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Computational Course Projects
and Undergraduate Research
B. K. Clark and Richard F. Martin, Jr.
Illinois State University
Contributors:
E. Rosa
D. Holland
R. Balfanz
N. Jurasek
Q. Su
R. Grobe
N. Nutter
B. Vleck
Resources: ISU Physics and its Peers
2004 - 2006
Institution
Top 10
ISU Physics
# of faculty
publications
per faculty
68
12
6.62
4.83
% of faculty
with publication
Top 10 Physics
64 %
83 %
average grant
amount
$ 569 K
$ 92 K
departments
Cal Tech, Harvard, Cornell, JHU, UC
Berkeley, NYU, Michigan, Duke, Stanford,
UIUC
From: “Chronicle of Higher Education” 1/12/2007
www.chronicle.com/stats/productivity
% of faculty
with grant
11 %
17 %
Undergraduate physics research at ISU
Nonlinear Dynamics
Nanoscience
Space Physics
Atomic, Molecular,
and Optical Physics
Biophysics
Annual Average Number of
Graduates 2002-2004
United States Air Force Academy 24
Harvey Mudd College 22
U. of Wisconsin – La Crosse 22
Illinois State University 20
Source: American Institute of Physics
ISU Computer Physics Sequence
1998-2007
Total graduates 35
Graduates per year 4
Computation Research Mentors 9
Advanced Computational Physics
Modules 7 (3 per year)
Number of physics graduates from 1980 to present
30
CPY: Computer
physics
25
20
PHY: Physics
ENG: 3/2
program
15
10
5
0
19
80
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PTE: Physics
Teaching
C PY
PTE
ENG
PHY
Computer Physics Curriculum
Frontiers in Physics
Physics for Scientists and Engineers I
Physics for Scientists and Engineers II
Physics for Scientists and Engineers III
Methods of Theoretical Physics
Mechanics I
Electricity and Magnetism I
Experimental Physics
Quantum Mechanics I
Thermal Physics
At least one from:
PHY 320 Mechanics II
PHY 340 Electricity and Magnetism II
PHY 384 Quantum Mechanics II
Elective Courses
One additional 300-level Physics
course.
Recommended Electives
Nonlinear Science
Molecular Dynamics Simulations
Methods of Computational Science
Advanced Computational Physics
Computational Research in Physics
Computer Science Courses
Programming for Scientists
Hardware and Software Concepts
Computer skills and techniques
Techniques introduced in the core for both computer physics and physics majors
2D graphics: 6-8
Mathematica: 2-3
Function Evaluation: 2+
Data analysis/curve fitting: 3
ODE – Euler, 2
ODE – 2nd Order Methods: 3
Monte Carlo (simple): 4
ODE – 4th order Runge-Kutta: 3
Fourier: 4-5
Integration: 1+
Over relaxation: 1-2
Graphical analysis of transcendental equations: 1-2
Molecular Dynamics: 1-3
Phy 320, 340, 384
Complex Analysis: 1
Monte Carlo (variational): 1
Eigenvalues: 2
Molecular Dynamics:1-3
Other elective courses
Surface-of-section: 1
Ray Tracing: 1
Matrix methods: 1
Fractal Dimension: 1
Computer skills and techniques
Methods of Computational Science
ODE – Adaptive/High Order: 1-3
Computational efficiency: 1
Integral equations by matrix inversion: 1
Theory of ODE techniques:1
Advanced Computational Physics
Split operator: 1
Finite element: 1
Neural network: 1
Molecular Dynamics: 1-3
Monte Carlo: 1-4
Computational Research in Physics
Mesh Method for Liouville eqn
Quadratic Programming and optimization
Matrix methods
Cellular Automata
Integral equations by matrix inversion
Neural network
Eigen analysis
Each student has seen one of these in the recent past
Students see at least three of these
Number indicates the number of times a student is likely to encounter a skill or
technique. Listings for advanced courses do not include all of the techniques a
student has previously encountered.
