Numerical modeling for NDT
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Numerical Modeling for Electromagnetic
Non-Destructive Evaluation

    A comprehensive approach to electromagnetic field modeling in relationton to non-destructive evaluation is presented for  the first time in this text. Its purpose is to allow students to model and represent fields that would not be solvable by any other means, in order to understand the various aspects of non-destructive testing of materials.
    The book begins reviewing electromagnetics in  general  before moving on to analytical methods ofcomputation.  This approach allowsthe contrast between analytical and numerical methods to be illustrated with the limitations of analytical methods being viewed as the starting point for numerical methods. The text then deals with the general  methods of finite differences and finite elements,  examines elliptical processes, introduces parabolic partial  differential equations and their applications in transient methods of testing, deals with  hyperbolic partial differential equations and their use in the modeling of wave propagation in the time domain and finally describes some numerical methods required for the application of  techniques previously outlined.
    Each chapter has a number of problems designed to review the material and expand upon it.  Most of the problems are simple
enough  to be  solved  as assignments,  but realistic enough to
allow extension  to testing geometries of interest to the reader.
    Consequently, this will be a valuable text for senior undergraduates and graduates of electrical engineering and engineering technology as well as researchers in NDE and electromagnetics. Practitioners of NDE and professional  engineers involved in design and production of  electromagnetic  devices who are  interested in expanding their understanding of  numerical solutions to NDE problems will also find this text useful.

Chapman and Hall, London, 1995, 511 pages
ISBN 0-412-46830-1

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   1 Elliptic, parabolicand hyperbolic processes 
   2 General approachesto solution of field problems 
   3 The analytic approach 
     3.1 Rangeof solution methods 
     3.2 Limitations 
   4 The numerical approachin NDT 
     4.1 Importance 
     4.2 Limitations 
   5 Numerical methods 
     5.1 Thefinite difference method (FDM) 
     5.2 Thefinite element method (FEM) 
     5.3 Boundaryelement methods(BEM) 
     5.4 CoupledFEM/BEM methods 
     5.5 Volumeintegral methods 

 1. The electromagnetic fieldequations
   1.1 Introduction 
   1.2 Maxwell's equationsin differential form 
   1.3 Maxwell's equationsin integral form 
   1.4 Constitutive relations 
   1.5 Electromagneticinterface conditions 
     1.5.1 Interfaceconditions for the electric field 
     1.5.2 Interfaceconditions for the magnetic field 
     1.5.3 Otherinterface conditions 
   1.6 Material properties 
     1.6.1 Conductivity 
     1.6.2 Permittivity 
     1.6.3 Permeability 
   1.7 Hysteresis 
   1.8 Magnetization 
     1.8.1 Minorhysteresis loops 
   1.9 Permanent magnets 
   1.10 The Poynting theorem 
     1.10.1The complex Poynting vector 
   1.11 Potential functions 
     1.11.1The electric scalar potential for static fields 
     1.11.2The magnetic scalar potential 
     1.11.3The magnetic vector potential 
     1.11.3The electric vector potential
   1.12 Gage condition 
   1.13 Field equationsin terms of potential functions 
     1.13.1Derivation for the magnetic vector potential 
     1.13.2Derivation for the electric vector potential 
   1.14 Derivation interms of scalar potentials 
     1.14.1Other scalar potential representations 
   1.15 Time-harmonicfields 
   1.16 Nonlinear fields 
   1.17 Plane waves andscattering 
     1.17.1The wave equation 
     1.17.2The time-harmonic wave equation 
     1.17.3The Helmholtz equations 
   1.18 Propagation ofwaves: plane waves 
   1.19 Propagation ofplane waves in lossymedia 
     1.19.1Losses in materials 
     1.19.2Propagation of waves in lossy dielectrics 
     1.19.3Propagation of waves in low loss dielectrics 
     1.19.4Propagation of waves in  conductors 
   1.20 Microwaves, waveguidesand resonantcavities 
     1.20.2Cavity resonators 
     1.20.3Energy relations in waveguides and cavity resonators 
   1.21 Skin depth 
     1.21.1Skin depth in planar materials 
     1.21.2Skin depth in tubularmaterials 
   1.22 Classificationof field equations 
   1.23 Problems 
   1.24 Bibliography 

2. Analytic methods of solution
   2.1 Introduction 
   2.2 Analytic methods 
     2.2.1 Rangeof solution methods 
     2.2.2 Rangeof solvable problems 
     2.2.3 Importanceof analyticsolutions for verification 
     2.2.4 Limitations 
   2.3 Separation of variables:solution toLaplace's equation 
     2.3.1 Separationof variables in Cartesian coordinates 
     2.3.2 Separationof variables in cylindrical coordinates 
   2.4 Example: skin effect 
     2.4.1 Skineffect in planar conductors 
     2.4.2 Skineffect in cylindrical conductors 
   2.5 Example: TM modesin a rectangular waveguide 
   2.6 Green's functionmethod 
     2.6.1 Example:fields and voltages in the remote field effect 
   2.7 Conformal mapping 
     2.7.1 General 
     2.7.2 TheSchwarz-Christoffel transformation 
   2.8 Other methods 
     2.8.1 Directintegration 
     2.8.2 Methodof images 
   2.9 Problems 
   2.10 Bibliography 

