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14 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Multifidelity methods for uncertainty quantification of a nonlocal model for phase changes in materials

This study is devoted to the construction of a multifidelity Monte Carlo (MFMC) method for the uncertainty quantification of a nonlocal, non-mass-conserving Cahn-Hilliard model for phase transitions with an obstacle potential. Here, we are interested in estimating the expected value of an output of interest (OoI) that depends on the solution of the nonlocal Cahn-Hilliard model. As opposed to its local counterpart, the nonlocal model captures sharp interfaces without the need for significant mesh refinement. However, the computational cost of the nonlocal Cahn-Hilliard model is higher than that of its local counterpart with similar mesh refinement, inhibiting its use for outer-loop applications such as uncertainty quantification. The MFMC method augments the desired high-fidelity, high-cost OoI with a set of lower-fidelity, lower-cost OoIs to alleviate the computational burden associated with nonlocality. Most of the computational budget is allocated to sampling the cheap surrogate models to achieve speedup, whereas the high-fidelity model is sparsely sampled to maintain accuracy. For the non-mass-conserving nonlocal Cahn-Hilliard model, the use of the MFMC method results in, for a given computational budget, about an order of magnitude reduction in the mean-squared error of the expected value of the OoI relative to that of the Monte Carlo method.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

A consistent and conservative volume distribution algorithm and its applications to multiphase flows using Phase-Field models

In the present study, the multiphase volume distribution problem, where there can be an arbitrary number of phases, is addressed using a consistent and conservative volume distribution algorithm. The proposed algorithm satisfies the summation constraint, the conservation constraint, and the consistency of reduction. The first application of the volume distribution algorithm is to determine the Lagrange multipliers in multiphase Phase-Field models that enforce the mass conservation, and a multiphase conservative Allen-Cahn model that satisfies the consistency of reduction is developed. A corresponding consistent and conservative numerical scheme is developed for the model. The multiphase conservative Allen-Cahn model has a better ability than the multiphase Cahn-Hilliard model to preserve under-resolved structures. The second application is to develop a numerical procedure, called the boundedness mapping, to map the order parameters, obtained numerically from a multiphase model, into their physical interval, and at the same time to preserve the physical properties of the order parameters. Along with the consistent and conservative schemes for the multiphase Phase-Field models, the numerical solutions of the order parameters are reduction consistent, conservative, and bounded, which are theoretically analyzed and numerically validated. Then, the multiphase Phase-Field models are coupled with the momentum equation by satisfying the consistency of mass conservation and the consistency of mass and momentum transport, thanks to the consistent formulation. Finally, it is demonstrated that the proposed model and scheme converge to the sharp-interface solution and are capable of capturing the complicated multiphase dynamics even when there is a large density and/or viscosity ratio.

42 ENGINEERING↗
DOE OSTI2021

On a nonlocal Cahn–Hilliard model permitting sharp interfaces

A nonlocal Cahn–Hilliard model with a non-smooth potential of double-well obstacle type that promotes sharp interfaces in the solution is presented. To capture long-range interactions between particles, a nonlocal Ginzburg–Landau energy functional is defined which recovers the classical (local) model as the extent of nonlocal interactions vanish. In contrast to the local Cahn–Hilliard problem that always leads to diffuse interfaces, the proposed nonlocal model can lead to a strict separation into pure phases of the substance. In this work, the lack of smoothness of the potential is essential to guarantee the aforementioned sharp-interface property. Mathematically, this introduces additional inequality constraints that, in a weak formulation, lead to a coupled system of variational inequalities which at each time instance can be restated as a constrained optimization problem. We prove the well-posedness and regularity of the semi-discrete and continuous in time weak solutions, and derive the conditions under which pure phases are admitted. Moreover, we develop discretizations of the problem based on finite element methods and implicit–explicit time-stepping methods that can be realized efficiently. Finally, we illustrate our theoretical findings through several numerical experiments in one and two spatial dimensions that highlight the differences in features of local and nonlocal solutions and also the sharp interface properties of the nonlocal model.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Phase-field simulations of fission gas bubble growth and interconnection in U-(Pu)-Zr nuclear fuel

Abstract The growth and interconnection of fission gas bubbles in the hotter central regions of U-(Pu)-Zr nuclear fuel has been simulated with a phase-field model. The Cahn-Hilliard equation was used to represent the two-phase microstructure, with a single defect species. The volume fraction of the bubble phase and surface area of the bubble-matrix interface were determined during growth and interconnection. Surface area increased rapidly during the initial stages of growth, then slowed and finally decreased as bubble interconnection began and coarsening acted to reduce surface area. The fraction of the bubbles vented to a simulation domain boundary, f V , was quantified as a measure of the microstructure’s interconnectivity and plotted as a function of porosity p . The defect species diffusivity was varied; although changes in diffusivity significantly affected the microstructure, the plots of f V vs. p did not change significantly. The percolation threshold p c was calculated to be approximately 0.26, depending on the assumed diffusivity and using an initial bubble number density based on experimental observations. This is slightly smaller than the percolation threshold for continuum percolation of overlapping 3D spheres. The simulation results were used to parameterize two different engineering-scale swelling models for U-(Pu)-Zr in the nuclear fuel performance code BISON.

