eddy stuff

Theory and Numerical Modelling of Lorentz Force Eddy Current Testing Dissertation zur Erlangung des akademischen Grades Doktoringenieur (Dr.-Ing.) vorgelegt der Fakult at f ur Elektrotechnik und Informationstechnik der Technischen Universit at Ilmenau von Herrn Dipl.-Ing. Mladen Zec geboren am 31.01.1983 in Biha c Datum der Einreichung: 27. November 2012 Datum der wissenschaftlichen Aussprache: 27. Februar 2013 Gutachter: 1. Univ.-Prof. Dr.-Ing. habil. Hannes T opfer 2. Univ.-Prof. Dipl.-Ing. Dr. techn. Oszk ar B r o 3. Prof. Dr.-Ing. habil. Stanis law Gratkowski urn:nbn:de:gbv:ilm1-2013000175 To my loving wife Nevena and daughter Viktoria ... Acknowledgements This work would not have been possible without the guidance and help of several people who contributed greatly in one way or another in the preparation and completion of the whole thesis. I owe my deepest gratitude to my colleagues and friends Dr. Hartmut Brauer and Dr. Marek Ziolkowski for their invaluable support and guidance over the past years. Without their sincere engagement, encouragement and criticism this work would not be possible. I would also like to thank my supervisor Prof. Hannes T opfer for his kind help and support whenever it was needed. The positive working environ- ment which he created not only facilitated the completeness of the thesis but also made the whole process more enjoyable and productive. Deepest gratitude I owe also to Prof. Andr e Thess for his trust and sincere encouragement. His valuable advices and talks inspired me greatly to work in this topic. Warmest thanks to all my colleagues of the Research Training Group. In particular to my friends Gautam Pulugundla for many open minded scien- tic discussions and for helpful English corrections and R. P. Uhlig for very productive team work and for providing all necessary experiential results. I am grateful to Konstantin Porzig for reading the thesis and for very useful comments and corrections. I would like to express my love and gratitude to my wife Nevena for her en- couragement and understanding during all these years. Without her endless support this thesis would not be possible Finally, I am forever indebted to my mother Nevenka, father Boro and sister Sanja for their dedication and encouragement during all my studies. Zusammenfassung Diese Arbeit beschreibt die theoretische und numerische Untersuchung des neuen zerstrungsfreien Materialprfverfahrens namens Lorentzkraft-Wirbels- tromprfung (LET). LET basiert auf der Messung der Lorentzkraft (LK), die auf ein Magnetsystem wirkt, welches sich in der Nhe eines leitfhigen Probek orpers bewegt. Um die zugrundeliegenden physikalischen Prinzip- ien vorzustellen, wird zuerst ein kanonisches Model untersucht, welches "kriechender Magnet" genannt wird. Durch die vereinfachte Kongura- tion, ist sowohl eine tiefgr undige analytische Beschreibung als auch eine eektive numerische Simulation m oglich. Um realistische LET-Kongurationen zu untersuchen, wird eine neue nu- merische Methodik basierend auf der Finiten Elemente Methode (FEM) entwickelt. Der sogenannte Logischer-Ausdruck-Ansatz (LEA) erlaubt eine schnelle und genaue Modellierung transienter Wirbelstromprobleme mit be- weglichen Teilen in einem statischen numerischen Netz. Die Ergebnisse werden mittels der bekannten "Sliding-Mesh"-Methode (SMT) veriziert und mittels Experiment validiert. Die Studie zeigt die Genauigkeit des LEA f ur alle Werte der magnetischen Reynoldszahl bei Verringerung der Gesamtsimulationszeit um mehr als das 10-fache. Die vorgeschlagene numerische Methodik wird daraufhin angewendet, um die Anwendbarkeit und das Einsatzpotential von LET zu ermitteln. Die Simulationen werden f ur zwei charakteristische LET-Kongurationen durch- gef uhrt, (i) einen defektfreien und (ii) einen Probek orper mit denierten De- fekten. Im Falle eines defektfreien Probek orpers wird beobachtet, dass LET f ur die kontaktlose Messung der elektrischen Leitfhigkeit des Probek orpers genutzt werden kann. Das "Lorentzkraft-Sigmometrie" genannte Verfahren ist robust gegen uber Anderungen des Lift-O-Abstandes und der Mag- netst arke. Die Studie des Probek orpers mit Defekt resultiert in Richtlinien zur eektiven Kraftmessung. Der Ein uss bestimmter Magnetformen und gr oen auf die Detektion und Rekonstruktion von Defekten wird ebenfalls untersucht. Die vorliegende Arbeit unterstreicht die Notwendigkeit einer dierentiellen LK-Messung. Daf ur wird ein neuer und g unstiger dierentieller LK-Sensor entwickelt, welcher auf etablierter Spannungsmessung beruht. Die spezis- che Modikation der urspr unglichen LET-Konguration wird "Dierentielle Lorentzkraft-Wirbelstromprfung" genannt. Abstract This thesis aims at the theoretical and numerical investigation of the novel non-destructive testing technique called Lorentz force eddy current test- ing (LET). LET is based on measurements of the Lorentz force acting on a magnet system moving in a close vicinity of an electrically conducting specimen. To provide insights into the fundamental principles of LET a canonical model, referred to as the creeping magnet problem, is rstly con- sidered. Due to its simplied conguration, this problem is amenable to rigorous analytic treatment and eective numerical simulation. To investigate realistic LET congurations, a novel numerical methodology based on the nite element method (FEM) is developed. The so-called logical expression approach (LEA) allows fast and accurate modelling of transient eddy current problems with moving parts on a xed computa- tional grid. The results are veried and validated using the known sliding mesh technique (SMT) and experiments, respectively. The study shows the accuracy of LEA for any value of the magnetic Reynolds number and also demonstrates the reduction of the total simulation time by more than 10 times. The proposed numerical methodology is later applied to test the feasibility and estimate the testing capabilities of LET. The simulations are performed for two characteristic LET testing congurations, namely (i) a conductor without defects, and (ii) a conductor with pre-dened material defects. In case of non-defective conductor, it is demonstrated that LET can be applied for contactless measurement of electrical conductivity of test specimens. The technique, termed Lorentz force sigmometry, is resistant to changes in lift-o distance and magnet strength. The study involving defective con- ductors provides guidelines regarding more eective force measurements. The in uence of the magnet shape and size on defects detection and recon- struction is analysed as well. The thesis at hand underlines the need for dierential Lorentz force mea- surements. Therefore, a new and low-cost dierential Lorentz force sensor, based on simple and well established voltage measurements, is designed. This specic modication of the initial LET conguration is termed dier- ential Lorentz force eddy current testing. Preface The work outlined in this dissertation was carried out over a period from January 2010 to December 2012, in the department of Advanced Electro- magnetics at the Ilmenau University of Technology, Germany. The project was nanced by the Deutsche Forchungsgemeinschaft (DFG) within the framework of the Research Training Group (RTG) "Lorentz Force Velocime- try and Eddy Current Testing" (Graduiertenkolleg "Lorentz-Kraft"). Expert supervision of the work was performed by Dr. Hartmut Brauer, Dr. Marek Ziolkowski and Prof. Hannes T opfer. All numerical results have been validated by experiments performed by Dipl.-Ing. Robert P. Uhlig. A detailed description of the used experimental setup is contained in the thesis of Mr. Robert P. Uhlig. Contents Contents vii List of Figures xi 1 Introduction 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Aims and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Theory of Lorentz Force Eddy Current Testing 9 2.1 Basic Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.1 Potential Functions . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.2 Fixed Frame of Reference . . . . . . . . . . . . . . . . . . . . . 18 2.2.3 Moving Frame of Reference . . . . . . . . . . . . . . . . . . . . 19 2.2.4 Magnetic Reynolds Number . . . . . . . . . . . . . . . . . . . . 20 2.3 Optimal A Formulation . . . . . . . . . . . . . . . . . . . . . . . 21 2.3.1 Modelling of Non-conducting Domains . . . . . . . . . . . . . . 22 2.3.2 Modelling of Conducting Domains . . . . . . . . . . . . . . . . . 23 2.3.3 Coupling of Potentials . . . . . . . . . . . . . . . . . . . . . . . 23 3 State of the Art in motion modelling using FEM 25 3.1 Brief Introduction to the Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.1.1 Nodal Finite Elements . . . . . . . . . . . . . . . . . . . . . . . 26 3.1.2 Edge Finite Elements . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2 Modelling Motion Using the Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4 Canonical Model: The Creeping Magnet 35 4.1 Analytical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.2 Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 vii Contents 5 Numerical Modelling of Lorentz Force Eddy Current Testing 47 5.1 Benchmark Problem Denition . . . . . . . . . . . . . . . . . . . . . . 47 5.2 Logical Expressions Approach . . . . . . . . . . . . . . . . . . . . . . . 49 5.2.1 Moving Magnet Approach . . . . . . . . . . . . . . . . . . . . . 51 5.2.2 Moving Defect Approach . . . . . . . . . . . . . . . . . . . . . . 52 5.3 Quasi-Static Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.4 Weak Reaction Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.5 Sliding Mesh Technique . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.6 Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.6.1 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5.6.2 Boundary and Interface Conditions . . . . . . . . . . . . . . . . 63 5.6.3 Meshing of the Conductor . . . . . . . . . . . . . . . . . . . . . 64 5.7 Numerical Verication . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.7.1 Verication of the Moving Magnet Approach . . . . . . . . . . . 65 5.7.2 Verication of the Moving Defect and Quasi-Static Approaches . 66 5.7.3 Verication of the Weak Reaction Approach . . . . . . . . . . . 67 5.7.4 Computational Requirements and Simulation Time . . . . . . . 68 5.8 Experimental Validation . . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.8.1 Metallic Sheets Approximation . . . . . . . . . . . . . . . . . . 71 5.8.2 Validation of the Moving Magnet Approach . . . . . . . . . . . 72 5.8.3 Validation of the Moving Defect Approach . . . . . . . . . . . . 73 5.8.4 Validation of the Isotropic Conductivity Model . . . . . . . . . . 73 6 Results and Discussion 87 6.1 Non-defective Conductor . . . . . . . . . . . . . . . . . . . . . . . . . . 87 6.1.1 Magnetic Reynolds Number Study . . . . . . . . . . . . . . . . 89 6.1.2 Lift-o Dependence Study . . . . . . . . . . . . . . . . . . . . . 91 6.1.3 Magnetization Strength Study . . . . . . . . . . . . . . . . . . . 92 6.1.4 Magnet Size Study . . . . . . . . . . . . . . . . . . . . . . . . . 92 6.1.5 Lorentz Force Sigmometry . . . . . . . . . . . . . . . . . . . . . 94 6.2 Defective Conductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.2.1 Magnet Shape Study . . . . . . . . . . . . . . . . . . . . . . . . 96 6.2.2 Magnet Size Study . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.2.3 In uence of the Magnetisation Direction and Intensity . . . . . 101 6.2.4 In uence of the Magnetic Reynolds Number . . . . . . . . . . . 102 6.2.5 Testing Depth Study . . . . . . . . . . . . . . . . . . . . . . . . 103 6.2.6 Lift-o Dependence Study . . . . . . . . . . . . . . . . . . . . . 104 6.2.7 Dierential Lorentz Force Eddy Current Testing . . . . . . . . . 105 7 Summary and Outlook 123 7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 7.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 7.2.1 Numerical Study . . . . . . . . . . . . . . . . . . . . . . . . . . 127 7.2.2 Theoretical Study . . . . . . . . . . . . . . . . . . . . . . . . . . 128 Appendix A 129 viii Contents Appendix B 131 Bibliography 133 ix List offigures x List of Figures 1.1 Illustration of Faraday's 1831 experiment demonstrating the principle of induction. The induced current is detected by the galvanometer. Courtesy of [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Illustration of Foucault's 1855 experiment with rotating copper disk demonstrating the eects of eddy currents. Courtesy of [2]. . . . . . . . 2 1.3 Schematic of F orster's earliest type eddy current testing device. (V) is the reference specimen, (P) test specimen, (J) galvanometer, (L) pho- tocell and (S) sorting mechanism. The illustration has been retrieved from [3, 4]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1 Principle sketch of the Lorentz force eddy current testing. . . . . . . . . 10 2.2 Typical motion involved in LET applications. . . . . . . . . . . . . . . 12 2.3 Denition of the xed frame of reference (K). . . . . . . . . . . . . . . 18 2.4 Denition of the moving frame of reference (K'). . . . . . . . . . . . . . 19 2.5 Truncation of the innite air domain. Mis the magnet domain, Ais the air domain and Cis the conductor domain. . . . . . . . . . . . . . 22 3.1 Sketch of a nite element mesh obtained by using 1storder nodal trian- gular elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 Sketch of a nite element mesh obtained by using 1storder edge trian- gular elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.3 FEM-based methods for modelling eddy current problems involving mo- tion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.1 Sketch of the creeping magnet problem. . . . . . . . . . . . . . . . . . . 35 4.2 Finite dierence solution of the creeping magnet problem. . . . . . . . 40 4.3 Path of the dipole for the pipe with and without a defect
- time shift,
- trajectory expansion ( = 0:05). . . . . . . . . . . . . . . . . 41 4.4 Numerical solution of the creeping magnet problem. . . . . . . . . . . . 42 4.5 Validation of the analytical and numerical models . . . . . . . . . . . . 45 4.6 Comparison of the analytical (A) and numerical (N) models in terms of velocity and Lorentz force perturbation ( Dm= 15 mm,= 1 mm) . . . 46 4.7 Comparison of the analytical (A) and numerical (N) models in terms of velocity and Lorentz force perturbation ( Dm= 15 mm,= 0:25 mm) . 46 5.1 Denition of the LET benchmark problem. The conductor contains three types of defects: long (" j"), wide ("") and cross ("+"). . . . . . 48 xi List of gures 5.2 Basic geometric primitives dened by logical expressions: (a) box (b) cylinder (c) sphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.3 Implementation of the moving magnet approach (MMA). The moving domain is dened in the air region. . . . . . . . . . . . . . . . . . . . . 52 5.4 Implementation of the moving defect approach (MDA). The moving do- main is dened inside the conductor. . . . . . . . . . . . . . . . . . . . 53 5.5 Implementation of the weak reaction approach (WRA). Only conducting region needs to be considered. . . . . . . . . . . . . . . . . . . . . . . . 57 5.6 Implementation of SMT. In the assumed xed frame of reference (K) the small air region including the magnet has been moved (Moving part). The rest of the computational domain was xed (Fixed part). . . . . . 59 5.7 Comparison of two SMT solutions of the LET benchmark problem for conductor free of defects. The eld continuity across the interface has been obtained by means of Lagrange multipliers and by the interpolation technique. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.8 Distribution of the magnetic eld Band induced current jatt= 0.
Xstart is the elongation of the initial position of the magnet which depends on the applied initial conditions . . . . . . . . . . . . . . . . . 76 5.9 Elongation of the initial position
Xstart . . . . . . . . . . . . . . . . . 76 5.10 Initial quasi-static solution ( A, ) for the transient simulation. . . . . 77 5.11 Truncation of the innite air domain. . . . . . . . . . . . . . . . . . . . 77 5.12 Distribution of the nite element mesh inside the conducting domain which is used in all numerical calculations for conductors with defects. The mesh consists of only tetrahedral elements. . . . . . . . . . . . . . 78 5.13 Verication of the moving magnet approach (MMA) with the sliding mesh technique (SMT). The long (" j") defect is located d= 2 mm below the surface of the conductor. . . . . . . . . . . . . . . . . . . . . . . . . 79 5.14 Verication of the moving defect approach (MDA) and quasi-static ap- proach (QSA). The comparison is performed for Rm= 0:1 (v= 0:103m/s). The long ("j") defect is located d= 2 mm below the surface of the con- ductor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.15 Verication of the moving defect approach (MDA) and quasi-static ap- proach (QSA). The comparison is performed for Rm= 10 (v= 10:3m/s). The long ("j") defect is located d= 2 mm below the surface of the con- ductor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.16 Verication of the weak reaction approach (WRA) with the moving mag- net approach (MMA). The long (" j") defect is located d= 2 mm below the surface of the conductor. . . . . . . . . . . . . . . . . . . . . . . . . 81 5.17 Experimental LET setup used for validation of the numerical results. (i) Solid conductor with linear surface defects (ii) Package of metallic sheets with articial defects. . . . . . . . . . . . . . . . . . . . . . . . . 82 5.18 Comparison between the anisotropic conductivity model (ACM) and isotropic conductivity model (ICM). The long (" j") defect is located at d= 2 mm and v= 0:5m/s. . . . . . . . . . . . . . . . . . . . . . . . . . 83 xii List of gures 5.19 Validation of numerical results obtained by the moving magnet approach (MMA) with experiments (EXP). The long (" j") defect is located in the second sheet from the top surface of the conductor ( d= 2 mm) which is modelled using ASM. . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.20 Validation of numerical results obtained by the moving defect approach (MDA) with experiments (EXP). The long (" j") defect is located in the second sheet from the top surface of the conductor ( d= 2 mm) which is modelled using ASM. . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.21 Validation of numerical results obtained for isotropic conductivity model (ICM) and moving magnet approach (MMA) with experimens (EXP). The test conductor is solid with idealised surface defects in form of linear slits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 6.1 Distribution of the nite element mesh inside the non-defective con- ductor used in the QSA modelling. The mesh consists of 2ndorder hexahedral elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.2 Lorentz force dependency on the value of Rm. Study is performed for conducting plate with thickness Hc. The corresponding velocity range isv= 0::40 m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6.22 In uence of the defect depth don the P2P Lorentz force perturbations. The calculations have been performed using MDA for Rm= 0:48 (v= 0:5 m/s) and lift-o distance z= 1 mm. . . . . . . . . . . . . . . . . . 104 6.26 Schematic of possible dierential Lorentz force sensor for DiLET appli- cations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 6.28 Comparison of dierential Lorentz force signals with respect to the in- duced voltages. Calculations have been performed for long (" j") defect located atd= 2 mm using MDA. The magnet is located at lift-o dis- tancez= 1 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 6.3 In uence of the induced (secondary) magnetic eld Bson the total mag- netic eld Bdistribution with respect to Rm. . . . . . . . . . . . . . . 111 6.4 Lorentz force dependency on the value of Rm. Study is performed for two conductors made from aluminium ( Al= 24:1 MS/m) and copper (Cu= 57:9 MS/m). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 6.5 Dependency of the lift-to-drag ratio on the value of Rm. . . . . . . . . . 112 6.6 Lift-o dependency of the Lorentz force. Testing velocity is v= 2 m/s. 113 6.7 In uence of the magnet magnetization strength jMjon the Lorentz force (Rm= 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.8 In uence of the magnet size (volume) magnetised with constant mag- netisation density Mon the Lorentz force ( Rm= 1). The used magnet has cylindrical shape ( DmHm=aa) and is placed at a lift-o distancez= 1 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 6.9 Numerically obtained calibration curves for lift-o distance z= 3 mm 114 6.10 Calculation of the point spread function (PSF). The relative position between the point-like defect and the permanent magnet is denoted by r. 115 xiii List of gures 6.11 Fictive point-like defect modelling used for fast calculation of the point spread function (PSF). r0is the relative position between the point-like defect and the permanent magnet. . . . . . . . . . . . . . . . . . . . . . 116 6.12 Three characteristic shapes of permanent magnets used for in LET sys- tems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.13 Typical shape of the PSF obtained for a spherical magnet having diam- eteraat lift-o distance z= 1 mm and Rm= 0:1. The depth of the point-like defect is d= 1:5 mm. . . . . . . . . . . . . . . . . . . . . . . 117 6.14 Direct comparison of normalized PSF for three typical shapes of perma- nent magnets. All magnets have the same characteristic size determined by length the parameter ai. The maximum perturbation of the spherical magnet
Fmax i;sphere (i2D;L) is used as a reference. . . . . . . . . . . . . 118 6.15 Direct comparison of normalized PSF for three typical shapes of perma- nent magnets. All magnets have the same volume Mwhich results in dierent characteristic length ai(i2a1;a2;a3). The maximum pertur- bation of the spherical magnet
Fmax i;sphere (i2D;L) is used as a reference.118 6.16 LET direct imaging of cross ("+") defect lying at d= 2 mm under the conductor surface. Cubic permanent magnet with a= 1;6;12 and 24 mm is located at lift-o distance z= 1 mm. . . . . . . . . . . . . . 119 6.17 In uence of the magnetisation direction Mon the shape of PSF for Rm= 0:1. The spherical magnet has a diameter a= 5 mm and it is placed at lift-o distance z= 1 mm. The depth of the point-like defect isd= 1:5 mm. The magnetisation direction along the z-axis has been used as a reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 6.18 Denition of the Lorentz force peak-to-peak (P2P) perturbations. The results have been obtained for the long (" j") defect. . . . . . . . . . . . 121 6.19 In uence of the magnetization strength jMjon the P2P Lorentz force perturbations. The cylindrical magnet is magnetised in the z-axis direc- tion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 6.20 In uence of the magnetic Reynolds number Rmon the characteristic Lorentz force perturbations. Calculations have been performed for the long ("j") defect located at d= 2 mm using MDA. The magnet is located at a lift-o distance of z= 1 mm. . . . . . . . . . . . . . . . . . . . . 122 B.1 In uence of the damping parameter of the transient generalized-  solver on the stationary value of Lorentz force ( F0 D;L) and on the resulting Lorentz force peak-to-peak perturbations (
FP2P D;L). The calculations have been performed for Rm= 1 (v= 1 m/s) and
X= 1 mm (dt= 0:5
103s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 xiv Chapter 1 Introduction The fundamental problem of motion of a solid electrical conductor in the presence of a magnetic eld is highly seminal in eld of theoretical physics. In fact it was this scientic thought experiment that had led Albert Einstein to develop the special the- ory of relativity in his famous 1905 Annus Mirabilis paper entitled Zur Elektrodynamik bewegter K orper . However, the actual foundations of Einstein's work were laid almost 70 years earlier in 1831 by Michael Faraday through his work on electromagnetic in- duction [5]. Faraday discovered that when an electrically conducting body crosses the static magnetic eld lines, or when it is exposed to time changing magnetic eld an electric current will ow. As noted by Faraday himself, this was only possible if there is a closed electrically conducting path through which the current can circulate (Fig. 1.1). Nowadays, these currents are widely known as eddy currents . However, the real physical nature of eddy currents was discovered much later in by French physicist Jean Bernard L eon Foucault in 1855. In his famous experiment Foucault discovered that the force required for the rotation of a copper disc becomes greater when being placed between the poles of a permanent magnet (Fig. 1.2). In the same time, he also noticed that the disc is heated by the currents induced inside the metal [6]. Due to his discovery eddy currents are sometimes referred to as Foucault currents . The tendency of eddy currents to oppose its source has been described mathemat- ically by Heinrich Lenz in 1834, whereas the heat produced by the current ow has been described by James P. Joule in 1840 [7]. The braking eect of eddy currents was fully understood much later by Hendrik Lorentz in 1892 [8]. Due to his contribution, the electromagnetic forces caused by the current ow or forces which act on the point charges moving across the magnetic eld lines are known as Lorentz forces . Eddy currents, apart for being fundamentally interesting, also nd versatile elds of applications. In fact their intrinsic eects, either desirable on undesirable, form 1 Chapter 1: Introduction Figure 1.1: Illustration of Faraday's 1831 experiment demonstrating the principle of induction. The induced current is detected by the galvanometer. Courtesy of [1]. Figure 1.2: Illustration of Foucault's 1855 experiment with rotating copper disk demonstrating the eects of eddy currents. Courtesy of [2]. 2 Chapter 1: Introduction the operational basis of many electromechanical devices. A particularly interesting application of eddy currents lies in the eld of non-destructive testing and evaluation (NDT&E). The so-called eddy current testing (ECT) involves the use of a time changing magnetic eld produced by an alternating current (AC) coils to induce eddy currents inside the material under test. If the material contains a defect or any other anomaly which make the spatial distribution of the electrical conductivity non-uniform, the path of the eddy currents will be perturbed. In return, the resulting impedance of the induction coil which generates the AC magnetic eld will be temporarily modied as well. Hence, by measuring the change in the impedance, the defect or anomaly can be detected. This working principle of ECT was rst demonstrated by David E. Hughes 1879 for comparing and material sorting on the basis of dierences in electromagnetic parame- ters such as conductivity and permeability. Hughes not only showed that the principle of electromagnetic induction can be used as a basis for non-destructive evaluation of dierent material properties but also demonstrated the utility of dierential measure- ments for increasing sensitivity. However, it took another 60 years for ECT to be used in industrial applications when in 1937 the German scientist Friedrich F orster applied this technique for metal sorting (Fig. 1.3) [3]. F orster used the basic principles of ECT to excite the test (P) and reference (V) specimens, whereas the so-called pickup coils were connected with a galvanometer (J) forming the secondary dierential loop. In case of any imbalance caused by test specimen, e.g. dierence in the electrical conduc- tivity or due to presence of a defect, the needle of the galvanometer de ects the light to the photocell (L) which activates the sorting mechanism (S). During his work F orster also developed precise theories for many basic eddy current tests and pioneered the impedance plane display in the 1940 [9]. Figure 1.3: Schematic of F orster's earliest type eddy current testing device. (V) is the reference specimen, (P) test specimen, (J) galvanometer, (L) photocell and (S) sorting mechanism. The illustration has been retrieved from [3, 4]. 3 Chapter 1: Introduction 1.1 Motivation Since the work of F orster, there have been many advances in the eld of ECT, which has led to its improved performance and new elds of application. Today ECT represents a particularly accurate and powerful NDT&E technique which is widely used for contactless defect detection as well as for measurement of the thickness, electrical conductivity and magnetic permeability of metallic materials [10, 11]. However, it is well known that ECT suers from a fundamental limitation of low electromagnetic eld penetration depth due to the skin eect [12, 13]. According to the denition, which assumes the AC magnetic elds and an electrically conducting half-space, the skin depth represents the depth at which the applied eld is attenuated to 1 =e37% of its maximum value. It is a function of the applied testing frequency, electrical con- ductivity and magnetic permeability of the specimen under test [13]. The applicability of the traditional version of ECT is therefore limited to the analysis of the immediate vicinity of the surface of a material, usually of the order of a millimeter. Attempts to overcome this fundamental limitation using low frequency dierential coils, giant magneto-resistance sensors (GMR) and superconducting quantum interference devices (SQUID) still have not nd a widespread application in the eld of NDT&E. Neverthe- less, the detection of defects up to 38 mm in a multilayer aluminium stack conductors with excitation frequency of 35 Hz and testing velocity of 6 mm/s has been reported [14], which is a remarkable result. More details about the current state of the art in the eld of ECT including other electromagnetic non-destructive testing (NDT) techniques can be found in [4, 15]. Another important drawback of ECT is a strong dependency in the so-called lift-o distance of the sensor and the surface conditions of the specimen [13]. All these issues are followed by a strong demand to develop new or investigate the possibility of applying hybrid NDT&E techniques which can overcome some of the limitations of already available techniques. In order to avoid or to reduce the intrinsic limitation of ECT in terms of the skin eect, an interesting method referred to as motion induced remote-eld eddy current testing (MIRFEC) has been investigated in [16]. In principle, MIRFEC is a direct current (DC) version of ECT, where currents are induced due to relative motion between the testing specimen and the applied source of the static magnetic eld, i.e an electromagnet or a permanent magnet. The perturbation of induced current is measured by a pickup coil placed relatively far away from the source. Using this approach a high speed material inspection (up to 5 m/s) and detection of surface material defects has been reported in [17]. Based on MIRFEC principles, authors in [18, 19] presented the so-called drag force method for determination of hysteresis losses, fatigue and residual stresses in ferro- 4 Chapter 1: Introduction magnetic electrically conducting materials. The extension of the drag force method for detection of defects by measuring the resulting Lorentz force perturbations has been rst reported in [20], where the technique has been termed as Lorentz force eddy current testing (LET). LET is based on the measurement of the Lorentz force acting on the magnet system moving relative to the conductor under test. In principle, LET diers from traditional eddy current testing in two respects, namely (i) in how the eddy currents are generated and (ii) in how their perturbation is detected. First, LET is based on generating eddy currents by setting a DC magnet system, usually a permanent magnet, into relative motion with respect to the material to be investigated. Under similar testing conditions, in terms of testing speed, this approach has the great virtue that the eddy currents penetrate the material to a much greater depth than with traditional ECT. Second, the perturbation of the eddy currents is detected through its in uence upon the Lorentz force acting both inside the material and on the permanent magnet. If the permanent magnet is swept across a defect, the Lorentz force acting upon it will experience a short breakdown whose detection is the key to successful implementation of LET. An advantage of the fact that the measurement is reduced to a force measurement is that LET can be conveniently applied during production processes since it is robust and relatively straightforward to implement. Based on all these considerations, it is therefore expected that LET can be used to detect defects lying deep within the material under test at relatively high speeds. The main aim of this work is therefore to provide fundamental insights and numerically evaluate the feasibility of the novel testing technique. 