Spectral Element Based Optimization Scheme for Damage Identification

Abstract

Due to its simplicity, impedance-based structural health monitoring (SHM) has gained a lot of interest in the SHM community. Impedance measurements have proved to be very effective in detecting the presence of damage. Damage identification, on the other hand, can hardly be done with a single impedance measurement, and an array of sensors is normally required to identify damage location and severity. However, impedance measurements contain valuable information about the fundamental frequencies of the structure, and when combined with modelling, more quantitative information on structural damages can be extracted. In this study, a single impedance measurement is used for damage detection and identification in a beam. Spectral element model is developed to calculate the structural impedance of the damaged beam, and then damage defining parameters, which are damage location, width and severity, are updated through an optimization scheme. The proposed technique is computationally efficient, as it requires solving a very small system of equations with only three optimization parameters.

Appendix

Matrices H(ω) and G(ω) appearing in Eqs. 3.11 and 3.12 are

$$ H\left(\omega \right)=\left[\begin{array}{cccccc} {e}^{i{k}_0{x}_1} & {e}^{-i{k}_0{x}_1} & 0 & 0 & 0 & 0 \\ 0 & 0 & {e}^{-i{k}_1{x}_1} & {e}^{-i{k}_2{x}_1} & {e}^{-i{k}_3{x}_1} & {e}^{-i{k}_4{x}_1} \\ 0 & 0 & {R}_1{e}^{-i{k}_1{x}_1} & {R}_2{e}^{-i{k}_2{x}_1} & {R}_3{e}^{-i{k}_3{x}_1} & {R}_4{e}^{-i{k}_4{x}_1} \\ {e}^{i{k}_0{x}_2} & {e}^{-i{k}_0{x}_2} & 0 & 0 & 0 & 0 \\ 0 & 0 & {e}^{-i{k}_1{x}_2} & {e}^{-i{k}_2{x}_2} & {e}^{-i{k}_3{x}_2} & {e}^{-i{k}_4{x}_2} \\ 0 & 0 & {R}_1{e}^{-i{k}_1{x}_2} & {R}_2{e}^{-i{k}_2{x}_2} & {R}_3{e}^{-i{k}_3{x}_2} & {R}_4{e}^{-i{k}_4{x}_2} \end{array}\right] $$

$$ \begin{array}{lll}G\left(\omega \right)&=\left[\begin{array}{ccc} i{Y}_BA{k}_0{e}^{i{k}_0{x}_1} & -i{Y}_BA{k}_0{e}^{-i{k}_0{x}_1} & 0\\ 0 & 0 & - GA{K}_1\left(i{k}_1+{R}_1\right){e}^{-i{k}_1{x}_1}\\ 0 & 0 & -i{Y}_BI{k}_1{R}_1{e}^{-i{k}_1{x}_1}\\ i{Y}_BA{k}_0{e}^{i{k}_0{x}_2} & -i{Y}_BA{k}_0{e}^{-i{k}_0{x}_2} & 0\\ 0 & 0 & - GA{K}_1\left(i{k}_1+{R}_1\right){e}^{-i{k}_1{x}_2}\\ 0 & 0 & -i{Y}_BI{k}_1{R}_1{e}^{-i{k}_1{x}_2} \end{array}\right. &\qquad \left.\begin{array}{ccc} 0 & 0 & 0 \\ - GA{K}_1\left(i{k}_2+{R}_2\right){e}^{-i{k}_2{x}_1} & - GA{K}_1\left(i{k}_3+{R}_3\right){e}^{-i{k}_3{x}_1} & - GA{K}_1\left(i{k}_4+{R}_4\right){e}^{-i{k}_4{x}_1} \\ -i{Y}_BI{k}_2{R}_2{e}^{-i{k}_2{x}_1} & -i{Y}_BI{k}_3{R}_3{e}^{-i{k}_3{x}_1} & -i{Y}_BI{k}_4{R}_4{e}^{-i{k}_4{x}_1} \\ 0 & 0 & 0 \\ - GA{K}_1\left(i{k}_2+{R}_2\right){e}^{-i{k}_2{x}_2} & - GA{K}_1\left(i{k}_3+{R}_3\right){e}^{-i{k}_3{x}_2} & - GA{K}_1\left(i{k}_4+{R}_4\right){e}^{-i{k}_4{x}_2} \\ -i{Y}_BI{k}_2{R}_2{e}^{-i{k}_2{x}_2} & -i{Y}_BI{k}_3{R}_3{e}^{-i{k}_3{x}_2} & -i{Y}_BI{k}_4{R}_4{e}^{-i{k}_4{x}_2} \end{array}\right]\end{array}$$