Download 2.1 Introduction

Seoul National
University
Introduction to
Finite Element Method
Chapter 5
Two-Dimensional Element
Calculations
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5.1 Introduction
 In chapter 4, we considered the variational formulation
and finite-element approximation of a class of elliptic
boundary-value problems in two dimensions. We now
examine the necessary element calculations of 2D
problems in greater detail.
 Our objectives here are twofold: first, to describe a
general and systematic method for calculation element
matrices; and second, to describe how these
techniques can be incorporated into a representative
finite-element code for two-dimensional problems. Our
treatment of both of these subjects parallels that for
the one-dimensional problems considered in chapters
2.
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5.2 Element Transformation
5.2.1 Master element
M

x   xi i ( , )

 x  x( , )

i 1
Te : 

M
 y  y ( , )  y 
yi i ( , )


i 1

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5.2 Element Transformation
Ne
Unknowns : u   a j j ( x, y )
j 1
Transformation and Solution approximation
could be different
NE < M : superparametric Transformation
NE = M : isoparametric Transformation
NE > M : subparametric Transformation
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5.2 Element Transformation
The expression of dx, dy in terms of  , will be
x
x

dx    d    d

y
y
dy 
 d 
 d



dx   x, x,  d 
or    
 d 
dy
y
y
   , ,   
In terms of x, y, d and d are expressed by
d  1  y,  x,  dx   , x  , y  dx 
d    y x  dy      dy 
  J  , ,     , x , y   
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5.2 Element Transformation
5.2.2 Construction of Transformation
We now come to the important issue of constructing the element transformations
Te .
We are guided by the following criteria:
1. Within each element, the functions
   ( x, y ) and    ( x, y ) must be
invertible and continuously differentiable
(else we could not use the convenient relations(5.2.7) in element calculations)
2.The sequence of mappings
{Te }must generate a mesh with no spurious gaps
between elements and with no element overlapping another.
3.
Each map Te should be easy to construct from the geometric data for the element.
4 . The functions    ( x, y ) and    ( x, y ) defining the element maps should
be easy to manipulate mathematically
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5.2 Element Transformation
We know that the coordinate variables s x, y may themse lves be viewed
simply as linear functions on  e . Hence, the finite element shape functions
can be utilized to construct the map Te ,
M
ˆ ( , ),
Te : x   x j 
j
j 1
M
ˆ ( , )
y   y j
j
j 1

ˆ j
ˆ j
 1 M

1 M

  yj
,
   xj
x J j 1

y
J j 1



ˆ j
ˆ j

1 M
 1 M
   yj
,
  xj
 (5.2.10)
x
J j 1

y J j 1


ˆ j  M
ˆ j   M
ˆ j  M
ˆ j 
M
J   x j
 y j
   x j
 y j

  j 1
   j 1
  j 1
 
 j 1
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5.2 Element Transformation
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5.2 Element Transformation
* Special analysis in one dimensional case for the appropriate mapping
Let look at the 1-D mapping with three points
x  0  1  a  2  1  3
1
2
 1   (1   )
therefore,
1
if a  ,
8
 2  1  2
1
 a (1   2 )   (1   )
2

1

   a  2   a
2
2

1
2
 3   (1   )
1
0
2
1
  1, 1  2a   0
2
  1, 2a  1 
dx
1
 (1  2a)   0
d
2
1
4
3
a
4
a
2
3
1
1 3
2
1
x   2        
8
2
8 8
3  24
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5.2 Element Transformation
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5. 3 Finite Element Calculations
In 2-D Finite Element formulation, we have


(ku  v  uv)d


 u v u v 
  b( x, y )u ( x, y )v( x, y ) d

  k ( x, y )

 x x y y 


Let
N
N
N
j 1
j 1
j 1
u    j  j ( x, y )    j  j ( x( , ), y ( , ))    j j ( , )
and
x
x

 d


d


dx



dx   x, x,  d 
or
dy    y y  d 

y
y
   , ,   
dy 
 d
 d 



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5.3 Finite Element Calculations
Recalling the expressions on the element matrix and vector from Chapter 4,
   e  je  e  je 

