This dissertation presents the development of numerical models based on lattice Boltzmann (LB) and cellular automaton (CA) methods for solving phase change and microstructural evolution problems. First, a new variation of the LB method is discussed for solving the heat conduction problem with phase change. In contrast to previous explicit algorithms, the latent heat source term is treated implicitly in the energy equation, avoiding iteration steps and improving the formulation stability and efficiency. The results showed that the model can deal with phase change problems more accurately and efficiently than explicit LB models. Furthermore, a new numerical technique is introduced for simulating dendrite growth in three dimensions. The LB method is used to calculate the transport phenomena and the CA is employed to capture the solid/liquid interface. It is assumed that the dendritic growth is driven by the difference between the local actual and local equilibrium composition of the liquid in the interface. The evolution of a threedimensional (3D) dendrite is discussed. In addition, the effect of undercooling and degree of anisotropy on the kinetics of dendrite growth is studied. Moreover, effect of melt convection on dendritic solidification is investigated using 3D simulations. It is shown that convection can change the kinetics of growth by affecting the solute distribution around the dendrite. The growth features of twodimensional (2D) and 3D dendrites are compared. Furthermore, the change in growth kinetics and morphology of Al-Cu dendrites is studied by altering melt undercooling, alloy composition and inlet flow velocity. The local-type nature of LB and CA methods enables efficient scaling of the model in petaflops supercomputers, allowing the simulation of large domains in 3D. The model capabilities with large scale simulations of dendritic solidification are discussed and the parallel performance of the algorithm is assessed. Excellent strong scaling up to thousands of computing cores is obtained across the nodes of a computer cluster, along with near-perfect weak scaling. Considering the advantages offered by the presented model, it can be used as a new tool for simulating 3D dendritic solidification under convection.

Dendrite growth is the primary form of crystal growth observed in laser deposition process of most commercial metallic alloys. The properties of metallic alloys strongly depend on their microstructure; that is the shape, size, orientation and composition of the dendrite matrix formed during solidification. Understanding and controlling the dendrite growth is vital in order to predict and achieve the desired microstructure and hence properties of the laser deposition metals. A two dimensional (2D) model combining the finite element method (FE) and the cellular automaton technique (CA) was developed to simulate the dendrite growth both for cubic and for hexagonal close-packed (HCP) crystal structure material. The application of this model to dendrite growth occurring in the molten pool during the Laser Engineered Net Shaping (LENS®) process was discussed. Based on the simulation results and the previously published experimental data, the expressions describing the relationship between the cooling rate and the dendrite arm spacing (DAS), were proposed. In addition, the influence of LENS process parameters, such as the moving speed of the laser beam and the layer thickness, on the DAS was also discussed. Different dendrite morphologies calculated at different locations were explained based on local solidification conditions. And the influence of convection on dendrite growth was discussed. The simulation results showed a good agreement with previously published experiments. This work contributes to the understanding of microstructure formation and resulting mechanical properties of LENS-built parts as well as provides a fundamental basis for optimization of the LENS process.

