Fundamentals of Finite Element Analysis.

This ppt will help reader specifically for the beginners to understand basics of Finite Element Analysis (FEA) course. It covers procedure, advantages, disadvantages and applications of FEA. It also covers few basic terms used in FEA.

1. FUNDAMENTALS OF FINITE
ELEMENT ANALYSIS
BY
Dr.VINAYAK H.KHATAWATE
Associate Professor,
SVKM’s Dwarkadas J. Sanghvi College
of Engineering, Mumbai.

2. INTRODUCTION
•METHODS TO SOLVE ANY ENGINEERING
PROBLEM
1.ANALYTICAL METHOD
2.NUMERICAL METHOD
3.EXPERIMENTAL METHOD

3. Analytical Method Numerical Method ExperimentalMethod
-Classical approach
-100 % accurate results
-Closed form solution
-Applicable only for simple
problems like cantilever &
simply supported beams etc.
-Complete in itself
-Mathematical representation
-Approximate, assumptions
made
-Applicable even if physical
prototype not available (initial
design phase)
-Real life complicated
problems
-Results can not be believed
blindly & must be verified by
experimental method or hand
calculation for knowing the
range of results
-Actual measurement
-Time consuming & needs
expensive set up
-Applicable only if physical
prototype is available
-Results can not be believed
blindly & min. 3 to 5
prototypes must be tested
Though analytical methods
could also give approximate
results if the solution is not
closed form, but in general
and broad sense, analytical
methods are considered as
closed form solutions i.e.
100 % accurate.
Finite Element Method:
Linear, Nonlinear, Buckling,
Thermal, Dynamic & Fatigue
analysis
Boundary Element Method:
Acoustics / NVH( noise
vibration and harshness)
Finite Volume Method:CFD
(Computational Fluid
Dynamics) &
Computational
Electromagnetics
Finite Difference Method:
Thermal & Fluid flow analysis
(in combination with FVM)
-Strain gauge
-Photo elasticity
-Vibration measurements
-Sensors for temp. &
pressure etc.
-Fatigue test

4. Procedure for solving any
analytical or Numerical problem
There are two steps
Step 1)Writing governing equation -Problem Definition or in other
words formulating the problem in the form of a mathematical
equation.
Step 2)Mathematical solution of governing equation.
Final result is summation of step 1 & step 2. Result will be 100 %
accurate when there is no approximation at either of the steps
(Analytical method).Numerical methods make approximation at step
1 as well as at step 2 & hence all the numerical methods are
approximate.

5. Analytical
Approximation"
Numericall
Approximation
Step 1Little or no
approximation
Approximate
Step 2 Little or no
approximation
Approximate
AccuracyHigh accuracy Approximate results

6. THE BASIC CONCEPT OF THE
FINITE ELEMENT METHOD
•The most distinctive feature of the finite element
method that separates it from others is the
division of a given domain into a set of simple
subdomains, called finite elements.
•Here the basic ideas underlying the finite
element method are introduced via following
simple example.
•Determination of the circumference of a circle
using a finite number of line segments

7. Approximation of the Circumference of a Circle
Consider the problem of determining the perimeter of a circle of radius R .
a) Circle of radius R
b) Uniform and nonuniform meshes used to represent the circumference of the circle.
c) A typical element.

8. STEPS IN FEA
1.Finite element descretization
2.Element equations.
3.Assembly of element equations and
solutions.
4.Convergence and error estimate.

9. 1.Finite element discretization.
•First, the domain (i.e., the circumference of the circle) is represented as a
collection of a finite number (n) of subdomains, namely, line segments.
This is called discretization of the domain.
•Each subdomain (i.e., line segment) is called an element.
•The collection of elements is called the finite element mesh.
•The elements are connected to each other at points called nodes.
•In the present case, we discretize the circumference into a mesh of five
(n = 5) line segments.
•Mesh may be either uniform or nonuniform mesh (see Fig )

10. 2.Element equations.
•A typical element (i.e., line segment, he ) is
isolated and its required properties, i.e., length,
are computed by some appropriate means. Let
he be the length of element he in the mesh. For
a typical element he, he is given by (see Fig. )
• he = 2R sin (1/2) θe
•where R is the radius of the circle and θe <π is
the angle subtended by the line segment. The
above equations are called element equations.

11. 3.Assembly of element equations
and solution.
•The approximate value of the circumference (or perimeter) of the
circle is obtained by putting together the element properties in a
meaningful way; this process is called the assembly of the element
equations. It is based, in the present case, on the simple idea that
the total perimeter of the polygon (assembled elements) is equal to
the sum of the lengths of individual elements:
n
•Pn =∑ he
e=1
•Then Pn represents an approximation to the actual perimeter, p. If
the mesh is uniform, or he is the same for each of the elements in
the mesh, then θe = 2 π /n, and we have
• Pn = n( 2R sin [π/n] )

12. 4.Convergence and error
estimate.
•For this simple problem, we know the exact
solution: p = 2 π R.
•We can estimate the error in the approximation and
show that the approximate solution Pn converges to
the exact p in the limit as n tends to infinite.
•Consider the typical element Ωe. The error in the
approximation is equal to the difference between
the length of the sector and that of the line segment.
i.e.,Ee=Se-he. (see Fig. C):
•where Se = R.θe is the length of the sector

14. 5.conclusion
•it is shown that the circumference of a
circle can be approximated as closely as
we wish by a finite number of piecewise-
linear functions.
•As the number of elements is increased,
the approximation improves, i.e., the error
in the approximation decreases.

