Civil Engineering Reference
In-Depth Information
y
∂f
∂
y
=
C
1
f
j
k
∂f
∂
x
=
C
2
l
i
x
Figure 3.7 Boundary conditions involving non-zero gradients of the unknown
For boundary condition (3.34) in Figure 3.7, the additional term is,
l
c
x
[
N
]
T
C
2
l
x
d
S
(3.38)
k
which is just a vector
0
0
1
1
C
2
c
x
(y
k
−
y
l
)
2
(3.39)
which would be added to the right hand side of the element equations. For a further
discussion of boundary conditions see Smith (1979).
In summary, boundary conditions of the type
φ
=
0arethemost
common and are easily handled in finite element analyses. The cases given by (3.33) and
(3.34) in which
∂φ/∂n
is fixed to a non-zero value that is either a constant or a linear
function of
φ
, are somewhat more complicated, but can be appropriately treated. Examples
of the use of all these types of boundary specification are included in the applications
Chapters 4 to 12.
constant or
∂φ/∂n
=
3.7 Programming using building blocks
The programs in subsequent chapters are constituted from over 70 “building blocks” in
the form of Fortran 95 functions and subroutines which perform the tasks of computing
and integrating the element matrices, assembling these into system matrices if necessary
and carrying out the appropriate equilibrium, eigenvalue or propagation calculations. In
Chapter 12, the message passing interface MPI libraries handle the necessary communica-
tion between processors.
It is anticipated that users will elect to pre-compile all of the building blocks and to
hold these permanently in a library. The library should then be automatically accessible to
