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Page 1: Fourier BEM: Generalization of Boundary Element Methods by Fourier Transform
Page 2: Fourier BEM: Generalization of Boundary Element Methods by Fourier Transform

Lecture Notes in Applied Mechanics

Series Editor

Prof. Dr.-Ing. Friedrich Pfeiffer

Volume 5

Page 3: Fourier BEM: Generalization of Boundary Element Methods by Fourier Transform

Springer-Verlag Berlin Heidelberg GmbH

Page 4: Fourier BEM: Generalization of Boundary Element Methods by Fourier Transform

123

Fourier BEM

Generalization ofBoundary Element Methodsby Fourier Transform

Fabian M. E. Duddeck

Page 5: Fourier BEM: Generalization of Boundary Element Methods by Fourier Transform

Dr.-Ing. Fabian M. E. DuddeckTechnische Universität MünchenLehrstuhl für BaumechanikArcisstraße 2180333 MünchenGERMANY

e-mail: [email protected]

This work is subject to copyright. All rights are reserved, whether the whole or part of thematerial is concerned, specifically the rights of translation, reprinting, re-use of illustrations,recitation, broadcasting, reproduction on microfilms or in any other way, and storage in databanks. Duplication of this publication or parts thereof is permitted only under the provisionsof the German Copyright Law of September 9, 1965, in its current version, and permission foruse must always be obtained from Springer-Verlag. Violations are liable for Prosecution underthe German Copyright Law.

http://www.springer.de

© Springer-Verlag Berlin Heidelberg 2002

The use of general descriptive names, registered names, etc. in this publication does not imply,even in the absence of a specific statement, that such names are exempt from the relevant protec-tive laws and regulations and free for general use.

The publisher cannot assume any legal responsibility for given data, especially as far as direc-tions for the use and the handling of chemicals and biological material are concerned. Thisinformation can be obtained from the instructions on safe laboratory practice and from themanufacturers of chemicals and laboratory equipment.

Cover design: design & production GmbH, Heidelberg

printed on acid-free paper

Library of Congress Cataloging-in-Publication Data

Duddeck, Fabian M.E., 1965–Fourier BEM : generalization of boundary element methods by Fourier transform /Fabian M.E. Duddeck.p.cm. – (Lecture notes in applied mechanics ; v. 5)Includes bibliographical references.

1. Boundary element methods. 2. Fourier transformations. I. Title. II. Series.

TA 347.B69 D83 2002 2002075987620’001’51535–dc21

Softcover reprint of the hardcover 1st edition 2002

ISBN 978-3-642-07727-2 ISBN 978-3-540-45626-1 (eBook)

DOI 10.1007/978-3-540-45626-1

Originally published by Springer-Verlag Berlin Heidelberg in 2002

ISBN 978-3-642-07727-2

Page 6: Fourier BEM: Generalization of Boundary Element Methods by Fourier Transform

Fourier - BEM

Generalization of Boundary Element Methods

by Fourier Transform

Dr.-lng. Fabian J\!I.E. Duddcck

March 10, 2002

Page 7: Fourier BEM: Generalization of Boundary Element Methods by Fourier Transform

Contents

1 Introduction

1.1 .\loti vation .

1.2 Outline of Content~

1.3 Biographical Context

2 Traditional BEM

2.1 Introduetion .

2. 2 Boundary lnt cgral ~quations ( l31E)

2.2.1 Reciprocity relation . . . . .

2.2.2 luversiou of t.hc differential operator .

2.2.:3 Galerkin and collocation BIE

2.:~ Complete System of DIE . . .

2.:3.1 Differential ion of l31E .

2.3.2 Symmetric system of BIEs

3 Distributional BEM

:3.1 Distributional Context. . . . . . . . . ..

:).2 Distributional De~cription of t.lte Domain

:3.2.1 The cutoff- distrilmtion

:).2.2 The normal vector

:.u Distributional DIE ....

:3

9

g

10

11

15

15

16

16

18

19

Ll

21

23

25

25

26

26

28

28

Page 8: Fourier BEM: Generalization of Boundary Element Methods by Fourier Transform

4

3.:3.1 Distributional reciprocity ..

:3.3.2 Distributional Galerkin BIE

Regulari;1;ed BIE

4 Fourier BEM

4.1 The Principal Idea

·1.2 Fourier Transform of RTE

4.2.1 Transformed Galer·kin I3IE

4.3 Ttansformcd fundamental solutions

4.3.1 Fundamental solutions

4.3.2 Green's functions . . .

4.4 Ttansfonned Trial and Test Functions .

4.4.1 Transform of the cut.off distributions

4.4 .2 Transform of t.lte trial functions

4.5 Construetion of the I3EIV[ rnatriees

5 Heat Conduction

5.1 Isotropic Ca.se .

5 .1.1 The Dirichlet pro hlen1

5.1.2 The -"Jeumann problem

5.1.3 The mixed boundary value problem

5. 1.4 Some comput at.ional aspects

5.2 Anisotropic Case ... ... ... .

5.2.1 The mixed boundary value problem

5.2.2 The Dirichlet. problem . . .

5.2. :3 The three-dimensional case .

28

:30

:)2

35

38

:38

:39

41

41

42

4:3

45

45

•15

50

55

57

58

58

09

61

Page 9: Fourier BEM: Generalization of Boundary Element Methods by Fourier Transform

COi'V1'LN1'S

6 Elasticity

6.1 Isotropic Case ............... .

6.2

6.1.1 The mixed boundar.Y value problem

6.1.2 The Dirichlet problem in two dimensions

Anisotropic Case . . . . .

6.2.1 The state of the art

6.2.2 General anisoLropic elasticity

6.2.3 Orthotropic elasticity in ~2

6.2.4 Tl'ansverse isotropic elasticity in R3

7 Plates

7.1 The Thin Plate

7.1.1

7.1.2

7.1.:3

7.1.4

7.1.5

Isotropic ca~e

Orthotropic case

General anisotropic. C.<J.'ie

The thin plate on a \\'inkler foundat.ion .

Combined bending and stressing of thin plates

7.2 Refined Plate Theories

7.2.1 The thick plate

7.2.2 Thick plates on Winkler foundations

8 Waves

8.1 Transient Problems

8.1.1

8.1.2

8.1.:3

8.1.4

8.1.5

8.1.6

Fourier BIE for transient problems

The clasl ie bar

Scalar waves in isotropic media

\Vaves in isotropic elastic media

Initial eonditions . . . .

Tr-ansient plate problenu;

J

63

5:3

6:3

66

67

67

68

69

71

73

73

7:3

85

87

88

89

90

90

93

95

9.)

95

96

99

100

102

102

Page 10: Fourier BEM: Generalization of Boundary Element Methods by Fourier Transform

G CONTE.'IVTS

8.1.7 Dynamic analysis for bended and stressed plates 10:3

8.1.8 Dynamie poroclasLieit.y 104

8.2 Stationary Problen1s 105

8.2.1 The stationarity assumption lOS

8.2.2 Scalar \Vaves in isotropic media lOG

8.2.3 Elast.ie waves in isotropic: media 107

9 Thermoelasticity 109

109

109

112

11:3

11:3

113

114

114

9.1 Coupled Therrnoelasticity

9.1.1 Coupled anisotropic thermoela.sticity

9.1.2 Coupled isotropic thcrmoclastic:ity .

9.2 Simplified Thermoelastic IVIodels .

9.2.1

9.2.2

9.2.3

9.2.4

Thermal stresses

Coupled quasi-sta.tie thermoelasticit.y

Uncoupled quasi-static thermoela.stic:ity .

St.al ionary thcnnorlas Lieit.:y·

10 Non-linearity

10.1 Physical ::--J on-lineari t.Y

10.1.1 Inelastic problems.

10.2 Geometrical :\on-linearity

10.2.1 Large dcflcet.ion of thin clastic plates

10.2.2 Dual reciprocity methods in Fourier space

11 Wavelets

11.1 Fundamentals of \Vavelet Theory

11.1.1 Dat.a compression by wavclct.s

11.1.2 The \vavelet transform .

11. L3 IVIult.iresolution analysis

115

115

115

119

119

122

125

120

125

127

1:30

Page 11: Fourier BEM: Generalization of Boundary Element Methods by Fourier Transform

CONTt;NTS

11.2 Wavelet Galerkin Discretization ........ .

11.2.1 .\'fat.rix compression in the original space

11.2.2 .\1at.rix compression in Fourier space

12 Conclusions

12.1 Hcsult.s .

12.2 Open Questions

A Glossary

A.1 Distribution theory

A.2 Boundary Element. .\{ethod .

B Special Distributions

C Integration of BEM matrices

C. 1 Anal,yt.ical integrations . . .

C.l.l Singular integrab by Fourier transform

C. 1.2 Additional regular Fourier pairs .

C.l.3 Addit.ional singular Fourier pairs

C.2 ~mnerical integrations . . . . . . . . ..

.~

I

1:34

134

136

139

1:39

1·10

143

11:)

148

153

161

161

161

161

162

164

Page 12: Fourier BEM: Generalization of Boundary Element Methods by Fourier Transform

Chapter 1

Introduction

1.1

Il !f a plus de 50 ans qw~ t'ingcnie~w Heaviside introduisit ses regles de calcul syrnbolique, dans un mbnoire audacieu:r rnl d~>.s calculs rrwthhrwtiques fort JWU justifies sorlt utili.~es pour la .mlution de prol!li'.mes de physique. ( . . . j Les ingenieurs utiliscnt (cc calcul S)tmboliqucj systbnatiqucmcnt, chacun avec sa con­ception personnellc, avec la conscience plus ou mains tmnquille; c'esl devenu une /:echniq·ue "qui n'esl pas ngoureuse rnais qui reussil bien''. L. Schwartz ( 1950), Theorie de8 Distributions, [Seh51jl

Motivation

The research in recent decades has established the boundary element method (I3E:.I) as a powerful tool in computational medmnics. Even dynamic and non-linear problems have been solved for fluid and solid mechanics. One of the remaining dra\vbacks is that the BEI\I as previously used is based on an explicit. knowledge of fundamental solut.ions. In many engineering problems, e.g. anisotropic media, \Ve do not know these fundamental solutions.

To overcome this drawback, an alt.ernat.ive BEJ\1 is present.ed here which is called here Fourier-Hb'J\d and is obt.ainecl by a spatial (and temporal)

1 It is more thau 50 years ago that the engineer Heavisid.e has iutroduced his symbolic calculu, in an audaciou' report in which a mathematical calculus was used for the solution of physical problems which was not at all justified. [ ... ] The engineers arc using [this symbolic calculus] in a systematic way, everyone with his own personal conception, with <~ more or less trauquil conscience; it has become a technique "which is not rigorous but rather succe:;sful" [translated by F.D.].

9

Page 13: Fourier BEM: Generalization of Boundary Element Methods by Fourier Transform

10 CHi\.YTJ::H 1. D'VTJWDUCTlON

Fourier transform of the traditional boundary integral equations ( l:HE). Only the Fourier transform of t.he fundamental solution is needed , which can be eonst.ruet.ed, in contrast. t.o t.he fundamental solution it.self, for · all Linear and homogeneous differential operators by a simple mat.rix inversion of t.he trans­formed diH'erential operator. lienee, the realm of applications of I3E:\I can be extended remarkably.

1.2 Outline of Contents

In Chapter 2. t.he main aspects of t.he traditional approach are summarized .. Attention here is focussed on direct Galcrkin methods though we embed some aspects of other variants of BK\1. The often rather mathematical idiom is ''translated" int.o a representation understandable to engineers and visualized by the prototypical example of the mixed boundary value problem for the heat conduction (Poisson equation).

In Chapter 3. the traditional BE:\·I is generalized by int.roducing a distribu­tional not.ation developed here for the first. Lime (distributions in the sense of Schwartz [Sch51]). Some special aspects arc taken from the more recent. the­ory of non-linear gem~ralized funet.ions introduee<L in the 80s by Colombeau [ Col84, Col85, Col92]. The mathematical tools for the Fourier transformed I3E:\I presented in the following part have been developed. ~ evertheless, Chapt.er 3 may be skipped by readers not familiar \Vit.h the theory of distribu­ticms alt.lwup;h it contains the mat.hema.tically consistent and hence simpler description of .i umps and singularities occurring in the context of the BE\1.

In Chapter 4, the Fourier-I3E:\il is presented. It introduces a new 1nethod for the assembly of stiffness matrices which works even if the fundamental solutions are not known . Hence, the approaeh presented here can be regarded as a generalization of the "classical'' BEl'vi. :\·Iany new fields of interest for an engineer are now accessible for analysis by a BE!\I-model.

Finally, several engineering exa.ruples are discussed t.o visualize the broad field of applinttions. IIeat conduction, anisotropic elasticity, thin and thick plates, thermo- and poro-elasticit.y, static and dynamic elast.ic problems are investigated . In addition, first. insights are given for applicat.ions to non-linear problems.

The integration of infinite and oscillating kernels is the main difficulty of the new approach. lienee, some ideas on optimized integration by wavelets are given in Chapter 11. This needs further investigation.

Page 14: Fourier BEM: Generalization of Boundary Element Methods by Fourier Transform

/ .. 3. RfOGR\PHTC.'\L CONTF:XT II

1.3 Biographical Context

lm;tc:a.d of giving a.n hisr.oric:;ll ovc~rvic~w of BBM which nm be found in t.hc: countless publications on DEM, e.g. the review papers [Ali97, I3es87, Des97, BMP98:, some biographical context is given here. Tt may ~erve to en1phm;ilc that BEM Hnds it.s place somewhere bet.ween applied m;u.hematics and eugi­neering mechanics. DEYI is the product. of a fruitful imeracrion her.ween these two discipliner-;. F'umier-BE.\'1 gmwralize;.; the tra.ditionlil syrmrwtric: Ga.krkin­DE!VL henee we restrict. the historical remarks on Fourier and Ga.lerkin.

