Environmental Engineering Reference
In-Depth Information
where, [
B
]
,
the matrix of the derivatives of the area coor-
dinates can be written as
equations. The global equations are solved while satisfying
compatibility at each node (Desai and Abel, 1972). Nodal
compatibility requires that a particular node shared by the
surrounding elements must have the same hydraulic head
in all of the elements (Zienkiewicz, 1971).
Equation 8.45 is nonlinear because the coefficients of per-
meability are a function of matric suction, which is related to
y
2
−
1
2
A
y
3
y
3
−
y
1
y
1
−
y
2
(8.46)
x
3
−
x
2
x
1
−
x
3
x
2
−
x
1
Either a hydraulic head or a flow rate must be specified
at boundary nodal points. Specified hydraulic heads at the
boundary nodes are called Dirichlet boundary conditions. A
specified flow rate across the boundary is referred to as a
Neumann boundary condition. The second term in Eq. 8.45
accounts for the specified flow rate measured in a direc-
tion normal to the boundary. For example, a specified flow
rate
v
w
in the vertical direction must be converted to a nor-
mal flow rate
the hydraulic head at each nodal point,
h
wn
. The hydraulic
heads are unknown variables in Eq. 8.45 which are solved
by using an iterative procedure. The coefficient of perme-
ability within an element is set to a value depending upon
the average matric suction at the nodal points. In this way,
the global flow equations are linearized and can be solved
simultaneously using a Gaussian elimination technique. The
computed hydraulic head at each nodal point is again aver-
aged to determine a new coefficient of permeability from
v
w
, as illustrated in Fig. 8.41. The normal
flow rate is converted to a nodal flow
Q
w
(Segerlind, 1984).
Figure 8.41 shows the computation of the nodal flows
Q
wi
and
Q
wj
at the boundary nodes
i
and
j
, respectively. A pos-
itive nodal flow signifies that there is infiltration at the node
or that the node acts as a “source.” A negative nodal flow
indicates evaporation or evapotranspiration at the node and
that the node acts as a “sink.” When the flow rate across
a boundary is zero (e.g., impervious boundary), the second
term in Eq. 8.45 disappears.
The finite element equation 8.45 can be written for
each element and assembled to form a set of global flow
¯
the permeability function
k
w
u
a
−
u
w
. The above steps are
repeated until the hydraulic heads and the coefficients of
permeability no longer change by a significant amount.
The hydraulic head gradients in the
x
- and
y
-directions
can be computed for an element by taking the derivative
of the element hydraulic heads with respect to
x
and
y
,
respectively:
i
x
i
y
[
B
]
h
wn
=
(8.47)
Figure 8.41
Vertical rainfall converted to applied moisture flow rate across sloping boundary
expressed as nodal flows.
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