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All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Inference of phase field fracture models

The phase field approach to modeling fracture uses a diffuse damage field to represent cracks. This representation mollifies singularities that arise in computations with sharp interface models and some of the resultant difficulties in the mathematical and numerical treatment of fracture. Phase field fracture models have proven effective at representing crack propagation, branching, and merging. Specific formulations, beginning with brittle fracture, have also been shown to converge to classical solutions. Extensions to cover the range of material failure, including ductile and cohesive fracture, lead to an array of possible models. There exists a large body of literature focusing on this class of models and on the impact of model form on the predicted crack evolution. However, there have not been systematic studies into how optimal models may be chosen. Here, we take a first step in this direction by developing formal methods for identification of the best parsimonious model of phase field fracture given full-field data on the damage and deformation fields. We consider some of the main models that have been used for the degradation of elastic response due to damage and its propagation. Our approach builds upon Variational System Identification (VSI), a weak form variant of the Sparse Identification of Nonlinear Dynamics (SINDy). Furthermore, in this first communication we focus on synthetically generated data but we also consider central issues associated with the use of experimental full-field data, such as data sparsity and noise.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2026

Effect of magneto-mechanical synergism in the process-structure correlation in Fe–C alloys: A phase-field modeling approach

Applied magnetic fields can alter phase equilibria and kinetics in steels; however, quantitatively resolving how magnetic, chemical, and elastic driving forces jointly influence the microstructure remains challenging. We develop a quantitative magneto-mechanically coupled phase-field model for the Fe–C system that couples a CALPHAD-based chemical free energy with demagnetization-field magnetostatics and microelasticity. Here, the model reproduces single- and multi-particle evolution during the α → γ inverse transformation at 1023 K under external fields up to 20 T, including ellipsoidal morphologies observed experimentally at 8 T. Chemically driven growth is isotropic; a magnetic interaction introduces an anisotropic driving force that elongates γ precipitates along the field into ellipsoids, while elastic coherency promotes faceting, yielding elongated cuboidal or “brick-like” particles under combined magneto-elastic coupling. Growth kinetics increase with C content, and decrease with field strength and misfit strain. Multi-particle simulations reveal dipolar interaction-mediated coalescence for field-parallel neighbors and ripening for field-perpendicular neighbors. Incorporating field-dependent diffusivity from experiment slows kinetics as expected; a first-principles-motivated anisotropic diffusivity correction is estimated to be small (<2%). These results establish a process-structure link for magnetically assisted heat treatments of Fe–C alloys and provide guidance for microstructure control via chemo-magneto-mechanical synergism.

Magnetic field↗
DOE OSTI2024

A phase-field diffraction model for thermo-hydro-mechanical propagating fractures

This paper introduces a novel diffraction based thermo-hydraulic–mechanical (THM) model for fracture propagation using a phase-field fracture (PFF) approach. The key innovation of the THM-PFF model lies in its integrated treatment of four solution variables—displacements, phase-field, pressure, and temperature—each governed by a combination of conservation of momentum (mechanics problem), a variational inequality (constrained minimization problem), mass conservation (pressure problem), and energy conservation (temperature problem). This leads to a new formulation of a coupled variational inequality system. A major advancement is the development of an extended fixed-stress algorithm, where displacements, phase-field, pressures, and temperatures are solved in a staggered sequence. An important aspect of this work is the global coupling of pressures and temperatures across the domain using diffraction systems, with diffraction coefficients defined by material parameters weighted by the diffusive phase-field variable. To ensure robust local mass conservation, we employ enriched Galerkin finite elements (EG) for both pressure and temperature diffraction equations. By enriching the continuous Galerkin basis functions with discontinuous piecewise constants, EG accurately represents solution and parameter discontinuities while preserving local mass and energy conservation—crucial aspects for THM problems and realistic behavior. Moreover, the use of a predictor–corrector local mesh adaptivity scheme is employed, allowing the model to handle small phase-field length-scale parameters while maintaining high numerical accuracy and reasonable computational cost. Furthermore, these new model and algorithmic developments represent significant advances in the field and have been substantiated through rigorous numerical tests.

