Question 10.1: The axisymmetric model is used to generate Tp and Tp∞ data f......

The sensitivity matrix, X, is constructed from the differences between the entries of the T_{p} vector, prepended with a “0” (just like the Δϕ values used to compute the sensitivity matrix for heat flux estimation). The first few values of this vector of differences are:

= \left\lfloor7.7962\\ 0\ \ 12.608\ \ 12.624\ \ 12.160\ \cdots\right\rfloor ^T (10.6)

These values are used to create the 20 × 20 matrix \mathbf{X}_{\phi_{p}}. The first few rows of this matrix are

{\bf X}_{\phi_{p}}= \begin{bmatrix} 7.7962 & 0 & 0 & 0 & 0 & 0 \cdots \\ 12.608 & 7.7962 & 0 & 0 & 0 & 0 \cdots \\12.624 & 12.608 & 7.7962 & 0 & 0 & 0 \cdots\\ 12.160 & 12.624 & 12.608 & 7.7962 & 0 & 0 \cdots\\ \vdots & \vdots &\ddots &\ddots &\ddots &\ddots \end{bmatrix}_{20\times 20} (10.7)

The \mathbf{X}_{\phi_{p}} matrix is used with first-order Tikhonov regularization to generate the filter matrix:

\mathbf{F}_{\phi_{p}}=\left(\mathbf{X}_{\phi_{p}}^{T}\mathbf{X}_{\phi_{p}}+\alpha_{\mathrm{Tik}}\mathbf{H}_{1}^{T}\mathbf{H}_{1}\right)^{-1}\mathbf{X}_{\phi_{p}}^{T} (10.8)

The first few entries of this matrix are:s

\mathbf{F}_{\phi_{p}}=\begin{bmatrix} 0.0584 & 0.0368 & − 0.0027 & − 0.0033 & 0.0002 & 0.0003\ \cdots \\ − 0.0576 & 0.0180 & 0.0369 & 0.0007 & − 0.0033 & − 0.0001 \cdots \\ − 0.0041 & − 0.0562 & 0.0207 & 0.0370 & 0.0005 & -0.0033 \cdots\\ 0.0056 & − 0.0003 & − 0.0563 & 0.0204 & 0.0370 & 0.0005 \cdots\\ \vdots & \vdots &\vdots &\vdots &\vdots &\vdots &\vdots & \end{bmatrix}_{20\times 20} (10.9)

The \mathbf{F}_{\phi_{p}} matrix multiplies into the “data” of \mathbf{T}_{p\infty}-\mathbf{T}_{p} in the inversion of Eq. (10.4)

\mathbf{T}_{p\infty\mathbf{-T}p}=\mathbf{X}_{\phi_{p}}\mathbf{H} (10.4)

to find the kernel vector H:

{\bf H}=\left({\bf X}_{\phi_{p}}^{T}{\bf X}_{\phi_{p}}+\alpha_{\mathrm{Tik}}{\bf H}_{1}^{T}{\bf H}_{1}\right)^{-1}{\bf X}_{\phi_{p}}^{T}\left\{{\bf T}_{p\infty}-{\bf T}_{p}\right\} =\mathbf{F}_{\phi_{p}}\{\mathbf{T}_{p\infty}-\mathbf{T}_{p}\} (10.10)

The result of these calculations is the following sequence:

Discussion:
Unlike many examples in this text, these calculations are carried out using dimensional data. Units on the entries of \mathbf{X}_{\phi_{p}} are Kelvin, and those on \mathbf{F}_{\phi_{p}} are K^{−1}. The correction kernel H is dimensionless.

Selection of the proper amount of regularization to use in the inversion is difficult. In this example, the value of \alpha_{T i k} = 10 was specified. This is a relatively large value in comparison to those used in solution of the IHCP in Chapter 7. In practice, the regularization should be large enough to produce a smooth H(t) curve. Experience suggests values from a rather broad range of the regularization parameter produce similar results.
Figure 10.4 displays results for H obtained using a range of regularization parameters. All of the values are much larger than used for IHCP solution. The results for the \alpha_{T i k}= 1 display irregularities in the result, indicating the regularization coefficient is too small. Results for \alpha_{T i k}=  100 are very smooth, but values are seen diminished compared to the nominal curve.
Another tool for selection of the regularization coefficient is consideration of a known heat flux input into the simulation model. See Example 10.2.