Modeling and Simulation - An Interdisciplinary, Project-Based First Year Course

Good Morning. My name is John Geddes and I am going to describe a year long course on modeling and simulation that my colleague Mark Somerville and I teach at Olin College of Engineering. It is an interdisciplinary, project-oriented course that satisﬁes graduation requirements in Single Variable Calculus, Multivariable Calculus, Mechanics, and Electricity and Magnetism. 1

Olin College of Engineering is located just outside of Boston, Massachusetts, and was founded in 1997 with a major gift from the F W Olin foundation. The first employees were recruited shortly thereafter, and the first students arrived in 2001. Olin College oers three engineering degrees, all of which were recently accredited by ABET. The college emphasizes hands- on and project-based learning, and all students receive full four-year tuition scholarships. The first class graduated in 2006, and we currently admit approximately 75 students every year. 2

The course I am going to describe this morning takes place in the ﬁrst year of the curriculum. It is part of a closely coordinated set of courses which all students take. It is interdisciplinary, and draws together ideas in mathematics, science, and computing. Students have several shared experiences and all courses frequently leverage concepts and skills developed in the other courses. 3

I want to tell you about this course by giving you a sense of what the student experiences are. As such, I’d like to tell you the story of the course. These are some of the characters in my story, and it’s worth telling you a bit about their backgrounds and personalities because this course was designed by our faculty, for our students. A typical Olin student is talented academically, has lots of extra-curricular interests, and has strong social skills. Virtually all students have seen calculus before and many have seen mechanics. In general, they are very independent-minded. The same description could apply to the faculty, with an added dose of ego! 4

I’ve told you who the people are, now let me tell you about the setting of this course. This course is taught in two locations. An auditorium, which is a pretty normal space. Here we meet with all 75 ﬁrst year students in order to deliver lectures. 5

And a set of studios where we meet with 25 students. At the beginning of the semester, a studio look like a typical classroom or lab. 6

By the end of the semester, the rooms look quite dierent. The tables are usually covered with paper or equipment, and there are dividers to create a sense of ownership. Students actually own this space - multiple courses are taught to the same students in this room. It can get incredibly messy during project time. 7

Now I want to give you a sense of the overall structure of this course, so that as we talk about student experiences you have a sense of where they are in the course. The theme for this course is modeling and simulation and it is a year long course. We cover both discrete and continuous systems, in both compartment and distributed form. 8

In the fall, we begin with discrete time compartment systems which involves concepts from sequences and dierence equations. In the second half of the semester, we work with continuous time compartment systems which involves fundamental concepts from dierential and integral calculus, some dierential equations, and rigid body mechanics. Students receive credit for single-variable calculus and mechanics on successfully completing this course. 9

In the spring we begin with continuous space distributed systems which involves concepts from vector calculus (multiple integrals, surface integrals etc.) and time-independent electric and magnetic ﬁelds. We end the year with distributed systems continuous in space and time which involves concepts in partial dierential equations and waves. Students receive credit for multi-variable calculus and electricity and magnetism on successfully completing this course. 10

In this presentation I’m going to describe in detail the student experience in the ﬁrst quarter of the course - the portion that concerns discrete time compartment systems. I will then give you a sense for what the students DO in the other parts of the course. 11

When students ﬁrst arrive in studio, this is what they see on their tables. It’s a simple game board, based on the beer game which was devised by John Sterman at MIT - it is typically used in MBA courses to teach students about supply chain eects. We have adapted and simpliﬁed it a little and we call it the production game. 12

Before we do anything else, we play the production game. Students are placed in teams of 4 and each one plays a dierent role - there is a retailer, a wholesaler, a distributor, and a factory. Each player picks up their mail, reads the product order, and tries to ﬁll it. They move items into shipping and then place an order for product which they pass down the line. The next player picks up their mail and so on. After the factory plays their turn, the items in inventory etc are recorded and the players are ready to play the next round. The team with the least costs at the end of the game are declared the winners. 13

As the game proceeds, ﬁghts start to break out. The combination of feedback and long delays between production and retailer leads to large piles of pennies and silly order strategies. After about thirty rounds, large oscillations in inventory set in and despite their best eorts students are unable to minimize the damage. 14

Students record data while playing the game and their ﬁrst assignment is to present their data graphically with an hypothesis about what happened. Each player gives a brief 5 minute presentation using a visual aid in their studio space. At this stage, the graphics are usually sloppy, the hypothesis are weak, and the presentations are poor. This sort of informal presentation using visual aids is repeated over and over during the semester and is a central theme in the course. As you can imagine, students tend to improve very quickly! 15

