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This paper provides a critical overview of experimental and computational studies conducted on additive manufacturing (AM) or 3D printing using bimetallic alloys. The review acknowledges the complexity introduced by mechanical interactions and significant interface anisotropies in multi-material AM, making the mechanisms of phase change and microstructure evolution more intricate. Various computational models, such as density functional theory (DFT), phase field, and finite element models, employed in the study of 3D printed bimetallic materials are discussed. The paper highlights the importance of future research in developing quantitative predictions that can simulate and forecast microstructure formation during the AM process. By incorporating computational modeling, this review underlines the potential for overcoming challenges associated with the intricate interactions between different materials in multi-material AM (MMAM). [ABSTRACT FROM AUTHOR]

This paper describes a new arc-length control procedure for tracing the equilibrium curve of brittle fracture problems modeled with a phase field approach. The balance equations of this model are solved with a staggered strategy. The control equation of the arc-length procedure determines the displacement increments during the mechanical stage. The arc-length parameter is interpreted as imposing a given increment of the driving force appearing into the micro-force balance equation. The innovative technique consisting of applying the control equation to the displacement degrees of freedoms (DOFs) of the mechanical stage offers an enhancement over earlier arc-length strategies that focused on controlling the damage DOFs in the micro-force balance equation stage. This advancement enables the phase field approach to handle and simulate a broader range of problems, as demonstrated in this paper. The arc-length parameter is stepwise adjusted to yield a pre-established maximum damage increment in each staggered scheme step. As a consequence, the crack tip advance can be strictly controlled in every step holding bounded the pseudo-time integration error, even using an explicit staggered strategy. This procedure entails moderate computational costs for tracing the complete equilibrium curve, including unstable responses, limit points, snap-backs, etc., with the subsidiary advantage that lack of convergence has never been detected in the tests presented in this paper. Additionally, line search techniques have not been necessary. The proposed arc-length procedure is easily implemented in standard finite element codes, and according to our numerical experiments, it does not significantly increase the computational burden of the original explicit staggered strategy. [ABSTRACT FROM AUTHOR]

This paper presents the L-stable spectral deferred correction (SDC) methods with low stages. These schemes are initiated by the Crank-Nicolson method. We adopt the linear stabilization approach for the phase field models to obtain the linear implicit SDC scheme. This is done by adding and subtracting the linear stabilization operators that are provided for the different phase field problems. Without loss of the low-stage property, the extrapolation technique is also used in the prediction step of the semi-implicit SDC method. Numerical experiments are given to validate the high-order accuracy and the energy decay property of the proposed semi-implicit SDC methods for the Allen-Cahn, Cahn-Hilliard, and molecular beam epitaxy equations. [ABSTRACT FROM AUTHOR]

We construct several examples related to the scaling limits of energy minimizers and gradient flows of surface energy functionals in heterogeneous media. These include both sharp and diffuse interface models. The focus is on two separate but related issues: the regularity of effective surface tensions and the occurrence of zero mobility in the associated gradient flows. On regularity, we build on the 2014 theory of Chambolle, Goldman, and Novaga to show that gradient discontinuities in the surface tension are generic for sharp interface models. In the diffuse interface case, we only show that the laminations by plane-like solutions satisfying the strong Birkhoff property generically are not foliations and do have gaps. On mobility, we construct examples in both the sharp and diffuse interface case where the homogenization scaling limit of the L² gradient flow is trivial, that is, there is pinning at every direction. In the sharp interface case, these are related to examples previously constructed for forced mean curvature flow in Novaga and Valdinoci's 2011 paper. [ABSTRACT FROM AUTHOR]

In this study, the phase field method has been used to study the effect of second phase particles with different shapes and different orientations on the grain growth of AZ31 magnesium alloy, after annealing at 350 °C for 100 min. The results show that the shape of the second phase particles would have an effect on the grain growth; the refinement effect of elliptical particles and rod-shaped particles was similar, and better than the spherical particles; the spatial arrangement direction of the second phase particles had no significant effect on the grain growth. On the other hand, when the microstructure of AZ31 magnesium alloy contained second phase particles with different shapes, the effect of mixing different shapes of second phase particles on the grain refinement was enhanced gradually with the decrease im the volume fraction of spherical particles. [ABSTRACT FROM AUTHOR]

