Parameter tuning for wavelet transform-based damage index using mixture design

Abstract

Composite structures are susceptible to sudden failures due their complex modes of failure. One of these modes is delamination, the detachment of the layers due to the rupture in the fiber-matrix interface. To avoid catastrophic failures, several Structural Health Monitoring techniques are employed. Damage indexes based on wavelet transform techniques are vastly explored and provide prominent results. However, they depend on user experience and are usually formulated by trial and error. The present study proposes a damage index to identify delaminations in a laminated composite beam, yet, the development is based on a well-defined methodology. The proposed damage index is composed of a weighted sum of discrete wavelet transform detail coefficients, obtained by applying the transform to the mode shapes of the structure. Mode shapes were obtained numerically and damage was simulated with a stiffness reduction in some elements of the model. A mixture design analysis and a multiobjective optimization were used for tuning the damage index parameters, which improved the accuracy of the damage index in 100% of the evaluated cases. The proposed method was capable of locating damage with substantial performance along the beam length and has the advantageous characteristic of being a no-baseline method.

Appendix A. Objective functions

The eighteen objective functions are listed below:

$$ \begin{aligned} y_{1} = & - \left( {0.0299 \cdot w_{1} + 0.1792 \cdot w_{2} + 1.3775 \cdot w_{3} + 0.0448 \cdot w_{4} + 1.9378 \cdot w_{5} } \right. \\ & + 12.8514 \cdot w_{6} - 0.2935 \cdot w_{1} \cdot w_{3} - 0.3551 \cdot w_{1} \cdot w_{5} - 0.5513 \cdot w_{1} \cdot w_{6} \\ & - 0.1605 \cdot w_{2} \cdot w_{3} - 0.508 \cdot w_{2} \cdot w_{5} - 0.7731 \cdot w_{2} \cdot w_{6} - 0.4503 \cdot w_{3} \cdot w_{4} - 1.4832 \cdot w_{3} \cdot w_{5} \\ & - \left. {2.6526 \cdot w_{3} \cdot w_{6} - 0.6991 \cdot w_{4} \cdot w_{5} - 0.9783 \cdot w_{4} \cdot w_{6} } \right) \\ \end{aligned} $$

$$ \begin{aligned} y_{2} & = - \left( {0.0308 \cdot w_{1} + 0.3694 \cdot w_{2} + 2.233 \cdot w_{3} + 0.4016 \cdot w_{4} } \right. \\ & \quad + 2.2483 \cdot w_{5} + 3.0408 \cdot w_{6} - 0.883 \cdot w_{1} \cdot w_{3} - 0.4789 \cdot w_{1} \cdot w_{5} - 0.5077 \cdot w_{1} \cdot w_{6} \\ & \quad - 0.4377 \cdot w_{2} \cdot w_{3} - 0.3396 \cdot w_{2} \cdot w_{4} - 1.1082 \cdot w_{2} \cdot w_{5} - 1.0835 \cdot w_{2} \cdot w_{6} - 1.603 \cdot w_{3} \cdot w_{4} - 3.8182 \cdot w_{3} \cdot w_{5} \\ & \quad - \left. {4.1004 \cdot w_{3} \cdot w_{6} - 0.1717 \cdot w_{4} \cdot w_{5} - 0.239 \cdot w_{4} \cdot w_{6} } \right) \\ \end{aligned} $$

$$ \begin{aligned} y_{3} & = - (0.0215 \cdot w_{1} + 0.4173 \cdot w_{2} + 0.6651 \cdot w_{3} + 0.0903 \cdot w_{4} + 3.998 \cdot w_{5} + 4.1752 \cdot w_{6} - 0.2709 \cdot w_{1} \cdot w_{3} \\ & \quad - 0.4318 \cdot w_{1} \cdot w_{5} - 0.3225 \cdot w_{1} \cdot w_{6} - 0.1791 \cdot w_{2} \cdot w_{3} - 0.2497 \cdot w_{2} \cdot w_{4} - 0.9211 \cdot w_{2} \cdot w_{5} \\ & \quad - 0.6543 \cdot w_{2} \cdot w_{6} - 0.5561 \cdot w_{3} \cdot w_{4} - 1.6234 \cdot w_{3} \cdot w_{5} - 0.6888 \cdot w_{3} \cdot w_{6} - 0.7407 \cdot w_{4} \cdot w_{5} \\ & \quad - 0.8298 \cdot w_{4} \cdot w_{6} - 0.7215 \cdot w_{5} \cdot w_{6} ) \\ \end{aligned} $$

