Cellular Automaton Finite Element Research Articles

Fracture modeling in dual-phase steel grades based on the random cellular automata finite element approach

The development of a parallel version of the fracture model dedicated for multi-phase materials based on a combination of the finite element model and random cellular automata approach is the overall goal of this study. Dual-phase (DP) steel, commonly used in the automotive industry, is selected as a case study for the present investigation. Firstly, various fracture modes that can occur during deformation in DP steel grade microstructures are presented from an experimental point of view. To consider explicitly microstructure features that play a significant role during initiation and subsequent failure propagation, the digital material representation concept is used. Then, details of the developed random cellular automata model, fully embedded within the finite element framework, are discussed. The cellular automata space definition, internal variables, state variables and transition rules replicating investigated fracture modes are presented in detail and discussed. The concept of data transfer and parallelization based on the Message Passing Interface methodology in such an innovative hybrid numerical model is also clearly presented. The final section of the paper is devoted to examples of obtained results highlighting model predictive capabilities.

Method for the explicit insertion of microstructure in Cellular Automata Finite Element (CAFE) models based on an irregular tetrahedral Finite Element mesh: Application in a multi-scale Finite Element Microstructure MEshfree framework (FEMME)

Multiscale models are needed in simulations of mechanical and thermal properties that consider the microstructural heterogeneity of materials, in which a coupling is essential between the numerical models that deal with the different scales. In the class of cellular automata-finite element (CA-FE) or CAFE models, the use of a regular FE mesh restricts the model versatility due to the limited variety of engineering problems that can be simulated with such meshes. A novel methodology is proposed to create the homogeneous cells of the CA model from a mesh that is formed by irregular tetrahedrons. The result is more versatile and capable of modelling complex geometries, improving its applicability. The problem is solved through a subdivision algorithm that creates homogeneous tetrahedral cells, using a methodology to insert microstructures that can be described by particles, pores or fibres. Its use is demonstrated in a multi-scale Finite Element Microstructure MEshfree (FEMME) framework to calculate the elastic strains caused by microstructural features (pores); the method is shown to have a significantly lower computational cost than finite element simulations of equivalent levels of discretization.

Impact of Electromagnetic Stirring on Grain Structure of Electroslag Remelting Ingot

A transient, two-dimensional axisymmetric model was developed to understand the effect of the electromagnetic stirring (EMS) on the grain morphology of the electroslag remelting ingot. The cellular automaton-finite element technique was employed to describe the nucleation and growth of the grain. The Joule heating and Lorentz force created by the current of the furnace, as well as the Lorentz force induced by the EMS device, are included. The effect of the EMS current on the grain structure was investigated using the model. A reasonable agreement between the experiment and simulation was obtained. The growth direction of the upper grain without the EMS is approximately 45° with respect to the vertical axis, while changes to the radial were caused by EMS. The grain was considerably refined by the EMS, and the average area of the grain decreased from 9.381 × 10−7 m2 to 6.781 × 10−7 m2 with the current of the EMS ranging from 0 A to 500 A. Both the local solidification time and second dendrite arm spacing decreased with the increasing stirring intensity. The metal pool depth, however, increased with the EMS, which definitely contributed to the macrosegregation formation. The upper ingot with EMS was darker than that without EMS in the experiment. The EMS technique should be used with caution.

Simulation of directional solidification of refined Al–7 wt.%Si alloys – Comparison with benchmark microgravity experiments

Formation of grain structure and segregation during solidification of refined Al–7wt.%Si alloys under microgravity is simulated using a two-dimensional axi-symmetrical Cellular Automaton-Finite Element (CAFE) solidification model. This fully integrated model resolves a complex interplay between heat transfer at the macroscale and grain structure evolution and microsegregation formation at the mesoscale. Direct comparison with benchmark experiments performed under microgravity conditions of the International Space Station is carried out. Boundary conditions used in the simulations are deduced from the experimental measurements of temperature. Qualitative agreement is achieved concerning CET (columnar-to-equiaxed transition) position, CET transition mode (sharp or progressive), distributions of grain elongation factor and equivalent diameter, and distribution of eutectic fraction. The influences of pulling velocity and thermal gradient on grain structure and intergranular segregation are correctly reflected and are discussed. Moreover, we take advantage of the simulation results to study the effect of otherwise inaccessible parameters such as the degree of constitutional undercooling and the amplitude of the undercooled region. Another originality of this paper is to conduct quantitative comparisons between experiments and simulations which allows discussion of the improvements needed in the simulation.

