Singular Integral Equations and Discrete Vortices [Reprint 2018 ed.] 9783110926040, 9783110460346

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Singular Integral Equations and Discrete Vortices

SINGULAR

INTEGRAL

EQUATIONS DISCRETE

AND

VORTICES

I.K. Lifanov

///VSP/// Utrecht The Netherlands, 1996

VSP BV P.O. B o x 3 4 6 3 7 0 0 A H Zeist The Netherlands

© V S P B V 1996 First p u b l i s h e d in 1 9 9 6 ISBN 90-6764-207-X

All rights reserved. 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 or otherwise, without the prior permission of the copyright owner.

Printed in The Netherlands

by Koninkiijke

Wöhrmann BV,

Zutphen.

CONTENTS Introduction

3

PART I . ELEMENTS OF THE THEORY OF SINGULAR INTEGRAL EQUATIONS

6

Chapter 1. One-dimensional singular integrals 1.1. Functions that satisfy the Holder condition on a curve and the notion of the Cauchy type integral 1.2. The limit value of the Cauchy integral 1.3. Representation of the Cauchy integral in the neighborhood of curve nodes 1.4. Singular integrals, depending on parameters. Poincare-Bertrand formula 1.5. Integral with the Hilbert kernel 1.6. Singular integrals from nonintegrable functions Chapter 2. One-dimensional singular integral equations 2.1. Solution of the Cauchy-Riemann problem in the class of absolutely integrable and nonintegrable functions 2.2. Solution of equations on piecewise-smooth curves in the class of absolutely integrable and nonintegrable functions 2.3. Equation on a segment, system of segments 2.4. Equations on closed smooth curve and with the Hilbert kernel Chapter 3. Singular integral equations with multiple Cauchy-type integrals 3.1. Multiple Cauchy integrals 3.2. Equations with multiple Cauchy-type integrals 3.3. Equations with real-value constant coefficients on a product of segments and circles 3.4. Equations with multiple integrals with Hilbert kernels

7 7 11 13 15 19 21 24 24 31 38 46 51 51 56 65 72

PART I I . REDUCING OF BOUNDARY PROBLEMS OF MATHEMATICAL PHYSICS AND SOME APPLIED FIELDS TO THE SINGULAR INTEGRAL EQUATIONS

Chapter 4. Boundary problems for Laplace and Helmholtz equations. Plane case 4.1. Some basic information from potential theory 4.2. The Dirichlet problem 4.3. The Neumann problem 4.4. Nodes of curves and solution singularities

77

78 78 89 98 105

vi

contents

Chapter S. Boundary problems for the Laplace and the Helmholtz equations. Spatial case 5.1. Some information from the potential theory 5.2. The Dirichlet problem 5.3. The Neumann problem Chapter 6. Stationary problems of aerohydrodynamics. Plane case 6.1. The statement of aerodynamic problems in the general case 6.2. Problems for airfoil, airfoil cascades 6.3. Problems for airfoil with ejection 6.4. Taking into account thickness of airfoil by carrying boundary conditions to the middle line 6.5. Taking into account of permeability of a thin airfoil surface Chapter 7. Stationary aerohydrodynamic problems. Spatial case 7.1. Modelling of flow body by vortex surface (layer) 7.2. Noncirculation flow around an arbitrary carrying surface 7.3. Stationary nonlinear problems 7.4. Circulation flow around a finite-span wing of rectangular form in plan Chapter 8. Nonstationary aerohydrodynamic problems 8.1. Spatial nonstationary nonlinear aerohydrodynamic problem 8.2. Problem for airfoil with angle points 8.3. Linear nonstationary problem for thin airfoil. The Chaplygin-Joukowski hypothesis 8.4. Problem with liquid boundary Chapter 9. Determination of aerohydrodynamic characteristics 9.1. Kinematic parameters. Spatial general case 9.2. Plane general case 9.3. Noncirculation flow. Joined masses Chapter 10. Some electrostatic problems 10.1. The principal plane electrostatic problem 10.2. One mixed boundary problem of electrostatics Chapter 11. Some problems of mathematical physics 11.1. The pair of summator equations 11.2. Diffraction of a scalar wave at a plane grid. The Dirichlet and Neumann problems for the Helmholtz equation Chapter 12. Problems in elasticity theory 12.1. Plane problems in elasticity theory 12.2. Contact problem of indentation of a uniformly moving punch into an elastic half-plane with heat generation 12.3. On the indentation of a pair of uniformly moving punches into an elastic strip

