1. i
Efficient Thermo-Mechanical Simulation of the
Directed Energy Deposition Additive Manufacturing Process
By
Anirudh Krishnakumar
A dissertation submitted in partial fulfillment of
the requirements for the degree of
Master of Science
(Mechanical Engineering)
at the
UNIVERSITY OF WISCONSIN-MADISON
2015
Date of final oral examination: 1/8/2016
The dissertation is approved by the following members of the Final Oral Committee:
Krishnan Suresh, Professor, Mechanical Engineering
Gregory F. Nellis, Kaiser Chaired Professor, Mechanical Engineering
Natalie Rudolph, Assistant Professor, Mechanical Engineering
3. iii
Abstract
There is significant interest today in various Additive Manufacturing (AM) processes, due to their
advantages over conventional subtractive processes. There is hence an increased demand to simulate and
understand AM technology. The study of the build, and an understanding of the relationship between the
final part properties and the process parameters is of significant interest. The most common AM
simulation method is finite element analysis (FEA).
However, there are three challenges that classic FEA-based AM simulation poses: (1) it is
computationally demanding since AM processes are inherently non-linear, involving multiple physics, (2)
repeated meshing and insertion of new elements during material deposition can pose significant
implementation challenges, and (3) setting up an AM simulation model using commercial FEA systems
can be tedious.
In this thesis, all three challenges are addressed through a proposed assembly-free finite element
framework.
This framework presents several advantages over current strategies: (1) the workspace is meshed only
once at the start of the simulation, (2) the global matrix is never assembled, making the addition and
deletion of elements trivial to implement, (3) the memory requirement is significantly reduced, thereby
accelerating computation, and (4) multi-physics and non-linear challenges (radiation, phase change, etc.)
are effectively incorporated into the system of equations without incurring a high computational burden.
The accuracy and effectiveness of the framework is demonstrated using benchmark studies. The
framework is then exploited to simulate the thermo-mechanical behaviour of the Directed Energy
Deposition Additive Manufacturing process.
In conclusion, the objective of this thesis is to provide an efficient computation tool to predict and capture
some aspects of an otherwise very complex thermo-mechanical manufacturing process.
4. iv
Acknowledgements
I am deeply grateful for the opportunity that has been given to me by my advisor Prof Krishnan Suresh.
He has helped me quickly adapt to the current research techniques and methods. I have been able to
effectively apply my skills, while learning novel finite element techniques. He has always given me the
confidence to explore my talents, and given me a lot of freedom and comfort during the course of my
graduate study. I always thought of 3D printing as a fascinating topic of study which I could only look at
from a distance, but thanks to him and the knowledge that I have gained with my research, I feel like an
integral part of the Additive Manufacturing community. I would also like to thank my committee
members, Prof Greg Nellis, and Prof Natalie Rudolph. I have been able to effectively apply their
teachings of the Heat Transfer and Additive Manufacturing courses in my research, and I could not think
of anyone better to be a part of my committee.
I extend many thanks to the ERSL members, Shiguang, Amir, Praveen, Chaman, Tej, Aditya, Anirban
and Alireza. They have been very helpful and were always available for my plethora of doubts and
questions. They have all made me a better programmer and a better mechanical engineer.
I thank my parents and my friend Sakshi Gupta for their constant support and for instilling a lot of
confidence in me. They always pray for my success and I would be nowhere without them.
Last but not the least; I would like to thank my close friends and music group SAAZ, for really improving
my stay here at Madison. They have been my shoulder, and have always provided me with an amazing
break from my graduate studies, making sure that I always have a smile on my face.
5. v
Table of Contents
Abstract................................................................................................................... iii
Acknowledgements................................................................................................. iv
List of Figures....................................................................................................... viii
List of Tables ............................................................................................................x
1 Overview.............................................................................................................1
1.1 An Introduction to Additive Manufacturing .................................................................................1
1.2 Directed Energy Deposition AM Process – LENS .......................................................................3
1.3 The Need for an Efficient Thermo-Mechanical Simulation..........................................................5
1.4 Thesis Overview ...........................................................................................................................6
2 FINITE ELEMENT SIMULATION AND ASPECTS TO CONSIDER .....7
2.1 Mathematical Model .....................................................................................................................7
2.1.1 Boundary Conditions – Description of the Laser Heat Source.............................................7
2.1.2 Boundary Conditions – Surface Losses ................................................................................9
2.2 Time Stepping.............................................................................................................................11
2.3 Melting and Solidification – Phase Change................................................................................12
2.4 Stress Computation.....................................................................................................................13
2.5 Materials Used ............................................................................................................................13
2.6 Inherent Challenges ....................................................................................................................14
2.6.1 Nonlinearities......................................................................................................................14
6. vi
2.6.2 Time stepping and Meshing – Material Deposition............................................................14
3 PROPOSED METHOD – ASSEMBLY FREE FRAMEWORK ...............16
3.1 Voxelization................................................................................................................................16
3.2 Assembly Free Analysis..............................................................................................................17
3.3 Melting and Solidification using Enthalpy Method....................................................................19
4 VERIFICATION STUDIES...........................................................................22
4.1 Static Analysis: Block Geometry: Effect of Diagonalization .....................................................22
4.2 Static Analysis: Curved Geometry: Effect of Voxel Mesh (With Nonlinearity) ........................23
4.3 Transient Analysis: Block Geometry: Phase Change .................................................................25
4.4 Transient Analysis: Block Geometry: Phase Change .................................................................26
4.5 Thermo-Elastic Simulation .........................................................................................................28
5 MODEL AND PROCESS SETUP.................................................................32
5.1 Work Material.............................................................................................................................32
5.2 Laser Properties ..........................................................................................................................32
5.3 Tool Path Generation for a given geometry................................................................................33
5.4 Material deposition in the LENS process....................................................................................34
6 CASE STUDIES – WALL WITH HOLES Build.........................................36
6.1 Model and process parameters....................................................................................................36
6.2 Thermal problem.........................................................................................................................37
6.3 Study and Verification of Mechanical Behavior.........................................................................38
6.3.1 Wall Deformation during build...........................................................................................39
7. vii
6.3.2 An alternative design to reduce Warping............................................................................40
7 CONCLUSION ................................................................................................44
7.1 Summary.....................................................................................................................................44
7.2 Future Work................................................................................................................................45
8 REFERENCES ................................................................................................46
8. viii
List of Figures
Figure 1: Additive manufacturing of this part leads to minimal wastage .....................................................2
Figure 2: Market Share of AM......................................................................................................................3
Figure 3: LENS Process by Optomec ...........................................................................................................4
Figure 4: Schematic representation of the substrate with the first layer of metal deposition .......................5
Figure 5: Double ellipsoid heat source model ..............................................................................................8
Figure 6: Uniform Voxelization of Geometry from Figure 1. ....................................................................16
Figure 7: Inactive Deposited Elements.......................................................................................................17
Figure 8: Assembly Free Framework..........................................................................................................21
Figure 9: Simple Block Problem.................................................................................................................22
Figure 10: (A) Curved Surface Geometry (B) Voxel Mesh........................................................................23
Figure 11: Thermal Distribution using; (A) Proposed Method and (B) ANSYS........................................24
Figure 12: Percent Deviation versus Number of Elements.........................................................................24
Figure 13: Computing Time versus Number of Elements. .........................................................................25
Figure 14: Variation of Temperature over Time (A) Proposed Method (B) ANSYS.................................26
Figure 15: (A) Boundary Conditions (B) Temperature Distribution. (Range: 1500K to 2326K)...............27
Figure 16: Temp at Center of Heated Side (A) Proposed Method (B) Comsol Multiphysics. ...................27
Figure 17: Thermo-Elastic Simulation........................................................................................................28
Figure 18: (A) Undeformed Model (B) Deformation/Extension (C) Stress Distribution...........................28
Figure 19: (A) Maximum Displacement (B) Stress Comparison between ANSYS and the proposed
method. .......................................................................................................................................................30
Figure 20: L-bracket build (Domain in Light blue) at (a) First Layer (b) Second Layer (c) Completed
Build............................................................................................................................................................33
Figure 21: Rocker build (Domain in Light blue) at 2 Instances .................................................................34
Figure 22: Sample Model build (a) Full Body (b) First Layer start (c) First Layer - Hole.........................35
9. ix
Figure 23: Wall with holes model...............................................................................................................36
Figure 24: Heat source Distribution during build .......................................................................................38
Figure 25: a) Deformation plot b) Stress plot .............................................................................................40
Figure 26: Wall with holes with Support Structures...................................................................................41
Figure 27: Mechanical Deformation plot after completion.........................................................................41
Figure 28: Mechanical Stresses plot after completion ................................................................................42
Figure 29: Mechanical Deformation Comparison.......................................................................................42
11. 1
1 OVERVIEW
1.1 An Introduction to Additive Manufacturing
Additive Manufacturing (AM) or 3D printing represents a class of manufacturing techniques for
fabricating components layer by layer, directly from a digital part description. The digital description is
usually a 3D CAD model which is used to create slices of the part that drives the AM system [1]. This is
in contrast to the conventional subtractive, joining and formative processes such as milling, drilling,
casting, forging, welding, fastening etc.