3. The finite difference method
   3.1 Introduction 
   3.2 The finite differenceapproximation 
     3.2.1 Finitedifference formulae: an intuitive approach 
     3.2.2 Finitedifference formulae: a systematic approach 
     3.2.3 Higherorder formulae 
     3.2.4 Forward,backward and central difference formulae 
   3.3 The finite differencegrid 
     3.3.1 Curvedboundaries 
   3.4 Explicit and implicitfinite difference methods 
     3.4.1 Implicitfinite difference method 
     3.4.2 Explicitfinite difference method 
   3.5 Finite differenceapproximation for time dependent equations 
   3.6 Inclusion of materialproperties 
   3.7 Problems 
   3.8 Bibliography

4. The finite element method
   4.1 Introduction 
   4.2 The finite elementapproximation 
     4.2.1 Arudimentary finite element approximation: the one
              dimensional Poisson's equation 
     4.2.2 Theanalytic solution 
     4.2.3 Approximatesolution 
     4.2.4 Collocation 
     4.2.5 Leastsquares approximation 
     4.2.6 Galerkin'smethod 
     4.2.7 Theenergy functional and variational methods 
   4.3 The finite elementmethod 
   4.4 The finite element 
     4.4.1 Triangularelements: direct derivation 
     4.4.2 Tetrahedralelements: direct derivation 
     4.4.3 Shapefunctions for eight-node isoparametric element 
     4.4.4 Shapefunctions for twenty-node isoparametric element 
   4.5 Finite elementformulation 
     4.5.1 Thefield equation 
     4.5.2 Choiceof finite elements 
     4.5.3 Functionalminimization 
     4.5.4 Finiteelement formulation with isoparametric elements 
     4.5.5 Assemblyof elemental matrices 
   4.6 The finite elementmesh 
     4.6.1 Eight-node,isoparametric element mesh generator 
     4.6.2 Shapefunctions 
     4.6.3 Meshassembly 
     4.6.4 Nodalpoint and element numbering 
     4.6.5 Reorderingof the mesh 
     4.6.6 Graphicrepresentationof the mesh 
     4.6.7 20-nodeisoparametric element mesh generator 
     4.6.8 Tetrahedralelement mesh generator 
   4.7 Two-dimensionalmesh generation 
   4.8 Pre-processingsoftware 
   4.9 Problems 
   4.10 Bibliography 

5. Elliptic partial differentialequations
   5.1 Introduction 
   5.2 The general ellipticpartial differential equation 
   5.3 Classes of problems 
     5.3.1 Staticelectric fields 
     5.3.2 Staticmagnetic fields 
     5.3.3 Quasi-staticfields: linear eddy currents 
     5.3.4 Thehomogeneous Helmholtz equation 
   5.4 Applications toNDT 
     5.4.1 Electrostatictesting 
     5.4.2 Magnetostatictesting 
     5.4.3 Sinusoidaleddy current testing 
     5.4.4 Testingin waveguides and resonant cavities 
   5.5 2-D, axisymmetric,and 3-D applications: differences 
     5.5.1 Two-dimensionalapplications 
     5.5.2 Threedimensional applications 
     5.5.3 Axisymmetricapplications 
   5.6 Bibliography 

6. Finite difference solutionof elliptic processes
   6.1 Introduction 
   6.2 Elliptic processes:applications in 2-D and 3-D electrostatics 
     6.2.1 Theproblem 
     6.2.2 Formulation 
     6.2.3 Boundaryand symmetry conditions 
     6.2.4 Solution 
     6.2.5 Presentationof results 
     6.2.6 Variationsin formulation 
   6.3 Magnetostatic applications 
     6.3.1 Thegoverning equation 
     6.3.2 Thefinite difference grid 
     6.3.3 Approximation 
     6.3.4 Solutionand results 
   6.4 Eddy Current applications 
     6.4.1 Governingequations 
     6.4.2 Othercalculations andextensions 
   6.5 Time-harmonic wavepropagation 
     6.5.1 Formulation 
     6.5.2 Discretization 
     6.5.3 Solution 
     6.5.4 Dielectricfilled cavity 
     6.5.5 Lossydielectric sample in a cavity 
   6.6 Nonlinear applications 
     6.6.1 Nonlinearequations for static fields 
     6.6.2 Themagnetization curve 
     6.6.3 Methodsof solution 
     6.6.4 Implementation 
   6.7 Problems 
   6.8 Bibliography 

