£¢

¥

¤

¥ ’˜

¦

¤

where is the element matrix for the element as de¬ned above. Also is an

¥

w

¤

Boolean connectivity matrix which maps the coordinates of the matrix into the

coordinates of the full matrix .

” ” ”n

uu u ™

n

¢

¤§ ¥ §¢

¡

¡© £ £ ¥ £ ¤ § ¨¡ ¢

§¥ © ©

Finite element mesh

6

Assembled matrix

4

4 5

3

2

2 3

1

1

¥ ¡¢

D T

A simple ¬nite element mesh and the pattern of

the corresponding assembled matrix.

¥ „

T

§¦

¥ ©

The assembly process can be illustrated with a very simple example. Con-

sider the ¬nite element mesh shown in Figure 2.8. The four elements are numbered from

bottom to top as indicated by the labels located at their centers. There are six nodes in this £¢

mesh and their labeling is indicated in the circled numbers. The four matrices asso-

ciated with these elements are shown in Figure 2.9. Thus, the ¬rst element will contribute

uCY

u

“¢ u

to the nodes , the second to nodes , the third to nodes , and the fourth to ¢ ¢

£ “¢

nodes . ¢

£ 6¢

£ 0¢ £ ¢ £ ¢¢

c ¢ 1)1)‘

¤ £¢

¥ ¡¢

D ¡T

The element matrices , for the

¬nite element mesh shown in Figure 2.8.

In fact there are two different ways to represent and use the matrix . We can form

h jQ ˜

¤

all the element matrices one by one and then we can store them, e.g., in an

rectangular array. This representation is often called the unassembled form of . Then the

matrix may be assembled if it is needed. However, element stiffness matrices can also

be used in different ways without having to assemble the matrix. For example, frontal

techniques are direct solution methods that take the linear system in unassembled form and

compute the solution by a form of Gaussian elimination. There are also iterative solution

techniques which work directly with unassembled matrices. One of the main operations

y

u„ d¥ ¥ ¦£ ”¥ ¤© © ¨

n ¥ u§ ¡ ¥

¢

¡© ¥ §¨© U© ¥ ¤¨¡

£©

¥¥

‚

¨ ¦

required in many iterative methods is to compute , the product of the matrix by

¦

an arbitrary vector . In unassembled form, this can be achieved as follows:

!& !&

‘ ’ w ‘ Y

t

C

£¢

¡

G B ¤ ¥ ¨ ¤ ¢ &¥

¨ ¦ ¦ ¦

01( 0(

¤

¦ ¦

Thus, the product gathers the data associated with the -element into a 3-vector

G

¥¥

¤

consistent with the ordering of the matrix . After this is done, this vector must be mul-

¥¥

¤ ¨