Diffraction systems↗
DOE OSTI2025

An end-to-end deep learning method for solving nonlocal Allen–Cahn and Cahn–Hilliard phase-field models

Here, we propose an efficient end-to-end deep learning method for solving nonlocal Allen–Cahn (AC) and Cahn–Hilliard (CH) phase-field models. One motivation for this effort emanates from the fact that discretized partial differential equation-based AC or CH phase-field models result in diffuse interfaces between phases, with the only recourse for remediation is to severely refine the spatial grids in the vicinity of the true moving sharp interface whose width is determined by a grid-independent parameter that is substantially larger than the local grid size. In this work, we introduce non-mass conserving nonlocal AC or CH phase-field models with regular, logarithmic, or obstacle double-well potentials. Because of non-locality, some of these models feature totally sharp interfaces separating phases. The discretization of such models can lead to a transition between phases whose width is only a single grid cell wide. Another motivation is to use deep learning approaches to ameliorate the otherwise high cost of solving discretized nonlocal phase-field models. To this end, loss functions of the customized neural networks are defined using the residual of the fully discrete approximations of the AC or CH models, which results from applying a Fourier collocation method and a temporal semi-implicit approximation. To address the long-range interactions in the models, we tailor the architecture of the neural network by incorporating a nonlocal kernel as an input channel to the neural network model. We then provide the results of extensive computational experiments to illustrate the accuracy, predictive capabilities, and cost reductions of the proposed method.

42 ENGINEERING↗
DOE OSTI2024

A Thermo‐Flow‐Mechanics‐Fracture Model Coupling a Phase‐Field Interface Approach and Thermo‐Fluid‐Structure Interaction

This work proposes a novel approach for coupling non-isothermal fluid dynamics with fracture mechanics to capture thermal effects within fluid-filled fractures accurately. This method addresses critical aspects of calculating fracture width in enhanced geothermal systems, where the temperature effects of fractures are crucial. The proposed algorithm features an iterative coupling between an interface-capturing phase-field fracture method and interface-tracking thermo-fluid-structure interaction using arbitrary Lagrangian–Eulerian coordinates. We use a phase-field approach to represent fractures and reconstruct the geometry to frame a thermo-fluid-structure interaction problem, resulting in pressure and temperature fields that drive fracture propagation. We developed a novel phase-field interface model accounting for thermal effects, enabling the coupling of quantities specific to the fluid-filled fracture with the phase-field model through the interface between the fracture and the intact solid domain. We provide several numerical examples to demonstrate the capabilities of the proposed algorithm. In particular, we analyze mesh convergence of our phase-field interface model, investigate the effects of temperature on crack width and volume in a static regime, and highlight the method's potential for modeling slowly propagating fractures.

fracture↗
DOE OSTI2025

Phase-field modeling of orientation-dependent crack growth in ductile single crystals with anisotropic elasticity

Crack growth in ductile single crystals (DuSCs) is orientation dependent due to the anisotropies of crystal plasticity and elastic tensor. This study develops a phase-field model incorporating both crystal plasticity and crack growth and proposes a general method to decompose the elastic energy into compressive and tensile parts to prevent crack growth under compression in the phase-field description. The phase-field model, in combination with three Euler angles, is employed to simulate orientation-dependent crack growth in DuSCs. The contributions from crystal plasticity and anisotropic elasticity are compared, and the former is found to dominate in the anisotropy of crack growth in copper single crystals. Furthermore, the simulation results demonstrate that crystal orientation strongly affects the heterogeneous distribution of plastic strain and the interaction between plastic strain and crack growth. High-throughput phase-field simulations are performed with exhaustive crystal orientations, and the results are explained based on the anisotropy of the Taylor factor.

