Geology Reference
In-Depth Information
4.2.6.8 Modeling the constants
,
and
.
The constants in
u
x
+ u
t
+ u = 0 are not arbitrary. Consider
= 0, for
which
u
t
+
u = 0
(4.2.53)
This has the exponential solution
u(0,t)
e
-( )t
(4.2.54)
which admits unrealistic “lifting” elevations when > 0. In general, the
coefficients are constrained by the mechanics of rock fracture; an example in
Chapter 8 shows that their relative values dictate smooth versus rough drilling.
4.2.7 Finite difference modeling.
Exact analytical solutions to practical engineering problems are rare, and
recourse to numerical solutions is often necessary.
Finite element
,
finite
difference
and
boundary integral
(a.k.a.,
panel
) methods have been successfully
used to solve complicated problems. In this section, we will discuss finite
difference techniques; these extremely powerful methods can be mastered with a
minimum amount of expertise in higher math. We will introduce and develop
the fundamental ideas, and rapidly progress to the formulation of state-of-the-art
algorithms for wave equations.
4.2.7.1 Elementary considerations.
Let us consider a differentiable function f(x) at three consecutive
equidistant locations x
i-1
, x
i
and x
i+1
, where i-1, i and i+1 are indexing
parameters. In this topic, we will assume that all grids are uniformly separated
by the constant gridblock distance x. Now, it is clear from Figure 4.2.5 that
the first derivative at an intermediate point A between x
i-1
and x
i
is
df(x
A
)/dx = (x
i
- x
i-1
)/ x (4.2.55)
while the first derivative at an intermediate point B between x
i
and x
i+1
is
df(x
B
)/dx = (x
i+1
- x
i
)/ x
(4.2.56)
Hence, the second derivative of f(x) at x
i
satisfies
d
2
f(x
i
)/dx
2
= {df(x
B
)/dx - df(x
A
)/dx}/
x
(4.2.57)
or
d
2
f(x
i
)/dx
2
= { f
i-1
- 2f
i
+ f
i+1
}/(
x)
2
+ O(
x)
2
(4.2.58)
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