Aagesen, Larry K. (ORCID:000000034936676X)↗
DOE OSTI2023

Phase Field Modeling of Corrosion for Design of Next-Generation Magnesium-Aluminum Vehicle Joints

In this study, WPI and its partners sought to understand galvanic corrosion in Mg-Al friction stir weld (FSW) joints. Unlike fusion welds, FSW is done in the solid state. This limits (but does not eliminate) the formation of brittle Al-Mg intermetallics, resulting in better joint strength and corrosion resistance. This study aimed to understand corrosion behavior of Mg-Al FSW joints by using a Cahn-Hilliard phase field electrochemistry model based on the work of Pongsaksawad et al. Its context challenge problem was an ultra-light door designed by Magna with 6061-Al sheet outer and ZEK100-Mg sheet inner panels, joined by an adhesive and FSW. Unlike most FSW joints, which weld through the softer material into the harder one, the Al sheet’s hem joint around Mg required the FSW tool to through the harder Al into the softer Mg, which was itself a challenge. Despite this challenge, PNNL succeeded in consistently making very strong FSW lap joints, by using a triflute tool and power control. Characterization showed mostly separate Al and Mg regions in the weld, usually with a hook protruding from Al into Mg, which likely contributed to high joint strength. Nanohardness mapping showed higher hardness in thin Mg swirls into the Al nugget, likely due to the intermetallics. The study used the industry standard SAE J2334 Cyclic Corrosion Test (CCT) to simulate corrosion conditions over the life of a vehicle, as well as linear polarization testing and ASTM G71 pitting corrosion testing. Corrosion reactions were: H 2 O + ½O 2 + 2e - → 2OH - at the Al cathode, and either Mg → Mg 2+ + 2e - or Mg + 2OH - → Mg(OH) 2 + 2e - at the Mg anode, with electron transfer through the joint. Mass loss increased with Al section length, supporting the hypothesis that galvanic corrosion in Al-Mg joints is limited by the cathodic reaction. Both pitting corrosion (first anode reaction) and hydroxide film formation (second reaction) were observed. The J2334 CCT test method was slightly modified, adding a chromium solution cleaning step after each week of corrosion testing, in order to remove Mg oxides and hydroxides and accurately measure mass loss over time. That said, FSW joints treated in this way did not did not exhibit significant reduction in lap shear strength of the joints vs. newly welded samples which had not undergone corrosion. Corrosion appeared to consist mainly of pitting in Mg in a way which did not directly affect the bond between the materials where the joints failed. A Cahn-Hilliard phase field model coupled with electrical potential described the electronically mediated galvanic corrosion electrochemical reactions described above in the four-component Al-Mg-H 2 O-(H 2 O+½O 2 ) system. This 2-D model successfully predicted Mg(OH)₂ formation at the anode in some circumstances, and Mg pitting corrosion in others, and showed the correct electric field directions in both cases. The model predicted Mg pit corrosion depth within a factor of two of measured pits.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
DOE OSTI2021

A phase-field method for boiling heat transfer

Here we present a phase field method for heat transfer in two-phase flow with boiling. The vapor/liquid interface evolution is modeled by the Cahn-Hilliard equation. The phase change rate is determined by accounting for the heat conduction balance on either vapor or liquid side of the interfacial area, depending on which side the temperature is assumed to be maintained at the saturation temperature during boiling. The velocity correction scheme proposed by Dong & Shen [27] is extended to solve the Navier-Stokes equations for a non-solenoidal velocity field, and the entropy viscosity method is employed for stabilization. The phase change model is verified by two-dimensional simulations of a vapor bubble growing in super-heated liquid and in film boiling. In both cases, mesh independence of the results is systematically performed. Subsequently, the method is applied to predict the growth of three-dimensional vapor bubble in a rectangular microchannel with boiling flow, achieving good agreement with experimental measurements and available simulation results using the level-set method. The numerical experiments demonstrate that the required mesh resolution for the phase field method is comparable with that of volume of fluid (VoF) and level-set methods.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Compositional patterning in irradiated alloys: Effective potentials and effective interfacial energy

