# The Finite Element Method ( Computational E&M: 490D )

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```					The Finite Element Method

(Computational E&M: 490D)
Contents
 Introduction

 Variational Methods

 Finite-Element Analysis of 1D Problems

 Finite-Element Analysis of 2D Problems

 FEM vs. FDM

16.02.2012         Finite Element Method   2

B.C.:         n6               n 8        n  14

  3.1415926

16.02.2012         Finite Element Method            3
1941: A. Hrenikoff, Solution of Problems in Elasticity by the Framework Method,
J.Appl.Mech., Trans. ASME, vol.9, pp. 169-175, 1941

1943: R. Courant, Variational Methods for the Solution of Problems of Equilibrium
and Vibration, Bull.Am.Math.Soc., vol. 40, pp. 1-43, 1943

1956: M. Turner, R. Clough, H. Martin, and L. Topp, Stiffness and Deflection Analysis
of Complex Structures, J.Aero.Soc., vol. 23, pp. 805-823, 1956.

1960: J.H. Argyris and S. Kelsey, Energy Theorems and Structural Analysis,
Butterworth Scientific Publications, London, 1960

The first usage of the term “finite element” was in 1960:
R. Clough, The Finite Element Method in Plane Stress Analysis, J.Struct.Div.,
ASCE, Proc. 2d Conf.Electronic Computation, pp. 345-378, 1960

16.02.2012                    Finite Element Method                          4
Introduction: The Basic Concept
n6

element    element
element   element

element          element                element            element

element   element
element    element

Finite-element mesh (1)                    Finite-element mesh (2)

16.02.2012                  Finite Element Method                     5
Introduction: The Basic Concept
1
Area   Height  Foot 
2

R                                           h
h  2b                                      R         2b
 2                                          2

                                           
b  R sin          h  R cos                    b  R tan
2               2                            2
16.02.2012                   Finite Element Method                      6
Introduction: The Basic Concept
Finite-element mesh (1)                         Finite-element mesh (2)

1     R2     2                                 2                
a             sin                               a           R tan
2

2       n                                                     n
Element equation

n                                                   n
A  a
1          (1)
i                             A    2
 a     (2)
i
i 1                                             i 1

1 R2   2                                         2                     
A  n sin                                         A            nR tan 2

2     n                                                                 n
16.02.2012                       Finite Element Method                           7
Introduction: The Basic Concept
Finite-element mesh (1)                  Finite-element mesh (2)

e1  Areasector  a1                  e2  Areasector  a 2

1 2
Areasec tor    R
2

 1    2                                    
e1  R   sin
2
                       e2  R  tan  
2

 n 2    n                                   n n

16.02.2012                  Finite Element Method                    8
Introduction: The Basic Concept
Finite-element mesh (1)                    Finite-element mesh (2)

1           n    2                     2                 
E          R    sin
2
               E           R  n tan   
2

    2     n                                     n    
1                                       2
R  A2
A              R   2

tan  y
2 sin  x
 2
1
lim A  lim R                             lim A               lim R    2
n                  y 0     y
n         x 0      x
2 cos  x                                             sec 2  y
 lim  R                                            lim  R 2
x 0         1                                      y 0         1
  R2                                                R2
16.02.2012                     Finite Element Method                            9
Introduction: The Basic Concept

Convergence of the finite-element solutions to the exact one   A0   R 2

16.02.2012                     Finite Element Method                          10
Introduction: Some Remarks
 One can discretize the domain into a mesh of more than one type of element.
If more than one type of element is used in the representation of the domain,
one of each kind should be isolated and its properties developed.

16.02.2012                    Finite Element Method                       11
Introduction: Some Remarks
 The governing equations are generally more complex. They are usually
differential equations. In many cases, the equations cannot be solved over an
element for two reasons. First, the equations do not permit the exact solution. It
is here that the variational methods come into play. Second, the discrete
equations obtained in the variational methods cannot be solved independent of
the remaining elements because the assemblage of the elements is subjected
to certain boundary and/or initial conditions. The discrete equations associated
with the finite-element mesh are solved only after the boundary and/or the
initial conditions have been imposed.

 For time-dependent problems, usually a two-stage formulation is followed. In
the first stage, the differential equations are approximated by the finite-element
method to obtain a set of ordinary differential equations in time. In the second
stage, the differential equations in time are solved exactly or further
approximated by either variational methods or finite-difference methods to
obtain algebraic equations, which are then solved for the nodal values.

