Computational Course Projects and Undergraduate Research

B. K. Clark and Richard F. Martin, Jr.

Illinois State University

Contributors:E. Rosa Q. SuD. Holland R. GrobeR. Balfanz N. NutterN. Jurasek B. Vleck

Resources: ISU Physics and its Peers2004 - 2006

Institution # of faculty publications % of faculty average grant % of faculty per faculty with publication amount with grant

Top 10 68 6.62 64 % $ 569 K 11 %ISU Physics 12 4.83 83 % $ 92 K 17 %

Top 10 Physics 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

Nonlinear DynamicsNanoscienceSpace PhysicsAtomic, Molecular,

and Optical PhysicsBiophysics

Undergraduate physics research at ISU

Annual Average Number of Graduates 2002-2004

United States Air Force Academy 24Harvey Mudd College 22

U. of Wisconsin – La Crosse 22Illinois State University 20

Source: American Institute of Physics

ISU Computer Physics Sequence 1998-2007

Total graduates 35Graduates per year 4

Computation Research Mentors 9Advanced Computational Physics

Modules 7 (3 per year)

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CPYPTEENGPHY

Number of physics graduates from 1980 to present

CPY: Computer physics

PTE: Physics Teaching

PHY: Physics

ENG: 3/2 program

Computer Physics Curriculum

At least one from:   PHY 320 Mechanics II   PHY 340 Electricity and Magnetism II   PHY 384 Quantum Mechanics II

Computer Science Courses Programming for Scientists Hardware and Software Concepts

Elective Courses  One additional 300-level Physics course.

Recommended Electives Nonlinear Science Molecular Dynamics Simulations

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 Methods of Computational Science

Advanced Computational Physics Computational Research in Physics

Computer skills and techniques

Techniques introduced in the core for both computer physics and physics majors2D graphics: 6-8Mathematica: 2-3

Function Evaluation: 2+Data analysis/curve fitting: 3

ODE – Euler, 2ODE – 2nd Order Methods: 3

Monte Carlo (simple): 4ODE – 4th order Runge-Kutta: 3

Fourier: 4-5Integration: 1+

Over relaxation: 1-2Graphical analysis of transcendental equations: 1-2

Molecular Dynamics: 1-3

Phy 320, 340, 384Complex Analysis: 1

Monte Carlo (variational): 1Eigenvalues: 2

Molecular Dynamics:1-3

Other elective coursesSurface-of-section: 1

Ray Tracing: 1Matrix methods: 1

Fractal Dimension: 1

Computational Research in PhysicsMesh Method for Liouville eqn

Quadratic Programming and optimizationMatrix methods

Cellular AutomataIntegral equations by matrix inversion

Neural networkEigen analysis

Each student has seen one of these in the recent past

Computer skills and techniques

Advanced Computational PhysicsSplit operator: 1Finite element: 1Neural network: 1

Molecular Dynamics: 1-3Monte Carlo: 1-4

Students see at least three of these

Methods of Computational ScienceODE – Adaptive/High Order: 1-3

Computational efficiency: 1Integral equations by matrix inversion: 1

Theory of ODE techniques:1

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 ISUComputational 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 networksComparison of classical and quantum physics

Bio-optical physicsFinite element analysis

Physics 390 Computational research in physics

Computational study of synchronization of coupled non-linear oscillator systemsGerrymandering and fractal dimensions of congressional districts

Cellular automaton investigation of the transition from non-flocking to flocking behaviorCentral 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 willUnderstand neural modelsBuild intuition for selecting net parametersReinforce 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 indexSee at least one associative or self-organizing network model

Neural Networks

Go over network design choices Results consistent with years of data analysis

Scientific ANN example: the Auroral Electrojet (AE) IndexFast decorrelation time so use causal inputsTrain with several different sets of input data to determine which sets

allow best predictionExample: Single hidden layer net, using backpropagation

[Gleisner & Lundstedt. 1997]

