The document provides an introduction to the finite element method (FEM). It explains that FEM is a numerical method used to approximate solutions to partial differential equations. It works by dividing a complex problem into smaller, simpler parts called finite elements. This allows for the problem to be solved computationally. The document outlines the basic steps of FEM, including preprocessing (modeling, meshing), solving, and postprocessing (analyzing results). It also discusses applications, history, software, element types, meshing, convergence, and compatibility conditions of FEM.

1. Introduction To FEM
Prepared By:Anubhav Singh

2. •FEM-An Introduction
Finite element method is a powerful tool for getting the numerical solution of a wide
Range of complex problems such as tedious mathematical formulations which are
Generally not possible by analytical methods in engineering.
Any Engineering problem can be solved by 3 methods:
Numerical method, Analytical method & Experiments.

3. •FEM-An Introduction
Finite element method is based on a simple idea of building a complicated
object
With simple blocks or dividing it into small & manageable sections.
It is used to determine the approximated solution for a partial differential
equations (PDE) on a defined domain (W). To solve the PDE, the primary
challenge is to create a function base that can approximate the solution.
There are many ways of building the approximation base and how this is
done is determined by the formulation selected. The Finite Element Method
has a very good performance to solve partial differential equations over
complex domains that can vary with time.
Why FEM?
Degree Of Freedom
In this example, an object is fixed at one end and a force is Force applied at
the point “P”. Due to the force, the object deforms and point P gets shifted to
new position P’.

4. •When Can We Say That We Know the Solution to The
Above Problem?
If and only if we are able to define the deformed position of each and every
particle completely.
The minimum number of parameters (motion, coordinates,
temperature, etc.) required to define the position and state of any
entity completely in space is known as degrees of freedom (dof)
The total DOFs for a given mesh model
is equal to the number of nodes
multiplied by the number of dof per
node.

5. All of the elements do not always have 6 dofs per node. The number of dofs
depends on the type of element (1D, 2D, 3D), the family of element (thin shell,
plane stress, plane strain, membrane, etc.), and the type of analysis. For
example, for a structural analysis, a thin shell element has 6 dof/node
(displacement unknown, 3 translations and 3 rotations) while the same element
when used for thermal analysis has single dof /node (temperature unknown).
The Finite Element Method only makes calculations at a limited (Finite)
number of points and then interpolates the results for the entire domain
(surface or volume).

6. Finite – Any continuous object has infinite degrees of freedom and it is
not possible to solve the problem in this format. The Finite Element
Method reduces the degrees of freedom from infinite to finite with the
help of discretization or meshing (nodes and elements).
Element – All of the calculations are made at a limited number of points
known as nodes. The entity joining nodes and forming a specific shape
such as quadrilateral or triangular is known as an Element. To get the
value of a variable (say displacement) anywhere in between the
calculation points, an interpolation function (as per the shape of the
element) is used.
Method - There are 3 methods to solve any engineering problem. Finite
element analysis belongs to the numerical method category.

9. Applications Of FEM In Engineering
Stress analysis on components and assemblies using FEA (Finite Element
Analysis);
• Thermal and fluid flow analysis Computational fluid dynamics (CFD);
• Kinematics;
• Mechanical event simulation (MES).
• Analysis tools for process simulation for operations such as casting, molding,
and die press forming.
• Optimization of the product or process.
A brief History Of FEM
•In 1943, Courant developed Variational method which became basis of FEM
•In 1960 FEM was termed by Clough & various engineering problems were
Solved in it.
•In 1967 first book on FEM was published.
•In 1970s most commercial FEM software packages were developed.
•In 1980s many pre & postprocessors were developed
•In 1990s analysis of large structural systems were developed by FEM.

10. FEM in Structural Analysis & computer
implementations
1) Pre PROCESSING
Creating the 3D Model
Setting up of Boundary Condition
Setting up Loads
Creating Nodes & Elements (MESHING)

11. FEM in Structural Analysis & computer
implementations
Boundary Conditions
Loads

12. FEM in Structural Analysis & computer
implementations
Meshing 2) SOLVING
The software solves the model with
given loads, Boundary conditions
and gives Max. Min. Stress Strain
through out the body.
In analysis cycle manual solving
takes Maximum time and with CAE
it takes minimum time.

13. FEM in Structural Analysis & computer
implementations
3) Post PROCESSING
Post Processing is used for displaying results
Type Of Results Required
Stress Von Mises
Max. Stress Theory Etc.
Strain

14. Available Commercial FEM Software Packages
•ANSYS
•SDRC/I-DEAS
•NASTRAN
•ABAQUS
•COSMOS
•ALGOR
•PATRAN
•HYPERMESH
•DYNA-3D
MESHING
•Meshing is an uniform network of elements & nodes. It is of two types:
1. Manual or structural meshing:In this method,The nodes are plotted by the user
who provides the coordinates & then,The elements are created by joining the
plotted nodes.
2. Automatic meshing:In this method,The user will provide the element size or
number of divisons as input & software will generate nodes & elements.