AM has been garnering significant attention lately due its inherent advantages over conventional
manufacturing processes [2]. One of the biggest advantage is the ability and ease of manufacturing
components with high geometric complexity. Additionally, most AM techniques drastically reduce the
time taken between the initial design and the final produced part, when compared to the conventional
manufacturing techniques. The number of process steps is reduced and manufacturing is carried out as a
single step compared to the multi-step nature of other conventional techniques.
AM processes can be completely automated, and hence there is no need for skilled labor and special
tooling equipment as in the case of traditional methods. This promotes iteratively redesign and
accelerated time to market. Finally material wastage is almost negligible compared to the subtractive
processes. For example, consider the part in Figure 1; while one can certainly make this part through
conventional subtractive processes, it will lead to significant wastage. AM, on the other hand, will
minimize wastage.
12. 2
There are few disadvantages with additive manufacturing as well. One of the major drawbacks is the low
production rates of AM; thus, conventional manufacturing has the edge for large production volumes of
simple geometries. The analysis and simulation of AM techniques is very complex as the material
transformations and scan rates are very fast and occur in the range of milliseconds. Finally in a few AM
methods, the printability analysis (ability to print thin regions of a complex body) is very cumbersome.
AM is used today to fabricate automotive parts such as fuel nozzles, architectural models, medical
implants, jewelry, ball bearing and gear assemblies, aerospace components etc. By 2020, General Electric
plans to produce over 100,000 parts using AM [3]–[6]. AM is most widely used as a Rapid Prototyping
tool as it produces the final product within a small time frame. It is also used for repairing expensive
equipment like Turbine Blades and Injection Mold tool inserts. The 3D printed component dimensions
can range from a few millimeters to a few meters. There is usually a 3 attribute reference frame to
categorize the manufacturability of all products – Production Volume, Complexity and Customization.
With increase in complexity and customization and decrease in production volume, AM is a better choice
than conventional techniques. It is also used in tandem with topology optimization design models as they
are usually convoluted and complex in nature [7], [8]. The percentage of market share of additive
manufacturing over the years is shown in the figure below.
Figure 1: Additive manufacturing of this part leads to minimal wastage
13. 3
1.2 Directed Energy Deposition AM Process – LENS
There are many types of AM processes [9], based on the choice of stock material, and bonding method.
The stock material can be in the form of powder bed, powder jet, wire feed, or material extrusion, while
the material bonding can be carried out by applying a liquid bonding agent, direct solidification, or by the
application of thermal energy in the form of lasers or electron beams. The most common classification is
given below. Each of these methods have their own process parameters like stock materials, build
volume, process speed, process analysis, part quality and performance.
Binder Jetting: Liquid bonding agent selectively deposited to join powder materials.
Direct Energy Deposition: Focused thermal energy used to fuse materials by melting.
Material Extrusion: Material selectively dispensed through nozzle or orifice.
Material Jetting: Droplets of build material are selectively deposited.
Powder Bed Fusion: Thermal energy selectively fuses region of powder bed.
Sheet Lamination: Sheets of material bonded to form an object.
Vat Photo-polymerization: Liquid photopolymer in a vat selectively cured by light activated
polymerization
Figure 2: Market Share of AM
14. 4
In this thesis, we focus on the laser engineering net shaping (LENS) method, a Directed Energy
Deposition process, which uses powder jet nozzles to feed the metallic stock material, and lasers to
induce melting and bonding. This metallic AM process was developed at the Sandia National
Laboratories, and is currently commercialized through Optomec Design Company. The most commonly
used materials in LENS include titanium alloys (Ti6Al4V), stainless steel alloys, aluminum, copper and
nickel alloys. The laser used in this process is usually Nd-Yag [10].
In the LENS process, one starts with a substrate. A high-powered laser beam is then used to create a
molten pool into which the metallic powder is simultaneously fed. The beam is moved along with the
powder feed jet to trace out the first slice of the layer of the part, as shown in the Figure below. The
process is repeated layer by layer, until the part is completed.
Figure 3: LENS Process by Optomec
15. 5
1.3 The Need for an Efficient Thermo-Mechanical Simulation
The transient thermal nature of an AM process largely determines the solidification rate, void formation
and residual stresses. Hence, there is an increasing need to simulate and understand the thermal behavior
of the process as it governs the final quality, functionality, performance and the microstructure
characteristics of the 3D produced part [11], [12]. A good understanding of the temperature and stress
distribution, and its dependency on the process parameters (feed-rate, laser energy, and thermal boundary
conditions), is therefore of paramount importance.
Over the years, several attempts have been made to improve the methods of DED simulation. The most
common method is transient finite element analysis. Additionally, an interesting technique to simulate the
DED build is by modeling the system similar to weld heat source model. However, the weld modeling
studies cannot be directly applied here, due to complications and nonlinearities that arise during the
course of the build. In other words, the „filler‟ material, makes up the majority of the part, compared to
the substrate, and hence many assumptions from the weld modeling studies cannot be ignored [13].
Due to complexities of such a process, it takes hours, or maybe days to simulate even a small build.
Results in [17] show that just a temperature history study for a small wall of dimensions 3cm * 1cm *
0.3cm can take upto 7000-8000 seconds with no parallel computing. There is also currently an absence of
any commercial interfaces to simulate such a process. Hence an efficient transient FEA tool is necessary
Figure 4: Schematic representation of the substrate with the first layer of metal deposition
16. 6
to comprehensively understand the thermal history as well as the thermal stresses and quickly simulate
the entire build. Additional challenges are discussed in the next chapter.
1.4 Thesis Overview
In this thesis, an Assembly-Free finite element framework is proposed as an efficient tool to simulate the
thermo-mechanical behavior of the LENS process. In order to better understand the framework, the next
chapter deals with various aspects one needs to consider in such a simulation. It covers the mathematical
model, corresponding equations and the inherent challenges to consider in this process. Chapter 3 then
describes an assembly free implementation of these different aspects, and the novel characteristics of this
thesis. Chapter 4 shows the accuracy of this framework using a variety of benchmark problems as well as
its application in nonlinear thermal systems, when compared to commercial applications. Chapter 5 and 6
deal with the setup of a generic DED AM problem and the assumptions, (substrate, tool path, laser
parameters) followed by a case study to obtain a TTSP (Time temperature stress position) report, during a
LENS process build.
