DPSM for Modeling Engineering Problems

DPSM FOR MODELING ENGINEERING PROBLEMS DOMINIQUE PLACKO AND TRIBIKRAM KUNDU WILEY-INTERSCIENCE A JOHN WILEY & SONS, IN...

Author: Dominique Placko | Tribikram Kundu

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DPSM FOR MODELING ENGINEERING PROBLEMS DOMINIQUE PLACKO AND TRIBIKRAM KUNDU

WILEY-INTERSCIENCE A JOHN WILEY & SONS, INC., PUBLICATION

DPSM FOR MODELING ENGINEERING PROBLEMS

DPSM FOR MODELING ENGINEERING PROBLEMS DOMINIQUE PLACKO AND TRIBIKRAM KUNDU

WILEY-INTERSCIENCE A JOHN WILEY & SONS, INC., PUBLICATION

Copyright ß 2007 by John Wiley & Sons, Inc. All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or comple teness of the contents of this book and speciﬁcally disclaim any implied warranties of merchantability or ﬁtness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional, where appropriate. Neither the publisher nor the author shall be liable for any loss of proﬁt or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com. Wiley Bicentennial Logo: Richard J. Paciﬁco Library of Congress Cataloging-in-Publication Data Placko, Dominique. DPSM for modeling engineering problems / by Dominique Placko and Tribikram Kundu. p. cm. ISBN 978-0-471-73314-0 (cloth) 1. Distributed point source method (Numerical analysis) 2. Engineering mathematics. 3. Ultrasonic waves–Mathematical models. 4. Electromagnetic devices–Design and construction–Mathematics. 5. Electrostatics–Mathematics. 6. Electromagnetism–Mathematical models. 7. Magnetism–Mathematical models. I. Kundu, T. (Tribikram) II. Title. TA347.D57P585 2007 620.001’51–dc22 2006038735

Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

We would like to dedicate this work to our wives: Monique and Nupur our daughters: Ame´lie, Anne-Laure, Ina, and Auni and our parents

CONTENTS

Preface

xv

Contributors

xix

Chapter 1 – Basic Theory of Distributed Point Source Method (DPSM) and Its Application to Some Simple Problems

1

D. Placko and T. Kundu

1.1 Introduction and Historical Development of DPSM, 1 1.2 Basic Principles of DPSM Modeling, 3 1.2.1 The fundamental idea, 3 1.2.1.1 Basic equations, 6 1.2.1.2 Boundary conditions, 8 1.2.2 Example in the case of a magnetic open core sensor, 9 1.2.2.1 Governing equations and solution, 9 1.2.2.2 Solution of coupling equations, 11 1.2.2.3 Results and discussion, 13 1.3 Examples From Ultrasonic Transducer Modeling, 16 1.3.1 Justiﬁcation of modeling a ﬁnite plane source by a distribution of point sources, 17 1.3.2 Planar piston transducer in a ﬂuid, 18 1.3.2.1 Conventional surface integral technique, 18 1.3.2.2 Alternative DPSM for computing the ultrasonic ﬁeld, 20 1.3.2.3 Restrictions on rs for point source distribution, 29 1.3.3 Focused transducer in a homogeneous ﬂuid, 31 vii

viii

CONTENTS

1.3.4 Ultrasonic ﬁeld in a nonhomogeneous ﬂuid in the presence of an interface, 32 1.3.4.1 Pressure ﬁeld computation in ﬂuid 1 at point P, 33 1.3.4.2 Pressure ﬁeld computation in ﬂuid 2 at point Q, 35 1.3.5 DPSM technique for ultrasonic ﬁeld modeling in nonhomogeneous ﬂuid, 38 1.3.5.1 Field computation in ﬂuid 1, 38 1.3.5.2 Field in ﬂuid 2, 42 1.3.6 Ultrasonic ﬁeld in the presence of a scatterer, 43 1.3.7 Numerical results, 45 1.3.7.1 Ultrasonic ﬁeld in a homogeneous ﬂuid, 45 1.3.7.2 Ultrasonic ﬁeld in a nonhomogeneous ﬂuid – DPSM technique, 50 1.3.7.3 Ultrasonic ﬁeld in a nonhomogeneous ﬂuid – surface integral method, 52 1.3.7.4 Ultrasonic ﬁeld in the presence of a ﬁnite-size scatterer, 53 References, 57 Chapter 2–Advanced Theory of DPSM—Modeling Multilayered Medium and Inclusions of Arbitrary Shape T. Kundu and D. Placko

2.1 Introduction, 59 2.2 Theory of Multilayered Medium Modeling, 60 2.2.1 Transducer faces not coinciding with any interface, 60 2.2.1.1 Source strength determination from boundary and interface conditions, 62 2.2.2 Transducer faces coinciding with the interface – case 1: transducer faces modeled separately, 64 2.2.2.1 Source strength determination from interface and boundary conditions, 65 2.2.2.2 Counting number of equations and number of unknowns, 68 2.2.3 Transducer faces coinciding with the interface – case 2: transducer faces are part of the interface, 68 2.2.3.1 Source strength determination from interface and boundary conditions, 69 2.2.4 Special case involving one interface and one transducer only, 71 2.3 Theory for Multilayered Medium Considering the Interaction Effect on the Transducer Surface, 76 2.3.1 Source strength determination from interface conditions, 78 2.3.2 Counting number of equations and number of unknowns, 80 2.4 Interference between Two Transducers: Step-by-Step Analysis of Multiple Reﬂection, 80 2.5 Scattering by an Inclusion of Arbitrary Shape, 83

59

CONTENTS

ix

2.6 Scattering by an Inclusion of Arbitrary Shape – An Alternative Approach, 85 2.7 Electric Field in a Multilayered Medium, 87 2.8 Ultrasonic Field in a Multilayered Fluid Medium, 91 2.8.1 Ultrasonic ﬁeld developed in a three-layered medium, 93 2.8.2 Ultrasonic ﬁeld developed in a four-layered ﬂuid medium, 94 Reference, 96 Chapter 3 – Ultrasonic Modeling in Fluid Media T. Kundu, R. Ahmad, N. Alnauaimi, and D. Placko

3.1 Introduction, 97 3.2 Primary (Active) and Secondary (Passive) Sources, 100 3.3 Modeling Ultrasonic Transducers of Finite Dimension Immersed in a Homogeneous Fluid, 100 3.3.1 Numerical results—ultrasonic transducers of ﬁnite dimension immersed in ﬂuid, 107 3.4 Modeling Ultrasonic Transducers of Finite Dimension Immersed in a Nonhomogeneous Fluid, 111 3.4.1 Obtaining the strengths of active and passive source layers, 112 3.4.1.1 Computation of the source strength vectors when multiple reﬂections between the transducer and the interface are ignored, 113 3.4.1.2 Computation of the source strength vectors considering the interaction effects between the transducer and the interface, 114 3.4.2 Numerical results—ultrasonic transducer immersed in nonhomogeneous ﬂuid, 116 3.5 Reﬂection at a Fluid–Solid Interface—Ignoring Multiple Reﬂections Between the Transducer Surface and the Interface, 117 3.5.1 Numerical results for ﬂuid–solid interface, 118 3.6 Modeling Ultrasonic Field in Presence of a Thin Scatterer of Finite Dimension, 118 3.7 Modeling Ultrasonic Field inside a Multilayered Fluid Medium, 120 3.8 Modeling Phased Array Transducers Immersed in a Fluid, 121 3.8.1 Description and use of phased array transducers, 121 3.8.2 Theory of phased array transducer modeling, 122 3.8.3 Dynamic focusing and time lag determination, 124 3.8.4 Interaction between two transducers in a homogeneous ﬂuid, 125 3.8.5 Numerical results for phased array transducer modeling, 126 3.8.5.1 Dynamic steering and focusing, 127 3.8.5.2 Interaction between two phased array transducers placed face to face, 129 3.9 Summary, 140 Reference, 141

97

x

CONTENTS

Chapter 4 – Advanced Applications of Distributed Point Source Method – Ultrasonic Field Modeling in Solid Media Sourav Banerjee and Tribikram Kundu

4.1 Introduction, 143 4.2 Calculation of Displacement and Stress Green’s Functions in Solids, 144 4.2.1 Point source excitation in a solid, 145 4.2.2 Calculation of displacement Green’s function, 147 4.2.3 Calculation of stress Green’s function, 148 4.3 Elemental Point Source in a Solid, 149 4.3.1 Displacement and stress Green’s functions, 150 4.3.2 Differentiation of displacement Green’s function with respect to x1 ; x2 ; x3 , 151 4.3.3 Computation of displacements and stresses in the solid for multiple point sources, 153 4.3.4 Matrix representation, 155 4.4 Calculation of Pressure and Displacement Green’s Functions in the Fluid Adjacent to the Solid Half Space, 157 4.4.1 Displacement and potential Green’s functions in the ﬂuid, 158 4.4.2 Computation of displacement and pressure in the ﬂuid, 159 4.4.3 Matrix representation, 161 4.5 Application 1: Ultrasonic Field Modeling Near Fluid–Solid Interface (Banerjee et al., 2007), 163 4.5.1 Matrix formulation to calculate source strengths, 164 4.5.2 Boundary conditions, 165 4.5.3 Solution, 165 4.5.4 Numerical results on ultrasonic ﬁeld modeling near ﬂuid–solid interface, 166 4.6 Application 2: Ultrasonic Field Modeling in a Solid Plate (Banerjee and Kundu, 2007), 180 4.6.1 Ultrasonic ﬁeld modeling in a homogeneous solid plate, 180 4.6.2 Matrix formulation to calculate source strengths, 181 4.6.3 Boundary and continuity conditions, 183 4.6.4 Solution, 185 4.6.5 Numerical results on ultrasonic ﬁeld modeling in solid plates, 185 4.7 Application 3: Ultrasonic Fields in Solid Plates with Inclusion or Horizontal Cracks (Banerjee and Kundu, 2007a, b), 198 4.7.1 Problem geometry, 198 4.7.2 Matrix formulation, 200 4.7.3 Boundary and continuity conditions, 201 4.7.4 Solution, 202 4.7.5 Numerical results on ultrasonic ﬁelds in solid plate with horizontal crack, 202 4.8 Application 4: Ultrasonic Field Modeling in Sinusoidally Corrugated Wave Guides (Banerjee and Kundu, 2006, 2006a), 204 4.8.1 Theory, 204

143

CONTENTS

xi

4.8.2 Numerical results on ultrasonic ﬁelds in sinusoidal corrugated wave guides, 210 4.9 Calculation of Green’s Functions in Transversely Isotropic and Anisotropic Solid, 218 4.9.1 Governing differential equation for Green’s function calculation, 218 4.9.2 Radon transform, 222 4.9.3 Basic properties of Radon transform, 223 4.9.4 Displacement and stress Green’s functions, 224 References, 225 Chapter 5 – DPSM Formulation for Basic Magnetic Problems

231

N. Liebeaux and D. Placko

5.1 Introduction, 231 5.2 DPSM Formulation for Magnetic Problems, 233 5.2.1 The Biot–Savart law as a DPSM current source deﬁnition, 233 5.2.1.1 Wire of inﬁnite length, 233 5.2.1.2 Current loop, 234 5.2.2 Current loops above a semi-inﬁnite conductive target, 235 5.2.3 Current loops above a semi-inﬁnite magnetic target, 236 5.2.4 Current loop circling a magnetic core, 237 5.2.4.1 Geometry, 237 5.2.4.2 DPSM formulation, 238 5.2.4.3 Results, 240 5.2.5 Finite elements simulation—comparisons, 241 5.3 Conclusion, 243 References, 244 Chapter 6 – Advanced Magnetodynamic and Electromagnetic Problems D. Placko and N. Liebeaux

6.1 Introduction, 247 6.2 DPSM Formulation Using Green’s Sources, 248 6.2.1 Green’s theory, 248 6.2.2 Green’s function in free homogeneous space, 249 6.3 Green’s Functions and DPSM Formulation, 249 6.3.1 Expressions of the magnetic and electric ﬁelds, 249 6.3.2 Boundary conditions, 253 6.4 Example of Application, 256 6.4.1 Target in aluminum ðs ¼ 50 Ms=mÞ, frequency ¼ 1000 Hz, 256 6.4.2 Target in aluminum ðs ¼ 50 Ms=mÞ, frequency ¼ 100 Hz, inclined excitation loop, 260 6.4.3 Dielectric target ðer ¼ 5Þ, frequency ¼ 3 GHz, 10 tilted excitation loop, 263 6.5 Conclusion, 270 References, 271

247

xii

CONTENTS

Chapter 7 – Electrostatic Modeling and Basic Applications

273

G. Lissorgues, A. Cruau, and D. Placko

7.1 Introduction, 273 7.2 Modeling by DPSM, 275 7.2.1 Digitalization of the problem, 275 7.2.2 DPSM meshing considerations, 276 7.2.3 Matrix formulation, 276 7.3 Solving the System, 278 7.3.1 Synthesizing electrostatic ﬁeld and potential, 278 7.3.2 Capacitance calculation, 279 7.4 Examples Based on Parallel-Plate Capacitors, 280 7.4.1 Description, 280 7.4.2 Equations, 281 7.4.3 Results of simulation, 282 7.4.4 Gap-tuning variable capacitor, 288 7.4.5 Surface-tuning variable capacitor, 288 7.5 Summary, 293 Reference, 293 Chapter 8 – Advanced Electrostatic Problems: Multilayered Dielectric Medium and Masking Issues

295

G. Lissorgues, A. Cruau, and D. Placko

8.1 Introduction, 295 8.2 Multilayered Systems, 296 8.3 Examples of Multimaterial Electrostatic Structure, 299 8.3.1 Parallel-plate capacitor with two dielectric layers, 299 8.3.2 Permittivity-tuning varactors, 301 8.4 Multiconductor Systems: Masking Issues, 302 8.5 Example of multiconductor system, 304 References, 305 Chapter 9 – Basic Electromagnetic Problems M. Lemistre and D. Placko

9.1 Introduction, 307 9.2 Theoretical Considerations, 308 9.2.1 Maxwell’s equations, 308 9.2.2 Radiation of dipoles, 308 9.2.2.1 Electromagnetic ﬁeld radiated by a current distribution, 308 9.2.2.2 Electric dipole, 309 9.2.2.3 Magnetic dipole, 310 9.2.3 The surface impedance, 311 9.2.4 Diffraction by a circular aperture, 314

307

CONTENTS

xiii

9.2.5 Eddy currents, 316 9.2.6 Polarization of dielectrics, 317 9.3 Principle of Electromagnetic Probe for NDE, 319 9.3.1 Application of dielectric materials, 319 9.3.2 Application to conductive materials, 320 9.3.2.1 Magnetic method, 320 9.3.2.2 Hybrid method, 323 9.4 Electromagnetic Method for Structural Health Monitoring (SHM) Applications, 327 9.4.1 Generalities, 327 9.4.2 Hybrid method, 327 9.4.3 Electric method, 330 References, 331 Chapter 10 – Advanced Electromagnetic Problems With Industrial Applications

333

M. Lemistre and D. Placko

10.1 Introduction, 333 10.2 Modeling the Sources, 334 10.2.1 Generalities, 334 10.2.2 Primary source, 335 10.2.3 Boundary conditions, 335 10.3 Modeling a Defect Inside the Structure, 339 10.4 Solving the Inverse Problem, 345 10.5 Conclusion, 347 References, 347 Chapter 11 – DPSM Beta Program User’s Manual

349

A. Cruau and D. Placko

11.1 Introduction, 349 11.2 Glossary, 350 11.2.1 Medium, 350 11.2.2 Object, 350 11.2.3 Interface, 351 11.2.4 Boundary conditions (BC), 351 11.2.5 Frontier, 351 11.2.6 Worksapce, 352 11.2.7 Scalar and vector physical values, 352 11.3 Modeling Preparation, 352 11.4 Program Steps, 352 11.5 Conclusion, 368 Index

369

PREFACE

Distributed point source method (DPSM) is a newly developed mesh-free numerical/ semianalytical technique that has been developed by the co-editors for solving a variety of engineering problems—ultrasonic, magnetic, electrostatic, electromagnetic, among others. This book gives the basic theory of this method and then solves the problems from different ﬁelds of engineering applications. In the last few decades, numerical methods such as the ﬁnite element method (FEM) and boundary element method (BEM) have been used for solving a variety of science and engineering problems. In the ﬁnite element technique, the entire problem geometry is discretized into a number of ﬁnite elements. In the boundary element technique, only the boundary of the problem geometry is discretized into boundary elements. In general, for complex geometries and inhomogeneous materials, FEM is comparatively a more efﬁcient technique. However, for homogeneous medium BEM is more efﬁcient; this is because it requires a fewer number of elements, distributed over the boundary only. Unlike in FEM or BEM in DPSM it is not necessary to discretize or mesh the problem geometry or its boundary. Instead, point sources are placed near the boundaries (not necessarily on the boundaries) and interfaces to build a model that can correctly take into account the radiation condition (reﬂection and/or transmission of ultrasonic waves or other types of energy) at these interfaces. Thus, DPSM is a mesh-free technique. DPSM can solve problems with both homogeneous and inhomogeneous media. For inhomogeneous medium two layers of sources on two sides of the interface are placed to model the ﬁeld on both sides of the interface. Because DPSM does not require the discretization of the entire problem geometry and only uses the point source solutions, it is generally much faster than the conventional numerical techniques. It needs the basic analytical solution for point sources; therefore, this method is xv

xvi

PREFACE

semianalytical. This method has been jointly developed by the two co-editors of this book through their collaborative research effort of several years that included their doctoral students in the United States and France. The co-editors have patented this technology; the worldwide patent has been published with international number W0 2004/044790 A1. They have also published a book chapter and a number of research papers on DPSM in scientiﬁc journals and conference proceedings. However, no book had been written on this subject hitherto. This book is the ﬁrst book on this technique; it describes the theory of DPSM in detail and covers its applications in ultrasonic, magnetic, electrostatic, and electromagnetic problems in engineering. The book chapter* on DPSM that is available in the current literature covers only one application—the ultrasonic modeling. Its other applications, such as for solving magnetic, electrostatic, and electromagnetic problems in engineering, have been published mostly in research papers over the past few years. For the convenience of the users, the detailed theory of DPSM and its applications in different engineering ﬁelds are published here in one book. The engineers and scientists can read this book to acquire a uniﬁed knowledge on DPSM. In Chapter 1, the authors present the basic principles of the method. First, the modeling of a transducer or a sensor in a free space is presented, starting from an example of U-shaped magnetic sensor with high-permeability core. Then, the method is extended to the problem geometry with one interface, assuming only one reﬂection at the interface. This is illustrated through some examples extracted from ultrasonic transducer modeling. In this chapter different DPSM source conﬁgurations are presented that include controlled-space radiation or controlled radiation source points and triplets. In Chapter 2, the advanced theory of DPSM is discussed: Multilayer, multisensor problems and the problem of scattering by inclusions of arbitrary shape are discussed. Some sample solutions are presented from the electric and ultrasonic ﬁelds that validate the theory presented in this chapter. In Chapter 3, the DPSM theory relevant to the ultrasonic problem modeling in the ﬂuid medium is presented, starting with the basic equations of ultrasonic problems. This chapter is written in such a manner that if a person wants to know the ultrasonic applications only and is not necessarily interested in the general theory of DPSM, he/ she can read this chapter skipping the ﬁrst two and still understand the materials. However, while developing the theory and going through the example problems, several references to Chapters 1 and 2 are made, and the readers can go back to the ﬁrst two chapters and read the speciﬁc sections of those chapters referred to in Chapter 3. Ultrasonic modeling in solid materials is much more complex. In the ﬂuid medium, only the ﬂuid pressure and the particle velocity normal to an interface is of interest. Boundary and interface conditions are deﬁned only on these two parameters, and only one kind of wave, the compressional wave or P-wave, can be present in a ﬂuid medium. On the contrary, in a solid medium both compressional (P) and * Placko, D., and T., Kundu, Chapter 2: Modeling of Ultrasonic Field by Distributed Point Source Method, in Ultrasonic Nondestructive Evaluation: Engineering and Biological Material Characterization, Ed. T. Kundu, Pub. CRC Press, pp. 143–202, 2004 ISBN 0-8493-1462-3.

PREFACE

xvii

shear (S) waves can be present, and across an interface three components of displacement and three components of stress must be continuous. Thus, the number of continuity conditions across an interface increases from two (for ﬂuid) to six (for solid). Similarly, the number of boundary conditions increases from one (for ﬂuid) to three (for solid) and mode conversion of waves from P to S and vice versa at the interface and boundary can occur for solids. For all these reasons the DPSM modeling of ultrasonic problems in the solid medium is much more complex. This problem is discussed in Chapter 4. Solutions of several example problems involving solid specimens are presented in this chapter along with the relevant theory. In Chapter 5, basic magnetic problems are presented: current elements and current loops, current loops above a semi-inﬁnite conductive target, current loops above a semi-inﬁnite magnetic target, current loops and magnetic cores, current loop circling a magnetic core, and current loop exciting a U-shaped magnetic core. These points are illustrated through some examples, and a comparison with mastered modeling techniques (ﬁnite elements) is presented. In this chapter, DPSM elemental current sources are introduced and validated for generating magnetic ﬁeld in a given subspace, under some particular assumptions such as inﬁnite conductivity for the targets. In Chapter 6, magnetodynamic and electromagnetic modeling is presented. As discussed Chapter 5, DPSM sources are elemental current sources but radiating now in a given medium with ﬁnite conductivity, permeability, and permittivity. So, instead of using simpliﬁed Green’s functions, we now use the general formulation, which helps to solve complex problems such as eddy current problems, illustrated in this chapter by some examples of different media in which an incident ﬁeld is created by a winding driven by an AC current, for a wide range of frequencies (100 Hz–3 Ghz). Then, the DPSM electromagnetic formulation allows, without heavy complexity, to obtain relevant modeling for problems of detection and Nondestructive Evaluation, such as crack detection in conductive materials with eddy current techniques, particularly used in industry. Some examples of this ability are given in Chapter 10, with modeling of small cracks in carbon composite plates. It is interesting to point out the problem in dependent universal nature of the DPSM, in which the boundary and interface conditions are fulﬁlled, in various applications, by taking care of the continuity of the potential and its ﬁrst derivative along the normal direction to the interface. For electromagnetic problems, we may note that the potential becomes a vector and the number of continuity conditions across an interface increases from two (for magnetostatic or electrostatic problems in which the potential is a scalar) to six, like in the ultrasonic problems solved in Chapter 4. DPSM automatically adapts the number of point sources required on each side of the interface to the corresponding number of equations in order to satisfy the boundary conditions in all cases. In Chapter 7, basic electrostatic problems are discussed. Through an introduction based on parallel-plate structures, the DPSM technique using electrostatic points are applied to the modeling of standard parallel-plate air-gap capacitors. In chapter 8, advanced electrostatic problems considering dielectric media or more complex structures with multiple electrodes and masking issues are studied with DPSM. MEMS (Micro-electro-mechanical systems) tunable capacitors are used to demonstrate that this method can solve engineering problems of interest.

xviii

PREFACE

Chapters 9 and 10 are devoted to Nondestructive evaluation (NDE) electromagnetic applications, mainly in the aerospace domain. Then, advanced electromagnetic techniques and devices for industrial applications are presented: modeling one emitter, modeling multiple emitters, and applications to electromagnetic NDE, with some examples illustrated by the principle of hybrid electromagnetic probes for NDE of carbon–ﬁber plates. These chapters also present a speciﬁc sensor for structural health monitoring: the HELP-Layer, with emphasis on the DPSM modeling of the detection of a crack, in Chapter 10. Chapter 11 gives the user’s Manual for the DPSM-based computer program for those readers who are interested in running the computer code.

D. Placko and T. Kundu

CONTRIBUTORS

R. Ahmad

M. Lemistre

University of Arizona USA

E´cole Normale Supe´rieure Cachan, France

N. Alnuaimi

N. Liebeaux

University of Arizona USA

Institut Universitaire de Technologie Cachan, France

Sourav Banerjee University of Arizona USA

A. Cruau Ecole Normale Supe´rieure Cachan, France

G. Lissorgues ESIEE, Noisy le Grand France

D. Placko Ecole Normale Superieure Cachan, France

T. Kundu University of Arizona USA

xix

1 BASIC THEORY OF DISTRIBUTED POINT SOURCE METHOD (DPSM) AND ITS APPLICATION TO SOME SIMPLE PROBLEMS D. Placko Ecole Normale Superieure, Cachan, France

T. Kundu University of Arizona, USA

1.1 INTRODUCTION AND HISTORICAL DEVELOPMENT OF DPSM In this chapter, the historical evolution of distributed point source method (DPSM) and its basic principles are presented. First, the magnetic ﬁeld generated by a magnetic transducer/sensor in a free space is obtained. A U-shaped magnetic sensor with high-permeability core is ﬁrst modeled (Placko and Kundu, 2001). Then, the method is extended to problem geometries with one interface (Placko et al., 2001, 2002). The source of the ﬁeld is denoted as ‘‘transducer’’ or ‘‘sensor,’’ and the interface between two media is sometimes called ‘‘target.’’ Observation points that are not necessarily on the interface are also called ‘‘target points.’’ Figure 1.1 shows the relative orientations of the transducer and interface. The interface or target can be an inﬁnite plane or it can have a ﬁnite dimension, acting as a ﬁnite scatterer. Only one reﬂection by the target surface is ﬁrst considered. The method is illustrated through some examples from electromagnetic and ultrasonic applications. In this chapter, different DPSM source

DPSM for Modeling Engineering Problems, Edited by Dominique Placko and Tribikram Kundu Copyright # 2007 John Wiley & Sons, Inc.

1

2

BASIC THEORY OF DISTRIBUTED POINT SOURCE METHOD (DPSM)

C A Medium 2

Medium 1

B D AB = Source or transducer or sensor CD = Interface or target

Figure 1.1 Problem geometry with interface.

conﬁgurations are considered including controlled-space radiation (CSR) sources and triplets, as it can be seen in the patent (Placko et al., 2002). DPSM modeling is based on a spatial distribution of point sources and can be applied to both two-dimensional (2D) and three-dimensional (3D) problem geometries. Mostly, 3D modeling is presented in this book for magnetic, acoustic, electrostatic, and electromagnetic ﬁeld problems. In the DPSM modeling technique the transducer surface and interface are replaced by a distribution of point sources, as shown in Figure 1.2a. One layer of sources is introduced near the transducer and a second layer near the interface. Point sources that model the transducer are called the ‘‘active’’ sources and those near the interface are called the ‘‘passive’’ sources. It should be noted here that a transducer generates a ﬁeld and an interface alters that ﬁeld by introducing reﬂected, transmitted, and scattered ﬁelds. If the interface is removed, the active point sources should still be present. However, if the active sources are turned off, then the passive point sources must be turned off as well because in the absence of active sources, the passive sources do not exist. Active and passive point sources can be distributed very close to the transducer face and interface, respectively, as shown in Figure 1.2a or away from them as shown in Figure 1.2b. It is also not

C

C A

Medium 2

A Medium 2

Medium 1

Medium 1

B

B (a)

D

D (b)

Figure 1.2 Synthesizing the ﬁeld by placing point sources: (a) close to the sensor and interface, (b) away from the sensor and interface.

1.2 BASIC PRINCIPLES OF DPSM MODELING

3

necessary for the layer of point sources to be parallel to the surface (transducer or interface) that is modeled by these sources. Strengths of the point sources are adjusted such that the boundary conditions on the transducer surface and continuity conditions across the interface are satisﬁed. This can be achieved by inverting some matrices. By adjusting the point source strengths, the total ﬁeld can be correctly modeled by different layers of point sources placed in different orientations. Naturally, for different orientations of the point sources, individual source strength vectors should be different. The total ﬁeld is computed by adding ﬁelds generated by all active and passive point sources. Note that unlike the boundary element or ﬁnite element techniques, in this formulation the discretization of the problem boundary or of the problem domain is not necessary. Like other numerical modeling schemes, accuracy of the computation depends on the number of point sources considered. This process of introducing a number of point sources can be called ‘‘mesh generation.’’ In this chapter, we study the effect of the spacing between two neighboring point sources on the accuracy of the ﬁeld computation and the optimum spacing for accurate numerical computation. It is shown here that for accurately modeling acoustic ﬁelds, the spacing between two neighboring point sources should be less than the acoustic wavelength (in fact, as we will see later, this condition has to be fulﬁlled for all kinds of waves, but the proof is given for the acoustic wave modeling). This restriction can be relaxed if we are interested in computing the ﬁeld far away from the point source locations. For example, if one is interested in computing the ﬁeld generated by a circular sensor of ﬁnite dimension in a homogeneous medium, the point source spacing must be a fraction of the wavelength if one is interested in computing the ﬁeld accurately adjacent to the transducer face. However, at a larger distance the ﬁeld can be computed accurately by considering fewer point sources of higher strength although it will not give good results near the transducer. Flat transducers or sensors with circular and rectangular cross-sections as well as point-focused concave transducers are modeled accurately by taking appropriate source spacing and are presented in this chapter. Figures 1.3–1.5 show the steps of DPSM evolution, improvements in elemental source modeling, and different problems that have been solved so far by this technique (Placko, 1984, 1990; Placko and Kundu, 2001, 2004; Placko et al., 1985, 1989, 2002; Ahmad et al., 2003, 2005; Dufour and Placko, 1996; Lee et al., 2002; Lemistre and Placko, 2004; Banerjee et al., 2006).

1.2 BASIC PRINCIPLES OF DPSM MODELING 1.2.1 The fundamental idea In this subsection, we ﬁrst describe the basic principle of this method, which is based on the idea of using multiple point sources distributed over the active part of a sensor or an interface. Active sources synthesize the transducer-generated signals in

4

BASIC THEORY OF DISTRIBUTED POINT SOURCE METHOD (DPSM)

Global formulation of multiple interfaces problems

Generalized multiple interface problems

Solved problems

General purpose interfaces

Diffraction problems

Sensors with target

Free space 3D sensors

Particular magnetic cases Dipolar models First approach to analytical 3D sensor problems

Virtual interface meshing

DPSM synthetis of interfaces and boundary conditions

First DPSM formulation for active surfaces

Extended electrical image method

Multiple reflections between 2 objects

June 2004

Oct 2003

June 2002

June 2001

May 2000

May 1990

April 1984

DPSM formulation improvement

Figure 1.3 DPSM birth and evolution.

Solved problems Global source & multilayered problems June 2004 Elemental current sources triplets

Spatial controlled radiation sources (SCR sources)

Electrostatic points and triplets

Magnetic triplets

March 2004

Oct 2003 June 2001 Multiple magnetic point sources (first DPSM approach) May 2000 Magnetic elemental point sources April 1984

Ultrasonic triplets June 2001 Multiple ultrasonic points sources June 2000 DPSM elemental sources improvement

Figure 1.4 DPSM source improvement.

Oct 2003

5

1.2 BASIC PRINCIPLES OF DPSM MODELING

Applications

Ultrasonic multilayered plate inspection

Eddy current tomography

June 2005

June 2004 Elemental current pipe detection

Electrostatic mems Oct 2003 Ultrasonic scattering problems Inductive sensors with target June 2001 Magnetic 3D sensors May 2000 Magnetic dipoles

June 2002 Ultrasonic sensors with target

May 2004

Help layer for structural health monitoring

Ultrasonic phased array sensors May 2004

March 2004

June 2001 Ultrasonic 3D sensors June 2000

April 1984

DPSM global progress

Figure 1.5 Different problems solved by DPSM.

a homogeneous medium, whereas the passive point sources distributed along the interface generate signals to model the reﬂection and transmission ﬁelds. For a ﬁnite interface the passive sources also model the scattered ﬁeld. Because the distributed point sources model the total ﬁeld, we call this method the ‘‘distributed point source method’’ or DPSM. It should be mentioned here that this technique is based on the analytical solutions of basic point source problems. Therefore, it can be considered as a semi-analytical technique for solving sensor problems that include magnetic, ultrasonic, and electrostatic sensors. For example, it is possible to compute the magnetic ﬁeld emitted by the open magnetic core of an eddy current sensor, or acoustic pressure in front of an ultrasonic transducer without discretizing the space by a large number of 3D ﬁnite elements. Magnetic and ultrasonic sensor examples are presented in this chapter to illustrate the method because these problems have some interesting properties as discussed later. It should be noted here that for a magnetic sensor, the magnetic potential remains constant on the sensor surface and the magnetic ﬂux varies from point to point, whereas for the acoustic sensor in a ﬂuid, the particle velocity remains constant on the sensor surface and the acoustic pressure varies. It requires an additional matrix inversion in the magnetic ﬁeld modeling, which is not necessary for the acoustic ﬁeld modeling. An elemental point source is shown in Figure 1.6. In a nonconductive medium, it involves both scalar potential and vector ﬁeld, the ﬁeld being proportional to the gradient of the potential. Each source is surrounded by a surface (‘‘bubble’’) on which

6

BASIC THEORY OF DISTRIBUTED POINT SOURCE METHOD (DPSM)

Potential contour lines Radiated field

Point source

Boundary condition bubble

Figure 1.6 Elemental point source.

the boundary conditions are applied. Because the boundary conditions are speciﬁed on the sensor surface for active point sources and on the interface surface for the passive point sources, the bubble surface should touch those surfaces such that the transducer surface or interface are tangents to the surface. Therefore, the point sources at the centers of the bubbles cannot be located on the transducer surface or on the interface. Reason for this restriction will be discussed later. 1.2.1.1 Basic equations The basic principle of the DPSM is illustrated in Figure 1.7. The implementation of the model simply requires the replacement of the active surface of the transducer by an array of point sources, so that the initial

Synthesized field

Array of point sources

Actual field

Transducer

Figure 1.7 Equivalent source radiation.

1.2 BASIC PRINCIPLES OF DPSM MODELING

7

Figure 1.8 Illustration of the controlled-space radiation source properties.

complexity associated with a complex ﬁnite shape of the transducer is changed into a superposition of elementary point source problems. One way of replacing the surface by an array of point sources is discussed below. The active surface of the transducer is discretized into a ﬁnite number of elementary surfaces dS, a point source is placed at the centroid of every elemental surface. The source strength and the radiation area of the sources are controlled. Unlike ordinary point sources, the sources used in DPSM do not necessarily radiate energy in all directions. For this reason these sources can be called CSR sources. For example, a source can be deﬁned to radiate only in the bottom or top half space, or right or left half space (see Fig. 1.8). In the generic derivation, symbols y and j are used to represent different parameters for different engineering problems as described below. For magnetic sensors, y and j represent the scalar magnetic potential and the ﬂux /m0 of the magnetic induction ðHÞ, respectively. For ultrasonic transducers, y and j represent the acoustic pressure P and the ﬂux of the particle displacement ðxÞ, respectively. Note that the particle velocity v ¼ dx dt. For electrostatic systems, y and j represent the scalar magnetic potential V and the ﬂux Q=e0 of the electric ﬁeld ðEÞ. The interaction function that relates the ﬁeld generated by the unit source (such as the elemental charge for electrostatic problems) to y is denoted by f. Table 1.1 shows the fundamental equations in different ﬁelds of engineering. It should be mentioned here that it is possible to obtain similar equivalent equations for problems from other ﬁelds of engineering such as thermal problems, for example. Nevertheless, it will be shown later that for electromagnetic waves the situation is slightly different because the sources in this case are elemental vectors of current, and in addition, the potential is often a vector and not a scalar, due to eddy currents generated in conductive media. It is interesting to note that the energy (or the power) radiated by such a system is the product of a scalar quantity and the ﬂux of a vector (or the time derivative of the ﬂux, for power). Let us denote the scalar quantity by yk and the ﬂux emitted by the point source k by jk. Figure 1.9 shows how the total ﬁeld at a given point is computed by adding ﬁelds generated by all the point sources. It also shows that because of

8

BASIC THEORY OF DISTRIBUTED POINT SOURCE METHOD (DPSM)

TABLE 1.1

Some physical values in DPSM modeling Surface energy

Surface power

Function f

! ~ E ¼ gradðVÞ

ÐÐ

~ ¼Q ~ E dS e0

dQS dD V ¼ V dt dt

1 Qs V 2

eiðkf rÞ 2pe0 r

! ~ ¼ grad Magnetostatic H ðÞ

ÐÐ

~ ¼ ~ dS H m0

ds dB ¼ dt dt

1 s 2

eiðkf rÞ 2pm0 r

~ ¼ ~ x dS

dx P dt

1 xP 2

iorn0 ikf r e 2pr

Electrostatic

d~ v F~v ¼ k ~ v¼r dt ! ¼ gradðPÞ

Ultrasonic

ÐÐ

[r is the distance of the point source, Kf is the wave number; for electrostatic systems, y is the scalar magnetic potential V; j is the ﬂux Q=e0 of the electric ﬁeld (E); for magnetic sensors, y is the scalar magnetic potential , j is the =m0 of the the magnetic induction (H); for ultrasonic problems, r is the ﬂuid density, P is the pressure, x is the particle displacement, is the ﬂux of particle displacement, v0 is the transducer velocity, o is the signal frequency.]

rotating symmetry, the elemental surface dS can be changed into a hemispherical surface dS with radius rS ðdS ¼ 2prS2 Þ. 1.2.1.2 Boundary conditions One needs to introduce the boundary conditions before solving the problem. For computing the values of the ﬂux jk for N sources, one needs N number of equations. These equations are obtained by introducing H Hs2 Hs3 M Hs1

rs1 rs3

rs2

r

Figure 1.9 Equivalent surface discretization.

1.2 BASIC PRINCIPLES OF DPSM MODELING

9

boundary conditions on the scalar quantity yk at N given points. One possible choice is to place these speciﬁc points (denoted by Pk ) at the apex of the hemispherical surfaces (then rS is normal to the surface). Clearly, greater is the number of points, smaller is the value of radius rS . Therefore, when N tends to inﬁnity, rS tends to 0, and Pk points tend to reach the surface on which the point sources are placed. Let us now illustrate this technique to model the magnetic ﬁeld generated by a magnetic sensor. 1.2.2 Example in the case of a magnetic open core sensor Under assumptions of very high permeability core, the implementation of the model simply requires discretization of the active surface (magnetic poles) of the core to obtain an array of point sources. Let us denote the scalar quantity by yk and the ﬂux emitted by the source k by jk. In the application described in this section, yk and jk represent the magnetic scalar potential and the ﬂux of magnetic induction, respectively. 1.2.2.1 Governing equations and solution Magnetic ﬁelds emitted by open magnetic cores (electrical motors, magnetic and eddy current sensors) are modeled in this section. Solving such problems without any approximation will be very difﬁcult in this domain of electromagnetic modeling. A magnetic sensor with a ‘U’-shaped open magnetic core (see Fig. 1.10) is considered as an illustrative example. Let us assume that the active part of a sensor is composed of two ‘‘poles’’—north and south poles of the magnetic transducer, see Figure 1.11.

I exc

N spires

Figure 1.10 Geometry of the magnetic sensor studied.

10

BASIC THEORY OF DISTRIBUTED POINT SOURCE METHOD (DPSM)

Hn1

M

Hn2

H

Hs2 Hs1

Rn2

Rs2 Rs1

Rn1

r r

South pole

North pole

Figure 1.11 Discretization of magnetic sensor poles.

The active surfaces are discretized into elementary surfaces dS. At the center of the ith elemental surface dS, a point source emitting a ﬂux ji is placed. From the conservation of the magnetic ﬂux, one can write that the summation of the ﬂux emitted by all point sources is equal to zero. jN þ jS ¼

X

X

jNi þ

pole N

jSi ¼ 0

ð1:1Þ

pole S

The magnetic potential at a given point M in the space is obtained by considering the contribution of all magnetic point sources. y¼

X

yi þ

pole N

X

ð1:2Þ

yj

pole S

General relations between the magnetic ﬁeld and the scalar magnetic potential are given by ~ ¼ dy H d~ r

ðð and

j¼

~ ~ dS m0 H

ð1:3Þ

yi ¼ ji 2pm0 r

ð1:4Þ

It yields in our case, for the ith point source ji ¼ Hi 2pm0 r 2

and

11

1.2 BASIC PRINCIPLES OF DPSM MODELING

Hence, Eq. (1.2) becomes y¼

X

jNi f ðrNi Þ þ

poˆ le N

X

jSj f ðrS j Þ

poˆ le S

where f ðrÞ ¼

1 2pm0 r

ð1:5Þ

After introducing Ci as the coordinate of the center of the source Si , Eq. (1.5) takes the following form: X

y ¼

jNi f ðM Ci Þ þ

poˆ le N

X

jS j f ðM Cj Þ

ð1:6Þ

poˆ le S

1.2.2.2 Solution of coupling equations At this step, the value of the magnetic ﬂux emitted by the point sources is unknown. An additional boundary condition is then introduced to obtain a new equation set. This is done by computing the magnetic potential yk at each peak point (Pk ) of the hemispherical surface (radius r), due to all sources Si (see Fig. 1.12). yk can be obtained from Eq. (1.6) X

yk ¼

poˆ le N

jNi f ðPk Ci Þ þ

X

jS j f ðPk Cj Þ

ð1:7Þ

poˆ le S

As the magnetic circuit of the sensor is composed of a material of high permeability, there is no difference in magnetic potential between the points of the same pole. Therefore, yk ¼ yN ¼ a for the north pole and yk ¼ yS ¼ b for the south pole, where yN yS ¼ a þ b ¼ N Iexc

ð1:8Þ

South pole

North pole CNi - PSj r

r

CNi - PNj Figure 1.12 Deﬁnition of distances for coupling matrix computation.

12

BASIC THEORY OF DISTRIBUTED POINT SOURCE METHOD (DPSM)

The number of equations thus obtained is equal to the number of point sources. Thus, one gets a square matrix for the coupling equations 0

yNp1

1

0

þa

1

B C B C B yNp2 C B þa C B C B C B C B C B yNp3 C ¼ B þa C B C B C B y C B b C @ Sp1 A @ A ySp2

b 22 3 2 FNp1Sc1 FNp1Nc1 FNp1Nc2 FNp1Nc3 64 6 FNp2Nc1 FNp2Nc2 FNp2Nc3 5 4 FNp2Sc1 6 F FNp3Nc2 FNp3Nc3 FNp3Sc1 ¼6 6 Np3Nc1 6 FSp1Sc1 4 FSp1Nc1 FSp1Nc2 FSp1Nc3 FSp2Nc1 FSp2Nc3 FSp2Nc3 FSp2Sc1

0 1 33 jN1 FNp1Sc2 7 Bj C FNp2Sc2 5 7 B N2 C C 7 B B FNp3Sc2 7 B jN3 C C 7 C 7 B C FSp1Sc2 5 B @ jS1 A FSp2Sc2 jS2 ð1:9Þ

This system of equations with four submatrices can be rewritten as: N ¼ FNN N þ FNS S S ¼ FSN N þ FSS S So that ¼F

ð1:10Þ

Inversion of Eq. (1.10) gives the magnetic ﬂux for all point sources ¼ F 1 ¼ G which gives 1 2" # jN1 GNp1Nc1 GNp1Nc2 GNp1Nc3 B jN2 C 6 GNp2Nc1 GNp2Nc2 GNp2Nc3 C 6 B C 6 G GNp3Nc2 GNp3Nc3 ¼B B jN3 C ¼ 6 Np3Nc1 @ j A 4 GSp1Nc1 GSp1Nc2 GSp1Nc3 S1 GSp2Nc1 GSp2Nc3 GSp2Nc3 j 0

"

HNp1Sc1 HNp2Sc1 H Np3Sc1 HSp1Sc1 HSp2Sc1

S2

# 3 0 þa 1 HNp1Sc2 C B HNp2Sc2 7 7 B þa C HNp3Sc2 7 B þa C 7 B C HSp1Sc2 5 @ b A HSp2Sc2 b

ð1:11Þ The condition on the ﬂux (Eq. 1.1) in combination with Eq. (1.11) gives a new equation to determine the values of a and b: jN þ jS ¼ a

X i;j

Gij b

X k;l

Hkl ¼ 0

ð1:12aÞ

1.2 BASIC PRINCIPLES OF DPSM MODELING

13

Figure 1.13 Illustration of the DPSM method with 9 sources on each pole.

After knowing the magnetic ﬂux values, the magnetic potential or the ﬁeld in the space in front of the magnetic circuit is computed from Eqs. (1.3) and (1.5). We can also compute some macroscopic parameters like the reluctance < by 1 < ¼ ðyN yS Þ j1 N ¼ N Iexc jN

ð1:12bÞ

1.2.2.3 Results and discussion Some results obtained with the DPSM model are presented and compared with the results obtained by the ﬁnite element method (Ansys 3D software) (ANSYS, 1999). Geometry of the sensor, with 9 point sources for each pole, is shown in Fig. 1.13. For this simulation, Figure 1.14a and b shows the magnetic potential at the surface of the north pole for 9 sources and 144 sources, respectively. Similar to Figure 1.13, in Figure 1.14a, a small number of sources (9 sources) is kept to illustrate the principle of the DPSM method. The boundary condition that the magnetic potential is constant at the apex of every hemispherical surface can be clearly seen in this ﬁgure. Same parameters are shown in Figure 1.14b when the number of point sources in each pole is increased to 12 12 ¼ 144. Dimensions along the x- and y-axes are given in millimeters. Figure 1.15a and b presents the normal component of the magnetic ﬁeld at the surface of the north pole for 9 and 144 sources, respectively.

14

BASIC THEORY OF DISTRIBUTED POINT SOURCE METHOD (DPSM)

Figure 1.14 Magnetic potential on north pole for (a) 9 and (b) 144 sources on each pole.

Figure 1.16a presents the Hx component of the magnetic ﬁeld at the level of the poles, obtained with the DPSM model. Figure 1.16b shows the variation of the magnitude of Hx along a line in the y-direction on the poles’ plane. Figure 1.16b shows the magnitude of Hx obtained by DPSM (dashed line) and ANSYS 3D simulation (continuous line). This ﬁgure shows that the DPSM results are in good agreement with the ANSYS simulation results. Furthermore, it should be pointed out that the DPSM synthesis of the magnetic ﬁeld has been made, in this simple example, by matching the boundary conditions on the scalar term (magnetic potential) at the surface of the transducer. It should be noticed here that the satisfaction of the boundary condition gives rise to some interesting conditions on the vector term (magnetic ﬁeld) because the scalar and vector terms are linked by a gradient relation (see Eq. (1.3)). Therefore, a ﬁrst-order development of the scalar term applied to the neighboring points of the boundary surface guarantees a matching of the ﬁrst-order derivative terms in any tangential direction to the boundary surface. In addition, it should be noted that the surface does not correspond to the wave front; therefore, every point, where the

Figure 1.15 Normal component of the magnetic ﬁeld on north pole for (a) 9 and (b) 144 sources.

15

1.2 BASIC PRINCIPLES OF DPSM MODELING

Figure 1.16 (a) Tangential component of the magnetic ﬁeld computed by DPSM; (b) comparison of results obtained from DPSM (dashed line) and ANSYS (continuous line) simulation.

scalar condition is matched, corresponds to different components of the wave. This interesting point can be easily observed in ultrasonic problems. In ultrasonic modeling (discussed below) like magnetic problems, the DPSM synthesis can be carried out by matching scalar boundary conditions (pressure). In the ultrasonic modeling pressure, matching induces a variation for the vector terms (velocity). This point is illustrated in Figure 1.17. It should be noted here that if the continuity conditions must be satisﬁed on vectors (see examples on ultrasonic problems in Chapter 4 and electromagnetic problems in Chapter 6) instead of scalars, then triplet point sources must be used instead of simple point sources.

For all neighboring points j:

Pj ≈ Pi +

∂Pi × drij + 0 ∂ri

POINT « i » Neighboring points « j »

dr 12

dr 13

dr 16

dr 14

dr 15 dr 17

Boundary condition points

ri

Transducer

Figure 1.17 Field synthesis properties and boundary conditions.

16

BASIC THEORY OF DISTRIBUTED POINT SOURCE METHOD (DPSM)

It will be shown in subsequent developments that for some boundary value problems, the boundary conditions may be speciﬁed on vector quantities. A simple such example is the ultrasonic transducer modeling. Ultrasonic transducer surfaces usually have a normal vibration speed, which induces a model in which the velocity vector is normal to the transducer surface. The next applications of DPSM will be illustrated with examples taken from the ultrasonic transducer modeling problems. For ultrasonic problems the point sources can be single elemental point source or triplet source. Although single elemental point sources are capable of satisfying only one boundary condition at a point, the triplet sources are capable of satisfying three boundary conditions (x, y, and z components of the velocity vector, for example) at one point.

1.3 EXAMPLES FROM ULTRASONIC TRANSDUCER MODELING Most derivations on ultrasonic transducer modeling presented in this section are taken from Placko and Kundu (2004). For step by step development of DPSM formulation for ultrasonic problems, readers are referred to Chapter 3 of this book, where the formulation has been derived starting from the basic equations of ultrasonic problems. Three most common ultrasonic wave fronts that are often used for modeling purposes are spherical, cylindrical, and plane. Spherical waves are generated by a point source in an inﬁnite medium, cylindrical waves are generated by a line source, whereas plane waves are generated by an inﬁnite plane, as shown in Figure 1.18. These waves can be harmonic or nonharmonic. Harmonic waves are generated from harmonic (time dependence ¼ eiot ) sources. The equation of the propagating spherical wave generated by a harmonic point source in a ﬂuid space is given by (Kundu, 2004) eikf r ð1:13:aÞ GðrÞ ¼ 4pr and the equation of a propagating plane wave in a ﬂuid is given by (Kundu, 2004) Gðx1 Þ ¼ eikf x1

F

ð1:13:bÞ

F

F S S

S

Figure 1.18 Point source (left), line source (middle) and inﬁnite plane source (right) generating spherical, cylindrical and plane wave fronts, respectively. Sources are denoted by S and the wave fronts by F.

1.3 EXAMPLES FROM ULTRASONIC TRANSDUCER MODELING

17

where, kf ð¼ o=cf Þ is the wave number of the ﬂuid and is deﬁned as the ratio of the angularfrequency (o) and the acousticwavespeedðcf Þ inthe ﬂuid,r istheradialdistance of the spherical wave front from the point source, and x1 is the propagation distance of the plane wave front from the plane source. In Eqs. (1.13a) and (1.13b) G can be either pressure (p) or wave potential (f). The wave potential–pressure relation is given by p ¼ ro2 f

ð1:13cÞ

If the wave sources of Figure 1.18 are located in a homogeneous solid instead of the ﬂuid medium, then only compressional waves are generated in the solid, and their expressions can be obtained by simply substituting kf by kP, where kP is the P-wave number of the solid. In the absence of any interface or boundary, the mode conversion does not occur and shear waves are not generated from the compressional waves. In many nondestructive evaluation (NDE) applications, elastic waves are generated by a source of ﬁnite dimension and the wave fronts are not spherical or cylindrical or plane. Diameters of the commercially available ultrasonic transducers that are most commonly used in NDE for ultrasonic wave generation vary from a quarter of an inch to one inch. Of course, in special applications the ultrasonic sources can be much smaller (in the order of microns for high-frequency acoustic microscopy applications) or much larger (several inches for large-structure inspection). To correctly predict the ultrasonic ﬁeld (displacement, stress, and pressure ﬁelds), generated by such ﬁnite sources, a semi-analytical modeling technique such as DPSM is needed. 1.3.1 Justiﬁcation of modeling a ﬁnite plane source by a distribution of point sources The pressure ﬁeld due to a ﬁnite plane source can be assumed to be the summation of pressure ﬁelds generated by a number of point sources distributed over the ﬁnite source area as shown in Figure 1.19. The ﬁnite source can be, for example, the front face of a transducer as shown in this ﬁgure. This assumption can be justiﬁed in the following manner: A harmonic point source that expands and contracts alternately can be represented by a point and a sphere as shown in Figure 1.20a. The point represents the contracted position and the sphere (circle in a 2D ﬁgure) represents the expanded position. When a large number of these point sources are placed side by side on a plane surface, then

Figure 1.19 Four point sources distributed over a ﬁnite source

18

BASIC THEORY OF DISTRIBUTED POINT SOURCE METHOD (DPSM)

(a)

(b)

(c)

Figure 1.20 Contracted (dark) and expanded (thin line) positions of the particles for (a) a point source, (b) distributed ﬁnite number of point sources, and (c) a very large number of point sources.

the contracted and expanded positions of the point sources are shown in Figure 1.20b. The combined effect of a large number of point sources, placed side by side, is shown in Figure 1.20c, where the contracted (dark line) and expanded (thin line) positions of a line source or the cross section of a plane source are seen. From Figure 1.20 it is clear that the combined effect of a large number of point sources distributed on a plane surface is the vibration of the particles in the direction normal to the plane surface. Nonnormal components of motion at a point on the surface, generated by neighboring source points, cancel each other. However, nonnormal components do not vanish along the edge of the surface. Therefore, the particles not only vibrate normal to the surface but also expand to a hemisphere and contract to the point along the edge, as shown in Figure 1.20c. If this edge effect does not have a signiﬁcant contribution on the total motion, then the normal vibration of a ﬁnite plane surface can be approximately modeled by replacing the ﬁnite surface by a large number of point sources distributed over the surface. 1.3.2 Planar piston transducer in a ﬂuid The pressure ﬁeld in a ﬂuid for the planar piston transducer of ﬁnite diameter, as shown in Figure 1.21, is computed ﬁrst. This problem can be solved in two ways as described below. 1.3.2.1 Conventional surface integral technique If one distributes the point sources over the transducer face, as discussed in Section 2.1, then the pressure ﬁeld x1

y x3 r x2

x

Figure 1.21 Point source y is on the transducer face, point x is where ultrasonic ﬁeld is computed

1.3 EXAMPLES FROM ULTRASONIC TRANSDUCER MODELING

19

at position x in the ﬂuid, due to the point sources at position y distributed over the transducer surface, can be given by integrating Eq. (1.13) over the transducer surface. ð expðikf rÞ pðxÞ ¼ B dSðyÞ 4pr

ð1:14aÞ

S

where B is proportional to the source velocity amplitude. The above integral can be written in the following summation form: pðxÞ ¼

N X B m¼1

4p

Sm

N expðikf rm Þ X expðikf rm Þ ¼ Am rm rm m¼1

ð1:14bÞ

However, from the Rayleigh–Sommerﬁeld theory (Schmerr, 1998), ð ior expðikf rÞ v3 ðyÞ dSðyÞ pðxÞ ¼ 2p r

ð1:15Þ

S

where v3(y) is the particle velocity component normal to the transducer surface; note that v1(y) ¼ v2(y) ¼ 0. For constant velocity of the transducer surface (v3 ¼ v0), Eq. (1.15) is simpliﬁed to pðxÞ ¼

ð iorv0 expðikf rÞ dSðyÞ 2p r

ð1:16Þ

S

A comparison between Eqs. (1.14a) and (1.16) gives B ¼ 2iorv0

ð1:16aÞ

Eq. (1.16) can be evaluated in closed form for a circular transducer of radius a for the following two special cases (Schmerr, 1998): (1) when x is located on the x3-axis qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pðx3 Þ ¼ rcf v0 expðikf x3 Þ exp ikf x23 þ a2

ð1:16bÞ

(2) when x is in the far ﬁeld. In other words, when r is much greater than the transducer radius pðx1 ; x2 ; x3 Þ ¼ iorv0 a2

expðikf RÞ J1 ðkf a sin yÞ R kf a sin y

ð1:16cÞ

R and y of Eqs. (1.16b) and (1.16c) are shown in Figure 1.22. J1 is the Bessel function of the ﬁrst kind.

20

BASIC THEORY OF DISTRIBUTED POINT SOURCE METHOD (DPSM)

x1 x R a

θ x3

x2

Figure 1.22 R and y denote the far ﬁeld point x

1.3.2.2 Alternative DPSM for computing the ultrasonic ﬁeld An alternative technique to compute the strength of the distributed point sources on the transducer surface is given in this section. Let the strength of the mth point source be Am such that the pressure at a distance rm from the point source is given by Eq. (1.17) (also see Eq. (1.14b)). pm ðrÞ ¼ Am

expðikf rm Þ rm

ð1:17Þ

If there are N point sources distributed over the transducer surface, as shown in Figure 1.23, then the total pressure at point x is given by N X

pðxÞ ¼

pm ðrm Þ ¼

m¼1

N X

Am

m¼1

expðikf rm Þ rm

ð1:18Þ

where rm is the distance of the mth point source from point x. Note that Eqs. (1.18) and (1.14b) are identical. From the pressure–velocity relation, it is possible to compute the velocity at x.

@p @vn ¼ iorvn ¼r @t @n

x1 x

ð1:19Þ

vm v3m

rm x3 m

x3m

x2

Figure 1.23 Velocity vm at point x due to the m-th point source

1.3 EXAMPLES FROM ULTRASONIC TRANSDUCER MODELING

21

Note that for eiot time dependence of velocity, its derivative can be obtained by simply multiplying vn by positive or negative io. For eiot time dependence, vn ¼

1 @p ior @n

ð1:20Þ

Therefore, the velocity in the radial direction, at a distance r from the mth point source, is given by Am @ expðikf rÞ Am ikf expðikf rÞ expðikf rÞ ¼ vm ðrÞ ¼ ior @r ior r r r2 Am expðikf rÞ 1 ¼ ikf ð1:21Þ ior r r and the x3 component of the velocity is Am @ expðikf rÞ Am x3 expðikf rÞ 1 ¼ v3m ðrÞ ¼ ik f ior @x3 ior r r2 r

ð1:22Þ

When contributions of all N sources are added, see Figure 1.23, then the total velocity in the x3 direction at point x is obtained. N X Am x3m expðikf rm Þ 1 v3 ðxÞ ¼ v3m ðrm Þ ¼ ikf ior rm2 rm m¼1 m¼1 N X

ð1:23Þ

where x3m is the x3 value measured from the mth source as shown in Figure 1.23. If the transducer surface velocity in the x3 direction is given by v0, then for all x values on the transducer surface, the velocity in the x3 direction should be equal to v0. Therefore, N X Am x3m expðikf rm Þ 1 ¼ v0 ikf ð1:24Þ v3 ðxÞ ¼ ior rm2 rm m¼1 By taking N points on the transducer surface, it is possible to obtain a system of N linear equations to solve for N unknowns ðA1 ; A2 ; A3 ; . . . ; AN Þ. However, difﬁculty arises when the point source location and the point of interest, x, coincide because then rm becomes zero and v3m, from Eq. (1.24), becomes unbounded. Note that if point sources and points of interest x are both located on the transducer surface, only then these two points may coincide and rm can be zero. To avoid this possibility, the point sources are placed slightly behind the transducer surface as shown in Figure 1.24. For this arrangement the smallest value that rm can take is rS. When point x is located on the transducer surface as shown in Figure 1.24, then its x3 component of velocity is matched with the transducer surface velocity v0. In Figure 1.24 one can see that point x is located at the apex of the small spheres touching the transducer surface and the point sources are placed at the centers of these small

22

BASIC THEORY OF DISTRIBUTED POINT SOURCE METHOD (DPSM)

x1 rm x x3

rS

Figure 1.24 Point sources are located at x3 ¼ rS while the transducer surface is at x3 ¼ 0.

spheres. In addition to matching the v3 component to v0, if one wants to equate the other two components v1 and v2 to zero, then for each point x on the transducer surface, there are a total of three conditions or equations to be satisﬁed, as shown in Eqs. (1.24) and (1.25). N X Am x1m expðikf rm Þ 1 ¼0 ikf v1 ðxÞ ¼ ior rm2 rm m¼1 ð1:25Þ N X Am x2m expðikf rm Þ 1 v2 ðxÞ ¼ ¼0 ikf ior rm2 rm m¼1 Thus, from N points on the sphere surfaces, 3N equations are obtained. Therefore, we get more equations than unknowns. To get the same number of unknowns as equations, the number of unknowns can be increased from N to 3N by replacing each point source by a triplet source. A triplet source is a combination of three point sources with three different strengths put together as shown in Figure 1.25. All sources are placed on the same plane at x3 ¼ rS parallel to the transducer surface. The three point sources of each triplet are located at the three vertices of an isosceles triangle that are oriented randomly, as shown in Figure 1.25, to preserve the isotropic material properties and prevent any preferential orientation. Thus, by solving a system of 3N linear equations (for triplet sources) or a system of N linear equations (for simple point sources), the source strengths Am associated with all point sources can be obtained. After getting Am, the pressure p(x) can be calculated at any point Triplet Source

Figure 1.25 Randomly oriented triplet sources

23

1.3 EXAMPLES FROM ULTRASONIC TRANSDUCER MODELING

from Eq. (1.18), on the transducer surface or away. The pressure ﬁeld obtained in this manner should be the same as that obtained from Eq. (1.16). Note that for a nonviscous perfect ﬂuid, only the normal velocity component ðv3 Þ at the ﬂuid–solid (transducer face) interface should be continuous. The velocity components parallel to the transducer face are not necessarily continuous because slippage may occur between the transducer face and the ﬂuid. However, for viscous ﬂuids such slippage is not possible and all the three velocity components should be continuous across the solid transducer face and ﬂuid interface. 1.3.2.2.1 Matrix formulation The matrix formulation for computing the source strengths is given below. The following formulation is presented for triplet sources when all the three velocity components are matched at the transducer face and ﬂuid interface. This is the case for viscous ﬂuids. However, for nonviscous perfect ﬂuids when only the normal velocity component needs to be matched, then simple elemental point sources should be used instead of triplet sources. Then v1 and v2 velocity components should be dropped from the following formulation. In that case the matrix and vector dimensions will be reduced from 3N to N. Eqs. (1.24) and (1.25) can be combined into the following matrix equation: VS ¼ MSS AS

ð1:25aÞ

where VS is the ð3N 1Þ vector of the velocity components at N number of surface points x, and AS is the ð3N 1Þ vector containing the strengths of 3N number of point sources. MSS is the ð3N 3NÞ matrix relating the two vectors VS and AS. From Eqs. (1.24) and (1.25) one can write fVS gT ¼ ½ v11

v12

v13

v21

v22

v23

...

vN1

vN2

vN3

ð1:25bÞ

Note that the transpose of the column vector VS is a row vector of dimension ð1 3NÞ. Elements of this vector are denoted by vnj, where the subscript j can take values 1, 2, or 3 and indicate the direction of the velocity component. Superscript n can take any value between 1 and N corresponding to the point on the transducer surface at which the velocity component is deﬁned. For most ultrasonic transducers, vnj ¼ 0 for j ¼ 1 and 2 (the velocity component parallel to the transducer face) and vnj ¼ v0 for j ¼ 3 (the velocity component normal to the transducer face). Then, Eq. (1.25b) is simpliﬁed to fVS gT ¼ ½ 0 0

v0

0

0

v0

... 0

0

v0

ð1:25cÞ

Vector AS of the source strengths is given by fAS gT ¼ ½ A1

A2

A3

A4

A5

A6

...

Að3N2Þ

Að3N1Þ

A3N

ð1:25dÞ

Note that the upper limits of Eqs. (1.24) and (1.25) are changed from N to 3N when triplet sources are considered, because then for every small sphere three point sources exist. Therefore, for N spheres 3N sources exist as shown in Figure 1.25.

24

BASIC THEORY OF DISTRIBUTED POINT SOURCE METHOD (DPSM)

Finally, the square matrix MSS is obtained from Eqs. (1.24) and (1.25). MSS ¼ 2 f ðx111 ; r11 Þ 6 6 f ðx1 ; r1 Þ 6 21 1 6 6 6 f ðx131 ; r11 Þ 6 6 6 f ðx2 ; r2 Þ 11 1 6 6 6 f ðx2 ; r2 Þ 6 21 1 6 6 6 f ðx231 ; r12 Þ 6 6 6 ... 4 N f ðx31 ; r1N Þ

f ðx112 ; r21 Þ

f ðx113 ; r31 Þ

f ðx114 ; r41 Þ . . .

...

1 f ðx11ð3N1Þ ; rð3N1Þ Þ

f ðx122 ; r21 Þ

f ðx123 ; r31 Þ

f ðx124 ; r41 Þ . . .

...

1 f ðx12ð3N1Þ ; rð3N1Þ Þ

f ðx132 ; r21 Þ

f ðx133 ; r31 Þ

f ðx134 ; r41 Þ . . .

...

1 f ðx13ð3N1Þ ; rð3N1Þ Þ

f ðx212 ; r22 Þ

f ðx213 ; r32 Þ

f ðx214 ; r42 Þ . . .

...

2 f ðx21ð3N1Þ ; rð3N1Þ Þ

f ðx222 ; r22 Þ

f ðx223 ; r32 Þ

f ðx224 ; r42 Þ . . .

...

2 f ðx22ð3N1Þ ; rð3N1Þ Þ

f ðx232 ; r22 Þ

f ðx233 ; r32 Þ

f ðx234 ; r42 Þ . . .

...

2 f ðx23ð3N1Þ ; rð3N1Þ Þ

...

...

...

f ðxN32 ; r2N Þ

f ðxN33 ; r3N Þ

f ðxN34 ; r4N Þ

...

...

... ...

... N f ðxN3ð3N1Þ ; rð3N1Þ Þ

1 f ðx11ð3NÞ ; r3N Þ

3

7 1 f ðx12ð3NÞ ; r3N Þ7 7 7 7 1 f ðx13ð3NÞ ; r3N Þ7 7 7 2 f ðx21ð3NÞ ; r3N Þ7 7 7 2 2 f ðx2ð3NÞ ; r3N Þ 7 7 7 7 2 2 f ðx3ð3NÞ ; r3N Þ 7 7 7 7 ... 5 N N f ðx3ð3NÞ ; r3N Þ

ð3N3NÞ

ð1:25eÞ

where f ðxnjm ; rmn Þ ¼

xnjm expðikf rmn Þ iorðrmn Þ2

ikf

1 rmn

ð1:25fÞ

In Eq. (1.25f), the ﬁrst subscript j of x can take values 1, 2, or 3 and indicate whether x is measured in the x1, x2, or x3 direction. The subscript m of x and r can take values from 1 to 3N depending on which point source is considered, and the superscript n can take any value between 1 and N corresponding to the point on the transducer surface where the velocity component is computed. As mentioned earlier in this formulation, from 3N point sources, three boundary conditions on the velocity are satisﬁed at every point of the N boundary points. However, for nonviscous ﬂuids the slippage between the transducer surface and the adjacent ﬂuid surface is possible. Therefore, it is not necessary to enforce the no-slip condition ðv1 ¼ v2 ¼ 0Þ on the ﬂuid particles that are adjacent to the transducer surface. If point x in Figure 1.24 is denoted by xn, indicating that this point is located on the nth boundary point, then the position vector connecting the mth point source and the nth boundary point is denoted by rmn , and its three components in x1, x2, and x3 directions are xnjm , j ¼ 1; 2; 3, in Eqs. (1.25e) and (1.25f). From Eq. (1.25a) one gets the point source strengths by inverting the matrix MSS . AS ¼ ½MSS 1 VS ¼ NSS VS

ð1:25gÞ

If point sources are located very close to the transducer surface (rS in Fig. 1.24 is small), then the point source strengths ðAS Þ should be approximately equal to the source strengths on the transducer surface. From Eqs. (1.14) and (1.16) we get, Am ¼

B 2iorv0 S Sm ¼ 4p N 4p

ð1:25hÞ

25

1.3 EXAMPLES FROM ULTRASONIC TRANSDUCER MODELING

In Eq. (1.25h) S is the transducer surface area. Note that this equation gives the same source strength for all values of m. Therefore, the vector AS, obtained from Eq. (1.25h), should have the following form: fAS gT ¼

iorv0 S ½1 2pN

1 1

... ...

1

ð1:25iÞ

After getting the source strength vector AS from Eq. (1.25g) or (1.25i), the pressure p(x) or velocity vector V(x) at any point (on the transducer surface or away) can be obtained from Eq. (1.18) (for pressure) or Eqs. (1.24) and (1.25) (for velocity components). If the points in the ﬂuid where the pressure and velocity vector are to be computed are called observation points or target points, then the pressure and velocity components, at these observation or target points, are obtained from the following matrix relation: PT ¼ QTS AS VT ¼ MTS AS

ð1:25jÞ

where PT is an ðM 1Þ vector containing pressure values at M number of target points and VT is a ð3M 1Þ vector containing three velocity components at every target point. The VT expression is similar to the VS expression given in Eq. (1.25b). The only difference is that its dimension is ð3M 1Þ instead of ð3N 1Þ. Matrix MTS will be the same as MSS of Eq. (1.25e) if the target points are identical to the transducer surface points where the velocity components are matched to obtain the point source strength vector AS in Eq. (1.25g). However, for computing the velocity ﬁeld at different points, the expression for MTS will be slightly different from the MSS expression given in Eq. (1.25e). Then its dimension will be ð3M 3NÞ as shown below: 2

1 1 Þ f ðx11ð3NÞ ;r3N Þ f ðx111 ;r11 Þ f ðx112 ;r21 Þ f ðx113 ;r31 Þ f ðx114 ;r41 Þ ... ... f ðx11ð3N1Þ ;rð3N1Þ

6 1 1 6 f ðx21 ;r1 Þ 6 6 1 1 6 f ðx31 ;r1 Þ 6 6 f ðx2 ;r2 Þ 6 MTS ¼ 6 11 1 6 f ðx2 ;r2 Þ 6 21 1 6 2 2 6 f ðx31 ;r1 Þ 6 6 ... 4 M f ðxM 31 ;r1 Þ

3

1 1 7 f ðx122 ;r21 Þ f ðx123 ;r31 Þ f ðx124 ;r41 Þ ... ... f ðx12ð3N1Þ ;rð3N1Þ Þ f ðx12ð3NÞ ;r3N Þ7 7 1 1 7 f ðx132 ;r21 Þ f ðx133 ;r31 Þ f ðx134 ;r41 Þ ... ... f ðx13ð3N1Þ ;rð3N1Þ Þ f ðx13ð3NÞ ;r3N Þ7 7 2 2 7 f ðx212 ;r22 Þ f ðx213 ;r32 Þ f ðx214 ;r42 Þ ... ... f ðx21ð3N1Þ ;rð3N1Þ Þ f ðx21ð3NÞ ;r3N Þ7 7 2 2 7 f ðx222 ;r22 Þ f ðx223 ;r32 Þ f ðx224 ;r42 Þ ... ... f ðx22ð3N1Þ ;rð3N1Þ Þ f ðx22ð3NÞ ;r3N Þ7 7 2 2 7 f ðx232 ;r22 Þ f ðx233 ;r32 Þ f ðx234 ;r42 Þ ... ... f ðx23ð3N1Þ ;rð3N1Þ Þ f ðx23ð3NÞ ;r3N Þ7 7 ... ... ... ... ... ... ... 5 M M M M M M M M M M f ðx32 ;r2 Þ f ðx33 ;r3 Þ f ðx34 ;r4 Þ ... ... f ðx3ð3N1Þ ;rð3N1Þ Þ f ðx3ð3NÞ ;r3N Þ

ð3M3NÞ

ð1:25kÞ

where f ðxnjm ; rmn Þ is identical to the expression given in Eq. (1.25f). Deﬁnitions of the subscripts j and x do not change from those in Eq. (1.25f). The superscript n of x and r can take any value between 1 and M depending on which target point is considered. Note that MTS is not a square matrix when M and N are different.

26

BASIC THEORY OF DISTRIBUTED POINT SOURCE METHOD (DPSM)

From Eq. (1.18) the matrix QTS can be obtained when there are 3N point sources and M target points as shown in Eq. (1.25l). 2

QTS

expðikf r11 Þ 6 r11 6 6 6 expðikf r12 Þ 6 6 r12 6 6 expðikf r13 Þ ¼6 6 6 r13 6 ... 6 6 6 ... 6 4 expðikf r1M Þ r1M

expðikf r21 Þ r21

expðikf r31 Þ r31

... ...

expðikf r22 Þ r22

expðikf r32 Þ r32

... ...

expðikf r23 Þ expðikf r33 Þ r23 r33 ... ... ... ... M expðikf r2 Þ expðikf r3M Þ r2M r3M

... ... ... ... ... ... ... ...

3 1 expðikf r3N Þ 7 1 r3N 7 7 2 expðikf r3N Þ7 7 2 7 r3N 7 7 3 expðikf r3N Þ 7 7 3 7 r3N 7 ... 7 7 7 ... 7 M 5 expðikf r3N Þ M r3N ðM3NÞ ð1:25lÞ

The deﬁnition of rmn is identical for Eqs. (1.25k) and (1.25j); it is the distance between the mth point source and nth target point. This alternative method and matrix formulation, discussed here, for computing the ultrasonic ﬁeld in a homogeneous ﬂuid was ﬁrst proposed by Placko and Kundu (2001); then it was extended to solve different ultrasonic problems by Placko et al. (2001, 2002) and Lee et al. (2002). This technique has been named by the authors as the distributed point source method or DPSM. The advantage of the DPSM technique is not obvious for this simple case of homogeneous medium. However, it will be evident later in this chapter when the ultrasonic ﬁeld, in the presence of a ﬁnite inclusion or scatterer, will be computed. Note that the DPSM technique, discussed in this section, is a general technique and is not restricted to the case of small value of rS (see Fig. 1.24). For small value of rS Eq. (1.25i) can be used; otherwise, Eq. (1.25g) will have to be used. When Eq. (1.25g) is used, then Eq. (1.25j) is modiﬁed to PT ¼ QTS NSS VS VT ¼ MTS NSS VS

ð1:25mÞ

Example 1.3.1 Give the modiﬁed expressions for VS (Eq. (1.25c)) and MSS (Eq. (1.25e)) for the case when the triplet sources are replaced by single point sources, located at the centers of the small spheres (see Figs. 1.23 and 1.24), and only the normal displacement components (normal to the transducer surface) at the apex (or collocation points) on the transducer surface are equated to the transducer surface velocity v0. Solution For N number of spheres distributed over the transducer surface, there will be N point sources and N collocation points. Therefore, the velocity vector VS of Eq. (1.15c) will

27

1.3 EXAMPLES FROM ULTRASONIC TRANSDUCER MODELING

have N entries instead of 3N entries. fVS gT ¼ ½ v0

v0

v0

:::::::

v0 ðN1Þ

ð1:25nÞ

The matrix MSS of Eq. (1.25e) will have a dimension of ðN NÞ instead of ð3N 3NÞ, because only the v3 component is to be matched. The ﬁnal form of MSS is given below. MSS ¼ 2 f ðx131 ; r11 Þ 6 6 f ðx2 ; r 2 Þ 6 31 1 6 6 6 f ðx331 ; r13 Þ 6 6 6 f ðx431 ; r14 Þ 6 6 6 ... 6 6 6 ... 6 6 6 ... 4 N f ðx31 ; r1N Þ

1 f ðx132 ; r21 Þ f ðx133 ; r31 Þ f ðx134 ; r41 Þ . . . . . . f ðx13ðN1Þ ; rðN1Þ Þ f ðx13N ; rN1 Þ

3

7 2 f ðx232 ; r22 Þ f ðx233 ; r32 Þ f ðx234 ; r42 Þ . . . . . . f ðx23ðN1Þ ; rð3N1Þ Þ f ðx23N ; rN2 Þ 7 7 7 3 3 3 3 3 3 3 3 3 3 7 f ðx32 ; r2 Þ f ðx33 ; r3 Þ f ðx34 ; r4 Þ . . . . . . f ðx3ðN1Þ ; rðN1Þ Þ f ðx3N ; rN Þ 7 7 7 4 f ðx432 ; r24 Þ f ðx433 ; r34 Þ f ðx434 ; r44 Þ . . . . . . f ðx43ðN1Þ ; rð3N1Þ Þ f ðx43N ; rN4 Þ 7 7 7 7 ... ... ... ... ... ... ... 7 7 7 ... ... ... ... ... ... ... 7 7 7 ... ... ... ... ... ... ... 5 N N N N N N N N N N f ðx32 ; r2 Þ f ðx33 ; r3 Þ f ðx34 ; r4 Þ . . . . . . f ðx3ðN1Þ ; rðN1Þ Þ f ðx3N ; rN Þ

ðNNÞ

ð1:25oÞ

where, from Eq. (1.25f) f ðxn3m ; rmn Þ

¼

xn3m expðikf rmn Þ iorðrmn Þ2

1 ikf n rm

ð1:25pÞ

Example 1.3.2 For a large number of point sources distributed along the transducer surface, as shown in Figures 1.23 and 1.24, evaluate the source strength vector AS using Eq. (1.25g) for the MSS and VS expressions given in Eqs. (1.25o) and (1.25n), respectively. Solution For a large number of distributed point sources, the radius of the individual spheres becomes small (see Fig. 1.24). As the number of point sources approaches inﬁnity, the radius of individual spheres reduces to zero. Therefore, rmn , the distance between the mth point source and nth collocation point (or apex point), becomes zero for m ¼ n. In other words, when the source is at the center of a sphere and the collocation point is at the apex of the same sphere, then the distance between the source and the collocation point is reduced to zero, as the number of point sources approaches inﬁnity.

