Finite Difference Methods
- Approximate the derivatives in the governing PDE
using difference equations.
- They are useful in solving heat transfer and fluid
mechanics problems.
The methods work well for 2-D regions with boundaries
parallel to the coordinate axes.
- However, they are cumbersome on curved and irregular
shaped boundaries.
-It is difficult to write a general purpose computer
program for the finite difference methods.
The Finite Element Method (FEM)
- Easily applied to irregular shaped objects
- Multiple materials (composites) are treated without
difficulty.
-Mixed BCs can be applied
- General purpose codes for whole classes of problems
can be written.
The FEM combines several mathematical concepts to
produce a system of linear or non-linear equations. It
has little value without a computer since equations range
from 20  20K or more.
There are 2 standard FEM treatment techniques:
a.) The Variational Method, and
b.) Weighted Residual Methods
Prof. J.M. Sullivan, Jr.
ME 515 Finite Element Lecture - 1
1
Using Calculus of Variation it can be shown that if:
1.) U = g(x) yields the lowest integration value of:
2
D  dU 
[
0 2  dx   QU]dx
L
then
2.) U is the solution of the D.E.
d 2U
D 2 Q  0
dx
with BCs U(0) = U0 and U(L) = UL
2
The term:
D  dU 
[ 
  QU]
2  dx 
is the approximate functional.
This solution strategy is NOT applicable for any D.E.
containing first derivative terms. The variational
approach originated in stress analysis which involved
even order derivatives only. (Fluid mechanics
traditionally use FDM for their solutions since the
variational formulation was not valid for the NavierStokes set of equations.)
Prof. J.M. Sullivan, Jr.
ME 515 Finite Element Lecture - 1
2
The Method of Weighted Residuals
Takes the governing equations
d 2U
D 2 Q  0 , for example.
dx
It approximates U with Û .
Since Û does not satisfy the D.E. an error or residual
results, i.e.,
d 2Uˆ
D 2  Q  R  0, necessaril y
dx
The method of WR requires that this residual, when
multiplied by a Weighting function, W, and summed
over the region be zero.
WR dx = 0
What we use for Û and W distinguishes the various forms
of MWR
b
Recall:
f g   f ( x) g ( x)dx
a
is the definition of an inner product
Hence, if WR dx = 0, it implies that R is orthogonal to W
L
The  R( x)W ( x)dx has i weight functions (1 for each node,
i
0
in our case, or degree of freedom in system, formally)
Prof. J.M. Sullivan, Jr.
ME 515 Finite Element Lecture - 1
3
The common WR methods are:
i
Collocation
ii
Subdomain
iii
Least Squares
iv
Galerkin
i.) Collocation =>
let Wi (x) = ( x- xi)
L
Then  R (x) δ (x  x )  0 for all x  x
i
i
0
= R(x) for x = xi
This requires that R(x) = 0 at specific points and
The number of points = number of undetermined
coefficients  a solution is available.
d 2U
D 2 Q0
dx
Ex:
Let D = EI ( resistance of the beam to deflection)
Q = - M (the moment)
M
L
Prof. J.M. Sullivan, Jr.
M B.C.’s U ( 0 ) =0
U(L)=0
ME 515 Finite Element Lecture - 1
4
The solution of U = y displacement;
2
EI d U  M(x)  0
dx 2
F.E.M.
Replace U with a guessed function, say
2Û
d
EI
 M(x)  R(x)  0
2
dx
N
U(x)  Û(x)   U N
j1 j j
Where Uj are undetermined coefficients and
Let Nj = basis function
 

= sin π jΔ   or many other possible basis fxns
L 

Ex. Lagrange Polynomials applied locally on each
element
n
(x  x i )
Nj  
i  j; n  # nodes in element
i 1 (x j  x i )
N j  0 for all nodes not in element
Exact Solution =
Prof. J.M. Sullivan, Jr.
U(x) 
Mo
2EI
XX  L
ME 515 Finite Element Lecture - 1
5
Collocation:
Trial fxn: U  Û =Uj Nj (implied summation)
 πX 

j  where X = j*x
= Uj sin 
j
 L 




Plug into O.D.E.
 EIU j π 
 L
2
πX
sin 
 L

j  - M
x



=
R(Xj)
Multiply by weighting fxn and integrate over Domain
L  
 R  X j  δ X  Xi  0
0