17. 7
2 FINITE ELEMENT SIMULATION AND ASPECTS TO CONSIDER
2.1 Mathematical Model
In order to perform the simulation, the thermal aspects of the AM process can largely be modeled via the
energy equation [1], [14]:
( ) ( )( )i i
i i p i i
u h u Hh k E H
Q
t x x x t xc (1)
where
: sensible heat ( )
: thermal conductivity
: specific heat
: internal energy source
: latent heat content
: density
: velocity of conserved mass
: x,y,z coordinates
p
p
i
E c T
k
c
Q
H
u
x
(2)
From the energy equation, one can extract just the transient thermal finite element governing equation, by
ignoring (for the purpose of this thesis) fluid flow [15], [16]. The latent heat term and melting and
solidification is discussed later in this chapter. Hence, Equation (1) simplifies to:
( )i p
i i
T dT
k Q c
x x dt
(3)
Once the temperature field is determined, the elasticity problem can be solved, via the thermal-expansion
coupling, as discussed later in the thesis.
2.1.1 Boundary Conditions – Description of the Laser Heat Source
Modeling the thermal history of the AM process is very similar to modeling a multi pass welding process.
Hence a double ellipsoidal heat source, similar to a weld laser model, can used to describe the heat source
18. 8
[11], [17]. In other words, the front half of the source is the quadrant of one ellipsoidal source and the rear
half is the quadrant of the second ellipsoidal source. The advantage of using a double ellipsoidal is that
front gradient can be made steep, and the rear gradient can be made gentler to exactly replicate a moving
laser heat distribution and the experimental analysis of the laser source. This cannot be achieved with a
simple ellipsoidal heat source model.
The power density distribution is given by:
22 2
2 2 2
3( )3 3
[ ]
6 3
c [c ,c ]
w
z v tx y
a b c
r f
fe
abc
Q (4)
Where:
: laser power(W)
: process efficiency
: scaling factor
: transverse depth(m)
: melt pool depth(m)
: longitudinal rear ellipsoid axis(m)
: longitudinal front ellipsoid axis(m)
: time(s)
: heat source veloci
r
f
w
P
f
a
b
c
c
t
v ty(m/s)
(5)
A schematic representation of the double ellipsoidal model is shown below:
Figure 5: Double ellipsoid heat source model
19. 9
The common lasers used in Nd-Yag lasers and fiber lasers. Optomec machines use the Nd-Yag lasers, but
recent machines use the fiber delivered lasers to create a focused beam [10]. The velocities of heat source
for laser based additive processes can vary from a 1mm/s to 200mm/s usually.
2.1.2 Boundary Conditions – Surface Losses
Surface losses due to convection and radiation play an important role. Initially, for the DED process, the
conduction dominates the heat transfer during the first few layers, and as the surface area of the 3D
printed part increases, surface losses play a larger role [13]. These surface losses are ignored by a few
models due to computational burden and the nonlinearity of the radiation boundary condition. But
ignoring these conditions lead to inaccurate predictions of the thermal history, especially as the part size
increases.
The modeling of convection is straightforward, as it is mostly linear. It is defined by
0
( )c
q h T T (6)
„h‟ being the heat transfer coefficient.
From a finite element implementation perspective, the „T‟ term is taken to the left hand side and
combined with the classic default stiffness matrix before solving for the temperature. The computation of
H matrix and the solving of the effective problem add to the computational effort. „h‟ is a convective
medium property and can be nonlinear if the convective medium is also changing temperature.
Radiation on the other hand is not so easy to implement. It is defined by
4 4
0
( )r
q T T (7)
„ε‟ is the emissivity (0 – 1) and „σ‟ is the Stefan Boltzmann Constant(5.67-8
W/m2
K4
). Here we assume the
view factor is accounted for in the emissivity. This is a nonlinear problem, and the nonlinear term shown
in this equation constantly affecting the stiffness matrix. Hence it cannot be solved like the convection,
and the solution has to be obtained iteratively.
20. 10
Boundary conditions hence include a combination of Dirichlet (fixed temperature), heat source,
convective, and radiative conditions. A detailed description can be found in [13]. Through the classic
finite element based Galerkin formulation [18], the governing equation and boundary conditions can be
collapsed into:
4 4
0 0
.
0
( ) ( )
j ji
p i j ij j i
i j
i
dT NN
c N N d k T d N Qd
dt x x
N h T T T T d
(8)
Where
2
8 2 4
: classic finite element shape functions
Q : laser heat source
: heat transfer coefficient(W/m K)
: emissivity [0,1]
: stefan boltzmann constant(5.67*10 W/m K )
N
h
If the non-linear radiative boundary condition is expressed as:
4 4 2 2
0 0 0 0
( ) ( )( )( )T T T T T T T T (9)
Then, the finite element discretization, Equation (8), together with Equation(9), can thus be reduced to:
[ ]{ } [ ]{ } { }C T K T F (10)
where:
[ ] [ ] [ ]T
p
C c N N d (11)
2 2
0 0
[ ] [ ] [ ][ ] [ ] [ ] ( )( )[ ] [ ]T T T
K B D B d h N N d T T T T N N d (12)
2 2
0 0 0 0
{ } [ ] [ ] [ ] ( )( ) [ ]
q
a
F N Qd q N d hT N d T T T T T N d (13)
[ ] [N]B (14)
21. 11
2.2 Time Stepping
To solve Equation(10), we rely on the Newmark Beta method for time-stepping [18], where:
1 1
{ } { } (1 )n n n n
T T t T T (15)
By multiplying Equation (10) with (1-β) at time step „n‟ and β at time step „n+1‟, and then, adding the
two will result in:
1 1 1
1 1
[ ] [ ] [ ] (1 )[ ] (1 ) ( )n n n n n n
C K T C K T F F
t t
(16)
Here the heat capacity matrix [C] is independent of time and temperature. The above equation can be
expressed in the form:
1 1
[ ]eff n eff n
K T F (17)
where:
1
[ ] [ ] [ ]eff
K C K
t
(18)
11
1
[ ] (1 )[ ] (1 ) ( )eff n n nn
F C K T F F
t
(19)
The stiffness matrix is temperature dependent due to the radiative boundary conditions, and the
dependency of the material properties on the temperature. Thus, at every time step one must solve a non-
linear set of equations. To ensure unconditional stability we use 1 here. In other words:
1
[ ] [ ] [ ]eff
K C K
t
(20)
11
1
[ ]eff n nn
F C T F
t
(21)
22. 12
2.3 Melting and Solidification – Phase Change
The phase change that occurs in the LENS process plays a critical role in actual formation of the material
and layer. The solidification rate affects the final thickness of the body, rate of formation, final precision
and residual stresses. Hence, an accurate calculation of the temperature is required, which can be obtained
from a post processing calculation to the finite element formulation.
The melting of Ti6Al4V is non-isothermal and it occurs over a temperature range of 55K. Consequently,
there are 3 phases that are seen in the process namely solid, liquid and melt (solid and liquid mixed). The
specific heat variation of the material during phase change can be given as follows [15], [16]:
(1 ) [ , ]
S MS
S L MS ML
L ML
C T T
C C C T T T
C T T
(22)
where:
: Specific Heat at Solid
C : Specific Heat at Liquid
: ( )
: Solidus Temperature
: Liquidus Temperature
S
L
ML
ML MS
MS
ML
C
T T
T T
T
T
(23)
Since the material is highly viscous, and solidification is assumed to occur as soon as the laser heat source
moves out of the area of melting, the fluid flow that occurs during the melting can be ignored [14]. This
drastically simplifies the computation of temperatures during the melting and solidification. A predictor-
corrector method is proposed in the next chapter, which can be repeated after the FEA calculation at every
time step.