The life of Jean Baptiste Joseph Fourier::J .Josc~ph Fourier was hom 21 VI a.n:h 1768 in i\ nxern~ as t.he ninth of t.wdve c:hi 1-

dren. His parents died early. In 1780 he sLanecl studying mathematics at Lhe Ecole Tior<Lle l\filitaire of 1\uxerre. Tn 178:3 he received tin;t. pri·le for his work on Bossuel.'s .Yleeauique Genera.le. Fourier combined a religious voeat.ion

.I.I:Ll. Fourier B.G. Ga.lmkiu

iu 1787 he nutnred l.he Benedietiuc abbey of St .. Bcnoit.-sur-Loirc l.o t.raiu for the priesthood with a malhematical oriemalion; he became a teacher at the ~;~~ole~ lioplic: M i lit.ain: of Auxc:rr·~~ in 1790. In 1793, lu~ lw1~anw involved in the revolutionary proce::.s and was imprisoned twice.

Later in 17!)4 Fourier was nominat.ed to st.udy <Lt the Eeole 1\ormale in P<Lris, a. tea.dwr \ra.iniug school. Among his l.cachers wm-c~ La.gra.ngt~, Laplac:t~, and Yionge. lie was appointed to a position at the Ecole Centrale de8 Tmvaux Publiqtws, which. wm; umkr the direction of l.aMre, Ca.nHrt. and \'Ionge, <1JH.I

2 PUJ·aphrm;e of an dc(;tronic artidc written by .J..J. 0 'Connor awJ J::J". H.obcrt,;on; http: II www-hbtory.rn(;:;. :;t-andrews.ae. uk/ hbtory / _\1a thema tida.m;! Fourier .htrnl.

Page 15: Fourier BEM: Generalization of Boundary Element Methods by Fourier Transform

12 CHi\.YTJ::H 1. D'VTJWDUCTlON

soon to be renamed Ecole Polytechnique. In 1797 he succeeded Lagrange in being appointed to the chair of analysis and mechanics.

In 1798 Fourier joined .\lapolcon's army as scientific adviser in it.s inva.'iion of Egypt.. He helped to establish edU(;ational facilities in Egypt., carried out archaeological explorations, and became secretary oft.he newly founded Cairo Im:ititute. Returning to Fta.nce in 18f.ll, Fourier resumed his post as ProfeHsor of Analysis at the Ecole Polytechnique.

On :-! apoleons iniliat.ive, F·ourier wa.'i nominated a.<; Prefect in Grenoble. ln 1 his position he was responsible for t.he drainage of 1 he swamps of Bourgoin, and for 1 he construction of a new highway from Grenoble t.o Turin.

During his time in Grenoble Fourier worked on the mathematical theory of heat .. He submitt.ed On the Propagation of Heat in S olid Bodie.s to the Paris Institute on 21 December 1807 where it. was criticized by the scientific com­mittee (Lagrange, Laplace, :'vionge and Lacroix). The first. object.ion, made by Lagrange and Laplace in 1808, was to Fourier's expansions of functions as trigonometrical series, what. we now call Fourier series. Furt.her clarification by Fourier still failed to convince t.hem. The second objection was made by I3iot against Fourier's derivation of the equations of transfer of heat. Fourier had not made reference to I3iot's 1804 paper on this topic. although I3iot's paper is certainly incorrect. Laplace, and later Poisson, had similar objec­tions.

Fourier wa.-; elected to the Academie des Sciences in 1817. ln 1822 Delambre died. who wa.5 secretar.v to the mathematical section of the Academie des Sciences, and Fourier together with I3iot and Arago applied for the post. After Arago withdrew, the eleet.ion gave Fourier an ea..sy win. Shortly after Fourier became seerctary, the Aeademy published his prize winning essay Theurie analytiqne de la chaleur in 1822.

During Fourier's la.st. eight. years in Paris he resumed his mathernat.ical re­searches and published. a number of papers, some in pure mathematics others on applied mathematical topics . However, his scientific life \vas not without problems, since his theory of heat. still provoked controversy. iliot claimed priority over Fourier, a claim which Fourier had lit.tle difficulty rofut.ing. Pois­son, however, attacked bot.h Fourier's mathematical tcdmiques and claimed t o have an alternative theory. Fourier wrote Ih~tm·ical Precis as a r eply to these claims but, although the work wa.s Hhown to various mathematicians, it w<ts never published. Fourier died 16 1\Ia.y 18:30 in Par is.

Page 16: Fourier BEM: Generalization of Boundary Element Methods by Fourier Transform

1.3. 1310GH .. 1PHJC .. 1L COXl'l:.'XT 13

The life of Boris Grigorievich Galerkin:·1

Galerkin wa.<; born 4 IVIareh 1871 in Polotsk, Belarus. From 1893 to 1899, he studied engineering and rnat.hernaties at. t.he Petersburg Technological Insti­t.ut.e (PTI). He start.ed Leaching in 1909 while working as an engineer, first a,t. the Kharkov Loc:ornotive Plant. and later a;; an engineering manager at the ~ orthern .\Iechanic.:al and Boiler Plant. His first publications range form practical engineering aspects for bridges and frames to theoretical mathe­matical areas. namely the method of approximate integration of differential equations known today as the Galerkin method.

In 1920 Galerkin was promoted to Head of Structural ::\Iechanics at. the PTI. By this time he also held two chairs, one in elasticit.y at the Leningrad In­stitute of Communications Engineers and one in structural mechanics at Leningrad Cniversity. In 1921 the St Petersburg 1\hthematical Society was reopened (it had closed in 1917 due to the Russian Revolution) as the Pet­rograd Physical and l'vlathmnat.ical Society. Galerkin played a major role in t.he Societ.;y along wit.h Stcklov, Sergi l3ernstein, Ftiedrnann and others.

Galer kin was a consultant in the planning and building of many of the Soviet Cnion's largest hydrostations. In 1929, in connection with the building of the Dnjepr dam and hydroelectric station, Galerkin investigated stresses in dams and breast walls with trapezoidal profile.

Other works for which Galer kin is famous are his studies of thin elastic plates. His major monograph on this topic Thin Elastic Plates was published in 19:37. From 1940 unt.il his death 1945, Galerkin was head of the Institute of .'vlechanic:s of the Soviet Academy of Sciences.

3 Paraphrase of an electrouic article \Vrit.t.en by .J.J. O'Connor and E.F. Robertson; ht,tp: I /www-hi~t.ory.rnc,.>t.-a.ndrew>. a c. uk/ hbt.ory I :via.t.hernat.icia.n> / Galerkin.ht.rnl.

Page 17: Fourier BEM: Generalization of Boundary Element Methods by Fourier Transform

Chapter 2

Traditional BEM Enrico IJ ett£

2.1 Introduction

BEM makes possible solutions to differential problems in complex geome­tries. Values, e.g. the displacements and volume forces, in the interior of an n-dimensional domain n c IRn, n = 2, 3 (Fig. 2.1) are related to exterior quantities like the displacements and the tractions at the boundary afl.

oO = l 'u U rt

Figure 2.1: Domain 0 with DiTichlet and Neumann boundary

The known and unknown boundary value::; are approximated by polynomial trial functions, whereas the interior remains undiscreti:1:ed. This results, in contrast to FEM, in a relatively low number of unknowns. An n-climonsional problem is reduced to one of the order n - 1. Then, an algebraic system of equations is obtained by a procedure of weighted residuals. However, the underlying matrices are in general asymmetric and totally populated, as a result. of the int.errelat.ion of all boundary values.

15

Page 18: Fourier BEM: Generalization of Boundary Element Methods by Fourier Transform

A A A

a) b)

c)

A

16 CH"\Pn;n 2. TlLiDLTlON.iL lJElvl

2.2 Boundary Integral Equations (BIE)

2.2.1 Reciprocity relation

Geometrica! restrictions The BE\1 can be applied to problems in arbitrary geometries. The only

condition neecled is that the boundary should be Lipschitzian. The Lipschitz­eondit.ion is requircd for t.hc Gauf3 t.hcorcrn whieh is t.he basie thcorcrn of all BE::\Is. A unit outcr normal vcct.or 1/ should bc ddincd cvcrywhcrc on the boundary an with t.hc execpt.ion of singlc point.s. Corncrs and cdgcs are allowed.

1\Iost of the cases which occur in engineering mechanics are problems posed in such Lipschit.z-domains, alt.lwugh t.hcre arc somc important. ca.scs, whcrc

8 a)

c)

Figure 2.2: Nun-Lipschitz dumains [JV!cL OOj.

the Lips(.;hit.;;~ wndition is not fulfilled (Fig. 2.2), cf. \kLean [\kLUO], pp. 89. In part a) of Fig. 2.2 .. the cusp at point. A is too sharp, i.e. the boundaries approach one another faster t han any polynomial. The eraek with zero crack width in Fig. 2.2b) is not. Lipsc:hit.zian bccause t.hc domain lics on both sidcs of thc boundary. And in Fig. 2.2c) wc cannot rcprcscnt. thc boundary in thc vicinit.y of thc point ,1 as a graph of a funct.ion. ln all t.hcsc c:ac;cs, wc 1nay divide the domain into subst.ructures. Especially for cracks with zero crackv.·itlth one ma.y find several rnotlificatiom; of the BE.\I in the Literature to overwme t his restridion, cf. the review of Alia.badi [Ali97].

48 CH"\PTJ::ll 5. HE/tT CONDUCT.ION

The in tegration of \Olume sources (Fig.5.2, right) leads for the first compo­uent of Fk t.o (Jo = 1)

1 l' ·• ' A

( 2 . ) 2 (Di (-i:) f x (:i:) U ( i) di: 7f . F:"

__ o_ __!:___ (P+i.i'l/2 - 1) e - . e - el i: eli· f l [' . ~ ( - ii:, 1) ( -'ii" 1) (2 )2 A . A A ( A'l A'') ' 1 " 2

1r · • Jli:2 x1 :c1:r'l -x1 - :c~

1 -. -(27r + 2 ln 2- 11) = -0.04417. 247f

The total vector on the right-hanel side is due to symmetry

(5 .8)

P = (-CJ.OH17, -CUHt17, -0.0Jt17, -0.0 1H n - O.CHt17, -O.CH t1 7-0.0,H17, -O.CH-117).

Thc unknown cocfficient.s for t.hr boundary flux arc obt.aincd by H-1 F :

e· = (o.2564, o.2.364, o.2564, o.2.564, o.2564, o.2.3 640.2.364, o.2.564). (.3.9)

As a first eontrol, we ta.ke another Gauss theorem, the heat equilibrium condition. The inte11,ral of the fiuxes a.t the boundary sta.nds in equilibrium with the total sources (L~ = 1/ 2 is th length of the i-th element)

ii j' 2__: L~e = 1.02544 d,; - J(:r) ch = 1. ~~ n

(5.10)

The coa.rse mesh leads to an acceptahle result, the global error is 2.5% .

.3362 .3362 .2564 .2564 .1667 .1 667

n D

25M ~ ~ .2564 .1 667 D ~ .1 667

.3362 o ] .3362

.2564 ~ ~ .2564 .3362 D 01 .3362

.1667 D 1 .1 667

tt E::~JOOE51 .2564 .2564 .1667 .1667

.3362 .3362

Figure 5.3: Flux at the boundary (left : '1 x 2 element:;; ri11,ht: 11 x ·1 element;;)

The results for a refinecl hounclary rnesh with four elernents on ea.ch side a.re shown in the right part of F ig .. ). :). T he global error of t he equilibriurn

Page 19: Fourier BEM: Generalization of Boundary Element Methods by Fourier Transform

2.2. 130UNLHHY li\i1E01UlL t_:qu .. lTJOiVS' (13LE) 17

Heat conduction as a paradigmatic example The ba.sic principles of traditional I3E.\I arc presented for the paradigmatic

example of the n-dimensional stationary heat. conduction which is described by the Pm:sson urrwtion

D. u(x) =- f(.r),

u(x) = ur(:r) , t(x) = tr(1,·),

" :z: E n c JR'.."' D. = L D~ I Dxi;

X E ru c DO;

:z: E 1\ can.

k=l

(2.1)

D. is the Laplace operator, u is the unknown quantit.y (temperature) and f denotes known volume sources in n. The fiux at the boundary lS

t = Atu = -a,u = -r1 · \ln. (2.2)

\1, r; arc the gradient and the outer unit normal; A~, = -11 · \1 is Lhc boundary operator. The partial derivatives ajaJ:k will be abbreviated by [)k· :r denotes the n-dirnensional vector and <h is the short. form for d:r 1 d:r 2 in the two­dirwmsional case and cl:J: 1 <h2 d:r:J in three dimensions. To obtain a well posed problem, half of the boundary data. eit.her 1/. on fu or t on rt, should be defined by boundary conditions, i.e. r u u ft = iJO.

The weak form of the differential equation (reciprocity) tvim;t of the nurnerieal methods are based on a weak form equivalent t.o

t.he strong form (2.1). They are obtained by the mu.lt.iplication by a second function and by the integration over t.he domain (weighting) . In the engi­neering literal ure this is known a.s Lhe principle of virLual work. The shifting of the differential operator to the weighting function by partial integration leads to additional boundary terms. Thus, the basic weak form for l3E_\1, t.he reciprocity relation is obtained

r n6.vdn = r D.u vdrl- r (~'llv - u~v) df. ./n .Jrl ./r ul/ ul/ (2 .:3)

l t is also known as Hetti's theorem. 1 and is equivalent to Green's fonnu.la2 . lf v = 'UIJ, u = u1 are regarded as t.wo dift'erent states (real and virtual) and if we introduce t.he notations D.'u1 = - h ; D.u.II = -.fii; t1 = -a{/u1; tii = -D,/uii, then the reciprocal character of equation (2.3) is obvious

r UJ fiid~~ = r /J UIJdO- r (ti UIJ- UJ tii) df . ./n ./n ./r

1 E. JJctti ll Nuovo C'icmcnlo 7- 8 (1872). 2 (~. (~reeu: An Essny on the Applicatwn of ./i,futhemutu:i~l Analysl8 to the Theones of

Electncdy and Mngnctism. :'\ottingham (1828) .