Meanwhile, in the auditorium, formal lectures and tutorials are delivered by faculty on relevant topics. For example, at this early stage I will lecture on sequences and dierence equations, and provide simple examples of compartment models. The teaching method in the auditorium uses typical active learning strategies - short lectures, conceptual questions, paired problem solving, etc. We also use undergraduate course assistants to circulate and help. 16

Back in studio, students then apply the concepts of sequences and dierence equations to the production game. Their goal here is to develop a formal description of the production game using the language of dierence equations. 17

Once students have developed the governing equations, they now develop a simulation of the production game. Our simulation environment is Matlab and we cover basic programming and simulation in the auditorium using short exercises. Students then apply the core ideas to simulating the production game itself. 18

Then, using this simulation tool, they investigate what actually happened during the production game. In this way, they validate their tool, better understand the game, and are now ready to use the simulation to do some work. Here you see a comparison between the date recorded during game play and date generated by the simulation - 90% of the errors turn out to be due to bad handwriting and faulty arithmetic during game play! 19

At this stage of the semester, which is about three weeks in, the course moves into a project-oriented phase during which students develop discrete time models of various phenomena, they pose a question, and develop a simulation in order to explore the answers. Students choose their own projects and we have seen all sorts of things. Some examples include a model of owl populations, a model of a SARS epidemic, a model of beetle populations, and extensions to the production game. 20

Here is an example of a question which Pam and Alex posed about owl populations. They found a published discrete model for owl populations, but wanted to adapt it for small populations of breeding owl pairs. 21

They developed a discrete stochastic model suitable for small population numbers. There are three classes of owl, and each one has a probability of dying. Adult pairs have a probability of mating and produce a juvenile which may die before it transitions to become a sub-adult. Sub-adults then become adults if they don’t die ﬁrst. 22

As with the production game, they develop governing equations and write a numerical simulation in order to explore their question. This is largely self-directed and takes place in studio. One of the interesting aspects of studio is the way it changes faculty- student interactions. We enter their space, respond to their questions, and ask them to defend their assertions. This can be hard to adapt to - we focus our attention on posing questions rather than providing answers. 23

During this entire process, students are constantly creating visual aids and making brief informal presentations. The ﬁnal deliverable is a formal poster which summarizes the results of their investigation. 24

In this case, for example, Pam and Alex ran lots of simulations using their small population model and compared their results to the deterministic large population model. They also investigated the impact of varying the number of initial breeding pairs. At this stage, graphics have improved considerably as well as hypothesis posing and testing. 25

At the end of this project, students are expected to publicly present and defend their work in a professional manner.This is roughly ﬁve weeks into the semester and students have had repeated practice at creating visual aids and giving presentations. There is usually an order of magnitude improvement over their initial attempts. 26

This is an overview of the piece of the course which I have just walked through. Students play the production game, they attend lectures on relevant topics, they develop a model using dierence equations, they create a simulation in order to validate their model, and they pose a question and explore it. They make frequent informal presentations using visual aids and this stage of the course ends with a formal poster presentation. From the student point of view there was a constrained phase consisting of introductory material and the case study, and a project phase in which they have more autonomy. 27

We repeat this pattern four times over the course of the year. After working on discrete time compartment systems, students then work in continuous time compartment systems (e.g., rigid body mechanics) We then move to dealing with time- independent fields – systems that are static in time, but continuous in space (e.g., electric fields due to charge distributions) And finally students learn about, and model, systems that are continuous in time and in space (e.g. propagating waves). I’m now going to show you some examples of the KINDS of things students DO in each of these phases. 28

First we will look at compartment systems that are continuous in time. We focus on rigid body dynamics described by a ﬁnite number of ordinary dierential equations for the most part, but students have some freedom to choose. Students ﬁrst investigate a seemingly simple case study - a block that is free to slide on a pivoting ramp. They create a model and a simulation and use it to explore some basic questions. In the open-ended phase they choose a system to investigate themselves. 29

For example, Greg and Kira decided to model and simulate an inverted pendulum on a cart. The pendulum is free to rotate, and the cart is free to move on the horizontal surface. Most projects begin with a general question like “can I control an inverted pendulum” but eventually become more speciﬁc like “what is the best PID control to use in order to control the inverted pendulum”. Other examples of projects include a caber toss, a ferris wheel, a trebuchet, a segway, an elephant on a drum .... 30

Students begin by developing a model and the relevant governing equations. For the inverted pendulum on a cart, this involves the motion of two bodies which leads to two coupled dierential equations. The focus here is on understanding the description of the model in terms of dierential equations, and not on analytical solution processes, none of which would be useful anyway. 31

