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Abstract In the engineering field, it is necessary to construct a numerical model that can reproduce multiphase flows containing three or more phases with high accuracy. In our previous study, by extending the conservative Allen–Cahn (CAC) model, which is computationally considerably more efficient than the conventional Cahn–Hilliard (CH) model, to the multiphase flow problem with three or more phases, we developed the conservative Allen–Cahn type multi-phase-field (CAC–MPF) model. In this study, we newly construct the improved CAC–MPF model by modifying the Lagrange multiplier term of the previous CAC–MPF model to a conservative form. The accuracy of the improved CAC–MPF model is evaluated through a comparison of five models: three CAC–MPF models and two CH–MPF models. The results indicate that the improved CAC–MPF model can accurately and efficiently perform simulations of multiphase flows with three or more phases while maintaining the same level of volume conservation as the CH model. We expect that the improved CAC–MPF model will be applied to various engineering problems with multiphase flows with high accuracy. Graphic abstract

The effect of natural convection on dendrite morphology is investigated through three-dimensional large-scale phase-field lattice Boltzmann simulations using a block-structured adaptive mesh refinement scheme with the mother-leaf method in a parallel-GPU environment. The simulations confirmed that downward buoyancy enhances the growth of the primary and secondary arms, and upward buoyancy delays the growth of those arms. In addition, the effect of natural convection on the solidification morphologies gradually decreased as the primary arm tip reached the top of the computational domain and finally stopped. Furthermore, in the longer simulation under purely isothermal diffusive conditions, detachment of the secondary arms owing to curvature-driven fragmentation was observed. A large-scale non-isothermal dendrite growth simulation was also conducted, wherein it was observed that the tip growth rate of the primary arm was delayed, and the secondary arm spacing was larger than that in the isothermal condition.

Permeability is a very important multiscale parameter providing information of dendritic structures in macroscale simulations of solidification with liquid flow during casting. In our previous study, it was demonstrated that the permeability of liquid flow normal to a columnar dendritic structure can be well approximated by the Kozeny–Carman (KC) equation with a coefficient kc = 9. In this study, the permeability of liquid flow parallel to a columnar dendritic structure is investigated using a method that combines both phase–field and lattice Boltzmann simulations. It is shown that the permeability of parallel flow can also be approximated by the KC equation with kc = 3. A permeability tensor is then proposed for a columnar dendritic structure. Validity of the permeability tensor is confirmed by simulations of liquid flow in arbitrary directions of a columnar dendritic structure.

In the phase-field simulation of dendrite growth during the solidification of an alloy, the computational cost becomes extremely high when the diffusion length is significantly larger than the curvature radius of a dendrite tip. In such cases, the adaptive mesh refinement (AMR) method is effective for improving the computational performance. In this study, we perform a three-dimensional dendrite growth phase-field simulation in which AMR is implemented via parallel computing using multiple graphics processing units (GPUs), which provide high parallel computation performance. In the parallel GPU computation, we apply dynamic load balancing to parallel computing to equalize the computational cost per GPU. The accuracy of an AMR refinement condition is confirmed through the single-GPU computations of columnar dendrite growth during the directional solidification of a binary alloy. Next, we evaluate the efficiency of dynamic load balancing by performing multiple-GPU parallel computations for three different directional solidification simulations using a moving frame algorithm. Finally, weak scaling tests are performed to confirm the parallel efficiency of the developed code.

An understanding of the morphology of growing dendrites in alloys is needed for an analysis of microsegregation, as well as an estimation of the permeability for macroscopic fluid dynamics. Quantitative phase-field simulations were used to study the growth process of three-dimensional (3D) equiaxed dendrites in an Al-1.0 mass%Cu alloy during continuous cooling. The dendrites were analysed using an interface shape distribution (ISD) map, which provides the probability of the local interface having a morphology with a given curvedness (C) and shape factor (S). Morphological changes in the microstructure can be measured sensitively from the change in the average value of the curvedness 〈C〉 relative to the solid volume fraction. The ISD map continued to change over time during continuous cooling, implying that it was not time-invariant. Furthermore, when microstructural changes occurred, similarities between the ISD maps were observed, independent of the cooling rates and system sizes.

