194
7 Integration of Functionals, PCM and Stochastic IntegralEquations
which, because it comprises only dot-products, is a scalar under rotations. Hence,
it can be evaluated in any coordinate system, such as the principal-axis system
described in the Introduction. This means that (7.61) is rigorously correct when
applied to the randomly-oriented crystallites of the numerical model that we have
described earlier in this section.
We’ll apply this result to a crystal of lower symmetry, such that in its principal
axis system its conductivity tensor becomes:
σ =
⎡
⎣
σ 11 0 0
0 σ 22 0
0 0 σ 33
⎤
⎦ .
(7.65)
Therefore, J = σ 11 E x a x + σ 22 E y a y + σ 33 E z a z , where a x , a y , a z are unit vectors
in the x, y, and z directions, respectively. The electric power dissipated per unit
volume is given by
P = E · J
= σ 11 E
2
x + σ 22 E
2
y + σ 33 E
2
z
= E
2
σ 11 sin
2 φ cos
2 θ + σ 22 sin
2 φ sin
2 θ + σ 33 cos
2 φ
,
(7.66)
where E is the magnitude of the electric-field vector, and θ, φ are the azimuthal
and polar angles in spherical coordinates, respectively.
Hence,
σ E = σ 11 sin
2 φ cos
2 θ + σ 22 sin
2 φ sin
2 θ + σ 33 cos
2 φ .
(7.67)
In this case, we require two variables, θ and φ, to define σ E , and the calculation of
the volume-fractions requires a separate step. If σ 11 < σ 22 < σ 33 , then, from (7.62)
V F =
σ E − σ 33
σ 11 − σ 33
.
(7.68)
If σ 11 = σ 22 in (7.67), then we recover the case of transverse isotropy in (7.61).
Appendix 2: The Fortran RANDOM_NUMBER Subroutine
The Fortran 90 RANDOM_NUMBER Subroutine [34] returns uniformly distributed pseudorandom number(s) over the range 0 ≤ x < 1. This is the subroutine
that we use to generate the random variables that are required in the KarhunenLoève expansion. As such, it is necessary that we demonstrate that its output is
7 Integration of Functionals, PCM and Stochastic IntegralEquations
which, because it comprises only dot-products, is a scalar under rotations. Hence,
it can be evaluated in any coordinate system, such as the principal-axis system
described in the Introduction. This means that (7.61) is rigorously correct when
applied to the randomly-oriented crystallites of the numerical model that we have
described earlier in this section.
We’ll apply this result to a crystal of lower symmetry, such that in its principal
axis system its conductivity tensor becomes:
σ =
⎡
⎣
σ 11 0 0
0 σ 22 0
0 0 σ 33
⎤
⎦ .
(7.65)
Therefore, J = σ 11 E x a x + σ 22 E y a y + σ 33 E z a z , where a x , a y , a z are unit vectors
in the x, y, and z directions, respectively. The electric power dissipated per unit
volume is given by
P = E · J
= σ 11 E
2
x + σ 22 E
2
y + σ 33 E
2
z
= E
2
σ 11 sin
2 φ cos
2 θ + σ 22 sin
2 φ sin
2 θ + σ 33 cos
2 φ
,
(7.66)
where E is the magnitude of the electric-field vector, and θ, φ are the azimuthal
and polar angles in spherical coordinates, respectively.
Hence,
σ E = σ 11 sin
2 φ cos
2 θ + σ 22 sin
2 φ sin
2 θ + σ 33 cos
2 φ .
(7.67)
In this case, we require two variables, θ and φ, to define σ E , and the calculation of
the volume-fractions requires a separate step. If σ 11 < σ 22 < σ 33 , then, from (7.62)
V F =
σ E − σ 33
σ 11 − σ 33
.
(7.68)
If σ 11 = σ 22 in (7.67), then we recover the case of transverse isotropy in (7.61).
Appendix 2: The Fortran RANDOM_NUMBER Subroutine
The Fortran 90 RANDOM_NUMBER Subroutine [34] returns uniformly distributed pseudorandom number(s) over the range 0 ≤ x < 1. This is the subroutine
that we use to generate the random variables that are required in the KarhunenLoève expansion. As such, it is necessary that we demonstrate that its output is