28

BASIC THEORY OF DISTRIBUTED POINT SOURCE METHOD (DPSM)

Note that in Eq. (1.25p) rmn appears in the denominator. Therefore, for small values of rmn Eq. (1.25p) can be simpliﬁed in the following manner: f ðxn3m ; rmn Þ

¼

xn3m expðikf rmn Þ iorðrmn Þ2

1 xn3m expðikf rmn Þ 1 xn expðikf rmn Þ ikf n n ¼ 3m 2 rm rm iorðrmn Þ iorðrmn Þ3

Note that all spheres have the same radius rm ¼ rS ¼ r; therefore, xn3m ¼ r. Substituting it into the above expression and expanding the exponential term in its series expansion, f ðxn3m ; rmn Þ

xn3m expðikf rmn Þ iorðrmn Þ3

r iorðrmn Þ3

ð1 þ ikf rmn þ . . .Þ

r iorðrmn Þ3 ð1:25qÞ

for m ¼ n, rmn ¼ rmm ¼ r. Substituting it into Eq. (1.25q), we get (no summation on m is implied) m f ðxm 3m ; rm Þ

r iorðrmm Þ3

r 1 3 iorr iorr2

ð1:25rÞ

Substitution of Eqs. (1.25q) and (1.25r) into Eq. (1.25o) yields 2

MSS

3

6 1 6 3 6 r 2 1 6 6 r1 3 ¼ 6 iorr2 6 r3 6 r1 6 ... 4 r r1N

3

r r21

1

3 r r23

. .. 3 r r2N

3 r r31

...

r r32

...

1 .

.. 3

... ... ...

3

r r3N

3 3 r r1 7

N 3 7 7 r 7 rN2

3 7 7 r 7 rN3 7 ... 7 5 1

ðNNÞ

It should be noted here that for m 6¼ n, rmn > r. Therefore, in the above matrix expression, the off-diagonal terms are smaller than the diagonal terms. With an increasing number of point sources as r approaches zero, all off-diagonal terms vanish and the above matrix simpliﬁes to 2

MSS

1 0 0 6 0 1 0 1 6 6 ¼ 0 0 1 6 iorr2 6 4... ... ... 0 0 0

3 ... 0 ... 0 7 7 7 ... 0 7 7 ... ...5 . . . 1 ðNNÞ

1.3 EXAMPLES FROM ULTRASONIC TRANSDUCER MODELING

Therefore, from Eq. (1.25g): 2

1 0 0 6 0 1 0 6 AS ¼ ½MSS 1 VS ¼ iorr2 6 0 0 1 6 4... ... ... 0 0 0

29

8 9 38 9 ... 0 > 1 > v0 > > > > > > > > > > > > > v ... 0 7 1 > = < = < 0 7 2 ¼ iorv v r ... 0 7 1 0 0 7> > > > > ...> . . . . . . 5> ...> > > > > > > ; : > ; : > v0 ... 1 1 ð1:25sÞ

Example 1.3.3 Prove that the coefﬁcients of Eqs. (1.25i) and (1.25s) are identical. Solution The total surface area from N hemispheres, associated with the N point sources, is equated to the transducer surface area S. Therefore, S ¼ 2pr 2 N ¼ 2pNr2 S ) r2 ¼ 2pN Substituting it into the coefﬁcient of Eq. (1.25s) gives ioov0 r 2 ¼

ioov0 S 2pN

1.3.2.3 Restrictions on rS for point source distribution It is evident from Figure 1.24 that as the number of point sources used to model the transducer surface is increased, rS is decreased. It is expected that with larger number of point sources, the computation time and accuracy both should increase. The question is what optimum number of point sources should produce reliable results? To answer this question the following analysis is carried out: For a very small transducer of surface area dS vibrating with a velocity of amplitude v0 in the x3 direction, the pressure at point x (at a distance r from the source at point y) can be computed from Eq. (1.16). pðxÞ ¼

iorv0 expðikf rÞ dS 2p r

ð1:26Þ

Using Eq. (1.20), the particle velocity in the radial direction can be computed from the above pressure ﬁeld. 1 @p 1 iorv0 ikf expðikf rÞ expðikf rÞ ¼ vr ¼ dS ior @r ior r r2 2p ¼

v0 ðikf r 1Þ expðikf rÞdS 2pr2

ð1:27Þ

30

BASIC THEORY OF DISTRIBUTED POINT SOURCE METHOD (DPSM)

and the velocity in the x3 direction v3 ¼

1 @p 1 @p @r v0 ðikf r 1Þ x3 y3 ¼ ¼ expðikf rÞdS ior @x3 ior @r @x3 2pr2 r

ð1:28Þ

where x3 and y3 are the x3 coordinate values of points x and y, respectively. If the point x is taken on the surface of the sphere of radius rS as shown in Figure 1.24, then r ¼ rS ¼ x3 y3 , and v3 of Eq. (1.28) is simpliﬁed to v3 ¼

v0 ðikf rS 1Þ dS expðikf rS ÞdS ¼ v0 ð1 ikf rS Þð1 þ ikf rS þ Oðkf2 rS2 ÞÞ 2prS2 2prS2

v0 ð1 þ kf2 rS2 Þ

dS 2prS2

ð1:29Þ

The right-hand side of Eq. (1.29) should be equal to v0 because the pressure computed in Eq. (1.16) is obtained from the transducer surface velocity v0 in the x3 direction. Hence, the velocity at x when x is taken on the transducer surface should be equal to v0 . The right-hand side of Eq. (1.29) is v0 when dS ¼ 2prS2 and kf2 rS2 1. Therefore, dS should be the surface area of a hemisphere of radius rS , and the second condition implies the following:

2 2pf rS 1: cf cf ) rS

2pf lf ) rS

2p

kf2 rS2 ¼

ð1:30Þ

where lf is the wavelength in the ﬂuid. Eq. (1.30) is used to compute the number of point sources in the following manner: Take a value of rS satisfying the condition (1.30), then compute the number of point sources N from the transducer surface area S from the relation N¼

S 2prS2

ð1:31Þ

Note that the spacing between two neighboring point sources is different from rS . If the point sources are arranged uniformly at the vertex points of squares of side length a, then each point source should be associated with an area of a2 of the ﬂat transducer face. This area is then equated to the hemispherical surface area of each point source to obtain a2 ¼ 2prS2

pﬃﬃﬃﬃﬃﬃ ) a ¼ rS 2p

ð1:31aÞ

31

1.3 EXAMPLES FROM ULTRASONIC TRANSDUCER MODELING

Substituting Eq. (1.30) into the above equation we get pﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ lf a ¼ rS 2p 2p 2p lf ) a pﬃﬃﬃﬃﬃﬃ 2p

ð1:31bÞ

1.3.3 Focused transducer in a homogeneous ﬂuid For a focused transducer, as shown in Figure 1.26, the ultrasonic ﬁeld in the ﬂuid can be modeled by distributing the point sources along the curved transducer face. O’Neil (1949) argued that for transducers with small curvature the Rayleigh–Sommerﬁeld integral representation (Eq. (1.16)) holds if the surface integral is carried out over the curved surface. Therefore, the DPSM technique, discussed in Section 1.3.2.2, holds good for the curved transducer face as well. In this case the point sources should be distributed over a curved surface, instead of a ﬂat surface. The integral representation of the pressure ﬁeld in the ﬂuid for a focused transducer should be the same as Eq. (1.16). This integral can be evaluated in closed form, for computing the pressure variation on the central axis of the transducer; in other words, for the on-axis pressure computation. The on-axis pressure ﬁeld is given by (Schmerr, 1998) qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rcv0 rcv0 expðikf x3 Þ expðikf x23 þ a2 Þ ¼ ½expðikf x3 Þ expðikf re Þ pðx3 Þ ¼ q0 q0 ð1:32Þ where q0 ¼ 1

x3 R0

ð1:33Þ

R0 is the radius of curvature of the transducer face, re is the distance of the point of interest from the transducer edge. At the geometric focus point, x3 ¼ R0 , the pressure is given by (Schmerr, 1998) pðR0 Þ ¼ ircv0 kf h expðikf R0 Þ

ð1:34Þ

re

R0

x a

x3 h

Figure 1.26 Focused transducer R0 is the radius of curvature of the transducer, a is its radius, focal point is denoted as x.

32

BASIC THEORY OF DISTRIBUTED POINT SOURCE METHOD (DPSM)

If, at x3 ¼ z, the on-axis pressure is maximum, then z should satisfy the following equation (Schmerr, 1998): kf d 2ðd þ zÞ sinðkf d=2Þ ¼ ð1:35Þ cos 2 ðd þ hÞq0 kf R0 where 1

d ¼ re z ¼ ½ðz hÞ2 þ a2 2 z

ð1:35aÞ

1.3.4 Ultrasonic ﬁeld in a nonhomogeneous ﬂuid in the presence of an interface If the ﬂuid, in front of the transducer, is not homogeneous but is made of two ﬂuids with an interface between the two, then the ultrasonic signal generated by the transducer will go through reﬂection and transmission at the interface as shown in Figure 1.27. In this case, the pressure ﬁeld in ﬂuid 1, at point P, can be computed by adding the contributions of the direct incident ray (R1) and reﬂected ray. To compute the pressure at point Q in ﬂuid 2, the contribution of only the transmitted ray needs to be considered. Acoustic wave speed and density of the two ﬂuids are denoted by cf and rf for ﬂuid 1, and cf2 and rf2 for ﬂuid 2, as shown in Figure 1.27. In Figure 1.27, point C is either on the transducer surface for Rayleigh–Sommerﬁeld integral representation of the pressure ﬁeld, or just behind the transducer surface (as shown in Figure 1.24) for the DPSM modeling, discussed in Section 1.2.2.2. We are interested in computing the acoustic pressure at point P in ﬂuid 1 and at point Q in ﬂuid 2. As shown in the ﬁgure, point P receives a direct ray (R1) from point C and a ray (R3) reﬂected by the interface at point T. Point Q can only receive a ray from point C after it is transmitted at the interface at point T. Position vectors of points C, T, P, and Q are denoted by y, z, x, and x, respectively, as shown in the ﬁgure. Because both points P and Q are the points where the pressure ﬁeld is to be computed, we use the same symbol x for denoting the positions of these two points although those are not at the same location. x1 C y(y1 , y2 , y3 )

R1

x2

S

R2 θ

θ

R3 P x(x1 , x2 , x3 ) Fluid 1 cf , ρf

T z(z1 , z2 , z3 ) θ2

x3 R3

Interface Fluid 2 cf2 , ρf2

Q x(x1 , x2 , x3 )

z0 = z 3

Figure 1.27 Transducer in front of an interface between two ﬂuids of different properties

33

1.3 EXAMPLES FROM ULTRASONIC TRANSDUCER MODELING

Now, the question is, when the coordinates (x1, x2, x3) and (y1, y2, y3) are known, then how to obtain the coordinates (z1, z2, z3) of T on the interface, where the ray is reﬂected or transmitted to reach point P or Q. This question can be answered from geometric considerations, as given below. 1.3.4.1 Pressure ﬁeld computation in ﬂuid 1 at point P Let vectors A and B represent CT and TP, respectively, in Figure 1.27; then, A ¼ ðz1 y1 Þe1 þ ðz2 y2 Þe2 þ ðz3 y3 Þe3 B ¼ ðx1 z1 Þe1 þ ðx2 z2 Þe2 þ ðx3 z3 Þe3

ð1:36Þ

Note that the magnitudes of vectors A and B are R2 and R3 , respectively. 1

R2 ¼ fðz1 y1 Þ2 þ ðz2 y2 Þ2 þ ðz3 y3 Þ2 g2 1

R3 ¼ fðx1 z1 Þ2 þ ðx2 z2 Þ2 þ ðx3 z3 Þ2 g2

ð1:37Þ

^ ¼ A and B ^¼ B Unit vectors A R2 R3 Unit vector ^ n normal to the interface is given by 8 9 8 9 < n1 = <0= n ¼ n2 n^ ¼ 0 : Note that for an inclined interface, ^ : ; : ; n3 1 From the problem geometry one can easily see that ^¼^ ^ ^ nA nB ^ ¼ ^ ^ ^ nA nB

ð1:38Þ ð1:39Þ

Let 8 9 < a1 = ^ ¼ a2 ; A : ; a3

8 9 < b1 = ^ ¼ b2 B : ; b3

ð1:40Þ

Substituting the above unit vector expressions in Eq. (1.38) one gets 2 3 2 3 e1 e2 e3 e1 e 2 e 3 det4 n1 n2 n3 5 ¼ det4 n1 n2 n3 5 a1 a2 a3 b1 b2 b3 or in an alternate representation 2

0 4 n3 n2

n3 0 n1

38 9 2 0 n2 < a 1 = n1 5 a2 ¼ 4 n3 : ; 0 a3 n2

n3 0 n1

38 9 n 2 < b1 = n1 5 b2 : ; 0 b3

34

BASIC THEORY OF DISTRIBUTED POINT SOURCE METHOD (DPSM)

or 2

1 0 0

0 4 1 0

38 9 2 0 0 < a1 = 0 5 a2 ¼ 4 1 : ; 0 a3 0

1 0 0

38 9 0 < b1 = 0 5 b2 : ; 0 b3

or 8 9 8 9 < a2 = < b2 = ¼ a b : 1 ; : 1 ; 0 0 or 8 z2 y2 9 8 z2 x2 9 > > = > = < R > < R3 2 ¼ > z1 y 1 > ; > ; : : z1 x1 > R2 R3 Similarly from Eq. (1.39) ½n1

8 9 < a1 = n3 a2 ¼ ½n1 : ; a3

n2

or ½0

8 9 < a1 = 0 a2 ¼ ½0 : ; a3

0

n2

0

ð1:41Þ

8 9 < b1 = n3 b2 : ; b3 8 9 < b1 = 0 b2 : ; b3

or z3 y3 z3 x3 ¼ R2 R3

ð1:42Þ

y1 ðx3 z3 Þ x1 ðz3 y3 Þ x3 2z3 þ y3 y2 ðx3 z3 Þ x2 ðz3 y3 Þ z2 ¼ x3 2z3 þ y3

ð1:43Þ

a3 ¼ b3 ) Solving the above equations z1 ¼

Note that if point C is on the x1 x2 plane, then y3 ¼ 0, and for ﬂuid 1, x3 is between 0 and z3 ; therefore, the denominator of Eq. (1.43) should never become zero. After obtaining z1 and z2, from Eq. (1.43), the lengths R2 and R3 can be easily obtained from Eq. (1.37). To evaluate R1, one does not need z1 and z2. It is simply equal to 1

R1 ¼ fðx1 y1 Þ2 þ ðx2 y2 Þ2 þ ðx3 y3 Þ2 g2

ð1:44Þ

35

1.3 EXAMPLES FROM ULTRASONIC TRANSDUCER MODELING

Then the pressure ﬁeld at point P can be obtained from the following equation: ð ð iorf vo expðikf R1 Þ iorf vo R expfikf ðR2 þ R3 Þg dS dS ð1:45Þ pP ðxÞ ¼ 2p 2p R1 R2 þ R3 S

S

In Eq. (1.45) the ﬁrst integral corresponds to the wave path CP and the second integral corresponds to the wave path CTP. Note that both these integrals are similar to the expression given in Eq. (1.16); the only difference is that in the second integral expression, the reﬂection coefﬁcient R has been included because this wave reaches point P after being reﬂected at the interface. The expression of the reﬂection coefﬁcient R is given in Eq. (1.208) of Kundu (2004). R¼

r2 cf2 cos y1 r1 cf1 cos y2 r2 cf2 cos y1 þ r1 cf1 cos y2

In this case, the incident angle is y, transmitted angle is y2, ﬂuid densities are rf and rf2 , and acoustic wave speeds in the two ﬂuids are cf and cf2 . Then, the transmitted angle y2 can be expressed in terms of the incident angle y using Snell’s law (see Eq. (1.204) of Kundu (2004)). qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ cos y2 ¼ 1 sin2 y2 ¼

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ ( 2 2 )12 cf2 sin y 2 cf2 cf2 þ cos y 1 ¼ 1 cf cf cf ð1:46Þ

Therefore, R takes the following form: r2 cf2 cos y rcf R¼

r2 cf2 cos y þ rcf

c2 c2 1 f22 þ f22 cos2 y cf cf c2 c2 1 f22 þ f22 cos2 y cf cf

12

12

ð1:47Þ

In the above equation cos y can be obtained from the relation given below. ^ is given by Dot product between the unit vectors ^ n and A ^ ¼ j^ ^ j cos y ¼ n1 a1 þ n2 a2 þ n3 a3 ^ nA njjA z3 y3 ) cos y ¼ n3 a3 ¼ a3 ¼ R2

ð1:48Þ

1.3.4.2 Pressure ﬁeld computation in ﬂuid 2 at point Q Let us deﬁne two vectors A and C, where A ¼ CT and C ¼ TQ; then, A ¼ ðz1 y1 Þe1 þ ðz2 y2 Þe2 þ ðz3 y3 Þe3 C ¼ ðx1 z1 Þe1 þ ðx2 z2 Þe2 þ ðx3 z3 Þe3

ð1:49Þ

36

BASIC THEORY OF DISTRIBUTED POINT SOURCE METHOD (DPSM)

Note that the magnitudes of vectors A and C are R2 and R3 , respectively. 1

R2 ¼ fðz1 y1 Þ2 þ ðz2 y2 Þ2 þ ðz3 y3 Þ2 g2 1

R3 ¼ fðx1 z1 Þ2 þ ðx2 z2 Þ2 þ ðx3 z3 Þ2 g2

ð1:50Þ

^ ¼ and C ^¼ Unit vectors A R3 R2 C

A

Unit vector ^ n normal to the interface is given by 8 9 8 9 <0= < n1 = n ¼ n2 n^ ¼ 0 . Note that for an inclined interface, ^ : ; : ; 1 n3 From the problem geometry shown in Figure 1.27, the following equations are obtained: ^ j ¼ sin y j^ nA ^ ¼ cos y ^ nA

ð1:51Þ

^ j ¼ sin y2 j^ nC ^ ¼ cos y2 ^ nC Let 8 9 8 9 < a1 = < c1 = ^ ¼ c2 ^ ¼ a2 ; C A : ; : ; a3 c3 From Eqs. (1.51) and (1.49) x3 z3 R3 z3 y3 a3 ¼ cos y ¼ R2 c3 ¼ cos y2 ¼

ð1:52Þ

and 2

e2

e1

6 ^ ¼ det6 n1 ^ nC 4

n2

c1

c2

2

0 6 ¼6 4 1 0

1 0 0

e3

3

2

0

7 6 6 n3 7 5 ¼ 4 n3

n3 0

c3 n2 n1 9 38 9 8 c2 > 0 > c1 > > > > > > = 7< = < ¼ 07 c c 5> 2 > > 1 > > > ; > : ; : > 0 c3 0

^ j2 ¼ c2 þ c2 ¼ sin2 y2 ) j^ nC 1 2

38 9 > > c1 > > 7< = 7 n1 5 c2 > > > : > ; 0 c3 n2

ð1:53Þ

1.3 EXAMPLES FROM ULTRASONIC TRANSDUCER MODELING

37

or sin2 y2 ¼

ðx1 z1 Þ2 þ ðx2 z2 Þ2 R23

ð1:54Þ

Similarly, 2 2 ^ j2 ¼ a2 þ a2 ¼ ðz1 y1 Þ þ ðz2 y2 Þ ¼ sin2 y j^ nA 1 2 R22

ð1:55Þ

^, ^ ^ are located on the same plane Because A n, and C ^ ¼ ^eS sin y ^ nA ^ ¼ ^eS sin y2 ^ nC

ð1:56Þ

^ , n^, and C ^. where ^eS is the unit vector normal to the plane containing A From Eq. (1.56) and Snell’s law (see Eq. (1.204) of Kundu(2004)) one can write ^ ^ ^ ^ nA nC ¼ sin y sin y2 ^ ^ ^ ^ nA nC ) ¼ cf cf2

ð1:57Þ

8 9 8 9 a2 > c2 > > > < = < = 1 1 a1 c1 ¼ > > cf > : ; cf2 > : ; 0 0 8 z y 9 8 x z 9 1> 1 1> > > 1 > > > > > c R > > c R > > > > f2 3 > f 2 = = < < z y x z 2 2 2 2 ¼ ) > > > > > > > cf2 R3 > > > > > > > c f R2 > > ; ; : : 0 0

ð1:58Þ

or

z1 and z2 can be obtained from Eq. (1.58) by minimizing the following error function: E¼

z1 y1 z1 x1 2 z2 y2 z2 x2 2 þ þ þ c f R2 cf2 R3 cf R 2 cf2 R3

ð1:59Þ

E can be minimized by some optimization technique such as simplex algorithm. In MATLAB code, ‘‘fminsearch’’ function can be used for this purpose. After evaluating

38

BASIC THEORY OF DISTRIBUTED POINT SOURCE METHOD (DPSM)

z1 and z2 , the pressure at point Q can be obtained from the following equation: ð iorf vo Tp expbikf R2 þ kf2 R3 Þc ﬃ dS pðx1 ; x2 ; x3 Þ ¼ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃsﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2p cf2 cf2 cos2 y S R2 þ R 3 R2 þ R 3 cf cf cos2 y2

ð1:60Þ

The numerator of the above integrand is similar to that in Eq. (1.16), the only difference is that in Eq. (1.60) the numerator has been multiplied by Tp, the transmission coefﬁcient at the interface, and the argument of the exponential term has two entries, corresponding to the ray paths CT and TQ. However, the denominator of Eq. (1.60) is much more complex in comparison to the one given in Eq. (1.16). Derivation of this denominator expression can be found in Schmerr (1998). The transmission coefﬁcient is given in Eq. (1.208) of Kundu(2004). Tp ¼

2rf2 cf2 cos y2 rf2 cf2 cos y1 þ rf cf cos y2

As mentioned above, in this case, the incident angle is y, angle of transmission is y2 , ﬂuid densities are rf and rf2 , and acoustic wave speeds in the two ﬂuids are cf and cf2 . Angle y2 can be expressed in terms of the incident angle y (see Eq. (1.46)); then, Tp takes the following form, Tp ¼

2rf2 cf2 cos y

12 c2f2 c2f2 2 rf2 cf2 cos y þ rf cf 1 2 þ 2 cos y cf cf

ð1:61Þ

1.3.5 DPSM technique for ultrasonic ﬁeld modeling in nonhomogeneous ﬂuid The steps discussed in Section 1.2.4 are based on the Rayleigh–Sommerﬁeld integral representation for the pressure ﬁeld computation in ﬂuids 1 and 2. In this section, an alternative technique based on DPSM, developed by Placko and Kundu (2001) for ultrasonic problems, is generalized to include the nonhomogeneous ﬂuid case. In the DPSM technique, the interface is replaced by a layer of equivalent point sources instead of tracing the rays from the transducer face to the point of interest in ﬂuid media 1 or 2, as done in Section 1.3.4. 1.3.5.1 Field computation in ﬂuid 1 The ﬁeld in ﬂuid 1 is computed by superimposing the contributions of the two layers of point sources distributed over the transducer face and the interface, respectively, as shown in Figure 1.28. The two layers of the sources are located at a small distance rS away from the transducer face and interface, respectively, such that the apex of the spheres (of radius rS ) touch the transducer face or interface, as shown in the ﬁgure.

1.3 EXAMPLES FROM ULTRASONIC TRANSDUCER MODELING

39

x1 x x3

Fluid 1

Fluid 2

rS rS

Figure 1.28 Point sources (at the center of small circles) for computing ultrasonic ﬁeld in ﬂuid 1

The strength of the point sources distributed along the transducer surface can be obtained from Eq. (1.25g) or (1.25i). For ﬁnding the strength of the point sources attached to the interface, velocity components at the interface due to the reﬂected waves at the interface are to be matched, as described below. As shown in Figure 1.29, any point P in ﬂuid 1 can receive only two rays, 1 and 2, from a single point source on the transducer surface. Ray 1 is the direct ray reaching P, and ray 2 arrives at P after being reﬂected at the interface. The total ultrasonic ﬁeld at point P can be obtained by superimposing the contributions of a number of point sources ðym Þ distributed over the transducer surface. The total ﬁeld at P, generated only by the reﬂected rays (ray number 2), from all the point sources on the transducer surface should be the same as the total contributions of all point sources distributed over the interface. Let us take a point P at xn on the interface. In Figure 1.29, the point P is shown very close to the interface. Let us assume that this point is now moved to the interface. Let there be N point sources ðym ; m ¼ 1; 2; . . . ; NÞ on the transducer surface and M points ðxn ; n ¼ 1; 2; . . . ; MÞ on the interface where the boundary conditions should be satisﬁed. If the boundary conditions are to be satisﬁed for the three components of velocity at all M points, then there are a total of 3M boundary conditions. It should be noted here that for nonviscous ﬂuids, matching of the normal velocity component only is sufﬁcient. In that case, from M points M boundary conditions will be obtained. x1 P 1 ym

Fluid 1

2

x3

xn

Fluid 2

rS

Figure 1.29 Point P can receive two rays, 1 (direct ray) and 2 (reﬂected from the interface), from a single point source ym :

40

BASIC THEORY OF DISTRIBUTED POINT SOURCE METHOD (DPSM)

Velocity components at M interface points due to ray 1 only (ignoring reﬂection) can be easily obtained from the N triplet or elementary simple point sources in the following manner (see Eq. (1.25j)): ViT ¼ MiTS AS

ð1:62Þ

where ViT is the ð3M 1Þ or ðM 1Þ vector of the velocity components at the target points ðxn Þ on the interface due to the incident beam only. AS is the ð3N 1Þ or ðN 1Þ vector of the point source strengths on the transducer surface. For the triplet source, there are three point sources inside every small sphere. The following formulation is given for triplet source only but can be easily modiﬁed to simple elemental sources used for nonviscous ﬂuids. Because the normal velocity component (v0 ) at the transducer surface is known, AS can be obtained from Eq. (1.25g) or (1.25i). MiTS is the ð3M 3NÞ matrix that relates the two vectors ViT and AS of Eq. (1.62). Note that the components of MiTS are identical to those for MTS given in Eq. (1.25k). In Eq. (1.62) and in subsequent equations, the superscripts and subscripts have the following meanings: Superscripts i – direct incident ray r – reﬂected ray t – transmitted ray Subscripts S – ultrasonic source or transducer points I – interface points T – target points or observation points (these points can be placed anywhere—in ﬂuid 1, ﬂuid 2, on the transducer surface, or on the interface). For the reﬂected ﬁeld computation at the interface, the velocity vector and the source strength vector are computed in a similar manner. VrT ¼ MrTS AS

ð1:63Þ

where VrT is the ð3M 1Þ vector of the three velocity components at the target points ðxn Þ on the interface due to the reﬂected beam only (ray 2 of Fig. 1.29). MrTS is the ð3M 3NÞ matrix that relates the two vectors VrT and AS of Eq. (1.63). Note that components of MrTS can be obtained by multiplying MiTS by appropriate reﬂection coefﬁcients for velocity ﬁelds. Next we would like to obtain the same VrT vector from the 3M point sources distributed along the interface. Within each sphere shown in Figure 1.29, three point

1.3 EXAMPLES FROM ULTRASONIC TRANSDUCER MODELING

41

sources or triplet (see Fig. 1.25) are placed; thus, from M spheres one gets 3M sources. Note that the interface point sources are located around the centers of the small spheres of Figure 1.29, and three sources of each triplet are placed parallel to the interface. Points xn are located on the surface of the small spheres. VrT at M points, generated by the 3M sources at the interface, can be written as VrT ¼ MiTI AI

ð1:64Þ

where VrT is the ð3M 1Þ vector, same as in Eq. (1.63). AI is the ð3M 1Þ vector of the strength of interface sources; this vector is unknown. MiTI is the ð3M 3MÞ matrix that relates the two vectors VrT and AI of Eq. (1.64). Note that the components of MiTI are similar to those for MSS given in Eq. (1.25e) and MTS of Eq. (1.25k). The two variables xnjm and rmn for MiTI computation can be obtained after knowing the point source coordinates zm ðm ¼ 1; 2; . . . ; 3MÞ distributed along the interface, and coordinates xn ðn ¼ 1; 2; . . . ; MÞ of the interface points. From Eqs. (1.63) and (1.64) AI ¼ ½MiTI 1 VrT ¼ ð½MiTI 1 MrTS ÞAS

ð1:65Þ

Eq. (1.65) gives the strength of the interface sources. After obtaining the interface source strengths, the ultrasonic ﬁeld at any set of target points xn ðn ¼ 1; 2; . . . ; NT Þ between the transducer face and interface can be obtained by adding the contributions of the incident waves from the two layers of point sources at the transducer face and interface as shown in Figure 1.28. In other words, the ﬁeld at any point can be obtained by adding the expressions given in Eqs. (1.62) and (1.64). VT ¼ ViT þ VrT ¼ MiTS AS þ MiTI AI

ð1:66Þ

The only difference between the two components of Eq. (1.66) and those in Eqs. (1.62) and (1.64) is in the deﬁnitions of xnjm and rmn for MiTI and MiTS . In Eqs. (1.62) and (1.64), the target points are located on the interface, whereas in Eq. (1.66) the target points are in between the transducer face and the interface. Therefore, the values of xnjm and rmn will change accordingly. As mentioned earlier rmn is the distance between the mth point source and nth target point, xnjm are the three components of rmn . 1.3.5.1.1 Approximations in computing the ﬁeld The approximation of the above section in deriving Eq. (1.66) is that the presence of the interface does not affect the source strength vector AS. Note that AS of Eq. (1.62) is computed from Eq. (1.25g) or (1.25i). With this assumption, the velocity vector computed on the transducer surface, using Eq. (1.66), will give a different value than v0 . If the interface is close to the transducer surface, then the transducer surface velocity will be signiﬁcantly different from v0 due to the interface effect. To make sure that the velocity vector on the interface is equal to a constant value (v0 ) in the x3 direction and zero in x1 and x2 directions, the following formulation is followed.

42

BASIC THEORY OF DISTRIBUTED POINT SOURCE METHOD (DPSM)

Similar to the previous section, it is again assumed that there are N triplet sources on the transducer surface and M sources along the interface. The velocity vector on the transducer surface, due to the point sources representing the transducer effect only, can be obtained from Eq. (2.49). ViS ¼ MiSS AS

ð1:66aÞ

where ViS is the ð3N 1Þ vector of the velocity components at the transducer surface. AS is the ð3N 1Þ vector of the point source strengths distributed over the transducer face, and MiSS is the ð3N 3NÞ matrix, identical to the one given in Eq. (1.25e). In the same manner, velocity components on the transducer surface due to the interface sources are given by, VrS ¼ MiSI AI

ð1:66bÞ

The above equation is obtained from Eq. (1.64), when the target points are placed on the transducer surface. Here, VrS is a ð3N 1Þ vector of the velocity components at N points on the transducer surface, AI is the ð3M 1Þ vector of the interface source strengths, and MiSI is the ð3N 3MÞ matrix, similar to the one given in Eq. (1.25k). Adding Eqs. (1.66a) and (1.66b) the total velocity at the transducer surface is obtained. VS ¼ ViS þ VrS ¼ MiSS AS þ MiSI AI

ð1:66cÞ

Substituting Eq. (1.65) into Eq. (1.66c): VS ¼ MiSS AS þ MiSI AI ¼ MiSS AS þ MiSI ½MiTI 1 MrTS AS ) VS ¼ ½MiSS þ MiSI ½MiTI 1 MrTS AS

ð1:66dÞ

or AS ¼ ½MiSS þ MiSI ½MiTI 1 MrTS 1 VS

ð1:66eÞ

where VS ¼ ½ 0

0

v0

0

0 v0

::::::::::::

0

0

v0 T

ð1:66fÞ

If AS is computed from Eq. (1.66e) instead of Eq. (1.25g), then the constant velocity at the transducer surface is guaranteed even when the interface is located very close to the transducer surface. 1.3.5.2 Field in ﬂuid 2 For ultrasonic ﬁeld computation in ﬂuid 2, only one layer of point sources, adjacent to the interface, is considered as shown in Figure 1.30. The total ﬁeld at x should be the superposition of ﬁelds generated by all these point sources, located at various distances from x, as shown by the dotted lines in Figure 1.30. Strengths of these sources are obtained, as before, by equating the velocity components computed by the point sources, distributed along the interface, to those obtained from the transmitted wave contribution.

43

1.3 EXAMPLES FROM ULTRASONIC TRANSDUCER MODELING

x1 x x3

Fluid 1

Fluid 2

rS

Figure 1.30 One layer of point sources (at the center of small circles) for computing ultrasonic ﬁeld in ﬂuid 2

Following similar analysis as outlined in section 2.5.1, strengths of the interface point sources in this case can be obtained from the relation (see Eq. (1.65)). AI ¼ ½MiTI 1 VtT ¼ ð½MiTI 1 MtTS ÞAS

ð1:67Þ

where AI is the ð3M 1Þ vector of the interface source strengths, MiTI is the ð3M 3MÞ matrix that relates the two vectors VtT and AI , and MtTS is the ð3M 3NÞ matrix relating the velocity vector VtT at the interface points and AS , the source strength vector for point sources, distributed along the transducer face. Note that in this case, the equations relating the interface velocity components to the transducer source strengths and interface source strengths are similar to Eqs. (1.63) and (1.64) and can be written as VtT ¼ MtTS AS

ð1:68Þ

VtT ¼ MiTI AI

ð1:69Þ

After computing the interface source strengths using Eq. (1.67), the ultrasonic ﬁeld at any new target points xn ðn ¼ 1; 2; . . . ; NT Þ on the right side of the interface (or in ﬂuid 2) can be obtained from Eq. (1.69). While computing the ﬁeld at new points, appropriate changes in the values of xnjm and rmn appearing in matrix MiTI should be taken into account. As mentioned earlier, rmn is the distance between the mth point source and nth target point, xnjm are the three components of rmn . 1.3.6 Ultrasonic ﬁeld in the presence of a scatterer DPSM technique is then applied to model ultrasonic ﬁeld near a scatterer of ﬁnite dimensions for which no closed-form analytical solution exists. Problem geometry showing the transducer and scatterer is given in Figure 1.31. To compute the ultrasonic ﬁeld in front of a scatterer (left of the scatterer), point sources are distributed along the transducer face and the solid–ﬂuid interface as well as along the imaginary interface (extending the front face of the solid scatterer, shown by the dotted line in Figure 1.31). Triplet sources are located around the centres of the

44

BASIC THEORY OF DISTRIBUTED POINT SOURCE METHOD (DPSM)

x1

xn Solid scatterer

Fluid ym

Fluid x3 zj

Figure 1.31 A ﬁnite solid scatterer immersed in a ﬂuid in front of a transducer face - two layers of point source contribute to the ultrasonic ﬁeld in between the transducer and the scatterer.

small spheres. Strength of the point sources on the transducer face is known from the normal velocity component v0 of the transducer surface (Eq. (1.25i) or (1.66e)). However, strength of the point sources distributed along the real and imaginary interface is not known. This is carried out in a manner similar to the one described in Section 2.5.1. The only difference here is that in Eq. (1.63), MrTS must be obtained by multiplying MiTS by appropriate reﬂection coefﬁcients. The technique to compute the reﬂection coefﬁcient for this case differs from the one given in Section 2.5.1. In the previous case, the same expression of the reﬂection coefﬁcient (Eq. (1.47)) was used for all interface points xn . However, for this problem geometry when the interface points xn are located on the scatterer surface, then the reﬂection coefﬁcient for a solid plate immersed in a ﬂuid (see Section 1.2.17 of Kundu(2004)) should be used. However, when the interface points xn are located on the dotted line, along the imaginary interface between two identical ﬂuids, then the reﬂection coefﬁcient should be zero. Except for this difference in the reﬂection coefﬁcient deﬁnition, the steps to compute the interface source strengths for these two problem geometries are identical, and the source strength vector can be obtained from Eq. (1.65) AI ¼ ½MiTI 1 VrT ¼ ð½MiTI 1 MrTS ÞAS For computing the ultrasonic ﬁeld behind the scatterer, or on the right side of the dotted line, the point sources should be taken as shown in Figure 1.32. Note that now some of the point sources are aligned with the right edge of the scatterer whereas the

x1

Solid scatterer

Fluid ym

xn Fluid x3 zj

Figure 1.32 A ﬁnite solid scatterer immersed in a ﬂuid in front of a transducer face - only the right layer of point sources contribute to the ultrasonic ﬁeld in the ﬂuid on the right side of the scatterer.

1.3 EXAMPLES FROM ULTRASONIC TRANSDUCER MODELING

45

rest are aligned with the imaginary interface along the left edge, and marked by the dotted line. Of course, for thin scatterer these two planes coincide. Following similar steps as in Section 2.5.2, the strengths of the interface point sources in this case can be obtained from Eq. (1.67), AI ¼ ½MiTI 1 VtT ¼ ð½MiTI 1 MtTS ÞAS where AI is the ð3M 1Þ vector of the interface source strengths, MiTI is the ð3M 3MÞ matrix that relates the velocity vector VtT at the interface points xn to the interface source strengths AI , and MtTS is the ð3M 3NÞ matrix that relates the velocity vector VtT at the interface points to the transducer source strength vector AS. Note that in this case, equations relating the interface velocity components to the transducer source strengths and interface source strengths are similar to Eqs. (1.63) and (1.64) and can be written as VtT ¼ MtTS AS VtT ¼ MiTI AI After computing the interface source strengths using Eq. (2.54), the ultrasonic ﬁeld at any new target points xn ðn ¼ 1; 2; . . . ; NT Þ on the right side of the interface can be obtained from Eq. (1.69). For computing the ﬁeld at new points, appropriate changes in the values of xnjm and rmn appearing in matrix MiTI should be taken into account. As mentioned earlier, rmn is the distance between the mth point source and nth target point and xnjm are the three components of rmn . 1.3.7 Numerical results Sections 1.3.1–1.3.6 describe the theory of the ultrasonic ﬁeld modeling by using the DPSM technique in homogeneous and nonhomogeneous ﬂuids. Based on this theory the authors have developed a number of MATLAB computer codes to model the ultrasonic ﬁelds generated by the ultrasonic transducers of ﬁnite dimension, which are immersed in a ﬂuid. In the simplest case, the transducer is immersed in a homogeneous ﬂuid. More complex problem geometries involve two ﬂuids with a plane interface and a solid scatterer of ﬁnite size immersed in a homogeneous ﬂuid. The numerical results clearly show how the ultrasonic ﬁeld decays as the distance from the transducer increases and the ﬁeld becomes more collimated as the size of the transducer increases. It also shows that the ﬁeld is reﬂected and transmitted at an interface, and how a ﬁnite size scatterer can give rise to the reﬂection and transmission as well as diffraction of the incident ﬁeld. 1.3.7.1 Ultrasonic ﬁeld in a homogeneous ﬂuid In this example the ultrasonic ﬁeld in front of a ﬂat circular, ﬂat rectangular, and concave circular transducer faces are generated. The transducer front face geometries are shown in Figure 1.33. The area of the ﬂat transducer face is 5.76 mm2 for both circular and square transducers. Note that a 2.7 mm diameter circular transducer gives an area of 5.76 mm2. A concave

46

BASIC THEORY OF DISTRIBUTED POINT SOURCE METHOD (DPSM)

Figure 1.33 Distribution of the point sources just behind the front face (see Fig. 1.24) of a ﬂat circular (top left) and ﬂat square (top right) transducer. Bottom ﬁgure - side view of the concave front face of a transducer

transducer face has different dimensions; its diameter is 12.7 mm (0.5 in.) and its radius of curvature is 8 mm, as shown in Figure 1.33. All of the dimensions in the ﬁgure are given in meter, but the scales are not necessarily the same in the horizontal and vertical directions. These three transducers are denoted as circular, square, and focused transducers. Note that the ﬂat transducer face is located on the xy plane. We would like to compute the ultrasonic ﬁeld in front of the transducer face in the xz plane or the yz plane. Both xz and yz planes are planes of symmetry and are perpendicular to each other. The ultrasonic pressure ﬁeld variations along the xz and yz planes in front of the transducer face are shown in Figure 1.34 for 5 MHz frequency of the transducers. The top-left and top-right images of Figure 1.34 are for the circular and square transducers, respectively. Note that the ﬁeld is less collimated for the square transducer. For both transducer geometries the ultrasonic ﬁeld has a number of peaks (or maxima) and dips (or minima) along the central axis (z-axis) of the transducer near the transducer face. The peaks and dips are a result of constructive and destructive interferences between the ﬁelds generated by different point sources on the transducer face. For the concave transducer the ﬁeld intensity increases as we approach the focal point. Note that the focal point is at a distance of 8 mm from the transducer face, whereas the plot is shown for a distance varying from 3 to 6 mm. It should be mentioned here that the focused transducer surface area is 21 times that of the ﬂat transducers. To maintain the same spacing between neighboring point

1.3 EXAMPLES FROM ULTRASONIC TRANSDUCER MODELING

47

Figure 1.34 Ultrasonic pressure ﬁelds generated by a circular (top left), a square (top right), and a concave circular (bottom) transducer. Transducer face geometries are shown in Fig.1.33. Transducer frequency is 1 MHz. The surface area of the ﬂat transducers is 5 mm2. The concave transducer has a radius of curvature of 8 mm, and the diameter of its periphery is 12.7 mm. The number of point sources is 259 for the top-left ﬁgure, 256 for the top-right ﬁgure, and 6470 for the bottom ﬁgure. The ultrasonic ﬁeld is plotted up to an axial distance of 6 mm. Note that the focal point for the concave transducer is at a distance of 8 mm, which is beyond the plotted region.

sources (see Section 1.3.2.3), the number of point sources for the focused transducer is made about 25 times that of the ﬂat transducers. Thus, the number of point sources for the focused transducer is 6470 whereas for the ﬂat transducers the two numbers are 256 and 259, respectively. Variations of the pressure ﬁeld along the z-axis, in front of the transducer face, are clearly shown in Figure 1.35. The top two ﬁgures of 1.35 are for the circular and square transducers, and the bottom ﬁgure is for the focused transducer. The analytical solutions (Eq. (1.16b) for the ﬂat circular transducer and Eq. (1.32) for the focused transducer) give results that are very close to the one obtained by the DPSM technique (Eq. (1.25j)), see Figure 1.35. Three peaks between 0 and 4 mm along the z-axis in Figure 1.35 correspond for both circular and square transducers to the three bright red dots in Figure 1.34 along the central axis of the ultrasonic beam. Example 1.3.4 Check if Eq. (1.30) is satisﬁed for the ﬂat circular cylinder with 259 point sources for 1 MHz signal.

48

BASIC THEORY OF DISTRIBUTED POINT SOURCE METHOD (DPSM)

Figure 1.35 Ultrasonic pressure ﬁelds generated by circular (left top), square (right top) and focused (bottom) transducers. Thin dashed curves in top left and bottom ﬁgures have been generated by the closed form expressions [Eq.(1.16b) for the ﬂat circular transducer and Eq.(1.32) for the focused transducer]. Continuous curves are obtained by the DPSM technique.

Solution The area for each point source ðAS Þ is computed from the surface area of the transducer face in the following manner: AS ¼

pr 2 p1:272 ¼ ¼ 0:01956 mm2 n 259

Because AS ¼ 2prS2 (see Fig. 1.24 and also the discussion on Eqs. (1.29) and (1.30)), we can write, rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 0:01956 ¼ 0:0558 mm rS ¼ 2p From the wavelength ðlf Þ, wave speed ðcf Þ, and frequency (f) relation, we get the wavelength in water for 1 MHz frequency lf ¼

cf 1:5 106 ¼ ¼ 1:5 mm f 106

1.3 EXAMPLES FROM ULTRASONIC TRANSDUCER MODELING

49

Figure 1.36 Ultrasonic pressure ﬁelds in the xz plane are generated by circular (top left), square (top right) and focused (bottom) transducers. The central axis of the transducer coincides with the z-axis.

From Eq. (1.30), rS

1:5 mm 2p

) rS 0:24 mm Because rS ¼ 0:0558 mm, the above condition is satisﬁed. Pressure ﬁeld variations in front of the transducer face along the xz plane for the three transducer geometries of Figure 1.33 are shown in Figure 1.36. In this ﬁgure one can clearly see how the pressure ﬁeld oscillates near the transducer face and decays laterally (in the positive and negative x directions) and axially (in the z direction) for the ﬂat transducers. For the focused transducer a clear peak can be observed near the focal point. Contour plots for the pressure ﬁeld variations in the xz plane for the same three transducers are shown in Figure 1.37. Figure 1.38 shows the effect of increasing the number of point sources. As more sources are considered, the computed ﬁeld becomes smoother. Because the oscillating velocity amplitudes at the transducer surface are different for the left and right columns of Figure 1.38 and so are the scales along the vertical axes, the numerical values in the two columns should not be compared. However, a comparison of the

50

BASIC THEORY OF DISTRIBUTED POINT SOURCE METHOD (DPSM)

Figure 1.37 Contour plots for the ultrasonic pressure ﬁelds in the xz plane are generated by circular (top left), square (top right) and focused (bottom) transducers. The same as Fig. 1.36 but contour plots are given here instead of surface plots.

relative variations of the pressure ﬁelds between the two columns clearly demonstrates the effect of the increasing number of point sources on the computed pressure ﬁeld. The effect of the presence of a small circular hole at the center of a 2.54 mm (0.1 in.) diameter ﬂat circular transducer is shown in Figure 1.39. The pressure ﬁeld in the xz plane (top-right plot of Fig. 1.39) is very similar to the one given in Figure 1.37 (top left). Therefore, a small hole at the center of a ﬂat circular transducer does not signiﬁcantly affect the generated pressure ﬁeld in the ﬂuid. The bottom two plots of Figure 1.39 show the pressure and normal velocity (Vz) variations in the xy plane, very close to the transducer surface. It should be noted here that an oscillating pattern is present in the pressure plot but not in the velocity plot. Theoretically, the velocity component should be a constant and equal to v0 on the transducer surface, see Eq. (1.16) and (1.25c). However, a small level of noise in the velocity plot exists due to the numerical error. 1.3.7.2 Ultrasonic ﬁeld in a nonhomogeneous ﬂuid – DPSM technique The pressure ﬁeld generated by a circular transducer placed parallel to the interface of two ﬂuids is computed. As before, the transducer frequency is set at 1 MHz and its diameter is 2.54 mm. The distance between the transducer face and the interface between two ﬂuids is 10 mm. The transducer is immersed in ﬂuid 1 (P-wave

1.3 EXAMPLES FROM ULTRASONIC TRANSDUCER MODELING

51

Figure 1.38 Pressure variations in the xz plane (top row) and xy plane (bottom row), close to the transducer surface for a rectangular transducer. Left (256 point sources) and right (1296 point sources) columns correspond to two different discretizations of the transducer surface.

speed ¼ 1.49 km/s, density ¼ 1 g/cc). The P-wave speed and density of ﬂuid 2 are set at 2 km/s and 1.5 g/cc, respectively. One hundred point sources are used to model the transducer surface and four hundred point sources (each point source is a triplet source) model the interface effect, see Figures 1.28 and 1.29. Point sources distributed over the interface, which are also called target sources, are distributed over a square area of 20 mm side length. Note that the interface source positions change (see Figs. 1.29 and 1.30) when computing the acoustic ﬁelds in ﬂuids 1 and 2. Pressure ﬁelds computed in the two ﬂuids are plotted in Figure 1.40. Note how the pressure variation in the xy plane is changed, as the distance of the observation (xy) plane from the transducer surface is increased from zero (middle-left ﬁgure) to 10 mm (bottom-left ﬁgure). Pressure variations in the xz plane in both ﬂuids are shown as a contour plot (top-right) and a surface plot (middle-right). Pressure along the z-axis is plotted in the bottom-right ﬁgure. Oscillations in the acoustic pressure in ﬂuid 1 are the effects of constructive and destructive interferences between two rays that can reach a point in ﬂuid 1—the ﬁrst ray travels from the transducer face to the point of interest and the second ray reaches the same point after being reﬂected at the interface, see Figure 1.27. Pressure and velocity variations in the two ﬂuids for an inclined transducer (inclination angle ¼ 20 ) are shown in Figure 1.41. The ﬂuid properties and the

52

BASIC THEORY OF DISTRIBUTED POINT SOURCE METHOD (DPSM)

Figure 1.39 Top left - Circular transducer with a small hole at the center is modeled by 1040 point sources; top right - pressure ﬁeld in the xz plane; bottom left - pressure ﬁeld in the xy plane, close to the transducer surface; bottom right - normal velocity component (Vz ) in the xy plane, close to the transducer surface.

transducer dimension are the same as those in Figures 1.40 and 1.41. The only difference between the problem geometries of Figures 1.40 and 1.41 is that in Figure 1.40 the transducer face is parallel to the interface and in Figure 1.41 it is inclined. From Snell’s law the transmission angle in the second ﬂuid can be computed. yT ¼ sin1

2 sinð20Þ 1:49

¼ 27:33

Incident and transmission angles, measured from the middle-left plot of Figure 1.41, give values close to 20 and 27.33 , respectively. Note that the Vz variation (bottomright) and the pressure variation (middle-right) in the two ﬂuids are similar. 1.3.7.3 Ultrasonic ﬁeld in a nonhomogeneous ﬂuid – surface integral method The ultrasonic ﬁeld in the nonhomogeneous ﬂuid can also be computed by the conventional surface integral technique instead of the DPSM technique. Unlike the DPSM technique, in the surface integral method the ﬂuid–ﬂuid interface is not modeled by the

1.3 EXAMPLES FROM ULTRASONIC TRANSDUCER MODELING

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Figure 1.40 Circular transducer in a non-homogeneous ﬂuid. Top left: 2.54 mm diameter transducer modeled by 100 point sources; middle left - acoustic pressure in the xy plane, close to the transducer surface (z ~ 0 mm); bottom left - acoustic pressure in the xy plane in ﬂuid 1 at the interface position (z = 10 mm); top right - contour plot of the pressure variation in ﬂuid 1 (z = 0 to 10 mm) and ﬂuid 2 (z = 10 to 20 mm); middle right - surface plot of the pressure variation in ﬂuid 1 (z = 0 to 10 mm) and ﬂuid 2 (z = 10 to 20 mm); bottom right - pressure variation along the z-axis in ﬂuid 1 (z = 0 to 10 mm) and ﬂuid 2 (z = 10 to 20 mm).

distributed point sources. Here, only the transducer surface is discretized into the distributed point sources. In this method, the pressure ﬁelds in ﬂuids 1 and 2 are computed by Eqs. (1.45) and (1.60), respectively. The theory of this computation is given in Section 2.4 whereas the theory of the DPSM computation is given in Section 2.5. Figure 1.42 shows the pressure ﬁeld along the z-axis in ﬂuids 1 and 2, computed by the surface integral technique. A comparison of Figure 1.42 with the bottom-right plot of Figure 1.40 shows a perfect matching between the results obtained by these two methods. 1.3.7.4 Ultrasonic ﬁeld in the presence of a ﬁnite-size scatterer Following the theory described in Section 1.3.6, a computer code has been developed to compute the ultrasonic pressure ﬁeld in the presence of a ﬁnite-size scatterer. This computer code is used to solve the problem of ultrasonic ﬁeld scattering by a ﬁnite-size steel plate, immersed in water. The problem geometry is shown in Figure 1.43. A ﬁnite-size thin steel plate (1 mm thick) is placed at the interface between the two ﬂuids—ﬂuid 1 and ﬂuid 2. The results are presented for the case in which both ﬂuids are water. Scattered ﬁelds are computed for a large plate (20 mm 20 mm, shown by the dashed line in Fig 1.43), and for a small plate (5 mm 5 mm shown by the solid line in Fig 1.43).

54

BASIC THEORY OF DISTRIBUTED POINT SOURCE METHOD (DPSM)

Figure 1.41 Top left: Inclined transducer face modeled with 100 point sources, angle between the interface and the transducer face is 20 ; top right - acoustic pressure in the xy plane in ﬂuid 1 at the interface position (z = 10 mm); middle left - contour plot of the pressure ﬁeld variation in ﬂuid 1 (z = 1 to 10 mm) and ﬂuid 2 (z = 10 to 20 mm); middle right - surface plot of the pressure ﬁeld variation in ﬂuid 1 (z = 1 to 10 mm) and ﬂuid 2 (z = 10 to 20 mm); bottom left - pressure variation along the z-axis in ﬂuid 1 (z = 1 to 10 mm) and ﬂuid 2 (z = 10 to 20 mm); bottom right surface plot of the velocity (Vz ) variation in ﬂuid 1 (z = 1 to 10 mm) and ﬂuid 2 (z = 10 to 20 mm).

The ultrasonic beam, generated by a 6.28 mm diameter cylindrical transducer, strikes the plate at angles yi ¼ 25 and 38.37 . Signal frequency is 1 MHz. The ultrasonic ﬁelds for these two striking angles are computed and plotted in Figures 1.44 and 1.45, respectively (Placko et al., 2003). Material properties for this computation are shown in Table 1.2.

Figure 1.42 Pressure variation along the z-axis in ﬂuid 1 (left ﬁgure) and ﬂuid 2 (right ﬁgure).

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55

Tp Steel plate

Fluid 2

Fluid 1

Rp

i

Transducer

Figure 1.43 A bounded ultrasonic beam from an inclined transducer strikes a ﬁnite steel plate immersed in water at an angle yi (numerical results are provided for ﬂuid 1 = ﬂuid 2 = water).

The plate is placed at a distance of 10 mm from the transducer face. Thirty-two point sources distributed slightly behind the transducer face, as shown in Figure 1.24, model the transducer. Note that in both Figures 1.44 and 1.45, scattered ﬁelds behind the steel plate are much stronger for the small plate. For the large steel plate, very little acoustic energy

Figure 1.44 Total ultrasonic pressure distributions (incident plus scattered ﬁelds) near a steel plate scatterer, immersed in water. Left and right columns are for large (20 mm 20 mm) and small (5 mm 5 mm) plates, respectively. Incident angle is 25 . In top and bottom rows, the same pressure ﬁelds are plotted in two different ways.

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BASIC THEORY OF DISTRIBUTED POINT SOURCE METHOD (DPSM)

Figure 1.45 Same as Fig.1.44, but these plots are for 38.37 angle of incidence

is transmitted into the ﬂuid, behind the plate, because of the large impedance mismatch between the steel plate and the water. It should also be noted that in addition to the transmitted ﬁeld, the reﬂected ﬁeld for the large plate is also relatively weak. The weak specular reﬂection for the large plate is more evident in Figure 1.45. Specular reﬂected beam means the reﬂected beam in the position predicted by the optics theory. The probable cause for a weak specular reﬂection by the large plate is that part of the ultrasonic energy generates leaky guided waves in the plate and propagates away from the striking zone. Therefore, less energy is specularly reﬂected by the larger plate. In Figure 1.46 we can see that for 38.37 incident angle, a guided wave mode is generated; thus, less energy is specularly reﬂected for this incident angle when the plate is large. This phenomenon of guided wave generation at the ﬂuid–solid interface is discussed in detail in chapter 4.

TABLE 1.2. Water and steel properties for the results presented in Figures 1.44 and 1.45 Material and Properties Steel Water

P-wave Speed (km/s)

S-wave Speed (km/s)

Density (g/cc)

5.96 1.49

3.26 -

7.93 1

57

REFERENCES 10.0

8.0

6.0

Cph km/s

4.0

2.0

25o 38.37o

0.0 0.0

1.0

2.0

3.0

4.0

5.0

Frequency (MHz) Figure 1.46 Dispersion curves for 1 mm thick steel plate (properties given in Table 1.2). Phase velocities corresponding to the two striking angles of Figs.1.44 (25 ) and 1.45 (38.37 ) are shown in the ﬁgure. Note that 38.37 incidence is capable of generating guided wave in the plate.

REFERENCES ANSYS Inc., ‘‘ANSYS 5.6 Online Help’’, copyright 1999. Ahmad, R., T., Kundu, and D., Placko, ‘‘Modeling of the Ultrasonic Field of Two Transducers Immersed in a Homogeneous Fluid Using Distributed Point Source Method’’, I2M (Instrumentation, Measurement and Metrology) Journal, Vol. 3, pp. 87–116, 2003. Ahmad, R., T., Kundu, and D., Placko, ‘‘Modeling of Phased Array Transducers’’, Journal of the Acoustical Society of America, Vol. 117, pp. 1762–1776, 2005. Banerjee, S., T., Kundu, and D., Placko, ‘‘Ultrasonic Field Modelling in Multilayered Fluid Structures Using DPSM Technique’’, ASME Journal of Applied Mechanics, Vol. 73 (4), pp. 598–609, 2006. Dufour, I., and D., Placko, ‘‘An Original Approach of Eddy Current Problems Through a Complex Electrical Image Concept’’, IEEE Transactions on Magnetics, Vol. 32 (2), pp. 348– 365, 1996. Kundu, T., Chapter 1: Mechanics of Elastic Waves and Ultrasonic NDE, in Ultrasonic Nondestructive Evaluation: Engineering and Biological Material Characterization, Ed. T. Kundu, Pub. CRC Press, 2004. Lee, J. P., Placko, D., Alnuamimi, N., and Kundu, T., ‘‘Distributed Point Source Method (DPSM) for Modeling Ultrasonic Fields in Homogeneous and Non-Homogeneous Fluid Media in Presence of an Interface’’, First European Workshop on Structural Health Monitoring, Ed. D. Balageas, Ecole Normale Superieure de Cachan, France, July 10–12, 2002. Lemistre, M., and Placko, D., ‘‘Proce´de´ et dispositif e´lectromagne´tique pour le controˆle de structures inte´gre´’’, Patent Application N BFF 040003 ENS Cachan/CNRS/ONERA, March 31, 2004, European extension March 31, 2005.

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Placko, D., ‘‘Dispositifs d’analyse de proﬁl utilisant des capteurs a` courants de Foucault’’, The`se de 3e`me cycle, University Paris XI, April 16, 1984 (in French). O’Neil, H.T., ‘‘Theory of Focusing Radiators’’, Journal of the Acoustical Society of America, Vol. 21 (5), pp. 516–526, 1949. Placko, D., Clergeot, H., and Monteil, F., ‘‘Seam tracking using a linear array of eddy current sensors’’, Proceedings ROVISEC 5, pp. 557–568, Amsterdam, October 1985. Placko, D., Clergeot, H., and Santander, E., ‘‘Physical Modeling of an eddy current sensor designed for real time distance and thickness measurement in galvanisation industry’’, IEEE Transactions on Magnetics, Vol. 25 (4), pp. 2861–2863, 1989. Placko, D., ‘‘Contribution a` la conception de capteurs inductifs pour la robotique industrielle’’, Diploˆme d’Habilitation a` Diriger des Recherches en Sciences, University Paris XI, May 22, 1990 (in French). Placko, D., and T., Kundu, ‘‘A Theoretical Study of Magnetic and Ultrasonic Sensors: Dependence of Magnetic Potential and Acoustic Pressure on the Sensor Geometry’’, Advanced NDE for Structural and Biological Health Monitoring, Proceedings of SPIE, Ed. T. Kundu, SPIE’s 6th Annual International Symposium on NDE for Health Monitoring and Diagnostics, March 4–8, 2001, Newport Beach, California, Vol. 4335, pp. 52–62, 2001. Placko, D., Kundu, T., and Ahmad, R., ‘‘Distributed Point Source Method (DPSM) for Modeling Ultrasonic Fields in Homogeneous and Non-Homogeneous Fluid Media’’, Second International Conference on Theoretical, Applied, Computational and Experimental Mechanics, Kharagpur, India, December 27–30, 2001, Pub. Globenet Computers, India, Published in CD, Paper No. 114, 2001. Placko, D., Liebeaux, N., and Kundu, T., ‘‘Presentation d’une methode generique pour La Modelisation des Capteurs de type Ultrasons, Magnetiques et Electrostatiques’’, Instrumentation, Mesure, Metrologie - Evaluation nondestructive, Vol. 1, pp. 101–125, 2001 (in French). Placko, D., Kundu, T., and Ahmad, R., ‘‘Theoretical Computation of Acoustic Pressure Generated by Ultrasonic Sensors in Presence of an Interface’’, Smart NDE and Health Monitoring of Structural and Biological Systems, Ed. T. Kundu, SPIE’s 7th Annual International Symposium on NDE for Health Monitoring and Diagnostics, March 18–21, 2002, San Diego, California, pp. 157–168, Vol. 4702, 2002. Placko D., Liebeaux N., Kundu T., ‘‘Proce´de´ pour e´valuer une grandeur physique representative d’une interaction entre une onde et un obstacle’’, Patent Application N 02 14108 ENS Cachan/CNRS/Universite´ d’Arizona, November 8, 2002. European and USA Extension November 10, 2003. Placko, D., and Kundu, T., Chapter 2: Modeling of Ultrasonic Field by Distributed Point Source Method, in Ultrasonic Nondestructive Evaluation: Engineering and Biological Material Characterization, Ed. T. Kundu, Pub. CRC Press, pp. 143–202, 2004. Placko, D., T. Kundu, and R. Ahmad, ‘‘Ultrasonic Field Computation in Presence of a Scatterer of Finite Dimension’’, Smart Nondestructive Evaluation and Health Monitoring of Structural and Biological Systems, Ed. T. Kundu, SPIE’s 8th Annual International Symposium on NDE for Health Monitoring and Diagnostics, March 3–5, 2003, San Diego, California, Vol. 5047, 2003. Schmerr L.W., Fundamentals of Ultrasonic Nondestructive Evaluation—A Modeling Approach, Pub. Plenum Press, 1998.

2 ADVANCED THEORY OF DPSM— MODELING MULTILAYERED MEDIUM AND INCLUSIONS OF ARBITRARY SHAPE T. Kundu University of Arizona, USA

D. Placko Ecole Normale Superieure, Cachan, France

2.1 INTRODUCTION In Chapter 1 the basic theory of distributed point source method (DPSM) has been presented, and some relatively simple example problems on magnetic and ultrasonic ﬁeld computations in a homogeneous medium have been solved. The problem becomes more complex when homogeneous medium is replaced by nonhomogeneous medium by either introducing a scatterer in the homogeneous ﬂuid or replacing the single ﬂuid medium by a medium with two ﬂuids with a plane interface between the two ﬂuids. Solutions of these two problems have been also presented in Chapter 1 after assuming that the presence of interface or the scatterer although affects the total ultrasonic ﬁelds does not affect the transducer surface condition. In other words, the transducer surface vibrates with the same uniform velocity v0 both in absence and presence of the interface and scatterer. This assumption is justiﬁed when the interface and scatterer are far away from the transducer.

DPSM for Modeling Engineering Problems, Edited by Dominique Placko and Tribikram Kundu Copyright # 2007 John Wiley & Sons, Inc.

59

60

ADVANCED THEORY OF DPSM—MODELING MULTILAYERED MEDIUM

The degree of complexity in modeling is signiﬁcantly increased if the interaction effect between the transducer and the interface or scatterer is to be considered or when the homogeneous medium is replaced by a multilayered medium. These two levels of complexity are considered in this chapter.

2.2 THEORY OF MULTILAYERED MEDIUM MODELING Multilayered medium in the presence of one or more ultrasonic transducers is modeled in this chapter. The problem geometry is shown in Figure 2.1. This ﬁgure shows two transducers Tand S separated by a nonhomogeneous medium consisting of n different materials and ðn 1Þ plane interfaces. Although in this ﬁgure the transducer faces are shown to be parallel to the interfaces, the technique also works when the faces are not parallel. 2.2.1 Transducer faces not coinciding with any interface Problem geometry shown in Figure 2.1 is ﬁrst modeled when transducer faces do not coincide with any interface. In Chapter 1 we have seen that one layer of source, called active source, can model a transducer whereas an interface needs two layers of sources, called passive sources. One layer of passive source is placed on each side of an interface. For the multilayered problem geometry

T n In–1 n–1 In–2

n–2

In–3

I3

3

I2

2

I1

1 S

Figure 2.1 Two transducers S and T having n different materials and ðn 1Þ interfaces I1 ; I2 ; . . . : : Iðn1Þ between them.

61

2.2 THEORY OF MULTILAYERED MEDIUM MODELING

AT T In–1 In–2 In–3

I3 I2 I1

n

An–1

n–1

An–2

n–2

An–3

A*n–1 A*n–2 A*n–3

A3 3

A*3

A2 2 A1 1 AS

A*2 A*1

S

Figure 2.2 Point sources (small circles) are introduced near transducers (S and T) and interfaces ðI1 ; I2 ; . . . ; In1 Þ.

shown in Figure 2.1, two layers of active sources AS and AT are introduced to model two transducers, and 2ðn 1Þ layers of passive sources are introduced for ðn 1Þ interfaces as shown in Figure 2.2. It should be noted here that as far as the source conﬁguration is concerned, there is no difference between the active and passive sources. The terminologies ‘‘active’’ and ‘‘passive’’ are used to distinguish between the point sources that model the ultrasonic transducers (or magnetic sensors) and interfaces. Two layers of passive sources are distributed over the entire region of every interface. In Figure 2.2 only a few passive sources are shown near each interface so that the ﬁgure does not look too cumbersome. AS and AT denote source strength vectors along the transducer surfaces S and T, respectively. Along each interface Im two sets of passive source vectors are denoted by Am (for sources located just above the interface) and Am (for sources located just below the interface). Total ultrasonic ﬁeld in each medium is obtained by superimposing the ﬁelds generated by two sets of sources located just above and below that medium as shown below. Medium 1: Summation of ﬁelds generated by AS and A1 . Medium 2: Summation of ﬁelds generated by A1 and A2 . Medium 3: Summation of ﬁelds generated by A2 and A3 .

62

ADVANCED THEORY OF DPSM—MODELING MULTILAYERED MEDIUM

Medium ðn 1Þ: Summation of ﬁelds generated by Aðn2Þ and Aðn1Þ . Medium n: Summation of ﬁelds generated by Aðn1Þ and AT . 2.2.1.1 Source strength determination from boundary and interface conditions The following boundary and interface conditions must be satisﬁed. On the transducer surfaces S and T, the velocity ﬁelds V are speciﬁed as VS0 and VT0 , respectively. Across the ðn 1Þ interfaces the pressure (P) and the z-direction velocity (V) must be continuous for ultrasonic problems. Similar continuity conditions must be satisﬁed for problems in other engineering ﬁelds as well. For those problems P and V will represent some other parameters, not necessarily pressure and velocity. For example, for electric ﬁeld problems P represents the vector of electric potentials and V represents the vector of electric ﬁeld normal to the interface or boundary. Because Vand P are related to the source strength vector A in the form V ¼ M A and P ¼ Q A, the boundary and continuity conditions give rise to the following equations: MSS AS þ MS1 A1 ¼ VS0 MTT AT þ MTðn1Þ Aðn1Þ ¼ VT0 M1S AS þ M11 A1 ¼ M12 A2 þ M11 A1 Q1S AS þ Q11 A1 ¼ Q12 A2 þ Q11 A1 M21 A1 þ M22 A2 ¼ M22 A2 þ M23 A3 Q21 A1 þ Q22 A2 ¼ Q22 A2 þ Q23 A3 : : : Mðn2Þðn3Þ Aðn3Þ þ Mðn2Þðn2Þ Aðn2Þ ¼ Mðn2Þðn2Þ Aðn2Þ þ Mðn2Þðn1Þ Aðn1Þ Qðn2Þðn3Þ Aðn3Þ þ Qðn2Þðn2Þ Aðn2Þ ¼ Qðn2Þðn2Þ Aðn2Þ þ Qðn2Þðn1Þ Aðn1Þ Mðn1Þðn2Þ Aðn2Þ þ Mðn1Þðn1Þ Aðn1Þ ¼ Mðn1Þðn1Þ Aðn1Þ þ Mðn1ÞT AT Qðn1Þðn2Þ Aðn2Þ þ Qðn1Þðn1Þ Aðn1Þ ¼ Qðn1Þðn1Þ Aðn1Þ þ Qðn1ÞT AT

ð2:1Þ

63

MSS

6 6 M1S 6 6 6 Q1S 6 6 6 0 6 6 0 6 6 6 0 6 6 6 0 6 6 6 ... 6 6 6 ... 6 6 6 ... 6 6 6 0 6 6 0 6 6 6 0 6 6 6 0 4 0

2

...

...

0

0

0

0

0

...

...

0

0

0

0

0

0

0

0

0

0

...

...

...

0

0

...

0

0

Q22

0

...

0

0

M22

Q12

Q11

Q11

0

M12

M11

M11

M21 Q21

0

0

MS1

0

0

0

0

0

...

...

...

M22 Q22 M32 Q32

0

0

0

...

0

0

0

0

0

...

...

0

0

0

0

0

...

...

0

0

0

0

0

...

...

...

Q34

...

Q33 ...

...

M34

M33

M33 Q33

Mn2n3 Qn2n3 0 0 0

... ... ... ...

...

...

...

...

0

0

0

0

0

0

0

...

...

...

...

0

0

...

...

...

...

Q23

0

0

0

0

0

0

0

0

M23

0

0

0

0

0

0

Qn2n2

Mn2n2

...

...

...

0

0

0

0

0

0

0

0

Mn2n2 Qn2n2 Mn1n2 Qn1n2

...

...

...

0

0

0

0

0

0

0

0

Qn1n1

Mn1n1

Qn2n1

Mn2n1

...

...

...

0

0

0

0

0

0

0

In matrix form the above equation can be written as (after denoting ðn–jÞ as nj, j ¼ 1; 2; 3 . . . :),

MTn1

Qn1n1

Mn1n1

0

0

...

...

...

0

0

0

0

0

0

0

3

8 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > <

ð2:2Þ

9 8 9 AS > > VS0 > > > > > > > 7 > > > > > > 0 7 A 0 > > > 1 > > > 7 > > > > > > 7 > > > > > 0 7 A1 > 0 > > > > > > 7 > > > > > > 7 > > > > > 0 7 A2 > 0 > > > > > > 7 > > > > > > 7 > > > > > 0 7 A2 > 0 > > > > > > 7 > > > > > > 7 > > > > 0 7 A3 > 0 > > > > > > > 7 > > > > > > 7 > > > 0 7 > A3 > 0 > > > > < = = 7 ... 7 . . . . . . ¼ 7 > > > 7 > > > > > > > > ... > > > > ... 7 ... > > > > > > > > 7 > > > > > > > > > 7 > > > > > > > ... 7 > . . . . . . > > > > > > > 7 > > > > > > > > > 7 > > > > > > 0 7 > An2 > 0 > > > > > > > 7 > > > > > > > > > 7 > > > > > 0 7 > An2 > 0 > > > > > > > > 7 > > > > > > > > > 7 > > > Mn1T 7 > > > > > An1 > > 0 > > > > > 7 > > > > > > > > > > > > > Qn1T 7 A 0 > > > > n1 > > 5 > > > > > ; > ; : : MTT AT VT0 0

64

ADVANCED THEORY OF DPSM—MODELING MULTILAYERED MEDIUM

The above matrix equation can be written in short form. ½MfAg ¼ fV0 g

ð2:3Þ

;fAg ¼ ½M1 fV0 g In this manner the source strength vector {A} is obtained. 2.2.2 Transducer faces coinciding with the interface – case 1: transducer faces modeled separately

In the next problem geometry, ðn þ 1Þ interfaces ðI0 ; I1 ; . . . : ; In Þ with ðn þ 2Þ materials ð0; 1; 2; . . . : ; n; ðn þ 1ÞÞ are considered. The transducer faces S and T coincide with the bottom ðI0 Þ and the top ðIn Þ interfaces, respectively, as shown in Figure 2.3. As shown in Figure 2.3, a number of source layers are introduced along ðn þ 1Þ interfaces I0 , I1 , I2 ; . . . : ; In . Note that similar to the previous problem geometry along each interface Im ð0 < m < nÞ, two sets of sources are introduced. These two sets of source strength vectors are denoted by Am (for sources located just above the interface) and Am (for sources located just below the interface). However, interfaces I0 and In have three sets of sources adjacent to it. The extra sets of sources (AS and AT ) appear due to the fact that transducer faces coincide with these interfaces. Although the interface I0 and the transducer surface S coincide (S is part of I0), we will treat I0

A*n

T An–1

In–1

n-2

An–3

In–3

An A*n

n n-1

An–2

In–2

A*n–1 A*n–2 A*n–3

A3

I3

3

A*3

A2

I2

2 A1

I1

1 A0

I0

AT

n+1

An In

A*0

S AS

0

A*2 A*1 A0 A*0

Figure 2.3 Transducers are placed along two interfaces I0 and In that separate two half spaces (denoted by media 0 and (n þ 1)) by n different material layers.