1.2 Aims and Objectives In this work, the above mentioned working principle of the novel NDT technique using Lorentz forces is extended to provide its thorough and comprehensive theoretical foundation. Furthermore, the information about the feasibility and testing capabili- ties of LET is numerically evaluated using the well-established nite element method (FEM). Due to relative displacement between the conducting parts an ecient FEM- based methodology is developed allowing fast and accurate modelling of general 2D/3D LET systems. To explain the basic working principles and to improve the understand- ing of the theory underlying general LET systems a simplied model that is amenable to rigorous analytic treatment is presented and solved. To reach the aims of the thesis, the following objectives are dened: 5 Chapter 1: Introduction (i) Provide a detailed theoretical foundation for the underlying physics of general LET systems. (ii) Dene and solve simplied but enlightening version of LET systems referred to as the creeping magnet problem. (iii) Develop a FEM-based methodology for performing 3D transient electromagnetic eld problems with moving parts. (iv) Perform 3D numerical simulations of a given LET benchmark problem to under- stand the in uence of varying testing parameters on the resulting Lorentz force perturbations. The parameters considered are: (a) defect depth and shape, (b) lift-o distance (c) shape and size of the magnet system (d) magnetisation of the magnet (e) electrical conductivity of the test conductor, (f) testing speed. 1.3 Thesis Overview To accomplish the aims and objectives of the project, the entire thesis has been divided into several chapters. In chapter 2 the basic working principles of LET and formulation of the theory describing general LET systems is presented. In chapter 3 the current state of the art for 2D/3D numerical modelling of eddy current problems with moving parts using FEM is presented. This is preceded by a brief introduction to FEM including its nodal and edge formulations. In chapter 4 a simplied LET model referred to as the creeping magnet is dened and solved, both analytically and numerically. Chapter 5 describes the new FEM-based numerical methodology developed for fast and accurate modelling of general 2D/3D LET systems including parts in motion. Additionally, several simplied numerical approaches that enable fast LET analysis are presented. This also includes the determination of limits of their applicability. In chapter 6 the developed numerical methodology is applied to the given LET benchmark problem in order to study the feasibility and testing capabilities of LET. Finally in chapter 7 some concluding remarks of the research and an outlook for future work in LET is presented. 6 Chapter 2 Theory of Lorentz Force Eddy Current Testing The main aim of this chapter is to explain the basic working principle and to formu- late the theory describing the novel electromagnetic non-destructive testing technique (sections 2.1 and 2.2). This contactless testing technique represents a modication of the conventional eddy current testing (ECT) and it is based on the measurement of the Lorentz force acting on the magnet system moving relative to the conductor under test. Since the exploited physical eects are eddy currents and Lorentz forces the technique is termed as, Lorentz force eddy current testing (LET) [20]. 2.1 Basic Principles The problem of an electrically conducting object moving in a static magnetic eld is well known and it is often used as a simple experiment to introduce the Faraday's law of induction [6, 21{23]. Let Bpbe a primary magnetic eld, i.e. the static magnetic eld which exists in space free of conductors. According to Faraday's law, when an electrically conducting object moves across static magnetic eld an electromotive force (EMF) is induced inside the material. The induced EMF, also known as motional EMF, according to Ohm's law further produces eddy currents jwhich follow circular paths in planes normal to the eld lines (Fig. 2.1). The direction of the induced current is determined by Lenz's law which states that the current will circulate in such a way that the induced or secondary magnetic eld Bsis opposite in polarity to the applied primary magnetic eld. This yields a total magnetic eld Bwhich is described as a sum of the two elds, B=Bp+Bs. Typically, in Lorentz force eddy current (LET) applications a permanent magnet is used as a source of Bp. Nevertheless, the same fundamental principle applies if the magnetic eld is created by a direct current (DC) 7 Chapter 2: Theory of Lorentz Force Eddy Current Testing Figure 2.1: Principle sketch of the Lorentz force eddy current testing. induction coil or by a complex magnet system which can be a combination of both (Fig. 2.1). Due to their interaction with the magnetic eld eddy currents further produce Lorentz force which acts inside the material and brakes the conductor in motion. The total value of the braking force FBwhich can be derived from the Lorentz force law [8] describing the force on a moving charged particle is given by FB=ZZZ CjBd ; (2.1) where Cis the volume of the conductor. This eect is well known and it is widely used in applications such as electromagnetic braking. However, it is less widely appreciated that by Newton's third axiom actio = reactio the force of equal intensity but opposite direction acts on the source of the applied static magnetic eld, i.e a permanent magnet or a DC induction coil. In case of a magnet this force is also referred to as the Kelvin force [24] F=ZZZ M(M
r)Bd =ZZZ CjBd ; (2.2) where Mis the magnetization density and Mis the volume of the used magnet. However, throughout this work more common term Lorentz force has been used. De- pending on the type of motion involved, Lorentz force in general comprises of all three components which are here referred to as Lorentz force drag ( FD), lift (FL), and side (FS) component (Fig. 2.1). This eect has been recently exploited in applications such as magnetohydrodynamics (MHD) for contactless measurement of liquid metal ows in metallurgy and it is known as Lorentz force velocimetry (LFV) [25, 26]. If the conductor contains a defect the ow of induced eddy currents will be tempo- rary perturbed (Fig. 2.1). According to Amp ere's law, the perturbation of the induced current will cause a change in the magnetic eld Bas well. In LET the perturbation of eddy currents is detected through its in uence upon the Lorentz force acting both 8 Chapter 2: Theory of Lorentz Force Eddy Current Testing inside the material and on the permanent magnet. For example, if the magnet is swept across a defect, the Lorentz force acting upon it will experience a short breakdown whose detection is the key to successful implementation of LET. If the material is free of defects the Lorentz force will remain unchanged. The principle of using motion to induce eddy currents, can be used to overcome or to reduce one of the major disadvantages of the traditional ECT which is limited by the strong frequency dependent skin eect [10, 27{29], and thus to surface and near surface inspections. Therefore, it is expected that LET can be used to detect defects lying deeper inside the conducting materials compared to the traditional ECT, when the inspection is performed at similar testing conditions. Some of the preliminary results demonstrating this ability can already be found in [15, 30, 31]. However, the main aim in this work is not to provide direct comparison between LET and traditional ECT. The main intention here is to present LET as a complementary NDT method that can, together with ECT or some other testing techniques, provide better inspection results. For practical realisation of LET it is necessary to provide the relative motion be- tween the magnet and the conductor under test. In order to formulate the theory describing the LET, a general assumption is made that the magnet is in motion rela- tive to a stationary conductor. 2.2 Governing Equations Lorentz force eddy current testing belongs to a special entity of dynamic electro- magnetic eld phenomena in which the eect of materials in relative motion occurs. This eect is well known and it represents the basic operating principle over a wide range of electromagnetic devices [22, 23, 32]. They mainly include dierent types of electromechanical systems in applications such as electrical machines, magnetic levi- tation, eddy current brakes, magnetohydrodynamics, inductive heating, NDT&E, etc. (a) conductor in translation (b) conductor in rotation Figure 2.2: Typical motion involved in LET applications. 9 Chapter 2: Theory of Lorentz Force Eddy Current Testing The operating frequency and velocities involved in these devices are quite modest com- pared to the speed of light c, thus a quasi-static electromagnetic eld theory is well suitable for their ecient analysis [23]. Relative motion involved in LET applications is typically in the form of rigid trans- lation or rotation, e.g. along one of the coordinate axis (Fig. 2.2). From the general theory of electrodynamics in moving media it is well known that the analysis of such a system can be performed by considering dierent frames of reference, which are either related to the moving (K') or to the xed part (K) of the assembly [21{23, 33{35]. If vis the translational velocity of the magnet described in xed reference frame K, then in the reference frame denoted by K' the magnet is stationary and the conductor is moving in opposite direction with velocity v(Fig. 2.2a). Similarly, if !is the angular velocity of the magnet in K, then in the reference frame K' the magnet is stationary and the conductor is rotating in opposite direction with angular velocity !(Fig. 2.2b). Although, it might seem that these two dierent LET congurations (conductor in translation and conductor in rotation) require separate analysis, the relation between the linear and angular velocity v=!rallows general LET investigation in terms of a three-component velocity vector v(Fig. 2.2b). Here, r=x
ex+y
ey+z
ezis the position vector of a moving material point. By introducing dierent frames of reference the electromagnetic eld analysis re- quires transformation equations which relate the eld quantities from one reference frame to another [22]. As already mentioned, the emphasis here is to a quasi-static electromagnetic eld phenomena in the domain of Newtonian mechanics ( vc). Thus, spatial and time coordinates between K and K' frames are related by Galilean transformations [22, 23] r0=rvtandt0=t (2.3) where randr0, are position vectors, and tandt0are time coordinates in K and K', respectively. The relationship between generally valid Lorentz transformations and its low-velocity approximation ( c!1 ) given by Galilean transformations has been dis- cussed in much more detail in [21{23, 33{35]. However, it is important to mention that the treatment of relative motion, either in its low-velocity range or in the relativistic range is based on the postulate that the laws of physics, and by that the Maxwell's equations, are form-invariant in any reference frame. These transformations are gen- eral and they can be applied to moving media of nite extent and of arbitrary shape [22, 23, 32, 36]. Based on the observations given above, the theory of LET has been described in two dierent, albeit physically equivalent frames of reference, namely the xed frame 10 Chapter 2: Theory of Lorentz Force Eddy Current Testing of reference and the moving frame of reference . In the xed frame of reference the coordinate system (K) is referred to a stationary conductor and in the moving frame of reference the coordinate system (K') is referred to a moving permanent magnet (Fig. 2.2). The working principle of LET has been introduced by considering several fundamental laws of electromagnetism. The generalization of these laws as well as their mathematical description is given by the set of Maxwell's equations which are supplemented by the constitutive relations. Even though there exist many excellent text books on the theory of classical electromagnetism, for the completeness of this work a short overview of the Maxwell's equations in their quasi-static approximation is presented. Maxwell's Equations Disregarding the term accounting for displacement currents, introduced by Maxwell in the modied Amp ere's law ( @D=@t=0), the quasi-static approximation of Maxwell's equations is as follows Faraday's law:rE=@B @t(2.4) Amp ere's law:rH=je+j (2.5) Gauss's law:r
B= 0 (2.6) where Eis the electric eld strength, His the magnetic eld strength, Bis the mag- netic ux density, and jeis the external current density, e.g. of an induction coil (Fig. 2.1). The induced eddy current density jinside the moving conductor has to full the continuity equation which is implied by the conservation of electric charge r
j= 0 (2.7) The system of partial dierential equations given by (2.4)-(2.7) is not complete. More- over, only two of the given equations are independent, i.e. the Faraday's law and the Amp ere's law [37]. In order to obtain a closed-form solution, Maxwell's equations are supplemented by the constitutive relations. These relations describe the material in the presence of an electromagnetic eld by relating the corresponding material properties and the electromagnetic eld quantities B=0(H+M) (2.8) j=E (2.9) 11 Chapter 2: Theory of Lorentz Force Eddy Current Testing where0anddenote for the magnetic permeability of vacuum and the electrical conductivity of the material, respectively. The equation (2.9) is widely known as Ohm's law. The magnetization density Mintroduced in (2.8) describes how the material is magnetized when placed in an external magnetic eld H. It can be interpreted as the volume density of the induced magnetic dipole moments inside the material. Permanent magnets, e.g. have a non-zero magnetization Mwhen there is no magnetic eld present. The amount of magnetization left behind is also called the magnetic remanence which is described as Br=0M. This means that the permanent magnets can be eciently modelled by a material having a specied distribution of magnetization density M or an equivalent amount of magnetic remanence Br. The extent to which a material is magnetized in general depends on the strength of the applied magnetic eld, i.e. M=mH, wheremis the magnetic susceptibility. As a result the constitutive relation (2.8) can be written as: B=0(1 +m)H=0rH (2.10) whererrepresents the relative magnetic permeability of the material. Based on this non-dimensional quantity materials are classied as diamagnetic, paramagnetic and ferromagnetic materials [37]. Due to their weak in uence on the resulting magnetic eld distribution ( r1) diamagnetic and paramagnetic materials are also referred to as non-magnetic, whereas ferromagnetic materials due to their high in uence on the magnetic eld distribution ( r1) are often referred to as magnetic materials. The material properties are usually non-linear, which means that they depend on the corresponding eld values and temperature T, i.e.r=r(H;T) and=(E;T). For isotropic materials they are scalar quantities, whereas for anisotropic materials they represent second-order tensor quantities [37]. It should be mentioned that the electric and magnetic properties of materials are very important factors in NDT&E as they characterize its mechanical and physical state, e.g. its composition and structure, pres- ence of defects, temperature, etc. However, they also restrict the type of the NDT&E method which can be applied. Without any loss of generality, in order to introduce LET and prove its basic principles all material properties are assumed to vary linearly with the applied magnetic eld and are temperature independent. Furthermore, the conducting objects used for the analysis are considered to be non-magnetic as well (r= 1). Maxwell's equations as expressed in the form given by (2.4)-(2.6) are rst formu- lated by J. C. Maxwell for stationary media. Nevertheless, according to the relativity postulate stated earlier in the text, these equations are form-invariant when described 12 Chapter 2: Theory of Lorentz Force Eddy Current Testing in the moving reference frame K', which is related to the moving magnet r0E0=@B0 @t0 (2.11) r0H0=j0 e+j0(2.12) r0
B0= 0 (2.13) The continuity equation (2.7) and the constitutive relations (2.8) and (2.9) are also form-invariant when described K' r0
j0= 0 (2.14) B0=0(H0+M0) (2.15) j0=E0(2.16) This is one of the main contributions of Minkowski who rst formulated the theory of electromagnetic elds in the presence of moving polarizable and magnetizable me- dia [22, 34, 35, 38]. Due to his contribution, transformations of electromagnetic eld quantities between dierent reference frames is often referred to as Minkowski trans- formation. In (2.11) to (2.16) the velocity of the moving media is not shown explicitly. It is contained implicitly in the transformation equations relating the space and time derivatives between K and K' reference frames. The transformation equations, which can be derived from the well-known Galilean transformations (2.3) are as follows [39] r0=r (2.17) @ @t0=@ @t+v
r (2.18) Applying (2.17) and (2.18) to Maxwell's equations described in K' (2.11)-(2.13), the equivalent system of equations is obtained r (E0vB0) =@B0 @t0 (2.19) rH0=j0 e+j0(2.20) r
B0= 0 (2.21) The Ohm's law for moving conductors is as follows: j0=E0=(E+vB) (2.22) 13 Chapter 2: Theory of Lorentz Force Eddy Current Testing 2.2.1 Potential Functions In order to write Maxwell's equations in simpler form usually potential functions are used. As it will be shown in the following LET analysis, they additionally provide simplications in calculation of elds, better understanding of underlying physical phe- nomena and in most cases they considerably decrease overall numerical computations. To describe general and time-dependent electromagnetic elds two dierent vector po- tentials are commonly used namely, the magnetic vector potential Aor current vector potential T. They are dened based on divergence-free property of the magnetic eld, r
B= 0, and divergence-free property of the induced current density, r
j= 0, which results in B=rA; (2.23) j=rT: (2.24) From the vector calculus, it is known that these equations are satised up to a di- vergence of AandT, respectively. To ensure the uniqueness of the solution, their divergence has to be additionally specied, which is known as gauging [40]. Due to the similarities of (2.23) and (2.24), it is sometimes a delicate problem to decide which potential formulation should be used for the analysis. The choice strongly depends on the particular type of the problem to be solved, i.e. on the used materials properties, electromagnetic eld sources (voltage or current sources), and on the applied gauging condition. A comprehensive study involving dierent potential formulations and the possibility of combining them in a single eld computation is given in [40{43]. It should be emphasised, that the use of current vector potential T inside the conducting regions requires always additional magnetic scalar potential to be applied, whereas the magnetic vector potential A, can be applied with or without an electric scalar potential V[42]. Apart from being the most widely used formulation, the possibility to disregard the additional electric scalar potential V, makes the magnetic vector potential Apreferable potential function to be used in future LET analyses. By using (2.23) together with Amp ere's law, the remaining eld quantity can be determined as E=@A @trV: (2.25) In the following sections the magnetic eld diusion equation which governs the physics underlying LET is derived. As dependent variables, the magnetic vector po- tential Aaugmented by electric scalar potential Vis used. The resulting A-Vformu- 14 Chapter 2: Theory of Lorentz Force Eddy Current Testing lation is generally valid and it does not result from any model simplication, neither it requires some specic numerical techniques for accurate analysis [40, 42]. Furthermore, depending on the denition of the frame of reference, two equivalent types of diusion equation are presented. 2.2.2 Fixed Frame of Reference In the xed frame of reference the global coordinate system K is associated with the stationary conductor, and the permanent magnet is moving with velocity v(Fig. 2.3). The set of Maxwell's equations which mathematically describes the LET problem in this reference frame is given by (2.4)-(2.6). According to the denition of the mag- netic vector potential (2.23), the Gauss's law (2.6) is implicitly satised, whereas the Faraday's law (2.4) and Amp ere's law (2.5) can be written as follows rE=r @A @trV ; (2.26) r1 0rAM =je+j: (2.27) The above set of equations includes the constitutive relation (2.8) and the denition of the electric scalar potential (2.25). Using the Ohm's law (2.9) and the simple sub- stitution in (2.26) and (2.27) the following transient magnetic eld diusion equation can be derived r1 0rAM =@A @t+rV +je: (2.28) This equation, together with the current continuity condition (2.7), represents the governing equation of LET systems, when described in the xed frame of reference. It includes the external source term in form of current density vector je, which can be used to describe additional DC or AC induction coils. The motion of the magnet is implicit in the time derivative of A, which has certain advantages when being applied to FEM discretization schemes (see Section 3.1)[28]. The previous analysis has shown that, using the denition of electromagnetic po- Figure 2.3: Denition of the xed frame of reference (K). 15 Chapter 2: Theory of Lorentz Force Eddy Current Testing tentials, the general LET system can be described only by AandV, instead of the vector eld quantities EandB. Thus, the number of equations to be solved is reduced from 6 to 4 for 3D LET investigations. In the 2D case only one component of the magnetic vector potential, i.e. Ay
eytogether with the electric scalar potential V needs to be applied. This reduces the number of equations to be solved from 4 to 2. 2.2.3 Moving Frame of Reference In the moving frame of reference the global coordinate system K' is associated with the moving permanent magnet. This makes the magnet stationary, whereas the conductor is moving with velocity v, i.e. opposite to the x-axis direction (Fig. 2.4). The set of Maxwell's equations which describes the LET problem in K' is given by (2.19)-(2.21). By following the same procedure as in the xed frame of reference and using the denition of electromagnetic potentials (2.23) and (2.25), Faraday's and Amp ere's law can be written as follows r (Evr A) =r @A @trV ; (2.29) r1 0rAM =je+j: (2.30) The superscript "0" has been omitted deliberately for simplicity. Additionally, to derive (2.29) and (2.30) the form-invariant constitutive relation (2.15) is applied as well. Using the Ohm's law for moving conductors (2.16) and simple substitution in (2.29) and (2.30), the following transient magnetic eld diusion-convection equation can be derived r1 0rAM =@A @t+rVvr A +je: (2.31) Together with the current continuity condition (2.14), this equation represents the governing equation of LET systems, when described in the moving frame of reference, K'. By direct comparison with the equivalent diusion equation derived in K, it can be seen that (2.31) includes an additional velocity term ( vr A). It describes Figure 2.4: Denition of the moving frame of reference (K'). 16 Chapter 2: Theory of Lorentz Force Eddy Current Testing the convection of the magnetic eld lines due to motion, which is in (2.28) implicit in the time derivative of A. The form of (2.31) is fairly complicated when compared to (2.28). However, computationally this can have certain advantages. When the eld conguration is time independent, i.e. for problems involving uniformly moving conductors with a constant cross-section normal to the direction of motion, only one stationary analysis can be performed to obtain an accurate steady-state solution [37]. In case of the diusion equation (2.28) a magnetic diusion transient response has to be calculated in order to obtain the same solution. This particular possibility makes (2.31) very useful, and therefore it is often applied in analytical studies of simplied eddy current problems with conductors in motion. 2.2.4 Magnetic Reynolds Number As already stated in the basic working principle of LET, eddy currents induced inside the moving conductor create their own secondary magnetic eld Bs(Fig. 2.1). It is reasonable to expect that this eld will perturb the primary magnetic eld Bpof the magnet. It can be shown that the extent of eld perturbation depends on several parameters associated with the conductor in motion, namely its velocity, electrical con- ductivity, relative permeability and its size [32]. Nevertheless, the number of dependent variables can be reduced to only one if the so-called magnetic Reynolds number Rmis used. It represents a non-dimensional quantity, which can be interpreted as a ratio of the induced and applied magnetic eld ( RmjBsj=jBpj) [44]. An equivalent deni- tion ofRmcan be obtained by normalizing the transient diusion-convection equation (2.31) which results in [32] ~r ~r ~A =Rm @~A @~t+~r(~v
~A)~v~r ~A! ; (2.32) where "" indicates the non-dimensional quantities and Rmrepresents the magnetic Reynolds number dened as Rm=0jvjL (2.33) A detailed derivation of (2.32) in terms of the magnetic vector potential is presented in Appendix A [32]. However, a particular attention in Rmdenition (2.33) should be given to the length-scale parameter, denoted by L. There is no general rule in choosing L. It depends strongly on the particular problem at hand and it should be carefully dened (see section 5.1). Using the obtained non-dimensional form of the diusion-convection equation (2.32), 17 Chapter 2: Theory of Lorentz Force Eddy Current Testing the magnetic eld perturbation can be analysed by only one parameter Rm. IfRm1 the convective and the transient terms on the right-hand side of (2.32) can be neglected. The diusion-convection equation is than considerably simplied and it describes only the stationary diusion process. In case of Rm1 the right-hand side in (2.32) domi- nates over the diusion term on the left-hand side. In this case the resulting magnetic eld will be strongly altered due conductor in motion (see section 6.1.1). 2.3 Optimal A Formulation The use of the magnetic vector potential Aand electric scalar potential Vin the whole computational domain, denoted by , represents the general LET solu- tion (Fig. 2.5). In the xed and the moving frame of reference this solution is well- described by (2.28) and (2.31), respectively. Nevertheless, by considering separately non-conducting Mand A, and conducting domains Cthe same solution can be obtained in more economic way [40, 45]. 2.3.1 Modelling of Non-conducting Domains In non-conducting domains ( Aand M), where there are no currents present, it is possible to use only magnetic scalar potential to represent general vector magnetic elds. Based on the resulting curl-free property of the magnetic eld ( rH=0), is described as H=r : (2.34) The obvious advantage of using scalar potentials to represent the electromagnetic eld quantities is that only one scalar function is sucient for their analysis. Using (2.34) the governing equations for both, xed and moving frame of reference ( KandK0) are Figure 2.5: Truncation of the innite air domain. Mis the magnet domain, Ais the air domain and Cis the conductor domain. 18 Chapter 2: Theory of Lorentz Force Eddy Current Testing simplied to Laplace type equation r
(0r +M) = 0: (2.35) The magnetization vector Monly applies to the domain of a permanent magnet. Due to this representation, the number of equations to be solved is reduced from three to one in both Aand M. 2.3.2 Modelling of Conducting Domains In the conducting regions, i.e. inside a moving conductor Cand in general inside the external current coil domain E, the magnetic vector potential Aalong with electric scalar potential Vhas to be applied. This represents the so-called A-Vformulation [42]. However, for transient eddy current analyses it is possible to modify the magnetic vector potential Aas A=A+Zt 0rVdt: (2.36) In this case, the electric scalar potential Vis excluded from (2.28) or (2.31), which results in @A @t=r1 0rA +je; (2.37) for the xed frame of reference K, and @A @tvr A =r1 0rA +je; (2.38) for the moving frame of reference K0. Due to this representation, the number of equations to be solved in Cis reduced from 4 to 3. In general, the modied vector potential Acan be used only in conducting domains with homogeneous electrical conductivity. If an abrupt change of electrical conductivity occurs, e.g. due to defects, it may also result in a discontinuity of the normal component ofAat the interface. This is because J=@A=@t. If this is satised, the obtained set of equations (2.35)-(2.37) and (2.35)-(2.38) represents the so-called optimal A- formulation with respect to the xed or the moving reference frame. 2.3.3 Coupling of Potentials The presented optimal A formulation involves the denition of dierent po- tential functions to obtain particular LET solutions. To provide the continuity of the normal and tangential components of the magnetic ux density B, these potentials 19 Chapter 2: Theory of Lorentz Force Eddy Current Testing have to be coupled on the appropriate interface. The continuity conditions are given as follows [42] rA
n10r