23. 13
2.4 Stress Computation
Once the temperature profile is obtained at each time step, the thermal stresses need to be computed. This
is carried out by coupling the thermal problem with elasticity. A thermal „force‟ vector is computed as
follows:
[ ] [ ] { }T
thermal
F dJ B D d (24)
where:
: Derivative of Jacobian
: From Eqn 2.14
: Element property matrix
: Coefficient of Thermal expansion (m/mK)
: Dummy vector [1 1 1 0 0 0]
dJ
B
D (25)
The thermal „force‟ is then added to any other elastic forces or boundary conditions in the system, and
solved as a simple static elastic problem as shown below:
[ ]elastic elastic thermal
K X F F (26)
The resultant {X} vector gives us x, y, z deformations of the system which arises as a result of the
previously defined thermal boundary conditions. The thermal stresses are subsequently calculated using
these deformations [18].
2.5 Materials Used
The LENS process uses alloys of titanium, nickel (IN 625, 600, 718, 690), stainless steel (316, 304L, 309,
17-4), cobalt, aluminum, copper, some refractory metals, tool steel (H13) and some composites and
specialty materials. From the literature, one of popular materials used is the titanium based alloy Ti6Al4V
[13]. However the process has now grown to include all kinds of materials.
24. 14
2.6 Inherent Challenges
2.6.1 Nonlinearities
As it has been mentioned, the model is complicated due to the high nonlinearity of the problem. It is seen
that the thermal conductivity of the material is not a constant value and varies drastically. For example,
the conductivity of Ti6Al4V varies from 6.6 W/m/ at 20 to 17.5 W/m/ at 870 . Similarly its
specific heat varies from 565 J/kg/ to 959 J/kg/ for the same temperature range. The values are
assumed constant above 870 degrees. Additionally the surface losses due to radiation are also nonlinear
which cannot be ignored and complicates the analysis further.
The nonlinearity of the thermal conductivity problem in powder bed fusion technologies is solved by
using an effective thermal conductivity using the Yagui and Kunni function. This takes into account the
effect of the gases present in the powder bed and surroundings. However the gas content is minimal here
in the powder feed. An effective thermal conductivity reduces the computational cost, but it doesn‟t very
accurately predict the thermal history. Otherwise, the Ti6Al4V conductivity values are linearly
interpolated to obtain the conductivity at different temperatures at different positions and times.
2.6.2 Time stepping and Meshing – Material Deposition
The temperature changes in the LENS process are large and very rapid (for example, the temperature can
increase from 20°
C to 1670°
C in milliseconds [13]) with large temperature gradients. This entails small
time steps, and a fine resolution of the geometry. Adaptive mesh techniques are often used [17], with fine
discretization near the heat source and on the top few layers. However, as the laser traverses, and as the
material is added to the top layer, constant re-meshing is required.
To account for material deposition, two methods, namely, quiet-element and inactive-element methods
are currently used [17]. In the quiet approach, all finite elements within the workspace are assembled into
the global stiffness matrix, but the elements yet to be deposited are assigned „void‟ material properties.
They are later assigned appropriate material properties, as and when needed. This method uses a constant
25. 15
mesh, and is simple to implement, but the incorrect selection of scaling factors can lead to ill-conditioning
of the Jacobian. The inactive method on the other hand, uses an evolving mesh, and includes only the
active elements for simulation. This leads to high simulation speeds, especially during the initial stages.
However, constant re-meshing and re-formulation of the finite element equation can be time consuming
and difficult to implement. A hybrid technique has also been used in [17].
To overcome these challenges, we propose a voxel-based, assembly-free method which does not require
the matrices to be assembled. This method can be used to deal with the various nonlinearities of the
system as well as the small time steps. It can also effectively model the material deposition and has
several advantages over the quiet and inactive methods.
26. 16
3 PROPOSED METHOD – ASSEMBLY FREE FRAMEWORK
3.1 Voxelization
Voxelization is a special form of finite element discretization where all elements are identical (hexahedral
elements). The most important advantage of voxelization is its robustness; voxelization rarely fails unlike
classic meshing. In addition, voxelization significantly reduces memory foot-print since the element
stiffness matrices are all identical; this directly translates into increased speed of analysis.
Voxels have been used in additive manufacturing, for example, to find effective mechanical properties
[19]. In this thesis, voxels are used as a computational unit. In the present approach, the entire workspace
is discretized once, at the beginning of the simulation (see Figure 7). As with the inactive element
method, the elements to be deposited are inactive. However, unlike the inactive approach, the global
stiffness matrix is never assembled. The challenges associated with „void‟ material assignment are
overcome through assembly-free analysis.
Figure 6: Uniform Voxelization of Geometry from Figure 1.
27. 17
3.2 Assembly Free Analysis
Assembly-free finite element analysis was proposed by Hughes and others in 1983 [20]. In recent years, it
has resurfaced due to the surge in fine-grain parallelization. The basic concept used in the analysis is that
the stiffness matrix is never assembled; instead, matrix operations are performed in an assembly-free
elemental level [21], [22]. For example, the typical Sparse Matrix Vector Multiplication (SpMV) is
typically implemented by first assembling the element stiffness matrices as follows:
{ } [ ] { }e
e
Kx K x (27)
In an assembly free method, this is implemented by first carrying out the multiplications at the element
level, and then assembling the results:
{ } [ ]e e
e
Kx K x (28)
Assembly free analysis is not advantageous when elements are distinct from each other (as in a classic
finite element mesh). However, with voxelization, since all elements are identical, only one elemental 8*8
stiffness matrix needs to be stored (for the thermal problem), and Equation (28) can be executed rapidly,
with reduced memory footprint.
Figure 7: Inactive Deposited Elements.
28. 18
In AM simulation, as elements are deposited, introducing them into the computation is also trivial. An
additional element (with identical stiffness matrix) needs to be inserted while evaluating Equation(28).
Similarly, the boundary conditions can be updated.
The SpMV in Equation (28) serves as the backbone of the classic Conjugate Gradient (CG) solver [21],
[22]. For the thermal problem, preconditioners are not needed since the stiffness matrix is typically well
conditioned.
In the current context of SpMV, the matrix is the effective stiffness matrix given by Equation(20). It
consists of the C matrix (Equation(11)) and the stiffness matrix term K (Equation(12)). The C matrix can
easily be treated in an assembly-free manner using a single copy of an element damping matrix. The K
matrix can be broken up into the following 3 matrices:
2 2
0 0
[ ] [ ] [ ] [ ]
where:
[ ] [ ] [ ][ ] (conduction)
[ ] [ ] [ ] (convection)
[ ] ( )( )[ ] [ ] (radiation)
D H R
T
D
T
H
T
R
K K K K
K B D B d
K h N N d
K T T T T N N d
(29)
The first term can be treated in an assembly-free manner using a single copy of an element stiffness
matrix. However, for the convection and radiation matrices in Equation (29) since the integral is over the
boundary, up to six different element stiffness matrices may have to be stored (to account for six different
faces of a voxel). To accelerate computations, these two matrices are diagonalized here; this is analogous
to the diagonalization (lumping) of the mass matrices [18]. The effective force vector given by Equation
(21) is also easy to evaluate in an assembly free manner.