65

2.2 THEORY OF MULTILAYERED MEDIUM MODELING

and S as two different interfaces. I0 is the passive region (no active source of energy) and S is the active region (source of energy is present). Similarly In and T will be treated as two different interfaces. Thus, in this problem there are ðn þ 1Þ passive interfaces I0 ; I1 ; I2 ; . . . ; In that have two sets of sources adjacent to every interface and only two sets of active surfaces S and T that correspond to two sets of additional sources AS and AT. Ultrasonic ﬁelds in most layers are obtained by superimposing the ﬁelds generated by two sets of sources as listed below. Medium 2: Summation of ﬁelds generated by A1 and A2 . Medium 3: Summation of ﬁelds generated by A2 and A3 . . . Medium ðn 1Þ: Summation of ﬁelds generated by Aðn2Þ and Aðn1Þ . In each half space, however, the ﬁeld is generated by only one layer of sources. Medium 0: Field is generated by sources A0. Medium ðn þ 1Þ: Field is generated by sources An . In the top and bottom layers (layer #1 and n), the ﬁeld is generated by three layers of sources. Medium 1: Summation of ﬁelds generated by A0, A1 , and AS . Medium n: Summation of ﬁelds generated by Aðn1Þ, An , and AT . 2.2.2.1 Source strength determination from interface and boundary conditions The following interface conditions must be satisﬁed. Across the middle ðn 1Þ passive interfaces ðI1 ; I2 ; . . . ; Iðn1Þ Þ, the pressure (P) and the z-direction velocity (V) must be continuous. These interfaces are called passive interfaces because no active source is present along these interfaces. Excluding the active surfaces S and T, the interfaces I0 and In have only the passive regions. Across these passive interfaces also, the pressure (P) and the z-direction velocity (V) must be continuous like other ðn 1Þ passive interfaces. On the transducer surfaces S and T, the velocity ﬁelds V are speciﬁed as VS0 and VT0. Therefore, on these surfaces the computed velocity must match with these values. Because V ¼ M A and P ¼ Q A, the boundary and continuity conditions give rise to the following equations: M0S AS þ M00 A0 þ M01 A1 ¼ VS0

M0S AS þ M00 A0 þ M01 A1 ¼ M00 A0 Q0S AS þ Q00 A0 þ Q01 A1 ¼ Q00 A0

on S on I0 on I0

66

ADVANCED THEORY OF DPSM—MODELING MULTILAYERED MEDIUM

M1S AS þ M10 A0 þ M11 A1 ¼ M11 A1 þ M12 A2 Q1S AS þ

Q10 A0

þ Q11 A1 ¼

Q11 A1

þ Q12 A2

on I1 on I1

M21 A1 þ M22 A2 ¼ M22 A2 þ M23 A3

on I2

Q21 A1

on I2

þ Q22 A2 ¼

Q22 A2

þ Q23 A3

: : Mðn1Þðn2Þ Aðn2Þ þ Mðn1Þðn1Þ Aðn1Þ ¼ Mðn1Þðn1Þ Aðn1Þ þ Mðn1Þn An þ Mðn1ÞT AT

on Iðn1Þ

Qðn1Þðn2Þ Aðn2Þ þ Qðn1Þðn1Þ Aðn1Þ ¼ Qðn1Þðn1Þ Aðn1Þ þ Qðn1Þn An þ Qðn1ÞT AT

on Iðn1Þ

Mnðn1Þ Aðn1Þ þ Mnn An þ MnT AT ¼ Mnn An

on In

Qnðn1Þ Aðn1Þ

on In

þ Qnn An þ QnT AT ¼

Qnn An

Mnðn1Þ Aðn1Þ þ Mnn An þ MnT AT ¼ VT0

on T

ð2:4Þ

The above equations can be rearranged in the following manner: M00 A0 þ M0S AS þ M01 A1 ¼ M00 A0

on I0

Q00 A0 þ Q0S AS þ Q01 A1 ¼ Q00 A0 M00 A0 þ M0S AS þ M01 A1 ¼ VS0 M10 A0 þ M1S AS þ M11 A1 ¼ M11 A1

on I0 on S þ M12 A2

Q10 A0 þ Q1S AS þ Q11 A1 ¼ Q11 A1 þ M21 A1 þ M22 A2 ¼ M22 A2 þ M23 A3 Q21 A1 þ Q22 A2 ¼ Q22 A2 þ Q23 A3

Q12 A2

on I1 on I1 on I2 on I2

: : Mðn1Þðn2Þ Aðn2Þ þ Mðn1Þðn1Þ Aðn1Þ ¼ Mðn1Þðn1Þ Aðn1Þ þ Mðn1Þn An þ Mðn1ÞT AT Qðn1Þðn2Þ Aðn2Þ

þ Qðn1Þðn1Þ Aðn1Þ

¼ Qðn1Þðn1Þ Aðn1Þ þ Qðn1Þn An þ Qðn1ÞT AT Mnðn1Þ Aðn1Þ

on Iðn1Þ

þ Mnn An þ MnT AT ¼ VT0

on Iðn1Þ on T

Mnðn1Þ Aðn1Þ þ Mnn An þ MnT AT ¼ Mnn An

on In

Qnðn1Þ Aðn1Þ

on In

þ Qnn An þ QnT AT ¼

Qnn An

ð2:5Þ

67

0

0 0

0

0 0

0

0

0 ::

:: ::

0

0 0

0 0

0

0 ::

:: ::

0

0 0

0 0

0 0

0 0

0

:: ::

0 ::

0

0 0

0 0

0

:: ::

::

Q21

M21

0 0

0 0

0

:: ::

Q22 ::

M22

M10 M1S M11 M11 M12 Q10 Q1S Q11 Q11 Q12

Q00 Q0S Q01 M00 M0S M01

M00 M00 M0S M01

6 Q 6 00 6 6 0 6 6 0 6 6 6 0 6 6 0 6 6 6 0 6 6 6 :: 6 6 :: 6 6 6 :: 6 6 0 6 6 6 0 6 6 0 6 6 4 0

2

:: ::

:: ::

::

0 0

0 0

0

:: ::

0 0

0 0

0

:: ::

:: ::

0 ::

0

0 0

0 0

0

:: ::

0 ::

0

0 0

0 0

0

:: ::

0 ::

0

0 0

0 0

0

:: ::

0 ::

0

0 0

0 0

0

:: ::

0 ::

0

0 0

0 0

0

:: :: 0 0

0 0

Mnn1 Qnn1

MnT QnT

Mnn Qnn

:: Qn1n2 Qn1n1 Qn1n1 Qn1T Qn1n :: 0 0 Mnn1 MnT Mnn

:: Mn1n2 Mn1n1 Mn1n1 Mn1T Mn1n

:: ::

Q22 Q23 :: :: :: ::

0 0

0 0

0

M22 M23 ::

0 0

0 0

0

In matrix form the above equations can be written as (after denoting (n–j) as nj; j ¼ 1; 2; 3 . . .) 3

9 8 9 8 A0 > > 0 > > > > > > > > > > > > > 0 > > > 0 7 A0 > 7 > > > > > > > > > 7 > > > > > > > > > > > V 0 7 A S > S0 > > > 7 > > > > > > > > > > >A > > > > > 0 7 0 7 > 1 > > > > > > > > 7 > > > > > > > > 7 > > > > 0 7 > A 0 > > > 1 > > > > > > > > > > 7 > > > > > > 0 7 > A 0 > > > 2 > > > > > > > 7 > > > > > > > > > 0 7 A 0 > > > > = 7 < 2= < 7 :: 7 A3 ¼ . . . > > > 7 > > > > > > ... > > :: 7 ... > > > > > > > > 7 > > > > > > > > 7 > > > > > > > . . . :: 7 > ... > > > > > > > > 7 > > > > > > > > > > > > > 0 7 A 0 n1 > > > 7 > > > > > > > > > 7 > > > > > > > An1 > > 0 > 0 7 > > > > > 7 > > > > > > > > > 7 > > > > 0 7 > A V > > T > T0 > > > > > > > > > > > > 7 > > > Mnn 5 > A 0 > > > n > > > > > ; ; : : Qnn An 0 ð2:6Þ 0

68

ADVANCED THEORY OF DPSM—MODELING MULTILAYERED MEDIUM

2.2.2.2 Counting number of equations and number of unknowns Let us assume that for every interface I1 , I2 ; . . . ; Iðn1Þ and so on, there are N sources above the interface and N sources below the interface. On the bottom (I0) and top ðIðnþ1Þ Þ interfaces, there are M1 and M2 sources on the two active transducer surfaces S and T, respectively. On the passive regions of these two interfaces, there are 2ðN M1 Þ and 2ðN M2 Þ sources, respectively. Thus, the total number of sources at the ðn þ 1Þ interfaces (including the active transducer surfaces) is 2ðn þ 1ÞN M1 M2 . Total number of equations is also 2ðn þ 1ÞN M1 M2 . Therefore, this system of equations can be uniquely solved. Matrix equation (2.6) can be written in short form. ½MfAg ¼ fV0 g

ð2:7Þ

;fAg ¼ ½M1 fV0 g

Note that the dimension of the M matrix is f2ðn þ 1ÞN M1 M2 g f2ðn þ 1Þ N M1 M2 g. The number of unknown source vectors is also f2ðn þ 1ÞN M1 M2 g. In this manner the source strength vector {A} can be computed. 2.2.3 Transducer faces coinciding with the interface – case 2: transducer faces are part of the interface In this section the modeling is carried out slightly differently. Unlike the previous section, here the active transducer sources substitute some of the passive interface sources as shown in Figure 2.4. Two layers of sources are introduced along each of the ðn þ 1Þ interfaces I0 , I1 , I2 ; . . . : :; In . Two sets of source strength vectors along the Im interface are denoted by Am (for sources located just above the interface) and Am (for sources located just below the interface). An

In In–1 In–2 In–3

I3 I2 I1

n+1

An

T

A*n n

An–1

n–1

An–2

n–2

An–3

A*n–1 A*n–2 A*n–3

A3 3

A*3

A2 2 A1 1

A*2 A*1

A0 I0 A*0

S

0

A*0

Figure 2.4 Transducers are placed along two interfaces I0 and In that separate two half spaces (denoted by media 0 and (n þ 1)) by n different layers. The difference between Figures 2.3 and 2.4 is how point source layers in the neighborhood of I0 and In interfaces are distributed.

69

2.2 THEORY OF MULTILAYERED MEDIUM MODELING

The total ultrasonic ﬁeld in each layer is obtained by superimposing the ﬁelds generated by two sets of sources as listed below: Medium 1: Summation of ﬁelds generated by A0 and A1 . Medium 2: Summation of ﬁelds generated by A1 and A2 . Medium 3: Summation of ﬁelds generated by A2 and A3 . . Medium n: Summation of ﬁelds generated by Aðn1Þ and An. In each half space, however, the ﬁeld is generated by only one layer of sources. Medium 0: Field is generated by sources A0. Medium ðn þ 1Þ: Field is generated by sources An . 2.2.3.1 Source strength determination from interface and boundary conditions The following interface conditions must be satisﬁed. Across the ðn 1Þ passive interfaces ðI1 ; I2 ; . . . ; Iðn1Þ Þ, the pressure (P) and the z-direction velocity (V) must be continuous. Interfaces I0 and In have both active and passive regions. In the active regions the transducer surfaces S and T are located. These regions are denoted as IS0 and ITn : The passive regions where the transducer faces are not present are simply denoted as I0 and In without any superscript. On the transducer surfaces IS0 and ITn , the velocity ﬁelds V are speciﬁed as VS0 and VT0 , respectively, and across the passive interfaces I0 and In, the pressure (P) and the z-direction velocity (V) must be continuous like other ðn 1Þ passive interfaces. Because V ¼ M A and P ¼ Q A, the boundary and continuity conditions give rise to the following equations: M00 A0 þM01 A1 ¼ VS0 M00 A0 þM01 A1 ¼ M00 A0 Q00 A0 þQ01 A1 ¼ Q00 A0 M10 A0 þM11 A1 ¼ M11 A1 þM12 A2 Q10 A0 þQ11 A1 ¼ Q11 A1 þQ12 A2 M21 A1 þM22 A2 ¼ M22 A2 þM23 A3 Q21 A1 þQ22 A2 ¼ Q22 A2 þQ23 A3 : : : Mðn1Þðn2Þ Aðn2Þ þMðn1Þðn1Þ Aðn1Þ ¼ Mðn1Þðn1Þ Aðn1Þ þMðn1Þn An

on Iðn1Þ

Qðn1Þðn2Þ Aðn2Þ þQðn1Þðn1Þ Aðn1Þ ¼ Qðn1Þðn1Þ Aðn1Þ þQðn1Þn An

on Iðn1Þ

Mnðn1Þ Aðn1Þ þMnn An ¼ Mnn An Qnðn1Þ Aðn1Þ þQnn An ¼ Qnn An Mnðn1Þ Aðn1Þ þMnn An ¼ VT0

on In

on IS0 on I0 on I0 on I1 on I1 on I2 on I2

on In on ITn ð2:8Þ

70

M00

0

fM00 g

0

M01

0

0

0

0

0

0

0

0

0

6 fQ g fQ g fQ g 0 0 0 0 6 00 01 00 6 6 0 0 0 M10 M11 M11 M12 6 6 0 Q10 Q11 Q11 Q12 0 0 6 6 6 0 0 0 M M M M 22 23 21 22 6 6 Q22 Q22 Q23 0 0 0 Q21 6 6 6 :: :: :: :: :: :: :: 6 6 :: :: :: :: :: :: :: 6 6 6 :: :: :: :: :: :: :: 6 6 :: :: :: :: :: :: :: 6 6 6 :: :: :: :: :: :: :: 6 6 0 0 0 0 0 0 0 6 6 6 0 0 0 0 0 0 0 6 6 4 0 0 0 0 0 0 0

2 0 0 0 0 0 :: :: :: :: ::

0 :: 0 0 :: 0 0 :: 0 0 :: 0 0 :: 0 :: :: :: :: :: :: :: :: :: :: :: :: :: :: ::

:: ::

::

:: ::

0 0

0

0 0

0

0 0

0 :: 0 0 :: 0

0

0

0 :: 0 Mn1n2 Mn1n1 0 :: 0 Qn1n2 Qn1n1

0

0 :: 0 0

0

3

9 9 8 8 A0 > > hVS0 i > > > > > > > > > 7 > > > > > > 0 0 0 A0 > 0 > 7 > > > > > > > > 7 > > > > > > > > > 7 > > > > 0 0 0 A 0 1 > > > 7 > > > > > > > > > 7 > > > > > > > > 0 0 0 A 0 7 > > > > 1 > > > > 7 > > > > > > > > 7 > > > > 0 0 0 A 0 > > 2 > > 7 > > > > > > > > 7 > > > > > > > 0 0 0 A 0 > > > > 7 > 2 > > > > > > > 7 > > > > > > > > > 7 :: :: :: A 0 > > > > 3 = = < 7 < 7 :: :: :: . .. 7 ... ¼ > > > 7 > > > > > > ... > 7 > :: :: :: . .. > > > > > > > > 7 > > > > > > > > 7 > > > > > > > > :: :: :: ... . .. 7 > > > > > > > > 7 > > > > > > > > > 7 > > > > :: :: :: ... 0 > > > > 7 > > > > > > > > 7 > > > > > > > > A Mn1n1 Mn1n 0 0 7 > n1 > > > > > > > 7 > > > > > > > > 7 > > > > Qn1n1 Qn1n 0 A 0 > > n1 > 7 > > > > > > > > > > > 7 > > > > > hVT0 i > > An > Mnn1 Mnn fMnn g 5 > > > > > > ; : ; : fQnn1 g fQnn g fQnn g An 0 ð2:9Þ 0

In matrix form the above equations can be written as (after denoting (n–j) as nj; j ¼ 1; 2; 3 . . .Þ

71

2.2 THEORY OF MULTILAYERED MEDIUM MODELING

The terms inside the curly brackets ðfgÞ are replaced by zero when the passive regions (I0 and In) of interfaces 0 and n are considered, and the terms inside the angular brackets ðhiÞ are replaced by zero when the active regions (IS0 and ITn ) of interfaces 0 and n are considered. The above matrix equation can be written as ½MfAg ¼ fV0 g

ð2:10Þ

;fAg ¼ ½M1 fV0 g In this manner the source strength vector fAg can be obtained. 2.2.4 Special case involving one interface and one transducer only

The problem geometry shown in Fig. 2.5 can be considered as a special case of Section 2.2.1 (or Fig. 2.2) when AT is removed and all interfaces other than I1 are eliminated. We are interested in this problem geometry for computing the reﬂection coefﬁcient R in terms of M and Q and thus get some insight about the physical meaning of M and Q matrices. For this problem geometry Eq. (2.1) is reduced to MSS AS þ MS1 A1 ¼ VS0 M1S AS þ M11 A1 ¼ M11 A1 Q1S AS þ Q11 A1 ¼ Q11 A1

ð2:11Þ

From the second and third equations of (2.11), one can write A1 ¼ ½M11 1 fM1S AS þ M11 A1 g

ð2:12Þ

A1 ¼ ½Q11 1 fQ1S AS þ Q11 A1 g Subtracting the above two equations one gets 0 ¼ ½M11 1 fM1S AS þ M11 A1 g ½Q11 1 fQ1S AS þ Q11 A1 g ) f½M11 1 M1S ½Q11 1 Q1S gAS þ f½M11 1 M11 ½Q11 1 Q11 gA1 ¼ 0 ) aAS þ bA1 ¼ 0 ) A1 ¼ b1 aAS

I1

ð2:13Þ

2

A1

1 AS

A*1

S

Figure 2.5 Point source distribution for the problem geometry involving one transducer and one interface.

72

ADVANCED THEORY OF DPSM—MODELING MULTILAYERED MEDIUM

where, a ¼ f½M11 1 M1S ½Q11 1 Q1S g b ¼ f½M11 1 M11 ½Q11 1 Q11 g

ð2:14Þ

Because from Eq. (2.11) we know that MSS AS þ MS1 A1 ¼ VS0 it can be written that MSS AS MS1 b1 aAS ¼ VS0 ) ½MSS MS1 b1 aAS ¼ VS0

ð2:15Þ

Velocity ﬁeld at the interface level in medium 2 can be written in terms of the transmission coefﬁcient matrix T as TM1S AS and in medium 1 in terms of the reﬂection coefﬁcient matrix R as M1S AS þ RM1S AS Therefore, TM1S AS ¼ M11 A1 and M1S AS þ RM1S AS ¼ M1S AS þ M11 A1 ) RM1S AS ¼ M11 A1

ð2:16Þ

Thus, A1 ¼ ½M11 1 TM1S AS and A1 ¼ ½M11 1 RM1S AS

ð2:17Þ

Note that the pressure continuity condition is not necessary to satisfy separately when the problem is formulated in terms of the reﬂection and transmission coefﬁcient matrices because the continuity of velocity guarantees the continuity of pressure also when the reﬂection and transmission coefﬁcient matrices are considered.

2.2 THEORY OF MULTILAYERED MEDIUM MODELING

73

From Eqs. (2.13) and (2.17), it can be written that A1 ¼ ½M11 1 RM1S AS ¼ b1 aAS ) ½M11 1 RM1S ¼ b1 a ¼ f½M11 1 M11 ½Q11 1 Q11 g1 f½M11 1 M1S ½Q11 1 Q1S g ) R ¼ b1 a ¼ M11 f½M11 1 M11 ½Q11 1 Q11 g1 f½M11 1 M1S ½Q11 1 Q1S g½M1S 1 ð2:18Þ From Eqs. (1.25m) and (1.25f), we get the elements for the Q matrix (relates pressure to the source strength) and the M matrix that relates the vertical component of velocity to the source strength in the following form: eikr r 1 x3 M¼ ðikr 1Þeikr ior r 3 Q¼

ð2:19Þ

where k ¼ oc , l ¼ cf ¼ 2pc o

~ and M ~ that are similar to Q and M Let us now deﬁne two matrices Q matrices of Eq. (2.18); the only difference is that while Q and M matrices take medium 2 properties (because they give pressure and velocity in medium 2 in ~ and M ~ matrices take medium 1 properties. terms of the source distribution A1 ), the Q ~ and M ~ matrices will give pressure and velocity ﬁelds due to Therefore, Q equivalent sources distributed in the A1 position when medium 2 is replaced by medium 1. Then, from Eq. (2.19) one can write ~ Q Q ¼ eik1 r eik2 r ik2 r ~e ¼Q ~ a2 )Q ¼Q ik r 1 e a1

ð2:20Þ

Similarly, ~ r1 M M r2 ¼ ik r ðik1 r 1Þe 1 ðik2 r 1Þeik2 r ik2 r ~ r1 ðik2 r 1Þe ¼ M ~ b2 )M ¼M ik r 1 r2 ðik1 r 1Þe b1

ð2:21Þ

74

ADVANCED THEORY OF DPSM—MODELING MULTILAYERED MEDIUM

~ 11 and M ~ 11 matrices are similar to Q11 and M11 matrices of Eq. (2.18); Note that Q ~ 11 and M ~ 11 matrices give pressure and velocity ﬁelds at the only difference is that Q the interface points due to the source distribution A1 whereas Q11 and M11 matrices give pressure and velocity ﬁelds at the interface points due to the source distribution A1 . Note that both the matrices take material properties corresponding to medium 1. Because A1 is located just above the interface whereas A1 is located just below the interface, the vertical velocity vectors at the interface points due to these two layers of sources will be of same magnitude but of opposite direction, whereas the pressure ﬁelds generated by these two layers of sources should be of same magnitude. Therefore, ~ 11 ¼ Q11 ¼ Q a1 Q 11 a2 ~ 11 ¼ M11 ¼ M b1 M 11 b2

ð2:22Þ

Substituting Eq. (2.22) into Eq. (2.18) we get 1 1 R ¼ M11 ½M11 1 M11 ½Q11 1 Q11 ½M11 M1S ½Q11 1 Q1S ½M1S 1 ( )1 b2 1 a2 1 ¼ M11 M11 M11 Q11 Q11 b1 a1 ( ) b2 1 a2 1 ~ ~ M11 M1S Q11 Q1S ½M1S 1 b1 a1

1

b1 a1 b1 ~ 1 a1 ~ 1 1 1 ¼ M11 ½M11 M11 þ ½Q11 Q11 ½M11 M1S ½Q11 Q1S ½M1S 1 b2 a2 b2 a2

1

b1 ~ 1 a1 ~ 1 ~ 11 b1 þ a1 ¼ M ½M11 M1S ½Q Q1S ½M1S 1 ð2:23Þ b2 a2 b2 a2 11

~ at the interface position where A is located is equivalent If the source distribution A 1 to the sources AS at the target position, then equating the velocity ﬁelds from the two sets of sources one can write ~ ¼ M1S AS ~ 11 A M ~ ¼ ½M ~ 11 1 M1S AS )A

ð2:24Þ

Similarly, equating the pressure ﬁeld we get ~ ¼ Q1S AS ~ 11 A Q ~ ¼ ½Q ~ 11 1 Q1S AS )A

ð2:25Þ

2.2 THEORY OF MULTILAYERED MEDIUM MODELING

75

~ and A are different although they are located at the same position. A ~ is Note that A 1 the equivalent source distribution when both the media are identical; in other words, ~ there is no real interface when medium 2 is replaced by medium 1, therefore for A present. However, for A1 there is a real interface present between media 1 and 2. Comparing Eqs. (2.24) and (2.25), we get ~ ¼ ½M ~ 11 1 Q1S AS ~ 11 1 M1S AS ¼ ½Q A ~ 11 1 Q1S AS ½AS 1 ~ 11 1 M1S AS ½AS 1 ¼ ½Q ) ½M

ð2:26Þ

~ 11 1 Q1S ~ 11 1 M1S ¼ ½Q ) ½M Substituting Eq. (2.26) into Eq.(2.23), we get

1

b1 ~ 1 a1 ~ 1 1 ~ 11 b1 þ a1 ½M11 M1S ½Q Q R ¼ M 1S ½M1S b2 a2 b2 a2 11 ~ 11 ¼ M

b1 a1 þ b2 a2

1

b1 ~ 1 a1 ~ 1 1 ½M11 M1S ½M M 11 1S ½M1S b2 a2

b1 a1 1 b1 a1 ~ 1 ~ ½M11 M1S ½M1S 1 ¼ M11 þ b2 a2 b 2 a2 ¼

b1 a1 þ b2 a2

b a1 ¼ 1þ b2 a2 ¼

1

1

ð2:27Þ

b1 a1 ~ ~ 1 M11 ½M11 M1S ½M1S 1 b 2 a2 b 1 a1 b 2 a2

a1 b2 a2 b1 a1 b2 þ a2 b1

Substituting a1, a2, b1 and b2 expressions from Eqs. (2.20) and (2.21) into Eq. (2.27) we get

R¼ ¼

a1 b2 a2 b1 eik1 r r1 ðik2 r 1Þeik2 r eik2 r r2 ðik1 r 1Þeik1 r ¼ a1 b2 þ a2 b1 eik1 r r1 ðik2 r 1Þeik2 r þ eik2 r r2 ðik1 r 1Þeik1 r r1 ðik2 r 1Þ r2 ðik1 r 1Þ r1 ðik2 r 1Þ þ r2 ðik1 r 1Þ

ð2:28Þ

76

ADVANCED THEORY OF DPSM—MODELING MULTILAYERED MEDIUM

For high frequency ðkm r 1Þ problems r1 ðik2 r 1Þ r2 ðik1 r 1Þ r1 ðik2 rÞ r2 ðik1 rÞ r1 k2 r2 k1 ¼ r1 ðik2 r 1Þ þ r2 ðik1 r 1Þ r1 ðik2 rÞ þ r2 ðik1 rÞ r1 k2 þ r2 k1 r1 o r2 co1 r1 c1 r2 c2 ¼ co2 ¼ r1 c2 þ r2 co1 r1 c1 þ r2 c2

R¼

ð2:29Þ

and for low frequency ðkm r 1Þ problems R¼

r1 ðik2 r 1Þ r2 ðik1 r 1Þ r1 ð1Þ r2 ð1Þ r1 r2 ¼ r1 ðik2 r 1Þ þ r2 ðik1 r 1Þ r1 ð1Þ þ r2 ð1Þ r1 þ r2

ð2:30Þ

Note that the reﬂection coefﬁcient for high-frequency case is similar to the plane wave reﬂection coefﬁcient for normal incidence. However, it is slightly different for the low-frequency case. The reason for this difference is that the formulation presented here is for spherical waves with one source point. Plane waves are modeled by superposition of a large number of source points.

2.3 THEORY FOR MULTILAYERED MEDIUM CONSIDERING THE INTERACTION EFFECT ON THE TRANSDUCER SURFACE The problem geometry considered here is identical to that shown in Figure 2.2. The difference in boundary conditions between this problem and the problem described in Section 2.1 is that in Section 2.1 the velocity at boundaries S and Tare speciﬁed as VS0 and VT0 , respectively, whereas in this case it is not speciﬁed. In the absence of all other interfaces and transducers, the velocity on S and T surfaces should be VS0 and VT0 , respectively. However, as soon as other interfaces and the second transducer are introduced, the velocity on surface S will be changed from VS0 , and in the same manner the velocity on surface T will change from VT0 . In this section this variation in the transducer surface velocity due to the transducer–interface interaction effect is considered. This problem is solved in two steps. In the ﬁrst step active transducer sources AS and AT are obtained from their surface velocity conditions. MSS AS ¼ VS0 ) AS ¼ ½MSS 1 VS0 and similarly, AT ¼ ½MTT 1 VT0

ð2:31Þ

77

2.3 THEORY FOR MULTILAYERED MEDIUM CONSIDERING

AT + An

In–1 In–2 In–3

I3 I2 I1

T = In

A*n

An–1

n n–1

An–2

n–1

An–3

A*n–1 A*n–2 A*n–3

A3 3

A*3

A2 2 A1 1

A0

A*2 A*1

S = I0 AS + A*0

Figure 2.6 Two layers of point source are introduced at every interface including active interfaces S and T that are identiﬁed as I0 and In interfaces, respectively.

In the second step two layers of sources are introduced at all interfaces including S and T interfaces as shown in Figure 2.6. With the introduction of these new sources, the total source strength below the S interface is AS þ A0 , and above the T interface it is AT þ An , see Figure 2.6. Ultrasonic ﬁelds in most layers are obtained by superimposing the ﬁelds generated by two layers of sources as listed below. Medium 2: Summation of ﬁelds generated by A1 and A2 . Medium 3: Summation of ﬁelds generated by A2 and A3 . . . Medium ðn 1Þ: Summation of ﬁelds generated by Aðn2Þ and Aðn1Þ . However, in top and bottom half spaces (Medium #1 and n, above surface S and below surface T), the total ﬁeld is generated by three layers of sources. Medium 1: Summation of ﬁelds generated by A0, A1 , and AS . Medium n: Summation of ﬁelds generated by Aðn1Þ, An , and AT . Inside the transducer medium (below surface S and above surface T), the ﬁeld is generated by only one layer of source.

78

ADVANCED THEORY OF DPSM—MODELING MULTILAYERED MEDIUM

Inside transducer S: The ﬁeld is generated by A0 sources only. Inside transducer T: The ﬁeld is generated by An sources only. 2.3.1 Source strength determination from interface conditions The following interface conditions must be satisﬁed. Across the middle ðn 1Þ passive interfaces ðI1 ; I2 ; . . . : : ; Iðn1Þ Þ; the pressure (P) and the z-direction velocity (V) must be continuous. Similarly, across interfaces S(I0) and TðIn Þ, the pressure (P) and the z-direction velocity (V) must be continuous like other ðn 1Þ interfaces. Because the velocity V ¼ M A and pressure P ¼ Q A, the continuity conditions give rise to the following equations: M00 A0 þ M0S AS þ M01 A1 ¼ M00 A0

on I0

Q00 A0 þ Q0S AS þ Q01 A1 ¼ Q00 A0

on I0

M10 A0 þ M1S AS þ M11 A1 ¼ M11 A1 þ M12 A2

on I1

Q10 A0 þ Q1S AS þ Q11 A1 ¼ Q11 A1 þ Q12 A2

on I1

M21 A1 þ M22 A2 ¼ M22 A2 þ M23 A3

on I2

Q21 A1 þ Q22 A2 ¼ Q22 A2 þ Q23 A3

on I2

: : Mðn1Þðn2Þ Aðn2Þ þ Mðn1Þðn1Þ Aðn1Þ ¼ Mðn1Þðn1Þ Aðn1Þ þ Mðn1Þn An þ Mðn1ÞT AT

on Iðn1Þ

Qðn1Þðn2Þ Aðn2Þ þ Qðn1Þðn1Þ Aðn1Þ ¼ Qðn1Þðn1Þ Aðn1Þ þ Qðn1Þn An þ Qðn1ÞT AT

on Iðn1Þ

Mnðn1Þ Aðn1Þ þ Mnn An þ MnT AT ¼ Mnn An

on In

Qnðn1Þ Aðn1Þ þ Qnn An þ QnT AT ¼ Qnn An

on In

ð2:32Þ

In matrix form the above equations can be written as (after denoting (n–j) as nj, j ¼ 1; 2; 3 . . .)

79

M00 6 Q00 6 6 0 6 6 0 6 6 0 6 6 0 6 6 ... 6 6 :: 6 6 :: 6 6 :: 6 6 ... 6 6 0 6 6 0 6 4 0 0

2

M00 Q00 M10 Q10 0 0 ... :: :: :: ... 0 0 0 0

M01 0 0 0 0 Q01 0 0 0 0 M11 M11 M12 0 0 Q11 Q11 Q12 0 0 0 M21 M22 M22 M23 0 Q21 Q22 Q22 Q23 . .. ... ... ... ... :: :: :: :: :: :: :: :: :: :: :: :: :: :: :: . .. ... ... ... ... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 ... :: :: :: ... 0 0 0 0

:: :: :: :: :: :: :: :: :: :: :: :: :: :: ::

ð2:33Þ

3 8 9 8 9 0 0 0 0 0 0 A0 > > M0S AS > > > > > > > > > > > A0 > > 0 0 0 0 0 0 7 Q0S AS > > > > 7 > > > > > > > > > > > > > 0 0 0 0 0 0 7 A M A 1 1S S > > > > 7 > > > > > > > > 7 > > > > 0 0 0 0 0 0 7 > A Q A > > S > 1S 1 > > > > > > > > 7 > > > > 0 0 0 0 0 0 7 > A 0 > > > 2 > > > > > > > > > > > > 7 0 0 0 0 0 0 7 > A 0 > > > 2 > > > > > > > > > > > > 7 .. . ... . .. ... .. . .. . 7 < A3 = < 0 = 7 :: :: :: :: :: :: 7 ... ¼ .. . > > > > > > ... > > :: :: :: :: :: :: 7 .. . > > > > > > > 7 > > > > > > > > > 7 > ... > :: :: :: :: :: :: 7 > > .. . > > > > > > > > > > > > 7 > > > > .. . ... . .. ... .. . .. . 7 > ... > > 0 > > > > > > > > 7 > > > > 0 7 > An1 > > Mn1T AT > 0 Mn1n2 Mn1n1 Mn1n1 Mn1n > > > > > > > > > 7 > > > > 0 7 > A 0 Qn1n2 Qn1n1 Qn1n1 Qn1n Q A T > > > n1T n1 > > > > > > > > 5 > > > > 0 0 0 Mnn1 Mnn Mnn A M A n > > > > nT T ; : ; : 0 0 0 Qnn1 Qnn Qnn An QnT AT

80

ADVANCED THEORY OF DPSM—MODELING MULTILAYERED MEDIUM

2.3.2 Counting number of equations and number of unknowns Let us assume that for each of the interfaces I1 , I2 ; . . . ; Iðn1Þ , there are N point sources above and N point sources below the interface. On the bottom (I0) and top ðIðn þ 1Þ Þ interfaces, there are 2M1 and 2M2 sources counting both layers of sources above and below the transducer surfaces S and T, respectively. Thus, the total number of unknown sources at the ðn þ 1Þ interfaces (including S and T surfaces) is 2½ðn 1ÞN þ M1 þ M2 ; in addition to these unknown sources, there are M1 þ M2 known sources that are AS and AT obtained from Eq. (2.31). The total number of equations is also 2½ðn 1ÞN þ M1 þ M2 because at each common point on the interface, two conditions are satisﬁed, and the total number of common points considered on all interfaces is ½ðn 1ÞN þ M1 þ M2 . Therefore, this system of equations can be uniquely solved. Matrix equation (2.33) can be written in short form. ½MfAg ¼ fV0 g

ð2:34Þ

;fAg ¼ ½M1 fV0 g

As mentioned above the dimension of M matrix is 2½ðn 1ÞN þ M1 þ M2 2½ðn 1ÞN þ M1 þ M2 . The number of unknown source vectors is also 2½ðn 1ÞN þ M1 þ M2 . Thus, the source strength vector fAg is uniquely obtained.

2.4 INTERFERENCE BETWEEN TWO TRANSDUCERS: STEP-BY-STEP ANALYSIS OF MULTIPLE REFLECTION Consider two transducers S and T placed face to face and separated by a homogeneous medium as shown in Figure 2.7. Each transducer surface can have some active region and some passive region. In the active region the surface velocity V0S (or V0T ) in the absence of any interface effect is speciﬁed. In the passive region the surface velocity is zero when it is not inﬂuenced by any other source.

AT0

AT1

AT2

AT3

AT4

AT5 T

S AS0

AS1

AS2

AS3

AS4

AS5

Figure 2.7 Multiple reﬂections between two transducers T and S placed face to face.

INTERFERENCE BETWEEN TWO TRANSDUCERS

81

AS0 and AT0 are point sources distributed over transducers S and T when the interaction effect between them is ignored. In other words, AS0 and AT0 represent point source distributions for modeling the ﬁeld generated by the transducer in a homogenous medium in the absence of any other transducer or scatterer. The AS0 generated ﬁeld is reﬂected by the transducer T. The reﬂected ﬁeld is modeled by introducing a new layer of sources AT1 distributed over the transducer T, as shown in Figure 2.7. Equating the velocity ﬁelds on surface T generated by sources AT1 and from the reﬂection of AS0 generated ﬁeld, the following equation is obtained: MTT AT1 ¼ RT MTS AS0 ) AT1 ¼ ½MTT 1 RT MTS AS0 ¼ BAS0

ð2:35Þ

The ﬁeld generated by AT1 is then reﬂected by transducer S. This ﬁeld is modeled by introducing a layer of sources AS2 distributed over the transducer S. Equating the velocity ﬁelds on surface S generated by sources AS2 and from the reﬂection of AT1 generated ﬁeld, the following equation is obtained: MSS AS2 ¼ RS MST AT1 ) AS2 ¼ ½MSS 1 RS MST AT1 ¼ CAT1 ¼ CBAS0

ð2:36Þ

Repeating these operations it is possible to write AT3 ¼ BCBAS0 AS4 ¼ CBCBAS0 AT5 ¼ BCBCBAS0 AS6 ¼ CBCBCBAS0

ð2:37Þ

Similarly, starting from sources AT0 and matching the ﬁeld at surface S from AS1 and the reﬂected ﬁeld from AT0 , one gets MSS AS1 ¼ RS MST AT0 ) AS1 ¼ ½MSS 1 RS MST AT0 ¼ CAT0

ð2:38Þ

and AT2 ¼ BCAT0 AS3 ¼ CBCAT0 AT4 ¼ BCBCAT0 AS5 ¼ CBCBCAT0

ð2:39Þ

82

ADVANCED THEORY OF DPSM—MODELING MULTILAYERED MEDIUM

Therefore, the total ﬁelds at the transducer levels S and T can be written as

AS AT

¼

AS0 þ AS1 þ AS2 þ AS3 þ . . .

AT0 þ AT1 þ AT2 þ AT3 þ . . . " CB 0 0 CBC 1 0 0 C ¼ þ þ þ 0 BC BCB 0 0 1 B 0 #

CBCB 0 AS0 þ þ ... AT0 0 BCBC

AS ¼ ½D0 þ D1 þ D2 þ D3 þ . . .A0 ¼ DA0 ) AT

ð2:40Þ

Therefore, the velocity and pressure ﬁelds can be obtained from

VS

¼

VT

PS PT

MSS

MST

¼

AS

¼ M0 A ¼ M0 DA0 ¼ MA0 MTT AT

QST AS ¼ Q0 A ¼ Q0 DA0 ¼ QA0 QTT AT

MTS

QSS QTS

ð2:41Þ ð2:42Þ

If the transducer velocity remains unchanged, then the source strength A0 can be obtained from Eq. (2.41).

1 VS ð2:43Þ A0 ¼ M VT If the active source area differs from the total transducer area and the velocity ﬁeld only on the active source area is speciﬁed, then one can write V¼

VS VT

VS ¼W VT

VS þW VT

ð2:44Þ

Note that W þ W should be an identity matrix. Equation (2.44) can be written as V¼

VS VT

AS0 AS0 ¼ WM0 D þ WM0 D AT0 AT0

ð2:45Þ

Note that

VS

VT Active

AS0 AT0

¼W

Active

¼W

VS

VT

AS0 AT0

ð2:46Þ

83

2.5 SCATTERING BY AN INCLUSION OF ARBITRARY SHAPE

Therefore,

VS VT

Active

¼ WM0 DW

AS0

þ WM0 DW

AT0

AS0 AT0

¼ WM0 DW

AS0 AT0

ð2:47Þ

Because AS0 and AT0 for the passive region are zero, the second term is zero. After getting rid of the zero valued columns and rows of WM0DW and deﬁne it as matrix E, the compacted relation is obtained as

AS0 VS ¼E VT Active AT0 Active ð2:48Þ

AS0 1 VS ) ¼E AT0 Active VT Active 2.5 SCATTERING BY AN INCLUSION OF ARBITRARY SHAPE To solve the scattering problem where the scatterer can have any arbitrary shape, two layers of sources are introduced—one layer (A1 ) is inside the scatterer and the second layer (A1 ) is outside the scatterer as shown in Figure 2.8. In addition to these two passive layers of source, two additional layers AS and AT of active sources are considered to model the two transducers. The total ﬁeld in medium 1 is obtained by the superposition of three layers of source AS , AT , and A1 whereas the ﬁeld in medium 2 is obtained from only one layer of sources, A1 . Applying the boundary conditions on the two transducer surfaces and continuity conditions across the interface between two media, one gets MSS AS þ MST AT þ MS1 A1 ¼ VS0 MTS AS þ MTT AT þ MT1 A1 ¼ VT0 M1S AS þ M1T AT þ M11 A1 ¼ M11 A1 Q1S AS þ Q1T AT þ Q11 A1 ¼ Q11 A1

ð2:49Þ

AT T

Medium 1 Medium 2 A1 A1*

AS

S

Figure 2.8 An inclusion of medium 2 is placed between two transducers AS and AT .

84

ADVANCED THEORY OF DPSM—MODELING MULTILAYERED MEDIUM

The above equation can be written in matrix form. 9 8 9 2 38 VS0 > 0 MST > MSS MS1 AS > > > > > > < = < = 6 M1S M11 M M1T 7 A1 0 11 6 7 ¼ 4 Q1S Q11 Q Q1T 5> A > > 0 > 11 > > ; > ; : 1> : MTS MT1 0 MTT AT VT0

ð2:50Þ

The above matrix equation can be solved for the unknown source strength vectors AS , AT , A1 , and A1 . Although the method outlined above, in principle can solve the problem, there are some practical problems associated with this approach that comes due to the shadow regions. Note that a point in medium 1 does not see all point sources of layer A1 ; it also may not see all point sources of the two transducers. In the previous examples with plane interfaces, covered in Sections 2.2–2.4, this problem did not arise because the point of interest could never be in the shadow region. To overcome this difﬁculty the point sources can be assumed to radiate energy in only one direction–in one half space while the other half space remains in dark, and if the inclusion guards the ray path from the transducer to the point of interest, then the contribution of that point source is ignored. These modiﬁcations are illustrated in Figure 2.9. Figure 2.9 shows three layers (AS , A1 , and AT ) of point source that contribute to the ﬁeld computed at point P. If the point sources only radiate in the front half space while the back half space remains in dark, then the radiation pattern from these three layers of point source will be as shown by the crescent shapes in the ﬁgure. Then, point P will receive radiation from all point sources AS between points A and B, from some of the point sources A1 between points C and D, and from all point sources AT between points E and F. However, in reality because of the presence of the scatterer between point P and some of the AT point sources that are located between points F and G, the point P will be excited by only the AT point sources that are located between points E and G. Thus, from the problem geometry it should be ﬁrst ﬁgured out which point sources should excite the ﬁeld point; then, the contributions of all other point sources for that ﬁeld point should be set equal to zero. Writing a computer code satisfying these additional conditions is not easy. An alternative approach for solving the inclusion G AT

T F

E D

Medium 2 A1 C P Medium 1 A

B S

AS

Figure 2.9 Point P in medium 1 is receiving radiation from point sources AS , A1 , and AT .

85

2.6 SCATTERING BY AN INCLUSION OF ARBITRARY SHAPE

problem is proposed in the next section that can avoid the problem associated with the point of interest falling in the shadow region, as discussed above.

2.6 SCATTERING BY AN INCLUSION OF ARBITRARY SHAPE – AN ALTERNATIVE APPROACH The problem geometry for a multilayered medium as shown in Figure 2.10 is considered ﬁrst. In Figure 2.10 medium 2 is bounded by parallel and nonparallel surfaces as shown. Fields in media 1, 2, and 3 are obtained by adding contributions of two layers of point sources for each medium as described below. Medium 1: Summation of ﬁelds generated by AS and A1 . Medium 2: Summation of ﬁelds generated by A1 and A2 . Medium 3: Summation of ﬁelds generated by A2 and AT . In Figure 2.10 source layers A1 and A2 have been shown by solid circles to distinguish these from A1 and A2 layers that have been shown by open circles. Satisfying boundary conditions at the two transducer surfaces and continuity conditions across the two interfaces between media 1 and 2, and between media 2 and 3, the following equations are obtained: MSS AS þ MS1 A1 ¼ VS0 MTT AT þ MT2 A2 ¼ VT0

M1S AS þ M11 A1 ¼ M11 A1 þ M12 A2

Q1S AS þ Q11 A1 ¼ Q11 A1 þ Q12 A2 M21 A1 þ M22 A2 ¼ M22 A2 þ M2T AT Q21 A1 þ Q22 A2 ¼ Q22 A2 þ Q2T AT

AT T A2

Medium 3

A*2 Medium 2 A

1

A*1 AS

Medium 1

S

Figure 2.10 A three-layered medium is placed between two transducers.

ð2:51Þ

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ADVANCED THEORY OF DPSM—MODELING MULTILAYERED MEDIUM

AT T A2 Medium 1

A*2 Medium 2 A1

A*1 Medium 1

AS

S

Figure 2.11 Problem geometry obtained from Figure 2.10 by reducing the thickness of medium 2 between parallel layers to zero and replacing medium 3 of Figure 2.10 by medium 1.

Note that in Figure 2.10, only a few points in a small region may fall in the shadow region while evaluating Eq. (2.51). Such small shadows are expected to produce only negligible error. It should also be noted here that when medium 3 properties are made equal to medium 1 properties and the distance between the parallel boundaries of medium 2 is reduced to zero, then the problem geometry in Figure 2.10 is changed to the one shown in Figure 2.11. Clearly, there is no difference between the problem geometries shown in Figures 2.8 and 2.11 except how the point source layers are deﬁned. In Figure 2.8 AS , AT , A1 , and A1 are the four layers of point sources. In Figure 2.11 AS and AT are identical to those deﬁned in Figure 2.8. However, source layer A1 of Figure 2.8 is divided into two layers A1 and A2 in Figure 2.11, and A1 of Figure 2.8 is divided into two layers A2 and A1 in Figure 2.11. Let us then examine how Eq. (2.51) is modiﬁed in the limiting case, shown in Figure 2.11. MSS AS þ MS1 A1 þ 0:A2 ¼ VS0

MTT AT þ 0:A1 þ MT2 A2 ¼ VT0 M1S AS þ M11 A1 þ 0:A2 ¼ M11 A1 þ M12 A2

ð2:52Þ

Q1S AS þ Q11 A1 þ 0:A2 ¼ Q11 A1 þ Q12 A2 M21 A1 þ M22 A2 ¼ M22 A2 þ 0:A1 þ M2T AT Q21 A1 þ Q22 A2 ¼ Q22 A2 þ 0:A1 þ Q2T AT or

A1 ¼ VS0 0 A2 A1 ¼ VT0 MTT AT þ ½ 0 MT2 A2 A1 ¼ ½ M11 M1S AS þ ½ M11 0 A2 MSS AS þ ½ MS1

A1 M12 A2

2.7 ELECTRIC FIELD IN A MULTILAYERED MEDIUM

A1 A1 ¼ ½ Q11 Q12 Q1S AS þ ½ Q11 0 A2 A2 A A 1 1 ¼ ½ 0 M22 þ M2T AT ½ M21 M22 A2 A2 A1 A1 ¼ ½ 0 Q22 þ Q2T AT ½ Q21 Q22 A2 A2

87

ð2:53Þ

Eq. (2.53) can be written in matrix form. 2

MSS ½ MS1 0 ½ 0 6 M1S M11 M 0 11 6 6 M 0 M 0 22 6

21 6 Q1S Q Q11 0 6 11 4 Q21 0 Q22 0 0 ½ 0 MT2 ½0

3 9 8 9 0 0 8 AS > VS0 > > > > > > > > > > 7 M12 A1 > 0 0 > > > > > > > 7> = = < < 7 M22 M2T 7 A2 0

¼ 7 Q12 0 A1 > 0 > > > > > > 7> > > > > > > > > > > Q22 A Q2T 5> 2 ; : 0 > ; : VT0 AT 0 MTT ð2:54Þ

If Eq. (2.54) is solved instead of Eq. (2.50), then problems associated with the shadow region can be avoided. The theory presented in the above sections is used in the subsequent chapters for solving engineering problems in the ﬁeld of ultrasonic, electromagnetic, electrostatic, eddy current, and so on. Detail analyses of those problems in different ﬁelds are given in those chapters. In this chapter only a few sample results are presented below to show the versatility and reliability of the multilayered modeling technique in solving ultrasonic and electrostatic problems.

2.7 ELECTRIC FIELD IN A MULTILAYERED MEDIUM Electric ﬁeld in air between two square electrodes of dimension 100 mm 100 mm separated by a distance of 300 mm is computed. The problem geometry is shown in Figure 2.12. Two straight lines at z ¼ 150 mm, marked by S and T, denote the electrode surface positions; the points behind the straight lines are the point source positions. The electrodes have þ3 V and 3 V potentials. To check how accurately the electric potential ﬁeld in the air between the two electrodes is computed when multilayered modeling scheme is followed, the air medium between the two electrodes is assumed to be composed of three different media with two interfaces I1 and I2 placed at z ¼ 76:25 mm, as shown in Figure 2.12. Two layers of points on two sides of each interface show point source positions, as mentioned in the above sections under the theory of multilayered modeling; every interface is modeled by two layers of point sources. All the three mediums are given the same permittivity value 9 e0 ¼ 10 36p As=Vm that of air. The transducer surface area (100 mm 100 mm) is modeled by 100 point sources whereas each interface area ð400 mm 400 mmÞ is

88

ADVANCED THEORY OF DPSM—MODELING MULTILAYERED MEDIUM

Figure 2.12 Problem geometry for the electrostatic problem–S and T are two electrodes, having voltages 3 V and þ3 V, respectively. Two interface locations are marked by I1 and I2.

modeled by 900 point sources on each side. Electric potential and electric ﬁeld in the air between the two electrodes are computed in two ways: (a) modeling the air medium as a three-layered medium having identical properties for all three layers and (b) modeling the air between the two electrodes as a homogeneous medium. If the multilayered formulation presented above is correct, then these two computations should give identical results. The capacitance between the two electrodes, computed from these two analyses, are 2.39 fF (for two-interface analysis) and 2.42 fF (for nointerface analysis). Note that the difference is less than 2%. By increasing the number of point sources used to model the electrodes and the interfaces, this difference can be reduced even further. Detail variations of electric potential and electric ﬁeld in the region between the two electrodes are shown in Figures 2.13–2.17. Figure 2.13a shows the electric potential variation on the xy plane (the central plane between the two electrodes; this plane is normal to the electrode surfaces) generated from the multilayered modeling. When the air medium is modeled as a homogeneous medium without any interface, then the computed potential ﬁeld on the xy plane, as expected, shows almost identical variation, see Figure 2.13b. Figure 2.13 proves that the multilayered modeling theory presented in this chapter gives correct results. Comparison between the results generated by the multilayered medium modeling and homogeneous medium modeling are presented in Figures 2.14–2.27. Contour plots of the electric ﬁeld vectors on the xy plane for the two cases are

2.7 ELECTRIC FIELD IN A MULTILAYERED MEDIUM

89

Figure 2.13 Electric potential variation on the xz plane between two electrodes from (a) multilayered medium modeling with two interfaces and (b) homogeneous medium modeling with no interface.

shown in Figure 2.14. A closer inspection of Figure 2.14a and b shows a small difference in the contours in the region where two interfaces have been placed for the multilayered medium modeling case. Figure 2.15 shows the continuity of the electric potential across the ﬁrst interface (I1) for the multilayered modeling case and its close match with the homogeneous medium modeling results. Continuity of the electric ﬁeld normal to the ﬁrst interface for the multilayered modeling case and its close match with the homogeneous medium modeling results are shown in Figure 2.16. Similar good matching between the results (both for electric potential

Figure 2.14 Contour plots showing electric ﬁeld vector variation on the xz plane between two electrodes from (a) multilayered medium modeling with two interfaces and (b) homogeneous medium modeling with no interface.

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ADVANCED THEORY OF DPSM—MODELING MULTILAYERED MEDIUM

Figure 2.15 Electric potential variation on the ﬁrst interface position (I1, see Fig. 2.12). Two curves show the potential variation on two sides of the interface, and the third curve shows the variation of the same parameter at the same position when the medium is modeled as a homogeneous medium.

Figure 2.16 Variation of the normal component of the electric ﬁeld along the ﬁrst interface position (I1, see Fig. 2.12). Two curves show the electric ﬁeld variation on two sides of the interface, and the third curve shows the variation of the same parameter at the same position when the medium is modeled as a homogeneous medium.

2.8 ULTRASONIC FIELD IN A MULTILAYERED FLUID MEDIUM

91

Figure 2.17 Electric potential variation between two electrodes along the central axis. Two curves are obtained from the multilayered medium modeling with two interfaces and homogeneous medium modeling with no interface.

and normal component of the electric ﬁeld) obtained from the two modeling approaches is also observed for the second interface but is not presented here. Figure 2.17 shows the variation of electric potential along the central axis connecting the two electrodes; hardly any difference between the two modeling results can be noticed in this plot.

2.8 ULTRASONIC FIELD IN A MULTILAYERED FLUID MEDIUM Banerjee et al. (2006) have developed MATLAB programs to model the ultrasonic ﬁeld using the DPSM formulation presented above. Results for two different cases, presented in that reference, are shown here. In case 1 two ﬂuids have been considered in a three-layered medium as shown in Figure 2.18a, and in case 2 four different ﬂuids have been considered as shown in Figure 2.18b. In the ﬁrst case the higher density ﬂuid has been placed in between two identical lower density ﬂuid half spaces. In the second case four layers of ﬂuid have been arranged such that their density monotonically increases from the bottom to the top. For convenience, in this section the x1 axis is called the x-axis and x3 axis is called the z-axis. Three different orientations of the two transducers have been considered as illustrated in Figure 2.19. In the ﬁrst orientation, transducers are placed face to face; transducer faces are parallel to the x-axis as shown in Figure 2.19a. In the second orientation, the transducer T has been shifted horizontally and inclined at an

92

ADVANCED THEORY OF DPSM—MODELING MULTILAYERED MEDIUM

Figure 2.18 (a) Three-layered and (b) four-layered ﬂuid structures inside which the ultrasonic ﬁeld is computed.

T

y x

T θ

D

T θ θ

S (a)

S

S

(b)

(c)

Figure 2.19 Three different orientations of transducers S and T, for which ultrasonic ﬁelds are computed.

angle y with respect to the z-axis, as shown in Figure 2.19b. Figure 2.19c illustrates the third orientation where both the transducers are inclined. The density and the P-wave speed of the ﬂuid considered in this study are given in Table 2.1.

TABLE 2.1. Fluid properties Fluids and Properties Acetone Ethyl benzol Water Glycerine

P-wave Speed (km/s) 1.17 1.34 1.48 1.92

Density (g/cc) 0.790 0.868 1.00 1.26

2.8 ULTRASONIC FIELD IN A MULTILAYERED FLUID MEDIUM

93

2.8.1 Ultrasonic ﬁeld developed in a three-layered medium Figure 2.20 shows the ultrasonic ﬁelds generated in the Water-Glycerin-Water (WGW, see Fig. 2.18a) structure for different transducer orientations, as shown in Figure 2.19. The transducers have 4 mm diameter. Ultrasonic ﬁelds generated in

Figure 2.20 Ultrasonic ﬁelds in the three-layered ﬂuid structure for different transducer orientations: (a) orientation of Figure 2.19c for 1 MHz transducers with only T on, (b) orientation of Figure 2.19c for 1 MHz transducers with both S and T on, (c) orientation of Figure 2.19b for 1 MHz transducers with both S and T on, (d) orientation of Figure 2.19c for 2.2 MHz transducers with only T on, (e) orientation of Figure 2.19c for 2.2 MHz transducers with both S and T on and, (f) orientation of Figure 2.19b for 2.2 MHz transducers with both S and T on.

94

ADVANCED THEORY OF DPSM—MODELING MULTILAYERED MEDIUM

WGW structure due to 1 MHz excitation have been presented in the left column of Figure 2.20 (Fig. a, b and c) and ultrasonic ﬁelds due to 2.2 MHz excitation have been presented in the right column, Figure 2.20d, e and f. The depth of the Glycerin layer is 20 mm for both 1 MHz and 2.2 MHz signals. One MHz transducers are kept at 10 mm distance from the Water-Glycerin interfaces, but 2.2 MHz transducers are placed at 20 mm distance from the same interfaces. Additional distance for the 2.2 MHz transducers is necessary to make sure that the interface is not placed within the near ﬁeld region (see Eq. (3.28)) of the transducers. Therefore, the total width (D in Fig. 2.19) of the WGW structure is 40 mm for 1 MHz transducers and 60 mm for 2.2 MHz transducers. It is well known that the pressure ﬁeld generated at a point in front of a transducer depends on the frequency of excitation and the distance of the point from the transducer face (see Chapters 1 and 3). As frequency increases the isobars gradually shift away from the transducer face. To generate approximately the same pressure value at the interface position, the distance D between the transducers has been varied when the transducer frequency is changed from 1 to 2.2 MHz. In Figure 2.20a–f, the transducer orientations shown in Figure 2.19b and c have been considered for the inclination angle y ¼ 30 , measured from the vertical axis. Figure 2.20a and d is generated with transducer Orientation shown in Figure 2.19c when only transducer T is on. Figure 2.20b and e is generated for the same transducer orientation but when both transducers S and T are on. In Figure 2.20b and e, the ultrasonic ﬁelds in glycerin form a nice pattern due to interaction between the transmitted and reﬂected ﬁelds. Note that this pattern is dependent on the signal frequency. For the WGW ﬂuid structure when the transducers are oriented as shown in Figure 2.19b with both the transducers on, at 1 MHz frequency a nice pattern is formed inside glycerin from the interaction between the transmitted ﬁelds. A strong interaction between two transmitted ﬁelds is also visible in Figure 2.20f for 2.2 MHz signals. 2.8.2 Ultrasonic ﬁeld developed in a four-layered ﬂuid medium In the second case four ﬂuids have been considered, and they are placed with monotonically increasing density as shown in Figure 2.18b. Analyses have been carried out for transducer orientation shown in Figure 2.19a with only T on, only S on, and both transducers S and T on. Computed ﬁelds are presented in Figure 2.21. Ultrasonic ﬁelds generated in acetone-benzol-water-glycerin (ABWG) structure by 1 MHz transducers have been presented in the left column (Fig. 2.21a–c), and the ultrasonic ﬁelds generated by 2.2 MHz transducers have been presented in the right column (Fig. 2.21d–f). For both the frequencies the thickness of benzol and water layers are taken as 20 mm each. For the same reason as discussed in Section 2.8.1, the transducers are placed 10 mm away from the interfaces for 1 MHz transducers and 20 mm away from the interfaces for 2.2 MHz transducers. Therefore, D of Figure 2.19 is equal to 60 mm for 1 MHz and 80 mm for 2.2 MHz transducers. In Figure 2.21 only the normal incidence of the wave ﬁeld from the transducers have been considered. Figure 2.21a and d shows the ultrasonic ﬁelds when transducer T is on with 1 MHz and 2.2 MHz signal frequency, respectively. The scattered ﬁeld

2.8 ULTRASONIC FIELD IN A MULTILAYERED FLUID MEDIUM

95

from the transducer S is visible near the central region of the bottom layer at 1 MHz frequency but it is not very clear at 2.2 MHz. Relative pressure in glycerin is much higher than acetone at 2.2 MHz in comparison to 1 MHz. When S is on and T is turned off, the scattered ﬁeld from transducer T can be clearly seen in the top layer at both the frequencies. Figure 2.21c and f has been generated when both S and T are on. We can

Figure 2.21 Ultrasonic ﬁelds in the four-layered ﬂuid structure for the transducer orientation, shown in Figure 2.19a for (a) 1 MHz transducers with only T on, (b) 1 MHz transducers with only S on, (c) 1 MHz transducers with both S and Ton, (d) 2.2 MHz transducers with only T on, (e) 2.2 MHz transducers with only S on and, (f) 2.2 MHz transducers with both S and T on.

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ADVANCED THEORY OF DPSM—MODELING MULTILAYERED MEDIUM

see that the ultrasonic ﬁeld generated in acetone is weaker compared to glycerin due to higher impedance of glycerin. At 2.2 MHz the pressure ﬁeld has several dips and peaks in benzol and it is not so prominent at 1 MHz. As expected, the pressure ﬁeld in water is more collimated at 2.2 MHz compared to that at 1 MHz. It is clear from the results presented in Sections 2.7 and 2.8 that the algorithm presented in this chapter is capable of producing electric and ultrasonic ﬁelds in multilayered media in the presence of multiple scatterers and sensors (electrodes and transducers) of ﬁnite dimension.

REFERENCE Banerjee, S., T., Kundu, and D., Placko, ‘‘Ultrasonic Field Modelling in Multilayered Fluid Structures Using DPSM Technique’’, ASME Journal of Applied Mechanics, Vol. 73 (4), pp. 598–609, 2006.

3 ULTRASONIC MODELING IN FLUID MEDIA T. Kundu, R. Ahmad, N. Alnuaimi University of Arizona, USA

D. Placko Ecole Normale Superieure, Cachan, France

3.1 INTRODUCTION Although the basic theory of distributed point source method (DPSM) has been presented in Chapter 1, some DPSM concepts and fundamental governing equations of ultrasonic ﬁeld modeling are repeated in this chapter for better understanding of DPSM modeling for ultrasonic problems. DPSM modeling is needed because it is often difﬁcult to solve ultrasonic problems in geometries that have boundaries of ﬁnite dimension. In the absence of any boundary, in an inﬁnite space, the problem of ultrasonic wave propagation can be easily solved. The solution of the elastic wave propagation in an inﬁnite isotropic ﬂuid or solid (Fig. 3.1a) is simple. It is the homogeneous solution of the wave equation. In a ﬂuid medium, the wave equation is given by r2 ff

1 € f ¼0 c2f f

ð3:1Þ

@ @ @ where r2 is the Laplacian operator. r2 ¼ @x 2 þ @x2 þ @x2 for three-dimensional (3D) 1 2 3 2 @2 @2 problems and r ¼ @x2 þ @x2 for two-dimensional (2D) problems. cf is the acoustic 2

1

2

2

2

DPSM for Modeling Engineering Problems, Edited by Dominique Placko and Tribikram Kundu Copyright # 2007 John Wiley & Sons, Inc.

97

98

ULTRASONIC MODELING IN FLUID MEDIA

x2

x2

o o

x1

x1 q

q

q

(b)

(a)

x2

x2 o

A

x1

o

x1 B

q

q

q

(d)

(c)

Figure 3.1 Ultrasonic waves propagating in different problem geometries: (a) ﬂuid full space, (b) ﬂuid half space, (c) ﬂuid quarter space, and (d) ﬂuid quarter space with the corner cut-off.

wave speed in the ﬂuid and ff is the wave potential. Two dots above ff indicate double derivative with respect to time. For harmonic time dependence ðeiot Þ, the above equation simpliﬁes to r2 ff þ kf2 ff ¼ 0

ð3:2Þ

where kf ¼ cof is the wave number and o is the wave frequency in radian per second. For simplicity, here we will restrict our discussion to 2D problems. Solution of Eq. (3.2) for the ultrasonic wave propagating in a two-dimensional space is given by ff ¼ Aeiðkx1 þZf x2 otÞ

ð3:3Þ

where A is the amplitude of the wave potential. k and Zf can be expressed in terms of the wave propagation direction y and the wave number kf . See Figure 3.1a for y. The relations between k, Zf , kf , and y are given by k ¼ kf sin y Zf ¼ kf cos y

ð3:4Þ

If the inﬁnite space of Figure 3.1a is now changed to a half space by introducing a plane surface boundary at x2 ¼ 0 as shown in Figure 3.1b, then Eq. (3.3) is not sufﬁcient to satisfy the stress-free boundary conditions at the plane boundary

99

3.1 INTRODUCTION

surface. For the half-space problem geometry of Figure 3.1b, it is necessary to satisfy the following boundary condition in addition to the governing equation given in Eq. (3.2): s22 ¼ rf o2 ff ¼ 0;

at x2 ¼ 0

) ff ¼ 0;

at x2 ¼ 0

ð3:5Þ

Clearly, solution (3.3) cannot satisfy both Eqs. (3.2) and (3.5). However, addition of one more term in the expression of ff of Eq.(3.3) can satisfy both the governing equation (3.2) and the boundary condition (3.5). With this additional term, the new expression of ff becomes ff ¼ Aeiðkx1 þZf x2 otÞ Aeiðkx1 Zf x2 otÞ

ð3:6Þ

Equation (3.6) represents two plane waves, one propagating in the positive x2 direction or in the upward direction and the other in the negative x2 direction or in the downward direction, as shown in Figure 3.1b. The downward wave is generated by the reﬂection phenomenon at the boundary. The problem becomes even more complex when two boundary surfaces are introduced at x2 ¼ 0 and x1 ¼ 0 as shown in Figure 3.1c because then additional boundary conditions are to be satisﬁed. For this quarterspace problem, it is necessary to satisfy the governing equation (3.2) and the following boundary conditions: s22 ¼ rf o2 ff ¼ 0;

at x2 ¼ 0

s11 ¼ rf o ff ¼ 0;

at x1 ¼ 0

) ff ¼ 0; ff ¼ 0;

at x2 ¼ 0 at x1 ¼ 0

2

ð3:7Þ

Equations (3.2) and (3.7) can be simultaneously satisﬁed when two more terms are added in the expression of ff as shown below. ff ¼ Aeiðkx1 þZf x2 Þ Aeiðkx1 Zf x2 Þ þ Aeiðkx1 Zf x2 Þ Aeiðkx1 þZf x2 Þ

ð3:8Þ

In the above equation the harmonic time dependence ðeiot Þ is implied. Four terms of Eq. (3.8) correspond to four plane waves propagating in different directions of the quarter space as shown in Figure 3.1c. If a third boundary surface AB is then introduced as shown in Figure 3.1d, then Eq. (3.8) is not sufﬁcient to satisfy the boundary condition along the boundary AB. Then, the solution becomes much more complex, and no closed-form simple analytical expression like Eqs. (3.3), (3.6), or (3.8) can satisfy all boundary conditions. As more boundary surfaces of ﬁnite dimension are introduced, the wave propagation problem becomes increasingly more difﬁcult and impossible to solve analytically. DPSM can be followed to solve such complex problems. This technique for solving

100

ULTRASONIC MODELING IN FLUID MEDIA

the ultrasonic problems is developed step by step in this chapter, and then solutions of some example problems are presented.

3.2 PRIMARY (ACTIVE) AND SECONDARY (PASSIVE) SOURCES A source of inﬁnite length generates the plane elastic wave given in Eq. (3.3). This source can be a planar transducer of inﬁnite width. This transducer is not shown in Figure 3.1a. Introduction of boundaries in Figure 3.1b and c produces more plane waves. Therefore, the introduction of boundaries, in essence, is similar to introducing more sources. We will call these additional sources as boundary sources or secondary or passive sources whereas the primary on active sources are the ultrasonic transducers that generate the original ultrasonic beams before these interact with the boundaries. It should be noted here that in the absence of primary sources, no secondary source can exist because no reﬂected beam can be generated if there is no incident beam. The secondary sources should have strengths such that the total ﬁeld (generated by the primary and secondary sources) satisﬁes all boundary conditions. Boundaries of inﬁnite length produce reﬂected waves with plane wave fronts as shown in Figure 3.1b and c; the equations corresponding to these reﬂected plane waves are given in Eqs. (3.6) and (3.8).

3.3 MODELING ULTRASONIC TRANSDUCERS OF FINITE DIMENSION IMMERSED IN A HOMOGENEOUS FLUID A boundary of ﬁnite dimension (AB in Fig. 3.1d) does not produce a perfect plane wave reﬂection. In the same manner, a source of ﬁnite dimension does not produce an ideal plane wave. As mentioned in the earlier chapters, in the DPSM technique, sources and boundaries of ﬁnite dimension are modeled by distributing a number of point sources along the ultrasonic transducer surface (primary or active sources) and the reﬂecting (or transmitting) boundaries (secondary or passive sources). Point source strengths are adjusted to satisfy the appropriate boundary conditions. The ultrasonic ﬁeld produced by a single point source is given by pðrÞ ¼ A

eikf r r

ð3:9Þ

where p(r) is the pressure at a radial distance r from the point source, A is the source strength, and kf deﬁned after Eq. (3.2) is the wave number in the ﬂuid. The point source displacement oscillates with time dependence eiot . The vibration of the point source is shown in Figure 3.2; it alternately expands and contracts giving a radial outward motion from the contracted position and inward motion from the expanded position. Let a number of these point sources be placed side by side along a plane surface, and every point source is placed very close to its neighboring point sources. When the

101

3.3 MODELING ULTRASONIC TRANSDUCERS

E

C

Figure 3.2 Schematic of a point source. Inward arrows show displacement direction from the expanded position, and outward arrows show displacement direction from the contracted position.

motions of these point sources are superimposed, then the displacement components normal to the plane surface are added whereas the components parallel to the surface cancel each other giving rise to a resultant displacement vibration normal to the surface, as shown in Figure 3.3. It should be noted here that if a point source has neighboring point sources on both sides, then their horizontal components of displacement cancel due to the presence of the neighboring sources. However, for point sources distributed along the periphery, not all horizontal components disappear. Therefore, point sources distributed along the periphery of the transducer surface show both vertical and horizontal components of displacement. If a large number of point sources are placed along a ﬁnite plane surface as shown in Figure 3.4a, then its combined effect is a vibrating plane whose expanded and contracted geometries are as shown in Figure 3.4b. The plane marked by the dashed lines is the contracted position. Note that the points on the periphery vibrate in all directions – vertical, horizontal, and inclined, whereas the inner points vibrate only vertically. A vibrating transducer surface is shown in Figure 3.4c where all points vibrate vertically. A comparison of Figure 3.4b and c shows very similar motions of the surface except near the surface periphery. In Figure 3.4b the peripheral points have both horizontal and vertical components of the vibratory motion whereas in Figure 3.4c all the points including the peripheral points have only the vertical

+

+

+

+

Figure 3.3 Effect of superposition of three point sources. After superposition for 2D problems, the middle point source shows vibration in the vertical direction only whereas the point sources on the two ends show vibrations that have both horizontal and vertical components.

102

ULTRASONIC MODELING IN FLUID MEDIA

(a)

(b)

(c)

Figure 3.4 (a) Point sources distributed over a planar surface, (b) resultant vibration of the planar surface generated by a large number of point sources distributed over the surface as shown in a, and (c) vibrating surface of a piston transducer. Note that the vibratory motions of b and c are similar (both are vibrating in the vertical directions) except for the peripheral points.

component of motion. Pressure ﬁelds generated by the vibratory motions shown in Figure 3.4b and c are expected to be close but not necessarily identical. If there are N number of point sources distributed over a plane surface S, as shown in Figure 3.5, then the total pressure ﬁeld at point x generated by all point sources is given by pðxÞ ¼

N X

pm ðrm Þ ¼

m¼1

N X m¼1

Am

expðikf rm Þ rm

ð3:10Þ

where Am is the source strength and rm is the distance of the observation point from the mth point source located at ym as shown in Figure 3.5. Source strength Am can be obtained by satisfying the boundary conditions speciﬁed on the transducer surface. For example, if the pressure p0 or the x3 direction velocity v0 on a speciﬁc point B (see Fig. 3.6) on the transducer surface is speciﬁed,

x1

ym

x3 rm

x2

x

Figure 3.5 Pressure ﬁeld at point x due to N number of point sources distributed over the transducer face.

103

3.3 MODELING ULTRASONIC TRANSDUCERS

x1

B x3

rS

Figure 3.6 Side view of a transducer showing the point sources located at a distance rS behind the transducer front face and a general point B on the transducer face.

then for that point we can write the boundary conditions in the following form (see Eqs. (1.18) and (1.24)), N X

N X

expðikf rm Þ rm m¼1 m¼1 N X Am x3m expðikf rm Þ 1 v0 ¼ ikf ior rm2 rm m¼1

p0 ¼

pm ðrm Þ ¼

Am

ð3:11Þ ð3:12Þ

Figure 3.6 shows the side view of a transducer whose front face is placed on the x1 x2 plane. Point sources are distributed at a distance rS behind the transducer face for avoiding the singularity problem, in other words to make sure that rm in Eqs. (3.11) and (3.12) does not become zero for any point B, as discussed in Chapter 1. If point B is taken on the apex of small circles associated with every point source, then there will be N number of points Bm (m ¼ 1; 2; 3; . . . ; N) for N point sources. Therefore, there will be N values of the pressure (Pm ) and normal velocity (Vm) at N points Bm. There are also N values of the source strength Am. These m values can be expressed in the vector form as shown below. PS ¼ ½P1 P2 . . . Pm . . . PN T VS ¼ ½V1 V2 . . . Vm . . . VN T

ð3:13Þ

AS ¼ ½A1 A2 . . . Am . . . AN T where subscript S is used to indicate that pressure (P), velocity (V), and source strength (A) are given at the transducer source level. Superscript T stands for transpose indicating that PS, VS, and AS are column vectors. From Eqs. (3.11) and (3.12), PS, VS, and AS can be expressed in the following form. PS ¼ QSS AS VS ¼ MSS AS

ð3:14Þ

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ULTRASONIC MODELING IN FLUID MEDIA

Double subscript SS indicates that matrices Q and M relate pressure (PS) and velocity (VS) at the transducer source level to the point source strengths (AS) that is also deﬁned at the transducer source level. Matrices QSS and MSS are deﬁned as (see similar Eqs. in (1.25l), (1.25e), and (1.25f) 2

QSS

expðikf r11 Þ expðikf r21 Þ 6 r11 r21 6 6 6 expðikf r 2 Þ expðik r 2 Þ f 2 1 6 6 r22 r12 6 6 6 expðikf r13 Þ expðikf r23 Þ ¼6 6 6 r13 r23 6 ... ... 6 6 6 ... ... 6 6 N 4 expðikf r Þ expðikf r N Þ 1 2 r1N r2N

expðikf r31 Þ r31

...

...

expðikf r32 Þ r32

...

...

expðikf r33 Þ r33 ...

...

...

...

...

...

...

...

expðikf r3N Þ r3N

...

...

3 expðikf rN1 Þ 7 rN1 7 7 2 7 expðikf rN Þ 7 7 rN2 7 7 3 7 expðikf rN Þ 7 ð3:15Þ 7 7 rN3 7 ... 7 7 7 ... 7 7 expðikf r N Þ 5 rNN

N

NN

and 2

MSS

f ðx131 ; r11 Þ

6 6 f ðx2 ; r 2 Þ 6 31 1 6 6 f ðx3 ; r 3 Þ 6 31 1 ¼6 6 6 ... 6 6 6 ... 4 N f ðx31 ; r1N Þ

f ðx132 ; r21 Þ

f ðx133 ; r31 Þ

... ...

f ðx232 ; r22 Þ

f ðx233 ; r32 Þ

... ...

f ðx332 ; r23 Þ

f ðx333 ; r33 Þ

... ...

...

...

... ...

...

...

... ...

f ðxN32 ; r2N Þ

f ðxN33 ; r3N Þ

... ...

f ðx13N ; rN1 Þ

3

7 f ðx23N ; rN2 Þ 7 7 7 3 3 7 f ðx3N ; rN Þ 7 7 7 7 ... 7 7 7 ... 5 f ðxN3N ; rNN Þ NN

ð3:16Þ

where, n f ðx3m ; rmn Þ

¼

n x3m expðikf rmn Þ

iorðrmn Þ2

1 ikf n rm

ð3:17Þ

In the above equations rmn stands for the distance between the mth point source n is the x3 component of rmn . and nth observation point, see Figure 3.7, and x3m n From Figure 3.7 it is clear that for MSS, x3m is equal to rS. Because for nonviscous perfect ﬂuid only x3 component of velocity should be continuous across the transducer–ﬂuid interface, the matrix MSS has a dimension of N N, and it relates only one component of velocity to the point source strength. If all the three components of velocity are to be continuous across the transducer–ﬂuid

105

3.3 MODELING ULTRASONIC TRANSDUCERS

x1

N Pn, Vn

n . . . m Am

rmn

x3

x3mn

2 1 rS

rmn

Figure 3.7 is the distance between the mth point source and nth observation point; xn3m is the x3 component of rmn .

interface as in the case for viscous ﬂuid or solid, then the dimension of MSS should increase as given in Eq. (1.25e). If the normal velocity on the ultrasonic transducer surface is speciﬁed, then the point source strength vector AS needed to model that transducer can be obtained from the second equation of Eq. (3.14). AS ¼ ½MSS 1 VS

ð3:18Þ

Generally, ultrasonic transducers immersed in a ﬂuid medium, when excited by an electric current, vibrate with a constant velocity v0. Therefore, VS ¼ ½v0

v0 . . . . . . . . . v0 T

ð3:19Þ

Hence, AS ¼ v0 ½MSS 1 ½1 1 1 . . . . . . 1T ¼ v0 NSS ½1 1 1 . . . . . . 1T

ð3:20Þ

where, NSS ¼ ½MSS 1

ð3:21Þ

Instead of velocity if the pressure at the transducer surface is speciﬁed, then the point source strength vector AS needed to model that transducer can be obtained from the ﬁrst equation of Eq. (3.14). AS ¼ ½QSS 1 PS

ð3:22Þ

106

ULTRASONIC MODELING IN FLUID MEDIA

x1

N rmn

. . m Am . .

x3mn

2

Source p oints

1 rS

M . . n . . . . . . . 4 3 2 1

Pn, Vn x1mn x3

Target p oints

Figure 3.8 Target points or observation points (1, 2, . . ., M) are located away from the transducer whereas the point sources (1, 2, . . ., N) are located at a distance rS behind the transducer face.