114 114 118 124 129 129 132 146 150 155 157 157 161 166 170 174 174 177 178 181 185 185 191 192 195 195 198 200 200 211 219 219 223 231

contents

P A R T I I I . CALCULATION OF SINGULAR INTEGRAL VALUES

Chapter 13. Quadrature formulas of the method of discrete vortices for one-dimensional singular integrals 13.1. Regularization method of the singular integral 13.2. The method of discrete vortices for a singular integral over a closed contour and with the Hilbert kernel 13.3. S ingular integral over a segment 13.4. Singular integral over a piecewise smooth curve 13.5. Unification of quadrature and difference formulas for singular integrals on a segment and with the Hilbert kernel 13.6. Singular integrals connected with the boundary problems for the Laplace and the Helmholtz equations Chapter 14. Quadrature formulas of interpolation type for one-dimensional singular integrals and operators 14.1. Singular integral with Hilbert kernel 14.2. Singular integral on a circle 14.3. Singular integral on a segment Chapter 15. Quadrature formulas for multiple and multidimensional singular integrals 15.1. Quadrature formulas of discrete vortex method type for multiple singular integrals 15.2. Quadrature formulas for multidimensional singular integrals 15.3. Quadrature formulas for hypersingular integrals 15.4. Quadrature formulas for singular integral of finite-span wing 15.5. Examples of calculating singular integrals Chapter 16. Proving the Poincare-Bertrand formula with the help of quadrature formulas 16.1. One-dimensional singular integrals 16.2. Multiple singular Cauchy integrals PART I V . NUMERICAL SOLUTION OF SINGULAR INTEGRAL EQUATIONS

Chapter 17. Equations of the first kind. The numerical method of discrete vortex type 17.1. Characteristic equation on a segment 17.2. Full equation on a segment 17.3. Equation on a system of non-intersecting segments 17.4. Equation on a circle

vii 235

236 236 238 244 252 256 259 265 265 270 272 277 277 281 286 293 298 302 302 306 310

311 311 325 330 334

viii

contents

17.5. Equation with Hilbert's kernel 17.6. The Dirichlet and Neumann problems for the Laplace equation 17.7. The Dirichlet and Neumann problems for the Helmholtz equation Chapter 18. Equations of the first kind. Interpolation methods 18.1. On one of the properties of the singular integral operator 18.2. General scheme of construction for numerical methods of interpolation type 18.3. Equation on a segment and system of segments 18.4. Equations on a circle and with the Hilbert kernel 18.5. The Neumann problem for the Helmholtz equation Chapter 19. Equations of the second kind. Interpolation methods 19.1. Equation with constant coefficients on a segment 19.2. Equation with constant coefficients on a circle 19.3. Equation with constant coefficients with Hilbert's kernel 19.4. Equation with variable coefficients on a segment 19.5. Equation with variable coefficients and the Hilbert kernel 19.6. Examples of numerical solution Chapter 20. Singular integral equations with multiple Cauchy integrals 20.1. Characteristic equation 20.2. On one integrodifferential equation

339

389 389 394

PART V . DISCRETE MATHEMATICAL MODELS AND CALCULATION EXAMPLES

397

Chapter 21. Discrete vortex systems 21.1. Basic concepts of the method of discrete vortices 21.2. Fundamental discrete vortex systems Chapter 22. Discrete vortex method for plane stationary problems 22.1. Thin airfoil, airfoil gridiron 22.2. Thick and permeable airfoils 22.3. Airfoil in the presence of external flow ejection Chapter 23. Method of discrete vortices for spatial stationary problems 23.1. Rectangular wing. Circulation flow 23.2. Rectangular wing. Non-circulation problem 23.3. Plane wing of an arbitrary form in plan 23.4. Non-circulation flow around an arbitrary carrying surface. Joined masses 23.5. Stationary linear and non-linear problems of aerodynamics

