The stress-flux correlations were numerically studied as early as in by Atherton and Czura [ 22 ]. They attributed the measured flux leakage over pipeline corrosion pits to stress-induced permeability changes and provided a 2D FEA to solve the inverse problem, i. The article [ 23 ] includes a study, in which apparently minor modifications of a model inclusion of the detector assembly produced dramatic qualitative changes of the results.

It was demonstrated by 2D FEA, that anomalous leakage fluxes in the vicinity of pipe defects were non-linearly dependent on a defect depth. Further on, the constant H-excitation in a model gave significantly different results from a more realistic constant-B source. In Atherton [ 24 ] a systematic study was performed towards the inverse problem solution of MFL over far side pipe grooves. The author demonstrated a strong linear correlation between anomalous radial flux densities calculated with 2D FEA and those obtained experimentally from 3D synthetic corrosion pits.

The following paragraphs summarise selected results of more recent research.

## Practical Design of Magnetostatic Structure Using Numerical Simulation

The 2D qualitative vs 3D quantitative models generated and computed in MagNet software were compared against an experiment. The study reproduced the typical stray field behaviour over a defect in a flat plate. Although material data description is missing in the paper, one may presume that linear magnetic permeability was defined. In Katoh et al. The simulation was relatively basic, and did not aim at solving any inverse problem. Another basic sensitivity analysis of the static magnetic circuit composed of a C-core magnet and the metallic specimen can be found in [ 26 ].

Gaunkar et al. They employed a DC excitation and did not take saturation effects into account. Babbar et al. They used MagNet software to perform a static solution with a nonlinear, anisotropic B H curves. Interestingly, apart from the electromagnetic analysis, they started with structural FEA, subsequently dividing the plate into 13 discrete regions, having varying magnetic properties correlated with stress state.

The MFL signal resulting from different stressed regions were studied separately and in combination with the shape effect. Some papers with a precisely defined industrial context are invoked in the following paragraphs. Christen et al. The set-up optimisation was sought, by studying the effect of a sensor lift-off and a magnetic excitation intensity.

Based on the FEA results, a postprocessing procedure was proposed for separating and eliminating disturbance introduced by the steel wrapping. In Gloria et al. The simulation was compared favourably against an experiment. Effects of alignment of nearby pits and stress-induced anisotropy were studied in a non-linear model. Kikuchi et al. In spite of using a 2D approach in a simulation, they observed a good correlation between experimental and FEA-derived initial magnetisation curves for layered plates. Coughlin et al. In this case the finite element 3D stress simulation was carried out, but no magnetic FEA.

Gao et al. The standard 3D MFL, in turn, was numerically studied in [ 33 ], and an efficient yet accurate inversion algorithm in 2D was produced. Some atypical usages of numerical simulation as a supportive tool for MFL are presented below. Snarskii et al. The demonstrated computation speed-up can be important when generating a large database of direct problem patterns, required for any inverse problem solution. Mahendran and Philip [ 35 ] discussed a new magnetic emulsion to enhance visual magnetic NDT. Moreover, they included a comprehensive review of analytical methods of reproducing the field distribution, and pointed to some FEA-based studies, including that by Katoh et al.

Mukhopadhyay and Srivastava [ 36 ] tested efficiency of a discrete wavelet transform for a noise reduction MFL signal from a series of defects in a buried pipeline, and claimed the superiority of the method over its alternatives. The observed trends and perspectives of further research are described in the last chapter of this preview. High-speed non-destructive inspection systems using MFL method are in great demand in online inspection and defect characterisation, especially in pipeline and rail track maintenance [ 20 ]. The main component of these systems is an autonomous mobile device containing magnetic field sources and sensors, usually named pipeline inspection gauge P.

Such a NDT set-up aims at deducing the shape and dimensions of a flaw, out of the voltage induced in a pick-up coil moving relatively fast e. An additional difficulty, in a signal interpretation and modelling, stems from the formation of eddy currents when an excitation source moves over or inside a pipeline. The first typical numerical approach to this problem is quasi-static, i. Another approach consists in representing the movement of the set-up over several time-steps, applying some type of electromagnetic coupling between the objects in the relative movement e. The start of finite element modelling of MFL with the velocity effect dates back to the s e.

