Concurrent Atomistic-continuum Method Research Articles

Transmitting multiple high-frequency phonons across length scales using the concurrent atomistic–continuum method

Coupled atomistic–continuum methods can describe large domains and model dynamic material behavior for a much lower computational cost than traditional atomistic techniques. However, these multiscale frameworks suffer from wave reflections at the atomistic–continuum interfaces due to the numerical discrepancy between the fine-scaled and coarse-scaled models. Such reflections are non-physical and may lead to unfavorable outcomes such as artificial heating in the atomistic region. In this work, we develop a technique to allow the full spectrum of phonons to be incorporated into the coarse-scaled regions of a periodic concurrent atomistic–continuum (CAC) framework. This scheme tracks phonons generated at various time steps and thus allows multiple high-frequency wave packets to travel between the atomistic and continuum regions. Simulations performed with this method demonstrate the ability of the technique to preserve the coherency of waves with a range of wavevectors as they propagate through the domain. This work has applications for systems with defined boundary conditions and may be extended to more complex problems involving waves randomly nucleated from an impact within a multiscale framework.

Moving window techniques to model shock wave propagation using the concurrent atomistic–continuum method

Atomistic methods have successfully modeled different aspects of shock wave propagation in materials over the past several decades, but they suffer from limitations which restrict the total runtime and system size. Multiscale methods have been able to increase the length and time scales that can be modeled but employing such schemes to simulate wave propagation and evolution through engineering-scale domains is an active area of research. In this work, we develop two distinct moving window approaches within a Concurrent Atomistic–Continuum (CAC) framework to model shock wave propagation through a one-dimensional monatomic chain. In the first method, the entire CAC system travels with the shock in a conveyor fashion and maintains the shock front in the middle of the overall domain. In the second method, the atomistic region follows the shock by the simultaneous coarsening and refinement of the continuum regions. The CAC and moving window frameworks are verified through dispersion relation studies and phonon wave packet tests. We achieve good agreement between the simulated shock velocities and the values obtained from theory with the conveyor technique, while the coarsen-refine technique allows us to follow the propagating wave front through a large-scale domain. This work showcases the ability of the CAC method to accurately simulate propagating shocks and also demonstrates how a moving window technique can be used in a multiscale framework to study highly nonlinear, transient phenomena.

Interference, scattering, and transmission of acoustic phonons in Si phononic crystals

The transient processes of phonon scattering and transmission in Si phononic crystals with periodic pores are simulated using the concurrent atomistic-continuum method. The nature of phonon transport is demonstrated to be dependent on the relation between the phonon wavelength, λ, the period length, p, and the neck width, n, of the phononic crystal. Three distinct regimes have been observed: (1) phonons with wavelengths equal to 2.9p or larger propagate predominantly ballistically, which is accompanied by specular reflection of varying extents, depending on the wavelength of the phonons; the properties of these phonons are consistent with the phonon dispersion relations of the phononic crystal modelled as a homogenous and harmonic system; (2) phonons with wavelengths < n are partly reflected by the pore boundaries and partly propagate ballistically in the solid regions of the necks; two types of vibrational modes are identified: the single crystal Si phonon modes, and the vibrational modes resulting from internal surface scattering; (3) phonons with wavelength close to p are most strongly scattered, and the internal surface-related modes dominate the transport; these modes have the slowest group velocities and lowest energy transmission, and the transport is predominantly diffusive even for specimens originally at zero temperature. For phononic crystals that contain a single crystal heater at their center, the interface between the single crystal heater and the phononic structure provides a strong resistance to short wavelength phonons, leading to predominantly diffusive phonon transport and an average energy flux that is two orders of magnitude lower than a same-sized single crystal specimen.

