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What are Multiscale Methods?. Russel Caflisch Mathematics & Materials Science Depts, UCLA. Introduction. Scientific problems involving multiple length and time scales Atomistic and continuum Nanosystems Protein folding
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What are Multiscale Methods? Russel Caflisch Mathematics & Materials Science Depts, UCLA Lk. Arrowhead, Dec. 13, 2005
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 • 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 • 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 • Molecular effects • 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 • 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
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