Identification of planar cracks by complete overdetermined data: inversion formulae

Abstract

The problem of determining a crack by overspecified boundary data is considered. When complete data are available on the external boundary, a reciprocity gap concept is introduced. This concept formalizes the comparison of the response of the safe body to the response of the cracked one of the same physical characteristics. If the crack is known (or assumed) to be planar, explicit inversion formulae are derived determining the host plane equation and the length of an emerging crack in two-dimensional (2D) situations. A reciprocity gap functional is designed and exploited to establish a complete identification result.Numerical trials of the identification methods proposed show very good accuracies and insignificant computational costs.

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

“…n = n true and τ = e 3 × n true were used in expressions (62) of the constant tensors A 11 , A 22 , A 31 . The reciprocity gap method [5] provides a means to identify the normal n = n true from complete overdetermined data on the boundary. Absent prior identification of n = n true , one may treat the angle ϕ such that n = (cos ϕ, sin ϕ) as an additional unknown.…”

Boundary integral formulae for the reconstruction of electric and electromagnetic inhomogeneities of small volume

“…By taking two boundary measurements we have thus directly obtained an approximation to the rescaled polarization tensor as well as the location of the inhomogeneity. The above formulas are in the same spirit as those derived in [2] for the identification of a single linear crack.…”

“…Integrals like Γ have already been used in several contexts to determine "interior" information from boundary data, most notably to determine the location of plane cracks, see for instance [2,3] or [4]. If, in (2.11), we replace u with the right hand side of (1.4), and γ ∂u ∂ν with g, then we obtain…”

Crack identification for transient heat operator by using domain decomposition method

“…These methods does not allow the recovring the crack, they give only qualitative results (information on the shape and the location of the crack). Further more, they are usually based on the essential assumption that the over determined boundary data are 'complete', which means that they are known on the whole outer boundary of the body (Andrieux and Ben Abda 1996 [4]; Brühl et al 2001 [14]; Baratchart et al 1999 [7]). …”