1. A New Approach in Processing Atomistic Simulations
L. Grinberg and G. E. Karniadakis
Presented by B. Caswell
Division of Applied Mathematics
Brown University
Providence, RI
Research is funded by NSF (OCI ), Computations have been performed at TACC and NICS.

2. Objective
We consider atomistic simulations of stationary and non-stationary 3D flow with molecular dynamic (MD) and dissipative particle dynamic (DPD) methods
One of the objectives in processing simulation data is to separate accurately the solution into the ensemble average and fluctuation components
We propose application of Window Proper Orthogonal Decomposition (WPOD) for analysis of data produced in atomistic simulations

3. Computed solution Ensemble average fluctuations ΩP
Add a scetch
Standard approach to compute an ensemble average solution works, but it typically requires long time integration and is not appropriate for non-stationary flows

4. Proper Orthogonal Decomposition (POD*)
Eigenvalues and Eigenvectors of C
Field V(t,x) can be decomposed into orthogonal temporal and spatial basis functions.
The temporal functions are obtained from the correlation matrix C. The lower POD modes represent events with high correlation length.
*Chatterjee A., An introduction to the proper orthogonal decomposition, Current Science, 2000.

5. Application of POD to non-stationary flow simulation: Window-POD (WPOD
POD is performed over a set of M snapshots. Each snapshot is computed by spatio-temporal averaging of velocity components over short time-intervals
POD window
Solution can be reconstructed at any time within the (sliding) POD window
*L. Grinberg et. al. Analyzing Transient Turbulence in a Stenosed Carotid Artery by Proper Orthogonal Decomposition, Ann. Biomed.l Eng., 2009.

6. DPD: Application of WPOD to stationary flow simulation
We consider a flow in a pipe driven by a constant force Fx, and perform DPD simulation.
To construct a snapshot, velocities are sampled over some (short) time period Δτ at points xp using the standard spatio-temporal averaging.
POD analysis based on 40 snapshots
Delta tau ~ 50 to 250 time steps, explain axis

7. DPD: Application of WPOD to non-stationary flow simulation
We consider a flow in a pipe driven by a time-varying force Fx(t), and perform DPD simulation.
To construct a snapshot, velocities are sampled over some (short) time period Δτ=50Δt at points xp using the standard spatio-temporal averaging.
Delta tau ~ 50 to 250 time steps
Δτ=50Δt
Δt=0.01 M=160

8. DPD: Application of WPOD to non-stationary flow simulation
Cross flow derivative for WSS
Check the coordinates
Computed with WPOD ensemble average and it derivative are compared to exact data.
The derivative is computed numerically using P=1 finite element discretization.

9. DPD: Application of WPOD to non-stationary flow simulation
PDF of fluctuations (u’)
L-2 error: computed with WPOD ensemble average and it derivative are compared to exact data
Npod = one snapshot data (standard averaging over time \Delta \tau
W’ instead of u’

10. MD: Application of WPOD to non-stationary flow simulation
We consider a flow between two plates driven by a time-varying force Fx(t), and perform MD simulation.
Flow between two plates and two obstacles. Flow is from left to right (in +x direction), simmulations have been performed with LAMMPS
M=80
Δτ=500Δt
Δt=0.0025

11. MD: Application of WPOD to non-stationary flow simulation
Flow between two plates and two obstacles. Flow is from left to right (in +x direction)

12. DPD: Application of WPOD to non-stationary flow simulation

13. DPD: Application of WPOD to non-stationary flow simulation
Add Nts, M, dt
M=160
Δτ=250Δt
Δt=0.001

14. DPD: Application of WPOD to non-stationary flow simulation

15. Summary and future plans
We propose to employ window proper orthogonal decomposition (WPOD) for processing data from atomistic simulations.
The WPOD has been applied in stationary and non-stationary 3D simulations with molecular dynamic and dissipative particle dynamic methods.
The new approach seems to be robust in separating the velocity field into ensemble average and fluctuation components.
We plan to employ the WPOD in coupled continuum-atomistic simulations to preprocess the data for interface conditions in multi-scale simulations.