Phase-field Equations Research Articles

Experimental and Numerical Study of Vacuum Resin Infusion for Thin-Walled Composite Parts

This paper considers a new approach to the modeling of the vacuum infusion process at the manufacturing of three-dimensional composite parts of complex shape. The developed approach and numerical methods focus on reliable prediction with needed accuracy and elimination of the unrecoverable defect of composite structure such as the dry spots. The paper presents some experimental results, which demonstrate two cases of dry spot formation in large aircraft composite panels, and analyzes the reasons for these defects arising. Our numerical technique is based on the vacuum infusion of the liquid resin into porous preform as the two-phase flow, which is described by the phase field equation coupled with the Richards equation describing the fluid motion in unsaturated soils with spatially varied pressure dependent porosity and saturation. This problem statement allowed correctly reconstructing the resin front motion and formation of inner and outer dry spots depending on its movement. For the rapid detection of preform zones that are suspicious for defect formation, two indicators calculated during process simulation are proposed and tested at the numerical experiments. The auxiliary program tool has been developed in the MATLAB environment to correctly detect the times of formation, localization and dimensions of the arising dry spots by using the results of the finite element model simulation.

An exact formulation for exponential-logarithmic transformation stretches in a multiphase phase field approach tomartensitic transformations

A general theoretical and computational procedure for dealing with an exponential-logarithmic kinematic model for transformation stretch tensor in a multiphase phase field approach to stress- and temperature-induced martensitic transformations with N martensitic variants is developed for transformations between all possible crystal lattices. This kinematic model, where the natural logarithm of transformation stretch tensor is a linear combination of natural logarithm of the Bain tensors, yields isochoric variant–variant transformations for the entire transformation path. Such a condition is plausible and cannot be satisfied by the widely used kinematic model where the transformation stretch tensor is linear in Bain tensors. Earlier general multiphase phase field studies can handle commutative Bain tensors only. In the present treatment, the exact expressions for the first and second derivatives of the transformation stretch tensor with respect to the order parameters are obtained. Using these relations, the transformation work for austenite ↔ martensite and variant ↔ variant transformations is analyzed and the thermodynamic instability criteria for all homogeneous phases are expressed explicitly. The finite element procedure with an emphasis on the derivation of the tangent matrix for the phase field equations, which involves second derivatives of the transformation deformation gradients with respect to the order parameters, is developed. Change in anisotropic elastic properties during austenite–martensitic variants and variant–variant transformations is taken into account. The numerical results exhibiting twinned microstructures for cubic to orthorhombic and cubic to monoclinic-I transformations are presented.

A fully decoupled, linear and unconditionally energy stable numerical scheme for a melt-convective phase-field dendritic solidification model

We consider in this paper numerical approximations of a melt-convective phase-field dendritic solidification model. A challenging numerical issue for solving this model is how to develop efficient time marching schemes that are not only unconditionally energy stable, but also linear and totally decoupled since the model is a highly coupled nonlinear system. We solve this issue by combining the modified projection scheme for the Navier–Stokes equations, the stabilized-Invariant Energy Quadratization method for the anisotropic phase field equation. Meanwhile, in order to obtain the fully decoupled feature, we introduce an auxiliary intermediate temperature variable to decouple the computation of the temperature from the phase field variable. We prove the unconditional energy stability of the developed scheme and further present various numerical simulations in 2D and 3D to demonstrate the accuracy and stability of the developed scheme.

Quantum-Phase-Field: From de Broglie–Bohm Double-Solution Program to Doublon Networks

Abstract Different forms of linear and non-linear field equations, so-called ‘phase-field’ equations, are studied in relation to the de Broglie–Bohm double-solution program. This defines a framework in which elementary particles are described by localised non-linear wave solutions moving by the guidance of a pilot wave, defined by the solution of a Schrödinger-type equation. First, we consider the phase-field order parameter as the phase for the linear pilot wave, second as the pilot wave itself, and third as a moving soliton interpreted as a massive particle. In the last case, we introduce the equation for a superwave, the amplitude of which can be considered as a particle moving in accordance to the de Broglie–Bohm theory. Lax pairs for the coupled problems are constructed in order to discover possible non-linear equations that can describe the moving particle and to propose a framework for investigating coupled solutions. Finally, doublons in 1 + 1 dimensions are constructed as self-similar solutions of a non-linear phase-field equation forming a finite space object. Vacuum quantum oscillations within the doublon determine the evolution of the coupled system. Applying a conservation constraint and using general symmetry considerations, the doublons are arranged as a network in 1 + 1 + 2 dimensions, where nodes are interpreted as elementary particles. A canonical procedure is proposed to treat charge and electromagnetic exchange.

