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http://creativecommons.org/licenses/by/4.0/ + http://creativecommons.org/licenses/by/4.0/ + + + + Software archive + 10.5905/ethz-1007-824 + + + GitHub review issue + https://github.com/openjournals/joss-reviews/issues/7068 + + + + 10.21105/joss.07068 + https://joss.theoj.org/papers/10.21105/joss.07068 + + + https://joss.theoj.org/papers/10.21105/joss.07068.pdf + + + + + + A fast atomistic approach to +finite-temperature surface elasticity of crystalline +solids + Saxena + Computational Materials +Science + 211 + 10.1016/j.commatsci.2022.111511 + 0927-0256 + 2022 + Saxena, S., Spinola, M., Gupta, P., +& Kochmann, D. M. (2022). A fast atomistic approach to +finite-temperature surface elasticity of crystalline solids. +Computational Materials Science, 211, 111511. +https://doi.org/10.1016/j.commatsci.2022.111511 + + + Nonequilibrium thermomechanics of gaussian +phase packet crystals: Application to the quasistatic quasicontinuum +method + Gupta + Journal of the Mechanics and Physics of +Solids + 153 + 10.1016/j.jmps.2021.104495 + 0022-5096 + 2021 + Gupta, P., Ortiz, M., & Kochmann, +D. M. (2021). Nonequilibrium thermomechanics of gaussian phase packet +crystals: Application to the quasistatic quasicontinuum method. Journal +of the Mechanics and Physics of Solids, 153, 104495. +https://doi.org/10.1016/j.jmps.2021.104495 + + + An adaptive quasicontinuum approach for +modeling fracture in networked materials: Application to modeling of +polymer networks + Ghareeb + Journal of the Mechanics and Physics of +Solids + 137 + 10.1016/j.jmps.2019.103819 + 0022-5096 + 2020 + Ghareeb, A., & Elbanna, A. +(2020). An adaptive quasicontinuum approach for modeling fracture in +networked materials: Application to modeling of polymer networks. +Journal of the Mechanics and Physics of Solids, 137, 103819. +https://doi.org/10.1016/j.jmps.2019.103819 + + + Quasicontinuum method + Miller + 2012 + Miller, & Tadmor, E. B. (2012). +Quasicontinuum method. +https://nanohub.org/resources/qcmethod + + + The quasicontinuum method. Modeling +microstructure on multiple length scales: A mixed continuum and +atomistics approach + Tadmor + 1996 + Tadmor, E. B. (1996). The +quasicontinuum method. 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      + + + + +Journal of Open Source Software +JOSS + +2475-9066 + +Open Journals + + + +7068 +10.21105/joss.07068 + +AQCNES: A Quasi-Continuum Non-Equilibrium +Solver + + + +https://orcid.org/0009-0003-6052-405X + +Bräunlich +Gerhard + + + + +https://orcid.org/0000-0002-5242-9103 + +Saxena +Shashank + + + + + +Weberndorfer +Manuel + + + + +https://orcid.org/0000-0002-5180-6149 + +Spinola +Miguel + + + + +https://orcid.org/0000-0003-3666-0257 + +Gupta +Prateek + + + + +https://orcid.org/0000-0002-9112-6615 + +Kochmann +Dennis M. + + + + + +Mechanics & Materials Laboratory, ETH Zürich, +Switzerland + + + + +Department of Applied Mechanics, Indian Institute of +Technology Delhi, India + + + + +Scientific IT Services, ETH Zürich, +Switzerland + + + + +4 +5 +2024 + +9 +101 +7068 + +Authors of papers retain copyright and release the +work under a Creative Commons Attribution 4.0 International License (CC +BY 4.0) +2022 +The article authors + +Authors of papers retain copyright and release the work under +a Creative Commons Attribution 4.0 International License (CC BY +4.0) + + + +C++ +atomistics +statistical mechanics +quasi continuum +finite temperature + + + + + + Summary +

      The behavior of macroscopic structures is determined by fast atomic + interactions at the nanoscales. Current atomic simulation techniques, + such as molecular dynamics (MD), are limited to a millions of atoms + and hence a few micrometers of domain length. Moreover, + finite-temperature vibrational frequencies of around tens of terahertz + restrict the time step of MD to femtoseconds, precluding the + simulation of problems of engineering interest. Consequently, there + has been a significant focus in recent decades on developing + multiscale modeling techniques to extend atomistic accuracy to larger + length scales and longer time frames. Existing techniques, such as the + quasicontinuum (QC) method, are restricted to spatial upscaling at + zero temperature, while temporal upscaling methods like the maximum + entropy (max-ent) approach are constrained to fully resolved atomistic + simulations at finite temperature.

