensity of the laser beam is represented by

a Gaussian function. The calculation time is 0.9ms

with time step 0.001ms. Because of the short

calculation time, the heating process can be assumed

to be instantaneous, so it is reasonable to assume that

the laser beam is fixed during this brief period. By

applying the Heaviside function, the laser beam

directly heats the surface of powders. The energy

balance can be calculated as equation below:

(42)

where the first term on the right side refers to

the input heat power by the laser beam. The second

and third term in the right side represent the heat loss

between melt pool and ambient gas. are the

Stefan-Boltzmann constant, radiation emissivity and

heat transfer coefficient, respectively. refers the

ambient temperature, which equals to 500K in this

model. By applying the delta function, the heat loss is

applied to the L/G interface.

Table 1: Thermal properties of solid and liquid phases [23]

Physical

Properties

Solid Liquid Unit

Specific heat {

483+0.215T, T≤ 1268K

412+0.180T, 1268<T≤1923

831 J

kg ⋅ K

Density 4420-0.154(T-500K) 3920-0.68(T-1923K) 𝑘𝑔

𝑚3

a b

8 Copyright © 2017 by ASME

Thermal

conductivity {

1.26+0.016T, T≤ 1268K

3.513+0.013T, 1268<T≤1923

-12.752+0.024T W

m ⋅ K

Surface

Tension 1.525-0.28×10-2(T-1941K)

N

m

In this case, instead of using the constant thermal properties,

temperature dependent properties are considered in both

solid and liquid phases as shown in Table 1, the value of

constant in this model is given in Table 2. The metal powers

are distributed in a rectangle with 1.5mm in length and 1 mm

in width. The laser beam is set at the point (0.25mm, 0.6mm)

with radius 3mm. Initially, the geometry of the powder bed

is shown in Figure 6. The laser beam intensity is 1000W.

Table 2: Constants in the simulation model

Definition Value Units

Initial temperature 500 K

Radius of metal powder 10-150 μm

Acceleration of gravity -9.8 m/s2

Power of laser beam 1000 W

Absorptivity coefficient 0.2 1

Stefan-Boltzmann constant 5.67×10-3 W/m2K4

Radiation emissivity 0.8 1

Finally, a mapped mesh with maximum element

size of 1.8×10-5 for the calculation domain. The

capillary force, buoyancy force and thermos-capillary

force are applied to the L/G interface.

Figure 6 shows the simulation results. At each

figure, the colored contours refer to the temperature;

the black contours are the plots for a level set function

at 0 level which can represent the interface between

metal and gas. The red arrow surface refers to the

velocity distribution. The label on the right side shows

the temperature value and its corresponding color. It is

observed that the highest temperature happens in the

area which is nearest to the center of the laser spot. By

the end of the simulation, the highest temperature

reaches to 2773K. The particles completely change

from a solid phase to liquid phase. The deformation of

melted particles can be observed as well.

t=0.07ms t=0.3ms

t=0.632ms t=0.7ms

t=0.744ms t=0.84ms

Figure 6: Evolution of the melt pool geometry, temperature contours, and velocity surface

9 Copyright © 2017 by ASME

Figure 7 shows the temperature and liquid faction

evolution during the phase change. From the results,

the phase change starts in the area closest to the laser

beam. Only regions higher than the melting point is

considered to be liquid. This is in accord with the

fraction function which has been discussed in Section

2.2. The phase change phenomenon while being

melted is simulated as a process of absorbing latent

heat. The phase change phenomenon while being

melted is simulated as a process of absorbing latent

heat. The latent heat can be regarded as a sensible heat

that is assimilated only within the range of transition

temperatures as shown in Figure 8. From left to right

side, the two labels represent the value of latent heat

and temperature contours, respectively. It is observed

the latent heat is non-zero in a range of temperature

equal to a melting point. Under or exceeding the phase

transition region, the value of latent heat equals to 0.

Figure 7: Relationship between temperature and liquid fraction

Figure 8: Latent heat when phase change

Convergence study As stressed before, the problem is a highly

nonlinear multiphysics one. There are three models

needed to be solved: the heat transfer, the fluid flow,

and level set. The heat transfer and fluid flow are

physics models, and the level set method is only used

to track the interface. In order to find a more efficient

way to solve the problem, the fully coupled solver, and

the segregated solver is employed and compared in

this work.

The direct way to solve the problem is by using

the fully coupled solver which solves the physics and

level set functions simultaneously. This approach is

straightforward but is more memory costing and time-

consuming.

The segregated approach is used instead, which

the basic idea is to solve the physics and level set

functions step by step. The computer solves one

specific model at one time and inherits the other

variables from last steps [24]. The comparison of the

convergence plots of each solver is shown in Figure 9.

10 Copyright © 2017 by ASME

Figure 9: Convergence plots of different solver

From the observation, by using the segregated solver,

the number of total computational time steps reduce

from 180 to 155, which means the approach is faster

than the fully coupled approach. Also, the convergence

plot for segregated solver is smoother than the full

direct couple solver. So we can conclude that the

segregated solver shows higher efficiency in solving

this multi-physics problem.

4 Summary In this study, a 2D numerical model for simulation

the melting process of metal-based SLM with a fixed

laser beam has been established. The numerical model

is validated in several aspects: the temperature field,

the velocity field and the interface between liquid and

gas phases. The model combines the traditional

transport phenomena (such as heat transfer and fluid

dynamics) with level set methods for free surface

tracking. The melting phase change is studied by the

enthalpy-porosity technique. The free surface flow is

studied by the level set approach. A segregated study

is used in this model. Compared with fully coupled

solver, the segregated solver shows higher efficiency

in solving this nonlinear multiphysics problem.

There are still several limitations in these model

and need to be improved by future research. In this

model, the effects of evaporation are neglected.

However, in the real AM process, the temperature in

the melt pool may exceed the evaporating point. So the

mass loss during evaporation should be taken into

account later. The wetting effects between melted

particles with un-melted particles are not studied.

Also, the thermal contacts between melt pool and the

substrate are not included. The above problems will be

further investigated in our future research. Also, the

re-initialization of the level set function needs to be

further improved to avoid the possible interface

shifting. Furthermore, in the current work, the total

number of particles is relatively small. The validation

of the proposed method with a more complex 3D

model will be further explored.

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