Comparative modeling of the disregistry and Peierls stress for dissociated edge and screw dislocations in Al

Many elementary deformation processes in metals involve the motion of dislocations. The planes of glide and specific processes dislocations prefer depend heavily on their atomic core structures. Atomistic simulations are desirable for dislocation modeling but their application to even sub-micron scale problems is in general computationally costly. Accordingly, continuum-based approaches, such as the phase-field microelasticity, phase-field dislocation dynamics (PFDD), generalized Peierls–Nabarro (GPN) models, and the concurrent atomistic–continuum (CAC) method, have attracted increasing attention in the field of dislocation modeling because they well represent both short-range cores interactions and long-range stress fields of dislocations. To better understand their similarities and differences, it is useful to compare these methods in the context of benchmark simulations and predictions. In this paper, we apply the CAC method and different PFDD variants – one of them is equivalent to a GPN model – to simulate an extended (i.e., dissociated) dislocation in Al with initially pure edge or pure screw character in terms of the disregistry. CAC and discrete forms of PFDD are also employed to calculate the Peierls stress. By conducting comprehensive convergence studies, we quantify the dependence of these measures on time/grid resolution and simulation cell size. Several important but often overlooked differences between PFDD/GPN variants are clarified. Our work sheds light on the advantages and limitations of each method, as well as the path towards enabling them to effectively model complex dislocation processes at larger length scales.

Phonon Transport Across Coherent and Incoherent Interfaces

Phonon-mediated heat transport in two-dimensional superlattices with coherent and incoherent interfaces is simulated by using the concurrent atomistic-continuum method. The energy transmission across superlattices with incoherent interfaces is found to be an order of magnitude lower than that with coherent interfaces. The simulation results provide a direct visualization of the transient processes of phonon propagation and scatterings, which facilitates an improved mechanistic understanding of phonon transport across multiple interfaces. This work finds that heat conduction in superlattices with coherent interfaces is dominated by coherent phonons, while the existence of defects that comprise the incoherent interfaces destroys the wave interference and changes the phonon transport nature from coherent to diffusive. In addition, phonon scattering by interface defects becomes stronger with decreasing phonon wavelength, and the phonon coherence is destroyed for phonons with 5 nm wavelength.

Sequential obstacle interactions with dislocations in a planar array

The strengthening of metals by nano-scale obstacles is mainly attributed to the impediment to glide dislocations offered by these obstacles. It is important to understand the mechanisms for dislocation bypass of obstacles having nano-scale dimension, including the atomic-scale structure changes sustained by both obstacles and dislocations after the bypass process. Recently, atomic-scale modeling has provided much insight into obstacle interactions involving a single dislocation. However, the more naturally occurring scenarios involving a sequence of encounters with arrays of moving dislocations are not as well understood owing to prohibitively large length scale requirements for atomistic models. In this study, we utilize a novel multiscale concurrent atomistic-continuum method to simulate a sequence of interactions between glide dislocations in an array with a spherical nano-obstacle (either a void or an impenetrable precipitate) in Al. In the case of a void, the bypassing array of dislocations progressively weakens the void until it splits the originally spherical void into two hemispheres. In the case of a large impenetrable precipitate, sequential dislocations in the array bypass via alternating mechanisms of Orowan looping and Hirsch looping. The residual dislocation loop created around the precipitate by the bypass of the first dislocation is completely removed by the passage of the subsequent dislocation. These mechanisms can benefit the design of materials that are reinforced with nanophase inhomogeneities to achieve ultra high strength.

Phonon spectrum and phonon focusing in coarse-grained atomistic simulations

This work investigates the accuracy, efficiency, and applicability of coarse-grained (CG) atomistic methods in simulation of phonon dynamics. First, we compute and compare phonon dispersion relations in CG models with those in atomically resolved models, using the concurrent atomistic-continuum (CAC). The CG atomistic models using the CAC method are shown to reproduce long-wavelength phonons with great accuracy, while capturing the dynamics of some short-wavelength phonons that are usually inaccessible to CG methods. We then present CG simulation results of the propagation of heat pulses in Si with the interaction between atoms being modelled with the Stillinger-Weber potential; the experimentally observed phonon-focusing patterns in the (1 0 0) and (1 1 1) planes of Si crystals are reproduced. The accuracy and efficiency of the CAC method in CG simulation of acoustic and optical phonon branches are quantified with respect to atomically-resolved molecular dynamics simulations. The applicability and limitations of concurrent multiscale methods in the simulation of phonon transport across atomistic-continuum interface are investigated. Possible ways to overcome the limitations are discussed.