After obtaining AS needed to model a speciﬁc transducer, the ultrasonic ﬁeld at any general point B, not necessarily on the transducer surface, can be obtained from the following equations: PT ¼ QTS AS VT ¼ MTS AS

ð3:23Þ

Note that Eq. (3.23) is similar to Eq. (3.14) except for the subscript T. Because the pressure and velocity ﬁeld is now computed at different points (not on the transducer surface, see Fig. 3.8), we call these new set of points as target points, and to distinguish these from the source points, the subscript T is used to represent pressure and velocity vectors at the target points. Because the point sources are still located at the transducer source position, the subscript of AS is not changed. QTS and MTS matrices are similar to QSS and MSS given in Eqs. (3.15) and (3.16), respectively; the only difference is that because the number of target points is M, the number of rows will change from N to M and the dimension of QTS and MTS will be M N instead of N N. If one is interested in computing all, three components of velocity at the target points, then the following equation should be used instead of the velocity expression given in Eq. (3.23) VT ¼ MTS AS

ð3:24Þ

107

3.3 MODELING ULTRASONIC TRANSDUCERS

where 2

MTS

f ðx111 ; r11 Þ 6 6 6 f ðx121 ; r11 Þ 6 6 6 f ðx1 ; r1 Þ 6 31 1 6 6 6 f ðx211 ; r12 Þ 6 ¼6 6 f ðx2 ; r2 Þ 6 21 1 6 6 6 f ðx231 ; r12 Þ 6 6 6 ... 6 4 M f ðxM 31 ; r1 Þ

f ðx112 ; r21 Þ

f ðx113 ; r31 Þ

f ðx114 ; r41 Þ

...

...

...

f ðx122 ; r21 Þ

f ðx123 ; r31 Þ

f ðx124 ; r41 Þ

...

...

...

f ðx132 ; r21 Þ

f ðx133 ; r31 Þ

f ðx134 ; r41 Þ

...

...

...

f ðx212 ; r22 Þ

f ðx213 ; r32 Þ

f ðx214 ; r42 Þ

...

...

...

f ðx222 ; r22 Þ

f ðx223 ; r32 Þ

f ðx224 ; r42 Þ

...

...

...

f ðx232 ; r22 Þ

f ðx233 ; r32 Þ

f ðx234 ; r42 Þ

...

...

...

...

...

...

...

...

...

M f ðxM 34 ; r4 Þ . . .

...

...

M M M f ðxM 32 ; r2 Þ f ðx33 ; r3 Þ

f ðx11N ; rN1 Þ

3

7 7 f ðx12N ; rN1 Þ 7 7 7 1 1 7 f ðx3N ; rN Þ 7 7 7 f ðx21N ; rN2 Þ 7 7ð3:25Þ 7 2 2 7 f ðx2N ; rN Þ 7 7 7 f ðx23N ; rN2 Þ 7 7 7 7 ... 7 5 M f ðxM 3N ; rN Þ 3MN

where xnjm expðikf rmn Þ 1 ik f xnjm ; rmn ¼ f rmn iorðrmn Þ2

ð3:26Þ

The velocity vector obtained from Eq. (3.24) will have the following form: h iT M M VT ¼ v11 v12 v13 v21 v22 v23 . . . vM 1 v2 v3

ð3:27Þ

It should be mentioned here that if the transducer surface velocity is known, then the ultrasonic ﬁeld in the ﬂuid can be computed by the DPSM formulation presented above or by Rayleigh–Sommerﬁed integral representation as given in Eq. (1.15). However, instead of the transducer surface velocity if the pressure at the transducer surface position is speciﬁed, then Rayleigh–Sommerﬁed integral representation will not work but DPSM will. The theory presented above for modeling ultrasonic ﬁeld generated by a transducer of ﬁnite dimension, immersed in a ﬂuid medium, is used in the following section for computing ultrasonic ﬁelds generated by piston transducers, immersed in water. 3.3.1 Numerical results—ultrasonic transducers of ﬁnite dimension immersed in ﬂuid Following the DPSM technique described above, the ultrasonic ﬁeld in a homogeneous ﬂuid (water) generated by a ﬂat circular transducer is computed. For such simple problem geometry, the closed form expression of the near ﬁeld zone length and the divergence angle of the emitted beam can be calculated in closed form (Kundu, 2000). Numerically computed values of these two parameters are

108

ULTRASONIC MODELING IN FLUID MEDIA

compared with the closed form analytical values to check the accuracy of the numerical results. From Kundu (2000),

NF ¼

D2 l2 D2 4l 4l

for l << D

ð3:28Þ

and 1

f ¼ 2 sin

l 1:2 D

ð3:29Þ

where NF is the near-ﬁeld length, the distance between the transducer face and the last peak of the pressure ﬁeld along the x3 axis (or z-axis), the axis of symmetry of the piston transducer. D is the transducer diameter; l is the wavelength of the ultrasonic signal, and f is the divergence angle. The analytical expression of the pressure in a homogeneous ﬂuid along the central axis of a circular transducer is given in Eq. (1.16b) qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ k j pðx3 Þ ¼ rcf v0 expðikf x3 Þ expðikf x23 þ a2 Þ Figure 3.9 shows acoustic pressure variations along the central axis (x3-axis or z-axis)) of circular transducers computed by the DPSM technique described above (solid curve) and the analytical expression (Eq. (1.16b) dashed curve). Coordinate z is measured from the transducer face. The transducer area is 2.24 mm2, its diameter ðDÞ ¼ 2:528 mm, and signal frequency ð f Þ ¼ 5 MHz. For the DPSM modeling 128 point sources are placed at z or x3 ¼ rS, while the transducer face is at x3 ¼ 0. For this problem geometry one can compute the point source, transducer surface distance qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pD2 ﬃ ¼ p2:528 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 0:08 mm. Because the wave speed in the surround¼ pDﬃﬃﬃﬃ rS ¼ 42pN 8N 8128 ing ﬂuid (water) is 1.49 km/s, wavelength l is equal to 0.298 mm for 5 MHz signal. From Eq. (3.28) NF is found to be 5.3 mm. This theoretical value of NF matched very well with the last peak obtained numerically. The matching between the DPSM computed curve and the analytical solution in Figure 3.9 is also excellent. Figure 3.10 shows acoustic pressure in front of a transducer with diameter D ¼ 6 mm and f ¼ 2.2 MHz, obtained by the DPSM technique with N ¼ 200. The theoretical divergence angle f for this transducer is 15.6 calculated from Eq. (3.29). The divergence angle in Figure 3.10 is measured and found to be 15 , very close to the theoretical prediction. These two checks – Figures 3.9 and 3.10 – validate the DPSM computed results. Figure 3.11 shows the ultrasonic beam (pressure ﬁeld) in water generated by circular transducers of 2.2 MHz natural frequency for three different transducer

109

3.3 MODELING ULTRASONIC TRANSDUCERS Acoustic pressure along Z-axis 4500 4000

DPSM model Exact formulation

NF = 0.0053 m

Pressure (micro Pa)

3500 3000 2500 2000 1500 1000 500 0

0

0.005

0.01

0.015

Z-axis (m)

Figure 3.9 Acoustic pressure variation along the central axis of a circular transducer. Coordinate z is measured from the transducer face. The transducer diameter (D) ¼ 2.528 mm and signal frequency ( f ) ¼ 5 MHz. Solid and dashed lines are generated by the DPSM technique and the analytical formula (Eq. (1.16b)), respectively. Near-ﬁeld zone (NF) predicted by Eq. (3.28) matches well with the last peak of the curve.

Figure 3.10 Ultrasonic pressure variation in water, generated by a 6 mm diameter circular transducer excited at 2.2 MHz frequency.

110

ULTRASONIC MODELING IN FLUID MEDIA

Figure 3.11 Ultrasonic pressure variation in water, generated by circular transducers of natural frequency 2.2 MHz and diameter (a) 2.6 mm, (b) 2.7 mm, and (c) 6 mm.

diameters. Figure 3.11a and b is generated by almost identical transducers; (a) is for 2.6 mm diameter and (b) is for 2.7 mm diameter transducers. However, the pressure ﬁelds in Figure 3.11a and b look quite different because the vertical scale is different in these two plots, but the horizontal scale is same. Figure 3.11a is plotted up to z ¼ 10 mm whereas Figure 3.11b is plotted up to z ¼ 40 mm. The near ﬁeld regions NF calculated from Eq. (3.28) are 2.33 mm for Figure 3.11a and 2.52 mm for Figure 3.11b. These values match well with those for the computed beam signals. For 6 mm diameter transducer, as shown in Figure 3.11c, as expected the beam is found to be more collimated, because the divergence angle is reduced for larger diameter transducer (see Eq. (3.29)). A few more cases of the ultrasonic pressure ﬁeld generated in a homogeneous ﬂuid are shown in Figures 1.34–1.39. Figure 1.33 shows the geometries of the three transducers that generate the ultrasonic beams that are presented in Figures 1.34–1.38. The three transducers are a ﬂat circular transducer, a ﬂat square transducer, and a focused transducer (concave circular transducer). These results have been discussed in Chapter 1, and that discussion is not repeated here.

111

3.4 MODELING ULTRASONIC TRANSDUCERS

3.4 MODELING ULTRASONIC TRANSDUCERS OF FINITE DIMENSION IMMERSED IN A NONHOMOGENEOUS FLUID Let the transducer be immersed in a ﬂuid medium consisting of two ﬂuids, with a plane interface between two ﬂuids located in front of the transducer, as shown in Figure 1.27. Clearly, the ultrasonic energy generated by the transducer will be reﬂected and transmitted by the interface. The ﬁeld in ﬂuid 1 is the summation of two ﬁelds, the incident ﬁeld from the transducer and the reﬂected ﬁeld from the interface. On the contrary, the ﬁeld in ﬂuid 2 is generated by the transmitted rays only. We can introduce three layers of point source AS, AI, and A*I, as shown in Figure 3.12, to model the incident ﬁeld, reﬂected ﬁeld, and transmitted ﬁeld, respectively. Figure 3.12 shows the point source layer AS placed near the transducer whereas AI and A*I are placed on two sides of the interface. Because AS models the ultrasonic ﬁeld generated by the transducer, it is the active or primary source layer, whereas AI and A*I model the reﬂected and transmitted ﬁelds, respectively, and thus are passive or secondary sources. Observation points or target points are shown by solid circles in Figure 3.12. The number of point sources in the active source layer AS is NS. Passive sources at the interfaces AI and A*I have the same number of point sources as shown in Figure 3.12, and it can be different from NS; let this number be NI. The number of target points in ﬂuid 1 or in ﬂuid 2 is denoted by M. If the source strength vectors AS, AI, and A*I are known, then the pressure and velocity ﬁelds at the target points in ﬂuids 1 and 2 can be obtained from the following equations that are similar to Eq. (3.23): In ﬂuid 1 PT ¼ QTS AS þ QTI AI

ð3:30Þ

VT ¼ MTS AS þ MTI AI and in ﬂuid 2 PT ¼ QTI AI

ð3:31Þ

VT ¼ MTI AI

Fluid 2

Observation p oints AI In terfac e

Fluid 1

3 layers of point source

Transducer

AS

A*I

S

Figure 3.12 Point source distributions (open circles) near the transducer face and the interface needed to model the ultrasonic ﬁeld in a nonhomogeneous ﬂuid medium. Observation points or target points are shown by solid circles in two ﬂuids.

112

ULTRASONIC MODELING IN FLUID MEDIA

It should be noted here that AS, AI, and A*I are vectors of dimensions (NS 1), (NI 1), and (NI 1), respectively, whereas PT and VT have the dimension (M 1). Therefore, MTS , MTI, and M*TI (as well as QTS , QTI, and Q*TI ) are not square matrices; their dimensions are (M NS) or (M NI). Q and M matrices of Eqs. (3.30) and (3.31) are given below. 2

QTS

expðikf r11 Þ 6 r11 6 6 6 6 expðikf r12 Þ 6 6 r12 6 6 3 6 ¼ 6 expðikf r1 Þ 3 6 r1 6 6 . .. 6 6 6 ... 6 6 4 expðikf r1M Þ r1M

expðikf r21 Þ r21

expðikf r31 Þ r31

... ...

expðikf r22 Þ r22

expðikf r32 Þ r32

... ...

expðikf r23 Þ r23 ...

expðikf r33 Þ r33 ...

... ...

...

...

... ...

expðikf r2M Þ r2M

expðikf r3M Þ r3M

... ...

... ...

3 expðikf rN1 S Þ 7 rN1 7 7 7 expðikf rN2 S Þ 7 7 7 rN2 7 7 3 7 expðikf rNS Þ 7 7 ð3:32Þ rN3 7 7 ... 7 7 7 ... 7 M 7 expðikf rNS Þ 5 rNMS MN S

and 2

MTS

f ðx131 ; r11 Þ

f ðx132 ; r21 Þ

f ðx133 ; r31 Þ

6 6 f ðx2 ; r 2 Þ f ðx2 ; r 2 Þ f ðx2 ; r 2 Þ 6 31 1 32 2 33 3 6 6 6 f ðx331 ; r13 Þ f ðx332 ; r23 Þ f ðx333 ; r33 Þ 6 ¼6 6 ... ... ... 6 6 6 6 ... ... ... 4 M M M M M f ðxM 31 ; r1 Þ f ðx32 ; r2 Þ f ðx33 ; r3 Þ

...

...

...

...

...

...

...

...

...

...

...

...

f ðx13NS ; rN1 S Þ

3

7 f ðx23NS ; rN2 S Þ 7 7 7 7 3 3 7 f ðx3NS ; rNS Þ 7 7 7 ð3:33Þ ... 7 7 7 7 ... 5 M M f ðx3NS ; rNS Þ MN S

The function f of Eq. (3.33) has been deﬁned in Eq. (3.17). Expressions of QTI, MTI, Q*TI, and M*TI are very similar to those given in Eqs. (3.32) and (3.33); only NS should be replaced by NI, and for Q*TI and M*TI, the properties of ﬂuid 2 should be taken; therefore, the wave number kf of ﬂuid 1 should be replaced by kf2. The next question is how to obtain the source strength vectors AS, AI, and A*I. 3.4.1 Obtaining the strengths of active and passive source layers Values of the source strength vectors AS, AI, and A*I should depend on the location of the interface relative to the transducer. However, if the interface is located far away from the transducer or at an angle such that the reﬂected beam from the interface does

113

3.4 MODELING ULTRASONIC TRANSDUCERS

not reach the transducer then AS is not affected by the interface. In other words, if the effect of the interface on the transducer surface is negligible, then the AS vector for the homogeneous ﬂuid problem (Eqs. (3.18) and (3.22)) and this problem will be the same. Note that in this case, the interface is only hit by the direct wave generated by the transducer; the reﬂected wave cannot strike the interface again after being reﬂected by the transducer face because the energy of the reﬂected wave that reaches the transducer face is negligibly small. Thus, it is much easier to model this problem because the multiple reﬂections of waves between the transducer face and the interface is not to be considered. The steps for computing AI and A*I for this simpler case is given below. 3.4.1.1 Computation of the source strength vectors when multiple reﬂections between the transducer and the interface are ignored As shown in Figure 1.29, two rays can reach the point P that is located close to the interface, ﬁrst one is the direct ray denoted by ray 1 and the second one is ray 2, which ﬁrst propagates from the transducer to the interface, is reﬂected by the interface and then reaches point P. The pressure ﬁelds due to rays 1 and 2 are modeled by QTSAS and QTIAI expressions, respectively. Note that the pressure ﬁeld generated by ray 2 can be alternately obtained by ﬁrst multiplying the elements of QTS terms by appropriate reﬂection coefﬁcient, thus forming QRTS matrix given in Eq. (3.34) and then multiplying this matrix with the AS vector. 2

QRTS

expðikf r11 Þ Rðy11 Þ r11

6 6 6 6 2 6 f r1 Þ 6 Rðy21 Þ expðik 2 6 r1 6 6 6 3 3 expðikf r1 Þ ¼6 6 Rðy1 Þ 3 6 r1 6 6 ... 6 6 6 ... 6 6 4 expðikf r1M Þ RðyM 1 Þ r1M

expðikf r21 Þ Rðy12 Þ r21

...

...

...

expðikf r22 Þ r22

...

...

...

Rðy22 Þ

expðikf r23 Þ r23 ...

...

...

...

...

...

...

...

...

...

...

expðikf r2M Þ RðyM 2 Þ r2M

...

...

...

Rðy32 Þ

expðikf rN1 S Þ Rðy1N Þ rN1

3

7 7 7 7 2 2 expðikf rNS Þ 7 7 RðyN Þ 7 rN2 7 7 3 7 3 expðikf rNS Þ 7 RðyN Þ 7 3 rN 7 7 7 ... 7 7 7 ... 7 M 7 5 expðik r Þ f NS RðyM NÞ rNMS MN

S

ð3:34Þ

where ynm denotes the angle of incidence at the interface of the ray traveling from the mth point source on the transducer surface to the nth target point close to the interface. The reﬂection coefﬁcient Rðynm Þ for the angle of incidence ynm is obtained from Eq. (1.47). Because the pressure ﬁeld generated by QTIAI and QRTS AS should be the same QTI AI ¼ QRTS AS ) AI ¼ ½QTI 1 QRTS AS

ð3:35Þ

114

ULTRASONIC MODELING IN FLUID MEDIA

From the above equation AI can be obtained after knowing AS. In the same manner A*I can be computed by equating the pressure ﬁelds generated by rays traveling from A*I sources directly to the target points in ﬂuid 2 and from AS sources after being transmitted at the interface. QTI AI ¼ QTTS AS

ð3:36Þ

) AI ¼ ½QTI 1 QTTS AS where, 2

QTTS

1 expðikf r21 Þ 1 expðikf r1 Þ Tðy12 Þ 6 Tðy1 Þ 1 r1 r21 6 6 6 2 6 expðikf r22 Þ 6 Tðy2 Þ expðikf r1 Þ Tðy22 Þ 1 2 6 r1 r22 6 6 6 3 6 expðikf r23 Þ f r1 Þ ¼ 6 Tðy31 Þ expðik Tðy32 Þ 3 6 r1 r23 6 6 ... ... 6 6 6 ... ... 6 6 6 M expðikf r1 Þ expðikf r2M Þ 4 TðyM TðyM M 1 Þ 2 Þ r1 r2M

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

3 expðikf rN1 S Þ 7 rN1 7 7 7 2 expðikf rNS Þ 7 7 Tðy2N Þ 7 rN2 7 7 7 3 3 expðikf rNS Þ 7 7 TðyN Þ 7 rN3 7 7 ... 7 7 7 ... 7 7 M 7 M expðikf rNS Þ 5 TðyN Þ rNMS MN Tðy1N Þ

S

ð3:37Þ In the above equation ynm denotes the angle of incidence at the interface of the ray traveling from the mth point source on the transducer surface to the nth target point close to the interface. The transmission coefﬁcient Tðynm Þ for the angle of incidence ynm is obtained from Eq. (1.61). Thus, after obtaining AS from Eq. (3.18) or (3.22), AI and A*I are obtained from Eqs. (3.35) and (3.36), respectively. Then ﬁnally, the pressure and velocity ﬁelds in ﬂuids 1 and 2 are obtained from Eqs. (3.30) and (3.31), respectively. It should be noted here that if the multiple reﬂections of the ultrasonic beams between the transducer face and the interface are ignored, then the pressure ﬁelds in ﬂuids 1 and 2 can be also obtained directly by ray tracing technique without introducing the point source layers AI and A*I, as discussed in Section 1.3.4. Equation (1.45) gives the pressure ﬁeld in ﬂuid 1 and Eq. (1.60) gives the pressure ﬁeld in ﬂuid 2. 3.4.1.2 Computation of the source strength vectors considering the interaction effects between the transducer and the interface When the transducer and the interface are placed close to each other, then the reﬂected energy from the interface that strikes the transducer surface may not be negligible; that energy is then reﬂected by the

115

3.4 MODELING ULTRASONIC TRANSDUCERS

transducer surface and hits the interface for the second time. Therefore, the net strength of the point sources distributed along the transducer surface is no longer the same as AS computed for a transducer immersed in the homogeneous ﬂuid. Because in this case AS cannot be computed from Eqs. (3.18) or (3.22), Eqs. (3.35) and (3.36) are also useless for computing AI and A*I because AS of these equations are not known. In this case AS, AI and A*I are to be computed simultaneously by satisfying the given boundary conditions on the transducer surface and continuity conditions across the interface as outlined in Chapter 2, Section 2.2.4. If the transducer surface velocity is deﬁned as VS0, then the velocity at the transducer surface computed from sources AS and AI should be equal to VS0; therefore, MSS AS þ MSI AI ¼ VS0

ð3:38Þ

Pressure and normal velocity components computed inside the two ﬂuids adjacent to the interface should be also continuous. Therefore, QIS AS þ QII AI ¼ QII AI

MIS AS þ MII AI ¼ MII AI

ð3:39Þ

Equations (3.38) and (3.39) can be combined to form the following matrix equation, from which the source strength vector can be computed as shown below. 2

MSS

6 6 QIS 4

MIS

MSI QII MII

9 38 9 8 AS > > VS0 > > > > > > = 7< = < 7 QII 5 AI ¼ 0 > > > > > > : ; : > ; > MII AI 0 0

8 9 2 MSS AS > > > = 6 < > Q ) AI ¼ 6 > > 4 IS > : > ; AI MIS

MSI QII MII

9 31 8 VS0 > > > > = 7 < 7 QII 5 0 > > > > ; : MII 0

ð3:40Þ

0

If the boundary condition on the transducer surface is given in terms of speciﬁed pressure instead of velocity, then Eq. (3.38) will be changed to QSS AS þ QSI AI ¼ PS0

ð3:41Þ

Equation (3.39) remains unchanged. Then, from Eqs. (3.39) and (3.41), the source strength vectors can be computed in the same manner. 8 9 2 QSS < AS = AI ¼ 4 QIS : ; AI MIS

QSI QII MII

9 31 8 0 < PS0 = QII 5 0 : ; MII 0

ð3:42Þ

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ULTRASONIC MODELING IN FLUID MEDIA

It should be noted here that compared to Section 3.4.1.1, here a bigger matrix needs to be inverted as is evident from Eqs. (3.40) and (3.42). However, the advantages of the analysis presented in this section are that explicit expressions of reﬂection and transmission coefﬁcients (Eqs. (1.47) and (1.61)) are not necessary here, and of course, the interaction effect between the transducer and the interface are properly considered here. Thus, this analysis is more general and is good for both cases, when the interface is close to the transducer as well as when it is far away. 3.4.2 Numerical results—ultrasonic transducer immersed in nonhomogeneous ﬂuid Ultrasonic ﬁelds computed in a two-ﬂuid medium using the theory presented in Section 3.4.1 (and 3.4.1.1, ignoring the multiple reﬂection effect) have been shown in Figures 1.40 (for normal incidence) and 1.41 (for oblique incidence) and not repeated here. It has been mentioned in the last paragraph of Section 3.4.1.1 that when the multiple reﬂection between the transducer surface and the interface is ignored, then after computing the source strength vector AS, the ultrasonic ﬁelds in ﬂuids 1 and 2 can be computed directly by ray tracing technique from Eqs. (1.45) and (1.60), respectively. Figures 3.13 and 3.14a show the pressure ﬁelds computed in this manner in a nonhomogeneous ﬂuid for normal incidence and oblique incidence, respectively (Nasser, 2004). In Figure 3.13 the interference between the incident and reﬂected beams in ﬂuid 1 creates alternate peaks and dips. Because only one transmitted beam is present in ﬂuid 2, such oscillations are absent in ﬂuid 2. Similar results were obtained in Figure 1.40. Figure 3.14b shows the pressure ﬁeld computed in the same ﬂuid structure when the interaction effect between the transducer and the interface is considered (Banerjee et al., 2006). The theory used to produce Figure 3.14b is presented in Section 3.4.1.2 Ultrasonic ﬁelds of Figure 3.14 have been generated by 2.4 mm diameter transducer vibrating at 2.2 MHz frequency and striking the interface at 30 angle of incidence. Fluid 1 is water (acoustic wave speed ¼ 1.49 km/s and density ¼ 1 g/cc) and ﬂuid 2 is an imaginary ﬂuid with acoustic wave speed ¼ 2 km/s and density ¼ 1.5 g/cc. Both ﬁgures match very well although the color scales used in the two computer codes are not identical. Such good matching is expected because strong multiple reﬂections between the inclined transducer and the interface do not occur; therefore, ignoring the interaction effect does not introduce signiﬁcant error. This ﬁgure also veriﬁes the reliability of both computer codes, one is developed using Eqs. (1.45) and (1.60) whereas the second one uses the theory of Section 3.4.1.2. In Figure 3.14a and b, the transmission angle is greater than the incident angle. It is expected because the acoustic wave speed in ﬂuid 2 is greater than that in ﬂuid 1. Therefore, from Snell’s law

sin y2 sin y1

¼ ccf2f1

(where cf1 and cf2 are P-wave speeds in ﬂuids 1 and 2, respectively), the transmission angle (y2 ) should be greater than the incident angle (y1 ). A weak reﬂection near the interface can be also noticed in Figure 3.14. Interference between the incident and reﬂected beams produce alternate peaks and dips in the ultrasonic beam in ﬂuid 1 near the interface.

117

3.5 REFLECTION AT A FLUID–SOLID INTERFACE Acoustic pressure in both fluid (n = 50) 9

Z - axis (mm)

8 7 6 5 4 3 2 1 –10 – 8

–6

–4

–2

0 2 4 X- axis (mm)

6

8

10

Figure 3.13 Ultrasonic pressure ﬁeld variation in ﬂuids 1 and 2 for normal incidence. The interface is at a distance of 5 mm from the 2.7 mm diameter transducer.

3.5 REFLECTION AT A FLUID–SOLID INTERFACE—IGNORING MULTIPLE REFLECTIONS BETWEEN THE TRANSDUCER SURFACE AND THE INTERFACE The analysis presented in Section 3.4.1.1 can be easily extended to a nonhomogeneous medium composed of a ﬂuid half space and a solid half space, if we are interested in

Figure 3.14 Incident, reﬂected, and transmitted ultrasonic beams for oblique incidence (incident angle ¼ 30 ), generated by a 2.4 mm diameter transducer vibrating at 2.2 MHz frequency. The transducer is immersed in ﬂuid 1, which is water (acoustic wave speed ¼ 1.49 km/s and density ¼ 1 g/cc), and ﬂuid 2 is an imaginary ﬂuid with acoustic wave speed ¼ 2 km/s and density ¼ 1.5 g/cc. Results are generated by (a) ray tracing technique (Eqs. 1.45 and 1.60) and (b) DPSM formulation, presented in Section 3.4.1.2.

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ULTRASONIC MODELING IN FLUID MEDIA

computing the ultrasonic ﬁeld in the ﬂuid medium only and the transducer is immersed in the ﬂuid half space. In the ray tracing approach, Eq. (1.45) can still be used with appropriate reﬂection coefﬁcient R. Reﬂection coefﬁcient for the ﬂuid–solid half space has been given in Eq. (1.228d) of Kundu (2004). Instead of ray tracing if the pressure ﬁeld inside the ﬂuid is to be computed adding the contributions of two layers of point source AS and AI (see Fig. 3.12), then one needs to know the values of AS and AI. The transducer source strength vector AS is computed in the same manner it is computed for transducers immersed in a homogeneous ﬂuid. The interface source strength AI is computed from Eq. (3.35), where the reﬂection coefﬁcient R used in matrix QRTS (see Eq. (3.34)) is the reﬂection coefﬁcient for the ﬂuid–solid half space (Eq. (1.228d) of Kundu, 2004). Thus, extending the ﬂuid–ﬂuid medium analysis to the ﬂuid-solid medium case is relatively simple when we are only interested in the ultrasonic ﬁeld computation in the ﬂuid medium, and the interaction effect (multiple reﬂections) between the transducer surface and the interface is ignored, as done in Section 3.4.1.1. The next chapter deals with the ultrasonic ﬁeld computation in the solid medium as well as in the ﬂuid medium, when the multiple reﬂection effect between the transducer surface and the interface or between any two interfaces (for multilayered solid modeling) is not ignored. 3.5.1 Numerical results for ﬂuid–solid interface Numerical results are generated by the ray tracing technique (Eq. (1.45)) with the reﬂection coefﬁcient R for the ﬂuid–solid interface (Nasser, 2004). Figure 3.15 shows the effect of transducer diameter on the reﬂected beam. Both beams in this ﬁgure strike the ﬂuid–solid interface, 10 mm away from the transducer at the same angle (30 ). In the ﬁgure the incident angles look different because the horizontal scales are different. The reﬂected beam for the 4 mm diameter transducer (Fig. 3.15b) is more collimated and stronger than the reﬂected beam for the 2.6 mm diameter transducer (Fig. 3.15a). Effect of the angle of incidence on the reﬂected beam strength is shown in Figure 3.16. All the four beams in this ﬁgure are generated by a 2.6 mm diameter transducer vibrating at 5 MHz frequency. The ﬂuid–solid interface is at a vertical distance of 10 mm from the transducer. The reﬂected beams are shown for four different incident angles 10 , 13 , 27 , and 30 . Reﬂected beam strength variation with the variation of incident angle is expected. However, what is interesting to note is that for a 3 variation in the incident angle, sometimes the reﬂected beam strength varied strongly as in the case for 10–13 , and sometimes it is hardly changed as in the case for 27–30 . This is because the ﬂuctuation in the plane wave reﬂection coefﬁcient for this ﬂuid–solid combination is much greater near 10 than near 30 (Rose, 2000).

3.6 MODELING ULTRASONIC FIELD IN PRESENCE OF A THIN SCATTERER OF FINITE DIMENSION The DPSM formulation presented in Section 3.4.1 and 3.4.1.1 for the nonhomogeneous ﬂuid medium can be easily extended to incorporate the effect of a thin scatterer

3.6 MODELING ULTRASONIC FIELD IN PRESENCE OF A THIN SCATTERER

119

Figure 3.15 Reﬂection of a 5 MHz ultrasonic beam striking a ﬂuid–solid interface at 30 angle (measured from the vertical axis). The interface is at a distance of 10 mm from the transducer; the transducer diameter is (a) 2.6 mm and (b) 4 mm.

Figure 3.16 Reﬂection of a 5 MHz ultrasonic beam generated by a 2.6 mm diameter transducer for different angles of incidence: (a) 10 , (b) 13 , (c) 27 , and (d) 30 . The interface is at a distance of 10 mm from the transducer.

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ULTRASONIC MODELING IN FLUID MEDIA

Fluid 1

Observation Points

AI

Scatterer

Transducer

Fluid 1

AS

A*I

S

Figure 3.17 Point source distributions near the transducer face, thin scatterer, and the imaginary interface in the ﬂuid beyond the scatterer, needed to model the ultrasonic ﬁeld in a ﬂuid medium in the presence of a thin scatterer of ﬁnite dimension. Observation points or target points are shown by smaller circles.

of ﬁnite dimension in the ﬂuid. The scatterer can be made of a solid or a ﬂuid medium. To model the scatterer we start with Figure 3.12 and slightly modify it, by placing a scatterer of ﬁnite dimension at the interface position and then removing the interface by making properties of ﬂuids 1 and 2 identical. The presence of the scatterer can be modeled by two source layers AI and A*I as shown in Figure 3.17, changing the reﬂection and transmission coefﬁcients of the passive layers (AI and A*I) appropriately. The dark spheres on two sides of the scatterer model ultrasonic waves generated by the reﬂected and transmitted waves produced by the scatterer. The white spheres beyond the scatterer position produce ultrasonic waves generated by the reﬂected and transmitted waves by the imaginary interface in the ﬂuid beyond the scatterer. Clearly, the reﬂection coefﬁcient corresponding to the white spheres should be zero (0) because no energy should be reﬂected by the imaginary interface and the transmission coefﬁcient should be 1. The reﬂection and transmission coefﬁcients (R and T ) for the point sources adjacent to the scatterer (dark spheres) can be obtained from Eq. (1.233) of Kundu (2004) if the scatterer is a thin solid plate. Therefore, to model the ultrasonic ﬁeld in a ﬂuid medium containing a scatterer of ﬁnite dimension as shown in Figure 3.17, the formulation presented in Section 3.4.1.1 can be followed. AI and A*I should be obtained from Eqs. (3.35) and (3.36), respectively. However, when computing the matrices QRTS and QTTS appropriate values of the reﬂection and transmission coefﬁcients in the elements of those matrices should be given in Eqs. (3.34) and (3.37), respectively. These values will be either 1 or 0 or the solution of Eq. (1.233) of Kundu (2004), as mentioned in the above paragraph. Numerical results for steel plate scatterers of two different dimensions immersed in water have been already presented in Figures 1.44 and 1.45 and are not repeated here. 3.7 MODELING ULTRASONIC FIELD INSIDE A MULTILAYERED FLUID MEDIUM This problem has been extensively analyzed in Chapter 2. Both cases – when the transducers are not placed at the interface (Figs. 2.1 and 2.2) and when those are

3.8 MODELING PHASED ARRAY TRANSDUCERS IMMERSED

121

placed at the interface (Figs. 2.3 and 2.4)–have been solved there. Numerical results have been presented in Figures 2.20 and 2.21 for two different multilayered ﬂuid structures shown in Figure 2.18 and for three different transducer combinations shown in Figure 2.19. Because this problem has been discussed in detail in Chapter 2, the theory and numerical results associated with the multilayered ﬂuid modeling have been presented in that chapter, those materials are not repeated here.

3.8 MODELING PHASED ARRAY TRANSDUCERS IMMERSED IN A FLUID Phased array transducers are multielement transducers, where different elements are activated with different time delays. The advantage of these transducers is that no mechanical movement of the transducer is needed to scan an object. Focusing and beam steering is obtained simply by adjusting the time delay. Ahmad et al. (2005) for the ﬁrst time used DPSM to model the ultrasonic ﬁeld generated by a phased array transducer in a homogeneous ﬂuid and to study the interaction effect when two phased array transducers are placed in the ﬂuid. Earlier discussions in Chapters 1 and 2 and Sections 3.1–3.7 deal with the modeling of the ultrasonic (and electric) ﬁelds generated by conventional transducers where all transducer points are excited simultaneously. In this section, combining the concepts of delayed ﬁring and DPSM, the phased array transducers have been modeled following the technique suggested by Ahmad et al. (2005). In addition to the single transducer modeling, the ultrasonic ﬁelds from two phased array transducers placed face to face in a ﬂuid medium is also modeled to study the interaction effect for phased array transducers. For DPSM modeling the transducer face is discretized, and point sources are placed in every elemental area. The point sources are excited with a certain phase difference (or time lag) to enhance the dynamic focusing. The results are presented to compare the phased array transducer and conventional planar transducer generated ultrasonic beams in different directions. The results also show the importance of considering the interaction effect between two phased array transducers when the total ﬁeld is computed. However, the scattering effect is found to be small in many cases and can be ignored. 3.8.1 Description and use of phased array transducers A phased array transducer is composed of many elements arranged in a certain pattern that emit acoustic energy at different times. The elements can be pulsed in certain sequence to control the beam angle, also known as the steering angle. During conventional ultrasonic nondestructive testing (NDT) experiments, one transducer mechanically moving from one point to the next or several transducers working simultaneously inspect an object. Mechanical movement of one transducer is a time consuming operation, and the use of multiple transducers is

122

ULTRASONIC MODELING IN FLUID MEDIA

an expensive operation. On the contrary, one phased array transducer can scan an object relatively quickly because it can inspect a wide range by emitting acoustic energy in controlled directions. Thus, efﬁcient scanning is possible by phased array transducers. Wooh and Shi (1999), Azar and Wooh (1999a,b), experimentally studied the steering and focusing behavior of phased array transducers in concrete structures. 3.8.2 Theory of phased array transducer modeling In the DPSM modeling of a conventional transducer where all point sources oscillate at the same phase (harmonic time dependence eiot ), the total ﬁeld at a point generated by N number of point sources is given by Eq. (3.10). However, for a phased array transducer, the point sources are excited at different times. Let the mth point source of strength Am be excited at time tm; then, the pressure at a distance rm from a speciﬁc point source is given by expðikf rm iotm Þ rm

pm ðrÞ ¼ Am

ð3:43Þ

where o is the angular frequency. In Eq.(3.43) the term tm can be expressed in terms of a time lag tm from a reference time t, tm ¼ t tm

ð3:44Þ

If there are N point sources distributed over the transducer surface, as shown in Figure 3.5, then the total pressure at point x is given by pðxÞ ¼

N X

pm ðrm Þ ¼

m¼1

N X

Am

m¼1

expðikf rm io ðt tm ÞÞ rm

ð3:45Þ

where rm is the distance of the mth point source from point x. The particle velocity in the radial direction, generated by the mth point source at a distance r, is given by Am @ exp ð i kf r i o ðt tm ÞÞ ior @r r Am ikf exp ð i kf rÞ exp ð i kf rÞ exp ð i o ðt tm ÞÞ ¼ ior r r2

vm ðrÞ ¼

or vm ðrÞ ¼

Am exp ð i kf rÞ ior r

i kf

1 r

exp ð i o ðt tm ÞÞ

ð3:46Þ

3.8 MODELING PHASED ARRAY TRANSDUCERS IMMERSED

123

After contributions of all N sources are added, the velocity components in the three orthogonal directions x1, x2 , and x3 are obtained.

v1 ðxÞ ¼

N X

v1m ðrm Þ ¼

m¼1

N X Am x1m expðikf rm Þ 1 exp ð i o ðt tm ÞÞ ik f ior rm2 rm m¼1

N X Am x2m expðikf rm Þ 1 exp ð i o ðt tm ÞÞ v2m ðrm Þ ¼ ikf v2 ðxÞ ¼ ior rm2 rm m¼1 m¼1 N X

v3 ðxÞ ¼

N X

v3m ðrm Þ ¼

m¼1

N X Am x3m expðikf rm Þ 1 exp ð i o ðt tm ÞÞ ik f ior rm2 rm m¼1 ð3:47Þ

On the transducer surface v1 ¼ 0 and v2 ¼ 0 because the transducer surface vibrates with a nonzero velocity in the x3 direction only. Note that for conventional transducers v3 ¼ v0 (the time dependence eiot is implied) and for phased array transducers v3 ¼ v0 expfioðt tm Þg. If the observation point x is placed on the transducer surface, then the three expressions of Eq. (3.47) should be equal to 0, 0, and v0 expfioðt tm Þg, respectively. The normal velocity component v0 expfioðt tm Þg should be continuous across the transducer– ﬂuid interface. However, x1 and x2 components of velocity at the interface between the solid transducer and the ﬂuid are not necessarily continuous because a perfect ﬂuid with zero viscosity can have slippage. Therefore, at N boundary points on the interface (where N spheres in Fig. 3.7 touch the interface), one knows N velocity components. After knowing the velocity vector VS, the source strength vector AS can be obtained from Eq. (3.18). The matrix MSS for the phased array transducer is similar to the expression given in Eq. (3.16); the only difference is in the deﬁnition of the function f ðxn3m ; rmn Þ as given below.

f ðxn3m ; rmn Þ

¼

xn3m expðikf rmn Þ iorðrmn Þ2

1 ikf n exp ð i o ðt tmn ÞÞ rm

ð3:48Þ

Eq. (3.48) is similar to Eq. (3.17) except for the additional exponential term in Eq. (3.48). Deﬁnitions of all subscripts and superscripts in these two equations are same. After computing the source strength, vector AS, the pressure PT, and velocity vector VT at any given number (M) of target points can be obtained from Eqs. (3.23) and (3.24). Matrix QTS of Eq. (3.23) has the following form for

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ULTRASONIC MODELING IN FLUID MEDIA

phased array transducers: 2 6 6 6 6 6 6 QTS ¼ 6 6 6 6 6 6 4

exp ðikf r11 i o ðt t11 ÞÞ r11

exp ðikf r21 i o ðt t21 ÞÞ r21

... ...

exp ðikf r12 i o ðt t12 ÞÞ r12

exp ðikf r22 i o ðt t22 ÞÞ r22

... ...

exp ðikf r13 i o ðt t13 ÞÞ r13

exp ðikf r23 i o ðt t23 ÞÞ r23

... ...

exp ðikf r1M i o ðt t1M ÞÞ r1M

... ...

...

... ...

...

... ...

exp ðikf r2M i o ðt t2M ÞÞ r2M

... ...

exp ðikf rN1 i o ðt tN1 ÞÞ 1 r3N

3 7 7

2 2 exp ðikf r3N i o ðt t3N ÞÞ 7 2 7 r3N

exp ðikf rN3 i o ðt tN3 ÞÞ rN3

... ...

exp ðikf rNM i o ðt tNM ÞÞ rNM

7 7 7 7 7 7 7 7 5 MN

ð3:49Þ

3.8.3 Dynamic focusing and time lag determination The main purpose of the phased array transducer is to be able to steer the beam and focus it at any point of interest by accurately controlling the time delay of excitation of the point sources. Figure 3.18 shows how the time delay can be determined to have a desired direction of the ultrasonic beam. The time lag tn can be obtained from the following equation, so that all signals arrive at point P at the same time. tn ¼

p Pn Pn1 ¼ c c

ð3:50Þ

where c is the velocity of the acoustic wave in the ﬂuid medium; Pn and Pn1 are the distances of the nth and (n 1)th point sources, respectively, from point P. If the P

Vp

P1

P2

Pn-1

Pn

dx

Hp

(n-1)*dx

Figure 3.18 Focal point P for a phased array transducer. VP and HP denote the position of point P relative to the transducer surface.

125

3.8 MODELING PHASED ARRAY TRANSDUCERS IMMERSED

distance between two consecutive point sources is dx, then Pn and Pn1 can be expressed as qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Pn ¼ Vp2 þ ½Hp þ ðn 1Þdx2 and Pn1 ¼ Vp2 þ ½Hp þ ðn 2Þdx2 ð3:51Þ where Vp and Hp are vertical and horizontal distances, respectively, of point P from one edge of the transducer face as shown in Figure 3.18. Note that the selection of the focal point automatically determines the steering direction; thus, both the focal point and the steering direction are determined from the above equations. 3.8.4 Interaction between two transducers in a homogeneous ﬂuid When two transducers are placed in a homogeneous ﬂuid, the ultrasonic ﬁeld can be modeled by superimposing two simpler problems as shown in Figure 3.19. xn

x1

x1

x3

x3

x2

x2

= xn

x1

x3 Transducer face

x2

Proble m I

Imaginary s catterer

xn Imaginary scatterer

+

x1

x3 Transducer face

x2

Problem II

Figure 3.19 Modeling two transducers placed in a homogeneous ﬂuid by superimposing two simpler problems: problem I and problem II.

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ULTRASONIC MODELING IN FLUID MEDIA

Fields due to these two simpler problems can be added to obtain the total ultrasonic ﬁeld at any point xn. Note that each transducer face will act as the source of energy as well as a reﬂecting surface. For simplicity we will call one of the transducers as the left-side transducer and the second one as the right-side transducer (see Fig. 3.19). As shown in Figure 3.19, the original problem is decomposed into two problems – in Problem I, the right transducer is replaced by an imaginary scatterer, and in problem II, the left transducer is replaced by the imaginary scattering surface (see the bottom two ﬁgures of Fig. 3.18). For the ﬁrst problem (the left ﬁgure), the ultrasonic ﬁeld at a point xn is the combined effect of the ultrasonic rays coming from the left side transducer face and the imaginary scatterer on the right side. For the second problem (the right ﬁgure), the acoustic ﬁeld at point xn is the combined effect of the right side transducer and the imaginary scatterer on the left side. For both the problems the transducer and the scatterer surfaces are modeled as distributed point sources. The total ultrasonic ﬁeld at point xn generated by the two transducers is then obtained by superimposing the two solutions. In mathematical notations, the modeling steps described above can be written as Pleft ¼ PiSðleftÞ þ PiIðright scattererÞ Pright ¼ PiSðrightÞ þ PiIðleft scattererÞ

ð3:52Þ

PTotal ¼ Pleft þ Pright where Pleft and Pright are the pressure ﬁelds generated by Problems I and II, respectively. PTotal is the combined effect for the two transducers. Note that PiI takes into account the interaction effect. If the ultrasonic ﬁelds generated by the two transducers are simply added without taking into account the interaction (or scattering) effect of one transducer on the other, then PiI will not appear in the formulation. 3.8.5 Numerical results for phased array transducer modeling Numerical results are presented in two parts. The ﬁrst part involves the modeling of the phased array transducer in a homogeneous ﬂuid. The second part focuses on studying the interaction effect when two phased array transducers are placed face to face. As mentioned earlier, a phased array transducer can easily steer an ultrasonic beam in different directions – the transducer surface does not have to be rotated or translated (moved) to rotate the ultrasonic beam. The term ‘‘steering angle’’ is used to denote the angle that the ultrasonic beam makes with the normal direction to the transducer face of the phased array transducer. However, for conventional transducers the transducer surfaces are rotated to obtain the desired steering direction, see Figure 3.20. For a phased array transducer, different segments of the transducer are activated at different time intervals. To model this phenomenon in the DPSM formulation, individual point sources are excited at a predetermined time interval. MATLAB

127

3.8 MODELING PHASED ARRAY TRANSDUCERS IMMERSED

Steering direction

θ

Steering angle

(z) x3

θ Steering angle Steering direction

(x) x1 (a )

(b)

Figure 3.20 (a) A phased array transducer with a steering angle y and (b) a conventional transducer rotated at an angle y.

codes, based on DPSM, have been developed to introduce the time dependent excitation of the point sources. 3.8.5.1 Dynamic steering and focusing As mentioned earlier, in a phased array transducer, desired steering direction and focusing point can be obtained by suitably selecting the time delay between several transducer elements as described in Section 3.8.5. In the ﬁrst investigation, phased array transducers that produce ultrasonic beams at angles 0 , 10 , 30 , and 45 are modeled. Figure 3.21 shows the pressure ﬁelds generated by a rectangular phased array transducer in a homogeneous ﬂuid (water). The area of the transducer surface is 2.5 mm2 and its length to width ratio is 2:1. The outermost layer of the transducer is taken as steel. The acoustic properties of steel and water are given in Table 3.1. The excitation frequency is 5 MHz. Purpose of the ﬁrst set of results (Figs. 3.21–3.23) is to compare the ultrasonic ﬁelds generated by phased array transducers and conventional planar transducers rotated at different steering angles. Figure 3.21a–d shows the acoustic pressure ﬁelds generated by a phased array transducer for steering angles equal to 0 , 10 , 30 , and 45 , respectively. Note that the scales in x and z directions are different. Figure 3.22a–d shows the acoustic pressures generated by a conventional planar transducer when the axis of the transducer surface is rotated by 0 (no rotation), 10 , 30 , and 45 angles, respectively. Often, phased array and conventional transducers are called as ‘‘dynamic’’ and ‘‘static’’ transducers, respectively. For both these transducers, the surface area is equal to 2.5 mm2 and the number of point sources is equal to 98. Fourteen rows of elements are distributed along the x direction representing 14 physical elements of the phased array transducer. Each physical element is discretized into seven point sources distributed along the y direction, giving rise to a total of 98 point sources. In Figure 3.21, the normal distance of the focal point from the transducer face is set at 10 mm in the z direction. The steering angles of 0 , 10 , 30 , and 45 are achieved by changing the horizontal distance of the focal point, keeping

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ULTRASONIC MODELING IN FLUID MEDIA

Figure 3.21 Ultrasonic pressure ﬁelds generated by a phased array transducer for steering angles equal to (a) 0 , (b) 10 , (c) 30 , and (d) 45 .

the normal distance ﬁxed at 10 mm. Figures 3.21a and 3.22a are pressure ﬁelds generated by the phased array and conventional static transducers for a steering angle of 0 . From these ﬁgures, it can be seen that the maximum pressure intensities are 8:2 109 and 7:6 109 units for phased array and static transducers, respectively. In the ﬁgures, it is also observed that the pressure beam produced by the phased array transducer is narrower or more collimated than that produced by the conventional static transducer. Similar conclusions are drawn from Figures 3.21b and 3.22b for 10 steering angle, 3.21c and 3.22c for 30 steering angle, and 3.21d and 3.22d for 45 steering angle. TABLE 3.1

Steel and water properties

Material and Properties Steel Water

P-wave Speed (km/s) 5.96 1.49

S-wave Speed (km/s) 3.26 -

Density (g/cc) 7.932 1

3.8 MODELING PHASED ARRAY TRANSDUCERS IMMERSED

129

Figure 3.22 Ultrasonic pressure ﬁelds for a conventional planar transducer for rotation angles equal to (a) 0 , (b) 10 , (c) 30 , and (d) 45 .

Such observation is expected because the phased array transducer is focused at a ﬁnite distance whereas the conventional planar transducer is focused at inﬁnity. Figure 3.23a–d shows the pressure ﬁelds along the steering directions for both phased array and conventional transducers for 0 , 10 , 30 , and 45 inclination angles, respectively. These plots show that near the transducer face, the pressure intensity variations for the two cases are somewhat close but not so beyond the near ﬁeld region. The phased array transducer produces stronger pressure beams than the static transducer in the region between the focal point (Z ¼ 10 mm) and the near ﬁeld distance NF. 3.8.5.2 Interaction between two phased array transducers placed face to face In the second set of results, the interaction effect between two phased array transducers placed in a homogeneous medium is studied. The transducer faces are placed parallel to each other, as shown in Figure 3.24. The central axes of the two

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ULTRASONIC MODELING IN FLUID MEDIA 9

9

Acoustic pressure along steering direction

x 10

9

8

7 Pressure (micro Pa)

Pressure (micro Pa)

Phased array Static transducer

8

7

6

5

6

5

4

4

3

3

2

Acoustic pressure along steering direction

x 10

9

Phased array Static transdu cer

2

0

0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 Z axis (m)

0.01

0

0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 Z axis (m)

(a) 10

x 10

9

(b)

Acoustic pressure along steering direction 14

x 10

9

Acoustic pressure along steering direction

Phased array

Phased array

Static transducer

9

10 Pressure (micro Pa)

Pressure (micro Pa)

Static transducer

12

8 7 6 5

8

6

4

4

2

3 2 0

0.01

0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 Z axis (m)

(c)

0.01

0 0

0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 Z axis (m)

0.01

( d)

Figure 3.23 Comparison of the acoustic pressures along the steering directions generated by the phased array transducer and the conventional transducer for steering angles equal to (a) 0 , (b) 10 , (c) 30 , and (d) 45 .

transducers are collinear. The surrounding medium is water. The direct ﬁeld generated by the ﬁrst transducer as well as the scattered ﬁeld from the second transducer is calculated. Note that each transducer surface is modeled as the energy-generating surface as well as the scattering surface or the reﬂecting surface for the ultrasonic ﬁeld generated by the other transducer. The steering angle for the phased array transducer is varied to study the interaction effect as a function of the steering angle. The interaction of the transducers for three different steering angles are presented. For the ﬁrst case, the steering angle is very small (y 0 ). For the second and third cases, the steering angle is increased to 10 and 30 , respectively. The reason for the incident angle to be close to 0 but not exactly equal to 0 is that during the discretization process of the transducer face, the

131

3.8 MODELING PHASED ARRAY TRANSDUCERS IMMERSED

Steering direction

Steering direction

θ

(Steering angle)

θ

(Steering angle)

(z) x3 (x) x1 Figure 3.24 Steering direction for two transducers.

point source placements have some randomness. Thus, for the 0 case all point sources are not necessarily placed in perfect symmetry with respect to the transducer axis going through the focal point. Therefore, the ﬁeld generated for the 0 case is not necessarily perfectly symmetric about the central axis of the transducer. Three different steering angles (y) are analyzed. When y is small (close to 0 ), bulk of the incident pressure beam hits the surface of the second transducer face. For 10 steering angle, a part of the beam hits the opposite transducer, and for 30 steering angle, the bulk of the incident beam travels through the ﬂuid medium without hitting the second transducer. We compared the acoustic pressure ﬁelds generated by the phased array transducers when the interaction or the scattering effect of the 2nd transducer is ignored and when it is included. Figure 3.25a and b shows the pressure ﬁelds generated by the phased array transducers for a small steering angle (y 0 ) when the interaction effect is ignored. Figure 3.26a and b shows the acoustic pressures generated by the same transducers when the interaction effect is included. Let us state here again what we mean by the interaction effect being ignored and included. The acoustic pressure ﬁeld in Figure 3.25a is generated by the bottom transducer when the top transducer is absent. In other words, in Figure 3.25a the ultrasonic waves generated by the bottom transducer propagate freely without being interrupted or scattered by the top transducer. The top transducer generates a similar beam that is simply a mirror image of the bottom transducer generated beam and is not shown here. Figure 3.25b is the superposition of uninterrupted ﬁelds generated by the bottom transducer (Fig. 3.25a) and the top transducer (not shown).

132

ULTRASONIC MODELING IN FLUID MEDIA x 10

–3

Acoustic pressure in XZ plane (ignoring interaction)

x 10

9

9 7 8 6

(a)

Z axis (m)

7 6

5

5 4

4 3

3

2 2 1 –6

(b)

Z axis (m)

x 10

–3

–4

–2

0 X a x i s (m )

2

4

6 x 10

–3

Combined acoustic pressure in XZ plane (ignoring interaction)

x 10

9

11

8

10

7

9

6

8

5

7

4

6

3

5

2

4

1

3 –6

–4

–2

0 X a x i s (m )

2

4

9

6 x 10

–3

Figure 3.25 Acoustic pressure in the xz (or x1 x3 ) plane for phased array transducers when the interaction effect is ignored. Transducers are placed at a distance of 10 mm. (a) shows pressure ﬁelds generated by the bottom transducer in the absence of the top transducer when the steering angle is 0 ; (b) shows the total ﬁeld, obtained by adding two ﬁelds generated by the bottom transducer (shown in Fig. 3.25a) and the top transducer (a mirror image of Fig. 3.25a).

3.8 MODELING PHASED ARRAY TRANSDUCERS IMMERSED

133

On the contrary, Figure 3.26a and b shows the pressure ﬁelds obtained when the interaction effect is considered. Figure 3.26a shows the pressure ﬁeld generated by the bottom transducer when the top transducer, placed at a distance of 10 mm, acts as a reﬂector. The pressure beam generated by the bottom transducer is partly reﬂected or scattered by the top transducer and thus affects the total pressure ﬁeld in the homogeneous medium. It should be mentioned here that for Figure 3.26a, the top transducer surface acts only as a scatterer and not as a wave generator. The top transducer generates a similar beam that is simply a mirror image of the bottom transducer generated beam and is not shown here. Figure 3.26b is the superposition of the ﬁeld shown in Figure 3.26a and a similar ﬁeld (simply a mirror image) generated by the top transducer. The steering angle is 0 for both Figures 3.25 and 3.26. In Figure 3.26a, the interaction effect (due to the beam reﬂection at the top transducer surface) can be clearly seen. In this ﬁgure the interference between the direct incident ultrasonic beam from the bottom transducer and the reﬂected beam from the top transducer causes alternate peaks and dips in the pressure ﬁeld, whereas in Figure 3.25a no such peaks and dips are observed. The difference between Figures 3.25b and 3.26b is comparatively less prominent; however, one can see from these two ﬁgures that in the central region, near z ¼ 5 mm, the beam in Figure 3.26b is slightly wider than that in Figure 3.25b. In Figure 3.27a the pressure variations along the steering direction are plotted for both cases, that is, when the interaction effect is considered and when it is ignored. In Figure 3.27a, the two lines almost coincide. When the difference between the two pressure ﬁelds is plotted, one can see in Figure 3.27b that the difference is more prominent near the transducer faces. Figure 3.27c shows the pressure difference as a percentage of the total pressure plotted as a function of the distance from the transducer surface. Note that the maximum difference between the two pressure ﬁelds is about 4% within 1 mm of the transducer face, and it is less than 0.5% when the distance from the transducer face is greater than 2.5 mm. The reason for the pressure difference being higher near the transducer face is that the reﬂected ﬁeld produced by the second transducer is relatively stronger near the reﬂecting surface, and it becomes weaker as the distance from the reﬂecting surface increases. Figure 3.28 shows the pressure ﬁeld when the steering angle is 10 and the interaction effect is considered. Figure 3.28a shows the pressure ﬁeld generated by the bottom transducer and then scattered by the top transducer. Small scattering effect can be noticed in the top part of the beam for z between 9 and 10 mm. Figure 3.28b shows the total effect when both top and bottom transducers are excited. The pressure ﬁelds generated after ignoring the interaction effect is very similar to the ﬁgures shown in Figure 3.28 although not identical and not shown separately here. The difference between these two pressure ﬁelds is plotted in Figure 3.29. Figure 3.29a shows the difference between the total pressure ﬁelds along the steering line when the interaction effect is considered (Fig. 3.28b) and when it is ignored (corresponding ﬁgure is not shown); Figure 3.29b shows the percentage difference between the two pressure ﬁelds along the steering line.

134

ULTRASONIC MODELING IN FLUID MEDIA

Figure 3.26 Acoustic pressure in the xz (or x1x3) plane for phased array transducers when the interaction effect is considered. Transducers are placed at a distance of 10 mm: Part (a) shows pressure ﬁelds generated by the bottom transducer when the top transducer acts as a scatterer or reﬂector, for 0 steering angle; part (b) shows the total ﬁeld, obtained by adding two ﬁelds generated by the bottom transducer (shown in part a) and the top transducer (a mirror image of part a).

3

x 10

10

Acoustic pressure along z-Axis without interaction With Interaction

(a)

Pres s ure (mic ro Pa)

2.5

2

1.5

1

0.5

0

3

0

0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 z-axis (m)

x 10

8

0.01

Difference in acoustic pressure along steering direction Difference

2.5

(b)

Pres s ure (mic ro Pa)

2 1.5 1 0.5 0 -0.5 -1 -1.5

0

0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 z-axis (m)

0.01

% Difference for 0 Degree 5 4

(c) % Di ffe ren c e

3 2 1 0 0

1

2

3

4

5

-1 -2 Distance (mm)

Figure 3.27 Acoustic pressures generated along the central axis of the transducers by two phased array transducers, placed face to face with steering angle ¼ 0 , when the interaction effect is considered and when it is ignored. Part (a) shows the two pressure ﬁelds that almost coincide; part (b) shows the difference between the two pressure ﬁelds; and part (c) shows the percentage difference between the two ﬁelds. Because of symmetry, the percentage difference is plotted from the transducer face to the central plane only.

136

ULTRASONIC MODELING IN FLUID MEDIA

Figure 3.28 Acoustic pressure in the xz (or x1x3) plane for phased array transducers when the interaction effect is considered. Transducers are placed at a distance of 10 mm. Part (a) shows pressure ﬁelds generated by the bottom transducer when the top transducer acts as a scatterer, for 10 steering angle; part (b) shows the total ﬁeld, obtained by adding two ﬁelds generated by the bottom transducer (shown in part a) and the top transducer (a mirror image of part a).

137

3.8 MODELING PHASED ARRAY TRANSDUCERS IMMERSED 8

8

Difference in acoustic pressure along steering direction

x 10

Difference 6

(a)

Pressure (micro Pa)

4 2 0

–2 –4 –6 –8

0

0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 z-axis (m)

0.01

% Difference for 10 Degree 15 10

(b)

% Difference

5 0 –5

0

1

2

3

4

5

–10 –15 –20 Distance (mm)

Figure 3.29 Pressure difference for 10 steering angle. Part (a) shows the difference between the total pressure ﬁelds along the steering line when the interaction effect is considered (Fig. 3.28b) and when it is ignored (not shown); part (b) shows the percentage difference between the two pressure ﬁelds along the steering line. Because of symmetry, the percentage difference is plotted from the transducer face to the central plane only.

138

ULTRASONIC MODELING IN FLUID MEDIA

Figure 3.30 Acoustic pressure in the xz (or x1x3) plane for phased array transducers when the interaction effect is considered. Transducers are placed at a distance of 10 mm. Part (a) shows pressure ﬁelds generated by the bottom transducer when the top transducer acts as a scatterer for 30 steering angle; part (b) shows the total ﬁeld, obtained by adding two ﬁelds generated by the bottom transducer (shown in part a) and the top transducer (a mirror image of part a).

139

3.8 MODELING PHASED ARRAY TRANSDUCERS IMMERSED x 10

5

8

Difference in acoustic pressure along steering direction Differenc e

4

(a)

Pressure (micro Pa)

3 2 1 0

–1 –2 –3 –4

0

0 . 0 01 0. 0 02 0 . 0 03 0. 00 4 0 . 0 05 0. 00 6 0 . 00 7 0. 00 8 0 . 0 09 z ax is (m )

0. 01

% Diffe re nce for 30 De gre e 15 10

(b)

% Difference

5 0 0

1

2

3

4

5

–5 –10 –15 –2 0 Distance (mm)

Figure 3.31 Pressure difference for 30 steering angle. (a) The difference between the total pressure ﬁelds along the steering line when the interaction effect is considered (Fig. 3.30b) and when it is ignored (not shown); part (b) shows the percentage difference between the two pressure ﬁelds along the steering line. Because of symmetry, the percentage difference is plotted from the transducer face to the central plane only.

140

ULTRASONIC MODELING IN FLUID MEDIA

Because of symmetry, the percentage difference is plotted from the transducer face to the central plane only. Note that the maximum difference between the two pressure ﬁelds is about 17.5% within 1.4 mm of the transducer face, and it is less than 0.2% when the distance from the transducer face is greater than 2.8 mm. From Figures 3.28 and 3.29, it is clear that the interaction from the second transducer surface has some effect on the total pressure distribution. It is interesting to note that although for the inclined beam comparatively less energy is being reﬂected by the second transducer, the difference between the two pressure ﬁelds is greater near the transducer face in Figure 3.29b in comparison to that in Figure 3.27c. For 0 steering angle probably some destructive interference between the two reﬂected beams reduces the interaction effect to some extent. However, in the central region Figure 3.27c shows higher value in comparison to that in Figure 3.29b. Figure 3.30 shows the pressure ﬁeld for 30 steering angle when the interaction effect is considered. These plots are similar to Figure 3.28; the only difference is that the steering angle for Figure 3.30 is 30 and for Figure 3.28 it is 10 . Figure 3.30a shows the pressure ﬁeld generated by the bottom transducer and scattered by the top transducer. Hardly any scattering effect can be observed in this ﬁgure because the pressure beam almost completely misses the second transducer. Figure 3.30b shows the total effect when both top and bottom transducers are excited. Pressure ﬁelds generated by ignoring the interaction effect looks almost identical to the ﬁgures shown in Figure 3.30 and not shown separately here. Although in naked eyes they look almost identical, there is some difference between these two pressure ﬁelds. This difference is plotted in Figure 3.31. Figure 3.31a shows the difference between the total pressure ﬁelds along the steering line when the interaction effect is considered (Fig. 3.30b) and when it is ignored (corresponding ﬁgure is not shown); Figure 3.31b shows the percentage difference between the two pressure ﬁelds along the steering line. Because of symmetry, the percentage difference is plotted from the transducer face to the central plane only. Note that the maximum difference between the two pressure ﬁelds is about 14% within 1 mm of the transducer face, and it is less than 0.1% when the distance from the transducer face is greater than 1.2 mm. In the central zone, the pressure difference is negligible. Therefore, even when the ultrasonic beam from one transducer apparently misses the other transducer, there is some signiﬁcant scattering effect near the transducer face, and it should not be ignored while computing the total ﬁeld if the total pressure ﬁeld near the transducer face is important.

3.9 SUMMARY In this chapter ultrasonic ﬁelds generated by the conventional and phased array transducers in homogeneous and nonhomogeneous ﬂuids have been modeled, both in the absence and presence of scatterers of ﬁnite dimension. Solution techniques for relatively complex geometries like multilayered ﬂuid structures have been also

REFERENCE

141

discussed. As long as one is interested in computing the ultrasonic ﬁeld inside the ﬂuid medium, the techniques presented in this chapter can be used to solve that problem. Computation of the ultrasonic ﬁeld inside a solid medium is more difﬁcult and has been discussed in the next chapter.

REFERENCE Ahmad, R., T., Kundu, and D., Placko, ‘‘Modeling of Phased Array Transducers’’, Journal of the Acoustical Society of America, Vol. 117, pp. 1762–1776, 2005. Azar, L., and Wooh, S.C., A Novel Ultrasonic Phased Arrays for the Nondestructive Evaluation of Concrete Structures, in Review of Progress in Quantitive Nondestructive Evaluation, Eds. D.O. Thompson and D.E. Chimenti, Pub. Plenum Press, New York, Vol. 18B, pp. 2153–2160, 1999a. Azar, L., and Wooh, S.C., Phase Steering and Focusing Behavior of Ultrasound in Cementitious Materials, in Review of Progress in Quantitive Nondestructive Evaluation, Eds. D.O. Thompson and D.E. Chimenti, Pub. Plenum Press, New York, Vol. 18B, pp. 2161–2167, 1999b. Banerjee, S., T., Kundu, and D., Placko, ‘‘Ultrasonic Field Modelling in Multilayered Fluid Structures Using DPSM Technique’’, ASME Journal of Applied Mechanics, Vol. 73 (4), pp. 598–609, 2006. Kundu, T., Chapter 12: Nondestructive Testing Techniques for Material Characterization, in Modeling in Geomechanics, Eds. M. Zaman, G. Gioda, and J. Booker, Pub. John Wiley & Sons, pp. 267–298, 2000. Kundu, T., Chapter 1: Mechanics of Elastic Waves and Ultrasonic Nondestructive Evaluation, in Ultrasonic Nondestructive Evaluation: Engineering and Biological Material Characterization, Ed. T. Kundu, Pub. CRC Press, 2004. Nasser, A., (2004) ‘‘Modeling Ultrasonic Transducers in Homogeneous and Non-Homogeneous Media Using DPSM Method’’, Ph.D. Dissertation, Department of Civil Engineering and Engineering Mechanics, University of Arizona, Tucson, USA. Rose, J., Elastic Waves in Solids. Pub. Cambridge University Press, New York, 2000. Wooh, S.C., and Shi, Y., A Design Strategy for Phased Arrays, in Review of Progress in Quantitive Nondestructive Evaluation, Eds. D.O. Thompson and D.E. Chimenti, Pub. Plenum Press, New York, Vol. 18A, pp. 1061–1068, 1999.

4 ADVANCED APPLICATIONS OF DISTRIBUTED POINT SOURCE METHOD – ULTRASONIC FIELD MODELING IN SOLID MEDIA Sourav Banerjee and Tribikram Kundu University of Arizona, USA

4.1 INTRODUCTION In recent years ultrasonic nondestructive testing has gained enormous popularity for nondestructive testing (NDT) of materials and structures. Often piezo-electric transducers are used to generate ultrasonic waves for NDT applications. Piezo-electric transducers are used both as transmitters to generate elastic waves and as receivers to receive the ultrasonic energy. In the previous chapters the ultrasonic wave propagation in ﬂuid media has been modeled by the distributed point source method (DPSM). In this chapter the DPSM modeling technique is extended to incorporate solid materials. The nature of the point sources is different for ﬂuids and solids. In a real-life ultrasonic problem, frequently both ﬂuid and solid media appear in the problem geometry. For example, during the NDT of solid materials, often solid specimens are immersed in a ﬂuid medium for inspection because the ﬂuid serves as a good coupling medium for ultrasonic waves. Ultrasonic waves travel from the transducer to the solid specimen through the coupling ﬂuid. Thus, in real-life applications the acoustic beams often strike the ﬂuid–solid interface; therefore, the interaction between the bounded ultrasonic beams and ﬂuid–solid interfaces needs to be properly modeled.

DPSM for Modeling Engineering Problems, Edited by Dominique Placko and Tribikram Kundu Copyright # 2007 John Wiley & Sons, Inc.

143

144

ADVANCED APPLICATIONS OF DISTRIBUTED POINT SOURCE

The problem involving a ﬂuid–solid interface is much more complex than the ﬂuid–ﬂuid interface problem considered in Chapter 2. The pressure and the three displacement components are the only four parameters of interest for ultrasonic ﬁeld computation in ﬂuid media. However, six stress components and three displacement components constitute a total of nine parameters that come into the picture when the ultrasonic ﬁeld inside a solid is considered. The ﬂuid–solid interface modeling has been a challenging problem in different ﬁelds of science and engineering. In many commercial ﬁnite element codes, ﬂuid is considered as a special case of solid that has a very small value of shear modulus. DPSM does not require such simplifying assumptions. In this chapter the development of the method has been ﬁrst presented step by step, and then it is used for solving a number of practical problems. The ultrasonic ﬁeld modeling for the following problem geometries is presented in this chapter: (1) a full space consisting of a solid half space and a ﬂuid half space with a plane ﬂuid–solid interface between the two half spaces, (2) a solid plate immersed in a ﬂuid in the presence and in the absence of internal cracks in the solid plate, (3) a multilayered solid plate immersed in a ﬂuid, and (4) a corrugated waveguide immersed in a ﬂuid. DPSM needs Green’s functions of the medium where the ultrasonic ﬁeld is to be computed. In Chapters 1 and 3 the elastodynamic Green’s functions have been presented for a ﬂuid medium. The calculation of elsatodynamic Green’s functions for a solid medium is presented in this chapter.

4.2 CALCULATION OF DISPLACEMENT AND STRESS GREEN’S FUNCTIONS IN SOLIDS The governing differential equation of motion, which is also known as the equilibrium equation in a solid medium, can be written as ::

sij; j þ Fi ¼ r ui

ð4:1Þ

where sij is the stress tensor and ui is the displacement vector at a point in the solid. Fi represents the body force per unit volume. i; j ¼ 1; 2; 3. For homogeneous solid media the density (r) remains constant; however, body force (Fi ) along with stress (sij ) and displacement (ui ) are, in general, functions of both space (xj ) and time (t). Therefore, the body force (Fi ) can be expressed as Fi ðxj ; tÞ. The constitutive law for any linear material can be written as sij ¼ Cijkl ekl

ð4:2Þ

where ekl is the strain tensor in the solid, k; l ¼ 1; 2; 3. Cijkl are elastic constants for any linear elastic isotropic or anisotropic material. For isotropic materials the stress–strain relation takes the following form: sij ¼ 2meij þ ldij ekk

ð4:3Þ

4.2 CALCULATION OF DISPLACEMENT AND STRESS GREEN’S FUNCTIONS

145

where l; m are the two Lame´ constants and dij is the Kronecker Delta. After substituting the expression of strain eij ¼ 12 ðui; j þ uj;i Þ into Eq. (4.3) and then that into Eq. (4.1), one gets the Navier’s equation of motion ::

ðl þ 2mÞuj;ij m 2ijk 2kmn un;mj þ Fi ðxp ; tÞ ¼ r ui

ð4:4aÞ

Eq. (4.4a) can be also written in the vector form ::

ðl þ 2mÞrðr uÞ mr ðr uÞ þ F ¼ r u

ð4:4bÞ

The Navier’s equation can be simpliﬁed further by Stokes-Helmholtz decomposition. In this decomposition, the displacement vector ﬁeld is expressed in terms of two vector potentials, or a scalar potential and a vector potential as shown below. u ¼ rðr UÞ r ðr WÞ ¼ rf r W

ð4:5Þ

where f is a scalar potential whereas U; W, and W are vector potentials. The longitudinal-wave speed or the P-wave speed in an isotropic homogeneous qﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃ , and the transverse-wave speed or the S-wave speed is cs ¼ mr. solid is cp ¼ lþ2m r Dividing both sides of the Navier’s equation by r, we get c2p ðrðr uÞÞ c2s ðr ðr uÞÞ þ

F :: ¼u r

ð4:6Þ

Substituting Eq. (4.5) in to Eq. (4.6), ::

::

rr ðc2p r2 U UÞ r r ðc2s r2 W WÞ þ

F ¼0 r

ð4:7Þ

Here the vector potentials U and W are both functions of space and time. In deriving Eq. (4.7) we used the vector identities r r A ¼ 0; r rA ¼ 0 for any vector A and rðr rðr UÞÞ ¼ rðr2 r UÞ ¼ rr r2 U and r r r r W ¼ r r ðrðr WÞ r2 WÞ ¼ r r ðr2 WÞ 4.2.1 Point source excitation in a solid Our objective is to obtain the displacements and stresses in a solid due to a point source excitation. The response of the solid to a point source excitation is called the Green’s function. When a point source is acting in the solid, then the body force term of Eq. (4.7) becomes a concentrated force with impulsive time dependence. This force can be represented by the Dirac delta function in space. Decoupling the independent variables, time (t) and space (x), the body force can be written as Fðx; tÞ ¼ T f ðtÞdðxÞ

or Fi ¼ Ti f ðtÞdðxj Þ

ð4:8Þ

146

ADVANCED APPLICATIONS OF DISTRIBUTED POINT SOURCE

where T is the force vector without the time dependence and space dependence. The general solution of Poisson’s equation ½r2 Z ¼ qðxÞ is given by (Auld, 1973). Z¼

1 4p

Z

qðyÞ dV jx yj

ð4:9Þ

V

R For qðxÞ ¼ dðxÞ, using the properties of Dirac delta function V dðxÞgðxÞdV ¼ gð0Þ, 1 . Therefore, for qðxÞ ¼ dðxÞ the Poission’s equation can be we can write Z ¼ 4pjxj 2 1 written as r 4pr ¼ dðxÞ (where r ¼ jxj for the source located at the origin). Substituting the expression of dðxÞ in Eq. (4.8) and using the vector identity r2 A ¼ rr A r r A we get F ¼ Tr2

f ðtÞ 4pr

f ðtÞ f ðtÞ ¼ T r r r r 4pr 4pr

ð4:10Þ

If we express the vector potentials in terms of two scalar potentials in the forms U ¼ Tf and W ¼ Tc and substitute Eq. (4.10) into Eq. (4.7), then we get :: :: f ðtÞ f ðtÞ 2 2 2 2 rr T cp r f f r r T cs r c c ¼ 0 ð4:11Þ 4prr 4prr For harmonic ðeiot Þ excitation, the time dependence and space dependence of all variables can be separated, fðxj ; tÞ ¼ fðxj Þeiot , cðxj ; tÞ ¼ cðxj Þeiot , and ui ðxj ; tÞ ¼ Ui ðxj Þeiot . Substituting these expressions into Eq. (4.11) we get 1 1 rr T c2p r2 f þ o2 f eiot r r T c2s r2 c þ o2 c eiot ¼ 0 4prr 4prr ð4:12Þ

Hence, we can write r2 f þ

o2 1 f¼ 2 cp 4prc2p r

ð4:13Þ

r2 c þ

o2 1 c¼ c2s 4prc2s r

ð4:14Þ

Particular solutions of Eqs. (4.13) and (4.14) are given by (Mal and Singh, 1991). f¼

1 eikp r 4pro2 r

ð4:15Þ

4.2 CALCULATION OF DISPLACEMENT AND STRESS GREEN’S FUNCTIONS

147

and c¼

1 eiks r 4pro2 r

ð4:16Þ

respectively, where kp ¼ cop and ks ¼ cos . 4.2.2 Calculation of displacement Green’s function Substituting Eqs. (4.15) and (4.16) into Eq. (4.5) we get u ¼ Ue

iot

1 eikp r 1 eiks r r rT eiot ð4:17Þ ¼r rT 4pro2 r 4pro2 r

Applying the vector identity ðr2 A ¼ rr A r r AÞ in the above equation, 1 eikp r 1 eiks r iot iot 2 e e þ r T u ¼ Ueiot ¼ r r T 4pro2 r 4pro2 r 1 eiks r eiot r rT 4pro2 r

ik r 1e s Applying Laplace operator on T 4pro and using Eq. (4.14) we get 2r u ¼ Ue

iot

ð4:18Þ

eikp r eiks r iot Teiks r 2 iot e k e ¼ rr T þ ¼ Gðx; 0Þ T eiot 4pro2 r 4pro2 r s ð4:19Þ

In index notation iks r ikp r 1 @2 e eiks r 2e T or k T i j @xi @xj r 4pro2 s r iks r ikp r 1 @2 e eiks r 2e d Tj ¼ Gij ðx; 0ÞTj Ui ¼ k ij @xi @xj r 4pro2 s r

Ui ¼

ð4:20Þ

When the point source is at y and the response at x is to be determined, then we can write ui ¼ Ui eiot ¼ Gij ðx; yÞTj eiot

ð4:21Þ

Here Gij ðx; yÞ is called the space-dependent Green’s function of displacement for the isotropic homogeneous solid. Substituting r ¼ jx yj, the displacement Green’s

148

ADVANCED APPLICATIONS OF DISTRIBUTED POINT SOURCE

function can be written as (Mal and Singh, 1991), ikp r ikp 1 1 e 2 kp Ri Rj þ 3Ri Rj dij 2 þ Gij ðx; yÞ ¼ r 4pro2 r r eiks r 2 iks 1 ks ðdij Ri Rj Þ ð3Ri Rj dij Þ 2 r r r

ð4:22Þ

i where Ri ¼ xi y r In matrix form

u ¼ Gðx; yÞT

ð4:23Þ

If the unit excitation force at y acts in the jth direction, then the displacement at x in the ith direction can be represented by Gij ðx; yÞ. 4.2.3 Calculation of stress Green’s function For isotropic homogeneous solids, the expression for stresses are given in Eq. (4.3). Substituting the expression for displacement ui ¼ Ui eiot ¼ Gij ðx; yÞTj eiot in the strain–displacement relation, eij ¼ 12 ðui; j þ uj;i Þ, we get 1 eij ¼ ðGik; j þ Gjk;i ÞTk 2

ð4:24Þ

The harmonic time dependence ðeiot Þ is implied and not shown for convenience. Substituting the expression for strains in Eq. (4.2), the stress Green’s function at x due to a concentrated time harmonic force at y can be obtained. For a general anisotropic material, the stress Green’s function can be written as 1 1 sij ðx; yÞ ¼ Cijkl ekl ¼ Cijkl ðGkq;l þ Glq;k ÞTq 2 2

ð4:25Þ

Calculation of the stress Green’s function for a general anisotropic material is not very easy. However, for isotropic homogeneous linearly elastic material, the expression for the stress Green’s function at x due to a concentrated harmonic force at y can be written as sij ðx; yÞ ¼ mðGik; j þ Gjk;i ÞTk þ ldij Gkq;k Tq

ð4:26Þ

sij ðx; yÞ ¼ ðmðGik; j þ Gjk;i Þdkq þ ldij Gkq;k ÞTq

ð4:27Þ

or

By rigorous differentiation, the expressions of all stress components given in Eqs. (4.26) and (4.27) have been obtained and presented in Section 4.3.2.