n1= 0; (2.39) 1 0rA n1r n1=0; (2.40) where the unit normal vector, directed from the conducting domain Cto surrounding air domain A, is denoted by n1(Fig. 2.5). 20 Chapter 3 State of the Art in motion modelling using FEM The intrinsic phenomena associated with electric and magnetic elds aect almost all aspects of our everyday life. This is followed by continuous development and design of more sophisticated electro-mechanical devices which result in better functionality, higher eciency and increased safety. Depending on the particular application, this task strongly relays on the accurate modelling of the electromagnetic elds within the device, i.e. it requires solution of the governing equations described by Maxwell under well-dened conditions. Whenever it is possible, the analytical solution of the resulting system of partial dierential equations (PDE) is most wanted [28, 37, 46{ 49]. Apart from providing exact and fast solutions, the obtained closed-form analytical expression also helps in better understanding of the underlying physical phenomena associated with the problem under investigation. Unfortunately, these solutions are not always available and they can be obtained only for some simplied device and eld congurations. Thus, in NDT&E applications the development and optimization of various testing techniques is performed using the numerical methods [28]. Due to its ability to handle complex geometries, anisotropic and inhomogeneous material properties, widely used numerical method in NDT&E applications is the nite element method (FEM) [48]. One of the main aims of this work is to present new FEM- based methodology for fast and accurate modelling of non-destructive eddy current problems with moving parts. For reasons of completeness in this section only a brief introduction to FEM is presented. A detailed description of the FEM can be found, e.g. in [28, 50]. This is followed by literature review on the existing methodology for numerical modelling of general eddy current problems with conductors in relative motion usually encountered in NDT&E applications. 21 Chapter 3: State of the Art 3.1 Brief Introduction to the Finite Element Method The nite element method (FEM) is a numerical technique which gives approximate solutions to PDE which are commonly used in engineering applications to describe the physical behaviour of a system [28, 50]. Unlike some other numerical techniques, e.g. nite dierence method, the FEM is used to approximate the solution rather than directly the partial derivatives appearing in the governing equations. Its fundamental idea is to divide the region under investigation into small well-described elementary domains called the nite elements. In 2D investigations they represent simple geo- metrical forms such as triangles and quadrilaterals, whereas in 3D these domains are typically in form of tetrahedra and hexahedra. The assembly of all elements is called the nite element mesh [50]. Within each nite element the unknown solution, i.e. the unknown scalar or vector potentials (see section 2) are usually approximated with low-order polynomial func- tions, which are in FEM applications widely known as shape functions. They can be dened as scalar or vector quantities. The scalar shape functions are associated with nodes of a nite element and they can be used to approximate both, scalar and vector potentials. The vector shape functions are associated with the corresponding edges of an element and they can only be used to approximate vector potentials. Depending on the particular choice, these elements are also referred to as nodal or edge nite elements. 3.1.1 Nodal Finite Elements In the optimal A formulation, the non-conducting regions of the LET problems are described using the magnetic scalar potential (2.35). In the nite element analy- sis (FEA) the scalar quantities are approximated using nodal nite elements [51{53]. Assuming the discretization of the computational domain as shown in Fig. 3.1 the unknown scalar potential can be estimated as follows (x;y;z ) a(x;y;z ) =NX k=1Nk(x;y;z ) a k; (3.1) whereNi(x;y;z ) is the resulting scalar shape function for particular node kandNis the total number of nodes forming the entire nite element mesh. The actual value of Nis determined by the order of the used polynomial approximation of the unknown potential within the element. This approximations is typically of rst or second order 22 Chapter 3: State of the Art Figure 3.1: Sketch of a nite element mesh obtained by using 1storder nodal triangular elements. [48, 50, 54]. The unknown scalar potentials a iat the mesh nodes are referred to as degrees of freedom (DoF). The exact analytical solution would completely satisfy (2.35). When using the approximate FEM solution given by (3.1) the residual Ris generated. r
(0r a+M) =R; (3.2) By forcing the residual within the whole discretized computational domain to zero, the required numerical procedure is obtained. This can be provided by using the general weighted residual method [48, 50, 54]. NX k=1ZZZ WkRkd = 0; (3.3) whereWkis the weighting function which belongs to node k[50]. Substituting (3.2) in (3.3) and using the vector identity r
(a) =a
r+r
a(3.3) results in NX k=12 4ZZZ r
(Wk0r a)d ZZZ 0r arWkd +ZZZ Wkr
Md 3 5= 0: (3.4) By applying the divergence theorem and the Galerkin's weighted residual procedure (Wk=Nk), the discretized weak form of (2.35) is obtained NX k=12 4I Nk0r andZZZ 0r arNkd 3 5=NX k=1ZZZ Nkr
Md :(3.5) where for simplicity the source term described by magnetization Mhas been placed on the right-hand side of (3.4) and represents the outer surface of and nthe corresponding unit normal vector (see Fig. 3.1). The equation (3.5) represents actually 23 Chapter 3: State of the Art a system of algebraic equations for Ndegrees of freedom (DoF). The resulting matrix of the system, also referred to as the stiness matrix, is symmetric and sparse [48, 50, 55]. To simplify the calculation of integrals in (3.5), instead of using the global coordinates x;yandz, the shape functions are usually dened in a local (normalised) coordinate systems [48, 50]. Using the nodal elements, vector quantities are discretized simply by considering each of its components separately. Applying this procedure at the material inter- faces, which are also the interfaces between dierent elements of the mesh, makes the approximated vector potential normally and tangentially continuous across that inter- face. When solving for modied magnetic vector potential A(see section 2.3) the FEM analysis would be restricted to isotropic and homogeneous materials [42]. To overcome this limitation, edge nite elements are widely applied. 3.1.2 Edge Finite Elements In case when anisotropic and inhomogeneous conducting materials are described only by a single magnetic vector potential, the applied FEM formulation must account for possible discontinuity of its normal component. This requirement can be easily fullled using the edge nite element formulation [42]. Otherwise, instead of using only modied vector potential Ato describe linear conductors in general LET systems, the electric scalar potential Vhas to be additionally applied (see section 2.2). In this case Vaccounts for discontinuity of induced eddy currents [56, 57]. Geometrically, the edge nite elements are identical to their nodal counterparts (see Fig. 3.2). Instead of using only nodes, edges of an nite element can also be used to approximate the unknown vector potential Aas A(x;y;z )A a(x;y;z ) =NX k=1Nk(x;y;z )A tk; (3.6) Figure 3.2: Sketch of a nite element mesh obtained by using 1storder edge triangular elements. 24 Chapter 3: State of the Art where A tkis the line integral of Aalong edge kandNkis the vector shape function at edgek(see Fig. 3.2). This makes only the tangential component of the unknown vector potential continuous across the element interfaces, allowing a discontinuity of its normal component [42, 56, 57]. By following a similar procedure as in the nodal FEM formulation, i.e. using the Galerkin's weighted residual procedure, the weak form of (2.37) and (2.38) can be obtained as NX k=12 4ZZZ Nk
@A a @td +I 1 0rA aNk
nd + +ZZZ 1 0(rA a)
(rNk)d 3 5=NX k=1ZZZ Nk
jed ;(3.7) in the xed reference frame Kand NX k=12 4ZZZ Nk
@A a @tvr A a d +I 1 0rA aNk
nd + +ZZZ 1 0(rA a)
(rNk)d 3 5=NX k=1ZZZ Nk
jed ; (3.8) in the moving reference frame K0. To obtain (3.7) and (3.8) vector identity b
ra= r
(ab) +a
r btogether with the divergence theorem has been applied [42]. For simplicity, the source terms associated with external current density have been placed on the right-hand side of (3.7) and (3.8). As in the case of nodal elements, to simplify the calculation of integrals in (3.7) or (3.8), instead of using the global coordinates x;yandz, the shape functions are usually dened in a local (normalised) coordinate system [48, 50]. Finally, to solve the LET problem using the optimal A formulation, nodal FEM formulation (3.5) has to be coupled with the corresponding edge FEM formulations (3.7) or (3.8) depending on the chosen frame of reference (K of K'). In case of linear conducting materials the coupling can be obtained directly on the outer surface of the conductor by satisfying the continuity conditions (2.40). It should be mentioned that the resulting stiness matrix in the xed frame of reference Kis symmetric, whereas in the moving reference frame K0due to the additional velocity term the stiness matrix is non-symmetric [48, 50, 55]. 25 Chapter 3: State of the Art 3.2 Modelling Motion Using the Finite Element Method The Lorentz force eddy current testing (LET) belongs to a special class of electro- magnetic eld phenomena in which the eect of materials in relative motion occurs (see section 2). In fact, this eect represents the basic operating principle over a wide range of electro-mechanical devices in application areas such as electrical machines, magnetic levitation systems, inductive heating, eddy current brakes, non-destructive testing and evaluation (NDT&E), etc. Due to its tremendous industrial relevance the application of FEM to this particular type of eld problems, also known as the moving eddy current problems [58], has undergone an extensive research over the past decades. As a result many dierent techniques for 2D/3D motion modelling using FEM have been developed, mostly referring to rotational electrical machines [58{61]. However, either due to their high computational requirements or diculties in their implementa- tion only a few techniques are nowadays commonly used in general-purpose commercial FEM codes. The main aim here is to give an overview of these techniques, whereas a more comprehensive study due to a large number of publications would be far beyond the scope of this work. Additional information concerning the particular technique can be found in the extensive reference list presented. In principle, independent on the actual type of motion (translation or rotation) all existing techniques for simulation of general moving eddy current problems can be classied into (i) xed grid methods and (ii) moving or time changing grid methods Figure 3.3: FEM-based methods for modelling eddy current problems involving motion. 26 Chapter 3: State of the Art (see Fig. 3.3) [59]. The xed grid methods are usually applied to 2D/3D static or time-harmonic eddy current problems involving uniformly moving conducting parts having invariant cross- section in the motion direction [27, 58, 59, 62{66]. The eld problem is formulated in the moving frame of reference (see section 2.2) where the additional velocity term ( v rA) is used to describe the induced eddy currents within the moving conducting part. This approach, also referred to as the quasi-static approach (QSA) [66], is very ecient in terms of computational time since only one stationary analysis needs to be performed to obtain an accurate steady state solution. For some simple device congurations several authors combine FEM with analytical solutions as well [65, 67]. Considerably reduction of the simulation time and increased accuracy has been reported. However, apart from simple geometries the analysis is restricted to stationary and time-harmonic problems. The convection of the magnetic eld lines introduced in the xed grid methods can cause spurious numerical oscillations when the resulting P eclet number (Pe=v
x) is large [68{70]. This can be avoided by performing the mesh renement within the conductor region or by technique known as the upwinding [71, 72]. The logical expression approach (LEA) presented in this work, extends the applicability of the xed grid methods to 2D/3D transient moving eddy current investigations (see Fig. 3.3). Furthermore, the moving conductors can have variable cross-section in the motion direction as well. Depending on the type of the object in motion LEA can be described in the xed or in the moving frame of reference. The moving grid methods are more general and they can be applied to simulate wide variety of electromechanical devices involving linear or rotational movement. In principle, from the model topology point of view all available techniques are quite similar. The main idea is to decompose the whole computational domain (the nite element mesh) into two parts associating them with the moving or with the xed part of the assembly [58, 59, 73]. Within each part the governing equations are solved in their own frame of references, whereas the relative displacement and the eld coupling is provided on the introduced interface [61, 74]. Depending on the actual interface, which can have constant or variable lift-o distance, to achieve the coupling many dierent techniques have been applied, each of them having certain advantages and disadvantages. One of the earliest techniques to model relative displacement of 2D induction ma- chines is based on mesh deformation [75]. During the movement this technique can produce elements with large aspect ratios leading to the loss of accuracy [61, 76]. Thus, its application is usually restricted for modelling 2D electromagnetic devices requiring small relative displacements [77{79]. 27 Chapter 3: State of the Art In order to provide arbitrary large displacements techniques based on the continuous re-meshing of the computational domain have been reported [80]. When re-meshing is performed only in the small air-gap region of the device, these techniques are referred to as the step-by-step nite elements or the moving band techniques [74, 80{84]. Due to time changing grids the obtained solution can be numerically noisy. Additionally, for 3D complex geometries the re-meshing procedure can also be very time-demanding. This is why these techniques are mostly applied to 2D eddy current problems. In order to avoid tedious re-meshing of the model several authors combine the advantages of the FEM and analytical expressions which provide the eld coupling without meshing the air-gap of the machine. The resulting technique is often referred to as the air- gap element or macro element method [76, 85{88]. Whenever they can be obtained, the analytical expressions are simply coupled with the FEM formulation providing high accuracy compared to all other methods available [58]. However, as reported in [88] the coupling procedure can reduce the sparsity of the resulting stiness matrix increasing the computational costs. Furthermore, due to diculties in obtaining the analytical expressions, the analysis is usually restricted to 2D eld problems [89]. On the same note, to avoid the re-meshing of the computational domain, hybrid methods based on the coupling of boundary element method (BEM) and FEM have been introduced. The BEM provides all the advantages to model linear unbounded air regions in which the conducting objects are free to move [70, 90{93]. The conducting regions, which in general can be non-linear and inhomogeneous, are eectively modelled by means of FEM. Similarly as before, coupled BEM-FEM technique can produce partially dense matrices. To reduce the computational costs several authors propose parallelization techniques based on a domain decomposition to BEM and FEM part [94, 95]. One of the widely applied moving grid methods to model relative displacements in general eddy current problems is the so-called sliding mesh technique (SMT) [96]. Similarly to all other methods, SMT also referred to as the moving mesh method [58] or slip surface method [97], requires two independent meshes to be dened. To provide the relative displacement, the meshes are simply slid relative to each other eliminating any need to alter their structure. The governing equations are solved independently in the xed reference frame of each moving part, thereby avoiding the convection (velocity) terms. Depending on the mesh distribution along the introduced interface, which can be conforming or non-conforming, the eld continuity can be preserved using several coupling techniques. In case of conforming meshes, the unknown potentials on each side of the sliding interface are made equal in the same way as a Dirichlet boundary conditions are imposed [58, 97{101]. However, the displacement is strictly controlled by the size of the nite elements in the motion direction and the time- 28 Chapter 3: State of the Art step size of the transient solver. To overcome this limitation non-conforming meshes along with Lagrange multipliers have been introduced [62]. The Lagrange multiplier approach introduces an additional set of variables on the sliding interface which ensures the continuity of the eld in a weak sense [102{105]. Unfortunately, the existence of additional variables considerably deteriorates the conditioning of the stiness matrix [105{107]. Furthermore, the matrix contains zeros on the main diagonal, which means that some standard iterative solvers, such as incomplete Cholesky conjugate gradient (ICCG) or conjugate gradient (CG), either have slow convergence or fail completely [79, 105, 107]. All these issues led to development of other coupling techniques for non-conformal meshes, such as interpolation method [80, 105] and the mortar element method [96, 106, 108{110]. Using the interpolation methods the eld continuity is only globally conserved across the interface. Additionally the coupling increases the bandwidth of the resulting matrix system increasing the computational costs for the same number of unknowns [105, 111, 112]. The mortar element method results in a positive denite stiness matrix, which makes it well suited for 3D moving eddy current problems. However, up to now this method has been only applied to solve 2D moving eddy current problems. Additional issues such as complex implementation and increase of non-zero elements in the resulting matrix system have also been reported as well [58, 107]. In order to solve electromagnetic eld problems with parts in arbitrary motion (variable air-gap sizes) the so-called overlapping or composite grid methods have been developed [58, 113{116]. The method applies an iterative algorithm to couple two sets of overlapped grids. The grid of the moving part is discretized with a ne grid, whereas the xed part is discretized with coarser grids. Due to longer simulation time and increased number of unknowns [115] the method is usually applied to 2D eld problems [58]. However, some 3D implementations using nodal nite elements have been reported [115, 117]. 29 Chapter 3: State of the Art 30 Chapter 4 Canonical Model: The Creeping Magnet In order to better understand the underlying physics of LET it is desirable to formulate and study highly simplied models which can be treated analytically. The denition of such a model is the main aim of the following analysis. Thus, the motion of a free-falling permanent magnet in an electrically conducting pipe, which contains an idealized defect, has been analysed (Fig. 4.1). This problem represents a slight modication of a popular educational experiment used to introduce the Faraday's law of induction, and it is a highly simplied yet enlightening version of the general LET system. The working principle is quite similar to the previously described LET setup (Fig. 2.1). Due to the motion of a magnet, eddy currents are induced inside the pipe walls. These currents produce their own secondary magnetic eld Bswhich is causing a braking Lorentz force on the magnet that is decelerating its free fall to a constant and relatively low velocity (usually in the order of several centimetres per second). The eect can be recognized by the signicantly longer falling time of the magnet compared (a) Simplied analytical model (b) Numerical model Figure 4.1: Sketch of the creeping magnet problem. 31 Chapter 4: Canonical Model: The Creeping Magnet with its free fall. This is the reason why the experiment is sometimes referred to as the creeping magnet. If a highly simplied defect is introduced inside the pipe walls, e.g. an axisymmetric cut, the induced eddy currents will be weakened. The braking force will decrease and the magnet will temporarily increase its falling velocity, resulting in a shorter falling time. As it will be demonstrated below, this problem is amenable to rigorous analytic treatment and ecient numerical simulation, which provides a number of unexpected phenomena useful for further development of LET. The investigation at hand is a combination of analytical theory, numerical simulation and experimental validation. 4.1 Analytical Solution The analytical theory consists of the derivation and solution of a non-linear ordinary dierential equation for the time-dependent position z(t) of the falling magnetic dipole, representing the permanent magnet (Fig. 4.1a). The conducting pipe is considered to be innitely long with thin walls compared with its radius ( R). Under this assumption the decay of the magnetic eld within the pipe walls can be neglected. Due to the fact that the creeping magnet problem results in low a magnetic Reynolds number, which is estimated as Rm=0v104(see section 2.2.4), the secondary magnetic eld Bsassociated with induced eddy currents has been neglected as well. Dierential Equation of the Creeping Magnet Motion The creeping magnet problem can be analysed in a very elegant way using the cylindrical coordinate system K' attached to the moving dipole. In this reference frame the dipole is xed and the pipe is moving with velocity v(t) in the direction of z0-axis (Fig. 4.1a). The primary magnetic eld Bpof a magnetic dipole is well-described and it is given by [118] Bp(r0) =0 4
( 3
m
r0
r0 r05m r03) ; (4.1) where r0is the position vector and mis the magnetic dipole moment. Assuming the dipole orientation along the z0-axis ( m=m
ez0) the axial symmetry of the problem allows considerable simplication of the analysis. The resulting magnetic eld has only two components and it is described as Bp(r0) =m0 4
( 3
r0z0 (r02+z02)5=2
er0+2z02r02 (r02+z02)5=2
ez0) : (4.2) 32 Chapter 4: Canonical Model: The Creeping Magnet Using Ohm's law for moving conductors (2.22), eddy currents induced inside the pipe walls follow circumferential paths. The density of the induced current is given by j=j'0e'0=30mv (t) 4
r0z0 (r02+z02)5=2: (4.3) Due to axial symmetry, the resulting Lorentz force (2.2) acts only in the direction of thez0-axis and brakes the free fall of the magnet. Using (4.3) it is described as Fz0=92 0m2v(t) 162
ZZZ pr02z02 (r02+z02)5d ; (4.4) where pis the volume of the pipe. This equation can be further simplied by assuming that the pipe has thin walls ( R), i.e. assuming that the magnetic eld has a constant value, Bp(r0=R) within the pipe walls. In this case the volume integral is simplied to only a line integral over the pipe length ( L!1 ) and it is given by Fz0=92 0m2v(t) 8R3
2 641Z 1z02 (R+z02)5dz0z0+hZ z0hz02 (R+z02)5dz03 75: (4.5) The rst integral in (4.5) can be solved analytically. Together with pre-factor it gives a constant braking force F0 z0acting on the dipole for a pipe free of defects F0 z0=452 0m2v 1024R4: (4.6) When the braking force is equal to the weight ( Vmg) of the magnet (4.6) can be used to estimate its equilibrium velocity ve ve=1024R4Vmg 452 0m2: (4.7) Here,is the density of the magnet, Vmis its volume, and grepresents the acceleration of gravity. The second integral describes the perturbation of the Lorentz force due to the presence of an axisymmetric defect. It has to be calculated for each relative position z0between the dipole and the defect. The total Lorentz force which brakes the free fall of the magnet is then described as Fz0=F0 z092 0m2v(t) 8R3
z0+hZ z0hz02 (R2+z02)5dz0: (4.8) 33 Chapter 4: Canonical Model: The Creeping Magnet This is a very important result since it shows that the relative Lorentz force per- turbation,Fz0=F0 z0, depends only on the geometry of the defect. Whereas, the force perturbation,
Fz0=Fz0F0 z0, depends linearly on the conductivity of the pipe, its velocity and thickness ( Rm), and quadratically on the strength of the applied mag- netic eld and magnetic permeability of the material. This eect can be used to verify numerical results of real 3D LET systems as well (see section 6.2). The dierential equation of motion, which gives the instantaneous position of the dipolez(t) within the pipe, can be obtained from Newton's second law of motion in the form Mz(t) =Fz0+Mg; (4.9) where z=d2z=dt2is the acceleration of the magnet. From (4.8) it can be observed that the braking Lorentz force Fz0depends on several parameters such as: electrical conductivity, thickness and radius of the pipe, weight of the magnet and size of the defect. To reduce the number of variables (4.9) can be normalized on the basis of R andve=gfor length and time, respectively. As a consequence, the time derivatives are calculated as d=g=vedt, whererepresents the non-dimensional time variable. The resulting non-dimensional form of dierential equation of motion (4.9) is given by +2 41128 5+Z 02 (1 +02)5d03 5_=; (4.10) where=z=R is the non-dimensional position coordinate, _=d=d is the non- dimensional velocity and =d2=d2is the non-dimensional acceleration of the falling magnet. Using (4.10) the analysis of the creeping magnet problem can be performed by considering only two non-dimensional parameters, namely the forcing parameter =ve2=(gR) and the defect size =h=R. The derivation of this equation constitutes a key result of the analytical model. To highlight the mathematical structure of the obtained equation it can be re-written as +f(; )
_=: (4.11) The function f(;) represents a position-dependent electromagnetic friction coe- cient which diers from its unperturbed value ( f= 1) only in the region of the defect (= 0). All geometrical data and material properties are contained in . This pa- rameter can be interpreted as a forcing parameter since it appears on the right-hand side of (4.11). If there is no defect, this equation reduces to +_=whose solution given by=describes steady as well electromagnetically damped motion of the 34 Chapter 4: Canonical Model: The Creeping Magnet (a) Dierent (= 0:01) (b) Dierent (= 0:05) Figure 4.2: Finite dierence solution of the creeping magnet problem. magnet with constant downward velocity . It is important to notice that represents the ratio between the squared unperturbed velocity v2 eand the the squared velocity gRwhich a freely falling body would attain after having traversed a height Rin the absence of electromagnetic damping. Hence, small values of correspond to low veloc- ity and strong electromagnetic damping, whereas higher values of indicate a higher velocity and weak damping, accordingly. The geometry of the defect is described by the parameter , where= 0 represents the pipe without defect. The solution of the (4.11) can not be obtained analytically in a closed form. Nev- ertheless, for small values of it is possible to further simplify (4.10) which is then amenable for completely analytical treatment. Due to the fact that this simplication does not contribute to any better understanding of LET, it will not be presented here. More details can be found in [119] where this analysis is referred to as the narrow defect approximation. The solution of the dierential equation of motion (4.10) can be obtained very eciently by implementing the nite dierence method (FDM), e.g. using the general purpose code MatlabR . It is illustrative to provide the solution of (4.10) in order to highlight the role of non-dimensional parameters andand test their in uence on the resulting velocity and Lorentz force perturbations. The results are presented in the form of _=f() showing the velocity of the falling magnet as a function of its temporal position. This representation is more convenient for the creep- ing magnet problem, than the seemingly more natural form used in LET investigations Fz0=f(). However, the same analysis in terms of magnet velocity applies for the braking Lorentz force as well. Fig. 4.2a shows the solution of (4.10) for dierent values of when the size of defect is xed to= 0:01, whereas Fig. 4.2b shows the in uence of for xed= 0:05. One apparently counter-intuitive feature is common to all solutions shown in Fig. 4.2. Based on qualitative reasoning it could be expected that the presence of the defect would lead to a temporary rise in velocity and that the curve _=f() would therefore have a 35 Chapter 4: Canonical Model: The Creeping Magnet bell-shape with a single maximum velocity close to the location of the defect, i.e. close to= 0. By contrast, all curves shown in Fig. 4.2 have two maxima rather than one. This indicates that the falling magnet experiences two phases of acceleration when passing the defect. The reason for this behaviour can be readily understood by invoking Fig. 4.4c in which the qualitative structure of the eddy currents is shown. It can be seen that the eddy currents induced by the moving magnet consist of two structures with opposite orientation. This leads to the fact that the Lorentz force has two minima rather than just one and that _=f() has two maxima as long as the defect is not bigger than the size of the falling magnet (2 h < Dm). The shape of _=f() is symmetric for small values of the forcing parameter. When increases, the velocity distribution becomes asymmetric due to the increasing in uence of inertia. Larger defects cause an asymmetry in velocity distribution as well (Fig. 4.2a). Here, the acceleration phase is dominating when the magnet is experiencing a decrease of the braking Lorentz force due to the presence of a defect. More important to notice is a very strong in uence of the defect size on the velocity, i.e. Lorentz force perturbation. This is a promising result, which can be used to reconstruct the shape of the defect by solving an inverse LET problem. The resulting trajectory of the dipole is shown in Fig. 4.3. When comparing it with the unperturbed trajectory represented by the solid line two features become apparent. In a given time the magnetic dipole travels a distance that is by
longer than in the unperturbed case. Observed, from a dierent perspective, the dipole arrives by
earlier at any position that is suciently far downstream of the defect. Here
 is referred to as the falling time dierence and
as the trajectory expansion with respect to defects free pipe. Since
can be measured, it is a particularly important quantity. It can be shown analytically [119] that
increases linearly with the defect width 2 , i.e.