398 398 399 408 408 415 420

346 349 352 352 356 358 365 365 368 368 371 373 375 378 379

425 425 427 432 436 440

contents

ix

Chapter 24. Method of discrete vortices in nonstationary problems of aerodynamics 447 24.1. Linear non-stationary problem for a thin airfoil 447 24.2. Non-linear non-stationary problem for an airfoil 449 24.3. Spatial non-linear non-stationary problem 454 24.4. Questions of regularization in the method of discrete vortices and numerical solution of singular integral equations 457 Chapter 25. Numerical method of discrete singularities in electrodynamic problems and elasticity theory 459 25.1. Main plane electrostatic problem 459 460 25.2. Problems of plane elasticity theory and punch theory References

I.K.LIFANOV

Singular integral equations and discrete vortices

1995

I.K.LIFANOV Method of singular integral equations and numerical experiment in mathematical physics, aerodynamics, elasticity theory and diffraction of waves In this b o o k the elements of the theory of singular integral equation solutions in the class of absolutely integrable and non- integrable functions and also the theory of simple and double layer potentials for Helmholtz equation are given. On the basis of these results a wide class of b o u n d a r y problems for Laplace and Helmholtz equations and problems of aerodynamics, electrodynamics and elasticity theory are reduced to b o u n d a r y singular and hypersingular integral equations. Some properties of these equations are investigated. F o r singular integrals and singular integral equations the discrete vortex type and interpolation type methods of computations and numerical solving are given, both in the class of absolutely integrable and in the class of nonintegrable functions. On the basis of these results the mathematical justification of the discrete vortex method of numerical solution of aerodynamic problems is given with examples of computations. O n the basis of a variety of numerical methods, the construction of discrete mathematical models is given for a wide class of stationary and non-stationary, linear and non-linear, plane and spatial aerodynamic problems, including flow a r o u n d bluff bodies (i.e. bodies which have edges and angles), and also for some plane problems of elasticity theory and electrostatics. These models can be the basis of numerical experiments in these applied domains. The results of computations of concrete problems are given. The b o o k is intended for specialists in numerical experiments in aerodynamics, elasticity theory, or diffraction of waves and also for mathematicians engaged in theory and numerical methods in singular integral equations. It may also be useful for post-graduate students and students of colleges because there are many formulations of unsolved mathematical problems.

Introduction In recent years, singular integral equations are finding more and more applications for numerical solution of various applied problems [ 1-20 ]. It should be noted that during analytical research in applications, some problems have been reduced long ago to singular integral equations because, for these equations, good theoretical basis in the one-dimensional case has been established (this basis was described fully enough in monographs [14, 21]. For the characteristic equations, the theory of obtaining all solutions in the class of absolutely integrable functions has been developed. This class of functions is the most natural for applied problems. Numerical methods of solving singular integral equations and methods involving singular integral calculations began to develop considerably later than the theoretical research, the first works appearing only in the 1930s [22, 23]. Reviews of state of numerical methods for solutions of these equations in different directions in the corresponding period one can see in works [ 5, 24 - 29]. It is interesting to mention the general tendency of developing numerical methods for integral equation solution. Numerical methods have been greatly developed for Fredholm integral equations of the 2nd type with "good" kernels. For these equations the following numerical methods have been developed: a) High exactness methods which can be applied only to a narrow class of equations when an unknown solution is interpolated by special polynomials or by partial row sums from eigenfunctions of corresponding operators. b) Methods based on applying quadrature formulas of rectangular type to an integral or analogous general quadrature formulas using one grid of points by which these formulas have been built. While developing numerical methods for singular integral equations the following problem has appeared. The singular integral is an integral which diverges in a common sense and one must interpret it in some special sense. That is why mathematicians decided that these quadrature formulas of rectangle type could not be applied to these integrals. Therefore, numerical methods of interpolation type have been developed at the first stage. However, these methods practically cannot be extended to the two dimensional singular integral equations which appear usually in different application problems (in aerodynamics, electrodynamics, elasticity theory) while solving spatial problems. But applied problems cannot wait until good mathematical theory of their numerical solution appears. Therefore, in the early 1950s, in aerodynamics, with the help of heuristic considerations and numerical experiments on a computer, the method of discrete vortices of numerical solution of corresponding singular integral equations on a segment (flow around thin airfoil) and on a rectangle (flow around a finite-span rectangular wing) has been elaborated by S.M.Belotserkovsky [30-33]. These equations appear in aerodynamics during natural modeling of a flow surface by a vortex layer. The main idea of the method of discrete vortices is as follows. A continuous vortex layer, which models a carrier surface and the track behind it, is replaced by a system of discrete vortices. On a carrier surface, one chooses the points called here and in the rest of this book the collocation points, in which the no-penetration conditions are satisfied (the sum of normal components of the velocities induced by the vortices and the incoming flow is equal to zero). The problem of finding unknown circulations of discrete vortex is reduced to a system of linear algebraic equations. The problem solution is not unique and can have singularities on edges and angles of a carrier surface. The required class of solution is defined by the physical context of the problem

