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CHAPTER 1

An Overview of the Finite Element Analysis 

 
1.1 Introduction 

Finite element analysis (FEA) involves solution of engineering problems using 
computers. Engineering structures that have complex geometry and loads, are either very 
difficult to analyze or have no theoretical solution. However, in FEA, a structure of this 
type can be easily analyzed. Commercial FEA programs, written so that a user can solve 
a complex engineering problems without knowing the governing equations or the 
mathematics; the user is required only to know the geometry of the structure and its 
boundary conditions. FEA software provides a complete solution including deflections, 
stresses, reactions, etc. 
 

In order to become a skillful FEA user, a thorough understanding of techniques for 
modeling a structure, the boundary conditions and, the limitations of the procedure, are 
very crucial. Engineering structures, e.g., bridge, aircraft wing, high-rise buildings, etc., 
are examples of complex structures that are extremely difficult to analyze by classical 
theory. But FEA technique facilitates an easier and a more accurate analysis. In this 
technique the structure is divided into very small but finite size elements (hence the name 
finite element analysis). Individual behavior of these elements is known and, based on 
this knowledge; behavior of the entire structure is determined. 
 

FEA solution of engineering problems, such as finding deflections and stresses in a 
structure, requires three steps: 
 

1.  Pre-process or modeling the structure 
2. Analysis 
3.  Post processing  

 
A brief description of each of these steps follows. 
 

Step1: Pre-process or modeling the structure 

Using a CAD program that either comes with the FEA software or provided by another 
software vendor, the structure is modeled. The final FEA model consists of several 
elements that collectively represent the entire structure. The elements not only represent 
segments of the structure, they also simulate it’s mechanical behavior and properties. 

Chapter 1 –An Overview of the Finite Element Analysis 

Regions where geometry is complex (curves, notches, holes, etc.) require increased 
number of elements to accurately represent the shape; where as, the regions with simple 
geometry can be represented by coarser mesh (or fewer elements). The selection of 
proper elements requires prior experience with FEA, knowledge of structure’s behavior, 
available elements in the software and their characteristics, etc. The elements are joined 
at the nodes, or common points. 
 
In the pre-processor phase, along with the geometry of the structure, the constraints, loads          
and mechanical properties of the structure are defined. Thus, in pre-processing, the entire 
structure is completely defined by the geometric model. The structure represented by 
nodes and elements is called “mesh”.  
 

Step 2: Analysis  

In this step, the geometry, constraints, mechanical properties and loads are applied to 
generate matrix equations for each element, which are then assembled to generate a 
global matrix equation of the structure. The form of the individual equations, as well as 
the structural equation is always,  
 

{F} = [K]{u} 

 
Where 
           {F} = External force matrix. 
 

[K] = Global stiffness matrix 

{u} = Displacement matrix 

The equation is then solved for deflections. Using the deflection values, strain, stress, and 
reactions are calculated. All the results are stored and can be used to create graphic plots 
and charts in the post analysis. 
 

Step 3: Post processing  

This is the last step in a finite element analysis. Results obtained in step 2 are usually in 
the form of raw data and difficult to interpret. In post analysis, a CAD program is utilized 
to manipulate the data for generating deflected shape of the structure, creating stress 
plots, animation, etc. A graphical representation of the results is very useful in 
understanding behavior of the structure 
. 
 

1.2  History of FEA  

Engineering applications of finite element analysis is approximately 40 years old. 
Evolution of FEA is tied with the development in computer technology. With the 
enhancement in computer speed and storage capacity, FEA has become a very valuable 
engineering tool. NASA is credited for developing comprehensive FEA software in 
1960’s, known as NASTRAN. Rights of the software were purchased by  
McNeal Schwendler Corporation, who refined it and commercially marketed it under the 
name, MSC-NASTRAN. The first college course in FEA was offered in 1970. In the 

ME 273 Lecture Notes © by R. B. Agarwal 

Introduction to Finite Element Analysis 

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Chapter 1 –An Overview of the Finite Element Analysis 

early 1970’s, application of FEA was limited to large corporations, who can afford 
expensive mainframe computers. However, in 1980’s, with the introduction of desktop 
computers, application of FEA became popular and indispensable engineering tool. In 
late 80’s, almost all the major FEA vendors introduced their software that can run on a 
PC. 
 

In the past ten years, there were several significant development in FEA, including: 
. 

• 

Introduction of P- elements. 

• 

Integration of sensitivity analysis and optimization capabilities. 

• 

Availability of faster and cheaper desktop computers to run FEA software that 
previously required mainframe computers. 

• 

Development of powerful CAD programs for modeling complex structures. 

• 

Making software user-friendly. 

1.3 How FEA works – Within software 

 
The following steps can summarize FEA procedure that works inside software: 
 

Using the user’s input, the given structure is graphically divided into small 
elements (sections or regions) so that each and every element’s mechanical 
behavior can be defined by a set of differential equations. 

The differential equations are converted into algebraic equation, and then into 
matrix equations, suitable for a computer-aided solution. 

The element equations are combined and a global structural equation is obtained.  

Appropriate load and boundary conditions, supplied by the user, are incorporated 
in to the structural matrix. 

The structural matrix is solved and deflections of all the nodes are calculated. 

A node can be shared   by several elements and the deflection at the shared node 
represents deflection of the sharing elements at the location of the node. 

