Integration of Functionals, PCM and Stochastic Integral Equations

Abstract

The mathematical theory of functional integration was developed in the fifties to be applied to problems in probability and quantum field theory. We’re not interested in the formal theory, but in its more practical form which can actually be used to generate numbers. In particular, it leads to the probabilistic collocation method (PCM) and other techniques for high-dimensional model representation (HDMR), which we intend to apply to eddy-current nondestructive evaluation (NDE), using the volume-integral code, VIC-3D®, as our vehicle.

Appendices

Appendix 1: The Numerical Model

In its principal-axis coordinate system, the electrical conductivity tensor for hexagonal crystals, such as pure titanium and its most common alloys, is given by

$$\displaystyle \begin{aligned} \boldsymbol{\sigma}=\left[\begin{array}{ccc}\sigma_{11}&0&0\\0&\sigma_{11}&0\\0&0&\sigma_{33}\end{array}\right]\ , {} \end{aligned} $$

(7.60)

where σ ₁₁ is the conductivity in the basal plane (plane of isotropy), and σ ₃₃ is the conductivity normal to the basal plane. Such a system is said to possess transverse isotropy, and the crystal class is labeled 6 mm. This notation means that the crystal contains a sixfold axis of rotational symmetry, as well as six mirror planes that contain that axis.

For pure titanium, σ ₁₁ = 2.205 × 10⁶ S/m, and σ ₃₃ = 2.083 × 10⁶ S/m. If ϕ denotes the angle between the electric field vector and the normal to the basal plane, then the conductivity in the ϕ −direction can be represented by the ellipsoid of revolution

$$\displaystyle \begin{aligned} \sigma(\phi)=\sigma_{33}\cos^2\phi + \sigma_{11}\sin^2\phi\ . {} \end{aligned} $$

(7.61)

We will use (7.61) in establishing our numerical model. See the next section for a further discussion and generalization of (7.61).

Consider a half-space host of pure titanium, whose crystal axis is oriented in the z −direction, normal to the surface of the half-space. Lying at the surface of this host is a rough patch of randomly oriented crystallites of titanium, as shown in Fig. 7.22. The use of a single conductivity value for the host would be rigorously correct if the coil were the only current source in the problem. The electric field induced into the host by this source lies within the basal plane, so only the basal-plane conductivity, 2.205 × 10⁶ S/m, enters the picture. The anomalous currents within the random crystallites, however, produce a scattered field within the host that is not confined to the basal plane, so the use of a single conductivity is an approximation.

Fig. 7.22

Illustrating the model setup for VIC-3D®. There is a second layer identical to the one shown immediately below the one shown. This is due to the fact that VIC-3D® requires a minimum of two cells in any direction of the grid

We can use any number of cells in the model in order to get a statistically reasonable answer. The value of the conductivity to be assigned each cell (or crystallite or grain) is determined by randomly choosing $\cos \phi$ in (7.61). We use a uniform distribution function for this purpose. The volume-fractions for the cells, from which the conductivities are determined, are computed off-line, and then imported into VIC-3D® in a routine manner. After simulating the impedance response of the bad patch, we can run several small slot responses to gain insight into how deleterious the grain noise is to detecting a crack in its presence.

The relationship between volume-fractions, V F, and conductivity of a cell is given by

$$\displaystyle \begin{aligned} \sigma = \sigma_{\mathrm{max}} + VF(\sigma_{\mathrm{min}}-\sigma_{\mathrm{max}})\ , \end{aligned} $$

(7.62)

where σ _max is the maximum conductivity in the region of interest (host plus anomaly), σ _min is the minimum conductivity, and 0 ≤ V F ≤ 1. As Fig. 7.22 indicates, σ _max = σ _host = 2.205 × 10⁶ in this setup.

Returning to (7.61), we can easily determine a relationship for volume-fractions:

$$\displaystyle \begin{aligned} \begin{array}{rcl} \sigma & =&\displaystyle \sigma_{\mathrm{max}}\sin^2\phi + \sigma_{\mathrm{min}}\cos^2\phi \\ & =&\displaystyle \sigma_{\mathrm{max}}(1-\cos^2\phi)+\sigma_{\mathrm{min}}\cos^2\phi \\ & =&\displaystyle \sigma_{\mathrm{max}} +\cos^2\phi(\sigma_{\mathrm{min}} - \sigma_{\mathrm{max}})\ , {} \end{array} \end{aligned} $$

(7.63)

from which we conclude that $VF = \cos ^2\phi$.

