# Closed-Form Solutions & the Case for God

First, let me give some background so you can maybe see why a nerd like me would make that connection. A closed-form solution, in this context, is where you can simply solve an equation  to find the unknown variable. For instance, in my practice of structural engineering, the deflection of a cantilever beam may be something I need to know as I’m sizing the beam. If the beam conforms to certain assumptions like a constant cross section, constant material stiffness, a uniform load, and so forth, I have a simple equation: $\Delta = \frac{wL^4} {8EI}$. If it’s a concentrated load at the end, there’s a slightly different equation. These equations are each derived from beam deflection theory for a specific boundary condition, like a cantilever, or a simple span beam, and they provide exact answers. We engineers like exact answers. It’s nice to be able to say “this beam will only deflect 1.21 inches under that load, which is still acceptable.” I like closed form solutions because they are directly solvable for the variable I’m looking for, but sometimes, even with tables of equations for dozens of different conditions, there are no closed-form solutions, or they are too complex to use, or it would take a while to derive the equation from scratch. An open-form solution like the approximation methods used in FEA is iterative and relies on the results of previous attempts. FEA models a component like a beam with lots of little pieces that can each respond differently, so I’m not quite as limited by simplifying assumptions. Think of a beam made out of lots of LEGO® bricks.  Each brick (a “finite element”) is connected to multiple other bricks, and the total strength of the beam depends on the behavior of all of those individual connections. In general, the smaller the bricks, the more accurately you can represent the beam. But as the number of LEGO® bricks increases, the time to calculate all of those interconnections increases exponentially. That type of solution gets complex pretty quickly, and requires a computer for any problem worth solving. But it also doesn’t produce an exact solution. It iterates, or repeats the calculations with different input values until the successive estimates begins to converge. In other words, it runs through the thousands of equations over and over until the results aren’t changing much with each pass, and are within a tolerance the user sets for what is “close enough”. And what is “close enough”? That’s going to vary with the user and the type of problem being solved. Also, another engineer could try solving the exact same problem with a different mesh size (i.e. bigger or smaller LEGOs) and arrive at different results since it’s not just the beam properties that determine the answer now, but the modeling choices like mesh sizes, convergence tolerances, and iteration method.  So why would I want to use a complicated, inexact, and sometimes difficult to verify process like that? FEA lets me solve things I couldn’t otherwise. Some problems get far outside the simplifying assumptions of our various formulas, and FEA (done correctly), is the best option for finding a solution, even if it isn’t exact.