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-0904288), Computations have been performed at TACC and NICS.

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

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

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.

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.

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

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

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.

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’

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

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)

DPD: Application of WPOD to non-stationary flow simulation

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

DPD: Application of WPOD to non-stationary flow simulation

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.