Modeling dislocations and heat conduction in crystalline materials: atomistic/continuum coupling approaches

ABSTRACTDislocations and heat conduction are essential components that influence properties and performance of crystalline materials, yet the modelling of which remains challenging partly due to their multiscale nature that necessitates simultaneously resolving the short-range dislocation core, the long-range dislocation elastic field, and the transport of heat carriers such as phonons with a wide range of characteristic length scale. In this context, multiscale materials modelling based on atomistic/continuum coupling has attracted increased attention within the materials science community. In this paper, we review key characteristics of five representative atomistic/continuum coupling approaches, including the atomistic-to-continuum method, the bridging domain method, the concurrent atomistic–continuum method, the coupled atomistic/discrete-dislocation method, and the quasicontinuum method, as well as their applications to dislocations, heat conduction, and dislocation/phonon interactions in crystalline materials. Through problem-centric comparisons, we shed light on the advantages and limitations of each method, as well as the path towards enabling them to effectively model various material problems in engineering from nano- to mesoscale.Abbreviations: AtC: atomistic-to-continuum; BCC: body-centred cubic; BDM: bridging domain method; CAC: concurrent atomistic–continuum; CADD: coupled atomistic/discrete-dislocation; DDD: discrete dislocation dynamics; DDf-MD: discrete diffusion-molecular dynamics; DOF: degree of freedom; ESCM: embedded statistical coupling method; FCC: face-centred cubic; GB: grain boundary; XFEM: extended finite element method; MD: molecular dynamics; MS: molecular statics; PK: Peach-Koehler; QC: quasicontinuum

PyCAC: The concurrent atomistic-continuum simulation environment

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Recent progress in the concurrent atomistic-continuum method and its application in phonon transport

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Ballistic-diffusive phonon heat transport across grain boundaries

The propagation of a heat pulse in a single crystal and across grain boundaries (GBs) is simulated using a concurrent atomistic-continuum method furnished with a coherent phonon pulse model. With a heat pulse constructed based on a Bose-Einstein distribution of phonons, this work has reproduced the phenomenon of phonon focusing in single and polycrystalline materials. Simulation results provide visual evidence that the propagation of a heat pulse in crystalline solids with or without GBs is partially ballistic and partially diffusive, i.e., there is a co-existence of ballistic and diffusive thermal transport, with the long-wavelength phonons traveling ballistically while the short-wavelength phonons scatter with each other and travel diffusively. To gain a quantitative understanding of GB thermal resistance, the kinetic energy transmitted across GBs is monitored on the fly and the time-dependent energy transmission for each specimen is measured; the contributions of coherent and incoherent phonon transport to the energy transmission are estimated. Simulation results reveal that the presence of GBs modifies the nature of thermal transport, with the coherent long-wavelength phonons dominating the heat conduction in materials with GBs. In addition, it is found that phonon-GB interactions can result in reconstruction of GBs.

Validation of the Concurrent Atomistic-Continuum Method on Screw Dislocation/Stacking Fault Interactions

Dislocation/stacking fault interactions play an important role in the plastic deformation of metallic nanocrystals and polycrystals. These interactions have been explored in atomistic models, which are limited in scale length by high computational cost. In contrast, multiscale material modeling approaches have the potential to simulate the same systems at a fraction of the computational cost. In this paper, we validate the concurrent atomistic-continuum (CAC) method on the interactions between a lattice screw dislocation and a stacking fault (SF) in three face-centered cubic metallic materials—Ni, Al, and Ag. Two types of SFs are considered: intrinsic SF (ISF) and extrinsic SF (ESF). For the three materials at different strain levels, two screw dislocation/ISF interaction modes (annihilation of the ISF and transmission of the dislocation across the ISF) and three screw dislocation/ESF interaction modes (transformation of the ESF into a three-layer twin, transformation of the ESF into an ISF, and transmission of the dislocation across the ESF) are identified. Our results show that CAC is capable of accurately predicting the dislocation/SF interaction modes with greatly reduced DOFs compared to fully-resolved atomistic simulations.

