Topics in Boundary Element

Transcription

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2 Topics in Boundary Element Research Edited by C. A. Brebbia Volume 7 Electrical Engineering Applications With 186 Figures and 11 Tables Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong

3 Editor: Dr. Carlos A. Brebbia Computational Mechanics Institute Wessex Institute of Technology Ashurst Lodge Ashurst, Southampton S04 2AA UK ISBN-13: DOl: / e-isbn-13: Library of Congress Cataloging-in-Publication Data (Revised for vol. 7) Topics in boundary element research. Includes bibliographies and indexes. Contents: v. 1. Basic principles and applications - [etc.] - v. 4. Applications in geomechanics - v. 7. Electrical engineering applications. 1. Boundary value problems. 2. Transients (Dynamics). 3. Vibration. I. Brebbia, C. A. TA347.B69T '.001' ISBN-13: This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its current version, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. Springer-Verlag Berlin, Heidelberg 1990 Softcover reprint of the hardcover I st edition 1990 The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Asco Trade Typesetting Ltd., Hong Kong 2161/ Printed on acid-free paper

4 Contributors Adey, R.A. Computational Mechanics Institute, (Chap. 3) Southampton, UK Aoki, S. Tokyo Institute of Technology, Japan (Chap. 4) Brebbia, C. A. Computational Mechanics Institute, (Chap. 3) Southampton, UK Bullock, J. L. Oak Ridge National Laboratory, USA (Chap. 7) Chuang, J. M. University of Windsor, Canada (Chap. 6) Deconinck, J. Free University, Brussels, Belgium (Chap. 8) de Mey, G. Ghent State University, Belgium (Chap. 1) Giles, G. Oak Ridge National Laboratory, USA (Chap. 7) Gray, L. J. Oak Ridge National Laboratory, USA (Chap. 7) Hirasawa, M. Tanagama University, Tokyo, Japan (Chap. 5) Kagawa, Y. Toyama University, Toyama 930, Japan (Chap. 9) Kishimoto, K. Tokyo Institute of Technology, Japan (Chap. 4) Lee, C.c. University of California, Irvine, USA (Chap. 2) Nakamura, M. Tanagama University, Tokyo, Japan (Chap. 5) Niku, S.M. Computational Mechanics Institute, (Chap. 3) Southampton, UK Palisoc, A. L. University of California, Irvine, USA (Chap. 2) Zamani, N. G. University of Windsor, Canada (Chap. 6)

5 Preface' The application of Boundary Elements in all fields of engineering and science has progressed at an accelerated rate since the first book on the method appeared in 1978 (i. e. Brebbia, C. A. "The Boundary Element Method for Engineers", Pentech Press, London, 1978). In particular, the advantages of boundary elements for potential problems are essential to solve a whole range of electrical engineering applications. Previous volumes in this series have focussed on the state of the art on other fields of applications while this Volume 7 discusses only problems related to electrical engineering. Chapter 1 discusses the numerous applications that BEM has in semiconductor device analysis. Although the technique is particularly useful for solving these problems, it is still comparatively unknown to many researchers in the field. The chapter gives an overview of the BEM research in semiconductors device modelling and points directions in which further research is needed. Lee and Palisoc (Chapter 2) concentrate on the thermal analysis of semiconductor devices. The removal of the generated heat from the devices is of primary importance as without adequate heat dissipation their reliability decreases. Modem integrated circuit technology concentrates many devices in a single chip and the increased heat needs to be dissipated effectively to achieve higher performance. BEM has proved to be a powerful technique for simulating the heat transfer mechanism of semiconductors devices and the chapter demonstrates how this can be achieved in many applications. One of the most successful BEM applications has been for the simulation of galvanic effects. This class of problem includes galvanic corrosion, cathodic protection methods and the inverse problem of electrodeposition commonly used in manufacturing. Chapter 3 concentrates on how a computer software system based on BEM can be used to accurately simulate those types of problems. Many examples are presented, related to the cathodic protection of offshore structures which has become a major field for BEM analysis. Chapter 4 also discusses the use of BEM for galvanic corrosion and cathodic protection. A boundary element modelling procedure is described for calculating electrogalvanic field responses due to multiple anode/cathodic interaction. The galvanic corrosion problem is reduced to solving a Laplace's equation with nonlinear boundary conditions, based on experimental electrochemical polarization curves. Numerical results demonstrate that the procedure can simulate accurately electrochemical effects related to the corrosion of engineering structures. In Chapter 5, the BEM for the computation of capacitances is applied using electrostatic fields to obtain the optimum choice for the measurement of displacements. The analytical results are compared against experiments to validate the

6 VIII Preface design of combshaped capacitance transducers which have important uses in many industrial applications. The field of Computer Aided Design has found numerous applications in the electroplating industry and one of them is the production of the plate profile as a function of time discussed in Chapter 6. The use of the BEM in this field is relatively recent and the chapter presents a detailed description of the theory and a series of relevant examples. Chapter 7 demonstrates the usefulness of the BEM to simulate electrodeposition problems. The computer modelling of electrochemical plating processes will contribute to make plating a more economical and useful engineering tool. The chapter proves that the BEM is a very convenient technique for plating simulation. In addition to the obvious benefits of reduced input and direct solution of the normal derivative, BEM codes can easily handle the non-linearity caused by the polarization boundary conditions. In industrial electrochemistry there is an increasing demand for high speed efficient processes. In order to achieve these objectives one needs a perfect insight into the interaction between electrode kinetics, cell geometry and mass and charge transport. For many practical problems the equations can be represented as a potential problem describing the charge or mass transport in the electrolytic solution plus some non-linear boundary conditions due to the electrochemical reactions of the electrodes. Chapter 8 reviews the powerful characteristics of the BEM to solve these potential problems which are typical of electrochemical systems. Inverse problems have recently received a great deal of attention in many branches of engineering. Chapter 9 discusses the application of the BEM to solve some of them. The problem of the identification of the parameters associated with the differential equation that governs the field is discussed with reference to tomography. This is an important and novel application of BEM and one that is still being developed further. The Chapter succeeds in describing the fundamentals of the approach and the type of problems which can now be simulated. The Editor hopes that the present volume of Topics in Boundary Element Research will help to widen the uses of the technique which has now reached a state of maturity in many electrical engineering applications. Southampton, April 1990 Carlos A. Brebbia Editor

