A Projection Transformation Method for Nearly Singular Surface Boundary Element Integrals

464 Pages·1992·7.191 MB·English

by  Dr. Ken Hayami (auth.)

Lecture Notes in
Engineering
Edited by C. A. Brebbia and S. A. Orszag
73
K. Hayami
A Projection Transfonnation Method
for Nearly Singular Surface
Boundary Element Integrals
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ACKNOWLEDGEMENTS
I wish to express my sincere appreciation and gratitude to my advisor
Dr. Carlos A. Brebbia, for his valuable guidance and encouragement throughout
the period of this study, which started with my stay at the Wessex Institute of
Technology, U. K.
I am grateful to Prof. W. S. Hall, Prof. M. Tanaka, Prof. J. C. F. Telles,
Dr. B. Adey, Dr. S. M. Niku, Dr. S. Amini, Dr. W. Tang, Dr. M. H. Aliabadi,
Prof. T. Honma, Dr. T. Takeda, Dr. L. Gray, Dr. N. Nishimura, Dr. Y. Iso, Prof.
M. Mori, Dr. M. Sugihara, Dr. K. Kishimoto and Dr. S. Akaboshi, among many
others, for stimulating discussions, valuable remarks and encouragements. I
would also like to thank Dr. W. Blain for his general assistance.
I am indebted to my company, NEC Corporation, in particular, Mr. Y. Kato,
Dr. K. Iinuma, Mr. A. Managaki, Mr. K. Nakamura, Dr. N. Harada, Mr. Y.
Nagai, Dr. E. Okamoto, Dr. S. Doi, Mr. K. Tsutaki and Mr. Y. Yanai, among
many others, for their support and understanding for the present work. I am
grateful to Mr. H. Matsumoto ofNEC Scientific Information System Development
Co. Ltd. for preparing tables for the Gauss-Legendre formula. I would also like to
thank Ms. M. Nakae, Mrs. K. Ishikura and Ms. T. Kitagawa, for their excellent
typing.
I would like to thank my sister, Mrs. Y. Mino for helping me with the proof
reading.
Last, but not the least, I would like to thank my wife Emiko, for her constant
support and encouragement.
TABLE OF CONTENTS
PART I THEORY AND ALGORITHMS
CHAPTER 1 INTRODUCTION 3
CHAPTER 2 BOUNDARY ELEMENT FORMULATION
OF 3-D POTENTIAL PROBLEMS
2.1 Boundary Integral Equation 8
2.2 Treatment of the Exterior Problem 14
2.3 Discretization into Boundary Elements 18
2.4 Row Sum Elimination Method 24
CHAPTER 3 NATURE OF INTEGRALS IN
3-D POTENTIAL PROBLEMS 29
3.1 Weakly Singular Integrals 30
3.2 Hyper Singular Integrals 40
3.3 Nearly Singular Integrals 53
CHAPTER 4 SURVEY OF QUADRATURE METHODS
FOR 3-D BOUNDARY ELEMENT METHOD 72
4.1 Closed Form Integrals 73
4.2 Gaussian Quadrature Formula 74
4.3 Quadrature Methods for Singular Integrals 76
(1) The weighted Gauss method 76
(2) Singularity subtraction and Taylor expansion method 77
(3) Variable transformation methods 78
(4) Coordinate transformation methods 80
VI
4.4 Quadrature Methods for Nearly Singular Integrals 81
(1) Element subdivision 82
(2) Variable transformation methods 84
(3) Polar coordinates 88
CHAPTER 5 THE PROJECTION AND ANGULAR & RADIAL
TRANSFORMATION (PART) METHOD
5.1 Introduction 89
5.2 Source Projection 93
5.3 Approximate Projection of the Curved Element 97
5.4 Polar Coordinates in the Projected Element 104
5.5 Radial Variable Transformation 113
(i) Weakly Singular Integrals 113
(ii) Nearly Singular Integrals 114
(1) . Singularity cancelling radial variable
transformation 114
(2) Consideration of exact inverse projection
and curvature of the element
in the transformation 121
(3) Adaptive logarithmic transformation (log-L2) 123
(4) Adaptive logarithmic transformation (log-Ll) 125
(5) Single and double exponential transformations 131
5.6 Angular Variable Transformation 139
(1) Adaptive logarithmic
angular variable transformation 140
(2) Single and double exponential transformations 143
5.7 hnplementation of the PART method 144
(1) The use of Gauss-Legendre formula 145
(2) The use of truncated trapezium rule 150
5.8 Variable Transformation in the Parametric Space 153
VII
CHAPTER 6 ELEMENTARY ERROR ANALYSIS
