Under the hood - part 02: Read me first
I propose a friendly learning method, based on successive attempts to understand a text statement, which is clarified by asking questions about its different parts.
-
Step 1: The action plan
The theory on numerical methods is well defined. That is not the problem. What we need is a friendly strategy to digest it and that is what we are going for with this tutorial.
It is not about "what to teach" but about "how to transmit it" so that the experience is not exhausting and, much less, frustrating.
To start, we will use a sentence that contains all the information about the operation of weighted residual methods (such as the finite element method) but that is compact and contains several terms that we surely do not know.
To advance, we only have to ask questions, again and again, about each thing that we do not have clear about that sentence. Thus, additional explanations will emerge (expansion process) until, in the end, we are able to read the original sentence and perceive all its content (synthesis process).
-
Step 2: The phrase
Well, let's go with the phrase that, in a short time, will allow us to visualize all the machinery under the hood: Weighted Residuals Method.
-
Step 3: First interpretation
There are only three words! How wrong can we get with three words? Let's see:
Weighted: it could be related to the action of determining the weight of something, weighing it, attributing a certain importance to it.
Residuals: in the same way, it gives the sensation of a surplus, which can sometimes be useless or unwanted.
Method: it is interpreted, at least colloquially, as something related to the way of doing things, as a procedure or standard. Something contrary to disorder.
Although the meanings of each word do not seem mysterious, the phrase itself does not convey clear ideas. It seems to refer to a way of doing things based on giving importance to certain surpluses, which no one knows what they are or where they come from.
-
Step 4: First clarifications
Since the phrase "condensed" doesn't make sense to us at the moment (although it will in the short term) we will start to "expand" it to delve deeper into its meaning.
Let's start by clarifying the term "residuals." In this particular phrase, it refers to the residuals of certain differential equations that model the behavior of some physical phenomenon that we are interested in studying.
That is, we are talking about weighing residuals of differential equations to discover how temperature, deformation, or any other phenomenon behaves in a certain case that we are interested in solving.
It seems that we only need to know what differential equations and their residuals are, and what is that weighting that will allow us to solve the unknowns about said physical phenomenon in that particular case.
-
Step 5: That's the game!
Although at this point we still have a lot to clarify about "what they are" and "how they work", we have become aware of "what they are for".
To learn almost anything, in a consistent manner, it is very good to answer four basic questions. The first two can take minutes... and are vital for motivation and focus, and the last two can involve hours, weeks, months or years... which we can go through without tiredness or frustration thanks to the motivation and focus obtained from the first ones.
1) What am I going to study? Sometimes the answer is immediate. A simple title, but a vital step for learning. Example: we are going to study "Weighted Residual Methods".
2) What is it for? The answer may be simple but it has enormous value to motivate study. Continuing with our example: we will study "Weighted Residual Methods" because they serve "to understand the behavior of physical phenomena in any case of interest".
3) How is it used? Answering this question may require a variable amount of time depending on the complexity of the "procedure" needed to apply the ideas "at the user level." That is, without the obligation to know the deep foundations that justify such a procedure. In our example, it would involve learning the "Finite Element Analysis" procedure in a particular software, ensuring competence through examples and cases of varying complexity.
4) Why does it work? This is the most complex question: what is under the hood? Answering it may take a lot of time and effort, which we will only be willing to invest if the previous answers have motivated us enough. The reward for such effort will be to evolve as simple "users" of procedures and technologies to become "analysts" in a first stage and, finally, "designers."
Under the hood: In the car analogy, it would involve going from being a driver (with the ability to drive the vehicle) to a mechanic (with the ability to analyze and solve vehicle problems, and even propose some improvements) and, finally, a designer (with the ability to create new types of vehicles).
This is a learning strategy that I formalized decades ago under the name 3Q+C, referring to the Spanish letters of the four questions: Qué, para Qué, Cómo y por Qué (What, What for, How and Why).
I'll just leave you with some example images, to avoid going off on a tangent. At another time it may be useful to return to this topic!
-
Step 6: Next Steps
To continue to delve deeper into the meaning of the phrase "Weighted Residual Methods" we will need to make sure we understand very well what a differential equation is.
As a "differential equation" is a particular type of equation, we must be sure to first understand what an equation is. In general mathematical terms, an equation is an equality that contains one or more unknowns.
In particular, a differential equation is one where the unknown is not a simple numerical value (like the classic "x" value of algebraic equations) but a function (expressing the relationship between something that varies independently, such as space and time, and something that depends on it, such as temperature, strain, stress, etc.).
This is our next challenge to start really looking "under the hood." I look forward to seeing you!
-
Step 7: Links
This tutorial comes from:
Under the hood - part 01: Motivation
and continues in:
