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A two-grid method is proposed to solve phase field models, arising from modeling the band gap optimization of photonic crystal design based on the Transverse Electric field (TE mode). In our approach, finite element methods on nested grids are used to solve different Partial Differential Equations (PDEs), say, the Hemholtz equation for the state constrain and the phase field equation for topology evolving, respectively. Dicretization on the coarse and fine grids are performed to balance the costs between the phase evolving and state equation solver. To fuse the two scheme together, additional correction step is necessary in two-grid settings. Furthermore, convergence analysis for the proposed two-grid scheme is established. Finally, numerical examples are presented to show the efficiency and robustness, and it is illustrated that the two-grid method improves the speed of computation while remain the computational accuracy.

ispartof: INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING vol:122 issue:15 pages:3919-3940 status: published

Cahn-Hilliard models are central for describing the evolution of interfaces in phase separation processes and free boundary problems. In general, they have non-constant and often degenerate mobilities. However, in the latter case, the spontaneous appearance of points of vanishing mobility and their impact on the solution are not well understood. In this paper we develop a singular perturbation theory to identify a range of degeneracies for which the solution of the Cahn-Hilliard equation forms a singularity in infinite time. This analysis forms the basis for a rigorous sharp interface theory and enables the systematic development of robust numerical methods for this family of model equations., Comment: Submitted to J. of Multiscale Modeling and Simulation

In this paper we propose a notion of irreversibility for the evolution of cracks in the presence of cohesive forces, which allows for different responses in the loading and unloading processes, motivated by a variational approximation with damage models, and we investigate its applicability to the construction of a quasi-static evolution in a simple one-dimensional model. The cohesive fracture model arises naturally via $$\Gamma $$ -convergence from a phase-field model of the generalized Ambrosio-Tortorelli type, which may be used as regularization for numerical simulations.

A multi-scale model integrating a macro-scale finite element model and micro-scale phase field models is proposed to simulate the grain growth in the molten pool of laser beam welding of the Inconel 718 alloy. The calculated welding profile using the macro-scale model agrees well with experimental measurements. According to the calculated macro-scale temperature field, the temperature gradient G, the solidification rate R in the molten pool are calculated. The micro-scale phase field models of macro-scale G and R are coupled are used to simulate the grain growth in the molten pool. Simulated columnar grains are dendritic, which gives agreements with experimental observations. The simulation results and experimental measurements show that the columnar grains spacing increases slightly from the fusion edge to the weld. A power-law function of λ s ∝(G × R) −0.433 is found from the simulation results, which is similar to the power-law function of λ s ∝(G × R) −0.431 from the experimental measurements. Simulated microstructural patterns of the competitive growth match experimental findings well. The spacing of favorable oriented grains is a bit larger than that of unfavorable oriented grains in the competitive growth.

The Pore Formation and Mobility Investigation at the International Space Station provided information on the morphological evolution during remelting and directional solidification under microgravity conditions for Succinonitrile-0.24 wt% water binary alloys. Unlike the terrestrial experiments where the growth is affected by natural convection, constrained diffusive growth is observed in the microgravity experiments. This study aims to provide an experimental benchmark of dendritic growth applicable for validation of theoretical and numerical dendrite growth models. The results of the experiment were compared with the cellular automata and the phase field models, which are two classes of numerical methods widely used by scholars in the field of dendritic solidification, in both two and three dimensions. The resulting morphologies and tip velocities from the models were compared with the Pore Formation and Mobility Investigation experimental results. The combination of experimental and simulation results shows fair agreement and together can be used as a benchmark solution for tip velocity and evolution of dendritic microstructures.