Modeling and Simulation of the Process for the Generation of Gradient Porous Structures From Immiscible Polymer Blends

Abstract There has been growing interest in integrating gradient porous structures into synthetic materials like polymers. One particular method for making gradient porous polymers is nonisothermal annealing of co-continuous phase structures of immiscible polymer blends under well-defined thermal boundary conditions. In this paper, we report a method to simulate this nonisothermal phase coarsening process for the generation of gradient-phase structures by the combined implementation of phase-field transport and momentum transport. Specifically, a phase-field equation is solved first to obtain a phase structure with phase size comparable with that of the blend to be annealed. This phase structure is then used as an initial geometry in a two-phase moving-interface flow simulation to gauge into the phase structure coarsening process. Several case studies were performed, and the results show that the controllable generation of gradient-phase structures can be enabled by well-designed geometry and thermal boundary conditions. Using 2D simulations, different types of gradient-phase structures experimentally observed were predicted. With increasing power in computation, the capability of 3D simulation may be unveiled for a more accurate prediction of the nonisothermal phase coarsening process and may ultimately evolve into a useful tool for the design and processing of gradient porous polymers.

Application of a Phase Field Model to Multicomponent Al-Cu-Si alloys

Abstract The solidification of metals and alloys and the resulting microstructures, which as a function of thermal and solutal parameters can evolve as planar, cellular and dendritic, are important from a practical point of view, since they strongly influence the properties and quality of the final product. In many practical situations it is impracticable to develop analytical solutions permitting reliable predictions of microstructural growth during unsteady-state solidification conditions. The Phase Field method has become very popular and effective in modeling complex solid/liquid interfaces due to its ability to simulate the interface kinetics and the formation and evolution of different morphologies along the solidification process. In this work, a numerical analysis of the microstructural evolution during the transient solidification of dilute alloys of the Al-Cu-Si system is developed, which uses a phase-field approach for the simulation of ternary alloys. The phase-field, energy and solute concentration equations were numerically solved for the correspondent ternary system, varying the mesh parameters, temperature and alloy composition. The analysis performed were confronted with existing theoretical models and the results obtained are in agreement with the solidification theory.

Optimal control theory and advanced optimality conditions for a diffuse interface model of tumor growth

We investigate a distributed optimal control problem for a diffuse interface model for tumor growth. The model consists of a Cahn–Hilliard type equation for the phase field variable, a reaction diffusion equation for the nutrient concentration and a Brinkman type equation for the velocity field. These PDEs are endowed with homogeneous Neumann boundary conditions for the phase field variable, the chemical potential and the nutrient as well as a “no-friction” boundary condition for the velocity. The control represents a medication by cytotoxic drugs and enters the phase field equation. The aim is to minimize a cost functional of standard tracking type that is designed to track the phase field variable during the time evolution and at some fixed final time. We show that our model satisfies the basics for calculus of variations and we present first-order and second-order conditions for local optimality. Moreover, we present a globality condition for critical controls and we show that the optimal control is unique on small time intervals.

Existence of Global-in-Time Weak Solutions for a Solidification Model with Convection in the Liquid and Rigid Motion in the Solid

We introduce a PDE problem modeling a solidification/melting process in bounded two- or three-dimensional domains, coupling a phase-field equation and a Navier--Stokes--Boussinesq system, where the latent heat effect is considered via a modification of the Caginalp model. Moreover, the convection in the nonsolid and solid regions is treated via a phase-dependent viscosity of the material that degenerates in the solid phase, letting only rigid motions in this phase. Then we prove the existence of global-in-time weak solutions (of a regularized problem in three-dimensional domains) based on a limit process of a sequence of dissipative problems furnished truncating the viscosity.