16.02.2012                    Finite Element Method                        12
Introduction: Some Remarks
 There are three sources of error in a finite-element solution:
(a) errors due to the approximation of the domain,
(b) errors due to the approximation of the solution, and
(c) errors due to numerical computation (e.g., numerical integration and
round-off errors in a computer).
The estimation of these errors, in general, is not a simple matter.

 The accuracy and convergence of the finite-element solution depends on
the differential equation solved (or the variational form used) and the
element used. The word "accuracy" refers to the difference between the
exact solution and the finite-element solution, and the word "convergence"
refers to the accuracy as the number of elements in the mesh is increased.

16.02.2012                 Finite Element Method                     13
Variational Methods
Method of Weighted Residual
d 2u
2
ux    u  0  0 u 1  0        0  x 1
dx
Trial function:
u  ax 1  x         u  0  0   u 1  0
Residual:
d 2u
R  2  u  x  2a  ax 1  x   x
dx
Because u is different from exact solution, R does not
vanish for all values of x within the domain.
16.02.2012               Finite Element Method               15
Method of Weighted Residual
1            d 2u
1           
I   wRdx   w  2  u  x  dx
0        0
 dx        
  w  2a  ax 1  x   x  dx
1

0

where w is test function (weight).

The resultant approximation of the solution differs
depending on the choice of the weight.

16.02.2012                Finite Element Method     16
Method of Weighted Residual
Some methods for the test function selection.
•Collocation Method
w    x  xi    xi is a point within the domain
•Least Squares Method
dR
w
da
•Galerkin’s Method
du
w
da
16.02.2012           Finite Element Method                      17
Methods of Weighted Residual
Exact Solution
u  0.5
0.0566
a  0.2222
u  0.2222x 1  x 
Collocation Method                            0.0556
w  2  x 1  x 
Least Square Method                           0.0576
u  0.2305x 1  x 
w  x 1  x 
Galerkin’s Method                             0.0568
u  0.2272x 1  x 
16.02.2012           Finite Element Method         18
Method of Weighted Residual
u  a1x 1  x   a2 x2 1  x 
d 2u
R  2  u  x  a1  2  x  x 2   a2  2  6 x  x 2  x 3   x
dx
•Collocation Method  w1    x  x1  w2    x  x2 
 w1  2  x  x 2
•Least Squares Method          
 w2  2  6 x  x 2  x3
 w1  x 1  x 

•Galerkin’s Method    

 w2  x 2 1  x 
16.02.2012              Finite Element Method               19
Weak Formulation
The problem has strong formulation:
d 2u
2
ux       u  0  0 u 1  0    0  x 1
dx
1         1   d 2u      
I   wRdx   w  2  u  x  dx  0
0        0
 dx        
Weak formulation of the same problem:
d u        
1
1                          du 
2
1                            dw du
I   w  2  u  x  dx            uw  xw  dx   w   0
0
 dx               0
 dx dx                   dx  0

16.02.2012                   Finite Element Method              20
Variational Method
2
u  0  0 u 1  0
d u
2
ux                                           0  x 1
dx

The variational expression:

 d u        
1
 du 
2
1
 J     2  u  x   udx    u 
0
 dx                   dx  0

   - is variational operator

16.02.2012                           Finite Element Method              21
Variational Method
 du d  u 
1                          
J                u u  x u  dx
                           
0 dx    dx
The variational operator is commutative with both
differential and integral operators, so
 1  du 2 1 2
1                 
 J        u  xu  dx
0  2 dx            
     2         
1  1  du         
2
1 2
J       u  xu  dx
0  2 dx            
     2         
16.02.2012                  Finite Element Method     22
Rayleigh-Ritz Method
2
u  0  0 u 1  0
d u
2
ux                               0  x 1
dx

Approximate solution:
u  ax 1  x 
1 2 1
                     
J  a  1  2 x   x 1  x  dx  a  x 1  x  dx
2            2        1
2                  2

2    0                               0

dJ
0       a  0.2272    u  0.2272x 1  x 
da
16.02.2012              Finite Element Method              23
Rayleigh-Ritz Method
Variational problem:                   B  v, u   l  v 
1
I  u   B  u, u   l  u 
2
Approximate solution has the form:
N
u N   c j j   0
j 1

