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Fast to run model for thermal fields during metal additive manufacturing simulations Upadhyay, Meet 2020

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  Fast to run model for thermal fields during metal additive manufacturing simulations  by  Meet Upadhyay  B.Tech., National Institute of Technology Karnataka, 2017  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Materials Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  September 2020  © Meet Upadhyay, 2020  ii  The following individuals certify that they have read, and recommend to the Faculty of Graduate and Postdoctoral Studies for acceptance, the thesis entitled: Fast to run model for thermal fields during metal additive manufacturing simulations  submitted by Meet Upadhyay  in partial fulfillment of the requirements for  the degree of Master of Applied Science in Materials Engineering  Examining Committee: Daan Maijer, Materials Engineering Supervisor  Steve L. Cockcroft, Materials Engineering Supervisory Committee Member  Chad W. Sinclair, Materials Engineering Supervisory Committee Member     iii  Abstract Additive Manufacturing simulations for thermal fields are computationally expensive because of the highly disparate length and time scales involved and can sometimes take days to run. Improving the speed of these simulations enables multiple virtual experiments to be run to understand the effects of various process parameters on heat buildup and can even be useful for in situ process control based on sensor measurements from the build area.  The goal of this work is to reduce the computational time of such simulations while maintaining sufficient physics fidelity to yield reliable results. The approach taken is to replace the FEM model with a Fast-to-run (FTR) model which exploits the cyclic nature of the process to predict the thermal fields during AM. In this approach, peak temperatures and melt pools dimensions in a substrate melted by a moving heat source are modelled. The dependence of the heat transfer patterns on the heat source location and characteristics and the initial conditions of the substrate is modelled using data from the FEM simulation. Simulation time using the FTR model has been reduced significantly compared to the FEM simulation based on the domain size and time simulated.  Finally, the FTR model is run on various complex scenarios. The effects of various hatching strategies are modelled and their maximum temperatures and melt depths are compared. Additionally, a slice of an impeller model is simulated using the FTR model to generate maximum temperature and melt depth maps, allowing the identification of hotspots and undermelted regions.       iv  Lay Summary The following work presents a Fast to Run (FTR) methodology to speed up temperature field and melt depth simulations for metal based additive manufacturing processes. Metal Additive Manufacturing is poised to revolutionize the aerospace and medical industries by producing critical parts quickly and affordably. However, the wide range of process parameters and process variability makes it difficult to consistently produce good quality parts. High resolution thermal simulations can provide an insight into the process but are however very computationally expensive. This methodology aims to provide the same resolution as conventional finite element approaches but at a fraction of the time. The FTR methodology can provide high resolution thermal maps and help identify areas of high temperatures before printing. The print parameters can then be adjusted to prevent the formation of these hotspots and the consequent defects.          v  Preface This thesis is the original, independent, and unpublished work by the author, Meet Upadhyay.                vi  Table of Contents Abstract ......................................................................................................................................... iii Lay Summary ............................................................................................................................... iv Preface ............................................................................................................................................ v Table of Contents ......................................................................................................................... vi List of Tables ................................................................................................................................. x List of Figures ............................................................................................................................... xi List of Symbols ............................................................................................................................ xv Acknowledgements .................................................................................................................... xvi Chapter 1 - Introduction .............................................................................................................. 1 1.1 Metal Additive Manufacturing ............................................................................................. 3 1.2 Process Parameters ................................................................................................................ 4 1.3 Reliability of Metal AM Technology ................................................................................... 6 1.4 Motivation ............................................................................................................................. 7 Chapter 2 - Literature Review ..................................................................................................... 9 2.1 Physical Phenomena ............................................................................................................. 9 2.1.1 Melting and Solidification ............................................................................................. 9 2.1.2 Fluid Phenomena ......................................................................................................... 10 2.2 Modelling Approaches ........................................................................................................ 11 2.2.1 Analytical Solutions ..................................................................................................... 12 vii  2.2.2 Thermal and Fluid Flow Phenomena Simulation in AM ............................................. 12 2.2.3 Computationally Efficient Models for Additive Manufacturing Simulations ............. 18 Chapter 3 - Scope and Objectives.............................................................................................. 24 Chapter 4 - Fast-to-Run Model Development .......................................................................... 27 4.1 Exploiting the Steady State ................................................................................................. 27 4.2 Model Independence ........................................................................................................... 29 4.3 Software Selection .............................................................................................................. 30 4.4 General Semi-infinite Model .............................................................................................. 30 4.4.1 Model Setup ................................................................................................................. 32 4.4.2 Semi-infinite Model ..................................................................................................... 37 4.4.3 Data Collection ............................................................................................................ 41 4.4.4 The Effect of Latent Heat ............................................................................................ 43 Chapter 5 - Relationship Formulation ...................................................................................... 45 5.1 Effect of Initial Temperature .............................................................................................. 45 5.1.1 Surface Temperature .................................................................................................... 45 5.1.2 Temperature Decay ...................................................................................................... 47 5.1.3 Melt Pool Depths ......................................................................................................... 50 5.2 Effect of Varying Power, Speed and Initial Temperature ................................................... 51 5.2.1 Power = 100W ............................................................................................................. 52 5.2.2 Power = 200W ............................................................................................................. 54 viii  5.2.3 Power = 300W ............................................................................................................. 56 5.3 Interpolation Based Approach ............................................................................................ 58 5.4 Comparing Quality of Fit .................................................................................................... 60 5.4.1 Graphical Comparison ................................................................................................. 60 5.4.2 Numerical Comparison ................................................................................................ 62 5.5 Summary ............................................................................................................................. 63 Chapter 6 - Validation – Line Scan Example ........................................................................... 64 6.1 Model .................................................................................................................................. 66 6.2 Mesh .................................................................................................................................... 66 6.3 Boundary Conditions .......................................................................................................... 68 6.4 Time Steps .......................................................................................................................... 68 6.5 FTR Flow ............................................................................................................................ 68 6.6 Comparison ......................................................................................................................... 71 6.6.1 Temperature Decay ...................................................................................................... 71 6.6.2 Temperature Contours ................................................................................................. 72 6.6.3 Maximum Temperature and Melt Depths .................................................................... 73 6.6.4 Run Time and Error Comparisons ............................................................................... 76 Chapter 7 - Applications ............................................................................................................ 77 7.1 Square Layer with Diagonal Hatching ................................................................................ 77 7.2 Effects of Hatching Strategy ............................................................................................... 78 ix  7.3 Increased Geometric Complexity – Impeller Example ....................................................... 85 Chapter 8 - Conclusion ............................................................................................................... 90 8.1 Summary ............................................................................................................................. 90 8.2 Limitations .......................................................................................................................... 91 8.3 Future Work ........................................................................................................................ 92 Chapter 9 - Bibliography............................................................................................................ 94             x  List of Tables Table 1.1: Advantages and disadvantages of Metallic Additive Manufacturing [3][4]. ................ 2 Table 1.2: Comparison of the LBAM and EBAM process parameters [7][9]. ............................... 6 Table 2.1: Summary of domain sizes and simulation times in various studies. ........................... 22 Table 3.1: Important Process Parameters and Build Properties .................................................... 25 Table 4.1: Beam parameters for general model. ........................................................................... 35 Table 4.2: Time stepping parameters. ........................................................................................... 36 Table 5.1: Comparison of error metrics. ....................................................................................... 63 Table 6.1: Time stepping information. ......................................................................................... 68 Table 6.2: Error comparison for 2-Heat Trace verification case .................................................. 76 Table 7.1: Process parameters. ...................................................................................................... 82 Table 7.2: Process parameters for impeller layer. ......................................................................... 88         xi  List of Figures Figure 1.1: Classification of metal additive manufacturing............................................................ 3 Figure 1.2: Typical Electron Beam Additive Manufacturing System schematic. .......................... 4 Figure 1.3: Process Parameters for Metal Additive Manufacturing ............................................... 5 Figure 2.1: Thermofluid phenomenon during Powder Bed Fusion processes. ............................. 10 Figure 4.1: Temperature contours at different time steps during scanning by heat source. ......... 28 Figure 4.2: Fast to Run model data flow....................................................................................... 29 Figure 4.3: Schematic of semi-infinite model showing domain and boundary conditions (a), Meshing and heat source motion (b). ............................................................................................ 32 Figure 4.4: Material Properties for Ti6Al4V [43]. ....................................................................... 34 Figure 4.5: Comparison of temperature decay curves with different beam run lengths. .............. 36 Figure 4.6: Temperature contours evolution over time. ............................................................... 38 Figure 4.7: Temperature over time response for nodes along the depth and width of the semi-infinite model. ............................................................................................................................... 39 Figure 4.8: Magnified temperature over time response for nodes along the depth of the semi-infinite model. ............................................................................................................................... 40 Figure 4.9: Magnified temperature over time response for nodes along the width of the semi-infinite model. ............................................................................................................................... 40 Figure 4.10: A typical contour map of temperatures generated by the semi-infinite model. ....... 41 Figure 4.11: Locations of data extraction. .................................................................................... 42 Figure 4.12: Typical melt pool profile. ......................................................................................... 42 Figure 4.13: Surface temperatures with and without latent heat. ................................................. 43 xii  Figure 4.14: Temperature vs time behaviour of the model with and without the effect of latent heat........................................................................................................................................................ 44 Figure 4.15: Melt pool profiles (a) with latent heat, (b) without latent heat. The beam travel direction is into the sheet. ............................................................................................................. 44 Figure 5.1: Effect of initial conditions on surface temperatures (a) arrow showing line along which nodal temperatures were extracted and (b) associated surface temperatures for P=200W, V=0.5m/s. ...................................................................................................................................... 46 Figure 5.2: Comparison between fit results and ABAQUS results for P=200W, V=0.5m/s. ....... 47 Figure 5.3: Temperature change over time on symmetry line for P=200W, V=0.5m/s. .............. 48 Figure 5.4: Temperature decay over time (a) and nodes corresponding to temperature curves (b) for P=200W, V=0.5m/s and Ti=973K. .......................................................................................... 48 Figure 5.5: Comparison between fit results and ABAQUS results for P=200W, V=0.5m/s. ....... 49 Figure 5.6: Variation of melt depth over distance from the beam center and initial temperature for P=200W, V=0.5m/s. ..................................................................................................................... 50 Figure 5.7: Temperature contour showing the melt pool boundary for P=200W, V=0.5m/s, Ti = 973K. ............................................................................................................................................. 