e
e
i
i
  bi  j  dxdy
k   k 

 x x
e
y y 
 

e
ij
(5.3.1a)
f i e   f ie dxdy
(5.3.1b)
Pije  
pie  je ds
(5.3.1c)
 ie  
ie ds
(5.3.1d)
e
 e2 h
 e2 h
1 e  E
1  i, j  N e
 e2 h : boundary
We must evaluate these integrals in (5.3.1) by using the concept of the master
element.
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5.3 Finite Element Calculations
The derivatives in the integration can
be written in terms of the coordinates variables ξ and η by
 j ( x, y )
x


 j ( , ) 

x

 j ( , ) 

x
1
( j , y,   j , y, )
J
M
M
1

   j ,  yi i ,   j ,  yi i , 
J 
i 1
i 1

 j ( x, y )
y
 ,   , x  , , x
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5.3 Finite Element Calculations
5.3.1 Master-Element Calculations
We begin by choosing a master element and the appropriate shape functions and
will calculate integrals of eq.(5.3.1) in coordinates ξ and η.
1. Element Maps : Element coordinate map Te is
Ne
ˆ ( , ),
Te : x   x j 
j
j 1
Ne
ˆ ( , )
y   y j
j
(5.3.2)
j 1
the element Ωe to which Te maps ̂ is completely determined by specifying
the x, y-coordinates (xj, yj) of all the nodal points of Ωe
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5.3 Finite Element Calculations
2. Transformation of Shape Functions
Let g be a scalar-valued function of x and y defined on an element Ωe .
Then we can convert g to a function ĝ of ξ and η defined on ̂ by setting
g ( x, y )  g ( x( , ), y ( , ))  gˆ ( , )
where x(ξ , η) and y(ξ , η) are now given by (5.3.2)
Similarly, if ĝ (ξ , η) is given on ̂ , then under the transformation of (ξ , η)
into (x,y),
gˆ ( , )  gˆ ( ( x, y ), ( x, y ))  g ( x, y )
Thus, element shape functions  je  je ( x, y) for Ωe are simply obtained
ˆ ( , ) by
from 
j
ˆ ( ( x, y), ( x, y)),
 je  
j
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5.3 Finite Element Calculations
e
The derivatives of  j are obtained by the chain rule:
ˆ  
ˆ 

j
j


;
x
 x  x
 je
According to (5.3.2),
ˆ ( , )
x N e 
k
  xk
,
 k 1

ˆ ( , )

y N e
k
  yk
,
 k 1

ˆ  
ˆ 

j
j


y
 y  y
 je
ˆ ( , )
x N e 
k
  xk
 k 1

ˆ ( , )

y N e
k
  yk
 k 1

(5.3.5)
(5.3.6)
Thus, using (5.2.7) and (5.3.6), equations (5.3.5) become
ˆ Ne
ˆ Ne
ˆ
ˆ 





 
j
j
k
k 
y

y




k
k










k 1
k 1


ˆ N e 
ˆ N e 
ˆ
ˆ 
 je

1 
 
j
j
k
k 


x

x




k
k
y
J ( , ) 









k

1
k

1


 je
1

x
J ( , )
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(5.3.7)
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5.3 Finite Element Calculations
3. Integration
Let g and ĝ be the functions defined in (5.3.3). Then it is clear that

e
g ( x, y )dxdy  ˆ gˆ ( , ) J ( , ) dd

where,
 Ne ˆ
gˆ ( , )  g   x j  j ( , ),
 j 1

ˆ
y j  j ( , ) 

j 1

Ne
The Gauss quadrature could be used for the integration
And ,if Gˆ ( , )  gˆ ( , ) J ( , ) is to be integrated over ̂ ,
we use the formula

