Math 215 Continuum-Molecular Modeling and Simulation Class meets Mo,We,Fr, 11:00-11:50 AM, Phillips 303, Fall Semester, 2004 Instructor: Sorin Mitran Office hours: Mo,We,Fr, 10:00-10:50 AM, Phillips 307 [Motivation] [Syllabus] [Grading] [Texts] [Homework] [Projects] [Mid-term] [Final] Motivation and objectives Typically physics furnishes the equations and boundary conditions describing the bulk behavior of matter. Mathematics then attempts to ascertain the existence of solutions and devise practical means to find these solutions. A great deal of success has traditionally been achieved in the sciences through this approach. The equations of fluid dynamics, elasticity, electrodynamics have all been linked to specific microscopic descriptions of matter. These equations have been studied and many methods have been found to find exact or approximate solutions. Yet, when we look at the materials encountered in our immediate surroundings we immediately find cases where the classical continuum level descriptions break down or are invalid. Complex fluids, solids with imperfections in their crystalline structure, plastic materials - all of these are outside the realm of the classical studies. The basic issue that arises again and again is that we do not know how to analytically pass from a realistic microscopic model of the material to a continuum level description. We shall investigate this issue in depth. First, we review some basic cases from classical physics to introduce the methods traditionally used to derive a continuum description from a molecular model. Then we shall present the basic difficulties faced in materials with complicated microscopic behavior. Finally we shall concentrate on computational approaches to solving these difficulties considering a number of practical situations. Syllabus Elementary kinetic theory Basic statistical physics Constitutive laws Specific models: solid rupture, plastic materials, polymer flows, biological systems Grading Policy Grading is based upon homework (20 points), midterm (10 points) and final (10 points) examinations and a final project (60 points). The homework is intended to verify comprehension of simple concepts. Each assignment shall contain 4 questions worth 0.5 grade points each. The weekly homework should require no more than 30 minutes to complete. The midterm and final examinations are given in the same vein - a simple verification of basic concepts. The projects form the most extensive coursework and shall require background research, numerical simulation and the drafting of a report. Grade points are translated to letter grades according to the following table: Clear Excellence H 81-100 points Entirely Satisfactory P 66-80 points Low Passing L 40-65 points Failed F 0-39 points Course Texts Lecture notes will be distributed periodically through this web site. The following texts are useful for background and more in depth discussions of the course material: Statistical Physics: Berkeley Physics Course, Vol. 5 (McGraw Hill) by F. Reif, 1967 Statistical Physics by L. Landau and D. Lifschitz, (Oxford) 1967 Kinetic theory : classical, quantum, and relativistic descriptions, by Richard Liboff, (Prentice Hall) Thermodynamics and Thermostatistics, Herbert Callen, 1985, Prentice Hall. Thermodynamics and Statistical Mechanics, W. Greiner, L. Heise, H. Stöcker, 1995, Springer A survey paper on multiscale computation by Achi Brandt: Multiscale Scientific Computation Technical reports from the Gauss Multiscale Computation Center of the Weizmann Institute SIAM Journal: Multiscale Modeling and Simulation Lecture notes Lecture Date Topic Notes Lecture Date Topic Notes 1, 8/25 A simple model of a string Lect01.pdf   2, 8/27 String rupture, C-M issues Lect02.pdf 3, 8/30 Dilute polymer flows Lect03.pdf  4, 9/1 Biological systems Lect04.pdf 5, 9/3 Boltzmann equation Lect05.pdf   9/8 Homework review   6, 9/10 Conservation equations Lect06.pdf  7, 9/13 Maxwell-Boltzmann distribution Lect07.pdf     8, 9/15 Chapman-Enskog expansion Lect08.pdf 9, 9/17 Analytical mechanics primer Lect09.pdf  10, 9/20 Liouville equation Lect10.pdf  11, 9/22 BBGKY hierarchy Lect11.pdf    9/24 Project ideas   12, 9/27 Thermodynamics Lect12.pdf 13, 9/29 Ideal gas, TD potentials Lect13.pdf 14,10/1 Statistical mechanics Lect14.pdf 15, 10/4 Multiscale computation       Homework Homework is given each Friday and due the next Friday. Solutions are posted at the end of the Due Date. Please download the homework from the website. HW # Due Problems Solution HW # Due Problems Solution 1 9/3 HW01.pdf 6 10/22 2 9/10 HW02.pdf 7 10/29 3 9/17 HW03.pdf 8 11/5 4 9/24 HW04.pdf 9 11/12 5 10/1 HW05.pdf 10 11/19   Projects Fracture mechanics Shell failure under dynamic bending - model project Shell failure under traction - model project Viscoelastic flow Mucus layer flow - Brandon Lindley Dilute polymer flow in a soap film - Nematic polymer films - Joohee Lee Biology Fungus growth - Eric Choate Subdiffusion Diffusion through tangled polymers - Zhi Lin Diffusion through regular lattices - Lingxing Yao Contact Lines Inmiscible fluids in porous media - James McClure Merging of contact lines - Xiaoyu Zheng Droplets Acoustic excitation and breakdown - model project