and
 Figure 4.3: Path of the dipole for the pipe with and without a defect
- time shift,
- trajectory expansion ( = 0:05). 36 Chapter 4: Canonical Model: The Creeping Magnet obey rigorous relations given by
= 2and
=2 (4.12) In principle, if there is only one defect within the pipe its size can be determined using (4.12) simply by time measurements. In order to localise and identify the structure of more than one defect, a time resolved force signal acting on the magnet (or the pipe mounting) needs to be measured. This actually is the basic idea of LET. 4.2 Numerical Solution The analytical model developed so far is suitable for fast determination of the falling time and the characteristic Lorentz force proles. It was derived using several simpli- cations which lead to lower accuracy than a full 3D electromagnetic eld computation would provide. The numerical simulation allows to overcome three limitations inher- ent in the analytical theory and to take into account (i) the nite size and shape of the falling magnet,(ii) the nite thickness of the pipe and (iii) the nite value of the magnetic Reynolds number. The analysis has been divided into two parts. The rst part comprises the consid- eration of a non-defective pipe. As a result the falling time of the magnet through the pipe is determined. The calculated falling time is used for the verication and valida- tion of the numerical simulations. In the second part an ideal defect is introduced into the pipe walls. The main aim of this analysis is to estimate the falling time dierence
with respect to the non-defective pipe. This can help to prove the linear relation between
and(4.12). Pipe Without Defects The geometry used for the numerical simulation of the creeping magnet problem is shown in Fig. 4.1b. The magnetic dipole is replaced with a permanent magnet of a spherical shape, and the conducting pipe is modelled as a hollow cylinder with nite wall thickness . If the magnet moves parallel to the z-axis, the induced eddy currents ow in the azimuthal direction (Fig. 4.1b). Thus, a 2D axisymmetric model can be applied (Fig. 4.4a). As in the presented analytical model the moving reference frame K' associated with the falling magnet has been applied. In this reference frame the magnet is xed, and the pipe is moving with velocity v, in thez0-axis direction. The governing equation is formulated as the diusion-convection equation (2.31), which includes an additional 37 Chapter 4: Canonical Model: The Creeping Magnet (a) Stationary model (b) Transient model (c) Induced currents Figure 4.4: Numerical solution of the creeping magnet problem. velocity term. Since there is no defect within the pipe and its length is assumed to be innite, a stationary form of (2.31) can be applied ( @A=@t=0). In order to account for the innite long pipe, the asymptotic boundary conditions (ABC) [120] have been applied on the outer model boundaries (Fig. 4.4a). This, considerably reduces the computational expenses due to smaller size of the resulting air domain. The computational algorithm is performed in two steps. Initially, for the zero brak- ing force (Fz0= 0) the velocity of the magnet is calculated using (4.9) and FDM implemented in MatlabR . In the second step, the resulting braking force Fz0is deter- mined using FEM and Comsol MultiphysicsR . In this step, since there is no defect in the pipe, the stationary form of (2.31) is used to calculate the induced currents and magnetic eld distribution, whereas the braking force is determined by (2.2). This process is repeated until the equilibrium velocity of the magnet veis reached. As a nal result, the falling time of the magnet is calculated using T=Te+(LLe) ve(4.13) whereTeis the time needed to reach the equilibrium state, and Leis the travelled distance during the time Te. In order to validate both, the proposed numerical procedure and the analytical prediction of the equilibrium velocity (4.7) an experiment has been performed [15, 119]. Hence, two dierent spherical permanent magnets (NdFeB) have been dropped through a non-defective copper pipe and its falling time has been measured with a stop watch. All necessary geometrical and material properties are given in Table 4.1. The obtained results are summarized in Table 4.2. The relative errors are calculated using eA;N= 100
38 Chapter 4: Canonical Model: The Creeping Magnet Table 4.1: Geometrical and material properties Parameter Value Pipe radius R 8 mm Pipe length L 1 m Wall thickness  1 mm Conductivity  44:5 MS/m Magnet diameter Dm 15 mm Magnet density  7588:85 Kg/m3 Magnetization density M 0:9054
106A/m Table 4.2: Experimental validation [15] DmtEXP tAtNEXPANeAeN 15 mm 11 :9660:092 s 14:690 s 11:663 s 0:095 0:063 0:099 22:7% 2:5% 10 mm 3:4260:092 s 4:353 s 3:497 s 1:73 1:014 1:67 27:1% 1:5% (tEXPtA;N)=tEXP, whereAandNdenote for the analytical and numerical estimation of the falling time, respectively. As expected, the analytical model overestimates the falling time for both used magnets. This is because the decay of the magnetic eld within the pipe walls has been neglected which resulted in higher braking Lorentz force. The numerical solution properly takes into account the eld decay, thus it agrees better with experiments. Pipe Containing an Idealized Defect The main aim of the following analysis is to verify and to validate used numer- ical procedure additionally proving the linear dependency between the falling time dierence
and the defect size 2 (4.12). Furthermore, various eects of the model simplications assumed in the analytical model are discussed and quantied as well. The entire dynamic behaviour of the velocity and the Lorentz force acting on the creeping magnet has been analysed using an ecient 2D-axisymmetric transient FEM model. The cylindrical coordinate system K is xed to the pipe and the magnet is Table 4.3: Experimental validation for a pipe with an ideal defect [15] 2 t EXP eAeN RPM= 7:5mm 2:66 10:230:08 s 22:45% 0:19% 5:33 10:200:09 s 22:89% 0:14% 8 10:150:11 s 23:45% 0:57% 10:66 10:000:11 s 25:30% 2:06% 16 10:030:07 s 24:95% 1:74% RPM= 5mm 2:66 2:810:06 s 32:73% 1:62% 5:33 2:790:06 s 33:96% 0:73% 8 2:820:06 s 32:42% 1:89% 10:66 2:800:06 s 33:57% 1:08% 16 2:740:06 s 36:15% 0:76% 39 Chapter 4: Canonical Model: The Creeping Magnet (a)Dm= 15 mm (b)Dm= 10 mm Figure 4.5: Validation of the analytical and numerical models moving with velocity vopposite to the z-axis (Fig. 4.4b). The governing equation is given by (2.28), where je= 0, since there are no external current excitations present. In order to provide relative motion between the magnet and the pipe the sliding mesh technique (SMT) is used (see chapter 3). Here, SMT is adapted for modelling linear displacement according to the given creeping magnet problem. The computational domain has been divided into two independent parts, namely the stationary and the moving part (Fig. 4.4b). The moving part comprises the magnet and the part of the surrounding air region, whereas the stationary part comprises the pipe with an idealised defect and the rest of the air region. The coupling interface has been placed in air between the magnet and the pipe. To assure the continuity of the magnetic eld across the interface, the continuity boundary conditions (2.39) and (2.40) along with the Lagrange multipliers have been used. Similarly, as in the case of a non-defective pipe, the asymptotic boundary condition (ABC) are applied on the outer boundaries of the stationary part. Initially, the magnet was placed far enough from the defect so that the equilibrium velocity vecan be reached before the defect region. After passing the defect the equilibrium velocity has been reached again so that the falling time of the magnet can be calculated according to (4.13). In order to validate the proposed models, both the analytical and the numerical one, experiments have been performed additionally [15, 119]. For given defect sizes the falling time has been measured with a stop watch. The obtained results are summarized in Table 4.3 (Fig. 4.5). All results show linear dependency between the falling time dierence
and the defect size 2 . The slope of the graph obtained analytically diers from the the ones obtained numerically and experimentally due to the dierence in the forcing parameter , which depends on the resulting equilibrium velocity (see Table 4.2). The numerical results agree better with experiments for both magnet sizes (Table 4.3). The estimation of the discrepancy is given in Table 4.3. It should be mentioned that the renement of the experiment to a more precise 40 Chapter 4: Canonical Model: The Creeping Magnet (a) Velocity ( A= 0:063 andN= 0:099) (b) Lorentz force ( F0 z= 0:1315 N ) Figure 4.6: Comparison of the analytical (A) and numerical (N) models in terms of velocity and Lorentz force perturbation ( Dm= 15 mm,= 1 mm) (a) Velocity ( A= 1:064 andN= 0:947) (b) Lorentz force ( F0 z= 0:1315 N ) Figure 4.7: Comparison of the analytical (A) and numerical (N) models in terms of velocity and Lorentz force perturbation ( Dm= 15 mm,= 0:25 mm) time measurement system is necessary, since uncertainties of time measurements using a stop-watch are quite high compared to the falling time changes caused by defects. Additionally, when passing by a defect and entering the pipe region again, the magnet hits the edge of the pipe which aects its falling time [15, 119]. Therefore, a better guidance for the magnet in the defect region needs to be provided as well. On the other hand, even though a simplied experiment has been applied, a linear dependency can still be observed which proves the predictions given by (4.12). It is illustrative to show the eect of the defect size 2 on the characteristic velocity proles and the resulting Lorentz force (Fig. 4.6). The results have been presented in form of _=i=f() andFz=F0 z=f(), where index i2fA;Ngdenotes for the analytically and numerically obtained forcing parameter. In spite of considerable sim- plication assumed in the analytical model, the obtained results qualitatively agree well. On the other hand the absolute values, in terms of the forcing parameter ido not match. The relative errors, compared to the experimental results are presented in Table 4.3. In order to better understand this error, and nd out which of the assumed simplications mostly in uence the obtained results, the analysis has been performed 41 Chapter 4: Canonical Model: The Creeping Magnet by reducing the thickness of the pipe to = 0:25mm. The obtained results are shown in Fig. 4.7. A considerable improvement of results can be observed. Moreover, the value of the forcing parameter obtained analytically is closer to the one obtained numerically (A= 1:064 andN= 0:947). These results are expected, since other assumed simpli- cations in terms of the low Rmapproximation and magnetic dipole representation of the spherical magnet do not in uence the accuracy of results. Although the presented creeping magnet model has no direct practical application, it helps in elucidating the basic laws of LET. Using (4.8) it has been shown that the relative Lorentz force perturbation caused by an idealized defect depends only on its geometry. This is an important feature of LET which can be used in order to localise and identify dierent types of defects. Furthermore, the linear dependency between the falling time dierence
and the defect size 2 was proved. This eect can be used in order to characterise the defect size as well. 42 Chapter 5 Numerical Modelling of Lorentz Force Eddy Current Testing The main aim of this chapter is to introduce new FEM-based methodology, which can be used to analyse and develop future LET systems. The particular emphasis is placed on the reduction of the overall computational requirements and the total simulation time, while maintaining the accuracy of the solution. Additional goals include development of simplied numerical models which enable fast 2D/3D LET analysis in conjunction with the validation of assumed simplications (sections 5.3 and 5.4). For comparison and validation of dierent approaches a benchmark problem which represents a typical LET conguration has been considered (section 5.1). 5.1 Benchmark Problem Denition In this section a typical LET benchmark problem representing a generic conductor with pre-dened articial defects, moving across the static magnetic eld is described (Fig. 5.1). The conductor under test is considered to be non-magnetic with the conduc- tivity denoted by and permeability equal to the permeability of vacuum =0. It has a rectangular cross-section determined by its width Wcand height Hc, whereas its length is denoted by Lc. A cylindrical permanent magnet described by magnetization Mis used as a source of the static (primary) magnetic eld. The diameter and the height of the magnet are denoted by DmandHm, respectively. The magnet is placed centrally above the conductor under test ( y= 0) at a lift-o distance z. For the analysis, three dierent types of articial defects, namely long (" j"), wide ("") and cross ("+") have been considered. Defects are placed centrally within the conductor at depthdbelow its surface. They are characterised by their width w, heighthand lengthl. The conductivity of the defect is denoted by d. 43 Chapter 5: Numerical Modelling of Lorentz Force Eddy Current Testing Figure 5.1: Denition of the LET benchmark problem. The conductor contains three types of defects: long ("j"), wide ("") and cross ("+"). In order to reduce the number of dependent variables, the magnetic Reynolds num- ber (Rm) has been used for the analysis (2.33). It involves the characteristic length- scale parameter Lwhose denition depends on the particular problem at hand and characterises the conductor in motion (see section 2.2.4). For the given benchmark problem (Fig. 5.1), there are several possibilities to dene L. In uid dynamics, in order to measure the eciency of dierent channel ows, all channel congurations are usually approximated with an equivalent circular pipe having the equivalent diameter Dedened as De=2A P=2Wc
Hc Wc+Hc; (5.1) whereAis the cross-section area of the conductor, and Pis its perimeter [32]. Following the same principle and choosing the radius of an equivalent cylinder ( De=2) as the characteristic length-scale for the given LET benchmark problem, the value of Rmhas been dened as follows Rm=0jvjWc
Hc Wc+Hc: (5.2) 5.2 Logical Expressions Approach In this section the logical expressions approach (LEA) that allows fast computa- tions of 2D/3D eddy current problems including parts in relative motion is presented [121]. Using the proposed methodology, the spatial coordinates of moving parts, either conducting or non-conducting, are modelled on a xed computational grid using logical expressions (LE). By applying the principles of Boolean algebra directly in nite ele- ment analysis (FEA) the shape of moving parts is determined on the y by calculating 44 Chapter 5: Numerical Modelling of Lorentz Force Eddy Current Testing Table 5.1: Logical expressions for basic geometric primitives Shape Logical expression (LE) Box (B) [ Xstart+vtxXstart+LB+vt] ? 1 : 0 Cylinder (C) [ ( x+Xstart+vt)2+y2R2 C] ? 1 : 0 Sphere (S) [ ( x+Xstart+vt)2+y2+z2R2 S] ? 1 : 0 Note:x;y;z { Cartesian coordinate system, Xstart { starting position of the moving part, v{ pre- scribed velocity of the displacement, t{actual simulation time. constraints given by LE and ltering the nite elements in those domains where LE are introduced. Fig. 5.2 shows three basic geometrical primitives, i.e. box, cylinder, and sphere which are modelled using LE. Figure 5.2: Basic geometric primitives dened by logical expressions: (a) box (b) cylinder (c) sphere. Independent to the type of reference frame used in LET analysis, modelling of moving parts using LE requires the existence of a homogeneous zone in which these expressions are applied. This zone is referred to as the moving domain and it is deter- mined by the shape of the moving part and its relative displacement L(Fig. 5.2). In order to introduce the motion and determine the shape of moving parts, the constraints given by LE are dened as time dependent. This step represents the basic idea of the logical expression approach [121]. Table 5.1 presents some sample logical expressions for basic primitive shapes moving with velocity valong the model x-axis (Fig. 5.2). Logical constructions "[ condition ]?1 : 0" in Table 5.1 should be interpreted as: if the condition in square brackets is valid then take 1. Otherwise if the condition is false take 0. The implementation of LEA in the existing nite element codes can be performed easily by multiplying material properties assigned to the moving domain with the appropriate LE. This modication does not require any major changes in the FEM code and it can be applied in any commercially available FEM software. To assign material properties to the moving domain not occupied by the moving part, a term multiplying these properties with the negation of the pre-dened logical expression, i.e. by 1LEshould be added as well. As a summary, any implementation of LEA requires the following modelling steps: 45 Chapter 5: Numerical Modelling of Lorentz Force Eddy Current Testing (i) creation of the moving domain (ii) denition of the corresponding LE (iii) modication of material properties of the moving domain by combining the prop- erties of the part under motion and the remaining region with appropriate logical expressions. This, the corresponding material properties have to be multiplied byLEior 1LEi, respectively. During the discretization of the moving domain, the element size in the moving direction
xhas to satisfy the following constraint: N
x=v
t (5.3) where
trepresents the size of the time step used during the solution process, and N is any integer number greater than zero. Then the outer surface of the moving part is forced to stay unchanged during its motion. Otherwise, its volume can uctuate in subsequent time steps introducing additional errors that are observed as non-physical Lorentz force oscillations. One of the obvious disadvantages of the proposed LEA is its limitation to the relatively simple shapes of the moving objects (Fig. 5.2). More complex shaped moving parts can be modelled by combining several moving domains and repeating the steps from (i) to (iii). Additionally, to model curved geometries using LEA the nite element mesh in the moving region should be relatively ne. In order to test the proposed LEA, LET benchmark problem (Fig. 5.1) been con- sidered. Depending on the denition of the frame of reference, the LET analysis has been performed using two dierent implementations of LEA, (i) in case of xed frame of reference, the logical expressions are used to model the motion of the permanent magnet, (ii) in case of moving frame reference, the logical expressions are used to model the relative motion of the defect. These two specic LEA implementations are referred to as the moving magnet approach (MMA) and the moving defect approach (MDA), respectively. 5.2.1 Moving Magnet Approach In the implementation of the moving magnet approach (MMA), the global coor- dinate system is associated with the conducting object (xed frame of reference) and the logical expressions are used to describe motion of the used cylindrical permanent magnet (Fig. 5.3). The magnet is moving with constant velocity valong the model 46 Chapter 5: Numerical Modelling of Lorentz Force Eddy Current Testing Figure 5.3: Implementation of the moving magnet approach (MMA). The moving domain is dened in the air region. x-axis in close vicinity of the conductor containing an articial defect below its sur- face. Using steps (i)-(iii) which describe the implementation of the LEA, the moving domain is dened in the surrounding air region (Fig. 5.3). The cross section of the moving domain is determined by the height Hmand the diameter Dmof the magnet, while its length depends on the starting position Xstartand its relative displacement L (Fig. 5.3). In general, the starting position Xstartis the distance of the moving object to the origin of the coordinate system at t= 0. It has to be large enough to avoid any in uence of the used initial conditions on the resulting Lorentz force perturbations. To model the motion of the magnet in the moving domain, the magnetisation M assigned to the moving domain has been modied as ~M=MLEC, whereLECis the logical expression for a cylinder (Table 5.1). When dening the particular logical expression (LE), it is important to keep the equivalent volume of the cylinder VLEC obtained by LE and the volume of the real cylinder given by Vcyl=(Dm=2)2Hm the same (VLEC=Vcyl). This can be achieved, e.g. by changing the diameter of the magnetDm. In the given LET conguration, the relative velocity between the magnet and the conductor under test is constant. Thus, the size of the nite element mesh in the moving direction
xis uniform (5.3). Applying the optimal A formulation, the governing system of equations in MMA is given by (2.35) and (2.37). According to section 3.1 this requires both nodal and edge nite element formulations in single computational domain. The nodal FEM formulation approximating the magnetic scalar potential is given by (3.5), whereas the corresponding edge formulation for the magnetic vector potential Ais given by (3.7). Due to the fact that (3.7) does not involve the additional velocity term the re- sulting system of equations remains symmetric. Another important feature of MMA is that the modication of the magnetization vector Mby time dependent LE contributes 47 Chapter 5: Numerical Modelling of Lorentz Force Eddy Current Testing as a source only to the right-hand side in the resulting FEM weak formulation (see sec- tion 3.1.2). Therefore, in MMA the resulting stiness matrix has to be assembled only once during the entire motion of the magnet which reduces the total computational time considerably. 5.2.2 Moving Defect Approach In the moving defect approach (MDA) the global coordinate system is assigned to the permanent magnet (moving frame of reference). In this reference frame the magnet is stationary and the conductor is moving in opposite direction with velocity valong thex-axis (Fig. 5.4). For the given LET problem, the use of LE to describe the motion of the whole conducting domain would be computationally very expensive. However, the implementation of LEA in the moving reference frame can be considerably simplied if only perturbations of Lorentz force caused by defects are required, which is in fact a typical LET problem. Thus, it is sucient to model only the movement of the defect relatively to the magnet, instead of modelling the motion of the whole conductor (Fig.5.4). In MDA the moving domain is dened entirely inside the conductor where LE are used to model the motion of the particular defect [121]. The shape and position of the moving domain is dened by the cross-section of the defect wh, its relative displacement L, and depth d(Fig.5.4). In order to simulate the defect motion, the electrical conductivity assigned to the moving domain is modied as ~ =(1LEB)+ dLEB, whereLEBis the logical expression given in Table 5.1, and dis the electrical conductivity of the defect. Due to the constant velocity, the size of the nite element mesh in the moving direction
xis uniform (5.3). Figure 5.4: Implementation of the moving defect approach (MDA). The moving domain is dened inside the conductor. 48 Chapter 5: Numerical Modelling of Lorentz Force Eddy Current Testing Similarly, to the previous MMA applying the optimal A formulation in MDA results in the system of governing equations given by (2.35) and (2.38). According to section 3.1 this requires both nodal and edge nite element formulations to be applied in the computational domain as well. The nodal FEM formulation approximating for the magnetic scalar potential is given by (3.5), whereas the corresponding edge formulation for the magnetic vector potential Ais given by (3.8). In contrast to MMA (3.7) the formulation used in MDA (3.8) involves the additional velocity term which makes the resulting system of equations non-symmetric. Additionally, the modication of the electrical conductivity by time dependent LE, introduced by MDA, modies the resulting stiness matrix as well. Therefore, the stiness matrix has to be re-assembled at every time step which increases the computational time compared to MMA. 5.3 Quasi-Static Approach The moving magnet approach (MMA) and the moving defect approach (MDA) assume no simplications for the given LET analysis. They oer accurate results for any relative testing velocity vbetween the magnet system and the conductor, and for any material and geometry parameters involved. Thus, they are valid for nite values of the magnetic Reynolds number (5.2), whether the conductor contains material defects or not. Conductor Free of Defects If LET conguration under investigation is time independent, i.e. it involves uni- formly moving conductors with a constant cross-section normal to the direction of motion (conductors free of defects) the analysis can be considerably simplied [37]. In principle, only one stationary can be performed to obtain an accurate steady state solution, e.g. the Lorentz force acting on the magnet system. This assumption requires the moving frame of reference (2.31), where the additional velocity term ( vr A) is used as a source of the induced eddy currents inside the conductor in uniform motion r1 0rAM =(rVvr A) +je; (5.4) r
(rVvr A) = 0: (5.5) The second equation results from the current conservation law (2.7) and it is an addi- tional equation for the electric scalar potential V. This system of equations takes the deformation of the magnetic eld lines correctly into account making it valid for any 49 Chapter 5: Numerical Modelling of Lorentz Force Eddy Current Testing value of the Rm. Conductor with Defects Although (5.4) and (5.5) provide fully correct solutions only for conductors without any material defects, their use can be still extended to NDT applications. In [66], it has been shown that for LET systems resulting in small Rm, they can be used for fast Lorentz force calculations on the moving magnet even for conductors with defects. Here, this method is referred to as a quasi-static approach (QSA). As a direct consequence of low magnetic Reynolds numbers ( Rm<1), the diusion time of the magnetic eld into the conducting object, estimated as Rm
L=v, whereLis the characteristic length-scale of the conductor (5.1), is small as well [24, 122]. This basically justies the instantaneous eld reaction ( @B=@t!0) to any perturbation of induced currents, which is assumed in QSA. Nevertheless, if this is not the case, the full transient form of (5.4) and (5.5) has to be considered, which in fact represents the governing equation of already presented moving defect approach (MDA). In the implementation of QSA, only the change in relative position between the magnet and the defect must be provided. This is done either by moving the magnet system relative to the defect, or vice-versa [66]. In any case, this requires a time con- suming re-meshing procedure of the entire model geometry for each new conguration. The re-meshing of the geometry can be avoided if the basic principle of the logical ex- pression approach (LEA) is combined with the quasi-static formulation given by (5.4) and (5.5). In this LEA implementation the time variable used in dierent LE is just a parameter which needs to be changed from one stationary solution to another providing the displacement of the moving part (magnet or defect). Basically, this means that the same geometry, used for the implementation of the LEA, i.e. MMA and MDA (Fig. 5.2 and Fig. 5.3), can be used for implementation of QSA as well. The only dierence introduced by QSA is in the governing equation in the conducting region, which is now in its stationary form and contains an additional scalar potential V. In regions free of eddy currents (surrounding air region and permanent magnet), the magnetic scalar potential formulation is used (section 2.3). 5.4 Weak Reaction Approach If the general assumption is made, that the secondary magnetic eld Bsassociated with induced currents jis low, compared to the primary magnetic eld ( BsBp) the QSA formulation (5.4) and (5.5) can be even further simplied. By neglecting the reaction eld ( Bs=0) the curl-free property of the total magnetic eld Bis preserved 50 Chapter 5: Numerical Modelling of Lorentz Force Eddy Current Testing Figure 5.5: Implementation of the weak reaction approach (WRA). Only conducting region needs to be considered. within the conductor region as well rB=r (Bp+Bs) =rBp=0: (5.6) In terms of the magnetic Reynolds number Rm(section 2.2) the constraint given by (5.6) is fullled when Rm1. In this case the electric potential Vand induced eddy currents are described by the Ohm's law for moving conductors (2.22) which results in j=(rV+vBp); (5.7) Applying the current conservation law r
j= 0 the following Laplace's equation for electric scalar potential is obtained r2V=r
(vBp) =Bp
(rv)v
(rBp) = 0: (5.8) The specic value of Vis dened by the boundary conditions for the given problem. They are obtained by making the normal component of the induced current density on the walls of the conductor to be zero (Fig. 5.5) [67] n
j= 0; (5.9) If the conductor contains a defect, this condition has to be imposed on its walls as well. The Laplace's equation (5.8) is actually the governing equation of LET systems when the reaction magnetic eld Bsof the induced eddy currents is neglected. In fact, this represents the so-called weak reaction approach (WRA). It should be mentioned that WRA is an extension of the method developed in [25, 67, 123] to moving conductors with articial defects. 51 Chapter 5: Numerical Modelling of Lorentz Force Eddy Current Testing In the numerical implementation of the WRA there are two possibilities to prescribe the primary magnetic eld Bp. If the used permanent magnets have simple shape, like of a sphere or a rectangle, Bpcan be obtained in the closed form analytically [124]. For a spherical magnet with radius rm, the magnetic eld outside the magnet region (r>rm) is described by magnetic point dipole formula (4.1). The magnetic moment m of an equivalent dipole is given by m=M
Vm, where Mis the magnetization density of used spherical magnet, and Vmits volume [118, 124]. In case of a rectangular magnet, the expression for Bpcan be found in [124]. For complex magnet systems, the primary magnetic eld distribution inside the conductor can be calculated numerically, e.g. by using the magnetic scalar potential (2.35). However, this requires time taking interpolation of the eld on the nite element mesh of a conductor. As in QSA, the relative displacement between the conductor and the magnet can be performed in two ways. In case, when the magnetic eld is described analytically it is most simple to move the magnet from one stationary solution to another and leave the conductor at a xed position. When the magnetic eld can only be obtained numerically, than it is most ecient to change the position of the conductor instead. In this case the eld distribution Bphas to be calculated only once whose solution is than interpolated for each new position of the conductor. This however, results in longer simulation time compared to the previous case where Bpcan be described analytically in the closed form. The main benet of WRA compared to QSA is that instead of both, the magnetic vector potential AandV, the solution of LET system involves only Vas the un- known quantity (5.8). This is a considerable reduction of the total number of degrees of freedom (DoF), as the number of equations to be solved is reduced from 4 to 1. Furthermore, due to the absence of the reaction eld Bsit is sucient to model only the region of the conductor (Fig. 5.5). 5.5 Sliding Mesh Technique The sliding mesh technique (SMT) is one of the widely applied FEM techniques for modelling relative motion in general eddy current testing problems. Similarly to all other moving grid methods, SMT (also referred to as the moving mesh method [58] or slip surface method [97]) requires the denition of two independent nite element meshes. In order to account for motion the meshes are simply slid relative to each other eliminating any need to alter their structure. However, this yields some kind of coupling procedure providing the continuity of the resulting magnetic eld across the interface where the meshes overlap. In the presented literature review (see section 52 Chapter 5: Numerical Modelling of Lorentz Force Eddy Current Testing Figure 5.6: Implementation of SMT. In the assumed xed frame of reference (K) the small air region including the magnet has been moved (Moving part). The rest of the computational domain was xed (Fixed part). 3.2) several coupling techniques have been reported. Nevertheless, only two of them are nowadays commonly applied in 3D electromagnetic eld computations. They are referred to as (i) Lagrange multipliers and (ii) eld interpolation techniques. The main aim here is to present one of the possible SMT implementations to solve the given LET benchmark problem (Fig. 5.1). The resulting SMT model will be used throughout the work in order to verify results and estimate the computational costs of the novel xed grid methods presented above. In any SMT implementation it is of main interest to minimize the coupling interface (boundary) between two independent meshes, i.e. the number of variables which need to be coupled. In order to minimize the size of the coupling surface resulting from the given 3D LET problem, only a small region around the magnet has been used as the moving part. The rest of the geometry was xed (Fig. 5.6). Additionally, the contact boundary is placed in the air region which has been described only with the scalar potential (see section 2.3). The coupling boundary actually consists of two surfaces, i.e. a source boundary which belongs to the moving part and a destination boundary which belongs to the xed part. Even though this is not directly required, the source and destination boundaries were meshed using identical nite element meshes, so that the meshes could conform during the entire motion. This helps to reduce possible errors due to non-matching grids. On the coupling boundary the eld continuity has been obtained by applying both, Lagrange multipliers and interpolation methods. On the rest of the boundary, which is not overlapping, the magnetic eld insulation boundary condition has been used (see section 5.6.2). The Lagrange multipliers method introduces an additional set of variables on the sliding interface which ensures the continuity of the eld in a weak sense [102{105]. Unfortunately, the existence of additional variables considerably deteriorates 53 Chapter 5: Numerical Modelling of Lorentz Force Eddy Current Testing (a) Drag Lorentz force component (b) Lift Lorentz force component Figure 5.7: Comparison of two SMT solutions of the LET benchmark problem for conductor free of defects. The eld continuity across the interface has been obtained by means of Lagrange multipliers and by the interpolation technique. the conditioning of the stiness matrix [105{107]. Furthermore, the matrix contains zeros on the main diagonal, due to which only direct solvers provided by Comsol MultiphysicsR could be used [79]. On the other hand, the interpolation technique does not introduce additional variables [80, 105]. The scalar potentials at node points on the destination side are made to be equal to the potentials at corresponding points on the source side. If the mesh is conforming, this should actually work well as long as the dependent variable which is being constrained is continuous between elements. If the mesh nodes are not conforming, then the interpolation at node points and inter-element boundaries can lead to instabilities, i.e. to oscillations of the Lorentz force. However, the matrix system remains well-conditioned for which iterative solvers behave well. It should be emphasised that irrespective of the approach being used to provide con- tinuity of the eld across the interface, the resulting stiness matrix of SMT is changing at each time step of the simulation time. This is because due to mesh displacement 54 Chapter 5: Numerical Modelling of Lorentz Force Eddy Current Testing Table 5.2: Computational requirements and simulation time Method Solv. type NoE DoF RAM Ttotal SMT (Lagr.) Direct 258 905 967 862 19:21 GB 17h : 7min SMT (Interp) Iterative 258 905 923 180 3:1 GB 9h : 56min the connections between dierent nite elements on the coupling boundary are chang- ing. For large displacements, this can also lead to considerable increase of the matrix bandwidth resulting in higher memory consumption, as reported in [105, 111, 112]. The results obtained by both, Lagrange multipliers and interpolation methods are presented in Fig. 5.7. As a benchmark problem the LET setup presented in Fig. 5.1 for the conductor free of defects has been used. The cylindrical magnet has been moved from left end of the conductor to the right starting at Xstart=160 mm. The lift-o distance was z= 5 mm. All other parameters are given in Table 5.3. For analysis, the generalized- solver with = 0 has been used. Although the same nite element mesh has been applied, the results obtained by the interpolation technique tend to oscillate. Due to lower force values, the oscillation is more obvious in the Lorentz force lift component, with respect to corresponding drag force component (compare Fig. 5.7a and Fig. 5.7b). This is mostly because the continuity of the eld across the interface can be only globally provided by the interpolation technique [105, 111, 112]. Furthermore, it is also possible that the nite element mesh on the coupling interface was not completely conforming. Important to mention is that higher values of lead to higher oscillations of the Lorentz force as well. The Lagrange multipliers, on the other hand provide smooth force signals in both force components independent on the value of. For that reason only SMT along with Lagrange multipliers was used to verify the results obtained by LEA. The computational costs resulting from both SMT implementations are summarized in Table 5.2. Since SMT together with Lagrange multipliers require direct solvers, its memory consumption was considerably higher than corresponding interpolation technique for the same number of elements (NoE). Due to additional variables on the coupling boundary the resulting degrees of freedom (DoF) are slightly higher as well. Consequently the total solution time Ttotalof SMT with Lagrange multipliers was 50% longer than with the interpolation technique. 