e
Nl
g ( x, y )dxdy  ˆ Gˆ ( , )dd   Gˆ ( l ,l ) wl  Eˆ

l 1
Eˆ is the quadrature error.
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5.3 Finite Element Calculations
4. Element Stiffness matrix and Load Vectors
The entries keij and fei in the element stiffness matrix and load vector defined
by (5.3.1a) and (5.3.1b) can be computed using (5.3.7) and the integration
rule (5.3.10)
If k=k(x,y), b=b(x,y) and f=f(x,y) are not constant, we may use their
interpolants,
Ne
k h ( x, y )   k j  ( x, y ),
j 1
e
j
Ne
bh ( x, y )   b j  je ( x, y ),
(5.3.11)
j 1
Ne
f h ( x, y )   f j  je ( x, y )
j 1
k j  k ( x j , y j ),
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5.3 Finite Element Calculations
5. Boundary Integrals
The calculation of Peij and γei of (5.3.1c) and (5.3.1d) is carried out by
integrating along those sides of the master element that are mapped onto the
sides  e2 h of Ωe along which natural boundary conditions are prescribed.
Let ˆ j denote the restriction of the master-element shape functions ̂ j to
side   1 :
ˆ (1, ),
ˆ j ( )  
j
j  1,2,, N e
(5.3.12)
It is clear that except for nodes j falling on the boundary ξ=1, ˆ j  0 .
We then have
P 
p  ds   pˆ ( )ˆi ( )ˆ j ( ) j ( ) d
 
  ds   ˆ ( )ˆi ( ) j ( ) d
e
ij
e
ij
 e2 h
 e2 h
e
i
e
i
1
e
j
1
(5.3.13)
1
1
where j is the Jacobian of the transformation of  into the arc-length
parameter s in the x, y-plane
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5.3 Finite Element Calculations
1/ 2
 x(1, )  2  y (1, )  2 
ds  

 d


      
 x(1, )   y (1, ) 
j ( )  



     
2
where
2
1/ 2



(5.3.14)
x( , ) and y ( , ) is defined in (5.3.2)
The one-dimensional Gaussian quadrature rule is needed for the integration
in boundary.
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5.3 Finite Element Calculations
5.3.2 Computational Aspests
1. Choose and Ωˆ and Ψˆ j , j  1,2,, N e and specify th e x, y - coordinate s
( x1 , y1 ), ( x2 , y 2 ), , ( xN , y N ) of nodal points of each element.
2. Specify a set of N l integratio n points ( l ,l ), l  1,2,, N l , and
ˆ
quadrature weights w for 
l
ˆ , 
ˆ /  , 
ˆ /  at the intergrati on points
3. Calculate the values of 
j
j
j
( l ,l ) (see (5.3.4)) .
4. Calculate the values of x  x( , ), y  y ( , ) of (5.3.2) and their
derivative s x /  , x /  , y /  , y /  , at the intergrati on points
( l ,l ) (see (5.3.6)) .
5. Calculate the values of the Jacobian J ( , ) and the functions  / x,
 / y,  / x,  / y at the integratio n points ( l ,l ) (see (5.2.7)).
6. Using (5.3.7) and the results of steps 3, 4, and 5, compute  je / x and
 je / y at each integratio n point ( l ,l ).
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5.3 Finite Element Calculations
7. Calculate the values of k , b, and f (or k h , bh , and f h ) at the integratio n
points ( l ,l ).
8. Using the results of steps 3 through 7, calculate the values of the
integrands in (5.3.1a), (5.3.1b) at the integratio n points ( l ,l ) and
multiply each by wl J ( l ,l ) .
9. Sum the numbers computed in step 8 in accordance with (5.3.10) to
obtain kije and f ije
A similar sequence of calculations can be performed to yield the
boundary integrals
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5.3 Finite Element Calculations
We note that the different shape functions for the coordinates mapping and the
solution approximation can be used the following mapping to coordinate mapping.
In other words, the order of the mapping functions can be higher or lower than the
order of the solution approximations as mentioned in the follows.
M
Te : x   x j j ( , ) ;
j 1
M
y   y j j ( , )
(5.3.15)
j 1
ˆ , k  1,2,, N
N e nodes and shape functions 
k
e
then, element maps are classified as follows:
M  N e : Te is a superparametric map
M  N e : Te is a isoparametric map
(5.3.16)
M  N e : Te is a suberparametric map
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5.3 Finite Element Calculations
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5.4 Two-Dimensional Finite Elements
The scheme for element calculations in the previous section can be employed for many
different kinds of finite elements. The specification of an element type entails a
prescription of the master element, the map Te, the shape functions, and the quadrature
rules. The information required to prescribe each of these items is given below.
1. Master element : the geometry of the master element ̂ and the
definitions of the element coordinate system ( , )
~
2. The map Te : the element coordinates ( j , j ) and shape functions  j for
the nodes, j  1,2, , M that define the map from ̂ to  e
~
3. Interpolation : the element coordinates ( j , j ) and the shape functions
j
N
e
for the nodes j  1,2,, N , that determine the approximation uhe   u ej  je
e
j 1
4. Quadrature rules : the number N l of integration points, the element
coordinates ( l ,l ) and weights wl (l  1,2,, Nl )
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5.4 Two-Dimensional Finite Elements
5.4.1 Quadrilateral elements
Finite elements with four sides are generally referred to as quadrilateral elements
~
* the shape function i for node i must assume a unit value at node i and vanish at all
other nodes of the element and along all sides of the element that do not contain node i
1) 4-node element : Fig.5.7 (a)
~
1
 1,  1
must vanish alongsides
and should be(1 at1,  1)
1
~
therefore the shape function
1 (is, )  (1   )(1   )
2) 9-node element : Fig.5.7. (b)
4
1
~
1 ( , )  (   2 )(   2 )
4
1
1
~
6 ( , )   (1   )(1   )(1   )  (   2 )(1   2 )
2
2
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5.4 Two-Dimensional Finite Elements
3) 8-node element (Fig. 5. 8 )
1
~
1 ( , )  (1   )(1   )( 1     )
4
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5.4 Two-Dimensional Finite Elements
Quadrature rules for quadrilateral elements are usually derived from one-dimensional
quadrature formulas by treating the integration over the master element as a double
integral.
1