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18 CH !\ PTP.R 2. TTV\ DTTTCH'{ ;\ L RF:M

2.2.2 Inversion of the differential operator

The fundamental solution as inverse of the differential operator From (~.3). the basic inLegral equation& for the DEM are con::.tructed by

an inversion of the differential operator ~- This <.:<tn be achieved by the con-

U(:r)

Figure 2.:1: The. /tmdamental sol~~lion U as response. to a single soww: r.~.

volulion with lhe fundamental solttlion U(x), i.e. we have lo replace v by U(:r y), wlwre U(:z: y) i~ the re~pon~G of tlw infinitG medium to a singk source .f = r'5 at x = y (Fig. 2.:J)

~ U(x- y) = -(~(x- y). :r, y E= R.". (2.4)

For t.lw l:a.pladan ~' we ohr.ain for r.wo- or r.hn!<!-dirrwnsiorw 1 probl.<~rn!-l :n LOO]

U(:T:) 1 Cu

:1: E R2 : (2.5) . ln l:rl :!~r

, 2JT

U(:r.) 1 I xEk;. (2.6)

-t~F h • '

Tlu.~ constant C0 in (2.5) is arbitrary. lt. r-cpn~sem.s rigid mode~ which are homo~cueous solut.ious of~- The fuiHla.uwnt,a.l solutions inherit their singular ehamcter from the Dirac-distribution. Cnfort.unately, these mutlytic f(muulas for il..tc fumlarnculal solut.ious ean only be found for simple cliiTcrential ope­rators, cf. :c:vw2, Kyt~G, 1\:ogOO]. 1\evertheless, as long as the coefficients of t.lw diff1~nmt.iHl operm.or arc consr.a.nL r.lw ~~xist.cncc of a. fnnclarrwnt.al solm.ion can always be assured.

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2.2. 130UNLHHY li\i1E01UlL t_:qu .. lTJOiVS' (13LE) 19

Somigliana identity The insertion of t.he fundamental solution U(1:- y) into (2.3) leads t.o

{ n(y) t:.U(:r- y) cHty .Ill

{ t:.u(y) U(1:- y) clny (2.7) }I!

+ { [n(y) T(x- y)- t(y) U(:r - y)] dfy, Jr~

>Yit.h T- At.U. On t.he Left-hand side, we subs! it.ut.e

t:.U(:r- y) = -6(:r- y), :r ,yEIR".

The Dirac-distribution 6 has in U the convolution property

{ u(y)8(.T- y) dHy = n(:r), ./I!

.TEn .

(2.8)

(2.9)

The relation (2.9) is not valid for boundary points, i.e. :1: tf. on. Together with (2.1) we obtain from (2.7) to (2.9) SrJ'fnigli.ana's irlentdyl for interior points X E n :

u(x) { f(y) U(x- y) rlOy .In (2.10)

+ { [t(y) U(:r- y)- n(y) T(x- y)] dfy. Jry

2.2.3 Galerkin and collocation BIE

Boundary integral equations (BIE) The transition to boundary int.egral equations (BlE) demands a limit pro­

cess :r --t 80, c:f. for example [Bon99]. Hereby the left-hand side of (2.10) is modified by a factor t.:(:r) E [0, 1] due t.o the special boundary relation

{ u(y) 6(1:- y) rlOy = n:(x)u(x), 1:, y Eon. (2.11) ./n

\Ve obtain the boundar·y integral eqv..ation

n:(x)u(x) { f(y) U(:r- y) dr2y ./n

+ { [t(y)U(:r-y) -u(y)T(:r-y)] dfy . ./r Y

(2.12)

For smooth boundaries, n:(:r) is equal to 1/2. Equation (2.11) is justified by a rigorous distributional argumentation in Chapter 3.2.

3 C. Somig-liana: Sopra l'e<luilibrio di Ull eoq>o elast.ieo isotrope. n NuU'I'() Cinrwnto 17- 19 (1886).

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20 CK\Pn;n 2. TlL\DLTlON.lL lJElvl

Approximation of the boundary quantities The known and unknown boundary quantities u, t arc approximated by a

sum of piecewise polynomial trial fum:timv; cp~,, cp~ with t.he coefficients ll.;, t' (Fig. 2.4)

!Vu ]\'t.

u(:r) ~ L u'o:,(x), t(:r) ~ L t'(l~(:r ).

F or con·· ergem:e reasons, the trial functions for the temperature u should be at least linear whereas for the flux t. it is sufficient. t.o take constant trial functions, d. for rxamplr [13on95a]l.

£.7 A ·7 ~A L&~

Figure 2.4: The line oTuwl ttrwdmtic trial fruu:tirms <f~,t.

Galerkin boundary in tegralequations An additional weighting with one of t.he trial functions for the fiux (2.13),

e.g. [Hol92, B.\1P98], leads to

.£,oi(:r)" (x)u(:r) df1, = l xoi (:r) h f(y) U(x - y) rlny df,, (2 .14)

+ ~- c;){(x) ~- [t(y) U(:r - y)- u(y) T( :r - y)] df11 df, . . r:~· . r -y

Here t.he t.rial functions of the fiux are chosen because t is t.he dual v aria.ble to u (dual in Lhe sense that t.he two variables are the two factors in the scalar product of work) and the ouLcr integrals have Lo be work terms. Introducing

4 The Galerkin approach weak ens t.hese coninuity rec1uirements; only C.:1 '"' is necessary for the hypersingular int.egra.l~ [KRR92. :YIR9G].

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2.3. COlvlPL.ETH S'YS'T.Elvl OF 131.E 21

(2.13) in (2.14) the discretized Galerkin HJH is obtained:

/. qj~(:z;)r{(:z;)u(:z;) dl', = ;· o~(x) { f(y) U(x- y) dl!ydl'"

• I 'x • I 'x .J 11

(2.15)

+ t ti /' oi(x) ~· q)~(y) U(:r- y) dfy <lfx i ' r:r . fy

Collocation boundary integral equations To avoid double integration in (2.15), Dirac-distributions (p{(:z;) = (i(:r-~J)

can be chosen as Lest functions with ~J E 811. Due to the sifting properLy L,n u(:r)rl(J:- ~J) dJ· = ·u(e) this leads to the discretized collocation DIE

r;.(~1 )a(e) = ;· .f(y) u(e- y) dny (2.16) . I

+ fti f o~(y)U(C-u)df11 - "tui fa:.(u)T(E;1 -u)df11 •

, .fr·. , ./r"

In contrast. to the Galerkin approach, the collocation method leads w non­symmetric matrices, i.e., the system looses its self-adjoint property, cf. [AS97]. It is not always clear where to put the collocation points and the theoretical reasoning, e.g. convergence proof:'> [l'v1P96], is much more difficult than for the Galer kin method. The coupling with the FE!\{ is not as straightforward a...:; for the Galerkin-I3EM.

2.3 Complete System of BIE

2.3.1 Differentiation of BIE

Somigliana identity for the flux For the :,;yrnrnetric Galerkin method, an additional I3IE i:,; needed for the

flux which can be obtained by applying the adjoint boundary operator A{ = -a ... on (2.10) , i.e. we have to differentiate with re:,;pect to the normal of the

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22 CK\Pn;n 2. TlL\DLTlON.lL lJElvl

j-t.h test function

A{u(x) = A i /' f(y) U(:r- y) dny (2 .17) ,[)

+Ai t t' ;· ~'>~(y) U(:r- y) df y- A{±: u; ;· q>:Jy) A:U(:r- y) df y• l • l"y t . 1"-y

where T was replaced by A~LI. The differentiat,ion and int,egration can be interchanged because of Lhe following equalit,y which is valid for all convolu­tional products even when the factors have severe singularities, d. [Dud99a.]'-',

A{ (u n.') = (Aiu) * v = ·u * (A{v). (2.18)

Thus, we obtain Sornigliana 's identity for the flux for interior points :z: E n

t(:r) = /' J(y) Ai U(J:- y) drly . !!

(2.19)

+ f t' ;· o!.(y) A{U(:r -,r;) dl'y- :t u·• ~- qi~Jy) A{A-~U(:z:- y) dl'v. i • I'~ i • I'~

Galerkin BIE for flux T(l get the DIE for boundary points x E DO, u hw; to be replaced by rut

on the right-hand side of (2.17). The differentiation of the product is then

Ai{r,;(J:)u{r)} = !i(x)Aiu(x) + u(.r)Ait.:(:r) . (2.20)

The value t~:(x) gives the ·'percentage" of Lhe Dirac-distribution c\'(x) relevant for the domain. When weighted with a test, function <>{, t,he second Lerm becomes infinite ( Lhe non-integrable singularity cancels with that. originating from the hyper singular part on the right-hand side as shown later). The Galerkin fliE for flux is fonnally :

.l,m~(:r)A{{Ir(:r)u(:r)} clfx = I"~(x) h f(u) AlU(x -u) clDydfx (2.21)

+ ; · \!l,,(x) f ti ; · 6;,(y)A1U(:r.- y) dl'ydr" • 1'.• ·i • I ' y

The fiux test functions ¢i were replaced by those of the temperature ¢{, to preserve duality. Otherwise the matrices will be asymmetric.

5 The only condition is : the support of oue of the two convolutional factors has t.o he mrnpact, [Hor90], p. 10:~. All trial functions have compa,ct support .

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2.3. COlvlPL.ETB S'YS'T.Elvl OF 131.E

Differentiation of the fundamental solution In t.he DIEs, t.he following differentiations of U arc needed :

1 :rk

-211 l:rl~' 1 ;r,k

---411 l:ri:>'

Similarly for the second derivatives in the JR2

and in t.he JR3 :

[

2x2 - '1'2 - ,,2 ·t "2 "' 3

3x~x1 3x:lxl

:3.1:t ;1'2

2:z:~ - :r3 - xf 3:r:lx~

2.3.2 Symmetric system of BIEs

The Galerkin BIEs (2.15. 2.21) lead to the algebraic system of B!Es

from (2.15) :

from (2.21) :

23

(2.22)

(2.23)

(2.24)

where known and unknown boundary values a.re still not distinguished. The vect.or and matrices are defined a~ follows :

F;; .- .{_c;>{(x) .l: f(y) U(x- y) dUy dr x;

·- ;· \b·i(:r ) ;· 0?~(y) U(:r:- y) dl"11 ell',: 1', , l ' y

.- /' q>{(:r) /' Q)~(y) A~U(:r - :y) dfydf": Jr;J_. .fry

·- h &i.(x) 1.:(:z:)o~(:1·) dfl, , r_,

(2.27)

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24 CK\Pn;n 2. TlL\DLTlON.lL lJElvl

and

1/ ·- 1 q)~(:r) j . .f(y) A{,U(:r- y) dny cll'1,; lx I!

(2.28)

·- f &{,(:r) { dl~(u)A{U(:r-y)dfydfx; .Jr.. ./r v

/' r!1Jr) /' 0~~(y) A~A:U(:r- y) dfy df,; .!I', .!I ' y

By reordering, where known boundary values are concatenated witJ1 the force vector, t.he hnal matrix s,ystem comparable to t.hat of the FE:\1 is obtained

(2.29)

for the evaluation of the unknown boundary quantities X. The system matrix Jl is fully populated and symmetric.

Page 27: Fourier BEM: Generalization of Boundary Element Methods by Fourier Transform

Chapter 3

Distributional BEM Olive·r Heat:i.side

3.1 Distributional Context T~l obtain the fourier transf(lrm of the 13IE derived above; all quantities have to bo extended from rl to H." ( t.lw Fourier l.ransfonnat,iou is defined on J{n and not on n). Formally. this can be done by defining a cutoff distribution,\ which is simply one in t.!w interior of n a.nd zero ont.sidt~. Then nll qna.m.it.it~S arc mult.iplied by x and fiually l.ransf(mned into Fourier space (windowed Fourier tmm;forrn). \ 1f ath cmm.i c.:ally this cxr.cnsiOJl and r.ran sfonnat.ion is jusr.itied only in the frame of t.he theory of dist.ribut.ionf>.

This r.heory is fundmnental f(lr integral equations. It simplifies r.he deriva­tion of t.!w n~dpmcil.y rdat.ion;.;, the emmet, t.n~atrncnl. of diseontinnit.ies a.nd singularilies, and lhe evaluaLion of Lhe free Lerms. The Fourier Lransform without. its distriJnJtion.al extension is nor. cornplcte enough to treat t.he :;im­plesL problems. Divergem. inlegrals are declared.

The main advantage of the theory of di~;t.ribut.iom; is tlmt it reestablishe:; dilierontiat,ion as a simple a!Hl consist(•nt procedure, all quamit,ies are dif­ferentiable even if they exhibit severe singularities or jumps, cf. Dieudonne :uie78].

Ilisl.orically t.he dist.rihut.iou t.heor_y was firf>t, iuit.iated heuristically by l.he engineer Heaviside1. Later in the fortie:;, SobolcY2 introduccxl the first math­(~Iwltica.lly rigorous WllC(~pl. which wa;; finally m;ta.blislwd by Sdrwa.rl.z in 1950

10. Hcavi.side: Ou opcrawr~ in physical mathematics. Pruc:. R.uy. Su1.:. 11 52. 50·1 .):l9 (l~!J:.l).

2S.L. Sobolev, Sur uu th6ureme d'aualy~e fonetioundle (.H.ut;t;iau; Frenda ~umrnm·y). Mat. Sb. 46 (1), 171 "197 (1938), and A.mer·. 1Hath. Sue. 'lhmsl. 34(2), :39 G8 (196:3).