Now consider the solution of the linear equation in Equation(17) . This equation must be solved at each
time-step. Since this is a non-linear equation, we rely on the iterative Newton Raphson method [23]. The
29. 19
0
1n
T is obtained from the CG solver, and this is used to calculate the 0
eff
F . The residual for the Newton
Raphson process is then defined as:
1 1 1
[ ]i i i i
n n eff n
R K T F (30)
The superscripts denote the iteration number of the Newton Raphson process. The subscripts denote the
current time step for which the temperature is required. One can now show that the tangent matrix used in
the Newton Raphson process is given by:
1
{ }
[ ] [ ] [ ] 4[ ]{{ } 3}
{ }
where
[ ] [ ] [ ]
i
i i
D H R n
T
R
R
J K K G T
T
G N N d
(31)
Using the methodology described above, the Jacobian can be computed in an assembly free manner,
followed by an assembly-free update of the temperature:
11
1 1
[ ] [ ]
i i
n n
T J RT (32)
3.3 Melting and Solidification using Enthalpy Method
The phase change problem is solved by studying the enthalpies of the material at each point of the body at
each phase. The enthalpies of the material at solid, liquid, melt phases and at the start and end of melting
are given below [15], [16]:
( )
( )
( ) ( )
2
( )
( )
( )
( )
( )
MS
M S MS
ML MS
L S MS S L
SOLID S
T
MS
MELT M MELT
ML MS T
LIQUID L L ML
H C T
T T
H C T L C C
H C T
L T T
H H C dT
T T
H H C T T
(33)
Where:
30. 20
: Enthalpy at T in J/kg
: Enthalpy at T in J/kg
: Latent Heat in J/K
: Expression for Specific Heat at T [ , ] in J/kgK
M MS
L ML
MELT MS ML
H
H
L
C T T
(34)
It should be noted that the HL and the HMELT equations contain both sensible heating and the latent heating
terms, due to the melting being non-isothermal in nature. At the time of initialization (time step n = 1), the
initial phase and the temperatures are known which can be used to calculate the enthalpies (H1) at all
nodes. Then the finite element equation is solved to obtain the new temperature (T*). However, the finite
element formulation assumes no phase change, treating the material at every point as solid, irrespective of
the temperature. In other words, „T*‟ would have been the temperature if the body remained in solid
phase. Using the temperatures (T*) which are obtained from the finite element solution, we can find the
new enthalpy (H2), using the following equation:
*
2 1 1
( )S
H H C T T (35)
The enthalpies obtained in the above equation (H2) tell us the exact amount of energy present in the body.
Using these new enthalpies (H2), we can correct the temperatures by substituting them back in equation
(33), and solving the equations to obtain T2. This predictor-corrector method can be repeated after the
FEA calculation at every time step.
All these above proposed techniques efficiently come together to create an effective Thermo-Mechanical
simulation tool in an Assembly Free Fashion.
31. 21
This can then be combined with the elastic FEA solver to compute the deformations and stresses. The
structural elastic FEA module used in this framework has been designed in a Deflated Conjugate Gradient
Assembly Free fashion [21], [22].. This makes the solver much faster than conventional methods,
especially for large structural problems. The advantages and accuracy of this method is well documented
in [21], [22]. Once the accuracy and benefits of these techniques are established in the next chapter, we
can then use this Assembly Free Framework to simulate the build of a sample LENS process, as well as
create a suitable interface for a quick setup of other samples.
Figure 8: Assembly Free Framework.
32. 22
4 VERIFICATION STUDIES
The assembly-free, voxel-based method was tested and verified for a few steady state and transient
thermal examples as follows. All FEA simulations were performed with an Intel(R) Core™ i7-5960X
CPU processor with 16.00 GB RAM and a Windows 7 Operating System.
The results and speeds were then compared with ANSYS and SolidWorks on the same machine, with the
same convergence criteria. All ANSYS and SolidWorks models were pre-meshed and only the time taken
to solve the FEA problem was compared. Conjugate gradient iterative solvers were used in the present
study. The CPU time taken for solving is obtained from the ANSYS and SolidWorks report, and is
compared against the proposed method.
4.1 Static Analysis: Block Geometry: Effect of Diagonalization
The block in Figure 9 is of dimensions (6m X 3m X 11m), and is made of Titanium with conductivity
K=19.9W/mK, and boundary conditions are as shown: fixed temperature on one-side, heat flux on the
other, while all other faces are subject to surface losses through convection and radiation with h =
30W/m2
K and emissivity of ε = 0.6; the ambient temperature being T0= 293K.
Since the geometry is a simple block, there is no error due to meshing. For about 2000 elements, ANSYS
predicts a minimum temperature of 292.18K and a maximum temperature of 500K, SolidWorks predicts
292.13 and 500, and the proposed method predicts 292.16 and 500. A variety of such problems were
Figure 9: Simple Block Problem.
33. 23
solved and results compared. The proposed method yielded results within 0.02% of commercial
implementations, suggesting that the error due to diagonalization is insignificant, and may be neglected.
4.2 Static Analysis: Curved Geometry: Effect of Voxel Mesh (With Nonlinearity)
Next, we will consider errors due to voxelization (meshing), using the part illustrated in Figure 10 that
exhibits curved surfaces. The body is an assembly of a hollow cylinder of length 2.5m, outer radius of
0.75m, and inner radius of 0.4m, with a block of dimensions (1m*1m*1m). The material is Alloy Steel
with conductivity K=50W/mK; the boundary conditions are as illustrated: fixed temp of 300K on the left
face, a convection boundary condition with h = 100W/m2
K and a medium temperature of 423K on the
inner cylindrical surface, and a radiation boundary condition, with ε = 0.4 and ambient temperature of
273K everywhere else.
The part was discretized using 38000 voxel elements in our method, while a conforming tetrahedral mesh
with the same number of elements was used both in ANSYS and SolidWorks. The temperature
Figure 10: (A) Curved Surface Geometry (B) Voxel Mesh.
34. 24
predictions from our method were within 0.14% of commercial implementations; the temperature
distributions are illustrated in Figure 11.
Further, Figure 12 illustrates the percentage deviation of the maximum temperature in the proposed
framework, when compared with the maximum temperature from the ANSYS as a function of the number
of elements (ANSYS). This is the typical deviation we have observed for curved geometry. It can be seen
that the voxel mesh does not capture the geometry for a small number of elements, but the error decreases
upon increasing the mesh size.
A) Temp Range: 400.459K - 413.27K B) Temp Range: 400.46K – 413.09K
Figure 11: Thermal Distribution using; (A) Proposed Method and (B) ANSYS.
Figure 12: Percent Deviation versus Number of Elements.
35. 25
For the above problem, Figure 13 illustrates the computing times as a function of the number of elements.
Here it can be seen that as the number of elements increase, the speed of the solver (meshing time not
included) in the proposed method is much faster compared to both ANSYS and SolidWorks.
The objective of the above tests is to show the efficiency of the proposed method to solve these complex
transient problems. For a large mesh sizes, the solution is fairly accurate when compared to ANSYS and
SolidWorks, but the solution is completed much faster than the commercial implementations.
4.3 Transient Analysis: Block Geometry: Phase Change
In this experiment, we will carry out a transient analysis on the geometry used in the previous section.
The boundary conditions are as before, with the initial temperature of the body set to 298K. A mesh of
38,000 elements was used as before. The transient maximum temperature plots are illustrated in Figure
14.
Figure 13: Computing Time versus Number of Elements.
36. 26
The deviation of the proposed method from ANSYS was less than 0.133 percent, while the time taken by
the proposed method was 10s compared to 400s by ANSYS.
4.4 Transient Analysis: Block Geometry: Phase Change
We next verify the phase change formulation using the block geometry part shown in Figure 15. The
material is Ti6Al4V, with specific heats 562 J/kgK (solid) and 572 J/kgK (liquid), latent heat of fusion
365000 J/kg, solidus temperature 1878 K, and liquidus temperature 1933 K. The thermal conductivity is
assumed to be constant at 7.2 W/mK for the purpose of this analysis. The boundary conditions are as
illustrated, fixed temperature of 1500K on the left side and a heat flux of 10000 W/m2
on the right side.
