Front Tracking Model Research Articles

Verification of a new cellular automata model of solidification using a case study on the columnar to equiaxed transition previously simulated using front tracking

Developing computational tools appropriate for modelling physical phenomena, such as alloy solidification, requires making optimisation decisions on the time and spatial scales that can be resolved with finite computational resources. Here we report on development of a model of grain nucleation and growth, based on a Cellular Automata (CA) approach, that can directly simulate columnar and equiaxed solidification, and their competition to trigger a columnar-equiaxed transition (CET), at the scale of a casting. Previous work has been to develop a Front Tracking (FT) model of columnar solidification and an equiaxed index capable of predicting the relative likelihood of CET formation in a casting case study. This FT model has been validated by extensive experimental studies. Here we apply the new CA model and compare the predictions to those of the FT model. The close agreement between the two models serves to verify the new CA model. The study also presents insights into the thermophysical phenomena affecting grain structure evolution in alloy solidification and confirms the validity and utility of the equiaxed index.

Multiphysics simulation of single pulse laser powder bed fusion: comparison of front capturing and front tracking methods

PurposeDuring thermal laser processes, heat transfer and fluid flow in the melt pool are primary driven by complex physical phenomena that take place at liquid/vapor interface. Hence, the choice and setting of front description methods must be done carefully. Therefore, the purpose of this paper is to investigate to what extent front description methods may bias physical representativeness of numerical models of laser powder bed fusion (LPBF) process at melt pool scale.Design/methodology/approachTwo multiphysical LPBF models are confronted: a Level-Set (LS) front capturing model based on a C++ code and a front tracking model, developed with COMSOL Multiphysics® and based on Arbitrary Lagrangian–Eulerian (ALE) method. To do so, two minimal test cases of increasing complexity are defined. They are simplified to the largest degree, but they integrate multiphysics phenomena that are still relevant to LPBF process.FindingsLS and ALE methods provide very similar descriptions of thermo-hydrodynamic phenomena that occur during LPBF, providing LS interface thickness is correctly calibrated and laser heat source is implemented with a modified continuum surface force formulation. With these calibrations, thermal predictions are identical. However, the velocity field in the LS model is systematically underestimated compared to the ALE approach, but the consequences on the predicted melt pool dimensions are minor.Originality/valueThis study fulfils the need for comprehensive methodology bases for modeling and calibrating multiphysical models of LPBF at melt pool scale. This paper also provides with reference data that may be used by any researcher willing to verify their own numerical method.

3-D front tracking model for interfaces with anisotropic energy

A new 3-dimensional front tracking model for grain boundaries (GBs) has been developed. It allows to efficiently simulate the motion of GBs in interaction with other GBs or particles of secondary phases. The model fully accounts for the dependence of the grain boundary energy on the local orientation of the boundary plane, in the sense that any functional dependence can be chosen. In principle, this model allows to study grain growth or recrystallisation considering any realistic description of the GB energy. The present paper describes the derivation of the mathematical concept from basic mechanics, in particular the consideration of the Herring torque, and some implementation details. The model is verified by comparison with analytical models. For some heuristic GB energy dependences, the equilibrium shape of a singular grain with a fixed volume embedded in another grain is calculated and discussed. As a first application, the Zener drag for realistic arrangements of about 1350 particles has been derived from simulations.

Influence of natural and forced gravity conditions during directional columnar solidification