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26 CH.\I'Tf<.R 3. IJISTIU/JC:·nON-\L HJ;,:;H

[Sch,-Jl]. Further developments can be found in the work of Gel'fand &:. Shilov lGSG4j. Here, reference is given most.ly to the standard book on linear diffe­rential operators by HiirnHwder :Hi.ir90].

S.L. Sobukv

(1908 89)

L. Schwanz

(*1915) l.II.-'1. Gd'fand

Cclf.J1;3)

F·ignn: 3.1: Founde·r.~ of tlu~ thr:ory of rhst·rilm.tirm8

Some additional ideas are taken from the more recent literature on non-linear gcw:ralizcd fmu:t.iom initiat.<:d by Colornlwau in hi~ books [Col84, Col85, Col92]. Recent. resuhs can be found iu [Bia.!JO, Obe92, ORDL Boi!JG]. Even the prodttet of two Dirac-distributions <l(:r) is now defined.

The reader not acquainted wit.h di~t.ributiou t.lwory may switch direet.ly t.o t.lw next. ehapter keeping in mind that. all qua.Jlt.it.ics a.ro cxtendl.'d frorn n w Rn aml t.ha.l. Fourier tra.nsformat.iou ca11 110w be applied Lo t.he BIE est a blishcd before.

3.2 Distributional Description of the Dornain

3.2.1 The cutoff- distribution

The cutoff distribution X as a description of the domain n The unknown quantity u. extends only over ll and not. over R.n. It may jump

across the boundary Ul t. In the t.heory of dist.ributions, this is de~cribed by a mult.iplieat.iun of H f.: cx.(R'') with t.lw (;'tdo.fJ di.>tribution X (chanu:l.nrist.ic dist.ribul.ion) of the domain n

u(:r) --7 x(:r)n(r)' :rEU :r E aU J ¢ n = n u nn.

(:U)

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3.2. DlSTlUlJUTIONilL DJ::SClUPTlOIV OF THJ:: DOAlillN 27

x equals one in the interior and vanishes outside the domain nand can be ex­pressed for smooth boundaries by a generalized rnult.i-dirnensional Heaviside­dist.ribut.ion, d. [Con74, GS64, Hiir90],

x(:r)- H(t!J(:r)), J: E .!R". (3.2)

~· describes as a hypersurface the boundary DO. Domains wit.h non­smooth boundaries are expressed b.Y products and sums of these Heaviside­dist.ribut.ions [Dud97b].

Example 3.2.1 cutoff distribution for the semicircle The cutoff distribution of a snnicircle in R2 of unit radins is (Fiy. 8.2)

x(:r) = H(:rt)H(l- :rf- x~). (3.3)

Figure :3.2: The cutoff distribution x of the semicircle

The edge and corner term "' of the cutoff distribution The value ti:(x) of x(:r), :r E DO direct.ly at the boundary Dn. i.e. at. edges

and corners, can be determined by a convolution \Vit.h the Dirac-distribution

r..·( .r) = X(:r) lxEi'l!l = r X(Y) 5(:t - Y) dy, }[tn y E JR"' X E DO'

This convolution is well defined but often difficult t.o compute. Hence, we refer t.o the non-linear theory of distributions, cf. [Col84, Col85, Col92. Bia90, Obe92, OR94, Boi96], where t.he Dirac-dist.ribut.lon is of! en replaced by a Dirac-convergent. sequence. \Ve define for example:

'( ) l' . ( ) L' k" kJixl " 0 X = llil cfJk X = llll -- e-k--HX · /:.' -'?X ?

:r E R". (:3.5)

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28 CH.iV.l'.EH 3. lJ15'11Ul3UTlON.iL lJElvl

The convolution (3.4) is now evaluated by

!'i;(:r)= ~- x(y)o(:r-y)dy= Lim~- x(y)o)k(x-y)dy. (:3.6) • JJ.n k >X, H_n

For curved boundaries, we approximate X locally by a tangential cutoff dis­tribution.

3.2.2 The normal vector

In the distributional context, the normal vector I/ can be generally defined by t.he gradient. of the hypersurface 7fl. \Ve have for smooth boundaries

The Heavlside-distribution is clilational invariant

II(ll') = II(al/!), a= a(:r) > 0,

which is also valid for the graclicnL

al!hough we have for !he Dirac-distribution, d. [GS64], p. 213,

6 (lf') ci6(lP)

[a[o(alf,); [a[ k+l(f b. ( a.'lji).

(:3. 7)

(3.9)

(3.10)

(3.11)

IIenee, in the distribut.ionaJ representation there is no need for normalization. Examples are given in the Appendix C.l.

3.3 Distributional BIE

3.3.1 Distributional reciprocity

Distributional differentiation of the product xu The generalized differentiation of the product xu Leads to terms in the inte­

rior of n and to additional boundary terms3 . ln our example of the Laplacian, we obtain

D.{xu} = xD.u + 2vx. vn + uD.x.

3 Leibniz' forruul<~ for the differentiation of products rewaius valid for distributions , Hormander [Hor90], equation (:3. 1.4).

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3.3. DlSTlUlJUTIONilL 131£; 29

An additional weighting with v leads t.o

The la,.:,l Lwo I cnm; arc simplified by I he following parcial integration

{ uuClxcly=- { \l{vu}·Vxdy=- { (u\lv+v\lu)·Vxdy . .I TR_n .I TR_n .I R:n

Therefore, we have deduced the weak form of (3.12) :

r. vil{xn} dy = r. V'{Clndy + r. [v\lu- uYu]. Vxdy . .1 JH.n } 11n .! JRn

The support of Vx is restricted to t.he boundary, hence the second integral on t.he right-hand side encompasses all boundary integrals. Here, we only have first derivatives of t.he cutoff distribution X· Thus. we do not. get additional corner terms.

Distributional reciprocity Equation (3.13) is almost the generalized Green's fornmla needed for the

BE:\I. \\'e have to integrate partially the integral on the Left-hand side to get the distrib·utional reciprocity, d. H(irma.nder [Hi)r90], theorem ::U.9. \Vith Clu. =-.f. we get

{ XU.ClV dlf = - { vxf d,1,1 - { . [v\lu.- u.Vv] · Vx dy. (3.14) }JI'..n .frtn Jnn

This procedure can be t.ransferred to arbitrary different.ial operawrs. The last integral of (3.14) represents t.hc boundary integrals, because the support of Vx is rc\'ltric:tcd t.o all.. The explicit cqLLivalcnccs arc ·with Vx = 8(V;)\l·t!J

f 6'(«')\l ~i; · \luvcly }]''' { u 6(1!)\ltf} · Vv dy

.!Rn

j. ~- Uu - I/ · \lu V dl\1 = - -;:;--V dl''11 ;

1' , 1' Ul/

h. ~r· Dv - u 1/ · Vvdr 11 = - u.-iJ dry , , r . r· IJ

(3.15)

(3.16)

which makes (:3.1'1) comparable to (2.:3). The main advantage of this repre­sentation i::; that the integrals extend fonnally over the entire JR.". Therefore, the Fourier transfonnation can be applied to the::;e integral equations.

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30 CH.iV.l'.EH 3. lJ15'11Ul3UTlON.iL lJElvl

3.3.2 Distributional Galerkin BIE

Distributional trial and test functions For the definition of trial functions in analogy to (2.13), we have to use spe­

cial cutoff distributions defined on the boundary panel. \Ve start by defining a c'lltoff di;;tr·ib'lltirm for· a refcrnu:c dcrncnt (Fig. 3.3)

x:u(.r ) := II(:ri)II(l- :r1) 6(x2), :r EN:~ ; x:u(.r ) := II(:r1 )II(l- :r1 )II(:r2)II(l- x2) 6(:r3), :r E N:l_

If a triangular mesh is preferred, ;ve define

( 3.17)

(3.18)

The trial functions are obtained by multiplying a CXJ(N")-function p0 (x),

Figure :3.:3: Supports of the cutoff distributions for the njerence elements

e.g. a polynomial, with these cmoff distributions

(3.19)

Arbitrary trial junctions 6~, t(1·) on arbitrary straight elements are obtained by translation and/or dilat.ion operators -

(3.20)

with the translation vector b; and Lhc dilation matrix ai. F'lnally, Lhc unknown and the known quantities on the boundaries are approximated by

Nt.

6(t!J)Vt~{r) · Vu(:r) :::::: L liot(:r). (3.21)

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3.3. DlSTlUlJUTIONilL 131£; 31

Distributional Galerkin BIE In (3.14), u is now replaced by the fundamental solution U(:r- y) a..s in

t.he derivation of (2.7). The additional weighting wit.h ¢>{ of the form (3.20) Leads t.o t.hc disl.ribul.ional equivalent. of !.he Galcrkin I3IE (2.14)

r (p{(:r)x(:r)u(:r) cb; }fRn

( ¢~(:r) ( x(y)J(y)U(:r- y) dycl:J: (3.22) }gn }Rn

+ /' q){(J:) I' vx(y). vu(y)U(:r - y) dyd:r • f.{n , Jktt

.l..~{(:r) L.:i(y)vx(y). vU(x- y) dydx.

with vx = 6(1/•)v·4.' for smooth boundaries. The term on the Left-ha.nd side was simplified due to 1

r 0){(:r) r x(y)n(y) r5(:r- y) dy d:r = r cp{(:r) x(J·)n(:r) cb:. }l}t-n JR~·,. lutn (3.23)

For smooth boundaries with x = H( ~!) , we have 6('u•)H(«:) = r5h':)/2 and therefore :

1. 11 6{ (x)H(t~·)u(x) d:r = -;- . 6{ (x)u(x) d:z:. ~ 2 ~

(3.24)

System of Galerkin BIEs To simplify the notation for the following chapters, the following abbrevi­

ations for the sealar product and the convolution a.re defined

scalar product: (a, b/ r a(:r) b(::r) dx; }f1.n

(3.25)

convolution: a* b = /' a(y) b(:~:- y) dy. • L'tn

(3.26)

And by notating u, = n(J}x(:r ), fl( = .f(J:)x(:I:) we get. the extended Galerkin I3IE for the temperature

Nc N u

(cp{, nx) = (0){,fl(*u)+Lti(c;){,~0~*u)-Lui(c;{c;)~,*A;y) (3.27)

4 The multiplication of the dist.ribut.ious d>[ and x is defined in t.he context of nou-liuear dbtributions, d. [Bia90, Boi96].

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32 CH.iV.l'.EH 3. lJ15'11Ul3UTlON.iL lJElvl

and for the flux :

- ( ¢-;_,, A{ux) (3.28) i.Vt. J.Vll

L {; (q'>{,, 0): * AiU) +Lui (0!,, 0;, * A{A~U).

3.3.3 Regularized BIE

The fundamental solution U is weakly singular, its derivative A~U strongly, and AiAtU h.y-persingular. \Veakly singular kernels can be integrated eas­ily, while particular regularization procedures are needed for the other ea.<;es that are often basell on Cauch:y's or Hadamard's Principal Values, cf. [H6r90]. Nevertheless, it can be shown that the BIE in total do not exhibit such singularities, i.e. that all non-integrable singularities r:ancel, eornpare [GKRR92, Bon9oa].

Regularization of the temperature BIE Due to u(y)J(:r-y) = n(:r)J(:~:-y), d. IIiirrnander [Wir90], example 3.1.2.,

we r:an write for the left-hand side of equation (:~.22)

r (~{(:~:) /. x(y)u(y) rl(J·- y) dyd:I: ./J!t-n • g.n

(3.29)

= l ¢{(:~:)n(:r) l x(y)6(:r- y) dy dJ: . .fH_n }i{n

By applying an addit,ional Green's formula (whir:h is an equilibrium r:ond.i­tion) we get with 6 = -!:.U = - divVU

!. x(y)6(x - y) dy • _Rn

= { vx(y) · vU(.r.- y) dy .lRn

- l x(y) divvU(:r- y) dy (:uo) .111{11-

l o(tf;)\71/'. vU(:r- y) dy . .frtn

Hence, by insert.ing (3.30) in (3.22), t.he Tegulariz ed Galerkin BIE for the temperature is obtained

0 (0){(:r) , fx(Y) * U) + (0){(x) , (vxvn * U)) (G){(:r) , (uVx*V'U)) + (0){(x), n(V\'*VU)).

(3.31)

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3.3. DlSTlUlJUTIONilL 131£; 33

The introduction of the trial functions leads to the regularized Galerkin Hlf'o' for t.hc temperature

N,

0 = (qj{, Ix * u; + L ti (o:,, ¢~ d i) (3.32)

with Ne = I'v't + I'v' 11 as the total number of elements and o;. as the constant trial function of the i-th element.. We have simplified ¢;u = uJ6{pJ with pJ &'i the polynomial of the trial function q'>~ . For i f- j all integrals arc a priori regular, and for i = j we obtained the regularized integrals

j ( 1 oi A' l ' .. J ( . i A-t l ' ) \ U 1J, • q)tl * t ; - fJ ([)c * t ; I ·

Regularization of the flux BIE The left-hand side of the fiux BlE (:3.28) can be written as

( Oi,A{{xu}d:r - ( <P;,xA{·ucLr+ ( <D-~nAixd:r. JR/1. }fEn l fJ}'·

The Last integral is divergent :

f' c/J~(:r)u(:I:)A{x(:r)cl:J; = j' q)~(:r) ·u(:r)o(«:)A{·d,d:r. .llltn H_n

Example 3.3.1 Singularity of the free term

(3.34)

(3.35)

To visualize the singular character of (3.S5}, we consider the semicircle of i'o'xample 3.2.1 with a test .function (~ = o(:z: 1 )H(:z;~)H(l- x 2 ). The singular integral is .for· the relevant part o.f the lwundary with ~/; = :1: 1 , I/ = [1, 0]

f' ~'J{,(:r) ·u(:r)o(«:)A(d,lh = ./.· .. n. tl(J:1)II(:r2)II(l- :r2)n(J:1 ,:I:2)S(:r1) d:r ./lltn. -"'

t 6(0) ./o tt(O, :r 2 ) d:r2 .

