We are interested in predic-ng the outcomes of long -me-‐scale thermal & diffusion processes with atomic detail, e.g. -‐ [Wagner et al, CMAME,2008] -‐ [Templeton et al,MSMSE,2010] We have developed methods for -me integra-on: -‐ frac-onal-‐step integra-on -‐ mul-ple -me-‐scale integra-on -‐ projec-ve integra-on as well as parallel implementa-ons of parallel replica dynamics, temperature accelerated dynamics, etc.
Time-‐scale issues in molecular dynamics-‐finite element coupling applica5ons R. Jones, J. Templeton, G. Wagner
Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. SAND No. 2011-XXXXP
Mul-ple Time-‐scale Integra-on
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1400
0 10 20 30 40 50 60 70 80 90 100
TE
MP
ER
AT
UR
E
TIME [ps]
LEFT RESERVOIRTUBE
RIGHT RESERVOIR
Two temperature model:
Assume short & long -mescales:
Expand and subs-tute:
Result : electrons interact with filtered/long -me-‐scale phonon evolu-on
Accelerated MD Methods
Accelerated MD methods focus on infrequent events that drive dynamics (e.g. atom-hopping) • Ignore long periods of thermal
vibration within energy basins
Saddle point
Local Minimum
Projec-ve Integra-on To model surface diffusion with the Equa-on-‐Free Method [I. Kevrekidis et al, 2003], fine/coarse scale descrip-ons are used to extend to longer -mescales
LIFT RESTRICT
MICRO-‐ SIMULATOR
tΔ
h(x,t+Δt) h(x,t)
Approximate: ( ) ( )( )1h h t t h tt t∂
≈ + Δ −∂ Δ
Project: ( ) ( ) hh t T h t Tt∂
+ Δ = + Δ∂
An Improved Li^ Operator: The Maximum Entropy Method
The lift operator must preserve higher-order system statistics (like spatial correlations) to replicate dynamics but this is problem specific & difficult in higher dimensions The goal is to constrain only the coarse scale values of interest, but allow other variables to come into correct equilibrium
Given “goal” average profile shape (top), generate many individual realiza-ons (bocom) that, when ensemble averaged, reproduce the goal profile while also preserving the correct dynamics (right).
Frac-onal Step Integra-on
6
�
J
∂
∂t(MIJθJ) =
�
J
KIJθJ + 2�
α
NIαvα ·�fMDα +
1
2fλα
�
mαv̇α = fMDα − mα
2
�
I
NIαλIvα
Atoms contribute to nodal heat equa-on
Heat at nodes affects MD energy through a spa-al thermostat