Computational Solid Mechanics↗
DOE OSTI2026

Benchmarking of massively parallel phase-field codes for directional solidification

We present a detailed benchmark comparing two state-of-the-art phase-field implementations for simulating alloy solidification under experimentally relevant conditions. The study investigates the directional solidification of Al-3wt%Cu under high-velocity solidification conditions and SCN-0.46wt% camphor under microgravity conditions from National Aeronautics and Space Administration (NASA) DECLIC-DSI-R experiments. Both codes, one employing finite-difference discretization with uniform mesh and GPU-acceleration (GPU-PF) and the other one employing finite-element discretization with adaptive-mesh and CPU-parallelization (PRISMS-PF), solve the same quantitative phase-field formulation that incorporates an anti-trapping current for the solidification of dilute alloys. We evaluate the predictions of each code for dendritic morphology, primary spacing, and tip dynamics in both 2D and 3D, as well as their numerical convergence and computational performance. While existing benchmark problems have primarily focused on simplified or small-scale simulations, they do not reflect the computational and modeling challenges posed by employing experimentally relevant time and length scales. Our results provide a practical framework for assessing phase-field code performance as well as validating and facilitating their application in integrated computational materials engineering (ICME) workflows that require integration with realistic experimental data.

36 MATERIALS SCIENCE↗
DOE OSTI2026

Phase-field model of freeze casting

Directional solidification of water-based solutions has emerged as a versatile technique for templating hierarchical porous materials. However, the underlying mechanisms of pattern formation remain incompletely understood. In this work, we present a detailed derivation and analysis of a quantitative phase-field model for simulating this nonequilibrium process. The phase-field model extends the thin-interface formulation of dilute binary alloy solidification with antitrapping to incorporate the highly anisotropic energetic and kinetic properties of the partially faceted ice-water interface. This interface is faceted in the basal plane normal to the ⟨0001⟩ directions and atomically rough in other directions within the basal plane. On the basal plane, the model reproduces a linear or nonlinear relationship between the interface growth rate and the kinetic undercooling that can be linked to experimental measurements. In both cases, spontaneous parity breaking of the solidification front is observed when the preferred growth direction is aligned with the temperature gradient. This phenomenon leads to the formation of partially faceted ice lamellae that drift laterally in one of the ⟨0001⟩ directions. Here, we demonstrate that the drifting velocity of the ice lamellae is controlled by the kinetics on the basal plane and converges as the thickness of the diffuse solid-liquid interface decreases. Furthermore, we examine the effect of the form of the kinetic anisotropy, which is chosen here such that the inverse of the kinetic coefficient varies linearly from a finite value in the ⟨0001⟩ directions to zero in all other directions within the basal plane, consistent with the assumption that the interface grows in local thermodynamic equilibrium in this plane. Our results indicate that the drifting velocity of ice lamellae is not affected by the slope of this linear relation, and the radius and undercooling at the tip of an ice lamella converge at relatively small slope values. Consequently, the phase-field simulations remain quantitative with computationally tractable choices of both the interface thickness and the slope assumed in the form of the kinetic anisotropy.