Phase field modeling of fracture has been in the focus of research for over a decade now. The field has gained attention properly due to its benefiting features for the numerical simulations even for complex crack problems. The framework was so far applied to quasi static and dynamic fracture for brittle as well as for ductile materials with isotropic and also with anisotropic fracture resistance. However, fracture due to cyclic mechanical fatigue, which is a very important phenomenon regarding a safe, durable and also economical design of structures, is considered only recently in terms of phase field modeling. While in first phase field models the material’s fracture toughness becomes degraded to simulate fatigue crack growth, we present an alternative method within this work, where the driving force for the fatigue mechanism increases due to cyclic loading. This new contribution is governed by the evolution of fatigue damage, which can be approximated by a linear law, namely the Miner’s rule, for damage accumulation. The proposed model is able to predict nucleation as well as growth of a fatigue crack. Furthermore, by an assessment of crack growth rates obtained from several numerical simulations by a conventional approach for the description of fatigue crack growth, it is shown that the presented model is able to predict realistic behavior., Deutsche Forschungsgemeinschaft, Projekt DEAL

Simple probabilistic models of fracture of 3D polycrystals involving transgranular and intergranular cracks are proposed, based on undamaged paths and appropriate percolation thresholds. In a second part, we propose theoretical extensions of phase field models for crack initiation and propagation to the case of locally heterogeneous and anisotropic fracture energy, and their implementation to get full field solutions by means of iterations of Fourier transforms to replace the standard finite element approach.

This version of the article has been accepted for publication, after peer review (when applicable) and is subject to Springer Nature’s AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: http://dx.doi.org/10.1007/s11831-021-09604-8. In this work, a critical comparison between three different numerical approaches for the computational modelling of quasi-brittle structural failure is presented. Among the many finite element approaches devised to solve the problem, both using continuous and discontinuous methods, the present study examines the relative performance of the XFEM, the mixed strain/displacement FE and phase-field models. These numerical techniques are selected as the current representatives of embedded, smeared and regularized models for analyzing the phenomenon of fracture with different mathematical descriptions for the cracking induced discontinuities in the displacement and strain fields. The present investigation focusses on the main differences of the formulation of these models both at the continuum and discrete level and discusses the main assets and burdens that ensue in their practical application. The relative advantages and difficulties related to their use in the computation of localized structural failure in engineering practice are evaluated against a 10-point checklist that cover the main challenges met by these models. The paper includes an extensive comparison of selected numerical benchmark problems analyzed with the three examined methods. Relative performance is assessed in terms of load capacity, force–displacement curves, crack paths, collapse mechanisms, cost-efficiency and other key issues. Financial support from the Spanish Ministry of Economy and Business via the ADaMANT (Computational Framework for Additive Manufacturing of Titanium Alloy) project (Proyectos de I + D (Excelencia) DPI2017-85998-P) is gratefully acknowledged. The support provided by the Spanish Ministry of Education to Mr. Gabriel Barbat via the FPU program is also acknowledged. The authors also acknowledge financial support from the Spanish Ministry of Economy and Competitiveness, through the “Severo Ochoa Programme for Centres of Excellence in R&D” (CEX2018-000797-S).

This paper proposes an effective computational tool for brittle crack propagation problems based on a combination of a higher-order phase-field model and a non-conforming mesh using a NURBS-based isogeometric approach. This combination, as demonstrated in this paper, is of great benefit in reducing the computational cost of using a local refinement mesh and a higher-order phase-field, which needs higher derivatives of basis functions. Compared with other approaches using a local refinement mesh, the Virtual Uncommon-Knot-Inserted Master-Slave (VUKIMS) method presented here is not only simple to implement but can also reduce the variable numbers. VUKIMS is an outstanding choice in order to establish a local refinement mesh, i.e. a non-conforming mesh, in a multi-patch problem. A phase-field model is an efficient approach for various complicated crack patterns, including those with or without an initial crack path, curved cracks, crack coalescence, and crack propagation through holes. The paper demonstrates that cubic NURBS elements are ideal for balancing the computational cost and the accuracy because they can produce accurate solutions by utilising a lower degree of freedom number than an extremely fine mesh of first-order B-spline elements., Comment: 40 pages, 35 figures