$$\begin{aligned} y_{4}= &{} - (0.0371 \cdot w_{1}\\&+ 0.2568 \cdot w_{2} + 0.8219 \cdot w_{3} + 0.2124 \cdot w_{4} \\&+ 1.5151 \cdot w_{5} + 8.2464 \cdot w_{6} - 0.42 \cdot w_{1} \cdot w_{3}\\&- 0.188 \cdot w_{1} \cdot w_{4} - 0.398 \cdot w_{1} \cdot w_{5} \\&- 0.714 \cdot w_{1} \cdot w_{6} - 0.568 \cdot w_{2} \cdot w_{3} - 0.332 \cdot w_{2} \cdot w_{4} - 0.438 \cdot w_{2} \cdot w_{5}\\&- 1.519 \cdot w_{2} \cdot w_{6} - 0.682 \cdot w_{3} \cdot w_{4} \\&- 1.374 \cdot w_{3} \cdot w_{5} - 2.65 \cdot w_{3} \cdot w_{6} - 1.67 \cdot w_{4} \cdot w_{5} - 2.448 \cdot w_{4} \cdot w_{6} - 0.209 \cdot w_{5} \cdot w_{6}) \end{aligned}$$

$$\begin{aligned} y_{5}= &{} - (0.0571 \cdot w_{1} + 0.2064 \cdot w_{2} + 1.5845 \cdot w_{3} + 0.1056 \cdot w_{4} \\&+ 1.4573 \cdot w_{5} + 15.0616 \cdot w_{6} - 0.656 \cdot w_{1} \cdot w_{5} \\&- 2.234 \cdot w_{1} \cdot w_{6} - 0.728 \cdot w_{2} \cdot w_{3} \\&- 0.492 \cdot w_{2} \cdot w_{5} - 2.654 \cdot w_{2} \cdot w_{6}\\&- 2.552 \cdot w_{3} \cdot w_{5} - 3.801 \cdot w_{3} \cdot w_{6} \\&- 1.092 \cdot w_{4} \cdot w_{5} - 2.059 \cdot w_{4} \cdot w_{6} - 10.689 \cdot w_{5} \cdot w_{6}) \end{aligned}$$

$$\begin{aligned} y_{6}= &{} - (0.068 \cdot w_{1}\\&+ 0.3108 \cdot w_{2} + 1.4867 \cdot w_{3} + 0.2659 \cdot w_{4}\\&+ 3.3538 \cdot w_{5} + 1.5728 \cdot w_{6} - 0.241 \cdot w_{1} \cdot w_{2}\\&- 0.652 \cdot w_{1} \cdot w_{3} - 1.698 \cdot w_{1} \cdot w_{5}\\&- 0.85 \cdot w_{1} \cdot w_{6} - 1.446 \cdot w_{2} \cdot w_{3} - 0.328 \cdot w_{2} \cdot w_{4} - 2.023 \cdot w_{2} \cdot w_{5}\\&- 1.344 \cdot w_{2} \cdot w_{6} - 0.881 \cdot w_{3} \cdot w_{4} - 4.073 \cdot w_{3} \cdot w_{5}\\&- 2.25 \cdot w_{3} \cdot w_{6} - 3.419 \cdot w_{4} \cdot w_{5} - 0.508 \cdot w_{4} \cdot w_{6} - 6.958 \cdot w_{5} \cdot w_{6}) \end{aligned}$$

$$\begin{aligned} y_{7}= &{} - (0.0546 \cdot w_{1} + 0.2716 \cdot w_{2} + 0.8035 \cdot w_{3} + 0.0622 \cdot w_{4} + 1.7467 \cdot w_{5}\\&+ 5.8198 \cdot w_{6} - 0.399 \cdot w_{1} \cdot w_{3} \\&- 0.716 \cdot w_{1} \cdot w_{5} - 1.327 \cdot w_{1} \cdot w_{6} - 1.018 \cdot w_{2} \cdot w_{3} - 0.501 \cdot w_{2} \cdot w_{5}\\&- 1.956 \cdot w_{2} \cdot w_{6} - 0.35 \cdot w_{3} \cdot w_{4} - 2.738 \cdot w_{3} \cdot w_{5} - 1.718 \cdot w_{3} \cdot w_{6}\\&- 0.991 \cdot w_{4} \cdot w_{5} - 1.3 \cdot w_{4} \cdot w_{6} - 7.073 \cdot w_{5} \cdot w_{6}) \end{aligned}$$

