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

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

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

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 c1 c2 c* Lk. Arrowhead, Dec. 13, 2005

Homogenization for Bubbly Liquids Sound speed Water 1500 m/s Air 350 m/s Bubbly liquid 50 m/s Lk. Arrowhead, Dec. 13, 2005

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

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

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

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

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

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

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

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

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

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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