**Chapter 1. Basic Theory of Distributed Point Source Method (DPSM) and its Application to Some Simple Problems (D. Placko and T. Kundu).** 1.1 Introduction and Historical Development of DPSM.

1.2 Basic Principles of DPSM Modeling.

1.2.1 The fundamental idea.

1.2.1.1 Basic equations.

1.2.1.2 Boundary conditions.

1.2.2 Example in the case of a magnetic open core sensor.

1.2.2.1 Governing equations and solution.

1.2.2.2 Solution of coupling equations.

1.2.2.3 Results and discussion.

1.3 Examples from Ultrasonic Transducer Modeling.

1.3.1 Justification of modeling a finite plane source by a distribution of point sources .

1.3.2 Planar piston transducer in a fluid.

1.3.2.1 Conventional surface integral technique.

1.3.2.2 Alternative distributed point source method (DPSM) for computing the ultrasonic field.

1.3.2.2.1 Matrix formulation.

1.3.2.3 Restrictions on rS for point source distribution.

1.3.3 Focused transducer in a homogeneous fluid.

1.3.4 Ultrasonic field in a non-homogeneous fluid in presence of an interface.

1.3.4.1 Pressure field computation in fluid 1 at point P.

1.3.4.2 Pressure field computation in fluid 2 at point Q.

1.3.5 DPSM technique for ultrasonic field modeling in non-homogeneous fluid.

1.3.5.1 Field computation in fluid 1.

1.3.5.1.1 Approximations in computing the field.

1.3.5.2 Field in fluid 2.

1.3.6 Ultrasonic field in presence of a scatterer.

1.3.7 Numerical results.

1.3.7.1 Ultrasonic field in a homogeneous fluid.

1.3.7.2 Ultrasonic field in a non-homogeneous fluid - DPSM technique.

1.3.7.3 Ultrasonic field in a non-homogeneous fluid - surface integral method.

1.3.7.4 Ultrasonic field in presence of a finite size scatterer.

References.

**Chapter 2. Advanced Theory of DPSM - Modeling Multi-Layered Medium and Inclusions of Arbitrary Shape (T. Kundu and D. Placko).**

2.1 Introduction.

2.2 Theory of Multi-Layered Medium Modeling.

2.2.1 Transducer faces not coinciding with any interface.

2.2.1.1 Source strength determination from boundary and interface conditions.

2.2.2 Transducer faces coinciding with the interface - Case 1: Transducer faces modeled separately.

2.2.2.1 Source strength determination from interface and boundary conditions.

2.2.2.2 Counting number of equations and number of unknowns.

2.2.3 Transducer faces coinciding with the interface - Case 2: Transducer faces are part of the interface.

2.2.3.1 Source strength determination from interface and boundary conditions.

2.2.4 Special case involving one interface and one transducer only.

2.3 Theory for Multi-layered Medium Considering the Interaction Effect on the Transducer Surface .

2.3.1 Source strength determination from interface conditions.

2.3.2 Counting number of equations and number of unknowns.

2.4 Interference between two Transducers: Step-by-Step Analysis of Multiple Reflection.

2.5 Scattering by an Inclusion of Arbitrary Shape.

2.6 Scattering by an Inclusion of Arbitrary Shape - An Alternative Approach.

2.7 Electric Field in a Multi-Layered Medium.

2.8 Ultrasonic Field in a Multi-Layered Fluid Medium.

2.8.1 Ultrasonic field developed in a three-layered medium.

2.8.2 Ultrasonic field developed in a four-layered fluid medium.

References.

**Chapter 3. Ultrasonic Modeling in Fluid Media (T. Kundu, R. Ahmad, N. Alnuaimi and D. Placko)**.

3.1 Introduction.

3.2 Primary and Secondary Sources.

3.3 Modeling Ultrasonic Transducers of Finite Dimension Immersed in a Homogeneous Fluid.

3.3.1 Numerical results - ultrasonic transducers of finite dimension immersed in fluid.

3.4 Modeling Ultrasonic Transducers of Finite Dimension Immersed in a Non-Homogeneous Fluid.

3.4.1 Obtaining the strengths of active and passive source layers.

3.4.1.1 Computation of the source strength vectors when multiple reflection between the transducer and the interface are ignored.

