Introduction
Hello and welcome to the theory introduction page, lets get started!
What is the finite element methodology and what can be used for??
The finite element methodology represents a technique in which a numerical approximation of the mass and stiffness characteristics of a structure is created in order to analyze and evaluate how an equivalent real structure would behave. In fact this methodology can be applied to structures or processes that can be mathematically characterized.
A numerical approximation? That sounds like I would need to know a lot of complicated math stuff!
Well that depends… Today there are a wide variety of finite element programs that incorporate the mathematic into their software, so you won’t have to do it. Although, here we are talking about engineering stuff… in order to understand it, you need to have some engineering knowledge. In fact the more you know… the better you will be at performing simulations and interpreting the results.
I have heard that with a finite element program, you can simulate anything and see how things will behave in real live.
As I said before the finite element models represents a mathematical approximation (keep in mind approximation) of a real structure, situation, process, etc. Therefore, by performing a correct simulation (keep in mind correct) you will be able to obtain an accurate enough estimation of what will happen in real live. It is quite common among engineers that simulate to state that their models represent reality… and that is never right! A properly made model in the best situation will be able to provide an accurate enough estimation of the real structure, process, etc.
Can you give me an example I’m not sure I fully understand.
Let’s say we have a rules embedded at one end having a weight attached at the free end, we want to know how much it will deflect. In real live you would go and just measure the deflection with a ruled, dial gauge or any measuring means you have. In the finite element world… first you will have to choose the element type to use (beams, shells, volumes) each one of this element types has its pros and cons on the complexity of the analysis and the accuracy of the results. But let’s say you want to do it the best way possible so you use the volume elements. The first problem is how to model the ruler and how accurate are you able to do it…. Let’s say the ruler is 1000 mm length, 30 mm width and 3 mm thickness.
The first problem comes here… is the ruler 1000 mm long or maybe 958 or 1004? Is it 30 mm wide of 30.5 or 29.8 or 30.9, etc? Finally is it 3 mm thick all over or maybe there are regions with 2.85 and 3.02 and 2.98 etc.
Then you have to assign the material properties and you know that you ruler is made out of aluminum… well problems again what kind of aluminum?? High grade, low grade, aerospace, what density, what stiffness, etc. Even if you have the properties from the manufacturer these properties vary from ruler to ruler and even along the length of the same ruler… there can be impurities, flaws of fabrication and so on.
But let’s say you deal with all these problems and decide to model you ruler as 1000 mm long, 30 mm wide, 3 mm thick and made of low grade aluminum. Then you simulate it and compare the results with what you have measured.
In 99.99% of the time (not to say 100%) your results will not be equal to those of the real structure and that is because the structures and processes are characterized by multiple random factor that your model is not taking into consideration or that cannot simply take them into consideration. Actually the more complex stuff you simulate the bigger the changed of your results varying from the real situations.
Well then if I’m not able to simulate the real live stuff this way, why would anybody even use them?
Good question… If you create a correct model (this implies a lot) you will be able to obtain a good enough approximation of the real process. For example, let’s say we measured that our ruler deflected on average (because even here there are variations) 15 mm and that our model estimated a deflection of 13.8 mm that means that our model presented a deviation of more or less 8%... at this point each individual need to estimate if that 8% is good enough or not… You can always improve your models by tuning up the data or your model but regularly that requires a lot of money and time. In the modern industry the finite element methodology has been more and more accepted and implemented and that is because it has some advantages to the classic engineering approach.
This next graphic will help you understand better:
This figure belong to Adams & Askenazi, 1999. In there you can see the development costs and knowledge about a product, during the concept, market and production fazes. With the classic engineering approach when you needed to develop a product it was necessary to have the idea, verify it, create a prototype and the start actually having some knowledge about it, this approach was quite complex and expensive, sometimes too expensive. With a finite element model you can have the idea, create the model, verify it mathematically (obtaining information about it) and then finally when you are much more confident about your last design actually proceed to build the prototype. This approach is much more productive and allows you to obtain better products.
It seems there is a lot more about the finite element methodology than I initially thought!!
Yes, there is a lot! And can be quite complex… but with the right tools and knowledge any good engineer can be a great analyst. I will recommend you some books that you could refer to in case you want to know more about it (Bibliography)
Clasification
Welcome to the theory  classification section, .....
I have heard about continuous and discrete structures but I have no idea what that means?
I'll explain it to you now ... If you make a classification of structures depending on how the distribution of its elements is done, you can distinguish two categories:
 Discrete: those structures formed by elements distinct from one another and joined together at certain points.
 Continuous: those structures in which if you select a part of the whole, the number of points of attachment with the rest of the structure represents an infinite number.
So what is the difference between them ??
The main difference between these two categories lays in the characterization of the deformations, displacement, etc, of the points when subjected to certain stresses.
 Discrete: With a finite number of parameters, you can define the deformation, displacement, etc, in an exact way. The deformation vector is defined by the previously mentioned parameters and the structure will be able to deform along as many directions as number of parameter the vector will contain.
 Continuous: in this case the deformation is not given by a vector composed of a finite number of parameters, but requires a vector function to indicate the deformation at any point.
The aim of the finite element method (FEM), is based on the discretization of a continuous medium in order to obtain the deformation and other parameters at any point. The model is created by imaginary lines and generated elements of the continuous medium with normalized geometric shapes called finite elements.
In the next picture you can see the discretization of a continuous medium into a discreet medium.
If I understood correctly, in the example there is a continuous medium and four discrete ??
Yes indeed. The four discrete ones, present different forms of discretization. With tetrahedral elements (pyramids) or hexahedrons with and without intermediate nodes per edge.
What are the nodes? I do not understand what are they and what do they do?
The entities known as "nodes" are the connection points between the finite elements. Displacements or other parameters of the nodes are the basic unknowns of the problems. These are independent unknowns since the rest of unknowns can be obtained as combination of others.
For each element a further interpolation functions that calculates the values ??of the parameters anywhere in the element need to be defined.
All this seem to be quite complicated ....
Well it depends ... It can be very complicated, but once you get the theoretical basis... everything will be much easier . I hope this information has been helpful ... I will recommend you a series of books if you want to obtain more information.
Types of Elements
I have heard that there are many element types, could you explain that to me?
Yes, I will , but only the main ones because there's a great variety of different elements.
Based on their geometrical characteristics there are three main groups:

