where [A] is the Airy matrix composed of m¼956 first stress invariant Airy expressions in terms of r andθ associated with the 956 selected source locations of the input values of Saswell as 2h ¼602 traction-free boundary conditions of the form of Eq. 21.7, {c} is the set of kunknown Airy coefficients and vector {d} includes the 956 measured values of Sas well as 602 zeros from the right hand side of Eq. 21.7. The linear least squares problem associated with Eq. 21.8 was solved using the backslash operator in MATLAB. With TSA data typically being unreliable at edges, the input values of S* selected originated at least two pixels (2.74 mm) away from the edge of the hole. The number of Airy coefficients, k, to retain was assessed by computing the rms for a series of different number of Airy coefficients. The rms represents the discrepancy between the calculated first stress invariant data {d0} and the measured TSA data. Plotting the rms values against multiple values for kyielded a value of 26 to be an appropriate number of Airy coefficients to utilize. Having evaluated all of the Airy coefficients, the first stress invariant and the separate stresses σr, σθ andσrθ are can now be made available at locations of the discarded TSA data using Eqs. 21.3 through 21.6. Figure 21.4 compares the normalized first stress invariant contours, S/σo, as reconstructed from using the present hybrid method with those based on the actual measurements. Beyond the improvement of the values of the first stress invariant at the edges of the cutout, data is now provided continuously over the structure and the availability of such information is no Fig. 21.3 TSA experimental setup and corresponding thermal image Fig. 21.4 Comparison of reconstructed normalized S (left) and that based on the actual measurement (right) 21 On Improving Thermoelastic Stress Analysis Data Near Edges of Discontinuities 161
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