50 Figure 5.8: Comparison of melt depth between fit results and ABAQUS results for P=200W, V=0.5m/s. ...................................................................................................................................... 51 Figure 5.9: Surface temperatures for Power = 100W and varying initial temperatures and speeds........................................................................................................................................................ 52 Figure 5.10: Temperature change over time for Power = 100W and varying initial temperatures and speeds. .................................................................................................................................... 53 Figure 5.11: Melt depth for Power = 100W and varying initial temperatures and speeds. .......... 53 xiii  Figure 5.12: Surface temperatures for Power = 200W and varying initial temperatures and speeds........................................................................................................................................................ 54 Figure 5.13: Temperature change over time for Power = 200W and varying initial temperatures and speeds. .................................................................................................................................... 55 Figure 5.14: Melt depth for Power = 200W and varying initial temperatures and speeds. .......... 55 Figure 5.15: Surface temperatures for Power = 300W and varying initial temperatures and speeds........................................................................................................................................................ 56 Figure 5.16: Temperature change over time for Power = 300W and varying initial temperatures and speeds. .................................................................................................................................... 57 Figure 5.17: Melt depth for Power = 300W and varying initial temperatures and speeds. .......... 57 Figure 5.18: Comparison of ABAQUS, curve fit and linear interpolation results for surface temperatures. ................................................................................................................................. 60 Figure 5.19: Comparison of ABAQUS and linear interpolation results for temperature change over time. .............................................................................................................................................. 61 Figure 5.20: Comparison of ABAQUS, curve fit and linear interpolation results for melt depth profile. ........................................................................................................................................... 61 Figure 6.1: FTR model process flow. ........................................................................................... 64 Figure 6.2: (a) Motion of heat source (b) Corresponding temperature profile. ............................ 66 Figure 6.3: Domain size and boundary conditions. ...................................................................... 67 Figure 6.4: Fine mesh domain size. .............................................................................................. 67 Figure 6.5: Information flow in the FTR model. .......................................................................... 70 Figure 6.6: Temperature decay comparison for Node 1 as shown in Figure 6.2(b) for 2-Heat Trace verification case. ........................................................................................................................... 71 xiv  Figure 6.7: Surface temperature at 0.015s calculated by (a) ABAQUS and (b) the FTR method for 2-Heat Trace verification case. ..................................................................................................... 72 Figure 6.8: Maximum surface temperature contours calculated by (a) ABAQUS and (b) the FTR method for 2-Heat Trace verification case. ................................................................................... 74 Figure 6.9: Maximum temperatures predicted by ABAQUS and the FTR Method mid-way along the heat trace (x = 5.3 mm) for the 2-Heat Trace verification case. ............................................. 75 Figure 6.10: Maximum melt depths predicted by ABAQUS and the FTR Method mid-way along the heat trace (x = 5.3 mm) for the 2-Heat Trace verification case. ............................................. 75 Figure 7.1: (a) Square contour and hatching pattern, (b) Temperature profile from FTR, (c) Melt Depth from FTR. ........................................................................................................................... 78 Figure 7.2: Scanning strategies. (a) Standard, (b) contour only, (c) hatch only, (d) five contours and (e) uni-directional hatch [47]. ................................................................................................ 80 Figure 7.3: Projected pore densities. (a) Standard, (b) contours only, (c) hatch only, (d) contour x 5, (e) uni-directional hatch and (f) hatch first (contour regions in darker grey) [47]. .................. 81 Figure 7.4: FTR simulated maximum temperature for different hatching strategies. .................. 83 Figure 7.5: FTR simulated maximum melt depth for different hatching strategies. ..................... 84 Figure 7.6: Centrifugal Impeller from Thingiverse [48]. .............................................................. 85 Figure 7.7: GCode for the impeller at layer height z = 4mm simulated in CAMotics [49]. ......... 87 Figure 7.8: Maximum temperatures for simulated layer z = 4mm. .............................................. 88 Figure 7.9: Maximum melt depths for simulated layer z = 4mm ................................................. 89    xv  List of Symbols η Beam efficiency v Beam speed (m/s) ρ Density (kg/m3) Ti Initial temperature (K) Dp Peak depth (m) Tp Peak temperature (K) P Power (W) Cp Specific Heat (J/kg/K) σ Standard deviation q Surface heat flux (W/m2) T Temperature (K) k Thermal conductivity (W/m/K) t Time (s) Q Volumetric heat source (W/m3)      xvi  Acknowledgements I would like to express my deepest gratitude to my supervisor, Prof. Daan Maijer for his incisive questions about and invaluable insight into the developing the methodology presented in this thesis. His unwavering support and patience during the duration of this project enabled me to produce quality work that I am proud to present.  I am extremely grateful to Prof. Chad Sinclair and Prof. Steve Cockcroft’s constructive criticism and practical suggestions during various group meetings and presentations over the course of the project. I am also very grateful William Sparling for his valuable contribution towards this project. His GCode generator was extremely helpful for validating my methodology and he has always been a great sounding board for ideas. Especially helpful to me during this time were Farzaneh Farhang-Mehr, Jun Ou, and Nisa Saadah, who provided me with valuable suggestions about my writing and technical assistance. I very much appreciate my fellow students from the Casting and Additive Manufacturing group for their suggestions, advice and assistance. They were always there when I needed help. I would also like to acknowledge the assistance of the office staff at the Materials Engineering Department, especially Michelle Tierney and Mary Jansepar for making my time as a graduate student at the department as smooth as possible. Finally, I would like to thank my friends and family for their constant encouragement and profound belief in my abilities. My success would not have been possible without their support.       Chapter 1 -  Introduction Additive Manufacturing (AM), initially known as Rapid Prototyping (RP), began as a method for design engineers to realise design concepts without heavily investing in subsequent manufacturing processes for prototypes. RP enabled the conversion of parametric CAD (computer-aided design) data to physical prototypes which could be tested to check if they met the design criteria. This saved not only time but also allowed the testing of multiple models/concepts [1]. The earliest methods of RP reached commercialisation in the 1990s [2] and involved processes such as photopolymer curing techniques using lasers and UV light sources (stereolithography), powder consolidation techniques like inkjet printing and Selective Laser Sintering (SLS), and filament extrusion techniques like Fused Deposition Modeling (FDM) [3]. As the speed, accuracy, and the  material choice of the processes improved, the focus of the industry shifted from Rapid Prototyping to ‘Rapid Manufacturing’ i.e. the process of manufacturing complete parts from a rapid prototyping device [4,5]. This led to the wider recognition of these processes under the umbrella term ‘Additive Manufacturing’. AM’s application areas have expanded into aerospace, medicine, architecture and more. This expansion has been aided by wide-scale development and innovation in AM processes.  Over the last twenty years, several AM processes have been developed to produce parts in a wide variety of metal, polymer and ceramic materials. Some of the main advantages and disadvantages of metallic AM processes are summarised in Table 1.1.  Metallic AM has the potential to transform industries where low volume manufacturing of complex parts occurs e.g. complex nozzle geometries for aerospace applications, patient-specific medical implants with excellent biocompatibility and custom tooling for automotive applications. AM processes also provide opportunities to consolidate multi-component parts and cut down on assembly time, allow 2  production of spare parts that are no longer in production. AM technology has the potential to keep a system functioning by being able to fabricate a critical part in days rather than weeks. However, the current AM technologies are less attractive for large scale productions, primarily because they cannot compete with conventional manufacturing technology in terms of cost and speed of production. Metallic additive manufacturing processes are found to be the most expensive due to the high cost of equipment and feedstock materials. Given that the cost of equipment is amortized over the number of parts produced, the major issues reducing the profitability of AM processes are recurring costs such as the cost of materials and maintenance. As third party feedstock vendors enter into the market [3], feedstock costs are expected to reduce due to an increased supply. Additionally, equipment vendors and researchers continue to work on the processes to produce parts with better material properties consistently.  Table 1.1: Advantages and disadvantages of Metallic Additive Manufacturing [3][4].   Significant design freedomMaterial savingsAutomated technologyLower inventory levels requiredLower tooling costs and ability to produce spare partsExpensive equipment and feedstockSlow process unsuitable for high volume productionLack of control over material propertiesLimited choices of material availableAdvantagesDisadvantages3  While the sales of commercial metal AM systems has seen significant growth, a major challenge preventing industry from adopting AM is the absence of a thorough understanding of the material, processes, properties and performance of the parts [5]. A concerted effort is being made by researchers in academia and industry to understand the process and develop defect free, structurally sound, and reliable metal AM parts.  1.1 Metal Additive Manufacturing Metal AM processes can be classified by both the method of binding/energy source and the type of feedstock material (refer to Figure 1.1) [6].   Figure 1.1: Classification of metal additive manufacturing. The most popular methods use laser or electron beams to deposit wire or consolidate powder on a substrate. These processes can be further classified into Powder Bed Fusion (PBF) and Direct Energy Deposition (DED) processes. Laser-DED uses a laser source to create a melt pool and a nozzle to feed powder or a wire into the melt pool as the laser traverses the bed, while a shielding gas prevents oxidation. Electron Beam-DED uses a similar principle, except wires are deposited into the melt pool. Laser-PBF uses lasers in an inert gas atmosphere to fuse thin layers of powders Metal Additive ManufacturingBinding MethodHeatLaserElectron BeamSonotrode Binding ResinType of FeedstockPowder WireSheet4  in layers to develop the 3D part. Electron Beam-PBF uses the same approach where an electron beam is used to consolidate power in a vacuum chamber to produce the parts. A schematic of an electron beam-PBF AM system is shown in Figure 1.2. The electron beam is focused and directed by electromagnetic coils in a vacuum chamber. The coils are used to raster the beam on the surface of the bed and thereby heat the powder on each layer. A first pass pre-heats / pre-sinters the powder on the bed to prevent electrostatic charging and repulsion of powder particles. A second pass is used to melt the powder in the areas of the bed which are part of the build. The pre-sintering step is performed at faster scan speeds [7]. The use of an electron beam heat source limits the application to printing conductive powders.  Figure 1.2: Typical Electron Beam Additive Manufacturing System schematic. 1.2 Process Parameters The typical process parameters that can be controlled during Metal Additive Manufacturing are shown in Figure 1.3. The four major parameters that control the heat input are the thermal 5  parameters – beam power, scan speed, beam diameter and preheat. The first three parameters directly affect the thermal cycle, e.g. temperatures and cooling rates experienced by the part, while the preheat affects the degree of powder sintering. Different modes are used in laser and electron beam processes to apply preheat to the build area. In laser processes, preheat is applied by maintaining the build plate at a constant temperature. In electron beam powder bed processes, the rapid scanning speeds allow the heating of the powder bed using a defocused beam. Consequently, electron beam systems can maintain a more uniform temperature as the added heat can be controlled every layer [7]. In simulations, preheating is usually incorporated by applying a constant preheat temperature to the substrate [8]. The scan strategy controls the distribution of heat over the layer, affecting temperatures on a macroscale. Powder size distributions and layer heights influence the competing factors: total build time and feature resolution [6]. While the main parameters of both laser and electron beam powder bed processes remain the same, the processes differ in some factors which are summarized in Table 1.2.  Figure 1.3: Process Parameters for Metal Additive Manufacturing Process ParamatersThermal ParametersBeam Power Scan SpeedBeam DiamaterPreheat TemperatureScan StrategyScan PatternsHatch SpacingIdle TimePowder PropertiesSize DistributionLayer Heights6   Table 1.2: Comparison of the LBAM and EBAM process parameters [7][9]. Property LBAM EBAM Powder size (µm) 10-60 50-150 Layer height (µm) 30-60 50-100 Preheat From base plate, up to 500 °C Electron Beam, 700-1100 °C Speed [7] 0.8-1.2 m/s 0.1-4 m/s Power [9] Laser power output (50-1000 W) Electron beam current and voltage (50-1000 W) Environment Inert gas Vacuum  1.3 Reliability of Metal AM Technology A variety of parameters, both controllable and inherent to the process, affect the final quality of the printed parts i.e. density, surface roughness, dimensional accuracy, and mechanical properties [10]. Studies have found part quality to differ between builds with the same process parameters, even within the same build at different locations [11], and across different machines from equipment manufacturers [12]. The same CAD model definitions and build parameters lead to different properties and success rates of parts across different machine platforms, between two machines of the same make and even in the same machine across different builds. Cross platform variations could occur due to errors in conversion of CAD models into triangulated 3D printing formats, differences in precision across machines and cross build variations occur due to material related issues such as differential shrinkage upon solidification and variations in thermal gradients leading to warpage and residual stresses [13]. Additionally, these variations are often random and 7  hard to predict. Coupling this variability with the high cost of powders and machine operating costs, it can be difficult to maintain part quality and ensure reliability and reproducibility of the additive manufacturing process. This lack of consistency is detrimental for the advancement of AM technology, especially for parts to meet the stringent regulations of the critical industries like aerospace and medicine. In a survey conducted by PWC, 47% of manufacturers reported that the barrier to the adoption of 3D printing is the “uncertain quality of the final product” [14].  1.4 Motivation To aid in understanding the AM process, it is important to develop models that can predict the three-dimensional, transient variation of variables such as temperature, fluid velocities and displacements; quantities that influence the microstructure and residual stresses generated in a part. These process variables are difficult to observe in situ because of the transient nature of the process. Additionally, unlike traditional manufacturing processes, measurements of temperature are typically limited to easily accessible locations like build surfaces or substrates and are often performed with non-contact techniques. Steep temperature gradients, excessive scattering, equipment noise, obstruction due to powder particles and the constant movement of the imaged area are just some of the challenges with these types of measurements [15]. Analytical and numerical modelling allows researchers to establish relations between the process parameters and the cooling rates and thermal gradients. Micro-scale simulations can accurately capture the flow of heat and molten metal in the melt pool and provide high resolution (both spatial and temporal) to understand the complex thermal-mechanical conditions existing in AM processes. Additionally, many virtual “experiments” can be performed using models in an automated manner by varying different parameters and assessing their effects and interactions with other parameters. Chapter 2 will review transport processes active in the melt pool, the approaches adopted by researchers to 8  model the interaction of heat sources with the feedstock and the effect of various controllable parameters on heat transfer at the micro-scale.   