Let's start with a bird's eye view on the finite-element method--FEM.
These days it's the workhorse of almost all mechanical engineers.
FEM can answer, for instance, what happens when you put a huge truck on a bridge.
To compute this deformation is application of FEM to the static case,
but FEM can also be applied in a dynamic setting,
for instance, to simulate the effect of a crash on a car body.
Here it's important to look at the process of buckling.
Whereas in the static case we're not really interested in how the truck was placed on the bridge.
It just has to be there.
I want to outline three fundamental ideas of the finite element method.
The first is discretization.
The continuous structures are approximated with the help of--guess what--finite elements--
elements of finite size, not infinitesimal size.
When we do so the first question is which geometry these finite elements should have.
Should they be tetrahedra?
Should they be cubes?
Or should they even be curvilinear?
The second fundamental idea is that of interpolation.
Given the finite elements, how do I compute a value at an arbitrary location?
For the static case, a really fundamental idea is that of minimization
of the potential energy.
Think about a ball that rolls on a terrain of mountains and valleys.
Eventually, it's going to come to rest in a valley.
In the static case, potential energy is minimized.
Let's have a closer look at that valley.
In the Gedankenexperiment, it splices this object a little further to the left
or a little further to the right.
Then the energy stays almost the same because we're at the bottom of the valley,
which means that to displace this object in this way require no work.
This is the concept of virtual work.
For all infinitesimal displacements that are allowed--
we can't go down and we can't go up, obviously--
the virtual work equals 0.
In mathematics, this way of posing the problem with the help of virtual work
is called a weak form.
The strong form would be to ask for all forces to compensate.
The weak form asks for the virtual work to be equal to 0 for all allowed virtual displacements.
This weak form results in a finite number of equations that we can solve on the computer.
This finite number, however, may range in the hundred thousands or even millions.