Phase field model for electric-thermal coupled discharge breakdown of polyimide nanocomposites under high frequency electrical stress

In contrast to conventional transformers, power electronic transformers, as an integral component of new energy power system, are often subjected to high-frequency and transient electrical stresses, leading to heightened concerns regarding insulation failures. Meanwhile, the underlying mechanism behind discharge breakdown failure and nanofiller enhancement under high-frequency electrical stress remains unclear. An electric-thermal coupled discharge breakdown phase field model was constructed to study the evolution of the breakdown path in polyimide nanocomposite insulation subjected to high-frequency stress. The investigation focused on analyzing the effect of various factors, including frequency, temperature, and nanofiller shape, on the breakdown path of Polyimide(PI) composites. Additionally, it elucidated the enhancement mechanism of nano-modified composite insulation at the mesoscopic scale. The results indicated that with increasing frequency and temperature, the discharge breakdown path demonstrates accelerated development, accompanied by a gradual dominance of Joule heat energy. This enhancement is attributed to the dispersed electric field distribution and the hindering effect of the nanosheets. The research findings offer a theoretical foundation and methodological framework to inform the optimal design and performance management of new insulating materials utilized in high-frequency power equipment.

Bifurcation Analysis Reveals Solution Structures of Phase Field Models

The phase field method is playing an increasingly important role in understanding and predicting morphological evolution in materials and biological systems.Here,we develop a new analytical approach based on the bifurcation analysis to explore the mathematical solution structure of phase field models.Revealing such solution structures not only is of great mathematical interest but also may provide guidance to experimentally or computationally uncover new morphological evolution phenomena in materials undergoing electronic and structural phase transitions.To elucidate the idea,we apply this analytical approach to three representative phase field equations:the Allen-Cahn equation,the Cahn-Hilliard equation,and the Allen-Cahn-Ohta-Kawasaki system.The solution structures of these three phase field equations are also verified numerically by the homotopy continuation method.

Phase field modeling of dendrite growth

Single dendrite and multi-dendrite growth for Al-2 mol pct Si alloy during isothermal solidification are simulated by phase field method. In the case of single equiaxed dendrite growth, the secondary and the necking phenomenon can be observed. For multi-dendrite growth, there exists the competitive growth among the dendrites during solidification. As solidification proceeds, growing and coarsening of the primary arms occurs, together with the branching and coarsening of the secondary arms. When the diffusion fields of dendrite tips come into contact with those of the branches growing from the neighboring dendrites, the dendrites stop growing and being to ripen and thicken.

Phase field modeling of the ring-banded spherulites of crystalline polymers: The role of thermal diffusion

The ring-banded spherulite is a special morphology of polymer crystals and has attracted considerable attention over recent decades. In this study, a new phase field model with polymer characteristics is established to investigate the emergence and formation mechanism of the ring-banded spherulites of crystalline polymers. The model consists of a nonconserved phase field representing the phase transition and a temperature field describing the diffusion of the released latent heat. The corresponding model parameters can be obtained from experimentally accessible material parameters.Two-dimensional calculations are carried out for the ring-banded spherulitic growth of polyethylene film under a series of crystallization temperatures. The results of these calculations demonstrate that the formation of ring-banded spherulites can be triggered by the self-generated thermal field. Moreover, some temperature-dependent characteristics of the ring-banded spherulites observed in experiments are reproduced by simulations, which may help to study the effects of crystallization temperature on the ring-banded structures.

Discussions on the non-equilibrium effects in the quantitative phase field model of binary alloys

All the quantitative phase field models try to get rid of the artificial factors of solutal drag, interface diffusion and interface stretch in the diffuse interface. These artificial non-equilibrium effects due to the introducing of diffuse interface are analysed based on the thermodynamic status across the diffuse interface in the quantitative phase field model of binary alloys. Results indicate that the non-equilibrium effects are related to the negative driving force in the local region of solid side across the diffuse interface. The negative driving force results from the fact that the phase field model is derived from equilibrium condition but used to simulate the non-equilibrium solidification process. The interface thickness dependence of the non-equilibrium effects and its restriction on the large scale simulation are also discussed.