      +

      The software introduced here, AQCNES, is a + C++-based framework that integrates the spatial-upscaling technique of + the quasicontinuum method with the statistical-mechanics-based + temporal upscaling technique known as Gaussian phase packets. Message + Passing Interface (MPI) is employed to enable massive parallelism, + which enhances the scalability of the software. This enables + computationally efficient and robust simulations of large atomistic + ensembles at finite temperature.

      +
      + + Statement of need +

      Commonly used atomic simulation techniques describe the entire + ensemble as a collection of particles, each having a position and + velocity in three-dimensional space. This fully refined spatial + representation in state space restricts the possible dimensions of the + ensemble to microscopic scales. Moreover, case studies of material + defects using atomistic simulations are often limited to Molecular + Statics (MS) or use unrealistically high loading rates + (Homer + et al., 2022; + Shenoy, + 2005). This is unavoidable because finite-temperature MD + simulations need long equilibration times and expensive + post-processing techniques + (Frenkel + & Ladd, 1984) to extract relevant thermodynamic + information. However, physically relevant material behavior is + observed at finite temperature. Therefore, research in the past + decades has focused on multiscale modeling techniques to advance + atomistic simulations to larger length scales and longer time scales + (Miller + & Tadmor, 2009; + Wernik + & Meguid, 2009).

      +

      Most research groups working in the broad field of upscaling + atomistic simulations have their proprietary codes. Two available + open-source codes are QuasiContinuum + (Miller + & Tadmor, 2012) based on the quasicontinuum (QC) method and + MultiBench + (Miller + & Tadmor, 2009), which is an implementation of fourteen + popular spatial upscaling methods. However, both are limited to + zero-temperature simulations for crystalline solids in two dimensions. + The MXE package in + LAMMPS + (Mendez + & Ponga, 2021) is an implementation of the temporal + upscaling technique max-ent for a fully + resolved atomic ensemble with no spatial coarse-graining. To the best + of the authors’ knowledge, there is no such open-source atomistic + simulation software that combines the aforementioned spatial and + temporal upscaling techniques.

      +

      AQCNES is a software that can predict + long-term behavior of large atomic ensembles using the spatio-temporal + upscaling of classical atomistic calculations. It enables the + calculation of material properties at finite (non-zero) temperature + from atomic scales, offering promising applications in solid-state + material science across scales. In the present implementation, full + atomistic resolution is needed for regions of local disorder as well + as for amorphous materials, although extension of the QC for amorphous + systems have been proposed + (Ghareeb + & Elbanna, 2020) and can be considered as a possible + extension. Hence, AQCNES is capable of + simulating both crystalline and amorphous materials in a temporally + upscaled fashion at zero and non-zero temperature, while the spatial + upscaling capability is limited to crystalline systems. It can also + simulate large atomic rearrangements induced by severe deformations in + multi-resolution domains by using an updated Lagrangian formulation + (Gupta + et al., 2021). This formulation uses the relaxed state after + every load step as the new reference configuration. An adaptive + neighborhood calculation strategy (similar to Tembhekar et al. + (2017)) + is adopted, where the neighborhoods are regenerated if the maximum + relative displacement of a neighbor with respect to a sampling atom + exceeds a given buffer radius. This unique combination of features + positions AQCNES as a versatile and powerful + tool in the realm of atomistic simulations.

      +
      + + Functionality +

      A technique known as Gaussian Phase Packets (GPP) + (Gupta + et al., 2021) is used to upscale in time. Instead of the + instantaneous phase-space coordinates (position + + and momentum + + ), statistical averages + + and variances + + of these coordinates are tracked for the entire ensemble in GPP, thus + separating the slow mean atomic motion from fast atomic vibrations. + The covariance matrices represented above are fully populated for the + most general case. However, in order to make the system of equations + more tractable, interatomic correlations are assumed to be zero. + Further, intra-atomic covariance matrices are assumed to be spherical + Gaussian clouds. This leads to + + where subscript + + i + runs over all atoms in the ensemble and + + 𝐈 + is an identity matrix in 3D space. Hence, the set of variables + + is solved for every atomic site. It can be shown that in the + quasistatic limit, mean momenta + + and thermal momenta + + βi + vanish for every atom. The mean positions + + and position variances + + {Σi:i=1,,N} + are obtained using the following equilibrium conditions + (Gupta + et al., 2021): +