Mesh refinement schemes for the concurrent atomistic-continuum method

Most concurrent multiscale methods that employ domain decomposition divide the simulation domain into atomistic and continuum subdomains such that nanoscale defects are described at atomistic resolution while a continuum treatment is employed elsewhere. An example is the concurrent atomistic-continuum (CAC) method. While dislocations can nucleate and migrate in the continuum domain in CAC, they are restricted to prescribed slip planes along interelement boundaries. In this paper, we extend the original CAC method by implementing mesh refinement schemes in a straightforward manner to accommodate a 3-D dislocation network and curved dislocation migration. CAC simulation results are validated by comparing to their equivalent fully atomistic simulations. Augmented with an adaptive mesh refinement scheme, a brittle-to-ductile transition in dynamic fracture response of single crystalline FCC Cu is accurately predicted using CAC simulations. Differing from most multiscale methods, the complexity of the geometric topologies during mesh refinement is avoided in this approach because CAC does not require interelement compatibility. The extended CAC method is expected to apply in multiscale modeling of more complex dislocation activities in crystalline materials up to μm scale.

Concurrent atomistic–continuum simulation of polycrystalline strontium titanate

This paper presents the new development of a concurrent atomistic–continuum (CAC) method in simulation of the dynamic evolution of defects in polycrystalline polyatomic materials. The CAC method is based on a theoretical formulation that extends Kirkwood’s statistical mechanical theory of transport processes to a multiscale description of crystalline materials. It solves for both the deformation of lattice cells and the internal deformation within each lattice cell, making it a suitable method for simulations of polyatomic materials. The simulation results of this work demonstrate that CAC can simulate the nucleation of dislocations and cracks from atomistically resolved grain boundary (GB) regions and the subsequent propagation into coarsely meshed grain interiors in polycrystalline strontium titanate without the need of supplemental constitutive equations or additional numerical treatments. With a significantly reduced computational cost, CAC predicts not only the GB structures, but also the dynamic behaviour of dislocations, cracks and GBs, all of which are comparable with those obtained from atomic-level molecular dynamics simulations. Simulation results also show that dislocations tend to initiate from GBs and triple junctions. The angle between the slip planes and the GB planes plays a key role in determining the GB-dislocation reactions.

A quasistatic implementation of the concurrent atomistic-continuum method for FCC crystals

In recent years, numerous partitioned-domain methods have been developed to describe dislocation behavior at length scales that are usually inaccessible to most classical atomistic methods. These methods retain full atomistic detail in regions of interest while using a continuum description to reduce the computational burden elsewhere. In most of these methods, however, lattice defects in the continuum are either implemented via constitutive relations, lattice elasticity with dislocation field interactions, or are not permitted at all. In such approaches, the transit of dislocations across the atomistic/continuum interface appeals to approximate heuristics intended to minimize the effects of the interface due to the change from atomistic to continuum degrees of freedom. The concurrent atomistic-continuum (CAC) method, originally developed for addressing dynamic dislocation behavior by Xiong et al. (2011), permits dislocations to propagate in a continuum domain that employs a piecewise continuous finite element description with interelement displacement discontinuities. The method avoids ghost forces at interface between atomistically resolved and coarse-grained domains. CAC has subsequently been used to investigate complex dislocation behavior in face-centered cubic (FCC) metals (Xiong et al., 2012b,a,c, 2015). In this paper, we propose a quasistatic 3-D method to carry out sequential energy-minimized simulations at 0 K. This facilitates study of structure evolution along minimum energy pathways, avoiding over-driven conditions of high rate molecular dynamics. Parallelization steps in code implementation are described. Applications are presented for the quasistatic CAC method in FCC metal plasticity. Comparisons are made with a fully-resolved atomistic method for generalized stacking fault energy, core structure and stress field of a single 60° mixed type dislocation, surface indentation, and 60° mixed type dislocation migration through the interface between atomistic and coarse-grained domains. It is shown that 3-D CAC simulations are useful in substantially reducing the number of degrees of freedom while preserving key characteristics of dislocation structure, stacking faults, and plasticity, including the net Burgers vector and long range fields of interacting dislocations.

Concurrent atomistic–continuum simulations of dislocation–void interactions in fcc crystals

Dislocation interactions with distributed condensed vacancy clusters in fcc metals were simulated via a concurrent atomistic–continuum method. Due to void strengthening, the dislocation lines are found to bow as a result of pinning on the original glide plane and undergo depinning through drawing out screw dipoles and forming prismatic loops on the secondary slip plane. We discovered an inertia-induced transition between Hirsch looping and void shearing mechanisms as the void spacing ranges from the scale of nm to hundreds of nm. Contrary to prior understanding, simulations suggest that large voids (∼5nm in diameter) can behave as weak barriers to dislocation motions under high strain-rate dynamic conditions.