       N            
B  i ,  c j j  0   l i         i  1, 2,   N
      j 1          
16.02.2012                 Finite Element Method                   24
Rayleigh-Ritz Method
2
 2  u  x 2  0 u  0  0 u 1  0
d u
0  x 1
dx
The bilinear and linear functionals are:
 dv du      
1                                      1
B  v, u            vu  dx         l  v     vx dx
2

0             
dx dx                             0

0  0 1  x 1  x         2  x2 1  x       N  x N 1  x 

uN  c1x 1  x   c2 x2 1  x        cN x N 1  x 
16.02.2012                Finite Element Method                    25
Rayleigh-Ritz Method
1
 d i  N    d j           N               1

  dx   c j dx         i  c j j  dx    i x dx
2
                                     
0        j 1               j 1             0

bij  B i ,  j  
1
li  l i    x  x  x           dx   i  3i  4
2     i    i 1                  1
0

sin x  2sin 1  x 
The exact solution is: u  x                          x 2
2

sin1
16.02.2012                 Finite Element Method                   26
Rayleigh-Ritz Method

The Ritz solution for N=3 coincides with the exact
solution up to 4 decimal points

16.02.2012     Finite Element Method                          27
Finite-Element Analysis
of 1D Problems
Basic Steps Involved in the FEM
•Discretization or subdivision of the domain

•Selection of the interpolation function

•Formulation of the system of equations

•Solution of the system obtained

•Postprocessing of the results

16.02.2012           Finite Element Method       29
1D FEM: Domain Discretization

16.02.2012   Finite Element Method   30
1D FEM: Element Equations

Quadratic functional associated with variational form
for the finite element:

16.02.2012          Finite Element Method          31
1D FEM: Element Equations
Substitution of the Ritz approximation into the
variational form yields:

16.02.2012           Finite Element Method       32
1D FEM: Interpolation Functions
Linear approximation:

In matrix form for one finite element:

16.02.2012           Finite Element Method   33
1D FEM: Interpolation Functions
After substitution and term collection:

16.02.2012           Finite Element Method   34
1D FEM: Interpolation Functions
Properties of the interpolation functions:

Finite-element model:

16.02.2012           Finite Element Method   35
Finite-Element Analysis
of 2D Problems
2D FEM: Variational Formulation

16.02.2012   Finite Element Method   37
2D FEM: Variational Formulation

The variation form is given by:

16.02.2012          Finite Element Method   38
2D FEM: F-E Formulation
The finite element model is presented by (after
substitution of the approximation function into the
variational form):

16.02.2012          Finite Element Method            39
2D FEM: Interpolation Functions
In 1D case:   n  r 1

In 2D case such correspondence is not unique.

3 nodes

4 nodes

5 nodes

16.02.2012         Finite Element Method         40
2D FEM: Interpolation Functions

16.02.2012   Finite Element Method   41
2D FEM: Interpolation Functions

16.02.2012   Finite Element Method   42
2D FEM: Interpolation Functions

n   p  1
2

16.02.2012   Finite Element Method                  43
2D FEM: Triangulation
Triangle is a 2D mesh generator and Delaunay Triangulator.
It was written by Jonathan Shewchuk (http://www.cs.berkeley.edu/~jrs/)

Winner of the 2003 James Hardy Wilkinson
Prize in Numerical Software

The homepage: http://www-2.cs.cmu.edu/~quake/triangle.html

16.02.2012               Finite Element Method                     44
2D FEM: Poisson Equation

Solution by linear triangular elements:

16.02.2012          Finite Element Method   45
2D FEM: Poisson Equation

All four elements are identical.
Thus, we can proceed with only
one element, say element 1

16.02.2012   Finite Element Method          46
2D FEM: Poisson Equation
For element 1:

16.02.2012      Finite Element Method   47
2D FEM: Poisson Equation

16.02.2012   Finite Element Method   48
2D FEM: Poisson Equation

16.02.2012   Finite Element Method   49
Finite Element Method
vs.
Finite Difference Method
FEM vs. FDM
top  100

Air                                            Air

Air
Silicon                                      Silicon

Silicon

bottom  100

19616 nodes
FEM, BELA Program                                       38984 elements