 d
G
(

,

)
d

d


G
(

,

)
d

ˆ
1 1

1
and approximate both the integration with respect to ξ and with respect to η by onedimensional quadrature rules of order N
N

G
(

,

)
d

d


G
(

,

)
w

n wm

ˆ
m 1  n 1

N
(5.4.1)
when (m,n) with the single index l with Nl=N2,
N
N


 G ( , )wn wm   G ( , ) wl
m 1  n 1
l 1

N
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2
(5.4.2)
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5.4.2 Triangular Elements
master element;
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ˆ ( , )  1    , 
ˆ ( , )   , 
ˆ ( ,)  

1
2
3
(5.4.3)
Coordinate map Te ;
3
ˆ ( , ),
x   x j
j
j 1
3
ˆ ( , )
y   y j
j
(5.4.4)
j 1
( x j , y j ) : Coordinate of j - th node j  1,2,3
and then,

1
( y3  y1 )( x  x1 )  ( x3  x1 )( y  y1 )
2 Ae

1
( y2  y1 )( x  x1 )  ( x2  x1 )( y  y1 )
2 Ae
(5.4.5)
Ae : area of  e
  2e ( x, y ),   3e ( x, y ), 1      1e ( x, y )
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(5.4.6)
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5.4 Two-Dimensional Finite Elements
Area Coordinates
The expressions  , and 1     in (5.4.5) and (5.4.6) may be conveniently
interpreted in terms of area ratios.
Let ( , ) be the coordinates of an arbitrary point in ̂ and (x,y) the corresponding
point in  e . Suppose that we construct straight lines joining ( , ) and ( x, y ) to the
vertices, and denote by âi and ai the subtriangle areas opposite node i in ̂
and  e, respectively. The terms  , and 1     in (5.4.6) are easily seen to be
ˆ , where  =1/2 is the area of the master element. We define
the area ratios aˆi / A
the area coordinates on ̂ as
i 
aˆi
Aˆ
i  1,2,3
(5.4.7)
or
 1  1    ,  2   ,  3  
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(5.4.8)
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5.4 Two-Dimensional Finite Elements
Since the transformation Te is linear, the Jacobian J is constant and is the ratio of
the areas of  e and ̂. This implies that the map distorts areas uniformly. Thus,
the meaning of area coordinates is preserved under linear transformations and we
have
aˆ
a
i 
i
Aˆ