Neural Networks

Predict Dow JonesTrain with 200 monthsPredict for 300 months

ANN TopicsBiological NNs, learning theoryNeuron models, training, limitationsLearning rules: Hebb, Delta, BackpropagationNet design: theorems, rules of thumb, testingPredictability of timeseriesBackpropagation for timeseries (AE and financial)Hopfield nets: character recognition

Written AssignmentsBasic neuron modelsLinear separability

ProgramsSingle neuron for NORDelta rule for XORBackpropagation 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 willSimulate non-interacting classical ensembles with a Monte Carlo

techniqueUse non-uniformly distributed random numbers, Box-Muller algorithm, rejection method, and Fast Fourier transformationUse Split-operator techniquesCreate an NCAR graphics animation

inputs (physics example) vs. self-prediction (financial example)

Comparison of Classical and Quantum Physics

Classical results: particle distribution in phase space at three times

Quantum results: wave function at three times

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.

Comparison of Classical and Quantum Physics

Written AssignmentsCalculate moments of a swarm of beesLiouville equation and the conservation of the norm of

ProgramsEvolution of a classical distribution of particlesEvolution of a quantum mechanical wave packet

Topics in Classical and Quantum TopicsDistribution functions, average values, higher momentsThe Liouville equation, multi-particle simulationsThe Schrödinger equation, exploiting linearity, decomposition into advantageous statesFree-time evolution using FFTSecond and Third order split-operator scheme with error estimates

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 willUnderstand the physics of x-ray and IR imagingUnderstand the micro- and macroscopic pictures of light/matter

interactionsApply the Boltzmann equation to light scattering using Monte

Carlo techniquesModel 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

Written AssignmentsX-ray shadow gram absorption coefficients1-D diffusion equation

Programs1-D Boltzmann equation via a Monte-Carlo algorithm

Photon density waves with constant and periodic time dependence

Topics in bio-optical physicsIntroduction to bio-optical physicsA matrix model of X-ray image reconstructionMicro- and macro-scopic views of light-tissue

interactionsThe Boltzmann equation (BE) for lightThe scattering phase functionsA bi-directional model of light scatteringA Monte-Carlo algorithm to solve the BEThe photon density wavesThe diffusion approximationImaging with mirrors

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 willUnderstand 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 AnalysisFundamental concepts

Nodes, elements, shape functionsCalculus of variationsFunctionalsHeat transferEmbedding equations

Finite Element Analysis

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.

Written AssignmentsCalculate various shape functionsDetermine single element equation matricesDetermine embedding equations

ProgramsSolve 1-D heat transfer along a fin (circular rod)

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

Computational Study of Synchronization ofCoupled Non-Linear Oscillator Systems

(a component of E. Rosa research program)

Phase difference: Δ12 = 1 - 2

Student: Brian Vlcek Advisor: E. Rosa

slow

medium

fast

ε12 = ε13 = 0.0 ε12 = 0.055 ε13 = 0.010

Computational Simulation: Chua Circuit Power Spectra

Neural Action Potential SimulationHodgkin-Huxley Neural Spiking Model

ε12 = ε13 = 0.01ε12 = ε13 = 0.0

Chicago Congressional Districts

Gerrymandering and Fractal Dimensions of Congressional Districts

Student: Nicholas Jurasek Advisors: B. Clark, D. Holland

Written in C++Uses SDL image library for image manipulationsIt has a very easy to use point and click interface.Very fast, can calculate the fractal dimension in seconds.

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

y = 1.2925x - 0.4338

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oxe

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y = 1.0999x + 2.5521

y = 1.309x + 1.4208

y = 1.4421x + 0.4872

y = 1.3369x + 0.0137

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Ln(S)

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 boidsW2, Collision Avoidance: don’t fly into other boidsW3, 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

vinitial

v1* w1

v2* w2

v3* w3

vfinal

Bird FlockingNo Organization

Typical behavior encountered for 16 boids

vinitial

v1* w1

v2* w2

v3* w3

vfinal

Vortex Fish

Swarm

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.

Transient Orbit

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Integrable Orbit

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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.

Current sheet algorithm

N

iitot zHfzHf

1

),,(),,(

Typical results for current sheet

Since individual particles are non-interacting, this is an ideal problem for parallel processing. (xgrid)