5.6 Numerical Implementation In this section the results obtained by applying the proposed numerical methodology to the LET benchmark problem shown in Fig. 5.1 are presented. All approaches have been implemented using the commercial nite element code Comsol MultiphysicsR 55 Chapter 5: Numerical Modelling of Lorentz Force Eddy Current Testing [79]. The verication of the obtained results has been performed using the sliding mesh technique (SMT). In order to provide direct control over the time-step size required by MMA and MDA, all simulations have been performed using the generalized- transient solver. The other possibility provided by ComsolR , namely the backward dierential formula (BDF) transient solver is based on a variable time-step size algorithm [79, 125, 126]. Thus, it could only be applied in SMT simulations. Nevertheless, the solution of LET problems using SMT and BDF as a transient solver required a relatively large number of time steps, which resulted in an extremely long simulation time. In order to provide a fair comparison the generalized- solver has been used in the implementation of SMT as well. The implicit and second order accurate generalized- transient solver was rst de- veloped for second-order systems arising in structural mechanics problems. Later its application has been extended to rst-order systems [127, 128]. One of its main advan- tages is the possibility to control the amount of high frequency numerical dissipation introduced by time discretization. This is provided through a single parameter which can take values between 0 and 1. If is chosen to be 0 the method damps the high frequency response in a single time step, whereas if is chosen to be 1 no numerical dissipation is introduced [127]. For MMA and SMT approaches a relatively strong dependency of the Lorentz force lift component with particular choice of has been observed (see Appendix B). This is because in the assumed velocity range ( Rm1) the lift force is around two orders of magnitude smaller than the corresponding Lorentz force drag component. The discrepancy was reduced when was approaching 1. How- ever, for large values of (>0:85) the solutions obtained by MMA, MDA and SMT tends to be unstable. This practically limited the application of the generalized-  solver to0:85. Fortunately, results obtained by MMA or SMT can be corrected in a simple way as proposed in Appendix B. If not stated otherwise, the material properties and geometrical parameters are dened as summarized in Table 5.3. To compare all approaches, the total Lorentz force acting on the cylindrical permanent magnet has been used (2.2). Due to the symmetry of the model with respect to the xz-plane, the side component FSof the Lorentz force vanishes, and for that reason it was not used for the analysis. The drag FDand the lift FLcomponents of the Lorentz force were compared quantitatively with a window normalized root mean square error (WNRMS error) c,c2fD;Lghas been used c=s 1
WZ
W=2
W=2 FcFRef c
2dx maxFRef cminFRef c100%; (5.10) 56 Chapter 5: Numerical Modelling of Lorentz Force Eddy Current Testing Table 5.3: Geometrical parameters and material properties of the LET problem Parameter Value [mm] Description Dm 15 Diameter of the magnet Hm 25 Height of magnet Lc 250 Length of conductor Wc 50 Width of conductor Hc 50 Height of conductor l 12 Length of defect h 2 Height of defect w 2 Width of defect d 2 Defect depth z 3 Lift-o distance L 2Xstart Length of moving domain
x 1 Mesh size in moving domain Property Value Description Br 1:17T Remanence of the magnet  30:61 MS/m Conductor conductivity d 0 MS/m Defect conductivity whereFcandFRef care Lorentz force values received by the method to be compared and by the reference method, and
Wis a window size for which we calculate the errors. 5.6.1 Initial Conditions In order to develop specic LET systems, it is sucient to study the Lorentz force perturbations caused by material defects. At the initial simulation time t= 0, there are actually two possibilities to prescribe Ai, i.e. either (i) the zero magnetic eld ( Ai=0) or (ii) the magnetic eld of the magnet diused in free-space ( Ai=Amag) can be used (Fig. 5.8a). In the rst case a considerable amount of simulation time is wasted on unnecessary calculations related to the magnetic diusion transient response until the steady state of the solution is reached [129]. In the second case, since at initial time (t= 0) there are no eddy currents induced inside the conductor ( j=0), the response time of the Lorentz force needs to be calculated. In principle, the former approach is more realistic, but it still requires additional computation. In order not to aect the Lorentz force perturbations caused by defects the initial position of the magnet, denoted byXstart, has to be increased by the distance
Xstart=r
v, whereris the response time of the Lorentz force for the given LET system. It is reasonable to expect that the 57 Chapter 5: Numerical Modelling of Lorentz Force Eddy Current Testing specic value of rdepends on the testing velocity vgeometric and material properties of the conductor under test. As an rst estimation of r, the magnetic eld diusion time,r=Rm
L=v, can be used. Assuming the testing velocity of v= 1 m/s and a conductor with the square cross-section ( Wc=Hc= 50 mm) with typical electrical conductivity of aluminium ( = 35 MS/m), the elongation of the initial position can be calculated as
Xstart=
v25 mm. To conrm this observation the response time of the Lorentz force to the step change in the testing velocity v(t) =v
h(t) has been calculated numerically. Thus, the LET benchmark problem presented in Fig. 5.1 has been used, where all geometry and material parameters are given in Table 5.3. The results for two dierent values of testing velocities are presented, i.e. v= 1 m/s andv= 10 m/s, which corresponds to Rm1 andRm10, respectively (Fig. 5.9). From the obtained results, it can be seen that for the fast estimation of
Xstart, the magnetic diusion time , can only be used for moderate velocities ( v= 1 m/s). For higher, testing velocities ( v= 10 m/s) this would overestimate the value of
Xstart. To avoid these unnecessary computations the quasi-static solution received from (5.4) and (5.5), for a defect-free conducting object ( Ai=Aqs), can be used as an initial condition (Fig. 5.8b). In principle, this makes the elongation of the initial position
Xstartbetween the magnet and the defect to zero. If the optimal A formulation is used (section 2.3), at the initial time ( t= 0), both the magnetic scalar potential and magnetic vector potential A, which are resulting from quasi-static solution have to be initialized, i.e. Ai=Aqsand i= qs(Fig. 5.10). In case when the magnet is initially placed next to the conductor under test, i.e. in the air region the initial conditions are simply obtained as Ai=Amag, (Fig. 5.8a). (a) Zero current as initial condition
Xstart>0 (b) Initial quasi-static solution
Xstart0 Figure 5.8: Distribution of the magnetic eld Band induced current jatt= 0.
Xstart is the elongation of the initial position of the magnet which depends on the applied initial conditions 58 Chapter 5: Numerical Modelling of Lorentz Force Eddy Current Testing (a) Initial velocity v= 1 m/s, F0 x= 0:771 N andF0 z= 0:119 N (b) Initial velocity v= 10 m/s, F0 x= 3:835 N andF0 z= 4:195 N Figure 5.9: Elongation of the initial position
Xstart Figure 5.10: Initial quasi-static solution ( A, ) for the transient simulation. 5.6.2 Boundary and Interface Conditions Boundary Conditions Lorentz force eddy current problems are characterised as open boundary problems. The easiest way to solve these problems is to truncate the innite computational domain by an articial outer boundary (Fig. 5.11). This approach is based on the assumption that the obtained solution is not altered by setting the potential or its normal derivative to zero (applying the Dirichlet or the Neumann boundary conditions) on the boundary which is suciently far away from the region of interest [130]. If the normal component of the magnetic eld Balong is set to zero, the so-called magnetic eld insulation boundary condition is obtained n
0r = 0; (5.11) where nis the unit normal vector on outer boundary surface (Fig. 5.11). If the tangential component of the magnetic eld Bis set to zero, the following condition has to be fullled = 0: (5.12) 59 Chapter 5: Numerical Modelling of Lorentz Force Eddy Current Testing Figure 5.11: Truncation of the innite air domain. Here 0is some prescribed value of the magnetic scalar potential, which is usually zero, 0= 0. In all considered approaches, the surrounding air domain has been modelled in form of a sphere along with magnetic insulation boundary conditions (Fig. 5.11). Based on the sensitivity study used to determine the in uence of the air domain size on the computed Lorentz force, a wide range of domain congurations were investigated. Finally, for the equivalent radius of a sphere the value of Re= 5Lehas been used, whereLeis the characteristic length shown in Fig. 5.11. Further increase of Redid not aect the obtained Lorentz forces. The fact that proposed LEA is implemented on the xed computational grid, enables the use of other techniques dealing with open boundary problems as well [130]. Nevertheless, techniques like asymptotic boundary conditions (ABC) or innite elements resulted in considerable increase of the simulation time compared to simple truncation technique. To further reduce the computational requirements, the symmetry of the model with respect to the xz-plane, allows only half of the model to be considered (Fig. 5.11). On the resulting outer and symmetry boundary of the model, the magnetic insulation boundary conditions (5.11) are applied. Additionally, if inside the conducting domain the electric scalar potential Vis used, the Dirichlet boundary condition V= 0 has to be applied as well. Interface Conditions For the given LET benchmark problem the surface of the conductor under test has been used as the coupling interface ( c) between two potentials associated with the optimal A formulation. (Fig. 5.11). This represents the most economical solution. The conditions providing the eld continuity across the interface have been described in section 2.3.3 and they are given by (2.40). 60 Chapter 5: Numerical Modelling of Lorentz Force Eddy Current Testing 5.6.3 Meshing of the Conductor In FEM, the overall accuracy of results is strongly in uenced by the spatial dis- cretization of the computational domain into nite elements [50]. A ner nite element mesh commonly gives better accuracy (see chapter 3). Since this would signicantly increase the memory consumption, the maximum number of nite elements is usually restricted by the computational resources available. Therefore, in any numerical model it is very important to control the distribution of the nite elements, providing higher mesh density in regions where higher magnetic eld gradients are present, e.g. in the close vicinity of the permanent magnet. To reduce a possible in uence of the domain discretization on the calculated values of Lorentz forces, a similar mesh distribution inside the conducting domain is used for the comparison of all considered numerical approaches. The schematic of the used nite element mesh is shown in Fig. 5.12. The mesh consists only of quadratic tetrahedral elements [48], where the particular distribution is based on a performed sensitivity analysis reducing the in uence of the mesh size on the resulting value of the Lorentz force. Similar mesh distribution is used in non-conducting domains (permanent magnet and air) as well, and it is not shown here for simplicity. The particular choice of tetrahedral elements considerably simplies meshing of complex domains including the conductor with defects. In case of simple geometries better control over the mesh distribution can be obtained by applying the hexahedral nite elements (see chapter 6). Nevertheless, for approximately the same number of DoF the analysis has been performed to make sure that the LET solution is not in uenced by the type of the nite elements applied. A comprehensive study on the in uence of dierent nite element shapes on the accuracy of the solution can be found Figure 5.12: Distribution of the nite element mesh inside the conducting domain which is used in all numerical calculations for conductors with defects. The mesh consists of only tetrahedral elements. 61 Chapter 5: Numerical Modelling of Lorentz Force Eddy Current Testing in [131]. 5.7 Numerical Verication In order to test the accuracy and determine the application range to which each of the proposed numerical approaches can be applied, simulations of the LET benchmark problem (Fig. 5.1) have been performed considering various values of Rm. Finally, the computational costs and simulation time for all proposed approaches are presented and discussed. 5.7.1 Verication of the Moving Magnet Approach First, the verication of the moving magnet approach (MMA) with reference to the sliding mesh technique (SMT) has been performed. As a benchmark problem the LET (a) Drag Lorentz force component (b) Lift Lorentz force component Figure 5.13: Verication of the moving magnet approach (MMA) with the sliding mesh technique (SMT). The long (" j") defect is located d= 2 mm below the surface of the conductor. 62 Chapter 5: Numerical Modelling of Lorentz Force Eddy Current Testing conguration shown in Fig. 5.1 has been used. All material and geometrical parameters are given in Table 5.3. The conductor has a nite length L= 250 mm and contains the long ("j") defect located symmetrically in its centre at d= 2 mm. The cylindrical magnet has been moved from left to right at constant velocity v= 0:5 m/s, starting atXstart=160 mm. During the solving process, the generalized- transient solver with a xed time step
t=
x=vand= 0:85 has been used. Since, both MMA and SMT would require the same constant correction factor (see Appendix B) the correction of obtained results has not been performed. The simulations resulted in a total number of 320 time steps (see Table 5.3). Fig. 5.13 shows the characteristic Lorentz force proles obtained by both, SMT and MMA. Furthermore, a small Lorentz force perturbation produced by the long (" j") defect in the middle of the force proles can bee observed. As expected, much stronger force perturbations are caused by the front and back walls of the solid conductor. Using SMT as a reference method and a window size of
W= 250 mm, the WNRMS errors (5.10) of the drag and lift Lorentz force components are equal to D= 0:03% andL= 0:06%. The small WNRMS error values additionally conrm a very good agreement between MMA and SMT. As already mentioned in section 5.2.1 the stiness matrix resulting from MMA is symmetric, well-conditioned and remains unchanged during the simulation time. This means that the time taking procedure involving the assembly of the stiness matrix needs to be performed only once, yielding a fast and stable simulation. Due to the additional variables in form of the Lagrange multipliers introduced on the coupling boundary, the structure of the stiness matrix in SMT is not symmetric and changes at each time step because of the mesh displacement (see section 5.5). Furthermore, the resulting matrix system contains zeros on the main diagonal which renders the SMT analysis to use of direct solvers only. All these issues along with the ease of imple- mentation makes MMA much faster than SMT for approximately the same number of DoF. For detailed comparison of the computational requirements and solution time see section 5.7.4. Since, both SMT and MMA solve a transient form of the magnetic eld diusion equation (2.37) the obtained results are valid for any nite value of the Rm. 5.7.2 Verication of the Moving Defect and Quasi-Static Ap- proaches In what follows the results obtained by MDA and QSA have been veried with respect to SMT. In order to test the validity of both approaches, the LET benchmark problem (Fig. 5.1) has been solved by considering two dierent values of Rm, i.e.,Rm= 0:1 andRm= 10, which correspond to relative velocities of v= 0:103 m/s and 10 :3 m/s, respectively (see Table 5.3). In the implementation of MDA the long (" j") defect has 63 Chapter 5: Numerical Modelling of Lorentz Force Eddy Current Testing (a) Drag Lorentz force component (b) Lift Lorentz force component Figure 5.14: Verication of the moving defect approach (MDA) and quasi-static approach (QSA). The comparison is performed for Rm= 0:1 (v= 0:103m/s). The long (" j") defect is located d= 2 mm below the surface of the conductor. (a) Drag Lorentz force component (b) Lift Lorentz force component Figure 5.15: Verication of the moving defect approach (MDA) and quasi-static approach (QSA). The comparison is performed for Rm= 10 (v= 10:3m/s). The long (" j") defect is located d= 2 mm below the surface of the conductor. been moved with a constant velocity from right end of the conductor to left end, starting atXstart= 55 mm (Fig. 5.4). The generalized- transient solver with a xed time step
t=
x=vand= 0:85 has been used for both values of Rm, resulting in total 110 time steps. Thus, the constant correction of the transient results obtained by SMT has been performed according to (B.1) and (B.2) (see Appendix B). As expected, all approaches, i.e., MDA, SMT, and QSA are in good agreement with each other for small values of Rm,Rm= 0:1 (see Fig. 5.14a). Using SMT as a reference method and the window size
W= 48 mm, WNRMS errors (5.10) calculated for MDA and QSA are equal to MDA D = 0:92%,MDA L = 1:68%, andQSA D= 1:20%, QSA L= 2:03%. For higher values of Rm(Rm= 10), the QSA produces erroneous Lorentz force perturbations due to the assumed instantaneous magnetic eld reaction ( @B=@t= 0). As a result, the WNRMS errors of QSA with reference to SMT are equal to QSA D= 28:14%,QSA L = 27:36%. The MDA, on the other hand agrees well with the SMT 64 Chapter 5: Numerical Modelling of Lorentz Force Eddy Current Testing solution for higher values of Rm(Fig. 5.15a). The obtained WNRMS errors are equal toMDA D = 1:57%,MDA L = 1:23%. It should be mentioned again that, SMT like MMA oer accurate LET results for any value of Rm. The reason for that is that both methods are described in the xed frame of reference in which the transient magnetic eld diusion equation without the additional velocity term applies (2.28). The MDA is described in the moving frame of reference (2.31) which is time dependent and contains the additional velocity term (vr A) as well. Due to a good agreement with SMT for any value of Rm, the presented analysis additionally conrms the validity of applying (2.31) for transient eddy current problems having conducting parts in non-uniform motion and variable cross-section in the moving direction (Fig. 5.15a). The QSA assumes instantaneous magnetic eld reaction ( @B=@t= 0). Strictly speaking this would restrict its appli- cability only to uniformly moving objects with constant-cross section in the moving direction. Nevertheless, as it has been shown above that QSA oers good results when applied for LET problems and conductors with defects under the condition Rm<1 (Fig. 5.14a). 5.7.3 Verication of the Weak Reaction Approach The closed-form analytical solution describing the magnetic eld distribution of a cylindrical permanent magnet could not be obtained [124]. Therefore, in order to verify the so-called weak reaction approach (WRA), the rectangular shape of the magnet has been considered instead. The closed-form analytical solution for a rectangular shape of the magnet can be found in [124] and it is not presented here for simplicity. The dimensions of the used rectangular magnet are the same as for the cylindrical magnet described before, i.e. Wm=Dm, whereWmandDmare its width and depth, respectively. The height of the magnet is given by Hm(see Table 5.3). Other geometry and material parameters are described by the LET benchmark problem (Fig. 5.1). Similarly as in the verication of MMA, in WRA the magnet has been moved from the left end of the conductor to its right end, i.e. starting at Xstart=160 mm at test- ing velocity v= 0:103 m/s (Rm= 0:1). The obtained Lorentz force proles are shown in Fig. 5.16. Due to a considerably shorter simulation time (section 5.7.1), instead of SMT as a reference method MMA has been applied. Furthermore, the rectangular shape of the magnet in MMA can be simply modelled by changing the corresponding logical expression (Table 5.1). For implementation of MMA, the generalized- tran- sient solver with a xed time step
t=
x=vand= 0:85 has been used. This yields the correction of MMA results according to (B.1) and (B.2) (see Appendix B). From the obtained results, a very good agreement in the drag component of the 65 Chapter 5: Numerical Modelling of Lorentz Force Eddy Current Testing (a) Drag Lorentz force component (b) Lift Lorentz force component Figure 5.16: Verication of the weak reaction approach (WRA) with the moving magnet approach (MMA). The long (" j") defect is located d= 2 mm below the surface of the conductor. Lorentz force can be observed (Fig. 5.16a). Using the window size of
W= 250 mm, the WNRMS error for the drag component obtained using WRA, with reference to MMA is equal to WRA D = 0:35%. In case of a lift force the agreement is deterio- rated (Fig. 5.16b). The WNRMS error for window size of
W= 250 mm is equal to WRA L = 1:87%. The relatively small error is expected since the value of Rmused for comparison was small Rm= 0:1. For higher values of Rmthe WNRMS would increase considerably. The main reason for such a behaviour is that WRA neglects any reaction of the induced eddy currents. Due to resulting symmetry of the current distribution within the conductor this makes the lift component of the Lorentz force acting on the magnet (its integral over the conducting domain) to zero. However, since the induced Lorentz force density within the conductor is not zero WRA actually calculates the per- turbation of the lift force component correctly. The MMA on the other hand accounts for the reaction eld of induced currents resulting in non zero lift force component. Due to this, the applicability of WRA is restricted to LET analysis resulting in low 66 Chapter 5: Numerical Modelling of Lorentz Force Eddy Current Testing values ofRm1. The obvious advantage of WRA over all other methods is that only one scalar po- tentialVis sucient to describe LET systems (see section 5.4). This considerably reduces the computational costs and total simulation time. Moreover, only the con- ducting domain needs to be modelled, which signicantly reduces model complexity and modelling time. 5.7.4 Computational Requirements and Simulation Time In the preceding sections the verication of the numerical results obtained by various numerical approaches (MMA, MDA, QSA and WRA) with respect to SMT together with Lagrange multipliers has been performed. For the analysis the LET benchmark problem dened in Fig. 5.1 has been applied, i.e. the resulting Lorentz force acting on the magnet has been used as a comparison parameter. The main aim in this section is to estimate the computational costs and quantitatively compare the simulation time of each of these methods (Table 5.4). As expected, the reference SMT method results in longest simulation time ( Ttotal= 18h : 28min). This is mostly because SMT is restricted to direct solvers only. To obtain the solution, these solvers use a large amount of available random access memory (RAM) and take relatively long time for matrix factorization which is required in each time step due to mesh displacement (see section 5.5). This results in a relatively long calculation time, even per single time step TSMT step =Ttotal=NT = 3min : 28s, where NTis the total number of time steps (see Table 5.4). In the implementation of SMT the multifrontal massively parallel sparse direct solver (MUMPS) has been used [132]. Even for approximately the same number DoF, MMA solves the given LET problem in much shorter time ( Ttotal= 1h : 35min). Apart the fact that iterative solvers can be used, this is mostly because the stiness matrix resulting from MMA is symmetric, well-conditioned and remains unchanged during the whole simulation time. Thus, the simulation time per time step is considerably reduced to TMMA step = 17:7s. Due to similarities in their implementation the computational costs resulting from MDA and QSA are very much alike. Nonetheless, since in QSA the time derivative of magnetic vector potential is set to zero ( @B=@t!0) the calculation time of QSA is Table 5.4: Computational requirements and simulation time Method Solv. type NoE DoF RAM NT T totalTstep=Ttotal=NT SMT Direct 257 343 922 110 20:1 GB 320 18h : 28min 3min : 28s MMA Iterative 303 613 1 066 535 4:7 GB 320 1h : 35min 17 :7s MDA Iterative 213 876 744 233 2:17 GB 80 41min : 32s 32 :2s QSA Iterative 213 876 869 056 2:59 GB 80 31min : 05s 23 :4s WRA Iterative 89 430 132 499 0:73 GB 320 26min : 10s 4 :9s 67 Chapter 5: Numerical Modelling of Lorentz Force Eddy Current Testing still shorter, with respect to MDA (see Table 5.4). Even though MDA and QSA require the same total number of nite elements (NoE), due to the additional scalar potential Vused inside the conducting domain the resulting number of degrees of freedom (DoF) in QSA is larger compared to MDA. Similarly as in SMT, due to motion of the defect the stiness matrix resulting from MDA and QSA is changing during the simulation which yields additional assembling procedure at every time step (see Sections 5.2 and 5.3). Moreover, the additional velocity term makes the stiness matrix non-symmetric as well. All this issues result in longer simulation time per time step compared to MMA (TMDA step = 32s andTQSA step= 23s). The WRA compared to all other approaches results in lowest computational costs and shortest simulation time (see Table 5.4). This is expected, since WRA requires only one scalar potential Vto be applied to solve 3D LET problems. The very short simulation time per single time step Tstep= 4:9s makes WRA preferable to all other approaches when the assumed simplications apply Rm1. 5.8 Experimental Validation In order to validate the obtained numerical results and to prove the feasibility of LET for the rst time, the LET benchmark conguration presented in Fig. 5.1 has been experimentally realised in [15]. Two dierent typed of test conductors have been considered, namely (i) the solid conductor with linear surface defects and (ii) an equiv- alent package of metallic sheets (Fig. 5.17). Each sheet is 2 mm thick and it is made of the same material as the solid conductor. This conguration allows to consider various types of defects (long " j" wide "", and cross "+") and to change their position within the conductor in an easy and exible way (Table 5.5). For completeness of this work the measuring system has been only brie y described here, whereas its detailed description can be found in [15]. In order to provide relative motion between the specimen and the Figure 5.17: Experimental LET setup used for validation of the numerical results. (i) Solid conductor with linear surface defects (ii) Package of metallic sheets with articial defects. 68 Chapter 5: Numerical Modelling of Lorentz Force Eddy Current Testing Table 5.5: Dimensions of the articial defects Defect type Long " j" Wide "" Cross "+" lwh 1222 mm 2122 mm 1222[2122 mm Table 5.6: LET measurement system - technical details [15] Parameter Linear drive Positioning stage yz3D-Force sensor Type belt driven step drive strain gauge Range 0 ::4m/s 3 310N Accuracy 100 m 30nm 15 1550mN Operation 3m 45 45mm 10kHz Table 5.7: Parameters of applied lters [15] Source of interferences Filter type Frequency [Hz] Sampling rate [Hz] Basement Notch 10 :9 44 Power supply Notch 50 200 Construction Band-stop 60 :::90 300 Uncertain Wavelet (Dmey) 50 :::230 continuous magnet, a linear belt-driven servo-drive has been used. The velocity range from 0 up to 4 m/s allows the the validation numerical simulations in the range of small and medium magnetic Reynolds numbers. The measurement frame is decoupled from the drive and carries the force sensor and the magnet system together with yz-positioning stage that provides a precise positioning in yandzdirections. A rear-earth neodymium perma- nent magnet (NdFeB) has been mounted directly on a three component force sensor that is built in strain gauge technology providing medium accuracy and high dynamics. The main technical details of this LET measurement system can be found in Table 5.6. It should be mentioned that the used force sensor was not an optimal choice for the given LET setup, as the accuracy in all force components is rather low (Table 5.6). This is because the overall resolution of the sensor is determined by the maximum measurement range, which is up to 3 N in the x-axis andy-axis direction, and 10 N inz-axis direction [15]. Additionally, due to the noisy laboratory environment which strongly in uences measured force signals the resulting signal-to-noise ratio is rather poor. Therefore, the raw force signals cannot be used directly. The signal quality can be signicantly improved by applying appropriate lter techniques whose parameters are summarized in Table 5.7. 5.8.1 Metallic Sheets Approximation As already mentioned, for arbitrary positioning of articial defects one of the test conductors used in experiments has been approximated by a package of metallic sheets, each 2 mm thick. From the basic principles of LET (section 2.1) it can be shown that eddy currents induced by relative motion form two-dimensional concentric loops in the planes normal to the magnetic eld lines [37] (in the yx-plane of the model). In order 69 Chapter 5: Numerical Modelling of Lorentz Force Eddy Current Testing (a) Drag force (b) Lift force Figure 5.18: Comparison between the anisotropic conductivity model (ACM) and isotropic conductivity model (ICM). The long (" j") defect is located at d= 2 mm and v= 0:5m/s. not to aect their distribution within the conductor the laminations used in experiments have been directed along the model z-axis (Fig. 5.17). If there are no defects within the conductor this approximation is completely justied since the z-component of induced eddy currents is zero, jz= 0 [133, 134]. Numerically, this would require no additional considerations and the laminated structure of the conductor could be modelled simply by applying an isotropic electrical conductivity to the conductor. ICM=jj: (5.13) This will be further referred as the isotropic conductivity model (ICM). In case of a defective conductor it is reasonable to expect that the distribution of the induced eddy currents around the defect is completely three-dimensional, i.e. jz6= 0. This actually means that the laminated structure of the test conductor used in experiments will aect the resulting perturbations of the Lorentz force. In order to test how strong this eect is a 3D numerical simulation based on MDA has been performed. The laminated structure of the conductor has been numerically handled by applying an anisotropic electrical conductivity in the form of a diagonal tensor with zero value at the position corresponding to z-axis ACM =0 0 00 0 0 0: (5.14) The so-called anisotropic conductivity model (ACM) is computationally more ecient than the modelling of each individual sheets would provide [135, 136]. Fig. 5.18 shows the perturbation of the Lorentz force caused by the long (" j") defect which is d= 2 mm below the surface of the conductor. The calculation has been performed for both, ICM 70 Chapter 5: Numerical Modelling of Lorentz Force Eddy Current Testing and ACM. By a direct comparison, a considerable increase in Lorentz force perturba- tions is obtained in case of ACM (Fig. 5.18). The maxima in force perturbation are increased by factors of 3 and 4 for the drag and the lift force components, respectively. As expected, suciently far away from the defect both ACM and ICM give identical results, which additionally conrms the observations made before. In order to exclude any in uence of possible discretization errors, identical nite element meshes are used for both ACM and ICM. The given analysis has shown that in order to obtain correct Lorentz force per- turbations for laminated conductors, an anisotropic conductivity model (ACM) has to be applied. In the following sections both, ICM and ACM have been validated with corresponding experiments (section 5.8). 5.8.2 Validation of the Moving Magnet Approach Similarly, as in the presented verication of the MMA approach for its validation it has been assumed that the conductor has a nite length ( L= 250 mm) and a long ("j") defect located at d= 2 mm. The cylindrical magnet has been moved from left end of the conductor to its right end ( Xstart=160 mm) at constant testing velocity v= 0:5 m/s and lift-o distance z= 1 mm (Fig. 5.3). During the solving process, the generalized- transient solver with a xed time step
t=
x=vand= 0:85 has been used, which resulted in total number of 320 time steps (see Table 5.3). Thus, the obtained force signals have been corrected according to (B.1) and (B.2) (see Appendix B) To approximate the laminated structure of the test conductor used in experiments, the anisotropic conductivity model (ACM) (5.14) has been applied. Fig. 5.19 shows the results obtained by the MMA and the corresponding experiments (EXP). A very good agreement can be observed in the whole range of the resulting Lorentz force perturbations. The calculated WNRMS errors (5.10), considering only perturbations caused by the defect, i.e.