Materials science↗
DOE OSTI2025

Gradient flow based phase-field modeling using separable neural networks

Allen–Cahn equation is a reaction–diffusion equation and is widely used for modeling phase separation. Machine learning methods for solving the Allen–Cahn equation in its strong form suffer from inaccuracies in collocation techniques, errors in computing higher-order spatial derivatives, and the large system size required by the space–time approach. To overcome these challenges, we propose solving the gradient flow of the Ginzburg–Landau free energy functional, which is equivalent to the Allen–Cahn equation, thereby avoiding the second-order spatial derivatives associated with the Allen–Cahn equation. A minimizing movement scheme is employed to solve the gradient flow problem, eliminating the complexities of a space–time approach. We utilize a separable neural network that efficiently represents the phase field through low-rank tensor decomposition. As we use the minimizing movement scheme to numerically solve the gradient flow problem, we thus, refer to the proposed method as the Separable Deep Minimizing Movement (SDMM) method. The evaluation of the functional in the minimizing movement scheme using the Gauss quadrature technique bypasses the inaccuracies associated with collocation techniques traditionally used to solve partial differential equations. A hyperbolic tangent transformation is introduced on the phase field prior to the evaluation of the functional to ensure that it remains strictly bounded within the values of the two phases. For this transformation, theoretical guarantee for energy stability of the minimizing movement scheme is established. Our results suggest that this transformation helps to improve the accuracy and efficiency significantly. The proposed method resolves the challenges faced by state-of-the-art machine learning techniques, outperforming them in both accuracy and efficiency. It is also the first machine learning method to achieve an order of magnitude speed improvement over the finite element method. In addition to its formulation and computational implementation, several case studies illustrate the applicability of the proposed method.

42 ENGINEERING↗
DOE OSTI2025

Massively parallel phase-field simulations targeting exascale

The interface thickness in the phase-field (PF) method limits its simulation scales. Consequently, large-scale PF simulations become prohibitively expensive for resolving the extremely fine microstructures that typically form during rapid solidification processing. This challenge is significant in predicting microstructure evolution in metal additive manufacturing and has been identified by the United States Department of Energy’s Exascale Computing Project. Here, to address this, we develop a multi-GPU and MPI-based massively parallel simulation code, utilizing state-of-the-art algorithms, software, and libraries, for large-scale three-dimensional (3D) PF simulations. We report the first GPU-parallel PF simulations on Frontier (currently the second TOP500 exascale cluster) and Summit machines, taking dendritic growth as an example problem. We evaluate the parallel performance of our implementation using scaling studies with more than 24 000 GPUs (among the largest known computations to date) and the acceleration performance using large-scale simulations of dendritic growth in 3D. Finally, massively parallel GPUs in these supercomputers enabled the first coupled multiscale simulations of laser melting and subsequent dendritic solidification on the scale of a full melt-pool, demonstrating the feasibility of performing PF simulations with a point total over 2 billion grid points within an acceptable time.

Exascale↗
DOE OSTI2025

Phase-field modeling and experiments of dynamic fracture in single crystal quartz

Predicting the onset and characteristics of brittle fracture is important for a wide range of engineering and geological material applications. In this paper, we study important aspects of brittle fracture in α-quartz by phase-field modeling and experiments using a top-down approach. In the modeling framework, the work term in the Griffith energy balance is replaced with internal energy contributions that represent surface energy, thermal energy, and elastic strain energy stored in defects. This allows parametrization of individual energy contributions in terms of internal state variables and keeps track of energy partitioning after the onset of fracture. The path and history dependence of fracture is included in evolution laws for internal state variables, e.g., entropy evolution, while the energy remains a true potential. In the experimental part, dynamic compression experiments coupled with X-ray phase contrast imaging are performed on cube-like samples with a hole. In the top-down analysis, dynamic compression and three point bending experiments from the literature are simulated with the developed phase-field damage model. In conclusion, the fitted model highlights the strain rate, size, and stress state dependence of damage nucleation and evolution in single crystal α-quartz.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Understanding coarsening of a post-corrosion microstructure in a molten salt by combining phase-field modeling and in situ tomography