Therefore, the singular character is pn~cisely o(O).

This singula.rit.y is transferred to the right-hand side to regularize the hyper­singular term. \Ve obtain in analogy to (3.30) and (3.31) :

L, %(x)u(:r)A{x(:z:) dx = h:" (~(x)u(x)Ai l, x(y) 6(x- y) dy d:z; (3.:36)

= { <ti-!J:r)u(:r)A~ f' i5(~:,) vt!J · vU(.z; - y) dydx . .!Rn .l};:n

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34 CH.iV.l'.EH 3. lJ15'11Ul3UTlON.iL lJElvl

Hence, the regularized HJH for the flux is

0 ;· (j~(:r) /' x(y )f(y) AiU(:r- y) clycl:r (:).37) • 1?.n .fuf.n

- /'. 6{, (:r) /' c5(v; l[t (y) - t(:r )] At U(:r - y) dy d:r .J}~" }JRn

+ ;· (>{,( :~; ) r b('ib)[n(y)- H (:~:)] AtA:u(:r- y) llyd:r. _it~H J JY!..U

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Chapter 4

Fourier BEM

4.1 The Principal Idea ,h:a·n Rnptistt: Jot;eph F'o·urier

.For S('Y<~ral physin-d probl<~ms, w<~ do not know Uu· funclanwnt.al ~olul.ion

(; rectnirr.d in ( :3.27) a.nd (:3.28). Bnt (if t.h(~ confficieul.s of the diff(~rr.I_ll.ial

operat-or are constant), we can always derive r.he Fourier t-ransform U(J) of U(x). Therefore, Fourier DE:\1 as a more general numerical approach is d<;rin~d in thit> chapter.

4.2 Fourier Transform of BIE

4.2.1 Transformed Galerkin BIE

The two fundamental theorem~ The ·n-dimr~nl-liona.l FrnLr-i(;r t·ransform F(u) = ·il of u E:: L 1

( H-") is ddined 1

as (i = y'=T) . . . .

il-(i') = ;· tt(x) e-·i<:~:,cr> dx. . . . . ~-~

( 4.1)

1 Scc Appendix ,\ f'lir a morc f\'Cnnal ddinilion 1nlid nJso for di;:.t.ributions.

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36 CHl1V.l'£1l 4. FOUlUJ:;H lJJ:;lvl

lt should be empha.<:,ized that the transformation has to be applied with respect. t.o all coordinates :I:k ~ x~;:, k = 1 ... n; C) eharaeLerizes a transformed

object., 4 denotes a pair of Fourier correspondence.

The basis of Fourier-BEI\'I are wm \Yell known theorems of the Fourier trans­formation, t.he theorem of Parseval

and the convolution theorem, cf. [Bra86],

t o(y)u(x- y) dy ./r{H

4

x,iEIR", (4.2)

(4.3)

Parseval's theorem (4.2) states the invariance of energy or work with respect. to t.he lkdimensional Fourier transfonuat.ion (:~:, :1: E IR:."). The convolution theorem ( 4.3) links a convolution in t.he original space to a simple umltipli­eat.ion in the transformed spaee. In the abbreviated notation introduced in (:).25), (:).26) we have for these two Fourier theorems :

( 6(:1:), u ( :.r)) (2!)" ( ~( -i:), u(i:)); ( 4.4)

Transform of the convolutional part The inner integrals in (3.27) arc convolutions. J3y applying ( 4.3) t.hcy arc

converted by the Fourier transformation to simple multiplications

fx * u F B fx (.'i') [}(i');

cY . t * u F B ~~(:i') [}(i'); (4.5)

¢~ * ~u F

B ~~ (.r) .4g'r(:r).

Transform of the scalar product The outer integrals in (:3.27) are scalar products (work), therefore we have

to apply Parseva.l 's theoren1 ('1.2) . Tugether with (L:i), we obtain as transform of (3.27) (the fact.or (21T) - n is cancelled)

(4.6)

.!.Vt. .:.Vl,

+ L ti ( 0;{( - :i:), ~U:r) D(:i:)) - Lui (or( - :i:), (b~(x) A'[D(:i:)). 'L 'l

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4.2. FOUWHU 11L\NS'FOHA1 OF HIE 37

Transform of the free term The free tcrrn, i.e. the left.-hand side of ( 4.6), can be simplified due to

where the sum is taken over all trial functions which have common sup­port with ¢>{. p:, are the polynmnial of these trial functions. r;, is the corner term discussed in Chapter :3.2. The polynomials p'· are transformed t.o Dirac­dist.ributions, hence the evaluation of the scalar product. on the left-hand side of ( 4.6) is trivial.

Example 4.2.1 Free term for smooth boundaries As an exarnple, we wrtsider the t·wo-dimensional Jw.lf-space :~:·1 ~ 0, :~:·2 E R

For s'i1nplicity, a constant test function is chosen (although not conform), e.g.:

F +-+ ~ 1 - 't ( -i;h 1) C\-;::- e - .

.L'2

F ux £s loc(J,lly, i.e. at the Ytlpport of 9t, r:onstnnt and eljwzl to u1II(:c1) +-+ tt1 (7f(5(:i: 1)- i/i: 1) 27f(5(i2 ). In the original 8fHJce, we lw:ue then fo.,· the free term.

( qj11 (:r), x(:r) u(x)) = u 1 .l" t)(:rl)H(:r2 )H(l - x:J H(xi) r:lx 1 d:r2

= tt1 [I~ 6(J:1)II(:c1)d:r1] [l'1

ld:r2 ] = tt1 I-:(5(:r 1 )II(:r 1 )d:~: 1 = ~

1

.

The last mtegral was evaluated by replacing the Dime-distribution wdh a Dirac-convergent series, cf (S. 5). In the tmnsfonned space we obtain

Here, we lwve n:gnlm·ized the la,o;t integral, i.e. we have taken a Canchy 's Pr·iru:ipal Valne at infi.n£ty: the singnla·rity at i:1 = +:x; cancels ·with that at x1 = - oc. The k ernell /:i:1 is odd, hence the integral over· the total real line is zero. We have eval·uatcd the free tenn as in the original space for smooth boundaries.

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38 CHl1V.l'£1l 4. FOUlUJ:;H lJJ:;lvl

Transform of the BIE for temperature and flux The tro.ru>forrn of the CiahTkin BIE given in (3.27) and (3.28) is

N,

\ (M ( -x), ilx) = \ (M ( -x), lx. ( i') () ( :i:)) + L ti \ 0;i (-:i:), d>~ ( :i:) D ( :i:)) i

(4.7)

.v, L ti (d),( -i), (;~(i) A{D(x))

·i ~, .. tl

+ Lu' \d),(-:!:), ~~Ji·) AiA~O(i)). (4.8)

4.3 Transformed fundamental solutions

4.3.1 Fundamental solutions

Transform of the fundamental solution The Fourier transformation of the differential equation convcrLs t.hc diffe­

rential operator P(D) to an algebraic expression P(i)

6.a(x) = - f(x) 1 --~~·· (--) r'( " ) - :!: U X = -. X , (4.9)

with the :;yrnbol P = -lil 2 = - 2..:~ :/;~.The Fourier-fundmnental :;olution (! i:; the response to a :;ingle unit force f(x) = c5(x) 4 }(.i') = 1. Therefore, we have to solve

6.U(x) = - c\(:r) 1"12r"r(-') 1 -:r j x= -, (-·UO)

which is achieved in the transformed space by the inversion of fj

(-t.ll)

Thib procedure can be applied to 11lllinear differential operators with conbtant coefficients. The Fourier-fundamental :;olution ib always known and often has a simple btructure. (r ib bingula.r in the point Iii = 0 and hence not uniquely

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4.3. 11LL\;'SFOHA1J::D FUND/u'v1J::f\i1/il.J SOLUTiONS 39

determined at. the origin, cf. the mathematical discussion in [DPOl]. The zeroes of the denominator correspond to additional homogeneous solutions of the differential equation

2l.u(:r) = 0 F

B

Transform of the derivatives of the fundamental solution

(4.12)

Due to our restriction to straight clements, the normal vector 11' is locally independent of x. Hence we get as transform of the funclament.al flux and of the hyper singular term

A~U = -zi' · 'VU 4 .A~(r = -z;' · i.d! ; (4.B)

AiA~U = IJ1 • 'V(r·" · 'VU) 4 AiA~D = 1/J • i:'i:(IJ' · ii)U.

4.3.2 Green's functions

Green's function for the half-space It is often advant.ageous t.o establish a BE:\if based on Green's functions

V(:~:), i.e. on special fundamental solutions which already fulfill some of the required boundary conditions. Discrct.ization is t.lwn only needed in the other part.:; of the boundary; fewer degreei:i of freedom are required, e.g. [Ridl6].

For half-:space problems, Green's functions in the original space can be eva­luated by the method of images if the fundamental solution is known. see e.g. [l3on95a]. J3ecause I he proposed r\mrier-131::~1 is espec.iall;.· of in teres! for cases where we do not. know this fundamental solution we need an alterna.l ive approach for the generat.ion of Green's functions which should be totally formulated in the transformed space. This method wa:s developed in Duddeck [Dud97b].

The condition that V should vanish for x1 < 0 is equivalent to a condition derived from the theorem of Paley-\Vicner ( cf. [Hiir90], theorem 7.3.1.). The transform of a quantity which vanishes in t.!w half-space :1: 1 < 0 ha.s no singularitie:; in the Lower complex half-plane 'S(:rl) ::; 0 of the transformed coordinates, d. [DucHl7h , \\'ol79]. lienee we are looking for additional tenus which eliminate the singularities of the fundamental solution U for 'S(:i'1 ) ::; 0.

Example 4.3.1 Two-dimensional half-space We consider a.s an exarnple the two-dirnensional half-space :r· 1 ~ 0. x 2 E IR

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40 CHl1V.l'£1l 4. FOUlUJ:;H lJJ:;lvl

with a .flu:r-free boundary uP.. The tempem.ture ~'(0, :z:2 ) at :z:1 = 0 can be nogardul as e:J:tcrrwl som·ccs de;wribul by

:F H

The corresponding temperature f >< U 4 }0 in transformed coordinates sub­tmcted from the fundamental solution U leads to Green's function for· the half-space ~· with still unknown tb :

1 - i:i.'ltJ~'(O, :i·z) :i:i + :i:~

The fundarnental solution for the infinite space C = li·l-2 i.s singular at i·l = ±ix:l. These singularities should cancel .for xl = -i l:i:~ I· Hen ce we have to annul the residue

This leads to the unknown temperature at the boundary 2•(0, :i:2 ) = IJ-2 1- 1 and finally to the transform of the Green's function for the half-space

This Fourier analogue of the method of images can be applied to all diffe­rential problems with constant coefficients in arbitrary dimensions and can as well be adapted to determine causal time-dependent fundamental .solution.s in the transformed .space, cf. [Dud97bj.

Green's function for the ball A~ second example, the Green's function f' for an n-dirnensional ball is

det.ermined where the temperature a.t t.he boundary is assumed to be zero. V is not only useful for circular problems. but for all bounded problems, if the domain q is entirely enclosed in the ball B(x) , Fig. 4.1. It. can be deduced from the Paley·- Wiener theorem LhaL C (:i:) do not have any singulariLics at all. Hence, no singular integrals have to be computed. The approach can be applied to all isotropic. problems.

Example 4.3.2 Two-dimensional disc The constant flux at the boundary of a two-dimensional ball with radius r0

as external source is. cf. {Dud97b},

f(:r) = 2 ro0'·(ro)6(r~ - :r7 - :r~) F H

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4.4. 11LL\;'SFOHA1J::D TlUilL LliVD Tt:ST FUNCTIONS 41

Figure 4.1: The ball B c JR.n which contains the total domain, i.e.: n ~ B.

·with .J 0 as Br:,o;sr:l',o; function of rFrdr:r z ero. The Gn~crt's function i,o; then :

(· _ (· [ _, -·· .,',( )J (. l·~·l)]- 1- 27frotb(ro).Jo(rol:rl) ~ - u 1 2n lo\0 ro . o 7o x - , ~ -~ . :rl +:rz

The singularitie.s. i.e. the residue.s. have to vanish {Paley- liViener)

[ [1- 211 ro1/'(ro)Jo(rol:i:l) A A J , J i . . ! Res Res , 2 , 2 .:r· 1 = ~:r2 ,x2 =0 =-;-+zKTo1/J(r0 )=0.

J:l + J:z 2

Hence, we qet for the .flux l!J(r0 ) = (2711'0)-1

• and jinallu .for Green's .function

~· (·'·) _ 1- Jo(rol.i'l) \ ,!, - A2 A2 '

J:l + J:2

The non-singular character of \7 and its dependency on the radius of the ball B is shown in F ig. 4.2. The larger the radius, the smaller is the extension of the Green'.s .function. The msidual appmach shown here is a generalization of an equilibrium condition. in the simple case discussed here, the jlu:r t~{r0 ) can as well be determined by enforcing the static equilibrium between the single unit source at the odgin and the circular )lux at the boundary

r. [6(:r) -1jJ(ro) o(ro - l:rlll dx = 1 - 27fro!)(ro) ~ 0. JP"

4.4 Transformed Trial and Test Functions

4.4.1 Transform of the cutoff distributions

The t.ransform of t.he cutoff dist.ribut.ion x0 is for the reference clement in JR2

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42

FiKure ·1.2: Greeu's funct,ion f(w l.he dise D wil.h the radius r 0 = 1 (left) aud ro - 2 (right) in the Fourier space .