4.3 ELEMENTAL POINT SOURCE IN A SOLID

149

4.3 ELEMENTAL POINT SOURCE IN A SOLID The basic theory of DPSM is described in Section 1.2.1. The nature of active and passive elemental point sources in ﬂuid media is discussed in section 3.2. The active point sources in a ﬂuid medium are those sources that are adjacent to the ultrasonic transmitter. This concept becomes more complicated when point sources in a solid medium are considered. In real ultrasonic applications, such as the ultrasonic nondestructive evaluation of materials and structures, the solid specimens generally remain in contact with a ﬂuid medium (coupling liquid) because the active medium (wave generator) is not necessarily always in direct contact with the solid specimen. Therefore, appropriate modeling of the ﬂuid–solid interface is important. The point sources distributed at the solid boundary of the specimen are called passive point sources because the solid boundary does not generate any energy; it simply reﬂects and transmits the incident waves to satisfy the appropriate continuity conditions at the boundary and interface The point sources distributed at the solid boundary should have three different force magnitudes in three different directions. In contrast, the point sources in ﬂuid media use only one value of the source strength to compute the pressure ﬁeld at any point. Force magnitudes ðTÞ of the point sources in the solid medium are equivalent to the source strengths of the point sources in the ﬂuid medium. Every point source in the solid medium has three force values or source strengths in three mutually perpendicular directions. The schematic of a point source in the solid medium is presented in Figure 4.1. Note that although in this ﬁgure, for clarity, three different spherical wave fronts are shown for the three point forces acting in three mutually perpendicular

Figure 4.1 Schematic of the point source for solid modeling – three point forces generate three different wave fronts that can have different strength distributions even when the wave fronts coincide.

150

ADVANCED APPLICATIONS OF DISTRIBUTED POINT SOURCE

directions, these three wave fronts can coincide and have different strength distributions along the wave front. Different wave fronts for propagating P-waves and Swaves are not shown separately. 4.3.1 Displacement and stress Green’s functions The displacement Green’s function presented in Eq. (4.22) can be written in the following form: Let ep ¼

eikp r ; r

es ¼

eiks r and rp ¼ r

ikp 1 2 ; r r

rs ¼

iks 1 2 r r

ð4:28Þ

Substituting the above expressions of Eq. (4.28) into Eq. (4.22) we get Gij ¼ Gpij þ Gsij

ð4:29Þ

where Gp11 ¼

i i 1 h 2 2 2 1 h 2

ep kp R1 þ 3R1 1 rp ; Gs11 ¼ es ks 1R21 3R21 1 rs 2 2 4pro 4pro ð4:30Þ

i

i 1 h 2 1 h 2 R R þ 3R R ¼ R R ep k rp ; Gs esðk R 3R rs Gp12 ¼ 1 2 1 2 12 1 2 1 2 p s 4pro2 4pro2 ð4:31Þ

h

i h

i 1 1 ep kp2 R1 R3 þ 3R1 R3 rp ; Gs13 ¼ es ks2 R1 R3 3R1 R3 rs Gp13 ¼ 2 2 4pro 4pro ð4:32Þ

i i 1 h 2 2 2 1 h 2

ep kp R2 þ 3R2 1 rpÞ ; Gs22 ¼ es ks 1R22 3R22 1 rs Gp22 ¼ 4pro2 4pro2

Gp23 ¼

1 4pro2

h

i epðkp2 R2 R3 þ 3R2 R3 rp ; Gs23 ¼

1 4pro2

ð4:33Þ

i esðks2 R2 R3 3R2 R3 rs

h

ð4:34Þ

i i 1 h 2 2 2 1 h 2

Gp33 ¼ ep kp R3 þ 3R3 1 rpÞ ; Gs33 ¼ es ks 1 R23 3R23 1 rs 4pro2 4pro2 ð4:35Þ

and Gpij ¼ Gpji ; Gsij ¼ Gsji for different values ofi and j

ð4:36Þ

151

4.3 ELEMENTAL POINT SOURCE IN A SOLID

Using the above expressions in Eq. (4.29) for the Green’s function, the total displacement at any point x in the solid due to a point force acting at y can be written as u1 ¼ G11 T1 þ G12 T2 þ G13 T3 u2 ¼ G21 T1 þ G22 T2 þ G23 T3

ð4:37Þ

u3 ¼ G31 T1 þ G32 T2 þ G33 T3 where at y, T1 ; T2 , and T3 are the magnitudes of the point forces acting along x1 ; x2 , and x3 directions, respectively. Similarly u1 ; u2, and u3 are the displacements at x along x1 ; x2 , and x3 directions, respectively. Hence, Eq. (4.23) can be expanded as shown below. u ¼ ½u1 u2 u3 T and T ¼ ½T1 T2 T3 T 2 3 G1 ðx; yÞ G1 ðx; yÞ ¼ ½ G11 6 7 7 Gðx; yÞ ¼ 6 4 G2 ðx; yÞ 5 and G2 ðx; yÞ ¼ ½ G21 G3 ðx; yÞ G3 ðx; yÞ ¼ ½ G31

ð4:38Þ G12

G13

G22

G23

G32

G33

ð4:39Þ

Thus, 8 9 2 G11 > < u1 > = 6 u2 ¼ 4 G21 > ; : > u3 G31

G12 G22 G32

38 9 G13 > < T1 > = 7 G23 5 T2 > ; : > G33 T3

ð4:40Þ

4.3.2 Differentiation of displacement Green’s function with respect to x 1 ; x2 ; x3 Let di denote the differentiation of a variable with respect to xi . @rp @ ikp 1 2 . Therefore, ¼ Thus, rpd1 ¼ @xi @xi r r 2R1 ikp R1 2R1 iks R1 2 ; rsd1 ¼ 2 rpd1 ¼ r3 r r3 r

ð4:41Þ

2R2 ikp R2 rpd2 ¼ 2 ; r3 r

2R2 iks R2 rsd2 ¼ 2 r3 r

ð4:42Þ

2R3 ikp R3 rpd3 ¼ 2 ; r3 r

2R3 iks R3 rsd3 ¼ 2 r3 r

ð4:43Þ

Ri has been deﬁned after Eq. (4.22).

152

ADVANCED APPLICATIONS OF DISTRIBUTED POINT SOURCE

After introducing the symbols eoij ¼ Ri Rj and etij ¼ 3eoij ¼ 3Ri Rj and keeping in mind that the repeated indices in the following expressions do not imply summation, one can write @ 2R3 2Ri ; ðR2i Þ ¼ i þ @xi r r 2Ri Rj Rk ; eoij dk ¼ r 6R3 6Ri ; etii di ¼ i þ r r 6Ri Rj Rk ; etij dk ¼ r eoii di ¼

2R2i Rj Rj þ ¼ eoji di; r r 2Ri Ri Rj eoii dj ¼ r 6R2i Rj 3Rj etij di ¼ þ ¼ etji di; r r 6Ri Ri Rj etii dj ¼ r

eoij di ¼

ð4:44Þ ð4:45Þ ð4:46Þ ð4:47Þ

Substituting the above expressions in the differentiation of displacement Green’s functions, we get 1 Ri ep 2 ½epðk eo di þ ð1 þ et Þrpdi þ rp et diÞ þ Gp ik R ep ð4:48Þ ii ii ii p i p ii 4pro2 r 1 Ri es 2 ½esðk eo di ð1 þ et Þrsdi þ rs et diÞ þ Gs ik R es Gsii di ¼ ð4:49Þ ii ii ii ii s i s 4pro2 r 1 Rj ep ½epðkp2 eoii dj þ ð1 þ etii Þrpdj þ rp etii djÞ þ Gpii ikp Rj ep Gpii dj ¼ ð4:50Þ 4pro2 r 1 Rj es ½esðks2 eoii dj ð1 þ etii Þrsdj þ rs etii djÞ þ Gsii iks Rj es Gsii dj ¼ ð4:51Þ 4pro2 r 1 Ri ep ½epðkp2 eoij di þ ðetij Þrpdi þ rp etij diÞ þ Gpij ikp Ri ep Gpij di ¼ ð4:52Þ 4pro2 r 1 Ri es ½esðks2 eoij di ðetij Þrsdi þ rs etij diÞ þ Gsij iks Ri es ð4:53Þ Gsij di ¼ 4pro2 r 1 Ri ep ½epðkp2 eoji di þ ðetji Þrpdi þ rp etji diÞ þ Gpji ikp Ri ep Gpji di ¼ ð4:54Þ 4pro2 r 1 Ri es ½esðks2 eoji di ðetji Þrsdi þ rs etji diÞ þ Gsji iks Ri es Gsji di ¼ ð4:55Þ 4pro2 r 1 Rk ep ½epðkp2 eoij dk þ ðetij Þrpdk þ rp etij dkÞ þ Gpij ikp Rk ep Gpij dk ¼ ð4:56Þ 4pro2 r 1 Rk es ½esðks2 eoij dk ðetij Þrsdk þ rs etij dkÞ þ Gsij iks Rk es Gsij dk ¼ ð4:57Þ 4pro2 r Gpii di ¼

Therefore, Gii di ¼ Gpii di þ Gsii di

ð4:58Þ

Gii dj ¼ Gpii dj þ Gsii dj

ð4:59Þ

Gij di ¼ Gpij di þ Gsij di Gji di ¼ Gpji di þ Gsji di

ð4:60Þ ð4:61Þ

Gij dk ¼ Gpij dk þ Gsij dk

ð4:62Þ

153

4.3 ELEMENTAL POINT SOURCE IN A SOLID

As noted earlier, in Eqs. (4.48)–(4.62) repeated indices do not mean summation and di represents differentiation with respect to xi . Thus, Gpij di ¼ @x@ i ðGpij Þ, where i; j ¼ 1, 2, or 3 but not added. Stress components ðsij Þ at point x due to a concentrated unit force acting in the xm direction at point y can be written as sm ij , where s133 ¼ ð2m þ lÞðG31 d3Þ þ lðG11 d1 þ G21 d2Þ s233 ¼ ð2m þ lÞðG32 d3Þ þ lðG12 d1 þ G22 d2Þ

ð4:63Þ

s333 ¼ ð2m þ lÞðG33 d3Þ þ lðG13 d1 þ G23 d2Þ s111 ¼ ð2m þ lÞðG11 d1Þ þ lðG21 d2 þ G31 d3Þ s211 ¼ ð2m þ lÞðG12 d1Þ þ lðG22 d2 þ G32 d3Þ s311

ð4:64Þ

¼ ð2m þ lÞðG13 d1Þ þ lðG23 d2 þ G33 d3Þ

s131 ¼ mðG31 d1 þ G11 d3Þ s231 ¼ mðG32 d1 þ G12 d3Þ

ð4:65Þ

s331 ¼ mðG33 d1 þ G13 d3Þ s132 ¼ mðG31 d2 þ G21 d3Þ s232 ¼ mðG32 d2 þ G22 d3Þ s332

ð4:66Þ

¼ mðG33 d2 þ G23 d3Þ

Thus, we have the expressions of stress components at any point x due to a concentrated force acting in any direction at another point y in the solid. This expression is very useful to calculate the total ﬁeld in the solid. For a group of concentrated forces acting at the boundary or at the interfaces, as modeled by the DPSM, the total ﬁeld is obtained by simple superposition. Sij ¼

M X m¼1

Sm ij ðx; ym Þ ¼

M X

½ðs1ij Þm T1m þ ðs2ij Þm T2m þ ðs3ij Þm T3m ¼

m¼1

M X

m sm ð4:67Þ ij T

m¼1

where 1 m 2 m 3 m sm and ij ¼ ðsij Þ ðsij Þ ðsij Þ

Tm ¼ ½T1m T2m T3m T

ð4:68Þ

M is the total number of point sources and m corresponds to the mth point source that has three force components T1m ; T2m , and T3m . 4.3.3 Computation of displacements and stresses in the solid for multiple point sources Let us consider a solid boundary in a two-dimensional space where multiple point sources are distributed along the boundary. When the point sources at the solid boundary are excited, then the response at any point inside the solid medium can be computed by simply superimposing the contributions of all the point sources. The

154

ADVANCED APPLICATIONS OF DISTRIBUTED POINT SOURCE

Figure 4.2 (a) Point sources distributed along the boundary of a solid medium. Total ﬁeld at point A is computed by adding the contributions of all point sources. (b) A point source for the solid modeling consists of three point forces acting in three mutually perpendicular directions.

solid half space and M number of point sources at the free boundary are shown in Figure 4.2. Our next objective is to compute the total response at A due to these point sources at the free boundary. The displacement at any point x due to the point source at y is given in Eq. (4.37). Therefore, the total displacement at point A due to M point sources distributed along the solid boundary can be written as u1 ¼ u2 ¼ u3 ¼

M X m¼1 M X m¼1 M X m¼1

m m m m m ðGm 11 T1 þ G12 T2 þ G13 T3 Þ ¼ m m m m m ðGm 21 T1 þ G22 T2 þ G23 T3 Þ ¼ m m m m m ðGm 31 T1 þ G32 T2 þ G33 T3 Þ ¼

M X m¼1 M X m¼1 M X

G1 m Tm

ð4:69Þ

G2 m Tm

ð4:70Þ

G3 m Tm

ð4:71Þ

m¼1

Similarly, the normal and shear stress components at point A can be written from Eq. (4.67) as S33 ¼ S11 ¼ S22 ¼

M X m¼1 M X m¼1 M X m¼1

½ðs133 Þm T1m þ ðs233 Þm T2m þ ðs333 Þm T3m ¼ ½ðs111 Þm T1m þ ðs211 Þm T2m þ ðs311 Þm T3m ¼ ½ðs122 Þm T1m þ ðs222 Þm T2m þ ðs322 Þm T3m ¼

M X m¼1 M X m¼1 M X m¼1

s33 m Tm

ð4:72Þ

s11 m Tm

ð4:73Þ

s22 m Tm

ð4:74Þ

155

4.3 ELEMENTAL POINT SOURCE IN A SOLID

S31 ¼ S32 ¼ S12 ¼

M X m¼1 M X m¼1 M X

½ðs131 Þm T1m þ ðs231 Þm T2m þ ðs331 Þm T3m ¼ ½ðs132 Þm T1m þ ðs232 Þm T2m þ ðs332 Þm T3m ¼ ½ðs112 Þm T1m þ ðs212 Þm T2m þ ðs312 Þm T3m ¼

m¼1

M X m¼1 M X m¼1 M X

s31 m Tm

ð4:75Þ

s32 m Tm

ð4:76Þ

s12 m Tm

ð4:77Þ

m¼1

where spij are deﬁned in Eqs. (4.63)–(4.66). Note that subscripts i; j and the superscript p can take values 1,2, or 3. 4.3.4 Matrix representation Computation of displacements and stresses at a single point in the solid are presented in the previous section. However, we are interested in the total ultrasonic ﬁeld inside the solid medium. Therefore, we need to compute the ﬁeld at multiple points, not at a single point only. For a group of observation points (we will call them target points), let us compute the displacement and stress ﬁelds inside the solid. Let us take N number of target points. The displacement and stress expressions at N target points due to M source points are given in Eqs. (4.78a)–(4.79f) in the matrix form; where ui are the displacements at N target points along the xi direction and sij are the stresses at N target points (i and j can take values 1, 2, or 3). The N target points and M source points are shown in Figure 4.3. u1T u2T u3T s33T

¼ DS1TS AS ¼ DS2TS AS ¼ DS3TS AS ¼ S33TS AS

ð4:78aÞ ð4:78bÞ ð4:78cÞ ð4:79aÞ

Figure 4.3 A schematic diagram of M source points along the boundary (or interface) and N target points inside a solid.

156

ADVANCED APPLICATIONS OF DISTRIBUTED POINT SOURCE

s11T ¼ S11TS AS

ð4:79bÞ

s22T ¼ S22TS AS s31T ¼ S31TS AS

ð4:79cÞ ð4:79dÞ

s32T ¼ S32TS AS s12T ¼ S12TS AS

ð4:79eÞ ð4:79fÞ

where h iT fAS gT ¼ ðT1 ÞT ðT2 ÞT ðT3 ÞT ðT4 ÞT . . . . . . ðTM2 ÞT ðTM1 ÞT ðTM ÞT

ð3M1Þ

ð4:80Þ ðTi ÞT is a ð1 3Þ vector, which is the transpose of Ti , deﬁned in Eqs. (4.38) and (4.68), where the superscript i designates the ith point source; for the mth point source, i ¼ m. Stress and displacement matrices of Eqs. (4.78a)–(4.79f) have the form as shown in Eqs. (4.81). S33TS and DS3TS matrices are written in detail in Eqs. (4.81a) and (4.81b), respectively, for illustration purpose. The ð1 3Þ vector sm ij used in Eq. (4.81a) has been deﬁned in Eq. (4.68). The additional subscript n of sm ij n indicates the nth target point. Note that n varies from 1 to N whereas the superscript m varies from 1 to M. Similarly, the element Gm i n of Eq. (4.81b) has been deﬁned in Eq. (4.39). The superscript m stands for the mth point source, and the second subscript n corresponds to the nth target point. 2

3 s33 1 1 s33 2 1 s33 3 1 s33 4 1 s33 5 1 ... ... s33 M2 1 s33 M1 1 s33 M 1 6 s33 1 s33 2 s33 3 s33 4 s33 5 ... ... s33 M2 s33 M1 s33 M 7 6 7 2 2 2 2 2 2 2 2 6 7 6 s33 1 3 s33 2 3 s33 3 3 s33 4 3 s33 5 3 ... ... s33 M2 3 s33 M2 3 s33 M2 3 7 6 7 6 ... ... ... ... ... ... ... ... ... ... 7 7 : S33TS ¼6 6 ... ... ... ... ... ... ... ... ... ... 7 6 7 6 1 7 6s33 N2 s33 2 N2 s33 3 N2 s33 4 N2 s33 5 N2 ... ...s33 M2 N2 s33 M1 N2 s33 M N2 7 6 1 7 4s33 N1 s33 2 N1 s33 3 N1 s33 4 N1 s33 5 N1 ... ...s33 M2 N1 s33 M1 N1 s33 M N1 5 s33 1 N s33 2 N s33 3 N s33 4 N s33 5 N ... ... s33 M2 N s33 M1 N s33 M N ðN3MÞ ð4:81aÞ and 2

DS3TS

G3 1 1 6 G3 1 2 6 6 G 1 6 3 3 ¼6 6 ... 6 1 4 G3 N1 G3 1 N

G3 2 1 G3 2 2 G3 2 3 ... G3 2 N1 G3 2 N

G3 3 1 G3 3 2 G3 3 3 :: G3 3 N1 G3 3 N

:: G3 M1 1 . . . G3 M1 2 . . . G3 M1 3 ... :: . . . G3 M1 N1 . . . G3 M1 M1

3 G3 M 1 G3 M 2 7 7 G3 M 3 7 7 ð4:81bÞ 7 ... 7 7 G3 M N1 5 G3 M N ðN3MÞ

157

4.4 CALCULATION OF PRESSURE AND DISPLACEMENT GREEN’S

s33 m n and G3 m n for the nth target point and mth point source can be written from Eqs. (4.68) and (4.39) in the following form: s33 m n ¼ ðs133 Þm ðs233 Þm ðs333 Þm n

and

m m G3 m n ¼ Gm 31 G32 G33 n

ð4:82Þ

Similarly, other stress and displacement matrices of Eqs. (4.78a)–(4.79f) can also be written in the form given in Eqs. (4.81a) and (4.81b), respectively. In these equations the subscript T stands for the target points inside the solid medium and S stands for the set of source points at the boundary of the solid medium.

4.4 CALCULATION OF PRESSURE AND DISPLACEMENT GREEN’S FUNCTIONS IN THE FLUID ADJACENT TO THE SOLID HALF SPACE The theory of ultrasonic wave propagation in a ﬂuid medium is given in Chapter 3. The pressure ﬁeld generated in the ﬂuid medium by ultrasonic waves is discussed in that chapter. The displacement ﬁeld in the ﬂuid needs to be computed to enforce the continuity conditions on the normal displacement component at the ﬂuid–solid interface. In this section our objective is to ﬁnd the displacement ﬁeld generated by ultrasonic waves in a perfect ﬂuid medium. A perfect ﬂuid is a homogeneous isotropic medium that cannot have shear stress and in which the normal stresses in all directions must be the same. Therefore, the stress or pressure (negative of hydrostatic stress) at any point can be written as sij ¼ ldij ekk ¼ pdij

ð4:83Þ

For dij ¼ 1, substituting the linear strain–displacement relation ekk ¼ 12 ðuk;k þ uk;k Þ ¼ uk;k in Eq. (4.83) we get lðu1;1 þ u2;2 þ u3;3 Þ ¼ lr u ¼ p

ð4:84Þ

The governing equation or the equation of motion in the ﬂuid can be written as ::

dij p; j þ Fi ¼ r ui

or

::

p;i þ Fi ¼ r ui

or

::

rp þ F ¼ r u

ð4:85Þ

Applying divergence on both sides of Eq. (4.85) and substituting Eq. (4.84), @ 2 ðr uÞ r :: ¼ p ð4:86Þ 2 @t l qﬃﬃ Assuming f ¼ r F and the wave velocity in ﬂuid cf ¼ rl, we can rewrite the above equation in the following form, r rp þ r F ¼ r

r2 p

1 :: p¼f c2f

ð4:87Þ

158

ADVANCED APPLICATIONS OF DISTRIBUTED POINT SOURCE

4.4.1 Displacement and potential Green’s functions in the ﬂuid Spherical bulk wave in a ﬂuid can be generated by a point source in an inﬁnite ﬂuid medium as shown in Figure 1.6. If the point source is harmonic, then it will generate harmonic spherical waves. When a point source is generating bulk waves in a ﬂuid, then the harmonic Dirac delta impulsive force will be the body force in Eqs. (4.85)– (4.87). For an inﬁnite ﬂuid medium with a point source, Eq. (4.87) can be written as r2 Gf

1 :: Gf ¼ dðx yÞeiot c2f

ð4:88Þ

where Gf is the pressure Green’s function in the ﬂuid at x due to the point source acting at y. Taking harmonic time dependence for Gf ½Gf ðr; tÞ ¼ Gf ðr; oÞeiot , the above equation takes the following form: r2 Gf ðr; oÞ þ

o2 Gf ðr; oÞ ¼ dðx yÞ c2f

ð4:89Þ

A particular solution of Eq. (4.89) can be written as (Schmerr, 1998) Gf ðr; oÞ ¼

eikf r ; 4pr

where kf ¼

o cf

ð4:90Þ

The pressure in the ﬂuid can also be expressed as a scalar potential function ff as written in Eq. (1.13c). The pressure–potential and the displacement–potential relations can be written as p ¼ s11 ¼ s22 ¼ s33 ¼ ro2 ff @ff ui ¼ @xi

ð4:91Þ ð4:92Þ

Therefore, the Green’s function for the wave potential can be expressed as ff ðr; oÞ ¼

eikf r 4pro2 r

ð4:93Þ

Taking derivatives of ff with respect to xi , three displacement components are obtained 1 1 eikf r ikf r ð4:94Þ ikf R1 e 2 R1 u1 ¼ r 4pro2 r 1 1 eikf r ikf r ð4:95Þ ik u2 ¼ R e R2 f 2 r2 4pro2 r 1 1 eikf r ikf r ð4:96Þ u3 ¼ ik R e R3 f 3 r2 4pro2 r i where Ri ¼ xi y r

4.4 CALCULATION OF PRESSURE AND DISPLACEMENT GREEN’S

159

4.4.2 Computation of displacement and pressure in the ﬂuid A large number of point sources distributed over a plane surface such as the transducer face generate vibration of particles in a direction normal to the plane surface. From the surface integral technique, the pressure ﬁeld at point x in front of a group of point sources at y is given in Eq. (1.14a), where B represents the source strength of the point sources. The integral form can also be written in the summation form and is given in Eq. (1.14b). In the DPSM technique, it has been assumed that each point source distributed over the surface has different source strengths as speciﬁed by Am, where m designates the mth point source and rm is the distance of the target point x from the mth point source. The pressure at any point at a distance rm from the mth point source with source strength Am is given in Eq. (1.17). If there are N point sources distributed over the ﬁnite surface, as shown in Figure 1.23, then the total pressure at point x is given in Eq. (1.18) From the pressure–velocity relation given in Eq. (1.20), it is also possible to obtain the velocity at x in all three directions due to N number of point sources placed at y. The velocity in the radial direction, at a distance r from the mth point source, is given in Eq. (1.21). When the contributions of all N sources are added, the total velocity in x1 , x2 , and x3 directions at point x can be written as

v1m ðrm Þ ¼

N X Am x1m expðikf rm Þ 1 ik f ior rm2 rm m¼1

ð4:97aÞ

v2m ðrm Þ ¼

N X Am x2m expðikf rm Þ 1 ik f ior rm2 rm m¼1

ð4:97bÞ

N X Am x3m expðikf rm Þ 1 v3 ðxÞ ¼ v3m ðrm Þ ¼ ikf ior rm2 rm m¼1 m¼1

ð4:97cÞ

v1 ðxÞ ¼

N X m¼1

v2 ðxÞ ¼

N X m¼1 N X

where xim is the shortest distance along the xi direction between the mth point source and the target point, as shown in Figure 4.4a. If the transducer face is inclined at an angle y, rotated about the x2 axis (Fig. 4.4a), the velocity in the direction perpendicular to the transducer face of the target points placed on the transducer face can be expressed as N X Am 1 ikf v1 ðxÞ sin y þ v3 ðxÞ cos y ¼ ior rm m¼1 x1m expðikf rm Þ x3m expðikf rm Þ sin y þ cos y ¼ v0 rm2 rm2

ð4:98Þ

where v0 is the speciﬁed velocity of the transducer face in the direction perpendicular to the face, as described in Section 2.2.2.

160

ADVANCED APPLICATIONS OF DISTRIBUTED POINT SOURCE

Figure 4.4 (a) Point sources distributed near the inclined transducer and the observation point (or target point) at x and (b) point forces are placed behind the transducer face at the center of the small spheres at a distance rs from the transducer face.

161

4.4 CALCULATION OF PRESSURE AND DISPLACEMENT GREEN’S

4.4.3 Matrix representation The velocity of M target points distributed on the transducer face due to point sources distributed just behind the transducer face (Fig. 4.4b) at a distance rs , can be written in matrix form as VS ¼ MSS AS

ð4:99Þ

where, VS is the ðM 1Þ vector of the velocity components, perpendicular to the transducer surface. If the velocity of the transducer face is given by v0, then VS can be written as fVS gT ¼ v10

v20

v30 . . . . . .

... ... ...

vM1 0

vM 0

T

ð4:100aÞ

where vn0 is the velocity of the nth target point. If AS is the ðN 1Þ vector of the source strengths, then fAS gT ¼ ½A1

A2

A3

A4

A5

A6

. . . . . . AN2

AN1

AN T

ð4:100bÞ

Every point source is placed inside a sphere, and the number of apex points of the spheres touching the transducer surface is the same as the number of point sources. When the target points are placed at the apex of the spheres of the point sources, then M is equal to N. Therefore, for the target points placed at the apex of the spheres of the point sources, MSS is a square matrix and is given in Eq. (1.25p). When the transducer is rotated about the x2 -axis, the elements of the MSS matrix can be written as m f ðxm tn ; rn Þ

¼

m xm tn expðikf rn Þ

iorðrnm Þ2

1 expðikf rnm Þ 1 m ikf m ¼ ikf m ðxm 3n cos y þ x1n sin yÞ rn rn iorðrnm Þ2 ð4:101Þ

For a general set of target points located not necessarily on the transducer face, the velocity due to the transducer sources can be written as VT ¼ MTS AS

ð4:102aÞ

where VT , the velocity vector ðM 1Þ, contains the normal velocity components of the target points distributed on the surface. The matrix MTS has elements that are similar to those of MSS , except for its size and three components of position vectors between the point sources and target points. The size of the matrix MTS is ðM NÞ, where M is the number of target points and N is the number of source points. Following the same concept, the pressure at any M number of target points due to N number of source points can be written as PT ¼ QTS AS

ð4:102bÞ

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ADVANCED APPLICATIONS OF DISTRIBUTED POINT SOURCE

where PT is the ðN 1Þ vector of pressure values at N target points, and QTS is the ðN MÞ matrix given in Eq. (4.102c). 2

QTS

expðikf r11 Þ expðikf r12 Þ 6 r11 r12 6 6 expðik r 1 Þ expðik r 2 Þ f 2 f 2 6 6 r21 r22 6 6 1 2 ¼6 6 expðikf r3 Þ expðikf r3 Þ 6 r1 r2 6 . .3. . .3. 6 6 ... ... 6 4 expðikf r 1 Þ expðikf r 2 Þ N N rN1 rN2

expðikf r13 Þ r13 expðikf r23 Þ r23 expðikf r33 Þ r3 . .3. ... expðikf rN3 Þ rN3

... ... ... ... ... ... ... ... ... ... ... ...

3 expðikf r1M Þ 7 r1M 7 expðikf r2M Þ 7 7 7 r2M 7 7 expðikf r3M Þ 7 7 7 r3M 7 ... 7 7 ... 7 M 5 expðikf rN Þ rNM ðNMÞ ð4:102cÞ

When the target points are located at the apex of the spheres of the point sources, Eq. (4.102b) takes the following form PS ¼ QSS AS

ð4:102dÞ

where QSS is an ðN NÞ matrix similar to QTS written in Eq. (4.102c). The deﬁnition of rnm is identical to that given in Chapter 1 after Eq. (1.25f). It is the distance between the mth point source and the nth target point. In the same manner, the matrix expression for displacements at T set of target points due to S set of source points in the ﬂuid can be written as U1T ¼ DF1TS AS

ð4:103aÞ

U2T ¼ DF2TS AS U3T ¼ DF3TS AS

ð4:103bÞ ð4:103cÞ

where 2

gðRi 11 ; r11 Þ 6 gðRi 1 ; r1 Þ 2 2 6 6 gðR 1 ; r1 Þ 6 i3 3 DFiTS ¼ 6 6 gðRi 14 ; r41 Þ 6 4 ... gðRi 1N ;rN1 Þ

gðRi 21 ; r12 Þ gðRi 22 ; r22 Þ gðRi 23 ; r32 Þ gðRi 24 ; r42 Þ ... gðRi 2N ; rN2 Þ

gðRi 31 ; r13 Þ gðRi 32 ; r23 Þ gðRi 33 ; r33 Þ gðRi 34 ; r43 Þ . .. gðRi 3N ; rN3 Þ

... ... ... ... ... ...

gðRi M1 ; r1M1 Þ 1 M1 M1 gðRi 2 ; r2 Þ gðRi M1 ; r3M1 Þ 3 M1 M1 gðRi 4 ; r4 Þ .. . gðRi M1 ; rNM1 Þ N

M 3 gðRi M 1 ; r1 Þ M 7 gðRi M 2 ; r2 Þ 7 M 7 gðRi M 3 ; r3 Þ 7 M M 7 gðRi 4 ; r4 Þ 7 7 5 ... M M gðRi N ; rN Þ ðNMÞ

ð4:103dÞ where

m gðRi m n ; rn Þ

Ri m n

" # m 1 1 eikf rn m m ikf rnm ¼ 2 m ikf Ri n e Ri n ; ro rn ðrnm Þ2 xi m yi m ¼ n m n ; and i takes the values 1,2, and 3. rn

ð4:103eÞ

4.5 APPLICATION 1: ULTRASONIC FIELD MODELING

163

Figure 4.5 Contributions of different point sources for computing ultrasonic ﬁelds in the ﬂuid (point D) and in the solid (point C) are shown by lines connecting the relevant point sources to the point of interest (C or D).

4.5 APPLICATION 1: ULTRASONIC FIELD MODELING NEAR FLUID–SOLID INTERFACE (BANERJEE ET AL., 2007) Let us consider a plane interface between a solid half space and a ﬂuid half space analogous to a river bed with water. A schematic diagram of the system considered for our analysis is shown in Figure 4.5, where the ﬂuid half space has been drawn below the solid half space. Only a few point sources have been shown along the interface in the diagram to keep it simple; however, in the actual model the point sources are distributed over the entire interface. There is a circular transducer immersed in the ﬂuid. A number of small point sources are distributed below the transducer face and on both sides of the interface. These sources should produce the total ultrasonic ﬁeld in ﬂuid and solid media. A1 is the source strength vector of the point sources that are placed above the solid–ﬂuid interface and generates the reﬂected ultrasonic ﬁeld in the ﬂuid. Similarly, A1 is the source strength vector of the sources that are distributed below the solid–ﬂuid interface and models the transmitted ﬁeld in the solid. The point sources that have been distributed below the transducer face have source strength vector AS. We intend to compute the ﬁeld in both solid and ﬂuid media. In Figure 4.5, two points C and D are shown for the illustration purpose. The ultrasonic ﬁeld at point C is the summation of the contributions of all point sources distributed below the interface. Similarly, the ultrasonic ﬁeld at point D is the summation of the contributions of the point sources distributed above the solid–ﬂuid interface and below the transducer face.

164

ADVANCED APPLICATIONS OF DISTRIBUTED POINT SOURCE

4.5.1 Matrix formulation to calculate source strengths The velocity and pressure ﬁelds in the ﬂuid medium and stress and displacement ﬁelds in the solid medium can be expressed in the matrix form when we take a group of target points instead of single points C or D of Figure 4.5. Referring to Eq. (4.99) we can write VS ¼ MSS AS

ð4:104aÞ

where VS is the ðN 1Þ vector of the velocity components at the transducer surface corresponding to N number of source points distributed slightly behind the transducer face and AS is the ðN 1Þ vector containing the strength of the transducer sources. The elements of MSS are given in Eq. (1.25p), and this matrix is deﬁned when all target points are distributed on the transducer surface. For a more general case when target points are not necessarily on the transducer surface, the velocity due to the transducer sources is given in Eq. (4.102a). Similarly, the velocity at any set of target points due to the interface sources (see Fig. 4.5) can be written as VT ¼ MT1 A1

ð4:104bÞ

The interface has M source points distributed on each side of the interface. Hence, A1 has ðM 1Þ elements. The pressure at any set of target points in the ﬂuid due to the transducer sources is given in Eq. (4.102b). The pressure at the same set of target points in the ﬂuid medium due to interface sources can be written as P1T ¼ QT1 A1

ð4:104cÞ

Therefore, at the speciﬁed target points, the total pressure ﬁeld generated by the transducer and interface sources is PT ¼ PT þ P1T ¼ QTS AS þ QT1 A1

ð4:104dÞ

Similarly, at any set of target points, the displacement along the x3 direction in the ﬂuid can be written as U3T ¼ DF3TS AS þ DF3T1 A1

ð4:104eÞ

It should be noted here that each point source that has been considered to calculate the transmitted ﬁeld in solid, as discussed in Section 4.3, has three different point forces in three different directions as unknowns. From Eq. (4.79a) the normal stress in x3 direction at the interface can be written as s3311 ¼ S3311 A1

ð4:104fÞ

Similarly, for shear stresses, from Eqs. (4.79d) and (4.79e), we can write s3111 ¼ S3111 A1

ð4:104gÞ

s3211 ¼

ð4:104hÞ

S3211 A1

4.5 APPLICATION 1: ULTRASONIC FIELD MODELING

165

where A1 is the source strength vector of dimension (3M 1) corresponding to the sources distributed below the ﬂuid–solid interface. 4.5.2 Boundary conditions Across the ﬂuid–solid interface, the normal displacement (u3 component) should be continuous. Also, at the interface negative of the normal stress ðs33 Þ in the solid and the pressure in the ﬂuid should be continuous. Shear stresses in the solid at the interface should vanish. If the normal velocity of the transducer face is assumed to be v0 , we can write the boundary conditions, on the transducer surface MSS AS þ MS1 A1 ¼ VS0

ð4:105Þ

on the interface Q1S AS þ Q11 A1 ¼ S3311 A1

ð4:106aÞ

DF31S AS þ DF311 A1 ¼ DS311 A1

ð4:106bÞ

S3111 A1 S3211 A1

¼0

ð4:106cÞ

¼0

ð4:106dÞ

Eqs. (4.105) – (4.106d) can be written in matrix form 2

MSS 6 Q 6 1S 6 6 DF31S 6 4 0 0

MS1 Q11 DF311 0 0

3 0 8 9 8 9 S3311 7 < VS0 = < AS = 7 7 A1 DS311 7 ¼ 0 : ; : ; 7 5 A1 ðN þ 4MÞ S3111 0 ðN þ 4MÞ S3211 ðN þ 4MÞðN þ 4MÞ ð4:107Þ

or ½MATfLg ¼ fVg

ð4:108Þ

4.5.3 Solution The source strength vector fg of the total system can be obtained from Eq. (4.108) as fLg ¼ ½MAT1 fVg

ð4:109Þ

After calculating the source strengths, the pressure, velocity, stress, and displacement ﬁelds can be calculated. For example, the pressure ﬁeld in the ﬂuid can be

166

ADVANCED APPLICATIONS OF DISTRIBUTED POINT SOURCE

written as PðFÞ ¼ QðFÞS AS þ QðFÞ1 A1

ð4:110Þ

where F is a set of target points inside the ﬂuid half space. Similarly, in the solid, the stress and displacement ﬁelds can be calculated as s33ðSÞ ¼ S33ðSÞ1 A1 s11ðSÞ ¼ S11ðSÞ1 A1 ð4:111Þ

...... u3ðSÞ ¼ u1ðSÞ ¼

DS3ðSÞ1 A1 DS1ðSÞ1 A1

where S is a set of target points inside the solid half space. 4.5.4 Numerical results on ultrasonic ﬁeld modeling near ﬂuid–solid interface Computer codes can be generated to model the ultrasonic ﬁelds near the ﬂuid–solid interface using the DPSM formulation presented above. Numerical results are presented for an aluminum solid half space over the water half space. For convenience in subsequent discussions x1 , x2 , x3 and x; y; z have been used interchangeably. Therefore, please keep in mind that x1 ¼ x; x2 ¼ y, and x3 ¼ z. During DPSM modeling while placing the point sources, one needs to satisfy the requirement of the maximum spacing allowed between neighboring point sources as discussed in Chapter 1. This distance depends on the wavelength of the ultrasonic signal. The wave speed and other material properties of aluminum and water are given in Table 4.1. TABLE 4.1

Material properties and critical angle calculation

Wave speed in water ðcf Þ

1.48 km/s

Density of water ðrf Þ

1 g/cc

P-wave speed in aluminum ðcp Þ

6.5 km/s

S-wave speed in aluminum ðcs Þ

3.13 km/s

Density of aluminum ðrs Þ

2.7 g/cc

First Lame´ constant of aluminum ðlÞ

rs ðc2p 2c2s Þ ¼ 61:17 GPa

Second Lame´ constant of aluminum ðmÞ

rs c2s ¼ 26:45 GPa l ¼ 0:349 2ðl þ mÞ 0:862 þ 1:14n c ¼ 2:923 km=s s 1þ n 1 cf ¼ 30:4196 sin cr

Poisson´s ratio of aluminum ðnÞ Rayleigh-wave speed in aluminum ðcr Þ Rayleigh critical angle ðyc Þ

4.5 APPLICATION 1: ULTRASONIC FIELD MODELING

167

Ultrasonic waves are generated by an ultrasonic transducer that is placed in water at different inclinations relative to the interface. The diameter of the transducer is 4 mm. In the following ﬁgures the gray scale bars are provided to give an idea about the magnitudes of the ultrasonic ﬁelds in the contour plots. Note that the gray scale bars are not identical in all ﬁgures. In Figure 4.6 the normal stresses s11 and s33 in aluminum are plotted in addition to the pressure ﬁeld in water for normal incidence of the ultrasonic wave at the interface. Ultrasonic ﬁeld in aluminum and water are shown in Figures 4.6a–f. Figures 4.6a, c, and e are generated with 1 MHz ultrasonic transducer and Figs. 4.6b, d, and f are is for 2.2 MHz transducer. Figures 4.6a–d show horizontal (s11 ) and vertical (s33 ) stresses. These two stress components are different in aluminum but same in water. The vertical displacement (u3 ) in aluminum is plotted in Figures 4.6e and f. Higher number of peaks and dips are visible for the 2.2 MHz signal in comparison to that for the 1 MHz signal. It can be seen that the beam is more collimated at 2.2 MHz as expected. Leaky guided waves in the ﬂuid are also modeled for critical inclination of the transducer. Formula for the critical angle calculation is shown in Table 4.1. For the

Figure 4.6 Computed ultrasonic ﬁelds for normal incidence of the wave at the interface. (a) the normal stress along x-axis (s11) in aluminum and pressure in water, signal frequency ¼ 1 MHz; (b) the normal stress along x-axis (s11) in aluminum and pressure in water, signal frequency ¼ 2.2 MHz; (c) the normal stress along z-axis (s33) in aluminum and pressure in water, signal frequency ¼ 1 MHz; (d) the normal stress along z-axis (s33) in aluminum and pressure in water, signal frequency ¼ 2.2 MHz; (e) the vertical displacement along z-axis (u3) in aluminum, signal frequency ¼ 1 MHz; (f) the vertical displacement along z-axis (u3) in aluminum, signal frequency ¼ 2.2 MHz.

168

ADVANCED APPLICATIONS OF DISTRIBUTED POINT SOURCE

Figure 4.6 (Continued)

4.5 APPLICATION 1: ULTRASONIC FIELD MODELING

169

Figure 4.6 (Continued)

critical angle (30.42 ) of incidence the ultrasonic ﬁelds in aluminum and water are shown in Figures 4.7a–f. For convenience the pressure in water and stresses in aluminum are plotted in the same ﬁgure. Figures 4.7a and b show the horizontal normal stress (s11 ) and Figures 4.7c and d show the vertical normal stress (s33 )

170

ADVANCED APPLICATIONS OF DISTRIBUTED POINT SOURCE

Figure 4.6 (Continued)

components. The vertical displacement component in the solid are plotted in Figures 4.7e and f. As before, Figs. 4.7a, c and e are generated with a 1 MHz transducer and Figs. 4.7b, d and f are generated with a 2.2 MHz transducer. It is well known that as the frequency of the signal increases, its wavelength decreases and the guided wave penetration depth decreases because it is proportional to the wavelength. Comparing Figures 4.7a and b, it can be seen that the thickness of the disturbance (penetration depth in the solid) due to the Rayleigh wave propagating along the interface is higher with 1 MHz signal compared to 2.2 MHz signal. The leaky wave in water is clearly visible in Figures 4.7a through d. The vertical displacement in the solid has been plotted in Figures 4.7e and f for 1 MHz and 2.2 MHz signals, respectively. We can see that the vertical displacements in Figures 4.7e and f are more conﬁned near the interface compared to Figures. 4.6e and f because the guided wave mostly excites the particles near the interface. It is interesting to note that even the null region predicted by Bertoni and Tamir (1973) in the reﬂected beam proﬁle can be seen in the computed results (see Fig. 4.7d). Figures 4.8a–f show the ultrasonic ﬁeld in aluminum and water, for 45.42 angle of incidence, which is 15 greater than the Rayleigh angle. Figures 4.8a, c, and e are generated with 1 MHz signals and Figs. 4.8b, d, and f are generated with 2.2 MHz signals. It is well known that when the angle of incidence of the signal is greater than the critical angle, then most of the incident energy is reﬂected by the interface. Therefore, very small amount of energy should be transmitted inside the solid half space. This phenomenon is clearly visible in Figures 4.8a–f. The pressure distribution in water and normal stresses s11 and s33 in aluminum are shown in Figures 4.8a and c, respectively, for 1 MHz signal frequency. Similarly, Figures 4.8b and d show those for

Figure 4.7 Computed ultrasonic ﬁelds for critical angle (30.42 ) of incidence of the wave at the interface. (a) the normal stress along x-axis (s11) in aluminum and pressure in water, signal frequency ¼ 1 MHz; (b) the normal stress along x-axis (s11) in aluminum and pressure in water, signal frequency ¼ 2.2 MHz; (c) the normal stress along z-axis (s33) in aluminum and pressure in water, signal frequency ¼ 1 MHz; (d) the normal stress along z-axis (s33) in aluminum and pressure in water, signal frequency ¼ 2.2 MHz; (e) the vertical displacement along z-axis (u3) in aluminum, signal frequency ¼ 1 MHz; (f) the vertical displacement along z-axis (u3) in aluminum, signal frequency ¼ 2.2 MHz. 171

172

ADVANCED APPLICATIONS OF DISTRIBUTED POINT SOURCE

Figure 4.7 (Continued)

4.5 APPLICATION 1: ULTRASONIC FIELD MODELING

Figure 4.7 (Continued)

173

174

ADVANCED APPLICATIONS OF DISTRIBUTED POINT SOURCE

Figure 4.8 Computed ultrasonic ﬁeld for 45.42 angle of incidence of the wave at the interface. (a) the normal stress along x-axis (s11) in aluminum and pressure in water, signal frequency ¼ 1 MHz; (b) the normal stress along x-axis (s11) in aluminum and pressure in water, signal frequency ¼ 2.2 MHz; (c) the normal stress along z-axis (s33) in aluminum and pressure in water, signal frequency ¼ 1 MHz; (d) the normal stress along z-axis (s33) in aluminum and pressure in water, signal frequency ¼ 2.2 MHz; (e) the vertical displacement along z-axis (u3) in aluminum, signal frequency ¼ 1 MHz; (f) The vertical displacement along z-axis (u3) in aluminum, signal frequency ¼ 2.2 MHz.

4.5 APPLICATION 1: ULTRASONIC FIELD MODELING

175

Figure 4.8 (Continued)

2.2 MHz signal frequency. In Figures 4.8a through d, a weak guided wave is observed although the incident angle is not the critical angle. The guided wave is observed because the incident beam is not perfectly collimated. Therefore, a small amount of energy still strikes the interface at the critical angle.

176

ADVANCED APPLICATIONS OF DISTRIBUTED POINT SOURCE

Figure 4.8 (Continued)

Figures 4.9a–f show the ultrasonic ﬁeld in aluminum and water, for 15.42 angle of incidence, which is 15 less than the critical angle. Figs. 4.9a, c, and e are generated with 1 MHz signals, and Figs. 4.9b, d, and f are generated with 2.2 MHz signals. When the angle of incidence of the signal at the ﬂuid–solid interface is less than the

4.5 APPLICATION 1: ULTRASONIC FIELD MODELING

177

Figure 4.9a Computed ultrasonic ﬁeld for 15.42 angle of incidence of the wave at the interface (a) The normal stress along x-axis (s11) in aluminum and pressure in water, signal frequency ¼ 1 MHz; (b) the normal stress along x-axis (s11) in aluminum and pressure in water, signal frequency ¼ 2.2 MHz; (c) the normal stress along z-axis (s33) in aluminum and pressure in water, signal frequency ¼ 1 MHz; (d) the normal stress along z-axis (s33) in aluminum and pressure in water, signal frequency ¼ 2.2 MHz; (e) the vertical displacement along z-axis (u3) in aluminum, signal frequency ¼ 1 MHz; (f) the vertical displacement along z-axis (u3) in aluminum, signal frequency ¼ 2.2 MHz.

178

ADVANCED APPLICATIONS OF DISTRIBUTED POINT SOURCE

Figure 4.9d (Continued)

4.5 APPLICATION 1: ULTRASONIC FIELD MODELING

Figure 4.9f (Continued)

179

180

ADVANCED APPLICATIONS OF DISTRIBUTED POINT SOURCE

critical angle, then relatively more energy (in comparison to Fig. 4.8) is transmitted into the solid half space as seen in Figures 4.9a–f. At 1 MHz signal the transmitted ultrasonic ﬁelds in the solid (Figs. 4.9a, c, and e) are much less than that for 2.2 MHz signal (Figs. 4.9b, d, and f). Strong transmitted ultrasonic ﬁelds are clearly visible in Figures. 4.9b, d, and f). The wave speed in aluminum is much higher than that in water. Therefore, when the ultrasonic beam hits the interface, it is expected that the transmitted ﬁeld should propagate in aluminum with a higher transmission angle than the angle of incidence in water. This phenomenon can be clearly seen in Figures 4.9b and d. 4.6 APPLICATION 2: ULTRASONIC FIELD MODELING IN A SOLID PLATE (BANERJEE AND KUNDU, 2007) In this section our objective is to model the ultrasonic ﬁeld generated by ultrasonic transducers of ﬁnite dimension in the vicinity of a solid plate when both the plate and the transducers are immersed in a ﬂuid. Thus, it numerically simulates the ultrasonic experiments for plate inspection. For modeling this problem by the DPSM technique, it is necessary to calculate stress and displacement Green’s functions for ﬂuid and solid media explicitly. Calculation of Green’s functions for the solid medium has been discussed in Section 4.3.2. The ultrasonic ﬁelds in the solid plate are calculated for critical angles corresponding to the symmetric and antisymmetric guided wave modes in the plate. 4.6.1 Ultrasonic ﬁeld modeling in a homogeneous solid plate Let us consider a plate immersed in a ﬂuid as shown in Figure 4.10. Two circular transducers are placed in the ﬂuid on two sides of the plate specimen. Behind the

Figure 4.10 Distribution of the point sources near the inclined transducers and the ﬂuid–solid interfaces of the plate.

4.6 APPLICATION 2: ULTRASONIC FIELD MODELING IN A SOLID PLATE

181

transducer faces and on both sides of the interfaces, a number of point sources are distributed. These sources, when superimposed, should produce the total ultrasonic ﬁeld in ﬂuid and solid media. The lower interface is denoted as ‘‘Interface 1.’’ A1 is the source strength vector of the point sources that are placed above Interface 1 and generate part of the ultrasonic ﬁeld in the ﬂuid below the plate. A1 is the source strength vector of the sources that are distributed below Interface 1. A1 models part of the transmitted ﬁeld in the solid plate. Similarly, A2 and A2 are the source strength vectors of the point sources that are distributed above and below Interface 2, respectively. Transducer faces have source strength vectors AS and AR . In Figure 4.10, three points (C, D, and E) are shown for the illustration purpose. The ultrasonic ﬁeld at point C is the summation of the contributions of all point sources with source strengths A1 and A2 distributed below and above Interfaces 1 and 2, respectively. Ultrasonic ﬁeld at point D is the summation of contributions of the point sources with source strengths A1 and AS distributed above the solid–ﬂuid Interface 1 and behind the front face of transducer S, respectively. Similarly, the ultrasonic ﬁeld at point E is the summation of contributions of the point sources with source strengths A2 and AR distributed below the solid–ﬂuid Interface 2 and behind the front face of transducer R, respectively. 4.6.2 Matrix formulation to calculate source strengths The velocity and pressure in ﬂuids can be expressed in matrix forms for multiple target points in solid and ﬂuid, as shown in Figure 4.11. Referring to Eq. (4.99) we can write VS ¼ MSS AS

ð4:112Þ

Where VS is the ðN 1Þ vector of the velocity components at the transducer surface corresponding to N number of point sources points distributed behind the transducer

Figure 4.11 A schematic diagram of M source points and N target points inside the solid and the ﬂuid of a ﬂuid–solid structure.

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ADVANCED APPLICATIONS OF DISTRIBUTED POINT SOURCE

face and AS is the ðN 1Þ vector containing the strength of the transducer sources. Following the steps similar to Eqs. (4.104c) through (4.104h), the pressure at any set of target points in the ﬂuid below the plate due to the transducer sources, can be written as PRsT1 ¼ QðT1ÞS AS

ð4:113Þ

The pressure at the same set of target points in the ﬂuid due to the interface sources can be written as PR1T1 ¼ QðT1Þ1 A1

ð4:114Þ

where T1 and T2 are two different sets of target points in the ﬂuid below and above Interfaces 1 and 2, respectively. Therefore, at those target points the total pressure ﬁeld is PRT1 ¼ PRsT1 þ PR1T1 ¼ QðT1ÞS AS þ QðT1Þ1 A1

ð4:115Þ

In the same manner for any set of target points in the ﬂuid above the plate can be written as

PRT2 ¼ PRsT2 þ PR2T2 ¼ QðT2ÞR AR þ QðT2Þ2 A2

ð4:116Þ

At any set of target points, the displacement in the x3 direction in the ﬂuid below and above the plate can be written as U3T1 ¼ DF3ðT1ÞS AS þ DF3ðT1Þ1 A1 U3T2 ¼ DF3ðT2ÞR AR þ

DF3ðT2Þ2 A2

ð4:117Þ ð4:118Þ

Every point source considered to calculate the transmitted ﬁeld in the solid has three different point forces in three mutually perpendicular directions as unknowns, as discussed earlier. From Eq. (4.79a), the normal stress in the x3 direction at the Interface 1 (the set of target points are called I1) due to A1 and A2 source strength vectors can be written as s33I1 ¼ S3311 A1 þ S3312 A2

ð4:119Þ

Similarly, for shear stresses, from Eqs. (4.79d) and (4.79e), it can be written as s31I1 ¼ S3111 A1 þ S3112 A2 s32I1 ¼ S3211 A1 þ S3212 A2

ð4:120Þ ð4:121Þ

where A1 and A2 are the source strength vectors of the sources distributed below and above the ﬂuid–solid interfaces, respectively. These vectors have (3M) elements.

4.6 APPLICATION 2: ULTRASONIC FIELD MODELING IN A SOLID PLATE

183

Similarly, from Eq. (4.79a), the normal stress component in the x3 direction at Interface 2 (the set of target points are called I2) can be written as s33I2 ¼ S3321 A1 þ S3322 A2

ð4:122Þ

and for shear stresses, from Eqs. (4.79d) and (4.79e), the following equations are obtained: s31I2 ¼ S3121 A1 þ S3122 A2 s32I2 ¼ S3221 A1 þ S3222 A2

ð4:123Þ ð4:124Þ

Considering the T1 and T2 target points located at I1 and I2 interfaces, respectively, the displacement component (see Eqs. (4.117) and (4.118)) along the x3 direction in the ﬂuid medium can be written as U3I1 ¼ DF3ðI1ÞS AS þ DF3ðI1Þ1 A1 U3I2 ¼ DF3ðI2ÞR AR þ

DF3ðI2Þ2 A2

ð4:125Þ ð4:126Þ

Similarly, from Eq. (4.78c) in the solid at I1 and I2 interfaces, the displacements along the x3 direction can be written as u3I1 ¼ DS3ðI1Þ1 A1 þ DS3ðI1Þ2 A2 u3I2 ¼

DS3ðI2Þ1 A1

þ DS3ðI2Þ2 A2

ð4:127Þ ð4:128Þ

4.6.3 Boundary and continuity conditions Across the ﬂuid–solid interfaces, the displacement normal to the interface should be continuous. Also, across the interfaces, the normal stress (s33) in solid and ﬂuid media should be continuous and the shear stresses at the interfaces must vanish. If the normal velocities of the two transducer faces are assumed to be VS0 and VR0 , then the velocity of the surface of the transducer (designated as S) can be expressed as MSS AS þ MS1 A1 ¼ VS0

ð4:129Þ

and on the surface of the transducer designated as R. MR2 A2 þ MRR AR ¼ VR0

ð4:130Þ

At the interfaces, from the continuity of the normal stress, Q1S AS þ Q11 A1 ¼ S3311 A1 S3312 A2 Q22 A2 þ Q2R AR ¼ S3321 A1 S3322 A2

ð4:131Þ ð4:132Þ

Continuity of the normal displacement across the two interfaces gives DF31S AS þ DF311 A1 ¼ DS311 A1 þ DS312 A2

DF322 A2

þ DF32R AR ¼

DS321 A1

þ DS322 A2

ð4:133Þ ð4:134Þ

184

ADVANCED APPLICATIONS OF DISTRIBUTED POINT SOURCE

and from the vanishing shear stress condition at the ﬂuid–solid interface, S3111 A1 þ S3112 A2 ¼ 0

ð4:135Þ

S3211 A1

ð4:136Þ

þ S3212 A2 ¼ 0

Eqs. (4.129)–(4.136) can be written in the matrix form 2

MSS

6 6 Q1S 6 6 6 DF31S 6 6 6 0 6 6 6 0 6 6 6 6 0 6 6 6 0 6 6 6 0 6 6 6 0 4 0

MS1

0

0

0

Q11

S3311

S3312

0

DF311

DS311

DS312

0

0

S3111

S3112

0

0

S3211

S3212

0

0

S3221

S3222

0

0

S3121

S3122

0

0

S3321

S3322

Q22

0

DS321

DS322

DF322

0

MR2

0

9 8 AS > > > > > > > > > > > > > > A 1 > > > > > > > > > = < A1 > > > > A1 > > > > > > > > > > > > > A > 2> > > > > > > ; : AR ð2Nþ8MÞ

¼

0 9 8 VS0 > > > > > > > > > > > > > 0 > > > > > > > > > > > > > 0 > > > > > > > > > > > > > > 0 > > > > > > > > > = < 0 > > > > > > > > > > > > > > > > > > > > > > > > > > :

0

3

7 7 7 7 0 7 7 7 0 7 7 7 0 7 7 7 7 0 7 7 7 0 7 7 7 Q2R 7 7 7 DF32R 7 5 0

MRR

ð2Nþ8MÞð2Nþ8MÞ

ð4:137Þ

> 0 > > > > > > 0 > > > > > > > 0 > > > > > > 0 > > > > > ; VR0 ð2Nþ8MÞ

or ½MATfLg ¼ fVg

ð4:138Þ

4.6 APPLICATION 2: ULTRASONIC FIELD MODELING IN A SOLID PLATE

185

4.6.4 Solution The source strength vector fg of the total system can be calculated from Eq. (4.138) fLg ¼ ½MAT1 fVg

ð4:139Þ

After calculating the source strengths, the pressure, velocity, stress, and displacement values at any point can be calculated. For example, the pressure ﬁeld in the ﬂuid can be written as PRðF1Þ ¼ QðF1ÞS AS þ QðF1Þ1 A1 PRðF2Þ ¼ QðF2ÞR AR þ

QðF2Þ2 A2

ð4:140Þ ð4:141Þ

where, F1 is a set of target points inside the ﬂuid medium below the plate and F2 is a set of target points inside the ﬂuid medium above the plate. Similarly, in the solid plate, the stress and displacement ﬁelds can be obtained from the following set of equations: s33ðSÞ ¼ S33ðSÞ1 A1 þ S33ðSÞ2 A2 s11ðSÞ ¼ S11ðSÞ1 A1 þ S11ðSÞ2 A2 ......

ð4:142Þ

u3ðSÞ ¼ DS3ðSÞ1 A1 þ DS3ðSÞ2 A2 u1ðSÞ ¼ DS1ðSÞ1 A1 þ DS1ðSÞ2 A2 where S is a set of target points inside the solid plate.

4.6.5 Numerical results on ultrasonic ﬁeld modeling in solid plates DPSM-based computer codes have been developed to model the ultrasonic ﬁeld in the solid plate by using the formulation presented above. It is well known from the Rayleigh–Lamb frequency equation that in a solid plate the Lamb wave propagates at certain frequencies with certain phase velocities. All wave modes inside the plate are dispersive. Fundamental symmetric and antisymmetric modes in the plates are modeled here. The DPSM modeling is used to model the ultrasonic ﬁeld in an aluminum plate immersed in water, whose dispersion curves are shown in Figure 4.12. Transducers with diameter 4 mm are used to generate the ultrasonic signals. The transducer locations are different for different frequencies. To obtain the maximum pressure of the striking beam at the plate surface position, the plate specimens are placed at different distances from the transducers of different frequency. The results (stress and displacement ﬁelds inside the plate) are presented in Figures 4.13 through 4.17.

186

ADVANCED APPLICATIONS OF DISTRIBUTED POINT SOURCE

Figure 4.12 Dispersion curves for the aluminum plate immersed in water.

Figure 4.13 Ultrasonic ﬁelds generated in the aluminum plate for normal incidence (a–d) and inclined incidence (e–h) of the ultrasonic beam. For inclined incidence the striking angle is 30.5 , which is the Rayleigh critical angle. Plots show the variations of normal stress components s11 (a,e), s33 (b, f), shear stress component s13 (c, g), and vertical displacement component u3 (d, h) in the plate.

4.6 APPLICATION 2: ULTRASONIC FIELD MODELING IN A SOLID PLATE

Figure 4.13 (Continued)

187

188

ADVANCED APPLICATIONS OF DISTRIBUTED POINT SOURCE

Figure 4.13 (Continued)

4.6 APPLICATION 2: ULTRASONIC FIELD MODELING IN A SOLID PLATE

Figure 4.13 (Continued)

189

190

ADVANCED APPLICATIONS OF DISTRIBUTED POINT SOURCE

Figure 4.13 (Continued)

The properties (the density and wave speeds) of the aluminum plate are given in Section 4.5 (Table 4.1). The aluminum plate is 50 mm long and immersed in water. Along the abscissa of Figure 4.12 the product of frequency and thickness of the plate in MHz mm is plotted. Along the ordinate, the phase velocity in km/s is plotted for different modes. It can be seen from this ﬁgure that for 10 mm thick plate and 1 MHz transducer (or for 1 mm thick plate and 10 MHz transducer), the Rayleigh 1:48 Þ ¼ 30:50 . waves are produced for an inclination angle of sin1 ðccrf Þ ¼ sin1 ð2:916 For normal incidence of the ultrasonic signal, the ultrasonic ﬁelds in the plate are shown in Figures 4.13a–d. For transducer inclination angle of 30.50 , the ultrasonic ﬁelds in the 10 mm thick aluminum plate on the left side of the point of strike are presented in Figures 4.13e–h. Normal stresses (s11 ; s33 ), shear stress (s31 ), and vertical displacement (u3 ) are plotted in Figure 4.13. The effect of guided waves on both interfaces are clearly visible in Figures 4.13e, f, g, and h. Then, a 1 mm thick aluminum plate is considered and 1 MHz transducers are used for generating guided waves in that plate. Figure 4.12 shows that the symmetric mode in the 1 mm thick aluminum plate propagates with a velocity of 4.733 km/s at 1 MHz frequency. Therefore, to generate this symmetric mode in the plate, transducers must 1:48 Þ ¼ 16:11 . The phase velocity be inclined at a critical angle of sin1 ðccfl Þ ¼ sin1 ð5:333 of the antisymmetric mode is 2.333 km/s. Therefore, to generate the antisymmetric mode, the transducers must be inclined at a critical angle of sin1 ðccaf Þ ¼ 1:48 Þ ¼ 39:4343 . Figures 4.14 and 4.15 show the ultrasonic ﬁelds in the sin1 2:333 1 mm thick aluminum plate for 39.43 and 16.11 angles of incidence, respectively. The antisymmetric mode generated in the plate is shown in Figure 4.14 and the symmetric mode is shown in Figure 4.15.

4.6 APPLICATION 2: ULTRASONIC FIELD MODELING IN A SOLID PLATE

191

Figure 4.14 Ultrasonic ﬁelds inside a 1 mm thick aluminum plate for A0 mode of excitation, generated by 1 MHz transducers inclined at 39.43 angle as shown in Figure 4.10. Four plots show the variations of (a) the vertical normal stress s33 , (b) the shear stress s31 , (c) the horizontal normal stress s11 , and (d) the vertical displacement u3 component.

192

ADVANCED APPLICATIONS OF DISTRIBUTED POINT SOURCE

Figure 4.14 (Continued)

4.6 APPLICATION 2: ULTRASONIC FIELD MODELING IN A SOLID PLATE

193

Figure 4.15 Ultrasonic ﬁelds inside a 1 mm thick aluminum plate for S0 mode of excitation, generated by 1 MHz transducers inclined at 16.11 angle as shown in Figure 4.10. Four plots show the variations of (a) the vertical normal stress s33 , (b) the shear stress s31 , (c) the horizontal normal stress s11 , and (d) the vertical displacement u3 component.

194

ADVANCED APPLICATIONS OF DISTRIBUTED POINT SOURCE

Figure 4.15 (Continued)

With the same critical angle of inclination and 0.5 MHz transducers, the antisymmetric and symmetric modes in a 2 mm thick aluminum plate are then generated. The ultrasonic ﬁelds for these two modes are presented in Figures 4.16 and 4.17. Figure 4.16 clearly shows the antisymmetric mode in the plate, and Figure 4.17 shows the symmetric mode.

4.6 APPLICATION 2: ULTRASONIC FIELD MODELING IN A SOLID PLATE

195

Figure 4.16 Ultrasonic ﬁelds inside a 2 mm thick aluminum plate for A0 mode of excitation, generated by 0.5 MHz transducers inclined at 39.43 angle as shown in Figure 4.10. Four plots show the variations of (a) the vertical normal stress s33 , (b) the shear stress s31 , (c) the horizontal normal stress s11 , and (d) the vertical displacement u3 component.

196

ADVANCED APPLICATIONS OF DISTRIBUTED POINT SOURCE

Figure 4.16 (Continued)

4.6 APPLICATION 2: ULTRASONIC FIELD MODELING IN A SOLID PLATE

197

Figure 4.17 Ultrasonic ﬁelds inside a 2 mm thick aluminum plate for S0 mode of excitation, generated by 0.5 MHz transducers inclined at 16.11 angle as shown in Figure 4.10. Four plots show the variations of (a) the vertical normal stress s33 , (b) the shear stress s31 , (c) the horizontal normal stress s11 , and (d) the vertical displacement u3 component.

198

ADVANCED APPLICATIONS OF DISTRIBUTED POINT SOURCE

Figure 4.17 (Continued)

4.7 APPLICATION 3: ULTRASONIC FIELDS IN SOLID PLATES WITH INCLUSION OR HORIZONTAL CRACKS (BANERJEE AND KUNDU, 2007a, b) 4.7.1 Problem geometry A plate containing an internal anomaly and immersed in a ﬂuid medium is considered here as shown in Figure 4.18. One can see from this ﬁgure that a cavity or an inclusion

4.7 APPLICATION 3: ULTRASONIC FIELDS IN SOLID PLATES WITH INCLUSION

199

Figure 4.18 Geometry of the isotropic plate with an anomaly (crack, cavity, or inclusion) immersed in water. Small circles show the distributed point sources used in the DPSM formulation.

is entrapped inside the solid plate. The cavity has stress-free boundaries whereas at the inclusion boundary, the continuity of the displacement and normal stress components must be enforced. Along the ﬂuid–solid interfaces, two layers of point sources are distributed, shown by the small circles in the diagram. Along the inclusion boundary, another two sets of point sources should be distributed. However, if the internal anomaly is a crack or cavity instead of an inclusion, then only one set of point sources is necessary inside the anomaly boundary, as shown in the ﬁgure. Two circular piston transducers are immersed in the ﬂuid and symmetrically placed on two sides of the plate specimen. A1 is the source strength vector of the point sources that are placed above the solid–ﬂuid interface (Interface I) and generate part of the ultrasonic ﬁeld in the ﬂuid below the plate. A1 is the source strength vector of the sources that are distributed below the ﬁrst solid–ﬂuid interface. It models part of the transmitted ﬁeld in the solid plate. Similarly, A2 and A2 are the source strength vectors of the point sources that have been distributed above and below the second ﬂuid–solid interface (Interface 2), respectively. Ac and Ac are the source strength vectors around the anomaly boundary. Transducer faces have source strength vectors AS and AR. In Figure 4.18, three points (C, D, and E) have been considered for the illustration purpose. The ultrasonic ﬁeld at point C inside the solid plate is the summation of the contributions of the point source A1 , A2 , and Ac . The ultrasonic ﬁeld at point D inside Fluid 1 is the summation of the contributions of the point sources A1 and AS. Similarly, the ultrasonic ﬁeld at point E inside Fluid 2 is the

200

ADVANCED APPLICATIONS OF DISTRIBUTED POINT SOURCE

summation of contributions of the point sources A2 and AR . It is possible to ignore the effect of the point sources that are located in the shadow region by introducing a ﬁctitious interface in the plate going through the anomaly and thus transforming the homogeneous plate into a plate made of two layers of identical material properties as discussed in Chapter 2, see Figure 2.10. However, the following results are obtained without introducing such ﬁctitious interface. 4.7.2 Matrix formulation The particle velocity and the pressure in ﬂuids can be expressed in the matrix form similar to the expressions presented in Sections 4.5 and 4.6. The particle velocity in ﬂuid 1 adjacent to the transducer face due to the transducer sources AS is expressed in Eq. (4.104a) or (4.112). The velocity due to the transducer sources at any set of target points in the ﬂuid is given in Eq. 4.104b. Similarly, Eqs. (4.113)–(4.118) hold good for this formulation. The only change in the formulation occurs when stress and displacement are calculated in the solid. The normal stress along the x3 direction (see Fig. 4.18) at Interface 1 (the set of target points along Interface 1 are denoted as I1) due to A1 , A2 , and Ac source strength vectors can be written as s33I1 ¼ S3311 A1 þ S3312 A2 þ S331c Ac

ð4:143Þ

Similarly, shear stresses can be written as s31I1 ¼ S3111 A1 þ S3112 A2 þ S311c Ac s32I1 ¼

S3211 A1

þ S3212 A2 þ

S321c Ac

ð4:144Þ ð4:145Þ

Vectors A1 , A2 , and Ac have 3M elements each, if the same number of point sources are used in A1 , A2 , and Ac . Similarly, the normal stresses along the x3 direction at Interface 2 (the target points are denoted as I2) can be written as s33I2 ¼ S3321 A1 þ S3322 A2 þ S332c Ac

ð4:146Þ

and for shear stresses, s31I2 ¼ S3121 A1 þ S3122 A2 þ S312c Ac s32I2 ¼ S32

21

A1

þ S3222 A2 þ S32

2c

Ac

ð4:147Þ ð4:148Þ

At target points I1 and I2 the vertical displacement components in ﬂuids 1 and 2 are similar to Eqs. (4.125) and (4.126), respectively. Similarly, in the solid at the same target points, the displacements in x3 direction can be written as u3I1 ¼ DS311 A1 þ DS312 A2 þ DS31c Ac u3I2 ¼

DS321 A1

þ DS322 A2 þ

DS31c Ac

ð4:149Þ ð4:150Þ

4.7 APPLICATION 3: ULTRASONIC FIELDS IN SOLID PLATES WITH INCLUSION

201

4.7.3 Boundary and continuity conditions If the normal velocities of the two transducer faces are known, then the particle velocities on the surface of the transducer designated as S can be computed and equated to the known value MSS AS þ MSI AI ¼ VS0

ð4:151Þ

and similarly the computed particle velocities on the surface of the transducer designated as R can be equated to the known velocity value for that transducer. MR2 A2 þ MRR AR ¼ VR0

ð4:152Þ

Other continuity conditions are achieved at the ﬂuid–solid interfaces, from the continuity of the normal stress Q1S AS þ Q11 A1 ¼ S3311 A1 S3312 A2 S331c Ac Q22 A2

þ Q2R AR ¼

S3321 A1

S3322 A2

S332c Ac

ð4:153Þ ð4:154Þ

and the continuity of the normal displacement gives DF31S AS þ DF311 A1 ¼ DS311 A1 þ DS312 A2 þ DS31c Ac

DF322 A2

þ DF32R AR ¼

DS321 A1

þ DS322 A2 þ

DS32c Ac

ð4:155Þ ð4:156Þ

From the vanishing shear stress condition at the ﬂuid–solid interface, we can write S3111 A1 þ S3112 A2 þ S311c Ac ¼ 0 S3211 A1 þ S3212 A2 þ S321c Ac ¼ 0

ð4:157Þ ð4:158Þ

The vanishing normal and shear stresses at a horizontal crack or cavity boundary gives S33CI1 A1 þ S33CI2 A2 þ S33CIc Ac ¼ 0

S31CI1 A1 S32CI1 A1

þ S31CI2 A2 þ þ S32CI2 A2 þ

S31CIc Ac S32CIc Ac

¼0 ¼0

ð4:159Þ ð4:160Þ ð4:161Þ

where CI indicates the target points on the crack surface. For any general shape of inclusions and cavities, the s33 ; s31 , and s32 stresses at every point on the inclusion or cavity boundary need to be transformed to the corresponding normal stress and parallel shear stresses at that point of the boundary. The Eq. (4.159) through (4.161) still holds good, but the matrices used in the equation will have different values due to the transformation operation. The boundary and continuity equations can be written in the matrix form similar to Eq. (4.137), except that new entries of matrix and vector elements corresponding to the Ac source strength vector appear in this equation.

