An Introduction to Multiscale Modeling

1. An Introduction to Multiscale Modeling Scientific Computing and Numerical Analysis Seminar CAAM 699

2. Outline • Multiscale Nature of Matter • Physical Scales • Temporal Scales • Different Laws for Different Scales • Computational Difficulties • Homogeneous Elastic String • Inhomogeneous Elastic String • Overview of Seminar Topics

3. Physical Scales • Discrete Nature of Matter • Multiple Physical (Spatial) scales Exist • Example: River • Physical Scale: km = 103 m http://ak.water.usgs.gov/yukon/index.php

4. Physical Scales • Water Drops • Physical Scale: • mm = 10-3 m • Water Cluster • Physical Scale: • 5 nm = 5 x 10-9 m http://www.btinternet.com/~martin.chaplin/clusters.html http://eyeofthefish.org/leaky-leushke/

5. Physical Scales • Water Molecule • Physical Scale: 0.278 nm = 2.78 x 10-10 m http://commons.wikimedia.org/wiki/File:Water_molecule.png

6. Temporal Scales • Multiple Time Scales in Matter • Time Scale of Interest Depends on Phenomenon of Interest • Fluid Time Scales: • River Flow: hours • Rain Drop Falling: 30-60 min • Water Molecule Interactions: fractions of a second

7. Different Scales, Different Laws • Governing Equations different for different scales • Example: Modeling a Fluid: • River Flow: Navier-Stokes Equations • Interactions between fluid particles: Newton’s Molecular Dynamics • Atomic, Subatomic Description of Fluid Molecule: Schrödinger’s equations

8. Model Choice • Could represent river as discrete fluid particles, and utilize molecular dynamics to model its flow • More details included in the model, the more accurate your model will likely be • What’s the problem??? Good Luck trying to do this computationally!!!

9. Computational Difficulties • Number of elements • Smaller Spatial Scale may warrant a smaller time scale in order to keep numerical methods stable • Example: CFL number for hyperbolic PDEs

10. Model Choice • Balance detail and computational complexity • Choice often made to model a material as a continuum • Goal is to then find a constitutive law that can explain how the material behaves • If the material is homogeneous, the continuum assumption is typically acceptable and constitutive laws can be found • Heterogeneous materials are more difficult to model, and motivate the need for multiscale models

11. Homogeneous Elastic String • Discrete Scale: Mass-Spring system • point masses of mass • Springs between each mass have spring constant • In zero strain state, springs are length • Derive Equation for Longitudinal Motion

12. Homogeneous Elastic String • Let be the displacement of mass from its zero strain state at time • Equation of motion for mass can be written using Newton’s Law: • The can be written as

13. Homogeneous Elastic String • Forces felt by mass come from mass and mass • Net force on mass difference in forces from the left and right mass

14. Homogeneous Elastic String • Full equation for mass

15. Homogeneous Elastic String • Elasticity Modulus • Linear Mass Density • Take Limit as

16. Homogeneous Elastic String • 1D Wave Equation • Continuum-Level model, limit of the microscopic (discrete) model • Wave speed determined by does NOT depend on location in the string • Hyperbolic PDE easy to simulate

17. Inhomogeneous Elastic String • Discrete Model, Mass-Spring system • Number of springs between each point mass can vary

18. Inhomogeneous Elastic String • point masses of mass • Springs between each mass have spring constant • In zero strain state, springs are length • displacement of mass • = number of springs between mass and at time

19. Inhomogeneous Elastic String • Equation of Motion for mass

20. Inhomogeneous Elastic String • Equation of Motion for mass • Take Limit as

21. Inhomogeneous Elastic String • Wave equation with locally varying wave speed: • To solve this wave equation you need to know but this is a microscopic quantity! (Local density of springs) • Micro quantity needed in a continuum equation

22. Inhomogeneous Elastic String Put another way in the form of a constitutive law (relation between stress and strain) Dividing by h and taking the limit h  0

23. Inhomogeneous Elastic String Equating these two quantities gives: Utilizing: Elastic properties of the spring vary spatially

24. Practical Example • Rupturing String: • Assume springs break if the segment length for some distance • Microscopic Model: • Continuum Model:

25. Practical Example • Begin with string in zero strain state attached at one end to wall • This string is stretched at the other end by a constant strain rate • 11 point masses, 100 springs between each pair of masses • When distance between masses exceeds then springs break

26. 1D Rupturing String

27. 1D Rupturing String Ruptured Strings Force Displacement Time Steps

28. Challenges • General problems with trying to couple a microscopic and continuum model • Number of elements in the microscopic scale • Carrying out the microscopic model for full continuum level time scale • If you only do a spatial or time sample of the microscopic evolution, how would you represent the micro-state at a later point in time?

29. Overview of Seminar Topics • Interesting Medical/Biological Problems that would benefit from multiscale modeling • Models of the Cytoskeleton • Continuum Microscopic (CM) Methods • Probability Theory, PDF Estimation • CM modeling with statistical sampling • Solution Methods to Non-Linear Systems of Equations • And more…..

30. References • E W, Engquist B, “Multiscale Modeling and Computation”, Notices of the AMS, Vol 50:9, p. 1062-1070