i
Ae
,
i  1,2,3
(5.4.9)
The subtriangles and coordinate lines are indicated in Fig. 5.12. It is important to
note that this property is not preserved if the map Te is nonlinear.
1. Let the  i  Coordinate of an arbitrary interior point P be (1 ,  2 ,  3 ) . The
coordinate line  i   i  constant is parallel to the side of the triangle opposite to
vertex i :  1  1 is a line parallel to side 2-3,  2   2 is parallel to 1-2, and
 3   3 is parallel to 1-2.
2. The coordinate lines  i  0 define the boundary of the triangle:  1  0 is the
side 2-3 opposite vertex 1,  2  0 is the side 1-3 opposite vertex 2, and  3  0
is the side 1-2 opposite vertex 3.
3. The triangular coordinates of the vertices 1, 2, and 3 are (1,0,0), (0,1,0) and
(0,0,1), respectively.
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High-Degree Shape Functions
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linear shape function (Fig. 5.13.a)
ˆ ( ,  ,  )   ,

i
1
2
3
i
i  1,2,3
(5.4.10)
quadratic shape function (Fig.5.13.b)
 1 (2 1  1), vertex node i  1,2,3
ˆ
i ( 1 ,  2 ,  3 )  
4 j k node i midside j  k for i  4,5,6
(5.4.11)
cubic Lagrange element (Fig. 5.13.c)
2
1
9

(


)(


 2 i i 3 i 3 ), vertex node i, i  1,2,3

 27   (  1 ) first tris ection node (i  4,5,6) side j  k
ˆ ( ,  ,  )   2 j k j 3

(5.4.12)
i
1
2
3
 27
1


(


) second trisectio n node (i  7,8,9) side j  k

j k
k
3
2
27 1 2 3 centroid node i  10
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Coordinate map Te ;
M
ˆ ( ,  ,  )
x( , )  x( 1 ,  2 ,  3 )   x j 
j
1
2
3
j 1
chain rule for calculating derivatives;
ˆ
ˆ  
ˆ 
ˆ 



i
i
i
3
1
2


 i
x
 1 x  2 x  3 x
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5.4 Two-Dimensional Finite Elements
Integration
The integrals on ̂ in terms of area coordinates;
 G( , 
ˆ