Pore-scale numerical modeling of coupled fluid flow and medium geometrical deformations in an unconsolidated porous medium

The interplay of fluid flow and medium grains' deformation/movement in unconsolidated porous media was numerically studied. The numerical simulations were done through coupling Cahn-Hilliard phase field and Navier-Stokes equations for fluid flow as well as stress-strain and Arbitrary Lagrangian/Eulerian mesh alteration equations for geomechanical effects, by the finite-element method. Single/two-phase flow through a real patterned micro-scale medium with/without grains' deformation and movements/rotation were studied. In single-phase models, the fluid velocity distribution was quite similar for the cases with rigid grains and that with only deformed grains. However, in an unconsolidated medium, the velocity magnitude and distribution were modified. The medium porosity had a linear trend with pressure, and was independent of the grains' movement/rotation. The models with deformed grains showed good agreement with Kozeny-Carman equation in permeability variation versus pressure. In the two-phase flow models, the velocity/displacement profiles, relative permeability end-points and capillary pressure were quantitatively compared in rigid and unconsolidated media versus medium wettability. The effect of the grains' deformation on the fluid distributions was negligible at low capillary numbers, especially in water-wet and neutral wetting conditions. However, the grains' movement/rotation considerably modified the flow regime at different grains' contact angles. At higher capillary numbers, the grains' deformation effect was more pronounced.

Reduction of discretisation-induced anisotropy in the phase-field modelling of dendritic growth by meshless approach

A numerical procedure is developed to assess and reduce the discretisation-induced anisotropy in the solution of the phase-field model for dendritic growth. The meshless radial basis function-generated finite difference (RBF-FD) method and the forward Euler scheme are used for the spatial and temporal discretisation of the phase-field equations, respectively. A second-order accurate RBF-FD method is ensured by the use of augmentation with monomials up to the second order, while shape-parameter-free polyharmonic splines of the fifth-order are used as the radial basis functions. The performance of the RBF-FD method is assessed on regular and scattered node distributions by observing the mean phase field, the size of the primary dendrite arm, and the growth velocity. The observables at different orientation angles are compared to assess the orientation dependence of the solution. We show for the first time that the use of the RBF-FD method on a scattered node distribution provides a robust approach for the solution of the phase-field model for dendritic solidification with respect to an arbitrary preferential growth direction.

Two fast and efficient linear semi-implicit approaches with unconditional energy stability for nonlocal phase field crystal equation

The phase-field crystal equation is a sixth-order nonlinear parabolic equation and have received increasing attention in the study of the microstructural evolution of two-phase systems on atomic length and diffusive time scales. This model can be applied to simulate various phenomena such as epitaxial growth, material hardness and phase transition. Compared with the classical local gradient flow and phase field models, the nonlocal models such as nonlocal phase-field crystal equation equipped with nonlocal diffusion operator can describe more practical phenomena for modeling phase transitions. We propose linear semi-implicit approach and scalar auxiliary variable approach with unconditional energy stability for the nonlocal phase-field crystal equation. The first contribution is that we have proved the unconditional energy stability for nonlocal phase-field crystal model and its semi-discrete schemes carefully and rigorously. Secondly, we found a fast procedure to reduce the computational work and memory requirement which the non-locality of the nonlocal diffusion term generates huge computational work and memory requirement. Finally, several numerical simulations are demonstrated to verify the accuracy and efficiency of our proposed schemes.

Effect of Temperature Gradient on the Grain Size Homogeneity of SEED Produced Semi-Solid Slurries by Phase-Field Simulation.

The distribution homogeneity of grain size affects the fluidity of the semi-solid slurry, which in turn affects the properties of the casting. One key factor affecting grain size uniformity resides in the nucleation number, which has been studied thoroughly, while the other factor is temperature gradient which has not been investigated yet. In this study, the microstructure evolutions under certain temperature gradients are investigated by experiment and simulation using a two-dimensional quantitative phase-field (PF) model. A parallel and adaptive mesh refinement algorithm is adopted to solve the nonlinear phase-field equations. The results indicate that temperature gradient can affect the size distribution of microstructure in the semi-solid slurry prepared by the SEED process. A higher temperature gradient (in the range of 0.230~0.657 °C/mm) along the radial direction is beneficial to the homogeneity of the grain size in a slurry.