While micro-scale simulations are an important tool for evaluating the link between process parameters and temperatures / thermal gradients, they are computationally expensive and are not feasible for simulating fabrication of macro-scale parts. Micro-scale simulations involve highly disparate length and time scales and can sometimes take days to run. Additionally, computation time increases significantly with increase in the temporal and spatial scales, i.e. it can take weeks even with high power computational resources to run microscale simulations for small scale geometries. Hence a significant challenge faced by researchers today is to develop a model at the overall scale of the part, at a reduced computational cost without sacrificing accuracy. Ideally, such models would be able to output quantities like distortions and residual stresses in real time. This would facilitate faster part design and process parameter selection, better control strategies and even layer re-processing mid-print.  Since parts can sometimes take weeks to manufacture, it’s important to identify and treat problem areas which can cause a production run to fail as and when they arise. Furthermore, as in situ monitoring technologies are developed for AM processes, measured data can be employed to predict and correct conditions conducive to defect formation in real-time, thus improving the reliability of printed products market confidence in the technology. Chapter 2 will also discuss attempts at reduced computation models are highlight research gaps.     9  Chapter 2 -  Literature Review 2.1 Physical Phenomena As noted, AM processes involve a wide range of physical phenomena from powder melting and consolidation, fluid flow within the melt pool, layer by layer material addition, heat transfer from the melt pool via different modes and distortion and residual stress accumulation. Some of the key phenomena influencing heat transfer in the part are discussed in this section. 2.1.1 Melting and Solidification Metal AM processes involve complex solidification phenomenon due to the repeated and localized heating and cooling of the build area and the interaction of the heat source with the feedstock material. For electron beam or laser-based powder bed processes, the beam of electrons/photons irradiates the powder, heating and subsequently melting it. A portion of the heat deposited in the sample melts the substrate and subsequently heats the build. Another portion of the heat is lost due to radiation, convection in laser processes, and evaporation of volatile elements  [16].  Temperature gradients and cooling rates vary both with time and location affecting solidification. This results in parts with heterogeneous microstructures and anisotropic properties [6]. While this might be a requirement for some applications – for example epitaxial growth in single crystal turbine blades – this anisotropy makes validation of parts more difficult. Progressive processing of each layer results in remelting and solidification and subsequent heat build-up in the part. Depending on the heat source parameters, this remelting can be multiple layers deep and must be carefully managed to ensure to complete densification of produced parts. In addition to the heat source parameters, thermofluid phenomena in the melt pool also affect the temperature field. This varying field determines the stresses and deformation of the part during a build and ultimately the 10  dimensions and residual stresses in the final part. Figure 2.1 shows some of the different heat transfer and fluid flow phenomena active in the powder bed AM process.   Figure 2.1: Thermofluid phenomenon during Powder Bed Fusion processes.   2.1.2 Fluid Phenomena AM processes can be visualized as small localized welding spots, where a focused heat source melts the material that will form the part being fabricated. While welding processes have been extensively studied and modelled and much of the understanding from the welding context can be applied to AM, the AM process is significantly more complicated. For instance, the heat source interacts differently with a powder bed, wire or solid metal. A powder bed adds further complexity due to the compaction of the layer height as the powder layer melts and voids between particles are filled by molten metal. The melt pool moves quickly across the build area and is affected by the changes in the heat source parameters and local differences in powder bed density and temperature across the substrate. As the powder melts, the free surface evolves, convection within the melt pool causes heat redistribution, surface tension gradients lead to Marangoni flow, recoil 11  pressure causes depression of the free surface, and evaporation of alloying elements from the melt pool applies a recoil pressure to the melt pool surface (Figure 2.1).  Convection in the melt pool is a phenomenon which is frequently ignored in favour of faster computation. A recent study [16], however, showed that ignoring liquid convection leads to unrealistic cooling rates in the proximity of the liquid region. The described model solved the mass, momentum and energy conservation equations with Marangoni boundary conditions applied to the momentum equations. It was found that the movement of the liquid reduces the temperature and the cooling rates were found to be 45-55% lower. Marangoni convection is a fluid flow phenomenon caused by the surface tension gradients that acts on the free surface of the melt pool. The variation of surface tension causes fluid to flow from an area of lower surface tension to an area of higher surface tension. This variation can be both due to temperature and compositional gradients [17]. The effect is to move the hotter molten metal towards the outer edges of the melt pool.  Directly below the heat source, the material can easily reach its boiling point, especially for more volatile alloying elements. While the heat is not normally high enough to cause ablation, but it can cause a recoil pressure to act on the surface of the liquid. This creates a depression on the surface of the liquid. This combined with the Marangoni convection can create a strong hydrodynamic flow in the melt pool [18]. As temperature varies over the depth of the melt pool, flow due to density differences can also occur and is termed as buoyancy-driven flow.  2.2  Modelling Approaches To model the thermo-fluid phenomena active in AM processes, various approaches have been taken by researchers. Analytical approaches based on the Rosenthal approach, empirical 12  approaches using process maps and micro- and mesoscale FEM and CFD approaches will be discussed in this section. 2.2.1 Analytical Solutions Exact analytical solutions are not feasible due to the complex nature of the process, but approximate models have been developed which can still provide useful information. The Rosenthal Solution, originally developed for simulating welding processes, can be used to analytically calculate the temperature gradients and cooling rates caused by moving heat sources [19]. The Rosenthal Solution has limited applicability to AM processes due to its many assumptions and simplistic approach. Of particular relevance to this study, Soylemez et al. [20] developed “fitted” Rosenthal models using a full-scale FEM solution. The model used thermal properties for the Rosenthal model at a temperature between the substrate temperature and melting temperature to match the melt pool dimensions predicted by an FEM model. Using this fitted model, the authors proceeded to generate process maps for beam power and velocity to predict melt pool dimensions in wire-based electron beam AM. While not exact, this approach demonstrated a methodology to predict the effect of beam parameters on the melt pool without the need to use computationally expensive FEM models for each prediction. This methodology serves as a good guideline for a simplistic line scan case but is inadequate for more complex hatching strategies and wider process windows because it does not account for heat conduction within the deposited material. 2.2.2 Thermal and Fluid Flow Phenomena Simulation in AM Thermal-fluid flow modelling simulates the flow of heat and metal in the melt pool and can be used to understand the effects of the characteristics and movement of the heat source on the melt pool. Efforts have been made to track the free surface, temperature gradients, cooling rates, 13  velocities and the movement of the solidification front. A variety of techniques have been used to model the AM process including FEM and CFD analyses of different physics and from micro to macroscale in terms of size of simulated domains.  2.2.2.1 Finite Element Modelling Approach for Thermal Field Simulation General purpose, commercial FEM packages, such as ABAQUS and ANSYS, can be used to solve the energy (heat) conservation equations to predict the 3D steady-state or transient temperature distributions. These distributions can provide important information about the size of the melt pool and the temperature distribution in the part. Shen et al. used ABAQUS to simulate the transient heat transfer problem with a moving Gaussian volumetric heat source distribution [21]. The effect of powder porosity and laser spot size on the melt pool size was studied. The emissivity and thermal conductivity of the powder were assumed to be functions of powder porosity and a user-subroutine was used to assign the properties depending on the temperature and cooling rate of the element. The model was validated with measured data from the LENS process for 316 stainless steel and compared with results reported in the literature. The melt pool was shown to become deeper and shorter with increasing porosity, but the width remained almost constant. As expected, the depth of penetration and maximum temperature decreased with increasing laser spot size, due to lower energy density.  Several researchers have shown that it is important to account for the evolution of material properties from the feedstock powder material to liquid metal and then to that of the bulk solid. Galati et al. [22] developed several user-subroutines in ABAQUS to simulate the AM process including the variation of material properties as a function of temperature. The subroutine UMATHT was used to change the material index when the temperature of the element reached the solidus temperature. The FILM subroutine was used to model the radiative heat loss from the build 14  domain via an effective heat transfer coefficient. USDFLD was used to update the density value and DFLUX was used to apply the surface flux. The maximum deviations from experimental values were found to be 25%.  A common strategy to simulate material addition used in FEM software is the element reactivation method. In this approach, the mesh is defined and the volumes which correspond to added material are deactivated (the material properties are set to ~10-6 times the bulk properties). These elements are then reactivated at the right properties to simulate material addition. Deactivated elements have negligible material properties and hence do not contribute towards the computational effort, hence saving time during simulations. Elements can be activated at different scales too. Additions of singular elements, single tracks, entire layers or even a stack of layers can be added. The computational effort decreases with increasing number of elements added because the time scale also increases correspondingly. This technique was used by Fox et al. using ABAQUS to predict the melt pool depth response to step changes in beam power and travel velocity during electron beam wire fabrication with Ti-6Al-4V [23]. The feedstock (e.g. bead) material was reactivated three elements ahead of the heat source to model heat conduction into the wire being fed. Latent heat was considered and temperature-dependent material properties were used. A Power-Velocity process map was developed, over a range of powers: 1-5 kW and travel velocities: 0-100 in/min. This map was used to predict how quickly the melt pool stabilizes when beam properties are changed – characterized in terms of response times and distances. Response times show little correlation; however, response distances show strong similarities when moving between same lines of constant melt pool area on the process map. The study concludes that response distance was governed by the initial and final melt pool sizes and not the initial and final power and velocity values.  15  Temperature distributions from thermal analyses have been used to develop process maps that link the solidified microstructure to the melt pool dimensions [24]. Gockel et al. used ABAQUS to simulate heat transfer in EB wire fabrication of Ti-6Al-4V. Constant melt pool sizes and constant melt pool aspect ratios led to similar grain morphology boundaries. Process maps of power, velocity, solidification rate and thermal gradient were developed to study grain morphology. Along a line on the process map for which the melt pool area remains constant increased power results in an increase in the population of equiaxed grains with constant grain size. Additionally, increasing the melt pool cross-sectional area increases the grain size. The transition from fully columnar to columnar plus equiaxed grains occurred as Power and Velocity were both increased. Hence, control of the melt pool cross-sectional area also approximately controls solidification cooling rate. 2.2.2.2 Computation Fluid Dynamics Approach for Heat and Fluid Flow Simulations FEM thermal-only analyses of AM have several limitations. Convection in the melt pool, which is not considered in these analyses, is a significant mode of heat transfer. The fixed mesh geometry typically used in these analyses fails to consider consolidation of the powder layer as the powder particles melt and coalesce. Coupled thermal-fluid models have been shown to more accurately capture these phenomena of AM processing.  Commercial packages like ANSYS and Flow 3D can be used to model the fluid flow in the melt pool. These packages simultaneously solve the 3D transient mass, momentum and energy conservation equations to predict the temperatures and velocities in the build domain. However, general purpose CFD algorithms have trouble predicting the effects of wetting, shrinkage and tracking the melt pool surface. Other schemes have been used to consider these phenomena such as Smoothed Particle Hydrodynamics (SPH), Level Set Method (LSM), Volume of Fluid (VOF), and Lattice Boltzmann Method (LBM).  16  SPH is a Lagrangian method which divides the fluid domain into a set of particles separated by a smoothing length over which their properties are evaluated by a kernel function. SPH has been used to model the fluid flow in laser beam wire fabrication of Ti-6Al-4V [25]. Surface tension was evaluated using a function for cohesion and adhesion effects based on the SPH methodology. To validate the model, experiments were performed were a wire was placed on the substrate and then scanned by the laser. Using a design of experiments approach nine combinations of power and power density were obtained. Properties such as scan width, stabilization distance, melt pool depth, deposit height, deposit width and contact angle were measured and showed good correlation with modelling data.  LBM uses imaginary particles to perform consecutive propagation and collision processes to model fluid flow. This method has been used to model the fundamental consolidation mechanisms during powder bed fusion processes [26]. A 2D model was used with stochastic powder application to consider the effect of individual powder particles. The influence of beam power, beam velocity and layer thickness on the vertical wall quality were studied and the Lattice Boltzmann method was used to track fluid flow in the melt pool and the free surface. Since these simulations considered the effects of individual particles, the mesh size was in the order of micrometres and the timescales were on the order of nanoseconds. The simulations were performed on a domain size of 1.25 mm x 1.0 mm, using a beam diameter of 350 µm and a corresponding process time of 6.6 x 10-4 s The VOF and LSM techniques are used to track the free surface of the molten fluid. The VOF approach computes and tracks the volume fraction of a particular phase in each cell rather than the interface itself. This approach was used to model the AM process for a titanium alloy [27]. The mass, momentum, and energy conservation equations along with the PLIC-VOF equations were 17  solved to obtain the free surface. Since material properties were highly temperature-dependent, the momentum and energy equations were highly coupled. Good agreement was observed between experimental results (peak pool temperatures and melt pool length). However, at higher power levels, the methodology proved to be inaccurate because a 2D model was used.  