3.4.1.2 Computation of the source strength vectors considering the interaction effects between the transducer and the interface .

3.4.2 Numerical results - ultrasonic transducer immersed in non-homogeneous fluid.

3.5 Reflection at a Fluid-Solid Interface - Ignoring Multiple Reflections between the Transducer Surface and the Interface.

3.5.1 Numerical results for fluid-solid interface.

3.6 Modeling Ultrasonic Field in Presence of a Thin Scatterer of Finite Dimension.

3.7 Modeling Ultrasonic Field inside a Multi-Layered Fluid Medium.

3.8 Modeling Phased-Array Transducers Immersed in a Fluid.

3.8.1 Description and use of phased array transducers.

3.8.2 Theory of phased array transducer modeling.

3.8.3 Dynamic focusing and time lag determination.

3.8.4 Interaction between two transducers in a homogeneous fluid .

3.8.5 Numerical results for phased array transducer modeling.

3.8.5.1 Dynamic steering and focusing.

3.8.5.2 Interaction between two phased array transducers placed face to face.

Reference.

**Chapter 4. Advanced Applications of Distributed Point Source Method - Ultrasonic Field Modeling in Solid Media (S. Banerjee and T. Kundu).**

4.1 Introduction.

4.2 Calculation of Displacement and Stress Green’s Functions in Solids.

4.2.1 Point source excitation in a solid.

4.2.2 Calculation of displacement Green’s function.

4.2.3 Calculation of stress Green’s function.

4.3 Elemental Point Source in a Solid.

4.3.1 Displacement and stress Green’s functions.

4.3.2 Differentiation of displacement Green’s function with respect to x1, x2, x3.

4.3.3 Computation of displacements and stresses in the solid for multiple point sources.

4.3.4 Matrix representation.

4.4 Calculation of Pressure and Displacement Green’s Functions in the Fluid Adjacent to the Solid Half-Space.

4.4.1 Displacement and potential Green’s functions in the fluid.

4.4.2 Computation of displacement and pressure in the fluid.

4.4.3 Matrix representation.

4.5 Application 1: Ultrasonic Field Modeling near Fluid-Solid Interface [Banerjee et al. 2006].

4.5.1 Matrix formulation to calculate source strengths.

4.5.2 Boundary conditions.

4.5.3 Solution.

4.5.4 Numerical results on ultrasonic field modeling near fluid-solid interface.

4.6 Application 2: Ultrasonic Field Modeling in a Solid Plate [Banerjee and Kundu 2006a].

4.6.1 Ultrasonic field modeling in a homogeneous solid plate.

4.6.2 Matrix formulation to calculate source strengths.

4.6.3 Boundary and continuity conditions.

4.6.4 Solution.

4.6.5 Numerical results on ultrasonic field modeling in solid plates.

4.7 Application 3: Ultrasonic Fields in Solid Plates with Inclusion or Horizontal Cracks [Banerjee and Kundu 2006b].

4.7.1 Problem geometry.

4.7.2 Matrix formulation.

4.7.3 Boundary and continuity conditions.

4.7.4 Solution.

4.7.5 Numerical results on ultrasonic fields in solid plate with horizontal crack.

4.8 Application 4: Ultrasonic Field Modeling in Sinusoidally Corrugated Wave Guides [Banerjee and Kundu 2006c].

4.8.1 Theory.

4.8.2 Numerical results on ultrasonic fields in sinusoidal corrugated wave guides.

4.9 Calculation of Green’s Functions in Transversely Isotropic and Anisotropic Solids.

4.9.1 Governing differential equation for Green’s function calculation.

4.9.2 Radon transform.

4.9.3 Basic properties of Radon transform.

4.9.4 Displacement and stress Green’s functions.

References.