      +

      where + + denotes the net force acting on atom + + i + having mass + + mi. + Infomation about momentum variances + + {Ωi:i=1,,N} + is obtained from the thermodynamic process which brings the system to + equilibrium. The governing equations + (1) of these + statistical parameters can be shown to yield configurations which + minimize a thermalized Helmholtz free-energy at the temperature of + interest. The reader is referred to Gupta et al. + (2021) for + further details. AQCNES uses FIRE + (Bitzek + et al., 2006) to relax the system of equations + (1) and thus obtain + a thermodynamically relaxed structure of the ensemble after a + quasi-static minimization. This approach is more robust and + computationally efficient + (Saxena + et al., 2022) than tracking individual atomic trajectories. + Moreover, it makes the simulation of realistic and quasistatic loading + scenarios possible, in contrast to the unrealistically high strain + rates accessible by MD simulations + (Vu-Bac + et al., 2014; + Zhao + et al., 2010).

      +

      For spatial coarse-graining, AQCNES uses QC + (Miller + & Tadmor, 2002; + Tadmor, + 1996). QC exploits the long-range order in crystalline + materials to explicitly model + + NhN + representative(rep-) atoms in the domain and uses + techniques from continuum-level finite-element modeling to obtain all + other atomic degrees of freedom as a function of those of the + representative atoms. Fully atomistic resolution is retained in the + vicinity of material defects, where the long-range order is broken. + The mean position + + and position variance + + Σi + of an atom + + i + are obtained by the following interpolation: +

      +

      where + + is the shape function for repatom + + a + evaluated at the position of atom + + i + in the reference configuration. This reduces the number of degrees of + freedom to solve from + + 4N + to + + 4Nh + for finite-temperature simulations. For energy and force calculation + in every minimisation iteration, a set of + + + NsN + sampling atoms is selected. The approximate Hamiltonian of the entire + system can then be written as: + + α=1Nswαα,

      +

      where + + wα + are the sampling atom weights and + + α + is the energy of the + + αth + sampling atom. A detailed explanation of optimally choosing sampling + atom locations and weights can be found in Amelang et al. + (2015).

      +
      + + Example applications +

      The following gives a summary of the mechanics and material science + applications where AQCNES has already been + used:

      + + +

      Surface Elasticity: Surfaces in solids are the + simplest extended defects and contribute to excess energy, leading + to an inherent stress associated with them. Their presence also + changes the elastic moduli of the solid as compared to those of + the bulk solid. AQCNES was used by Saxena + et al. + (2022) + to perform a detailed case study of three differently + crystallographically oriented surfaces for FCC and BCC metals as a + function of temperature. The results were also compared against + those obtained from state-of-the-art thermodynamic integration + techniques + (Freitas + et al., 2016), showing convincing accuracy. The comparison + of computational times taken by MD and + AQCNES (see + [table:sim_times]) + shows that there is an approximately fifty-fold computational + speedup and a significant reduction in the computational resources + needed for computing the excess surface free energy for an + ensemble at finite temperature. + [fig:surfaces] + shows the surface free energy, stresses, and elastic constants for + the (001), (011), and (111) surfaces in BCC iron computed using + AQCNES and their comparison with MD + simulations and ab-initio results by Schönecker + et al. + (2015). + The embedded atom method (EAM) potential developed by Chamati et + al. + (2006) + was used for AQCNES and MD simulations.

      +
      +
      + + + +

      Simulation times for a surface free energy computation for + the (001) surface in Fe at 300 K with AQCNES and thermodynamic + integration in LAMMPS (adapted from Saxena et al. (2022)).