In LSM, the interface is captured and tracked by the level set function, defined as a signed distance from the interface. This technique was used to develop a 3D transient model during Electron Beam Free Form Fabrication of Ti-6Al-4V [28]. In this model, a drop was assumed to detach from the titanium wire when the gravity force exceeds the surface tension. The wire was assumed to be vertical. Using this approach, the radius of the droplet was evaluated, and the temperature was experimentally determined as 2000K. The fluid flow of the melt pool was mainly considered to be driven by recoil pressure due to evaporation, surface tension, thermal-capillary forces and the impacting force of the molten droplets. The free surface was then evaluated by solving the level set equation. The analyses tools described in this section can predict mesoscale phenomenon during the melting and consolidation of the powder but are very computationally expensive. Since the simulations consider effects due to individual particles, the mesh size is on the order of micrometers and the timescales are on the order of nanoseconds. These quantities are many orders of magnitude smaller than the size of a typical part produced by AM (cm ~ m) and time to manufacture such parts (days ~ weeks). To use such a model to simulate the printing of industrial-scale part would require a significant computational effort. Hence, such models cannot be used to simulate large-scale geometries and there is a need for computationally efficient solutions.  18  2.2.3 Computationally Efficient Models for Additive Manufacturing Simulations Different approaches have been taken by researchers to reduce computational effort and model macroscale geometries. Researchers have addressed the limited applicability of the Rosenthal model by complimenting it with the superposition principle. In [29], the simulation was broken down into two initial value problems. The first problem used the Rosenthal method to predict the steep temperature gradients near the heat source and the second problem approximates the temperature fields in regions away from the heat source. In these regions, the temperature fields vary relatively slowly and are quickly predicted using FEM. Both the solutions are superimposed to approximate the solution produced by a full FEM model. While the results obtained are close to the FEM results, the model uses a very simplistic approach where material properties do not vary with temperature. Additionally, the authors estimate that this calculation method would require 5.7 months on a single CPU or 4 hours on a 1000 core cluster to complete for a 21 cm3 part at a build rate of 5.4 mm3/s; a computational cost which is not feasible for most manufacturers. Another approach proposes an enriched analytical solution for simulation of AM processes by improving the Rosenthal solution for Gaussian sources [30]. The solution used a recursive scheme to approximate the effect of temperature-dependent properties. When the heat sources moves along an edge, the model simulates the effect of boundary and internal reflection of heat by using a mirrored heat source across the boundary and superimposing the effects of both the heat sources. However, the model is more suited to slower processes such as direct energy deposition and does not consider the effects of latent heat release. Researchers have also tried to modify existing FEM formulations to reduce computational effort. A line heat input model was proposed by Irwin [31] for PBF AM by averaging the Goldak heat source over its path. The heat source path was divided into smaller segments that were applied at 19  each increment. The thermo-mechanical model formulation yielded reasonably accurate results for macro-scale displacements (~10% error) and reduced computational time by 90%. However, this technique sacrifices the resolution of the thermal maps of the build state and overlooks any hot spots that might occur. The lack of a smooth and high-resolution thermal map also means this method could not be used to predict microstructures based on cooling rates.  Ding et al. developed a Lagrangian instead of an Eulerian model to simulate the AM process using a steady-state approach and resulted in an 80% computational saving. A transient 3D thermo-elastic–plastic model (Eulerian) and an advanced steady-state thermal model (Lagrangian) were developed in this study [32]. A four-layer wall was simulated by both these methods. The steady-state model applied a mass flow rate per area based on the welding speed and cooling time to simulate the movement of the heat source. These thermal results were then used to simulate distortions and residual stresses which were then verified experimentally using laser scanners and neutron diffraction measurements, respectively.  Another common technique used is to coarsen the mesh in areas where the solution is not changing to reduce the degrees of freedom (DOFs) and consequently the computational time. Denlinger et al. [33] introduced elements layer by layer to mimic the printing process and activated material properties based on the pattern traced by the heat source. After processing, lower layers in the build were merged to reduce the total number of equations to be solved. This strategy was used to model the distortion of a 3.6m long part with 107 build layers produced using wire-feed additive manufacturing. A new FE framework was introduced in [34], which performs remeshing to update the discretized geometry at regular intervals for the thermal predictions and also maintains a high density mesh  that the thermal field is mapped onto at the end of every time step. After the heat source passes over a region, its mesh is coarsened to reduce the degrees of freedom of the model. 20  The high-resolution thermal field is then interpolated to the coarse mesh as a pre-defined field for the next time step. At the end of each discretization step, the results are mapped back to the fine mesh. A one-half foot thin-walled cylindrical part was simulated using this framework which was then printed with the DMD process, though the authors claim the method is also applicable to powder bed, powder fed and wire-based processes. The stress and strain predictions showed a close agreement with the dense mesh FEM solution, but both the numerical results deviated considerably in some areas from the experimental results. A part-scale model for fast prediction of thermal distortion which uses STL files as an input was developed in [35] and [36]. The thermal analysis was performed using a thermal circuit network (TCN) model. An STL processor was used to create slices, modelled as thermal volumes. These were modelled as thermal capacitances and resistances connected to a TCN for thermal analysis. The model was then tested for two geometries: a disc and a rectangular bar printed horizontally. The model predicted the distortion of the geometries in terms of their radius of curvature (ROC) and compared these predictions to experimental results. The ROC of the disc differed by 8% while the ROC of the bar was predicted to be twice that of the obtained experimental results.  Another approach gaining popularity is that of “superlayers”. In this technique, the geometry is divided into large layers – superlayers – with heights on the order of 10 to 20 printed layers. The super layer is activated with an initial temperature equal to the liquidus temperature of the material or a constant heat is applied over the entire superlayer before allowing it to cool down. This approach avoids having to track the location of the heat source on the surface and reduces the number of layer additions to significantly reduce computational expense. This approach was used in [37] and [38] to evaluate residual stress and distortion evolution for a double cantilever geometry. Commercial software packages such as ANSYS Additive [39] have implemented this 21  technique to predict macro-distortions and stresses for both powder bed fusion (PBF) and direct energy deposition (DED) processes. However, this approach does not account for the varying temperature fields resulting from different heat source scanning strategies and employs large time steps/mesh sizes that do not capture the steep temperature gradients involved in these processes and the resulting inelastic strains. Additionally, this method fails to capture the progressive remelting characteristic of AM processes.  It is known that scanning strategies have a significant impact on the temperature distribution, distortion and cracking for metal AM fabrication [40][41]. To tackle this, an equivalent heat source approach based on a microscale scan model has been used to determine the temperature field for a mesoscale hatch model [42]. A Gaussian distribution was used on a single track to obtain a stable melt pool. The temperature field was then extracted and converted into an equivalent heat input. A coupled thermal-mechanical analysis was then performed for a small domain (5 x 5 x 0.15 mm) with a fixed hatch pattern. This approach was used to predict the mesoscale residual stress field which was then applied to the macroscale part. However, since this approach averages out the heat input over a model with a small surface volume and uses it for a large-scale model, it cannot be used to describe local hot spots or heat buildup during progressive layer additions.  Table 2.1 summarizes the rapid simulation models described in the literature review. The time required to run the fast model and the conventional model shown in columns 6 and 7 refer to the total (wall clock) time based on the references unless otherwise specified.     22  Table 2.1: Summary of domain sizes and simulation times in various studies. Reference Physics Numerical Method Domain Size (mm3) Time simulated Time to run fast model Time to run conventional model 17 CFD SIMPLE algorithm 10 x 3.1 x 4 0.15 s   19 CFD ALE3D 1 x 0.1 x 0.04 585 µs   22 Thermal FEM Macroscale 3D ~s   23 Thermal FEM 10 x 10 x10  ~s   26 CFD SPH 51×12.7× 6.35 12 s   27 CFD LBM - 2D 1 x 2  2.5 ms   28 CFD LSM 200 x 30 x 15  3 s   29 Thermal Super-position based FEM 21 cm3 - 445000 elements 1 hour  4000 CPU hours  33 years 30 Thermal Enriched Analytical Method 15 x 10 x 10 0.6 s 0.1 s 15 min 31 Thermo-mechanical FEM 18 x 18 x 1.5 0.11 s 109 s 1490 s  32 Thermo-mechanical FEM 500 x 60 x 20 60 s 15 hr 75 hr 33 Thermo-mechanical FEM 3810 x 457 x 25.4 300 s 114 hr  34 Thermo-mechanical FEM 42 x 15 x 5 ~min 26.7 hr 45 hr 35 Thermo-mechanical Thermal Circuit Network 45 x 45 x 5 ~s 4 min  37 Thermo-mechanical Superlayers 75 x 15 x 12 4.2 s   38 Thermo-mechanical FEM  45 x 5 x 6 ~s   42 Thermo-mechanical FEM 45 x 22 x 1.15 10 s    23  In this chapter, the published studies on various simulation methods, including computationally efficient approaches, was reviewed and analyzed. The simulation times for multi-physics models that incorporate heat transfer, fluid flow and mechanical deformation mechanisms are very high and hence require significant computational resources. Such requirements make simulations at the part scale, increasingly provide diminishing returns due to simulation times being many times longer than the time taken to print a full part. However multi-physics simulations are important tools to aid in understanding melt pool dynamics, in-situ temperature gradients and evolution of strains and residual stresses. The computationally efficient models discussed in this literature review have been successful at reducing the simulation times to varying degrees of success. These approaches provide computational efficiency at the cost of spatial and temporal resolution and fidelity to the process. An opportunity exists to develop quick simulation schemes driven by data from multi-physics models. This should allow high resolution predictions of the local effects of the heat source while reducing the requirement for computational resources.        24  Chapter 3 -  Scope and Objectives It has been established that there is a need for computationally efficient and high-resolution thermal maps which account for the contributions of the various physical phenomenon dominant in AM processes. Using such simulations, optimal process parameters for each build can be determined on a realistic time scale. On a lower level such simulations can also form an integral part of in situ feedback control system that can process current thermal maps to predict the conditions in the next layer. One possible way to address this need is to develop a technique that utilises information from detailed AM simulation models, which incorporate various heat and mass transfer phenomena, and provide the same information with the same resolution but at a lower computational cost.  The method developed in this study aims to address this challenge. Dubbed the Fast-To-Run (FTR) model, this tool aims to eliminate the need to run a micro-scale FEA thermal model for a part which can take days/weeks/months to run, depending on the size of the part, by providing the same key build state (output) predictions in a fraction of the time (e.g. on the order of seconds). The FTR model will use data from a semi-infinite, microscale FEA model to “simulate” processing via additive manufacturing by correlating the resulting build state to process parameters like beam properties, beam paths and initial conditions of the build plate. The FTR model will then effectively replace the FEA model and provide the same information but at a significantly reduced computational cost. This will be achieved performing the following steps: 1. Running FEM simulations in ABAQUS over a range of processing conditions to understand the thermal response.  2. Developing relationships between process parameters and localized build states (Table 3.1).  25  3. Developing a framework which incorporates these relationships with beam motion to predict build states on a layer-by-layer basis.  Build states refer to snapshots of an important quantity or measure that is significant for determining the quality of the completed part. The build states that have been focused on in this work are: • Peak temperatures on the build layer surface: Peak temperatures are important because they can indicate hotspots and areas that might experience excessive evaporation losses leading to changes in alloy composition.  • Melt pool depth relative to all locations on the build layer surface: A map of the melt pool depths can be used to highlight areas of overmelting or undermelting.  • Temperature at all locations on the build layer surface at the end of processing a layer: This measure is very important because it functions as the initial condition for the new layer when new material is added.   Table 3.1: Important Process Parameters and Build Properties Process Parameters Build States Beam speed and path (Hatch and contour) Peak surface temperatures Initial conditions Rate of temperature change Beam parameters Temperature of processed layer Powder type and dimensions  Melt pool depth Predicting these quantities will not only help design better printing strategies by preventing excessive or insufficient heat application but may also help in real-time process control by understanding the effect of the heat signature of a printed layer on the next layer to be printed. It 26  must be noted that all these quantities are easily predicted using a full physics FEM model. However, as already mentioned before, full physics models are too cumbersome to be used even on the scale of just centimeters.              27  Chapter 4 -  Fast-to-Run Model Development 4.1 Exploiting the Steady State Prior to providing the details of developing a fast-to-run model, it is important to provide an overview of the Fast-to-Run (FTR) methodology and how it relates to traditional FEM/CFD based approaches. As mentioned previously, validated and verified FEM/CFD based simulations can provide detailed and accurate predictions of materials processing operations through careful application of appropriate boundary and loading conditions. However, in the context of additive manufacturing, the processing conditions are such that there are sections of the build where the melt pool achieves steady state. An example of this is when an electron beam traverses a straight-line path with no changes in direction or speed. When quasi-steady-state has been achieved, it is not necessary to calculate the temperatures at every location and at every instant, rather one could calculate the temperature distribution as a function of the distance from the beam centerline and depth in the sample. Additionally, the output of thermal FEM simulations, i.e., temperatures, can be understood as a function of the boundary conditions and applied loads. If a simple relationship can be derived between the input parameters and output temperatures, the problem of computing the solution as a function of time at millions of nodes can be reduced to a few variables and the computational speed can be greatly increased.  To demonstrate the FTR model concept, this study focuses on the simple case of re-melting a substrate by melting it with a heat source. For this case, the beam parameters are assumed to be constant over the duration of the scan and the material properties do not change across the domain (due to the absence of powder and any density difference that might occur because of powder distribution and densification during processing). This facilitates the calculation of a quasi-steady state temperature distribution and enables this to be exploited to train an FTR model. Figure 4.1 28  shows the temperature contours at two different time steps for a situation where a heat source scans the symmetry boundary of a simple domain. The temperature profiles along the dotted lines (perpendicular to beam motion) are the same at different positions along the sample. The dotted line starts at the location where the domain experiences the maximum temperature. As the heat source moves, this set of temperatures continues to sweep along the path of the heat source. Thus, for the simple case where a beam moves in a straight line across the domain, the maximum temperatures experienced by this domain can be easily described. Similar observations can be made about the temperature decay following the heat source and the depth into the substrate that melting occurs. These quantities are the key build states that the FTR model aims to predict.    