1. What are Multiscale Methods?
Russel Caflisch
Mathematics & Materials Science Depts, UCLA
Lk. Arrowhead, Dec. 13, 2005

2. Scientific problems involving multiple length and time scales
Introduction
Scientific problems involving multiple length and time scales
Atomistic and continuum
Nanosystems
Protein folding
Numerical and analytical methods combining multiple length and time scales
Lk. Arrowhead, Dec. 13, 2005

3. Outline Intro Homogenization Multigrid Method Multiple Physics
Multiple length & time scales
Multiple physics
Homogenization
Electrical conductivity
Bubbly liquids
Multigrid Method
Numerical method for combining solution on different scales
Multiple Physics
Crack propagation – Quasi-continuum method
Molecular effects in fluids – HMM method, IFMC method
Legacy codes - Equation-free method
Conclusions
Lk. Arrowhead, Dec. 13, 2005

4. Homogenization for Electrical Conductivity
Conductivity c* of a wire made of two materials with conductivity c1 and c2
c* = <c-1>-1
<c-1> = (L1 c1-1 + L2 c2-1) /L
Averaging of c is harmonic, not arithmetic L
c c2 c*
Lk. Arrowhead, Dec. 13, 2005

5. Homogenization for Bubbly Liquids
Sound speed
Water m/s
Air m/s
Bubbly liquid 50 m/s
Lk. Arrowhead, Dec. 13, 2005

6. Sound speed Sound speed c
c = (mass density) -1 (compressibility) -1
Density and compressibility average arithmetically
c* = (<mass density>) -1 (<compressibility>) -1
Liquid → large mass density
Bubbles → large compressibility
c* is small for bubbly liquid
Lk. Arrowhead, Dec. 13, 2005

7. Multigrid Method Solution of elasticity on a numerical grid
Multigrid strategy
Construct grids at multiple resolution
Eliminate errors of wavelength k = r-1 on grid of size r
Lk. Arrowhead, Dec. 13, 2005

8. Multigrid Method Computational speed Accelerated computation
Fast on coarse grid
Slow on fine grid
Accelerated computation
Largest errors at small k, eliminated quickly on coarse grid
Easy to remove errors at large k, requires few find grid solves
Lk. Arrowhead, Dec. 13, 2005

9. Problems with Multiple Physics: Crack Propagation
Crack propagation is multiscale
Atomistic
Mesoscale
Continuum scale
Lk. Arrowhead, Dec. 13, 2005

10. Problems with Multiple Physics: Crack Propagation
Quasi-continuum method
Near crack tip use atomistic physics
Away from crack tip use continuum elasticity
Domain decomposition
Boundary between the two regions is sensitive
Crack and grain bdry
Miller, Tadmor, Phillips, Ortiz (1998)
Lk. Arrowhead, Dec. 13, 2005

11. Sensitivity to Boundary Conditions (BCs)
Waves propagation
Naïve BCs reflection off of interface between continuum and atoms
Correct BCs eliminate anomalous reflection
Anomalous wave reflection at interface for naïve BCs
No wave reflection at interface for correct BCs
Lk. Arrowhead, Dec. 13, 2005

12. Problems with Multiple Physics: Molecular Effects in Fluids
fluid/solid bdry
In regions where mean free path ~ characteristic length
HMM (Engquist & E)
Perform molecular simulations in localized regions to determine molecular effects
Solver fluid equations throughout, without inputs from molecular simulations
Molecular regions
Continuum grid
Lk. Arrowhead, Dec. 13, 2005

13. Problems with Multiple Physics: Molecular Effects in Fluids
IFMC (RC & Pareschi)
Represent molecular velocity distribution as combination of Maxwell-Boltzmann distribution and collection of particles
Simulate particles by Monte Carlo
Simulate fluid component by fluid mechanics
density
velocity
Lk. Arrowhead, Dec. 13, 2005

14. Problems with Multiple Physics: Legacy Codes
E.g. fluid simulations with complex equations of state
Details of physics and algorithms may be complex or forgotten
Equations free method
Coarse graining without equations
Kevrekidis and co-workers
Perform small number of fine scale simulations
computationally expensive
Extrapolate from fine scale to get coarse scale evolution
Lk. Arrowhead, Dec. 13, 2005

15. Lk. Arrowhead, Dec. 13, 2005

16. Conclusions
Multiscale methods needed for many current problems in science and technology
Mathematical methods capturing multiscale features are being developed and deployed
Many issues remain
When do these work?
How to quantify their accuracy?
How to handle rare events?
Lk. Arrowhead, Dec. 13, 2005