Let Xi = L/2; Then
 EIU j π 
 L
2
sin π  - M0 = 0
2
Solve for the unknown coefficient Uj
 M0L2
Uj
EIπ 2
which multiplies the basis function for the final
approximation of Û
Prof. J.M. Sullivan, Jr.
ME 515 Finite Element Lecture - 1
6
 M 0L2 πX j 
Û 
sin 

2
 L 
EIπ


ii) Subdomain
- Assume only 1 domain i.e.
Wi
1
L
0
etc.
2
1
Same R(Xj) as with collocation
L
L
 R jWidx   R jdx  0
0
0

πX
L  EI

j
2
  2 U jπ sin L  M dx  0
0  L

x

πX 
L
L

j
EI
2
 U π  sin
dx -  Mdx  0
j
L
2


L
0 
0

Prof. J.M. Sullivan, Jr.
ME 515 Finite Element Lecture - 1
7
Cos(X / L) L
j
 (1 1)  2
 /L
0
 2 EIπ U j  M0L  0
L
M0L2
Uj
2EI
M L2 πX j 
Û   0 sin 

2EI
 L 

Prof. J.M. Sullivan, Jr.

ME 515 Finite Element Lecture - 1
8
iii) Galerkin:
Let Wi=Nj
L
L
πx 
 R jWidx   R jsin  L i dx


0
0
 πX


π Xj
L  EI


(i

j)
2



 dx = 0
U π sin
 M0 sin 
=  

 L2 j

L
L


0



M0L2
Solving , U j  4
 3EI
Prof. J.M. Sullivan, Jr.
same as Variational Method
ME 515 Finite Element Lecture - 1
9
iv) Least Squares:
2

π Xj
L  EI
L
L
2


 R jWidx   R jR idx    2 U jπ sin L  M0  dx  0
0 L
0
0

Error=
U j2
2
 EIπ 2 

L
2 
L


2
4M0EIπ

U j  M02L
L
Minimize
2
Error  U L EIπ 2   4M0EIπ  0
j  L2 
L
U


j
M0L2
;
U j  4
3
 EI
Prof. J.M. Sullivan, Jr.
πX
Û  U j sin 
 L

ME 515 Finite Element Lecture - 1
j 



10
Variational Method:

L  EI  du 2
=      M0U dx
0 2  dx 


N
approximate U(x)  Û(x)   U N
j1 j j


2
 πX  


πX
L  EI 
π


j
j


=     j U cos
   M0  j U j sin 
dX
j

 L 
L  L 
0  2 






L

 πX  

 

j

 

cos
2







L  

U π 
 πX j 

j
 EI 

 L

  1 X  1 sin 2 
  M U
  

π
 L 
π
0
j
 2  L   2 j 4






 


L





0
EI U 2π 2L 

4L2 j
2M LU
0 j
π
Minimize:
  EI π 2 U  2M 0L  0
π
j
U  2L
j
Prof. J.M. Sullivan, Jr.
ME 515 Finite Element Lecture - 1
11
4M0L2
Uj 
π 3EI
4M0L2 πX j 
Û  
sin 

3
 L 
π EI

Prof. J.M. Sullivan, Jr.

ME 515 Finite Element Lecture - 1
12
FEM
SOLUTION
STRATEGY
Variational Approach
Method of Weighted Residuals
Conservative Systems
Use PDE
Potential Energy of system
Stationary
Guess a solution to U
Apply a weighting fxn
PE  0 k  1, N
U k
Integrate
Functional
Require residual to
be zero globally
Integrate
Algebraic set of Equations
Apply B. C.s
Matrix Solver
Solution
Prof. J.M. Sullivan, Jr.
ME 515 Finite Element Lecture - 1
13