Time-resolved in-situ X-ray tomography and high-performance phase-field simulations are state-of-the-art approaches to clarifying dendrite solidification. However, major issues persist, such as the insufficiency of spatiotemporal resolution in experiments and lack of material properties in simulations. To overcome these issues, in this study, we developed a data assimilation system using an ensemble Kalman filter based on the phase-field lattice Boltzmann method as a simulation model for the dendrite solidification of binary alloys with liquid flow. The validity of the developed system was confirmed through twin experiments to infer the kinematic viscosity in a two-dimensional dendrite growth problem with forced convection.

In solidification of metals and alloys under earth’s gravity, the growing free equiaxed dendrites either settle or float because of the difference between the solid and liquid densities. In this study, we developed a two-dimensional computational method that can efficiently simulate the growth of free dendrite settling over a long distance. In the developed method, the growth of the dendrite is expressed by a phase-field method, the liquid flow is computed by a lattice Boltzmann method, and the settling of dendrite is expressed by equations of motion. A moving-frame algorithm is employed to track the dendrite settling over a long distance. The computation is accelerated using a multi-level mesh method and GPU computing. The validity of the developed method was evaluated through simulation of the settling of a single circular particle. Specifically, using the developed method, we simulated the growth of a single dendrite in an isothermal undercooled melt of a binary alloy under gravity, and evaluated the time evolutions of the growth velocity of the primary arms, settling velocity, and crystal orientation of the dendrite. The results confirmed that the developed computational method can efficiently express the dendrite growth and settling over a long distance at a reasonable computational cost.

Morphological change of equiaxed dendritic structure in fcc-based alloys associated with transition in preferred growth direction was investigated by means of three-dimensional quantitative phase-field simulations. The anisotropy parameters of interfacial energy e1 and e2 were systematically changed to investigate the growth direction and growth morphology. The growth morphologies can be classified into four types; 〈1 0 0〉 growth, 〈1 0 0〉-like hyperbranched growth, 〈1 1 0〉-like hyperbranched growth and 〈1 1 0〉 growth morphologies. An orientation selection map for these morphologies was constructed in a space spanned by e1 and e2. Furthermore, dependence of the orientation selection map on solidification condition and alloy system was investigated. When initial supersaturation is large and/or partition coefficient of alloy is small, the region of 〈1 0 0〉 growth, typical growth direction of fcc alloy, diminishes in the orientation selection map.

Computer simulation is the most promising approach for the systematical permeability prediction of liquid flow in the mushy zone for various solidification conditions. In this study, we propose a permeability prediction method by using large–scale simulation of phase–field and lattice Boltzmann methods. Using the proposed method, permeability predictions are performed for the flow normal to the columnar solidification structures with multiple dendrites/cells. In addition, the prediction is also performed for columnar solidification structures with periodically arranged dendrites/cells to evaluate the permeability in the full range of solidification fraction and investigate the effect of primary arm array on the permeability. As a result, it is concluded that the dimensionless permeability using a specific interface area of columnar solidification structures can be well approximated by that of a regular hexagonal array of cylinders. Moreover, it is confirmed that the columnar dendrite/cell structure with a periodic regular hexagonal array provides realistic permeability predictions for columnar solidification structures.

Radiation necrosis with massive hemorrhage is a rare complication of radiotherapy. We report the case of a male patient who had undergone radiotherapy therapy 18 years earlier and presently underwent gamma knife radiosurgery for a metastatic brain tumor in his right occipital lobe. The patient showed aberrant behavior with left homonymous hemianopsia and a gradual deterioration of cognitive function after radiotherapy. A CT scan showed the presence of an intracerebral hematoma over the right occipital lobe with mass effect, and small spotty enhancements on the lesion when enhanced on gadolinium contrast-enhanced MRI. Intraoperative findings revealed necrosis of the occipital surface and a hematoma in the occipital lobe. Pathological findings showed damage to the walls of the sinusoidal capillaries and vitreous degeneration of the inner membrane with a spongiform hemangioma. After surgery, the cerebral edema resolved, and the patient's clinical symptoms improved. The cause of the radiation necrosis and bleeding in this patient was assumed to be due to the breakdown of the congested walls of the sinusoidal capillary vessels.

The phase-field (PF) method is a method for interface tracking that can treat complex topological changes relatively easily because of its diffuse interface model. Therefore, as computational resources become increasingly powerful, it is a promising interface tracking method for multiphase flow problems. In this study, we developed a novel multi-PF (MPF) model for expressing flows that consist of an arbitrary number of phases. Here, we adopt the conservative Allen–Cahn (CAC) model instead of the Cahn–Hilliard model, which is often employed in multiphase flow problems, to reduce computational cost and increase the flexibility of the model for various problems. Six sets of two-dimensional simulations were performed to confirm the validity of the developed model for simulating the static and dynamic phenomena of the multiphase flow problem while conserving the volume of each phase. The results confirm that the developed CAC-type MPF model can accurately express the contact angle of a droplet placed on a solid substrate and the motion of bubbles interacting with liquid–liquid interfaces.