Figure 4.1: Temperature contours at different time steps during scanning by heat source.  Using this simple approach, the effects of input process parameters on the build state (maximum temperatures, temperature history, and maximum melt depth) can be assessed. The quasi-steady-state condition may be predicted with a model over a wide range of parameters to understand the response of the build states to the various parameters and develop relationships between the input and output parameters. The analysis of the data from various simulations will guide the Maximum 29  development of these relationships (Figure 4.2). The next sections will detail the development of the FEM model in ABAQUS and discuss the dependence of the results on the input parameters.   Figure 4.2: Fast to Run model data flow 4.2 Model Independence The FTR methodology can be applied to any model where such quasi-steady-states exist. If a CFD approach is used to simulate heat transfer and fluid flow during AM instead of the thermal-only prediction of an FEM approach, the temperatures would reach a quasi-steady-state which could then be analyzed to develop appropriate relationships. Hence, for this study it is not important for the numerical model to provide exact correlations with experimental results. The model needs to provide spatially and temporally high-resolution data so relationships can be formulated. As a result, an ABAQUS thermal-only model with a fine mesh is used to generate the data necessary to demonstrate the FTR methodology. This model considers heat diffusion and uses temperature dependent properties. This model describes the complex non-linear behaviour of melting and solidification surrounding an intense heat source without the large computational requirements 30  associated with more detailed fluid flow models. The FTR approach seeks to replace the numerical model and hence becomes agnostic of the numerical model. Additionally, different physics-based approaches can provide different output parameters and their prediction can be added to the FTR model in a similar fashion. For example, a CFD simulation can provide fluid flow velocities, pressure changes, compositional changes and many other quantities. By understanding how these properties vary under the process range, more relationships can be obtained for these other build states. Hence, the applicability of this approach can be wide and far ranging.  4.3 Software Selection For this study, Abaqus (version 6.17), a commercial, general purpose finite element method (FEM) simulation package, was selected to simulate the heat transfer occurring in the EBAM process. Abaqus excels in solving non-linear problems and offers the capability to incorporate additional functionality through user-written subroutines. This flexibility was used to describe the heat source and process parameters. The material response to the process parameters was simulated for a wide range of process conditions representing those experienced in industry. Furthermore, the software provides a good platform to visualize results and generate data for further processing.   4.4 General Semi-infinite Model To generate the data necessary to formulate the FTR model, the heat transfer response of a characteristic, semi-infinite geometry was studied using an ABAQUS simulation. The general model is a representation of the substrate that is being affected by a beam, sized to be widely applicable. The geometry is chosen to satisfy a semi-infinite domain assumption with respect to the heating effects of the beam. This involves ensuring that the heat pulse resulting from beam heating does not reach the edges of the domain in the time simulated (Section 4.4.2). The transport phenomena considered in the semi-infinite model used for this study were limited to consider only 31  heat diffusion, i.e. a transient thermal-only model (ignoring radiation), in a powder bed process. The governing partial differential equation solved by ABAQUS is the transient heat diffusion equation in three dimensions and is presented in Equation (4.1).  𝜕𝜕𝜕𝜕𝜕𝜕�𝑘𝑘𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕� + 𝜕𝜕𝜕𝜕𝜕𝜕�𝑘𝑘𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕� + 𝜕𝜕𝜕𝜕𝜕𝜕�𝑘𝑘𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕� − 𝜌𝜌𝐶𝐶𝑝𝑝𝑑𝑑𝜕𝜕𝑑𝑑𝑑𝑑+ 𝑄𝑄 = 0                     …𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑑𝑑𝐸𝐸𝐸𝐸𝐸𝐸 4.1 where 𝜕𝜕 is the temperature (K), 𝑘𝑘 is the thermal conductivity (W/m/K), 𝜌𝜌 is the density (kg/m3), 𝐶𝐶𝑝𝑝 is the specific heat (J/kg/K), 𝑑𝑑 is the time (s) and 𝑄𝑄 is a volumetric heat source term (W/m3) associated with the latent heat of solidification and 𝜕𝜕, 𝜕𝜕 and 𝜕𝜕 are directions (m). Equation (4.1) is solved to predict the temperatures in the semi-infinite model geometry, given material properties and boundary and initial conditions.  A continuum approach has been taken and changes in volume and material properties due to powder densification are not considered in this model. Heating conditions representing Electron Beam AM processing have been implemented. A symmetry condition along the centerline is used to reduce the model size and the heat source is moved along the edge as shown in Figure 4.3. Additional complexity is introduced in the model in the form of temperature dependent material properties. The initial conditions and process parameters are also varied to study the behaviour of the model over a larger process window. The data generated using this simplified model will be used to demonstrate the FTR modeling approach. Future versions of the FTR model could be trained using data from a multi-physics model considering heat diffusion, convection and fluid flow.   32  4.4.1 Model Setup 4.4.1.1 Meshing The domain (10 x 10 x 5mm), shown in Figure 4.3, is meshed with the highest density of elements in the region directly affected by the beam to provide data with high spatial resolution. The dimensions of the fine mesh region are 1.02mm wide, 0.465mm deep and 10mm long. The mesh transitions to a coarse mesh outside of this region to reduce the total number of elements in the problem. The dense region is meshed with 8-noded hexahedral elements (DC3D8) and maximum element edge lengths of 30𝜇𝜇m. This is followed by a transition region of 4-noded tetrahedral elements (DC3D4) that links to the coarse hexahedral mesh with maximum element edge lengths of 1 mm.    Figure 4.3: Schematic of semi-infinite model showing domain and boundary conditions (a), Meshing and heat source motion (b). 4.4.1.2 Boundary Conditions The heat flux describing the beam heating conditions is incorporated using the DFLUX user-subroutine applied to the fine mesh region’s top surface. The heat source is defined in Section 4.4.1.4. The beam parameters used are described in Table 4.1. The remaining surfaces are subjected to adiabatic boundary conditions.  (a)  (b)  33  4.4.1.3 Material Properties The material selected for this study was Ti6Al4V based on its wide spread use in metal additive manufacturing. Temperature dependent material properties of Ti6Al4V were adapted from [43] with a reduced data point frequency to reduce ABAQUS interpolation times and hence improve calculation speed. The density, conductivity and specific heat are shown in Figure 4.4. The decrease in specific heat at 1000 K is related to the α-β transition of the alloy. The latent heat of melting / solidification is equal to 286 kJ/kg. Since this modelling approach was deliberately kept simple, an assumption was made that the latent heat is linearly distributed over the temperature range of 1868 K to 1898 K.   It is important to note that in reality latent heat release is in fact not linear [43]. However, for this case, which involves repeated cycling of temperatures and solid-liquid-solid phase changes, the assumption of a linear latent heat release is sufficient to satisfy the heat balance. Additionally, the added complexity of a non-linear latent heat release would require a much smaller maximum temperature change to resolve the non-linearity, increasing the computational time significantly. Hence, as the primary interest of this study is the peak temperatures, the maximum melt depths and the final temperatures, the exact distribution of the latent heat is considered less important and is assumed to be linear.  Another concern is the dependence of density on temperature. In the domain of a thermal problem, the volume remains constant and hence a reduced density of the domain implies a loss of mass from the domain, which does not correlate with the physics of the problem. However, at the same time a constant density would not take into account the phase change due to the heat input by the beam. The overall temperatures would be reduced as the thermal mass is now much higher than the case of less dense liquid domain. Once again, this is a choice that is up for much debate. Since 34  the volume of liquid is much smaller than the size of the domain, and the priority is a more accurate understanding of the temperatures and, consequently, the melt depths, the density is chosen to vary with temperature.   Figure 4.4: Material Properties for Ti6Al4V [43]. 4.4.1.4 Heat Source Definition – DLFUX The ABAQUS subroutine DFLUX was used to define the distribution and motion of the heat source on the top surface of the domain. The heat source was defined as a normal distribution scaled by the product of the power and efficiency and applied to the top surface of the domain shown in Figure 4.3 and traversed along the edge of the domain. The heat input at any location (𝜕𝜕,𝜕𝜕) when the centre of beam is at (𝜕𝜕t,𝜕𝜕t) is defined by Equations (4.2 and 4.3). The beam centre has been positioned on the center of each track. 𝐸𝐸 = 𝑃𝑃 × 𝜂𝜂2𝜋𝜋𝜎𝜎2 𝑒𝑒−� 𝑅𝑅2𝜎𝜎2�                                                …𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑑𝑑𝐸𝐸𝐸𝐸𝐸𝐸 4.2 where 𝑅𝑅 = (𝜕𝜕 − 𝜕𝜕t)2 + (𝜕𝜕 − 𝜕𝜕t)2                      …𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑑𝑑𝐸𝐸𝐸𝐸𝐸𝐸 4.3                           The beam parameters that affect the heat input to the domain are the beam power, 𝑃𝑃 (W), beam efficiency, 𝜂𝜂,  and the beam width determined by 𝜎𝜎 (m). The beam is assumed to move along the Thermal Conductivity (W/m/K) Specific Heat (J/kg/K) Density (kg/m3 ) 35  edge of the domain with a beam speed, 𝑣𝑣 �ms�. These controllable parameters affect the maximum surface temperature achieved and the size of the heat affected zone. For this study, the beam parameters employed are summarized in Table 4.1. The parameters selected represent the values usually used in EBAM processes [9]. Extreme parameters, especially high power values were not selected because the time steps required are very small to resolve the steep temperature gradients. Table 4.1: Beam parameters for general model. Parameter Value Power, 𝑃𝑃 100, 200, 300W Efficiency, 𝜂𝜂 1 Beam Width, 𝜎𝜎 250 x 10-6 m Beam Speed, 𝑣𝑣 0.25, 0.5, 0.75 m/s 4.4.1.5 Initial Conditions A uniform initial temperature is defined in the model as the starting temperature to represent the state of the powder layer following preheating prior to melt / sinter processing this location in the layer. This initial temperature can represent either the state of the powder following preheating, after powder addition, prior to layer processing or the local temperature of the powder affected by heat buildup from processing locations in proximity. The initial temperature has been varied over the range of 973K to 1373K (e.g. 700℃ to 1100℃) in 100 K increments. This is meant to cover the range of data required by the FTR model for further predictions and could be expanded if necessary. 4.4.1.6 Load Steps The model runs with three load steps (Table 4.2). The first load step considers the surface heating of the domain by the moving beam. The time was chosen such that the maximum temperatures stabilise while limiting the simulation time. Hence, the beam only runs halfway along the domain. 36  The beam was also run the complete length of 10 mm to confirm that the added heat will not affect the cooldown. The comparison is shown in Figure 4.5 and there is no change in temperature of the node at 5 mm (half way down the model edge) where the data is extracted from and hence this choice of beam run length is justified to reduce computational time. In this step, the maximum temperature change per increment is limited to 5 K to resolve the sharp gradients. The second step captures the cooldown right after the beam is switched off and has the same restriction on temperature change. Finally, the cooldown to a stabilized temperature is captured in the third step. This temperature would act as an initial condition for a subsequent pass on this layer. The time step size and maximum temperature change per increment are relaxed here to allow for faster computation as the gradients and cooling rates are lower.  Figure 4.5: Comparison of temperature decay curves with different beam run lengths. Table 4.2: Time stepping parameters. Parameter Step 1 Step 2 Step 3 Step Duration (s) 7x10-3 2x10-2 2.8x10-1 Initial time increment (s) 7.5x10-6 1x10-3 2x10-5 9001100130015001700190021002300250027000 0.05 0.1 0.15 0.2 0.25 0.3Temperature (K)Time (s)Beam run 5 mmBeam run 10 mm37  Smallest allowable time increment (s) 1x10-7 1x10-7 2x10-7 Maximum allowable temperature change per increment (K) 5 5 50 4.4.2 Semi-infinite Model To limit the computational requirements of the data generation process, it was important to minimize the size and number of degrees of freedom of the domain representing the process as much as possible. The size of the domain must be chosen in a manner such that the domain size remains small, but it needs to be large enough that the effects of heating by the beam do not reach the boundaries of the domain during the time of simulation. To verify the semi-infinite property of the general model, temperatures were extracted at various nodes to observe their response over time to the beam heat input.  The process parameters used for this run are P = 200 W, 𝜕𝜕𝑖𝑖 = 1373 K and 𝑣𝑣 = 50 mm/s. The contours in Figure 4.6 show the evolution of temperature over the time of the simulation. In this case, the beam is active and runs along the length of the domain for 0.02 s followed by a further cooldown of 0.28 s. During this time, the heat pulse barely reaches the limits of the domain. To verify this quantitatively, temperatures at nodes shown in Figure 4.7 were plotted and studied. These nodes were picked along the depth and the width of the domain to observe how quickly the pulse travels along the domain. Node 1 lies on the edge of the domain and experiences the maximum temperatures as it aligns with the beam center. Figure 4.7 shows the temperature variations over time for all nodes.  38   Figure 4.6: Temperature contours evolution over time.   t = 0.04s t = 0.40 t = 0.32 s 39   Figure 4.7: Temperature over time response for nodes along the depth and width of the semi-infinite model. The temperatures shown in Figure 4.7are studied in more detail in Figure 4.8 and Figure 4.9 by looking at smaller temperature ranges near the initial temperature of 1373 K. The vertical dotted line represents the duration of time that the beam is heating the model surface. Hence, during this time period there is no effect of the heat pulse on the outer boundaries of the domain (Nodes 4 and 7). However, at Nodes 2 and 5, an effect is observed. These nodes are the edges of the fine meshed region of the domain. Even at the middle of the domain (Nodes 3 and 6), the effect of the heat pulse is to raise the temperature by merely ~5 K from the initial temperature. The edge nodes 4 and 7 are barely affected by the heat pulse during the entire run time. Thus, this representative case shows that the heat pulse will not affect the edges of the geometry during the run time and hence this domain satisfies the semi-infinite condition.  1300.001500.001700.001900.002100.002300.002500.002700.002900.000.0000 0.0400 0.0800 0.1200 0.1600 0.2000 0.2400 0.2800 0.3200Temperature (K)Time (s)N: 1 N: 2 N: 3 N: 4 N: 5 N: 6 N: 740   Figure 4.8: Magnified temperature over time response for nodes along the depth of the semi-infinite model.  Figure 4.9: Magnified temperature over time response for nodes along the width of the semi-infinite model.  1370.001375.001380.001385.001390.001395.001400.000.0000 0.0400 0.0800 0.1200 0.1600 0.2000 0.2400 0.2800 0.3200Temperature (K)Time (s)N: 1N: 2N: 3N: 41370.001375.001380.001385.001390.001395.001400.000.0000 0.0400 0.0800 0.1200 0.1600 0.2000 0.2400 0.2800 0.3200Temperature (K)Time (s)N: 1N: 5N: 6N: 741  4.4.3 Data Collection A typical contour map of temperatures predicted by the model is shown in Figure 4.10. Temperature data representing the state of the domain can be extracted from the .odb files and then used to populate the database for training the FTR model. The data collected will be discussed in detail in the next chapter and the observed trends will be employed to develop relationships for the FTR model.   Figure 4.10: A typical contour map of temperatures generated by the semi-infinite model. 4.4.3.1 Temperatures Figure 4.11 shows a contour plot of temperature on the surface of the semi-infinite model when the beam is midway along its traverse of the edge. The surface temperatures are extracted at the line shown in Figure 4.11(a) that begins at the location of maximum temperature and perpendicular to the direction of the beam motion. The variations of temperature with respect to time are captured at the nodes highlighted in Figure 4.11(b). An example of a typical response can be observed in Figure 4.13 and Figure 4.14 (solid lines).   300 µm 42                   Figure 4.11: Locations of data extraction. 4.4.3.2 Melt Depths An iso-surface contour is used to find the melt depth and a typical plot is shown in Figure 4.12. A user defined temperature scale was used to highlight the boundary between the solid and liquid regions. The images are then processed in a digitizer to determine the melt pool profile using the mesh grid as a scale. Engauge digitizer software [44] was used for this process. The scale of the x and y dimensions of the image were defined by selecting two known points on each edge. The rightmost corner was chosen as the origin for these images. Using the known grid dimensions of 30 µm width and 35µm height, the depth is extracted at each vertical gridline.  