Numerical simulation of solidification structure of 6.5 wt-%Si steel ingot slab

The solidification structure of 6.5%Si steel ingot was simulated based on the cellular automaton–finite element (CAFE) method and compared with an actual casting. The results showed that both the average grain size and proportion of columnar crystals agreed well with the experimental ones. The parameters suitable for simulating the microstructure of 6.5%Si steel ingot were determined. The influence of superheat and cooling conditions on the solidification structure was also studied in detail. The results show that the proportion of columnar crystals decreases with decreasing superheat, and the average grain size becomes smaller. The microstructure under the condition of slow cooling is finer than of air cooling.

Prediction of microstructural features and forming of friction stir welded sheets using cellular automata finite element (CAFE) approach

The main aim of the present work is to predict the microstructural features like grain size and dislocation density in the weld zone during friction stir welding (FSW) of similar (Al6061T6/Al6061T6) and dissimilar (Al6061T6/Al5086O) Aluminium grades using Cellular Automata Finite Element (CAFE) approach. The FSW process is not modelled with the stirring action, instead heat flux, strain-rate and strain are incorporated by analytical models. The grain size is controlled through cellular automata (CA) cells and dislocation density is related to this by two different (analytical and empirical) models. After FSW, four different methods are proposed for predicting the tensile behaviour of weld zone and the efficiency of these methods is evaluated through validations. The results indicate that the thermal, strain-rate, and strain models are accurate enough in their predictions when compared with existing results. The grain size predictions from CAFE model, which include the transition rule, are also consistent with the literature results, both for similar and dissimilar material combinations. The analytical model shows better dislocation density prediction than empirical model when compared with the experimental data. Of all the methods proposed for tensile behaviour prediction, the CAFE model that includes dislocation density evolution using the second model is efficient and accurate. The stress–strain data predicted from an averaged flow stress of many CA cells is also encouraging. Through these results, it has been demonstrated that the CAFE approach along with few validated analytical models can be used to predict the micro-features and forming aspects during FSW consistently.

Investment casting of nozzle guide vanes from nickel-based superalloys: part II – grain structure prediction

AbstractThe control of grain structure, which develops during solidification processes in investment casting of nozzle guide vanes (NGVs), is a key issue for optimization of their mechanical properties. The main objective of this part of the work was to develop a simulation tool for predicting grain structure in the new generation NGVs made from MAR-M247 Ni-based superalloy. A cellular automata - finite element (CAFE) module is employed to predict the three-dimensional (3D) grain structure in the as-cast NGV. The grain structure in the critical sections of the experimentally cast NGV is carefully analyzed, the experimental results are compared with the modeling outcomes, and the model is calibrated via tuning parameters which govern grain nucleation and growth. The grain structures predicted by the calibrated model show a very good accordance with the real ones observed in the critical sections of the as-cast NGV. It is demonstrated that the calibrated CAFE model is a reliable tool for the foundry industry to predict grain structure of the as-cast NGVs with very high accuracy.

Effect of Secondary Cooling Conditions on Solidification Structure and Central Macrosegregation in Continuously Cast High-Carbon Rectangular Billet

AbstractSolidification structures of high carbon rectangular billet with a size of 180 mm × 240 mm in different secondary cooling conditions were simulated using cellular automaton-finite element (CAFE) coupling model. The adequacy of the model was compared with the simulated and the actual macrostructures of 82B steel. Effects of the secondary cooling water intensity on solidification structures including the equiaxed grain ratio and the equiaxed grain compactness were discussed. It was shown that the equiaxed grain ratio and the equiaxed grain compactness changed in the opposite direction at different secondary cooling water intensities. Increasing the secondary cooling water intensity from 0.9 or 1.1 to 1.3 L/kg could improve the equiaxed grain compactness and decrease the equiaxed grain ratio. Besides, the industrial test was conducted to investigate the effect of different secondary cooling water intensities on the center carbon macrosegregation of 82B steel. The optimum secondary cooling water intensity was 0.9 L/kg, while the center carbon segregation degree was 1.10. The relationship between solidification structure and center carbon segregation was discussed based on the simulation results and the industrial test.

