Hopefully by means of this post I get to clarify some of the topics, trying to remove excessive algebraic and mathematical verbosity that is so annoyingly present everywhere.
The main background of FEM is that of structural engineering in the 60s. Engineers are very practical people, so initially they devised a system which allowed them to set algebraic equations where the relations between different points or "nodes" of their structures could be set.
These algebraic relations were further proven to be of many different types, not only structural, so also thermal relations could be formulated and many others: all one needed to analyze was a proper discretizacion of the space in the form of a mesh with nodes to relate to each other.
STEP 1: Discretization
In this basic yet classic example, I have chosen to divide my sample structure in only 5 elements where nodes are clearly identifiable in the meetings between beams. There are four nodes.
|Degrees of Freedom|
|A sample structure (mesh) and its topology table|
STEP 2: Element characterization and shape functions
In this step is where FEM formulation and literature get really really awkward and nasty. In fact, this is the core of everything and where FEM differenciates from other ways of solving PDEs.
|Different types of Finite Elements|
|2DOF Beam element matrix|
|3DOF Beam Element matrix|
|3DOF Timoshenko Beam element matrix|
|Global Matrix Assembly|