4

Method

of singular

integral

equations

and selected by choice of mutual location of the discrete vortex set and collocation points. (B-condition of the method of discrete vortices). At the boundaries of the surface, where the solution must be infinitely large, the discrete vortices are placed in the nearest position to the edges. If the solution must be finite, the collocation points are placed in the nearest position to the edges. Besides that, the sums, which substitute the singular integrals in the theory of a carrier surface, must correspond to the integrals in the sense of the Cauchy principal value. For this purpose, internal calculation points must be located at the centers between the vortices on a surface (or tend to these locations). In this form, the method of discrete vortices was originally formulated in 1955 in the doctoral dissertation of S.M.Belotserkovsky [32]. After this, the systematic application of this method began in aerodynamics [ 4 - 7, 28, 32, 34, 35 ]. More detailed information about this method is given in Chapters 22 and 23 of this monograph. Geometrical representation of mutual location of discrete vortex and collocation point sets is given in Figures 22.1, 23.1 and 23.2. The method of discrete vortices has been criticized [ 36 ] because it uses the calculation of the singular integral with the help of special quadrature sums of rectangular type and while choosing a solution class, an obvious extraction of singularity on the edges is not used. Therefore, S.M.Belotserkovsky with J.E.Polonsky involved the author, who was engaged at that time in general topology, in the mathematical justification of the principal ideas of the method of discrete vortices. This has been done in the author's works [37-48] . In 1981, the author passed his doctoral dissertation, which was based on this justification. After that the author had the notion to apply the ideas of the method of discrete vortices to the theory of elasticity and electrodynamics [ 24, 48-66 ]. These first results were described in [ 5,28] . It is interesting to mention that even in these first works, as well as in the following ones, the mutual influence of applied problems and mathematical research has been shown. So, problems of flow around an airfoil with ejection [67, 68] provoked consideration of solutions for singular integral equations in the class of functions which have in some point a singularity of 1/(x-q) type, i.e. in the class of non-integrable functions. On the other hand, investigation of such a class of solutions led to the solving of a new applied problem - the organisation of blowless flow of a corporal airfoil with the help of ejection [69,70]. L.N.Poltavsky gave [71] the mathematical justification of the method of discrete vortices in the linear non - stationary problem for a thin airfoil. He also proved for this problem [ 72 ] the hypothesis of Chaplygin Joukowski - Kutta which is that the intensity of a joined vortex layer tends to zero in moving to the edge from which the vortex shroud is coming. Once this hypothesis for bodies with angles was accepted, the problem of flow around "blunt" bodies was numerically solved with the help of the method of discrete vortices [ 73 - 77 ]. In these works it was also recognized that the method of discrete vortices is the method of numerical solution for singular ( hypersingular) integral equations. These are produced while solving the boundary Neumann problem for the Laplace equation with the help of the double layer potential relative to the gradient of jump of this potential on a flow surface. The results of calculation for blunt bodies with the results of the book [5] are published in U.S.A. [28], Recently, it has been recognized recently [78] that one can derive analogous methods of numerical solution for hypersingular integral equations, obtained while solving the Neumann problem for the Helmholtz equation, with the help of the double layer potential and, therefore, one can solve numerically the problems of wave diffraction. So, the method of discrete vortices and the method of discrete singularities (which is a generalization of the first) have found wide application in solving applied problems [ 2, 4-7, 33, 62, 65, 66],