**Chapter 5. DPSM Formulation for Basic Magnetic Problems (N. Liebeaux and D. Placko).**

5.1 Introduction .

5.2 DPSM Formulation for Magnetic Problems.

5.2.1 The Biot-Savart law as a DPSM current source definition.

5.2.1.1 Wire of infinite length.

5.2.1.2 Current loop.

5.2.2 Current loops above a semi-infinite conductive target.

5.2.3 Current loops above a semi-infinite magnetic target.

5.2.4 Current loop circling a magnetic core.

5.2.4.1 Geometry.

5.2.4.2 DPSM formulation.

5.2.4.3 Results.

5.2.5 Finite Element Simulation - Comparisons.

5.3 Conclusion.

References.

**Chapter 6. Advanced Magnetodynamic and Electromagnetic Problems(D. Placko and N. Liebeaux).**

6.1 Introduction.

6.2 DPSM Formulation using Green’s Sources.

6.2.1 Green’s theory.

6.2.2 Green’s function in free homogeneous space.

6.3 Green’s Functions and DPSM Formulation.

6.3.1 Expressions of the magnetic and electric fields.

6.3.2 Boundary conditions.

6.4 Example of Application.

6.4.1 Target in aluminum (σ= 50 Ms/m), frequency = 1000 Hz.

6.4.2 Target in aluminum (σ= 50 Ms/m), frequency = 100 Hz, inclined excitation loop.

6.4.3 Dielectric target (εr = 5), frequency = 3 GHz, 10° tilted excitation loop.

6.5 Conclusion.

References.

**Chapter 7. Electrostatic Modeling and Basic Applications (G. Lissorgues, A. Cruau and D. Placko).**

7.1 Introduction.

7.2 Modeling by DPSM.

7.2.1 Digitalization of the problem.

7.2.2 DPSM meshing considerations.

7.2.3 Matrix formulation.

7.3 Solving the System.

7.3.1 Synthesizing electrostatic field and potential.

7.3.2 Capacitance calculation.

7.4 Examples Based on Parallel-Plate Capacitors.

7.4.1 Description.

7.4.2 Equations.

7.4.3 Results of simulation.

7.4.4 Gap-tuning variable capacitor.

7.4.5 Surface-tuning variable capacitor.

7.5 Summary.

References.

**Chapter 8. Advanced Electrostatic Problems: Multi-Layered Dielectric Medium and Masking Issues (G. Lissorgues, A. Cruau and D. Placko).**

8.1 Introduction.

8.2 Multi-Layered Systems.

8.3 Examples of Multi-Material Electrostatic Structure.

8.3.1 Parallel-plate capacitor with two dielectric layers.

8.3.2 Permittivity-tuning varactors.

8.4 Multi-Conductor Systems: Masking Issues.

8.4.1 Example of multi-conductor system.

References.

**Chapter 9. Basic Electromagnetic Problems (M. Lemistre and D. Placko).**

9.1 Introduction.

9.2 Theoretical Considerations.

9.2.1 Maxwell’s equations.

9.2.2 Radiation of dipoles.

9.2.2.1 Electromagnetic field radiated by a current distribution.

9.2.2.2 Electric dipole.

9.2.2.3 Magnetic dipole.

9.2.3 The surface impedance.

9.2.4 Diffraction by a circular aperture.

9.2.5 Eddy currents.

9.2.6 Polarization of dielectrics.

9.3 Principle of Electromagnetic Probe for NDE.

9.3.1 Application to dielectric materials.

9.3.2 Application to conductive materials.

9.3.2.1 Magnetic method.

9.3.2.2 Hybrid method.

9.4 Electromagnetic Method for Structural Health Monitoring Applications.

9.4.1 Generalities.

9.4.2 Hybrid method.

9.4.3 Electric method.

References.

**Chapter 10. Advanced Electromagnetic Problems with Industrial Applications (M. Lemistre and D. Placko).**

10.1 Introduction.

10.2 Modeling the Sources.

10.2.1 Generalities.

10.2.2 Primary source.

10.2.3 Boundary conditions.

10.3 Modeling a Defect Inside the Structure.

10.4 Solving the Inverse Problem.

10.5 Conclusion.

**Chapter 11. DPSM Beta Program User’s Manual (A. Cruau and D. Placko).**

11.1 Introduction.

11.2 Glossary.

11.3 Modeling Preparation.

11.4 Program Steps.

11.5 Conclusion.

Index.