W= 48 mm are equal to MMA D = 5:6% andMMA L = 4:9%, for the drag and the lift Lorentz force components, respectively. For the whole length of the conductor, i.e.
W= 250 mm the calculated WNRMS errors are equal to MMA D = 2:8% andMMA L = 1:5%. In the experimental force signals a constant drift can be observed in the Lorentz force drag component (Fig. 5.19a). It is caused by a small tilt in the mounting of the specimen leading to a constant change in the lift-o distance between the magnet and the conductor. Due to low force values which lie in the uncertainty range of the used force sensor the tilt is not so expressed in the lift force component (Fig. 5.19b). Nevertheless, the results further underline the necessity of applying the ACM to the conductor, whenever the metallic sheets approximation of the solid conductor is used. 71 Chapter 5: Numerical Modelling of Lorentz Force Eddy Current Testing (a) Drag Lorentz force component (b) Lift Lorentz force component Figure 5.19: Validation of numerical results obtained by the moving magnet approach (MMA) with experiments (EXP). The long (" j") defect is located in the second sheet from the top surface of the conductor ( d= 2 mm) which is modelled using ASM. 5.8.3 Validation of the Moving Defect Approach In order to validate the moving defect approach (MDA) the wide (" ") and cross ("+") defects have been considered additionally (Fig. 5.17). Furthermore, the same LET benchmark problem (Fig. 5.1) and material properties as for the MMA have been used (Table 5.3). In the implementation of the MDA the defects are moving from right side of the magnet to its left side starting at Xstart= 55 mm (Fig. 5.4). As a transient solver the generalized- solver with a xed time step
t=
x=vand= 0:85 has been used. Since, MDA is not depending on the particular value of damping coe- cient(see Appendix B) no correction of the obtained results was necessary. Due to shorter displacement ( L= 2
Xstart) compared to MMA, the simulation resulted in total number of 110 time steps (see Table 5.3). Due to the laminated structure of the test conductor the anisotropic electrical conductivity model (ACM) has been applied as well. Fig. 5.20 shows the Lorentz force perturbations only in the vicinity of defects, 72 Chapter 5: Numerical Modelling of Lorentz Force Eddy Current Testing (a) Drag force (b) Lift force Figure 5.20: Validation of numerical results obtained by the moving defect approach (MDA) with experiments (EXP). The long (" j") defect is located in the second sheet from the top surface of the conductor ( d= 2 mm) which is modelled using ASM. since the front and back walls of the conductor are far away from defects and do not in uence the Lorentz force perturbations (Fig. 5.19). The calculated WNRMS errors of the drag and lift Lorentz force components are equal to "" D= 16:7%,"" L= 18:4%, and"+" D= 5:5%,"+" L= 5:4%, for wide ("") and cross ("+") defects, respectively. The WNRMS errors have been calculated with reference to the corresponding experi- mental signals (EXP) and window size of
W= 48 mm. Higher discrepancy between obtained results has been observed for the wide (" ") defect compared to the cross ("+") defect. This is due to the fact that wide (" ") defects produce relatively small force perturbations which are strongly in uenced by the low signal-to-noise-ratio of the measurement signal (section 5.8). Beside validation of the numerical results, the analysis also helps to determine the in uence of the shape of the defect on the resulting Lorentz force perturbations. For the given size of the defects, the strong dependency of the Lorentz force perturbations on the actual shape of defects can be observed. This is a promising result since it can be used to identify and localise defects by solving the corresponding inverse problem. 5.8.4 Validation of the Isotropic Conductivity Model For the validation of MMA and MDA, experiments with the laminated conductor has been used (Fig. 5.17). This conguration has been numerically handled by applying the anisotropic conductivity model (ACM), which resulted in a very good agreement with measured force signals. In order to additionally validate the isotropic conductivity model (ICM), described by (5.13), MMA has been applied to the solid test conductor with linear surface defects (Fig. 5.17). The size and the material properties of the conductor are the same as used for the metallic sheets analysis (see Table 5.3). Fig. 5.21 shows the obtained Lorentz force perturbations for the whole solid conductor, i.e. 73 Chapter 5: Numerical Modelling of Lorentz Force Eddy Current Testing (a) Drag Lorentz force component (b) Lift Lorentz force component Figure 5.21: Validation of numerical results obtained for isotropic conductivity model (ICM) and moving magnet approach (MMA) with experimens (EXP). The test conductor is solid with idealised surface defects in form of linear slits. taking its front and back walls into account. Similar to the previous ACM analysis the results agree well with experiments (EXP). The calculated WNRMS errors of the drag and lift Lorentz force components, with respect to the corresponding experimental signals (EXP) and a window size of
W= 250 mm, are equal to ICM D = 4:7% and ICM L = 6:8%. The same constant drift in the experimental force signals is observed again (Fig. 5.19 and Fig. 5.21). This only conrms the observation made before that the drift is caused by a small tilt in the mounting of the specimen leading to a constant change in the lift-o distance between the magnet and the conductor. 74 Chapter 6 Results and Discussion In this chapter results of 3D nite element simulations of a solid electrically con- ducting bar moving in the vicinity of a permanent magnet are presented. In particular, parametric studies have been carried out in order to quantify eects of various param- eters on the total value of the Lorentz force acting on the magnet. The simulations have been performed for two characteristic LET testing congurations, namely (i) for a conductor without defects and (ii) for a conductor with pre-dened material defects. In case of a non-defective conductor, the parametric studies have been used to quantify the eect of the lift-o distance, magnetization of the used magnets, and the magnetic Reynolds number on the resulting Lorentz force (section 6.1). The study involving defective conductors provides guidelines regarding more eective force measurements. The in uence of the magnet shape and size on defects detection and reconstruction is analysed as well (section 6.2). 6.1 Non-defective Conductor Considering a conductor free of any material defects, the actual problem under investigation is described in Fig. 5.1. If not stated otherwise all geometrical and ma- terial parameters described in Table 5.3 apply here as well. Since the length of the conductor is much larger than the diameter of the used cylindrical permanent magnet (LcDm), and the conductor is free of defects, the quasi-static formulation given by (5.4) and (5.5) is well suited for the analysis. In the assumed moving frame of reference the additional velocity term ( vr A) describes the motion of the conductor enabling only one stationary analysis to be performed for an accurate steady state solution. In the non-conducting domains (air and magnet) the magnetic scalar potential (2.35) is used. Due to identical geometry of the problem, the same boundary and interface conditions described in section 5.6.2 also apply. In order to obtain more ecient mesh 75 Chapter 6: Results and Discussion Figure 6.1: Distribution of the nite element mesh inside the non-defective conductor used in the QSA modelling. The mesh consists of 2ndorder hexahedral elements. Table 6.1: Mesh convergence study ( Rm= 10) NoE DoF FD(N)FL(N) 4370 255711 3 :8438 4:1884 6704 301499 3 :8392 4:1920 9638 361805 3 :8371 4:1938 13220 440826 3 :8362 4:1947 17498 537158 3 :8356 4:1952 22520 649873 3 :8353 4:1955 distribution inside the non-defective conductor it is more appropriate to use hexahedral instead of tetrahedral nite elements (Fig. 6.1). The size of successive elements along the edges of the conductor has been controlled by means of the geometric progression. The applied common ratio of elements was rx= 1=6,ry= 1=5, andrz= 1=7 along thex,yandz-axis, respectively. The total number of elements (NoE = 22520) used for the analysis is based on the performed sensitivity study providing the converged Lorentz force values independent on the actual mesh distribution (Table 6.1). Due to more complex geometries, the magnet and the surrounding air region are meshed with quadratic tetrahedral elements having ner distribution in the close vicin- ity to the magnet. The resulting mesh distribution has not been shown here for sim- plicity. 6.1.1 Magnetic Reynolds Number Study To illustrate the validity of QSA for any value of the magnetic Reynolds number (Rm) when conductor is free of defects, a semi-analytical model described by Lee and Menendez [134] has been used. The method provides exact expressions for Lorentz force acting on a current carrying coil moving above a conducting sheet of arbitrary thickness which are calculated by means of Fourier transform approach. To approximate the 76 Chapter 6: Results and Discussion Figure 6.2: Lorentz force dependency on the value of Rm. Study is performed for conducting plate with thickness Hc. The corresponding velocity range is v= 0::40 m/s. cylindrical shape of the magnet, an array of nparallel octagon shaped coils which are carrying the current I0equal tonI0=BrHm=0were dened. Here Br=M0 andHmare the remanence and the height of the used permanent magnet, respectively. This semi-analytical approach has been implemented in MatlabR and enables fast force calculations [137]. In order to account for an innite plate, in the implementation of QSA the conductor is considered to be long and wide enough compared to the size of the magnet, i.e. Lc;Wc>> Dm;Hc. The comparison between numerical and semi- analytical solutions for a wide range of the resulting magnetic Reynolds numbers, i.e. Rm= 0::20 (v= 0::40 m/s) is presented in Fig. 6.2. A very good agreement between both approaches is obtained in the whole range of Rm. In order to better understand the physical background of such a force behaviour it is useful to distinguish three characteristic regions presented in Fig. 6.2, namely (i) low- Rm, (ii) moderate- Rm, and (iii) high-Rmrange. In the low-Rmrange (Rm<1) the dependency of the Lorentz force with Rmis well described. The drag component of the Lorenz force varies linearly with Rm(FDRm), whereas the corresponding lift force component varies quadratically with Rm(FL R2 m) [23, 133, 134]. This is mostly because the induced (secondary) magnetic eld Bsis low compared to the applied (primary) magnetic eld Bp(Fig. 6.3a). As a consequence, the total magnetic eld Bis equal to Bpdue to which induced eddy currents have symmetric distribution and penetrate deep inside the conducting domain. Based in this observation, it is therefore expected that, under the similar testing conditions, LET could provide deeper defects detection compared to the traditional ECT technique. 77 Chapter 6: Results and Discussion (a) Low-Rmrange where BsBp(Rm= 0:01). (b) Moderate- Rmrange where Bs<Bp(Rm= 1) (c) High-Rmrange where BsBp(Rm= 100) Figure 6.3: In uence of the induced (secondary) magnetic eld Bson the total magnetic eld B distribution with respect to Rm. 78 Chapter 6: Results and Discussion (a) Drag component (b) Lift component Figure 6.4: Lorentz force dependency on the value of Rm. Study is performed for two conductors made from aluminium ( Al= 24:1 MS/m) and copper ( Cu= 57:9 MS/m). In the moderate- Rmrange (1< Rm<20) the induced secondary magnetic eld starts to in uence the total magnetic eld distribution. This so-called motional skin eect makes the distribution of induced eddy currents asymmetric and consequently reduces their penetration depth (Fig. 6.3b). As a result the non-linear dependency of drag and lift force components with Rmis observed (Fig. 6.2). In the high- Rmrange (Rm>20) the motional skin eect is pre-dominant. The magnetic eld is almost completely pushed out from the interior of the conductor inducing the current only close to the surface of the conductor (Fig. 6.3c). Thus, the drag component of the Lorentz force decreases with increasing Rm, whereas the corresponding lift force still continues to rise with Rm(Fig. 6.2). It should be mentioned that, for extreme values of Rmboth forces reach their asymptotic limits [133]. The characteristic Lorentz force to Rmdependency (Fig. 6.2) has been additionally conrmed experimentally [15, 138]. To reach moderate values of Rm, two dierent con- ductors namely, from aluminium ( Al= 24:1 MS/m) and copper ( Cu= 57:9 MS/m) have been used. In both cases the conductor had the same geometry with a nite cross- section described by parameters given in Table 5.3. The analysis also helps to validate the results of QSA used in all numerical calculations. As expected, the obtained re- sults conrm observations made above and clearly show the linear and the quadratic dependency of drag and lift Lorentz force components with Rm, respectively. It should be mentioned that the actual limit of low- Rmrange (Rm1) strongly depends on the denition of the characteristic length scale L(section 2.2.4). Nevertheless, the exper- iments additionally justify the choice of L=Hc=2 for the LET benchmark problem under investigation (Fig. 5.1). In order to measure the eciency of magnetic levitation (MAGLEV) systems the so-called lift-to-drag ratio of the Lorentz force is often used [133, 134]. Based on the previous analysis, in low- Rmrange the lift-to-drag force ratio in LET depends linearly 79 Chapter 6: Results and Discussion Figure 6.5: Dependency of the lift-to-drag ratio on the value of Rm. onRm, i.e.FL=FDRm(Fig. 6.5). Moreover, it can be observed that the linearity is preserved even for moderate values of Rm(Rm3) which in case of a copper conductor (Cu= 57:9 MS/m) corresponds to testing velocities of v1:7 m/s. 6.1.2 Lift-o Dependence Study It is well known that the magnitude of the eddy currents induced inside the mov- ing conductor vary with the relative position between the magnet and the conductor [37, 139]. For simplied LET congurations, when the magnet is described with a point dipole formula (4.1) and when the conductor has the form of an innite plate, it can be shown analytically that the drag component of the Lorentz force varies monotonically with the so-called lift-o distance z, i.e.FDz3[25, 26]. For realistic LET systems such as given LET problem (Fig. 5.1) the lift-o dependency can not be determined in the closed form analytically [67]. Thus, in this section, the eect of zon the resulting Lorentz force acting on the magnet has been studied numerically. The validation of re- sults with corresponding experiments has been performed as well (see Fig. 6.6a). Apart from a good agreement between numerical end experimental predictions it is more im- portant to notice a similar dependency of both Lorentz force components on variations (a) Drag and and lift components (b) Lift-to-drag ratio Figure 6.6: Lift-o dependency of the Lorentz force. Testing velocity is v= 2 m/s. 80 Chapter 6: Results and Discussion inz. For low lift-o distances ( z < 5 mm) the obtained decay is exponential, with small constant decay rate of FL;De0:34z. It has been shown before that actual the decay rate is not depending on the testing velocity or the size of the used magnets [138]. For larger distances the power decay of FD;Lz3:1for both force components is reached (Fig. 6.6a). From the obtained results an obvious conclusion can be made that by using the lift-to-drag ratio of the Lorentz force ( FL=FD) the eect of variations inzcan be considerably reduced or even completely avoided. This assumption has been demonstrated in Fig. 6.6b both, numerically and experimentally. 6.1.3 Magnetization Strength Study The magnetization strength jMjof the permanent magnets is usually unknown or it is given in a wide range by the manufacturer. In practice, jMjis determined by measurements of the external magnetic eld and then deducing jMjfrom this eld. However, for design and practical realization of future LET systems it is necessary to estimate the sensitivity of the Lorentz force on possible variations in jMj. For the given LET benchmark problem, the eect of the magnetization strength on both Lorentz force components has been calculated numerically (Fig. 6.7a). For the analysis it is assumed that the magnet is uniformly magnetized in the z-axis direction, i.e. M=Mez. From the obtained results it can be observed that both Lorentz force components scale quadratically with jMj. This observation was also conrmed in [25, 26], where instead of realistic permanent magnets the point dipole approximation has been used. This means that the characteristic lift-to-drag ratio of Lorentz force is independent on the actual magnetization of the used permanent magnets (Fig. 6.7b). (a) Drag and lift force components (b) Lift-to-drag ratio Figure 6.7: In uence of the magnet magnetization strength jMjon the Lorentz force ( Rm= 1). 81 Chapter 6: Results and Discussion 6.1.4 Magnet Size Study In the previous section it has been shown that the absolute Lorentz force acting on the magnet increases quadratically with the magnetization strength jMjof the magnet (see Fig. 6.7). The main aim here is to study how the size of the magnet, namely its volume aects the intensity of the Lorentz force. Similar studies have been performed in magnetohydrodynamics (MHD) applications, in particular for Lorentz force velocimetry (LFV) investigations where Lorentz forces are used to measure the ow rates of liquid metals and even glass melts [25, 26]. Due to relatively low electrical conductivities, the main interest in LFV is to increase the intensity of the Lorentz force acting on the magnet system while minimising its weight [140]. In LET systems this information can help to choose the optimal force sensor or the magnet size for specic NDT&E application. For the analysis, the LET benchmark problem (Fig. 5.1) and cylindrical permanent magnet with DmHm=aahave been used. The magnet is magnetised along the z-axis with magnetization density M=Mezgiven in Table 5.3. From the obtained results the linear dependency between both Lorentz force components and the resulting volume of the used magnet ( V==4a3) has been observed (Fig. 6.8). Similar linear dependency of the drag Lorentz force component with respect to the volume of the magnet has been reported for LFV applications involving liquid metals [141]. (a) Drag force component (b) Lift force component Figure 6.8: In uence of the magnet size (volume) magnetised with constant magnetisation density Mon the Lorentz force ( Rm= 1). The used magnet has cylindrical shape ( DmHm=aa) and is placed at a lift-o distance z= 1 mm. 82 Chapter 6: Results and Discussion 6.1.5 Lorentz Force Sigmometry Based on the previous observations the working principle of LET has found its practical application for contactless measurement of the specic electrical conductivity of a solid or an electrically conducting uids [137, 142]. The technique is termed Lorentz Force Sigmometry (abbreviated LoFoS) where the neologism Sigmometry is derived from the Greek letter sigma, often used to denote the electrical conductivity. The main aim of this section is to give a brief theoretical description of LoFoS, whereas the measurement procedure can be found in [15, 137]. Using the denition of the magnetic Reynolds number (2.33) and assuming the low-Rmregion (Rm<1) it has been shown that dependency of Lorentz forces with Rm are as follows FDRm; (6.1) FLR2 m: (6.2) An obvious idea to calculate the electric conductivity using only the drag or the lift Lorentz force component would lead to higher measurements uncertainties due to the high force dependency on the variations in lift-o distance and the strength of the used magnet [137] (Fig. 6.6a and Fig. 6.7a). To overcome these disadvantages a modied approach based on the lift-to-drag ratio has been applied. As a result of performed parametric studies which additionally conrm (6.1) and (6.2) it has been shown that the ratioFL=FDdepends linearly on Rm FL FDRm: (6.3) Under the assumption that the conductivity is homogeneous, isotropic and the testing speed is constant, the lift-to-drag force ratio and the electrical conductivity are related as =sFL FD; (6.4) where the calibration coecient sdepends on the geometry of the magnet system, on the translational velocity and (weakly) on the distance between the plate and the magnet (Fig. 6.6b). Since Sisa priori unknown, the LoFoS system has to be cali- brated. For simplied LoFoS congurations, to obtain the necessary calibration curves a semi-analytical model presented in section 6.1.1 can be used. When conductors have nite dimensions or measurements are based on more complex magnet systems a nu- merical model can be applied. For the given benchmark problem the calibration curves obtained numerically for dierent testing velocities are presented in Fig. 6.9a. In the 83 Chapter 6: Results and Discussion (a) Calibration curves (b) Estimation of using LoFoS Figure 6.9: Numerically obtained calibration curves for lift-o distance z= 3 mm wide range of considered (unknown) electrical conductivities the results show fairly linear dependency between and the lift-to-drag Lorentz force ratio FL=FD. Never- theless, this is only true if the condition Rm<1 is satised, i.e. in the low- Rmregime. For higher Rmvalues the non-linearity caused by the motional skin eect has to be considered. In order to prove the theoretical observations experimentally, the electrical con- ductivity of two solid specimens of known electrical conductivity, namely aluminum (Al= 20:4 MS/m) and copper ( Al= 57:92 MS/m) have been obtained by LoFoS. The given conductivities have been measured with the eddy current device Sigmatest 2.069 (Institut Dr. Foerster GmbH & Co. KG) [15, 137]. For measurements of the lift-to-drag ratio of the experimental setup described in section 5.8 has been used. By applying LoFoS conductivities of Al= 21:59 MS/m and Cu= 60:08 MS/m have been obtained (Fig. 6.9b). The combined measurement uncertainties of uAl= 9:82 MS/m anduCu= 3:46 MS/m are based on the uncertainty of the force sensor of dFD= 15 mN anddFL= 50 mN (Table 5.6). These uncertainties can be reduced when the measure- ments are repeated several times or by applying a more accurate force sensor[15, 137]. The presented analysis showed that LoFoS is able to provide the electrical conduc- tivity of a unknown specimen. In a nutshell, LoFoS has the following advantages (i) LoFoS is contactless, (ii) LoFoS can be applied continuously during production pro- cesses, (iii) LoFoS is a method that enables the user to measure conductivity beyond the surface and (iv) the proposed data processing makes LoFoS resistant to changes in the lift-o distance and the strength of the magnetic eld source. LoFoS is suitable for specimen of any kind of physical condition, where the only limitation is given by the minimal measurable Lorentz force components of a multi-component force sensor [15]. 84 Chapter 6: Results and Discussion 6.2 Defective Conductor In this section several parametric studies based on the LET benchmark problem and the conductor with defects (Fig. 5.1) have been performed. The main aim is to quantify the eect of various testing parameters on the resulting perturbations of the Lorentz force, such as: magnet shape and size, magnetization M, lift-o distance z, depth of defects dand the resulting magnetic Reynolds number Rm. The analysis provides reference results to better understand the feasibility and testing capabilities of LET. In order to provide validation of results the calculations have been performed assuming the laminated structure of the conductor under test. 6.2.1 Magnet Shape Study It is reasonable to expect that the density and spatial distribution of eddy currents induced inside a conductor in motion, i.e. the resulting Lorentz force depends on the geometry of the used magnets. In the eld of optics, the so-called point spread functions (PSF) are typically used in order to describe the quality of an imaging instrument [143]. They were rst introduced by Sir Airy in 1835 to describe the best focused spot of light that a perfect circular lens can produce [144]. In principle, PSF can be understood as the three-dimensional diraction pattern of light emitted from a point source, which is transmitted to an imaging plane of an instrument [143]. In traditional eddy current testing (ECT) a similar approach based on the PSF has been applied to test the sensitivity and performance of dierent pick-up coil congurations [145, 146]. In contrast to its rst application in optics in ECT, PSF is dened as the spatial change of the impedance of the pick-up coil which results from a raster surface scan of the test conductor containing a small point-like defect (Fig. 6.10a). The point-like defects in NDT applications should be understood as defects whose (a) Traditional eddy current testing (b) Lorentz force eddy current testing Figure 6.10: Calculation of the point spread function (PSF). The relative position between the point-like defect and the permanent magnet is denoted by r. 85 Chapter 6: Results and Discussion Figure 6.11: Fictive point-like defect modelling used for fast calculation of the point spread function (PSF). r0is the relative position between the point-like defect and the permanent magnet. size is small, so that the applied eld Bcan be considered to be uniform over its entire volume [146, 147]. Additionally, the dimension of defects must be small compared with the distance from the defect to the nearest boundary of the conductor under test and small compared to the size of the used sensor (see Fig. 6.10). It should be mentioned that the shape of the particular PSF is not only a function of the coils geometry, but it also depends on the current lift-o distance zand depth of the defect d[146]. Using the same principle as in traditional ECT, the sensitivity of the Lorentz force perturbations on the particular shape and size of the used magnet has been studied by means of PSF. In LET systems, PSF can be dened as the spatial distribution of the Lorentz force perturbation caused by raster surface scanning of the test conductor containing a point-like defect (Fig. 6.10b). In contrast to ECT which have only one PSF for particular pick-up coil conguration [145, 146] in LET systems each Lorentz force component has dierent and independent PSF. They can be calculated by applying the Lorentz force formula (2.2) for each relative position between the magnet and the point-like defect. Since PSF describes the perturbation of the Lorentz force only due to defects, its calculation involves two separate analysis, i.e. for defective and non- defective conductor
F(r) =ZZZ C Dj(r)B(r)d |{z} Defective cond.+ZZZ Cj(r)B(r)d |{z} Non-defective cond.; (6.5) where Cis the volume of the test conductor and Dthe volume of the small point-like defect. For each relative position between the magnet and the defect the solution of (6.5) would involve separate FEM calculations, which in 3D LET investigations can be very time consuming. Fortunately, in contrast to ECT the calculation of PSF in LET systems can be considerably simplied (Fig. 6.11). In order to solve (6.5) for each relative position between the magnet and the point-like defect defect, it is more ecient to move the 86 Chapter 6: Results and Discussion defect instead ( r!r0). Furthermore, applying the superposition principle an equiva- lent model of the conductor can be used (Fig. 6.11). The model assumes ctive current inside the point-like defect jfwhich has the same amplitude but opposite direction as the current in the non-defective conductor at r0, i.e.jf(r0) =j(r0). This makes the total current inside the real point-like defect to zero (Fig. 6.11). Taking all this into account the left integral in (6.5) can be calculated as ZZZ C Dj(r0)B(r0)d =ZZZ Cj(r0)B(r0)d ZZZ Dj(r0)B(r0)d ; (6.6) Applying (6.6) in (6.5) PSF can be obtained by