202

ADVANCED APPLICATIONS OF DISTRIBUTED POINT SOURCE

4.7.4 Solution The vector of source strengths fg of the total system can be obtained by solving the system of linear equations, fLg ¼ ½MAT1 fVg

ð4:162Þ

After calculating the source strengths, the pressure, velocity, stress, and displacement values at any point can be obtained from the equations similar to Eqs. (4.151) through (4.161), as written in Eqs. (4.110) and (4.111). 4.7.5 Numerical results on ultrasonic ﬁelds in solid plate with horizontal crack The numerical results are generated for an isotropic aluminum plate containing a thin cavity or crack when the plate is placed under water. The detail problem geometry showing the crack position and the transducer orientation is given in Figure 4.19. The material properties of this aluminum plate are shown in Table 4.1. Figure 4.20 shows the ultrasonic ﬁeld developed in the aluminum plate with the horizontal cavity for the ultrasonic transducers placed normal to the interface as shown in Figure 4.19. Figures 4.20a, b, and c show s11, s13 , and u3 distributions, respectively, in the plate. Stress concentrations near the cavity boundary can be seen in these ﬁgures. Figures 4.13e, f, g, and h show the ultrasonic ﬁeld developed in the aluminum plate (in the absence of any cavity) when the transducers are inclined at 30.50 , the Rayleigh

Figure 4.19 Schematic diagram of an isotropic aluminum plate with a cavity or a crack (not to scale).

4.7 APPLICATION 3: ULTRASONIC FIELDS IN SOLID PLATES WITH INCLUSION

203

Figure 4.20 Ultrasonic ﬁelds developed in a 10 mm thick aluminum plate with a horizontal cavity/crack for the normal incidence of the striking beam: (a) s33 distribution (b) s13 distribution, and (c) u3 distribution.

204

ADVANCED APPLICATIONS OF DISTRIBUTED POINT SOURCE

Figure 4.20 (Continued)

critical angle. Figures 4.21a, b, c, and d show s11, s13 , s33 , and u3 distributions, respectively, in the defective aluminum plate for the transducers inclined at the Rayleigh critical angle, 30.50 . Comparing Figures 4.13e–h and 4.21a–d, it can be concluded that even when the crack or cavity is parallel to the guided wave propagation direction, the ultrasonic ﬁeld is signiﬁcantly altered by the anomaly both near the crack and away from it. Figure 4.22 shows the stress ﬁeld distribution in the plate along the perpendicular bisector of the cavity. It can be seen from this ﬁgure that the boundary conditions at the cavity boundary are perfectly satisﬁed. 4.8 APPLICATION 4: ULTRASONIC FIELD MODELING IN SINUSOIDALLY CORRUGATED WAVE GUIDES (BANERJEE AND KUNDU, 2006, 2006a) 4.8.1 Theory In this section the application of DPSM is extended to symmetrically corrugated, sinusoidal waveguides. On two sides of the waveguide, ﬂuid 1 and ﬂuid 2 are used as the coupling ﬂuids that help to transmit ultrasonic waves from the ultrasonic transducers to the waveguide (see Fig. 4.23). The sinusoidal wave guide is just like the plate wave guide considered in Section 4.6; the only difference is that for the sinusoidal waveguide, the interface is not a plane and has speciﬁed sinusoidal geometry. Therefore, there is no change in the general formulation presented in

4.8 APPLICATION 4: ULTRASONIC FIELD

205

Figure 4.21 Ultrasonic ﬁelds developed in a 10 mm thick aluminum plate with a horizontal cavity/crack for the Rayleigh angle of incidence: (a) s11 distribution, (b) s13 distribution, (c) s33 distribution, (d) u3 distribution.

206

ADVANCED APPLICATIONS OF DISTRIBUTED POINT SOURCE

Figure 4.21 (Continued)

Section 4.6. However, the stresses at the ﬂuid–solid interface must be transformed to the local normal and parallel directions relative to the point of interest on the interface to apply the continuity conditions across the interface. Similar to Section 4.6, a total of four sets of point sources are distributed along the two interfaces of the wave guide

4.8 APPLICATION 4: ULTRASONIC FIELD

207

Figure 4.22 Stress proﬁles along the depth of the 10 mm thick plate at the center line (normal bisector) of the cavity or crack.

Figure 4.23 Sinusoidally corrugated waveguide placed between two transducers: Circles show the point source distribution for the DPSM analysis.

208

ADVANCED APPLICATIONS OF DISTRIBUTED POINT SOURCE

and additional two sets of point sources are placed behind the front faces of the two transducers as shown in Figure 4.23. AS ; AR ; A1 ; A2 ; A1 , and A2 are the source strength vectors for the sources distributed near the transducer faces and two interfaces (see Fig. 4.23). The period of corrugation of the sinusoidal wave guide is D and the depth of corrugation is equal to e. Because the boundary surfaces of the sinusoidal waveguide are nonplaner, at every point of the interface, normal stress and normal displacement relative to the surface need to be deﬁned to satisfy the continuity conditions across the interface. The direction cosine of the sinusoidal wave guide at any point on the surface can be deﬁned as n ¼ ðn1 e1 þ n3 e3 Þ. Projections of unit normal (n) on x1 and x3 axes are given in Eqs. (4.163) and (4.164), respectively. 2pe 2px1 sin D D ð4:163Þ n1 ¼ " #12 2 2pe 2px1 þ1 sin2 D D n3 ¼ "

2pe D

2

1 #12 2px 1 þ1 sin2 D

ð4:164Þ

When the point source is acting at y in an isotropic solid, the stresses developed at point x are written in Eq. (4.63) through (4.66). Assuming a point force acting along the xj direction, stresses at point x on the boundary of the sinusoidal waveguide can be written as 2

sj11

6 rj ¼ 4 sj21 sj31 The transformation matrix at point x is 2 n3 6 T¼4 0 n1

sj12

sj13

3

sj22

7 sj23 5

sj32

sj33

ð4:165Þ

3 0 n1 7 1 0 5 0 n3

ð4:166Þ

Therefore, transformed stresses at point x is 2

n3

0

6 j r0 ¼ Tsj TT ¼ 6 40

1

n1

0

n1

32

s11j

76 j 6 0 7 54 s21 n3 s31j

s12j

s13j

32

n3

s22j

76 6 s23j 7 54 0

s32j

s33j

n1

0 n1

3

2

s011j

7 6 0j 6 07 5 ¼ 4 s21 0 n3 s031j 1

s012j

s013j

3

s022j

7 s023j 7 5

s032j

s033j

ð4:167Þ

209

4.8 APPLICATION 4: ULTRASONIC FIELD

To deﬁne the boundary conditions at point x, one normal stress (s033 ), perpendicular to the sinusoidal boundary surface and two shear stresses parallel to the boundary surface (s031 ; s032 ), are needed. When transformation is applied to every point at the interface, the matrix elements in Eq. (4.81a) will change accordingly by replacing primed stresses in Eq. (4.67). Displacements at point x generated by a point source acting at point y in an isotropic solid is given in Eq. (4.22). The displacements at x due to the point force acting at y along the xj direction are denoted as G1j ; G2j , and G3j . Considering the same point force along the xj direction, the normal displacement of the sinusoidal solid surface at x can be written as unj ¼ G1j n1 þ G2j n2

ð4:168Þ

Considering a set of M point sources distributed on the interface, the normal displacement at point x on the sinusoidal surface can be written as un ¼

M X

ððG11 n1 þ G31 n3 Þm T1m þ ðG12 n1 þ G32 n3 Þm T2m þ ðG13 n1 þ G33 n3 Þm T3m Þ

m¼1

¼

M X

ð4:169Þ

Gnm Tm

m¼1

where T is described in Eq. (4.8). Let T be a set of target points in the solid. Normal displacements at these points (T) on the sinusoidal surface can be written in the following form: unT ¼ DSnT1 A1 þ DSnT2 A2

ð4:170Þ

Similar to Eq. (4.81b), the matrix DSnTS can be written as 2

DSnTS

Gn11

6 Gn1 6 2 6 6 Gn13 ¼6 6 ... 6 6 4 Gn1N1 Gn1N

Gn21

Gn31

::

GnM1 1

Gn22 Gn23

Gn32 Gn33

... ...

GnM1 2 GnM1 3

... Gn2N1

:: Gn3N1

... ...

:: GnM1 N1

Gn2N

Gn3N

...

GnM1 N

GnM 1

3

7 7 7 7 7 ... 7 7 7 5 GnM N1 GnM 2 GnM 3

GnM N

ð4:171Þ

ðN3MÞ

In a ﬂuid medium, the displacement components at point x generated by a point source at y are given in Eqs. (4.94) through (4.96). Using the direction cosines (ni ) of the normal vector to the corrugated surface, the displacement component normal to the corrugated interface in the ﬂuid at point x can be written as ufn ¼ u1 n1 þ u3 n3

ð4:172Þ

210

ADVANCED APPLICATIONS OF DISTRIBUTED POINT SOURCE

Following the same rule in the presence of the transducers (see Fig. 4.23), the displacement of the ﬂuid at Interfaces 1 and 2 can be written as UnI1 ¼ ððDF3ðI1ÞS Þn3 þ ðDF1ðI1ÞS Þn1 ÞAS þ ððDF3ðI1Þ3 Þn3 þ ðDF1ðI1Þ1 Þn1 ÞA1 ð4:173Þ UnI2 ¼ ððDF3ðI2ÞR Þn3 þ ðDF1ðI2ÞR Þn1 ÞAR þ ððDF3ðI2Þ2 Þn3 þ ðDF1ðI2Þ2 Þn1 ÞA2 ð4:174Þ or UnI1 ¼ DFnðI1ÞS AS þ DFnðI1Þ1 A1

ð4:175Þ

UnI2 ¼ DFnðI2ÞR AR þ DFnðI2Þ2 A2

ð4:176Þ

Similar to Eq. (4.103d), matrix DFnTS is constructed DFn 2TS 1 1 gðRi 1 ; r1 Þ 6 6 gðRi 12 ; r21 Þ 6 6 gðR 1 ; r 1 Þ i3 3 6 ¼6 6 gðRi 14 ; r41 Þ 6 6 ... 4 gðRi 1N ; rN1 Þ

gðRi 21 ; r12 Þ

gðRi 31 ; r13 Þ

...

gðRi M1 ; r1M1 Þ 1

gðRi 22 ; r22 Þ

gðRi 32 ; r23 Þ

...

gðRi M1 ; r2M1 Þ 2

gðRi 23 ; r32 Þ

gðRi 33 ; r33 Þ

...

gðRi M1 ; r3M1 Þ 3

gðRi 24 ; r42 Þ

gðRi 34 ; r43 Þ

...

gðRi M1 ; r4M1 Þ 4

...

...

...

...

gðRi 2N ; rN2 Þ

gðRi 3N ; rN3 Þ

...

gðRi M1 ; rNM1 Þ N

M gðRi M 1 ; r1 Þ

3

M 7 gðRi M 2 ; r2 Þ 7 7 M 7 gðRi M 3 ; r3 Þ 7 7 M 7 gðRi M 4 ; r4 Þ 7 7 ... 5 M M gðRi N ; rN Þ ðNMÞ

ð4:177Þ

where m gðRi m n ; rn Þ

1 ¼ ro2

""

# " # # m m 1 eikf rn 1 eikf rn m ikf rnm m m ikf rnm m ikf R3 n e R3 n n3 þ m ikf R1 n e R1 n n1 rnm rn ðrnm Þ2 ðrnm Þ2

xi n yi n Ri m and the subscript i take the values 1,2, and 3. n ¼ rnm Then, one can obtain the matrix equation that can be solved to obtain the source strength vectors following the steps similar to those described in Section 4.6.3. m

m

4.8.2 Numerical results on ultrasonic ﬁelds in sinusoidal corrugated wave guides The numerical results are presented for the corrugated aluminum waveguides with Lame´ constants l and m equal to 54.55 GPa and 24.85 GPa, respectively, and density equal to 2.7 g/cc. P-wave and S-wave speeds (cp ¼ 6220 m=s and cs ¼ 3040 m=s) in the material are obtained from the above elastic constants. Four different waveguides are considered in the analysis. The dimensions of the waveguides are given in Table 4.2.

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4.8 APPLICATION 4: ULTRASONIC FIELD

TABLE 4.2

Waveguide Waveguide Waveguide Waveguide

1 2 3 4

Waveguide geometry (See Fig. 4.23) 2h

e

D

e=D

10 10 10 10

0.5 1 1.5 2

10 10 10 10

0.05 0.1 0.15 0.2

Ultrasonic ﬁelds in four different waveguides are generated by the DPSM technique. Normal incidence of the ultrasonic beam on a corrugation peak of the waveguide is considered ﬁrst and then the transducers are inclined at two different angles. Results for these three different orientations of the transducers are presented. Figure 4.24 shows different transducer orientations. The transducer frequency is set at 1 MHz. Figures 4.25 and 4.26 show the horizontal (u1 ) and vertical (u2 ) displacement ﬁelds, respectively, inside the waveguides. In these two ﬁgures the displacement ﬁelds are presented for waveguides 2, 3, and 4 (see Table 4.2 for dimensions). Figures 4.25a, b, and c show the u1 displacement for normal incidence (transducer orientation is shown in Fig. 4.24a) in waveguides 2, 3, and 4, respectively. Figures 4.25d, e, and f show the u1 displacement for 30 striking angle (transducer orientation is shown in Fig. 4.24b) in waveguides 2, 3, and 4, respectively. Similarly, Figures 4.26a, b, and c show the u2 displacement for normal incidence in waveguides 2, 3, and 4, respectively, and Figures 4.25d, e, and f show the u2 displacement for 30 striking angle in waveguides 2, 3, and 4, respectively. It can be seen from Figures 4.25 and 4.26 that the ultrasonic waves in waveguide 2 (Figs. 4.25d and 4.26d) propagate in the forward direction or in other words, in the same direction as the horizontal component of the striking beams. In waveguide 4 (Figs. 4.25f and 4.26f), ultrasonic waves in the waveguide propagate in the backward

Figure 4.24 Three different transducer orientations: (a) orientation – I: normal incidence (b) orientation – II: 30 inclination (c) orientation – III: 45 inclination.

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Figure 4.25 Horizontal displacement ﬁelds in three different corrugated waveguides (#2, 3, and 4) for two different angles of strike (0 and 30 ): (a) in waveguide 2 for normal incidence, (b) in waveguide 3 for normal incidence, (c) in waveguide 4 for normal incidence, (d) in waveguide 2 for 30 inclination angle, (e) in waveguide 3 for 30 inclination angle, (f) in waveguide 4 for 30 inclination angle. For inclined incidence, more energy is observed in the backward direction (x > 0) in (f) (large corrugation) whereas the opposite trend is noticed in (d) (small corrugation). Table 4.2 gives waveguide dimensions.

4.8 APPLICATION 4: ULTRASONIC FIELD

213

Figure 4.25 (Continued)

direction, or in other words, opposite to the direction of the striking beams. In waveguide 3 (Figs. 4.25e and 4.26e) the wave propagates in both directions. The phenomenon of the wave propagation in the backward direction in waveguides 4 and 3 is called ‘‘back propagation.’’ The back propagation phenomenon can be more clearly seen in Figure 4.27. Figure 4.27 shows amplitudes of u1 displacement near the

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ADVANCED APPLICATIONS OF DISTRIBUTED POINT SOURCE

Figure 4.25 (Continued)

central plane of the waveguides. In this ﬁgure the displacement variations in all four waveguides listed in Table 4.2 are shown. These displacement ﬁelds are generated for three different transducer orientations as shown in Figure 4.24. Figure 4.27 clearly shows the back propagation of ultrasonic waves (Figs. 4.27d and f) for large

4.8 APPLICATION 4: ULTRASONIC FIELD

215

corrugation (e=D ¼ 0:2 and 0.15) and forward propagation (Figs. 4.27c and e) for small corrugation (e=D ¼ 0:05 and 0.1) when the ultrasonic beam strikes the plate at an angle. The e=D ratio was carefully changed between 0.1 and 0.15 to ﬁnd out for what value of this ratio the back propagation starts to dominate. It is found that for the inclined incidence of the ultrasonic bounded beam on a corrugation peak when e=D 0:11, the ultrasonic waves propagate in both directions with almost equal strength. For e=D > 0:11 the back propagation dominates, and for e=D < 0:11 the forward propagation dominates. When the signal frequency in Figures 4.25, 4.26, and 4.27 was changed from 1 MHz to 2 MHz, the details of the ﬁgures changed to some extent, however, the general conclusion about the forward and backward propagation phenomenon did not change. For 2 MHz plots also (not shown here), it was observed that for e=D > 0:11, the back propagation dominates and for e=D < 0:11 the forward propagation dominates.

Figure 4.26 Vertical displacement ﬁelds in three different corrugated waveguides (#2, 3, and 4) for two different angles of strike (0 and 30 ): (a) in waveguide 2 for normal incidence, (b) in waveguide 3 for normal incidence, (c) in waveguide 4 for normal incidence, (d) in waveguide 2 for 30 inclination angle, (e) in waveguide 3 for 30 inclination angle, (f) in waveguide 4 for 30 inclination angle. As expected, symmetric displacement ﬁelds are observed for normal incidence ((a), (b), and (c)). However, for inclined incidence, more energy is observed in the backward direction (x > 0) in (f) (large corrugation) whereas the opposite trend is noticed in (d) (small corrugation). Table 4.2 gives waveguide dimensions. Readers should not be confused by the notation u2 in this ﬁgure for vertical component of displacement; it is same as the displacement component in the x3 direction of Figure 4.23.

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Figure 4.26 (Continued)

4.8 APPLICATION 4: ULTRASONIC FIELD

Figure 4.26 (Continued)

217

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ADVANCED APPLICATIONS OF DISTRIBUTED POINT SOURCE

Figure 4.26 (Continued)

4.9 CALCULATION OF GREEN’S FUNCTIONS IN TRANSVERSELY ISOTROPIC AND ANISOTROPIC SOLIDS In the previous sections, the DPSM formulations and numerical results are presented for isotropic materials. The DPSM technique can be used for anisotropic solids also if the Green’s functions for anisotropic solids are known. Calculation of the Green’s function is one important component in the DPSM formulation. Several researchers have calculated the displacement Green’s functions in anisotropic solids, but efﬁcient calculation of stress Green’s functions in anisotropic solids have not been reported yet. In this section, the most efﬁcient technique to calculate the Green’s functions in anisotropic solids available today is discussed. If stress Green’s functions in anisotropic solids can be efﬁciently calculated, then the DPSM technique can easily be extended to anisotropic solids.

4.9.1 Governing differential equation for Green’s function calculation Referring to Eqs. (4.1) and (4.2), the governing partial differential equation of motion for a solid material can be written as Cijkl ðxn Þ

@2 @ 2 ui ðxn ; tÞ uk ðxn ; tÞ þ Fi ðxn ; tÞ ¼ rðxn Þ @xj @xl @2t

ð4:178Þ

4.9 CALCULATION OF GREEN’S FUNCTIONS IN TRANSVERSELY ISOTROPIC

219

where Cijkl ðxn Þ are the material constants, ui are displacement components, Fi given in Eq. (4.8) denotes the body force per unit volume, and i; j; k; l, and n take values 1,2, and 3. In the subsequent formulation, all the subscripts correspond to usual index notation in three-dimensional space as in Section 4.2. dij is the Kronecker delta symbol. In calculations of the Green’s functions, inhomogenity of the material can be incorporated as done by Manolis et al. (2005). Let us consider the material to be homogeneous and Cijkl ¼ Cjikl ¼ Cklij ¼ Cikjl to satisfy the symmetry condition. The external force in Eq. (4.8) can be considered as an impulsive force at the origin with magnitude T. Substituting this Fi in Eq. (4.178) we get Cijkl

@2 @ 2 ui ðxn ; tÞ uk ðxn ; tÞ þ Ti f ðtÞdðxn Þ ¼ r @2t @xj @xl

ð4:179Þ

The Green’s function for different materials can be solved from the above equation following different techniques. In the following sections, brief descriptions of the most convenient ways to calculate Green’s functions are presented for transversely isotropic

Figure 4.27 Vertical displacements near the horizontal central planes of four different corrugated waveguides (dimensions are given in Table 4.2 ) for three different striking angles (shown in Fig. 4.24): (a) normal incidence in waveguides 1 and 2, (b) normal incidence in waveguides 3 and 4, (c) 30 incidence in waveguides 1 and 2, (d) 30 incidence in waveguides 3 and 4, (e) 45 incidence in waveguides 1 and 2, (f) 45 incidence in waveguides 3 and 4. As expected, parts (a) and (b) show symmetric response for normal incidence. For inclined incidence, parts (c) and (e) show strong wave propagation in the forward direction for small corrugation (waveguides 1 and 2), whereas parts (d) and (f) show strong backward direction wave propagation for large corrugation (waveguides 3 and 4). Readers should not be confused by the notation u2 in this ﬁgure for vertical component of displacement, it is same as the displacement component in the x3 direction of Figure 4.23.

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Figure 4.27 (Continued)

and general anisotropic materials. Irrespective of the methods, without any body force the plane wave solution to the governing differential equation generates Christoffel equation. For any material, Christoffel equation can be written as (Auld, 1990) ik gk ¼ rc2 gi

ð4:180Þ

where ik ¼ Cijkl nj nl ; gi is the amplitude vector, and nm is the directional vector.

221

4.9 CALCULATION OF GREEN’S FUNCTIONS IN TRANSVERSELY ISOTROPIC

Figure 4.27 (Continued)

The governing differential equation with Dirac delta force can be written as @2 @ 2 Gip ðxn ; tÞ Gkp ðxn ; tÞ þ dip dðtÞdðxn Þ ¼ r @xj @xl @2t @2 @2 rdik 2 Gkp ðxn ; tÞ ¼ dip dðtÞdðxn Þ or Cijkl @xj @xl @t

Cijkl

where Gij ðxn ; tÞ is the displacement Green’s function in the solid.

ð4:181Þ

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ADVANCED APPLICATIONS OF DISTRIBUTED POINT SOURCE

Figure 4.27 (Continued)

Transversely isotropic and general anisotropic solid materials have one difference. Transversely isotropic material has a preferential direction and thus one of the two quasi transverse waves in anisotropic solids is purely transverse horizontally polarized wave (Auld, 1990). The calculation of displacement Green’s function in general anisotropic materials is much more complicated than that for isotropic materials. Solution from the Fourier transform of the governing differential equation is computationally inefﬁcient and not of much use for the DPSM formulation. Several researchers came up with different ideas to calculate Green’s functions for anisotropic solids in the past decades. All these methods have one thing in common — the governing partial differential equation is transformed to a one-dimensional wave equation by applying Radon transform (John, 1955; Dean, 1983). The beauty of Radon transform is that it transforms the partial differential equation to a onedimensional ordinary differential equation. After solving the one-dimensional problem, the solution follows from the application of the inverse Radon transform to solve for the time domain displacement Green’s functions in anisotropic solids. 4.9.2 Radon transform An overview of Radon transform relevant to our application is presented in this section. More detail on Radon transform can be found in John (1955), Helgason (1980), Dean (1983), among others. Let us consider an Euclidean space <3 , with x1 ; x2 , and x3 being the coordinates of a point in space. If f ðxÞ is an arbitrary function in <3 , the Radon transform of the function can be written as Z ^f ðs; nÞ ¼ Rff ðxÞg ¼ f ðxÞdðs n xÞdx ð4:182Þ where n is a unit vector.

4.9 CALCULATION OF GREEN’S FUNCTIONS IN TRANSVERSELY ISOTROPIC

223

Radon transform is an integral transform that represents the integration of a function over all the planes deﬁned by n x ¼ s. This concept is reminiscent toward the integration of a wave function kernel over the surface of a unit sphere to account for all possible directions of wave propagation in any solid medium. 4.9.3 Basic properties of Radon transform The homogeneity property of Radon transform can be written as ^f ðas; anÞ ¼ jaj1^f ðs; nÞ

ð4:183Þ

The Radon transform is linear and the linearity property can be written as Rðaf1 ðxÞ þ bf2 ðxÞÞ ¼ a^f1 ðs; nÞ þ b^f2 ðs; nÞ

ð4:184Þ

Transformation of derivatives are given by

@^f ðs; nÞ @s 2 @ f @ 2^f ðs; nÞ ¼ ni n j R @xi @xj @s2 R

@f @xi

¼ ni

ð4:185Þ ð4:186Þ

The inverse Radon transform from the above equation can be written as ( f ðxÞ ¼ R

1

@ 2^f ðs; nÞ @s2

!)

1 ¼ 2 8p

Z jnj¼1

@ 2^f ðs; nÞ @s2

dðnÞ

ð4:187Þ

s¼nx

where dðnÞ is an elemental surface of the unit sphere deﬁned by jnj ¼ 1. Let us consider a unit sphere. At polar angle y and azimuthal angle f, the unit normal direction can be written as n ¼ fsin y cos f; sin y sin f; cos yg. Therefore, n x ¼ x1 sin y cos f þ x2 sin y sin f þ x3 cos y and Eq. (4.187) can be rewritten as ( !) @ 2^f ðs; nÞ 1 f ðxÞ ¼ R @s2 1 ¼ 2 8p

Z2p

Zp df

0

0

@ 2^f ðx1 sin y cos f þ x2 sin y sin f þ x3 cos y; nÞ sin y dy @s2 ð4:188Þ

This integration can be performed numerically, analytically, or approximately by ﬁnding the leading order behavior of the integral depending on the function inside the integral equation.

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4.9.4 Displacement and stress Green’s functions Applying the Radon transform, the transformed one-dimensional wave equation can be written as @2 @2 ^ ik 2 rdik 2 Gkp ðs; tÞ ¼ dip dðsÞdðtÞ @s @t

ð4:189Þ

^ kp ðs; tÞ ¼ 0 for t < 0, where ik ¼ Cijkl nj nl . and the initial condition is G Here, ik is not a diagonal matrix but a symmetric positive deﬁnite matrix. The eigen values of this matrix are the same as the roots from Christoffel’s equation (Eq. (4.190)). Three roots are three wave slowness values (ðcM Þ1 ) (Auld, 1990) corresponding to one quasi-longitudinal and two quasi-transverse waves. For a particular direction of propagation on the plane wave, there are three distinct roots with three different eigen vectors that correspond to the amplitude vectors of the wave. The eigen vectors for the transversely isotropic material can be written explicitly (Tverdokhlebov and Rose, 1988), whereas for general anisotropic material the Christoffel’s equation need to be solved for a particular direction of propagation. Let eigen vectors or projection operators (Wang and Achenbach, 1994) or polarization vectors (Tverdokhlebov and Rose, 1988) for any material be designated as gM k , where Mð 3Þ corresponds to different modes as described earlier. Applying the concepts ofPgeneralized Helmholtz decomposition, the eigen vectors have the M properties 3M¼1 gM i gk ¼ dik (Brodia and Williamson, 1989) . Solving the onedimensional wave equation after applying the above-mentioned properties and carrying out the inverse Radon transform, the displacement Green’s function in time domain can be obtained. Considering the source at the origin, the time domain displacement and stress Green’s functions can be written as (Wang and Achenbach, 1993)

Gij ðx; tÞ ¼

sqij ðx; tÞ ¼

3 HðtÞ X 8p2 M¼1

Z jnj¼1

3 Cijkl HðtÞ X 8p2 M¼1

M gM i gj @ ½dðcM t nk xk Þdðnp Þ rc2M @t

Z jnj¼1

M 2 nl gM k gq @ ½dðcM t np xp Þdðnm Þ rc3M @t2

ð4:190Þ

ð4:191Þ

The kernels in Eqs. (4.190) and (4.191) present the plane wave generation at plane np xp at t ¼ 0 along direction n. Hence, the integration over a unit sphere (jnj ¼ 1) is nothing but the superposition (John, 1955; Dean, 1983) of all possible plane waves propagating in all directions. These integrations are 2D surface integrals over the surface of a sphere and can be carried out numerically. However, we are interested in the frequency domain Green’s functions. By taking the Fourier transform of Eqs. (4.190) and (4.191), the frequency domain displacement and stress Green’s

225

REFERENCES

functions can be obtained. Gij ðx; 0; oÞ ¼

þ

Z

i 8p2 1 8p2

jnj¼1

Z

jnj¼1

Zþ1 sqij ðx; 0; oÞ ¼ 1

2

3 k gM gM X M i j M¼1

2rc2M

3 gM gM X i j M¼1

rc2M

expðiðkM np xp ÞÞdðnm Þ

dðnp xp Þdðnm Þ

Z 3 6Cijkl HðtÞ X 4 8p2 M¼1

jnj¼1

3 M 2 nl gM g 7 k q @ ½dðcM t np xp Þdðnm Þ5eiot dt rc3M @t2

ð4:192Þ

ð4:193Þ

where jxj ¼ r. The frequency domain stress Green’s function can be calculated numerically integrating Eq. (4.193).

REFERENCES All publications referred in the text as well as some additional references related to this topic are listed below. Achenbach, J.D., and H.I., Epstein, ‘‘Dynamic Interaction of a Layer and a Half-space’’, Journal of the Engineering Mechanics Division, Proceedings of ASCE, Vol. 93, pp. 27–42, 1967. Ahmad, R., T., Kundu, and D., Placko, ‘‘Modeling of Phased Array Transducers’’, Journal of the Acoustical Society of America, Vol. 117, pp. 1762–1776, 2005. Auld, B.A., Acoustic Fields and Waves in Solids, Vols. I & II, John Wiley & Sons, New York, 1973. Auld., B.A., ‘‘Acoustic Fields and Waves in Solids’’, John Wiley & Sons, New York, Vol. I & II., 1990 Banerjee, S., and T., Kundu, ‘‘Elastic Wave Propagation in Symmetrically Periodic Sinusoidal Waveguide’’, Health Monitoring and Smart Nondestructive Evaluation of Structural and Biological Systems, Proceedings of SPIE, Vol. 5394, pp. 89–98, 2004. Banerjee, S., ‘‘Elastic Wave Propagation in Corrugated Wave Guides’’, PhD Dissertation, University of Arizona, Tucson, Arizona, USA, 2005. Banerjee, S., and T., Kundu, ‘‘Elastic Wave Propagation in Sinusoidally Corrugated Wave Guides’’, Journal of the Acoustical Society of America, Vol. 119 (4), pp. 2006–2017, 2006. Banerjee, S., and T., Kundu, ‘‘Symmetric and Anti-symmetric Rayleigh-Lamb Modes in Sinusoidally Corrugated Waveguides: An Analytical Approach’’, International Journal of Solids and Structures, Vol. 43, pp. 6551–6567, 2006a. Banerjee, S., T., Kundu, and N., Alnuaimi, ‘‘DPSM Technique for Ultrasonic Field Modelling near Fluid-Solid Interface’’, Ultrasonics, in press, 2007. Banerjee, S., and T., Kundu, ‘‘Ultrasonic Field Modelling in Plates Immersed in Fluid’’, International Journal of Solids and Structures, under reivew, 2007. Banerjee, S., and T., Kundu, ‘‘Scattering of Ultrasonic Waves by Internal Anomalies in Plates Immersed in a Fluid’’, (Health Monitoring and Smart Nondestructive Evaluation of Structural and Biological Systems, Ed. T. Kundu, SPIE’s 11th Annual International Symposium on NDE for Health Monitoring and Diagnostics, San Diego, California, February 26 – March 2, 2006 SPIE Vol. 6177) Optical Engineering Journal, in press, 2007a.

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Banerjee, S. and T. Kundu, ‘‘Semi-Analytical Modeling of Ultrasonic Fields in Solids with Internal Anomalies Immersed in a Fluid’’, Wave Motion, under review, 2007b. Bertoni, H. L., and T., Tamir, ‘‘Uniﬁed Theory of Rayleigh-Angle Phenomena for Acoustic Beams at Liquid-Solid Interface’’, Applied Physics Letters, Vol. 2, pp. 157–172, 1973. Biondi, B., Solving the Frequency-Dependent Eikonal Equation, 62nd Annual International Meeting, Society of Experimental Geophysics, Expanded Abstracts, pp. 1315–1319, 1992. Bostro¨m, A., ‘‘Passbands and Stopbands for an Electromagnetic waveguide with a periodically varying cross section’’, IEEE Transactions on Microwave Theory and Techniques, Vol. 31, pp. 752–756, 1983. Bostro¨m, A., ‘‘Propagating, Damped, and Leaky Surface Waves on the Corrugated TractionFree Boundary of an Elastic Half-Space’’, The Journal of the Acoustical Society of America, Vol. 85, pp. 1549–1555, 1989. Brekhovskikh, L.M., Waves in Layered Media, Academic Press, 1960. Brodia., J., Williamson, S.G., ‘‘A comprehensive introduction to linear algebra’’, AddisonWesley, Redwood city, 1989. Burridge, R., P., Chandwick, and A.N., Norris, ‘‘Fundamental Elastodynamic Solutions for Anisotropic Media With Ellipsoidal Slowness Surfaces’’, Proceedings of the Royal Society of London, Vol. 440A, 655–681, 1993. Cerveny´, V., and I., Psencik, ‘‘Gaussian Beams and Paraxial Ray Approximation in ThreeDimensional Elastic Inhomogeneous Media, Journal of Geophysics, Vol. 53, pp. 1–15, 1983. Cerveny´, V., ‘‘The Application of Ray Tracing to the Numerical Modeling of Seismic Waveﬁelds in Complex Structures, Seismic Shear Waves, Part A, Theory, pp. 1–124, 1985. Cerveny´, V., ‘‘Ray Tracing Algorithms in Three-Dimensional Laterally Varying Layered Structures’’, in Seismic Tomography Ed. G. Nolet, Riedel Publishing Co., pp. 99–134, 1987. ¨ ber die Fortpﬂanzung von Sto¨ssen durch elastiche feste Ko¨rper’’, Annals Chrinstoffel, E.G., ‘‘U of Discrete Mathematics, Vol. 8, pp. 193–243, 1977. Deans., S.R., ‘‘The Radon Transform and Some of Its Applications’’, Wiley-Interscience Publication, 1983. Declercq, N. F., J., Degrieck, R., Briers, and O., Leroy, ‘‘Diffraction of Homogeneous and Inhomogeneous Plane Waves on a Doubly Corrugated Liquid/Solid Interface’’, Ultrasonics, Vol. 43 (8), pp. 605–618, 2005. Dravinski, M., and T.K., Mossessian, ‘‘Scattering of Plane Harmonic P, SV, and Rayleigh Waves by dipping layers of arbitrary shape’’, Bulletin of the Seismological Society of America, Vol. 77, pp. 212–235, 1987. Duff, G.F.D., ‘‘The Cauchy Problem for Elastic Waves in an Anisotropic Medium’’, Philosophical Transactions of the Royal Society of London, Vol. 252A, pp. 249–273, 1960. El-Bahrawy, A., ‘‘Stopbands and Passbands for Symmeetric Rayleigh-Lamb Modes in a Plate with Corrugated Surfaces’’, Journal of Sound and Vibration, Vol. 170(2), pp. 145– 160, 1994a. El-Bahrawy, A., ‘‘Point Force Excitation of Surface Waves Along the Doubly Corrugated Traction-Free Boundary of an Elastic Half-Space’’, The Journal of the Acoustical Society of America, Vol. 96, pp. 3167–3176, 1994b. Ewing, W.M., Jardetsky, W.S., and F., Press, Elastic Waves in Layered Media, McGraw-Hill, New York, 1957. Fokkemma, J.H., ‘‘Reﬂection and transmission of Elastic Waves by the Spatially Periodic Interface Between Two Solids (Theory of Integral-Equation Method)’’, Wave Motion, Vol. 2, pp. 375–393, 1980.

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Graff, K.F., Wave Motion in Elastic Solids, Dover Publication, New York, 1991. Hah, Z. G., and K.M., Sung, ‘‘Effect of Spatial Sampling in the Calculation of Ultrasonic Fields Generated by Piston Transducers’’, Journal of the Acoustical Society of America, Vol. 92, pp. 3403–3408, 1992. Harris, G. R., ‘‘Review of Transient Field Theory for a Bafﬂed Planar Piston’’, Journal of the Acoustical Society of America, Vol. 70, pp. 10–20, 1981. Helgason, S., ‘‘The Radon Transform’’, Birkhauser, BostonBasel Stuttgart, 1980. Ingenito, F. and B.D., Cook, ‘‘Theoretical Investigation of the Integrated Optical Effort Produced by Sound Field Radiated from Plane Piston Transducers’’, Journal of the Acoustical Society of America, Vol. 45, pp. 572–577, 1969. Jensen, J. A., and N. B., Svendsen, ‘‘Calculation of Pressure Fields from Arbitrary Shaped, Apodized, and Excited Ultrasound Transducers’’, IEEE Transactions on Ultrasonics, Ferroelectric and Frequency Control, Vol. 39, pp. 262–267, 1992. John, F., ‘‘Plane Waves and Spherical Means Applied to Partial Differential Equations’’, Interscience, New York, 1955. Kennett, B.L.N., Seismic Wave Propagation in Stratiﬁed Media, Cambridge University Press, London, 1983. Kundu, T., Ed., Chapters 2 and 4 Ultrasonic Nondestructive Evaluation: Engineering and Biological Material Characterization, CRC Press, Boca Raton, Florida, 2004. Kundu, T., J.P., Lee, C., Blase, and J., Bereiter-Hahn, ‘‘Acoustic Microscope Lens Modeling and its Application in Determining Biological Cell Properties from Single and MultiLayered Cell Models’’, Journal of the Acoustical Society of America, Vol. 120 (3), pp. 1646–1654, 2006. Lee, J. P., Placko, D., Alnuamaini, N., and T., Kundu, ‘‘Distributed Point Source Method (DPSM) for Modeling Ultrasonic Fields in Homogeneous and Non-Homogeneous Fluid Media in Presence of an Interface’’, First European Workshop on Structural Health Monitoring, Ecole Normale Superieure de Cachan, France, Ed. D. L. Balageas, Pub. DEStech, PA, USA, pp. 414–421, 2002. Lerch, T. P., L. W., Schmerr, and A., Sedov, ‘‘Ultrasonic Beam Models: An Edge Element Approach’’, Journal of the Acoustical Society of America, Vol. 104, pp. 1256– 1265, 1998. Li, H. U, and K., Negishi, ‘‘Visualization of Lamb Mode Patterns in a Glass Plate’’, Ultrasonics, Vol. 32(4) pp. 243–256, 1994. Liu, G.R., and Xi, Z.C., Elastic Waves in Anisotropic Laminates, CRC Press, 2002. Lockwood, J. C., and J. G., Willette, ‘‘High-Speed Method for Computing the Exact Solution for the Pressure Variations in the Near Field of a Bafﬂed Piston’’, Journal of the Acoustical Society of America, Vol. 53, pp. 735–741, 1973. Mal, A. K., and Singh, S. J., Deformation of Elastic Solids, Prentice Hall: Englewood Cliffs, New Jersey, 1991. Manolis. G.D., Rangelov., T.V., Dineva., P.S., ‘‘Elastodynamic Fundamental Solutions for 2D Inhomogeneous Anisotropic Domains’’, Proc. 5th GRACM International Congress on Computational Mechanics, 2005. Morse, P. M., and Ingard, U. K., Theoretical Acoustics, McGraw-Hill, New York, 1968. Nayfeh, A.H., and O. A., Kandil, ‘‘Propagation Waves in Cylindrical Hard-Walled Ducts With Generally Weak Undulations’’, AIAA Journal, Vol. 16, pp. 1041–1045, 1978. Nayfeh, A.H., Wave Propagation in Layered Anisotropic Media with Applications to Solids, Elsevier, Amsterdam, 1995. Newberry, B. P., and R. B., Thompson, ‘‘A Paraxial Theory for the Propagation of Ultrasonic Beams in Anisotropic Solids’’, Journal of the Acoustical Society of America, Vol. 85, pp. 2290–2300, 1989.

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Payton, R.G., ‘‘Two-Dimensional Anisotropic Elastic Waves Emanating From Point Source’’, Proceedings of the Cambridge Philosophical Society, Vol. 70, pp. 191–210, 1971. Placko, D., and T., Kundu, ‘‘A Theoretical Study of Magnetic and Ultrasonic Sensors: Dependence of Magnetic Potential and Acoustic Pressure on the Sensor Geometry’’, Advanced NDE for Structural and Biological Health Monitoring, Proceedings of SPIE, Ed. T. Kundu, SPIE’s 6th Annual International Symposium on NDE for Health Monitoring and Diagnostics, March 4–8, Newport Beach, California, Vol. 4335, pp. 52–62, 2001. Placko, D., T., Kundu, and R., Ahmad, ‘‘Theoretical Computation of Acoustic Pressure Generated by Ultrasonic Sensors in Presence of an Interface’’, Smart NDE and Health Monitoring of Structural and Biological Systems, SPIE’s 7th Annual International Symposium on NDE and Health Monitoring and Diagnostics, San Diego, CA, Vol. 4702, pp. 157–168, 2002. Placko, D., T., Kundu, and R., Ahmad, ‘‘Ultrasonic Field Computation in Presence of a Scatterer of Finite Dimension’’, Smart NDE and Health Monitoring of Structural and Biological Systems, SPIE’s 8th Annual International Symposium on NDE and Health Monitoring and Diagnostics, San Diego, CA, Vol. 5047, pp. 169–179, 2003. Placko, D., and Kundu, T., ‘‘Chapter 2: Modeling of Ultrasonic Field by Distributed Point Source Method’’, Ultrasonic Nondestructive Evaluation: Engineering and Biological Material Characterization, Ed. T. Kundu, Pub. CRC Press, Boca Raton, Florida, pp. 144–201, 2004. Placko, D., N., Liebeaux, and T., Kundu, ‘‘Presentation d’une method generique pour La Modelisation des Capteurs de type Ultrasons’’, Magnetiques at Electrostatiques, Instrumentation, Mesure, Metrologie (I2M Journal): Evaluation Nondestructive, Vol. 1, pp. 101– 125, 2001. PZFlex Software, 2001, Version:1-j.6, Weilinger Associates, Inc. Rayleigh, J.W.S., The Theory of Sound, Dover, New York, Vol 2., 1945. Rayleigh, L., Theory of Sound, Dover, New York, Vol. 2, pp. 162–169, 1965. Rose, J.L., Ultrasonic Waves in Solid Media, Cambridge University Press, New York, 1999. Scarano, G., N., Denisenko, M., Matteucci, and M., Pappalardo, ‘‘A New Approach to the Derivation of the Impulse Response of a Rectangular Piston’’, Journal of the Acoustical Society of America, Vol. 78, pp. 1109–1113, 1985. Schmerr, L.W., Fundamental of Ultrasonic Nondestructive Evaluation-A Modeling Approach, Plenum Press, New York, 1998. Schmerr, L.W., ‘‘A Multigaussian Ultrasonic Beam Model for High Performance Simulations on a Personal Computer’’, Materials Evaluation, pp. 882–888, 2000. Schmerr, L.W., H.J., Kim, R., Huang, and A., Sedov, ‘‘Multi-Gaussian Ultrasonic Beam Modeling’’, Proceedings of the World Congress of Ultrasonics, pp. 93–99, WCU 2003, Paris, September 7–10, 2003. Sha, K., J., Yang, and W.S., Gan, ‘‘A Complex Virtual Source Approach for Calculating the Diffraction Beam Field Generated by a Rectangular Planar Source’’, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, Vol. 50, pp. 890–895, 2003. Spies, M., ‘‘Transducer-Modeling in General Transversely Isotropic Media Via Point-SourceSynthesis Theory’’, Journal of Nondestructive Evaluation, Vol. 13, pp. 85–99, 1994. Spies, M., ‘‘Elastic wave propagation in Transversely Isotropic Media II: The generalized Rayleigh-Function and an Integral Representation for the Transducer Field Theory’’, Journal of the Acoustical Society of America, Vol. 97, pp. 1–13, 1995. Spies, M., ‘‘Transducer Field Modeling in Anisotropic Media by Superposition of Gaussian Base Functions’’, Journal of the Acoustical Society of America, Vol. 105, pp. 633–638, 1999.

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5 DPSM FORMULATION FOR BASIC MAGNETIC PROBLEMS N. Liebeaux Institut Universitaire de Technologie, Cachan, France

D. Placko Ecole Normale Superieure, Cachan, France

5.1 INTRODUCTION It has been underlined in Chapter 1 that the distributed point source method (DPSM) method was ﬁrst introduced as a new technique for nondestructive testing (NDT). Previously ultrasonic waves (US) NDT or eddy currents (EC) NDT was of interest. In Chapters 3 and 4, the DPSM basics and advanced theory for US NDT have been described. Among other techniques, such as X-rays and thermal imaging, EC NDT still remains a good alternative, when electrically conductive materials are to be tested. EC NDT has many advantages: it usually has a good reliability; it is low cost and quite easy to use. An EC automatic testing process can be quick and efﬁcient. EC NDT is based upon the Faraday’s induction law, stating that eddy currents appear in an electrically conductive material as soon as this material is placed in a time-varying magnetic ﬁeld. According to the Lenz’s law, these eddy currents create an induced magnetic ﬁeld, which tends to nullify the inductive magnetic ﬁeld, as it is shown in Figure 5.1.

DPSM for Modeling Engineering Problems, Edited by Dominique Placko and Tribikram Kundu Copyright # 2007 John Wiley & Sons, Inc.

231

232

DPSM FORMULATION FOR BASIC MAGNETIC PROBLEMS

I(t) z

coil

V(t) Excitation magnetic field Induced magnetic field

Electrically conductive target

Eddy currents circulation

Figure 5.1 Problem geometry with interface.

This ﬁgure represents the simplest conﬁguration of an EC problem. A sensor, often made of an excitation coil, is placed over a conductive media, usually known as the target. The excitation must be time varying in order to create induced currents in the target. The excitation can be harmonic, made of a single frequency (monofrequency excitation) or a few frequencies (simultaneous multifrequency excitation). The impedance of the sensor is measured; this impedance depends on the EC circulation in the target. Thus, impedance variations can lead to interesting properties of the target, such as its conductivity, the detection of a defect inside the target (which can be seen as a local conductivity variation), or the lift-off (the sensor/target distance). The impedance measurement is always achieved by a synchronous detection, in order to get back to the real and imaginary parts of the impedance. For multifrequency excitation, one has as many synchronous detections as testing frequencies; the electronic measurement circuit is therefore more complex. The interaction between the sensor (creating the excitation ﬁeld) and the target (modifying this ﬁeld) is an electromagnetic phenomenon perfectly well known since James Clerk Maxwell published its theory and its famous equations in 1865. As a matter of fact, any modeling of sensor/target interaction is theoretically studied as a particular case of the Maxwell equations. Unfortunately, a geometry as simple as the one described in Figure 5.1 is not very useful. Practical cases are much more complicated. For example, one can deal with symmetryless targets, with multiple excitation coils, with shielding, and so on. These examples show that analytical solutions to Maxwell equations cannot always be obtained. Nowadays, modeling computer-assisted methods, such as the ﬁnite elements (FE) method, the boundary elements (BE) method, or the boundary integral (BI) method, are widespread and the associated softwares cannot be avoided. In this chapter, the DPSM basics for magnetic problems are presented, by introducing current sources.

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5.2 DPSM FORMULATION FOR MAGNETIC PROBLEMS 5.2.1 The Biot–Savart law as a DPSM current source deﬁnition This law was ﬁrst described in 1820 by French physicists Jean–Baptiste Biot and Fe´lix Savart. The Biot–Savart law is as fundamental for magnetostatic as the Coulomb law is for electrostatic (Perez, 1997). It gives the magnetic ﬁeld induced by a ﬂowing line DC current: ~ ðMÞ ¼ dH

I ~ ~ r dl ^ 3 4p r

ðAÞ

~ the current ~ stands for the magnetic ﬁeld, I the current through the line, dl where H element, and ~ r the distance vector between the observation point M and the element ~ It is important to notice that the dH ~ ðMÞ is deﬁned by a cross product, meaning that dl. dl ~ and ~ ~ ðMÞ is orthogonal to dl dH r. If the DPSM is to be applied on magnetic systems, current sources must be deﬁned the same way point sources have been introduced in Chapter 1. The DPSM matrix formulation will still be correct. According to the Biot–Savart law and DPSM formulation, a current source is represented in Figure 5.2. This current source can be physical (when representing the current ﬂowing through an excitation coil) or ﬁctitious (when representing assumptions on boundary conditions). The next three sections will check the validity of these current sources, using MATLAB software. 5.2.1.1 Wire of inﬁnite length A current I is ﬂowing through an inﬁnite wire. The amplitude H of the magnetic ﬁeld created by the wire, at a distance r, is given by H¼

I 2p r

ðBÞ

Potential contour lines Radiated field H Vector from the element to the field r Observation point M I.dl Current source

Boundary condition bubble

Figure 5.2 A current source.

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DPSM FORMULATION FOR BASIC MAGNETIC PROBLEMS

Magnetic field created by an infinite wire 350 DPSM Analytical formula

300

Hy component

250

200

150

100

50

0

0

1

2

3

4

5 X (mm)

6

7

8

9

10

Figure 5.3 Magnetic ﬁeld component Hy for an inﬁnite wire.

In the MATLAB simulation, the inﬁnite wire is placed on the z-axis. Its ‘‘inﬁnite’’ length is 100 mm, from z ¼ þ50 mm to z ¼ 50 mm. Figure 5.3 represents the amplitude H of the magnetic ﬁeld calculated on the x-axis, from x ¼ 0. The inﬁnite ~ Because the elements are located on wire has been cut into 1000 current elements dl dl. the z-axis, and the observation point on the x-axis, the magnetic ﬁeld vector has only the y component, Hy. Error analysis shows that the relative mean square error is less than 0.7% when the number of elements is greater than 1000. For 100 elements, the error is 1.8%, which is acceptable. 5.2.1.2 Current loop This is also a ‘‘classic’’ exercise found in every electromagnetic book. The current I ﬂows through a perfect circle loop of radius r. The amplitude H of the on-axis magnetic ﬁeld, at the elevation z, is given by

H¼

I:r 2 3

2p ðr 2 þ z2 Þ2

ðCÞ

Results are plotted in Figure 5.4. The radius r is equal to 10 mm for plotting. For 50 current elements, the relative mean square error is 0.3%.

235

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On-axis magnetic field Hz component created by a current loop 50 DPSM Analytical formula

45 40

Hz component

35 30 25 20 15 10 5 0 –40

–30

–20

–10

0 Z (mm)

10

20

30

40

Figure 5.4 On-axis magnetic ﬁeld component Hz for a current loop.

5.2.2 Current loops above a semi-inﬁnite conductive target Previous sections have validated the introduction of DPSM current sources for generating magnetic ﬁelds in free space. This section introduces a semi-inﬁnite inﬁnitely conductive target below a DPSM current sources loop, as shown in Figure 5.5. DPSM current loop V(t)

I(t)

z Inductive magnetic field

Induced magnetic field

Semi-infinite electrically conductive target

Surface eddy currents circulation

Figure 5.5 Current loop above a semiinﬁnite conductive target.

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DPSM FORMULATION FOR BASIC MAGNETIC PROBLEMS

current loop Unit magnetic field in the YZ plane 5 0 –5

z-axis

–10 –15 –20

target

–25 –30 –35 –50

–40

–30

–20

–10

0 y-axis

10

20

30

40

50

Figure 5.6 Current loop above a semiinﬁnite conductive target.

In this case, eddy currents are induced in the target. Due to inﬁnite conductivity, these currents ﬂow at the target surface and not within the target. The surface total magnetic ﬁeld is hence tangential and has no normal component. The inﬁnitely conductive target is said to have an reﬂection coefﬁcient equal to þ1. Figure 5.6 represents the magnetic ﬁeld in the symmetry plane x ¼ 0. In order to plot arrows of same length, the magnetic ﬁeld has been normalized. As expected, the magnetic ﬁeld is tangential at the target surface. The target thickness does not matter because no magnetic ﬁeld penetrates it. For this simulation, a 1 A current ﬂows through a 3 cm radius current loop. The target is located at the altitude of 3 cm, where the loop is located at the 0 altitude. The target is 10 cm long by 10 cm wide. 5.2.3 Current loops above a semi-inﬁnite magnetic target In this application, the target attracts the magnetic ﬁeld, and only its normal component exists at the target surface. Geometry is the same as described in 5.2.2, but the reﬂection coefﬁcient is now equal to 1 (inﬁnite magnetic permeability). As expected, the magnetic ﬁeld plotted in Figure 5.7 has no tangential component at the target surface. The DPSM method has now the ability to deal with inﬁnite permeability magnetic targets. The next section will look into DPSM simulation of current loops and magnetic cores.

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current loop Unit magnetic field in the YZ plane 5

0

–5

z-axis

–10

–15

target –20

–25

–30 –60

–40

–20

0 y-axis

20

40

60

Figure 5.7 Current loop above a semiinﬁnite conductive target.

5.2.4 Current loop circling a magnetic core 5.2.4.1 Geometry This section presents a comparison between a DPSM modeling and an ANSYS modeling (ANSYS1 is a ﬁnite element software). The geometry is described in Figure 5.8, where all dimensions are indicated. This modeling is noteworthy for two main reasons: The studied geometry is a closed surface totally meshed, which makes it different from the previous examples. The boundary condition, ﬁxed on the magnetic scalar potential, varies linearly on the z-axis, between the south pole and the north pole. To achieve this modeling using DPSM, the bar is divided into three surfaces: The north pole surface, that is, the upper disk and the upper one third of the bar. The south pole surface, that is, the lower disk and the lower one third of the bar. The middle one-third part of the bar.

1

ANSYS ia a product of the Ansoft company.

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DPSM FORMULATION FOR BASIC MAGNETIC PROBLEMS

z

North pole

magnetic bar μr = ∞

θ

Iexc

r

o h

height = 3 cm r2 = 4 mm

r2 South pole

Figure 5.8 Magnetic bar circled by current loops.

5.2.4.2 DPSM formulation The bar can be divided into three active surfaces: the upper disk, the lower disk, and the cylinder surface. We have placed N ¼ 500 point sources, which have been equally spaced on the three surfaces of the bar. The boundary conditions applied, described in ﬁgure 5.9, are the following: The upper disk point source potential U is set to a ¼ þ0:5 A. The same condition is applied on the point sources located on the upper one third of the cylinder. The lower disk point source potential L is set to b ¼ 0:5 A. The same condition is applied on the point sources located on the lower one third of the cylinder. The point sources located between þh=6 and h=6 (the middle of the cylinder where current density excitation is applied) are characterized by their potential C ðzÞ ¼ gðzÞ gradually increasing from þ0:5 A to 0:5 A, in order to take into account the real excitation conditions due to the excitation windings.

+0.5 A

z-axis Some points sources on the upper disk and the cylinder surfaces

Magnetic scalar potential θ

h/3

r2 –0.5 A

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5.2 DPSM FORMULATION FOR MAGNETIC PROBLEMS

The active surfaces are discretized into elementary surfaces dS. At the center Ck of the kth elemental surface dS a point source emitting a ﬂux jk is placed. From the conservation of the magnetic ﬂux, one can write that the summation of the ﬂux emitted by all point sources is equal to zero. X jCk ¼ 0 ð5:1Þ j¼ active surface

The magnetic potential at a given point M in the space is obtained by considering the contribution of all magnetic point sources. X yC k ð5:2Þ y¼ active surface

General relations between the magnetic ﬁeld and the magnetic scalar potential are given by Z Z ~ ~ dS ~ ¼ 1 dy and j ¼ ð5:3Þ H m0 H m d~ R It yields in our case, for the kth point source Hk ¼

jk 2pm0 R2

and

yk ¼

jCi f ðPk Ci Þ

with

jk 2pm0 R

ð5:4Þ

So, yk ¼

X

f ðRÞ ¼

all sources

1 2pm0 R

ð5:5Þ

For example, if we consider one point source on each lower disk (L) and upper disk (U) and three point sources on the cylinder surface (C), Eq. (5.5) is rewritten below using a matrix notation, with p sufﬁx for peak and c sufﬁx for center: 0 1 0 3 0 1 1 2 yUp1 FUp1Uc1 FUp1Lc1 FUp1Cc1 FUp1Cc2 FUp1Cc3 þa jUc1 B yLp1 C B b C 6 FLp1Uc1 FLp1Lc1 FLp1Cc1 FLp1Cc2 FLp1Cc3 7 B jLc1 C B C B 7 B C C 6 B yCp1 C ¼ B þgðzÞ C ¼ 6 FCp1Uc1 FCp1Lc1 FCp1Cc1 FCp1Cc2 FCp1Cc3 7 B jCc1 C B C B 7 B C C 6 @ yCp2 A @ þgðzÞ A 4 FCp2Uc1 FCp2Lc1 FCp2Cc1 FCp2Cc2 FCp2Cc3 5 @ jCc2 A yCp3 FCp3Uc1 FCp3Lc1 FCp3Cc1 FCp3Cc2 FCp3Cc3 jCc3 þgðzÞ ð5:6Þ This system of equations with nine submatrices can be rewritten as U ¼ FUU U þ FUL L þ FUC C L ¼ FLU U þ FLL L þ FLC C C ¼ FCU U þ FCL L þ FCC C

ð5:7Þ

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DPSM FORMULATION FOR BASIC MAGNETIC PROBLEMS

So that ¼F

ð5:7bÞ

Inversion of Eq. (5.7) leads to the expressions of ji . 0

2 1 GUp1Uc1 jUc1 B jLc1 C 6 GLp1Uc1 B 6 C 6 C ¼B B jCc1 C ¼¼ 6 GCp1Uc1 @ jCc2 A 4 GCp2Uc1 GCp3Uc1 jCc3

GUp1Lc1 GLp1Lc1 GCp1Lc1 GCp2Lc1 GCp3Lc1

GUp1Cc1 GLp1Cc1 GCp1Cc1 GCp2Cc1 GCp3Cc1

GUp1Cc2 GLp1Cc2 GCp1Cc2 GCp2Cc2 GCp3Cc2

3 0 1 GUp1Cc3 þa B C GLp1Cc3 7 7 B b C B þg C GCp1Cc3 7 7 B C GCp2Cc3 5 @ þg A GCp3Cc3 þg ð5:8Þ

The condition on ﬂux (Eq. (5.1)) in combination with Eq. (5.8) gives a new equation to determine the values of a, b, and gðzÞ. jU þ jL þ jC ¼ a

X i;j

Gij b

X k;j

Hkl þ g

X

Imn ¼ 0

m;n

5.2.4.3 Results Figure 5.10 represents the magnetic scalar potential y outside the bar, where x varies from r2 to 3 r2 , and z from h=2 to þh=2. The decrease of y is noticeable, for large values of x: the y limit being 0 for x value approaching inﬁnity. On the cylinder surface, two equal steps, with a linear decrease joining them, stand for the boundary condition applied on y, on the point sources located on the outside cylinder surface.

Figure 5.10 Magnetic scalar potential y outside the bar.

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5.2 DPSM FORMULATION FOR MAGNETIC PROBLEMS

70

60

50

40 o r

h 30

2 20

10

0 –0.06

–0.04

–0.02

0

0.02

0.04

0.06

Figure 5.11 Hz component as a function of height z (position 1).

5.2.5 Finite elements simulation—comparisons The geometry has been studied using ANSYS ﬁnite elements (FE) software (ANSYS), in order to compare the DSPM results. Three positions are then examined. The graph of Figure 5.11 displays a good likeness between both curves. The average error never exceeds 2 A/m. As expected, with inﬁnite magnetic permeability mr, induction ﬁeld H is null. As soon as the height z gets closer to þh=2 or h=2, that is, 1:5 cm, the point where H is calculated is nearer to the point sources located on the upper/lower disks. That means that distance R (see Eq. (2.4)) is becoming smaller as the H ﬁeld approaches inﬁnity. Position 2 is along an axis parallel to z-axis, for a radius value of r ¼ r2 þ 1 mm. Figures 5.12 and 5.13 represent components Hx and Hz for this radius. Average errors between the DPSM curves and the FE results are 1.5 A/m and 2.23 A/m, respectively. Position 3 is obtained for z ¼ 0. Hx and Hy components are zero; the only existing component is the Hz component, represented in Figure 5.14. Average error is 0.7 A/m. It can be interesting to notice that DPSM model gives higher results around the position z ¼ 5 mm. This is precisely an area where the DPSM mesh is not sufﬁcient enough to produce accurate results because of the radius value of the hemispherical source. The cylinder point sources are located on the outside cylinder surface, with a radius value of 0.54 mm. Eq. (5.4) clearly demonstrates that the point source and the apex must not be the same. Because ANSYS positions are used in the DPSM model to get a realistic comparison, the points in the area x ¼ 5 mm are much too close to the

242

DPSM FORMULATION FOR BASIC MAGNETIC PROBLEMS

150

100

50

0 o

h –50

r 2

–100

–150 –0.06

–0.04

–0.02

0

0.02

0.04

0.06

Figure 5.12 Hx component of the induction ﬁeld H as a function of height z (position 2).

60 40 20

o r 2

0 h –20 –40 –60 –80 –0.06

–0.04

–0.02

0

0.02

0.04

0.06

Figure 5.13 Hz component of the induction ﬁeld H as a function of height z (position 2).

243

5.3 CONCLUSION

10 0 –10 –20 –30 o r 2

h

–40 –50 –60 –70 –80 –90 0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Figure 5.14 Hz component of the induction ﬁeld H along the x-axis (position 3).

point sources, which gives higher values of ﬁeld and magnetic scalar potential than in the FE simulation.

5.3 CONCLUSION The DPSM model has been used to calculate the magnetic scalar potential y and the induction ﬁeld H created by a magnetic bar. The localization of point sources on each surface has allowed us to synthesize the value of y and H in the entire surrounding space. It has been demonstrated that this original modeling shows good accuracy, compared to the ﬁnite element simulation. The DPSM method is efﬁcient and quicker because only active surfaces are meshed. Finite element methods require the meshing of all the area, or volume, under test and can lead to time-consuming simulations in the case of complex geometries. In our case, as soon as closed surfaces are deﬁned, and boundary conditions expressed on them, the DSPM method can be applied. The results are obtained no matter how complex the geometry is: The only time-consuming operation lies in the inversion of matrix F, as deﬁned in Eqs. (5.7b) and (5.8). So, results can be obtained very fast as soon as closed surfaces are roughly meshed and no high precision is needed. Furthermore, this method is closer to the electromagnetic phenomena involved and leads to a better understanding of the problem because no harsh algorithms are involved. Even if the DSPM method does not cover every target/sensor conﬁguration, it presents a few advantages and can be seen as a powerful tool in NDE.

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5.4 REFERENCES ANSYS Inc.: ‘‘ANSYS Online Help’’, copyright 2006. Dufour I., and D., Placko, ‘‘An Original Approach of Eddy Current Problems Through a Complex Electrical Image Concept’’, IEEE Transactions on Magnetics, Vol. 32(2), pp. 348–365, 1996. Kundu, T., Chapter 1: Mechanics of Elastic Waves and Ultrasonic NDE, in Ultrasonic Nondestructive Evaluation: Engineering and Biological Material Characterization, Ed. T. Kundu, Pub. CRC Press, 2004. Lemistre, M., and D., Placko, ‘‘Proce´de´ et dispositif e´lectromagne´tique pour le controˆle de structures inte´gre´’’. Patent Application N BFF 040003 ENS Cachan/CNRS/ONERA, March 31, 2004, European extension March 31, 2005. Liebeaux, N., ‘‘Contribution a` la mode´lisation de capteurs e´lectromagne´tiques. Application au Controˆle Non Destructif par courants de Foucault’’, The`se de 3e`me cycle, universite´ Paris XI, november 22th, 2002. Pe´rez, J. P., Carles, R., and Fleckinger, R., ‘‘E´lectromagne´tisme, fondements et applications’’, 3e`me e´dition, MASSON, Paris, 1997. ISBN 2-225-83037-1 Placko, D., ‘‘Dispositifs d’analyse de proﬁl utilisant des capteurs a` courants de Foucault’’, The`se de 3e`me cycle, University Paris XI, April 16, 1984 (in French). Placko, D., H., Clergeot, and E., Santander, ‘‘Physical Modeling of an Eddy Current Sensor Designed for Real Time Distance and Thickness Measurement in Galvanisation Industry’’, IEEE Transactions on Magnetics, Vol. 25(4), pp. 2861–2863, 1989. Placko, D., ‘‘Contribution a` la conception de capteurs inductifs pour la robotique industrielle’’, Diploˆme d’Habilitation a` Diriger des Recherches en Sciences, University Paris XI, May 22, 1990 (in French). Placko, D., and T., Kundu, ‘‘A Theoretical Study of Magnetic and Ultrasonic Sensors: Dependence of Magnetic Potential and Acoustic Pressure on the Sensor Geometry’’, Advanced NDE for Structural and Biological Health Monitoring, Proceedings of SPIE, Ed. T. Kundu, SPIE’s 6th Annual International Symposium on NDE for Health Monitoring and Diagnostics, March 4–8, 2001, Newport Beach, California, Vol. 4335, pp. 52–62, 2001. Placko, D., T., Kundu, and R., Ahmad, ‘‘Distributed Point Source Method (DPSM) for Modeling Ultrasonic Fields in Homogeneous and Non-Homogeneous Fluid Media’’, Second International Conference on Theoretical, Applied, Computational and Experimental Mechanics, Kharagpur, India, December 27–30, 2001, Pub. Globenet Computers, India, Published in CD, Paper No.114, 2001. Placko, D., N., Liebeaux, and T., Kundu, ‘‘Presentation d’une methode generique pour La Modelisation des Capteurs de type Ultrasons, Magnetiques et Electrostatiques’’, Instrumentation, Mesure, Metrologie–Evaluation nondestructive, Vol. 1, pp. 101–125, 2001, (in French). Placko, D., T., Kundu, and R., Ahmad, ‘‘Theoretical Computation of Acoustic Pressure Generated by Ultrasonic Sensors in Presence of an Interface’’, Smart NDE and Health Monitoring of Structural and Biological Systems, Ed. T. Kundu, SPIE’s 7th Annual International Symposium on NDE for Health Monitoring and Diagnostics, March 18–21, 2002, San Diego, California, Vol. 4702, pp. 157–168, 2002.

5.4 REFERENCES

245

Placko, D., N., Liebeaux, and T., Kundu, ‘‘Proce´de´ pour e´valuer une grandeur physique representative d’une interaction entre une onde et un obstacle’’. Patent Application N 02 14108 ENS Cachan/CNRS/Universite´ d’Arizona, November 8, 2002. European and USA Extension November 10, 2003. Placko, D., and T., Kundu, Chapter 2: Modeling of Ultrasonic Field by Distributed Point Source Method, in Ultrasonic Nondestructive Evaluation: Engineering and Biological Material Characterization, Ed. T. Kundu, Pub. CRC Press, pp. 143–202, 2004. Placko, D., Kundu, T., and Ahmad, R., ‘‘Ultrasonic Field Computation in Presence of a Scatterer of Finite Dimension’’, Smart Nondestructive Evaluation and Health Monitoring of Structural and Biological Systems, Ed. T. Kundu, SPIE’s 8th Annual International Symposium on NDE for Health Monitoring and Diagnostics, March 3–5, 2003, San Diego, California, Vol. 5047, 2003. Schmerr, L. W., Fundamentals of Ultrasonic Nondestructive Evaluation: A Modeling Approach, Pub. Plenum Press, 1998.

6 ADVANCED MAGNETODYNAMIC AND ELECTROMAGNETIC PROBLEMS D. Placko Ecole Normale Superieure, Cachan, France

N. Liebeaux Institut Universitaire de Technologie, Cachan, France

6.1 INTRODUCTION The previous chapter was dealing with basic magnetostatic problems, with some assumptions for solving eddy current (EC) problems only in the case of inﬁnitely conductive targets. Thus, no magnetic ﬁeld or potential could be calculated in the target because no incident ﬁeld penetrates it. Important work was previously carried out (Dufour and Placko, 1996) to analytically solve the case of the interaction of a magnetodynamic ﬁeld with a metallic target by introducing a new concept of complex electrical images, but results were limited to simple geometries. One of the many advantages of the EC technique is the ability to detect cracks or ﬂaws inside a target, and it is widely used in industry. The formulation presented in Chapter 5 assumes well-known reﬂection coefﬁcients. It is not the case for ﬁeld computation in presence of a conductive non-homogeneous material. This chapter deals with an extension of the DPSM formulation, where Green’s sources are introduced in order to solve electromagnetic problems without heavy approximations, with the possibility to compute ﬁelds and potentials in all the media

DPSM for Modeling Engineering Problems, Edited by Dominique Placko and Tribikram Kundu Copyright # 2007 John Wiley & Sons, Inc.

247

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ADVANCED MAGNETODYNAMIC AND ELECTROMAGNETIC PROBLEMS

of the problem, based on the calculation of the magnetic vector potential ~ A. Then, electric and magnetic ﬁelds are derived from it and computed inside a target.

6.2 DPSM FORMULATION USING GREEN’S SOURCES 6.2.1 Green’s theory Green’s theory is one of the many numerical computation techniques used in the electromagnetism ﬁeld to solve nonhomogeneous wave equations (Ney) such as A þ k 2~ A ¼ m ~ J r2~

ð6:1Þ

where ~ A stands for the magnetic vector potential, ~ J the current density, and k2 ¼ o2 me josm

ð6:2Þ

e ¼ e0 er with e0 ¼ 1=36p 109 m ¼ m0 mr with m0 ¼ 4p 107 We may note that for a conductive medium, osm o2 me, which leads to s oe.ﬃ ﬃ qﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃ osm 2 2 . k ﬃ josm and k ﬃ ð1 jÞ 2 , which let the skin depth appear: d ¼ osm In these conditions, we have diffusion equations. On the contrary, if the medium has no conductivity, k2 ﬃ o2 me and pﬃﬃﬃﬃﬃ k ﬃ o em ¼ o=c, c being the wave celerity, we have propagation equations, and we can deﬁne the wavelength: l ¼ 2p=k ¼ 2pc=o. Green’s theory allows an integral formulation of the problem, which ﬁts nicely in the DPSM formulation where all the source contributions are summed on the observation point. Let Gð~ r ;~ r 0 Þ be the response to a Dirac pulse excitation to an Helmhotz equation and its associated boundary conditions, which can be written as r ;~ r 0 Þ þ k2 Gð~ r ;~ r 0 Þ ¼ dð~ r ~ r 0Þ r2 Gð~

ð6:3Þ

So the particular solution is r2 jð~ r Þ þ k2 jð~ r Þ ¼ f ð~ rÞ

ð6:4Þ

With j unknown, and f is the excitation. j is given by Z Z Z jð~ rÞ ¼

Gð~ r ;~ r 0 Þ f ð~ r 0 Þ dv0

ð6:5Þ

V

Gð~ r ;~ r 0 Þ is the Green’s function or elementary solution linked to the Eq. (6.4), and f is the source inside the volume v.

249

6.3 GREEN’S FUNCTIONS AND DPSM FORMULATION

6.2.2 Green’s function in free homogeneous space We now consider a free homogeneous space, where there is a current density ~ J ð~rÞ in a volume v. The magnetic vector potential is then the solution of Eq. (6.1). We can obtain from Eq. (6.3) the Green’s function in spherical coordinates (Ney). 1 @ 2 @GðRÞ ð6:6Þ R þ k2 GðRÞ ¼ dðRÞ R2 @R @R where R ¼k ~ r ~ r 0 k is the distance between the observation point and the excitation point. The Green’s function solution of Eq. (6.6) is GðRÞ ¼

ejkR 4pR

ð6:7Þ

This solution can be seen as a spherical wave, decreasing in R1 , and having a singularity at location point R ! 0. Now, the particular solution to excitation ~ J ð~r Þ can be obtained by applying the convolution integral (6.5). This solution, with the right side of Eq. (6.1) and the Green’s function (6.7), can be written as Z Z Z Z Z Z m ejkR ~ ~ ~ dv0 J ð~ r 0 Þ GðRÞ dv0 ¼ J ð~ r0 Þ ð6:8Þ Að~ rÞ ¼ m R 4p V

V

Magnetic and electric ﬁelds ~ B and ~ E can be deduced from Eq. (6.8). Furthermore, this equation represents the contribution of all the sources in the volume v0 at the observation point~ r . There is an obvious similarity between this Green’s formulation and the DPSM formulation, with the exception that DPSM sources are point sources. We can notice that in the formulation (6.8), ~ J ð~r0 Þ is the current density, whose unit is 2 A/m .

