Divide and conquer.
But first, let's back up and discuss what is being conquered. FEA is applied to many types of problems, such as temperatures in consumer electronics, airflow around aircraft, and magnetic fields in electric motors. By far the most common application is structural FEA -- determining how a solid body responds to various forces. The structural problem amounts to writing down some "governing equations" that describe the material and how it behaves, and then solving those equations for the physical part being analyzed subject to how it is held and loaded. This can be done on paper for some simple part shapes. The resulting "closed form solution" is another equation that provides the answer in terms of the basic variables, such as the part's dimensions.
But reality intervenes, and most parts are too complicated to solve in closed form. FEA comes to the rescue by providing a "numerical solution" for each individual problem. This is a large gathering of numbers approximating the desired answers, such as displacements and stresses, across the part. But each solution is unique to a specific case; there is no simple answer in equation form.
Now then, how does FEA divide and conquer the problem to provide the numerical solution? The answer lies in the name, "Finite Element Analysis".
"Analysis" is obvious: the part is being analyzed under certain conditions.
"Element" describes a small section of the part. In fact, the governing equations mentioned above can generally be derived by considering a small section, writing the equations for what's happening in that section, and then mathematically allowing the size of the section to become infinitesimal, or infinitely small. In FEA, each section is called an "element", and the elements are not made infinitely small.
"Finite", then, refers to the countable number of elements used to represent the structure. The elements are of finite, measurable size. A computer can handle the computations on this finite number of elements.
Each element acts on its neighboring elements. FEA assembles the equations from all the elements into one large matrix equation, and the computer is used to determine the numerical solution. A key concept of FEA is this: if the elements are made small enough and are spread advantageously across the part, the numerical solution can closely approximate reality.
An experienced analyst can prepare the finite element model such that it accurately predicts the part's behavior, and can ensure that the solution algorithms do not interject significant errors. Results from the less-experienced are often suspect, and identifying them as so can be a tremendous advantage in the courtroom.