Figure 4.12: Typical melt pool profile. (b)  (a)  𝜕𝜕 O 𝜕𝜕 60 µm 43  4.4.4 The Effect of Latent Heat An important effect observed in the temperature history and the melt pool profile is the distortion due to the latent heat release that occurs during solidification. When the temperature of a location in the domain cools to the liquidus temperature (1898 K), the temperature decrease becomes stagnant as latent heat is released. This is seen as a slight distortion near the liquidus temperature in Figure 4.13 and plateau in Figure 4.14. The beam parameters used for this example are Power = 200 W, speed = 25 m/s and initial temperature =1373 K. The melt pool profile is also much smoother and deeper, as seen in Figure 4.15. Hence, the effect of latent heat distorts the curves and make relationships harder to formulate.   Figure 4.13: Surface temperatures with and without latent heat.  44   Figure 4.14: Temperature vs time behaviour of the model with and without the effect of latent heat.   Figure 4.15: Melt pool profiles (a) with latent heat, (b) without latent heat. The beam travel direction is into the page.  (a)  (b)  150 µm 45  Chapter 5 -  Relationship Formulation  5.1 Effect of Initial Temperature As discussed earlier, the temperatures and melt depths were extracted for the domain over the range of process parameters. Initially, only the effects of changing the initial temperature were considered while the power and speed were kept constant at 200 W and 0.5 m/s.  The semi-infinite model was run with initial temperatures from 973K to 1373K with 100 K increments. The effect of the initial conditions on the surface temperatures and the temperature decay is discussed in the following subsections. The effect of changing the initial temperature has been captured by fitting the ABAQUS data over the range of initial temperatures while keeping the number of coefficients in the fitted relationships to a minimum.  5.1.1 Surface Temperature Figure 5.1(a) shows a contour plot of temperature on the surface of the semi-infinite model when the beam is midway along the edge. The effects of increasing the initial temperature on the surface temperatures as a function of radial distance from the center of the beam are shown in Figure 5.1(b).  The overall effect of increasing initial bed temperature is to increase surface temperatures. Due to the latent heat and temperature dependent material properties, a 100 K increase in the initial temperature does not cause a 100 K increase in peak temperatures. From Figure 5.1(b), it is apparent that the surface temperatures are approximately Gaussian in nature, mirroring the definition of the heat source. The arrow on contour plot corresponds to the order of temperatures on the graph. It is worth noting that near the solidification range (1868 to 1898K) the temperature curves are distorted due to the release of latent heat (range shown by dotted lines).  46   Figure 5.1: Effect of initial conditions on surface temperatures (a) arrow showing line along which nodal temperatures were extracted and (b) associated surface temperatures for P=200W, V=0.5m/s. The quantitative effect of changing the initial conditions is to offset the temperature profiles with a scaling factor. Thus, a Gaussian curve, given as Equation 5.1, was fit to this data to provide a relationship between the temperature experienced by the node and its distance from the center of the heat source (i.e. the model edge) and its initial temperature. Equation 5.1 expresses the Gaussian relationship, where the temperature 𝜕𝜕𝑟𝑟 is the temperature at a Euclidean distance 𝑟𝑟 from the beam center. 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 (Full Width Half Maximum) is the distance from the center where the temperatures achieve half their maximum value. 𝜕𝜕𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 is the peak temperature defined by Equation 5.2 where it is expressed as a function of the initial temperature 𝜕𝜕𝑖𝑖 and 𝜕𝜕0, the temperature the equation was originally defined at i.e. 973 K.  𝜕𝜕𝑟𝑟 = 𝜕𝜕𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 exp �− � 𝑟𝑟𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹�2�                             …𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑑𝑑𝐸𝐸𝐸𝐸𝐸𝐸 5.1 𝜕𝜕𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝  =  −4.974 × 10−4(𝜕𝜕𝑖𝑖 − 𝜕𝜕0)2  −  0.1579(𝜕𝜕𝑖𝑖 − 𝜕𝜕0)  +  2485    …𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑑𝑑𝐸𝐸𝐸𝐸𝐸𝐸 5.2 A comparison of the fit between ABAQUS values and the fit from Equation 5.1 and 5.2 are shown in Figure 5.2.                 (a)                                                                                                                  (b)                                  47   Figure 5.2: Comparison between fit results and ABAQUS results for P=200W, V=0.5m/s. 5.1.2 Temperature Decay The time-dependent temperature history was also extracted from semi-infinite model. Figure 5.3 shows the temperature history at a 3 mm down the (x-direction) center line of the heat source for each initial temperature. The effect of changing initial temperature is to offset the temperature history like the effect of distance on the surface temperature. When looking at the temperature decay in the direction transverse to heat source movement, it is evident that the temperatures of all nodes decay rapidly to the initial condition (substrate temperature) in about 0.05 s (refer to Figure 5.4(a)). Additionally, in Figure 5.3, the effect of the latent heat is seen as the plateauing of the temperature curves near the melting point. The melting range is indicated by the dashed lines.  48   Figure 5.3: Temperature change over time on symmetry line for P=200W, V=0.5m/s at x=3mm.  Figure 5.4: Temperature decay over time (a) and nodes corresponding to temperature curves (b) for P=200W, V=0.5m/s and Ti=973K. Assuming, the temperature reaches a maximum at 0 s, the temperature decay behaviour for the center line as a function of initial temperature and time can be modelled as a combination of a                (a)                                                                                                                            (b)                                  N1 N7 49  power law relationship between 0 to 0.006 s followed by another relationship for 0.006 s onwards. The two equations capture the significantly differently temperature gradients in these regions.  𝜕𝜕𝑑𝑑𝑝𝑝𝑑𝑑𝑝𝑝𝑑𝑑 = �0.1654 𝑑𝑑−1.46 +  𝜕𝜕𝑖𝑖 , 𝑑𝑑 ≤ 0.006 𝑠𝑠 1.905 𝑑𝑑−0.98 + 𝜕𝜕𝑖𝑖 ,              𝑑𝑑 > 0.006 𝑠𝑠                …𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑑𝑑𝐸𝐸𝐸𝐸𝐸𝐸 5.3 The comparison of this fit with the ABAQUS results are shown in Figure 5.5. While this fit offers a convenient way to calculate the temperature decay, it is unable to model the plateau due to the latent heat and the temperature rise. However, for this case the plateau is less than 0.002 s for the highest initial temperature (1373K) and for the purposes of this study not significant to capture. For locations away from the centerline, the gradients are not as steep (Figure 5.4(a)) as those at the centerline and can be modelled by a power law relationship whose coefficients are calculated using the peak temperature calculated from Equation 5.1 and the temperature at 0.04 s calculated by Equation 5.3. After 0.04 s, Equation 5.3 can be used as usual as all temperatures over varying distances from the centerline follow the same curve.   Figure 5.5: Comparison between fit results and ABAQUS results for P=200W, V=0.5m/s. 50  5.1.3 Melt Pool Depths The melt pool depth profile has been extracted from the semi-infinite model for each initial temperature. Figure 5.6 shows the variation of melt pool depth as a function of distance from the beam centerline and initial temperatures. There is irregularity in the melt depth values extracted as a function of distance due to the release of latent heat at the boundary between the liquid and solid and a plateauing effect similar to that described earlier. Upon further inspection, a similar effect can be observed around the edges of the melt pool for Ti = 973 K in Figure 5.7.  Figure 5.6: Variation of melt depth over distance from the beam center and initial temperature for P=200W, V=0.5m/s.   Figure 5.7: Temperature contour showing the melt pool boundary for P=200W, V=0.5m/s, Ti = 973K. 450 µm 51   The maximum depth is predicted at the centerline of the beam and the melt pool depth decreases with distance from the centerline. A fitting procedure was performed to link melt pool depth with distance from the beam centerline. This relationship is expressed by Equation 5.4 and 5.5. 𝐷𝐷𝑝𝑝 is the maximum depth as a function of the initial temperature and other symbols have the same definition as those in previous figures. A comparison of the fit is shown in Figure 5.8.  𝐷𝐷𝑒𝑒𝐷𝐷𝑑𝑑ℎ = −0.1077𝑟𝑟2 + 𝐷𝐷𝑝𝑝                                …𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑑𝑑𝐸𝐸𝐸𝐸𝐸𝐸 5.4 𝑤𝑤ℎ𝑒𝑒𝑟𝑟𝑒𝑒 𝐷𝐷𝑝𝑝 =  0.97𝑒𝑒1.4×10−3( 𝑇𝑇𝑖𝑖−𝑇𝑇0)                        …𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑑𝑑𝐸𝐸𝐸𝐸𝐸𝐸 5.5  Figure 5.8: Comparison of melt depth between fit results and ABAQUS results for P=200W, V=0.5m/s. 5.2 Effect of Varying Power, Speed and Initial Temperature Once the relationships were established for the case of fixed power and speed and varying initial temperatures, further simulations were performed to observe the effect changing all the other two 52  variables while keeping one constant. The Figure 5.9 to Figure 5.17 show the results for these simulations.    5.2.1 Power = 100W The power level of 100W is on the lower end of the range of powers studied. As can be seen from Figure 5.9 and Figure 5.10, near the melting temperature, there is a flattening of the surface temperature curves. At lower temperatures and higher speeds, the heat input is not enough to cause melting. In Figure 5.11, it is seen that for the cases where 𝜕𝜕𝑖𝑖 =  973K  and speed ≥ 50 mm/s, and where 𝜕𝜕𝑖𝑖 =  1173K  and speed = 75 mm/s, melting does not occur. These conditions hence form the lower extreme boundary for the working of the FTR model. The emphasis is placed on training the model for conditions where melting does occur.   Figure 5.9: Surface temperatures for Power = 100W and varying initial temperatures and speeds.  53   Figure 5.10: Temperature change over time for Power = 100W and varying initial temperatures and speeds.  Figure 5.11: Melt depth for Power = 100W and varying initial temperatures and speeds. 54  5.2.2 Power = 200W A smaller subset of the 200W data was studied in the previous section, i.e. for speed = 50 mm/s.  Figure 5.12 and figure 5.14 show the data over the complete process window. Over this range, the irregularities increase with longer plateaus especially in Figure 5.13. The procedure used previously proves to be tedious to model the effect of four independent variables. Additionally, the melt depths in Figure 5.14 are also more irregular as the profile gets flattened again because of latent heat release.   Figure 5.12: Surface temperatures for Power = 200W and varying initial temperatures and speeds. 55   Figure 5.13: Temperature change over time for Power = 200W and varying initial temperatures and speeds.  Figure 5.14: Melt depth for Power = 200W and varying initial temperatures and speeds. 56  5.2.3 Power = 300W  Increasing the power level and reducing the speed also leads to sharper peaks in temperature both spatially and temporally and widening and deepening of melt pools. For the case of Power = 300W, the results are like the ones seen for the other power levels in relation to the irregularities.  However, at process conditions of higher heat input especially at speed = 25 mm/s, (Figure 5.15 and Figure 5.17) these irregularities smoothen out because the temperatures are significantly higher than the melting point. At the same time the plateaus during the cooling are much longer as seen in Figure 5.16.   Figure 5.15: Surface temperatures for Power = 300W and varying initial temperatures and speeds. 57   Figure 5.16: Temperature change over time for Power = 300W and varying initial temperatures and speeds.  Figure 5.17: Melt depth for Power = 300W and varying initial temperatures and speeds. 58  5.3 Interpolation Based Approach A careful analysis of the figures in Section 5.2 reveals that the simple approach used in the previous sections to model the temperatures and melt depths will not be enough to model the same quantities over this wider process window. At the extreme edges of this window, i.e., low speeds, higher power and higher initial temperatures, more heat is added into the domain and the effect of latent heat is much more pronounced. Additionally, at the other extreme, high speeds, lower power and lower initial temperatures, the added heat is sometimes not enough to melt the surface. The approach followed in Section 6.1 works well for limited process parameters, but following the same approach for varying power, speed and temperature proves to be inaccurate. Thus, a more general approach has been taken by performing multivariate linear interpolation over the collected data. The computational effort is however more than the simple approach and will be quantified and studied in detail in the next chapter.  The interpolation-based approach uses 3 matrices to store the data for i) the temperature along the width of the beam – location dependent temperature data, ii) the temperature rise/decay over time for the central node – time dependent temperature data, and iii) the maximum melt depth along the width of the beam – location dependent melt depth data. The data shown in Figure 5.9 to  5.17 is stored in three different matrices. For each matrix, four input variables must be provided to access the data point in the respective matrix, i.e. location dependent temperature data, time dependent temperature data or location dependent melt depth data. The location dependent temperature data matrix is the combination of the data shown in Figure 5.9, Figure 5.12 and Figure 5.15. The values stored are the temperatures on the surface of the domain, along the beam width and the axes are the process parameters, beam power, beam speed and initial temperature, and the distance from the center of the beam. In this context, axes refer to numerical axes of the matrix and not the axes 59  of the 3D system (x, y and z).The time dependent temperature data matrix is the combination of the data shown in figures 5.10,  5.13, and  5.16. The values stored are temperatures of the edge node (at beam center) varying with time and the axes are the process parameters and time since the beam was directly over the node, i.e. when the node achieved the peak temperature. The third 4D matrix has the maximum melt depth values normalized over the height of a typical EBAM powder layer – 70 µm, shown in Figures 5.11, 5.14 and 5.17. The axes are the process parameters and the distance from the center of the beam.  The interpolation approach is programmed in Python 3.7 [45] and uses the class RegularGridInterpolator from the Scipy library [46]. The class uses n-linear interpolation to determine the value of the dependent variable for the given tuple of n independent variables. The program essentially functions as a look up table for the generated data. The method works by finding the 2n points near the point of interest and then successively interpolating along each independent variable axis to determine the dependent quantity for the point of interest. The point of interest is expressed as a tuple of the process parameters and either the distance from the center of the beam or the time since the beam affected the node. For example, if one would like to find the temperature of a node when the beam power is 200 W, beam speed 0.5 m/s and initial temperature 973 K and 2 x 10-4 m away from the center of the beam, the corresponding tuple would be [200, 0.5, 973, 2.5 x 10-4]. In this case the interpolator class would find 24 = 8 points around the point of interest and provide 2026 K as the temperature output.  If the data point is outside the range of training data, the closest highest or lowest data point is used instead.   60  5.4 Comparing Quality of Fit  5.4.1 Graphical Comparison The FTR program was populated with training data and then tested on unseen verification data points, namely for P = 200W, speed = 50mm/s and 𝜕𝜕𝑖𝑖 = 1073 K and 1273 K. For the surface temperatures predictions shown in Figure 5.18, the ABAQUS data was closely approximated by the linear interpolation approach. The interpolated fit was even closer than the simple curve fit presented in Section 5.1 and it also captured the plateauing of the temperatures near the melting point of Ti-6Al-4V. For the temperature change with time, shown in Figure 5.19, the linear interpolation approach was able to predict the behavior observed for 𝜕𝜕𝑖𝑖 = 1073 𝐾𝐾 and 1273 𝐾𝐾 very closely. When compared to the simple curve fit (Figure 5.5), it was also able to capture the rise in temperature and better predict the plateau. The comparison for the melt depth data is shown in Figure 5.20. The simple curve fit and the linear interpolation perform quite similarly for this quantity.   Figure 5.18: Comparison of ABAQUS, curve fit and linear interpolation results for surface temperatures. 61   Figure 5.19: Comparison of ABAQUS and linear interpolation results for temperature change over time.   Figure 5.20: Comparison of ABAQUS, curve fit and linear interpolation results for melt depth profile. 