15. Types Of Elements
1. 1-d Element-Stress in these elements act in one direction & the
loading is uniaxial.It’s shape is a straight line.Examples-Bar,Beam,Rod.
The modeling is simple & results are accurate.
2. 2-D Element/Plane/Shell Element-Here stresses act in 2 directions.If
the thickness of a component is less than or equal to 1/10th of
dimension of component,Then a 3-D problem can be assumed as 2-D
problem. It’s shape can be a 3 noded triangular or 4 noded
quadirateral.

16. Types Of Elements
Element behaviour in 2-D:
1. Plane Stress-Stress is in XY Plane only.
2. Plane strain-Strain is in XY Plane only.Used in simulation of
components having infinite length or very long parts.Ex-Ship body &
Railway track.
3. Axi-Symmetric-This is applied while simulating revolved sections.Ex-
Pressure vessels.
4. Plane stress with thickness-Ex-Plate with a hole helps in
understanding stress concentrations.
3. 3-D Elements/Solid/Volume Elements-Here stresses act in all 3
directions.It’s shape can be 4 noded tetrahedral or 6 noded
hexahedral.

17. Higher order elements or parabolic elements
Post discretization in FEA, all the elements are assigned a function (a polynomial)
which would be used
to represent the behavior of the element. Polynomial equations are preferred for this
as they can be easily
differentiated and integrated. Order of an element is the same as the order of the
polynomial equation
used to represent the element.
•A Linear element or First order element will have nodes only at the corners. This is
something like the
Edge Centered Cubic Structure.
•However, a Second order element or Quadratic element will have mid side nodes in
addition to nodes at
the corner (edge + body + face centered cubic structure).

18. Higher order elements or parabolic elements
A linear element in the above diagram clearly has two nodes per edge and hence
needs only a Linear
equation to be assigned to represent the element behaviour.
However, a Quadratic element needs a quadratic equation to describe its behaviour
as it has three nodes.
For elements in which you would like to capture curvature, higher order polynomials
are preferred. First
order elements cannot capture curvature.
The order of the element has nothing to do with geometry. In the below diagram, for
the same triangle,
first order as well as second discretization can be done but second order has good
chances of capturing
curvature.

19. Higher order elements or parabolic elements
To accurately capture complex curvatures, very high order polynomials are needed
but they come at the
cost of increased computational time. Hence, its better to have a trade off between
degree of accuracy
and computational time.
Now, lets talk of number of nodes between first and second order elements

20. Convergence
Convergence means obtaining solution to closest value of accuracy.Most linear
problems don’t need an iterative solution procedure.In non-linear problems
convergence is an important term.
H & P Mesh convergence
H convergence-
To check convergance,difference
between two Consecutive result values
should be between 1 & 2%.

21. H-Method
More accurate information is obtained by increasing the number
of elements.The name for the h-method is borrowed from mathematics. The
variable h is used to specify the step size in numeric integration. If a part is
modeled with a very course mesh, then the stress distribution across the part
will be very inaccurate. In order to increase the accuracy of the solution,
more elements must be added. This means creating a finer mesh. As an
initial run, a course mesh is used to model the problem. A solution is
obtained. To check this solution, a finer mesh is created. The mesh must
always be changed if a more accurate solution is desired. The problem is run
again to obtain a second solution. If there is a large difference between the
two solutions, then the mesh must be made even finer and then solve the
solution again. This process is repeated until the solution is not changing
much from run to run.

22. P Method
The p in p-method stands for polynomial. Large elements and complex
shape functions are used in p-method problems. In order to increase the
accuracy of the solution, the complexity of the shape function must be
increased. Increasing the polynomial order increases the complexity of the
shape function.The mesh does not need to be changed when using the p-
method.As an initial run, the solution might be solved using a first order
polynomial shape function. A solution is obtained. To check the solution the
problem will be solved again using a more complicated shape function. For
the second run, the solution may be solved using a third order polynomial
shape function. A second solution is obtained. The output from the two
runs is compared.If there is a large difference between the two solutions,
then the solution should be run using a third order polynomial shape
function. This process is repeated until the solution is not changing much
from run to run.

23. The simple explanation is:In h method we decrease the size of existing
elements ,Going towards more finer mesh & then comparing solution
differences between Two optimal sized meshes & when we observe
uniformness in solutions,We say,Convergence is achieved.
In p-method,We don’t go for fine mesh but change the order of
elements(introducing One or two mid-size nodes between each
elements.This method is suitable where Curvature in geometries exist
because more nodes tend to create a element representing more
accurate shape of a curvature.

24. Compatibilty Conditions
When a continuum is divided into numerous elements the elements
deforms due to application of load.This condition sees to it that the
deformation should not be discontinuous or the deformed elements should
not overlap.
If the convergence conditions & compatibility conditions are satisfied then
the element is termed as conforming or compatible elements.
If convergence conditions are satisfied & compatibility conditions are not
Satisfied then the elements are called as non-conforming elements.

25. Thanks