M / CAM 393C: Topics in Computational Modeling

Spring 2006 (57590)

General Information

Instructor: Oscar Gonzalez (og@math.utexas.edu) 471-1138 Office hrs: RLM 10.122, W 4-5 and by appointment Location: RLM 9.166, MWF 2-3pm Text: (none required) Web: http://www.ma.utexas.edu/~og/TEACHING/M393C/m393c.html Links: schedule, references

Description

This course will cover various topics in the theory of rigid bodies and elastic filaments with applications. The goal is to derive, analyze and interpret the fundamental equations in the theory of rigid bodies and elastic filaments, discuss applications of these equations in the mechanical modeling of DNA and other physical systems, and develop numerical methods for their approximation. The modelling of coupled problems such as the sedimentation of an elastic filament in an incompressible Stokes fluid and various optimal packing problems for filaments may also be discussed as time permits.

Audience

Graduate students in a variety of fields including math, physics, engineering and chemistry.

Prerequisites

Students will be expected to have a basic knowledge of linear algebra, multivariable calculus, differential equations and elementary physics. It is also expected that students will have some basic programming skills and some familiarity with mathematical software such as Matlab. Efforts will be made to accommodate students with different backgrounds.

Grading

No exams will be given. Course grades will be based entirely on homework sets assigned throughout the semester. Students who do not plan to do the homeworks should register on a Credit/No-Credit basis.

Tentative schedule

The following list is a tentative schedule for the course. The actual material to be covered will depend on student interest and time.

1. Vectors, frames, rotations in 3D

• vectors, components, index notation
  
• frames, rotations, the set SO_3
  
• parametrization of rotations
  
• kinematics of rotations, angular velocity
  

2. Modelling of rigid bodies

• configurations, kinematics
  
• momentum balance laws
  
• numerical methods
  
• steady-spinning states, stability
  
• equilibrium states
  
• chains of rigid bodies
  

3. Modelling of elastic filaments

• configurations, kinematics
  
• momentum balance laws
  
• elastic stress-strain laws
  
• material constraints
  
• numerical methods
  
• equilibrium states
  
• link, twist, writhe
  

4. Coupled fluid-body problems

• Stokes fluid equations
  
• sedimentation problem for a rigid body
  
• Stokes resistance matrices
  
• steady-states, stability
  
• sedimentation problem for an elastic filament
  

References

Vector algebra, rotations in 3D
• Vector analysis and cartesian tensors, D.E. Bourne and P.C. Kendall, Chapmann and Hall, 1992.
• Rational kinematics, J. Angeles, Springer, 1988.
• A survey of attitude representations, M.D. Shuster, J. Astronautical Sciences (1993) Vol 41 pp. 439-517.
Mechanics of a rigid body
• Classical mechanics, H. Goldstein, 2nd Edition, Addison-Wesley, 1980.
• Spacecraft attitude dynamics, P.C. Hughes, Wiley, 1986.
Steady-states, stability in ODEs
• Differential equations, dynamical systems and linear algebra, M.W. Hirsch and S. Smale, Academic Press, 1974.
• Differential equations and dynamical systems, L. Perko, Springer-Verlag, 2001.
Rigid body chains, DNA
• Theory of sequence-dependent DNA elasticity, B.D. Coleman et. al., J. Chem. Phys. (2003) Vol 118 pp.7124-7140.
• Understanding DNA, C.R. Calladine and H.R. Drew, Academic Press, 1997.
• DNA structure and function, R.R. Sinden, Academic Press, 1994.
Mechanics of a filament
• Nonlinear problems of elasticity, S.S. Antman, Springer, 1995.
Coupled fluid-body problems
• Dynamics of a rigid body in a stokes fluid, O. Gonzalez, A. Graf and J. Maddocks, J. Fluid Mechanics (2004) Vol 519 pp.133-160.
• Low reynolds number hydrodynamics, J. Happel and H. Brenner, Martinus Nijhoff Publishers, 1983.