7 Contents SEMICONDUCTOR DEVICE ANALYSIS Introduction The Basic Semiconductor Equations... Two Well Known Approximations.... A Particular Application: The MOS Transistor Some Further Applications Conclusion THERMAL ANALYSIS OF SEMICONDUCTOR DEVICES Introduction Review of Previous Works The Boundary Element Method.. The Boundary Integral Formulation Modeling Using the Boundary Element Method Accuracy of the Boundary Element Method.. Applications Two and Three Dimensional Models of Semiconductor Devices The Effect of Die-Bond Voids on Device Performance Summary... Acknowledgement APPLICATIONS OF BOUNDARY ELEMENTS IN CORROSION ENGINEERING Introduction.... Corrosion Prediction Techniques..... Theoretical Foundations of Cathodic Protection and Galvanic Corrosion..... Mathematical Aspects. Numerical Solution.. Electrode Kinetics.. Coupling of the System Polarisation.. Systems Approach

8 x Example 1: Comparison of BE Results with Cathodic Protection Experiment on a Plate. Example 2: Cathodic Protection of Underground Pipelines Example 3: Analysis of Galvanic Corrosion (A Comparison with FE Analysis). Example 4: Analysis of a Galvanic Corrosion Due to a Chemical Cleaning Process. Example 5: Analysis of a Jacket Type Offshore Platform Conclusions. Acknowledgements Contents APPLICATION OF BEM TO GALVANIC CORROSION AND CATHODIC PROTECTION Introduction. Electrochemical Aspects Mathematical Model Boundary Element Formulation Iterative Solution Procedures Infinite Problems Applications. Comparison with Theoretical Results Effect of Externally Impressed Polarization Comparison with Experimental Data Estimation of Maximum Current Density. Three-Dimensional Problems Infinite Problems Concluding Remarks Acknowledgements CAPACITANCE OF TRANSDUCERS FOR DISPLACEMENT MEASUREMENT. 87 Introduction. 87 Calculation of Capacitance by the Boundary Element Method 89 Green Solution and Capacitance 89 Boundary Element Method 90 Two-Dimensional Case 92 Calculation of Conductance or Resistance in an Infinite Area by the Boundary Element Method 92 Numerical Results 93 Curve of Change of Capacitance 93 Effect of the Gap Between the Two Electrodes 94 Effect of Width of Teeth 95 Effect of Filing a Dielectric Between Electrodes 97

9 Contents XI Experimental Results.... Experimental Equipment... Curve of Change of Capacitance Effect of the Gap Between the Two Electrodes Effect of Width of Teeth Conclusions.. Acknowledgement ELECTROPLATING.. Introduction..... Mathematical Formulation Boundary Element Formulation Solution of the Nonlinear System Development of a One-dimensional Model Numerical Examples Example 1 Example 2 Example 3 Conclusions Acknowledgement SIMULATION OF AN ELECTROCHEMICAL PLATING PROCESS Introduction. Equations. Boundary Element Method Experiment. Plating Cell. Tank Discretization Node Movement Parameters Results. Conclusions. Acknowledgements : 8.2:: Z ' ELECTROCHEMICAL CELL DESIGN Introduction.. Fundamental Equations Transport of Charge Transport of Mass Boundary Conditions lnsulators

10 XII Electrodes Electrode Reactions Resistances Involved by Coatings Concentration Overpotentials Resistive Electrodes 8.4 Types of Distributions, the Wagner Number. 8.5 Electrode Shape Change and Moving Boundaries 8.6 Solution of the Potential Model Discretization ofthe BEM Two-Dimensional Problems Axisymmetrical Problems Solution of the Non-linear System Solution of Resistive Electrodes Repetitive Geometries Examples Comparison with Measurements 8.7 Solution of Electrode Shape Changes Discretization with Respect to Time Electrode Shape Change Next to an Insulator Electrochemical Machining Examples Comparison with Measurements Contents INVERSE PROBLEMS AND SOME APPLICATIONS 9.1 Introduction. 9.2 Fields of Application of Inverse Problems 9.3 Application of Boundary Element Methods 9.4 Numerical Simulation The Inverse Electrocardiography Problem Fundamentals of an Electrocardiogram Boundary Element Model Numerical Experiments On Ill-Conditioning and Regularization Problems Boundary Determination in Impedance Plethysmography On Impedance CT and Plethysmography Influence Coefficient Approach and Lead Theory Approach Numerical Experiments Some Mathematical Background Other Approaches and Applications Improvement of the Boundary Determination Capability Efficiency and Convergence Boundary Determination Based on Dual Complementary Formulation.' Lead Theory in Single Medium Field Dual and Complementary Expressions Boundary Element Discretization

11 Contents XIII Dual and Complementary Energy Expressions Numerical Simulation Electric Potential-nondestructive Testing Impedance Plethysmography SUBJECT INDEX , 207