Prediction of microstructure evolution during hot forging using grain aggregate model for dynamic recrystallization

In this study, dynamic recrystallization during nonisothermal hot compression test was numerically simulated by finite element analysis using new grain aggregate model for dynamic recrystallization. This model was developed based on mean field approach by assuming grain aggregate as representative element. For each grain aggregate, changes of state variables were calculated using three sub-models for work hardening, nucleation, and nucleus growth. A conventional single parameter dislocation density model was used to calculate change of dislocation density in grains. For modeling nucleation, constant nucleation rate and nucleation criterion developed by Roberts and Ahlblom were used. It was assumed that the nucleation occurs when the dislocation density of certain grain reaches a critical nucleation criterion. Conventional rate theory was used to model nucleus growth. The developed dynamic recrystallization model was validated by comparing with isothermal hot compression of pure copper. Then, the finite element analysis was conducted to predict the local changes of microstructure and average grain size by using the grain aggregate model. The predicted results were compared with nonisothermal hot compression results. The simulation results were in reasonably good agreement with experimentally obtained microstructures and the calculation time was much shorter than cellular automata-finite element method.

Coupled Cellular Automaton (CA) – Finite Element (FE) Modeling of Directional Solidification of Al-3.5 wt% Ni Alloy: A Comparison with X-ray Synchrotron Observations

Development of dendritic grain structure and mesa-segregation leading to localized intergranular eutectic is simulated during directional solidification of an Al - 3.5 wt% Ni alloy with two pulling velocities (4 mu m s(-1) and 8 mu m s(-1)) using a two-dimensional (2D) Cellular Automaton Finite Element (CAFE) model. 2D CAFE simulations are compared with in situ and real-time experiments characterized by means of X-ray radiography at the European Synchrotron Radiation Facility (Grenoble, France). Two nucleation models are used. The first model is established by listing all measured positions and orientations of experimentally-observed nucleation events. The undetermined value of the nucleation undercooling is then set to 0 degrees C for all nucleation sites. The other model considers a stochastic model with a Gaussian distribution of the nucleation site as a function of the undercooling. It is derived from series of experiments. The influences of the nucleation models and liquid convection on the grain structure characteristics are numerically investigated. A good agreement with experimental observations is achieved concerning the evolutions of both the dendritic and the eutectic growth fronts (i.e., the size of the mushy zone). The predicted grain size and elongation are compared with measurements for the two nucleation laws. Simulations using the list of nucleation events reach better agreement with the longitudinal profiles of the grain equivalent diameter and elongation factor compared to the stochastic nucleation model. Direct tracking of the eutectic growth front as well as three-dimensional analyses are found to be required for improvement of the predictions.

Optimized parallel computing for cellular automaton–finite element modeling of solidification grain structures

A numerical implementation of a three-dimensional (3D) cellular automaton (CA)–finite element (FE) model has been developed for the prediction of solidification grain structures. For the first time, it relies on optimized parallel computation to solve industrial-scale problems (centimeter to meter long) while using a sufficiently small CA grid size to predict representative structures. Several algorithm modifications and strategies to maximize parallel efficiency are introduced. Improvements on a real case simulation are measured and discussed. The CA–FE implementation here is demonstrated using 32 computing units to predict grain structure in a 2.08 m × 0.382 m × 0.382 m ingot involving 4.9 billion cells and 1.6 million grains. These numerical improvements permit tracking of local changes in texture and grain size over real-cast parts while integrating interactions with macrosegregation, heat flow and fluid flow. Full 3D is essential in all these analyses, and can be dealt with successfully using the implementation presented here.

Method of Stray Grain Inhibition in the Platforms with Different Dimensions During Directional Solidification of a Ni-Base Superalloy

A model of typical turbine blade shape with different platforms was designed to study the method of stray grain inhibition in the platforms by both experimental investigation and a ProCAST simulation based on a Cellular Automaton Finite Element model. The results show that stray grains with random orientations increasingly nucleate and grow in these platforms with the increase of platform dimension. To inhibit these stray grains, assistant bars are designed to introduce the primary grain into these platforms according to the theory of seeding method. The results indicate by the help of these assistant bars, the primary grain is introduced to the large dimensional platforms by developing secondary and tertiary dendrites, which form “crisscross” structure in these platforms. And, the stray grains in the large dimensional platforms are successfully and effectively inhibited. Further more, it is found that the thermal condition and the solidification sequence are changed in the large dimensional platforms by the use of assistant bars, which result in the introduction of primary grain and the inhibition of stray grains. Besides, the simulation results are in accordance with experimental findings.