In a recent study, the present authors reported an analysis of a transient process of directional solidification of TiAl alloys in the absence of convection (Battaglioli et al., 2017). The adopted front tracking model, also coupled with indirect methods for predicting columnar to equiaxed transition (CET), showed how the development of different grain regions, namely axial columnar, radial columnar and equiaxed, depends on process parameters such as temperature distribution and applied cooling rates, as well as on properties such as the degree of inoculation of the melt and nucleation undercooling required for equiaxed growth. In this paper, the previous front tracking model is significantly developed, by including the solution of Navier-Stokes equations in order to predict thermal convection in the liquid region as well as in the columnar mush (treated as an isotropic porous medium). This improvement is introduced with the aim to investigate TiAl alloys solidification under different gravity conditions. Accordingly, the simulation setup employed in the study reproduces the one used in experimental campaigns carried out on the MAXUS 9 sounding rocket (microgravity) and on ESA’s Large Diameter Centrifuge (hypergravity), within the framework of ESA’s GRADECET (Gravity Dependence of CET in TiAl alloys) project. The ability of the model in predicting thermal convection is demonstrated by considering several case studies. Results show that the evolution of fluid flow patterns in the samples depends on the external forces considered, combined to the transient change of axial and radial temperature gradients that occur during the solidification process. A parametric study of the directional solidification process on a centrifuge is performed by changing the value of the centrifuge arm and rotation rate and results are compared to predictions derived from non-dimensional considerations.

Axisymmetric front tracking model for the investigation of grain structure evolution during directional solidification

A recent numerical investigation by the present authors Battaglioli et al. (2017) had shown that the heat and mass transfer associated with steady-state Bridgman furnace solidification is dependent on advection of sensible and latent heat, and also on the axisymmetric geometry of the crucible. The present work extends the previous one by considering more complex non-steady solidification scenarios, where transient (limited-duration) Bridgman solidification is subsequently combined with a controlled power-down cooling process. A significant advancement with respect to the previous model is the inclusion of a front tracking method for the simulation of columnar growth. In the front tracking method the columnar mush region is demarcated by a series of markers that advance at a growth rate governed by the solutal undercooling at the dendrite tips. A classic micro-segregation law is used to govern the evolution of solid fraction in the columnar mush and the release of latent heat. The inclusion of the front tracking method has provided greater insights into how heat fluxes and thermal conditions can influence the final grain structure. Firstly, we show that by analysing the trajectories of the markers, it is possible to predict the transition from axial columnar growth (directional solidification) to unwanted radial columnar growth due to significant radial heat fluxes in the sample. Secondly, we show that model can simulate an undercooled liquid region ahead of the columnar front where equiaxed grains could possibly nucleate and grow, leading to a columnar to equiaxed transition (CET). Two indirect CET prediction methods from literature have been included to the model to assess the likelihood of a CET occurring in the as-cast grain structure of the alloy. The model was employed to simulate experimental scenarios from literature involving a γ-TiAl alloy where the samples were subjected to different transient cooling conditions in a Bridgman furnace. Examples of axial columnar growth, radial columnar growth, and CET are discussed in detail with elucidation from the model.

Columnar and Equiaxed Solidification of Al-7 wt.% Si Alloys in Reduced Gravity in the Framework of the CETSOL Project

During casting, often a dendritic microstructure is formed, resulting in a columnar or an equiaxed grain structure, or leading to a transition from columnar to equiaxed growth (CET). The detailed knowledge of the critical parameters for the CET is important because the microstructure affects materials properties. To provide unique data for testing of fundamental theories of grain and microstructure formation, solidification experiments in microgravity environment were performed within the European Space Agency Microgravity Application Promotion (ESA MAP) project Columnar-to-Equiaxed Transition in SOLidification Processing (CETSOL). Reduced gravity allows for purely diffusive solidification conditions, i.e., suppressing melt flow and sedimentation and floatation effects. On-board the International Space Station, Al-7 wt.% Si alloys with and without grain refiners were solidified in different temperature gradients and with different cooling conditions. Detailed analysis of the microstructure and the grain structure showed purely columnar growth for nonrefined alloys. The CET was detected only for refined alloys, either as a sharp CET in the case of a sudden increase in the solidification velocity or as a progressive CET in the case of a continuous decrease of the temperature gradient. The present experimental data were used for numerical modeling of the CET with three different approaches: (1) a front tracking model using an equiaxed growth model, (2) a three-dimensional (3D) cellular automaton–finite element model, and (3) a 3D dendrite needle network method. Each model allows for predicting the columnar dendrite tip undercooling and the growth rate with respect to time. Furthermore, the positions of CET and the spatial extent of the CET, being sharp or progressive, are in reasonably good quantitative agreement with experimental measurements.