A conservative diffuse interface method for two-phase flows with provable boundedness properties

Central finite difference schemes have long been avoided in the context of two-phase flows for the advection of the phase indicator function due to numerical overshoots and undershoots associated with their dispersion errors. We will show however, for an incompressible flow, in the context of a specific diffuse interface model, one can maintain the boundedness of the phase field while also taking advantage of the low cost and ease of implementation of central differences to construct a non-dissipative discretization scheme for the advective terms. This is made possible by combining the advection and reinitialization steps of a conservative level set scheme introduced by Olsson and Kreiss (2005) [1] to form a phase field equation similar to that of Chiu and Lin (2011) [24]. Instead of resorting to specialized upwind methods as in these articles, we prove that the boundedness of the phase field is guaranteed for certain choices of the free parameters (ϵ and γ) for a specific central difference scheme that we propose. The proposed discretely conservative and bounded phase field equation, which is free of any reinitialization or mass redistribution, possesses desirable properties that can be leveraged in the coupled finite difference discretization of the two-phase momentum equation. Additionally, as compared to the state-of-the-art conservative and bounded two-phase flow methods, the proposed method boasts competitive accuracy-vs-cost trade-off, small memory requirements, ease of implementation, providing a viable alternative for realistic two-phase flow calculations. Given that the proposed method uses cheap, local numerical stencils with uniform loading throughout the computational domain, its implementation is expected to achieve superior parallel scalability compared to the commonly employed two-phase flow methods such as level set and VOF.

Formation of stress- and thermal-induced martensitic nanostructures in a single crystal with phase-dependent elastic properties

In the present paper, the effect of phase-dependent elastic properties on martensitic phase transformations (PTs) in a single crystal is investigated using the phase field approach. The simplest phase dependence of elastic properties is defined by different Young’s moduli for austenite (A) and martensite (M), and its effect is investigated for thermal- and stress-induced propagation of an A–M interface. The phase dependence of elastic properties is then included using the quadratic elastic energy with two constants different for A and martensitic variants. The coupled system of phase field and elasticity equations is solved using the nonlinear finite element method, and various examples of PTs are studied. A planar A–M interface propagation is studied under different thermal and mechanical loadings. It is revealed that the effect of phase-dependent elastic properties is more pronounced when thermal strain is included due to the interplay of elastic, transformational and thermal strains. The thermal-induced growth of a martensitic nucleus and the effect of periodic boundary conditions on the nucleus growth are investigated for both phase-independent (PI) and phase-dependent (PD) elastic properties with thermal strain and without it. Martensitic PTs with two variants are studied under different loadings using a one-fourth model and symmetric boundary conditions to reduce the effect of stress concentrations. Martensitic PTs with two variants are also studied in the presence of two circular holes for both the PI and PD elastic properties. This pronounces the significant effect of heterogeneous stress concentration and the size on the PTs. The effect of phase-dependent elastic properties is also studied on twinning in a martensitic grain embedded inside an austenitic matrix under overcooling. The obtained results reveal a significant effect of phase-dependent elastic properties on different types of martensitic PTs and remarkably change the interpretation of structural transformations at the nanoscale.

An improved phase field method by using statistical learning theory-based optimization algorithm for simulation of martensitic transformation in NiTi alloy

An improved phase field method by using statistical learning theory based optimization algorithm is developed for solving the phase field equations through building simple relationships between the key phase field variables and the phase evolution driving force, and using statistical analysis of mass computed data during phase field simulation. Phase field simulation results of growth of R phase and the B2–R phase transformation in a Ni-rich Ni50.5Ti49.5 alloy by using the proposed statistical strategy algorithm are compared with that using the conventional numerical algorithm, which demonstrates that with coupling the statistical learning theory, i.e., by means of the optimization algorithm, the credible simulated microstructure is obtained while maintaining high accuracy, and meanwhile the computational time has been significantly reduced.

Efficient modified stabilized invariant energy quadratization approaches for phase-field crystal equation

The phase-field crystal equation is a sixth-order nonlinear parabolic equation and can be applied to simulate various phenomena such as epitaxial growth, material hardness, and phase transition. We propose a series of efficient modified stabilized invariant energy quadratization approaches with unconditional energy stability for the phase-field crystal model. Firstly, we propose a more suitable positive preserving function strictly in square root and consider a modified invariant energy quadratization (MIEQ) approach. Secondly, a series of efficient and suitable functionals H(ϕ) in square root are considered and the modified stabilized invariant energy quadratization (MSIEQ) approaches are developed. We prove the unconditional energy stability and optimal error estimates for the semi-discrete schemes carefully and rigorously. A comparative study of classical IEQ, MIEQ, and MSIEQ approaches is considered to show the accuracy and efficiency. Finally, we present various 2D numerical simulations to demonstrate the stability and accuracy.