6.3 GREEN’S FUNCTIONS AND DPSM FORMULATION 6.3.1 Expressions of the magnetic and electric ﬁelds According to the DPSM formulation, the term ~ J represents a ﬁnite sum of Ns point sources. ~ J ð~ r0 Þ ¼

Ns X

~ r 0 ~ ri Þ Ji dð~

ð6:9Þ

i¼1

If Eq. (6.9) is inserted into Eq. (6.8), we have: ~ Að~rÞ ¼ m

Ns X i¼1

~ Ji

Z Z Z v

Gð~ r ~ r 0 Þ dð~ r0 ~ ri Þdv ¼ m

Ns X i¼1

~ Ji Gð~ r ~ ri Þ ð6:10Þ

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ADVANCED MAGNETODYNAMIC AND ELECTROMAGNETIC PROBLEMS

We may note that in Eq. (6.10), in comparison with Eq. (6.8), ~ Ji is equivalent to the r 0 Þ, which has been integrated on a small elemental volume dv, and current density ~ J ð~ then it represents an elemental source vector (triplet source) with three x, y, and z components, whose units are current multiplied by a length, or Am. In Cartesian coordinates, the vector potential calculated at a point of coordinate~ r is written as m X ejkRi Jxi 4p Axð~ rÞ Ri i jkRi X m e ~ Jyi Að~ r Þ ¼ ¼ Ayð~ rÞ ð6:11Þ R 4p i i ejkRi m X Azð~ rÞ Jzi 4p i Ri with

x Cxi ~ r ~ ri ¼ y Cyi Ri ¼ ~ z Czi

and k~ Ri k ¼ Ri ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðx Cxi Þ2 þ ðy Cyi Þ2 þ ðz Czi Þ2

using the point source coordinates ðCxi ; Cyi ; Czi Þ and the P observation point coordinates ðx; y; zÞ. The DPSM formulation add the contribution of all the point sources at a given array of observation points P. For example, let us assume that we have Np observation Að~ rj Þ is composed of points. So, for every value of rj , the magnetic vector potential ~ ! ! ! ~ is given by three vectors : Ax ; Ay , and Az . For example, Ax 2 3 ejkR11 ejkR1Ns 6 R R1Ns 7 6 7 11 6 . 7 .. 6 .. 7 m . ~ 6 7 ~ ~ ~ Ax ¼ 6 . ð6:12aÞ 7 J x , Ax ¼ ½Wxx J x .. 4p 6 . 7 . . 6 jkRNp1 7 4e ejkRNpNs 5 RNp1 RNpNs where Rji is the distance between the excitation point i and the observation point j. Finally, we will obtain, for the vector A containing all the x, y, and z coordinates of A, 2 3 2 3 ~ Wxx 0 0 J~ x Ax ~ A ¼ ~ J ð6:12bÞ Wyy 0 5 4 J~ y 5 ¼ W ~ Ay ¼ 4 0 ~ 0 0 Wzz J~ z Az

251

6.3 GREEN’S FUNCTIONS AND DPSM FORMULATION

We may also note that if the elemental triplet Jxi , Jyi , Jzi has its components not exactly located at the same point ~ ri , the matrix Wxx, Wyy, and Wzz may be slightly different. So, we can obtain the magnetic ﬁeld ~ Bð~ r Þ at a given point ~ r by ! ~ Bð~ r Þ ¼ rot ~ Að~ rÞ and @Az @Ay @y @z @Ax @Az ! ~ rot Að~ rÞ ¼ @z @x @Ay @Ax @x @y So m X ejkRi ejkRi Jxi Jxi 4p Ri Ri i Ns jkRi X jkRi X m e 4p e ~ Jyi ~ Að~ rÞ ¼ Að~ r Þ ¼ , Jyi Ri m Ri 4p i i¼1 jkRi X jkRi e m e Jzi Jzi 4p i Ri Ri Let the ~ Að~ rÞ curl coordinates be rotAx ! ~ rot Að~ r Þ ¼ rotAy rotAz with rotAx ¼

@Az @Ay ; @y @z

rotAy ¼

@Ax @Az ; @z @x

rotAz ¼

@Ay @Ax @x @y

For only one i source, the partial differentiation leads to @Ri y Cyi @ðejkRi =Ri Þ ð1=Ri þ jkÞ ðy Cyi Þ ejkRi ¼ ) ¼ @y Ri @y R2i

ð6:13Þ

252

ADVANCED MAGNETODYNAMIC AND ELECTROMAGNETIC PROBLEMS

So 4p @Azi Jzi ejk:Ri 1 ¼ ðy Cyi Þ Jzi ai ¼ j k m @y Ri R2i 4p @Ayi Jyi ejk:Ri 1 ¼ ðz Czi Þ Jyi ai ¼ j k m @z Ri R2i ejkRi 1 With ai ¼ 2 j k þ Ri Ri So, for the rotAx coordinate, we have: X 4p rotAx ¼ ai ððz Czi Þ Jyi ðy Cyi Þ Jzi Þ m i Idem for the rotAy and rotAz coordinates: X 4p ai ððx Cxi Þ Jzi ðz Czi Þ Jxi Þ rotAy ¼ m i X 4p ai ððy Cyi Þ Jxi ðx Cxi Þ Jyi Þ rotAz ¼ m i and ﬁnally a m i ½ðz Czi Þ Jyi þ ðy Cyi Þ Jzi 4p X ai m ~ ½ðx Cxi Þ Jzi þ ðz Czi Þ Jxi Bð~ rÞ ¼ 4p i a m i ½ðy Cyi Þ Jxi þ ðx Cxi Þ Jyi 4p ejkRi 1 . Always with ai ¼ 2 j k þ Ri Ri

ð6:14aÞ

We can also write this relation in the matrix formulation, like Eq. (6.12b), for an array of Np observation points. 2 ~ 0 Bx ~ B ¼ ~ By ¼ 4 Bzx ~ Byx Bz

Bzy 0 Bxy

3 2 3 Byz J~ x Bxz 5 4 J~ y 5 ¼ X ~ J 0 J~ z

ð6:14bÞ

where Bab is bloc matrix, in which index a and b, respectively, indicate the derivation axis and the orientation axis of the elemental point source component.

6.3 GREEN’S FUNCTIONS AND DPSM FORMULATION

253

It is interesting to note that in static case ðo ¼ 0Þ, we ﬁnd exactly the same result as those obtained with Biot and Savart equation. m i d~l ^ ~ R ~ B¼ 3 4p R 2 3 Jx i dx by taking ~ J ¼ Jy ¼ i d~l ¼ 4 i dy 5 Jz i dz

ð6:14cÞ

Finally, for problems without electrostatic components, we may obtain the electric ﬁeld ~ E according to @~ A ~ ¼ jo~ A E¼ @t

ð6:15aÞ

By using a different gauge, for example Lorentz gauge, we should obtain ~ E ¼ jo ~ Aþ

1 ! ~ gradðdivðAÞÞ o2 me

ð6:15bÞ

Equations (6.14) and (6.15) are the expression of the magnetic and electric ﬁelds in the DPSM theory. The Green’s theory has been adapted to the DPSM formulation, in order to use the DPSM concept of point sources. The next section will present the boundary conditions required in order to cross the surfaces in the DPSM/Green’s formulation. We may point out that the originality of DPSM formulation consists in placing on each side of an interface, arrays of virtual point sources (triplets current sources in these electromagnetic cases) to synthesize the ﬁelds in the media separated by the interface. With global matrix formulation introduced in the previous sections, the problem is not limited to simple geometries (like the example presented in Section 6.4), and can include several interfaces, media, or 3D objects. 6.3.2 Boundary conditions According to the DPSM general theory illustrated in other chapters of this book in ultrasonics, magnetostatics, and electrostatics domains, the boundary conditions are satisﬁed by matching a potential and its ﬁrst derivative along the normal~ n to the boundary surface (also see Chapter 1 for general presentation of the method). In the particular case of electromagnetic problems, this formulation leads to match the magnetic vector ~ potential ~ A and its ﬁrst derivative along the normal~ n to each point of the interface: m1 @@~An . For example, in the case described in Figures 6.1 and 6.2, the magnetic potential A~1 is computed in medium 1 by superimposing the effects of sources ~ JS and ~ JA1 ,

254

ADVANCED MAGNETODYNAMIC AND ELECTROMAGNETIC PROBLEMS DPSM current loop I(t)

V(t) z

Inductive magnetic field

Induced magnetic field

Surface eddy currents circulation

Semi-infinite electrically conductive target

Figure 6.1 Geometry.

when the magnetic potential A~2 in medium 2 is uniquely due to the sources ~ JA2 . So we can write 8 < A~1 ¼ A~2 ð6:16Þ 1 @ A~ 1 @ A~ : 1¼ 2 n n m1 @~ m2 @~ Now we can write the expression of the variations of the vector potential ~ A along the normal to the interface ~ n, in general case. 2 3 @Ax @Ax @Ax 6 @x @y @z 7 2 3 6 7 nx ~ 7 @ Að~ rÞ 6 @Ay @Ay @Ay 5 7 4 ð6:17Þ ¼6 6 @x @y @z 7 ny @~ n 6 7 n z 4 @Az @Az @Az 5 @x @y @z Js DPSM virtual excitation sources

Z = 0 : loop plane Medium 1 : air JA2 DPSM virtual sources Z = –20mm : target surface JA1 DPSM virtual sources Medium2 : target (semi-infinite thickness)

Figure 6.2 DPSM geometry meshing.

6.3 GREEN’S FUNCTIONS AND DPSM FORMULATION

255

In the particular case presented in Figure 6.1,~ n exactly ﬁts with z-axis; so the previous equation becomes 2

3 2 3 @Ax @Ax @Ax ai m @Ax 6 @x @y @z 7 2 3 6 4p ½ðz Czi Þ Jxi 7 @z 7 6 7 0 a m 6 X ~ 7 @ Að~ rÞ 6 @Ay @Ay @Ay @Ay i 6 7 7 405 ¼ 6 ½ðz Czi Þ Jyi ð6:18aÞ ¼6 ¼ 7 6 @x @y @z 7 4p 6 7 @~ z @z 6 7 i ai m 1 4 @Az 5 4 @Az @Az @Az 5 ½ðz Czi Þ Jzi 4p @z @x @y @z @~ Að~ rÞ ~ r Þ: ¼ dnA dnAð~ @~ n With same derivations and notations as for Eq. (6.12b) or (6.14b), and for a given array of Np observation points, we can write in this particular case where ~ n ¼~ z: Let us call

2 ~ Yzx ! dnAx dnA ¼ dn~ Ay ¼ 4 0 dn~ 0 Az

0 Yzy 0

3 2 3 0 J~ x y 5 ¼ Y ~ J 0 5 4 J~ Yzz J~ z

ð6:18bÞ

! Figure 6.2 presents the conﬁguration of the problem, with sources J S (located in medium 1), the interface between medium 1 and 2 with both associated DPSM layers ! ! J A1 and J A2 (respectively oriented outward for radiating in medium 1 and inward for radiating in medium 2). Boundary conditions may be rewritten now for the DPSM formulation: ! ! * Potential A1S and ﬁrst derivative dnA1S induced on the face 1 of the interface, ! created by the sources JS (inductor): ! ! A1S ¼ W1S JS ! ! dnA1S ¼ Y1S JS

ð6:19aÞ

! ! * Potential A11 and ﬁrst derivative dnA11 induced on the face 1 of the interface, ! created by the sources JA1 (outward DPSM layer): ! ! A11 ¼ W11 JA1 ! ! dnA11 ¼ Y11 JA1

ð6:19bÞ

! ! * Potential A22 and ﬁrst derivative dnA22 induced on the face 1 of the interface, ! created by the sources JA2 (inward DPSM layer): ! ! A22 ¼ W22 JA2 ! ! dnA22 ¼ Y22 JA2

ð6:19cÞ

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ADVANCED MAGNETODYNAMIC AND ELECTROMAGNETIC PROBLEMS

! ! The values of A and dnA along the interface must satisfy the following equations: ! ! ! W1s JS þ W11 JA1 W22 JA2 ¼ 0 ! ! ! 1=m1 Y1s JS þ 1=m1 Y11 JA1 1=m2 Y22 JA2 ¼ 0

ð6:20aÞ

Because JS is well known, we can also write JS ¼ I JS , with I as ð3Ns 3NsÞ identity matrix. So, one gets 2

3 2 ~ I JS 4~ 5 4 ¼ W 0 1S ~ 1=m1 Y1S 0

0 W11 1=m1 Y11

3 2 3 ~ 0 JS W22 5 4 ~ JA1 5 ~ 1=m2 Y22 JA2

ð6:20bÞ

Like for all the DPSM problems already solved in the previous sections of this book, unknown parameters ~ JA1 and ~ JA2 will be now obtained by inversion of the previous equation, in which the global matrix is always square, because W1S is a matrix of dimension 3Np 3Ns, W11 is a matrix of dimension 3Np 3Np, W22 is a matrix of dimension 3Np 3Np, so the global matrix is square and of dimension 3ðNs þ Np þ NpÞ 3ðNs þ Np þ NpÞ. Because all point source values are now well known, we will compute ﬁelds in given discretized areas of space, to observe the results of our modeling. Let us now present the results for different conﬁgurations of the simple (but interesting) problem presented in Figure 6.1. We may also note that this apparently simple problem is already very difﬁcult to solve analytically (Dodd and Deed, 1968), even if the inductor stays parallel to the plane interface. In other cases (tilted inductor for example), the complexity increases for analytical solutions, but remains the same for DPSM modeling, as we will see in further illustrations. 6.4 EXAMPLE OF APPLICATION The DSPM/Green’s formulation is applied on the geometry described in Figure 6.1. The DPSM simulation will be performed for different parameters of the target and frequency values. The geometry meshing is presented in Figure 6.2. The current loop is a circle of radius 50 mm, in the Z ¼ 0 XoY plane, meshed by N ¼ 25 current elements ðJs Þ. The target is a semi-inﬁnite medium from height Z ¼ 20 mm to Z ¼ 1, having dimensions of 300 300 mm. The interface is ~ Jxi ~ meshed by Np ¼ ð22 22Þ ¼ 484 triplets of elemental sources Ji ¼ ~ Jyi on each ~ side Jzi 6.4.1 Target in aluminum (r ¼ 50 Ms=m), frequency ¼ 1000 Hz

! The Figures 6.3 and 6.4 represent the top view of DPSM virtual current sources J A1 ! and J A2 , which, respectively, radiate in medium 1 and medium 2. Fields in medium 1

6.4 EXAMPLE OF APPLICATION

! Figure 6.3 Real part of J A1 DSPM sources along x- and y-axis.

! Figure 6.4 Real part of J A2 DSPM sources along x- and y-axis.

257

258

ADVANCED MAGNETODYNAMIC AND ELECTROMAGNETIC PROBLEMS

Figure 6.5 H components’ real parts in the target.

! are obtained by the sum of ﬁelds created by loop excitation sources Js and DPSM J A1 ! sources. Fields in medium 2 are only created by DSPM J A2 sources. For information, ! the excitation loop geometry (sources J S ) is recalled in red color. Figures 6.6 and 6.7 show the real and imaginary parts of the magnetic ﬁeld along three planes in the target located at height –20 mm (i.e. the target surface), –25 mm (5 mm below the surface) and –30 mm (–10 mm below the surface).

Figure 6.6 H components’ imaginary parts in the target.

6.4 EXAMPLE OF APPLICATION

259

Figure 6.7 A modulus real part contour lines.

Figure 6.7 represents equipotential contour lines, which give an interesting information about the magnetic ﬁeld chart. We can see that the penetration of the magnetic ﬁeld in the target is reduced, due to the eddy currents effect. Figure 6.8 represents the same geometry as in the interesting case where all other parameters are taken identical to those of the air (no conductivity). This case is equivalent to the case where there is no target. It is very interesting to point out that the DPSM layers perfectly synthesize the ﬁelds in medium 2: One can see that the top and bottom contour lines are symmetrical, in spite of the DPSM ! ! A function interface layers J A1 and J A2 . On the contrary, we can check that the ~

Figure 6.8 A modulus real part contour lines – air target.

260

ADVANCED MAGNETODYNAMIC AND ELECTROMAGNETIC PROBLEMS

~ Figure 6.9 @ A modulus real part contour lines – air target. @~ n

matches across the boundary, according to Subsection 6.3.2. Figure 6.9 shows that the ~ vector @@~An is also matched across the surface. 6.4.2 Target in aluminum (r ¼ 50 Ms=m), frequency ¼ 100 Hz, inclined excitation loop The excitation loop is now tilted with a 10 angle, along a rotation axis parallel to the y-axis for x ¼ 50 mm. The frequency is now decreased to 100 Hz. Figures 6.10

Figure 6.10 Real part of ~ JA1 DSPM sources along x and y axes.

6.4 EXAMPLE OF APPLICATION

261

! Figure 6.11 Real part of J A2 DSPM sources along x and y axes.

! ! and 6.11 represent the top view of the DPSM virtual current sources J A1 and J A2 , which radiate in medium 1 and medium 2, respectively, like Figures 6.3 and 6.4 in the previous case. Figures 6.12 and 6.13 show the magnetic ﬁeld inside the target, along the three slices deﬁned previously for Figures 6.5 and 6.6.

Figure 6.12a H components’ real parts in the target.

262

ADVANCED MAGNETODYNAMIC AND ELECTROMAGNETIC PROBLEMS

Figure 6.12b H modulus’ real part in the target.

Figures 6.14 and 6.15, respectively, show equipotential contour lines of both vectors ~ ~ ~ A and @@~An (which corresponds to @@zA in this particular case with a plane interface along XoY). Like in the previous case, we can check that the ~ A function matches across the boundary, according to Subsection 6.3.2. The ﬁgure shows that it is also matched across the surface.

Figure 6.13a H components’ imaginary parts in the target.

6.4 EXAMPLE OF APPLICATION

263

Figure 6.13b H modulus’ imaginary part in the target.

6.4.3 Dielectric target ðer ¼ 5Þ, frequency ¼ 3 GHz, 10 tilted excitation loop The excitation loop is also tilted with a 10 angle in this new case, along a rotation axis parallel to the y-axis for x ¼ 50 mm (the bottom side of the excitation loop remains

Figure 6.14 ~ A modulus’ real part contour lines in the target.

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ADVANCED MAGNETODYNAMIC AND ELECTROMAGNETIC PROBLEMS

~ Figure 6.15 @ A modulus’ real part contour lines in the target. @~ n

at z ¼ 0). The frequency is now increased to 3 GHz, with a nonconductive and dielectric medium 2. Figures 6.16 and 6.17 represent the top view of the DPSM JA2 , which radiate in medium 1 and medium 2, virtual current sources ~ JA1 and ~ respectively, like Figures 6.10 and 6.11 in the previous case.

Figure 6.16 Real part of ~ JA1 DSPM sources along x and y-axes.

6.4 EXAMPLE OF APPLICATION

265

Figure 6.17 Real part of ~ JA2 DSPM sources along x and y axes.

Figures 6.18 and 6.19 show the magnetic ﬁeld inside the target, along the three slices deﬁned previously for Figures 6.12 and 6.13. Figures 6.20 and 6.21 show the electric ﬁeld inside the target. Figures 6.22a and shows the vector ~ A modulus’ real part in the target and its ~ contour lines, respectively. Figure 6.23 shows the equipotential contour lines of @@~An

Figure 6-18a H components’ real parts in the target.

266

ADVANCED MAGNETODYNAMIC AND ELECTROMAGNETIC PROBLEMS

Figure 6-18b H modulus’ real part in the target. ~

(which corresponds to @@zA in this particular case with a plane interface along XoY). Like in the previous case, we can check that the ~ A function matches across the boundary, ~ according to Subsection 6.3.2. The ﬁgure shows that @@~An is also matched across the surface.

Figure 6-19a H components’ imaginary parts in the target.

6.4 EXAMPLE OF APPLICATION

Figure 6-19b H modulus’ imaginary part in the target.

Figure 6.20a E components’ real parts in the target.

267

268

ADVANCED MAGNETODYNAMIC AND ELECTROMAGNETIC PROBLEMS

Figure 6.20b E modulus’ real part in the target.

Figure 6.21a E components’ imaginary parts in the target.

6.4 EXAMPLE OF APPLICATION

Figure 6.21b E modulus’ imaginary part in the target.

Figure 6.22a ~ A modulus’ real part in the target.

269

270

ADVANCED MAGNETODYNAMIC AND ELECTROMAGNETIC PROBLEMS

Figure 6.22b ~ A modulus’ real part contour lines in the target.

~ Figure 6.23 @ A modulus’ real part contour lines in the target. @~ n

6.5 CONCLUSION This chapter has presented an extension of the DPSM formulation, where Green’s sources are introduced to solve electromagnetic problems, allowing the possibility of computing ﬁelds in all media or parts of the problem, including three-Dimensional (3D) objects. Similar to other technologies for which DPSM modeling has already been applied (magnetostatic, electrostatic, or ultrasonic problems), interfaces are substituted by layers of elemental sources, which synthesize the ﬁelds in the given

REFERENCES

271

media of the problem. The values of these sources are computed from the boundary conditions, in order to match the magnetic vector potential ~ A and its ﬁrst derivative ~ along a direction ~ n normal to the interface at each point: m1 @@~An . So, the interface does not have to be plane or unlimited. Moreover, by introducing the concept of space controlled radiation sources presented in Chapter 1 (Figure 1.8), real 3D geometries with shadow problems can be easily solved. We may emphasize that according to the DPSM technique, all elemental sources always radiate in homogeneous medium, leading to particularly simple Green’s functions. So the global matrix, which is to be inverted in order to ﬁnd the values of each source component, is always square and almost never rank deﬁcient. Some illustrations of these possibilities will be presented in Chapters 9 and 10, with applications to detection of cracks in carbon composite plates for aeronautic industry.

REFERENCES Bowler, J. R., H. A., Sabbagh, and L. D., Sabbagh, ‘‘The Reduced Impedance Function for CupCore Eddy-Current Probes’’, IEEE Transactions on Magnetics, Vol. 25 (32), pp. 2646–2649, 1989. Dodd, C. V., and W. E., Deed, ‘‘Analytical Solutions to Eddy Current Probe-Coil Problems’’, Journal of Applied Physics, Vol. 39(6), pp. 2829–2838, 1968. Dodd, C. V., and W. E., Deed, ‘‘Integral Solutions to Some Eddy Current Problems’’, International Journal of Non Destructive Testing, Vol. 1, pp. 29–90, 1969. Dufour, I., and D., Placko, ‘‘An Original Approach of Eddy Current Problems Through a Complex Electrical Image Concept’’, IEEE Transactions on Magnetics, Vol. 32 (2), pp. 348–365, 1996. Placko, D., N., Liebeaux, and T., Kundu, ‘‘Proce´de´ pour e´valuer une grandeur physique representative d’une interaction entre une onde et un obstacle’’, Patent Application N 02 14108 ENS Cachan/CNRS/Universite´ d’Arizona, November 8, 2002, European and USA Extension November 10, 2003. Placko, D., Liebeaux N., Kundu T., ‘‘Proce´de´ universel de mode´lisation des interactions entre au moins une onde et au moins un objet, la surface de chaque objet de´ﬁnissant une interface entre au moins deux milieux’’. Patent Application N 05 13219. ENS Cachan/CNRS/ Universite´ d’Arizona/Universite´ Paris Sud XI, December 23, 2005, international extension in december 2006. Kriezis, E. E., T. D., Tsiboukis, S. M., Panas, and J. A., Tegopoulos, ‘‘Eddy Currents: Theory and Applications’’, Proceedings of the IEEE, Vol. 80(10), pp. 1559–1585, 1992. Ney M., ‘‘Bases de l’e´lectromagne´tisme’’, Techniques de l’Inge´nieur, ref. E1020, 2003.

7 ELECTROSTATIC MODELING AND BASIC APPLICATIONS G. Lissorgues ESIEE, Noisy le Grand, France

A. Cruau and D. Placko Ecole Normale Superieure, Cachan, France

7.1 INTRODUCTION As presented in previous chapters, the distributed point source method (DPSM) can be illustrated through different examples coming from electromagnetic and ultrasonic applications. The basic principle is based on the idea of using point sources distributed over the active part of a sensor or an interface, and the equations rely on analytical solutions of basic point source problems. Consequently, it is similar to a semi-analytical technique used to solve sensor problems like magnetic, ultrasonic, or electrostatic sensors (in Chapter 1 Table 1.1 presents the physical values typically used in the DPSM method). Electrostatic modeling is an important domain of physics, encountered in many microsystem devices, especially if electrostatic actuation is required. This chapter describes how the DPSM equations can be applied to the electrostatic domain. Hereafter basic applications will be detailed, mainly based on the classical parallel-plate capacitor conﬁguration, which is widely used in electrostatic sensors and actuators. From a very general point of view, an electrostatic problem can be described by a structure containing n perfectly conductive objects with their surfaces Si (with i

DPSM for Modeling Engineering Problems, Edited by Dominique Placko and Tribikram Kundu Copyright # 2007 John Wiley & Sons, Inc.

273

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ELECTROSTATIC MODELING AND BASIC APPLICATIONS

S1 p1

S4 p4

S5 p5

qt1

qt5=0 S3 p3

qt4=0

S2 p2

qt3

qt2

Figure 7.1 Electrostatic equilibrium of a group of metallic active (continuous line) and passive (dashed lined) surfaces (with n ¼ 5 and m ¼ 3).

varying from 1 to n) of any shape in an homogenous medium, m of which being called active surfaces while connected to an electrostatic potential pi (with i varying from 1 to m). The other (n m) surfaces are isolated while their total electrostatic charge written as qti is null (with i varying from (m þ 1) to n) and can be called passive surfaces (Fig. 7.1). The electrostatic equilibrium of this structure corresponds to the following equations, each surface Si being at a constant potential pi and carrying an equivalent charge qti (7.1). 8 p1 ¼ d1;1 qt1 þ þ d1;m qtm þ d1;mþ1 qtmþ1 þ þ d1;n qtn > > > > . > > .. > > > > < p ¼ d qt þ þ d qt þ d m m;1 1 mm m mmþ1 qtmþ1 þ þ dmn qtn > p ¼ d qt þ þ d qt mþ1;1 1 mþ1;m m þ dmþ1;mþ1 qtmþ1 þ þ dmþ1;n qtn > mþ1 > > > > .. > > . > > : pn ¼ dn;1 qt1 þ þ dn;m qtm þ dn;mþ1 qtmþ1 þ þ dn;n qtn ð7:1Þ The dij coefﬁcients (with i and j varying from 1 to n) are derived from the charge densities si on Si surfaces (7.2), with rij being the distance between the elementary surfaces dSi and dSj , and e being the dielectric permittivity of the homogeneous surrounding medium. Z Z 1 s i sj dSi dSj ð7:2Þ di; j ¼ rij 4 p e qti qtj Si

Sj

Then, the potential pm can be written as pm ¼

n X i¼1

1 4 p e qti

Z Z Si

Sm

si sm dSi dSm rim

ð7:3Þ

275

7.2 MODELING BY DPSM

Unfortunately, such a system is difﬁcult to solve directly because of the surface integrals. It requires an equivalent digitized model close to the real conﬁguration. At this point, the DPSM can offer an interesting alternative. Indeed, as previously explained in Chapter 1, the DPSM is able to model the electrodes of any electrostatic system as a group of distributed point sources that emit an electrostatic ﬁeld in the surrounding medium. As meshing in DPSM is less complicated than that with ﬁnite element methods, initial complexity of a 3D electrostatic problem is then transferred to a superposition of elementary problems.

7.2 MODELING BY DPSM 7.2.1 Digitalization of the problem Any metallic electrode surfaces Si can be considered as a group of Ni unit charges aik (k ranging from 1 to Ni ). To obtain the potential on this electrode, because of its uniformity on surface, we can consider only one charge of this electrode. On this charge the potential pi is approximated by a double sum that replaces the integral surface (Eq (7.4)). j n X X 1 ajx 4 p e j¼1 x¼1 rjx

N

pi ¼

ð7:4Þ

Where ajx are charges on each surface Sj of the n conductive objects in inﬂuence with Si . Therefore, using DPSM modeling, surfaces are divided into a ﬁnite number N of elementary surfaces dSk in the center of which a point source Ck of charge ak has been placed (Fig. 7.2). In this method, boundary conditions are applied on the apex of hemispheres centered on Ck points (deﬁned in Chapter 1) called peak points and labeled as Pk . These points stand for the electrode surface.

Figure 7.2 Distribution of charges on the electrode surface with associated points and electrostatic ﬁeld E(O).

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ELECTROSTATIC MODELING AND BASIC APPLICATIONS

Clearly, greater is the number of points, smaller is the distance between a source and its peak. Hence, when the number N of sources tends to inﬁnity, the radius r tends to 0 and Pk points come closer to Ck points, and the model tends to the real case where charges are placed at the surface of a conductor. Note that the point sources Ck are located wherever needed depending on the shape of the whole conductive object surface. With DPSM formulation, we can synthesize the electrostatic potential P(O) and ﬁeld E(O) at an observation point O located at distances rk from charges ak placed at source points Ck (7.5). Considering the Coulombian ﬁeld hypothesis, the P(O) expression shows a typical function fk similar to other DPSM problems. The Coulombian ﬁeld hypothesis considers that the distance rk between the charge at the point source and the point O where potential and ﬁeld are calculated is much higher than the charge dimensions, which is veriﬁed when the charge is point like in DPSM.

PðOÞ ¼

N X 1

X ! ak E ðOÞ ¼ N

1 4 p e rk

1

! rk ak 4 p e rk2

ð7:5Þ

and fk ¼

1 4 p e rk

7.2.2 DPSM meshing considerations As explained above, the ﬁrst step of the DPSM method consists of choosing relevant surfaces to be meshed; for electrostatic problems regions with concentrated free charges are considered as surfaces. Points Pk always represent the oriented surface of conductors, and points Ck will be therefore situated slightly behind to keep the gap constant, whatever number of points is used.To illustrate this meshing principle, we can work on the classical parallel-plate capacitor of Figure 3. As DPSM applied to electrostatics only considers conductive surfaces, the parallel-plate capacitor is reduced to the schematic of Figure 3b, where the dielectric substrate is not represented because the electric ﬁeld is only calculated into the air gap. Meshing will be required on the two electrode surfaces in regard. If more precision is needed, DPSM allows putting more point sources where the ﬁeld is more important, at the corners and peaks, for example. 7.2.3 Matrix formulation To solve the problem we have to ﬁnd all charges ak placed in the structure using boundary conditions, which are potentials pk ﬁxed on the electrodes at points Pk . We can write these conditions using matrices and the previous

277

7.2 MODELING BY DPSM

Conductive membrane Microspring and its anchor

Lx = 200μm

Air gap g = 5μm Ly=200μm

Substrate of relative permittivity εr z y

DPSM modeling

x

Figure 7.3 Sketch of a micromechanical parallel-plate actuator translated in DPSM modelling.

functions fk (7.6). 3 2 Q p1 1;1 .. 6 .. 7 6 6 6 . 7 6 . 6 7 6 p 7 6 Qm;1 6 m 7 6 6 7¼6 6 6 pmþ1 7 6 Qmþ1;1 6 7 6 .. 7 6 . 4 . 5 6 4 .. pn Qn;1 2

Q1;m .. . Qm;m Qmþ1;m .. . Qn;m

Q1;mþ1 .. . Qm;mþ1 Qmþ1;mþ1 .. . Qn;mþ1

3 2 3 Q1;n A1 .. 7 6 .. 7 . 7 . 7 7 6 7 7 6 6 Qm;n 7 6 Am 7 7 76 7 Qmþ1;n 7 Amþ1 7 7 6 6 7 .. 7 .. 7 7 6 4 . 5 . 5 An Qn;n

with 2

1 6 4 p e ri1; j1 6 6 .. Qi;j ¼ 6 . 6 4 1 4 p e riN i ; j1

3 1 4 p e ri1;jN j 7 7 7 .. 7 . 7 5 1 4 p e riN i ; jN j

ð7:6Þ

If all potentials pk are known, inverting the matrix Q gives the distribution of charges on the electrodes. Therefore potential P(O) and electrostatic ﬁeld E(O) can be calculated in every point of the medium above the surface, in the semispace of emission of point sources.

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ELECTROSTATIC MODELING AND BASIC APPLICATIONS

7.3 SOLVING THE SYSTEM 7.3.1 Synthesizing electrostatic ﬁeld and potential Because inverting the matrix Q gives the distribution of charges on the electrodes, Eq. (7.6) induces the following relation (7.7) where the Ci,j coefﬁcient represents the inﬂuence of the i conductor over the j conductor. 2

3 2 C 1;1 A1 6 .. 7 6 .. 6 . 7 6 . 6 7 6 6 Am 7 6 Cm;1 6 7 6 6 Amþ1 7 ¼ 6 Cmþ1;1 6 7 6 6 .. 7 6 . 4 . 5 4 .. An C n;1

C 1;m .. .

Cm;m C mþ1;m .. .

Cm;mþ1 C mþ1;mþ1 .. .

C n;m

3 2 3 C1;n p1 .. 7 6 .. 7 6 7 . 7 7 6 . 7 6 pm 7 C m;n 7 76 7 6 7 Cmþ1;n 7 7 6 pmþ1 7 6 .. 7 .. 7 . 5 4 . 5 pn Cn;n

C 1;mþ1 .. .

C n;mþ1

summarized by A ¼ Q1 p ¼ C p

ð7:7Þ

Knowing that electrostatic potentials are constant on whole conductive surfaces, each vector Pi can be factorized as in Eq. (7.8), where C i;j is the column vector made of the sum of in-line (horizontal) terms of Ci,j. Therefore, we can add elements of the charge vectors Ai to ﬁnd the total charges qti (equal to the sum of ai;j , j varying from 1 to Ni ), with ci6¼j being the inﬂuence coefﬁcients and cii the capacitance coefﬁcients (7.9). 2

Ai ¼ ½ C i;1

2

3

2

c1;1 qt1 6 . 7 6 . 6 .. 7 6 .. 6 7 6 6 7 6 6 qtm 7 6 cm;1 6 7¼6 6 qt 7 6 6 mþ1 7 6 cmþ1;1 6 7 6 6 .. 7 6 .. 4 . 5 4 . qtn cn;1

C i;m

C i;mþ1

C i;n

c1;m .. .

c1;mþ1 .. .

cm;m cmþ1;m .. .

cm;mþ1 cmþ1;mþ1 .. .

cn;m

cn;mþ1

3 p1 6 . 7 6 .. 7 6 7 6 7 6 pm 7 6 7 6 7 6 pmþ1 7 6 7 6 .. 7 4 . 5

pn 2 3 c1;n 3 p1 6 . 7 .. 7 6 . 7 . 7 7 6 . 7 7 6 7 cm;n 7 6 pm 7 76 7 6 7 cmþ1;n 7 7 6 pmþ1 7 7 6 7 .. 7 6 .. 7 4 5 5 . . pn cn;n

ð7:8Þ

ð7:9Þ

279

7.3 SOLVING THE SYSTEM

Referring to the (n m) passive surfaces of Figure 1, their total electrostatic charges qtk (k varying from m þ 1 to n) are null, and the subsystem isolated from (7.9) can be written as (7.10). Consequently, we can solve this last system because we can extract the (n m) potentials pk from the ﬁxed potentials pj (j varying from 1 to m) due to the boundary conditions (7.11). 2 3 2 0 cmþ1;1 6.7 6 6 . 7 6 .. 6.7 6 . 6 7 6 6 7 6 6 0 7 ¼ 6 ck;1 6 7 6 6 7 6 6 .. 7 6 .. 6.7 4 . 4 5 cn;1 0 2

cmþ1;mþ1 6 .. 6 6 . 6 6 6 ck;mþ1 6 6 .. 6 . 4 cn;mþ1

2

3 p1 cmþ1;m cmþ1;mþ1 cmþ1;n 6 . 7 6 . 7 . 7 .. .. .. 7 7 6 7 . . . 7 6 7 7 6 p 6 7 m 7 7 ck;m ck;mþ1 ck;n 7 6 6 7 6 pmþ1 7 7 7 7 .. .. .. 7 6 6 . . . 5 6 .. 7 7 4 . 5 cn;m cn;mþ1 cn;n pn 2 3 3 cmþ1;n cmþ1;1 cmþ1;m 6 . .. 7 .. 7 6 . 7 2 pmþ1 3 7 2 p1 3 6 . . 7 . 7 6 7 6 7 6 7 7 7 .. 7 .. 7 7 ¼ 6 ck;n 7 6 ck;m 7 6 6 ck;1 . 4 5 6 7 7 4 . 5 6 . .. 7 .. 7 6 . 7 7 pn pm . 5 . 5 4 . 3

cn;n

cn;1

cn;m ð7:10Þ

summarized by Ctpp Pp ¼ Ctpa Pa

ð7:11Þ

If Pp represents the potentials on passive surfaces, whereas Pa the potentials on active surfaces, the matrix Ctpp and Ctpa , respectively, correspond to the inﬂuence coefﬁcients of passive surfaces over themselves and inﬂuence coefﬁcients of active surfaces over the passives, leading to Eq. (7.12), and the problem is solved because all constant potentials are calculated. 1 Ctpa Pa Pp ¼ Ctpp

ð7:12Þ

7.3.2 Capacitance calculation Two isolated conductors connected to potentials p1 and p2 6¼ p1 will inﬂuence each other and generate charges qt1 and qt2 , respectively. These charges will only depend on their respective geometrical position and on the difference between p1 and p2 .

280

ELECTROSTATIC MODELING AND BASIC APPLICATIONS

The capacitance capa of this system of two conductors is deﬁned by relation (7.13), which only relies on the geometry of such system. qt ¼ capa u ¼ capa ðp2 p1 Þ

ð7:13Þ

If p1 ¼ p2 , then qt1 ¼ qt ¼ qt2 , else if qt1 6¼ qt2 , the charge qt therefore corresponds to the charge, which should be transferred from conductor 1 to conductor 2 so that equilibrium is reached (Purcell, 1963). We have seen in Eq. (7.9) that when we double sum by blocks the inversed matrix, we obtain inﬂuence coefﬁcients ci;j , which only depend on the geometry and the expression is valid whatever the environment is. Using the previous deﬁnition of capacitance with two conductors i and j out of n, we can extract from (7.9) the total charges qti and qtj to write the following relation (7.14), where charge qt corresponds to the common potential p on conductors i and j. Therefore, comes Eq. (7.15) giving the capacitance capai;j between the two conductors i and j. 2 3 p1 6 . 7 6 .. 7 6 7 6 7 6 7 6p7 qti ci;1 ci;i ci;j ci;n 6 . 7 qt þ ¼ 6 .. 7 ð7:14Þ 7 qtj cj;1 cj;i cj;j cj;n 6 þqt 6 7 6p7 6 7 6 . 7 6 . 7 4 . 5 pn Then,

ci;i ci;i pi þ ci;j pj qt þ ¼ cj;i pi þ cj;j pj cj;i þqt

p cj;j p ci;j

qt ¼ capai;j ðpi pj Þ And ﬁnally, capai;j ¼

ci;i cj;j ci;j cj;i ci;i þ ci;j þ cj;i þ cj;j

ð7:15Þ

7.4 EXAMPLES BASED ON PARALLEL-PLATE CAPACITORS 7.4.1 Description This structure is composed of two plane electrodes facing each other. In the cases we will study, we chose rectangular shape for electrodes (Fig. 7.4). They are meshed by point sources considering the number of points on the shortest side called nmin. The number of points on the largest side is calculated based on nmin to

281

7.4 EXAMPLES BASED ON PARALLEL-PLATE CAPACITORS

Area 1 with N1 source points

z Area 2 with N2 source points

y

x

Figure 7.4 DPSM Model of a parallel-plate actuator.

respect the following rule: Hemispheres cannot penetrate each other. This rule allows obtaining a meshing of good quality, even in high-aspect ratio conditions. It means that at the most hemispheres can touch on one dimension (their diameter is equal to smallest side of elementary surface) and are at a larger distance on the largest dimension as shown in Figure 7.5. The second rule that we have decided to satisfy for meshing is that the gap between electrodes, determined by the distance between apex points, has to be at least equal to the minimum radius of hemispheres as shown in Figure 7.6. The last point to notice is that for symmetrical electrodes; symmetrical meshing is more suitable. 7.4.2 Equations The system is described by vectors of point sources C1 and C2 and apex points P1 and P2 placed at the physical surface of electrodes. At point sources elemental charges A1 and A2 are placed and potential P1 and P2 are applied on electrodes. The a

b

r Electrode

Elemental surface Hemisphere

2πr2 = ab a = 2r⇒b = πr

Figure 7.5 Bulbs adapted distribution on surface.

282

ELECTROSTATIC MODELING AND BASIC APPLICATIONS

gap > r

Sources Hemisphere radius r = distance between sources and electrode surface

Figure 7.6 Minimum distance between facing electrodes.

equations giving the inﬂuence between electrodes use coupling matrices Qij are shown below. P1 ¼ Q11 A1 þ Q12 A2 P2 ¼ Q21 A1 þ Q22 A2 Written in matrix expression: Q11 P1 ¼ P2 Q21

Q12 Q22

ð7:16Þ

A1 A2

ð7:17Þ

This gives the following charge values:

A1 A2

Q11 ¼ Q21

Q12 Q22

1

P1 P2

C11 ¼ C21

C12 C22

P1 P2

ð7:18Þ

Finally, we obtain capacitance matrix by summing in both dimensions of matrix Cij .

At1 At2

¼

c11 c21

c12 c22

p1 p2

ð7:19Þ

And the capacitance between electrodes is capa ¼

c11 c22 c12 c21 c11 þ c22 þ c12 þ c21

ð7:20Þ

7.4.3 Results of simulation The chosen dimensions are 200 mm 200 mm for a gap of 5 mm or 20 mm (cases a and b). The medium in the gap is homogeneous, and ﬁrst, we put air characterized by a permittivity of 8:54 1012 F/m, and in the case c, glass of relative permittivity 4 will be used (gap is 5 mm). The number of points for each electrode is 900. The global

7.4 EXAMPLES BASED ON PARALLEL-PLATE CAPACITORS

283

matrix is then 1800 1800. The capacitances calculated are 74 fF for case a, 23.4 fF for case b, and 296 fF for case c. Note that case c value is four times that of case a due to relative permittivity. To observe ﬁeld and potential, a bias of 3 V is applied between electrodes (þ3 V and 0 V). The observation is done in the transverse plane yoz placed in the middle of the structure. Figure 7.7 represents the transverse component Ez of electrostatic ﬁeld

Figure 7.7 Transverse component of the electrostatic ﬁeld (Ez): (a) for a 20 mm wide gap; (b) for a 5 mm wide gap.

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ELECTROSTATIC MODELING AND BASIC APPLICATIONS

Figure 7.8 Lateral component of the electrostatic ﬁeld (Ey): (a) for a 20 mm wide gap; (b) for a 5 mm wide gap.

7.4 EXAMPLES BASED ON PARALLEL-PLATE CAPACITORS

285

Figure 7.9 Electrostatic potential : (a) for a 20 mm wide gap; (b) for a 5 mm wide gap.

286

ELECTROSTATIC MODELING AND BASIC APPLICATIONS

Figure 7.10 Charges (punctual sources values) distribution on one electrode : (a) for a 20 mm wide gap; (b) for a 5 mm wide gap.

287

7.4 EXAMPLES BASED ON PARALLEL-PLATE CAPACITORS

(major component) and Figure 7.8 the lateral component Ey for both cases a and b (Ex is negligible). For the 5 mm gap structure, the ﬁeld is higher because the electrodes are closer and the capacitance is larger. Figure 7.9 shows the potential in the gap. It should be noted that the boundary conditions (0 and 3V) are veriﬁed on electrode surfaces. From these ﬁgures one can conclude that for a smaller gap, more source points are needed; small peaks at the surfaces show their location, and 5 mm gap structure has higher peaks and is less smooth than 20 mm gap structure. Figure 7.10 contains the charge distribution on electrode surface (x and y axes correspond to the increments of points, not to coordinates). The peak effect can be seen clearer; charge values are higher at the corners. Next, simulations have been performed to illustrate the importance of the meshing criteria. We have calculated the variations in the capacitance value obtained depending on the kind of meshing (number of points representing the electrodes) and compared the results to the theoretical capacitance value obtained using the analytical expression. Results are summarized in the table of Figure 7.11. This table conﬁrms the accuracy of the DPSM for the electrostatics simulations (results are very close to the theoretical analytical values for the classical parallelplate conﬁguration). The number of meshing points can be quite small, which induces short computing time. Nonetheless, the meshing highly depends on the geometry as in other simulation methods, and for the highest shape ratio, the greatest number of points is required.

Dimensions (μm)

Capacitance versus number of points 77

nx = ny = 3·(i+1) i from 0 to 11

value

75

DPSM

74

Comments

Square shape, in case of gap g << L/2 600 points needed with DPSM to fit the theoretical value with accuracy below 1% Only 100 points needed to fit with accuracy around 5%

theoretical

76

C (fF)

Lx = 200 ; Ly = 200 g=5

73 72 71 9

81

225

441

729

1089

number of points on one electrode

21.85

Lx = 200 ; Ly = 50 g=5 C (fF)

nx = 3·(i+1) ny = 12·(i+1) i from 0 to 6

Rectangle shape, g = L/10 300 points needed with DPSM to fit the theoretical value with accuracy below 1%

21.65 21.45 21.25 21.05 20.85

36

324

900

1764

number of points on one electrode

nx = 2·(i+1) ny = 20·(i+1) i from 0 to 6

C (fF)

Lx = 200 ; Ly = 10 g = 10

Rectangle shape, in case of gap g > L/2 (here g = L) 1000 points needed with DPSM to fit the theoretical value with accuracy below 1%

4.3 4.25 4.2 4.15 4.1 4.05 4 40

360

1000

1960

number of points on one electrode

Figure 7.11 Table for meshing precision comparison.

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ELECTROSTATIC MODELING AND BASIC APPLICATIONS

Figure 7.12 (a) Gap-tuning parallel-plate capacitor; (b) capacitance tuning.

7.4.4 Gap-tuning variable capacitor The structure we have simulated in the previous paragraph has now a progressively decreasing gap. This is the simplest case of variable capacitor (or varactor). In this simulation the gap is decreased (to 2 mm) without taking into account the mechanical and electrostatic forces. That is why we can look at capacitance tuning for a displacement greater than one third of the gap (1.66 mm). Normally, gap-tuning varactors are limited to that position because they collapse when the gap tends to be smaller; stability cannot be reached anymore. Figure 7.12 represents the varactor sketch and the capacitance change versus the transverse displacement. 7.4.5 Surface-tuning variable capacitor Another way of tuning a capacitor is by decreasing the common area between electrodes. This is generally obtained by displacing a movable electrode laterally (Fig. 7.13a). In the study presented in this paragraph, a simple structure is ﬁrst modeled, and ﬁnally, results for a multiﬁnger varactor are given. Multiﬁnger (a)

(b) Common area

Length of fingers : very superior to width

Movable electrode

Width = maximal displacement for actuators

Figure 7.13 Principle of surface-tuning varactors: (a) simple structure and (b) fragmented.

7.4 EXAMPLES BASED ON PARALLEL-PLATE CAPACITORS

289

Figure 7.14 Capacitance tuning for (a) simple structure and (b) fragmented varactor.

290

ELECTROSTATIC MODELING AND BASIC APPLICATIONS

Figure 7.15 Electrostatic ﬁeld: (a) initial position and (b) ﬁnal position.

7.4 EXAMPLES BASED ON PARALLEL-PLATE CAPACITORS

291

Figure 7.16 Electrostatic potential: (a) initial position and (b) ﬁnal position.

structures (Fig. 7.13b) allow having a high-tuning ratio with a minimum displacement that means minimum power consumption. The ﬁrst structure is the 5 mm gap square (200 mm 200 mm) used in the preceding paragraph. The top electrode is displaced by 220 mm. The capacitance

292

ELECTROSTATIC MODELING AND BASIC APPLICATIONS

Figure 7.17 Charge distribution: (a) initial position and (b) ﬁnal position.

REFERENCE

293

tuning curve of Figure 7.14a shows the important decrease of capacitance with increasing misalignment of electrodes. The second structure is fragmented, 200 mm long with four fragments of 50 mm width for each electrode (meshed with 1040 points for each electrode 260 per fragment with 10 points on the width). The displacement is 50 mm to decrease capacitance. The results are presented in Figure 7.14b. The capacitance tuning ratio is less important ( 2 instead of 7 in Fig. 7.14a), and the initial capacitance is higher because of increased fringing effects due to holes. Figures 7.15–7.17 show the electrostatic ﬁeld, the electrostatic potential, and charge distribution for the two extreme positions.

7.5 SUMMARY This chapter has demonstrated that DPSM can solve electrostatic basic systems composed of conductive objects in one homogeneous medium. In particular, it proves its efﬁciency in accurately calculating capacitance with a small number of points. More complex systems with several dielectric media or screen effect between conductors have been discussed in the next chapter.

REFERENCE Purcell E.M., ‘‘Electricity and Magnetism’’, Berkeley Physics Course, Vol. 2, McGraw-Hill, 1963.

8 ADVANCED ELECTROSTATIC PROBLEMS: MULTILAYERED DIELECTRIC MEDIUM AND MASKING ISSUES G. Lissorgues ESIEE, Noisy le Grand, France

A. Cruau and D. Placko Ecole Normale Supe´rieure, Cachan, France

8.1 INTRODUCTION In the following sections we describe more complicated electrostatic systems that contain several different media or materials in a real three-dimensional (3D) conﬁguration. Naturally, it better represents the problems that physicists or engineers encounter to build sensors, actuators, and microsystems in general. The ﬁrst case that we will pay attention to will be the multilayered structures. They represent the devices characterized by stacked layers of dielectrics between two conductors standing for electrodes (compared to ultrasonic problems where there are targets and emitters separated by ﬂuid or solid media). One example is shown in Figure 8.1. Second case of systems is composed of those based on multiconductor geometries (more than two conductive objects are in the studied medium) where one or more conductors can act as masks for others so that no interaction takes place between masked conductors. The example shown in Figure 8.2 demonstrates how two conductors can be partially or totally hidden from each other by another DPSM for Modeling Engineering Problems, Edited by Dominique Placko and Tribikram Kundu Copyright # 2007 John Wiley & Sons, Inc.

295

296

ADVANCED ELECTROSTATIC PROBLEMS

conductor

Boundaries between media

3rd medium

2nd medium 1st medium

conductor

Figure 8.1 Example of a multilayered electrostatic system with three different dielectric media.

Two fragmented parallel electrodes Fragmented conductor masking the fragmented electrodes from each other

Figure 8.2 Example of an electrostatic system containing masking conductors.

conductor. This difﬁculty can be overcome using DPSM; the technique is explained in Section 8.4. 8.2 MULTILAYERED SYSTEMS Introducing a dielectric layer between two conductors greatly changes all the electrostatic parameters due to the different electric ﬁeld distribution in space. To be able to handle this problem geometry, DPSM uses its powerful surface-reduced technique to model the phenomenon. It means that all different media are reduced to their interfaces, which are then meshed to allow numerical computing and ﬁnd the quasianalytical solution, as it has been demonstrated in the preceding chapters for other physical domains. Figure 8.3 is a sketch of this transformation from a real 3D problem (of Fig. 8.1) to a DPSM model interpretation. After the surfaces are meshed, how should electrostatic ﬁeld cross the frontier of a dielectric layer? DPSM uses the fundamentals of physics to solve this issue by using the boundary conditions at interfaces given by the relations (8.1) where 1 and 2 refer to two different media. N stands for normal ﬁeld and T for tangential ﬁeld. DN1 DN2 ¼ rs DT1 ¼ DT2

ð8:1Þ

297

8.2 MULTILAYERED SYSTEMS

a

b

Workspace frontiers

A4

Frontier

P4 Fixed BC

A3* P3

Workspace

A3

A2* P2 Fixed BC

A2

P1

Bounded interfaces standing for media boundaries

Frontier

A1

Figure 8.3 DPSM modeling of Figure 8.1: (a) 3D view and (b) cut view with DPSM sources layer position.

The electric ﬂux density D is the electric ﬁeld multiplied by the relative permittivity of the medium, and rs are the free charges at the interface. For a perfect dielectric, where there are no trapped charges, rs is null. In DPSM, the conditions (8.1) can be written as relations (8.2), using vectors containing values at peak points. er1 EN1 ¼ er2 EN2 V1 ¼ V2 ðcontinuity of potentialÞ

ð8:2Þ

The normal ﬁelds EN are obtained by a scalar product of the electric ﬁelds and the point normal vectors of the surface (one direction is chosen) at each peak point. It allows to have a nonplanar surface. In Chapter 2, it has been explained that for each boundary condition applied on a surface (modeled by a layer of so-called peak points), one layer of point sources is needed (it can be triplets when it concerns a ﬁeld boundary condition instead of a scalar one). For the multilayered electrostatic systems case, there are two equations (8.2) to solve by virtual charges; it means that each dielectric interface (between two dielectric media) needs two layers of point sources. They are placed on each side of the interface to emit in both directions and to link both media (Fig. 8.4a). When a DPSM interface (meaning two layers of point sources surrounding a layer of peak points) is introduced between two conductors, it creates a discontinuity in the electric ﬁeld. As explained in previous chapters, physical values are calculated in both regions separately with the adequate sources. In Figure 8.4a, an example is drawn where A1 and A2 * emit in medium 1 and A3 * and A2 emit in medium 2, and ﬁnally, A3 and A4 emit in medium 3. In the example of Figure 8.3, the equations are then formulated in system (8.3) (refer to Chapter 7 for notations). The sign ‘0’ represents

298

ADVANCED ELECTROSTATIC PROBLEMS

Figure 8.4 Capacitor with air gap where a DPSM interface has been inserted: (a) DPSM points and (b) electric ﬁeld and potential isoclines.

matrices full of zero meaning that no interaction takes place between the vectors Pi and Aj . 2P 3 1

6 6 6 6 6 6 6 4

2Q

11

6 0 7 7 6 Q21 7 0 7 6 6 M21 7¼6 0 7 6 0 7 6 0 5 4 0 P4 0

0 Q22 M22 Q32 M32 0

Q12 Q22 M22 0 0 0

0 0 0 Q33 M33 Q43

0 Q23 M23 Q33 M33 0

0 3 2 A1 3 6 7 0 7 7 6 A2 7 6 7 0 7 6 A2 7 7 76 7 Q34 7 6 A3 7 7 6 7 M34 5 4 A3 5 Q44

ð8:3Þ

A4

To check that all continuity conditions at the interface are satisﬁed and that there is no interference in the calculation of electrostatic values, there is a simple test to perform: Observe ﬁeld and potential distributions in a structure where the interface separates two identical media. In this case, no discontinuity should appear at the interface. The test results for the structure of Figure 8.4a is shown in Figure 8.4b. There is a good match for electric potential, but there is a discontinuity in the electric ﬁeld that has to be solved to enable the accurate use of DPSM for electrostatic multilayered systems.

8.3 EXAMPLES OF MULTIMATERIAL ELECTROSTATIC STRUCTURE

299

Figure 8.5 Sketch explaining electric ﬁeld bad recovery at interface.

This perturbation in the electric ﬁeld map is due to the mathematical law in 1/r 2 of the electric ﬁeld, where r is the distance between electric ﬁeld source and observation point. Figure 8.5 is a simple sketch where the discontinuity is created because the sources of interface are too close to peak points carrying the boundary conditions of continuity. To obtain a satisfying recovery of electric ﬁelds at interface, the sources should be moved away from the hemisphere centers and be placed at a distance of much more than the radius r of the hemisphere. Experience demonstrated that 2 r is sufﬁcient in the majority of cases. As explained earlier, the sources can be placed anywhere, and the constraints related to wavelength that decided for hemisphere radius are not relevant here. Results of the displacement of sources are shown in Figure 8.6. There is no discontinuity in electric ﬁeld, and the interface is invisible when the surrounding media have the same relative permittivity. The DPSM discretization is well conﬁgured now, and the following sections will develop some typical multilayered applications modeled with DPSM.

8.3 EXAMPLES OF MULTIMATERIAL ELECTROSTATIC STRUCTURE 8.3.1 Parallel-plate capacitor with two dielectric layers In this example, the structure of the capacitor is the same as in the preceding chapter (Section 7.4.3), a 5 mm gap parallel-plate varactor with a square area of 200 mm side length discretized with 900 points. Furthermore, in order to stay as close as possible to real microsystems, one layer of dielectric lying on bottom electrode should be

300

ADVANCED ELECTROSTATIC PROBLEMS

Figure 8.6 Capacitor with air gap where a DPSM interface has been inserted: (a) adapted DPSM modeling and (b) electric ﬁeld and potential isoclines.

considered as the passive layer preventing any short circuit during actuation. An insulator of relative permittivity 4 (glass for instance) with a thickness of 2 mm is chosen. The interface has a 300 mm side length and is meshed with 900 points with a distance between peak points and source points of twice the radius of hemispheres. The capacitance is equal to 106.74 fF compared to the analytical approximated value 106.08 fF obtained with the basic formula (8.4) for two capacitors, one with an air gap of gair ¼ 3 mm and the other with a glass gap gglass. This expression is used in combination with an improvement of the typical approximated capacitance given by expression (8.5)(Bao, 2000) and only valid when the gap is much less than half of the minimum length (our case). Cðgair Þ Cðgglass Þ Cðgair Þ þ Cðgglass Þ e0 er S 2g pL 1þ ln ¼ g pL g

Ceq ¼

ð8:4Þ

CðgÞ

ð8:5Þ

8.3 EXAMPLES OF MULTIMATERIAL ELECTROSTATIC STRUCTURE

301

Figure 8.7 Electrostatic (a) ﬁeld, (b) ﬂux density z components, and (c) potential.

Figure 8.7 represents the electric ﬁeld, the electric ﬂux density, and the potential between the two electrodes. Figure 8.8 shows the charge distribution on the interface and electrodes. 8.3.2 Permittivity-tuning varactors One family of tunable capacitor is based on the tunability of a ferroelectric material (BST for example: barium strontium titanate with different concentration for each species) placed in the gap of a parallel-plate capacitor (other structures with interdigitated comb ﬁngers are also found in the literature depending on the process used for deposition of all layers). When the dielectric lying on the counter electrode has a relative permittivity tunable while a voltage bias is applied between electrodes (like ferroelectric materials), the method is used to get the plot of capacitance versus relative permittivity. Figure 8.9 is an example of tuning for the previously studied capacitor having the same geometry. This kind of study is also useful for a permittivity degrading with time, humidity, or temperature variation. The departure point is a database of permittivity values depending on one or several parameters.

302

ADVANCED ELECTROSTATIC PROBLEMS

Figure 8.8 Charge distribution on (a) top electrode, (b) bottom electrode, and (c) ﬁrst side of interface; (d) second side.

C (fF)

Capacitance versus relative permittivity

107 106.5 106 105.5 105 104.5 104 103.5 4

3.9

3.8

3.7

3.6

3.5

3.4

Er

Figure 8.9 Capacitance variation with relative permittivity.

8.4 MULTICONDUCTORS SYSTEMS: MASKING ISSUES The main issue in multiconductors structures is to solve the equations taking into account the masking areas. Figure 8.10a shows an example of one conductor (represented by two faces called f31 and f32) screening two others (conductors 1 and 2).

303

8.4 MULTICONDUCTORS SYSTEMS: MASKING ISSUES a

Conductor 2 ≡ face f2

Conductor 1 ≡ face f1 f1

f32

f2

b

A1

f31

P1

P31 S3 S32 1

P2

S2

P32

Masked area Conductor 3 ≡ 2 faces Masked area by 3 on 2 in by 3 on 1 in f31 and f32 regards of 1 regards of 2

Figure 8.10 (a) three-conductor system with masking zones and (b) DPSM cut view.

Only the surfaces facing each other are considered in this sketch because the interactions mainly happen in the facing areas of conductors. Notice that the dark zones created on conductors 1 and 2 depend on the orientations of all conductors relative to each other. The global matrix standing for the system is written in (8.6). 3 2 3 2 3 2 Q11 Q12 Q131 A1 0 P1 7 6 7 6 7 6 6 P2 7 6 Q21 Q22 0 Q232 7 6 A2 7 6 7¼6 76 7 ð8:6Þ 6P 7 6Q 6 7 0 Q3132 0 7 4 31 5 4 311 5 4 A31 5 0 Q322 P32 0 Q3231 A32 Figure 8.10b demonstrates that the matrices Q12 and Q21 contain some null coefﬁcients due to the screening effect of conductor 3. To identify these coefﬁcients, a test ! should be automated on Pij Cij vectors to check if they cross a point source layer. One example is shown in Figure 8.10b. Remaining matrices are calculated as usual by the electrostatic analytical formula (Chapter 7). The problem is then solved by inverting the global matrix. Furthermore, when one conductor is moving, one crucial point appears. The distance between points should be as small as the displacement step. Figure 8.11 explains the reason of that limitation. a

b 1 1 3 3 2

⇒ For Q12 and Q21 there is no difference between both positions : no new source is activated. On the another hand, Q13, Q31, Q24, and Q42 changed

2 ⇒ All matrices are modified by the displacement : all interactions are well defined

Figure 8.11 The DPSM meshing of a three-conductor system, one moving between the others:

304

ADVANCED ELECTROSTATIC PROBLEMS

Suspension folded arm

Actuation electrodes

V-shaped comb varactor

Anchor

RF pads

Figure 8.12 SEM view of the V-shaped comb varactor.

8.5 EXAMPLE OF MULTICONDUCTOR SYSTEM The structure presented here relies on the displacement of an isolated conductive comb in the air gap of a V-shaped comb capacitor. The shape is obtained by frontside wet etch of silicon bulk. It has been fully described in an earlier publication (Cruau et al., 2004). Figure 8.12 is a picture taken with a scanning electron microscope. This device has been modeled by DPSM, and Figure 8.13 is an example of how the structure has been approximated (no source points for the conductor placed in the middle of moving combs).

Figure 8.13 The DPSM partial modeling of the V-shaped comb varactor.

REFERENCES

305

Figure 8.14 Plots of capacitance variation for three values of the distance d, compared to the approximation of two half structure in series.

The simulation has been worked out with only one ﬁnger for each comb of width 10 mm and length 100 mm, the minimum distance between ﬁxed combs dis having three values 50 mm, 100 mm, or 150 mm. Figure 8.14 shows the result of the study by comparing capacitance tuning plots. The conclusion is that with a larger distance dis the capacitance is closer to the case of two couple of combs in translation plug in series. Yet, it seems really important to take the inﬂuence between ﬁxed combs into account because the gap with the ideal case is about 50%. The second key point is that for larger dis the tuning ratio r is greater: r ¼ 2.32 at 50 mm, 2.5 at 100 mm, and 2.6 at 150 mm. This numerical study can help us to design the varactor.

REFERENCES Bao M. H., ‘‘Micromechanical Transducers: Pressure Sensors, Accelerometers and Gyroscopes’’, Handbook of Sensors and Actuators, Tome 8, S. Middelhoek, Elsevier, pp. 144–146, 2000. Cruau, A., Lissorgues, G., Nicole, P., Placko, D., Ionescu, A. M., ‘‘V-shaped Micromechanical Tunable Capacitor for RF Applications’’, DTIP 2004 Symposium on Design, Test, Integration and Packaging of MEMS/MOEMS, Montreux, 2004.

9 BASIC ELECTROMAGNETIC PROBLEMS M. Lemistre E´cole Normale Supe´rieure, Cachan, France

D. Placko E´cole Normale Supe´rieure, Cachan, France

9.1 INTRODUCTION A new family of electromagnetic techniques allowing to obtain information on the health of the structures made of composite materials (CFRP and GFRP) has been developed recently. These techniques consist of determining the state of the health of a structure by measuring its two main electric parameters: the electric conductivity and/or the dielectric permittivity; damages induce locally signiﬁcant variation of these two parameters. The goal of Chapters 9 and 10 is to expose these methods and to explain how it is possible to build a model using distributed point source method (DPSM) in order to solve the inverse problem. Chapter 9 is particularly dedicated to basic electromagnetic problems. The basic theory of electromagnetism is ﬁrst reviewed; it is necessary to understand fully the electromagnetic techniques recently developed. Then various techniques are exposed and their applications in the domain of SHM are given.

DPSM for Modeling Engineering Problems, Edited by Dominique Placko and Tribikram Kundu Copyright # 2007 John Wiley & Sons, Inc.

307

308

BASIC ELECTROMAGNETIC PROBLEMS

9.2 THEORETICAL CONSIDERATIONS 9.2.1 Maxwell’s equations All electromagnetism can be explained by Maxwell’s equations; there are four equations that one can write in the time domain, as follows: r e r~ B¼0

r~ E¼

q~ B qt ~ J q~ E B¼ þ v2 r ~ e qt r~ E¼

ð9:1aÞ ð9:1bÞ ð9:1cÞ ð9:1dÞ

In these equations, ~ E, ~ B, and ~ J represent the electric ﬁeld, the magnetic induction, and the current density, respectively, r a charge density, and t a time variable; v is the 1 Þ, where e and m are, respectively, the absolute dielectric propagation velocity ðv2 ¼ em permittivity and the absolute magnetic permeability in the medium, deﬁned as follows: e ¼ e0 er

and

m ¼ m0 mr

ð9:2Þ

In this relation, e0 and m0 are the proportionality coefﬁcients due to the unit system used; in the international system (SI), e0 ¼ 8:84 1012 F/m and m0 ¼ 4p 107 H/m; er (relative dielectric permittivity) characterizes the medium toward electric ﬁeld; mr (relative magnetic permeability) characterizes the medium toward magnetic ﬁeld; in the free space, e ¼ e0 , m ¼ m0 , and v ¼ c ¼ 3 108 m/s. Remark ~ to indicate the magnetic ﬁeld; its relation to magnetic induction ~ One can use H B is ~ ~ ~ can indicate the electric induction with D ~ ¼ e~ B ¼ m H ; likewise D E. The ﬁrst two Eqs. (9.1a) and (9.1b) are static equations: They deﬁne the electric ﬁeld ~ E and the magnetic induction ~ B in terms of the characteristics of the medium. The next two Eqs. (9.1c) and (9.1d) are ‘‘dynamic’’ equations: They show the ~ (or its induction interdependence between ~ E and ~ B. A fact: The magnetic ﬁeld H ~ B) has no real identities; it is uniquely the ‘‘relativistic’’ consequence of a displacement of electric charges. 9.2.2 Radiation of dipoles 9.2.2.1 Electromagnetic ﬁeld radiated by a current distribution According to ~ and electric the deﬁnition of the magnetic potential vector ~ A, the magnetic ﬁeld H

309

9.2 THEORETICAL CONSIDERATIONS

ﬁeld ~ E outside of a conductor having a current density ~ J, are given by ~ ¼ 1r~ A H m 1 ~ ~ ¼ 1 ð~ E¼ rH AÞ joe jome

ð9:3aÞ ð9:3bÞ

The magnetic potential vector is then given by Z

~ A¼

mð~ J ejkr Þ dv 4pr

ð9:4Þ

vol

where r represents the distance between the observation point and the current element pﬃﬃﬃﬃﬃ ~ J dv (volume current density), k being the wave number, k ¼ o me. After transformation in the time domain t, the relation (9.4) yields ~ A¼

Z

m~ J cos o t cr dv 4p

ð9:5Þ

vol

where c is the propagation velocity in the considered medium. 9.2.2.2 Electric dipole Let us consider an inﬁnitesimal current element ~ I dl (see Fig. 9.1). According to the relation (9.4), the magnetic potential vector at point p is given by jkr

me ~ AðpÞ ¼ ð~ I dlÞ a 4pr

ð9:6Þ

with a as the length of the dipole. In spherical coordinates, the relation (9.3a) yields ~ 1 I dla 2 jkr j ~ þ k sin ye HF ¼ kr k2 r 2 4p z p q

r (Idl)a

x y

F

Figure 9.1 Electric dipole.

ð9:7Þ

310

BASIC ELECTROMAGNETIC PROBLEMS

Likewise, for the electric ﬁeld, ~ I dla 2 1 1 jkr ~ k cos ye j 3 3 Er ¼ Z 0 4p k2 r2 k r ~ I dla 2 1 1 1 ~ Ey ¼ Z0 k sin yejkr j þ 2 2 j 3 3 4p kr k r k r Z0 represents the wave impedance in the free space, deﬁned as follows: rﬃﬃﬃﬃﬃ jEj m0 Z0 ¼ ¼ ¼ 120p ðohmÞ e0 jHj

ð9:8Þ ð9:9Þ

ð9:10Þ

Relations (9.7) and (9.8) are effective in near-ﬁeld conditions (i.e., the distance r < the wavelength l). For conditions of far ﬁeld ðr lÞ, the terms having 1/r2 and 1/r3 can be neglected, and these relations yield ~ ~ F ¼ j I dlak sin y ejkr H 4pr ~ I dlak ~ sin y ejkr Ey ¼ Z 0 j 4pr ~ Er ﬃ 0

ð9:11aÞ ð9:11bÞ ð9:11cÞ

9.2.2.3 Magnetic dipole A circular current element (i.e., an elementary loop) can be considered as a magnetic dipole (see Fig. 9.2). This current element produces an ~ are permuted; identical radiation as the electric dipole. However, the vectors ~ E and H the relations of magnetic and electric components for far ﬁeld and near ﬁeld are given as follows: Near ﬁeld kpa2~ 1 1 I 2 ~ j 2 2 k sin y ejkr E F ¼ Z0 kr k r 4p 2~ 2kpa I 2 1 1 jkr ~ H r ¼ Z0 k cos y e j 2 2 3 3 4p k r k r 2~ ~ y ¼ Z0 kpa I k2 sin y ejkr 1 þ j 1 1 H kr k2 r 2 k3 r 3 4p

ð9:12aÞ ð9:12bÞ ð9:12cÞ

Far ﬁeld ~ ~y EF ¼ Z0 H ~r ¼ 0 H 2~

~y ¼ pa I k2 sin y ejkr H 4pr

ð9:13aÞ ð9:13bÞ ð9:13cÞ

311

9.2 THEORETICAL CONSIDERATIONS

z p q r a

x

I

y

F

Figure 9.2 Magnetic dipole.

9.2.3 The surface impedance Let us consider a plane structure made of a material having the following electrical properties:

magnetic permeability m ¼ m0, dielectric permittivity e ¼ e0 er, electric conductivity s, thickness d.

The structure is illuminated by a plane wave with an inclined incidence (see Fig. 9.3). ~i , respecEi and H Electric and magnetic components of the incident wave ~ pi are ~ ~r being the components of the refracted wave ~ pr ; y1 and y2 are the tively, ~ Er and H angles of incidence and refraction, respectively; N1 is the refractive index of the external medium and N2 the refractive index of the structure. y1 and y2 are linked by the following relation: N1 sin y1 ¼ N2 sin y2

ð9:14Þ

Ei Hi

pi

θ1

θ2

Er pr

Hr

N1

N2

Figure 9.3 Illumination of a conductive material.

312

BASIC ELECTROMAGNETIC PROBLEMS

N1 is the index of the external medium; in free space N1 ¼ 1. Taking into account the complex relative permittivity of the material er ¼ e0r j e00r , N2, the index of the material, can be written as follows: N2 ¼

pﬃﬃﬃﬃ er ¼

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ s e0r j oe0

ð9:15Þ

If one considers the material as a good conductive material (i.e., s joe), N2 can be reduced to rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ s N2 ¼ j oe0

ð9:16Þ

Then the relation (9.14) becomes sin y2 ¼

1 sin y1 N2

ð9:17Þ

or

cos y2 ¼

1 1 N2

!1=2

2 2

sin y1

oe0 2 1=2 sin y1 ¼ 1þj s

ð9:18Þ

For good conductive materials, neglecting the second term of the parentheses, the relation (9.18) yields cos y2 ﬃ 1

ð9:19Þ

Therefore, the wave penetrates through the material perpendicular to the surface of the structure ðy2 ¼ 0Þ. Figure 9.4 shows the ﬁeld’s conﬁguration inside the structure. d

E2 x z y

n1

Ea

E1

na

Ha H1

H2 n2

Eb

nb Hb

Figure 9.4 Fields conﬁguration inside a structure of thickness d.

313

9.2 THEORETICAL CONSIDERATIONS

~ ðzÞ can be written as follows: The two ﬁelds ~ EðzÞ and H ~ EðzÞ ¼ ~ Ea expðgzÞ þ ~ Eb expðþgzÞ

ð9:20aÞ

expðgzÞ expðþgzÞ ~ ðzÞ ¼ ð~ H n~ Ea Þ þ ð~ n~ Eb Þ Z Z

ð9:20bÞ

with Z being the wave impedance inside the material deﬁned by the following relation: rﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃ m0 mf p j þ 1 ¼ ð j þ 1Þ Z¼ ¼ e s sd

ð9:21Þ

and g the propagation constant given by g ¼ jo

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ j þ 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃ e m0 ¼ ðj þ 1Þ mf ps ¼ d

ð9:22Þ

where ~ n is the unit vector in the z direction (i.e., the thickness of the material), f the frequency of the incident wave, and d the skin depth deﬁned by the following relation 1 d ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ mf ps

ð9:23Þ

Electric and magnetic ﬁelds tangential to the structure, on each one of its faces ~ E1 , ~2 (see Fig. 9.4) are given by the following equations: ~ E2 , H H1 and ~ Z ~1 þ Z ~ ~2 ~ n1 H n2 H thðgdÞ shðgdÞ Z ~ ~1 þ Z ~ ~2 ~ E2 ¼ n1 H n2 H shðgdÞ thðgdÞ

~ E1 ¼

ð9:24aÞ ð9:24bÞ

One can deﬁne a surface current density ~ Js, which is the integral of the volume current density ~ J in the thickness of the structure. One can write the boundary equations as follows: ~1 H ~2 Þ ¼ ~ ~2 H ~1 Þ ~ n1 ð H n2 ð H Js ¼ ~

ð9:25Þ

For low frequencies such as d d, that is, jgdj 1, one can write the following approximations: shðgdÞ ﬃ gd

and

thðgdÞ ﬃ gd

ð9:26Þ

Relations (9.24a) and (9.24b) yield 1 ~ E2 ¼ ~ Etg ¼ ~ Js E1 ¼ ~ sd

ð9:27Þ

314

BASIC ELECTROMAGNETIC PROBLEMS

V I a b d R=

a V = I σ bd

if

a=b

→

R=

1 σd

= Zs

Figure 9.5 Equivalent electric diagram of the surface impedance.

Etg being the electric ﬁeld tangential to the surface of the structure, it is now possible to deﬁne the surface impedance Zs as ~ Etg ¼ Zs~ Js

ð9:28Þ

with Zs ¼

1 sd

ð9:29Þ

One can see that at low frequencies, the current density is distributed uniformly in the material; then one can neglect the skin effect and resonances inside the material. The condition jgdj 1 (or d > d) deﬁnes the domain of application of the concept of surface impedance; the material is then called ‘‘electrically thin.’’ In the case of d > d, the material is called ‘‘electrically thick.’’ Losses by attenuation inside the material and by reﬂection are then the main phenomena. The surface impedance is given in ‘‘square ohm’’ (symbol c): it is the impedance of a square sample electrically thin, having a conductivity s. One can represent the equivalent electric diagram to the surface impedance by Figure 9.5. 9.2.4 Diffraction by a circular aperture For an incident plane wave, Bethe (1944) has given an approximate analytical representation of diffracted ﬁelds by a small circular aperture in a structure that is considered as inﬁnitely conductive (a good conductive metal such as aluminum can be considered as inﬁnitely conductive), having an inﬁnite surface (i.e., large compared to the diameter of the aperture), with the following assumptions: The size of the aperture is small compared to the wavelength of the incident ﬁeld. The ﬁelds are calculated at a large distance compared to the size of the aperture. ~cc , which are Bethe (1944) has given the concept of ‘‘short-circuit ﬁelds’’ ~ Ecc and H representing the ﬁelds on the aperture loaded by a perfectly conductive material.

315

9.2 THEORETICAL CONSIDERATIONS

Figure 9.6 Incident ﬁelds and equivalent dipoles.