Simulation of size effects by a phase field model for fracture

In phase field fracture models the value of the order parameter distin- guishes between broken and undamaged material. At crack faces the order param- eter interpolates smoothly between these two states of the material, which can be regarded as phases. The crack evolution follows implicitly from the time inte- gration of an evolution equation of the order parameter, which is coupled to the mechanical field equations. Among other phenomena phase field fracture mod- els are able to reproduce crack nucleation in initially sound materials. For a 1D setting it has been shown that crack nucleation is triggered by the loss of stability of the unfractured, spatially homogeneous solution, and that the stability point depends on the size of the considered structure. This work numerically investi- gates to which extend size effects are reproduced by the 2D phase field model. Exemplarily, a finite element study of the hole size effect is performed and the simulation results are compared to exnerimental data.

Experimental studies and phase field modeling of microstructure evolution during solidification with electromagnetic stirring

Thixocasting requires manufacturing of billets with non-dendritic microstructure.Aluminum alloy A356 billets were produced by rheocasting in a mould placed inside a linear electromagnetic stirrer.Subsequent heat treatment was used to produce a transition from rosette to globular microstructure.The current and the duration of stirring were explored as control parameters.Simultaneous induction heating of the billet during stirring was quantified using experimentally determined thermal profiles.The effect of processing parameters on the dendrite fragmentation was discussed.Corresponding computational modeling of the process was performed using phase-field modeling of alloy solidification in order to gain insight into the process of morphological changes of a solid during this process.A non-isothermal alloy solidification model was used for simulations.The morphological evolution under such imposed thermal cycles was simulated and compared with experimentally determined one.Suitable scaling using the thermosolutal diffusion distances was used to overcome computational difficulties in quantitative comparison at system scale.The results were interpreted in the light of existing theories of microstructure refinement and globularisation.

Phase field modeling of multiple dendrite growth of Al-Si binary alloy under isothermal solidification

Phase field method offers the prospect of being able to perform realistic numerical experiments on dendrite growth in metallic systems.In this study,the growth process of multiple dendrites in Al-2-mole-%-Si binary alloy under isothermal solidification was simulated using phase field model.The simulation results showed the impingement of arbitrarily oriented crystals and the competitive growth among the grains during solidification.With the increase of growing time,the grains begin to coalesce and impinge the adjacent grains.When the dendrites start to impinge,the dendrite growth is obviously inhibited.

Numerical simulation for GMAW with a new model based on phase field model

In this paper,a numerical investigation about the metal transfer of GMAW is investigated based on the phase field model.Be different of most published work,we take the thermocapillary effect and mixture energy into the process of phase transfer and interface change which is different from volume of fluid( VOF) method.We discretize the whole model with a continuous finite element method and we also apply a penalty formulation to the continuity condition enhancing the stability of the pressure.Metal transfer of GMAW with constant and pulse current is computed as numerical examples which agrees well with the data of high-speed photography.The result shows that the computing process of the phase field model is stability and it has a higher precision in predicting the diameter of droplet.

Prediction of mushy zone permeability of Al-4.5wt%Cu alloy during solidification by phase field model and CFD simulation

Liquid permeability of the mushy zone is important for porosity formation during the solidification process. In order to investigate the permeability of the mushy zone, an integrated model was developed by combining the phase field model and computational fluid dynamics (CFD) model. The three-dimensional multigrain dendrite morphology was obtained by using the phase field model. Subsequently, the computer-aided design (CAD) geometry and mesh were generated based on calculated dendrite morphologies. Finally, the permeability of the dendritic mushy zone was obtained by solving the Navier-Stokes and continuity equations in ANSYS Fluent software. As an example, the dendritic mushy zone permeability of Al-4.5wt%Cu alloy and its relationship with the solid fractions were studied in detail. The predicted permeability data can be input to the solidification model on a greater length scale for macro segregation and porosity simulations.

Crack nucleation and propagation simulation in brittle two-phase perforated/particulate composites by a phase field model

Fracture is a very common failure mode of the composite materials,which seriously affects the reliability and service-life of composite materials.Therefore,the study of the fracture behavior of the composite materials is of great significance and necessity,which demands an accurate and efficient numerical tool in general cases because of the complexity of the arising boundary-value or initial-boundary value problems.In this paper,a phase field model is adopted and applied for the numerical simulation of the crack nucleation and propagation in brittle linear elastic two-phase perforated/particulate composites under a quasi-static tensile loading.The phase field model can well describe the initiation,propagation and coalescence of the cracks without assuming the existence and the geometry of the initial cracks in advance.Its numerical implementation is realized within the framework of the finite element method(FEM).The accuracy and the efficiency of the present phase field model are verified by the available reference results in literature.In the numerical examples,we first study and discuss the influences of the hole/particle size on the crack propagation trajectory and the force-displacement curve.Then,the effects of the hole/particle shape on the crack initiation and propagation are investigated.Furthermore,numerical examples are presented and discussed to show the influences of the hole/particle location on the crack initiation and propagation characteristics.It will be demonstrated that the present phase field model is an efficient tool for the numerical simulation of the crack initiation and propagation problems in brittle two-phase composite materials,and the corresponding results may play an important role in predicting and preventing possible hazardous crack initiation and propagation in engineering applications.