$$\begin{aligned} y_{8}= &{} - (0.1003 \cdot w_{1}\\&+ 0.3035 \cdot w_{2} + 0.8288 \cdot w_{3} + 0.1503 \cdot w_{4} + 3.418 \cdot w_{5}\\&+ 11.292 \cdot w_{6} - 1.897 \cdot w_{1} \cdot w_{5}\\&- 3.888 \cdot w_{1} \cdot w_{6} - 1.102 \cdot w_{2} \cdot w_{3} \\&- 2.278 \cdot w_{2} \cdot w_{5} - 4.057 \cdot w_{2} \cdot w_{6} \\&- 2.818 \cdot w_{3} \cdot w_{5} - 5.611 \cdot w_{3} \cdot w_{6} \\&- 2.375 \cdot w_{4} \cdot w_{5} - 3.433 \cdot w_{4} \cdot w_{6} - 22.061 \cdot w_{5} \cdot w_{6}) \end{aligned}$$

$$\begin{aligned} y_{9}= &{} - (0.675 \cdot w_{1} \\&+ 2.828 \cdot w_{2} + 9.902 \cdot w_{3} + 1.055 \cdot w_{4} + 7.272 \cdot w_{5} + 25.174 \cdot w_{6} - 3.1 \cdot w_{1} \cdot w_{2} \\&- 3.71 \cdot w_{1} \cdot w_{3} - 3.96 \cdot w_{1} \cdot w_{5}\\&- 13.32 \cdot w_{1} \cdot w_{6} - 16.75 \cdot w_{2} \cdot w_{3}\\&- 11.4 \cdot w_{2} \cdot w_{5} - 8.08 \cdot w_{2} \cdot w_{6} - 8.73 \cdot w_{3} \cdot w_{4}\\&- 4.41 \cdot w_{3} \cdot w_{5} - 41.49 \cdot w_{3} \cdot w_{6}\\&- 9.15 \cdot w_{4} \cdot w_{5} - 6.74 \cdot w_{4} \cdot w_{6} - 45.13 \cdot w_{5} \cdot w_{6}) \end{aligned}$$

$$\begin{aligned} y_{10}=\, & {} 0.56 \cdot w_{1} + 2.666 \cdot w_{2} + 5.262 \cdot w_{3} + 1.299 \cdot w_{4} + 6.992 \cdot w_{5} \\& + 17.667 \cdot w_{6} - 3.102 \cdot w_{1} \cdot w_{2} - 2.012 \cdot w_{1} \cdot w_{3} - 1.044 \cdot w_{1} \cdot w_{4} \\& - 3.968 \cdot w_{1} \cdot w_{5} - 8.579 \cdot w_{1} \cdot w_{6} \\& - 11.706 \cdot w_{2} \cdot w_{3} - 1.335 \cdot w_{2} \cdot w_{4} - 11.013 \cdot w_{2} \cdot w_{5} - 5.524 \cdot w_{2} \cdot w_{6} 5.813 \cdot w_{3} \cdot w_{4} \\& - 4.217 \cdot w_{3} \cdot w_{5} - 22.312 \cdot w_{3} \cdot w_{6} - 9.448 \cdot w_{4} \cdot w_{5} - 5.2 \cdot w_{4} \cdot w_{6} - 33.857 \cdot w_{5} \cdot w_{6} \end{aligned}$$

$$\begin{aligned} y_{11}= &{} 0.674 \cdot w_{1} + 3.287 \cdot w_{2} + 5.874 \cdot w_{3} + 1.638 \cdot w_{4} + 9.444 \cdot w_{5}\\&+ 16.513 \cdot w_{6} - 3.699 \cdot w_{1} \cdot w_{2}\\&- 2.103 \cdot w_{1} \cdot w_{3} - 1.296 \cdot w_{1} \cdot w_{4}\\&- 4.711 \cdot w_{1} \cdot w_{5} - 8.666 \cdot w_{1} \cdot w_{6} - 12.437 \cdot w_{2} \cdot w_{3} - 1.604 \cdot w_{2} \cdot w_{4} \\&- 13.212 \cdot w_{2} \cdot w_{5} - 5.681 \cdot w_{2} \cdot w_{6} - 6.603 \cdot w_{3} \cdot w_{4} - 6.065 \cdot w_{3} \cdot w_{5}\\&- 22.575 \cdot w_{3} \cdot w_{6} - 11.035 \cdot w_{4} \cdot w_{5} - 3.468 \cdot w_{4} \cdot w_{6} - 35.56 \cdot w_{5} \cdot w_{6} \end{aligned}$$