Compositional patterning (CP) in binary alloys during energetic particle irradiation is studied using a kinetic model that considers two competing kinetic processes, a thermally activated one promoting macroscopic phase separation (MPS) of the concentration field c(r,t) and a forced one resulting in finite-range random atomic mixing. The forced mixing is modeled by a Gaussian relocation distribution with a characteristic distance R. A series of approximate kinetic models are introduced by expanding the mixing function into a series of n terms, thus replacing the non-local evaluations of the concentration field c(r'-r,t) by local derivatives of c(r,t). This approach makes it possible to obtain exact effective potentials and build steady-state diagrams for each order-n model. Phase field (PF) simulations using these order-n models reveal that near the onset of patterning, phase evolution is accurately described using an order-3 model, which changes smoothly from an extended Cahn-Hilliard free energy in the MPS regime to a one-mode Swift-Hohenberg functional in the CP regime. Deeper into the patterning regime, higher-order models are required to achieve convergence, yielding square-like concentration profiles characteristic of a strong segregation regime. These higher-order effective free energies are analogous to multimodal Swift-Hohenberg functionals. Here, a new definition for the effective interfacial energy is proposed in the CP regime, since the interfacial area is no longer an excess quantity in that regime, precluding the use of the standard thermodynamic definition of interfacial energy.

36 MATERIALS SCIENCE↗
DOE OSTI2025

A fluctuating hydrodynamics model for nanoscale surfactant-laden interfaces

A multispecies diffuse interface model is formulated in a fluctuating hydrodynamics framework for the purpose of simulating surfactant interfaces at the nanoscale. The model generalizes previous work to ternary mixtures, employing a Cahn-Hilliard free energy density combined with incompressible, isothermal fluctuating hydrodynamics where dissipative fluxes include both deterministic and stochastic terms. The intermolecular parameters in the free energy are chosen such that one species acts as a partially miscible surfactant. From Laplace pressure measurements, we show that in this model the surface tension decreases linearly with surfactant concentration, leading to Marangoni convection for interfaces with concentration gradients. In the capillary wave spectrum for interfaces with and without surfactant, we find that for the former, the spectrum deviates significantly from classical capillary wave theory, presumably due to Gibbs elasticity. In non-equilibrium simulations of the Rayleigh-Plateau instability, deterministic simulations showed that the surfactant delays pinching of a fluid cylinder into droplets. However, stochastic simulations indicate that thermal fluctuations disrupt the surfactant's stabilizing effect. Similarly, the spreading of a patch of surfactant, driven by Marangoni convection, was found to be partially suppressed by thermal fluctuations.

Capillary waves↗
DOE OSTI2025

Flux effects in precipitation under irradiation simulation of Fe-Cr alloys

Radiation-enhanced precipitation of Cr-rich α’ in irradiated Fe-Cr alloys, which results in hardening and embrittlement, depends on the irradiating particle and the displacement per atom (dpa) rate. Here, we utilize a Cahn-Hilliard phase-field based approach, that includes simple models for nucleation, irradiating particle and rate dependent radiation-enhanced diffusion and cascade mixing to simulate α’ evolution under neutrons, heavy ions, and electron irradiations. Different irradiating particles manifest very different cascade mixing efficiencies. The model was calibrated using neutron data. For cascade inducing neutron/heavy-ion dpa rates at 300 °C between 10-8 and 10-6 dpa/s the model predicts approximately constant number density, decreasing radius, decreasing α’ Cr composition, and lower α’ volume fraction. The model then predicts a dramatic transition to no α’ formation above approximately 10-5 dpa/s, while electron irradiation, with weak mixing, had little effect at dpa rates up to 10-3 dpa/s. These model predictions are consistent with experiments. We explain the results in terms of the flux dependence of the radiation enhanced diffusion, cascade mixing, and their ratio, which all vary significantly in relevant flux ranges for neutron and cascade inducing ion irradiations. These results show that both cascade mixing and radiation enhanced diffusion must be accounted for when attempting to emulate neutron-irradiation effects using accelerated ion irradiations.

Ke, Jia-Hong↗
DOE OSTI2021

Anisotropic Cahn-Hilliard free energy and interfacial energies for binary alloys with pairwise interactions

The original Cahn-Hilliard derivation of the contribution of compositional inhomogeneity to the free energy of a binary alloy with pairwise interactions is extended to include higher-order inhomogeneity terms. For alloys on a cubic lattice, the coefficient of the first inhomogeneity is a second-rank tensor and reduces to a scalar, but it is shown that the second order and the third order inhomogeneity terms are weighted by fourth-rank and sixth-rank tensors, thus resulting in anisotropic contributions. Furthermore, each interaction shell generates a unique set of inhomogeneity coefficients that is determined by the set of vectors connecting an atom to its neighbors on that shell. These coefficients are calculated for fcc and bcc alloys with interactions up fourth nearest neighbors. Phase field simulations based on these extended Cahn-Hilliard free energies are performed to measure interface free energies along specific crystallographic directions as a function of temperature, and to obtain the equilibrium shape of precipitates. Interface free energies, and the resulting anisotropies, are compared to those obtained by discrete models and Monte Carlo simulations.