      + + + + + + + + + + + + + + + + + + + + + + + + + + + +
      No. of atomsNo. of procs.No. of iterationsSimulation time (min)
      AQCNES2624257679.79
      LAMMPS16400128 + + 4.8107422
      +
      +
      + +

      (a) Surface free energy density + + + (ξ) + and average surface stress + + (τavg) + vs. temperature + + (T) + and (b) polar compliance plots (in Å + + 2/eV) + for the (001), (011), and (111) surfaces in BCC iron (adapted from + Saxena et al. + (2022)). +

      + +
      + + +

      Grain Boundaries: Grain boundaries (GBs) are + regions of crystallographic mismatch between two differently + oriented grains. They significantly influence the mechanical and + thermal properties of polycrystalline materials. Hence, + investigating GB properties via atomic simulations is of + scientific interest in the material science community. + AQCNES has been used to find relaxed + energies of + + [001] + and + + [011] + symmetric-tilt GBs in copper as a function of temperature for a + range of tilt angles + (Spínola + et al., 2024). Different metastable states have been + explored for each temperature and tilt angle. In addition, the + lowest-energy metastable state was subjected to a quasistatic + displacement-driven shear to obtain the shear coupling factor of + all grain boundaries. AQCNES could also + identify the Helmholtz free energy of bicrystals, for which the + standard thermodynamic integration techniques failed due to hops + of the system from one metastable state to another. + [fig:grain_boundaries] + summarizes the excess grain boundary free energy density and the + shear coupling factors for the lowest energy metastable state of + + + [001] + tilt axis grain boundaries in copper. The EAM potential developed + by Mishin et al. + (2001) + was used for these simulations.

      +

      + +

      (a) Excess grain boundary free energy density + + + (γgb) + and (b) shear coupling factor + + (β) + vs. tilt angle for the + + [001] + symmetric tilt grain boundaries in copper at different + temperatures (adapted from Spínola et al. + (2024)). +

      + + +

      +
      +
      + + +

      Nanoindentation: + Nanoindentation is a widely used technique to probe the mechanical + properties of materials and nanostructures. + AQCNES has been used to simulate the + three-dimensional thermo-mechanically-coupled nanoindentation of + copper + (Gupta + et al., 2021). Two layers of spatial coarse graining were + used to simulate a cube of side length + + + 0.077µm + with + + 0.2 + million representative atoms. The complicated microstructure of + prismatic dislocation loops below the nanoindenter could be + observed in the finite-temperature simulations at + + + 300 + and + + 600K. + The temperature dependence of the critical indenter force before + dislocation nucleation could also be captured. + [fig:nanoindentation] + shows the microstructure below the indenter and the indenter force + for different temperatures.

      +

      + +

      (a) Dislocation loops generated as a result of + nanoindentation beneath the spherical indenter at a depth of 1 + nm. (b) Indenter force vs. depth for isothermal + nanoindentation at different temperatures (adapted from Gupta + et al. + (2021)).

      + + +

      +
      +
      +
      + + Dependencies and API documentation +

      The project uses + CMake + as its build system generator. The following third party libraries are + required and located using CMake’s + find_package.

      + + +

      Boost + (components: program_options, mpi, serialization): version + 1.67

      +
      + +

      Eigen: + version 3.4

      +
      + +

      MPI: + version 3.1

      +
      + +

      PETSc: + version 3.15

      +
      + +

      CGAL: + version 5.4

      +
      + +

      PnetCDF: + Version 1.12

      +
      + +

      nlohmannjson: + Version 3.10

      +
      + +

      Scotch/ParMETIS

      +
      + +

      qcmesh: + Version 1.0

      +
      + +

      VTK: + Version 9.3

      +
      +
      +

      The dependency versions mentioned above are not strictly the + minimum versions required, but the ones which have been tested to work + well. The code quality of the project is analyzed in a CI pipeline + which runs inside the development docker + container + and on the ETH HPC cluster + Euler. + The pipeline covers the following checks:

      + + +

      Building the docker container and publishing it to + registry.gitlab.ethz.ch.

      +
      + +

      Building documentation and publishing it to + qc.mm.ethz.ch.

      +
      + +

      Code compilation and running unit/integration tests.

      +
      + +

      Presence of a license header for each source file.

      +
      + +

      Consistency of code formatting using + clang-format + for C++ and + ruff + for Python.

      +
      + +

      Static code analysis (linter) using + clang-tidy + for C++ and + ruff + for Python.

      +
      +
      +

      Detailed documentation of the API can be found on the + AQCNES + website, + and simple examples to get started can be found + here. + Importantly, the use of AQCNES is not limited + to these examples and the ones listed in this contribution.

      +
      + + Acknowledgements +

      The support from the European Research Council (ERC) under the + European Union’s Horizon 2020 research and innovation program (grant + agreement no. 770754) is gratefully acknowledged. We also acknowledge + contributions to the code from Stefan Zimmerman and Anny Wang.

      +
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