Observations on computer physics at ISU
Computational physics is on an equal footing with experimental and
theoretical physics at ISU. This will probably be the norm in another
generation.
When computational techniques were introduced across the program in
the late 80’s and early 90’s, faculty teaching each course chose which
techniques to include in their respective courses.
Some general discussion occurred in an attempt to make sure that
students encountered a broad range of techniques and techniques
deemed critical, in particular.
The computer physics sequence started in 1998. Methods of
Computational Science is designed to provide a theoretical framework for
computational techniques. Computational Research in Physics evolved
from an earlier course and provides students with a mixture of theoretical
physics and related cutting edge computational techniques.
The program at ISU focuses on students writing their own computer code.
There are some exceptions. In a few instances a faculty member
provides a working code and students must make some changes. The
department also uses Mathematica.
Most faculty actively contribute to the integration of computer physics into
the curriculum. Some have been encouraged to boost the level of
computer physics in core courses.
In informal discussion with 3rd semester physics students, teacher
education students generally prefer less computer physics integration,
computer physics students really like it, and traditional physics students
fall somewhere in between. By graduation, each student has selected the
course of study most appealing to him or her, and each program is
responsible for about a quarter of our graduates.
Computer physics and traditional physics students generally agree that
computational physics has helped them to more clearly understand
equations and systems.
The Computer physics sequence at ISU thrives in part because 75% of
faculty classify themselves as computational (at least in part), providing a
strong base. All faculty support the program. Computational physics is
more financially accessible to under-funded state schools than
experimental physics.
Physics 388
Advanced computational physics
Neural networks
Comparison of classical and quantum physics
Bio-optical physics
Finite element analysis
Physics 390
Computational research in physics
Computational study of synchronization of coupled non-linear oscillator systems
Gerrymandering and fractal dimensions of congressional districts
Cellular automaton investigation of the transition from non-flocking to flocking behavior
Central current sheet ion distribution functions
Neural Networks
Interdisciplinary field active since 1940’s
Used regularly in science and engineering for prediction,
optimization, data mining, etc.
Pedagogical goals: students will
Understand neural models
Build intuition for selecting net parameters
Reinforce basic timeseries analysis (e.g. power
spectrum & autocorrelation function)
Understand when to train with causal inputs
(physics example) vs. self-prediction (financial example)
Write ANN code to do self-prediction with Dow Jones index
See at least one associative or self-organizing network model
Neural Networks
Scientific ANN example: the Auroral Electrojet (AE) Index
Fast decorrelation time so use causal inputs
Train with several different sets of input data to determine which sets
allow best prediction
Example: Single hidden layer net, using backpropagation
[Gleisner & Lundstedt. 1997]
Go over network design choices
Results consistent with years
of data analysis
Neural Networks
ANN Topics
Biological NNs, learning theory
Neuron models, training, limitations
Learning rules: Hebb, Delta, Backpropagation
Net design: theorems, rules of thumb, testing
Predictability of timeseries
Backpropagation for timeseries (AE and financial)
Hopfield nets: character recognition
Predict Dow Jones
Train with 200 months
Predict for 300 months
Written Assignments
Basic neuron models
Linear separability
Programs
Single neuron for NOR
Delta rule for XOR
Backpropagation for time series, DJIA
Comparison of Classical and Quantum Physics
(based on research program of R. Grobe and Q. Su)
Classical and quantum physics are employed to describe
many phenomena. Understanding their range of
applicability is important in developing students physics
intuition
Pedagogical goals: students will
Simulate non-interacting classical ensembles with a Monte Carlo
technique
Use non-uniformly distributed random numbers, Box-Muller
algorithm, rejection method, and Fast Fourier
transformation
Use Split-operator techniques
Create an NCAR graphics animation
inputs (physics example) vs. self-prediction (financial
example)
Comparison of Classical and Quantum Physics
Students calculate spreading of a classical ensemble of particles and wave
function that describes the equivalent quantum mechanical picture. The
particles experience a constant (linear) force.