Three new industrial applications of irreversible phase field models for crack growth are presented in this study. The phase field model for irreversible crack growth in an elastic material is derived as a unidirectional gradient flow of the Francfort–Marigo energy with the Ambrosio–Tortorelli regularization, which is known to be consistent with classic Griffith fracture theory. The obtained compact parabolic-elliptic system of PDEs including two regularization parameters for space and time enables us to simulate various kinds of crack behaviors with standard finite element software, without any geometric restriction on the crack path. We extend the irreversible phase field model to thermal cracking in solder and to cracking in a viscoelastic material, keeping the compact forms of the PDEs and the energy consistency. On the other hand, for hydrogen-assisted cracking in metal, we propose a compact phase field model focusing on a kinematic jamming effect of the hydrogen by a weak coupling approach. Several numerical experiments for these three models show not only their reasonableness and usefulness but also flexible extendability of the phase field approach in fracture mechanics.

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 exponential form of nonlinear free energy potential is more effective and applicable than the traditional SAV method which is very popular to construct energy stable schemes. The first contribution is that the auxiliary variable without square root removes the bounded from below restriction of the nonlinear free energy potential. Then, we prove the unconditional energy stability for the semi-discrete schemes carefully and rigorously. Another contribution is that we can discrete the auxiliary variable combined with the nonlinear term totally explicitly. Such modification is very efficient for fast calculation. Furthermore, the positive property of $r$ can be guaranteed which is very important and reasonable for the models' equivalence. Besides, 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 E-SAV 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.

In this paper, a quasi-brittle damage model for micro-architectural materials is presented within the framework of isogeometric analysis to exploit the high-order continuity of the non-uniform B-spline basis functions. The constitutive relation depends not only on the strain field, but also on their first and second strain gradient terms. The simplified second-gradient elasticity formulation from Mindlin’s theory is employed with corresponding micro-architecture-related length scales to capture the material nonlocality and size effects. The strain-based damage is modelled by a nonlocal independent field coupled to the displacement field. Influences of the two types of nonlocalities (manufactured micro-architectures and damage-induced micro-defects) on the response of structures, as well as the damage initiation and propagation, are analysed through numerical experiments. A formula to determine the micro-defect-related length scale from macroscopic measurements is proposed, boosting the accuracy and applicability of the model. In addition, relevant open problems and further developments of this damage model are discussed.

Compressive-shear fracture is commonly observed in rock-like materials. However, this fracture type cannot be captured by current phase field models (PFMs), which have been proven an effective tool for modeling fracture initiation , propagation, coalescence , and branching in solids. The existing PFMs also cannot describe the influence of cohesion and internal friction angle on load–displacement curve during compression tests. Therefore, to develop a new phase field model that can simulate well compressive-shear fractures in rock-like materials, we construct a new driving force in the evolution equation of phase field. Strain spectral decomposition is applied and only the compressive part of the strain is used in the new driving force with consideration of the influence of cohesion and internal friction angle. For ease of implementation, a hybrid formulation is established for the phase field modeling. Then, we test the brittle compressive-shear fractures in uniaxial compression tests on intact rock-like specimens as well as those with a single or two parallel inclined flaws. All numerical results are in good agreement with the experimental observation, validating the feasibility and practicability of the proposed PFM for simulating brittle compressive-shear fractures.

Mechanical behavior of layered materials and structures greatly depends on the mechanical behavior of interfaces. In the past decades, the failure in such layered media has been studied by many researchers due to their critical role in the mechanics and physics of solids. This study aims at investigating crack-interface interaction in two-dimensional (2-D) and three-dimensional (3-D) layered media by a phase field model. Our objectives are fourfold: (a) to better understand fracture behavior in layered heterogeneous systems under quasi-static load; (b) to introduce a new methodology for better describing interfaces by a regularized interfacial transition zone in the context of variational phase field approach, exploring its important role; (c) to show the accuracy, performance and applicability of the present model in modeling material failure at the interfaces in both 2-D and 3-D bodies; and (d) to quantitatively validate computed crack path with respect to experimental data. Phase field models with both perfectly and cohesive bonded interfaces are thus derived. A regularized interfacial transition zone is introduced to capture characteristics of material mismatch at the interfaces. Numerical examples for 2-D and 3-D layered systems with experimental validation provide fundamentals of fracture behavior in layered structures. The obtained results shed light on the behavior of crack paths, which are drastically affected by the elastic modulus mismatch between two layers and interface types, and reveal the important role of the proposed interfacial transition zone in phase field modeling of crack-interface interactions.