An array of columnar dendrites is important to the microsegregation and permeability of interdendritic liquid flow. In this study, an array of tilted columnar dendrites growing during the directional solidification of a binary alloy was investigated via large-scale phase-field simulation. The main conclusion is that the hexagonal array is the dominant array for reasonably tilted dendrites as well as dendrites that grow along the temperature gradient. It is also concluded that the array ordering gradually improves with growth for all tilt angles, and the ordering rate is faster for a small tilt angle.

Competitive grain growth during the directional solidification of a polycrystalline binary alloy is investigated by performing systematic three-dimensional large-scale phase-field simulations with the GPU supercomputer TSUBAME2.5 at the Tokyo Institute of Technology. Contrary to two-dimensional investigations, in which an unusual growth of unfavorably oriented (UO) grains has been observed frequently in preference to favorably oriented (FO) ones, the grain selection in the present three-dimensional simulations follows essentially the Walton and Chalmers model, in which FO grains are predominant. The UO dendritic grains persist for longer times than the UO cellular grains, and the FO dendritic grains remain smaller than the FO cellular ones. The change in the number of grains follows a power law, with an exponent that is much lower than that of the Kolmogorov's model describing a purely geometrical growth selection.

Grain growth kinetics under the anisotropic grain boundary properties is investigated by large-scale and long-time molecular dynamics (MD) simulations of contentious processes of nucleation, solidification and grain growth in a submicrometer-scale system. Microstructures obtained via homogeneous nucleation from undercooled melt iron consists of approximately 1500 grains and the number of grains decreases to one tenth of the number via the grain growth process. The grain growth exponent obtained from the MD simulation deviates from the ideal value since anisotropic effects in the grain boundary properties are inherently included in MD simulations. It is confirmed that the decrease of the reduced mobility (i.e., the product of the intrinsic grain boundary mobility and the grain boundary energy) is a dominant factor for the deviation from the ideal grain growth. The deviation from the Mackenzie function for the distribution of the disorientation angle between neighboring grains implies that the preferential selection of grain boundaries with small grain boundary energies occurs during the grain growth. This enhances the anisotropy in grain boundary properties and therefore decreases the reduced mobility of the grain boundary. Moreover, a multi-phase-field simulation starting from a MD configuration results in an ideal grain growth when a constant value of the reduced mobility is employed, which validates the discussion on the reduced mobility. The new insight in this study is achieved for the first time owing to a multi-graphics processing unit (GPU) parallel computation over 50 days for one case using 128 GPUs on the GPU-rich supercomputer.

In the formation of a typical equiaxed structure during the solidification of metals and alloys, multiple equiaxed dendrites typically grow with motion, collision, and coalescence and subsequently grain growth occurs after the formation of grain boundaries. In this study, we develop a phase-field lattice Boltzmann model that can simulate these complex formation processes involving equiaxed structures. In this model, multiple dendrites are represented by employing multiple phase-field variables, and the formation of grain boundaries is modeled by simply introducing an interaction term between the phase-field variables. Liquid flow is computed using the lattice Boltzmann method, and the motion of a solid is described by solving the equations of motion. Collision-coalescence representation in the present model was validated by performing simulations of collisions between two circular objects. Furthermore, grain growth was validated through static and dynamic conditions in a simple three-grain system. Good agreements with theoretical solutions were obtained for both cases. Finally, using the developed model, a series of formation processes of multiple-dendrite growth with motion, collision, and coalescence and the subsequent grain growth are successfully performed for the first time.

The present study proposes multi-scale topology optimization for polycrystalline microstructures applying a multi-phase field method. The objective function is to maximize the heat compliance of macrostructure and the equality constraint is the material volume of constituents in an alloy consisting of two components with different heat conductivity. Two-scale steady-state heat conduction problem based on a homogenization method is conducted. The Allen-Cahn non-conserved time evolution equation with the additional volume constraint scheme is employed as the optimization strategy for updating the crystal configuration. In the time evolution equation, sensitivities of objective function with respect to phase-field variables are considered to relate topology optimization to the multi-phase field method. It is verified from a series of numerical examples that the proposed method has great potential for the development of material design underlying polycrystalline structure.