Prediction of phonon properties of 1D polyatomic systems using concurrent atomistic–continuum simulation

In this work, we present the simulation results of phonon properties, including phonon dispersion relations, group velocities, phonon relaxation time and mode-specific thermal conductivity, of low-dimensional crystals using the newly developed concurrent atomistic–continuum (CAC) method. With significantly less number of degrees of freedoms than all-atom molecular dynamics, the CAC method predicts the phonon properties of one-dimensional (1D) polyatomic crystals at finite temperatures with the full anharmonicity of the atomic interactions being incorporated. Complete phonon branches of polyatomic crystals are obtained by CAC through the phonon spectral energy density analysis. It is shown that CAC allows medium to long-wavelength phonon transport from atomic to coarsely meshed finite element region without the need of special numerical treatment. The frequency-dependent phonon group velocities are explicitly measured in the simulation. Sub-THz phonon lifetimes in 100 -mm-long polyatomic chains containing 400 million of atoms are predicted. Analysis of the phonon mode-specific thermal conductivity shows that the medium to long-wavelength acoustic phonons are contributing to the majority of the heat transport in 1D crystal, suggesting that the long-wavelength phonons effectively act as thermal carriers in 1D system. This work provides a possible explanation for the anomalous heat transport observed in low-dimensional materials.

Nucleation and growth of dislocation loops in Cu, Al and Si by a concurrent atomistic-continuum method

Submicron-sized samples with 42,000 finite elements containing up to ∼86 million atoms have been simulated using a concurrent atomistic-continuum method. The simulations reproduce not only nucleation and growth of semicircular dislocation loops in Cu and Al, but also hexagonal shuffle dislocation loops in Si, with the loop radius approaching ∼75 nm. Details of leading and trailing partial dislocations connected by intrinsic stacking faults, dislocation loop coalescence through annihilation, and formation of junctions are reproduced.

Coarse-grained atomistic simulations of dislocations in Al, Ni and Cu crystals

This paper presents the application of a recently developed concurrent atomistic continuum (CAC) methodology in coarse-grained (CG) atomistic simulations of dislocation nucleation and migration in face-centered cubic (fcc) Al, Ni and Cu crystals, using an EAM force field and an adaptive mesh refinement strategy. The CAC method is based on recently developed atomistic field theory (AFT) that frames the problem in terms of a continuum field representation of lattice and sublattice atomic arrangements. Four sets of CG models with different finite element mesh refinement are constructed to test the accuracy and efficiency of the CG method relative to full molecular dynamics (MD). Simulation results show that the CG method is able to reproduce key phenomena of dislocation dynamics in initially defect free fcc crystals, including strain bursts caused by dislocation nucleation and migration, the structure of leading and trailing partial dislocations separated by equilibrium stacking faults in Al, formation of intrinsic stacking fault ribbons in Ni and Cu, and 3D migration of curved dislocations, all comparable with results of MD simulations. CG simulations have also revealed that the yield strength of Al depends on the thickness of the specimens as well as the operative deformation mechanisms. The effects of mesh size and adaptive mesh refinement on CG simulation results are analyzed and discussed.

Atom collocation method

This paper presents a new concurrent atomistic-continuum method called the atom collocation method (ACM). By adopting the framework of continuum collocation method, ACM aims at overcoming the current difficulties in interfacial mismatch, adaptive analysis, and parallel implementation of existing atomistic-continuum methods. The proposed ACM is truly meshfree and generalizes the full atomistic description, which naturally yields a perfectly compatible atomistic/continuum interface that eliminates any ghost forces. A unique feature of ACM is that the collocation atoms can be turned on or off freely at any time without the need to reconstruct interpolation functions, which greatly enhance the ability to perform adaptive analysis. The proposed ACM is applied to solve problems involving point defect and crack propagation as well as surface, edge, and corner effects and demonstrates excellent accuracy and efficiency compared to molecular statics.