Classical results: particle distribution
in phase space at three times
Quantum results: wave function
at three times
Comparison of Classical and Quantum Physics
Topics in Classical and Quantum Topics
Distribution functions, average values, higher moments
The Liouville equation, multi-particle simulations
The Schrödinger equation, exploiting linearity, decomposition into
advantageous states
Free-time evolution using FFT
Second and Third order split-operator scheme with error estimates
Written Assignments
Calculate moments of a swarm of bees
Liouville equation and the conservation of the norm of r
Programs
Evolution of a classical distribution of particles
Evolution of a quantum mechanical wave packet
Bio-Optical Physics
(based on research program of Q. Su and R. Grobe)
One of the youngest fields and expected to play a significant
role in the “century of life sciences”
Non-invasive optical diagnostic techniques are expected to
have great impact on the economics of medicine and help
provide early detection of cancers
Pedagogical goals: students will
Understand the physics of x-ray and IR imaging
Understand the micro- and macroscopic pictures of light/matter
interactions
Apply the Boltzmann equation to light scattering using Monte
Carlo techniques
Model the propagation of light through a turbid medium
Bio-Optical Physics
Transmission and reflection
of modulated beam
Beam spread for constant intensity
Beam spread for modulated intensity
Bio-Optical Physics
Topics in bio-optical physics
Introduction to bio-optical physics
A matrix model of X-ray image reconstruction
Micro- and macro-scopic views of light-tissue interactions
The Boltzmann equation (BE) for light
The scattering phase functions
A bi-directional model of light scattering
A Monte-Carlo algorithm to solve the BE
The photon density waves
The diffusion approximation
Imaging with mirrors
Written Assignments
X-ray shadow gram absorption coefficients
1-D diffusion equation
Programs
1-D Boltzmann equation via a Monte-Carlo
algorithm
Photon density waves with constant and
periodic time dependence
Finite Element Analysis
Powerful numerical method for solving problems in physics
and engineering such as: fluid flow, heat transport, structural
mechanics (torsion, elasticity, etc.)
Frequently used in engineering for modeling problems such
as the structural framework of automobiles and aircraft,
groundwater flow, and heat flow.
Easily generalized to handle 1D, 2D and 3D problems with
complicated boundaries, sources, sinks, and multiple
materials.
Finite Element Analysis
Pedagogical goals: students will
Understand the theoretical basis for the finite element method, i.e.
minimization of a functional on a grid. (Calculus of Variations.)
Understand how to set up the element grid in 1 and 2 dimensions.
Write a 1-D finite element code for calculating the temperature in a
fin with various boundary conditions (e.g. insulated/non insulated)
and with varying materials.
Topics in Finite Element Analysis
Fundamental concepts
Nodes, elements, shape functions
Calculus of variations
Functionals
Heat transfer
Embedding equations
Finite Element Analysis
Written Assignments
Calculate various shape functions
Determine single element equation matrices
Determine embedding equations
Programs
Solve 1-D heat transfer along a fin (circular rod)
Example results for a 1-D
uninsulated rod of radius 1cm
and length 10 cm. The
ambient air temperature is 30
C and the thermal
conductivity of the material is
75 W/cm-C. There is a
continuous heat input of 450
W/cm2 on the left end of the
rod. Calculation in done
using 10 elements.
Computational Study of Synchronization of
Coupled Non-Linear Oscillator Systems
(a component of E. Rosa research program)
Student: Brian Vlcek
Advisor: E. Rosa
Chua Circuit and Chaotic
Attractors
dx/dt = (G(x2-x1)-y(x1))/C1
dy/dt = (G(x1-x2)-x3)/C2
dz/dt = -x2/L
G = 1/R
Phase difference: Δj12 = j1 - j2
Computational Simulation: Chua Circuit Power
Spectra
ε12 = ε13 = 0.0
ε12 = 0.055 ε13 = 0.010
fast
medium
slow
Neural Action Potential Simulation
Hodgkin-Huxley Neural Spiking Model
ε12 = ε13 = 0.0
ε12 = ε13 = 0.01
Gerrymandering and Fractal Dimensions of
Congressional Districts
Student: Nicholas Jurasek
Advisors: B. Clark, D. Holland
Written in C++
Uses SDL image library for image manipulations
It has a very easy to use point and click interface.