F(r0) =ZZZ Dj(r0)B(r0)d : (6.7) Since, inside the point-like defect the magnetic eld B(r0) and ctive current j(r0) is essentially uniform, the integral (6.7) can be substituted by simple multiplication
F(r0) = (j(r0)B(r0))
D; (6.8) which needs to be performed for each relative position r0between the magnet and the defect. This can be obtained by ease evaluating the Lorentz force density f=jB inside the conductor under test at the xy-plane determined by depth of the defect d
F= (jB)
D=f
D: (6.9) The particular shape of PSF also depends on the lift-o distance zof the magnet. As it will be shown later, typically symmetrical shapes of PSF can be signicantly deformed when magnetic Reynolds number ( Rm) is high. Without any loss of generality, to test the sensitivity of the Lorentz force per- turbations on the particular shape and size of magnets the low- Rmregime has been considered ( Rm= 0:1). The eect of Rmhas been investigated in a separate section using realistic articial defects described in experiments (see section 6.2.4). PSF have been calculated for spherical, cylindrical and cubic magnets which are typically applied in LET. All magnets have been placed at the xed lift-o distance z= 1 mm and mag- netised along z-axis with magnetization density Mdened in Table 5.3. To provide a comprehensive comparison the analysis has been performed for (i) magnets having the same size determined by the length parameter a(Fig. 6.12a) and (ii) magnets having the same characteristic volume M(Fig. 6.12b). Figure 6.13 shows the characteristic 87 Chapter 6: Results and Discussion (a) Constant size a (b) Constant volume M Figure 6.12: Three characteristic shapes of permanent magnets used for in LET systems. (a) Drag comp. (Sphere) (b) Side comp. (Sphere) (c) Lift comp. (Sphere) Figure 6.13: Typical shape of the PSF obtained for a spherical magnet having diameter aat lift-o distancez= 1 mm and Rm= 0:1. The depth of the point-like defect is d= 1:5 mm. shape of PSF for all three Lorentz force components of a spherical magnet. The radius of the point-like defect used for PSF calculations was rD=a=30. For generality and to provide better comparison, the obtained results have been scaled to the resulting maximum force perturbation
Fmax i;sphere (i2D;L) of a spherical magnet. The shapes of PSF resulting from cylindrical and cubic magnets are almost identical to those shown in Fig. 6.13. Thus, they are not presented for simplicity. More informative direct com- parison between the considered magnet shapes is provided by taking a linear cut of the corresponding PFS at y= 0 along the x-axis, i.e. in the motion direction (Fig. 6.13). Since aty= 0 the side component of the Lorentz force FSis zero, it was not used for comparison. When magnets have the same characteristic size a(Fig. 6.12a) the strongest Lorentz force perturbations have been obtained for a cubic magnet (Fig. 6.14). The increase in the force perturbation with respect to the spherical magnet is 2:5 and3 times for drag and lift Lorentz force components, respectively. The force perturbation re- sulted from a cylindrical shape was lower compared to cubic magnet shape as well (see Fig. 6.12a). When magnets have the same volume M(Fig. 6.12b) similar results have been 88 Chapter 6: Results and Discussion (a) Drag component (b) Lift component Figure 6.14: Direct comparison of normalized PSF for three typical shapes of permanent magnets. All magnets have the same characteristic size determined by length the parameter ai. The maximum perturbation of the spherical magnet
Fmax i;sphere (i2D;L) is used as a reference. (a) Drag component (b) Lift component Figure 6.15: Direct comparison of normalized PSF for three typical shapes of permanent magnets. All magnets have the same volume Mwhich results in dierent characteristic length ai(i2a1;a2;a3). The maximum perturbation of the spherical magnet
Fmax i;sphere (i2D;L) is used as a reference. obtained. Nevertheless, in contrast to the previous case the dierence in the maxima of force perturbations was not so expressed (Fig. 6.15b). The force perturbations re- sulting from cubic and cylindrical magnets are almost identical for both Lorentz force components, but still 1:5 larger with respect to the spherical magnet . Thus, independent on the type of the defect to be detected, in order to increase the resulting Lorentz force perturbations it is advisable to use larger magnets of cubic shape. On the other hand, in the LET applications based on absolute force measure- ments, larger magnets will result in higher absolute forces which scale linearly with the resulting volume of the magnet (see section 6.1.4). Even though this increases the force perturbations as well they are still 2 or even 3 orders of magnitude lower than the absolute forces acting on the magnet. This actually means that, it could be very dicult task to resolve the resulting force perturbations using the absolute force sen- sors today commercially available. Furthermore, if the size of the magnet is increased the weight of the whole magnet system which is carried by the force sensor needs to 89 Chapter 6: Results and Discussion be taken into account [15]. Nonetheless, independent on the shape and type of defects PFS based approach described above can be used to test and optimize dierent magnet congurations to further increase force perturbations. 6.2.2 Magnet Size Study Increasing the size of the magnet with respect to the size of defects would result in higher Lorentz force perturbations. This is of great interest if only the detection of defects within the conductor is required. However, in non-destructive evaluation (NDE) applications it is not only enough to detect defects, but the shape reconstruction is required as well [11, 148]. As it will be shown below, with increasing the magnet size the information content of the actual shape of defects is considerably reduced. Apart from allowing the fast estimation of the intensity of force perturbation for dierent shapes of magnets, PSF are very useful to determine the sensitivity of force signals on the actual shape of defects as well. To this end, based on Fig. 6.13 for each Lorentz force component an eective radius of the PSF can be dened as the distance from the center of the magnet to the point in space where the intensity of the force perturbation drops to zero (
F(rPSF i) = 0). The eective radius is denoted by rPSF i wherei2 D;S;L represents the drag, side and lift Lorentz force component, respectively. For the characteristic shapes of the magnets typically used in LET (sphere, cylinder and cube) all three force components have approximately the same eective PSF radius rPSF ia(see Fig. 6.13). This is expected since all magnets under investigation have the same characteristic size determined by a. In principle this means that, independent on the particular shape of defects, PSF allow to estimate the information content of the force signals by only one parameter given by a. If defects are larger than the characteristic size of the magnet a, the force perturba- tion will be strongly in uenced by the shape of the defect. This means that the force signal carries enough information about the particular defect which should simplify its reconstruction and localization. When defects are small compared to the size of the magnet ( w;h;l < a ) the resulting force perturbations will have the similar form as the PSF shown in Fig. 6.13. Thus, the information about the defect shape will be considerably reduced or even lost completely. For example, if there is more than one defect in the area determined by parameter a, it would be very dicult to determine whether the force is perturbed by only one larger defect or by several smaller defects. Strictly speaking when w;h;l < a , the resulting force perturbations have always the same form determined by the shape of the particular PSF, i.e. by the size and shape of the used magnets. A very intuitive example conrming the observations mentioned above can be pre- 90 Chapter 6: Results and Discussion sented in form of direct defect imaging. Thus, the LET benchmark problem dened in Fig. 5.1 has been applied and solved using weak reaction approach (WRA) approach (see section 5.4). Due to the strongest force perturbation the analysis has been per- formed for the cubic permanent magnet with a= 1;6;12 and 24 mm. For imaging the cross ("+") defect lying at d= 2 mm under the surface of the conductor has been considered. As expected, for relatively small magnet sizes ( a= 1 mm) the Lorentz force perturbations are strongly in uenced by the shape of the defect. Due to dierent characteristic PSF shapes (see Fig. 6.13) each Lorentz force component carries dierent and independent information about the particular defect (Fig. 6.16a, 6.16b and 6.16c). This is of great interest in NDE where defects have to be reconstructed. Further- more, due to the characteristic Mexican hat shape of the PSF (Fig. 6.13a), the drag force component can even provide the direct image of the cross ("+") defect without additionally solving an inverse problem. The quality of an image is not only in uenced by the eective radius of the PSF (rPSF ia), i.e. the size of the magnet, but it also depends on the depth of the defectdand lift-o distance z. Defects lying deeper inside the material and magnets at higherzwill result in somehow blurry images of the defect as seen in Fig. 6.16a. It is important to notice that, due to the small size of the magnet the intensity of the force perturbation is very low. Increasing the size of the magnet the intensity of Lorentz force perturbations increases, while their information content about the particular defect shape is considerably reduced (Fig. 6.16). As mentioned above, when magnet is larger than the used cross ("+") defect ( a= 24 mm) the obtained image is mostly determined by the shape of the magnet (Fig. 6.16j, 6.16k and 6.16l). In this case the shape reconstruction of defects would be very dicult or even impossible to perform. 91 Chapter 6: Results and Discussion (a) Drag component (b) Side component (c) Lift component (d) Drag component (e) Side component (f) Lift component (g) Drag component (h) Side component (i) Lift component (j) Drag component (k) Side component (l) Lift component Figure 6.16: LET direct imaging of cross ("+") defect lying at d= 2 mm under the conductor surface. Cubic permanent magnet with a= 1;6;12 and 24 mm is located at lift-o distance z= 1 mm. 92 Chapter 6: Results and Discussion 6.2.3 In uence of the Magnetisation Direction and Intensity Magnetisation Direction In the previous section the eect of the magnet size and shape on the resulting Lorentz force perturbations has been discussed. The main aim in this section is to investigate the eect of the magnetisation vector on the intensity of Lorentz force perturbations. To study the eect of the magnetization direction PSF procedure has been applied again. Thus, the same point-like defect rd=a=30 and material parameters described in section 6.2.1 have been used. Without any loose of generality the magnet is considered to be of spherical shape with a= 5 mm. In order to simplify the comparison between dierent orientations of M, the corresponding PSF have been normalised with reference toM=Mez(see Fig. 6.17). From the obtained results, a very strong dependency of the Lorentz force perturbations with respect to the actual magnetization direction has been observed. The particular direction of Mnot only aects the shape of the PSF for each force component, but the intensity of the force perturbations is considerably reduced as well. For this test example the amplitude of the PSF can be up to 4 times lower for both x- andy-axis orientations of M, with respect to its z-axis direction. Based on this results, it is desirable to design LET systems with magnets magnetized along thez-axis, i.e. normal to the direction of motion. 93 Chapter 6: Results and Discussion (a) Drag comp. ( M=M
ex) (b) Side comp. ( M=M
ex) (c) Lift comp. ( M=M
ex) (d) Drag comp. ( M=M
ey) (e) Side comp. ( M=M
ey) (f) Lift comp. ( M=M
ey) (g) Drag comp. ( M=M
ez) (h) Side comp. ( M=M
ez) (i) Lift comp. ( M=M
ez) Figure 6.17: In uence of the magnetisation direction Mon the shape of PSF for Rm= 0:1. The spherical magnet has a diameter a= 5 mm and it is placed at lift-o distance z= 1 mm. The depth of the point-like defect is d= 1:5 mm. The magnetisation direction along the z-axis has been used as a reference. 94 Chapter 6: Results and Discussion Magnetization Intensity Another important parameter which aects the force perturbations is the strength of magnets. It is well known that the absolute values of the Lorentz force increase quadratically with jMj(see section 6.1.3). For working with LET systems it is very useful to know how the force perturbations are aected by the particular value of jMj. Thus, the LET benchmark problem with all material and geometrical parameters given in Table 5.3 has been solved using the MDA approach (see section 5.2.2). For this analysis Rm= 0:48 (v= 0:5 m/s) and the long (" j") defect lying d= 2 mm under the surface of the conductor has been considered. The cylindrical magnet is placed at z= 1 mm above the conductor. To simplify the comparison of the resulting characteristic force perturbation pro- les (see Fig. 5.20), the so-called peak-to-peak (P2P) force perturbations can be used (Fig. 6.18). The P2P Lorentz force perturbation for each force component, denoted (a) Drag component (b) Lift component Figure 6.18: Denition of the Lorentz force peak-to-peak (P2P) perturbations. The results have been obtained for the long (" j") defect. (a) Drag P2P perturbation (b) Lift P2P perturbation Figure 6.19: In uence of the magnetization strength jMjon the P2P Lorentz force perturbations. The cylindrical magnet is magnetised in the z-axis direction. 95 Chapter 6: Results and Discussion asFP2P DandFP2P Lcan be described as FP2P D= max(FDF0 D)min(FDF0 D); (6.10) FP2P L= max(FLF0 L)min(FLF0 L); (6.11) whereF0 DandF0 Lrepresent the drag and the lift Lorentz force components for conductor free of defects, respectively. Since the magnet was positioned centrally in the middle of the conductor under test the side component FSof the Lorentz force was zero. The obtained results show, that the P2P perturbations of both Lorentz force components, as well as their corresponding absolute values are increasing quadratically with the strength of applied magnets (Fig. 6.19). Thus, to increase the testing depth of LET systems stronger magnets should be applied. 6.2.4 In uence of the Magnetic Reynolds Number A detailed analysis describing the in uence of the magnetic Reynolds number ( Rm) on the absolute Lorentz forces acting on the magnet for non-defective conductor is given in section 6.1.1. It has been observed that dierent Lorentz force components (a) Drag component (low- Rm) (b) Lift component (low- Rm) (c) Drag component (high- Rm) (d) Lift component (high- Rm) Figure 6.20: In uence of the magnetic Reynolds number Rmon the characteristic Lorentz force perturbations. Calculations have been performed for the long (" j") defect located at d= 2 mm using MDA. The magnet is located at a lift-o distance of z= 1 mm. 96 Chapter 6: Results and Discussion (a) Drag component (low- Rm) (b) Lift component (low- Rm) (c) Drag component (high- Rm) (d) Lift component (high- Rm) Figure 6.21: In uence of the magnetic Reynolds number Rmon the P2P Lorentz force perturbations. Calculations have been performed for long (" j") defect located at d= 2 mm using MDA. The magnet is located at lift-o distance z= 1 mm. have a dierent dependency on Rm. Moreover, when Rmis small (Rm<1) it is observed that the drag component of Lorentz force depends linearly on Rm, and that the corresponding lift force depends quadratically on Rm, i.e.FDRmandFLR2 m (see Fig. 6.2). As already mentioned in section 6.2.2, when defects are smaller than the used magnet, the characteristic perturbation of the Lorentz force in LET systems is strongly determined by the shape of the resulting point-spread functions (PSF). Fig. 6.13 shows the characteristic PSF for all three Lorentz force components which has been calculated for lowRmvalues (Rm= 0:1). Thus, apart from testing the in uence of Rmon the resulting P2P force perturbations, the additional aim here was to study the eect of Rm on the characteristic shape of the Lorentz force perturbations, i.e. on the characteristic shape of the PSF. Therefore, the LET benchmark problem (see section 5.1) has been solved using MDA by considering the long (" j") defect placed at d= 2 mm and lift-o distance of the magnet z= 1 mm. In low-Rmrange (Rm<1) the characteristic shapes of the Lorentz force per- turbations are practically independent on the actual value of Rm(see Fig. 6.20a and Fig. 6.20b). Nonetheless, the intensity of the P2P Lorentz force perturbations for both force components is increasing linearly with Rm, i.e.FP2P D;LRm. 97 Chapter 6: Results and Discussion In the moderate- Rmrange (Rm>7) the characteristic shapes of the Lorentz force perturbations are strongly in uenced by the value of Rm. The so-called bell shape of the drag force perturbation has now very similar bipolar form to the lift force compo- nent in low- Rmrange (compare Fig. 6.20c and Fig. 6.20b). The shape of the lift force perturbation in the moderate- Rmrange tends to the shape of the corresponding drag force perturbation in the low- Rmrange. However, the resemblance is not so expressed as in the drag force component (compare Fig. 6.20d and Fig. 6.20a). In contrast to low- Rmrange the resulting P2P Lorentz force perturbations in moderate-Rmrange are decreasing with Rm. For the given benchmark problem the decrease of the P2P force perturbations starts earlier for the drag force component (Rm=7) compared to the corresponding lift force component ( Rm=15). The decrease of the P2P force perturbations is caused by the so-called motional skin eect which occurs in the moderate- and high- Rmrange [149]. Due to this eect the induced currents are being pushed towards the surface of the conductor which than reduces the force perturbations (see Fig 6.3). The actual value of Rmfor which the P2P force perturbations start to decrease is therefore a function of the actual depth of the defect. This eect can be exploited in order to determine the actual depth of defects. A similar strategy has been applied in traditional ECT systems, where defects depth has been estimated by performing material testing with dierent excitation frequencies. This technique is also referred to as multi-frequency ECT [150]. 6.2.5 Testing Depth Study The rst estimation of the testing capabilities of LET can be obtained by varying the depthdof the defects within the test conductor. Therefore, the LET benchmark problem has been applied and the depth of all three articial defects, i.e. cross ("+"), long ("j") and wide ("") has been changed in range of d= 0 to 12 mm. In section 5.1 the depth of defects has been dened as a distance between the upper surface of the conductor and the upper surface of the defect (see Fig. 5.1). Thus, the zero depth (d= 0 mm) should be understood as the defect lying on the top surface of the conductor under test. Fig. 6.22 shows the resulting Lorentz force P2P perturbations as a function of defect depth obtained by the moving defect approach (MDA). In order to validate the obtained results LET problem has been solved assuming a testing velocity ofv= 0:5 m/s (Rm= 0:48) and a lift-o distance of z= 1 mm. Due to the fact that in experiments the solid conductor is approximated with a set of aluminum sheets (each 2 mm thick) the conductor has been modelled using the anisotropic conductivity model (ACM) described in section 5.8.1. Apart from a good agreement between numerical and experimental results, it is 98 Chapter 6: Results and Discussion (a) Drag component (b) Lift component Figure 6.22: In uence of the defect depth don the P2P Lorentz force perturbations. The calculations have been performed using MDA for Rm= 0:48 (v= 0:5 m/s) and lift-o distance z= 1 mm. more important to determine the actual dependency between the resulting P2P Lorentz force perturbations and d. When defects are relatively close to the magnet ( d<3 mm), an exponential decay of P2P force perturbations for both force components is observed. The obtained decay constant is rather small (
FDe0:37dand
FLe0:35d) and it is independent on the particular shape of the used defects (Fig. 6.22). For lager defect depths (d>7 mm) the exponential decay increases to power decay of d2:9, for both force components. This is expected since the magnetic eld relatively far away from the magnet has the decay rate of d3, i.e. the eld distribution corresponds more to the eld created by a magnetic point dipole (see section 4.1). This also means that in the low-Rmrange (Rm<1) the testing capabilities of LET systems are only limited by the natural diusion of the applied magnetic eld Bwhich is in order of d3. Due to the limited accuracy of the force sensor used in experiments, defects lying up tod= 6 mm under the surface of the conductor could be detected ( v= 0:5 m/s). The detection of defects lying deeper inside the material requires a force sensor with higher precision. 6.2.6 Lift-o Dependence Study Another important issue of LET is the dependency of the Lorentz force on the lift-o distance of the magnet. In order to increase the total force acting on the magnet small lift-o distances zare preferable (see Fig. 6.6a). The disadvantage of decreasing the lift-o distance is the increase of the sensitivity to surface dependent lift-o distance changes, e.g. surface roughness. For non-defective conductors (see section 6.1.2) it has been observed that the intensity of both force components have similar exponential dependency on variations in z. Thus, the so-called lift-to-drag ratio of the Lorentz force has been introduced which is only weakly dependent on variations in z. In order to study the sensitivity of the Lorentz force perturbations caused by ma- 99 Chapter 6: Results and Discussion (a) Drag component (b) Lift component Figure 6.23: In uence of the lift-o distance on the P2P Lorentz force perturbations. The calculations have been performed for Rm= 0:48 (v= 0:5 m/s) and depth of defect d= 2 mm. Figure 6.24: Dependency of the P2P lift-to-drag ratio of the Lorenz force on the changes in the lift-o distance z. terial defects on changes in zthe peak-to-peak (P2P) force perturbations (6.11) have been applied. Thus, an additional parametric study based on the LET benchmark problem has been performed. The lift-o distance of the magnet has been changed in the range of z= 1 mm to 15 mm. To test if there is any in uence of the defect shape on the P2P force variations with z, the simulations have been performed for all three types of articial defects, i.e. cross ("+"), long (" j") and wide ("") defect (Fig. ??). As in the case of the non-defective conductor, for low values of z(z < 3 mm) the P2P perturbations of both Lorentz force components have similar exponential decay with increasingz(FP2P De0:37zandFP2P Le0:35z). Even though, for higher values ofz(z > 7 mm) the power decay rate of FP2P D;Lz3is expected, due to the nite cross-section of the conductor its edges additionally in uence the force perturbations. As a result the power decay is slightly increased to FP2P Dz3:4andFP2P Lz3:3. Similarly to the conductor free of defects, it is therefore expected that the so-called lift-to-drag ratio of the P2P perturbations is only weakly or completely independent onzvariations. Fig. ??shows the sensitivity of the lift-to-drag ratio of the Lorentz force P2P perturbations resulting from given variations in z. Compared to single force components (Fig. ??) a considerable decrease in the lift-o distance dependency can be observed. Furthermore, it has been shown previously in [30] that the decay rates of 100 Chapter 6: Results and Discussion Figure 6.25: Lift-to-drag ratio of the Lorentz force obtained for the solid (isotropic) test conductor with idealised surface defects in form of linear slits. the Lorentz force with respect to variations in zare independent on the actual depth of defects as well [121]. To prove the above statements an experiment has been performed [15, 31]. A solid electrically conducting bar with 4 articial surface defects in form of linear slits of dierent sizes has been used (see section 5.8). The cylindrical magnet, moving with constant velocity v= 0:5 m/s, has been placed at z= 1 mm and z= 3 mm above the surface of the conductor. The obtained results show that the resulting lift-to-drag ratio of the total Lorentz force ( FL=FD) is practically independent on variations in z(Fig.??). In contrast to single force components, the perturbation of the lift-to- drag ratio is therefore only in uenced by the presence of defects which is extremely important in NDT&E applications. The same experimental conguration has been also solved numerically by applying the fast MMA approach (see section 5.2.1). Apart of good agreement with experiments it additionally conrms the observations made above. 6.2.7 Dierential Lorentz Force Eddy Current Testing The previous parametric studies have conrmed that LET represents a promising NDT&E technique which could provide fast and relatively deep testing of electrically conducting materials. Even though the used absolute force sensor was not optimal for the presented experimental setup (see section 5.8), its resolution of 15 mN for drag and side force components and 50 mN for the corresponding lift force component could still provide the detection of defects lying up to 6 mm deep under the surface of the conductor (see Fig. 6.22) [15, 30, 31]. Additionally to the low resolution, the relatively low dynamics of the force sensor (up to 10 KHz), which is based on the strain-gauge 101 Chapter 6: Results and Discussion Figure 6.26: Schematic of possible dierential Lorentz force sensor for DiLET applications. technology limited the LET investigations for conductors with defects to velocities up tov= 0:5 m/s orRm= 0:48 (see section 5.8 and [15] for more informations). Based on the LET analysis presented above, the testing should be performed with higher velocities v(Rm) and using larger quadratically shaped permanent magnets placed at low lift-o distances z. Apart from increasing the force perturbations this would also increase the absolute values of the Lorentz force which can be even few orders of magnitudes larger than the perturbations caused by defects (compare Fig. 6.4 and Fig. ??). Unfortunately, precise measurements of small force variations in rela- tively large range of applied forces is very dicult [15]. Thus, in LET systems there is a strong demand for the dierential force measurements. Similar tendency is observed in traditional ECT systems as well. In ECT, various dierential pick-up probe cong- urations oering higher testing sensitivity have already been designed and successfully implemented [146, 151]. Currently there are several possibilities to obtain the dieren- tial force signals resulting from LET systems. In principle, it is also possible to measure the acceleration or the velocity of the magnet [152]. However, the application of com- mercially available dierential force sensors would lead to higher spatial integration requirements and considerably higher costs. This would be even more important when designing sensor arrays for LET which could simplify and advance defects detection and reconstruction. The main aim in this section is to propose the simple and low-cost modication of LET setup which could be used for measurements of dierential Lorentz force signals caused by material defects. The proposed modication aects only the used magnet where three independent and passive pick-up coils have been winded on its outer surface (see Fig. 6.26). The principal idea is to use voltages induced in the additional coil system and correlate the voltage signals with the corresponding dier- ential Lorentz force signals acting on the magnet. In fact, the resulting magnet system can be directly applied to the existing LET experimental setup, i.e. it can be used as a complete dierential Lorentz force sensor. Since, this specic modication of LET systems allows dierential Lorentz force measurements, the proposed technique has been termed as dierential Lorentz force eddy current testing, abbreviated DiLET. The main idea of using simple coil system to obtain the dierential Lorentz force 102 Chapter 6: Results and Discussion signals can be traced back to Lorentz (Kelvin) force law (2.2). Assuming a cubic permanent magnet uniformly magnetised along the z-axis ( M=M