Deflection at any other point in the element is calculated by interpolation of all 
the node points in the element. 

An element can have a linear or higher order interpolation function. 

The individual element matrix equations are assembled into a combined structure 
equation of the form {F}=[k]{u}.  

 
As defined earlier,  
 

{F} = Column matrix of the externally applied loads. 
 
[k] = Stiffness matrix of the structure, which is always a symmetric matrix. 
This matrix is analogues to an equivalent spring constant of several connected 
springs. 
 

ME 273 Lecture Notes © by R. B. Agarwal 

Introduction to Finite Element Analysis 

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Chapter 1 –An Overview of the Finite Element Analysis 

{u} = Column matrix representing the deflection of all the node points, that 
results when the load {F} is applied. 

1.4  How FEA works – User’s interaction 
 

The above described software procedure is mostly transparent to the user. A user has the 
following interaction with the software, through user’s computer. 

Create the geometry, representing the structure: A CAD modeling software is 
used to create the structure’s geometry. 

Provide the material properties, loads, constraints, etc. 

Analyze the result data. 

 
1.5  Convergence – Assuring Optimum Mesh Size 
 

How do we determine the exact number of elements for a structure and make sure that the 
FEA mesh is optimum? There is no exact answer to this question; however, if we keep 
refining a mesh until the variation in the result is less than a specified value, we will 
reach the desirable mesh density. Convergence refers to this process, where we optimize 
the mesh to arrive at the desired results. In general, there are three types of convergences: 
 

1.  Von-Mises Stress (VMS) convergence

: Mesh is refined until the percentage 

variation in VMS is less than 1, 5, 10 or any given value selected by the user. 
VMS convergence should be avoided if there are stress concentration points, 
convergence will be difficult to achieve. 

2.  Strain Energy Convergence: 

Mesh is refined until the percentage variation in 

the average strain of elements is less than a chosen value. Strain convergence is a 
better criterion for optimizing an FEA mesh. Stress concentrations points do not 
significantly influence the average strain energy of elements and variation in 
strain energy is influenced by mesh size or polynomial order of the elements only.

3. Deflection Convergence: 

It is similar to the above convergences, except, node 

deflection values are used for the convergence criterion.

 
1.6  H- versus P- elements  

In FEA, there are two types of elements:  

1.  H-elements and,  
2. P-elements 
 

H-element is the original and “classic” element. The name is derived from the field of 
numerical analysis, where the letter ‘h’ is used for the step size, to achieve convergence 
in the analysis. The h-element is always of low order, usually, linear or quadratic. When a 
finite element mesh is refined to achieve convergence, the procedure is called h-
convergence. For h-elements, convergence is accomplished at the expense of excessively 

ME 273 Lecture Notes © by R. B. Agarwal 

Introduction to Finite Element Analysis 

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Chapter 1 –An Overview of the Finite Element Analysis 

large number of elements. The high stress concentration regions require a very fine mesh, 
thereby increasing the number of elements. Finite elements used by commercial programs 
in the 1970s and 80s, were all h-elements. However, with improvement in computer 
power and efficiency, a much more useful, p-elements were developed.  
 
P-elements are relatively new, developed in late 1980s and offer not only the traditional 
static analysis, they provides option of optimizing a structure. P-elements can have edge-
polynomial as high as 9

th

 order, unlike the low order polynomials of h-elements. The high 

polynomial edge order of p-elements makes it possible to model a curved edge of a 
structure with accuracy. Therefore, fewer elements can be used to achieve convergence. 
In FEA, the number of elements in the mesh usually remains fixed; convergence is 
achieved by increasing the polynomial order of the p-elements, rather than refinement of 
the mesh. For optimization, as the dimensions of the structure being analyzed are 
changed, the number of elements remains constant. Only, the polynomial order of the 
elements is changed as needed.  
 
 

1.6 Bottom-up and Top-down approach 

 
When modeling a structure (creating an FEA model), bottom – up approach refers to 
creation of model by defining the geometry of the structure with nodes and elements. 
These nodes and elements represent the physical structure. When an FEA model is 
created by this procedure, it is known as a bottom-up approach. This is the original 
procedure for creating FEA mesh, and requires a substantial investment in time and skill. 
When this method is employed, most of analyst’s time is devoted to creation of the mesh, 
and only a fraction of time is spent for analysis and results interpretation.  
 
In FEA, a top-down procedure refers to creation of FEA mesh by first building a solid 
model, using a 3-D CAD program, and then dividing the model into nodes and elements. 
Thus, the top-down method requires building of a geometric model of the structure and 
then using it to create an FEA mesh. The advantages of the top-down approach are 
obvious; we don’t have to define the geometry of individual elements in the structure, 
which can be very time consuming. Obviously, a 3-D model requires high-end computer 
hardware, along with familiarity with the modeling software. 
 
 

1.7  Discretization or Division of a structure into small elements 

 
In FEA, an engineering structure is divided into small elements. These elements coincide 
with the geometry of the structure and represent the geometry and the mechanical 
properties in the regions. 
 
Selection of elements to represent the structure is a matter of engineering judgement and 
prior experience with FEA procedure. A sound advice for beginners is: keep the elements 
size small enough to yield good results and yet large enough to reduce computational 
time. Smaller elements are desirable where the results are changing rapidly (change in 

ME 273 Lecture Notes © by R. B. Agarwal 

Introduction to Finite Element Analysis 

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