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…
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)…
With that, let’s get started on some basics…
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…)
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.
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).
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…
But, I hear you saying…
Asked (maybe), and answered…
Hopefully you’re convinced, so let’s go…
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.
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).
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!
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.