In this paper, we study the phase field models with fractional-order in time. The phase field models have been widely used to study coarsening dynamics of material systems with microstructures. It is known that phase field models are usually derived from energy variation so that they obey some energy dissipation laws intrinsically. Recently, many works have been published on investigating fractional-order phase field models, but little is known of the corresponding energy dissipation laws. We focus on the time-fractional phase field models and report that the effective free energy and roughness obey a universal power-law scaling dynamics during coarsening. Mainly, the effective free energy and roughness in the time-fractional phase field models scale by following a similar power law as the integer phase field models, where the power is linearly proportional to the fractional order. This universal scaling law is verified numerically against several phase field models, including the Cahn-Hilliard equations with different variable mobilities and molecular beam epitaxy models. This new finding sheds light on potential applications of time fractional phase field models in studying coarsening dynamics and crystal growths.

International audience; We propose a framework to represent a partition that evolves under mean curvature flows and volume constraints. Its principle follows a phase-field representation for each region of the partition, as well as classical Allen-Cahn equations for its evolution. We focus on the evolution and on the optimization of problems involving high resolution data with many regions in the partition. In this context, standard phase-field approaches require a lot of memory (one image per region) and computation timings increase at least as fast as the number of regions. We propose a more efficient storage strategy with a dedicated multi-image representation that retains only significant phase field values at each discretization point. We show that this strategy alone is unfortunately inefficient with classical phase field models. This is due to non local terms and low convergence rate. We therefore introduce and analyze an improved phase field model that localizes each phase field around its associated region, and which fully benefits of our storage strategy. To demonstrate the efficiency of the new multiphase field framework, we apply it to the famous 3D honeycomb problem and the conjecture of Weaire-Phelan's tiling.

Phase field models recently gained a lot of interest in the context of tumour growth models. Typically Darcy-type flow models are coupled to Cahn–Hilliard equations. However, often Stokes or Brinkman flows are more appropriate flow models. We introduce and mathematically analyse a new Cahn–Hilliard–Brinkman model for tumour growth allowing for chemotaxis. Outflow boundary conditions are considered in order not to influence tumour growth by artificial boundary conditions. Existence of global-in-time weak solutions is shown in a very general setting.

Microstructure evolution in the electrodes of solid oxide fuel cell is an important degradation mechanism which reduces active sites for redox reaction and the electric conductivity. Phase field models for microstructure evolution simulation are usually expensive for large scale simulations. In this work, a reduced-order coarsening model is developed using dynamic discrepancy reduced modeling, which reduces the model order by inserting Gaussian process stochastic functions into the dynamic equations of Ostwald ripening. The reduced order model has been calibrated on a dataset generated by a phase field model that has been well validated to experiments. A validating dataset has also been generated with which the model prediction show good agreement. This model is further applied to predict long term microstructure evolution in different SOFC electrodes. This work is the first attempt of building a degradation model of SOFC using data science techniques.

Solute trapping is an important phenomenon in rapid solidification of alloys, for which the continuous growth model (CGM) of Aziz et al. [1] is a popular sharp interface theory. By modulating the so-called anti-trapping current and using asymptotic analysis, we show how to quantitatively map the thin interface behavior of an ideal dilute binary alloy phase field model onto the CGM kinetics. We present the parametrizations that allow our phase field model to map onto the sharp interface kinetics of the CGM, both in terms of partition coefficient k ( V ) and kinetic undercooling. We also show that the mapping is convergent for different interface widths, both in transient and steady state simulations. Finally we present the effect that solute trapping can have on cellular growth in directional solidification. The presented treatment for solute trapping can be easily implemented in different phase field models, and is expected to be an important feature in future studies of quantitative phase field modeling in quasi-rapid solidification regimes, such as those relevant to metal additive manufacturing.