Conservative phase-field method with a parallel and adaptive-mesh-refinement technique for interface tracking.

Based on Fick's second law and Cahn-Hilliard theory, a conservative phase-field model is developed to track interface. The phase-field variable changes in a hyperbolic tangent behavior across the diffuse interface over which the interface curvature can be easily calculated. Different from the frequently used lattice-Boltzmann-based discrete method, the phase-field equation is discretized using a fourth-order Runge-Kutta method. Accordingly, the present numerical scheme alleviates the programming burden, reduces the memory usage, but maintains a high numerical accuracy. To achieve large-scale interface tracking, a parallel and adaptive-mesh-refinement algorithm is developed to reduce the computing overhead. Various cases of the interface evolutions under steady flow fields indicate that the proposed numerical scheme can capture the interface with high accuracy. Furthermore, the robustness of the numerical scheme is validated by simulating the Rayleigh-Taylor instability, and good agreement with previous work is achieved.

A stabilized phase-field method for two-phase flow at high Reynolds number and large density/viscosity ratio

Simulating two-phase flows in realistic industrial-complexity conditions remains an open problem. We present a phase-field method based on the Cahn-Hilliard equation that is able to simulate two-phase flow at high Reynolds number and at large density and viscosity ratios. We employ the entropy-viscosity method (EVM), applied both on the Navier-Stokes equations and phase-field equation, to stabilize the simulation in conjunction with an EVM-based artificial interface compression method (AICM) that maintains the sharpness of the interface. We implement this method based on a hybrid spectral-element/Fourier (SEF) discretization and demonstrate second-order accuracy in time and spectral convergence rate in space for smoothed fabricated solutions. We first test the accuracy and robustness of the stabilized SEF-EVM solver by solving the so-called three-dimensional “LeVeque problem” and compare against other available methods. Subsequently, we simulate a rising air bubble in a water container and find that the method is robust with respect to various parameters of the phase-field formulation. Lastly, we apply the method to simulate the onset and subsequent evolution of an air/oil slug in a long horizontal pipe using realistic parameters and incorporating gravity and surface tension effects. This is a particularly difficult flow to simulate with existing methods in realistic conditions and here we show that the new stabilized phase-field methods yields results in good agreement with the experimental data.

Convergence to equilibrium of global weak solutions for a Cahn–Hilliard–Navier–Stokes vesicle model

In this paper, we consider a model describing the dynamics of vesicle membranes within an incompressible viscous fluid in 3D domains. The system consists of the Navier–Stokes equations, with an extra stress tensor depending on the membrane, coupled with a Cahn–Hilliard phase-field equation associated with a bending energy plus a penalization related to the area conservation (volume is exactly conserved). This problem has a dissipative in time free energy which leads, in particular, to prove the existence of global in time weak solutions. We analyze the large-time behavior of the weak solutions. By using a modified Lojasiewicz–Simon’s result, we prove the convergence as time goes to infinity of each (whole) trajectory to a single equilibrium. Finally, the convergence of the trajectory of the phase is improved by imposing more regularity on the domain and initial phase.

CD-MPM

We present two new approaches for animating dynamic fracture involving large elastoplastic deformation. In contrast to traditional mesh-based techniques, where sharp discontinuity is introduced to split the continuum at crack surfaces, our methods are based on Continuum Damage Mechanics (CDM) with a variational energy-based formulation for crack evolution. Our first approach formulates the resulting dynamic material damage evolution with a Ginzburg-Landau type phase-field equation and discretizes it with the Material Point Method (MPM), resulting in a coupled momentum/damage solver rooted in phase field fracture: PFF-MPM. Although our PFF-MPM approach achieves convincing fracture with or without plasticity, we also introduce a return mapping algorithm that can be analytically solved for a wide range of general non-associated plasticity models, achieving more than two times speedup over traditional iterative approaches. To demonstrate the efficacy of the algorithm, we also develop a Non-Associated Cam-Clay (NACC) plasticity model with a novel fracture-friendly hardening scheme. Our NACC plasticity paired with traditional MPM composes a second approach to dynamic fracture, as it produces a breadth of organic, brittle material fracture effects on its own. Though NACC and PFF can be combined, we focus on exploring their material effects separately. Both methods can be easily integrated into any existing MPM solver, enabling the simulation of various fracturing materials with extremely high visual fidelity while requiring little additional computational overhead.