Introduction

5

This is the method of numerical solution of singular integral equations and boundary integral equations of potential theory for the Laplace and the Helmholtz equations. But monographs [ 14, 21 ] in which good description of the theory of singular integral equations is given have become a bibliographic rarity and of solving singular integral equations in the class of non-integrable functions are absent in these monographs. There is no convenient description of potential theory in aerodynamic literature as applied to aerodynamic problems, nor use of singular integral equations in this field as boundary integral equations of potential theory. Moreover, many new results of theoretical and calculation character have appeared but these results have been published only in journal articles [ 69, 70, 78 - 82 ]. That is why the author decided to write this monograph which consists of five parts. In the first part, the elements of the theory of the singular integral and singular integral equations in the class of absolutely integrable and non-integrable functions are given. Also, the same equations with multiple integrals of the Cauchy and Hilbert types are considered. In the second part, the elements of potential theory for the Helmholts equation are given. It is shown how one can reduce the Dirichlet and Neumann problems for the Laplace and Helmholts equations to singular integral equations. The singularities of solutions of there equations in the angle points are investigated in the plane case. Some stationary and non-stationary, plane and spatial problems of aerohydrodynamics are formulated as boundary problems and then reduced to boundary singular integral equations. Some singularities of solutions for these equations are shown. Also, the same way is used for pair sum equations and some plane problems from the theory of elasticity. In the third part, methods of calculation are given for different one-dimensional and two-dimensional singular integrals, which appear in the second part. For the first time, quadrature formulas of discrete vortex pair type in the plane case and closed vortex frame type in the spatial case for singular integrals, which appear in problems of wave diffraction, are described. Sample calculations of some singular integrals are given. In the fourth part, the quadrature formulas obtained in the third part are applied to numerical solution of singular integral equations of the 1st and 2nd kind with constant and variable coefficients. These equations are considered both for equations with Cauchy and the Hilbert kernels. In the fifth part, discrete mathematical models of some problems in aerodynamics, electrodynamics and the theory of elasticity, described in the second part, are given on the basis of the results in the third and fourth parts. On the basis of these models one can conduct numerical experiments in these fields. In conclusion, the author would like to express deep gratitude to S.M.Belotserkovsky for his inntroducing the author to this field of mathematics which has given rise to this monograph. The author would also very much like to express his appreciation to I.I. Lifanov, O.N. Lifanova and A.S. Skotchenko for their great help in preparing and illustrating the book.

Parti Elements of the theory of the singular integral equations. Many applied problems of the mechanics of a solid medium (in aerodynamics, the elasticity theory, electrodynamics and other Fields) can be reduced to singular integral equations in a natural manner. The plane problems, which are often good model problems, are reduced to one-dimensional singular integral equations. For these equations, the full and well described theory is given in monographs [14,21]. But these monographs, on the one hand, are now a bibliographical rarity, and from the other contain extensive material in which it is very difficult for an engineer to orientate himself. Therefore, in this book, some of the principal information from the theory of one-dimensional singular integrals and equations with these integrals is given briefly. This information is convenient for an engineer to use.

Chapter 1 One-dimensional singular integrals. 1.1. Functions that satisfy the Holder condition on a curve and the notion of the Cauchy type integral. Below we shall consider only curves located on a plane. The system of coordinates OXY on a plane is supposed to be right hand Cartesian. A curve will be called smooth if it is simple, i.e. does not intersect itself. A line L will be called smooth and unclosed (an arc) if it can be presented in the following parametric form: x=x(s),

y=y(s),

sa 0 it has the form

O**(/0) eH

in the chosen neighborhood of the point c, 0 0 is however small.

Proof.