15. 6.Advantages of the Finite Element Method
•Can readily handle complex geometry:
•The heart and power of the FEM.
•Can handle complex analysis types:
•Vibration
•Transients
•Nonlinear
•Heat transfer
•Fluids
•Can handle complex loading:
•Node-based loading (point loads).
•Element-based loading (pressure, thermal, inertial
forces).
•Time or frequency dependent loading.
•Can handle complex restraints:
•Indeterminate structures can be analyzed.

16. Advantages of the Finite Element Method (cont.)
•Can handle bodies comprised of nonhomogeneous materials:
•Every element in the model could be assigned a different set of
material properties.
•Can handle bodies comprised of nonisotropic materials:
•Orthotropic
•Anisotropic
•Special material effects are handled:
•Temperature dependent properties.
•Plasticity
•Creep
•Swelling
•Special geometric effects can be modeled:
•Large displacements.
•Large rotations.
•Contact (gap) condition.

17. 7.Disadvantages of the Finite Element Method
•A specific numerical result is obtained for a specific problem. A
general closed-form solution, which would permit one to
examine system response to changes in various parameters, is
not produced.
•The FEM is applied to an approximationof the mathematical
model of a system (the source of so-called inherited errors.)
•Experience and judgment are needed in order to construct a
good finite element model.
•A powerful computer and reliable FEM software are essential.
•Input and output data may be large and tedious to prepare and
interpret.

18. Disadvantages of the Finite Element Method (cont.)
•Numerical problems:
•Computers only carry a finite number of significant digits.
•Round off and error accumulation.
•Can help the situation by not attaching stiff (small) elements
to flexible (large) elements.
•Susceptible to user-introduced modeling errors:
•Poor choice of element types.
•Distorted elements.
•Geometry not adequately modeled.
•Certain effects not automatically included:
•Buckling
•Large deflections and rotations.
•Material nonlinearities .
•Other nonlinearities.

19. 8.FEA Application Areas
•Automotive industry
•Static analyses
•Modal analyses
•Transient dynamics
•Heat transfer
•Mechanisms
•Fracture mechanics
•Metal forming
•Crashworthiness
•Aerospace industry
»Static analyses
»Modal analyses
»Aerodynamics
»Transient dynamics
»Heat transfer
»Fracture mechanics
»Creep and plasticity analyses
»Composite materials
»Aeroelasticity
»Metal forming
»Crashworthiness
•Architectural
»Soil mechanics
»Rock mechanics
»Hydraulics
»Fracture mechanics
»Hydroelasticity

20. •Primary Variable (PV)
•Secondary Variable (SV)
9.Types of Variables in FEA:

21. 21
Field Variables:
Primary Variables (PV)
These are the Dependent
Variables
y = f(x)
•These are the Variables derived
from PVs
•Mostly, derivatives of PVs: y`, y``
etc.
•Less accurate than PVs
•Obtained only after finding out
PVs
Secondary Variables (SV)

22. 22
10.Domain:
ADomainisaFieldwithendboundaryconditions
u(0) L u(L)
u
x
u = f(x)
Scalar/Vector Field with End
Boundary Conditions
L—Extent of Domain

23. 23
Definition of Boundary Conditions:
Specifiedvaluesoffieldvariables(orrelatedvariablessuch
asderivativesoffieldvariables)ontheboundariesofthe
FieldorDomain.

24. 24
u(0)=1 u(L)=2 Essential Non-Homogeneous BCs
u(0)=0 u`(L)=1 Mixed Non-Homogeneous BCs
u(0)=0 u(L)=0 Essential Homogeneous BCs
u`(0)=0 u`(L)=0 Natural Homogeneous BCs
Homogeneous
BCs (BCs = 0)
u(0)=0 u`(2)=1 BCs:PV-SV
Mixed BCs
u`(0)=0 u`(2)=1 BCs:SV-SVNatural BCs
(Neumann BCs)
u(0)=0 u(4)=2 BCs:PV-PVEssential BCs
(Dirichlet BCs)
Classifications of Boundary Conditions (BCs):
Non-homogeneous
BCs (one or both
BCs 0)

25. 11. Process Flow in a Typical FEM
Analysis
Start
Problem
Definition
Pre-processor
•Creates geometry of
nodes and elements
(ex: ANSYS,
Hypermesh, etc.)
•Assigns material
property data.
•Assigns boundary
conditions (loads and
constraints.)
Processor
•Generates
element shape
functions
•Calculates master
element equations
•Calculates
transformation
matrices
•Maps element
equations into
global system
•Assembles
element equations
•Introduces
boundary
conditions
•Performs solution
procedures
Post-processor
•Prints or plots
contours of stress
components.
•Prints or plots
contours of
displacements.
•Evaluates and
prints error
bounds.
Analysis and
design decisions
Stop
Step 1, Step 4
Step 6
Steps 2, 3, 5

26. 12.Information Available from Various Types of FEM
Analysis
•Static analysis
•Deflection
•Stresses
•Strains
•Forces
•Energies
•Dynamic analysis
•Frequencies
•Deflection (mode
shape)
•Stresses
•Strains
•Forces
•Energies
•Heat transfer analysis
»Temperature
»Heat fluxes
»Thermal gradients
»Heat flow from
convection faces
•Fluid analysis
»Pressures
»Gas temperatures
»Convection coefficients
»Velocities

27. Thank You!