Finite Element Method Through Paintings

How many “sections” are there in the following image?

I’m guessing you probably said 3 — orange, walkway, blue.

Now try to describe the middle section.

There is a walkway with two people, a man and a woman, and they are holding an umbrella. The leaves above are brownish.

That just divided that one section into 5 more sections. And obviously, you could keep going.

The point here is that you’re able to subdivide a large image into sections that share some similarity. The smaller the section, the more similar the things in it will be and the more accurately you’ll be able to describe it.

Now try this:

Look at the following painting for 30 seconds. Then scroll down and see how much you remember.

What’s goes through the center of the painting?

What color(s) are the roofs on the houses on the left?

What shape is the underside of the bridge?

How many birds are there?

Who is the painter?

Is there a chimney on the building on the right side?

Try again for 30 seconds with this next image. Except this time, I’m going to tell you to focus on the large green building on the right.

Is the main entrance of the green building on the left or right side?

How many windows are there on the second level of the green building?

At what level in the green building is the single brown window?

What color is the clock on the central tower?

What direction is the sunlight coming from?

Is the closest person standing up?

Which of the questions were you able to answer? Did it change when I told you what to focus on? My guess is that in general, you were able to answer the first three questions very easily and that the final three you either didn’t get or weren’t sure about.

In the first example, the easy questions were about overarching things about the image. You had to have a vague sketch of the entire landscape. In the second example, the easy questions were about the green building even though they were more specific. You had to concentrate on one particular area very closely and remember more detail, but that came at the expense of remembering other aspects of the image.

The takeaway here is that since the amount you can remember is (approximately) proportional to the time spent looking at the image, you can either choose generality or specificity.

And believe it or not, those two basic principles are the foundation of Finite Element Method (FEM), a common mathematical tool used by engineers to understand their systems.

So how exactly does this translate? As a reminder, the two points were:

  • You can divide an image into sections that are similar; smaller sections = more accurate
  • You can either focus on general information or specific information from a smaller area

Sectioning is really important. It provides a framework to understand and explain something.

A computer fully understands an image by storing an array of pixel colors, but to understand an image, especially in a reasonable amount of time, you don’t need to know every exact pixel color. You can work in larger sections and sum many pixels together. In a way, it’s kind of like averaging. In the first image, all of the different shades of red and yellow and orange can just be summarized as orange.

Then you can take all of the individual sections that you’ve summarized to understand the entire image.

In FEM, instead of breaking apart an image, you’re discretizing a function representation. The original function is often a complex partial derivative and so the goal is to turn it from a smooth continuous shape into finite sized chunks (instead of infinitesimally sized ones). This step is called creating a mesh.

Two examples of discretization

And instead of summarizing the sections with a color, you use a very simple function to describe the shape of the space.

Then you combine all of the simple functions that describe each finite element into one master function for the entire structure.

Now you may wonder, why would you ever want to do this?

  • It’s a good approximation method
  • It allows you to focus the computational power on the parts that matter

The governing equations are extremely complicated. Imagine having to create and solve a single equation that determined how a bridge would bend when a car drove over it or how a tropical cyclone will move over the ocean depending on atmospheric conditions. That would be like having to memorize the picture as a whole without separating the people from the trees from the buildings.

This method takes a continuous function and breaks it into finite elements so a computer can solve it (this is the same principle by which computers solve definite integrals as well).

Second, you can focus on the important parts of the system.

When modeling a head on car crash, the impacts on the back end of the car are much less interesting so you don’t need to precisely model what happens there. You can focus your computational energy on having a highly accurate prediction for the front end.

If your job is to predict the weather, it doesn’t really matter whether the temperature is 67 or 68˚F above a cornfield in Illinois but it is highly important to understand how the exact temperature and humidity levels will change the dynamics of an oncoming hurricane.

By breaking the model into elements, you can have the most accurate representations (functions) of the important elements and less accurate representations of the others. This is like enabling you to remember specific information about the green building without needing to also go equally in depth into every other element of the image.

All in all, this is a very high level summary of FEM, an extremely important tool in engineering modeling and simulations.

Steps:

  • Discretize — break the shape into finite elements
  • Summarize — find a simple function that represents each element
  • Combine — add together all of the simple functions into one large one to explain the entire shape

Benefits:

  • Approximation is necessary
  • Have different resolutions for each element, i.e. some can be more accurate than others to save computational power/time

This was a very simplified explanation, so if you want more technical detail about how this process works, you can read my other article here.

Thank you for reading and I hope this was an intuitive explanation without being filled with formulas! Please feel free to reach out if you’re interested in talking more about materials science.

At 17 years old, I love learning and am interested in materials science, education, and environmental sustainability.