A Phase-field Model to Simulate Recrystallization in an AZ31 Mg Alloy in Comparison of Experimental Data

A model has been established to simulate the realistic spatio-temporal microstructure evolution in recrystallization of a magnesium alloy using the phase field approach. A set of rules have been proposed to decide the real physical value of all parameters in the model. The thermodynamic software THERMOCALC is applied to determine the local chemical free energy and strain energy, which is added to the free energy density of grains before recrystallization. The Arrhenius formula is used to describe boundary mobility and the activity energy is suggested with a value of zinc segregation energy at the boundary. However, the mobility constant in the formula was found out by fitting to a group of grain size measurements during recrystallization of the alloy. The boundary range is suggested to decide the gradient parameters in addition of fitting to the experimental boundary energy value. These parameter values can be regarded as a database for other similar simulations and the fitting rules can also be applied to build up databases for any other alloy systems. The simulated results show a good agreement with reported experimental measurement of the alloy at the temperatures from 300 to 400℃ for up to 100 min but not at 250℃. This implies a mechanism variation in activity energy of the boundary mobility in the alloy at low temperature.

Computer Simulation of Microscopic Stress Distribution in Complex Microstructure Using a Phase Field Model

Microscopic stress distribution in a metallic material which has complex microstructure is simulated using a phase field model.The fundamental equations which take into account the coupling effects among phase transformation,temperature and stress/strain are used,while thermal effects are neglected to focus on the volumetric change due to phase transformation in this paper.A two-dimensional square region is considered,and the evolution of microscopic stress and the resultant residual stress distribution are calculated using the finite element method.As the phase transformation progresses and grains grow larger,stress is generated around the growing interface.When a grain collides with another one,specifically large stress is observed.Residual stress is finally distributed in the microstructure formed,and apparently large stresses are retained along the grain boundaries. Subsequently,dependency of the stress distribution on microstructure pattern is investigated.First,variously sized square grains are tested,and it reveals that the maximum stress tends to decrease as the grain size becomes smaller.Next,the shapes of the grains are varied.As a result,the stress distribution is remarkably affected,while the maximum stress value does not change so much.More complicated grain arrangement is finally tested with eight or nine grain models.Then,it is revealed as a common feature that large stress is generated along the grain boundaries and that the stress distribution is dependent on the grain arrangement.

MICRO-DESCRIPTION OF THE SOLUTE-FIELD AND THE PHASE-FIELD MODEL FOR ISOTHERMAL PHASE TRANSITION IN BINARY ALLOYS

A new phase field method for two-dimensional simulations of binary alloy solidification was studied. A model basing on solute conservative in every unit was developed for solving the solute diffusion equation during solidification. Two-dimensional computations were performed for ideal solutions and Ni-Cu dendritic growth into an isothermal and highly supersaturated liquid phase.

EFFICIENT LINEAR SCHEMES WITH UNCONDITIONAL ENERGY STABILITY FOR THE PHASE FIELD MODEL OF SOLID-STATE DEWETTING PROBLEMS

In this paper,we study linearly first and second order in time,uniquely solvable and unconditionally energy stable numerical schemes to approximate the phase field model of solid-state dewetting problems based on the novel"scalar auxiliary variable"(SAV)approach,a new developed efficient and accurate method for a large class of gradient flows.The schemes are based on the first order Euler method and the second order backward differential formulas(BDF2)for time discretization,and finite element methods for space discretization.The proposed schemes are proved to be unconditionally stable and the discrete equations are uniquely solvable for all time steps.Various numerical experiments are presented to validate the stability and accuracy of the proposed schemes.

Phase Field Models Versus Parametric Front Tracking Methods: Are They Accurate and Computationally Efficient?