The simulation is carried out with a time step size of 5000 seconds for 60 time steps, with an initial
temperature of 298K. The boundary conditions and the final resulting contour are shown below.
Figure 14: Variation of Temperature over Time (A) Proposed Method (B) ANSYS.
37. 27
The results were compared against Comsol Multiphysics for the same geometry at a particular point,
chosen at the center of the flux surface. The plot of temperature versus time is shown below. The solid
and liquid phases are in good agreement within a 0.1% error; however a very slight instability can be seen
in the melt phases due to choices in time stepping.
Figure 15: (A) Boundary Conditions (B) Temperature Distribution. (Range: 1500K to 2326K)
Figure 16: Temp at Center of Heated Side (A) Proposed Method (B) Comsol Multiphysics.
38. 28
4.5 Thermo-Elastic Simulation
Next, we consider the thermal and elastic FEA solver coupling; using the part illustrated in Figure 17 that
also exhibits curved surfaces. The height of the body is 0.53m (length of the longer components), and the
thickness is 0.1m. The lengths of the shorter componets are 0.31m. The material is Alloy Steel with
conductivity K=60.5W/mK, Young‟s modulus of 2E11 Pa, and coefficient of thermal expansion of 1.2e-
5/K; the boundary conditions are as illustrated: fixed temp of 298K on the left extreme corner, and a heat
flux of 20000 W/m2
on the right extreme corner as shown.
As expected the thermal boundary conditions result in an outward expansion of the body as shown. The
resulting displacement and stress plots are shown below.
Figure 17: Thermo-Elastic Simulation.
Figure 18: (A) Undeformed Model (B) Deformation/Extension (C) Stress Distribution.
39. 29
The results were compared with ANSYS in a transient setting with 25 time steps of 1000 seconds each.
The resulting max deformations and stresses were plotted at each time-step and the plots were overlapped
to show the difference. The maximum displacement is 0.8721mm and the maximum stress obtained was
4.9364e6 Pa. The displacements growths are almost overlapping which show the accuracy of the
proposed method compared to the commercial implementations. The stresses are however not that
similar. A close look at the resultant stresses show that the pattern is the same, but all the values are lower
in the proposed method by a constant factor. This means that the proposed method is under estimating the
stresses. ANSYS is perhaps computing the stresses at gauss points, as opposed to the proposed method,
which computes the stresses at the center of element. An advantage with the proposed framework is that it
could be easily coupled as opposed to ANSYS which requires a manual coupling of the thermal and
structural analysis at every time step. Finally the structural analysis solver speed in the proposed method
is much faster than that of ANSYS, due to the deflation based CG solver that is used as mentioned before.
40. 30
As the experiments demonstrate, the method is sufficiently accurate (to within 0.15% of commercial
implementations while computing temperatures and displacements), but significantly faster (with a speed-
up ranging from 2 to 50). The output of the thermo-mechanical simulation is time, temperature,
displacement, and stress for each computational node/element within the model. The next 2 chapters will
Figure 19: (A) Maximum Displacement (B) Stress Comparison between ANSYS and the proposed
method.
42. 32
5 MODEL AND PROCESS SETUP
In this chapter, we setup the model for our AM simulation. Once it has been setup, we can perform a
thermo-mechanical analysis at each laser location (considered as a time step in the transient problem) The
4 key aspects to be considered to setup the model are
Materials Used
Laser Properties
Tool Path Generation for Given Geometry
Material Deposition
5.1 Work Material
As mentioned in the previous chapters the material used for the simulations will be Ti6Al4V. A
conductivity of 7.2 W/mK is assumed as the average thermal conductivity during the build for the
purpose of the simulations. The surrounding medium is air, at room temperature, with convection
coefficient h = 35W/m2
K and emissivity ε = 0.6. The surrounding and initial temperature is set to 298K.
For the purpose of these simulations, the effect of the build on the surrounding medium temperatures is
ignored. The geometry used is discussed in detail in the next section. It is built on a sample 10* 10 cm
substrate, made of Ti6Al4V also, and the bottom of the substrate block is maintained at 298K.
5.2 Laser Properties
The Nd-Yag laser distribution described in section 2.1.1 is used over here. The laser parameters are
chosen as follows [17]: a = 1.5mm, b = 3mm, c1 = 1 mm, c2 = 2mm. These parameters were chosen in the
literature to replicate the range upto which 5% of the heat from a 325W laser can be felt[17]. Since it is a
double ellipsoidal model, equation (4) has a scaling factor of 0.66 ahead of the laser spot, and 1.33 behind
the laser spot replicate the heat source distribution shown in(4). Finally the velocity used is 10mm/s, and
the analysis is carried out at every 0.1 time step. It can be seen that the velocity also affects the location of
43. 33
the laser heat distribution and the location of the peak heat value. In order to figure out which elements
are subjected to the heat source, at every laser location, a cuboidal bounding box is defined, where
“hidden elements” are deposited and subjected to heat. The bounding box ranges are +c1 and –c2 in the
direction of laser, +a and –a in the side, and 0 and –b in depth. Only the elements in the bounding box that
feel more than 5% of the peak heat are deposited.
5.3 Tool Path Generation for a given geometry
For the purpose of these simulations, a generic tool path generation tool was designed. The objective is
take a meshed body, and search for points or mesh elements and generate a simple tool path that in the +x
and –x directions as the y coordinates increase from minimum to maximum. Once all the points are
generated in one layer, the z coordinate is increased for the search. The laser speed is also taken into
consideration while generating points, which replicates the material deposition in a real life scenario. The
framework then takes the points, and generates a laser heat distribution at each of those points. Hence one
can see that this would work even if the tool path is supplied externally, in the form of a g-code. A simple
illustration of the tool path generation is shown for a sample L Bracket and a rocker mechanism.
Figure 20: L-bracket build (Domain in Light blue) at (a) First Layer (b) Second Layer (c)
Completed Build.
44. 34
Observe that the tool path jumps over holes and cavities during the build.
5.4 Material deposition in the LENS process.
The above tool path generation examples were shown to depict the way a body can be built given the
domain and the mesh. However, in the case of the LENS process, the substrate is already present and the
body above the substrate is built. In order to effectively depict that, we take the platform and the required
body, assemble them and mesh them together. Subsequently, we “hide” all the elements above the
substrate. This is carried out by setting a value to the pseudo density parameter (0, 1) [8], [24], [25] for
each element. Due to the assembly free framework, as mentioned in chapter 3, we only need to store an
8*8 elemental stiffness matrix for the thermal study and a 24*24 elemental stiffness matrix for the
thermo- mechanical study. Additionally we store the Thermal and Structural Force vectors, for each node,
and the pseudo densities. Upon solving, the conductivities are multiplied by these scalar quantities, and
they affect the solution only if they are present (pseudo density = 1). Naturally, one can expect the
solution time to increase during the build, the increase in time is not as much as traditional methods where
one would have to re-mesh repeatedly and store and access huge matrices at every laser location.
An example of a build on the substrate is shown in the Figure 22. The body shown here is 5.5 cm tall, and
has a 5cm X 7cm base.
Figure 21: Rocker build (Domain in Light blue) at 2 Instances
45. 35
It is important to note that the element size should be smaller than the distance between 2 adjacent points
in a laser path, in order to deposit more than 1 element at a time. For the purpose of the above tool path
generation, 400,000 elements were used. For the thermo-mechanical simulation, a much finer mesh will
be used, to get as close to the material powder size (50-100 microns) as possible.
Figure 22: Sample Model build (a) Full Body (b) First Layer start (c) First Layer - Hole
46. 36
6 CASE STUDIES – WALL WITH HOLES BUILD
6.1 Model and process parameters
A new model shown in Figure 23 used to perform the following case studies. The substrate is 3.81 cm X
0.762 cm square block with a thickness of 0.254cm. The wall is a 2.54cm *0.254cm base with a height of
0.89cm. The hole diameters are 0.2cm, 0.4cm, and 0.6cm.