Shannon and Jackson [ 39 ] or Rodger et al. The effect of magnetizer velocity on MFL was subsequently simulated in by Nestleroth and Davis [ 41 ] using axisymmetric FEA, followed by works of Katragadda et al. Ireland and Torres [ 45 ] presented 2D finite element simulations in which a section of a pipe is magnetized along its circumference under both static and moving tool conditions. They employed the standard quasi-static approach with velocity equation component. They highlighted difficulties associated with maintaining a stable magnetic circuit when the set-up is moving.

They noted, that a magnetic field profile is extremely complex under both stationary and dynamic tool conditions, and the repeatability of MFL defect patterns can be poor. In another work [ 46 ] the same authors presented a preliminary research on a circumferential magnetizing set-up for pipeline inspection. Both 2 and 3D models were generated, and the B H nonlinearity was taken into account. Several limitations and complexities of the method were indicated, including the difficulty to magnetize a pipe circumferentially and the dependence of the signal on the angular position of the set-up.

Li et al. The authors discussed the signal-to-noise ratio and the optimum configuration of sensors. Nestleroth and Davis [ 47 ] studied pairs of permanent magnets rotating around the central axis of a pipe in proximity of its surface. The generated magnetic field was measurably disturbed over a defect. A finite element simulation, validated with experimental data, allowed for investigating a design space with various parameters such as a geometry, material properties, and excitation frequency.

The mentioned works are a representative, but non-exhaustive selection from amongst numerous studies on MFL with a moving source.

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New research results regularly emerge e. Judging from the number of publications and commercial implementations, the magnetic Barkhausen effect is the most important among micromagnetic phenomena applicable in ENDT. There is a significant amount of experimental articles aiming at elucidating the nature of MBN [ 50 , 51 , 52 ]. The micromagnetic modelling approaches to MBN representation were recently reviewed by Zapperi and Durin [ 53 ], and some results from stochastic Monte Carlo models were published as well [ 54 ]. However, relatively little has been done so far to develop macromagnetic or multiscale modelling schemes.

## numerical treatment — с английского на русский

FEA plays an irreplaceable role in a comprehensive description of a time- and space-distribution of the magnetic induction within a studied object plate, tube , which is impossible with any available analytical or experimental method. Additional, special degrees of freedom D. Symmetry planes and initial conditions require special modelling approach.

A time-transient scheme is preferred over a harmonic calculation, which usually leads to an incorrect representation of skin-effects due to the assumed B H linearity. Augustyniak et al.

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Another sensitivity analysis of the static magnetic circuit composed of a C-core magnet and the metallic specimen can be found in [ 26 ]. The Japanese group [ 59 ] focused on standardized samples used to assess the average induction within an object magnetized dynamically with a C-core set-up. They built a 3D model of the C-core and plates, made of different steels, and concluded, that the standard shim is applicable only to materials exhibiting a high magnetic permeability.

Interestingly, they made use of a pseudo-nonlinear harmonic FEA instead of a fully-nonlinear time-transient solution, and found acceptable comparison against experiment. A recent work [ 57 ] presents a detailed analysis of a time- and space-distribution of the nonlinear magnetic field inside a steel plate magnetized with a double-core electromagnet with a separate control of AC excitation currents on both branches.

These results, obtained with transient electromagnetic FEA are complementary to the previous research by Nagata [ 60 ], who applied the method of boundary elements for calculation of magnetic fields and eddy currents induced by a pair of orthogonal C-cores. Another step after having determined the magnetic field time- and space-distribution is reproducing the characteristics of Barkhausen effect or magnetoacoustic emission.

Spanish team [ 61 ] focused on angular anisotropy of MBN in pipes due to hot-rolling. Microscale FEM simulations of the magnetic flux density in an idealized steel sample containing the ferrite matrix and the pearlite bands were performed. Pulsed MFL belongs to the methods under development. It bears some analogy to the acoustic borehole logging, i. Researchers from the University of Huddersfield [ 62 ] put forward a pulsed magnetic flux leakage technique PMFL for a crack detection and characterization. They indicated the limitations of a DC MFL, and suggested the superiority of PMFL, where the probe is driven with a square waveform and the rich frequency components can provide information from different depths due to the skin effects.

They applied 2D harmonic FEA to demonstrate the feasibility of detection of defects by magneto-optic films. The optical effect was produced by either an AC or DC excitation. Eddy current testing ECT is one of the most effective techniques for detecting cracks and flaws in conducting materials. In ECT devices, an alternate current flows in an exciting coil placed near a specimen suspected to have a flaw. Induced eddy currents affect a signal detected by the surrounding pick-up coils, influenced by a position, shape, and other characteristics of defects or variations of material properties [ 64 ].