Three-dimensional growth morphologies of equiaxed dendrites growing under forced convection, with their preferred growth direction inclined from the flow direction, were investigated by performing large-scale phase-field lattice Boltzmann simulations on a graphical-processing-unit supercomputer. The tip velocities of the dendrite arms with their preferred growth directions inclined toward the upstream and downstream directions increased and decreased, respectively, as a result of forced convection. In addition, the tip velocities decreased monotonically as the angle between the preferred growth direction and the upstream direction increased. Here, the degree of acceleration of the upstream tips was larger than the degree of deceleration of the downstream tips. The angles between the actual tip growth directions and the preferred growth direction of the dendrite arms exhibited a characteristic change with two local maxima and two local minima.

Utilizing the data assimilation and multi-phase-field grain growth model, this study proposes a novel framework of measuring anisotropic (nonuniform) grain boundary energy and mobility. The framework can evaluate a large number of boundary properties from typical observations of grain growth without requiring specifically designed experiments or calculations. In this method, by optimizing the multi-phase-field model parameters such that the simulation results are in good agreement with the observation data, the energies and mobilities of multiple individual boundaries are directly and simultaneously estimated. To validate the method, numerical tests on boundary property estimation were performed using synthetic microstructure dataset generated from grain growth simulations with a priori assumed property values. Systematic tests on simple tricrystal systems confirmed that the proposed method accurately estimates each boundary energy and mobility within an error of only several % of their assumed true values even for conditions with strong property anisotropy and grain rotation. Further numerical tests were conducted on a more general multi-grain system, showing that our method can be successfully applied to complicated polycrystalline grain growth. The obtained results demonstrate the potential of the proposed method in extracting a large dataset of grain boundary properties for arbitrary boundaries from actual grain growth observations.

The formation process of equiaxed structures during alloy solidification is a complicated multiphysics problem as it includes solid–liquid phase transformations, transport of solute and heat, liquid flow, solid motion, solid–solid interactions, and grain growth after solidification completion. In this study, a multi-phase-field (MPF) model with the double-obstacle (DO) potential is introduced to our previous model [T. Takaki et al., Comput. Mater. Sci., 147 (2018) 124–131] to express the formation process of equiaxed structures accurately and efficiently, from the growth of multiple mobile dendrites to the grain growth with grain boundary anisotropy after solidification. The accuracy of the resulting MPF-lattice Boltzmann (MPF-LB) model with the DO potential is evaluated by comparison with the results obtained using a model with the double-well potential through simulations of purely diffusive dendrite growth, the growth of an immobile dendrite with forced convection, and the growth of a mobile dendrite under shear flow. Finally, the model is applied to both large-scale polycrystalline solidification and semi-solid deformation processes.

Continuous processes of homogeneous nucleation, solidification and grain growth are spontaneously achieved from an undercooled iron melt without any phenomenological parameter in the molecular dynamics (MD) simulation with 12 million atoms. The nucleation rate at the critical temperature is directly estimated from the atomistic configuration by cluster analysis to be of the order of 10 34 m −3 s −1 . Moreover, time evolution of grain size distribution during grain growth is obtained by the combination of Voronoi and cluster analyses. The grain growth exponent is estimated to be around 0.3 from the geometric average of the grain size distribution. Comprehensive understanding of kinetic properties during continuous processes is achieved in the large-scale MD simulation by utilizing the high parallel efficiency of a graphics processing unit (GPU), which is shedding light on the fundamental aspects of production processes of materials from the atomistic viewpoint.

Numerical studies of the effects of anisotropic (misorientation-dependent) grain-boundary energy and mobility on polycrystalline grain growth have been carried out for decades. However, conclusive knowledge has yet to be obtained even for the simplest two-dimensional case, which is mainly due to limitations in the computational accuracy of the grain-growth models and computer resources that have been employed to date. Our study attempts to address these problems by utilizing a higher-order multi-phase-field (MPF) model, which was developed to accurately simulate grain growth with anisotropic grain-boundary properties. In addition, we also employ general-purpose computing on graphics processing units to accelerate MPF grain-growth simulations. Through a series of simulations of anisotropic grain growth, we succeeded in confirming that both the anisotropies in grain-boundary energy and mobility affect the morphology formed during grain growth. On the other hand, we found the grain growth kinetics in anisotropic systems to follow parabolic law similar to isotropic growth, but only after an initial transient period.