1
2
,  3 )d 2 d 3
( 1  1   2   3 )
by Gauss quadrature rule
 G( , 
ˆ

1
Nl
2
,  3 )d 2 d 3   wl G( l )
(5.4.14)
l 1
 l  (( 1 )l , ( 2 )l , ( 3 )l )
wl : quadrature weight
N l : order of quadrature formula
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5.5 Coding of two-dimensional finite
element calculations
5.5.1 Parameter description.
Control Parameters
COMMON/RCON/TITLE(20), NNODE, NELEM, NMAT, NPOINT, NBC1, NBC2
TITLE = alphanumeric array containing problem title
NNODE = number of nodal points in problem
NELEM = number of elements in problem
NMAT = number of materials in problem
NPOINT = number of point loads in problem
NBC1 = number of nodes at which essential boundary conditions are prescribed
NBC2 = number of element sides along which natural boundary conditions are
prescribed
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Nodal-Point Data
COMMON/CNODE/X(2,450),U(450)
X = nodal point x- and y- coordinates
U = nodal point values of finite element approximation
Element Data
COMMON/CELEM/NE(100),MAT(100), NODES(9,100)
NE = number of nodes in element
MAT = material number of element
NODES = node numbers of nodes in element
Material Data
COMMON/CMATL/PROP(3,5)
PROP = material properties k,b, and f for up to five materials
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Boundary-Condition Data
COMMON/CBC/NODBC1(70),VBC1(70),NELBC(40),NSIDE(40),VBC(40,2)
NODBC1 = node numbers of nodes at which essential boundary conditions
are prescribed
VBC1 = values of essential boundary conditions
u  uˆ
û
NELBC = element numbers of elements on which natural boundary conditions are
prescribed
NSIDE = side number of element on which natural boundary conditions are
prescribed
VBC = values of
p and  (or ̂ ) for natural boundary conditions
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Matrix Storage
COMMON/CMATRX/GK(450,21),GF(450)
GK = global stiffness matrix K
GF = global load vector F
Numerical Quadrature Data
COMMON/CINT/XIQ(9,2), XIT(7,2), WQ(9), WT(7)
XIQ =
XIT =
 ,
 ,
coordinates of integration points for square master element
coordinates of integration points for triangular master element
WQ = integration weights for square element
WT = integration weights for triangle element
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5.5.2 Preprocessor routine – Subroutine RNODE
N1
= node number of first node along the line.
NUM = number of nodes along the line.
INC
= increment in node number.
X1,Y1 = x,y-coordinates of first node.
XN,YN = x,y-coordinates of last node.
SUBROUTINE RNODE
COMMON/CCON/TITLE(20), NNODE
COMMON/CNODE/X(2,450)
READ.NREC
DO 20 IREC=1,NREC
READ,N1,NUM,INC,X1,Y1,XN,YN
NUM1=NUM-1
XNUM=NUM1
X(1,N1)=X1
X(2,N1)=Y1
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IF (NUM.EQ.1) GO TO 20
DX=(XN-X1)/XNUM
DY=(YN-Y1)/XNUM
DO 10 N=1,NUM1
XN=N
IN=N1+N*INC
X(1,IN)=X1+XN*DX
10 X(2,IN)=Y1+XN*DY
11 CONTINUE
C…PRINT NODAL POINT COORDINATES
WRITE(6,101)
DO 30 N=1,NNODE
30 WRITE(6,102) N,X(1,N),X(2,N)
RETURN
101 FORMAT(35H NODE NO XCOORDINATE Y-COORDINATE)
102 FORMAT(I7,2E16.4)
END
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5.5 Coding of two-dimensional finite
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5.5.2 Preprocessor routine – Subroutine RELEM
NUM = number of elements in the sequence.
INC = increment in nodal numbers from element to element in sequence.
NEE = number of nodes in each element.
MATE = material property number for the elements.
NODE(I) = nodal point numbers of nodes in first element in the sequence.
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SUBROUTINE RELEM
COMMON/CCON/TITLE(20),NNODE,NEL
EM
COMMON/CELEM/NE(100),MAT(100),NO
DES(9,100)
DIMENSION NODE(9)
C…READ ELEMENT DATA
READ.NREC
NEL=0
DO 20 IREC=1,NREC
READ,NUM,INC,NEE,MATE,(NODE(I),I=