62  5.4.2 Numerical Comparison The quality of the two fits was also measured numerically by comparing two important error metrics, Mean Square Error (MSE) and Mean Absolute Percentage Error (MAPE). MSE (Equation 5.5) calculates the average of the square of the difference between the ABAQUS results (𝜕𝜕) and the fit results (𝜕𝜕�). The error grows quadratically and hence this metric penalizes those predictions more that stray far from the ABAQUS values. It is also important to note that the unit of the MSE is the square of the unit of the measured quantity. MAPE (Equation 5.6) is the average percentage of the absolute difference between the ABAQUS results and the fit predictions. This metric quantifies how far the predicted data deviates from the ABAQUS data.  𝐹𝐹𝑀𝑀𝐸𝐸 = 1𝐸𝐸�(𝜕𝜕 − 𝜕𝜕�)2                                 …  𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑑𝑑𝐸𝐸𝐸𝐸𝐸𝐸 5.5 𝐹𝐹𝑀𝑀𝑃𝑃𝐸𝐸 = 100%𝐸𝐸��𝜕𝜕 − 𝜕𝜕�𝜕𝜕�                                 …  𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑑𝑑ion 5.6  The calculated MSE and MAPE values for the simple curve fit and linear interpolation approaches are summarized in Table 5.1. The linear interpolation approach performs significantly better for the temperature predictions. There is very little difference in the MSE and MAPE values between the two schemes for the melt depth data. The depth data is more irregular and has the least number of data points, which makes predictions more difficult. The irregularity in the melt depth data is caused by the irregular contour results due to the release of latent heat as discussed in Section 5.1.3.   63  Table 5.1: Comparison of error metrics. Predicted Quantity Fit Type Curve Fit Linear Interpolation MSE MAPE MSE MAPE Surface temperature 362.03 0.855 39.86 0.166 Temperature change with time 169.32 0.297 15.05 0.054 Melt depth 4.68 6.534 6.96 6.061  5.5 Summary Relationships have been formulated from the data collected from the FEM model run over the process window. Initially, an attempt was made to build the relationship using an explicit curve fitting approach. While the equations could be fit to limited and well-behaved data, such an exercise proved to be tedious to perform over the entire process window. Thus, an interpolation-based approach was developed which was able to predict the temperatures and melt depths but requires operating on the entire range of data. Both the approaches performed well in terms of accuracy, with the interpolation-based approach being overall better as it was able to model more features especially temperature rise and plateau in the time dependent temperature data. The interpolation-based approach is expected to be more computationally expensive but more accurate. In the following chapter, a combination of the schemes will be used in an effort to minimize computational time and maximize accuracy.      64  Chapter 6 -  Validation – Line Scan Example The data and relationships collected from the semi-infinite model establishes the effects of the beam on the peak temperature, temperature history, and melt depth as a function of distance from the beam center. The initial temperature at a location is determined by the initial bed temperature (temperature of a layer following preheating) and any transient, local temperature change resulting from heating caused by the heat source as it moves across the domain. Figure 6.1 shows the flow of information in the FTR model. The semi-infinite model and the equations generated in Section 4 form the first dashed box labelled ABAQUS.   Figure 6.1: FTR model process flow. The FTR predictive model takes the data generated by ABAQUS as an input in addition to the beam location data and initial / local temperature. The beam location data provides the location of the beam at every time step and is extracted from either hard-coded beam paths or based on G-code information. Combined with the temperature of the current location, the FTR model is then able to interpolate the key quantities discussed earlier, namely temperature at the end of processing, peak surface temperatures and maximum melt depth experienced.   65  To verify the FTR approach, a simple processing case where a heat source is moved over a substrate bed along two tracks was defined. The heat source path is shown schematically in Figure 6.2(a). The heat source moves in a line along the bottom half of the domain prior to returning along a line in the top half of the domain (refer to path and the directions indicated by the arrows in Figure 6.2(a)). The temperature history at Nodes 1 and 2 are shown in Figure 6.2(b). As the beam approaches the nodes, a temperature spike is observed which depends on the proximity of the heat source to the node. This can be seen in the difference of the temperature spikes at Node 1 and Node 2. The temperature spike is followed by a temperature decay where the nodes quickly achieve the same temperature and cooling rate as the heat diffuses away. As the beam returns, the spike is again observed, though at this time, Node 2 exhibits the highest temperature. As expected, the temperature at the nodes is dependent on their location with respect to the heat source and on the time relative to the motion of the heat source. This spike-cooldown response is the characteristic heat transfer response that can be described by the FTR model.                    (a)  66   (b) Figure 6.2: (a) Motion of heat source (b) Corresponding temperature profile. 6.1 Model To evaluate the accuracy and efficiency of the FTR model for the line scan case, the model was created with the same mesh density as the semi-infinite model (Section 4.4.1.1). The material properties are the same as shown in Figure 4.4. The model represents a small Ti6Al4V substrate where the electron beam is rastered in the manner shown in Figure 6.2(a). The entire domain was subjected to the initial temperature of 973 K and the electron beam was applied to the surface of the fine mesh region. The DFLUX parameters are the same as those described in Section 4.4.1.4. 6.2 Mesh The domain (45 x 45 x 5mm), shown in Figure 6.3 and Figure 6.4, is meshed with the highest density of elements in the region directly affected by the beam to provide data with high spatial resolution. The dimensions of the fine mesh are 6 mm wide, 0.21 mm deep and 10.8 mm long. The mesh transitions to a coarse mesh outside of this region to reduce the total number of elements in 9001100130015001700190021002300250027000 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04Temperature (K)Time (s)Series1 Series267  the problem. The dense region is meshed with 8-noded hexahedral elements (DC3D8) and maximum element edge lengths of 30𝜇𝜇m. This is followed by a transition region of 4-noded tetrahedral elements (DC3D4) that links to the coarse hexahedral mesh with maximum element edge lengths of 3 mm.    Figure 6.3: Domain size and boundary conditions.  Figure 6.4: Fine mesh domain size. 5 mm 45 mm 45 mm 10.8 mm 6 mm 0.21 mm 68  6.3 Boundary Conditions The heat flux defined by DFLUX acts on the fine mesh region with the track lengths being 5mm each. The heat source definition is the same as that defined in Section 4.4.1.4. The beam parameters used in this case were P = 200 W and V = 0.5 m/s. The remaining surfaces are subjected to the adiabatic boundary conditions.  6.4 Time Steps The model runs in two steps each corresponding to the two tracks described by the beam. The time is calculated based on the distance to be travelled (10 mm) and the speed (0.5 m/s). The maximum temperature change is limited to 5 K to resolve the latent heat release.  Table 6.1: Time stepping information for FEM simulation. Parameter Step 1 Step 2 Step Duration (s) 2x10-2 2x10-2 Initial time increment (s) 7.5x10-6 7.5x10-6 Smallest allowable time increment (s) 1x10-7 1x10-7 Maximum allowable temperature change per increment (K) 5 5 6.5 FTR Flow The FTR model was written in Python 3.7 incorporating the data and interpolation scheme shown in Chapter 5. First a domain representing the surface where the beam will be operating (e.g. the fine mesh region in Figure 6.3) was created as a mesh array of points and the temperature at each point was initialised to Ti = 973 K.  At each time step of 0.00025 s, the location of the beam is determined from a pre-set path function. The path function can be in the form of a function that outputs beam location with the time as 69  input, list of locations of the beam at different time steps or GCode information supplemented with beam speed and power information. In this case a function defined by Equation 6.1 was used. Here 𝑣𝑣 is the speed, 𝑑𝑑 is the time at which beam location is desired, (𝜕𝜕0,𝜕𝜕0) is the initial beam location, 𝑑𝑑𝑤𝑤 is the track width, 𝑙𝑙𝑝𝑝𝑒𝑒𝑝𝑝𝑒𝑒𝑝𝑝𝑒𝑒𝑒𝑒 is the length of the cuboidal element, (𝜕𝜕𝑒𝑒 ,𝜕𝜕𝑒𝑒) is the final calculated location. The two cases shown in the equation are for the two different tracks.  𝐵𝐵(𝑑𝑑) =⎩⎪⎨⎪⎧ 𝐸𝐸𝑖𝑖 𝑑𝑑 < 0.02 𝑠𝑠, �𝜕𝜕0 + 𝑣𝑣 ×  𝑑𝑑𝑙𝑙𝑝𝑝𝑒𝑒𝑝𝑝𝑒𝑒𝑝𝑝𝑒𝑒𝑒𝑒 ,𝜕𝜕0� 𝐸𝐸𝑖𝑖 𝑑𝑑 ≥ 0.02 𝑠𝑠, ��𝜕𝜕0 + 0.02𝑣𝑣 − 𝑣𝑣 × (𝑑𝑑 − 0.02)𝑙𝑙𝑝𝑝𝑒𝑒𝑝𝑝𝑒𝑒𝑝𝑝𝑒𝑒𝑒𝑒 � , ( 𝜕𝜕0 + 𝑑𝑑𝑤𝑤)� = (𝜕𝜕𝑒𝑒 , 𝜕𝜕𝑒𝑒)  …𝐸𝐸𝐸𝐸𝐸𝐸 6.1 The surface temperatures are then calculated at a subset of the surface points surrounding this location (±750 µm in each direction).  The size of the subset or distance from the center of the beam to update the surface temperatures was determined from the surface temperature data (Figure 5.1). Then, the maximum melt depths are calculated and stored. Every time a node is considered for calculation the program saves the current time step. This allows the program to check if the current node has been affected by the beam at an earlier time step. If it has been affected, the change of temperature as a function of time data is used to determine the current temperature of the location. The current temperature is then used as the input in the interpolation function to calculate the new, higher temperature due to the beam’s presence. The time input is the difference in the current time and the time the beam affected the node last. The new temperature is calculated for the node and if it is found to be higher than the previously stored temperature, the maximum melt depth is also re-calculated and stored. This process can be seen in the dotted rectangle in 70  Figure 6.5 and is repeated until all nodes near the center of the beam (±750 µm in each direction) are updated.  At this point, the program moves forward in time and extracts the next location of the beam and the process repeats. The program terminates when all the time steps and the corresponding beam locations are iterated over. The entire FTR process flow can be summarized using the flowchart shown in Figure 6.5.   Figure 6.5: Information flow in the FTR model. 71  6.6 Comparison 6.6.1 Temperature Decay The temperatures predicted using the FTR model at a location midway between the two heat source traces are compared to the ABAQUS predictions for the verification case in Figure 6.6. The temperatures calculated with the FTR method follow the ABAQUS prediction closely for the most part. Specifically, the FTR method accurately predicts the maximum temperature (maximum ~4% error) for each rise in temperature. The FTR method is able to accurately predict the substrate temperature prior to the second heat source pass, but slightly under predicts the substrate temperature after the second heat source pass.  The FTR method also over predicts the temperature during the decay after the heat source has passed.  Figure 6.6: Temperature decay comparison for Node 1 as shown in Figure 6.2(b) for 2-Heat Trace verification case. 9001100130015001700190021002300250027000 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04Temperature (K)Time (s)ABAQUS FTR72  6.6.2 Temperature Contours The temperatures calculated using the FTR method can be plotted in a contour plot at each time. This feature can be used to capture a snapshot of the temperatures at the end of processing a layer. Figure 6.7 shows the comparison between the temperature contours produced by ABAQUS and FTR models at 0.015 s.   Figure 6.7: Surface temperature at 0.015s calculated by (a) ABAQUS and (b) the FTR method for 2-Heat Trace verification case. a) ABAQUS b) FTR 15 mm 73  6.6.3 Maximum Temperature and Melt Depths Contour plots of the final maximum temperatures of the substrate are shown in Figure 6.8. The effects of the two heating passes (traces) is clearly evident in the maximum temperatures experienced by the substrate.  Along the path of the 1st trace, the maximum temperature varies with distance from the 1st trace centerline and is uniform along most of the length of the 1st trace on the side opposite the 2nd trace. Higher temperatures are observed to occur initially during the 2nd trace because of the elevated initial temperatures due to the heat from the first trace. The initial temperatures are outside of the range of temperatures on which the FTR method was trained on and they are not easily described by the relationships formulated. This is a limitation for the FTR method and is the cause for the difference in the predicted maximum temperatures at start of the second trace in Figure 6.8(b). Additionally, the FTR method cannot account for the beginning of the electron beam motion as it always assumes a steady state. While the ramp up of power due to machine control lag can be programmed, the material response lag has not been estimated in this study.   Figure 6.9 shows the cross-sectional profiles of peak temperature and Figure 6.10 shows the maximum melt depth at a location mid-way along the heat source traces (x=5.3mm). The FTR method accurately predicts the peak surface temperature in proximity to the heat source path.  The FTR method effectively captures the effect of the heat buildup as the heat source returns which can be seen with the higher peak temperatures and melt depths shown for the second heat source trace. However, away from the beam center it over predicts the temperature. The FTR model interpolates the temperature as a curve based as a function of the initial temperature. The over prediction is likely caused by heat from the 1st trace diffusing into the volume affected by the 2nd trace. The melt depths are more difficult to accurately capture due to the irregularity of the curves. 74  However, the maximum depth for the 2nd trace calculated by the FTR method is very close to that predicted by ABAQUS (maximum ~3% error).   Figure 6.8: Maximum surface temperature contours calculated by (a) ABAQUS and (b) the FTR method for 2-Heat Trace verification case. a) ABAQUS b) FTR 15 mm 75   Figure 6.9: Maximum temperatures predicted by ABAQUS and the FTR Method mid-way along the heat trace (x = 5.3 mm) and perpendicular to beam travel direction for the 2-Heat Trace verification case.  Figure 6.10: Maximum melt depths predicted by ABAQUS and the FTR Method mid-way along the heat trace (x = 5.3 mm) and perpendicular to beam travel direction for the 2-Heat Trace verification case. 9001100130015001700190021002300250027000 0.0005 0.001 0.0015 0.002Temperature (K)Distance (m)ABAQUS FTR Beam Trace 1 Beam Trace 200.20.40.60.811.20.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016Layers melted (normalized)Horizontal distance (m)ABAQUS FTR Beam Trace 1 Beam Trace 276  6.6.4 Run Time and Error Comparisons The ABAQUS model of the 2-Heat Trace verification case required 191 hours (~8 days) of computation time on twelve Xeon CPUs (2292 CPU hours), whereas the FTR method required 300s using Python 3.7 on a Dell XPS 8930 to compute the maximum temperatures and melt depths for all the time steps, resulting in a dataset with the same surface spatial and temporal resolutions as the ABAQUS model. The time savings are on the order of 23000. The MSE and MAPE errors for the 2-Heat Trace verification case are summarised in Table 6.2. The FTR method also performs fairly accurately with the melt depths as the worst metric with an average error of 6.9% (Table 6.2). Table 6.2: Error comparison for 2-Heat Trace verification case Predicted Quantity Error Metrics MSE  MAPE Surface temperature  990.4 1.385% Temperature change with time  124.2 0.537% Melt depth  12.83 6.872%      77  Chapter 7 -  Applications In the previous chapter the FTR approach was verified by comparing it to an ABAQUS simulation of a simple 2-heat trace case on a small domain. In this chapter, some more realistic situations will be explored using the FTR model. 7.1 Square Layer with Diagonal Hatching A common build strategy for parts is to perform a single contour followed by a 45˚ hatch. Figure 7.1 shows the contour and hatch pattern applied to a 2 cm square shape (a) and the FTR predictions of maximum temperature and melt depth profiles, (b) and (c) respectively. The outer contour was generated with a speed 0.28 m/s, followed by hatching at 0.5 m/s at a power of 300 W. The hatch spacing used is 0.2 mm. The initial plan of action was to compare the FTR results to an ABAQUS simulation. However, the order of time required to simulate a hatch pattern so dense would be on the order of months. Hence, only the FTR results are shown to highlight the features it can simulate. In Figure 7.1 (b) and (c), the effect of the slower speed on the boundary contours can be seen. The maximum temperatures and melt depths are higher: 3975 K and 261 µm in the contour regions versus 3810 K and 200 µm in the hatch regions. Additionally, in the upper right and lower left corners and near all four sides the maximum melt depths and temperatures are higher too as the beam retraces its path. The areas near the corners experience a higher average temperature and melt depths because the beam retraces its path much quicker than at the sides due to the shorter hatch lengths. Hence, the FTR model can also be used to identify hotspots which can lead to defects. Also, since the maximum temperatures and melt depths are both a function of the same variables, Ti, distance from the heat source center and time, the contours are very similar but differ in terms of scale. 