These ﬁelds are deﬁned in the following manner: ~ E0 Ecc ¼ 2~

and

~ cc ¼ 2H ~0 H

ð9:30Þ

~0 are the orthogonal electric component and the tangential magnetic where ~ E0 and H component of the incident ﬁeld, respectively. The diffracted ﬁelds by the aperture are the sum of the radiated ﬁelds by an electric dipole having a moment ~ Pe and a magnetic dipole having a moment ~ Pm . These two dipoles model the orthogonal and tangential ﬁelds, respectively, as shown in Figure 9.6. Let us introduce the concept of ‘‘polarizability’’: the dipolar moments are related to short circuit ﬁelds by the following relations: ~ Ecc Pe ¼ eae~

~ cc and ~ Pm ¼ am H

ð9:31Þ

where ae and am are the electric polarizability and the magnetic polarizability, respectively, given by an aperture of radius a. The polarizability depends on the geometry of the structure and the aperture. For a plane aperture of any geometry, the electric polarizability is a scalar number; the magnetic polarizability is a tensor of rank 1. For a plane circular aperture of radius a, the magnetic polarizability is a diagonal tensor having the two diagonal terms equal. 2 ae ¼ a 3 3 amxy ¼ amyx ¼ 0 4 amxx ¼ amyy ¼ am ¼ a3 3

ð9:32aÞ ð9:32bÞ ð9:32cÞ

With Bethe’s assumptions Casey (1981) has calculated the moment of the magnetic dipole for a plane circular aperture having a radius a, loaded by a nonperfect conductive material such as carbon epoxy. Casey deﬁnes a resistance of ‘‘electrical gasket’’ Rg, that is, the contact resistance between the conductive structure and the material loading the aperture, Rg is given by a product ohm meter (length of contact between material and structure). Casey’s method involves the solution of an integral equation. Two solutions are proposed; the ﬁrst one being exact leads to a semianalytical expression of the magnetic dipole including a coefﬁcient that must

316

BASIC ELECTROMAGNETIC PROBLEMS

be calculated numerically. The second one, obtained by an approximate method, leads to the dipolar moment ~ Pm under the form of the transfer function of a ﬁrst-order low-pass ﬁlter. ~ Pm 1 ¼ ~ Pm0 1 þ j ffc

ð9:33Þ

In this expression, ~ Pm0 is the magnetic dipolar moment given by a free aperture, f represents the frequency of the incident wave, fc is the cut-off frequency given by the following relation: Rg 3 Zs 1þ ð9:34Þ fc ¼ aZs 8m0 a It should be noted here that the cut-off frequency is a function of the surface impedance Zs, and thus the conductivity s of the material (see relation (9.29)). This cut-off frequency is called ‘‘Casey’s frequency.’’ 9.2.5 Eddy currents If a conductive material is subjected to a variable magnetic induction ~ B, the Maxwell’s Eq. (9.1c) shows that an electric ﬁeld ~ E arises inside the material inducing a current density ~ J such that ~ J ¼ s~ E. This current density generates a magnetic B. The magnetic induction ~ Bi can be induction ~ Bi , which is in opposite direction to ~ written with the simpliﬁed Maxwell’s Eq. (9.1d). ~ J v2 r ~ Bi ¼ e

ð9:35Þ

The magnitude of the electric ﬁeld ~ E inside a conductive material and the current density ~ J decrease following an exponential law. rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pmsf ~ ~ d J ¼ J0 e ð9:36Þ where ~ J0 is the surface current density, f the frequency of excitation, and d the thickness of the penetration. This is the skin-effect phenomenon. d Let us write ~ J ¼~ J0 ed . So, d represents the distance where the current density is attenuated by a factor of e1 (i.e., 1/2.718. . ., 1=3). This distance d is called skin depth, deﬁned by the following relation: 1 d ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pmsf

ð9:37Þ

~ One can deﬁne a characteristic frequency fs as ~ J ¼ Je0 (i.e., d ¼ d); this frequency fs is called ‘‘skin frequency.’’

317

9.2 THEORETICAL CONSIDERATIONS

9.2.6 Polarization of dielectrics All materials that have a conductivity s 1020 S/m at room temperature ð 300 KÞ are called dielectrics. At large scale, dielectrics seem to be electrically neutral. However, at microscopic scale, dielectrics show an ‘‘assembly’’ of elementary electric dipoles having a random space orientation. For a large number of dipoles, one can consider dielectrics as statistically neutral. If a dielectric material is subjected to an electric ﬁeld, elementary dipoles tend to orient in the direction of the incident electric ﬁeld; one can deﬁne a polarization vector per unit volume ~ P as: ~ P ¼ Nq~ d

ð9:38Þ

with q representing elementary charges (per atom or molecule), separated by a distance ~ d, and N the number of atoms (or molecules) per unit volume. The product q~ d represents the elementary dipolar moment ~ p for each atom (or molecule); this p ¼ a~ E0 . In this relation, the dipolar moment is related to the local electric ﬁeld ~ E0 as ~ proportionality factor a is called ‘‘polarizability’’: it is a function of electrical characteristics of the medium (i.e., the dielectric) and more precisely of its dielectric relative permittivity er, and it is deﬁned by the Clausius–Mossoti’s relation: a¼

3 er 1 4pN er þ 2

ð9:39Þ

The polarization phenomenon in dielectric medium results from three different sources: electronic polarization, ionic polarization, and orientation polarization. These three sources admit polarizability coefﬁcients ae , ai , and a0 , respectively; the real part of the coefﬁcient a being the sum of the three real parts of elementary polarizability. Electronic polarization arises because the center of local electronic charge cloud E0 . around the nucleus is displaced under the action of the electric ﬁeld ~ Pe ¼ Nae~ Ionic polarization occurs in ionic materials because the electric ﬁeld displaces E0 . positives ions and negatives ions in opposite directions ~ Pi ¼ Nai~ Orientation polarization can occur in materials composed of molecules that have permanent electric dipoles. The alignment of these dipoles depends on 2the temperap , where p is ture and leads to an ‘‘orientational polarizability’’ per molecule a0 ¼ 3KT the permanent dipolar moment per molecule, K is the Boltzmann constant, and T is the temperature. Because of the different nature of these three polarization processes, the response of a dielectric solid to an applied electric ﬁeld will strongly depend on the frequency of the ﬁeld. The resonance of the electronic excitation takes place in the ultraviolet part of the electromagnetic spectrum; the characteristic frequency of the ions vibration is located in the infrared, whereas the orientation of dipoles requires ﬁelds of much lower frequencies (below 109 Hz). This response to electric ﬁeld of different frequencies is shown in Figure 9.7.

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BASIC ELECTROMAGNETIC PROBLEMS

Figure 9.7 Frequency dependence of different contributions to polarizability.

For a low-frequency excitation (i.e., 1 kHz–10 MHz), one can consider that the response of the dielectric medium is quasi-static. So, it is possible to describe the electric ﬁeld inside the dielectric by the following equation: r~ E¼

rl þ rp e0

ð9:40Þ

where rl is the free charge density and rp an apparent density of charges due to the polarization phenomenon. The ﬁeld ~ E can be considered as the resultant of two components, the ﬁeld resulting from the free charges ~ Ef and the ﬁeld resulting from E¼~ Ef þ ~ Ep . Then one can write: the polarization phenomenon ~ Ep such that ~ r~ E ¼ r~ Ef þ

rp e0

ð9:41Þ

One can deﬁne a polarization vector ~ P as rp ¼ r ~ P (Feynman, 1986), Eq. (9.41) can be rewritten as ~ P ~ E ¼ Ef þ e0

ð9:42Þ

The vector ~ P is linked with the incident ﬁeld ~ Ei by the following relation: ~ Ei P ¼ we e 0 ~

ð9:43Þ

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9.3 PRINCIPLE OF ELECTROMAGNETIC PROBE FOR NDE

Measurement point

Excitation electrodes Incident field A

Ei

P

Dielectric material

Induced polarization vector

Figure 9.8 Excitation and measurement conﬁguration applied to a dielectric material.

where we is the electric susceptibility and is linked with the relative electric permittivity er by the following expression: e r ¼ 1 þ we

ð9:44Þ

Taking into account relations (9.42), (9.43), and (9.44), one can write: ~ E¼~ Ef þ ðer 1Þ~ Ei

ð9:45Þ

However, in most of the dielectrics, the term due to the free charges can be neglected and the value of the electric ﬁeld ~ E is reduced to the polarization term Ei . ðer 1Þ~

9.3 PRINCIPLE OF ELECTROMAGNETIC PROBE FOR NDE 9.3.1 Application to dielectric materials Let us consider a dielectric material having an electric relative permittivity er, in the presence of an electric ﬁeld ~ Ei , the material being considered in a macroscopic manner (i.e., quasi-isotropic). The electric ﬁeld ~ Ei induces a phenomenon of polarization inside the material, characterized by the vector ~ P. The total electric ﬁeld ~ ET , measured at point A (see Fig. 9.8) can be written as ~ P ~ Ei þ ¼ ~ Ei þ ðer 1Þ~ Ei ET ¼ ~ e0

ð9:46Þ

After subtraction of the incident electric ﬁeld ~ Ei , one can obtain directly the value of er . Note that the frequency of excitation must be lower than the cut-off frequency of the orientation polarization phenomenon (i.e., 10 MHz).

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BASIC ELECTROMAGNETIC PROBLEMS

Excitation electrodes

Differential measurement electrodes

Figure 9.9 Differential electric probe for detection of damages in dielectric materials.

This method can be applied to all dielectric composite materials such as GFRP, sandwich, among others. An electric probe has been designed (Lemistre, 2001) allowing to detect some defects in dielectric composites. This probe performs a differential measurement between two adjacent zones (see Fig. 9.9). Figure 9.10 shows an example of detection performed on a sample of glass epoxy sandwich having a lack of foam in the middle of the lower part. The right view shows the electric image obtained by scanning the material. 9.3.2 Application to conductive materials 9.3.2.1 Magnetic method The conductive materials such as metallic structures having a conductivity s between 107 S/m and 108 S/m will not be considered here. For those materials the effectiveness of classical methods using eddy currents is well established and is not necessary to demonstrate here. Here we will consider composite materials having a mean conductivity s about 104 S/m such as CFRP, the so-called ‘‘non perfectly conductive materials.’’ One has seen, in Subsection 9.4.2 (relation (9.33)) that the magnetic dipolar moment ~ Pm of an aperture loaded by a nonperfectly conductive material can be written in the form of a transfer function of a ﬁrst-order low-pass ﬁlter. The dipolar

Figure 9.10 Example of defect detection in dielectric materials.

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9.3 PRINCIPLE OF ELECTROMAGNETIC PROBE FOR NDE

r

Material Emission loop

a

x Reception loop

d

Figure 9.11 Excitation with near magnetic ﬁeld.

moment being directly proportional to the magnetic ﬁeld, the terms ~ Pm and ~ Pm0 can be ~ ~ replaced by H and H0 , respectively, in relation (9.33) ~ 1 H ¼ ~ H0 1 þ j ffc

ð9:47Þ

~ the magnetic ~0 is the magnetic ﬁeld measured through a free aperture and H where H ﬁeld measured through a loaded aperture by a conductive material. The cut-off frequency fc being a function of the surface impedance Zs , one has direct access to the value of the conductivity s of the considered material by using the relation (9.34). Let us consider now, a local excitation by a near magnetic ﬁeld (e.g., with a Hertz loop) and a local measurement of the resulting magnetic ﬁeld, as shown in Figure 9.11; one can omit the inﬁnite conductive plane and the contact resistance Rg . ~ and H ~0 : An analytical calculus gives a new transfer function between H ~ ð f ; rÞ H ¼ ~ H0 ð f ; rÞ

r 2 32 Z 1 u2 r 1þ J1 ðuÞeua du f a 0 u þ jf

ð9:48Þ

c

with r the distance between the two loops, a the radius of the emission loop, and J1 the Bessel function of ﬁrst-order, the cut-off frequency fc has the following expression: fc ¼

1:4 pm0 sae

ð9:49Þ

where a represents the radius of the emission loop and e the thickness of the material. The distance between the emission loop and the material d has no inﬂuence on the transfer function; it is possible to put the material immediately ‘‘after’’ the reception loop as shown in Figure 9.12, that is, allowing to test the material only on a single face.

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BASIC ELECTROMAGNETIC PROBLEMS

r

Reception loop

x

a Emission loop

Material

d

Figure 9.12 Excitation and measurement on a single face.

However, in this case, the transfer function following terms T1 þ T2 :

T1 ¼ 1

r 2 32 1þ a !32 2d r 2 1þ a

~ H ~0 H

is given by the sum of the two

r 2 32 Z 1 T2 ¼ 1 þ a 0

u2 uþj

f fc

J1 ðuÞe

u

2d r a du

ð9:50Þ One can state that when f tends toward inﬁnity, the function T1 þ T2 tends toward the constant T1 . So, if the distance d increases, the value of the constant T1 becomes close to the value of the transfer function before attenuation (i.e., f < fc ); this is the problem of the ‘‘lift-off’’ well known in the eddy current techniques. The goal of this kind of analysis is not necessarily to measure the exact value of the conductivity of a material under test, but to detect damages inducing a local variation of this conductivity. A differential probe (Lemistre, 1998, 2001), measuring the ‘‘contrast’’ of the conductivity between two adjacent zones, has been designed (see Fig. 9.13). This probe compares the magnitude of the magnetic ﬁeld between the two zones (1 and 2, see Fig. 9.13); when the two magnetic ﬁelds are different, the voltage Vout is nonzero. The excitation frequency f is set between two characteristic frequencies—the skin frequency fs and the Casey’s frequency fc (relation (9.49)). The excitation frequency must be smaller than the skin frequency fs but greater than the Casey’s frequency fc, in order to have a signiﬁcant variation of the measured magnetic ﬁeld due to the variation of the local conductivity s. ( relation (9.50)). Figure 9.14 shows the evolution of these two frequencies as a function of the thickness of a quasi-isotropic carbon epoxy multilayer structure. By scanning a structure one can build an image where damaged areas appear clearly; an example is given in Figure 9.15. This ﬁgure shows the magnetic image (i.e., s contrast) performed on a quasi-isotropic carbon epoxy multilayer sample of

9.3 PRINCIPLE OF ELECTROMAGNETIC PROBE FOR NDE

323

2 1 Emission loop

Differential reception loop

Figure 9.13 Differential magnetic probe.

60 mm 60 mm 2 mm dimension including a delamination and ﬁber break. The delamination has been performed by a calibrated impact with energy of 3 J. 9.3.2.2 Hybrid method One can consider that a carbon epoxy structure is made up of two different media: a conductive medium, the carbon ﬁbers (conductivity s 104 S/m), and a dielectric medium, the resin (relative dielectric permittivity er 4). The local electric ﬁeld ~ El induced inside a conductive structure by a magnetic induction ~ B can be represented by the following Maxwell’s equation: r~ El ¼

q~ B qt

ð9:51Þ

This electric ﬁeld itself induces a current density ~ J ¼ s~ El. However, in a carbon epoxy Jd . The ﬁrst term ~ Jc medium, the current density ~ J is the sum of two terms, ~ J ¼~ Jc þ ~ (conductive current density) is due to the conductivity of the carbon ﬁbers; the second one ~ Jd (displacement current density) is a transient term due to the polarization phenomenon in the resin epoxy. Let us consider the quasi-static hypothesis, frequency

Figure 9.14 Evolution of fs and fc as a function of the thickness e.

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BASIC ELECTROMAGNETIC PROBLEMS

Figure 9.15 Magnetic image for a carbon epoxy sample with a delamination.

below 107 Hz (i.e., below the cut-off frequency of the orientational polarization phenomenon); taking into account Eq. (9.42), the measured local electric ﬁeld ~ Em can be written as follows: ~ Jc ~ P ~ Em ¼ þ s e0

ð9:52Þ

where ~ P is the polarization vector due to the resin epoxy. With the relation (9.45) one can write: ~ Jc ~ El ðer 1Þ Em ¼ þ ~ s

ð9:53Þ

So the measured electric ﬁeld ~ Em is a function of the conductivity s of the medium and also of its relative dielectric permittivity er. This technique based on the magnetic induction (i.e., eddy currents) and on the analysis of the resulting electric ﬁeld is called the ‘‘hybrid method.’’ The possibility to have simultaneous access to the main electric properties of the medium presents a great potential for carbon epoxy structures used in aeronautical domain, for detecting main damages that are found in these structures. These damages can be classiﬁed into three categories: The damages having a mechanical origin: They are provoked by impacts, inducing delaminations and generally ﬁber breakage. These critical damages are common for carbon (ﬁbers); so in terms of electrical properties, they uniquely induce a local variation of the conductivity.

325

9.3 PRINCIPLE OF ELECTROMAGNETIC PROBE FOR NDE

Differential dipole

Electric field measurement

Inductive current

a

b

Coaxial cable

Ex

50 Ω

Inductive current

Ey

Ey

Electric field measurement

Z = Z0 = 50 Ω

c

Ex

d

Figure 9.16 Electromagnetic probe for hybrid technique.

The thermal damages: They arise either from proximity with a hot body or from an electrical impact (i.e., spark, lightning). In the ﬁrst case one can detect them uniquely by conductivity variation. In the second case, if the burn is light, there is not much s variation but only er variation due to the pyrolysis phenomenon of the resin. Nevertheless, this kind of damage generally affects the two electrical parameters. The damages resulting from liquids ingress (water, oil, fuel, etc.): They can be critical due to the fact that the liquid can start a chemical reaction and weaken the structure. This kind of damage uniquely induces er variation due to the presence of a new medium having other dielectric permittivity (i.e., the liquid). From this hybrid concept, a new probe has been designed (Lemistre, 2004). This probe can be considered as a combination between the electric probe and the magnetic probe, including some improvements. Figure 9.16a presents a photograph of the

326

BASIC ELECTROMAGNETIC PROBLEMS

probe and Figure 9.16b shows a schematic view of the probe, designed with a dielectric parallelepiped ð3 cm 3 cm 1 cmÞ including on one face the inductive coil and on the opposite face the differential dipole, for the measurement of the electric ﬁeld. The inductive coil appears as a Moebius loop made with a hard coaxial cable (see Fig. 9.16c). This geometry allows forming a double loop of induction with preservation of real impedance equal to the characteristic impedance of the coaxial (i.e., 50 ), for all used frequency domain, from 100 kHz to 10 MHz. The measurement unit appears as double-crossed dipole (see Fig. 9.16d) that allows to perform differential measurements with a sensitivity to two orthogonal components of the tangential Ey . electric ﬁeld ~ Ex and ~ The operating frequency f is not submitted to the same imperative as magnetic measurement. It is only necessary to operate with a frequency below the orientational polarization phenomenon cut-off (i.e., f < 107 Hz, see Subsection 9.2.6). However, the magnitude of the electric ﬁeld induced inside the material is directly proportional to the frequency of the inductive magnetic ﬁeld (Eq. (9.51)); it is preferable to use a frequency as high as possible. Nevertheless, a too high frequency does not allow penetration through the total thickness of the material such as carbon epoxy multilayer, due to the skin effect. For this kind of material, the good frequency domain is between 100 kHz and 10 MHz. Figure 9.17a shows the electromagnetic image of a quasi-isotropic sample of carbon epoxy ð200 mm 150 mm 4 mmÞ with various burns. These results are compared with a C scan ultrasonic image shown in Figure 9.17b. Only the electromagnetic hybrid investigation is able to detect light burns.

Figure 9.17 Investigation of a composite sample with various burns.

9.4 ELECTROMAGNETIC METHOD FOR SHM APPLICATIONS

327

9.4 ELECTROMAGNETIC METHOD FOR STRUCTURAL HEALTH MONITORING (SHM) APPLICATIONS 9.4.1 Generalities In the domain of SHM, the goal is to determine at all times, the state of the health of the structure in use. So it is necessary to integrate multiple sensors inside the material during the manufacturing process, in order to obtain the so-called ‘‘smart structure.’’ Generally, this process consists of integrating an active layer including various sensors (e.g., PZT sensors (Giorgiutiu, 2002; Ihn, 2001; Lemistre, 2003; Osmont, 2001) or optical ﬁbers (Leblanc, 1995; Guemes, 2004) between two layers of the composite structure. In the case of electromagnetic methods, which are adaptations of the methods used in NDT/NDE ﬁeld, the active layer is a ﬂexible layer including wire networks, having 100 mm to 200 mm of thickness. This layer can be bonded on the inner face of the structure or inserted between two layers of a multilayer structure. As for NDT/NDE applications, there are two main possible techniques: the hybrid technique and the electric technique. The method is selected based on the kind of material to be inspected. The electric method being based on the variation of the dielectric permittivity is used for dielectric structures such as glass epoxy structures and sandwich structures. Because the hybrid method is based on both the conductivity variation and the dielectric permittivity variation, it is theoretically possible to use it for all kinds of materials. However, the electric method being better adapted for dielectric structures, it will be preferred in this case. The hybrid method gives the best results when it is used with composite structures that have both conductive medium and dielectric medium.

9.4.2 Hybrid method The hybrid method is the most interesting electromagnetic method; it allows building an active and fully integrated SHM system. Its simultaneous measurement of both electric parameters of a structure (i.e., conductivity and permittivity) allows it to determine the origin of a damage and fully characterize various damages, and it can compute the new values of local conductivity and local dielectric permittivity in the damaged zone by using DPSM. This technique of simulation allows solution of the inverse problem, which is fully explained in Chapter 10. The SHM system using the hybrid electromagnetic method is composed of two principal elements: an active layer called HELP-Layer1 (Hybrib ELectromagnetic Performing Layer) including a circuit inducing eddy currents inside the structure and a circuit for the measurement of the resulting electric ﬁeld, and an electronic system to generate the excitation signal, to acquire the resulting signal, to perform the data reduction, to solve the inverse problem, and to build the images of structures where the damages appear clearly (Lemistre, 2002). The HELP-Layer1 includes two orthogonal wire networks printed on a 200 mm thick ﬂexible substrate. One of these networks, being short circuited, represents a

328

BASIC ELECTROMAGNETIC PROBLEMS

Kapton

200 μm

E field sensors Inductive Lines

Figure 9.18 Geometry of the HELP-Layer1.

network of induction loops; the other one, which is open circuited, can be considered as a network of capacitances; it is sensitive at the electric ﬁeld. The geometry of the HELP-Layer1 is shown in Figure 9.18. Figure 9.19 represents an instrumented structure with the HELP-Layer1 bonded on the inner face. A damaged structure and the image of the damaged structure obtained with this method are given in Figures 9.20 and 9.21, respectively. The image (Fig. 9.21) is obtained with a 16-ply orthotropic carbon epoxy plate to the [02, 902]2s with dimensions 610 mm 305 mm 2 mm. The plate has been damaged by six various defects: a 4 J impact (I2) inducing a severe delamination with ﬁber breaks, a 2 J impact (I1) inducing a light delamination, and four local burning performed by ‘‘high energy sparks’’ (30 V, 5 A) with various duration, given energies of 40 J (B2), 80 J

Figure 9.19 An instrumented structure with the HELP-Layer1.

9.4 ELECTROMAGNETIC METHOD FOR SHM APPLICATIONS

329

Figure 9.20 Damaged structure.

(B3), 120 J (B1), and 400 J (B4) (see Fig. 9.20). The magnetic ﬁeld sensor network is excited by a frequency of 700 kHz. A computer simulation of the system by DPSM allowing to calculate each component of the electric ﬁeld shows that a local variation of the conductivity affects uniquely the component of the electric ﬁeld that is parallel to the induction wire. On the contrary, a local variation of the dielectric permittivity affects both components. If one assumes that a damage having a mechanical origin, such as an impact, induces a ﬁber break phenomenon that is inducing a variation of the conductivity, then other kinds of damages, such as burning or liquid ingress, induce a variation of dielectric

Figure 9.21 Electromagnetic image of the damaged structure.

330

BASIC ELECTROMAGNETIC PROBLEMS Measurement network

A B

Ey

2-D HELP-Layer®

Ex

V A,B = f(E y )

V A = VB = f (Ex ) Inducing network

Figure 9.22 Electric ﬁeld components measurement process.

permittivity. So, it is possible to determine the origin of damages from the measurement of both components of the resulting electric ﬁeld. Practically, the ﬁrst measurement (i.e., ~ Ey , component parallel to the induced current) will be performed between two wires in differential mode; in this case, one will obtain the electric ﬁeld component orthogonal to the measurement wires (a pair of wire being considered as an elementary capacitance). The second measurement (i.e., ~ Ex , component orthogonal to the induced current) will be performed by uniquely using one wire in the common mode; in this second case, one will obtain the electric ﬁeld component parallel to the measurement wires, one wire being considered as an elementary capacitive antenna (see Fig. 9.22). A computation using a DPSM model allows to determine the kind of damages (i.e., impact, burn, etc.) and to determine the new values of conductivity and permittivity on the damaged zones that one can link with the mechanical performances of the structure and gives the solution of the inverse problem (Lemistre 2004b). 9.4.3 Electric method This method dedicated to dielectric structures such as GFRP, sandwich, among others is not usable in aeronautic domain. This is because the purely dielectric materials are used precisely because they are dielectric and very often these materials have a function of protection of various antennas (communications, navigation systems, RADAR, etc.). It is obvious that these materials cannot be masked by a conductive structure such as wire networks. If this technique gives good results in the ﬁeld of applications of NDT/NDE, it is not adaptable for an SHM system in the domain of aeronautics. However, in the naval domain, where many structures are made of glass epoxy, this technique is fully applicable and very interesting. The main difference of the electric version of the HELP-Layer1 is that the network of induction is not short circuited in order to induce directly an electric ﬁeld inside the structure. So, the resulting electric ﬁeld measured gives only an information about the contrast of the relative dielectric permittivity of the medium, induced by damages.

REFERENCES

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REFERENCES Bethe, H.A., ‘‘Theory of Diffraction by Small Holes’’, Physical Review, Vol. 7–8, pp. 367–377 (1944). Casey, K.F., ‘‘Low Frequency Electromagnetic Penetration of Loaded Apertures’’, IEEE Transaction on Electromagnetic Compatibility, Vol. EMC-23 (4), pp. 367–377, 1981. Feynman, R.P. Electromagne´tisme, Vol. 1–2, InterEditions, Paris, 1986. Giorgiutiu, V., et al., ‘‘Piezoelectric Wafer Embedded Active Sensors for Aging Aircraft Structural Health Monitoring’’, 1st EWSHM, pp. 41–62, 2002. Guemes, A., et al., ‘‘Fiber Optic Sensors for Hydrogen Cryogenic Tanks’’, Proceedings of the 2nd EWSHM, 2004. Ihn, J.B., and F.K., Chang, ‘‘Built-in Diagnostics for Monitoring Cracks Growth in Aicraft Structures’’, 3rd IWSHM, pp. 284–295, 2001. Leblanc, M., and R.M., Measures, ‘‘Fiber Optic Bragg Intra-Grating Strain Gradient Sensing’’, Proceedings of SPIE 2444, pp. 136–147, 1995. Lemistre, M., R., Gouyon, and D., Balageas, ‘‘Electromagnetic Localization of Defects in Carbon Epoxy Materials’’, Proceeding of SPIE, Vol. 3399, pp. 89–96, 1998. Lemistre, M.B., and D.L., Balageas, ‘‘Electromagnetic Structural Health Monitoring for Composite Materials’’, Structural Health Monitoring, The Demands and Challenges, Ed. Fu-Kuo Chang, CRC Press, pp. 1281–1290, 2001. Lemistre, M., and D., Balageas, ‘‘A New Concept for Structural Health Monitoring Applied to Composite Materials’’, Structural Health Monitoring, DEStech publications, Ed. D.L. Balageas, pp. 493–507, 2002. Lemistre, M.B., and D.L., Balageas, ‘‘Hybrid Electromagnetic Acousto-Ultrasonic Method for SHM of Carbon/Epoxy Structures’’, 4th IWSHM, pp. 153–160, 2003. Lemistre, M., and A., Deom ‘‘De´tection de bruˆlures dans les composites a` base de carbone’’, Nouvelles me´thodes d’instrumentation, Ed. H. Lavoisier, Vol. 2, pp. 305–312, 2004. Lemistre, M., and D., Placko,‘‘Evaluation of the Performances of an Electromagnetic SHM System for Composite, Comparison Between Numerical Simulation, Experimental Data and Ultrasonic Investigation’’, Proceedings of SPIE, Ed. T. Kundu, Vol. 5394, pp. 148–156, 2004. Osmont, D.L., et al., ‘‘Piezoelectric Transducer Network for Dual-Mode Detection, Localization and Evaluation of Impact Damages in Carbon/Epoxy Composites Plates’’, Proceedings of SPIE, Vol. 4073, pp. 130–137, 2000.

10 ADVANCED ELECTROMAGNETIC PROBLEMS WITH INDUSTRIAL APPLICATIONS M. Lemistre and D. Placko E´cole Normale Supe´rieure, Cachan, France

10.1 INTRODUCTION In the structural health monitoring (SHM) domain, the challenge is to link the mechanical properties of a structure to the characteristics of damages detected. In the case of an electromagnetic SHM system such as the HELP-Layer1 (Lemistre and Placko, 2003a), it is necessary to link the mechanical properties to the electrical properties (i.e., electric conductivity s and dielectric permittivity e) due to the presence of damages, in order to determine the potential of use of the instrumented structure. In other words, to solve the inverse problem (Lemistre et al., 2003); (Lemistre and Placko, 2004) is our goal. The ﬁrst step necessary (but not sufﬁcient) to solve the inverse problem is to have at our disposal a model able to simulate the behavior of the SHM system; the distributed point source method (DPSM) is particularly adapted to this part of the problem. In this chapter, we will see at ﬁrst how to compute the electric ﬁeld resulting from a magnetic induction inside a complex medium such as a CFRP1 structure, including a part of conductive material (the carbon ﬁbers) and a part of dielectric (the resin). Next,

1

Carbon Fiber Reinforced Plastic

DPSM for Modeling Engineering Problems, Edited by Dominique Placko and Tribikram Kundu Copyright # 2007 John Wiley & Sons, Inc.

333

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ADVANCED ELECTROMAGNETIC PROBLEMS WITH INDUSTRIAL APPLICATIONS

we will compute the incidence of a variation of s and/or e induced by a damage. At last, we will build an algorithm to determine the real variation of s and/or e due to a real damage detected by the HELP-Layer1 system.

10.2 MODELING THE SOURCES 10.2.1 Generalities In Chapter 9 we have seen that the HELP-Layer1 system includes two crossed conductive networks (see Fig. 9.18). One of these networks having in charge to induce eddy currents inside the carbon structure appears under the form of parallel lines, with a spacing of 20 mm, short circuited at one end. In order to induce signiﬁcant eddy currents inside the structure (i.e., a signiﬁcant electric ﬁeld), it is necessary to inject a current of 1 A to each line alternately. The frequency of excitation is chosen taking into account the depth of the structure (i.e., the skin effect, see Section 9.2.5). To simplify the problem, only one element of the HELPLayer1 of 60 mm 60 mm is modeled including only one inductive line; the structure is set 0.1 mm above the HELP-layer1. The current source is an inductive line of 60 mm length located in the x0y plane at x ¼ 30 mm and y ¼ 0–60 mm, meshed by 60 current elements I dl (see Fig. 10.1). This current source will be called ‘‘DPSM primary sources JAp’’ (see Fig. 10.2). The formulation used is the DPSM/Green formulation described in Chapter 6. The ﬁrst computation consists of calculating the electromagnetic ﬁeld on the ~ ) or more precisely the magnetic vector potential surface of the structure (~ E and H ~ A1 , in medium 1 (free space) computed by superimposing the effect of current sources JAp and JA1 (DPSM virtual sources). The magnetic vector potential in medium 2

Structure

60 40

z

HELP-Layer

y 20

x 0 20

40

60

Inductive line

Figure 10.1 Geometry of modeling for primary source.

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10.2 MODELING THE SOURCES

DPSM virtual sources JA1

Medium 2 (structure)

z x DPSM secondary sources JAs Medium 1 (free space)

DPSM primary sources (JAp) HELP-Layer®

Figure 10.2 DPSM meshing geometry.

(the structure) ~ A2 is uniquely deﬁned due to the current sources JAs and called ‘‘DPSM secondary sources.’’ 10.2.2 Primary source The magnetic vector potential A given by a current I at distance r can be written by the following relation: ð A¼

m0 expðj k0 rÞ I dl 4pr

ð10:1Þ

where Idl denotes each current element, r the distance of observation, and k0 ¼ oc the wave number in free space. Close to the surface of the structure the matrix of Ax,y called QA can be written as follows: QA ¼

X m expðj k0 rx;y Þ 0 I dln 4 p rx;y n

ð10:2Þ

The vector potential being collinear to the inductive current I, components Ax and Az are equal to zero. Figure 10.3 shows the magnitude of the component Ay computed at the structure location. 10.2.3 Boundary conditions In Chapter 6, we have seen that there are continuity on the vector potential and its ﬁrst derivative along z-axis: 8 > A2 A1 ¼ ~ <~ 1 @~ A1 1 @~ A2 > ¼ : m1 @z m2 @z

ð10:3Þ

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Figure 10.3 Ay component of the magnetic vector potential, computed at the location of the structure.

In the case of harmonic excitation, one can compute the electric ﬁeld by the following relation: ~ E ¼ jo~ A

ð10:4Þ

Due to the fact that there is no z component (i.e., normal component to the surface), there is continuity on the only component (y) of the electric ﬁelds ~ E1 and ~ E2 (in medium 1 and in medium 2, respectively), one can write E1 þ ~ E2 þ ~ P1 Þ ¼ ðe0~ P2 Þ ðe0~

ð10:5Þ

P2 denote polarization vectors in media 1 and 2, respectively. If where ~ P1 and ~ assuming that there is no polarization phenomenon in free space, Eq. (10.4) yields E1 ¼ e~ Ee e0~

ð10:6Þ

~ where ~ Ee denotes an equivalent electric ﬁeld such as ~ Ee ¼ ~ E2 þ eP0 , e being the absolute dielectric permittivity (e ¼ e0 er ). Taking into account the conductivity r of the medium 2, e can be written as follows: 0

e ¼ e0 e r j

s o

ð10:7Þ

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10.2 MODELING THE SOURCES

In terms of current density ~ J, one can write the expression (10.4) as follows: ~ JT e0 ~ E1 ¼ e s

ð10:8Þ

The total current density ~ J T includes the conductive current density ~ J c and the ~ displacement current J p due to the polarization process. However, the polarization vector inside the dielectric part of the structure ~ P ¼ a~ Ey is not collinear to the electric ﬁeld due to the fact that the polarizability a is a tensor. So, the polarization vector in the x0y plane can be written as follows: ~ a E Px xx axy ~ x ð10:9Þ ¼ ~ ~ ayx ayy Py Ey The ~ Ex component of the excitation ﬁeld in medium 1 being equal to zero, relation (10.9) yields ~ Px ¼ axx ~ Py ayx

axy 0 ayy ~ Ey

ð10:10Þ

The polarization vector components are equal to ~ Ey Px ¼ axy ~ ~ ~ Py ¼ ayy Ey

ð10:11Þ

Therefore, the y component of the electric ﬁeld inside the structure will be equal to the sum of the conductivity effect s in the carbon ﬁbers plus the y component of the polarization vector ~ Py =e0, the x component being uniquely equal to the x component of E1;y is the y component of the electric ﬁeld in medium 1 the polarization vector ~ Px =e0. If ~ ~ (i.e., free space), E2x the x component due to the polarization in medium 2 (i.e., the structure), and ~ E2y the y component due to the conductivity of the carbon, one can write ~ Py ~ E1y E2y þ ¼ ~ e0 ~ Px E2x ¼ e0

ð10:12Þ

ayy ~ E2y ~ ~ ¼ E1y E2y þ e0 E2y ~ E2x ¼ axy e0

ð10:13Þ

or

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ADVANCED ELECTROMAGNETIC PROBLEMS WITH INDUSTRIAL APPLICATIONS

As we have seen in Chapter 9, Subsection 9.2.6, the polarizability a can be expressed by the following relation: a ¼ e0 ðer 1Þ

ð10:14Þ

~ E2y ¼ ~ E1y E2y þ ðer;yy 1Þ ~ ~ ~ E2x ¼ ðer;xy Þ E2y

ð10:15Þ

So, relations (10.13) become:

Unfortunately, inside this complex medium that includes a part of conductive material and a part of dielectric material, it is impossible to separate the component ~ E2y due E2y due to the polarization; we to the conductivity and the component ðer;yy 1Þ ~ only know the sum of these two components ~ E2ye , the equivalent component y of the ﬁeld in medium 2. If we assume that there is no conductivity in the dielectric part of medium 2 and taking into account the relation (10.7), we can write the following approximation:

0 s ~ ~ E2ye er j ¼ E1y E2ye er ¼ ~ o 0 ~ E2x ¼ ~ E2ye er

ð10:16Þ

Figure 10.4a and b presents the real part and the imaginary part respectively of the y component of the electric ﬁeld on the surface of the structure ~ E2y . Figure 10.5a and b shows the same parameters for the x component of the electric ﬁeld ~ E2x . One can E2y components are practically identical except that remark that the ~ E2x and the ~ ~ E2x > ~ E2y ; the electric ﬁeld due to the polarization is the major contribution; this is

Figure 10.4 y component of the electric ﬁeld on the surface of the structure: (a) real part and (b) imaginary part.

10.3 MODELING A DEFECT INSIDE THE STRUCTURE

339

Figure 10.5 x component of the electric ﬁeld on the surface of the structure: (a) real part and (b) imaginary part.

due to the fact that the conductivity r is relatively high (104 Sm1), so the electric ﬁeld due to the conductivity phenomenon is very weak.

10.3 MODELING A DEFECT INSIDE THE STRUCTURE If we assume that the defect is located at 1 mm depth inside the structure generating a variation of one electric parameter of the structure (i.e., r or e). The ﬁrst step will be to construct the secondary sources JAs inside the structure. Figure 10.6 shows the

Figure 10.6 Vector potential at 1 mm depth inside the structure.

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DPSM virtual sources JA2

Medium 3

JAs2 Surface of the structure

Medium 2

1 mm

z x

DPSM secondary sources JAs

Figure 10.7 New DPSM sources.

magnitude of the magnetic vector potential located at 1 mm depth inside the structure. After that, we perform the same process as presented in Subsection 10.2.1, with new virtual sources JA2 and new secondary sources JAs2, medium 2 being the structure and medium 3 the damaged zone of the structure (see Fig 10.7). Note that we do not

Figure 10.8 Electric ﬁeld resulting from a damaged area by s variation, computed at the interface structure/damage: (a) y component, real part, (b) y component, imaginary part, (c) x component, real part, and (d) x component, imaginary part.

10.3 MODELING A DEFECT INSIDE THE STRUCTURE

341

Figure 10.9 Vector ﬁeld of induced current: (a) damaged structure and (b) sound structure.

Figure 10.10 Electric ﬁeld resulting from a damaged area by er variation, computed at the interface structure/damage: (a) y component, real part, (b) y component, imaginary part, (c) x component, real part, and (d) x component, imaginary part.

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Figure 10.11 Vector ﬁeld of induced current: (a) damaged structure and (b) sound structure.

Figure 10.12 Contour lines of the modulus of the electric ﬁeld resulting from a damage generating a s variation: (a) y component at the level of the damage, (b) x component at the level of the damage, (c) y component at the measurement surface, and (d) x component at the measurement surface.

10.3 MODELING A DEFECT INSIDE THE STRUCTURE

343

solve the global problem but a new elementary problem with new DPSM sources JAs2. These sources are computed taking into account a local variation of r and/or er. Figure 10.8a–d shows the electric ﬁeld resulting from a damage located at x ¼ 30–40 and y ¼ 20–30. The damage is simulated by a local variation of conductivity: sound zone r1 ¼ 104 Sm1, damaged zone r2 ¼ 5 103 Sm1. Figure 10.9a presents the vector ﬁeld of the induced current in this damaged structure; for comparison Figure 10.9b shows the vector ﬁeld of the induced current in a sound (defect-free) structure. Figure 10.10a–d presents the electric ﬁeld resulting from a damage simulated by a local variation of relative dielectric permittivity (sound area er ¼ 4, damaged area er ¼ 2), the damaged area having the same location as the previous case. Figure 10.11a and b presents the vector ﬁeld of induced current.

Figure 10.13 Contour lines of the modulus of the electric ﬁeld resulting of a damage generating a er variation: (a) y component at the level of the damage, (b) x component at the level of the damage, (c) y component at the measurement surface, and (d) x component at the measurement surface.

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The last step of the simulation consists of propagating the ﬁeld due to the sources JAs2 up to the surface of the structure, at the location of the measurement surface, by the same process as the previous one. Figure 10.12a–d shows the contour lines of the modulus of the electric ﬁeld at the structure/damage interface (a and b) and at the surface of the structure (c and d), for a damage resulting a r variation. Figure 10.13a–d presents the same views, resulting an er variation. If we take the difference between the electric ﬁeld generated by a sound structure and a damaged structure (at the measurement surface), we clearly see the damage. Figure 10.14a–d shows the results obtained for each kind of damage. These ﬁgures represent for each kind of damage, the effect on the two measured components of Ey . One can see that in the case of a damage having generated a electric ﬁeld ~ Ex and ~ variation of the electric conductivity r, the y component of the electric ﬁeld is Ex , respectively where dominating. In Figure 10.14a and b, showing ~ Ey and ~ the magnitudes are given in V mm1, the maximum value of the y component is

Figure 10.14 Modulus of the difference between the electric ﬁelds generated by a sound structure and a damaged structure: (a) variation of s, y component, (b) variation of s, x component, (c) variation of er , y component, and (d) variation of er, x component.

345

10.4 SOLVING THE INVERSE PROBLEM

103 V mm1 whereas the maximum value of the x component is only 1:3 107 V mm1. On the contrary, in the case of a damage having generated a variation of the dielectric permittivity e , the major contribution on the electric ﬁeld is given by the x component (see Fig. 10.14c and d), the maximum value of the y component is only 2:5 108 V mm1 so the x component is 3 103 V mm. This is a very important result because it allows us to determine the kind of damage (i.e., the origin of the damage, see Chapter 9, Subsection 9.3.2.2).

10.4 SOLVING THE INVERSE PROBLEM The preceding remarks allow us to develop an algorithm in order to solve the ﬁrst part of the inverse problem (i.e., computation of the new values of r and er and determination of the kind of damage), see Figure 10.15. The ﬁrst step helps us to determine if the modulus of the x component of the electric ﬁeld measured by the HELP-Layer1 system Exm is signiﬁcant (i.e., > 103 V.m1). If the Exm component is not signiﬁcant, the damage has mechanical origin (delamination, ﬁber breaking, and crack) so one computes the modulus of the Ey component (resulting from the difference: sound structure – damaged structure) with variation of s value and compares it to the experimental Eym component. When the computed Ey value is equal to the experimental Eym value, the last value of r is the local conductivity of the structure due to the damage. If the Exm component is signiﬁcant, the damage has no mechanical origin; then, it may be due to burning, liquid ingress and so on. One can compare the computed Ex value with the experimental Exm value and determine the local permittivity er due to the damage, by the same process. Then, one can perform the

x

Exm

y Non mechanical damage

Y

Significant Exm

Measured electric fields

N

Mechanical damage

Comput. Ex

Comput. Ey

εr Variation

σ Variation N N

Ex = Exm

Y

Ey = Eym

Y

Ey m εr Value

σ Value

Figure 10.15 Algorithm allowing to solve a part of the inverse problem.

σ Value

346

ADVANCED ELECTROMAGNETIC PROBLEMS WITH INDUSTRIAL APPLICATIONS

TABLE 10.1.

Equivalent r and er for impact delaminations

Impact

Measured ﬁelds

Deduced properties

energies(J)

Ex (V m)

Ey (V m)

r (S m)

er

0.75 2 2.5 3 4

NS NS NS NS NS

NS NS 1:1 102 1:6 102 5:0 102

X X 8 103 7 103 4:7 103

X X X X X

same process with the Ey components, in order to determine the possible local variation of r. The ﬁrst evaluation concerns a quasi-isotropic plate of 2 mm thickness [452 =02 =452 =902 s , including various delaminations generated by calibrated impacts (impact energies of 0.75 J, 2 J, 2.5 J, 3 J, and 4 J). The HELP-Layer1 system measures the two components of the electric ﬁeld (i.e., Exm and Eym); the simulation program computes the values er and r on the damaged area corresponding to the same electric ﬁeld component variations. For a sound structure, the electrical parameters are er ¼ 4 and r ¼ 104 S.m1. Table 10.1 (NS ¼ non significant values and X ¼ same values as a sound structure) shows the results obtained. One can see that a delamination resulting from an impact having energy lower than 2.5 J does not provoke a ﬁber breaking phenomenon and consequently cannot induce a variation of r. This damage will not be detected. The threshold of detection for this kind of defect is a delamination resulting from an impact of about 2.5 J for a 2 mm thick plate of such a composite. The second evaluation concerns a 2 mm thick orthotropic plate [02 =902 =02 =902 ]s, including various burning generated by electric sparks, for electric energies of 10 J, 40 J, 80 J, and 120 J. The ﬁeld variations measured are given in Table 10.2. Assuming the same sound material properties as in the ﬁrst case, the simulation leads to the values of er and r as given in this table. Here, an extra hypothesis was necessary. It was assumed that in the case of light burning, the low variation of the conductivity of the resin is masked by the relatively high conductivity of the carbon and that in the case of hard burning inducing a ﬁber-breaking phenomenon, the increase of r in the resin due to the pyrolysis and the decrease of r in the carbon ﬁber bundles compensate one another. Based on these considerations, it has been assumed that only the permittivity er was affected. In fact, it would have been more accurate if we could let the two TABLE 10.2. Electric energies (J) 10 40 80 120

Equivalent r and er for electric burning Measured ﬁelds Ex (V m) 0.07 0.26 0.53 0.80

Deduced properties

Ey (V m) 3

1:8 10 6:0 103 1:3 102 2:0 102

r (S m)

er 3.93 3.80 3.56 3.39

REFERENCES

347

parameters er and r vary. However, this would require the establishment of a new experimental procedure allowing the identiﬁcation of both parameters. This improvement of the method is one objective for future developments.

10.5 CONCLUSION This chapter dedicated to an industrial application shows that the DPSM (Placko et al., 2003) can be used for simulating complex applications. In this case we considered a complex medium like a composite material. It was necessary to make various approximations. We have not solved the global problem (Placko et al., 2005) but a succession of simple problems by using new sources at each step of the computation (i.e., at each new medium). We have used Born’s hypothesis that the damage does not affect the incident ﬁeld for determining the terms of the polarization tensor. In spite of these approximations, the results obtained are highly comparable to the experiences. The DPSM thus allows us to solve an important part of the inverse problem with a great facility. The time of computation including several iterations is of the order of 1 min; calculation was performed with a good laptop computer.

REFERENCES Lemistre, M. B., and D., Placko, HELP-Layer System, 2003, French Patent N FR0403310, March 30, 2004, US Patent N US2005/0228208A1, March 20, 2005. Lemistre, M. B., D., Placko, and N., Liebeaux, ‘‘‘Simulation of an Health Monitoring Concept for Composite Materials, Comparison with Experimental Data’’, Proceedings of SPIE, Vol. 5047, pp. 130–139, 2003. Lemistre M., and D., Placko,‘‘Evaluation of the Performances of an Electromagnetic SHM System for Composite, Comparison Between Numerical Simulation, Experimental Data and Ultrasonic Investigation’’, Proceedings of SPIE, Ed. T. Kundu, Vol. 5394, pp. 148–156, 2004. Placko, D., N., Liebeaux, and T., Kundu, ‘‘Proce´de´ pour e´valuer une grandeur physique representative d’une interaction entre une onde et un obstacle’’. Patent Application N 02 14108 ENS Cachan/CNRS/Universite´ d’Arizona, November 8, 2002, European and USA Extension November 10, 2003. Placko, D., N., Liebeaux, A., Cruau, and T., Kundu, ‘‘Proce´de´ universel de mode´lisation des interactions entre au moins une onde et au moins un objet, la surface de chaque objet de´ﬁnissant une interface entre au moins deux milieux’’, French Patent N 05 13219 ENS/ CNRS/UNIV of Arizona, De´cembre 23, 2005.

11 DPSM BETA PROGRAM USER’S MANUAL A. Cruau and D. Placko Ecole Normale Supe´rieure, Cachan, France

Welcome to the beta version manual of the distributed point source method based program (called DPSM program in this document). It has been written in MATLAB 6.1 language to be used as an executable program for MATLAB 6.1. So, in this document MATLAB stands for the version 6.1. These few pages contain explanations, advice, and guidelines to succeed in carrying out good modeling with the DPSM program. Let us begin the discussion by having a brief overview of how DPSM program is built and how it works.

11.1 INTRODUCTION The DPSM program has been written to model 3D systems containing only planar objects superposed in the z direction (further implementations could be more complex). Its basic structure has been patented in January 2006. It is characterized by the following steps:

choice of the physical domain of the interactions happening in the system, deﬁnition of the properties of the media composing the system, creation and point meshing of objects delimiting the media, association of two media per object (above and beside), determination of the type of boundary conditions for each object,

DPSM for Modeling Engineering Problems, Edited by Dominique Placko and Tribikram Kundu Copyright # 2007 John Wiley & Sons, Inc.

349

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DPSM BETA PROGRAM USER’S MANUAL

building the global matrix containing all coupling relations between objects and which depends on physical domain, boundary conditions, media properties, and geometric positions of objects relative to each other, inversion of the global matrix, obtaining values of the point sources by multiplying this inverse by the boundary conditions vector, calculation and/or visualization of the physical values wanted by the user. These steps are described in detail in Section 11.4. To better understand the following discussion, one has to know some speciﬁc deﬁnitions that are needed to keep DPSM program as clear and simple as its theoretical functioning.

11.2 GLOSSARY This glossary is classiﬁed in the order of appearance of notions in the program run. 11.2.1 Medium It is only deﬁned by its homogeneous physical properties. To deﬁne it, the user has to give its characteristics related to the physical domain studied (e.g., relative permeability for magnetism). A medium is always at least partially closed because it is contained in a closed volume object (CVO), it is limited by at least one interface or the workspace frontiers.

11.2.2 Object Two types of objects are considered in a DPSM modeling: Closed volume object (CVO): The surface of this type of object is closed (e.g., a sphere or a cube). This surface is considered as an interface between an external medium and an internal one, each one having different properties. If the user is not interested in the phenomena happening in the object (in the internal medium) or outside the object (in the external medium), ﬁxed boundary conditions should be put on the interface with sources oriented in the medium that is to be considered (the sources will be placed on the side not considered). When the internal space of an object is ignored, the object volume can be equal to zero (‘‘ﬂat electrode’’ approximation). Open volume object (OVO): The surface of this type of object must be open; this interface is a limit between two semi-inﬁnite media. Because an OVO has to have ﬁnite dimensions, it is limited in space by lateral frontiers deﬁned by the user.

351

11.2 GLOSSARY

To deﬁne an object, the user has to give the following:

its dimensions and meshing characteristics, its position in the system, its surface boundary conditions, the two media separated by the object.

11.2.3 Interface Surfaces of all the objects are called interfaces. In this program version, an interface delimits only two media. DPSM uses a layer of point sources only on one or both sides of those interfaces. 11.2.4 Boundary conditions (BC) They are essential to solve a problem, and two types of BC are used in the program. Intrinsic boundary conditions (IBC): They stand for continuity conditions of scalar and vector physical values when they cross an interface between two media of different properties. The DPSM principle veriﬁes those IBC, thanks to the two groups of point sources placed on two sides of an interface. Boundary conditions ﬁxed by the user (UBC): They translate user knowledge about scalar and vector physical values on an interface. The DPSM principle veriﬁes those UBC by using just one group of point sources placed on one side of an interface. They are energy sources of the system (e.g., the bias between electrodes of a capacitor). It means that a system has to have at least one object with UBC. 11.2.5 Frontier They are created when an OVO is introduced in the system by the user, or when an UBC is put on one side of a CVO interface (see Fig 11.1). They limit the workspace.

OVO

CVO direction of UBC

z y

4 frontier planes infinite in z direction

1 frontier plane infinite in x and y directions

x

Figure 11.1 Examples of frontiers.

352 TABLE 11.1

DPSM BETA PROGRAM USER’S MANUAL

Scalar and Vector parameters in different domains of Physics

Physics Domain

Scalar Value

Vector Value

Electrostatics Magnetostatics Ultrasound

Electric scalar potential Magnetic scalar potential Acoustic pressure

Electric ﬁeld Magnetic ﬁeld Velocity

11.2.6 Workspace All objects and media are necessarily contained by this space. No physical value can be calculated out of the workspace. It can be unbounded; for example, when a biased conductive sphere is studied in the air, the system is totally open. It is limited when frontiers have been deﬁned by unidirectional UBCs for CVO or by the presence of one OVO. For example, when the gap of a parallel-plate capacitor is modeled, the workspace is only limited in two directions. It can be completely closed; for example, when only the interior of a CVO is modeled. 11.2.7 Scalar and vector physical values The physical values associated with physics domain treated by the program are shown in Table 11.1. Now that those notions are well deﬁned, the modeling of a system can begin.

11.3 MODELING PREPARATION First, any accurate modeling should begin with a necessary preparation. It means that the user must know what he or she wants to observe and how to create correctly a modeled system. In particular, the user must

list all media and their related properties, list all objects and their boundary conditions, place them in a 3D space, be aware of speciﬁc constraints for each type of physics domain (wavelength for ultrasound propagation), be aware of DPSM program constraints (on media names, for example). These last constraints are explained in the following section (Fig. 11.2). 11.4 PROGRAM STEPS When you have the folder containing all executable programs (.c,. h, and the dll master program), copy it on your hard drive. Open MATLAB and enter the path of the folder into the directory ﬁeld to access all necessary libraries. Launch the program by

353

11.4 PROGRAM STEPS

Figure 11.2 DPSM program ﬂowchart.

typing demodpsm in the command window. The ﬁrst window that appears gives a few lines of introduction and speciﬁes that all units are in SI (Fig. 11.3). Remind checking your data units before entering them. Then you are invited to enter a name for the output ﬁle, the program writes in .m format. This ﬁle contains all the simulation steps and the associated result (Fig. 11.4). Now the modeling really begins. The next explanations follow the ﬂowchart numbering of Figure 11.2.

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Figure 11.3 DPSM welcome window.

(1) - Physics domain choice You have three choices (by clicking on one domain): electrostatics, ultrasonic, and magnetostatics. Electrostatics has been chosen to be the default case: The word is selected. If you close the window, the default case is taken by the program (Fig. 11.5). (2) - Media properties deﬁnition The window has a ﬁrst line for the name of the medium and second for speciﬁc properties (and third for ultrasound). Some important comments are added in parentheses. If a wrong parameter is entered (typically a negative value or a zero), an error message appears and you have to restart. The ﬁrst time the window appears, it is ﬁlled with the default medium name and parameters (Fig. 11.6). Each physics domain has a default medium;

Figure 11.4 Entering the results ﬁle in .m format.

Figure 11.5 Choosing the physical domain.

11.4 PROGRAM STEPS

355

Figure 11.6 First medium deﬁnition: (a) for electrostatics or magnetostatics and (b) for ultrasonic.

electrostatics and magnetostatics: ‘‘air’’ with ‘‘1’’ for relative permittivity or relative permeability, respectively. ultrasound: ‘‘water’’ with ‘‘1000’’ for density and ‘‘1490’’ for velocity of waves. On next windows, lines are empty. If you click on OK or Cancel or close window, the list of media is considered to be closed (a notice message appears and you have to click on OK to continue). If you ﬁll the lines, you create another medium and the same window will appear till you ﬁnish your list (OK, cancel, or close window). When your system has the same medium (same properties) in two separated volumes on your structure, you have to give them two different names; otherwise the program will collapse (unable to write the global matrix) or refuse your media at step (7b). (3) - Frequency entry This step and the next step (#4) are only checked when ultrasound has been chosen. You have to enter a frequency: a default value of 1 MHz is already written in the window. If you click OK , or Cancel , or simply close the window, this value is taken by the program. If you enter a negative number or a zero, an error message appears and you have to restart. (4) - Calculation of the maximum distance between points This operation is automatically done by the program. No window appears but the result is written in MATLAB window and in the results ﬁle.

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Figure 11.7 (a) frequency input and (b) results ﬁle with maximum distance between source points.

The maximum distance between the meshing points is calculated with the wavelengths li (with i going from 1 to the number of media) and chosen to be equal to min(li )/3. It is useful for objects meshing (Fig 11.7). (5) - CVO creation First, the window asks you to choose between a Rectangle and a Circle for your ﬁrst CVO. Rectangle is the default. It is selected by clicking on OK or close window. If you choose to draw a rectangle, the geometry building of CVO window opens with the following content: name of object (its number by default), center coordinates, dimension in x direction (Lx ), dimension in y direction (Ly ), the number of points on smallest side (in ultrasound this does not appear here), and the multiplying factor. This last entry gives the factor for multiplying the distance between the source points (that radiate ﬁeld directionally) and the peak points (where BC is applied). By moving away sources (with factor 2.5), you obtain a better ﬁeld link at an interface with IBC. This phenomenon is explained in Chapter 8. The DPSM allows you to put sources where you want so the program gives you this possibility. For circles the only differences are that dimensions are replaced by external and internal radii (to create a hole in the disk). They still have to be positive, and internal radius can be zero. Also, the number of points is the total number you want for the circle (mesh will approach as close as possible to this objective).

11.4 PROGRAM STEPS

357

Figure 11.8 (a) ﬁrst CVO shape choice and (b) rectangle deﬁnition window for ultrasound.

In ultrasound, the minimum for the number of meshing points is calculated with maximum distance obtained by step (4) and object dimensions, and it serves as the default value. Therefore, a second window is used to enter it. If you cancel this second step window, you restart from the choice of shape (Figs. 11.8 and 11.9).

Figure 11.9 (a) next CVO shape choice and (b) circle deﬁnition window for electrostatics or magnetostatics.

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If you click Cancel or close the geometry-building window: If it is the ﬁrst object, you are asked to conﬁrm your choice (default: Yes ¼ no CVO has been created and simulation is over, No ¼ restart), for the next objects that simply ends the CVO creation (notice window). For the next objects you also have the possibility to choose None and end the CVO creation (notice window). If you forget to ﬁll one line or enter a wrong parameter, for example, dimensions or number of points zero or negative, or factor smaller than 1, an error message appears and the window is cleared: you restart by choosing between rectangle and circle. If all your parameters are good, the program asks you if you want to duplicate your object. Answering Yes opens a window with the number of duplicates (1 gives you two rectangles or circles at the end in one object), and then the vector of duplication with default values of 1.5*Lx for x direction and zeros for y and z directions. If you click on cancel or close window the program asks if you want to cancel duplication: No drives you to restart, Yes takes you to structure drawing. By clicking No at ﬁrst duplication question, or validating your duplication, you are driven to Figure 11.1 plotting a 3D view of your structure (Fig. 11.10). First you are asked to choose a color (MENU window with, Red , Green and Blue buttons); then the program draws peak points (the chosen color) and source points (two colors near one of the peak points) of both sides of the object. If you want to manipulate ﬁgure don’t click OK on notice window. When you have ﬁnished looking at your structure and checking it (distance between objects should be larger than distance between peaks and sources), click OK (See Fig 11.11).

Figure 11.10 (a) duplication, (b) parameters and (c) cancelation.

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359

Figure 11.11 (a) color menu, (b) ﬁgure of the structure and (c) notice window for checking the structure.

Then a window asks if you are satisﬁed if Yes you continue creating objects; If no, the program erases the last object (click OK on Notice window) and redraws the structure (See Fig. 11.12). (6) - OVO creation If your system contains interfaces that cannot be modeled by CVO, which means that those are not transducers but boundaries between media carrying IBC, you have to create OVOs (only rectangular shape available in this version).

Figure 11.12 (a) user check window and (b) notice before erasing.

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Figure 11.13 (a) ﬁrst OVO window and (b) parameters.

First, a window asks you if you want to create OVO. By answering No you go to step (7). Else the geometry building of OVO window opens. It contains name of object (its number by default), center coordinates, dimension in x direction (Lx ), dimension in y direction (Ly ), the number of points on the smallest side (not in ultrasound; see step (5)), and the multiplying factor. Same comments as in step (5) can be done for wrong parameters and canceling or closing window (Fig. 11.13). The difference here is that for the next OVO-building windows, only the name, altitude (z coordinate), and the number of meshing points are needed (in ultrasonic the minimum is written as default answer). The ﬁrst OVO has already created the lateral frontiers (in x and y directions) of the workspace (Fig. 11.14). (7) - Choice of B C types (a) A window appears for each object created; it contains two choices: standard or scalar. Standard stands for IBC and scalar stands for UBC using scalar physical value. For scalar UBC you can choose between normal ﬁeld (typically used in ultrasound) or potential (typically used in electrostatics); then in a third window you are asked for the direction of the scalar UBC: top, bottom, or both t & b (Fig. 11.15). This last option is not really practical because the DPSM program is not able to manage the observation of physical values in this case, but it can be used for step (10). At the end, when all BC-type objects have been entered, a message asks if you are satisﬁed by BC: Just check them by reading the summary written in MATLAB window, then click on Yes or No (to restart deﬁning BC types). Yes is the default answer.

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Figure 11.14 (a) following OVO window and (b) OVO deﬁnition end notice.

(b) A window appears for each object created: It asks to enter the names of the medium above and the medium beside object. If you enter the same name twice, an error message appears and you have to restart: A medium cannot be split in two parts by one object. This operation links objects and media and allows writing the global matrix. If you do a mistake once, an error message appears and you can restart, twice and your whole simulation is canceled. When top or bottom have been chosen in

Figure 11.15 (a) BC type, (b) scalar BC choice, and (c) directions for scalar BCs.

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Figure 11.16 (a) BC veriﬁcation window, (b) restart BC deﬁnition, and (c) linking objects to their surrounding media.

step (7a), the second line (beside object) or ﬁrst line (above object) is ﬁlled by default with a name beginning with ‘‘0’’ and followed by a number, indicating that this space is out of the user preoccupation (Figure 11.16c). (8) - Global matrix writing The system is now ready for solving (Fig 11.17). You have to click OK on the appearing window to continue. It ﬁrst consists in writing the global matrix summarizing all interactions. This is done by the program itself by using the BC types and placement of media entered in step (7). The complete way of ﬁlling the global matrix is not given here. If you want more details, refer to the previous chapters of this book. Yet, it is important for you to be able to understand and check your global matrix. DPSM is based on the creation of

Figure 11.17 Window announcing the global matrix writing.

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363

a matrix containing all the interactions (depending on the physics domain) between objects. The program gives you a matrix deﬁnition that describes its structural content. You can ﬁnd ‘‘Z,’’ which means this is a matrix full of zeros (no interaction between the two object sources). ‘‘M’’ stands for scalar BC on normal ﬁeld and ‘‘Q’’ for scalar BC on potential. ‘‘i’’ and ‘‘j’’ in brackets are the object numbers. The sufﬁx ‘‘e’’ means that this interaction concerns the point sources placed above the object j and radiating underneath it. Q{i,j} means there is an interaction between sources beside object j and the peak points of object i. The best way of checking your structure is to check the global matrix because it summarizes all relations between the objects, also detailing sources involved. (9) - Solving This is the power of the method: It just consists of the global matrix inversion. (10) - Calculation of physical values For the actual DPSM program, it only concerns capacitance calculation when the physics domain is electrostatics. A message warns you that if you want to skip this step you can click cancel on the next window (Fig. 11.18). Click OK to continue. Then the program asks you for two object names. The objects you entered have to carry complementary (two facing directions) scalar UBC on potential (to be conductive). If no error message appears you will be invited to restart. Two errors will result in the cancelation of the calculation. You can enter several objects: A window appears until you click on cancel to end the capacitance calculation. Results are written on MATLAB window and in the output ﬁle.

Figure 11.18 (a) explanation for capacitance calculation, (b) capacitor electrode names, and (c) skipping calculation.

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(11) - Entering numerical values of scalar BC When you choose scalar BC types, you have to give the numerical values to be able to calculate point source values and to observe physical values of your system. See Table 11.1 to know what scalar value corresponds to your physics domain. For every scalar BC entered, a window opens and waits for a ﬁgure. If you enter a wrong parameter twice (for one wrong entry you can restart after validating the warning message) or let the case empty, the default value zero is taken (a message warns you). At the end, you have to conﬁrm your entries by answering Yes . Else you restart the whole process (Fig. 11.19). (12) - Observation domain deﬁnition You can plot scalar and vector physical values in planes (one dimension with minimum equal to maximum) or on lines (two dimensions with minimum equal to maximum) and also the distribution of source values on objects. A message asks you if you want to plot physical values of your system. No directs you to the question on change of parameters. Yes causes the opening of a window containing domain name, X minimum and maximum and its number of meshing points, Y minimum and maximum and its number of meshing points, and Z minimum and maximum and its number of meshing points. If you enter a wrong parameter (maximum and minimum reversed, 3D space, etc.) a message instructs you to restart (Fig. 11.20).

Figure 11.19 Windows to apply numerical BC for plotting.

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Figure 11.20 (a) observation of ﬁeld and potential and (b) parameters.

(13) - Plot, check, and save results The program plots the physical values in the space you have just deﬁned. If you want to manipulate ﬁgures do not click on the OK button of the notice window. When you are ready to continue, just click on Ok . If you want to keep those ﬁgures, you must save (MATLAB format) or export (standard drawings formats) them. You can also save plots later: They are still on screen till you restart the program (this clears data and ﬁgures to free the memory). A message asks you if you want to deﬁne another

Figure 11.21 (a) Wait to give time to save plots and (b) ask for other observation planes.

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Figure 11.22 (a) do you want to change the structure?; (b) choice in tuning parameters.

observation domain: Yes takes you to step (12), No drives you to do a loop on the program (arrows on the ﬂowchart) (Fig. 11.22). Looping allows you to change some parameters. A window asks you if you want to change parameters: No ﬁnishes the program run and closes the output ﬁle, Yes takes you to another window. It gives you three choices: boundary conditions, media properties, or structure geometry (Fig. 11.22). Clicking on Boundary conditions: This opens a window with two cases: scalar BC values or BC type. First one is the default answer because it is the more useful change. When you have done your changes, the program recalculates global matrix and values of sources (if a scalar BC has been entered, else it asks you if you want to give numerical values to those scalar BC). It also proposes you to plot your results (Fig. 11.23). Clicking on media properties: This gives two options: if you are studying ultrasound you can change frequency ﬁrst. Answer Yes and you are asked to reenter frequency and the program does a loop. No drives you to choose (MENU window) between changing one existing medium properties in your list and adding new ones,

Figure 11.23 Window proposing to change BC types or values.

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Figure 11.24 (a) menu to choose the medium to be tuned and (b) change medium properties.

Figure 11.25 (a) choice of the object to be, (b) displaced or removed, and (c) displacement parameters.

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Figure 11.26 End of DPSM program run.

If you are studying electrostatics or magnetostatics, you are directly in this conﬁguration: If you click on one medium you are asked to enter a new permittivity (or permeability or velocity and/or density for ultrasounds) and the program recalculates your system. If you click on add media the same window as in step (2) opens (it works similarly) (Fig. 11.24). Clicking on Structure Geometry: This allows you to displace or erase one object. Displacement is done with a vector direction (x; y; z) and a step number n. It runs the program n times, and if you have calculated capacitances in your ﬁrst simulation, you can automatically calculate capacitances tuning and plot it (Figure 11.25). Erasing is only possible when you have more than one object. It causes the program to restart from BC type deﬁnitions. All those changes can be done several times (make sure that you restart from the last conﬁguration): until you answer No and quit the program (Fig. 11.26).

11.5 CONCLUSION This beta version is supposed to be the ﬁrst of a long and fruitful series of DPSM programs. There are probably some errors remaining and certainly some cases to be developed and integrated in the code. If you want to warn the authors of some ‘‘bugs’’ and help DPSM team to improve its product, you are invited to contact us. The DPSM staff hopes you will ﬁnd the DPSM program helpful for your research and teaching activities. If you encounter any difﬁculty, have any questions, or want to collaborate in DPSM activities please contact: Prof. Dominique Placko SATIE laboratory ENS de Cachan 61 avenue du Pre´sident Wilson France þ33 1 47 40 55 81 [email protected]

INDEX

Back Propagation, 213 BC (Boundary Conditions), 277, 295, 349, 358, 361, 362, 364 Biot and Savart equation, 231, 253 Boltzman constant, 317 Boundary Condition, 62, 65, 69, 183, 201 boundary conditions, 236, 253, 255, 271

Conductive p271, 276 Conductor 277 Continuity Condition, 183, 201 Corrugated Wave Guide, 204, 210 Crack, 198, 202 Critical Angle, 166, 170, 171, 180, 190 CSR (Controlled Space Radiation) Source, 2, 7 current density 250 current loop, 232, 233, 256 current source, 231 Cut-off frequency, 316–321 CVO(closed volume object), 348, 354

Capacitance 276, 277, 280, 298, 351 Capacitor 278, 297, 302 Casey’s frequency, 316–322 CFRP, 307–320 CFRP, 333 Charge, 272, 273 Circular Transducer, 45 Clausius-Mossoti’s relation, 317 Concave Transducer, 46 conductive medium, 233, 248

Delamination, 324–328 Delamination, 345 detection of cracks 271 Dielectric 293, 299 dielectric target 263 Differential probe, 322 diffusion equation 248 Digitalization, 273 Dipolar moment, 316–320 Dipole (electric), 309–317

3D 293, 356 accuracy 285 Active Source, 2, 100, 112 active, 272 Anisotropic Solid, 218

DPSM for Modeling Engineering Problems, Edited by Dominique Placko and Tribikram Kundu Copyright # 2007 John Wiley & Sons, Inc.

369

370 Dipole (magnetic), 310–315 Displacement Green’s Function, 144, 147, 150, 157, 158, 224 Distribution 275, 276, 362 Divergence Angle, 108 Domain, 352 DPSM formulation 247–249, 255, 270 DPSM layers 255, 259 DPSM modeling 270 DPSM simulation 256 DPSM virtual current sources 264 DPSM, 327, 329–330 DPSM, 333–347 Dynamic Focusing, 124, 127 Dynamic Steering, 127 eddy current, 229, 234, 247 Eddy Current, 9 Eddy currents, 316, 320, 322–324, 327 Eddy currents, 334 electric ﬁeld 247, 249, 253, 265 Electric Field, 8, 87 Electric ﬂux density, 295 Electrical gasket, 315 Electrode 280 electromagnetic problems 253, 270 electrostatics 253, 270 elemental source vector 250 elemental source 256 Equilibrium, 272 equipotential contour lines 259, 262 excitation loop 258, 263 Faraday’s law, 229 Fiber breakage, 324–329 Fiber breaking, 345 Field, 274 ﬁnite elements (FE) (software) (simulation), 235, 239, 241 Flowchart, 351 Fluid-Solid Interface, 163, 166, 180 Focused Transducer, 31 Fragmented 294 Frequency, 353 Frontier, 349 Gap 280, 286, 296, 302 Geometry 277, 366 GFRP, 307–330

INDEX

global matrix (formulation) 253, 256, 271 Green’s formulation 249 Green’s formulation, 334 Green’s functions 248, 249, 271 Green’s sources 247, 248, 270 Guided Wave, 170 Harmonic Waves, 16 Helmotz equation 248 HELP-Layer1, 327–330 HELP-Layer1, 333–346 Hemisphere, 273, 279, 297 Historical Development, 1 Homogeneous 272 Homogeneous Fluid, 31, 45, 100, 125 Hybrid Method (technique), 323–327 IBC(Intrinsic boundary conditions) 349, 358 Inclusion, 83, 85 Inﬂuence, 276–278 Interaction Effect, 76, 114, 129 Interface, 295, 349 Interface Condition, 62, 65, 69, 78 interface 259 Interference, 80 inward DPSM layer 255 Lift-off, 322 Looping, 351, 364 (magnetic) (vector) potential, 235, 237, 247–250, 253, 254, 271 (metallic) target 247, 256 magnetic core, 235 magnetic ﬁeld, 234, 237, 247, 249, 259, 261, 265 Magnetic Induction, 8 Magnetic Potential, 8 Magnetic Sensor, 9 magnetodynamic ﬁeld 247 magnetostatics 253, 270 Mask, 293, 300 MATLAB, 347, 350, 353, 358, 361 Matrix, 274, 301, 360 Maxwell’s equations, 230 Maxwell’s Equations, 308–323

371

INDEX

Medium (media), 272, 295, 348, 352, 359, 364 Meshing 274 Micromechanical 275 Moebius loop, 326 Multiconductor, 293, 300 Multiﬁnger 286 Multilayered Medium, 60, 76, 87, 91, 120 Multilayered, 293 Multiple Point Sources, 153 Multiple Reﬂection, 113, 117 Navier’s Equation, 145 NDT/NDE, 327–330 Near Field Length, 108 Near Field Zone, 107, 109 non destructive testing (NDT), 229, 229 Nonhomogeneous Fluid, 32, 38, 51, 53, 111, 116 Numerical values, 362 Object, 348 Observation 274, 362 Open Magnetic Core, 9 outward DPSM layer 255 OVO (open volume object), 348, 357 P Wave Speed, 145 Parallel-plate 278, 297 Passive 272, 277 Passive Source, 2, 100, 112 Peak effect, 285 Peak point, 273, 295 Permeanility (magnetic), 308 Permittivity 272, 280, 295, 299 Permittivity (dielectric), 308–330 Permittivity (dielectric), 333–345 Phased Array Transducer, 121, 122, 126, 129 Physical values, 295, 350 Piston Transducer, 18 Plot 363 Point Source in Solid, 145, 149 point source 249, 256 Polarizability (electric), 315 Polarizability (magnetic), 315 Polarizability coefﬁcients, 317 Polarization (electronic), 317 Polarization (ionic), 317

Polarization (orientation), 317–326 Polarization vector, 317–324 Polarization, 336–338 Potential, 272–274 Potential Greeen’s Function, 158 Potential vector (magnetic), 308–309 Potential vector (magnetic), 334–340 Preparation, 350 Pressure Field, 17, 33, 35 Pressure Green’s Function, 157 Primary Source, 100 Program, 347 Propagation constant, 313 propagation equation 248 PZT, 327 Radon Transform, 222 Rayleigh Wave, 170 Rayleigh-Sommerﬁeld Theory, 19 Reﬂected Ray, 40 reﬂection coefﬁcient, 234 S Wave Speed, 145 Scatterer, 43, 54, 118 Screening, 300, 301 Secondary Source, 100 SHM, 327–330 SHM, 333 Short-circuit ﬁeld, 314 Signal Frequency, 8 Simulation, 280, 303 Skin Depth, 313–316 Skin Effect, 316 Skin effect, 334 Skin Frequency, 316–322 Snell’s law, 52 Solid Half-Space, 157 Solid Plate, 180, 185, 198, 202 Solving, 276, 361 Source Radiation, 6 Source Strength, 62, 65, 69, 78, 113, 114, 164, 181 Sources, 273, 295 Space Controlled Radiation Sources 271 Square Transducer, 45 Steps, 350 Stokes-Helmholtz Decomposition, 145 Stress Green’s Function, 144, 148, 150, 224 Surface Impedance, 311–321

372 Surface Integral Technique, 19 Susceptibility (electric), 319 synchronous detection, 230 Synthesizing, 276 target in aluminium 260 Target Points, 40 Time Lag, 124 Transducer Velocity, 8 Transmitted Ray, 40 Transversely Isotropic Solid, 218 triplet (current) (source) 250, 253, 256 Triplet Source, 22 Tuning, 286

INDEX

UBC(User boundary conditions), 349, 354, 357, 358, 361 Ultrasonic Field, 31, 91, 94, 107 Ultrasonic Transducer, 16, 100, 111, 116 ultrasonics 253, 270 Varactor (variable capacitor) p286, 299, 302 virtual point sources 253 Wave Equation, 97 Wave impedance, 313 Wave Number, 8 Workspace, 350