The length of the wall makes the part similar in size to a small metallic industrial component, like a fuel
nozzle.
This model was chosen to study the temperature variations, and the effect of thermal stresses. Due to the
absence of any support structures, it would be interesting to observe the behavior of this model, during the
build. In order to reduce any numerical errors, a fine mesh of 100000 voxel elements for this small model
has been used for these simulations. This mesh gives an element size of 230 microns. As mentioned
before, the framework can be used to read any set of laser locations from arrays or files provided by a
user as well. However, the simulations here are carried out on top the tool path generation tool. Hence the
laser properties are changed at the simulation interface, as opposed to being an integral component of the
input file (g-code).
Figure 23: Wall with holes model
47. 37
6.2 Thermal problem
The body is subjected to heat from the laser source during the build as the powder is being deposited. In
order to consistently distribute the heat, only the elements that feel more than 5% of the peak heat are
deposited (brought back) at each time step. The laser velocity is 10mm/s. To make sure that we don‟t skip
elements during the simulation, a relatively small time-step of 0.1s is assumed. Therefore, for the body in
Figure 23, about 56 time steps occur in the first few layers. A layer height of 0.001 m is assumed. Since
the body is 0.0089m tall, one would require close to 9 layers for the complete simulation of such a
problem. Another issue is that the solver becomes slightly slower due to increasing numbers of elements
in each time step. However the decrease in speed in negligible compared to that of traditional techniques
where constant re-meshing would be required.
We first simplify the build, using just a Dirichlet boundary condition (fixed temperature) at the base of the
substrate. The base of the substrate is maintained at 298K (room temperature). The laser of power 325W,
with the distribution is given in the previous chapter. The metallic 3D printers today have the substrate
surface usually heated a little, and higher powered lasers, when the material deposition occurs. Since we
use Ti6Al4V, we should be seeing temperatures much higher than the real life build, due to the absence of
convection and radiation. For this case, we have also excluded the melting and solidification, so the
temperatures are calculated neglecting any phase change. It can be understood that the phase change
would also lower the peak temperature due to latent heat processes that occur during melting and
solidification. As a result, for this example, peak temperatures at a location are close to 2700 - 2800K
away from the corners and about 2900-3100K near the corners. These values are close to what one would
expect for AM using Ti6Al4V without any ambient conditions. The change in temperatures occurs as the
laser spends more time while turning at a corner, which leads to more power supplied for a small control
volume, as opposed to when the laser is moving straight. Suitably the laser power needs to be adjusted at
the corners, which are usually accounted for in the input file. However the tool path generation tool
assumes a constant laser power for now.
48. 38
The next step is to incorporate the melting and solidification and the ambient conditions. The enthalpy
method in chapter 3 is used to make a correction to the output temperature. The surrounding medium is
assumed to be air, with a heat transfer coefficient of 35W/mK and emissivity of 0.6. All the surfaces of
the wall are exposed to the air in the model. The temperature of air is assumed to be room temperature,
298K. The effects of the build on the air are neglected. Consequently the peak temperatures around 2000-
2200K is seen in the straight portions, and 2300 – 2500K is seen in the corners. Once again these
temperatures are just above the melting temperatures of 1933K which are when we want the laser to move
to the next point, to allow cooling and solidification. In a real life scenario, there would be better cooling
using a forced convection model to lower the temperatures. An instance of the temperature distribution is
shown in Figure 24. As expected the double ellipsoid model is shifts towards the right, as the laser moves
in that direction, and leaves a trail of heat in its path. The pure thermal simulation for this body took 12
minutes.
6.3 Study and Verification of Mechanical Behavior
Warpage and deformation that occurs due to thermal expansion during an AM build is one of the most
important reasons for the rejection of many components. Since it is an unavoidable material property,
design changes are made in order to reduce the effect of thermal expansion. Currently the use of support
structures, heat treatment after the manufacturing, an increase in the AM chamber temperature and a
correct choice of materials are the most popular ways to reduce thermal expansion and deformation.
Temperature Range: 298K to 2332K
Figure 24: Heat source Distribution during build
49. 39
Another method is to design the body slightly different, such that we would return to its desired shape
during the build. The effect of gravity is also ignored in this example.
6.3.1 Wall Deformation during build.
A mesh of 100000 elements was used to obtain the deformation and stresses at every point. This slows
down the framework drastically since there are more degrees of freedom to solve for at each step. The
objective of this experiment was to visualize the deformation tendencies during the build. It was seen at
the initial time-steps that the deformation was very minimal, due to well defined geometry on the
substrate. Once the model was being built above the holes, it was seen that the values drastically
increased. There was a wave shaped bend which can be seen, followed by an expansion above the hole in
thickness. This is consistent with what one would expect for such a problem. The deformed shape can be
seen in Figure 25. A plot of the stresses is also seen in Figure 25. An interesting thing to note is that the
stress values are beyond the yield stresses. This means that there are portions during the build that break
and deform, have cracks and they fuse back together when they are cooled. Some portions at the base of
the wall in the stress plot are red in color, which indicate that those could be zones of warping. This
simulation took 37 minutes.
50. 40
6.3.2 An alternative design to reduce Warping.
The deformed model in Figure 25 cannot be used as it was initially planned. The hole portions have been
heavily weakened, and the stresses should have been “conducted” away from the part. One way to do that
is using support structures, and we will use this idea, to verify the effect of thermal expansion on our
body. As seen in Figure 26, the model is modified to include two support structures as shown. The
thicknesses of the support structures were 0.25 and 0.08 cm. For the purpose of simplicity, support
structures of the same Ti6Al4V material were used.
Figure 25: a) Deformation plot b) Stress plot
51. 41
The simulation was carried out again, with the same parameters as before. It was seen that the support
structures do restrict the movement of the cantilever tip. In other words, the tip is made steady with a
physical connection to the base. Another important aspect to observe is that the temperature distribution is
also affected by the support structures. Due to the structures, there is now more area for convection and
radiation, and more volume for convection and radiation.
It was seen that the deformation in the body is more spread out and reduced by a one third. Another
observation is that the parts with the holes are well balanced and resistant to expansion as shown in Figure
27 .
Figure 26: Wall with holes with Support Structures
Figure 27: Mechanical Deformation plot after completion
52. 42
The stress plot is shown in Figure 28. It can be seen that the stresses are reduced near the hole, but the
corners at the base of the body are still susceptible to warpage.
As expected, the deformation is drastically reduced when support structures are present. A plot of the
comparison of the displacement history between the two parts is shown in Figure 29.
Figure 28: Mechanical Stresses plot after completion
Figure 29: Mechanical Deformation Comparison
53. 43
These results quantitatively validate the effectiveness and the ability of the proposed framework in
predicting both the thermal and the mechanical behavior of a variety of complex builds. It also predicts
those zones that are susceptible to crack propagation, yielding and warpage. This could thus help us with
reorienting the part before it is built, to prevent the aforementioned problems.
54. 44
7 CONCLUSION
7.1 Summary
This thesis was aimed at providing information and insight into some of the critical aspects of the LENS
AM process. A description of the LENS process, followed by the different aspects like the mathematical
model, and the corresponding equations were well documented. Subsequently, a four part framework was
proposed which uses the concept of Assembly Free FEA to deal with the above aspects. It was shown that
the assembly free framework is much faster and fairly accurate when compared to commercial
implementations like ANSYS and SolidWorks in solving non-linear, thermos-mechanical problems.
Finally the proposed framework was couple with the thermo-mechanical module, and its application in
the AM LENS model and setup was shown.