ez) the Lorentz force acting upon it can be described as F=wmZ 0lmZ 0hmZ 0(M@ @z)Bdxdydz; (6.12) wherewm,lmandhmrepresents the width, length and height of the magnet, respec- tively (see Fig. 6.26). Dierentiating (6.12) with respect to time and by considering all three force components separately the dierential force signals can be described as @FD @t=M@ @twmZ 0lmZ 0hmZ 0@Bx @zdxdydz; (6.13) @FS @t=M@ @twmZ 0lmZ 0hmZ 0@By @zdxdydz; (6.14) @FL @t=M@ @twmZ 0lmZ 0hmZ 0@Bz @zdxdydz; (6.15) If dimensions of the magnet ( wm;lm;hm) are small enough the following approximation applies @Bi @z=Bz=hm iBz=0 i hm(6.16) wherei2fx;y;zg. By using (6.16) directly in (6.13) - (6.15) the dierential Lorentz force acting on the magnet can be well approximated as @FD @tM hmwmZ 0lmZ 0hmZ 0@Bz=hmx @t@Bz=0 x @t dxdydz; (6.17) @FS @tM hmwmZ 0lmZ 0hmZ 0" @Bz=hmy @t@Bz=0 y @t# dxdydz; (6.18) @FL @tM hmwmZ 0lmZ 0hmZ 0@Bz=hmz @t@Bz=0 z @t dxdydz; (6.19) In DiLET the pick-up coils are moving together with the primary magnetic eld Bp, i.e. 103 Chapter 6: Results and Discussion @B=@t=@(Bp+Bs)=@t=@Bs=@t. Using the reasonable assumption @Bz=hm i=@t<< @Bz=0 i=@t, wherei2fx;y;zg(6.17) - (6.19) can be simplied as @FD @tM hmwmZ 0lmZ 0hmZ 0@Bz=0 x @tdxdydz; (6.20) @FS @tM hmwmZ 0lmZ 0hmZ 0@Bz=0 y @tdxdydz; (6.21) @FL @tM hmwmZ 0lmZ 0hmZ 0@Bz=0 z @tdxdydz: (6.22) From (6.20) - (6.22) it can be observed that in order to obtain the dierential Lorentz force signals it is sucient to measure the time variation of the magnetic ux density (@Bi=@t) atz= 0, i.e. at the bottom surface of the magnet. As it will be shown, this can be very eectively accomplished by a simple pick-up coil system presented in Fig. 6.26. Voltages induced in each of the coils can be calculated as follows V(Cx)=NxX 12 4wmZ 0hmZ 0@Bx @tdydz3 5Nx hmwmZ 0lmZ 0hmZ 0@Bx @tdxdydz; (6.23) V(Cy)=NyX 12 4wmZ 0hmZ 0@By @tdxdz3 5Ny wmwmZ 0lmZ 0hmZ 0@By @tdxdydz; (6.24) V(Cz)=NzX 12 4wmZ 0hmZ 0@Bz @tdxdy3 5Nz lmwmZ 0lmZ 0hmZ 0@Bz @tdxdydz; (6.25) whereV(Cx),V(Cy)andV(Cz)are the voltages induced in the x,yandz-axis oriented coils, respectively. The total number of windings of each coil is denoted by Nx,Nyand Nz, respectively. By direct comparison of (6.23) - (6.25) with (6.20) - (6.22) the dierential Lorentz force acting on the magnet, which is caused by the presence of material defects can be well approximated as @FD @tM hmhm NxV(Cx)(6.26) @FS @tM hmwm NyV(Cy)(6.27) @FL @tM hmlm NzV(Cz)(6.28) 104 Chapter 6: Results and Discussion (a) Drag component (b) Voltage induced in Cx (c) Lift component (d) Voltage induced in Cz Figure 6.27: Direct comparison of the dierential Lorentz force signals acting on the square magnet (a= 1cm) for dierent values of Rm. Calculations have been performed for long (" j") defect located atd= 2 mm using MDA. The magnet is located at lift-o distance z= 1 mm. In order to test the proposed DiLET methodology the LET benchmark problem presented in Fig. 5.1 (see section 5.1) has been solved using MDA. Thus, the cylindrical permanent magnet has been replaced by a cubic magnet with nite dimensions wm= lm=hm= 1 cm placed at lift-of distance z= 1 mm. All other geometrical and material parameters are described in Table 5.3. According to DiLET experiment which is currently in preparation (see Fig. 6.26), the total number of windings in each coil was xed toNx=Ny=Nz= 300. Furthermore, an additional parametric study assuming the wide range of testing velocities up to v11 m/s (Rm= 10) has been performed to estimate the working range of this specic DiLET implementation. Fig. ??and Fig. ?? show the dierential Lorentz force signals calculated directly by time dierentiation of (2.2), whereas Fig. ??and Fig. ??show the corresponding voltages induced in coils oriented along x-axis andz-axis. From the obtained results a very good correlation between the dierential Lorentz force signals and the corresponding voltages can be observed (Fig. ??). Due to the axial symmetry of the model y= 0 (see Fig. 5.11) the side component of the Lorentz force ( FS) is equal to zero. To compare the results quantitatively the so-called Pearson's correlation coecient (PCC) has been used [153]. PCC, here denoted as riwherei2fD;Lg, is dened as the covariance of 105 Chapter 6: Results and Discussion (a) Pearson's correlation coecient (b) Calibration coecients Figure 6.28: Comparison of dierential Lorentz force signals with respect to the induced voltages. Calculations have been performed for long (" j") defect located at d= 2 mm using MDA. The magnet is located at lift-o distance z= 1 mm. the two variables under comparison divided by the product of their standard deviations rD=cov@FD @t;VCx
@FD @t
(VCx)andrL=cov@FL @t;VCz
@FL @t
(VCz): (6.29) The value of PCC should be interpreted as follows. If ri= +1 then the two variables under comparison have perfect increasing linear correlation. If ri=1 the correspond- ing variables have perfect decreasing linear correlation [153]. As riapproaches to zero the variables are completely uncorrelated. Fig. 6.28a shows the PCC values for both dierential Lorentz force components as a function of Rm(testing velocity v). It can be observed that even for relatively large size of magnets ( w=l=h= 1 cm), the induced voltages are in very good correlation to the corresponding dierential Lorentz force signals. Moreover, in the testing range of up to Rm= 5 the calculated PCC was more than 0 :99. Since magnetisation of the magnet is considered to be along the z-axis (M=M
ez), the voltage induced in the z-axis oriented coil ( VCz) correlates well with the corresponding dierential lift force component in the whole range of Rm. The presented analysis conrms the validity of simplications assumed in order to derive (6.26) and (6.28). Nevertheless, to determine dierential force signals directly from voltages induced in the corresponding coils an additional calibration of the pro- posed dierential force sensor has to be performed. For this the P2P perturbations of the obtained dierential force signals have been compared directly with the cor- responding P2P perturbations of the induced voltages (see Fig. 6.28b). As directly follows from (6.26) - (6.28) the dependency between P2P values is linear. Moreover, due to the cubic shape of the magnet ( wm=lm=hm) and coils with the same number of windings ( Nx=Ny=Nc), the calibration curves for both force components are almost identical. 106 Chapter 7 Summary and Outlook In this chapter the major results from the present study are summarised and con- cluded. Additionally, suggestions for future work involving numerical and theoretical study of LET problems are presented. 7.1 Summary The non-destructive testing technique, namely Lorentz force eddy current testing (LET) was thoroughly investigated in this work. In particular, the focus of the work was on the theoretical and numerical analysis of the problem of motion of a defective conductor in the presence of a permanent magnet. The following objectives have been accomplished: A creeping magnet problem was considered to provide insights into the funda- mental principles of LET. This problem refers to the Faraday experiment of a magnet falling through an electrically conducting pipe. However, in this work, the motion of a free-falling permanent magnet in a pipe containing an idealized defect was considered. This problem represents a highly simplied yet enlight- ening version of Lorentz force eddy current testing. A combination of analytical theory, numerical simulation and experimental validation was used to investigate this problem. The analytical theory consists of the derivation and analysis of a non-linear ordinary dierential equation for the time-dependent position of the falling magnet. The analytical theory allowed a rigorous prediction of the relation between the width of the defect and the change in falling time. The numerical simulation helped to overcome three limitations inherent in the analytical theory and to take into account (i) the nite size of the falling magnet, (ii) the nite width of the pipe and (iii) the nite value of the magnetic Reynolds number de- scribing the deformation of the magnetic eld lines. The predictions were tested 107 Chapter 7: Summary and Outlook by a comparison with a series of experiments. It was observed that the dierence between the measured values of the change in falling time and the predictions of the numerical and analytical models is in the range between 0 :14% and 36 :15%, respectively. Therefore, it can be concluded that the theory properly captures the essence of Lorentz force eddy current testing in spite of its apparent simplicity. Unlike the creeping magnetic problem, a real LET problem deals with completely three-dimensional and arbitrary defects. This requires eective numerical simu- lations to solve 3D transient LET problems involving parts in relative motion. Therefore, a new logical expressions approach (LEA) was proposed that permits the fast analysis of moving eddy current problems on a xed computational grid. The proposed method provided accurate results for both small and large magnetic Reynolds numbers ( Rm). Depending on the type of the object in motion, two dierent LEA implementations were presented, namely, the moving magnet ap- proach (MMA) and the moving defect approach (MDA). The MMA is very time ecient, but it is restricted to simple magnet shapes. The MDA extends the applicability of the LE approach to complex magnet systems whilst maintaining their computational eciency and accuracy. To additionally reduce the simulation time several fast numerical models have been developed based on the certain model simplications. In the so-called low- Rmregime it has been shown that the instantaneous magnetic eld reaction to any change in eddy current distribution can be considered. This resulted in the quasi-static approach (QSA). If conductors are free of defects, with QSA only one stationary analysis can be performed, providing the steady Lorentz forces acting on the magnet. If the conductor contains a defect, QSA can only be applied to LET problems within the regime of low Rm, i.e.Rm<1. On the other hand, if the secondary magnetic eld associated with the eddy currents is low, compared to the primary magnetic eld, the QSA can be even further simplied. This is referred to as the weak reaction approach (WRA). It enables general 3D LET systems to be described by only one scalar electric potential, resulting in a considerable reduction of computational requirements. Nevertheless, the application range of the WRA is restricted to very low values of Rm(Rm1). All numerical results were veried with the sliding mesh technique (SMT), which requires the use of computationally expensive direct solvers. The results of the simulations also agreed well with the experiments, which additionally underlines the ability of the LET to detect deep lying defects with relatively high testing velocities. 108 Chapter 7: Summary and Outlook In order to understand the feasibility and testing capabilities of LET systems, the benchmark problem was dened and solved by the proposed numerical method- ology. In particular, parametric studies were performed to quantify the eect of various parameters on the total value of the Lorentz force acting on the magnet. Simulations were performed for two characteristic LET testing congurations, namely (i) a conductor without defects and (ii) a conductor with pre-dened material defects. In case of a non-defective conductor, the parametric studies have been used to quantify the eect of the lift-o distance, size and magnetiza- tion of the magnet, and the magnetic Reynolds number on the resulting Lorentz forces. It has been observed that all force components have the same dependency in terms of magnet size, its magnetization and lift-o distance. Furthermore, a linear dependency between the Lorentz force and the volume of the magnet has been observed. In case of magnetization intensity the dependency is quadratic. However, the dependency of the Lorentz force on the lift-o distance is non- monotonic. In particular, two regimes were identied. For low lift-o distances force decay exponentially with distance with small decay constant. By increas- ing the lift-o distance further, a regime where the force decays cubically has been observed. This corresponds to the actual decay of the magnetic eld of the permanent magnet itself. For a conductor with material defects identical studies were performed to determine the in uence of the given parameters on the Lorentz force. However, unlike in the case of a non-defective conductor, only Lorentz force perturbations were evaluated. Furthermore, the eect of the shape, size and position of the defect within the conductor on the characteristic Lorentz force perturbations were investigated. The analysis provided reference results to understand the feasibility and testing capabilities of LET. Another important parameter considered in the work was the magnetic Reynolds number,Rm. The dependency of individual force components on Rmis dier- ent. In the low- Rmregime it has been observed that the drag force is linearly dependent on Rm, whereas the corresponding lift force increases quadratically. This behaviour resulted, rather serendipitously, in a so-called Lorentz force sig- mometry (LoFoS). LoFoS is based on the lift-to-drag radio of the Lorentz force and enables contactless measurement of electrical conductivity of the conductor under test. The proposed technique LoFoS has the following advantages: (i) it is contactless, (ii) it can be applied continuously during production processes, (iii) it enables the user to measure the conductivity beyond the oxidised surface and (iv) the proposed data processing makes LoFoS resistant to changes in lift-o distances, velocity and strength of the magnetic eld source. LoFoS is suitable 109 Chapter 7: Summary and Outlook for specimen of any kind of physical condition. The only limitation is given by the minimal measurable Lorentz force components of the multi-component force sensor. To increase the testing range of LET, i.e. to detect defects deeper inside the material, the testing should be performed with low lift-o distances, large cubic magnets and higher Reynolds numbers. However, if the size of the magnet is larger than the size of the defect then the information content contained in the Lorentz force signals is reduced. In this case the shape of the force perturbation is merely an image of the magnet itself. In principle, this means that the solution of the inverse problem providing the identication of the defect would be very dicult or even impossible to perform. Therefore, the magnet size needs to be closely monitored and should be chosen based on the required testing resolution. Generation of high Lorentz force perturbations has a rather unfortunate side ef- fect of increasing the absolute force as well. In such cases, the absolute values of Lorentz force is a few orders of magnitudes larger than the perturbations caused by defects. Unfortunately, precise measurements of such small force variations in relatively large range of applied forces is very dicult. Therefore, there is a strong demand for dierential force measurements in LET systems. To this end, in this work a simple and low-cost modication to the initial LET conguration was proposed enabling three component dierential Lorenz force measurements. Such a technique, has been referred to as Dierential Lorentz Force Eddy Cur- rent Testing (DiLET). The proposed modication aects only the magnet system, whose outer surface was wound by three independent and passive pick-up coils. The principal idea is to use voltages induced in the additional coil system and correlate the obtained voltage signals with the corresponding dierential Lorentz force signals acting on the magnet. In fact, the resulting magnet system can be directly applied to the existing LET experimental setup as a complete dieren- tial Lorentz force sensor. Numerical results demonstrated that the correlation between the dierential Lorentz force and voltage induced in corresponding coils was more than 0 :99 forRm<5. The correlation of the force component in the direction of the magnetization of the magnet was independent of Rmin the given range (Rm<12). 110 Chapter 7: Summary and Outlook 7.2 Outlook Theoretical and numerical modelling of LET problems is fairly well investigated in this work. However, some important questions remain still unanswered. Therefore, in this section, the focus is on addressing these questions to provide a good foundation for any future work involving LET. To this end, the section is divided into two parts, namely (i) numerical study and (ii) theoretical study. 7.2.1 Numerical Study Suggestions for future numerical work concerning LET problems are as follows Extension of numerical analysis of LET benchmark problem by combining nite element and boundary element numerical procedures. This can help to addi- tionally reduce the computational costs. Furthermore, the modelling can be considerably simplied since only the conducting domain needs to be considered. Logical expression approach (LEA) employed in this work requires permanent control over the time-step size used in the numerical simulations. In the com- mercial code Comsol MultiphysicsR this was only possible by using the implicit generalized- transient solver. However, it was observed that this solver is rather sensitive to the a priori specied damping factor . Therefore, implementation of the LET problem (with and without LEA) in other commercial codes util- ising dierent transient solvers would increase the robustness of the numerical methodology. Verication of the numerical results has been obtained by applying the SMT to- gether with Lagrange multipliers. According to many numerical tests performed with the commercial FEM code Comsol MultiphysicsR , the resulting system of equations could only be solved by applying direct solvers. Another technique, namely the eld interpolation technique, enabled the use of iterative solvers as well. However, even the nite element grid on the interface was conforming, the obtained solution was numerically noisy. To nd out more about this error, it is advisable to solve the given LET problem by applying SMT and the interpolation method in other commercial codes. 111 Chapter 7: Summary and Outlook 7.2.2 Theoretical Study The suggestions for future theoretical work in LET problems are as follows In this work only one permanent magnet was included in the analysis. However, it was apparent from the parametric studies that in order to increase the testing capabilities of general LET systems, strong and highly localised magnetic eld distributions are required. Therefore, any future work could involve the investi- gation of complex magnet systems including an optimised array of magnets with ferromagnetic shielding materials or even superconductors. Most industrial applications employ ferromagnetic materials and therefore, there is a strong need to extend LET for testing such materials as well. However, such an endeavour creates additional challenges due to the non-linearity in magnetic permeability. In particular, extremely high reluctance forces between these ma- terials and the permanent magnets makes the actual measurement of Lorentz forces very dicult or even impossible. This problem could be overcome by DiLET methodology. This, however requires thorough numerical and experi- mental investigation of DiLET systems proposed in this work. 112 Appendix A To simplify the analysis of the general LET systems, the magnetic Reynolds number Rmhas been extensively used throughout the work. In [32] Rmis derived using the magnetic eld diusion-convection equation described only by the magnetic ux density B. Since, here the potential functions AandVhave been used instead, an equivalent derivation of Rmbased on [32] is performed. Assuming a linear and isotropic conducting material the magnetic eld diusion- convection equation given by (2.31) can be written as 1 0r (rA) =@A @t+r(v
A)vr A ; (A.1) where the electric scalar potential has been replaced by V=v
A[62]. Since Rmis used to describe the eect of the secondary magnetic eld only inside the conducting region, the external current density and the magnetisation of a permanent magnet are equal to zero ( je=0andM= 0). Based on the scales of length L, velocityjvj, timeL=jvjand vector potential jAj, which are characteristic for the considered conductor in motion, equation (A.1) can be written as follows 1 0L2jAj~r ~r ~A =1 LjAjjvj @~A @~t+~r(~v
~A)~v~r ~A! ;(A.2) where the non-dimensional quantities have been dened as r=1 L~r; (A.3) @ @t=jvj L@ @~t; (A.4) A=jAj~A; (A.5) v=jvj~v: (A.6) 113 Appendix A Using the given denition of the magnetic Reynolds number [32] Rm=0jvjL (A.7) the nial non-dimensional form of the magnetic eld diusion-convection equation (2.31) can be obtained ~r ~r ~A =Rm @~A @~t+~r(~v
~A)~v~r ~A! : (A.8) 114 Appendix B The MMA and SMT are dened in the xed frame of reference K (see Section 2.2 and Section 5.2). In this reference frame the governing equation of LET system voids the additional velocity term which accounts for the induced eddy currents inside the conductor under test. In K the eddy currents are actually described by j=@A=@t, which is strongly in uenced by the applied time-discretisation scheme. In order to provide direct control over the time-step size the generalized- transient solver has been applied in both, MMA and SMT implementations. However, it has been observed that the particular choice of the control parameter strongly aects the stationary values of the Lorentz force. In order to numerically estimate the in uence of , the stationary value of the Lorentz force, i.e. its step response has been calculated using MMA. Thus, the LET benchmark problem for conductor free of defects has been applied with all geometrical and material parameters given in Table 5.3. The cylindrical magnet was placed in the middle of the conductor, whose length Lwas considered to be long enough not to in uence the obtained solution. The results have been compared with respect to quasi-static solution (5.4) and (5.5), which has been extensively validated (a) Stationary value of the Lorentz force (b) Peak-to-peak (P2P) force perturbations Figure B.1: In uence of the damping parameter of the transient generalized- solver on the stationary value of Lorentz force ( F0 D;L) and on the resulting Lorentz force peak-to-peak perturbations (
FP2P D;L). The calculations have been performed for Rm= 1 (v= 1 m/s) and
X= 1 mm (dt= 0:5
103s) 115 Appendix B by experiments (see Section 6.1). From the obtained results, a very strong dependency of the stationary Lorentz force lift component F0 Lonhas been observed (Fig. B.1a). For the given LET parameters, the discrepancy of the lift force was up to 30% ( = 0) and it was monotonically reducing to 0:7% (= 0:95). The dependency of the corresponding drag force component F0 Dwas very weak ( <0:2%) in the whole range of applied. This is mostly because in the assumed velocity range the drag force component is almost two orders of magnitude larger than the corresponding lift force. However, for higher values of (>0:85) LET solution obtained by MMA and SMT tends to be unstable. This practically limited the application of the generalized-  solver to0:85. Fortunately, peak-to-peak (P2P) Lorentz force perturbations for both force components are independent on the used value of (Fig. B.1b). The actual quantication of the discrepancy between transient and quasi-static solution with respect to could not be performed in the given time. Nevertheless, taking into consideration all the observations listed above and Fig. B.1 simple and constant correction of the transient Lorentz force signals obtained by MMA and SMT can be used Fcorr D=FD
FD; (B.1) Fcorr L=FL
FL; (B.2) whereFcorr D;Lrepresents the corrected values of the Lorentz forces, FD;Lis the transient solution obtained by MMA or SMT and
FD;L=F0 D;LFQS D;Lis the constant correction factor (see Fig. B.1a). It should be mentioned that the proposed correction of the numerical results should only be performed when a direct comparison with experiments is required. Otherwise, since the characteristic perturbation of the Lorentz force is not in uenced by values of , the LET systems can be studied using unchanged MMA or SMT solutions. The MDA is dened in the moving frame of reference K' (see section 2.2). In this reference frame the induced eddy currents are described by the additional velocity term j=(vr A). Due to this, the transient LET solution obtained by the MDA was independent on particular values of damping parameter . 116 Bibliography [1] E. E. Brooks and A. W. Poyser, Magnetism and Electricity: A manual for Students in Advanced Classes , 2nd ed. Longmans, Green in London and New York, 1914. [2] K. Rankin, Electrical Instalations , 2nd ed. London : Caxton, 1909. 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Klein, Applied General Statistics . Prentice-Hall, 1968. 126 Erkl arung Ich versichere, dass ich die vorliegende Arbeit ohne unzul assige Hilfe Dritter und ohne Benutzung anderer als der angegebenen Hilfsmittel angefertigt habe. Die aus anderen Quellen direkt oder indirekt  ubernommenen Daten und Konzepte sind unter Angabe der Quellen gekennzeichnet. Bei der Auswahl und Auswertung folgenden Materials haben mir die nachstehend aufgef uhrten Personen in der jeweils beschriebenen Weise unent- geltlich oder entgeltlich geholfen. 1. Niemand. Explizit erw ahnen m ochte ich, dass einige Teile der Ergebnisse publiziert wurden (siehe Liste Publikationen) und entsprechende Abschnitte der Dissertation auf diesen Publika- tionen beruhen und nur wenig abgewandelt in der Arbeit  ubernommen wurden. Weitere Personen waren an der inhaltlich-materiellen Erstellung der vorliegenden Arbeit nicht beteiligt. Insbesondere habe ich hierf ur nicht die entgeltliche Hilfe von Vermittlungs- bzw. Beratungsdiensten (Promotionsberater oder anderer Personen) in Anspruch genom- men. Niemand hat von mir unmittelbar oder mittelbar geldwerte Leistungen f ur Ar- beiten erhalten, die im Zusammenhang mit den Inhalten der vorgelegten Dissertation stehen. Die Arbeit wurde bisher weder im In- noch Ausland in gleicher oder  ahnlicher Form einer Pr ufungsbeh orde vorgelegt. Ich bin darauf hingewiesen worden, dass die Unrichtigkeit der vorstehenden Erkl arung als T auschungsversuch bewertet wird und gem a x7 Abs. 10 der Promotionsordnung den Abbruch des Promotionsverfahrens zur Folge hat. Ilmenau, den 27.11.2012 Mladen Zec