.l:'or the quadrat,ic boundary element, in llt1 we ~et

x.o(;~:)

:\(J (i.)

H(:r1 )H(l- :r1 )H(:r~)H(l- J:2 )<i(:ra)

~ (e-i:i:- 1) ~ (e-i:i:2 - 1); :I:] ;!;:.!

mul for t.lw r.ria.ngula.r boundary denu~m. in JR3

,'( o(:r) H(:rt)H(:rz)H(I- :r1- :rz)1.>(:r3)

( •l.lfi)

.• 0 <~ :~ (x) :i:l ~:i::l [:~~~ (1 e iii) .1~~ (1 e ix~)]. (4.16)

4.4.2 Transform of the trial functions

.t:.:lmm~ms of a.rbit,nuy polynomial d1~grcP arc cousl.mdc1l \'ia. a umlt.iplicat,ion by p0 ( ;1:) in the original space or an analytical convolution in the t.ransformed span~

( 4.17)

:\II polynomials rP ( :r) are r.mnsformed t.o I )i me di~tri but.ions

p0(x) = L L Ckfl:~. J)'\r) = (::!J; )" 2:::2::: ckri16(1.)(:i:k). k k

lienee, Lhe evaluation of Lhe convolmions in (4.17) is Lrivial

1 7)0 * vo - """' """' c ,,it fl..,: 0 (211 )n '" ~ ~ ~- /,;.\ · k I

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4.5. COi\iSTIWCTlON OF 1HJ:: Hl'.'M lvl/t11UCJ::S 43

Table 4.1 shows the trial functions for some two-dimensional line elements. Three-dimensional elements arc const.ruct.ed analoguously.

constant: ¢}l = x0

linear: ¢1 = (1 - xt)xu

¢>2 = X1)(0

par<:Lbolic: ¢1 = (2:cf - :3:r1 + 1)y0

¢>2 = (·i:rf - •i:r1) y 0

¢>:~ = (2:cf- :r, )y0

4 01 = (1- iDd:\·0

4 ¢~~ = W1~o

F H

4 ¢~~ = ( - 'Wf- •1iD)k0

4 ¢) = ( -2Df- iDt ).\ 0

Table :1.1: Trial and test fnnctions of the R_:!

General transformed straight elements are obtained as shown in (3.20) by the transformed dilation and translation operators

('1.18)

with [u.i]-T as the transposed inverse matrix of a'. For straight clements and for arbitrary polynomial trial functions p0 (:~:), ! he transformed expressions <:Lre analytically known for ~2 and ~3 .

4.5 Construction of the BEM matrices

Algebraic system of equations The discretized Fourier BlE ( 4. 7) and ( 4.8) Lead t.o an algebraic syst.em of

j equations identical to t.hat obtained in the original space (2.26) :

from (4.7) : pJ ~ Hfi i ~c·Ji ·i . u + ~ u t - ~ 'u U : (4.19)

from ( 4.8) :

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44 CHl1V.l'£1l 4. FOUlUJ:;H lJJ:;lvl

But now, the matrices are computed in the transformed space. \Ve have :

F~

F.j t

1 / A A A \ (211)" \ rpi( -:i:), f(i:) U(:i:) I;

(2:)" ( ~{(-i·), J>~(:l-) {J(i));

(2:)" (d~(-i:), ¢~(:[) A~(T(:r));

·- (2:)" (J~( -i·),p~,(i"));

r2!)" ( ~( -n, len AiU(i:));

(2!)" ( ~(x), ¢~(:!:) A.~O(:i:J); (2!)" ( ~(i), (;~(±) A{A~D(x)):

( 4.20)

Due to the equivalence of the work terms in the original space and t.he trans­formed space which is stated by Parseval's theorem (4.2), all the vectors and matrices of (4.19) have the same values as would be obtained by a traditional BE.\I approach. Therefore, the further algorit.lun of the BE.\if algorithm can be taken without any modification from the standard BE.\1. This is not. dis­cussed here, see [Bon99].

Seperation of known and unknown boundary values The final RE\1 algorithm is obtained by rearranging the matriees obtained

in (4.19). The parts related t.o the known boundary values have to be shifted to the right-hand side, i.e. to the force vector, of the equation. The reordered system can be written in the following form

LA.i'X'=.P. ( 4.21)

with the s;yrmnetric and fully populated matrix .~1 and the vector of unknown degrees of freedom X.

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Chapter 5

Heat Conduction

5.1 Isotropic Case

5.1.1 The Dirichlet problem

We com;ider first. a pla,IH:' domain n which i~ heated by f:it.ati011<1l'Y interior

sonrcc·s f. Tho temperature at. tho boundaries is kept to 7:CnJ. \Ia.t.lwma.tieally, this leads to Liw Dirichkt, probkrn of t.he Poisson ('quation. Due to tlw lack of siu~ularit.ies aud it,s g·eneral simplicity, tl.J.is Dirichlet, problem is best S11ited t,o

demonst.rat.ing t.he equivalence between traditional 13ENI and Fourier DC:rvt Wit.hont lo~s of generality, thl.' isot.ropic c:omluctivity /{ is set. t.o one.

The Fourior-Galerkin boundary integral equations ;\s a. fin;! awl l)(~ndunark Pxarnplt~, l.lw Dirichlet pwbkm for Uw Poi;;soll

equallon

Su( :r)

u(x)

- f(:r),

Ur·- 0.

x E U;

;1: E f',

( 0.1)

i~ solved in a quadratic two-dimen~ional domain il - [0, 1] x :o, 1]. ;\t the bonmlarier.; the ternpora.t.nn· 11 is kopt. to zero. The interior is r.;ubjoct.cd t.o

sta.l.ionnry heat sources f. Tlw boumla.ry nn is dividc;d imo c'ight, dcmenl.s, two for each side (Fig .. :i.l). \Ve prefer thi~ coarse mesh for the sake of clarity of the validat.ion. The fundamental solution and its transform tiJr t.he Laplacian 2. arc

I. 1 I I 1 c' - 21f Jl '/ J; 1

-10

l_i 1

(3.2)

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46 CK\Pn;u5. HEAT CONDUCTION

u=O 1

'//, = 0

1

Figure 5.1: Quadratic domain with eight. boundary elements

T aking imo account that t.he temperature is vanishing at the boundaries, the general system of BIE can be reduced to

0

4 0

N,

( •J j' [T\ ~ l ( •j •i F\

C/\' X * ; I + 6 t ¢ 1.' C/Jt * u I

N

(1~i(-:L-),fJi) + f:t' (0;{(-x),ou!).

A uniform heat source f is assumed oved~ (Fig.G.2, right) :

4

J'x(x)

j~(i·)

(5.3)

(5.4)

Figure 5.2: Constant. trial function (left.) and uniform heal. source (right.)

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5.1. lSOTlWPIC CASt; 47

Equivalence test of BIE in Fourier space The constant. t.rial and t.est functions for t.he flux t, Fig.5.2, left. ,

c/Jf H(:I: 1 )H (1 - 2::c: 1) J(:c2) F f--7

., 1 ti''t ii:il(c-i>'l/2 - 1),

rp~ H(2:r:1 - l)H(l - :q)o(:1:2) :F f--7 (~~ i:i:[ 1(e iX1 -e iil/2) ,

¢{ H(xz )H(1- 2x2)J( x1 - 1) 4 '3 ¢t ii::!l(e-i,~d2- 1)e-i:h,

(l)t H(2x2 - 1)H(1 - J-'2)5(~:1 - 1) F rN ii::!L(c-ih _ c- iX2l :t)c-·iX1, f--7

r/JC H(:~:J)H(1- 2:r:J)6(:~:~- 1) F (~'( i:i:[1(e ixl/2 _ 1)e iX:z f--7 '

q,r II(2x 1 - 1 )II(1 - X1)J(l:2 -1) 4 '' (i rf>t il'\ 1(0 -ii1 _ c-ii1/2)c-i:i::2,

r/J[ H(:~:2)H(1- 2:r:2)6(:~:l) F f--7 (~I i:i::/ 1(e ix,/2 _ 1),

¢~ H(2x2- 1)H(1 - x2)8(x1) 4 '8 ¢l ii::!l(e-i"'" _ e-ii:2/'l) .

(5.5)

The integration of t.he first matrix ent.r_y in the original space is

1·1/21·1/2 1 1·1/21·1/2 H 11 U(J: 1 - y 1) dy1 rl:r1 = ;:- In J(:rt - Y1 )2 dy1 rb: 1

0 0 2r. 0 0

1 1·1/'2 ----:;: [In l :~: 1- 1/212 - 2- 4:~:1 (In IJ: 1 - 1/21 -ln l:r1 I)] d:J:1 8" 0

1 - -(2ln 2 + :3) = -0.08726. (5.6)

16 1f

The corresponding integration in Fourier space leads to the same value

( ~ '") d . I

The benchmark example The total matrix II is obtained analytically either in t.he original or in

Fourier space as :

-.08726 -.0:_1210 -.0()!);_14 .002:_14 .0007!1 .00484 -.04222 .00!);_14 -.08726 -.CH222 - .009:H .DOt8'1 .00079 -.0093{ .0023,1

-.08726 - .03210 .D023'1 - .0093'1 .00079 .OCHs.-1

II = - .08726 -.00934 - .04222 .00484 .00079

-.08726 - .o:J21o - .00934 -.04222 sym. - .08726 .002 :34 -.00934

-.08726 -.03210 -.08726

Page 50: Fourier BEM: Generalization of Boundary Element Methods by Fourier Transform

.2564

.2564

.2564.2564

.2564 .2564

.2564

.2564

.3362

.3362.1667

.1667

.1667

.1667.3362

.3362

.1667

.1667

.1667

.1667

.3362

.3362

.3362

.3362

48 CH"\PTJ::ll 5. HE/tT CONDUCT.ION

The in tegration of \Olume sources (Fig.5.2, right) leads for the first compo­uent of Fk t.o (Jo = 1)

1 l' ·• ' A

( 2 . ) 2 (Di (-i:) f x (:i:) U ( i) di: 7f . F:"

__ o_ __!:___ (P+i.i'l/2 - 1) e - . e - el i: eli· f l [' . ~ ( - ii:, 1) ( -'ii" 1) (2 )2 A . A A ( A'l A'') ' 1 " 2

1r · • Jli:2 x1 :c1:r'l -x1 - :c~

1 -. -(27r + 2 ln 2- 11) = -0.04417. 247f

The total vector on the right-hanel side is due to symmetry

(5 .8)

P = (-CJ.OH17, -CUHt17, -0.0Jt17, -0.0 1H n - O.CHt17, -O.CH t1 7-0.0,H17, -O.CH-117).

Thc unknown cocfficient.s for t.hr boundary flux arc obt.aincd by H-1 F :

e· = (o.2564, o.2.364, o.2564, o.2.564, o.2564, o.2.3 640.2.364, o.2.564). (.3.9)

As a first eontrol, we ta.ke another Gauss theorem, the heat equilibrium condition. The inte11,ral of the fiuxes a.t the boundary sta.nds in equilibrium with the total sources (L~ = 1/ 2 is th length of the i-th element)

ii j' 2__: L~e = 1.02544 d,; - J(:r) ch = 1. ~~ n

(5.10)

The coa.rse mesh leads to an acceptahle result, the global error is 2.5% .

.3362 .3362 .2564 .2564 .1667 .1 667

n D

25M ~ ~ .2564 .1 667 D ~ .1 667

.3362 o ] .3362

.2564 ~ ~ .2564 .3362 D 01 .3362

.1667 D 1 .1 667

tt E::~JOOE51 .2564 .2564 .1667 .1667

.3362 .3362

Figure 5.3: Flux at the boundary (left : '1 x 2 element:;; ri11,ht: 11 x ·1 element;;)

The results for a refinecl hounclary rnesh with four elernents on ea.ch side a.re shown in the right part of F ig .. ). :). T he global error of t he equilibriurn

Page 51: Fourier BEM: Generalization of Boundary Element Methods by Fourier Transform

x1

0.3

1.0

x1

1.0

0.01

0.001

series solution

two elements per side

four elements per sideten elements per side

Page 52: Fourier BEM: Generalization of Boundary Element Methods by Fourier Transform

50 CK\Pn;u5. HEAT CONDUCTION

in both directions are taken: k = 1..12, l = 1..12). ln the right part, the mean error wit.hin each clement (the difference bet.ween the t.wo solutions) is plot.!ed. A discretization of 10 rlcment.s per side is sufficent. to reduce t.he local error to approximately 1'/c .. A tot.al number of 122 = 144 eigenfunctions are related here to '1 x 10 = ·10 boundary element:;.

The postprocessing : interior values for the temperature The general BIE for the evaluat.ion of the int.erior values of the t.emperature

are:

Nt

( (/>{z, f X * U) + L ti ( (j){l' q'J~ * U)

i.Yt

4 ( ~i~( -x), fix) = ( ~fl( - :[·),fir)+ L ti ( q)il( -:L'), cp~(r) ·

The simplest choice for the int.erior test functions oi1, the Dirac-distributions 6(x- t;i), leads to a point-wise evaluation of the interior temperature u(:r = .;.i) duP to ((5(:r- (i), ux(:r )) = tt(~j), ~j E [2 :

1'lt

n(e) = (o(x- e), fx * U) + L e (6(:r- e), 9~ * U)

Fig.0.G (left) give:; the result for the example of the coarsest mesh (4 x 2 elements). The error, i.e. the difference to the series solution obtained with with 12 x 12 eigenfunctions, for this very coarse mesh is already· reasonable small (right graph of Fig.5.5). This means chat. a total of eight. boundary clements already gives a result which differs by only 5% from an eigenfunction solution (144 eigenfunctions).