Solidification Structure and Macrosegregation of Billet Continuous Casting Process with Dual Electromagnetic Stirrings in Mold and Final Stage of Solidification: A Numerical Study

Coupling macroscale heat transfer and fluid flow with microscale grain nucleation and crystal growth, a mixed columnar–equiaxed solidification model was established to study the SWRT82B steel solidification structure and macrosegregation in 160 mm × 160 mm billet continuous casting with dual electromagnetic stirrings in mold and final stage of solidification (M-EMS and F-EMS). In the model, the phases of liquid, columnar, and equiaxed were treated separately and the initial growing equiaxed phase, which could move freely with liquid, was regarded as slurry. To obtain the equiaxed grains nucleation and columnar front evolution, the unit tracking method and the columnar front tracking model were built. The model was validated by magnetic induction intensity of stirrer, billet surface temperature, and carbon segregation. The equiaxed phase evolution and the solute transport with effect of fluid flow and grains transport were described in this article. The results show that the equiaxed phase ratio will not increase obviously with higher current intensity of M-EMS, while the negative segregation near the strand surface becomes more serious. The negative segregation zone near the billet center and the center positive segregation come into being with the effect of equiaxed grains sedimentation and liquid thermosolutal flow. It is also found that the liquid solute transport in the F-EMS zone becomes the main factor with higher current intensity rather than the solidification rate, and therefore, the final billet center segregation decreases first and then turns to rise with the current intensity. The optimal current intensities of M-EMS and F-EMS proposed for SWRT82B billet continuous casting are 200 and 400 A, respectively.

An improved Front-Tracking technique for the simulation of mass transfer in dense bubbly flows

Direct numerical simulation results of mass transfer in dense bubble swarms using a Front-Tracking (FT) model will be presented, where the effect of the gas hold-up has been investigated. The FT method is particularly suited for bubble swarm simulations, since bubbles do not coalesce artificially, but traditional FT techniques often suffer from artificial volume loss of the bubbles. For this reason, a specialized remeshing technique is presented to counteract any occurring volume defects, while keeping all physical undulations on the bubble surfaces unharmed.For the simulation of gas-to-liquid mass transfer, a species transport equation (convection–diffusion–reaction) was coupled to the FT hydrodynamics solver, which was solved on a superimposed refined mesh for higher accuracy. The velocity components have been interpolated to the refined grid using a higher-order solenoidal method. Enforcement of the Dirichlet condition for the concentration at the gas–liquid interface is achieved with an immersed boundary method, enabling the description of gas to liquid mass transfer. Careful validation of the newly implemented model shows satisfactory results.The liquid side mass transfer coefficient in dense bubble swarms, with gas fractions between 4% and 40%, has been investigated using the new model. The simulations have been performed in a 3D domain with periodic boundaries, mimicking an infinite swarm of bubbles. The results indicate that the liquid-side mass transfer coefficient rises only slightly with increasing gas fraction.

On the Origin of Grid Anisotropy in the Simulation of Dendrite Growth by a VFT Model

A virtual front tracking model, based on solute and heat diffusion in two dimensions, is chosen to capture the full microstructural behavior of dendritic solidification in a binary alloy. We use a simple method of calculation, easy to perform, with relatively high stable time step, to simulate the dendrite growth in an Al-8 wt pct Mg alloy for which no numerical simulation has been carried out in the past. Local equilibrium at the liquid solid interface and the buildup of solute ahead of the interface are solved, and the dendrite growth process is simulated in isothermal solidification conditions. We show that the artificial grid anisotropy originates from the four cell neighborhood method adopted for capturing the moving front. By a correct neighborhood configuration, a grid independent set of results and expected phenomena are reproduced for a free dendrite growing either aligned or inclined with the grid. The dendrite morphology and orientation, and the growth velocity are explored via physical simulation parameters such as undercooling and surface tension anisotropy.