Simulation of grain selection during single crystal casting of a Ni-base superalloy

Abstract The grain selection process during single crystal casting of a Ni-base superalloy DD403 in spiral grain selector is simulated by a macro-scale ProCAST coupled a meso-scale Cellular Automaton Finite Element (CAFE) model. And the simulation results are validated by experimental observations. It is found that the grain orientations can be optimized in starter block and the length of starter block could be reduced to about 26.0 mm. A single crystal can be rapidly selected after the solidification front climbing 400° along the spiral passage. It is proposed that the coupling effect of the heat flow direction, the preferred growth direction and the geometrical restriction of spiral wall makes contribution to the crystal selection in spiral passage. An investigation into the grain orientation selection in grain selectors with different geometries reveals that crystal orientation can be optimized by increasing the length of starter block or decreasing its width. However, no obvious relationship is found between the crystal orientation and the parameters of spiral passage.

Analysis of Stray Grain Formation in Single Crystal CMSX-4 Superalloy

Abstract Modern single crystal (SX) turbine blades are fabricated by directional solidification using a grain selector. The grain selection process was investigated by numerical simulation and verified by the experiment. A coupled ProCAST and cellular automaton finite element (CAFE) model was used in this study. According to the latest literature data, we designed the grain selector. Simulation confirmed an optimal grain selection efficiency of the applied selector geometry. The obtained experimental results reveal the possibility of stray grain formation in SX castings with a designed selector, in contrast to the simulation results.

Influence of natural convection during upward directional solidification: A comparison between in situ X-ray radiography and direct simulation of the grain structure

This paper presents a comparison between the in situ and real time observation of a directional solidification experiment carried out at the European Synchrotron Radiation Facility and a direct simulation of grain structure formation in the sample using a two-dimensional cellular automaton–finite element (CA-FE) model. In situ characterization of the columnar-to-equiaxed transition in a refined Al–3.5wt.% Ni alloy is achieved by synchrotron X-ray radiography. Two main characteristics are derived from the radiographs for each nucleated grain, namely the nucleation position in the sample and the orientation of the main trunk relative to the vertical temperature gradient. These data are then used as input for the CA-FE simulations. The CA-FE model takes into account the effects of macroscopic transport of heat, liquid momentum and solute mass on the development of the dendritic grain structure and vice versa. The influence of convection on macroscopic shape of the growth front, the grain structure, the microstructure distribution and macro-segregation is determined by a comparison between the experimental observations and results from the numerical simulations with and without fluid flow. Good qualitative agreement is obtained and limitations that are linked to the two-dimensional approximation and the need for direct tracking of the eutectic grain structure are pointed out.

Microstructure Simulation of Al-Cu Alloy Based on Inverse Identified Interfacial Heat Transfer Coefficient

The interfacial heat transfer coefficient (IHTC) is taken as one of the most important factors affecting the accuracy of the simulation. In the present paper, the IHTC variation with temperature was obtained by an inverse heat conduction method. Then, a 3D cellular automaton-finite element method was adopted to predict the microstructure of an Al-Cu alloy based on the identified IHTC. It was found that the IHTC was of prime importance for the precise simulation of solidification microstructure, especially in the grains distribution. In addition, the simulated results using the IHTC variation with temperature were found to exhibit a better agreement with the experimental results than those using the constant value.

Direct Simulation of a Solidification Benchmark Experiment

A solidification benchmark experiment is simulated using a three-dimensional cellular automaton—finite element solidification model. The experiment consists of a rectangular cavity containing a Sn-3 wt pct Pb alloy. The alloy is first melted and then solidified in the cavity. A dense array of thermocouples permits monitoring of temperatures in the cavity and in the heat exchangers surrounding the cavity. After solidification, the grain structure is revealed by metallography. X-ray radiography and inductively coupled plasma spectrometry are also conducted to access a distribution map of Pb, or macrosegregation map. The solidification model consists of solutions for heat, solute mass, and momentum conservations using the finite element method. It is coupled with a description of the development of grain structure using the cellular automaton method. A careful and direct comparison with experimental results is possible thanks to boundary conditions deduced from the temperature measurements, as well as a careful choice of the values of the material properties for simulation. Results show that the temperature maps and the macrosegregation map can only be approached with a three-dimensional simulation that includes the description of the grain structure.