$$\begin{aligned} y_{12}= &{} 0.739 \cdot w_{1} + 2.797 \cdot w_{2} + 5.39 \cdot w_{3} + 1.837 \cdot w_{4} + 9.63 \cdot w_{5}\\&+ 17.743 \cdot w_{6} - 3.666 \cdot w_{1} \cdot w_{2}\\&- 1.944 \cdot w_{1} \cdot w_{3} - 1.78 \cdot w_{1} \cdot w_{4}\\&- 4.559 \cdot w_{1} \cdot w_{5} - 9.216 \cdot w_{1} \cdot w_{6} \\&- 11.706 \cdot w_{2} \cdot w_{3} - 1.479 \cdot w_{2} \cdot w_{4} - 12.001 \cdot w_{2} \cdot w_{5} - 5.989 \cdot w_{2} \cdot w_{6} \\&- 7.021 \cdot w_{3} \cdot w_{4} - 5.354 \cdot w_{3} \cdot w_{5} - 22.118 \cdot w_{3} \cdot w_{6} - 12.489 \cdot w_{4} \cdot w_{5} - 4.933 \cdot w_{4} \cdot w_{6} - 34.535 \cdot w_{5} \cdot w_{6} \end{aligned}$$

$$\begin{aligned} y_{13}= &{} 0.791 \cdot w_{1} + 2.295 \cdot w_{2} + 6.545 \cdot w_{3} + 2.026 \cdot w_{4} + 6.804 \cdot w_{5}\\&+ 20.625 \cdot w_{6} - 3.296 \cdot w_{1} \cdot w_{2} - 2.055 \cdot w_{1} \cdot w_{3} - 2.139 \cdot w_{1} \cdot w_{4}\\&- 3.303 \cdot w_{1} \cdot w_{5} - 9.847 \cdot w_{1} \cdot w_{6} \\&- 10.84 \cdot w_{2} \cdot w_{3} - 1.47 \cdot w_{2} \cdot w_{4} - 8.735 \cdot w_{2} \cdot w_{5} - 5.455 \cdot w_{2} \cdot w_{6} - 8.497 \cdot w_{3} \cdot w_{4}\\&- 4.11 \cdot w_{3} \cdot w_{5} - 25.533 \cdot w_{3} \cdot w_{6} \\&- 11.327 \cdot w_{4} \cdot w_{5} \\&- 4.076 \cdot w_{4} \cdot w_{6} - 30.006 \cdot w_{5} \cdot w_{6} \end{aligned}$$

$$\begin{aligned} y_{14}=\; & {} 0.943 \cdot w_{1} + 2.65 \cdot w_{2} + 6.206 \cdot w_{3} + 2.455 \cdot w_{4} + 8.717 \cdot w_{5}\\&+ 23.778 \cdot w_{6} - 3.959 \cdot w_{1} \cdot w_{2} \\&- 1.764 \cdot w_{1} \cdot w_{3} - 2.504 \cdot w_{1} \cdot w_{4}\\&- 4.65 \cdot w_{1} \cdot w_{5} - 11.666 \cdot w_{1} \cdot w_{6} - 11.19 \cdot w_{2} \cdot w_{3} - 1.443 \cdot w_{2} \cdot w_{4} \\&- 11.308 \cdot w_{2} \cdot w_{5} - 7.767 \cdot w_{2} \cdot w_{6}\\&- 8.3 \cdot w_{3} \cdot w_{4} - 5.609 \cdot w_{3} \cdot w_{5} - 26.905 \cdot w_{3} \cdot w_{6} \\& - 14.536 \cdot w_{4} \cdot w_{5} - 4.546 \cdot w_{4} \cdot w_{6} - 42.162 \cdot w_{5} \cdot w_{6} \end{aligned}$$