More Physics

Let σ be the conductivity tensor of a material in an arbitrary coordinate system, so that it will have (in general) nonzero off-diagonal entries. In this coordinate system we still have J = σ ⋅E, where J is the electric current density, and E is the electric field. The electric power density dissipated within the material is P = E ⋅J = E ⋅σ ⋅E, which defines an ellipsoid in E-space.

By definition, the conductivity in the direction of the electric field, E, is given by

$$\displaystyle \begin{aligned} \begin{array}{rcl} \sigma_E& =&\displaystyle {\displaystyle\frac{P}{\mathbf{E}\cdot\mathbf{E}}} \\ & =&\displaystyle {\displaystyle\frac{P}{\vert\mathbf{E}\vert^2}}\ , {} \end{array} \end{aligned} $$

(7.64)

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:

$$\displaystyle \begin{aligned} \boldsymbol{\sigma}=\left[\begin{array}{ccc}\sigma_{11}&0&0\\0&\sigma_{22}&0\\0&0&\sigma_{33}\end{array}\right]\ . {} \end{aligned} $$

(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

$$\displaystyle \begin{aligned} \begin{array}{rcl} P& =&\displaystyle \mathbf{E}\cdot\mathbf{J} \\ & =&\displaystyle \sigma_{11}E_x^2 + \sigma_{22}E_y^2 + \sigma_{33}E_z^2 \\ & =&\displaystyle E^2\left(\sigma_{11}\sin^2\phi\cos^2\theta +\sigma_{22}\sin^2\phi\sin^2\theta +\sigma_{33}\cos^2\phi\right)\ , {} \end{array} \end{aligned} $$

(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,

$$\displaystyle \begin{aligned} \sigma_E=\sigma_{11}\sin^2\phi\cos^2\theta +\sigma_{22}\sin^2\phi\sin^2\theta +\sigma_{33}\cos^2\phi\ . {} \end{aligned} $$

(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)

$$\displaystyle \begin{aligned} VF={\displaystyle\frac{\sigma_E-\sigma_{33}}{\sigma_{11}-\sigma_{33}}}\ . {} \end{aligned} $$

(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 Karhunen-Loève expansion. As such, it is necessary that we demonstrate that its output is consistent with the requirements of the expansion, namely that the random variables are uncorrelated and have a unit variance.

First, we note that the variance of a random variable that is uniformly distributed over [0, 1) is 1/12, so that we must multiply the output of RANDOM_NUMBER by $\sqrt {1}2$ in order to generate a unit-variance random variable. Secondly, we transform the range of the output to [−0.5, 0.5) in order to generate a zero-mean random variable, which will be useful in our later work.

In order to demonstrate that the output of RANDOM_NUMBER is uncorrelated, we perform the following experiment. We generate a 32-element random vector which is the output of RANDOM_NUMBER, and identify the 17th and 32nd elements as two typical random variables. We repeat this experiment 10, 100, 1000, 10,000 and 100,000 times to generate five sample spaces. We then compute the means and variances of each of the two random variables, as well as their covariance. The results, obtained using the usual equations of statistics [18],

$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathrm{MEAN(17)}& =&\displaystyle {\displaystyle\frac{1}{N}}\sum_{i=1}^NRV_{17}(i) \\ \mathrm{MEAN(32)}& =&\displaystyle {\displaystyle\frac{1}{N}}\sum_{i=1}^NRV_{32}(i) \\ \mathrm{COV}(17,32)& =&\displaystyle {\displaystyle\frac{1}{N}}\sum_{i=1}^N(RV_{17}(i)-\mathrm{MEAN(17)})\times (RV_{32}(i)-\mathrm{MEAN(32)}) \\ \mathrm{VAR}(17)& =&\displaystyle {\displaystyle\frac{1}{B}}\sum_{i=1}^N(RV_{17}(i)-\mathrm{MEAN(17)})^2 \\ \mathrm{VAR}(32)& =&\displaystyle {\displaystyle\frac{1}{B}}\sum_{i=1}^N(RV_{32}(i)-\mathrm{MEAN(32)})^2\ , {} \end{array} \end{aligned} $$

(7.69)

are shown in Table 7.4. It is clear that the required conditions are met, especially the very small covariance and unit variances, with increasing sample size. Thus, we can confidently use the Fortran RANDOM_NUMBER Subroutine to generate numbers that are consistent with the statement of the Karhunen-Loève expansion.

Table 7.4 Convergence of the Fortran RANDOM_NUMBER Subroutine