Atomistic-to-Continuum Dynamics

Mitchell Luskin

School of MathematicsUniversity of Minnesota

June 28, 2012

Collaborators W.-K. Kim (UMN), C. Le Bris (CERMICS), T. Lelievre(CERMICS), C. Ortner (Warwick), D. Perez (LANL), A.Shapeev (UMN), G. Simpson (UMN), E. Tadmor(UMN), A.F. Voter (LANL)

Brittle Fracture

The accurate and efficient computation of fracture requirescoarse-grained dynamics as well as coarse-grained space.

[Kermode, Albaret, Sherman, Bernstein, Gumbsch, Payne, Csanyi, deVita; Low speed fracture instabilities in a brittle crystal, Nature, 2008]

Coarse-grained Dynamics

Assume that we are given a smooth mapping

S : Rd → N

which to a configuration in Rd associates a state number (e.g., anumbering of the wells of the potential E , which can be the position ofa defect).

The goal of coarse-grained dynamics is to generate very efficiently atrajectory (St)t≥0 which has (almost) the same law as (S(Xt))t≥0,where Xt can be given, for example, by Langevin dynamics:{

dXt = M−1Pt dt,

dPt = −∇V (Xt) dt − γM−1Pt dt +√

2γβ−1dWt .

The Potential of Mean Force

The Potential of Mean Force EPMF (yA, θ) reproduces theequilibrium properties of observables O(yA) at temperature θthat depend only on the positions yA in an atomistic region A :

EPMF (yA, θ) = −kBθ ln

[∫Ce−βE(y

a,y c ) dy c].

We need to approximate this high-dimensional integral.

The Hot-QC Energy

The Hot-QC Energy (Dupuy, Tadmor, Legoll, Miller, and Kim)approximates EPMF (yA, θ) by using a local harmonicapproximation and Cauchy-Born coarse-graining in the continuumregion C :

EQC (y , θ) :=∑j∈AEaj (y) +

∑T∈T

vT W (∇y |T , θ).

Accelerated Dynamics (Voter)

Hyperdynamics utilizes a bias potential to accurately acceleratethe state-to-state dynamics of rare events such as defect motion.

The ratio of any two escape rates out of A depends only on thedividing surfaces for these paths, where the bias potential isrequired to be zero; thus hyperdynamics correctly acceleratesescapes in Hot-QC.

Hyper-QC Model Problem(Kim, Tadmor, Luskin, Perez, Voter)

Lennard-Jones potential with one weak bond, Langevin thermostat(θ=0.03), NN interaction, strain=0.00375, # of atoms=1000.

Hot-QC Accuracy and Efficiency

15

Result(2) – TST Time (All-Atom Model vs. Hot-QC)

Shorter chains consisting of all atoms and subject to the same strain.

Hot-QC models even with only 100 atoms can reproduce the TST time of the original system (1000 atoms), but all-atom models with a smaller number of atoms do not!

Hot-QC models even with only 100 atoms can reproduce the transitionstate time (TST) of the original system (1000 atoms), but all-atommodels with a smaller number of atoms do not!

The Bias Potential and Boost Factor

1 1.1 1.2 1.3 1.4

Weak Bond Length0

0.02

0.04

0.06

0.08

0.1

0.12

Pote

ntia

l Ene

rgy

Bias Potential

Vmax

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.

Vmax

0

5

10

15

20

25

Boo

st F

acto

r

Fully-Atomistic ModelHot-QC Dynamic Model (N

atom = 100)

Boost Factor(Temperature = 0.03, Strain = 0.00375)

Hyper-QC Accuracy and Efficiency

(Clock Time)

Hyper-QC method can reproduce the original TST time with both asmaller number of atoms and shorter simulation times.

Accelerated Dynamics References

[1] Arthur F. Voter. Parallel replica method for dynamics ofinfrequent events. Phys. Rev. B, 57(22):13985–13988, Jan 1998.

[2] D. Perez, B.P. Uberuaga, Y. Shim, J.G. Amar, and A.F. Voter.Accelerated molecular dynamics methods: introduction andrecent developments. Annual Reports in ComputationalChemistry, 5:79–98, 2009.

[3] Claude Le Bris, Tony Lelievre, Mitchell Luskin, and Danny Perez.A mathematical formalization of the parallel replica dynamics.Monte Carlo Methods Appl., to appear. arXiv:1105.4636.

[4] Gideon Simpson and Mitchell Luskin. Numerical Analysis OfParallel Replica Dynamics. arXiv:1204.0819v2.

KITP Lecture and Discussion

Kavli Institute for Theoretical Physics, UCSB

Physical Principles of Multiscale Modeling, Analysis and Simulation inSoft Condensed Matter

Theory and Computation of Parallel Replica Dynamics

June 7, 2012

http://online.kitp.ucsb.edu/online/multiscale c12/luskin/

Hot-QC and HyperQC References

[1] L. M. Dupuy, E. B. Tadmor, F. Legoll, R. E. Miller, and WooKyun Kim. Finite temperature quasicontinuum, in preparation.

[2] Woo Kyun Kim, Ellad Tadmor, Mitchell Luskin, Danny Perez, ArtVoter. Accelerated Finite-Temperature QuasicontinuumSimulations Using Hyperdynamics, in preparation.