Very fast, can calculate the fractal dimension in seconds.
Chicago Congressional Districts
Box Counting Algorithm
The program loads in a BMP image, then displays it on the screen.
The user then clicks on the boarder Color.
A district color is then selected.
The program then breaks the image into boxes and looks in each box to see if
it contains both the border color and the district color.
If a box meets both conditions it is on the perimeter of the district, and it is
counted.
The number of boxes is then plotted against the box dimension.
The boxes are then decreased in size by a factor of 2 and the process is
repeated.
After several iterations the slope of the emerging line is calculated and this
becomes the fractal dimension.
Resulting Fractal Dimensions
Ln(S)
Ln(N)
0
0
.693147
0
1.38629
1.38629
2.07944
2.19722
2.77259
3.17805
Von Koch Snowflake
3.5
y = 1.2925x - 0.4338
Ln(Nboxes)
3
2.5
2
1.5
1
0.5
0
0
1
2
Ln(S)
3
Cellular automaton based investigation of the
transition from non-flocking to flocking behavior
Student: Ryan Balfanz
Advisors: D. Holland, B. Clark
Flocking can be simulated from a few simple “microscopic”
rules (Reynolds)
W1, Flock Centering: head to the center of the other boids
W2, Collision Avoidance: don’t fly into other boids
W3, Velocity Matching: approach the average velocity of the other
boids
By applying the rules to each boid, a macroscopic behavior
emerges
What causes the transition between non-flocking and
flocking motion?
Typical behavior encountered for 16 boids
W1
W2
W3
Observed Behavior
0
0
0
No Organization
1
1
1
Bird Flocking
10
100
-1
Vortex
10
100
1
Fish
10
100
0
Swarm
No Organization
vinitial
vfinal
v1 * w 1
v2* w2
v3* w3
Bird Flocking
Typical behavior encountered for 16 boids
Vortex
Fish
Swarm
vinitial
vfinal
v1* w1
v2* w2
v3* w3
Central current sheet ion distribution functions
(a component of D. Holland research program)
Student: Nathan Nutter
Advisor: D. Holland
Ions interacting with the magnetotail have complicated
trajectories resulting in a relative redistribution of particles
as compared to their incident distribution.
Integrable Orbit
Transient Orbit
Chaotic Orbit
1.5
1.0
0.5
6.0
4.0
0.0
2.0
0.0
-2.0
-4.0
-50.0
-40.0
-6.0
-30.0
-20.0
-8.0
4.0
z
2.0
-0.5
z
-0.2
0.0
0.2
0.4
0.6
x
0.0
-1.0
-2.0
-1.5
0.8
1.0
0.5
1.0
1.2
0.0
-0.5
-1.0
y
-4.0
-50.0
-40.0
-6.0
-30.0
-20.0
-8.0
-10.0
-10.0
x
6.0
0.0
10.0
-1.0
0.0
1.0
2.0
3.0
y
4.0
5.0
6.0
x
0.0
10.0
-3.0
-2.0
-1.0
0.0
1.0
y
2.0
3.0
4.0
z
Current sheet algorithm
Use a test particle code to push a distribution of particles through a
model magnetic field.
Each particle should contribute equal phase space “weight” in the
uniform magnetic field region.
Create single particle distribution by putting the particle in its proper
energy/pitch angle/z-position bin at each time step. fi (H,,z)
Divide the single particle distribution by the total number of “counts” in
the top grid cell so that each particle contributes unit density to the total
distribution.
Sum the single particle distributions to get an overall distribution.
N
f tot ( H ,  , z )   f i ( H ,  , z )
i 1
Typical results for current sheet
Since individual particles are
non-interacting, this is an ideal
problem for parallel processing.
(xgrid)