Alloys corroding in molten salt have been observed to form bicontinuous, nanoporous microstructures via dealloying, which subsequently undergo coarsening due to facile transport in high-temperature conditions. In this work, we describe a methodology to elucidate the underlying transport mechanisms during coarsening of a bicontinuous microstructure via quantitative comparisons between phase-field simulations and four-dimensional in situ experiments, in this case X-ray nanotomography of the coarsening of a dealloyed 80 wt% Ni-20 wt% Cr microwire in molten KCl-MgCl 2 at 800°C. We conduct phase-field simulations initialized from experimental data to model coarsening via three different transport mechanisms: surface diffusion, solid bulk diffusion, and liquid bulk diffusion. These simulations reproduce key features of the experiment, such as the densification of the outer layer of the dealloyed wire and the reduction in radius over time. We quantitatively compare different microstructural characteristics between the simulations and experiment and extract temporal scaling factors that optimally match the time scales of the simulations to that of the experiment. This allows us to evaluate morphological similarity between the simulations and experiment and relate the experimental coarsening kinetics to fundamental material properties. We find that surface diffusion is most likely to be the dominant coarsening mechanism, and its kinetics imply a surface diffusivity of D S = 8.9 x 10 -20 m 3 /s, which is within the range of reported values for Ni-vacuum interfaces at 800°C. However, the difference between the experiment and the surface diffusion simulation increases substantially at late times, suggesting that other mechanisms, such as the dissolution of residual Cr, may be at play.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Phase-field modeling of aging-induced microstructure evolution in pentaerythritol tetranitrate thin films and ramifications for shock initiation

Aging of energetic materials may change performance and affect their safety and reliability, but the relationship between microstructure changes induced by aging and consequent performance changes has not been fully established. This work presents results of phase-field method simulations used to model microstructure evolution of vapor-deposited pentaerythritol tetranitrate (PETN) thin films. Simulated aging is shown to induce grain coarsening and substantial changes of the configuration of porosity in the film: Specifically, we show that porosity tends to concentrate in large pores to a greater degree in aged films, a state that is arrived at by closure or consolidation of small pores. To evaluate the performance of the as-deposited and aged films, we perform two-dimensional hydrocode flyer-film impact simulations that incorporate the phase-field output microstructures directly, permitting us to connect features therein to changes in reactivity, a key metric of energy output for shock initiation. The results demonstrate that declining sensitivity obtained for the simulated aged films can be correlated with the loss of fine-structured pores relatively early in the aging process, while long-term microstructure evolution that gradually alters the shape of large, branching pores is less impactful. Finally, we discuss commonalities and discrepancies between our simulation results and high-throughput initiation experiments on shock initiation of aged PETN thin films.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Phase-Field Modeling of Damage Evolution in Ceramic Matrix Composite (CMC) and Environmental Barrier Coating (EBC)

Ceramic matrix composites (CMCs) protected by environmental barrier coatings (EBCs) present a promising materials solution for next generation gas turbines. Developments of more robust and efficient EBCs and mechanically tougher CMCs are thus of significant technological importance. Here we develop a phase-field modeling framework that incorporates the thermally grown oxide (TGO), recognized as a critical factor for degradation and failure of EBCs. We simulate crack growth in the TGO and the potential extension into the bond coat / CMC substrate. The model efficiently takes account of the large inelastic deformation induced by the severe volume expansion of TGO, thanks to our recently developed, so-called incremental realization of inelastic deformation (IRID) algorithm. A phase-field model is built for damage evolution in CMCs including crack growth and interfacial sliding. The effects of fiber layout and interfacial sliding on the macroscopic toughness of CMCs are revealed by large-scale simulations and compared to experiments.

advanced energy systems and materials↗
DOE OSTI2025

Coupled momentum balance and phase-field solver with fenicsx module

Code solves momentum balance and phase-field equations simultaneously. The differential equations are solved on a discretized domain with appropriate boundary and initial conditions using finite element method. Primary purpose of the code is to simulate brittle fracture under dynamic loading. Constitutive equations are that of linear elasticity with degradation of stress due to fracture. Small strain formulation is used.

