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NARESUAN UNIVERSITY
FACULTY OF ENGINEERING
The Finite Element Method
by Dr.Udomrerk
Introduction
The finite element method is a very powerful method for the numerical solution of
a wide range of the physical problems that may be characterized by energy
theorems, functionals or differential equations with a prescribed set of boundary
conditions. These problems may be as diversed as structural, elasticity, heat
transfer, fluid flow, magnetic field, soil-structure interaction, and fluid-structuresoil interaction problems.
Finding a solution that satisfies a differential equation throughout a domain (or
region) as well as the boundary conditions is very difficult (if not impossible) for
all but the most elementary problems. The fundamental concept of the finite
element method is to approximate any continuous quantity, such as,
displacement, stress function, temperature or pressure, etc. by a discrete model
comprising defined over a finite number of sub-domains (or elements) as to
satisfies the differential equation in “average sense” at a finite number of points
(or nodes) in the domain.
Example:
Cantileve Beam
Plane Frame
References
1. Bathe, K.J. (1982). Finite element procedures in engineering analysis. Prentice Hall.
2. Chaung, Y.K. and Yao, M.F. (1979). A practical introduction to finite element analysis.
Pitman.
3. Grandin, H. (1986). Fundamentals of the finite element method. Macmillan.
4. Segerlim, L.J. (1984). Applied finite element analysis. Wiley.
5. Smith, I.M. (1982). Programming the finite element method – with application
to geomechanic. Wiley.
6. Zienkiewicz, O.C. (1977, 1983). The finite element method. McGraw – Hill
Solution for Boundary Value Problems
The best way to solve any physical problem governed by a differential equation is
to obtain the analytical solution. There are many situations, however, where the
analytical solution is difficult or impossible to obtain. In this part of the notes, the
numerical solution of boundary value problem is illustrated using a simple one –
dimensional problem where the analytical solution can be obtained.
>>> Analytical Solution
Consider the following 1-D problem of a rod subjected to a distributed load.
T(x) = Cx
x
L
Consider a segment of the rod of length, dx
T(x)
P
P
dP
.dx
dx
dx
The governing differential equation may be obtained by considering the
equilibrium of forces as follows
dP 

.dx   P  T ( x).dx  0
P
dx


or
dP
 T ( x)  0
dx
One-dimensional axial force
(1)
P X A
(2)
where  X = axial stress
A = cross-sectional area
Stress-strain relationship
 X  E. X  E
du
dx
(3)
Substitute Eq.(2) and (3) into (1)
d 
du 
 AE
  T ( x)  0
dx 
dx 
(4)
This is the general form of the differential equation in which A&E may very with x.
If A&E are constants, then
AE
d 2u
dx  T ( x)  0
dx 2
(5)
The boundary conditions for this problem
At
x=0, u=0
du
0
x=L,
dx
( zero strain and hence stress-free)
Consider problem where T(x) varies linearly with length, i.e.
T(x) = C . x
where C= constant
Eq.(5) becomes
AE
d 2u
 Cx
dx 2
Integrating once,
 AE
d 2u
dx    Cxdx
dx 2
AE
du
Cx 2

 A1
dx
2
(6)
Integrating Eq.(6)
du
Cx 2
dx   
 A1dx
dx
2
Cx3
AEu  
 A1 x  A2
6
 AE
(7)
Evaluating boundary conditions to give A1& A2
At x=0, u=0.
or
At x=L,
Eq.(7) gives
0 = 0+0+ A2
A2= 0
du
 0.
dx
Eq.(6) gives
CL2
 A1
2
CL2
A1 
2
0
or
Thus general solution to this problem is ( from Eqn 7 )
u
1  CL2
C 
x  x3 

AE  2
6 
(8)
>>> Numerical Solution (continuous trial function over whole domain)
1. Weighted Residual Method
1 CL2 
x2 
ˆ
a) Collocation Method : u 
x

2 AE 
2L 
b) Subdomain Method: uˆ 
1 CL2 
x2 
x



2 AE 
2L 
c) Least-Squares Method: uˆ 
d) Galerkin Method : uˆ 
1 CL2 
x2 
x

2 AE 
2L 
5 CL2 
x2 
x



8 AE 
2L 
>>> Comparison Between methods
It can be seen that the collocation, subdomain, and least-square methods all
gave the same solution for this particular problem. In general, this will not be so
far all problems.
Summary
Taking A = 1, E =1, L =1 and c =1, the solutions for different x values are
X
Exact
Least-Sqart Galerkin
0
0.2
0.4
0.6
0.8
1.0
0
0.099
0.189
0.264
0.315
0.333
0
0.090
0.160
0.210
0.240
0.250
0
0.112
0.200
0.262
0.300
0.312
The approximate numerical solutions can be improved by taking polynomials can
be improved by taking polynomials of higher degrees.
2. Principle of Minimum Potential Energy
The solution is uˆ 
5 CL2 
x2 
x


 which is similar to Galerkin Method.
8 AE 
2L 
Final Exam ( 2006 ) 304652 Finite Element Analysis
1. Determine the approximate solution on a simple beam shown in Fig. A. By using
a) Weighted Residual Method and b) Principle of Minimum Potential Energy.
W / unit length
L