The signal is usually analysed in terms of complex impedance plane trajectory. The method applies equally to ferromagnetic and paramagnetic materials, including aluminium alloys. ECT techniques are widely but not exclusively used for the characterization of safety-critical components, e. A FEA solution of an EC problem requires a single harmonic linear run for each frequency of interest.

A modelling can be axisymmetric, 2D or 3D. Material data consists of linear magnetic permeabilities and electrical conductivities of all the regions. The resulting vector fields primary: J—current density and secondary: B—magnetic induction are composed of real and imaginary parts, convertible into a magnitude or phase. Unlike in a harmonic structural analysis, resonances do not occur, so sampling of frequency at relatively large intervals is acceptable.

A mesh density has to be refined within a sub-surface of an object in order to adequately represent the skin depth effect. More discussion on numerical approaches to eddy currents can be found in [ 6 ].

## numerical treatment

First attempts to reproduce eddy current characteristics with FEA date back to late 70ties [ 66 ]. In the 80ties, intense research was conducted, led by Ida, Lord, Udpa and others [ 67 , 68 , 69 , 70 ]. Industrial case-studies were developed, e. Aming at an optimized sensor configuration, Robaina et al. Apart from the experiment, they produced an ANSYS model of the coils, calculating the electric potential inside the coil wires, the resulting magnetic flux, and finally the eddy current density inside the copper sample.

Nonlinearities were neglected because of a relatively low field and current density values. The aim was to optimize dimensions of a coil and an excitation current to have the highest level of a signal as possible. It is interesting to note, that the eddy currents were excited by a pulsed voltage excitation, and not a harmonic steady excitation. In a study dedicated to ECT defectoscopy in airplane maintenance, Rosell and Persson [ 77 ] performed an experimental eddy current inspection of small fatigue cracks in Ti—6AL—4V sample and compare it against a finite element model.

They noted, that as static loads were applied across the crack faces, electrical connections arise within the crack, which has a strong, detectable influence on the eddy current signal. The work focused on optimum method of incorporating this electrical contact effect into the simulation. Apart from defectoscopy, the EC set-ups can be used to evaluate some material properties, including electrical conductivity, magnetic permeability, porosity, and tensile strength. In another example, Ma et al. The response was proportional to a change of mutual inductance of coils.

The ultimate goal of both the experiment and the accompanying simulation was a reliable porosity evaluation with eddy currents. In Augustyniak et al. The proposed amplified differential signal was shown to carry information on the amount of magnetic ferrite phase forming on the tube during service.

Sablik and Augustyniak [ 80 ] proposed an application of EC sensors in the context of steel mills. Both the computation and experiment show an increase of the amplitude of the third harmonic with an increasing tensile strength of the plate. Remote field eddy current testing RFECT is a major, industrially important variation of the standard eddy-current defectoscopy. The choice of a frequency reduces the skin effect and enables detection of flaws localized at the opposite side of the examined object, usually a pipe.

In the ties, RFECT phenomena were already experimentally observed successfully analysed using the finite element method. In Kim et al. Kasai et al. They modified the sensor and examined the signal itself as well as the signal-to-noise ratio.

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They determined algorithm parameters using proprietary 3D numerical FEA, and validated the concept with experimental measurements conducted on a small test tank. FEA was successfully employed in non-standard EC techniques, including pulsed eddy-current defectoscopy [ 87 ] and stochastic approaches [ 88 ]. A permanent magnet in the simulation had a nonlinear anhysteretic B H relationship.

Three examples have been selected for thes review. Zhang et al. The problem at hand concerned pulsed eddy currents produced in a ferromagnetic plate with a flat bottom hole. Miorelli et al. Finally, Xin an Lei [ 13 ] addressed a difficult problem of incorporating minor hysteresis loops into the calculation, in the context of measurement of pipeline wll thickness.

Fetzer et al. They argued, that representation of the surrounding air with boundary elements greatly reduced computational effort. Another alternative approach consisted in coupling FEA with some non-standard algorithms. For example, Ida [ 91 ] has recently proposed the coupling of FEM with surface impedance boundary conditions SIBC , allowing elimination of the mesh in a conductor beyond the skin depth zone, thus increasing the speed of the solution without compromising accuracy.