Using the simulation tools developed earlier, students go on to simulate the motion of the pendulum on the cart by numerically solving the dierential equations. Typically, they will begin by validating their model and simulation. Here, for example, Greg and Kira examined the various contributions to the energy of the uncontrolled pendulum on a cart. In addition to conﬁrming that total energy is conserved, they gain valuable insight into the dynamics of the system. 32

After validation, students now DO some work with their simulation tool. Here Greg and Kira demonstrates that the inverted pendulum on a cart is controllable using some sort of PID control. In this particular case they went onto explore the eect of varying PID control on the state of the pendulum and cart. As with the discrete time phase of the course, this stage ends with a formal poster presentation in which they present and defend their work. 33

During the second half of the year we move onto systems that are distributed. First, we begin with systems that are distributed in space. Here we develop concepts and skills in ﬁelds and multiple integrals in the context of electricity and magnetism. Again, I want to give you a sense for the sorts of projects students DO. 34

In this example, Bill and Ilari decided to model and simulate an electrostatic precipitator for use a coal burning power plant. This involves deﬁning a plate geometry, computing the electrostatic ﬁeld and then tracking the motion of ions as they travel through the system. During the spring semester, we switch from oral and visual communication to written communication. Students write short paper fragments over and over again before the project ends with a formal written deliverable. 35

Bill and Ilari investigated the eficiency of several precipitator designs. In the first, a wire is held at -10V while two plates are held at ground. In the second, two wires are held at -10V while the two plates are held at ground. In the final design, one plate is held at ground while the other is held at -10V. Bill and Ilari wrote a simulation to compute the electrostatic field due to these configurations and then tracked the motion of ions as they pass through the system for dierent initial velocities. 36

Here they present their simulation results as the rate of particle capture versus particle velocity. They demonstrate that the simple two plate design is best and that there is an optimal velocity in order to maximize the capture rate. 37

We end the course with systems that are continuous and distributed in space and time. This involves the major vector theorems from multivariable calculus, as well as some partial dierential equations. Again, students complete a case study in electricity and magnetism, but we then open it up to waves in general. 38

In this example, Rachel and Ilari decided to investigate the tones produced by a real piano and compare them to the tones produced by a mathematical model of a piano. They again work in studio in a self-directed manner, and produce a written paper as the ﬁnal deliverable. 39

In this project, Rachel and Ilari did quite a bit of background reading and decided to implement a model that they found in the literature. A piano hammer impacts the string which sets up a propagating wave. It then resonates with the bridge, sending vibrations through the piano. The model takes the form of a partial dierential equation which captures the motion of a damped, propagating wave subject to forcing from the piano hammer. While this may look out of reach for these students, it follows naturally from the earlier case studies and projects and they have little dificulty in adapting their simulation tools to handle this. 40

Here they present their simulation results and compare them against a real piano. They made a recording of an actual piano and then decomposed the signal into its Fourier components. Here they show the relative strength of each fundamental frequency for each of the dierent notes. Then using relevant parameter values and an accepted model for the hammer impact, they ran a numerical simulation for each note and also decomposed the signal into its Fourier components. They concluded their paper by comparing and contrasting the results and by discussing the various ways in which the model could be improved. 41

That ends my description of the course. I’d like to spend the remaining time reﬂecting on this experience. I’m not going to pretend that we are conducting a careful, pedagogical experiment here - rather we have developed this course based on a deep-seated belief in the power of modeling and simulation. We also believe that oral and written communication are fundamental competencies that leads to a deeper knowledge base. 42

I mentioned earlier that this course was developed by our faculty for our students. I believe that, like politics, all education is local. What works for our students won’t necessarily work for your students and vice versa. We developed this course using an user-oriented curriculum design process - the details are ﬁnely tuned for our students. 43

Having said that, I’d like to make a couple of observations. Our focus on student self-direction at this early stage leads to a group of fearless students. They are simply not intimidated, and will take on anything. They are not boxed into dealing with models that are simple enough to admit simple analytical solutions - real problems with realistic models just aren’t like that. 44

It is also incredibly motivational. The limits of simulation become quite apparent to them and they are very ready for alternative approaches and models. In future classes, they come in hungry for knowledge. In a nonlinear dynamics and chaos course which I teach I usually have trouble containing them - I think we all know the beneﬁts of dealing with a motivated group of students. 45

And, in conclusion, I have to say that I believe that they understand the fundamental concepts better and can better apply them - but again this is based on my own experience over the years. I’d be delighted to take some questions. 46

Modeling and Simulation - An Interdisciplinary, Project-Based First Year Course

by Mark Somerville on Mar 24, 2009

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Modeling and Simulation - An Interdisciplinary, Project-Based First Year Course — Document Transcript