Sensitivity analysis of dendritic growth kinetics in a Bridgman furnace front tracking model

A directional solidification experiment of a Ti-Al-Nb-B-C alloy by power down method is simulated using a Bridgman furnace front tracking model. The effect of varying the dendritic growth parameters; C, the columnar dendrite growth coefficient, and n, the undercooling exponent, is investigated. A matrix of growth coefficients and undercooling exponents - at three levels each, based around a growth law for Ti-46wt.%Al - is applied in simulations, and the effect on columnar dendrite tip temperature, tip velocity, and tip temperature gradient is observed. The simulation results show that the dendrite tip velocity and temperature gradient at the tip are practically unaffected by the use of different growth parameters. However, the predicted columnar dendrite tip undercooling did vary to give the required dendrite tip velocity. This finding has implications for the analysis of microstructural transitions, such as the Columnar to Equiaxed Transition (CET). In conclusion, it is suggested that, for transient solidification conditions, a CET prediction criterion based on tip undercooling is preferable to one that uses growth velocity.

Modelling the creation and destruction of columnar and equiaxed zones during solidification and melting in multi-pass welding of steel

The authors present a novel meso-scale (mm–cm) numerical model of multi-pass welding of stainless steel based on the front tracking formulation. During a typical multi-pass notched butt weld, multiple cycles of melting and solidification occur throughout the weld pool, where previously solidified equiaxed zones are destroyed by melting and subsequently resolidify in columnar form. These thermal cycles result from the thermal effects of the current deposition of liquid filler metal being superimposed onto those of the previous pass. The advancement of the liquidus isotherm is employed to characterize melting, while both columnar and equiaxed dendritic growth in the undercooled melt are considered in the model of solidification. Columnar solidification is simulated using a front tracking method where computational markers are employed to explicitly define the advancing columnar front. Competing equiaxed solidification is modelled using a volume averaging approach. During the first and second welding passes of a case study, the weld pool solidifies as purely columnar. This is due to the high thermal gradient present in the weld pool, which results in a very small region of undercooled liquid ahead of the columnar front and subsequently low levels of equiaxed grain nucleation/growth. Towards the end of the third and final pass the equiaxed grains have sufficient dwell time in the undercooled liquid to form a coherent network and present a physical barrier to the advancing columnar front. A Columnar to Equiaxed Transition (CET) occurs and the remainder of the weld pool solidifies in a fully equiaxed microstructure. The results are compared with Hunt’s analytical model of CET and the agreement is reasonably good. The occurrence of CET throughout the weld pool is directly predicted via this front tracking model without the need to rely on estimated thermal gradients and solidification rates. Further simulations indicate that CET can be promoted, even in early passes, by increasing the grain refiner level and/or by pre-heating the parent metal.

Numerical simulation of prepreg resin impregnation effect in vacuum-assisted resin infusion/prepreg co-curing process

A new finite element formulation was developed to simulate the prepreg resin impregnation effect in vacuum-assisted resin infusion/prepreg co-curing process. The numerical model combined the fiber compaction and resin flow model in prepreg part with resin impregnation front tracking model in dry fiber fabric. The influences of processing parameters, including various temperature and time, on the impregnation height in dry fiber fabric and change of fiber volume fraction in prepreg stack were analyzed by numerical model and micrographs in experiment. The accuracy of numerical model was validated by the good agreement between simulation and experimental results. These results are greatly helpful to optimize processing parameters and improve the manufacturing efficiency in co-cured vacuum-assisted resin infusion process.

Order verification of a Bridgman furnace front tracking model in steady state

Numerical model verification is a prerequisite to model validation. However, verification can be difficult if an analytical solution is not available for the normally complex problem that the numerical model is setting out to address. This article aims to formally verify a Bridgman furnace solidification front tracking model code in a steady state scenario. To do this an order verification procedure is applied using an established analytical solution from the literature. The model is verified as first order accurate in space for steady state Bridgman solidification.

