FEM     

     Many engineering applications cannot be modeled and therefore analyzed simply through differential equations and other mathematical methods.  This is due to their real world complications such as complex geometry, boundary conditions, loading conditions, and material composition.  In such cases, numerical methods must be employed in order to analyze a system for its desired properties.  Finite Element Methods (FEM) is one such numerical method in which a continuous function with a domain of infinite degrees of freedom is transformed into a set of discrete continuous regions known as finite elements.  Approximate functions for each of these finite elements can then be derived and totaled by superposition in order to obtain a global function that can easily be solved and analyzed.  The advantages to using FEM is that it can solve for bodies made of different materials, bodies with irregular shapes, and bodies that are subjected to many different loading conditions and boundary conditions.  If a portion of the material is subjected to high stress concentrations then the element size can be decreased to get a better approximation.  The element size is varied due to level of accuracy and cost of the system.