The physics-informed neural networks (PINNs) has been widely utilized to numerically approximate PDE problems. While PINNs has achieved good results in producing solutions for many partial differential equations, studies have shown that it does not perform well on phase field models. In this paper, we partially address this issue by introducing a modified physics-informed neural networks. In particular, they are used to numerically approximate Allen-Cahn-Ohta-Kawasaki (ACOK) equation with a volume constraint. Technically, the inverse of Laplacian in the ACOK model presents many challenges to the baseline PINNs. To take the zeromean condition of the inverse of Laplacian, as well as the volume constraint, into consideration, we also introduce a separate neural network, which takes the second set of sampling points in the approximation process. Numerical results are shown to demonstrate the effectiveness of the modified PINNs. An additional benefit of this research is that the modified PINNs can also be applied to learn more general nonlocal phase-field models, even with an unknown nonlocal kernel. [ABSTRACT FROM AUTHOR]

In this work, we consider the outer Stefan problem for the short-time prediction of the spread of a volatile asset traded in a financial market. The stochastic equation for the evolution of the density of sell and buy orders is the Heat Equation with a space–time white noise, posed in a moving boundary domain with velocity given by the Stefan condition. This condition determines the dynamics of the spread, and the solid phase [ s − (t) , s + (t) ] defines the bid–ask spread area wherein the transactions vanish. We introduce a reflection measure and prove existence and uniqueness of maximal solutions up to stopping times in which the spread s + (t) − s − (t) stays a.s. non-negative and bounded. For this, we define an approximation scheme, and use some of the estimates of Hambly et al. (2020) for the Green's function and the associated to the reflection measure obstacle problem. Analogous results are obtained for the equation without reflection corresponding to a signed density. [ABSTRACT FROM AUTHOR]

In this paper, we establish a novel linear relaxation method with regularized energy reformulation for phase field models, which we name the RRER method. We employ the molecular beam epitaxy (MBE) model and the phase-field crystal (PFC) model as test beds, along with several coupled phase field models, to illustrate the concept. Our proposed RRER strategy is applicable to a broad class of phase field models that can be derived through energy variation. The RRER method consists of two major steps. First, we introduce regularized auxiliary variables to reformulate the original phase field models into equivalent forms where the free energy is transformed under the auxiliary variables. Then, we discretize the reformulated PDE model under these variables on a staggered time mesh for the phase field models. We incorporate the energy reformulation idea in the first step and the linear relaxation concept in the second step to derive a general numerical algorithm for phase field models that is linear and second-order accurate. Our approach differs from the classical invariant energy quadratization (IEQ) and scalar auxiliary variable (SAV) approaches, as we don't need to take time derivatives for the auxiliary variables. The resulting schemes are linear, i.e., only linear algebraic systems need to be solved at each time step. Rigorous theoretical analysis demonstrates that these resulting schemes satisfy the modified discrete energy dissipation laws and preserve the discrete mass conservation for the PFC and MBE models. Furthermore, we present numerical results to demonstrate the effectiveness of our method in solving phase field models. • We propose a linear relaxation method with regularized energy reformulation for phase field models. • Our approach avoids time derivatives for auxiliary variables, aligning closer with the original PDEs and energy laws. • The proposed schemes for the MBE and PFC models are linear and unconditional energy stable. • The proposed approach is general and applicable to many phase field models and general gradient flow models. [ABSTRACT FROM AUTHOR]

Free energy functionals of the Ginzburg--Landau type lie at the heart of a broad class of continuum dynamical models, such as the Cahn--Hilliard and Swift--Hohenberg equations. Despite the wide use of such models, the assumptions embodied in the free energy functionals frequently either are poorly justified or lead to physically opaque parameters. Here, we introduce a mathematically rigorous pathway for constructing free energy functionals that generalizes beyond the constraints of Ginzburg--Landau gradient expansions.We show that the formalism unifies existing free energetic descriptions under a single umbrella by establishing the criteria under which the generalized free energy reduces to gradient-based representations. Consequently, we derive a precise physical interpretation of the gradient energy parameter in the Cahn--Hilliard model as the product of an interaction length scale and the free energy curvature. The practical impact of our approach is demonstrated using both a model free energy function and the silicon--germanium alloy system. [ABSTRACT FROM AUTHOR]