Simulating dendritic growth with natural convection is challenging because of the size of the computational domain required when compared to the dendrite scale. In this study, a phase-field-lattice Boltzmann model was used to simulate dendritic growth in the presence of natural convection due to a difference in solute concentration. To facilitate and accelerate the large-scale simulation, a parallel computing code with multiple graphics processing units was developed. The effects of the computational domain size as well as those of gravity on the dendritic morphologies were examined by performing two-dimensional free dendritic growth simulations with natural convection. The effects of the gravity direction on the dendrite spacing and morphology were also investigated by simulating unidirectional solidification from multiple seeds.

Solid-liquid interfacial properties out of equilibrium provide the essential information required for understanding and controlling solidification microstructures in metallic materials. However, few studies have attempted to reveal all interfacial properties out of equilibrium in detail. The present study proposes an approach for simultaneously estimating all interfacial properties in a pure metal below the melting point on the basis of the Bayesian inference theory. The solid-liquid interfacial energy, interfacial mobility, and anisotropy parameters in pure Fe are estimated by combining molecular dynamics simulation with phase-field simulation using an ensemble Kalman filter, which is a data assimilation technique. Furthermore, the temperature dependences of all interfacial parameters are computed and discussed. In summary, the proposed multiscale approach integrates atomistic and microstructural simulations within the framework of data science and it has considerable potential for a wide variety of applications in materials engineering.

The surface of a hot-dip galvanized layer is characterized by a beautiful solidification pattern of large Zn grains with a {0001} texture, which is called the spangle. The formation mechanism for the {0001} spangle texture has not yet been conclusively clarified. In this study, dendrite growth in a thin coating layer with Zn-0.2 wt%Al was systematically simulated in two dimensions using the phase-field method. As a result of the simulations, the {0001} texture is not formed when considering only the anisotropy of the solid–liquid interface energy. On the other hand, when considering the anisotropy of mobility, a clear {0001} texture is observed. Thus, it is concluded that the formation of the {0001} spangle texture may result from mobility anisotropy.

Oil with a chlorinated extremely-pressure additive (EP-additive) is effective to prevent galling in cold forming for stainless steel. However environmental issues have pressured the manufacture to replace this oil with high performance oil without chloride. Particularly, sulfur-based EP-additives are accepted as a practicable replacement for chlorinated EP-additives. Thus, the authors analyzed the structures of organic-sulfur compounds. Moreover, the effects of the molecular structures on the anti-galling performance are estimated by a cup internal ironing test. This test has been devised to classify the performance of sample oils by the ironing load and the damage on the workpiece after the test. Consequently, sulfurized olefin was superior to sulfurized ester and sulfurized fatty oil in cold ironing for stainless steel. It was also clarified that sulfur EP-additives generated iron sulfide and sulfate on the surface of stainless steel by using the surface analyses of X-ray photoelectron spectroscopy.

Based on the multi-phase-field (MPF) model reported by Steinbach et al., we constructed a higher-order MPF model in a previous study that contains a higher-order term and an additional kinetic parameter to represent the properties of triple junctions (TJs); this model was observed to be suitable for the simulation of 2D grain growth with anisotropic grain-boundary (GB) energy and mobility, which are strongly dependent on the misorientation angle (Δθ). In the current study, we attempt to improve the accuracy of 3D MPF simulations of anisotropic grain growth by extending this higher-order MPF model such that it accounts for the properties of quadruple junctions as well as those of TJs. In addition, using the extended higher-order MPF model, a series of grain-growth simulations are performed for a 3D columnar structure while considering the anisotropic GB properties, through which the accuracy of the model is examined in detail. The results confirm that the extended higher-order MPF model enables the anisotropic GB properties to be handled accurately for wider-ranging Δθ than in previous models.

The multi-phase-field (MPF) model proposed by Steinbach et al. has several advantages when it comes to numerically simulating the grain growth, recrystallization, and multiple phase transitions. In this study, in order to improve the accuracy of MPF simulations using the anisotropic grain-boundary energy and mobility, which depend strongly on the misorientation angles, we account for the triple-junction properties in the MPF model. Further, two-dimensional simulations of grain-boundary migrations in three-grain systems as well as simulations of abnormal grain growth in a polycrystalline system are performed using the proposed model, in order to confirm its validity. The results show that the proposed model allows for the introduction of the anisotropic energy and mobility with high accuracy for a wider range of misorientations, in contrast to the conventional model.