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Multiscale Simulation Methods

Multiscale Simulation Methods. Russel Caflisch Math & Materials Science Depts, UCLA. Outline. Simulations Equations often unavailable or cumbersome New multiscale strategies needed Perron-Cluster Cluster Analysis Automatically identifies metastable states Example of clustering algorithm

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Multiscale Simulation Methods

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

  2. 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 IPAM, Sept 13, 2005

  3. 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 IPAM, Sept 13, 2005

  4. Perron Vectors • Stochastic matrix T • Nonegative entries • Rows sum to 1 • Assume eigenvalue 1 has multiplicity k • Invariant sets • Invariant set of dimension di • Invariant measure -> eigenvector Xi with eigenvalue 1 • Matrix X=(X1 ,…, Xk ) • Characteristic “functions” • Let χi with χ=(0,…,0,1,…1,0,…0)t with di 1’s • For χi ∙ χj =0 for different i,j • Matrix of e-vectors χ=(χ1 ,…, χk ) • Coordinate transformation A • χ=XA IPAM, Sept 13, 2005

  5. PCCA • Stochastic matrix T • Nonegative entries • Rows sum to 1 • Assume k eigenvalues close to 1 • Nearly-invariant measures • eigenvector Xi with eigenvalue near 1 • Matrix X=(X1 ,…, Xk ) • 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 IPAM, Sept 13, 2005

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  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 IPAM, Sept 13, 2005

  9. 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 IPAM, Sept 13, 2005

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  11. Application of Equation-Free Multiscale Method • Diffusion in a random medium • Comparison to Monte Carlo solution IPAM, Sept 13, 2005

  12. 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 IPAM, Sept 13, 2005

  13. 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) IPAM, Sept 13, 2005

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

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

  16. 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 IPAM, Sept 13, 2005

  17. 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 IPAM, Sept 13, 2005

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

  19. 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 IPAM, Sept 13, 2005

  20. Thermalization Approximation • Spatially Homogeneous Problem • Wild expansion • fk 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 IPAM, Sept 13, 2005

  21. 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 IPAM, Sept 13, 2005

  22. 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 (+). IPAM, Sept 13, 2005

  23. 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 IPAM, Sept 13, 2005

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

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

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

  27. 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 IPAM, Sept 13, 2005

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