5.1.2 The Neumann problem

Next., t.he equivalem'c of the traditional approach and Fourier ilE!Vl is studied when singularities (in this ease hypcrsingularitics) arc present. The rigorous distributionaJ forrnulation of the DIE developed in Chapter :3 enables the accurate handling of all ::;ingularities \vhich occur in the ~ingle integral:;. It ::;how::; in total that all non-integrable :;ingularitie::; vanish if the terms are

Page 53: Fourier BEM: Generalization of Boundary Element Methods by Fourier Transform

0.06

0

-0.003

0

+0.003

:3.1. TSOTnOPTC C.'\SF: 51

temperature error

Figure 5/i: ne~ult of t.lw Dirichlet. problem (2 elements per 1.iide); left : the temperat.ure ·a in the imerior; right : l.he differeuee to t.he series solul.iou

combined adequately 1 . 1\o regulariL.ation is needed either in the original space or in thuril~r r-;pacl~.

Fourier-Galerkin boundary integral equations As an example for a ::'\ eumanu problem, we are considering

.6.u(r:) t(;l:)

-f(:r), tr- ll:

:r E: n: X~[,

(fi.lii)

in t.lw qua.dmt.il; r.wo-dinumsiomli domain n - [0, 1] X [0, 1]. Hc~re, r.lw flux l at t,he houudaries is set t,o zero. The iuner plaue is heated as in Chapter 5.1.1. The syrnmetri<.: Gctlerkin TITf: for thi~ 1\eumann problem is

4 (6?,(-J:),A{{ii~J) = (0!,(-:i:)J~:.A.{(J)- f:u•(o·!.(-:i:),<;:,.A.{.A;r!). i

The differential boundary operators are (vJ. ti are the normal vectors of the lc~st. fundion (/>-~ a.nd Uw t.ria.l fundions (/>~p n~spc~ct.ivdy)

(5.17)

1Compare l:htngi & Guiggiaui lC:KHR92, J:o·COlj anti MikhlinlYlP86J.

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52 CK\Pn;u5. HEAT CONDUCTION

Hence, the fundamental fiux is obtained as

(5.18)

and as the hypersingular term (the explicit term in Lhe original space is suppressed here, iL is too lengLhy)

F B

(' ,] A{A;.u = z;J. v(1). vu)

Aj _Ai(; = (v{:h + IJ~i:2)(11fi:1 + IJF2) I t . xi + i:~ .

The Left-hand side of (5.16) can be simplified (ux = x(.r)u(.r) and A{u tr = 0) :

(5.20)

This free terrn is singular: its singularity is of type c\(0) which will be shown in t.lte next paragraph.

The singular integrals The entries for the matrix are obtained from (i.j = 1, 2, ... , i\J

(5 .21)

Because of the hypersingularity of the kernel A{~U at x = 0 there are some pairs oftest. and trial functions which lead to infinite entries. The appropriate combination of these singularities, which is derived here, shows ! hal finally all combined integrals arc at, mosL weakly singular and can be cvalua! cd with standard rouLinc~"l. The only condition required is ! hal con! inuous Lest, and trial functions are dwsen (they should belong at least to C'0 ). This continuity is already achieved by linear polynomials, cf. [I3on99].

An example for pairs for which the kernels are singular is visua.li;wd in Fig.5.6; the corresponding distributional notations are :

<!>t, = (1- 2x1)II(:rt)II(1- 2:rl)6(:r2)

rj>~ = (2xt)II(:rt)II(1- 2:r·1)r5(:r2)

rj>~ = (1- 2x2 )II(:r2 )II(1- 2:r2 )6(:ri)

4 Jt, = :r1 2 (2 - i:i'1 - 2e_.;;, , ;~):

4 ¢~ = :r 1 2 (ie- ·ii:Ii2 (i:i\ + 2) - 2);

4 ¢~ = :r2 2 (2 - i:'i·2 - 2e-L"21~) .

Page 55: Fourier BEM: Generalization of Boundary Element Methods by Fourier Transform

5.1. lSOTnOPlC C11SJ:J 53

tt:>.'it function trial functions

Figure 5.G: The pairs of t.ria.l and test functions with infinite matrix entries

ln t.hc original space we get. for the first. singular ent.r;y

I 11 ' 1 A1A1Lt\-- (1- ?··) - y 1 d· d-· 1 11/2 l-1/2 1 2 \ (/Ju • 0?u * t t · / - . ~ ~.!- 1 (·, , )2 !J1 -X 1 2,. . () . 0 .L I - Ul

~ t12

l((2x1 - 1)(ln l:r1- 1/ 21-ln lx1l)) + 1- ~J d:r1 " .Jo 2.r 1

1 ·. . 1 1 ~ - ( ) -(2ln2 + 1) +lim;- ln c + -6(0). 5.22 471 £---+0 271 6

The first part of the last integral, i.e. J;l11~((2:r1 - l)(lnlx1 - 1/21- ln lx11)) d:r1 in (5.22), is weakly singular at :r1 = 0 and therefore integrable. The last

term J;~12 1/ (2:r 1) d:r1 is a homogeneous distribution, ef. [II" or9~ which is only defined as a Hadamard finite pan. The a.ddit.ional infinite part 6(0)/6, which is normally neglected by Hadamard's concept., was obtained by an in t.egration via dist.ributional F ourier transform. This result is more easily obtained in the transformed space :

1 / ~ 1 ' ~ 1 ' 1 ' 1 'c) (21r)2 \c\,(-x) , 0\At ~L

_1 __ l i'~ (2 + ii1 - 2~iXJ ~~) (2,~ i :h- 2e (2~)2 '1.4( '1'" + .,.2) .,. • R " · ·1 · ' 1 · '2

1 1 1 - ( 2 ln 2 + 1) + lim ;- ln o: + --:;-6 ( 0) . 47r ,,_,Q 27r 6

iiJ / 2) (Li1 <ti2

(5.23)

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54 CK\Pn;u5. HEAT CONDUCTION

The second singular value is

(5.24)

and the third is

(;).25)

Another pair of two singularities originate from the free term (5.20). \Ve have

- ((1- 2.rl)II(:rt)II (l- 2:rt)r5(:r2), -u(:r1, :r2)o(x2)) ·1/ 2

-ii(O) l (1- 2xdu(:c1 , 0) cb:1 ,

where the local cutoff distribution x = H(:c~) and the local boundary cliffe­renLial operator Ac = //1 V = -EJ~ were Laken, which results in A~x = -J(:r~). Locally, "u(:r 1, 0) WRl'i approximated by the polynomial ansatz functions on the boundary, i.e. by:

The free term on t.he right-hand side leads t.o a singular value related Lo u 1

F f-t Ru =

u

and to another singular value related to u2

(5 .26)

(5.27)

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5.1. lSOTlWPIC CASt; 55

The easiest way to compute the free term in Fourier space is t.o rearrange the factors in t.he scalar product. \Ve define :

;p (1- 2:r 1)2H(:r 1)H(l- 2J:t)6(:r2)

:F f--1 01

<Y -i1

3 (ii~- '1:i: 1 - 8i + 8e ih/2):

(1- 2:rl)2x1H(x1)H(l- 2:r!}6(x2 )

T, ~.;·2 _,y.-:l (2·A1. +8;+(?->· _o.z·)~-i:i, /2) " '" ·"1 ·. 1 ,. -·<·1 0 <.. •

Additionally we need A£x = F{-6(:1:2 )} = -2Ttcl(i1).

These computations arc presented explicitly t.o emphasize t.hat. all non­integrable singularities cancel. INc have to combine

1 - -(2ln2 + Tt)u1

: 87f 1 2 --u.

47f

The finalmaLrix [G' ' ] is of rank Nu- 1, Lhe problem is not unique.

5.1.3 The mixed boundary value problem

(5.28)

Compared to tJ1e ~eumann problems, t.he more general ease of mixed boun­dary value problems do not lead to additional singularities. The off-diagonal mat.rix has no singular entries. Therefore, we restrict the presentation to BIE and the explicit. evaluation of Lhc free terms.

The system of Galerkin boundary integral equations The system of Galerkin BIE for the mixed boundary value problem

.6..-u(:r) - j(:r), X En; n(x ) = Ur , X E fu, (5.29) t(:r) = tr, .<: E f t

is in operator notation in the original space

(5.30)

( r9u, fx * A;u) (!J.:31) iV L Nll

+ ""t' ( c~1 d * AJrr)- ""u' (c!J.l c~i 8 AJA' [T ) L.....t : ul,1 t_ t.V L...,.; .·11 ~ ·1 ll L t '

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56 CK\Pn;u5. HEAT CONDUCTION

and in Fourier space

(o.:n)

~"\Tt, !Vu

+ I>i\i~J-:i·),~;~AUi) - l:u;\i~J-:L·), c;j~A:{A:O),

The additional free terms The left-hand side of the Galerkin displa<.:ement DIE (i).:.m) for a <.:onstant

test function q)~ = H(:rl)H(1- 2:rt)6(x2) 4 01~ = ii'~ 1 (e-·ii:l/~- 1) leads to

1. 1 ;·l/~ (¢1~,nx) = .H(:t· 1)H(1-2:t: 1)rl(:t·2 )u(:r)x(:r)d:t:= 2 n(:t· 1,0)d:r 1•

~ 0

The relevant. local par! of the cutoff disLrilmLion X for t.he domain rl is equal !o H(:r2 ) . The kernel was simplified because of ri"(:r2)H(:r2 ) = 6(:r~)f2, c.f. ChapLcr 3. With the local diseretizaLlon n(:t:1 , 0) = u1 (1- 2:~:1) + u2 (2:r1 ) t.he free tenn is separated into two parts

1 [' / 2 1

2 ./o (1- 2:r 1) (Lr1 = 8: (5.34)

1 ll/2 1 ? (2xl) d:r1 = -

8.

-.o

The free term of the traction BIE is according to (5.20)

(~D-;, ,A{{ux}) - (&;,,xA{u) + (0?;,,nAix).

The second term was discussed in (5.26) and (5.27). The first is equal to

(d.. xAju\ = ((oi. xt\. . u, , t. I . u , . I (5.:35)

For the first Linear test function (1 - 2:~: 1 )II (:~: t) II(1 - 2:t: 1 )6(:r2 ) and the loc.al approximation f(>r the traction t ~ t 1 = const. we get

R:~l R1:~ . ( ) 1 111/2 1 ·t = "t = ;- 1 - 2:rl ( ·"1 = -.

2 0 8 (5.36)

lienee, there are no additional singularities. The same results for the free terms can be obtained in Fourier space.

Page 59: Fourier BEM: Generalization of Boundary Element Methods by Fourier Transform

5.1. lSOTlWPIC CASt; 57

5.1.4 Some computational aspects

In thi~ ~edion some comments on the analytical and numerical evaluation of the integrals in Fourier space are given. The Fourier kernels are in general highly osl:illant and demand particular integration routines. Additionally, the integrals have to be evaluated over IR". A t.nmcation necessary for numerical routines should be done carefully taking into account the fact that the kernels decrease in t.he worst. cases by only l:i:l-2

• The imaginary part. of all kernels cancel.

Numerical computations in Fourier space In the following, the numerical evaluation of one single matrix entry is

discussed. As an example, we compute the value u 1(G 11 + G 1:J- R~ 1 ) which was determined analytically in (5.28). The kernel leading to the singular free term H~ 1 is concentrated on a line

71( ' )Al ' 1 X(') 0 -x "'-tX =- 31fu X1 ,

which is defined by a Dirac-distribution and hence cannot easily· be repre­sented numerically. It. is the counterpart. of the constant part of G 1

.1 + G13

which is the singularity at. infinity, i.e. we compute rmmerically (1/1 is t.he normal vector corresponding to t.he test. function)

with :

The integral / 11 is regular , it. is neither locally nor globally singular and can be evaluated numerically, cmnpare Fig.5.7,

1 - 256r.2 '

lim (kll - lim /(11 ) :i:·--+oo 1.1 1 ·i·--+ x

Analytic integrations in Fourier space The analytic integration algorithnu; of the mathematical computer sys­

tem .\IAPLE [Wat97] are not sufficiently capable of handling complicated expression:; containing Dirac-, IIeaviside- or ~ignum-distrilmtioru;. \Vherea;;

Page 60: Fourier BEM: Generalization of Boundary Element Methods by Fourier Transform

58 CH;\ PTP.R, .'). HF;\ T CO:VOUCTTO;v

Figure 5. 7: Left: The not eombined singular kernel in Fourier sp<tee; right: The wmbiucd n~gula.r kcrnd for 1 11 in FouriPr spa.n~

the "fourier'' -routine of the same program does not have t.hese problems be­cause it. is based ou the distributional defiuit.iou of the Fourier t.rausfonn, ef. lDud97b]. Consequently the latter routine is used here w replace the required int.egmt.ions. i\ st.mula.rd integration over R.1 is defined m;

r d{q) dtj = lim F{ d{q)} =lim;· :X. ai(v) e'YY dij: } -'-" · • • • !i-->0 · • • ·1i-+U -oc · · · ' ·

(i'>.J7)

a successive applicalion renders lhe mulli-dimensional integrations.

