The steady GF for a rectangular parallelepiped satisfies the following
equations:
+
+
=
- (x - x^{})(y - y^{})(z - z^{})
(12)
0 < x < L; 0 < y < W; 0 < z < H
k_{i}
+ h_{i}G
=
0 for faces i = 1, 2,..., 6
(13)
The units of the 3D GF are
[meters^{-1}]. The 3D GF
may be stated in one of four alternative forms, discussed below.
Triple-Summation Form for the 3D Parallelepiped
The GF may be found by separation of variables in all three coordinates in
the form
G_{3D}(x, y, z | x^{}, y^{}, z^{})
=
x
x
Here
N_{x}(),
N_{y}() and
N_{z}() denote the norms
of the m^{th} x-direction eigenfunction, the n^{th} y-direction
eigenfunction, and the p^{th} z-direction eigenfunction, respectively.
Eigenfunctions X_{m}(x) and norm
N_{x}() are given
in Tables 2 and 3; related y -and z -axis quantities can be found from
these tables by substituting appropriate values; for example in the z-direction
substitute z, H, and
in place of x, L, and
.
The triple summation GF also arises naturally from
transient GF which
are integrated over time. The triple sum form of the 3D GF generally has poor convergence
properties; the double-summation GF
discussed next is recommended for numerical computation.
Double-Summation Form for the 3D Parallelepiped
The GF stated below contains a double summation that involves two
eigenfunctions, their norms, and a kernel function P, as follows:
G_{3D}(x, y, z | x^{}, y^{}, z^{}) = P_{np}(x, x^{})
(14)
Kernel functions P_{np} are identical to the functions already
introduced in equation (14) except that for the 3D case, replace
eigenvalue
by
defined by
=
+ . That the above double-summation
form satisfies equation (15) can be demonstrated by direct substitution;
however the identity for the Dirac delta function, equation (7), is
needed for both
(y - y^{}) and for
(z - z^{}).
The n = 0 and p = 0
terms are only needed when Y22Z22 is part of the GF
number; in this special case the
= 0 kernel function is needed;
it is P_{0}, described earlier for the 2D and 1D GF.
Alternative Double-summation Forms for the Parallelepiped
In the GF given above the kernel functions P_{np} are placed along the
x-axis. For a given parallelepiped there are three different double-sum
series expansions for the GF, depending on whether the kernel functions are
placed along the x-, y-, or z-axis. Each of the alternative forms of
the GF, although a different series expansion, represent the same
unique solution. The convergence behavior of alternative forms of
the 3D GF are generally complementary. Where one converges slowly,
another may generally be found that converges rapidly.
Next:Pseudo GF. Rectangular coordinate, Up:Rectangular Coordinates. Finite Bodies, Previous:Rectangle, steady GF, single-Kevin D. Cole 2002-12-31