16.02.2012                      Finite Element Method                          51
FEM vs. FDM
9.000e+001 : >1.000e+002
8.000e+001 : 9.000e+001
7.000e+001 : 8.000e+001
6.000e+001 : 7.000e+001
5.000e+001 : 6.000e+001
4.000e+001 : 5.000e+001
3.000e+001 : 4.000e+001
2.000e+001 : 3.000e+001
1.000e+001 : 2.000e+001
0.000e+000 : 1.000e+001
-1.000e+001 : 0.000e+000
-2.000e+001 : -1.000e+001
-3.000e+001 : -2.000e+001
-4.000e+001 : -3.000e+001
-5.000e+001 : -4.000e+001
-6.000e+001 : -5.000e+001
-7.000e+001 : -6.000e+001
-8.000e+001 : -7.000e+001
-9.000e+001 : -8.000e+001
<-1.000e+002 : -9.000e+001
Density Plot: V, Volts

16.02.2012   Finite Element Method                                     52
FEM vs. FDM

FDM, grid 100x50

16.02.2012   Finite Element Method            53
FEM vs. FDM
top  100
Air                                                Air

Air

Silicon                                            Silicon

Silicon
bottom  0

2807 nodes
FEM, BELA Program                                           5408 elements

16.02.2012                      Finite Element Method                       54
FEM vs. FDM
9.500e+001 : >1.000e+002
9.000e+001 : 9.500e+001
8.500e+001 : 9.000e+001
8.000e+001 : 8.500e+001
7.500e+001 : 8.000e+001
7.000e+001 : 7.500e+001
6.500e+001 : 7.000e+001
6.000e+001 : 6.500e+001
5.500e+001 : 6.000e+001
5.000e+001 : 5.500e+001
4.500e+001 : 5.000e+001
4.000e+001 : 4.500e+001
3.500e+001 : 4.000e+001
3.000e+001 : 3.500e+001
2.500e+001 : 3.000e+001
2.000e+001 : 2.500e+001
1.500e+001 : 2.000e+001
1.000e+001 : 1.500e+001
5.000e+000 : 1.000e+001
<0.000e+000 : 5.000e+000
Density Plot: V, Volts

16.02.2012   Finite Element Method                                   55
FEM vs. FDM
•33

FDM, grid 10x10

16.02.2012   Finite Element Method            56
FEM vs. FDM

FDM, grid 50x100

16.02.2012   Finite Element Method            57
FEM vs. FDM
Air        top  100                      Air

Air

Silicon                                    Silicon

Silicon

bottom  100
6617 nodes
FEM, BELA Program                                          12950 elements

16.02.2012                         Finite Element Method                    58
FEM vs. FDM
9.000e+001 : >1.000e+002
8.000e+001 : 9.000e+001
7.000e+001 : 8.000e+001
6.000e+001 : 7.000e+001
5.000e+001 : 6.000e+001
4.000e+001 : 5.000e+001
3.000e+001 : 4.000e+001
2.000e+001 : 3.000e+001
1.000e+001 : 2.000e+001
1.421e-014 : 1.000e+001
-1.000e+001 : 1.421e-014
-2.000e+001 : -1.000e+001
-3.000e+001 : -2.000e+001
-4.000e+001 : -3.000e+001
-5.000e+001 : -4.000e+001
-6.000e+001 : -5.000e+001
-7.000e+001 : -6.000e+001
-8.000e+001 : -7.000e+001
-9.000e+001 : -8.000e+001
<-1.000e+002 : -9.000e+001
Density Plot: V, Volts

16.02.2012   Finite Element Method                                     59
FEM vs. FDM

FDM, grid 100x100

16.02.2012   Finite Element Method            60
FEM: Example – Circle

top  100
Air                                     Air

Air
Silicon                                  Silicon

Silicon

bottom  100

744 nodes
1437 elements

16.02.2012                          Finite Element Method                   61
FEM: Example – Circle
9.000e+001 : >1.000e+002
8.000e+001 : 9.000e+001
7.000e+001 : 8.000e+001
6.000e+001 : 7.000e+001
5.000e+001 : 6.000e+001
4.000e+001 : 5.000e+001
3.000e+001 : 4.000e+001
2.000e+001 : 3.000e+001
1.000e+001 : 2.000e+001
-1.421e-014 : 1.000e+001
-1.000e+001 : -1.421e-014
-2.000e+001 : -1.000e+001
-3.000e+001 : -2.000e+001
-4.000e+001 : -3.000e+001
-5.000e+001 : -4.000e+001
-6.000e+001 : -5.000e+001
-7.000e+001 : -6.000e+001
-8.000e+001 : -7.000e+001
-9.000e+001 : -8.000e+001
<-1.000e+002 : -9.000e+001
Density Plot: V, Volts

16.02.2012   Finite Element Method                                62

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