<title>Reduced order computational methodology for damage detection in structures</title>

Abstract

This paper explores the feasibility of detecting damage within structures such as air foils by application of eddy current based techniques and reduced order modeling. To identify the geometry of a damage, an optimization algorithm is employed which requires solving the forward problem numerous times. Therefore, the forward algorithm must be solved with extremely fast and accurate solution methods. In constructing these forward methods, we employ reduced order Proper Orthogonal Decomposition (POD) techniques.The POD technique is a method which creates an "optimal" ordered basis in the sense that information captured in the first few basis elements is maximized. One then uses a fixed number (based on a quantitative formula for percentage energy captured) of the first few basis elements, called the reduced POD basis, in the forward algorithm. Since one uses only a small number of basis elements, one is able to create a fast forward algorithm that accurately represents the relevant information.In this paper, for illustrative purposes and proof-of-concept, we consider rectangular "cracks" parameterized by a vector parameter q representing the length, thickness, depth, center, etc. of the damage. We attempt to recapture the parameters of a damage assuming we have access to the magnetic flux density B . Our analysis uses simulated data perturbed with normally distributed noise to represent corrupted experimental data. When recapturing the length and thickness of a damage using the component of the magnetic flux density orthogonal to the eddy current flow in the sample, the methods are shown to be efficient and robust even with data containing 10% relative noise.

Real time computational algorithms for eddy-current-based damage detection

“…where c 1 is the constant in lemma 3.1 and K is the constant in lemma 3.3. We can conclude using (23) that…”

“…The mathematical model we use as the basis of our efforts employs Maxwell's equations in phasor form [4] and is derived carefully in [23]. The system equations are…”

“…The boundary value problem is given in terms of two unknowns A and φ satisfying ( 12) and ( 13) with the additional condition given by ( 14) which, as we shall see, uniquely determines the solution when coupled with appropriate boundary conditions. However, we can reduce the above system to a single integro-differential equation by noting (see [23] for details) that the term ∇φ is piecewise constant across all regions. Therefore, ∇φ can be written in terms of A in (13).…”

“…Therefore, ∇φ can be written in terms of A in (13). Hence we can combine all the equations as done in [23] to obtain the equivalent form of the boundary value problem…”

Identification of material damage in two-dimensional domains using the SQUID-based nondestructive evaluation system

“…Numerical experiments are summarized in section 6 to demonstrate the efficacy of the proposed method. Related efforts on similar problems are given in [3,4,15] where eddy-current-based techniques are used and the focus is model reduction techniques as opposed to the use of the method of maps employed here.…”