x
x
x
x
x
x
x
x
a
b
c
d
e
f
g
h
a′
b′ c′ d′ e′
f′
g′ h′
a′′
b′′ c′′ d′′ e′′
f′′
g′′ h′′
a′′′
b′′′ c′′′ d′′′ e′′′
f′′′
g′′′ h′′′
a
b
c
d
e
f
g
h
a′
b′
c′ d′ e′
f′
g′
h′
a′′ b′′ c′′ d′′ e′′
f′′
g′′ h′′
a′′′ b′′′ c′′′ d′′′ e′′′
f′′′
g′′′ h′′′
[A]=
.
.
.
.
.
.
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.
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.
h
a
b
c
d
e
f
g
a′ b′
c′
d′ e′
f′
g′
h′
h′′
a′′ b′′
c′′
d′′ e′′
f′′
g′′
h′′′
a′′′ b′′′
c′′′
d′′′ e′′′
f′′′
g′′′
x
x
x
x
x
x
x
x
Figure 31. Coefficient matrix for the Newton-Raphson iteration procedure.
m+1 n+1 m n+1
dj = dj + ∆dj .
(146)
The values of the variables are updated in this way for the `m + 1' iteration and
solved again until a specified tolerance is met. The tolerance is expressed in a least
squares form
(
)
N -1
N
2
Tol = ∑ m+1F-mF = ∑ (∆' s) .
2
(147)
j=1
j=1
When the tolerance is met, the final `m + 1' iteration values are used as the values of
the dependent variables for time `n + 1'. The time is then incremented and the
iteration procedure begins again. For the first iteration at a time step (`m = 0'), the
initial values of the variables are taken to be equal to the final value of the variables
at the previous time step.
Ice cover stability, solution methods, and boundary conditions
The conservation of ice momentum, as expressed in eq 102 or 115, is actually an
equation expressing a unique balance of ice momentum and external forces acting
on an ice jam element. Through assumption of the passive-pressure failure criteria
and adopting the Rankine equation for Kp (the passive pressure coefficient), the
equation is valid for a static element of an ice jam at its limit of stability. The equa-
tion is also valid for an ice jam element failing and in motion. In the former case, if
the forces on a static element exceed the passive pressure limit, application of the
equation (in concert with the other conservation equations) results in downstream
ice movement. For an ice element in motion, application of the equation for a change
in the forces may result in changes in ice velocity or thickness. For many cases, the
49
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