This work is devoted to the development of a dissipative spatial discretization of a phase field model describing the dynamics of a multicomponent multiphase isothermal viscous compressible fluid with a resolved interface and effects associated with it (surface tension, wetting). Mass densities of mixture components are used as order parameters. The total Helmholtz free energy incorporates wall free energy providing wetting effects. The model under consideration is preliminary regularized, which allows one to increase time step in explicit time-marching methods. A finite-difference semi-discrete method is proposed and a discrete counterpart of the dissipativity theorem is proven. All the statements remain valid in the absence of regularization as well. 3D simulations of two-phase two-component and three-phase three-component fluids using explicit Euler method are performed. The simulation results confirm theoretical predictions. [ABSTRACT FROM AUTHOR]

In the present work, a thermodynamically consistent damage phase field formulation is adapted to include the effect of preferential directions in the damage evolution. A scalar damage variable is associated with each predefined preferential direction of crack propagation. Any other direction is penalized by a parameter (β ≫ 1) that represents the degree of anisotropy of the fracture. When β = 0, the isotropic case is recovered. When there is more than one preferential direction, the material is considered totally fractured when any of the damage variables reaches value one. Simulations using the developed model show that it can reproduce the expected crack propagation pattern for materials with one and two preferential directions. In particular, the model was successful in simulating a zigzag crack pattern commonly obtained in double cantilever beam of spinel MgAl2O4 crystals. The model is fully dynamic in the sense that describes the actual time evolution of the unknown variables, in particular of damage growth. Moreover, anisotropic mechanical response can be easily included in the model by modifying the elasticity tensor. [ABSTRACT FROM AUTHOR]

In this paper, we consider an exponential scalar auxiliary variable (E-SAV) approach to obtain energy stable schemes for a class of phase field models. This novel auxiliary variable method based on the exponential form of the nonlinear free energy potential is more effective and applicable than the traditional SAV method, which is very popular in constructing energy stable schemes. The first contribution is that the auxiliary variable without square root removes the bounded-frombelow restriction of the nonlinear free energy potential. Then we prove the unconditional energy stability for semidiscrete schemes carefully and rigorously. Another contribution is that we provide a total and explicit discretization of the auxiliary variable combined with the nonlinear term. Such a modification is very efficient for fast calculations. Furthermore, the positivity preserving property of r can be guaranteed, which is very important and reasonable for the models' equivalence. In addition, for complex phase field models with two or more unknown variables and nonlinear terms, we construct a multiple E-SAV (ME-SAV) approach to enhance the applicability of the proposed ESAV approach. A comparative study of classical SAV and E-SAV approaches is considered to show the accuracy and efficiency. Finally, we present various 2D numerical simulations to demonstrate the stability and accuracy. [ABSTRACT FROM AUTHOR]

Abstract A formal asymptotic analysis of two classes of phase field models for void growth and coarsening in irradiated solids has been performed to assess their sharp-interface kinetics. It was found that the sharp interface limit of type B models, which include only point defect concentrations as order parameters governed by Cahn-Hilliard equations, captures diffusion-controlled kinetics. It was also found that a type B model reduces to a generalized one-sided classical Stefan problem in the case of a high driving thermodynamic force associated with the void growth stage, while it reduces to a generalized one-sided Mullins-Sekerka problem when the driving force is low in the case of void coarsening. The latter case corresponds to the famous rate theory description of void growth. Type C models, which include point defect concentrations and a non-conserved order parameter to distinguish between the void and solid phases and employ coupled Cahn-Hilliard and Allen-Cahn equations, are shown to represent mixed diffusion and interfacial kinetics. In particular, the Allen-Cahn equation of model C reduces to an interfacial constitutive law representing the attachment and emission kinetics of point defects at the void surface. In the limit of a high driving force associated with the void growth stage, a type C model reduces to a generalized one-sided Stefan problem with kinetic drag. In the limit of low driving forces characterizing the void coarsening stage, however, the model reduces to a generalized one-sided Mullins-Sekerka problem with kinetic drag. The analysis presented here paves the way for constructing quantitative phase field models for the irradiation-driven nucleation and growth of voids in crystalline solids by matching these models to a recently developed sharp interface theory.

A homogenized phase field model for phase transitions in porous media is considered. By making use of the method of formal asymptotic expansion with respect to the interface thickness, a sharp interface limit problem is derived. This limit problem turns out to be similar to the classical Stefan problem with surface tension and kinetic undercooling.

