Welcome to the Finite World!

Also known as, let’s get started talking about computational mechanics and, in particular, Finite Element Analysis (or FEA).

Now this section is going to be rather different than the previous sections. I might be accused of being a little lazy here, but to hell with it – I have a slide deck of some 200+ slides from a first-year graduate-level class on FEA In Engineering that I’m really freaking proud of. And so, the plan is really simple.

  • You get the slides as I presented them back in the day, and
  • You get some commentary sprinkled within the slide deck

Sound good. Well I don’t really care, that’s what I’m gonna do so let’s drive on…

Simple enough, right?
Also pretty straightforward, methinks. FYI, here is a link to the reference text.

While we’re on the subject, let’s talk a bit more about the text (as it was in 2007-8 when I was teaching this)…

The text turned-out to be quite popular despite it being new and, at the time, rather rife with errors.

With that, let’s get started on some basics…

90k-foot view of FEA

There are some key points here that I’d like to take some time to ruminate on, because if you don’t understand this part then you may as well give-up understanding the method at any level. For starters a term or two – PDEs == Partial Differential Equations. The response of the system is described by (rather than governed by, of course – kinda had the cart before the horse there) PDEs, and these are generally unsolvable in closed-form (as in, precisely, mathematically, without approximation). So what we need is some sort of methodology that takes these PDEs and “solves” them. FEA is an example of such a methodology. Though the development of FEA predates the availability of computers it is almost perfectly suited for computational approaches.

Aside: for anyone who may not like my approach to defining all these terms with fine detail, I cannot more highly recommend Mortimer Adler’s “How To Read A Book”, the first 180-pages or so in particular. Better yet, join Online Great Books. You’ll thank me later. Seriously.

The bottom-line is that these definitions are absolutely essential toward making sure we’re communicating rather than talking past each other.

(Climbing back down off my soapbox…)

A very brief history of the development of FEA

Bottom line here is that this is a very mature methodology that continues to be under constant development and enhancement. It’s also a great market to be in, by they way.

My abilities to prognosticate were pretty good in this regard, actually

Recognizing that this slide is about 15-years old I say I pretty much nailed this one. Also, this remains the key development areas now, especially multi-physics (the linking of different physical domains – such as fluid-structure – into a single analysis).

The 75k-foot view of the method

Any FEA code, no matter how complex, expensive, etc. is going to follow this general formula. And we’re going to start with the simplest systems you can think of, namely…

Digging-in…

But, I hear you saying…

The questions I know you’re asking

Asked (maybe), and answered…

Trusses, baby. They’re literally everywhere

Hopefully you’re convinced, so let’s go…

Trusses are also known as rods and bars

And so we introduce our first “finite element”. The rod/bar/truss element is what is known as a structural element, in other word, an element that mimics the behavior of structural elements (such as the truss supports of bridges) without explicitly providing a detailed model the geometry of the structural member.

That’s detail, which will hopefully be much clearer by the end of this post.

The behavior of the rod element

In other words, the physical structural element is modeled as a line in our computational world, and we use very basic statics principles to determine the internal forces in the element. Just as in our physical structure, we must satisfy Continuity (which you can think of as equilibrium) and Compatibility (that the structure is continuous, no gaps or holes). Meanwhile the behavior of the member under load is described by a Constitutive relation (in this case, a linear relation between the stress and strain in the member).

Introductory fun with matrices!

If you’re going to dig at all into FEA you had best get comfortable with matrices, matrix manipulations, and linear algebra. We assemble the force and displacement vectors, and use the constitutive relations between the forces and displacements (or stresses and strains) to develop an assembled matrix that relates the two vectors. Viola! we have our first element stiffness matrix!

I can tell you’re losing interest, so…

Hopefully that slide motivates you to continue to press on. FYI, p. 15 in the above slide refers to the first edition of the referenced text.

We’ll leave it at that for now, and leave this topic for some time as we really need to get going deeper on some of our other subjects. Please come back to this post as needed to understand this very high-level instruction. It will be a great help as we dig-in.

Leave a comment

Your email address will not be published.