78   Figure 7.1: (a) Square contour and hatching pattern, (b) Temperature profile from FTR, (c) Melt Depth from FTR. 7.2 Effects of Hatching Strategy The simulation in the previous section showed how hatching can affect temperatures and melt depths achieved during the build and brings up the question of the effects of different hatching strategies on part quality. In this section, the effects of different contouring and hatching strategies are studied and compared to a study which correlated the effects of hatching strategies to porosity 2 cm (a) (b) (c) 79  distributions [47]. In this study, it was found that the location of the pores formed during Selective Electron Beam Melting (SEBM) showed a strong correlation with the beam strategy used, i.e., extent of contouring or infill by hatching. The samples were characterised using XCT imaging and were built on an Arcam S12 SEBM machine using Arcam Ti6Al4V feedstock powder. In the Arcam process, a 2D section slice is melted by outer contours and inner hatching. The outer contours are melted slower and with a lower power as compared to inner hatches. The beam actually moves much faster to maintain 10-50 melt pools over the contour using the Arcam MultiBeam technology. During hatching, the melt pool speeds are faster and the beam power higher. The beam is also more focussed during contouring and less focussed during hatching. Combining the above effects, the energy density is higher during contouring than during hatching. The porosity characterized via XCT was mainly small spherical gas pores that the authors suggest results from less opportunity for gas to escape at lower energy densities. Consequently, the number of pores was significantly larger in hatched regions. As a comparison to the FTR model, the metric that correlates the best with input energy density is the maximum temperatures reached by the domain. A higher energy density correlates to either a higher applied power or a lower speed of the heat source, both of which have an effect of increasing the peak temperatures, as seen in Section 5.2. In the study, eight combinations of contours and hatching were studied of which six were simulated by the FTR model. The other two involved the proprietary turning function by Arcam which changes the speed of the beam at the end of a hatch line and were excluded. The simulated scanning strategies are shown in Figure 7.2. The strategies are: (a) the standard strategy which performs three contours first followed by snaking hatches, (b) contours only, (c) hatching only, (d) five contours followed by snaking hatches, (e) three contours followed by uni-directional hatches and 80  snaking hatches first followed by three contours (not shown in Figure 7.2). The samples printed were 10mm wide, 10mm long and 25 mm high. The porosity imaged by the XCT scan of the entire volume was projected on to an x-y plane to better visualize the pore densities. The results are shown in Figure 7.3 (a-f). It is apparent from Figure 7.3 that the density of porosity in the hatched regions is much higher than in contour regions.  Figure 7.2: Scanning strategies. (a) Standard, (b) contour only, (c) hatch only, (d) five contours and (e) uni-directional hatch [47]. The scanning strategies were simulated with the FTR model to compare the temperature and melt depth maps to the porosity measurements. The aim was to verify if a correlation existed, even if it was a qualitative one. The conditions used in the study were outside the process window that the FTR model was trained for. However, the closest values were selected and are compared in Table 7.1.  81    Figure 7.3: Projected pore densities. (a) Standard, (b) contours only, (c) hatch only, (d) contour x 5, (e) uni-directional hatch and (f) hatch first (contour regions in darker grey) [47]. (e) 82  Table 7.1: Process parameters. Parameter Tammas-Williams et al. FTR Model Power, 𝑃𝑃 300-342 W 300 W Substrate Temperature, ℃ 730 730 Beam Width, 𝜎𝜎 Variable Constant Beam Speed, 𝑣𝑣 0.28 (hatch), 0.324 (contour) m/s 0.28 (hatch), 0.324 (contour) m/s The results of the FTR simulations for the maximum temperature and the maximum melt depths for the various hatching strategies are shown in Figure 7.4 and Figure 7.5, respectively. It can be immediately seen that the slower speeds during the contouring lead to higher maximum temperatures and higher melt depths. Comparing them to Figure 7.3, the contour regions show lower number of porosity defects. Thus, it can be inferred that the areas exposed to higher energy densities are less likely to develop porosity defects, but at the same time these heating conditions could lead to other defects like key-holing or compositional changes due to evaporation losses. When the number of contours is increased, the area exposed to higher energy density also increases. In the contour only configuration the energy density is high throughout the region. The heat deposited continues to build up as the contour traces reach the center. Consequently, the depth of melt also continues to increase and the porosity density is lower.  When hatching is used, the temperatures are on average lower in that region. However, when the heat source reverses direction along an adjacent track, the energy density is higher. The overall lower energy density in the region leads to higher porosity. When the region is hatched in one direction, limiting the effects of prior heating by the beam, the temperature distribution is more uniform. However, when the heat source reaches the outer contours, the lack of overlap results in reduced maximum temperature and may lead to incomplete fusion defects. The concentration of 83  defects towards the left end in Figure 7.3 (d) can be compared to the lower maximum temperature and melt depths in Figure 7.4 Figure 7.5 (Uni-direction hatch).  Figure 7.4: FTR simulated maximum temperature for different hatching strategies. 84   Figure 7.5: FTR simulated maximum melt depth for different hatching strategies. This set of simulations and their comparison to porosity studies reveals that even a simple thermal simulation can help correlate heat input to defect formation. Areas of lower energy density do not allow the trapped gases to escape, leading to gas porosities. Additionally, low energy input can lead to incomplete fusion both along different hatch lines and across the depth of the part. This can 85  lead to unmelted powder particles and additional porosities that can act as stress-risers and produce a poor-quality part.  7.3 Increased Geometric Complexity – Impeller Example The next step to expand and assess the capabilities of the FTR model was to apply this approach to a part with more geometric complexity, including a variety of section thickness / features. The object selected was the centrifugal impeller shown in Figure 7.6 [48]. The geometry used was published on Thingiverse, a sharing-platform for open-source hardware design files. Apart from the complex curvature of the fins of the impeller, the design also incorporates weight reducing holes or voids; features which would be interesting to simulate and understand.   Figure 7.6: Centrifugal Impeller from Thingiverse [48]. 86  The build envelope of the impeller is 60 mm x 60 mm x 10 mm. The layer at height z = 4 mm was chosen to be simulated using the FTR methodology because it exhibits voids for weight reduction and thin wall geometries in the impeller blades. Unfortunately, commercial slicers are not suitable for generating GCode for laser/electron beam processes because they are primarily designed for filament based additive manufacturing processes which tend to operate at a fixed filament temperature, extrusion rate, and print-head speed during printing. They do not have the functionality to modify the beam speed, nor can they set the additional parameters such as current/power or beam focus needed for this type of AM. As part of the development of the Large Electron Beam Additive Manufacturing facility at UBC, an in-house Python code has been developed to generate GCode capable of controlling the system during printing experiments. The impeller geometry was processed with this tool to generate the GCode for each layer. The GCode for the layer to be simulated with the FTR method can be visualized in CAMotics, an open source, 3D GCode simulator [49]. The visualization, shown in Figure 7.7, is colour coded where the green lines indicates the path of the electron beam with the beam on, i.e. G1 - linear interpolation and the blue lines show the path where the beam is off. A unique feature being developed for this facility is the taper out function, which reduces the beam power to zero as the beam approaches the outer contour region to prevent over-melting. The contour is rastered or performed after hatching the layer to even out any irregularities due to the taper out locations. The hatch spacing was maintained at 0.7 mm to encourage melt pool overlap and the hatch angle was arbitrarily rotated by 15˚ from the x-axis. The parameters used for the FTR model are shown in Table 7.2. The maximum temperature and melt depths calculated by the FTR method are shown in Figure 7.8 and Figure 7.9 respectively. As seen previously in Section 7.2, the contour areas with a higher power show a higher temperature and higher melt depth. In Figure 7.8, the effects of the 87  taper out approach can be seen on the outer contour traces. Upon inspecting the temperature and melt depth maps, some problem areas can be easily identified. Some of the print areas that form the blades of the impeller (top right and bottom left) appear to be under-melted. The contour trace of the beam would successfully outline the shape, but the in-fill resulting from the hatch operation would be unsuccessful because of the orientation of these blade regions relative to the hatch orientation. Additionally, in general the hatch spacing appears to be insufficient to allow overlap and complete melting over the hatched region. The areas between the hatches are under-melted which could lead to lack of fusion defects. This can be seen in Figure 7.9, where the melt depths between hatches and especially in some blade areas are between 0 – 40 µm which is less than the typical layer height of ~ 70 µm for EBAM.   Figure 7.7: GCode for the impeller at layer height z = 4mm simulated in CAMotics [49]. 88  Table 7.2: Process parameters for impeller layer. Parameter Values Power, 𝑃𝑃 300 W (contour), 250 W (hatching) Substrate Temperature, ℃ 700 Beam Width, 𝜎𝜎 250 µm Beam Speed, 𝑣𝑣 0.5 m/s The FTR model provides a rapid means to simulate key AM process outcomes at the layer level for complex geometric profiles. This capability enables easy identification of problem areas by inspecting the maximum temperatures and melt depths. One can quickly identify regions that have a much higher or much lower energy density than required which can cause over or under melting issues. Additionally, various hatching strategies can be quickly simulated to assess which one performs the best and / or could lead to the least number of defects.   Figure 7.8: Maximum temperatures for simulated layer z = 4mm. 89   Figure 7.9: Maximum melt depths for simulated layer z = 4mm          90  Chapter 8 -  Conclusion 8.1 Summary In this work, a Fast to Run simulation approach was developed to primarily model the build states such as temperatures and melt depths during the rastering of a layer of an additively manufactured part by a heat source. The FTR model uses an interpolation based approach to output the build states based on process parameters such as beam location, initial temperature of the bed and the beam parameters. The data for interpolation was generated by simulating a representative geometry in ABAQUS over a range of beam powers, beam speeds, and initial temperatures. Finally, a framework was developed in Python to read beam locations, calculate temperatures and depths and store temperature decay information if a node was to be affected by the beam again. Additional functionality to interpret GCodes for beam location was also added. It was shown that the FTR model is a powerful tool that can provide data with high spatial and temporal resolution quickly with a minimal loss in accuracy. The computational savings are tremendous, performing 23000x faster for the example shown in Chapter 6. The simulated time for the impeller example in Section 7.3 are 100x longer with larger domains. The time savings for these cases are significant.  The model is able to achieve such a magnitude of time savings because it takes advantage of the fact that the heat dissipates fairly quickly in metal AM processes because of the large domain and small beam size. The domain hence reaches a near uniform temperature distribution which serves as an effective estimate for initial temperature.  Additionally, it is seen that while the computational time for the ABAQUS model increases significantly with the increase in complexity, the computational time of the FTR model is only dependent on the number of nodes in the domain. In the initial stages of the project the two-track example in Chapter 6 was simulated in ABAQUS with constant material properties. When the properties are constant, the simulation time in 91  ABAQUS was much lesser than that of the simulation with temperature dependent properties. However, the time taken by the FTR model was roughly the same. Hence, there is a much higher time saving for the model with temperature dependent material properties in Chapter 6. Thus, increasing the complexity of the physics of the FEM/CFD model might increase the time to train the FTR model, as data generation might take longer to perform. However, this would not increase the time to actually run the FTR model and perform predictions and this is a significant point in favour of this methodology.  Using the FTR methodology, one is quickly able to identify problematic regions where the energy density experienced by the domain may lead to over-melting or under-melting. These regions may experience a variety of defects such as gas porosities, lack of fusion pores, and change in alloy composition due to species evaporation at high temperatures. 8.2 Limitations The FTR model is however not without its limitations. There are some situations that it is currently unable to handle. For example, in the unlikely scenario where the beam retraces its path in the time taken to reach the uniform condition, i.e. less than 0.05s according to Figure 5.4(a), the temperatures would escalate to unrealistically high values. In such a situation, the FTR model will not be able to accurately predict the temperatures as the initial temperatures will be outside the range of process parameters simulated in ABAQUS and also because the increase in peak temperature does not correlate well to the initial temperature.  The FTR model is also not able to predict temperatures when the beam is just started and the temperatures are in the transient region as it assumes a steady state. This delay relates to the delay that the material faces as it heat ups from a “cold” state. The delay can be quantified but has not 92  been currently incorporated in the FTR model. Though this window is really small, on the order of microseconds, it might pose significant challenge in a situation where the beam is started and stopped frequently or when a speed function is used to change beam speed on the corners or contours. Additionally, this window will change at higher energy inputs and different materials.  In the FTR model, melt depth prediction is closely linked to the initial bed temperature and serves well for simple situations. However, melt depths are more closely tied to interior temperatures of the domain which are harder to predict especially with heat flowing in due to the effect of the multiple tracks. This could be a reason for some of the disparity between ABAQUS melt depths and predicted melt depths. A better technique might be to predict the average temperature of the surface and link the melt depth predictions to this metric.  Additionally, the model physics remain relatively simple using only conduction-based heat transfer. Relationships developed when including radiation and fluid flow might not be as straightforward. However, the FTR approach presented here represents a framework that should be capable of incorporating results from more physically accurate thermal-fluid models.  8.3 Future Work Currently, the interpolation approach is not very efficient, having to interpolate for every node for every time step. A promising alternative are machine learning methods which can generate a computationally inexpensive function to evaluate the temperature and melt depth based on the temperature, location and beam parameters. Additionally, a larger dataset can be easily incorporated and processed by machine learning. Using advanced data techniques will not only speed up data processing, but also the calculation of the build states.  93  The next step would be to expand the process window to allow simulations over a wider range of process variables. The Electron Beam system at UBC works at much slower speeds and lower power densities as it was refurbished from an Electron Beam Welding system. Incorporating that range of beam properties would help verify the FTR results against experiments. This would require more simulations at different process parameters in ABAQUS and will also help increase the size of the database for the machine learning models.  As mentioned earlier, the ABAQUS model is relatively simplistic. Incorporating the effects of radiation, marangoni flow and evaporation would improve its fidelity to experimental results and consequently improve the FTR model. Taking it further, the entire database could be populated using results from a CFD package. Apart from incorporating more phenomenon such as evaporation, recoil pressure, marangoni flow, buoyancy and diffusion, a CFD approach would also provide with more output variables like velocities, species concentrations, pressure, and details of the liquid-solid and liquid-atmosphere interface. A similar approach to that shown in this work can be used to model these variables using the FTR approach.  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