The size and morphology of the dendrite growth patterns are simulated for laser powder deposition of Ti-Nb alloys under steady-state and transient growth conditions. A phase field model using an adaptive grid technique was employed to simulate the steady-state growth of dendrites on rather small domains, in which fixed local solidification conditions are present. For simulation of dendrite growth patterns at transient conditions, a cellular automaton model was used along with a virtual front tracking technique on larger domains, containing various initial orientations of the solid-liquid (SL) interface. To obtain the required input thermal data, i.e., the temporal distribution of temperature, a finite element analysis was performed along with a novel numerical approach for the real-time addition of new deposition material in each time step, thus building the deposition geometry momentarily. Using the output of the thermal model, the motion and morphology of the SL interface was determined through tracking the isotherm of the solidification temperature. First, in this study, the appropriate set of processing parameters was found through an optimization process using a new concept, laser supplied energy Es, which combines the effects of the energy and powder density in the process. With the developed analytical/experimental procedure, crack and pore-free coatings of Ti-Nb with continuous beads were produced by examining the effects of a few sets of processing parameters, including laser power, laser scan velocity, laser beam diameter and powder feed rate. The results of the thermal model for the optimized set of parameters matched with the thermocouple temperature measurements with only ~5% deviation. The thermal model was able to predict realistic profiles for the temporal development of deposition geometry, thus predicting meaningful morphologies of the SL interface. The model output was easily treated for extraction of local processing parameters, such as the temperature gradient and solidification velocity. These data are very useful when simulating the dendrite growth patterns at steady-state conditions in directional solidification of selected regions in the microstructure. In order to define transient growth conditions, the simulated distribution of temperature can be also directly fed into the microstructure model at each solution time step. Phase field simulations of steady-state growth of dendrites during directional solidification showed a remarkable agreement with the experimental observations for the local dendrite arm spacing across the microstructure. Also qualitatively agreeing with the experiment, the simulated dendrite spacing exhibited a minimum around the mid-height region of the microstructure, which is explained by the counter effect of the temperature gradient and solidification velocity along the height of the sample. On a large domain containing different initial orientations of the SL interface, cellular automaton simulations for transient growth patterns of dendrites could reproduce most qualitative features observed in the microstructure. The dendrite arm spacing gradually decreased from the top of the microstructure. The competition was won by the dendrites growing in areas with higher cooling rates, i.e., in the regions closer to the top of the microstructure. The secondary arms of the primary dendrites, which are initially inclined on the vertical axis, grew extensively only along the overall growth direction and eventually became primary arms in some cases.

The size and morphology of the dendrite growth patterns are simulated for laser powder deposition of Ti-Nb alloys under steady-state and transient growth conditions. A phase field model using an adaptive grid technique was employed to simulate the steady-state growth of dendrites on rather small domains, in which fixed local solidification conditions are present. For simulation of dendrite growth patterns at transient conditions, a cellular automaton model was used along with a virtual front tracking technique on larger domains, containing various initial orientations of the solid-liquid (SL) interface. To obtain the required input thermal data, i.e., the temporal distribution of temperature, a finite element analysis was performed along with a novel numerical approach for the real-time addition of new deposition material in each time step, thus building the deposition geometry momentarily. Using the output of the thermal model, the motion and morphology of the SL interface was determined through tracking the isotherm of the solidification temperature. First, in this study, the appropriate set of processing parameters was found through an optimization process using a new concept, laser supplied energy Es, which combines the effects of the energy and powder density in the process. With the developed analytical/experimental procedure, crack and pore-free coatings of Ti-Nb with continuous beads were produced by examining the effects of a few sets of processing parameters, including laser power, laser scan velocity, laser beam diameter and powder feed rate. The results of the thermal model for the optimized set of parameters matched with the thermocouple temperature measurements with only ~5% deviation. The thermal model was able to predict realistic profiles for the temporal development of deposition geometry, thus predicting meaningful morphologies of the SL interface. The model output was easily treated for extraction of local processing parameters, such as the temperature gradient and solidification velocity. These data are very useful when simulating the dendrite growth patterns at steady-state conditions in directional solidification of selected regions in the microstructure. In order to define transient growth conditions, the simulated distribution of temperature can be also directly fed into the microstructure model at each solution time step. Phase field simulations of steady-state growth of dendrites during directional solidification showed a remarkable agreement with the experimental observations for the local dendrite arm spacing across the microstructure. Also qualitatively agreeing with the experiment, the simulated dendrite spacing exhibited a minimum around the mid-height region of the microstructure, which is explained by the counter effect of the temperature gradient and solidification velocity along the height of the sample. On a large domain containing different initial orientations of the SL interface, cellular automaton simulations for transient growth patterns of dendrites could reproduce most qualitative features observed in the microstructure. The dendrite arm spacing gradually decreased from the top of the microstructure. The competition was won by the dendrites growing in areas with higher cooling rates, i.e., in the regions closer to the top of the microstructure. The secondary arms of the primary dendrites, which are initially inclined on the vertical axis, grew extensively only along the overall growth direction and eventually became primary arms in some cases.