Again,

as

in

theorem

1.4.1,

it

is

enough

Q ( i 0 , r + / i ) - Q ( i 0 , r ) , which we represent as function

to

consider

the

difference

(i0,t) in the form o f sum

Q u + fl2. Because

cp"(i,x) e / / ( a , p )

on

[a.ijxr

and

function

belongs to the class H o n [a, c], we get |Q2|

h^Ofr),

(1.6.3)

[a,b].

While research on the problems of plate bending with thin hard inclusions and the problems of contacts of plates and shells with linear press tools [83,84,85], it is necessary to consider integrals of the form ,oG(_u)>

where -m-1 < ReA+ < -tm, .) using the formula

The One-Dimensional singular integrals

) { t - a f f{t)dt

=

23

](t-a)

dx +

a

(1.6.9)

(k + l) where K>-n-1,

(X

+

2)

(n-\)l(X

+ n)'

/ ' " ' ( f ) eC[a,fc] and the integral in the right side is absolutely

integrable. It can be mentioned that formula (1.6.9) is equivalent to the formula obtained from (1.6.8) by using n times the formula of integration by parts. But it is more convenient to use formula (1.6.9) for considering the integral of form (1.6.4). If, in integral (1.6.4), we substitute t on z, then we obtain the function (z), which for Re/T or Re X~ is greater than -1. This function is treated in the sense of an analytical extension and as it is shown in [88], it behaves as i?|(l + z)' l "j, A; = min for z —> i 1, and as o{z 1) for z —» oo. This function has limit values on (-1,1), for which the Sokhotsky-Plemeli formulas (1.2.3) or (1.2.4) hold. Really, it is enough to mention the equality 1

l+T

1

1+z t - z

r-z

1

(1.6.10)

1+ z

and represent the function ®(z) in the form [88]:

4>(z) =

1 ( l - z f ( l + z)"" 2m Ji

T-7

^ ''

where

2m

(1 - z )

m

fo{\

-z)k

f , { \ + z)k

(1.6.11)

The divergent integrals in q(z) do not depend on z and their regularization values can be calculated as in (1.6.8). Therefore, q(z) is an analytical function for z * ±1, having in the points z = ±1 poles of the orders m and n correspondingly. If one denotes p = max(/w,w), then it is natural to suppose that p 0 ( r ) & H p \ a , b \ or

X ( z ) .

Then, the general solution of problem (2.1.1), vanishing at infinity, by the chosen points CQ ,..., cQ will be given by the formula

Let now the curve L be a segment [-1,1]. We shall take a r g G ( - l ) = 0 , where 0 " e(-27t,0) (suppose that function G(t) in (2.1.1) satisfies this condition). Then, function e ,

where y(z) is given in (2.1.3), one can present in the form

29

One-dimensional singular integral equations

(2.1.25)

r e"(z) = ( z - l p T * «M

„*M 2ti e = 0+*)-

where À = [argG(/)]'^ is an increment of the function G(t) argument on [-1,1] when l changes from -1 to 1; r,(z), T2(z) are functions, bounded in the neighborhoods

of

points -1 and 1, correspondingly. Following [84, 85, 88], let us consider the function -„-I

y(f

(2.1.26)

where [ ] denotes the integral part of a number, m and n are integers. This function is the solution of problem (2.1.1); it may be represented as ( ^ ( l m z ) ^ when z—>±1, where X*

-m-\

=1 Z (l + z) M

equations

N>1-

(2.1.30)

Let us consider now the inhomogeneous Cauchy-Riemann problem.

Problem. Find a piecewise-holomorphic function (z) with the boundary curve L, vanishing at infinity, i.e. (co) = 0 (just such functions are interesting for us), by the boundary condition < r ( í ) = G(í)cT(f) + g(í) o n L ,

(2.1.31)

where G(t) and g(t) are given on L functions of the class H0, G(t) * 0 on L. Let X(z) be the canonical solution of the class h(c,,...,cq) of homogeneous problem (2.1.1) in the class of absolutely integrable functions, i.e.

Then equality (2.1.31) one can write in the form:

**(0

*-(')_

g(')

(2.1.32)

* + ( 0 the solutions of the given class are obtained by the formula

where / ^ ( z ) is an arbitrary polynomial of degree, not higher than ae-1 ( / ^ ( z ) = 0 when x =0); when ae