From atoms to the macroscale: the nonlocal quasicontinuum method
Various computational techniques have been established that describe and predict the mechanics and physics of materials on many different length and time scales, from atomistic methods such as molecular statics and dynamics all the way up to the largest length scale where continuum mechanics and the finite element method are powerful modeling tools. Each method is appropriate for a particular, limited range of length and time scales.
Figure: computational and experimental techniques for a variety of length and time scales.
Unfortunately, many recent technologies fall into a gap between length scales, where none of the aforementioned methods is applicable alone. Shown below are such examples which include nanoporous or nanocrystaline metals (where microstructural features are on the order of a few nanometers, thus requiring atomistic resolution, but representative volumes are too large to be modeled by molecular dynamics). Further examples are micro- and nano-electro-mechanical systems (MEMS and NEMS, respectively) whose feature sizes may require atomistic accuracy but the overall components are too large for MD. Finally, structural solids as the nanolattices have an enormous potential, yet their inelastic deformation and failure mechanisms fall right into the same gap between atomistics and the continuum.
Figure: technologies falling into the gap between atomistic techniques and continuum methods.
Therefore, we have developed tools that bridge across scales from atomistics to the continuum, by merging techniques of molecular statics and dynamics with those of the much larger continuum scale. The ultimate goal is a computational toolbox that enables us to understand the basic microstructural mechanisms in crystalline solids, including dislocation interactions, dislocation emission, adsorbtion and transmission at grain boundaries, dislocation-vacancy interactions, grain boundary mechanisms, slip-twinning interactions, ductile failure arising from nanovoid nucleation and coalescence, and the influence of free surfaces, to name but a few. The focus of our research lies equally on the development of efficient modeling tools and on the application of such tools to study the deformation behavior of metals and ceramics.
Figure: summation rules for the quasicontinuum method (small open circles are lattice sites, solid black circles denote representative atoms, and large open circles are sampling atoms, as described below).
Nonlocal QC with Optimal Sampling Rules
The quasicontinuum (QC) method coasre-grains a crystalline atomistic ensemble by the selection of a small number of representative atoms (or repatoms, these carry degrees of freedom) and by interpolating the positions and motion of all remaining lattice sites from the set of repatoms. To this end, we have explored both the traditional affine and a new meshless interpolation schemes. A critical challenge lies in the calculation of the total energy of the atomic ensemble, which is required for the determination of forces on repatoms and resulting atomistic motion. Instead of summing over all lattice sites, we employ summation or sampling rules which approximate the total Hamiltonian by a weighted sum over a small set of sampling atoms, whose selection process is crucial. Choosing many such sampling atoms increases the accuracy but destroys the efficiency; choosing very few sampling atoms leads to high efficiency but looses the sought atomistic accuracy. Shown above are various traditional summation rules.
Figure: non-physical residual force artifacts appearing in coarse-grained meshes in the undeformed configuration because of a non-uniform discretization.
Summation or sampling rules not only introduce the obvious approximation errors but they also give rise to non-physical force artifacts in non-uniform QC meshes. While a perfect crystal in the undeformed ground state should be free of net forces on any of its interior atoms, the coarse-grained domain seemingly experiences such forces (even in the undeformed ground state), which are non-physical and artifacts of the sampling rule used to approximate the eneryg of the coarse-grained ensemble. The figures above show such residual forces arising in the undeformed ground state of a coarse-grained QC mesh from different sampling rules. We have introduced a new suite of sampling rules which reduce force artifacts to a minimum by carefully selecting sampling atoms and associated weights in a systematic fashion. The leftmost image above (new scheme of 1st order) shows the residual force artifacts when using the proposed new sampling rules - although not zero everywhere, they are orders of magnitude smaller than in all other schemes, and they vanish almost everywhere not only in the undeformed ground state but also in any affinely-deformed configuration in centrosymmetric lattices.
Figure: relative energy and displacement errors per atom in a crystal non-uniformly coarse-grained by the QC method using different summation rules (including two of our new summation rules).
We have carried out a comprehensive simulation benchmark campaign to study the influences of the approximation errors and force artifacts from the various summation rules. The above graphics show the errors arising in an example boundary value problem: uniaxial extension of a cube with a spherical nano-void at its center and coarse-grained by the QC method. Obviously, the errors introduced by using our new sampling rules are (like the force artifacts) orders of magnitude smaller than those of many comparable schemes. The new sampling rules also provide excellent capabilities to account for the relaxation effects of free surfaces.
Figure: snapshots of a nanoindentation simulation (a spherical indenter is punched quasistatically into a copper single-crystal); shown are atoms with their centrosymmetry parameter which identifies dislocations underneath the indentation site.
We have developed a fully-nonlocal 3D quasicontinuum scheme based on the new sampling rules, which can simulate, among others, the evolution of microstructural defects with locally atomistic accuracy. To this end, the model uses techniques of mesh adaptation to refine the QC mesh locally where required (e.g. around a moving dislocation). The QC scheme has been implemented in a massively-parallel, cache-aware code to make optimal use of high-performance computing resources. Our code can easily handle millions of representative atoms (in turn representing many orders of magnitude more actual lattice sites). Among others, we have priority access to an in-house cluster (partly own by our group), which we use for simulation runs. The below image shows an example QC mesh before and after indentation with a spherical indenter.
Figure: QC representation before and after indentation with a spherical indenter (in two dimensions).