1,NEE)
N1=NEL+1
NEL=NEL+NUM
IF(NEL.GT.100) GO TO 99
DO 20 N=N1,NEL
NINC = (N-1)*INC
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DO 10 M=1,NEE
10 NODES(M,N) = NODE(M)+NINC
NE(N)=NEE
20 MAT(N)=MATE
C…PRINT ELEMENT DEFINITIONS
WRITE(6,101)
DO 30 N=1,NELEM
NEN=NE(N)
30 WRITE(6,102)
N,NE(N),MAT(N),(NODES(I,N),I=1,NEN)
RETURN
99 WRITE(6,103)
101 FORMAT(19H ELEM NO NE MAT, 20X,
12HNODE NUMBERS //)
102 FORMAT(I5,I7,I6,5X,9I5)
103 FORMAT(37H ELEMENT NUMBER
EXCEEDS MAXIMUM VALUE)
STOP
END
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5.5 Coding of two-dimensional finite
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5.5.3 Preprocessor routine – Subroutine RBC
 Point loads: one record per point load
N = the node number of the node at which the point load is applied.
V = the value of the point load.
 Essential boundary conditions u  uˆ : each record defines a sequence of nodes with the
same boundary condition.
N1 = node number of first node in the sequence.
NUM = number of nodes in sequence.
INC = increment in node number.
V = value of û in the essential boundary condition.
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 Natural boundary conditions  k (u / n)  pu   : each record defines a sequence of
elements that have the natural boundary condition prescribed along one side.
N1 = element number of first element in the sequence.
NUM = number of elements in the sequence.
INC = increment in element numbers.
NS = side number of element along which boundary condition is prescribed
P = value of the coefficient p in (4.2.18d)
V = value of  in (4.2.18d)
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SUBROUTINE RBC
COMMON/CCON/TITLE(20),NNODE,NEL
EM,NMAT,NPOINT,NBC1,NBC2
COMMON/CBC/NODBC1(70),VBC1(70),N
ELBC(40),NSIDE(41),VBC2(2,40),NPT(10),
VPT(10)
C…READ POINT LOADS
READ,NREC
NPOINT=NREC
IF(NREC.EQ.0) GO TO 20
DO 10 I=1, NREC
READ,N,V
NPT(I)=N
10 VPT(I)=V
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C…READ ESSENTIAL BOUNDARY
CONDITION DATA
20 READ,NREC
NBC1=0
IF(NREC.EQ.0) GO TO 40
DO 30 J=1, NREC
READ,N1,NUM,INC,V
DO 30 I=1,NUM
NBC1=NBC1+1
N=N1+(I-1)*INC
NODBC1(NBC1)=N
50 VBC1(NBC1)=V
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C…READ NATURAL BOUNDARY
CONDITION DATA
40 READ,NREC
NBC2=0
IF(NREC.EQ.0) GO TO 60
DO 50 J=1,NREC
READ,N1,NUM,INC,NS,P,V
DO 50 I=1,NUM
NBC2=NBC2+1
N=N1+(I-1)*INC
NELBC(NBC2)=N
NSIDE(NBC2)=NS
VBC2(1,NBC2)=P
50 VBC2(2,NBC2)=V
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C…PRINT BOUNDARY CONDITION DATA
.
.
(THE REMAINDER OF THIS ROUTINE IS
LEFT AS AN EXERCISE)
.
.
.
END
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5.5 Coding of two-dimensional finite
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5.5.4 Processor Routines – subroutine ELEM
FIGURE 5.17 Flowchart of boundary condition calculations
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5.5 Coding of two-dimensional finite
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N e (and shape functions) in the element
X(I,J) = nodal-point coordinates, x j and y j , j  1, 2, ..., N e
NOTE : X (1, J )  x j and X ( 2, J )  y j
NL = number N l of integral points
N = number of nodal points
XI(J,L) = coordinates (in the master-element coordinate system)
l
and
l
,
l  1, 2, ..., Nl
NOTE :  l  XI (1, L) and l  XI (2, L)
W(L) = integration weights wl , l  1, 2, ..., N l
PSI(I) = values of the shape functions ˆ ( , ), i  1, 2, ... , N
l
e
at an integration point
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DPSI(I,J) = values of the shape function gradients ˆ l
/ 
i  1, 2, ..., N e at an integration point
NOTE : ˆ l /  = DPSI(l,1) ˆ l / 
and
ˆ l / 
= DPSI(l,2)
DXDS(I,J) = values the entries in the Jacobian matrix J of at an integration point
NOTE :
x / 
= DXDS(1,1)
x / 
