1. IPAM, Sept 13, 2005 Multiscale Simulation Methods Russel Caflisch Math & Materials Science Depts, UCLA

2. IPAM, Sept 13, 2005 Outline Simulations –Equations often unavailable or cumbersome –New multiscale strategies needed Perron-Cluster Cluster Analysis –Automatically identifies metastable states –Example of clustering algorithm Equation-Free Multiscale Method of Kevrekidis –Using simulation results to form approximate model on fine scale –Extend to coarse scale Interpolated fluid/Monte Carlo method for rarefied gas dynamics –Combine particle and continuum descriptions of gas in single hybrid method Conclusions

3. IPAM, Sept 13, 2005 Perron-Cluster Cluster Analysis (PCCA) Objective: Identify modes for reduced order description of a complex system –E.g. metastable states for bio molecule Method: clustering methods –Principal component analysis (PCA) Principal orthogonal decomposition (POD) –Independent component analysis (ICA) –PCCA Nonlinear method Similar to Laplace projection

4. IPAM, Sept 13, 2005 Perron Vectors Stochastic matrix T –Nonegative entries –Rows sum to 1 –Assume eigenvalue 1 has multiplicity k Invariant sets –Invariant set of dimension d i –Invariant measure -> eigenvector X i with eigenvalue 1 –Matrix X=(X 1,…, X k ) Characteristic “functions” –Let χ i with χ=(0,…,0,1,…1,0,…0) t with d i 1’s –For χ i ∙ χ j =0 for different i,j –Matrix of e-vectors χ=(χ 1,…, χ k ) Coordinate transformation A –χ=XA

5. IPAM, Sept 13, 2005 PCCA Stochastic matrix T –Nonegative entries –Rows sum to 1 –Assume k eigenvalues close to 1 Nearly-invariant measures –eigenvector X i with eigenvalue near 1 –Matrix X=(X 1,…, X k ) Find –transformation matrix A, –characteristic matrix χ –χ ≈ XA Robust algorithm for finding A, χ –Deuflhard, Dellnitz, Junge & Schutte (1999) –Deuflhard & Weber (2004) –Schutte to speak in Workshop IV Project dynamics onto subspaces given by A, to find reduced order approximation

6. IPAM, Sept 13, 2005

8. Equation-Free Multiscale Method For many processes, equations are not readily available –dynamics specified by an algorithm, difficult to write as a set of equations –Legacy computer code Multiscale modeling and simulation must proceed without use of equations –Method by Kevrekidis and co-workers –Kevrekidis to speak in Caltech workshop following Workshop IV

9. IPAM, Sept 13, 2005 Equation-free multiscale method Perform small number of fine scale simulations –computationally expensive Evolution of coarse-grained variables determined by projection –Sensitive to choice of coarse-grained variables –Polynomial expansion used –Perhaps PCCA would be of use

10. IPAM, Sept 13, 2005

11. Application of Equation-Free Multiscale Method Diffusion in a random medium Comparison to Monte Carlo solution

12. IPAM, Sept 13, 2005 Overview of RGD Rarefied gas dynamics (RGD) –RGD required when collisional effects are significant –Key step (i.e. computational bottleneck) in many material processing and aerospace simulations –Direct Simulation Monte Carlo (DSMC) is dominant computational method Boltzmann equation for density function f –ε = Knudsen number = mean free path / characteristic length scale Applications –Aerospace –Materials processing –MEMS

13. IPAM, Sept 13, 2005 Equilibrium and Fluid Limit of Boltzmann Maxwellian equilibrium –Q(f,f)=0 implies f=M(v;ρ,u,T) Equilibration –Consider f=f(v,t) spatially homogeneous –Entropy –Boltzmann’s H-theorem –H-theorem implies f →M as t →∞ Fluid Limit (Hilbert, Grad, Nishida, REC) –ε→0 –f(v,x,t)→ M(v;ρ,u,T), with ρ= ρ(x,t), etc. –ρ,u,T satisfy Euler (or Navier-Stokes)

14. IPAM, Sept 13, 2005 Rarefied vs. Continuum Flow: Knudsen number Kn

15. IPAM, Sept 13, 2005 Collisional Effects in the Atmosphere

16. IPAM, Sept 13, 2005 DSMC DSMC = Direct Simulation Monte Carlo –Invented by Graeme Bird, early 1970’s –Represents density function as collection of particles –Directly simulates RGD by randomizing collisions Collision v,w →v’,w’ conserving momentum, energy Relative position of v and w particles is randomized –Particle advection –Convergence (Wagner 1992) Limitation of DSMC –DSMC becomes computationally intractable near fluid regime, since collision time-scale becomes small

17. IPAM, Sept 13, 2005 Hybrid method IFMC=Interpolated Fluid Monte Carlo –Combines DSMC and fluid methods –Representation of density function as combination of Maxwellian and particles ρ, u, T solved from fluid eqtns, using Boltzmann scheme for CFD N = O(1- α) α = 0 ↔ DSMC α = 1 ↔ CFD –Remains robust near fluid limit

18. IPAM, Sept 13, 2005 IFMC for Spatially Homogeneous Problem Implicit time formulation Thermalization approximation Hybrid representation

19. IPAM, Sept 13, 2005 Implicit time formulation From Pareschi’s thesis –related to Bird’s “no time counter” (NTC) method Collision operator –Rewrite with constant negative term (“trial collision” rate) Implicit time formulation of Boltzmann –New time variable –Boltzmann equation becomes

20. IPAM, Sept 13, 2005 Thermalization Approximation Spatially Homogeneous Problem –Wild expansion –f k includes particles having k collisions Thermalization approximation –Replace particles having 2 or more collisions in time step dt by Maxwellian M –Resulting evolution over dt

21. IPAM, Sept 13, 2005 Hybrid Representation and Evolution Hybrid representation –g= {particles}, M=Maxwellian, β= equilibration coefficient Evolution equations –From thermalization approximation –Equations for β and g –g eqtn has Monte Carlo (DSMC) representation

22. IPAM, Sept 13, 2005 Relxation to Equilibrium Spatially homogeneous, Kac model Similarity solution (Krook & Wu, 1976) Comparison of DSMC(+) and IFMC(◊) At time t=1.5 (top) and t=3.0 (bottom). Number of particles (top) and number of collisions (bottom) for IFMC with dt=0.5(◊) and dt=1.0 (+).

23. IPAM, Sept 13, 2005 IFMC for Spatially Inhomogeneous Problem Time splitting Collision step as above –Because of disequilibrium from advection, start with β=0 Convection step: 2 methods –Move particles by their velocity –Move continuum part: 2 methods Direct convection of Maxwellians Use of Euler or Navier-Stokes equations for convection

24. IPAM, Sept 13, 2005 Computational Results Shock Leading Edge problem –Flow past half-infinite flat plate Flow past wedge

25. IPAM, Sept 13, 2005 Comparison of DSMC (blue) and IFMC (red) for a shock with Mach=1.4 and Kn=0.019 Direct convection of Maxwellians ρ u T

26. IPAM, Sept 13, 2005 Comparison of DSMC (contours with num values) and IFMC (contours w/o num values) for the leading edge problem. ρ T u v

27. IPAM, Sept 13, 2005 Conclusions and Prospects Hybrid method for RGD that performs uniformly in the fluid and near-fluid regime Applications to aerospace, materials, MEMS Current development –Generalized numerics, physics, geometry –Test problems