abstract: The formation of dendrites in materials is usually seen as a failure-inducing defect in devices. Naturally, most research views dendrites as a problem needing a solution while focusing on process control techniques and post-mortem analysis of various stress patterns with the ultimate goal of total suppression of the structures. However, programmable metallization cell (PMC) technology embraces dendrite formation in chalcogenide glasses by utilizing the nascent conductive filaments as its core operative element. Furthermore, exciting More-than-Moore capabilities in the realms of device watermarking and hardware encryption schema are made possible by the random nature of dendritic branch growth. While dendritic structures have been observed and are well-documented in solid state materials, there is still no satisfactory theoretical model that can provide insight and a better understanding of how dendrites form. Ultimately, what is desired is the capability to predict the final structure of the conductive filament in a PMC device so that exciting new applications can be developed with PMC technology. This thesis details the results of an effort to create a first-principles MATLAB simulation model that uses configurable physical parameters to generate images of dendritic structures. Generated images are compared against real-world samples. While growth has a significant random component, there are several reliable characteristics that form under similar parameter sets that can be monitored such as the relative length of major dendrite arms, common branching angles, and overall growth directionality. The first simulation model that was constructed takes a Newtonian perspective of the problem and is implemented using the Euler numerical method. This model has several shortcomings stemming majorly from the simplistic treatment of the problem, but is highly performant. The model is then revised to use the Verlet numerical method, which increases the simulation accuracy, but still does not fully resolve the issues with the theoretical background. The final simulation model returns to the Euler method, but is a stochastic model based on Mott-Gurney’s ion hopping theory applied to solids. The results from this model are seen to match real samples the closest of all simulations. / Dissertation/Thesis / Masters Thesis Electrical Engineering 2016