Gear update for FE dynamics, predictor/corrector requires:
Velocity-‐Verlet update for MD, two-‐step update requires:
7
θ̇FE = M−1IJ KIKθK
v̇α = m−1α fMD
α
∆θI = M−1IJ
�
α
NIαeα∆Vα
��tn+1 −M−1
IJ
�
α
NIαeα∆Vα
��tn
NI∆EI =�
α
NIα
�∆tvα · fα +
∆t2
2m−1
α fα · fα + φ�α
��tn+1 − φ�
α
��tn
�
tn
tn+1/2
tn+1
tn+1,∗
∆θFE
∆θMD
∆vα
∆λ
Update Lagrange mul-pliers a^er the predic-on phase by solving the non-‐linear equa-on:
8
∆t�
α
NαI Kα
�
JNα
J λJ − ∆t2
4
�
α
NαI Kα
��
J
NαJ λJ
���
K
NαKλK
�= Rc
I
∆tλ < 1
d
dtNIα =
d
dtNI (xα) = ∇NI · vα
7
θ̇FE = M−1IJ KIKθK
v̇α = m−1α fMD
α
∆θI = M−1IJ
�
α
NIαeα∆Vα
��tn+1 −M−1
IJ
�
α
NIαeα∆Vα
��tn
NI∆EI =�
α
NIα
�∆tvα · fα +
∆t2
2m−1
α fα · fα + φ�α
��tn+1 − φ�
α
��tn
�
tn
tn+1/2
tn+1
tn+1,∗
∆θFE
∆θMD
∆vα
∆λ
Apply resul-ng force at three -me increments: , , and
7
θ̇FE = M−1IJ KIKθK
v̇α = m−1α fMD
α
∆θI = M−1IJ
�
α
NIαeα∆Vα
��tn+1 −M−1
IJ
�
α
NIαeα∆Vα
��tn
NI∆EI =�
α
NIα
�∆tvα · fα +
∆t2
2m−1
α fα · fα + φ�α
��tn+1 − φ�
α
��tn
�
tn
tn+1/2
tn+1
tn+1,∗
∆θFE
∆θMD
∆vα
∆λ
7
θ̇FE = M−1IJ KIKθK
v̇α = m−1α fMD
α
∆θI = M−1IJ
�
α
NIαeα∆Vα
��tn+1 −M−1
IJ
�
α
NIαeα∆Vα
��tn
NI∆EI =�
α
NIα
�∆tvα · fα +
∆t2
2m−1
α fα · fα + φ�α
��tn+1 − φ�
α
��tn
�
tn
tn+1/2
tn+1
tn+1,∗
∆θFE
∆θMD
∆vα
∆λ
7
θ̇FE = M−1IJ KIKθK
v̇α = m−1α fMD
α
∆θI = M−1IJ
�
α
NIαeα∆Vα
��tn+1 −M−1
IJ
�
α
NIαeα∆Vα
��tn
NI∆EI =�
α
NIα
�∆tvα · fα +
∆t2
2m−1
α fα · fα + φ�α
��tn+1 − φ�
α
��tn
�
tn
tn+1/2
tn+1
tn+1,∗
∆θFE
∆θMD
∆vα
∆λ
8
∆t�
α
NαI Kα
�
JNα
J λJ − ∆t2
4
�
α
NαI Kα
��
J
NαJ λJ
���
K
NαKλK
�= Rc
I
∆tλ < 1
d
dtNIα =
d
dtNI (xα) = ∇NI · vα
7
θ̇FE = M−1IJ KIKθK
v̇α = m−1α fMD
α
∆θI = M−1IJ
�
α
NIαeα∆Vα
��tn+1 −M−1
IJ
�
α
NIαeα∆Vα
��tn
NI∆EI =�
α
NIα
�∆tvα · fα +
∆t2
2m−1
α fα · fα + φ�α
��tn+1 − φ�
α
��tn
�
tn
tn+1/2
tn+1
tn+1,∗
∆θFE
∆θMD
∆vα
∆λ
7
θ̇FE = M−1IJ KIKθK
v̇α = m−1α fMD
α
∆θI = M−1IJ
�
α
NIαeα∆Vα
��tn+1 −M−1
IJ
�
α
NIαeα∆Vα
��tn
NI∆EI =�
α
NIα
�∆tvα · fα +
∆t2
2m−1
α fα · fα + φ�α
��tn+1 − φ�
α
��tn
�
tn
tn+1/2
tn+1
tn+1,∗
∆θFE
∆θMD
∆vα
∆λ
7
θ̇FE = M−1IJ KIKθK
v̇α = m−1α fMD
α
∆θI = M−1IJ
�
α
NIαeα∆Vα
��tn+1 −M−1
IJ
�
α
NIαeα∆Vα
��tn
NI∆EI =�
α
NIα
�∆tvα · fα +
∆t2
2m−1
α fα · fα + φ�α
��tn+1 − φ�
α
��tn
�
tn
tn+1/2
tn+1
tn+1,∗
∆θFE
∆θMD
∆vα
∆λ
7
θ̇FE = M−1IJ KIKθK
v̇α = m−1α fMD
α
∆θI = M−1IJ
�
α