DEVELOPMENT OF CELLULAR AUTOMATON MODELS AND SIMULATION METHODS FOR SOLIDIFICATION OF ALLOYS

Dendritic structure is the most frequently observed solidification microstructure of alloys. It has a dominant effect on the mechanical properties of alloys. The formation of the dendritic microstructure has attracted extensive attentions. It has been demonstrated that numerical simulation is a powerful tool for studying the microstructure formation during the solidification of alloys. Various models, such as the front-tracking (FT) model, the phase-field (PF) model and the cellular automaton (CA) model have been proposed to simulate the formation process of dendrite. Compared with other methods, CA is an effective numerical simulation method with high calculation efficiency and clear physical meaning. It is more suitable to be applied to simulate the formation kinetics of the dendritic microstructure of alloys. It has been widely applied in the investigation of the solidification of alloys. This paper makes a detailed introduction to the common process of CA modeling and simulation, the constructing method of CA model and the calculation method for some key parameters such as nucleation rate of nuclei, growth velocity of dendrite, etc. A review of the development of the CA models for the solidification of alloys is carried out. The features and applications of the existing CA models are critically assessed. The applications of the CA models in the investigations of the practical solidification process are summarized. The problems to be solved and the future development of CA models are also pointed out.

Columnar-to-Equiaxed Transition in Solidification Processing of AlSi7 Alloys in Microgravity the CETSOL Project

This paper gives an overview of the experiments on-board the International Space Station (ISS) performed so far by the CETSOL team. Al-7 wt% Si alloys with and without grain refiners were solidified in microgravity. Detailed grain structure analysis showed columnar growth in case of non-refined alloy, but the existence of a columnar to equiaxed transition (CET) in refined alloy. One main result is a sharp CET when increasing the solidification velocity and a progressive CET for lowering the temperature gradient. Applying a front tracking model this behavior was confirmed numerically for sharp CET. Using a CAFE model both segregation and grain structures were numerically modeled and show a fair agreement with the experimental findings.

Front tracking approach to modeling binary alloy solidification

Purpose – The purpose of this paper is to endorse the idea of using a special post-calculating front tracking (FT) procedure, along with the enthalpy-porosity front tracking (EP-FT) single continuum model, in order to identify zones of different dendritic microstructures developing in the mushy zone during cooling and solidification of a binary alloy. Design/methodology/approach – The 2D and 3D algorithms of the FT approach along with different crystal growth laws were implemented in macroscopic calculations of binary alloy solidification with the identification of different dendrite zones developing during the process. Findings – Direct comparison of results predicted by the FT model with that based on the concept of the critical value of the solid volume fraction shows the sensitivity of the latter on an arbitrary assumed value of the dendrite coherency point (DCP). Moreover, for a carefully chosen DCP value the second model provides results that are close to those given by the FT-based approach. It is also observed that the macro-segregation pattern obtained by the proposed method is hardly influenced by chosen dendrite tip kinetics. Originality/value – To the best authors’ knowledge, for the first time the 3D FT model has been used along with the enthalpy porosity approach to simulate the development of zones of different dendrite morphology during binary alloy solidification. And, a weak influence of assumed different dendrite tip kinetics on the macro-segregation pattern has been proved, what justifies this underlying assumption of the EP-FT method.

Numerical Method of Fabric Dynamics Using Front Tracking and Spring Model

AbstractWe use front tracking data structures and functions to model the dynamic evolution of fabric surface. We represent the fabric surface by a triangulated mesh with preset equilibrium side length. The stretching and wrinkling of the surface are modeled by the mass-spring system. The external driving force is added to the fabric motion through the “Impulse method” which computes the velocity of the point mass by superposition of momentum. The mass-spring system is a nonlinear ODE system. Added by the numerical and computational analysis, we show that the spring system has an upper bound of the eigen frequency. We analyzed the system by considering two spring models and we proved in one case that all eigenvalues are imaginary and there exists an upper bound for the eigen-frequency This upper bound plays an important role in determining the numerical stability and accuracy of the ODE system. Based on this analysis, we analyzed the numerical accuracy and stability of the nonlinear spring mass system for fabric surface and its tangential and normal motion. We used the fourth order Runge-Kutta method to solve the ODE system and showed that the time step is linearly dependent on the mesh size for the system.