We analyze a control problem for a phase field system modeling the solidification process of materials that allow two different types of crystallization coupled to a Navier-Stokes system and a nonlinear heat equation, with a reduced number of controls. We prove that this system is locally exactly controllable to suitable trajectories, with controls acting only on the motion and heat equations. [ABSTRACT FROM AUTHOR]

• A comprehensive review on phase-field models for fracture and fatigue is presented. • The theory fundamentals and implementation approaches are provided and compared. • All the phase field models for fatigue failure are summarized and discussed. • The future research directions of phase field fatigue models are pointed out. Phase field fracture models have demonstrated great capacities in simulating crack nucleation, propagation, branching, and joining in brittle and ductile materials subjected to external stimulations. Because of their great flexibility, phase field fracture models can incorporate various material properties including anisotropy, elastoplasticity, viscoelasticity, hyperelasticity, piezoelectricity, etc. Recently, the models have been extended to fatigue, one of the most common material failure mechanisms in structural engineering. The purpose of this paper is to provide a comprehensive review of recent work on phase field models for fatigue damage and failures. Following a brief introduction to the development of phase field fracture models as well as the theories and models of fatigue, the fundamentals of the models for elastic and elasto-plastic cases are presented, including basic theories, formulas, tension–compression splits, and numerical implementations. Then, the emphasis is placed on different aspects of the phase field models for fatigue fracture involving the approaches of introducing fatigue damage, the descriptions of crack nucleation and propagation, the coupling of cyclic plasticity and phase field models, and the acceleration algorithms. Finally, the future research directions of phase field models for fatigue fracture and the major challenges to be conquered in their engineering applications are discussed. [ABSTRACT FROM AUTHOR]

Fractional differential operators provide an attractive mathematical tool to model effects with limited regularity properties. Particular examples are image processing and phase field models in which jumps across lower dimensional subsets and sharp transitions across interfaces are of interest. The numerical solution of corresponding model problems via a spectral method is analyzed. Its efficiency and features of the model problems are illustrated by numerical experiments. [ABSTRACT FROM AUTHOR]

Epitaxy, a special form of crystal growth, is a technically relevant process for the production of thin films and layers. It can generate microstructures of different morphologies, such as steps, spirals or pyramids. These microstructures are influenced by elastic effects in the epitaxial layer. There are different epitaxial techniques, one being liquid phase epitaxy. Thereby, single particles are deposited out of a supersaturated liquid solution on a substrate where they contribute to the growth process. This article studies a two-scale model including elasticity, introduced in Eck et al. (Eur Phys J Special Topics 177:5-21, 2009) and extended in Eck et al. (2006). It consists of a macroscopic Navier-Stokes system and a macroscopic convection-diffusion equation for the transport of matter in the liquid, and a microscopic problem that combines a phase field approximation of a Burton-Cabrera-Frank model for the evolution of the epitaxial layer, a Stokes system for the fluid flow near the layer and an elasticity system for the elastic deformation of the solid film. Suitable conditions couple the single parts of the model. As the main result, existence and uniqueness of a solution are proven in suitable function spaces. Furthermore, an iterative solving procedure is proposed, which reflects, on the one hand, the strategy of the proof of the main result via fixed point arguments and, on the other hand, can be the basis for a numerical algorithm. [ABSTRACT FROM AUTHOR]

A concurrent multiscale method is developed to model time-dependent heat transfer and phase transitions in heterogeneous media and is formulated in a way such that the energy being exchanged between scales is conserved. Ensuring this energetic consistency among scales enables the implementation of high fidelity physics-based models at critical locations within the coarse-scale to temporally and spatially resolve highly complex and localized phenomena. To achieve this, only Neumann boundary conditions are applied over the fine scale domain, ensuring a conservative formulation. The coarse-scale solution is used to reconstruct these Neumann boundary conditions on the fine scale, which are then used to evolve a separate system of governing equations. The results on the fine scale are then sent back to the coarse scale through an energy-based homogenization scheme. Transient simulations for the heat equation are implemented with the proposed method to demonstrate its accuracy in energy conservation and effectiveness, including the coupling of a phase field model at the fine scale to a coarse-scale heat equation. [ABSTRACT FROM AUTHOR]