La nucleació és un procés molt comú pel qual la separació de fases es dóna en sistemes líquids multicomponents. Els models de camps de fase són habitualment emprats per estudiar la separació de fases, però són insuficients per estudiar els processos de nucleació, degut a què aquestes transicions són activades tèrmicament i requereixen superar una barrera d’energia. Per superar aquesta dificultat, hem combinat els models de camps de fase amb el “string method” (mètode de la corda), un esquema numèric capaç de trobar camins de transició i estats crítics, per estudiar la nucleació en sistemes binaris i ternaris. El nostre model és capaç de capturar la nucleació homogènia i heterogènia tant en sistemes binaris com ternaris. Validem la nostra aproximació comparant-la amb la teoria de nucleació clàssica, vàlida en el límit de interfases abruptes, i examinen i discutim les discrepàncies entre aquestes dues aproximacions. En el futur aquest model pot ser emprat per estudiar configuracions més complexes, algunes de les quals són interessants des d’un punt de vista de la biologia perquè simulen la separació de proteïnes en el fluid intracel·lular. La nucleación es un proceso muy común por el cual la separación de fases se da en sistemas líquidos multicomponentes. Los modelos de campos de fase son habitualmente empleados para estudiar la separación de fases, pero son insuficientes para estudiar los procesos de nucleación, debido a que estas transiciones son activadas térmicamente y requieren superar una barrera de energía. Para superar esta dificultad, hemos combinado los modelos de campos de fase con el “string method” (método de la cuerda), un esquema numérico capaz de hallar caminos de transición y estados críticos, para estudiar la nucleación en sistemas binarios y ternarios. Nuestro modelo es capaz de capturar la nucleación homogénea y heterogénea tanto en sistemas binarios como ternarios. Validamos nuestra aproximación comparándola con la teoría de nucleación clásica, válida en el límite de interfases abruptas, y examinamos y discutimos las discrepancias entre estas dos aproximaciones. En el futuro este modelo puede ser empleado para estudiar configuraciones más complejas, algunas de las cuales son interesantes desde un punto de vista de la biología porque simulan la separación de proteínas en el fluido intracelular. Nucleation is a very common process by which phase separation takes place in multicomponent liquid mixtures. Phase field models are usually applied to study phase separation, but they are insufficient to study nucleation events, since these transitions are thermally activated and require overcoming an energy barrier. To overcome this difficulty, we combined phase field models with the string method, a numerical scheme capable of finding transition pathways and critical states, to study nucleation in binary and ternary systems. Our model is able to capture homogeneous and heterogeneous nucleation both in binary and ternary systems. We validate our approach by comparing it with classical nucleation theory, valid in a sharp interface limit, and further examine and discuss the discrepancies between these two approaches. In the future this model can be applied to study more complex configurations, some of which are interesting from a biological point of view because they simulate protein separation in the intracellular fluid.

We develop a three-phase field model for the simulation of eutectic and eutectoid transformations on the basis of a nondiagonal model obeying Onsager relations for a kinetic cross coupling between diffusion and the phase fields. This model overcomes the limitations of existing phase field models concerning the fulfillment of local equilibrium boundary conditions at the transformation fronts in the case of a finite diffusional contrast between the phases. We benchmark our model in the well understood one-sided case with diffusion only in the parent phase against results from the literature. In addition to this solidification scenario, the case of solid-state transformations with diffusion in the growing phases is investigated. Our simulations validate the relevance of the theory developed by Ankit et al. [Acta Mater. 61, 4245 (2013)], that describes in a single frame the two limiting regimes where diffusion mainly takes place whether in the mother phase or in the growing phases. In both the one-sided and two-sided cases, we verify the necessity of the kinetic cross coupling for quantitative phase field simulations.