Meso-scale modelling of directional solidification and comparison with in situ X-ray radiographic observations made during the MASER-12 XRMON microgravity experiment

Computational modelling of advanced solidification processes has made considerable advances over the last half century, with ever increasing levels of modelling complexity. There is, therefore, an increasing need for state of the art experimental investigation to provide suitable validation for these model predictions. In situ X-ray radiography has become a powerful tool for solidification experimentation. Using either synchrotron or microfocus X-ray sources, thin samples, encased in X-ray transparent crucibles, can be directionally or isothermally solidified, allowing for direct real time observation of dynamic solidification phenomena. This paper presents the results of a meso-scale Front Tracking simulation of a directional solidification experiment, performed using an Al–20wt.% Cu alloy, carried out under microgravity conditions on board the MASER 12 sounding rocket. The sample was mounted in a Bridgman type gradient furnace and solidified using a prescribed cooling regime with a constant gradient, thus promoting directional solidification in the field of view. The actual thermal gradient in the sample was found to be lower than the nominal thermal gradient, as set/recorded by thermocouples embedded in the heater elements. The adjusted thermal data were supplied as inputs to the Front Tracking model and good agreement was then observed between the model predictions and the in situ observations. The extent and amplitude of the undercooled zone ahead of the columnar front was predicted based on analytical growth kinetics laws and the results were also compared to analytical models of columnar-to-equiaxed transition (CET) prediction.

Direct Numerical Simulations of gas–liquid–solid three phase flows

Gas–liquid–solid three phase flows are encountered in for example the Fischer–Tropsch process for the production of synthetic fuels in bubble slurry columns. To predict the hydrodynamics in large slurry bubble columns a multi-scale modeling approach can be used, which accounts for the large variation in time and length scales. In this paper, the smallest scale model has been developed using the Front Tracking model of Dijkhuizen et al. (2010b) and the Immersed Boundary model of Kriebitzsch (2011). In the Front Tracking model, each bubble is tracked separately. Furthermore, the Immersed Boundary method introduces the particle–fluid and the particle–particle (via a hard sphere model) interactions in the model. The resulting hybrid Front Tracking Immersed Boundary model is able to simulate dense three-phase flows and accounts for swarm effects in a fundamental manner. From our simulations we found that the relative drag coefficient of bubbles in three-phase flows seems to increase with increasing solids volume fraction. However, longer averaging periods are needed to derive a fully predictive correlation for the relative drag coefficient with respect to the solid volume fraction.

INVESTIGATION OF EFFECT OF INTERFACE ENERGY ANISOTROPY ON DENDRITIC GROWTH IN UNIDIRECTIONAL SOLIDIFICATION BY FRONT TRACKING SIMULATION

The dendritic growth with the different solid/liquid (S/L) interface energy anisotropies in the unidirectional solidification has been investigated using the self-consistent front tracking model. It is found that, for a given solidification condition, there were two kind of interface shape solutions with the different spacing Peclect number ranges. The interface shape with the small spacing Peclect number range was similar with cellular tip, and that with the large spacing Peclect number range referred to dendritic tip. The higher S/L interface energy anisotropy was in favor of the widening of the dendritic growth solution range. There was a certain power exponential relationship between the dendritic tip marginal stability parameter σ ∗ and the S/L interface energy anisotropic parameter E4. A modified Fisher dendritic tip solution, which considered the effect of S/L interface energy anisotropy, was obtained as follows: RIMS =2 .5646( ΓD L Vk 0ΔT0 ) 0.5 E −0.1905 4 , ΔT0 = mC0(k0 − 1)/k0. The undercooling in front of the S/L interface decreased with increasing the anisotropic parameter. The primary dendritic spacing mainly depended on the interaction of solute diffusion field between the adjacent dendrite, and the S/L interface energy had little influence on