The scale-bridging character of the deformation of crystalline solids requires full-physics models that are based on an accurate description on the lowest (atomic) scale while studying the behavior of bodies and systems on much bigger scales. In particular, under extreme conditions such as those commonly found in space applications, predictive full-physics models are essential to understand deformation and failure mechanisms and to ensure the safe design of technological components. The QC method is one such approach that replaces the phenomenological models commonly found in continuum mechanics by representations that are solely based on interatomic potentials. This allows for simulations with predictive capabilities.
Movie: adaptive remeshing of a 3D nanocube modeled by our fully-nonlocal QC scheme during nanoindentation with a spherical indenter coming in from above.
One of the key components of any coarse graining technique is the ability to adaptively change the mesh size and to deploy full atomistic resolution where and when it is required in a simulation. This enables us to set up the initial configuration of a simulation with as few representative atoms as possible and thereby increase computational efficiency. We have developed model adaptivity in both 2D and 3D versions of the non-local QC formulation, which includes both automatic neighborhood updates and mesh refinement.
Figure: example of automatic mesh adaptivity in 2D for a dynamic void growth simulation in a copper single-crystal (shown are all sampling atoms color-coded by centrosymmetry to highlight dislocations with progressing model refinement).
As 2D simulations are less computationally demanding than 3D ones (e.g., with respect to memory requirements), we can employ a global remeshing technique to solve mesh refinement in 2D as a first step. At every refinement step, we identify elements that are to be refined and add additional repatoms to the simulation by a longest-edge bisection algorithm. Once we have inserted new repatoms, the previous mesh is discarded and a new mesh is generated using the locations of the rep atoms as nodes.
Movie: adaptive remesh refinement during a 2D nanoindentation test (the top left region of each image shows the mesh, the right region the sampling atoms color-coded by centrosymmetry, and the bottom left region all types of sampling atoms (using the second-order optimal summation rule).
For massively-parallel 3D simulations, we are currently developing a local remeshing algorithm. Localizing the region of refinement allows us to limit sections of the simulations during a refinement step. The basis of our remeshing technique a discrete version of the Constrained Advancing Front (CAF) algorithm.
Polycrystals and Defect Interactions
To study deformation mechanisms of the various nano-structures mentioned above, an important computational tool needed is the ability to study polycrystals. Our QC implementation is equipped to accurately model grains and grain boundaries with full atomistic resolution near the grain boundary and coarsening away from it. Using this tool, we simulate polycrystals in both 2D and 3D. Coincident Site Lattice (CSL) based grain boundaries are commonly studied using atomistic modeling and observed experimentally. These are planar grain boundaries where the grains are mis-oriented at specific angles that gives rise to coincidence between some of the atoms of the different grains at the interface. This gives rise to interesting grain boundary properties which in turn affect the bulk material behavior. We use the fully non-local QC method to study dislocation motion and interactions in CSL grain boundaries at scales that are beyond the reach of classical atomistic techniques.
Figure: examples of CSL bicrystal simulations using the 3D fully-nonlocal QC method.
In addition to our 3D QC implementation that uses affine interpolation, we have developed a novel meshless QC framework based on local maximum-entropy interpolation schemes. Here, finite element interpolation is replaced by nonlocal shape functions which allows for improved adaptation functionality (but also comes with increased computational expenses). Overall, we have a suite of versatile atomistic-to-continuum-coupling techniques (all developed and implemented in-house in our own code) which are being extended to even more complex conditions and applied to a variety of technological and scientific applications from nano-electronics to novel nanoscale materials.
You can watch online a three-hour recording of a presentation given by Prof. Kochmann at RWTH Aachen (Germany) as part of its EU Regional School. The presentation discusses atomistic-to-continuum coupling techniques with a particular focus on the QC method, its challenges and opportunities, as well as example applications and open questions (please note that the recording is from 2014). More recent QC-related recordings include Prof. Kochmann's plenary lectures given at COMPLAS 2015 and PARTICLES 2015.
- I. Tembhekar, J. S. Amelang, L. Munk, D. M. Kochmann. Automatic adaptivity in the fully-nonlocal quasicontinuum method for coarse-grained atomistic simulations, Int. J. Numer. Meth. Engng., pre-published online (September 28, 2016).
- J. S. Amelang, G. N. Venturini, D. M. Kochmann. Summation rules for a fully-nonlocal energy-based quasicontinuum method, J. Mech. Phys. Solids 82 (2015), 378-413.
- J. S. Amelang, D. M. Kochmann. Surface effects in nanoscale structures investigated by a fully-nonlocal energy-based quasicontinuum method, Mech. Mater. 90 (2015), 166-184.
- D. M. Kochmann, G. N. Venturini. A meshless quasicontinuum method based on local maximum-entropy interpolation, Mod. Sim. Mat. Sci. Eng. 22 (2014), 034007.
- M. Espanol, D. M. Kochmann, S. Conti, M. Ortiz. A Γ-convergence analysis of the quasicontinumm method, Multiscale Model. Simul. 11 (2013), 766–794.
- J. S. Amelang, G. N. Venturini, D. M. Kochmann. Microstructure evolution during nanoindentation by the quasicontinuum method, Proc. Appl. Math. Mech. 13 (2013), 553-556.
- D. M. Kochmann, J. S. Amelang, M. I. Espanol, M. Ortiz. From atomistics to the continuum: a mesh-free quasicontinuum formulation based on local max-ent approximation schemes, Proc. Appl. Math. Mech. 11 (2011), 393-394.
We gratefully acknowledge the support from the National Science Foundation (NSF) through award CMMI-123436 as well as (previous) support form the National Nuclear Security Administration (U.S. Department of Energy) through Caltech's PSAAP center.