### Finite Element Method

When it comes to engineering stress analysis, complex geometries such as the one shown above – not really complex though considering the average complexities that engineers have to deal in their everyday world – often hinder an effective analysis on internal stress distributions within engineering materials. There are several analytical methods out there. However, the method can easily be overwhelmed by geometrical complexities making them almost useless in most of the cases.

The Finite Element Method (FEM) was initially developed in attempt to solve stress distribution of lattice structured airplane wings. Instead of attempting to find an exact solution, the object to be analyzed is divided into subsections in such a way that the boundary conditions around each subsection can be simplified and therefore become manageable in terms of stress calculations. This is called "meshing" since we are creating a virtual representation of the actual geometry with a mesh-like structure. Once this initial preparation is done, the next step that follows is basically a repetition of simple stress modeling within each element (subsections) and properly inter-relating results between a network of elements, ultimately creating a distribution of stress within the entire body.FEM is a very powerful way as it is robust and versatile. But, best of all, the idea of "breaking down to little pieces, analyze, then reattach them together," makes it applicable pretty much every geometrical setup.

FEM is not only for a simple analysis of engineering stress within a body. For example, the Laplace equations for thermal conduction processes are notorious for its insolvability if the geometry changes even a slightest bit from, say, only few known geometries where analytical solutions are known (I know off the top of my head only four geometries where an exact analytical solution is known). FEM says, no worry, I just have to break it down into manageable sizes and reattach them together!

The little picture above shows an internal stress distribution of a metal arm geometry which is subjected to a torque. One can easily identify the location of maximum stress and minimum. The figure was created using Abaqus.