تجزیه و تحلیل حساسیت از ارزیابی غیر مخرب از میکرو ترک ها با استفاده از سنسورهای GMR

Micro-cracks in a magnetized ferromagnetic material cause stray fields that can be observed using giant magnetoresistive (GMR) sensors. This work investigates the applicability of GMR sensors to the non-destructive evaluation of micro-cracks via the observation of stray fields. For this purpose, our measurement setup is assessed using a fast new sensitivity analysis based on adjoint states, employing the finite-element method. A model for the GMR sensor is developed and verified. We are able to resolve micro-cracks with an opening of 3 μm and a depth of 30 μm. GMR positioning inaccuracies are analyzed.

Magnetic stray fields are caused by a change in the distribution of the magnetic properties of a magnetized ferromagnetic object. Their observation in magnetic flux leakage investigations can allow the non-destructive detection and evaluation of material defects [1]. The assessment of micro-cracks can be important in the safety testing of ferromagnetic components. Quantitative magnetic field measurements can be achieved using the giant magnetoresistive (GMR) effect to produce small elements highly sensitive to magnetic fields [2]. GMR sensors measure a projection of the magnetic vector field in one spatial direction. Despite their small size, GMR sensing areas cannot be considered point-like with respect to micro-cracks [3]. Therefore, a realistic sensor model is required for simulations. Inaccuracies in the measurement setup often degrade the quality of detection and the results of the non-destructive evaluation. The aim of this study is to characterize our experimental setup for measurements of magnetic flux leakage using GMR sensors. Sensitivity analysis can provide information about a specific setup׳s detection and identification abilities and indicate possible improvements [4]. It provides gradient information necessary for optimization procedures used for solving inverse problems. For our future evaluation, we are interested in computing the magnetic properties that produce the measurement data. The underlying inverse problem is under-determined; it follows that the number of unknown magnetic parameters greatly exceeds the number of measurements. For the magneto-static case, the problem can be formulated with regard to a few parameters that define the defect geometry. Then an analytical solution of the corresponding sensitivity equations can be computed quickly and accurately [5]. Magnetic flux leakage analysis with respect to the experimental design can be found in [6], [7] and [8]. In [7], a sensitivity analysis for sensor lift-off variations based on analytic stray field equations was performed. For the electromagnetic case, the reciprocity theorem allows quick and accurate sensitivity analysis with only two solutions of the forward problem [9] and [10]. Probabilistic uncertainty and sensitivity analysis is a popular area of research, see for instance [11]. In this paper, we reformulate deterministic sensitivities in terms of adjoint states, as introduced in [12]. We focus particularly on the sensitivity analysis of resolving micro-cracks using GMR measurements. The sensitivity analysis involves a finite-element forward model in order to represent material damages in terms of changes in the distribution of the magnetic permeability. An advantage of the adjoint state formulation is the small number of required finite-element approximations: one simulation for the forward problem and one simulation for eachcorresponding measurement are required for each updated magnetic material parameter distribution. Therefore, quicker computation is achieved in comparison with finite-difference quotient analysis. The reformulation of sensitivities is achieved analytically, and we expect to reduce the overall numerical approximation error. We apply the new adjoint states sensitivity method to the analysis of magnetic flux leakage measurement using GMR sensors. We investigate the detection limits for micro-cracks with respect to measurement uncertainties such as sensor position and orientation. The properties of the linearized inverse problem, which describe the reconstruction of the micro-cracks, are assessed in terms of model resolution analysis and condition numbers.

An adjoint method was introduced to compute the sensitivity distribution inside the observed material region for a given GMR magnetometer sensor. The sensitivity analysis demonstrated the detectability of the sample micro-crack with our experimental setup, where increased sensitivity values were observed at the defect position. The solution of the forward and adjoint problem is flexible because the parametrization of the source material region allows also to represent defects with arbitrary shapes. We used condition numbers of sensitivity matrices in order to analyze GMR positioning uncertainties, which likely occur for our measurements. We found that the combined information from three slightly varied GMR sensor orientations can minimize the condition of the ill-posed problem. Thus, using a GMR sensor array at varied orientations may yield better information about the magnetic permeability. This finding is in agreement with the results reported by [21]. GMR tilting might be much less problematic than a GMR x-position variation, which is more than 4 μm far from the center of the testing probe. Future tasks include the use of the forward and the adjoint model in the reconstruction of the magnetic permeability. This nonlinear optimization can be realized through solving the linearized forward problem sequentially. For more realistic forward calculations, a nonlinear permeability tensor can be considered. This requires detailed information about the investigated ferromagnetic material. The method can be applied to other sensor devices, e.g., Hall sensors and various probe designs.