CAFE modeling, neural network modeling, and experimental investigation of friction stir welding

The main aim of the present work is to develop a cellular automata finite element –artificial neural network (CAFE-ANN) hybrid model to predict the evolution of grain size and yield strength during friction stir welding. The CAFE model was developed by linking CA cells with elements in ABAQUS, a finite element code. Then a neural network was developed and trained appropriately by using the data obtained from the validated CAFE model that predicts the grain size and yield strength. The ANN results are validated. Finally it has been demonstrated that ANN can be used as a ‘virtual machine’ to examine the effect of rotational speed, welding speed, axial force and shoulder diameter on the variation in grain size and yield strength. The temperature and strain-rate distribution predicted by the thermal and strain-rate models are consistent with the existing results. The CAFE model grain size predictions are also accurate as compared to experiments. The grain size and yield strength predictions from ANN coincide well with the experimental values and CAFE model predictions. It has been demonstrated that CAFE-ANN hybrid model can be used as a ‘virtual machine’ to predict and analyze the effect of above said parameters on the grain size and yield strength evolution. The predicted influence is found to agree with some of the available results. In few cases, a slight deviation in the trend is witnessed as compared to literature results.

Modeling dendritic solidification of Al–3%Cu using cellular automaton and phase-field methods

We compared a cellular automaton (CA)–finite element (FE) model and a phase-field (PF)–FE model to simulate equiaxed dendritic growth during the solidification of cubic crystals. The equations of mass and heat transports were solved in the CA–FE model to calculate the temperature field, solute concentration, and the dendritic growth morphology. In the PF–FE model, a PF variable was used to identify solid and liquid phases and another PF variable was considered to determine the evolution of solute concentration. Application to Al–3.0wt.% Cu alloy illustrates the capability of both CA–FE and PF–FE models in modeling multiple arbitrarily-oriented dendrites in growth of cubic crystals. Simulation results from both models showed quantitatively good agreement with the analytical model developed by Lipton–Glicksman–Kurz (LGK) in the tip growth velocity and the tip equilibrium liquid concentration at a given melt undercooling. The dendrite morphology and computational time obtained from the CA–FE model are compared to those of the PF–FE model and the distinct advantages of both methods are discussed.

Cellular automata finite element (CAFE) model to predict the forming of friction stir welded blanks

The main aim of the present work is to develop a cellular automata coupled finite element (CAFE) model to predict the grain size distribution during friction stir welding (FSW) and the influence of weld defects (like voids) on the forming of FSW sheets. In order to achieve this, FSW process is simulated just by applying thermal and strain-rate analytical models on the elements constituting the FSW blanks by using ABAQUS6.8 finite element (FE) code with the help of DFLUX sub-routine. A uniform grid mesh within each finite element for cellular automata to accommodate grain size is generated for the calculation. The sheet used is Al6061-T6 with a dimension of 300×100×2.1mm. The final grain size and yield strength are predicted by generating transition rules, relating them to temperature, strain-rate and strain evolved during friction stir welding. The computed results are compared with experimental results for validation. The forming (true stress–strain) behavior was predicted for FSW blanks containing defects by two different methods and their accuracy was compared. The thermal and strain-rate models followed in this work predict the temperature and strain-rate distribution across the weld region consistently. The results are matching well with the available data. The grain size and yield strength distribution predictions through CAFE model are agreeing well with the experimental results for all the FSW conditions. The true stress–strain behavior of a FSW blank obtained from CA model (containing grain size and strength history after FSW) coincides well with the stress–strain behavior predicted from FE simulations. The CAFE model, with a stress concentration factor, successfully predicts the influence of defects on the tensile behavior. The forming behavior obtained from both the methods (METHOD 1 and METHOD 2) is coinciding well with each other, except at the initial stages of forming.