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

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

Physical Scales Discrete Nature of Matter – Multiple Physical (Spatial) scales Exist – Example: River – Physical Scale: km = 10 3 m

Physical Scales Water Cluster Physical Scale: – 5 nm = 5 x m leushke/ aplin/clusters.html Water Drops Physical Scale: – mm = m

Physical Scales Water Molecule Physical Scale: nm = 2.78 x m Water_molecule.png

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: min – Water Molecule Interactions: fractions of a second

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

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!!!

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

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

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

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

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

Homogeneous Elastic String Full equation for mass

Homogeneous Elastic String Elasticity Modulus Linear Mass Density Take Limit as

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

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

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

Inhomogeneous Elastic String Equation of Motion for mass

Inhomogeneous Elastic String Equation of Motion for mass Take Limit as

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

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

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

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

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

1D Rupturing String

Displacement Force Ruptured Strings Time Steps

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?

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…..

References E W, Engquist B, “Multiscale Modeling and Computation”, Notices of the AMS, Vol 50:9, p