Fast identification of cracks using higher-order topological sensitivity for 2-D potential problems

Abstract

To cite this version:Marc Bonnet. Fast identification of cracks using higher-order topological sensitivity for 2-D potential problems. Engineering Analysis with Boundary Elements, Elsevier, 2011, 35, pp.223-235 Abstract This article concerns an extension of the topological sensitivity (TS) concept for 2D potential problems involving insulated cracks, whereby a misfit functional J is expanded in powers of the characteristic size a of a crack. Going beyond the standard TS, which evaluates (in the present context) the leading O(a 2 ) approximation of J, the higher-order TS established here for a small crack of arbitrarily given location and shape embedded in a 2-D region of arbitrary shape and conductivity yields the O(a 4 ) approximation of J. Simpler and more explicit versions of this formulation are obtained for a centrally-symmetric crack and a straight crack. A simple approximate global procedure for crack identification, based on minimizing the O(a 4 ) expansion of J over a dense search grid, is proposed and demonstrated on a synthetic numerical example. BIE formulations are prominently used in both the mathematical treatment leading to the O(a 4 ) approximation of J and the subsequent numerical experiments.

Performance analysis of multi-frequency topological derivative for reconstructing perfectly conducting cracks

“…Accordingly, the extension of this research to the crack with Neumann 12 boundary condition will be an interesting work. We believe that applying higher-order terms in the asymptotic expansion formula [9] give a higher order topological derivative [13]. The calculation and analysis of higher-order topological derivative will be a valuable research topic.…”

“…This concept was originally developed for the shape optimization problem, but its application to rapid shape reconstruction has only recently been proven. Related works can be found in [5,10,12,13,14,24,25,27,30,31,38] and references therein. One of the advantages of topological derivative concept is that it does not require a large amount of many incident field data; however, a reduction in the amount of this data has been reported to result in poor resolution of the reconstructed shape.…”

Topological sensitivity based far-field detection of elastic inclusions

“…By passing the limit δ d → 0, the topological derivative of the cost functional J[u inc ](z S ) is obtained and the required expression (20) for I TD [u inc ] follows immediately.…”

“…Many promising computational and mathematical frameworks adaptive to different imaging and experimental setups have been developed to address these inverse problems over a span of last few decades (see, e.g., [6][7][8][9][10][11][12][13][14][15][16]). In particular, topological sensitivity frameworks have received significant attention for the reconstruction of location, shape or constitutive parameters of anomalies due to their simplicity and robustness (see, e.g., [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33]).…”

Linear sampling method for imaging an extended crack using limited-range far-field data

“…Moreover, iteration-based schemes need the calculation of Fréchet derivative, appropriate regularization terms, and a priori information about the unknown crack. To avoid these difficulties, alternative methods have been developed, for example, MUltiple SIgnal Classification [12], [13], [14], [15], [16], [17], [18], [19], [9], topological derivatives [20], [21], [22], [23], [24], [25], Kirchhoff and subspace migration [26], [27], [28], [29], [30], [31], [32], [33], [34], and linear sampling methods [35], [36], [37], [38], [39], [40]. Among them, the linear sampling methods have been successfully applied for reconstructing shapes of various inhomogeneities.…”