36 MATERIALS SCIENCE↗
DOE OSTI2024

A-B Transition in Superfluid $^3$He and Cosmological Phase Transitions

First-order phase transitions in the very early universe are a prediction of many extensions of the Standard Model of particle physics and could provide the departure from equilibrium needed for a dynamical explanation of the baryon asymmetry of the Universe. They could also produce gravitational waves of a frequency observable by future space-based detectors such as the Laser Interferometer Space Antenna. All calculations of the gravitational wave power spectrum rely on a relativistic version of the classical nucleation theory of Cahn-Hilliard and Langer, due to Coleman and Linde. The high purity and precise control of pressure and temperature achievable in the laboratory made the first-order A to B transition of superfluid $^3$He ideal for test of classical nucleation theory. As Leggett and others have noted, the theory fails dramatically. The lifetime of the metastable A phase is measurable, typically of order minutes to hours, far faster than classical nucleation theory predicts. If the nucleation of B phase from the supercooled A phase is due to a new, rapid intrinsic mechanism that would have implications for first-order cosmological phase transitions as well as predictions for gravitational wave production in the early universe. Here we discuss studies of the A-B phase transition dynamics in $^3$He, both experimental and theoretical, and show how the computational technology for cosmological phase transition can be used to simulate the dynamics of the A-B transition, support the experimental investigations of the A-B transition in the QUEST-DMC collaboration with the goal of identifying and quantifying the mechanism(s) responsible for nucleation of stable phases in ultra-pure metastable quantum phases.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2020

Phase field benchmark problems targeting fluid flow and electrochemistry

Here, we continue our development of phase field model benchmark problems with the addition of a third set, complimenting our previously developed problems for diffusion, precipitation, dendritic growth and linear elasticity. These benchmark problems are being jointly developed by the Center for Hierarchical Materials Design (CHiMaD) and the National Institute of Standards and Technology (NIST) along with input from the phase field community. The first problem in this third set targets Stokes flow, with a particular emphasis on flow around an obstruction placed asymmetrically in the domain. While Stokes flow is not traditionally in the canon of phase field problems, it is a class of problems gaining importance in areas such as filtration and water purification. The second problem deals with coupled Cahn-Hilliard diffusion and electrostatic forces, which is an important area in energy storage and battery sciences. We present our own solutions and discuss sources of numerical errors for the Stokes problem as well as simple checks to avoid fundamental issues in the coupled diffusion-electrostatics problem. The latter problem contains some subtleties that we expand on in an Appendix.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Bifurcation Analysis Reveals Solution Structures of Phase Field Models

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

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

MEUMAPPS (C++ Version)

Many materials, metal alloys in particular, have features on the on micrometer or nanometer scale that have a large impact on the properties of the material. These features are known as the microstructure of the material. Understanding why and how the microstructure forms in a material is of fundamental scientific interest as well as of significant technological interest. The capability to predict microstructure evolution in a material allows the intentional design of microstructures and hence the intentional design of material properties. The phase-field method is one of the leading methods for predicting microstructure evolution. One of the most significant problems for phase-field models is their computational expense. Even limited phase-field simulations can easily require thousands of CPU core-hours to complete, which significantly limits their use. This code provides both a general framework for creating scalable, GPU-accelerated phase-field model applications as well as several applications themselves. The code is capable of using hundreds of GPUs efficiently, which greatly reduces the time required to perform simulations. The code is written with an emphasis on performance portability, that is the ability for the code to run efficiently on a number of different computing architectures without modification of the source code. The performance portability of this code is primarily enabled through the use of two libraries, Kokkos (performance portable data structures and execution patterns) and heFFTe (performance portable distributed 3D fast Fourier transforms). The code consists of a core library, applications, and tests. The core library includes shared functionality between applications. This includes interfaces with fast Fourier transform (FFT) libraries such as heFFTe, data structures based on Kokkos, file input and output capabilities, and a solver for infinitesimal strain mechanical equilibrium problems. Five applications are included in the code. The flagship application is the MEUMAPPS-SS application, which implements the Kim-Kim-Suzuki phase-field model for precipitation for an arbitrary number of phases and components in a metal alloy. Five simpler applications are also included that solve the Eshelby inclusion problem, Allen-Cahn equation, the coupled Allen-Cahn and diffusion equations, and the Cahn-Hilliard equation. The code includes two applications to solve the Cahn-Hilliard equation, one with constant-step-size first-order time integration and the second with adaptive high-order time integration.

DeWitt, Stephen [Oak Ridge National Lab. (ORNL), O↗