 Beam elements (unidimensional):
These elements are defined only in one direction throughout two nodes I, J, sometimes it might be necessary to define an additional 3rd node K for the element orientation.

 Shell elements(bidimensional):
These elements are defined along two directions, depending on their mathematical definition they can have a minimum of 3 nodes and generally the maximum number of nodes is 8.

 Solid elements (threedimensional)
These elements are defined along all three directions of the space. Depending on their mathematical definitions they can't have a minimum of 6 nodes and generally can reach up to 20 nodes.
But why do we need different types of elements, what's the advantage of complicating this stuff?
Well this question is quite complicated….There are multiple reasons for that and one of them is to simplify the models and reduce the computational requirements.
I will explain it to you in a very simplified manner, although this topic is very complex. The volume type elements represent the most general elements throughout which you can perform any type of simulations, nevertheless they also represent the elements that require the most computational resources.
The Shell type elements represent a particular case derived from the volume type elements in which one of the dimensions is discarded (thickness).
These elements allow you to analyze shell type elements, like metal sheets, membranes, etc. Also, due to the fact that they have a smaller number of nodes, they also require less computational resources.
Furthermore, the beam type elements represent an even more particular case in which two of the dimensions are discarded. These elements can be used to simulate beams or tubular structures, they represent the element type that requires the least amount of computational resources since they have the smallest number of nodes.
Indeed all of these seem to be quite complex
It's true that there are multiple aspects that need to be taken into consideration and a very significant amount of mathematics behind all of this stuff…. In case you would like to know more about them, I recommend you these following books.
Bibliografy
In this section we want to provide a brief but useful list with a recommended bibliography of finite element books.
In the theory section we have presented, the most fundamental aspects of the finite element methodology with the aim of helping the most basic user to understand a little bit o this methodology.
As we mentioned along the document, the finite element methodology presents a significant amount of mathematics and an important degree of complexity. One of the fundamental aspects to be able to obtain good and reliable simulations lies in the necessity of having a good theoretical background. As any engineer knows... you don't need to know everything in order to be good, you just need to know where to find the information and know how to interpret it.
In the next paragraphs we will present you some of the best finite element books you can find in the world. We hope that this recommended bibliography will be useful for you and your professional activity.
Authors  Book name  Observations 

Vince Adams and Abraham Askenazi  Building Better Products with Finite Element Analysis  A great book that explains why the finite element methodology is necessary and how can be used in real live applications. Really good at explaining very useful tips and tricks it has a very reduced theory mathematical amount 
Olek C Zienkiewicz , Robert L Taylor , J.Z. Zhu  The Finite Element Method: Its Basis and Fundamentals  This book is one of the best and most comprehensive. It presents in detail all of the theoretical aspects of the finite element methodology. The authors of this book are among the most important in the finite element world and they have many more publications. 
J. S. Przemieniecki  Theory of Matrix Structural Analysis  If you need to get into the core of the finite element methodology, understand were things come from, how they been discovered and developed... this is your book. Mostly recommended for researchers and very specialized users. 
S. S. Quek , G.R. Liu  Finite Element Method: A Practical Course  This book presents a very interesting analysis of the finite element methodology from the practical point of view but also having a substantial amount of theory. Both its authors have other specialized publications on finite element analysis. 
David V. Hutton  Fundamentals of Finite Element Analysis  This book treats most of the theoretical aspects of the finite element methodology in a very appealing way and with proper illustrations that make it very easy to read. 
J Reddy  An Introduction to the Finite Element Method  This author represents an important reference of the finite element methodology world. Any time you will search for a book among the top authors Reddy will be present. 
Vladimir D. Liseikin  Grid Generation Methods  If for some reason you need to access the core of the meshing methodologies and mathematical aspects of it, you will find out that there is another full topic with tons of mathematical theory, and very big books... If that is your situation you might want to look at this great book about grid generation methods. 
Joe F. Thompson, Bharat K. Soni, Nigel P. Weatherill  Handbook of Grid Generation  Again if your need are related to the meshing aspects. This very comprehensive book might have all the answers for you questions... At least if they are mathematical it probably will! 