Here, we pay homage to our hero in this difficult endeavor.
That's right, it's Shrek! That series of movies did more for the science of physics modeling, than 30 years of engineering. Those guys just recently won an Oscar for particle dynamics, which is used for simulating water, smoke, clouds, explosions, etc. They use big clusters of Linux computers, and each run probably uses more power than any seismic analysis. Unfortunately, all the smart guys are either working for Hollywood, or doing crash analysis for cars, and there isn't much literature available.
So, with Shrek in mind, we will take a totally different route for seismic analysis, starting with a clean slate, and using modern computing techniques. With an iPod being more powerful than the computers I used 30 years ago, we have no need for the old 'compute saving' shortcuts.
Now we start. Think of very small chunk of concrete, suspended in the air. This particle is subjected to various forces, such as gravity. It has a defined mass, measured in kilograms. Newton gave us a very great gift, to calculate what this particle will do next: F = M a, or a = F/M. That means the acceleration of the particle (defined as a 3D vector), will merely be a vector sum of all the forces divided by the mass (keeping consistent units is a bitch!).
Although this particle would love to be in a Hollywood explosion, for us, it is embedded in a civil structure, and will only fly in an earthquake, if it is made of Montreal Mafia concrete! But, assuming a normal structure, this 'hunk' has restraining forces. We model this as simple springs, all around the hunk, so that if it moves to the right, the spring force increases to push it back to the left.
In fact, this is all we really need to model the physics, except that we are now going to thousands of hunks. If the conglomeration (structure) is as boring as a Toronto suburb, then it's not moving. For that, we need a 'forcing function', such as would be provided by an earthquake.
The Shrekians perfected a marvelous technique of explicit time domain modeling. This what I used with my own computer code 30 years ago, and had I been smarter, I'd be rich! With it, we make the time steps very small, so that we can predict exactly what our hunk will do in the next time step, by summing the forces in this time step. This type of calculation is 'explicit' in that we don't have to do any fancy recursion or anything, and thus lends itself to be solved by large computer clusters.
In the next article, I shall finally open up a computer program, that I have never seen before, and have no great confidence in myself, to get it working!