Zecevic, Milovan↗
DOE OSTI2026

Phase field dislocation dynamics formulation coupled with Fourier based micromechanics solver and its application to grain boundary–dislocation interactions

A new phase field dislocation dynamics (PFDD) formulation for homogeneous and heterogeneous materials is presented, which couples micromechanical solvers and the time-dependent Ginzburg–Landau equation. The strain fields are obtained from the micromechanical solver by solving the Lippmann–Schwinger equation and then used to define energy terms to model the evolution of the dislocations. Grain boundary (GB)–dislocation interactions are studied using the coupled PFDD formulation and by describing GBs as inclusions. GB energy and stiffness tensors are computed from molecular statics simulations, and a newly proposed lattice energy term that is dependent on the GB energy is considered in the calculations. Interaction of a screw dislocation with minimum energy and metastable states of low and high angle ⟨110⟩ symmetric tilt grain boundaries are studied. We show good agreement between predictions from our PFDD formulation and molecular dynamics simulations of grain boundary–dislocation interactions.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Phase-field model of alloy solidification far from chemical equilibrium at the solid-liquid interface

We further develop a recently introduced phase-field model of far-from-equilibrium alloy solidification under additive manufacturing conditions [K. Ji et al., Phys. Rev. Lett. 130, 026203 (2023)]. This model utilizes enhanced solute diffusivity within the spatially diffuse interface region to quantitatively capture solute trapping with a larger interface width, thereby making simulations on experimentally relevant length and timescales computationally feasible. The main developments presented here include testing the robustness of different variational formulations, extending the model to concentrated alloys by incorporating solid and liquid free energies from thermodynamic databases, as illustrated for hypoeutectic Al-Ag alloys with CALPHAD, extending convergence tests as a function of interface width to 3D, and carrying out simulations in both 2D and 3D to examine existing theories of microstructure development. Our results indicate that the simplest variational formulation that interpolates the bulk free-energy density between its solid and liquid forms is the most robust. Remarkably, for hypoeutectic Al-Ag alloys, this formulation yields a high-velocity nonequilibrium phase diagram that is independent of interface width, thereby demonstrating that the framework of enhanced solute diffusivity can be nontrivially extended to concentrated alloys. Other variational formulations have restricted ranges of materials or processing parameters that can be reliably modeled. We use 2D simulations to construct high-velocity microstructure selection maps for dilute Al-Cu alloys. The results validate the important role of latent heat rejection at the interface and extend the limited predictions of linear stability analysis [A. Karma and A. Sarkissian, Phys. Rev. E 47, 513 (1993)] and sharp-interface 1D simulations to fully nonlinear regimes. Furthermore, 3D simulations, carried out using a computationally tractable axisymmetric cellular/dendritic interface shape, demonstrate a good convergence similar to that observed in 2D as a function of interface width. Full 3D simulations, in turn, reveal that the standard theory of absolute stability is a good predictor of the upper critical velocity beyond which steady-state growth becomes unstable, despite the different morphological manifestations of this instability in 2D and 3D.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Phase-field modeling of diffusion bonding in 316H stainless steel: Impact of processing conditions on grain morphology and bonding quality

A novel multi-phase, multi-component phase‐field model is presented to study the diffusion bonding of 316H stainless steel. Combined with targeted experimental investigations, this model simulates the bond-growth process and predicts the bonding quality. Unlike previous models, our approach captures the simultaneous evolution of voids and grain structures, while quantifying bonding quality using defined bonding ratio. A comprehensive analysis of bond process control is performed by changing temperature, pressure and surface roughness observing the resulting bond structure, which is consistent with experimental observations and analytical predictions. Temperature is determined to be the dominant factor, with the transition from a flat to a robust bond occurring between 1000 °C and 1050 °C. At the ideal bonding temperature of 1050 °C, a surface roughness exceeding 0.6 μm or an applied stress below 4 MPa results in poor bonding quality. Beyond this, higher pressures and smoother surfaces reduce void size, accelerate void shrinkage, and lead to improved bond integrity. This diffuse-interface model can be extended to other material systems if supplied with appropriate thermodynamic and kinetic data. In conclusion, this makes it an effective modeling platform for optimizing high-temperature diffusion bonding and developing reliable bonded components such as compact heat exchangers.

Diffusion bonding↗