International audience; Phase-field models of brittle fracture can be regarded as gradient damage models including an intrinsic internal length. This length determines the stability threshold of solutions with homogeneous damage and thus the strength of the material, and is often tuned to retrieve the experimental strength in uniaxial tensile tests. In this paper, we focus on multiaxial stress states and show that the available energy decompositions, introduced to avoid crack interpenetration and to allow for unsymmetric fracture behavior in tension and compression, lead to multiaxial strength surfaces of different but fixed shapes. Thus, once the length scale is tailored to recover the experimental tensile strength, it is not possible to match the experimental compressive or shear strength. We propose a new energy decomposition that enables the straightforward calibration of a multi-axial failure surface of the Drucker-Prager type. The new decomposition, which hinges upon the theory of structured deformations, encompasses the volumetric-deviatoric and the no-tension models as special cases. Preserving the variational structure of the model, it includes an additional free parameter that can be calibrated based on the experimental ratio of the compressive to the tensile strength (or, if possible, of the shear to the tensile strength), as successfully demonstrated on two data sets taken from the literature.

A vector order parameter phase field model derived from a grand potential functional is presented as an alternative approach for modeling polycrystalline solidification of alloys. In this approach, the grand potential density is designed to contain a discrete set of finite wells, a feature that naturally allows for the growth and controlled interaction of multiple grains using a single vector field. We verify that dendritic solidification in binary alloys follows the well-established quantitative behavior in the thin interface limit. In addition, it is shown that grain boundary energy and solute back-diffusion are quantitatively consistent with earlier theoretical work, with grain boundary energy being controlled through a simple solid-solid interaction parameter. Moreover, when considering polycrystalline aggregates and their coarsening, we show that the kinetics follow the expected parabolic growth law. Finally, we demonstrate how this vector order parameter model can be used to describe nucleation in polycrystalline systems via thermal fluctuations of the vector order parameter, a process that cannot be treated consistently with multiphase or multi-order-parameter based phase field models. The presented vector order parameter model serves as a practical and efficient computational tool for simulating polycrystalline materials. We also discuss the extension of the order parameter to higher dimensions as a simple method for modeling multiple solid phases.

Materials 14(7), 1787 (2021). doi:10.3390/ma14071787 special issue: "Special Issue "Strong Coupling of Thermo-Chemical and Thermo-Mechanical States in Applied Materials" / Special Issue Editor: Prof. Ingo Steinbach, Guest Editor", Published by MDPI, Basel

We propose a phase-field theory for enriched continua. To generalize classical phase-field models, we derive the phase-field gradient theory based on balances of microforces, microtorques, and mass. We focus on materials where second gradients of the phase field describe long-range interactions. By considering a nontrivial interaction inside the body, described by a boundary-edge microtraction, we characterize the existence of a hypermicrotraction field, a central aspect of this theory. On surfaces, we define the surface microtraction and the surface-couple microtraction emerging from internal surface interactions. We explicitly account for the lack of smoothness along a curve on surfaces enclosing arbitrary parts of the domain. In these rough areas, internal-edge microtractions appear. We begin our theory by characterizing these tractions. Next, in balancing microforces and microtorques, we arrive at the field equations. Subject to thermodynamic constraints, we develop a general set of constitutive relations for a phase-field model where its free-energy density depends on second gradients of the phase field. A priori, the balance equations are general and independent of constitutive equations, where the thermodynamics constrain the constitutive relations through the free-energy imbalance. To exemplify the usefulness of our theory, we generalize two commonly used phase-field equations. We propose a ‘generalized Swift–Hohenberg equation’—a second-grade phase-field equation—and its conserved version, the ‘generalized phase-field crystal equation’—a conserved second-grade phase-field equation. Furthermore, we derive the configurational fields arising in this theory. We conclude with the presentation of a comprehensive, thermodynamically consistent set of boundary conditions.

Materials / Materials Diversity Preservation International 14(3), 654 (2021). doi:10.3390/ma14030654 special issue: "Special Issue "Strong Coupling of Thermo-Chemical and Thermo-Mechanical States in Applied Materials" / Special Issue Editor: Prof. Ingo Steinbach, Guest Editor", Published by MDPI, Basel