Understanding the fundamental mechanisms of solidification and the relative significance of different parameters governing these mechanisms is of vital importance for controlling the evolution of microstructure during solidification, and consequently, for improving the efficacy of a casting process. Towards achieving this goal, the present work attempts to study the effect of convection and shrinkage on solidification and microstructure formation primarily through the development of computational models which are complemented with experimental investigations and analytical solutions. Convection strongly influences the solutal and thermal distribution adjacent to the solidification interface and affects the growth rate and morphology of dendrites. To investigate this, a numerical model based on the enthalpy method is developed for binary alloy dendrite growth in presence of convection. The model results are validated with corresponding predictions using level-set method and micro-solvability theory. Subsequently, the model is applied for studying the effect of convection on the growth morphology of single dendrites. Results show that the presence of flow significantly affects the thermo-solutal distribution and consequently the growth rate and morphology of dendrites. Parametric studies performed using the model predict that thermal and solutal Peclet number and melt undercooling strongly influence the tip velocity of dendrites. Additionally, an analytical model is developed to quantify the effect of convection on dendrite tip velocity through the definition of an equivalent undercooling. An expression for this equivalent undercooling is derived in terms of the flow Nusselt and Sherwood numbers and the analytical equivalent undercooling values are compared with corresponding predictions obtained using the numerical model. Subsequently, the interaction of multiple dendrites growing in close proximity is studied. It is observed that the presence of neighbouring dendrites strongly influences the thermo-solutal distribution in the domain leading to significant changes in growth pattern. The effect of seed density on the growth morphology is investigated and it is observed that a higher initial seeding density leads to more spherical dendritic structure. Comparison with results from chilled casting of Al-6.5% Cu alloy with and without grain refiners show qualitative similarity in both the cases. The next part of the thesis presents a eutectic solidification model developed using the general enthalpy-based framework for dendritic solidification. New parameters and rules are defined and suitable modifications are made to incorporate the physics of eutectic solidification and account for the additional complexities arising due to the presence of multiple solid phases. The model simulates the presence of buoyancy driven convection and its interaction with the solidification process. i The model predictions are found to be in good agreement with the Jackson-Hunt theory. At first, the model is applied to simulate regular eutectic growth in a purely diffusive environment and it is observed that the model predicts the variation in interface profile with change in lamella width similar to those observed in experimental studies on eutectic solidification. Subsequently, a few case studies are performed to demonstrate the ability of the model in handling complex scenarios of eutectic growth such as width selection, lamella division and presence of solutal buoyancy. It is observed that solutal buoyancy gives rise to flow cells ahead of the eutectic interface facilitating the transfer of solute between the two phases. Apart from forced and natural convection, another important factor affecting solidification is the presence of shrinkage. Currently, solidification shrinkage is mostly modelled using empirical relations and criteria functions. In the present work, a phenomenological model for shrinkage driven convection is developed by incorporating the mechanism of solidification shrinkage in an existing framework of enthalpy based macro-scale solidification model. The effect of shrinkage flow on the free surface deformation is accounted for by using the volume-of-fluid method. The results predicted by the model are found to be in excellent agreement with analytical solutions for one-dimensional solidification with unequal phase densities. A set of controlled experiments are designed and executed for validating the numerical model. The experiments involve in-situ X-ray imaging of casting of pure aluminium in a rectangular cavity. The numerical predictions for solidification rate, free surface movement and temperature profiles are compared with corresponding experimental results obtained from the in-situ X-ray images and thermocouple data. Subsequent case studies, performed using the model, show significant influence of applied heat flux and mould geometry on the formation of shrinkage cavities. The shrinkage flow model provides the foundation for development of a generalized model to accurately predict the formation and morphology of internal porosity. The validated macro-scale shrinkage model is extended to the microscopic scale to study the influence of shrinkage flow on the growth rate of dendrites. Results demonstrate that shrinkage driven convection towards the dendrite strongly influences the solutal and thermal distribution adjacent to the solidification interface and consequently decreases the growth rate of the dendrite. Additionally, an analytical model is developed to quantify the effect of shrinkage driven convection through the definition of an equivalent undercooling for shrinkage flow. The present models provide significant physical insight into various mechanisms governing the process of solidification. Moreover, due to their similar framework, the individual models have the potential to be an effective foundation for the development of a generalized multi-scale solidification model incorporating the presence of important phenomena such as shrinkage and convection.

<p>This study is focused on the prediction of residual stresses and microstructure development of steel and aluminum alloys during laser-based manufacturing processes by means of multi-physics numerical modeling.</p> <p>A finite element model is developed to predict solid-state phase transformation, material hardness, and residual stresses produced during laser-based manufacturing processes such as laser hardening and laser additive manufacturing processes based on the predicted temperature and geometry from a free-surface tracking laser deposition model. The solid-state phase transformational model considers heating, cooling, and multiple laser track heating and cooling as well as multiple layer tempering effects. The residual stress model is applied to the laser hardening of 4140 steel and to laser direct deposition of H13 tool steel and includes the effects of thermal strain and solid-state phase transformational strain based on the resultant phase distributions. Predicted results, including material hardness and residual stresses, are validated with measured values.</p> <p>Two dendrite growth predictive models are also developed to simulate microsegregation and dendrite growth during laser-based manufacturing processes that involve melting and solidification of multicomponent alloys such as laser welding and laser-based additive manufacturing processes. The first model uses the Phase Field method to predict dendrite growth and microsegregation in 2D and 3D. It is validated against simple 2D and 3D cases of single dendrite growth as well as 2D and 3D cases of multiple dendrite growth. It is then applied to laser welding of aluminum alloy Al 6061 and used to predict microstructure within a small domain. </p> The second model uses a novel technique by combining the Cellular Automata method and the Phase Field method to accurately predict solidification on a larger scale with the intent of modeling dendrite growth. The greater computational efficiency of the this model allows for the simulation of entire weld pools in 2D. The model is validated against an analytical model and results in the literature.

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