We critically compare the practicality and accuracy of numerical approximations of phase field models and sharp interface models of solidification.Here we focus on Stefan problems,and their quasi-static variants,with applications to crystal growth.New approaches with a high mesh quality for the parametric approximations of the resulting free boundary problems and new stable discretizations of the anisotropic phase field system are taken into account in a comparison involving benchmark problems based on exact solutions of the free boundary problem.

Mass and Volume Conservation in Phase Field Models for Binary Fluids

The commonly used incompressible phase field models for non-reactive,binary fluids,in which the Cahn-Hilliard equation is used for the transport of phase variables(volume fractions),conserve the total volume of each phase as well as the material volume,but do not conserve the mass of the fluid mixture when densities of two components are different.In this paper,we formulate the phase field theory for mixtures of two incompressible fluids,consistent with the quasi-compressible theory[28],to ensure conservation of mass and momentum for the fluid mixture in addition to conservation of volume for each fluid phase.In this formulation,the mass-average velocity is no longer divergence-free(solenoidal)when densities of two components in the mixture are not equal,making it a compressible model subject to an internal constraint.In one formulation of the compressible models with internal constraints(model 2),energy dissipation can be clearly established.An efficient numerical method is then devised to enforce this compressible internal constraint.Numerical simulations in confined geometries for both compressible and the incompressible models are carried out using spatially high order spectral methods to contrast the model predictions.Numerical comparisons show that(a)predictions by the two models agree qualitatively in the situation where the interfacial mixing layer is thin;and(b)predictions differ significantly in binary fluid mixtures undergoing mixing with a large mixing zone.The numerical study delineates the limitation of the commonly used incompressible phase field model using volume fractions and thereby cautions its predictive value in simulating well-mixed binary fluids.

Comparison of Cellular Automaton and Phase Field Models to Simulate Dendrite Growth in Hexagonal Crystals

A cellular automaton （CA）-finite element （FE） model and a phase field （PF）-FE model were used to simulate equiaxed dendritic growth during the solidification of hexagonal metals. In the CA-FE model, the conservation equations of mass and energy were solved in order to calculate the temperature field, solute concentration, and the dendritic growth morphology. CA-FE simulation results showed reasonable agreement with the previously reported experimental data on secondary dendrite arm spacing （SDAS） vs cooling rate. In the PF model, a PF variable was used to distinguish solid and liquid phases similar to the conventional PF models for solidification of pure materials. Another PF variable was considered to determine the evolution of solute concentration. Validation of both models was performed by comparing the simulation results with the analytical model developed by Lipton-Glicksman-Kurz （LGK）, showing quantitatively good agreement in the tip growth velocity at a given melt undercooling. Application to magnesium alloy AZ91 （approximated with the binary Mg-8.9 wt% AI） illustrates the difficulty of modeling dendrite growth in hexagonal systems using CA-FE regarding mesh-induced anisotropy and a better performance of PF-FE in modeling multiple arbitrarily-oriented dendrites growth.

Numerical Simulation of Two-Dimensional Dendritic Growth Using Phase-Field Model

In this article, we study the phase-field model of solidification for numerical simulation of dendritic crystal growth that occurs during the casting of metals and alloys. Phase-field model of solidification describes the physics of dendritic growth in any material during the process of under cooling. The numerical procedure in this work is based on finite difference scheme for space and the 4th-order Runge-Kutta method for time discretization. The effect of each physical parameter on the shape and growth of dendritic crystal is studied and visualized in detail.

Phase-Field Modeling for the Three-Dimensional Space-Filling Structure of Metal Foam Materials

Phase-field modeling for three-dimensional foam structures is presented. The foam structure, which is generally applicable for porous material design, is geometrically approximated with a space-filling structure, and hence, the analysis of the space-filling structure was performed using the phase field model. An additional term was introduced to the conventional multi-phase field model to satisfy the volume constraint condition. Then, the equations were numerically solved using the finite difference method, and simulations were carried out for several nuclei settings. First, the nuclei were set on complete lattice points for a bcc or fcc arrangement, with a truncated hexagonal structure, which is known as a Kelvin cell, or a rhombic dodecahedron being obtained, respectively. Then, an irregularity was introduced in the initial nuclei arrangement. The results revealed that the truncated hexagonal structure was stable against a slight irregularity, whereas the rhombic polyhedral was destroyed by the instability. Finally, the nuclei were placed randomly, and the relaxation process of a certain cell was traced with the result that every cell leads to a convex polyhedron shape.