The identification of optimal structures in reaction-diffusion models is of great importance in numerous physicochemical systems. We propose here a simple method to monitor the number of interphases formed after long simulated times by using a boundary ux condition as a control parameter. We consider as an illustration a 1-D Allen-Cahn equation with Neumann boundary conditions. Numerical examples are provided and perspectives for the application of this approach to electrochemical systems are discussed. [ABSTRACT FROM AUTHOR]

This letter introduces novel linear relaxation schemes for solving the phase field models, particularly the Allen–Cahn (AC) type and Cahn–Hilliard (CH) type equations. The proposed schemes differ from existing schemes for the phase field models in the literature. The resulting semi-discrete schemes are linear by discretizing the AC and CH models on staggered time meshes. Only a linear algebra problem needs to be solved at each time marching step after the spatial discretization. Furthermore, our proposed schemes are shown to be unconditionally energy stable, i.e., the numerical solutions respect energy dissipation laws without restriction on the time steps. Several numerical examples are provided to illustrate the power of the proposed linear relaxation schemes for solving phase field models. [ABSTRACT FROM AUTHOR]

Models for branched networks are often expressed as the minimization of an energy Mα over vector measures concentrated on 1-dimensional rectifiable sets with a divergence constraint. We study a Modica-Mortola type approximation Mεα, introduced by Edouard Oudet and Filippo Santambrogio, which is defined over H¹ vector measures. These energies induce some pseudo-distances between L² functions obtained through the minimization problem min{Mεα (u): ∇ · u = f+ - f-}. We prove some uniform estimates on these pseudo-distances which allow us to establish a Γ-convergence result for these energies with a divergence constraint. [ABSTRACT FROM AUTHOR]

A homogenized phase field model for phase transitions in porous media is considered. By making use of the method of formal asymptotic expansion with respect to the interface thickness, a sharp interface limit problem is derived. This limit problem turns out to be similar to the classical Stefan problem with surface tension and kinetic undercooling. [ABSTRACT FROM AUTHOR]

We consider a porous medium with a pore space that is completely filled by three different phases: two immiscible fluids (say water and oil) and a solid phase. One fluid phase contains dissolved ions, which can precipitate at the pore boundary to form the solid phase. The reverse process of dissolution, is also possible. Consequently, the solid phase changes in time; its variation is not known a priori. The second fluid contains no solute and has no interaction with the solid phase. Starting from a standard sharp interface model for the pore-scale dynamics we develop a diffuse interface approach that accounts for the time-dependent spatial distribution of the three species and the overall concentration of the solute. Basic analytical results for this model are presented, including the well posedness of the phase field component of the model. Next we apply matched asymptotic techniques to show that the diffuse interface model converges to the sharp interface one. Further, homogenization is applied to derive a new two-scale model that is valid at the Darcy scale. This leads to a parabolic reaction-diffusion system in a medium with variable, concentration dependent porosity. The diffuse interface approach allows describing the variation in the porosity as phase field type equations at the pore-scale. The last part of the article presents an efficient numerical scheme to approximate the solution of the two-scale model. This scheme has as starting point the algorithm in (Redeker, M. & Eck, C. (2013) A fast and accurate adaptive solution strategy for two-scale models with continuous inter-scale dependencies. J. Comput. Phys., 240, 268-283.). After some test cases validating the method, we finally present computations for several realistic scenarios. The results demonstrate the interdependence of the change of the pore structure due to precipitation/dissolution and the evolution of the averaged, Darcy scale concentration of the dissolved ions in the one fluid. [ABSTRACT FROM AUTHOR]

We obtain a cohesive fracture model as Γ-limit, as ε → 0 , of scalar damage models in which the elastic coefficient is computed from the damage variable v through a function f ε of the form f ε ( v ) = min ⁡ { 1 , ε 1 2 f ( v ) } , with f diverging for v close to the value describing undamaged material. The resulting fracture energy can be determined by solving a one-dimensional vectorial optimal profile problem. It is linear in the opening s at small values of s and has a finite limit as s → ∞ . If in addition the function f is allowed to depend on the parameter ε , for specific choices we recover in the limit Dugdale's and Griffith's fracture models, and models with surface energy density having a power-law growth at small openings. [ABSTRACT FROM AUTHOR]