This post will summarise the EngSci courses in Semester One.
ENGSCI 211: Mathematical Modelling II
A mathematical model uses maths to describe any sort of phenomena or system (i.e. a thing in the real world).
You want to elegantly describe the motion of a glider in a thermal field? Mathematical models can describe it for you. You want to describe the vibrations created by an earthquake? Mathematical models can help you.
But why would we want to use mathematical modelling in the first place? Mathematical models provide information. Models helps us understand phenomena. With this information, you can begin to play around with different factors to control what happens. You can also predict unknown events based on what we already know.
Mathematical modelling is an information game. To win, use this information to predict and control outcomes.
Understand, predict, control. That’s the overarching theme. Let’s run through the different modules.
Ordinary Differential Equations
You’ve already dealt with ODEs in ENGSCI 111. To recap, ODEs are useful for modelling systems and phenomena that are smooth with respect to a continuous variable (for example, time).
Another quick refresher. A non-homogenous ODE is one where not all terms involve the ‘y’ variable, the thing you were solving for: y’’ + y’ – 4y = 6 is non-homogenous because of the 6). One of the constraints you had to deal with in ENGSCI 111 was that you couldn’t solve non-homogenous ODEs. In ENGSCI 211, you get the tools to solve non-homogenous ODEs. You also learn how to rewrite ODEs so that computers can solve them for you. Computers can solve ODEs that are pesky to solve by hand.
Recall that you used the Taylor and Maclaurian series to simplify unwieldy mathematical models into nice simple ones.
Where Taylor and Maclaurian simplify polynomial models, Fourier is used to simplify periodic functions and phenomena that repeats itself (like sound waves or MRIs). With fancy footwork, you can use Fourier series to simplify ANY phenomena, regardless of whether it repeats itself or not.
Algebraic equations describe phenomena. Trying to solve 100 equations without matrices gets annoying. Matrices describe equations so that computers do the heavy lifting. Matrices also allows us to change certain numbers in the matrix (read: control inputs and outputs, study its effects, make predictions).
The step up to ENGSCI 211 is threefold. Firstly, we learn to write these matrices in a specific way (i.e. Jacobian and Gauss-Seidel) to fully unleash the power of a computer. Second, ENGSCI 211 includes special types of matrices, called eigenvectors, that transform matrices in special ways and enable clever designs. Finally, we learn a mathematical judo move called ‘diagonalisation’. Diagonalisation rewrites first order ODEs into matrix form so that they are elegantly solved.
With the calculus you’ve seen so far, you can play with mathematical models that use only two variables (i.e. y = 3(x-2)(x+9)). You’ve been stuck in 2D world.
Multivariable calculus enables play with more than two variables (i.e. z = 3x + 7y). We’ve gone from 2D to 3D. Instead of calculating areas under curves, we’re now calculating volumes underneath surfaces. Instead of finding high and low points on curves, we’re now finding high and low points on surfaces, like finding peak temperatures in a geothermal field and finding the most rapid temperature change in that field.
Data analysis is a type of modelling that give us information on whether X changes Y. It’s similar to statistics.
Next course: BIOMENG 221: Mechanics of Engineered and Biological Materials
Think of ‘materials’ as the different ingredients that you’d used to build something. If you were cooking a Thai curry, you’d pay attention to the different spices you used so that you made something that is punchy and not gradual. Similarly, you need to know how different materials act under certain forces so that you get the right ingredient mix and the right design.
Mechanics of Materials starts with learning about the different types characteristics that a material might have. Does it stretch easily? Does it get thinner if I stretch it, like pulling Blu-tack into a string? How hard do I need to pull/push/twist it before it breaks or changes shape? Instead of pulling it lengthwise, what if I applied a tangential force to it, similar to how a hurricane nudges a house sideways? Does it matter which direction I pull this object in? What if I change the temperature? What if I pull it slowly?
By asking these questions and using experiments, we can begin to understand the stiffness of a material object and how it might change shape. Just like Mathematical Modelling II, Mechanics of Materials starts off as an information game and ends with using models to describe the behaviour of these materials. Can we describe this material as brittle? Ductile? Elastic? Plastic? Viscoelastic? Can we treat this object as a pressure vessel or as a beam?
Using this information and these models, we can understand more about the material’s strength. Strength is important to know because strength describes how much force the object can handle before it breaks. It also provides insight into the type of breaking the object would undergo. Does it crack? Does it fracture? Where does it fracture? In which direction? Or is it not a clean break? Does it get mangled as?
Once we have information on what we’re dealing with, we can turn the information game into a ‘predict, control, design, and improve’ game. The latter game is important because this game improves the design of structures and objects. This is your duty as an engineer.
BIOMENG 221 involves lots of mathematical derivations. This is awesome because you get a deeper appreciation of why certain things happen – something you wouldn’t get in Civil or Mechanical. There’s also a fair amount of mathematical modelling (read: ENGSCI 211) in this course, especially ODEs.
Next course: ENGSCI 233: Computer Systems and Computational Techniques
Computers are a form of intelligent machines. Being able to interact with and operate intelligent machines to understand and solve complex problems is a rare and valuable skill. Rare and valuable skills earn top dollar.
This course begins to teach you this rare and valuable skill.
Computer Systems is all about understanding how computers operate and how to interact with its physical hardware.
Computational Techniques is about writing the software and processes that enable computers to perform tasks for us.
Similar to the other courses, ENGSCI 233 begins as an information game. The information game starts with understanding three things: How do computers store information? How does software affect the computer’s hardware? How can things go wrong when using computers?
Once we received this inbound, we can start pitching outbound. We can use our knowledge to design circuits that use information to make decisions and actions. We can move thousands of data files quickly and efficiently. We can organize and search for information on the computer. We can create mathematical models (!) when we have only a few measurements for the phenomena. We can perform high-level calculations for these mathematical models We can set up the computer to solve ODEs (!). We can run simulations (read: recreate and understand physical phenomena and predict unknown events) to solve scientific, engineering, and humanities problems.
Computers are powerful machines. Learning to make full use of them will help you become more efficient and productive, especially when you’re playing on a large scale.
Just like ENGSCI 211 and BIOMENG 221, this course follows the same pattern: understand, predict, control, improve.
This doesn’t include ENGSCI 255 – Modelling in Operations Research. I take ENGSCI 255 next semester. Otherwise, congratulations! You now know what Engineering Science is as much as I do.