NIαeα∆Vα
��tn+1 −M−1
IJ
�
α
NIαeα∆Vα
��tn
NI∆EI =�
α
NIα
�∆tvα · fα +
∆t2
2m−1
α fα · fα + φ�α
��tn+1 − φ�
α
��tn
�
tn
tn+1/2
tn+1
tn+1,∗
∆θFE
∆θMD
∆vα
∆λ
7
θ̇FE = M−1IJ KIKθK
v̇α = m−1α fMD
α
∆θI = M−1IJ
�
α
NIαeα∆Vα
��tn+1 −M−1
IJ
�
α
NIαeα∆Vα
��tn
NI∆EI =�
α
NIα
�∆tvα · fα +
∆t2
2m−1
α fα · fα + φ�α
��tn+1 − φ�
α
��tn
�
tn
tn+1/2
tn+1
tn+1,∗
∆θFE
∆θMD
∆vα
∆λ
7
θ̇FE = M−1IJ KIKθK
v̇α = m−1α fMD
α
∆θI = M−1IJ
�
α
NIαeα∆Vα
��tn+1 −M−1
IJ
�
α
NIαeα∆Vα
��tn
NI∆EI =�
α
NIα
�∆tvα · fα +
∆t2
2m−1
α fα · fα + φ�α
��tn+1 − φ�
α
��tn
�
tn
tn+1/2
tn+1
tn+1,∗
∆θFE
∆θMD
∆vα
∆λ
7
θ̇FE = M−1IJ KIKθK
v̇α = m−1α fMD
α
∆θI = M−1IJ
�
α
NIαeα∆Vα
��tn+1 −M−1
IJ
�
α
NIαeα∆Vα
��tn
NI∆EI =�
α
NIα
�∆tvα · fα +
∆t2
2m−1
α fα · fα + φ�α
��tn+1 − φ�
α
��tn
�
tn
tn+1/2
tn+1
tn+1,∗
∆θFE
∆θMD
∆vα
∆λ
7
θ̇FE = M−1IJ KIKθK
v̇α = m−1α fMD
α
∆θI = M−1IJ
�
α
NIαeα∆Vα
��tn+1 −M−1
IJ
�
α
NIαeα∆Vα
��tn
NI∆EI =�
α
NIα
�∆tvα · fα +
∆t2
2m−1
α fα · fα + φ�α
��tn+1 − φ�
α
��tn
�
tn
tn+1/2
tn+1
tn+1,∗
∆θFE
∆θMD
∆vα
∆λ
Sintered Pd
M. Ong, Sandia
Coarsening after heating to 200 C for 5 hr
GaN nanowire with top contacts A. Talin, Sandia
Photoluminescence and Joule heating under power
Methods have been implemented in Sandia’s LAMMPS [S.Plimpton,, A. Thompson, P. Crozier] code: • Nudged Elastic Band (NEB), Parallel Replica Dynamics (PRD),
Temperature Accelerated Dynamics (TAD) • Innovations in parallelization of methods • New features can be used with any LAMMPS interatomic potential • Open source release to broad research community See http:/lammps.sandia.gov (and http://www.sandia.gov/~sjplimp/spparks.html)
MD/FE thermal coupling requires a stable interleaving of integrators for the molecular and coarse scale dynamics
Given the rela-ve thermal proper-es of electrons and phonons, an efficient coupling scheme needs to operate on two disparate -mescales
Laser hea-ng of a CNT connect to thermal reservoirs
[R.Jones et al, IJNME, 2010] [J. Templeton et al, in prepara5on]
(1) Use fine-‐scale simula-ons as “computa-onal experiments” to get approximate course-‐scale -me deriva-ves (2) Project coarse scales in -me over longer -mestep, and reini-alize fine scales
[G. Wagner et al, Int. J. Mul*scale Comp. Eng., 2010] [J. Deng et al, in prepara5on]