= DXDS(1,2)
y / 
= DXDS(2,1)
y / 
= DXDS(2,2)
DSDX(I,J) = values of the entries of the inverse Jacobian matrix J 1 at an
integration point
NOTE :
 /  x
= DSDX(1,1)
 /  x
= DSDX(1,2)
 / x
= DSDX(2,1)
 / y
= DSDX(2,2)
XK,XB,XF = values of the coefficients in the differential equation for an element
EK = element stiffness array,
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EF = element load vector array
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5.5 Coding of two-dimensional finite
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DO 20 K=1,N
SUBROUTINE ELEM(X,N,EK,EF,NL,XI,W,MAT)
20 DXDS(I,J)=DXDS(I,J)+DPSI(K,J)*X(I,K)
DIMENSION X(2,N),EK(N,N),EF(N),XI(2,NL),W(NL) C…..CALCULATE DSDX...EQUATION (5.2.7)
DIMENSION
DETJ=DXDS(1,1)*DXDS(2,2)DPSIX(9),DPSIY(9),DXDS(2,2),DSDX(2,2)
DXDS(1,2)*DXDS(2,1)
DIMENSION PSI(9),DPSI(2,9)
IF(DETJ.LE.0) GO TO 99
C…..INITIALIZE ELEMENT ARRAYS
DSDX(1,1)=DXDS(2,2)/DETJ
DO 10 I=1,N
DSDX(2,2)=DXDS(1,1)/DETJ
EF(I)=0.0
DSDX(1,2)=-DXDS(1,2)/DETJ
DO 10 J=1,N
DSDX(2,1)=-DXDS(2,1)/DETJ
10 EK(I,J)=0.0
C…..CALCULATE D(PSI)/DX…EQUATION (5.3.5)
CALL GETMAT(XK,XB,XF,MAT)
DO 3C I=1,N
C…..BEGIN INTEGRATION POINT LOOP
DPSIX(I)=DPSI(I,1)*DSDX(1,1)+DPSI(I,2)*DSDX(
DO 50 L=1,NL
2,1)
CALL SHAPE(XI(1,L),N,PSI,DPSI)
30
DPSIY(I)=DPSI(I,1)*DSDX(1,2)+DPSI(I,2)*DSDX(
C…..CALCULATE DXDS…EQUATION (5.3.6)
2,2)
DO 20 I=1,2
DO 20 J=1,2
DXDS(I,J)=0.0
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C…..
ACCUMULATE INTEGRATION POINT VALUE OF INTEGRALS
FAC=DETJ*W(L)
DO 40 I=1,N
EF(I)=EF(I)+XF*PSI(I)*FAC
DO 40 J=I,N
40 EK(I,J)=EK(I,J)+FAC*(XK*(DPSIX(I)*DPSIX(J)+DPSIY(I)*DPSIY(J))+XB*PSI(I)*PSI(J))
50 CONTINUE
C.....
CALCULATE LOWER SYMMETRIC PART OF EK
DO 60 I=1,N
DO 60 J=1,I
60 EK(I,J)=EK(J,I)
RETURN
99 WRITE(6,110) DETJ,X
110 FORMAT(13H BAD JACOBIAN,E10.3/9E10.3/9E10.3)
STOP
END
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5.5.4 Processor Routines – subroutine ELEM
SUBROUTINE APLYBC
COMMON/CCON/TITLE(20),NODE,NELEM,NMAT,NPOINT,NBC1,NBC2
COMMON/CBC/NODBC1(70),VBC1(70),NELBC(40),NSIDE(41),VBC2(2,40),NPT(10),VP
T(10)
COMMON/CELEM/NE(100),MAT(100),NODES(9,100)
COMMON/CMATRX/GK(450,21),GF(450)
COMMON/CNODE/X(2,450)
DIMENSION NOD(3), XBC(2,3),PE(3,3),GAME(3)
C…..
APPLY POINT LOADS
IF (NPOINT.EQ.0) GO TO 20
DO 10 I=1,NPOINT
N=NPT(I)
10 GF(N)=GF(N)+VPT(I)
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C.....
APPLY ESSENTIAL BOUNDARY CONDITIONS
20 IF(NBC1.EQ.0) GO TO 40
BIG=1.E30
DO 30 I=1,NBC1
N=NODBC1(I)
GK(1,N)=BIG
30 GF(N)=BIG*VBC1(I)
C…..
APPLY NATURAL BOUNDARY CONDITIONS
40 IF (NBC2.EQ.0) GO TO 70
DO 60 I=1,NBC2
C…..
PICK OUT NODES ON SIDE OF ELEMENT
NEL=NELBC(I)
NS=NSIDE(I)
NC=4
IF(NE(NEL).EQ.6) NC=3
NOD(1)=NS
NOD(2)=NS+NC
NOD(3)=NS+1
IF(NS.EQ.NC) NOD(3)=1
Seoul National University
Aerospace Structures Laboratory
5.5 Coding of two-dimensional finite
element calculations
C…..PICK OUT NODAL COORDINATES
DO 50 J=1,3
NJ=NOD(J)
NOD(J)=NODES(NJ,NEL)
NJ=NOD(J)
XBC(1,J)=X(1,NJ)
50 XBC(2,J)=X(2,NJ)
C…..CALL BCINT TO CALCULATE BOUNDARY INTEGRALS PE AND GAME
CALL BCINT(VBC2(1,I),VBC2(2,I),XBC,PE,GAME)
C…..CALL ASSEMB TO ADD PE TO GK AND GAME TO GF
CALL ASSMB(PE,GAME,3,NOD,GK,GF,450)
60 CONTINUE
70 RETURN
END
Seoul National University
Aerospace Structures Laboratory
5.5 Coding of two-dimensional finite
element calculations
5.5.4 Post Processor
 u
u 
h
, h 
Print of values of the finite element approximation u h and the flux  h  k 
 x y 
the calculation of  h follows the scheme for the calculation of ˆ l / x and ˆ l / y
but now used for the evaluation of u / x and u / y . There is significant difference
in that the values of ui are now known.
Thus, u /  and u /  can be calculated directly as
u N e ˆ j

uj
 j 1 
Seoul National University
and
u N e ˆ j

uj
 j 1 
Aerospace Structures Laboratory