Similarly, Sabbagh and co-workers put forward an eddy-current NDT modelling scheme based on volume-integral equations [ 92 , 93 , 94 ]. The approach proved to be very successful in the computation of flaw responses in a number of simple geometries, but exhibited limitations in description of complicated surfaces. More details on ECT methodologies with some references to numerical modelling can be found in [ 95 ].

The method can be considered as an extension of the standard ultrasonic defectoscopy particularly well suited for non-contact wave generation. EMATs work on nonmagnetic conducting materials Lorentz force , or ferromagnetic materials combination of Lorentz force, magnetostriction and magnetization forces. Optimization of EMATs in ferromagnetic materials is often accomplished using computational simulations that account for all the mentioned three main types of transduction mechanism.

However, modelling the magnetization force is the least understood part of EMAT simulation and various authors often use controversial methods that lead to contradictory predictions. Although EMATs are broadband transducers and can function with pulsed excitation, they are often excited with a narrow band tone burst to maximize the signal-to-noise ratio, and the majority of simulations deal with a sinusoidal single-frequency excitation. FE analysis of EMATs represents elastic wave propagation resulting from local application of magnetic fields. It is usually composed of weakly coupled electromagnetic and structural ultrasound analyses.

The electromagnetic FEA involved determination of a distribution of a static bias magnetic field, followed by a transient or harmonic calculation of time-varying B, eddy currents and Lorentz forces. First numerical representations of EMAT date back to the 70ties, with the works by Thompson [ 96 ] and Kawashima [ 97 ].

An exhaustive work by Mirkhani et al. Wang et al. The FEA results corroborated with an experiment. Dhayalan [ ] presented a numerical analysis of multimode Lamb waves interacting with artificial defects and compares these calculations with measurements on a thin aluminium plate. Kim et al. They used a topology optimization scheme over the permeable area of a transducer, in order to maximize both the static bias induction and the time-varying excitation field.

Dutton et al. Effects of pole separation and lift-off were investigated. Both a simulation FEMLAB and an experiment consistently indicated an increase of the achieved magnetic induction in the region of interest by a factor of 1. Huang et al. Zhou et al. In spite of extensive published research, some fundamental controversies remained, including that defined by Ribichini et al. The question of major mechanism of elastic wave creation in ferromagnetic objects was asked: is this predominantly the Lorentz force, or rather magnetostriction?

Using a COMSOL FE model, they demonstrate inconsistencies in previous studies and conclude that the Lorentz force usually dominates, except for highly magnetostrictive materials, such as nickel, iron—cobalt alloys or ferrite oxides. However, waves of higher length including microwaves can successfully be simulated, which was demonstrated in some papers [ , , ]. Electromagnetic wave propagation is the key phenomenon in electromagnetic emission EME , sharing many features similar with the acoustic emission AE.

In case of a crack initiation and propagation, a temporary electric charge imbalance is a source of an electromagnetic wave measurable within a distance of a few millimetres [ ]. Gade et al. They claimed, that during the crack growth, an electromagnetic wave of acoustic frequency could be detected within a few mm from the surface of the studied epoxy resin object, and that EME could be a useful complementary tool for classic AE NDT. It has to be underlined, that problems involving electromagnetic waves are more efficiently solved using other numerical approaches than FEA, such as Method of Moments MoM or finite difference in time domain FDTD [ ].

NDT methods making use of an electric field as a primary factor are electrical impedance tomography EIT and electrical resistance tomography ERT , being a particular case of the former. They do not involve any magnetic phenomenon and can be solved with FE e. The source field is produced by a direct current DC or by an AC source at very low frequency. Potential drops or local electric field measurements are often preferred to impedance or local magnetic field measurements, which are typical for ECT.

Szczepanik and Rucki [ ] performed a field analysis and investigated electrical models of multi-electrode impedance sensors. Both the full FEA modelling and the simplified representation were successfully validated by the experiment. Kahunen et al. They used an in-house software for both the forward and inverse analysis.

Acceptable solutions of the inverse problem in some benchmark set-ups were presented. Albanese et al. They carried out two benchmark simulations and experiments. ERT was used to retrieve the shape of a thin crack, and subsequently the geometry of a column immersed in water. Genetic algorithms were used to solve the inverse problem. Daneshmand [ ] and his co-author modelled the EIT to solve a forward problem in a biomedical context a human thorax.