$$\begin{aligned} y_{15}=\; & {} 0.919 \cdot w_{1} + 3.657 \cdot w_{2} + 4.306 \cdot w_{3} + 2.461 \cdot w_{4} + 11.275 \cdot w_{5} \\& + 13.033 \cdot w_{6} - 5.239 \cdot w_{1} \cdot w_{2} \\& - 1.145 \cdot w_{1} \cdot w_{3} - 2.724 \cdot w_{1} \cdot w_{4}\\& - 6.029 \cdot w_{1} \cdot w_{5} - 8.704 \cdot w_{1} \cdot w_{6} - 11.692 \cdot w_{2} \cdot w_{3} - 2.124 \cdot w_{2} \cdot w_{4} \\& - 15.5 \cdot w_{2} \cdot w_{5} - 5.65 \cdot w_{2} \cdot w_{6}\\& - 6.866 \cdot w_{3} \cdot w_{4} - 7.225 \cdot w_{3} \cdot w_{5} - 17.217 \cdot w_{3} \cdot w_{6} \\& - 17.885 \cdot w_{4} \cdot w_{5} - 2.114 \cdot w_{4} \cdot w_{6} - 38.207 \cdot w_{5} \cdot w_{6} \end{aligned}$$

$$\begin{aligned} y_{16}= &{} 0.915 \cdot w_{1} + 5.061 \cdot w_{2} + 5.199 \cdot w_{3} + 2.116 \cdot w_{4} + 15.209 \cdot w_{5}\\&+ 19.01 \cdot w_{6} - 6.066 \cdot w_{1} \cdot w_{2} - 1.597 \cdot w_{1} \cdot w_{3} - 1.953 \cdot w_{1} \cdot w_{4} \\&- 8.323 \cdot w_{1} \cdot w_{5} - 11.368 \cdot w_{1} \cdot w_{6}\\&- 15.984 \cdot w_{2} \cdot w_{3} - 2.317 \cdot w_{2} \cdot w_{4} - 22.406 \cdot w_{2} \cdot w_{5} - 8.116 \cdot w_{2} \cdot w_{6}\\&- 6.969 \cdot w_{3} \cdot w_{4} - 8.509 \cdot w_{3} \cdot w_{5} - 22.185 \cdot w_{3} \cdot w_{6} - 19.484 \cdot w_{4} \cdot w_{5} \\&- 4.506 \cdot w_{4} \cdot w_{6} - 57.544 \cdot w_{5} \cdot w_{6} \end{aligned}$$

$$ \begin{aligned} y_{{17}} & = 0.795 \cdot w_{1} + 5.374 \cdot w_{2} + 10.137 \cdot w_{3} \\ & \quad + 1.524 \cdot w_{4} + 15.213 \cdot w_{5} + 30.597 \cdot w_{6} \\ & \quad - 5.36 \cdot w_{1} \cdot w_{2} - 4.06 \cdot w_{1} \cdot w_{3} - 8.71 \cdot w_{1} \cdot w_{5} \\ & \quad - 16.3 \cdot w_{1} \cdot w_{6} - 23.22 \cdot w_{2} \cdot w_{3} - 2.4 \cdot w_{2} \cdot w_{4} \\ & \quad - 22.19 \cdot w_{2} \cdot w_{5} - 12.61 \cdot w_{2} \cdot w_{6} - 9.28 \cdot w_{3} \cdot w_{4} \\ & \quad - 10.93 \cdot w_{3} \cdot w_{5} - 40.79 \cdot w_{3} \cdot w_{6} - 16.43 \cdot w_{4} \cdot w_{5} \\ & \quad - 9.08 \cdot w_{4} \cdot w_{6} - 74.29 \cdot w_{5} \cdot w_{6} \\ \end{aligned} $$

$$\begin{aligned} y_{18}= &{} 0.0511 \cdot w_{1} \\&+ 0.1917 \cdot w_{2} + 1.559 \cdot w_{3} + 0.0814 \cdot w_{4} \\&+ 1.0488 \cdot w_{5} + 6.9707 \cdot w_{6} - 0.586 \cdot w_{1} \cdot w_{3}\\&- 0.269 \cdot w_{1} \cdot w_{5} - 1.218 \cdot w_{1} \cdot w_{6} \\& - 0.995 \cdot w_{2} \cdot w_{3} - 0.44 \cdot w_{2} \cdot w_{5} - 1.506 \cdot w_{2} \cdot w_{6} - 0.603 \cdot w_{3} \cdot w_{4} \\&- 1.656 \cdot w_{3} \cdot w_{5} - 4.584 \cdot w_{3} \cdot w_{6} 0.276 \cdot w_{4} \cdot w_{5} \\& - 1.359 \cdot w_{4} \cdot w_{6} - 3.435 \cdot w_{5} \cdot w_{6} \end{aligned}$$