The simulation of this process is very important, as it holds the key information regarding material
strength, residual stresses, component usability and a comparison with traditional manufacturing methods.
While this framework is in no means a complete simulation of the process, it does efficiently and
accurately capture the transient thermal behavior, non-linearities, thermal expansion, melting and
solidification, material properties and their relationship with the AM process parameters. The effect of
warping and ambient conditions were also shown using the proposed framework.
All the data generated, namely the Temperature, Stresses, Displacements, and Laser locations at every
time step are stored in files, and can be used to generate a TTSP report as discussed in the earlier chapters.
However, these are just a few facets of an otherwise much larger and complex multi-physics or even
multi- science process. A look into the other aspects and future work and add-ons to this framework are
proposed in the next section.
55. 45
7.2 Future Work
As mentioned before, the proposed framework is in no way a complete analysis of the entire process.
Aspects like fluid flow [15], [16], void formulation, forced convection models [13], gravity, are all
important factors that can be added to the simulation. The computed data can then be reused in order to
optimize the laser velocities and power, to obtain the required body. An ultimate goal would be to use
Machine learning tools to predict the required process parameters for any given body and process.
Material science engineers, take the above data to study microstructure properties and crack propagation
characteristics when compared to traditional subtractive techniques [1]. Since additive manufacturing
makes multi-material design and manufacturing easier, it is even more critical to understand the
crystalline structures during and after the build.
Another important application of finite element analysis is for the optimization of tool path. Many AM
processes give rise to anisotropic parts. Once the part is made, it is simulated to study the strengths in
different directions. Based on the application, it could be beneficial to suggest different tool paths, to
improve the material properties in some directions. It also gives us a good method to calculate an
optimized thickness of the material deposition, and parameters like bead overlap and air space between
powder particles.
Finally the above framework could also be useful in the simulation of other AM processes like Powder
Bed Fusion (PBF). Since PBF is also a metallic process, a few changes in the setup would be required to
perform the simulation. Convection and Radiation don‟t play important roles in PBF due to the powder
bed arrangement; hence conduction dominates the heat transfer process. However in this case, elements
should replicate or be finer than the powder particles, in order to clearly differentiate between the build
part and the surrounding material.
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8 REFERENCES
[1] R. Martukanitz, P. Michaleris, T. Palmer, T. DebRoy, Z.-K. Liu, R. Otis, T. W. Heo, and L.-Q.
Chen, “Toward an integrated computational system for describing the additive manufacturing
process for metallic materials,” Additive Manufacturing, vol. 1–4, pp. 52–63, Oct. 2014.
[2] I. Gibson, D. W. Rosen, and B. Stucker, Additive Manufacturing Technologies. Springer, 2010.
[3] B. P. Conner, G. P. Manogharan, A. N. Martof, L. M. Rodomsky, C. M. Rodomsky, D. C. Jordan,
and J. W. Limperos, “Making sense of 3-D printing: Creating a map of additive manufacturing
products and services,” Additive Manufacturing, vol. 1–4, pp. 64–76, Oct. 2014.
[4] P. Witherell, S. Feng, T. W. Simpson, D. B. Saint John, P. Michaleris, Z.-K. Liu, L.-Q. Chen, and
R. Martukanitz, “Toward Metamodels for Composable and Reusable Additive Manufacturing
Process Models,” J. Manuf. Sci. Eng., vol. 136, no. 6, pp. 061025–061025, Oct. 2014.
[5] K. V. Wong and A. Hernandez, “A Review of Additive Manufacturing,” ISRN Mechanical
Engineering, vol. Article ID 208760, 2012.
[6] T. J. Horn and O. L. Harrysson, “Overview of current additive manufacturing technologies and
selected applications,” Sci. Prog., vol. 95, no. 3, pp. 255–282, 2012.
[7] A. M. Mirzendehdel and K. Suresh, “A Pareto-Optimal Approach to Multimaterial Topology
Optimization,” Journal of Mechanical Design, vol. 137, no. 10, 2015.
[8] K. Suresh, “Efficient Generation of Large-Scale Pareto-Optimal Topologies,” Structural and
Multidisciplinary Optimization, vol. 47, no. 1, pp. 49–61, 2013.
[9] F42 Committee, “Terminology for Additive Manufacturing Technologies,” ASTM International,
2012.
[10] O. Marketing, “Metal 3d Printing Materials,” Optomec Additive Manufacturing, 26-Jan-2015. .
[11] J. Goldak, A. Chakravarti, and M. Bibby, “A new finite element model for welding heat sources,”
MTB, vol. 15, no. 2, pp. 299–305, Jun. 1984.
57. 47
[12] R. Ye, J. E. Smugeresky, B. Zheng, Y. Zhou, and E. J. Lavernia, “Numerical modeling of the
thermal behavior during the LENS® process,” Materials Science and Engineering: A, vol. 428, no.
1–2, pp. 47–53, Jul. 2006.
[13] J. C. Heigel, P. Michaleris, and E. W. Reutzel, “Thermo-mechanical model development and
validation of directed energy deposition additive manufacturing of Ti–6Al–4V,” Additive
Manufacturing, Jan. 2015.
[14] J. G. Michopoulos, S. Lambrakos, and A. Illopoulos, “Multiphysics Challenges for Controlling
Layered Manufacturing Processes Targeting Thermomechanical Performance,” ASME, vol. 1A.
[15] V. R. Voller, “An overview of numerical methods for solving phase change problems,” in Advances
in Numerical Heat Transfer, vol. 1, CRC Press, 1996.
[16] H. Hu and A. Argyropoulos, “Mathematical modelling of solidification and melting: a review,”
Modelling Simul. Mater. Sci. Eng, vol. 4, pp. 371–396, 1996.
[17] P. Michaleris, “Modeling metal deposition in heat transfer analyses of additive manufacturing
processes,” Finite Elements in Analysis and Design, vol. 86, pp. 51–60, Sep. 2014.
[18] R. D. Cook, D. S. Malkus, M. E. Plesha, and R. Witt, Concepts and Applications of Finite Element
Analysis, 4th ed. John Wiley & Sons, 2002.
[19] S. Park, D. W. Rosen, S. Choi, and C. E. Duty, “Effective Mechanical Properties of Lattice Material
Fabricated by Mateial Extrusion Additive Manufacturing.,” in Proceedings of the ASME 2014
International Design Engineering Technical Conferences & Computers and Information in
Engineering Conference, Buffalo, NY, USA.
[20] T. J. R. Hughes, I. Levit, and J. Winget, “An element-by-element solution algorithm for problems of
structural and solid mechanics,” Comput Meth Appl Mech Eng, vol. 36, no. 2, pp. 241–254, 1983.
[21] P. Yadav and K. Suresh, “Limited-Memory Deflated Conjugate Gradient for Solid Mechanics,” in
Proceedings of the ASME 2014 International Design Engineering Technical Conferences &
Computers and Information in Engineering Conference, Buffalo, NY, USA, 2014.
58. 48
[22] P. Yadav and K. Suresh, “Large Scale Finite Element Analysis Via Assembly-Free Deflated
Conjugate Gradient,” J. Comput. Inf. Sci. Eng, vol. 14, no. 4, pp. 041008–1: 041008–9, 2014.
[23] J. N. Reddy, An introduction to nonlinear finite element analysis. Oxford University Press, 2004.
[24] K. Suresh, “A 199-line Matlab code for Pareto-optimal tracing in topology optimization,” Structural
and Multidisciplinary Optimization, vol. 42, no. 5, pp. 665–679, 2010.
[25] A. Krishnakumar and K. Suresh, “Hinge-Free compliant mechanism design via the Topological
Level-Set,” Journal of Mechanical Design, vol. 137, no. 3, 2015.