Claimed cost-effectiveness of a simple portable measurement set-up has to be weighted against low reliability, leading to false-positive or false-negative findings. In spite of the official industrial standardisation ISO , there is a shortage of fundamental studies, especially those demonstrating its aptness to detect and quantify stress concentrations. Interestingly, the magnetic FEA, useful as it proved in the standard MFL research, has rarely been used to understand, calibrate and optimize the methodology of passive stray field technique.

The only published applications of numerical magnetic analyses in this context are either non-conclusive [ ], or bring forth arguments against the concept [ , , ]. Zurek [ ] presented a preliminary study of contactless magnetostatic stress measurements on an ring made of a plain carbon steel.

The results were partially successful, because a limited number of strain gauges did not allow the exact determination of stress and strain changes on the external ring surface. Usarek et al. A simple 3D model generated spatial variation of normal and tangential B component in agreement with the experimental data. The article concluded, that a reliable analysis of the stray magnetic field signal required balanced consideration of several factors, such as geometry of the element, plastic strain, and both internal and external stresses.

In a follow-up study, Usarek et al. Experimental, as well as magnetostatic FEA, showed that geometrical effect, related to the notch presence, as well as degradation of magnetic permeability in the yielded zone, were mainly responsible for the specific MFL distribution above the sample. In Augustyniak and Usarek [ ] the magnetostatic 3D FEA was applied to study the influence of concurrent factors geometrical discontinuity, magnetic permeability and remnant magnetisation on the field strength gradient, claimed by some MMM proponents as correlated with elevated residual stress.

The sensitivity study revealed, that the field gradient did not carry enough information to allow solving of an inverse quantitative problem, so determination of stress levels with a magnetic passive stray field alone is impossible. The effect of residual stresses in the studied dog-bone sample was found to be play minor role. Numerical analysis served in this case to demonstrate the ambiguity of signal recorded in the passive MFL, and helped to better define the scope of applicability of the method.

However, application of numerical analysis in each ENDT method exhibits some unique tendencies. In static MFL, the range of simulation roles is broad: studying the stress or defect influence on stray field [ 27 ], optimising a set-up [ 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 ], and finally providing a reference solution for some special alternative numerical scheme [ 34 ]. One of the most advanced works in the field was that by Babbar [ 27 ], with a non-trivial geometry, non-uniform stress distribution and non-linear B H curve, and an excellent agreement between FEA and experiment.

In MFL with velocity effects the defectoscopy of pipelines and its numerical representation is the grossly dominating topic in scientific papers. One of most remarkable achievements was published by Park and Park [ 44 ], who put forward a compensation scheme for the signal from a moving PIG, of direct practical use for the operators. Research on MFL with AC excitation usually aims at a realistic description of the dynamics of magnetic induction within a magnetic circuit composed of a C-core magnet and a studied ferromagnetic object.

In ECT, the leading topic is the design and optimisation of new sensor configurations. Most of the studies feature a well-defined industrial context aviation, power plants, steel plants. Among several papers seeking to optimise a transducer, that by Ribichini [ ] is remarkable for addressing and resolving the fundamental physical controversy concerning the source of the ultrasonic waves. The break-through presented in [ ] a function allowing to quantitatively deduce a residual stress from a stray field level was later shown to be an overinterpretation [ ].

In NDT, the direct problem consists of the evaluation of a field perturbation at measurement probe locations, for a given exciting field and sample characterization position, shape, material properties, stress state and defects. In the inverse problem, one has to find parameters of a sample assuming measurements and a forcing field as known quantities. Fast, accurate, and reliable simulation tools are necessary for a success of any inversion procedure.

There are three fundamental obstacles making the inverse problem solution in industrial ENDT challenging. The first one is the possible non-monotonic character of some measured parameters. The second consists in coexistence of several influence factors of a comparable impact on the measured signal. Relevant examples are the confounding conductivity and permeability factors in EC, or superposition of magnetic anisotropy and stress state in MBN [ 50 ].

The third problem is the complexity of any in-situ measurements, involving uncertainties higher than those associated with laboratory setting. These issues are discussed by [ ], in the context of eddy current defectoscopy, and by Karhun et al. The numerical solution of the inverse problem is usually not unique, and it is sensitive to modelling errors and a measurement noise. Some attempts were unsuccessful, for example that by Roskosz [ ], because of excessive number of influence variables as compared to the amount of measured data, which made the function inversion impossible [ ].

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