7. Finite elementformulation
   7.1 Introduction 
   7.2 Choice of formulationsand finite elements (2-D and 3-D) 
   7.3 Formulation usingan energy functional: variational approach 
     7.3.1 Theequations 
     7.3.2 Theformulation 
     7.3.3 Thefinite element 
   7.4 Formulation usingGalerkin's method 
     7.4.1 Thefield equations 
     7.4.2 Approximatingand weighting functions 
   7.5 Examples: staticapplications 
     7.5.1 Electrostaticformulations 
     7.5.2 Poisson'sequation: the electric scalar potential 
     7.5.3 Laplace'sequation: calculation of current densities 
     7.5.4 Magnetostaticformulations 
   7.6 Examples: eddycurrent applications 
     7.6.1 Two-dimensionalapplications 
   7.7 Examples: axisymmetricapplications 
     7.7.1 Formulation 
     7.7.2 Modelingof absolute and differential eddy current probes 
     7.7.3 Theremote field effect 
   7.8 Examples: three-dimensionalapplications 
   7.9 Extensions andmodifications 
     7.9.1 Solutionof Helmholtz's equation 
     7.9.2 Solutionof Helmholtz's equation as a deterministic problem
     7.9.3 Nonlinearapplications
   7.10 Problems 
   7.11 Bibliography 

8. Boundary integral, volume integraland combined formulations
   8.1 Introduction 
   8.2 Boundary integralmethods 
   8.3 The method of moments:an intuitive approach 
   8.4 Integral equations 
     8.4.1 Boundaryelement methods 
     8.4.2 Volumeintegral equations 
     8.4.3 Finiteelement/boundary element formulation 
   8.5 Finite elementimplementation 
     8.5.1 Finiteelement approximation in ? 
     8.5.2 Boundaryelement approximation on S 
     8.5.3 Boundaryelement equations 
     8.5.4 Couplingof the finiteelement and boundary element matrices
     8.5.5 Example:scattering due to lossy dielectrics 
   8.6 Integral equationsfor static fields 
     8.6.1 Formulation 
     8.6.2 Methodof moments implementation 
     8.6.3 Formulationfor magnetostatic fields 
     8.6.4 Implementation 
   8.7 Problems 
   8.8 Bibliography 

9. Parabolic partial differentialequations
   9.1 Introduction 
   9.2 The general parabolicpartial differential equation 
   9.3 Transient finiteelement formulation 
     9.3.1 Governingequations 
     9.3.2 Approximationin space 
     9.3.3 Approximationin time 
     9.3.4 Example:transient field in a coil 
   9.4 Transient finitedifference formulation 
     9.4.1 Governingequations 
     9.4.2 Finitedifference formulae 
     9.4.3 Boundaryand interfaceconditions
     9.4.4 Initialconditions
     9.4.5 Solution 
   9.5 Three-dimensionalsolutions 
   9.6 Finite differencetime domain methods 
     9.6.1 Formulation:two-dimensional applications 
     9.6.2 Scaling 
     9.6.3 Boundaryconditions 
     9.6.4 Formulationfor axisymmetric geometries 
     9.6.5 Boundaryconditions inaxisymmetric geometries 
     9.6.6 Implementationof field equations: 2-D geometries 
     9.6.7 Implementationof radiation boundary condition: 2-D
     9.6.8 Implementationof field equations on interfaces: 2-D
     9.6.9 Implementationof field equations:axisymmetric 
   9.7 Examples 
     9.7.1 Twodimensional solutions 
     9.7.2 Axisymmetricsolutions 
   9.8 Problems 
   9.9 Bibliography 

10. Hyperbolic partial differentialequations 
   10.1 Introduction 
   10.2 The general hyperbolicpartial differential equation 
     10.2.1Finite difference modeling for standing waves 
   10.3 The finite differencetime domain method 
     10.3.1The field equations 
     10.3.2The scaled equations 
     10.3.3Boundary conditions 
     10.3.4Finite difference approximation 
     10.3.5Axisymmetric applications 
     10.3.6Three-dimensional applications 
   10.4 Examples 
     10.4.1Scattering from conductors 
     10.4.2Testing of composite materials 
   10.5 Problems 
   10.5 Bibliography 

11. Miscellaneousnumerical methods
   11.1 Introduction 
   11.2 Numerical integration 
     11.2.1Gaussian quadrature: Gauss-Legendre integration 
   11.3 Numerical differentiation 
   11.4 Solution of linearsystems of equations 
     11.4.1Solution of linear systems of equations: direct methods 
     11.4.2Solution of linear systems of equations: iterative methods 
     11.4.3Solution of linear systems of equations: other methods 
   11.5 Solution of nonlinearsystems of equations 
     11.5.1The Newton-Raphson method 
   11.6 Methods of solutionfor eigenvaluesand eigenvectors 
     11.6.1The Jacobi and Givenstransformations 
     11.6.2The QR and QZ methods 
   11.7 Insertion of Dirichletboundary conditions 
   11.8 Bibliography 

Subject index


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