Numerical Modeling for Electromagnetic Non-Destructive Evaluation, 1994

This text on numerical methods applied to the analysis of electromagnetic nondestructive testing (NOT) phenomena is the first in a series devoted to all aspects of engineering nondestructive evaluation. The timing of this series is most appropriate as many university engineering/physics faculties around the world, recognizing the industrial significance of the subject, are organizing new courses and programs with engineering NOE as a theme. Additional texts in the series will cover electromagnetics for engineering NOE, microwave NOT methods, ultrasonic testing, radiographic methods and signal processing for NOE. It is the intended purpose of the series to provide senior-graduate level coverage of the material suitable for university curricula and to be generally useful to those in industry with engineering degrees who wish to upgrade their NOE skills beyond those needed for certification. This dual purpose for the series reflects the very applied nature of NOE and the need to develop suitable texts capable of bridging the gap between research laboratory studies of NOE phenomena and the real world of certification and industrial applications. The reader might be tempted to question these assertions in light of the rather mathematical nature of this first text. However, the subject of numerical modeling is of critical importance to a thorough understanding of the field-defect interactions at the heart of all electromagnetic NOT phenomena.

1 Elliptic, parabolic and hyperbolic processes.- 2 General approaches to solution of field problems.- 3 The analytic approach.- 4 The numerical approach in NDT.- 5 Numerical methods.- 1. The electromagnetic field equations.- 1.1 Introduction.- 1.2 Maxwell’s equations in differential form.- 1.3 Maxwell’s equations in integral form.- 1.4 Constitutive relations.- 1.5 Electromagnetic interface conditions.- 1.6 Material properties.- 1.7 Hysteresis.- 1.8 Magnetization.- 1.9 Permanent magnets.- 1.10 The Poynting theorem.- 1.11 Potential functions.- 1.12 Gage condition.- 1.13 Field equations in terms of potential functions.- 1.14 Derivation in terms of scalar potentials.- 1.15 Time-harmonic fields.- 1.16 Nonlinear fields.- 1.17 Plane waves and scattering.- 1.18 Propagation of waves: plane waves.- 1.19 Propagation of plane waves in lossy media.- 1.20 Microwaves, waveguides and resonant cavities.- 1.21 Skin depth.- 1.22 Classification of field equations.- 1.23 Problems.- 1.24 Bibliography.- 2. Analytic methods of solution.- 2.1 Introduction.- 2.2 Analytic methods.- 2.3 Separation of variables: solution to Laplace’s equation.- 2.4 Example: skin effect.- 2.5 Example: TM modes in a rectangular waveguide.- 2.6 Green’s function method.- 2.7 Conformal mapping.- 2.8 Other methods.- 2.9 Problems.- 2.10 Bibliography.- 3. The finite difference method.- 3.1 Introduction.- 3.2 The finite difference approximation.- 3.3 The finite difference grid.- 3.4 Explicit and implicit finite difference methods.- 3.5 Finite difference approximation for time dependent equations.- 3.6 Inclusion of material properties.- 3.7 Problems.- 3.8 Bibliography.- 4. The finite element method.- 4.1 Introduction.- 4.2 The finite element approximation.- 4.3 The finite element method.- 4.4 The finite element.- 4.5Finite element formulation.- 4.6 The finite element mesh.- 4.7 Two-dimensional mesh generation.- 4.8 Pre-processing software.- 4.9 Problems.- 4.10 Bibliography.- 5. Elliptic partial differential equations.- 5.1 Introduction.- 5.2 The general elliptic partial differential equation.- 5.3 Classes of problems.- 5.4 Applications to NDT.- 5.5 2-D, axisymmetric, and 3-D applications: differences and similarities.- 5.6 Bibliography.- 6. Finite difference solution of elliptic processes.- 6.1 Introduction.- 6.2 Elliptic processes: applications in 2-D and 3-D electrostatics.- 6.3 Magnetostatic applications.- 6.4 Eddy Current applications.- 6.5 Time-harmonic wave propagation.- 6.6 Nonlinear applications.- 6.7 Problems.- 6.8 Bibliography.- 7. Finite element formulation.- 7.1 Introduction.- 7.2 Choice of formulations and finite elements (2-D and 3-D).- 7.3 Formulation using an energy functional: variational approach.- 7.4 Formulation using Galerkin’s method.- 7.5 Examples: static applications.- 7.6 Examples: eddy current applications.- 7.7 Examples: axisymmetric applications.- 7.8 Examples: three-dimensional applications.- 7.9 Extensions and modifications.- 7.10 Problems.- 7.11 Bibliography.- 8. Boundary integral, volume integral and combined formulations.- 8.1 Introduction.- 8.2 Boundary integral methods.- 8.3 The method of moments: an intuitive approach.- 8.4 Integral equations.- 8.5 Finite element implementation.- 8.6 Integral equations for static fields.- 8.7 Problems.- 8.8 Bibliography.- 9. Parabolic partial differential equations.- 9.1 Introduction.- 9.2 The general parabolic partial differential equation.- 9.3 Transient finite element formulation.- 9.4 Transient finite difference formulation.- 9.5 Three-dimensional solutions.- 9.6 Finite difference time domain methods.- 9.7Examples.- 9.8 Problems.- 9.9 Bibliography.- 10. Hyperbolic partial differential equations.- 10.1 Introduction.- 10.2 The general hyperbolic partial differential equation.- 10.3 The finite difference time domain method.- 10.4 Examples.- 10.5 Problems.- 10.5 Bibliography.- 11. Miscellaneous numerical methods.- 11.1 Introduction.- 11.2 Numerical integration.- 11.3 Numerical differentiation.- 11.4 Solution of linear systems of equations.- 11.5 Solution of nonlinear systems of equations.- 11.6 Methods of solution for eigenvalues and eigenvectors.- 11.7 Insertion of Dirichlet boundary conditions.- 11.8 Bibliography.