6.1 The Use of Error Estimate for
Gauss-Legendre Quadrature Formula 155
6.2 Case {3=2 160
(Adaptive Logarithmic Transformation: fog-L2)
6.3 Case {3= 1 Transformation 166
6.4 Case {3=3 Transformation 169
6.5 Case {3 = 4 Transformation 172
6.6 Case {3 = 5 Transformation 185
6.7 Summary of Error Estimates for (3=1-5 188
6.8 Error Analysis for Flux Calculations 190
CHAPTER 7 ERROR ANALYSIS USING
COMPLEX FUNCTION THEORY
7.1 Basic Theorem 204
7.2 Asymptotic Expression for the Error Characteristic 207
Function <P n(z)
7.3 Use ofthe Elliptic Contour as the Integral Path 208
7.4 The Saddle Point Method 210
7.5 Integration in the Transformed Radial Variable: R 212
7.6 Error Analysis for the Identity Transformation: R(p) = p 214
c
(1) Estimation of the size (J of the ellipse q 217
(2) Estimation of max I j(z) I 218
Z E£q
(3) Error estimate En(j) 220
7.7 Error Analysis for the fog-L Transformation 221
2
(1) Case: 0 = odd 223
(2) Case : 0 = even 227
(i) Contribution from the branch line f +, f- 229
c
(ii) Contribution from the ellipse q 234
(iii) Contribution from the small circle CE 238
(iv) Summary 239
VIII
7.8 Error Analysis for the eog-Ll Transformation 241
(1) Error analysis using the saddle point method 244
(2) Error analysis using the elliptic contour: Cq 249
(i) Estimation of max I f(z) I 250
z ECq
(ii) Estimation of a 252
(iii) Error estimate En(j) 253
7.9 Summary of Theoretical Error Estimates 257
PART II APPLICATIONS AND NUMERICAL RESULTS
CHAPTER 8 NUMERICAL EXPERIMENT PROCEDURES
AND ELEMENT TYPES
8.1 Notes on Procedures for Numerical Experiments 263
8.2 Geometry of Boundary Elements used for
Numerical Experiments 265
(1) Planar rectangle (PLR) 265
(2) 'Spherical' quadrilateral (SPQ) 267
(3) Hyperbolic quadrilateral (HYQ) 270
CHAPTER 9 APPLICATIONS TO WEAKLY SINGULAR INTEGRALS
9.1 Check with Analytial Integration Formula for
Constant Planar Elements 273
9.2 Planar Rectangular Element with Interpolation
Function rp jj 282
9.3 'Spherical' Quadrilateral Element with
Interpolation Function rpij 294
(1) Results for Isrpjju*dS 294
(2) Results for Is rpijq* dS 303
9.4 Hyperbolic Quadrilateral Element with
Interpolation Function rpij 313
(1) Results for Is rpij u* dS 313
(2) Results for Is rp jj q* dS 321
IX
9.5 Summary of Numerical Results for
Weakly Singular Integrals 329
CHAPTER 10 APPLICATIONS TO NEARLY SINGULAR INTEGRALS
10.1 Analytical Integration Formula for Constant
Planar Elements 331
10.2 Singularity Cancelling Radial Variable Transformation
for Constant Planar Elements 335
10.3 Application of the Singularity Cancelling
Transformation to Elements with Curvature
and Interpolation Functions 353
(1) Application to curved elements 353
(2) Application to integrals including interpolation
functions 368
10.4 The Derivation of the eog-L2 Radial Variable
Transformation 377
(1) Application of radial variable transformations
p dp =r 'P dR «(3* a) to integrals Js lira dS over
curved elements 377
(2) Difficulty with flux calculation 392
10.5 The eog-Ll Radial Variable Transformation 395
10.6 Comparison of Radial Variable transformations
for the Model Radial Integral la.o 397
(1) Transformation based on the Gauss-Legendre rule 408
(i) Identity transformation 408
(ii) eog-L2 transformation 409
(iii) eog-Ll transformation 410
(2) Transformation based on the
truncated trapezium rule 411
(i) Single Exponential (SE) transformation 411
(ii) Double Exponential (DE) transformation 412
(3) Summary 412
x
10.7 Comparison of Different Numerical Integration
methods on the 'spherical' Element 413
(1) Effect of the source distance d 414
(2) Effect of the position of the source projection Xs 430
10.8 Summary of Numerical Results for Nearly Singular
Integrals 438
CHAPTER 11 APPLICATIONS TO HYPERSINGULAR INTEGRALS 441
CHAPTER 12 CONCLUSIONS 445
REFERENCES 451
PART I
THEORY AND ALGORITHMS