5.2 Anisotropic Case

5.2.1 The 1nixed boundary value problem

The differential equation and its fundamental solution The anisot.ropic heat eomluet.ion is desr:rihed by the pa.rt.ial differeutial

equation -div(Kvu)- f which is in tensor not<ttion (k,l- 1,2)

(5.:38)

'Jhe eonduetivity t.ensor H = [J(kl] is a symmel.rie teusor of order two. ll.s eigenvectors point int.o the two prineipal direetions of the eonduetivity. If K 1l1~p1~llds not. on .1:, the 1lifl'1~rcnt.ial op1~n1.t.or is simply

P(a) - P..'n du - '1Kui1l'l - K'l'2()'l'l

Kuxi + Ku.'i:d:'l + K:nx~.

Page 61: Fourier BEM: Generalization of Boundary Element Methods by Fourier Transform

5.2. iliVlSOTROPlC CilS'J::

The fundamental solution is hence (11{1 = detlC r 2 = :z:1'K-1x)

u

4 (!

1 1 '2 --IKI- I lnf 47f

1

59

(5.!10)

The tenn in the original space can be found for example in [I3on99]. The unknown flux at the boundary is obtained from

F H (5.!11)

Therefore, the other fundamental terms nel:essary for t.he Galerkin I3IE are

T (5.!12)

/; AiA~ (J = -lJ~[(kl;i;lv!nKmn·'i',[r,

where u·i , z) are the normal vectors of the i-th trial function and the j-th test function, respect.ively.

The system of Galerkin BIE in Fourier space In analogy to the isotropic ease, the Galerkin I3IE are in Fourier spaee

N,

(6i(-i:),ftx) = (6{(-.f:),fx[f)+ L:r'(Ji ( -i:), ~~O) i

/\/u

""' i ( ]_j( ~ ) c; A'.;t'') - ~ u (;Jj, -:r 'rf:'u t ' ; (5.43)

·'-'t

( :J ( A) j' A~j[",.) ""' i ( ii ( .') ~iA~j(T) - rf:'t -J; • X t. / - ~ t Wj, -:r , rilt t.U

~Vl1

+Lui ( ~{(-i) , <7)~A{~{r). (5.!1-t)

Attention should be paid to the free term A{ilx as discussed in the isotropic case. ALL singularities cancel.

5.2.2 The Dirichlet problem

For the Dirichlet problem ('u = ur = 0) the system of BIE reduces to

N,

0 = (0;{(- i:), fxDJ + Lt' ((p{(-:i:),~bf.DJ (5.45)

Page 62: Fourier BEM: Generalization of Boundary Element Methods by Fourier Transform

60

The fiux at, the boundaries is approximaced by ·1 X ·1 con:>Lant elements. The tensor of conductivity is chosen a3

[ . . [ .'3 2 ] 1\/r:l. = 2 :3 : (5.46)

TIH~ principal din~cr.ions charact.<~riz<·d by t.lw Pig!'m'<~ct.ors ~ 1 ,1 of r.lw m<·dinm

lie in r.lw diagunahi of tJw carl.t•sian coonlin:Ht~ ~ysl.t•nt. Tlw Pigcuvahu:s an~ 0'1,:l = 1, i'>, i.e. t.he medium has in direct.ion 6 a conductivity which is five times higher than that in the ot-her direct-ion. The analyt-ical integration of the fir~t. pair of trial and tc~t. fnnct.ion:-:;, (1;)1,o1} fron1 (5.5), leads t.o a result.

analogue t.o ( 5. G) :

TIH~ t•xplicit. symnwr.ri<: mat.rix is g;i\·t~n h<'l'<'. lr. may st'n't' as a. bc'JH:hmark,

-.0390.3 -.01436 -.00872 -.00813 -.00425 .00006 -.00145 -.01559 .0.390.3 .02·197 .00872 .OOOOG .00571 .005 IG .00115

-.03!!03 - ()1136 -.OOH.j .OO.ji(j ,()().j7l .00006

H -.0.3903 -.Oliii\9 -.00145 .00[)[)fi -.OO-l2ii = -.0.3!)03 -.01436 -.00872 -.00813

~ym. -.03!.103 -.02197 -J)()/:\72 -.o:~mn -.OU:lfi

-.03903

ln Fig.5.8 tlw l.t:IUJH~rat,un~ dut' t.l) a single som-e<~ in l.lw center of r.lw sqHaw is given. The auisot.ropie r:hara.(·ter is elra.rly rdket.ed iu this r(~:mlL

()

Figure 0.8: The temperature u due t.o a sing-le cent-ered somTe m au anisotropic medium (4 x 4 elements)

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5.2. iliVlSOTROPlC CilS'J:: 61

5.2.3 The three-dimensional case

The fundamental solution for the ]Rl is

TT- 1 I r.··l-1/2 1 u--L\. '-471 r

(S.47)

with IKI = detK,r2 = x 1K- 1:r and k ,l = 1,2,:3.

Page 64: Fourier BEM: Generalization of Boundary Element Methods by Fourier Transform

Chapter 6

Elasticity

6.1 Isotropic Case Claude L.M.H. Na·vie1·.

6.1.1 The mixed boundary value problem

Na.vier's differential equation and its fundamental solution The strong form of .\lavicr's differential equation for linoa.r dat;tostmies is

in t.lm:e dinwn~iDns (.r, :i: E:: R_:l}

"F 0

11 Bun,, - ().- 11) rhru1

I' :t:dtii.k (.\ I /'·) :i'~r;:Z:(u,

-h,

k k, l = 1: 2, :1:

(6.1)

"11.!-, ft., an· thC' l.lace componrnt.s of t.!w displa.c(•meuts arul volttrn(• forces n;­spectively. tt.). are Lame's constant. parameters for t.he material which are related t.o Young's moduhu; E and t.he Poisson ratio ;; by

E11 E /\- · · , Jr - ( . . (G.2)

(l+t;)(l-2t;)' 21+ll)

\Vc a:;t;unu.: geometrical linearity (i.o. small displacements ami t;rnall defor­mations) and physical linearity (Hooke's liuear stress-strain rdat,iou). The fundamental solution is t.he response to a unit f(lW:' in the direction Xm

(G.3)

S~:m is l<n.mecker's symbol. l~y rearranging (6.1} and inscrt.ing (6.:3} t.lw rnorc suil.abiP matrix form is obtaim:d (I is the idcnt.il.y matrix)

P(D) u- -~o(x)I fl

' ' 1 Pi:r) lJ- --I.

p (6.4)

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64 CH/tPTE.'H 6. t.'LilSTlCLTY

with the three-dimensional clifferent.ial operator

F H

P(a)

P(x) (6.5)

where the two parameters for t.hc material arc now t.he wave vclocit.ics cP, c, of the elw;tic medium (d = c;- c~)

2 ,\ + 2p cP = ---,

p (6.6)

The funda.rnental solution(; in Fourier space itj now derived by simple inver­::;ion fron1

[ 2 .' 2 2 .- 2 r:f:i: I :/;.2 c8 .D + c1.r1

cix2i'l c~xz + di-~ ci:h:/'1 cfhx2

IIenee U is

[7:3cl _ [l.T. ]3d _ ' - · km -

di-l:i-3 Dn di·'2:i'3 (;21

c~i·2 + cix~ l [ l.fn

cf:i; 1 i2

c;i~ + c:TJ:~ ~ ' A c1:r3:r2

012 (Ju

l [ 1 0 0 l (rzz (;2:3 0 1 0 (r32 (!:3:3 0 0 1

(6 .7)

For t.he two-dimensional case the Fourier fundamental solution is simply

(6 .8)

The fundamental solutions for the original space in two and three dimensions can be found in the literature, e.g. [GF97]. They are wit.h l:rl = (2:::= :r~) 1/2

L , 2d / k rn

(6.9)

(6.10)

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6.1. lSOTlWPIC CASt; 65

The system of Galerkin BIE The quant.it.y dual to the boundary displacmnent.s, the t.ract.ion vector t , is

derived from the strain-displacmnent. relation (compat.ibilit:y)

(6.11)

and the stress-strain relation

(6.12)

The traction vector t' = 1i' · () of the i-th boundary element with the normal 11; is derived directly from the displacements u by

AII(Ornnrnrlkt + fl l!{ (Dt.:lll + 8ruiJ (>.u'fJJm + JU!.;"uk)um + ttu;numllk

i(>.uk:/:·m + JW;nxk}iim + ipv:n.imflk.

Hence the fundament.al traction f = A~t/ is in Fourier space

The boundary operator~ is obtained by solving T' = ~0", it is

with c~ = c; - 2c:; = Aj p.

dui i2 + c;l!z:l:l c;l/i · [:i; [ + ci112:f 2

C~ll;~:f·2 + C~l{I:)

(6.13)

(6.14)

\Ve can now establish the system of the vectorial Galerkin BIE in Fourier space for the displacements and the tractions:

(6.16)

Nt N u

+ L e ( ¢{( - 1:). (J>~(J) - L u' ( ¢{( -i), (b~,~(J): i i

- \0:(-X).A~iix) =- \~(-x),/xAiD) (6.17)

In an abbreviated notation t.his is :

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x2

x2

1x x2

1x x2

1x

1x

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6.2. iliVlSOTROPlC CilS'J:: 67

finds its equivalent in the Galerkin BlE obtained from (6.16)

(6.21)

The indices i, j run over the total number of degrees of freedom N1 .•

As example, a simple square n = [0, 1] X [0, 1] is stwlied. The material is described by Poisson's ratio i! = 1j:3 and Young's modulus E = 2.1 · 105

[l'viPa]. The tractions t = (t 1, t 2 ) on the boundaries are approximated by const.ant trial functions oi I' q>L. In the first case two elements and in the second eight elements were chosen for each side. Hence t.he total number of degrees of freedom are N, = 2 x 4 x 2 = 16 and Nt = 2 x 4 x 8 = 64. As volume force a constant uniform Loading in :r:! direction is taken, i.e . .f1 = 0, f:! = 1 for X E n. The results for the displacements u = (uL, U;!) in the interior are given in Fig.6.1. The boundary conditions u 1 = u2 = 0 are still only very approximatively fulfilled in the ea.se of two elements per side, though sufficiently well by the finer model with eight elements on each side.

6. 2 Anisotropic Case

6.2.1 The state of the art

For general anisotropy, it is not possible to find an analytic expression for t.he fundamental solution U in the original space. The approaches which can be found in t.hr literature, e.g. [K6g00, SG98, Sch94b], arc ba."icd on t.hc rnunerical evaluation of n. l~nfortunately this is too time commrning to he r;ornhined directly within a standard I3Ervl algorithm. Therefore, a sufficiently r;ornplete table of numerical evaluated values of the fundamental solution ha;,; t.o be established in advance, cf. [\VC78].The actual BEI\I problem is then solved by cubic interpolation from these tabulated precalculated values. The main drmvbacks are the high storage necessary for the tabulated values and t.hr dependence of t.hr accuracy on Lhc degree of anisot.ropy·. The fundamental solution must, be sufficicnlly smooch whieh is only t.hr ca."ir in staLk problems. The interpolation remains doubtful in the vieinity of the singularity.

Alternatively, t.he values of the fundamental soluLion arc constructed from Lhc isotropic ca.:;e modified by a perturbation series, cf. [GGK96]. This method is only adequate for weak anisotropy. Another approach splits the integrand into the isotropic and the anisotropic part. The Latter is handled Like a volume

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68 CH/tPTE.'H 6. t.'LilSTlCLTY

term, for example via the dual reciprocity method [P B\V92]. The choice of the appropriate number and location of internal points remains diffieulL.

The method proposed by [l\~is88, SG98] is based upon the calculus of the residues of the inverse symbol. Nishimura's approach is also based on the Fourier fundamental solution but. not on Parseva.l's theorem. The main prob­lem seems to be the numerical location of the roots and the following numer­ical integration.

6.2.2 General anisotropic elasticity

The Galerkin BIE for general anisotropy The stress tensor for general anisotropic media is, cf. [KogOO],

(6.22)

The 81 components of the elasticity tensor Ckcrnn can be deduced in the general case from 21 different. material parameters. The equilibrium equation

(6.23)

leads with f k = Dkp6(:r) to the part.ia.l different.ia.l equation for t.he determi­nation of t.he fundamental solution U

:F H (6.24)

As will be shown in the examples, the evaluation of the fundamental solution in Fourier space is straightfonvard. The fundamental tract.ions a.t a. boundary with the normal v; a.re given by

(6.25)

which leads to the definition of the boundary trac:Lion operator

(6.26)

Finally, the formal notation for the Ga.lerkin DIE is in analogy to (6.18)

with A, J3 as defined in (6.19).

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6.2. /l .. i\ilSOTlWPlC C.iSt; 69

6.2.3 Orthotropic elasticity in IR.2

The fundamental displacement and the fundamental traction For orthotropie materials in the plane st.rain case, the relevant components

of the elasticity tensor Cklmn are Cuu, Cu22, C2222- C1212- The t.wo equations in (6.22) lead Lo, cf. [GP97J,

()II Culli}lttl +C1miJ2u.2:

~c1 2 12(D2 u.1 + a1 u-2);

Inserted in (6.2:1), t.he differential operator is obtained as

P(U) '1111 11 + 2 1212 22 1122 + 2 ' 1212 12 l C, u· 1C' D (C' 1 C' )iJ-

(Cun + 4C1212)D12 C2222D22 + 1C'tmDu

(6.28)

(6.29)

P(x) (C1 m + tC1212)i:ti:2 ] Cnn:i'~ + ~C1mxT .

The fundamental solution in Fourier space is D = P 1(i)

wit.h t.he detenninanl.

For this simple anisotropy, it is possible to derive a fundamental solution even in the original space1

. It. is here taken from [GP97, CS83]

[

I /2 _ 2 . . _1/'2 . '2 • , , l U = ~ o.1 A 2 ln r 1 _-- n 2 .4.1 Ln r2 _ , __ A-1_.4.2 (82 - &;) _ _ _

i-i - ' .-1; 24_'2 ·' _-1; 24_'2 ' !- s: m. o 2 " 2 Ln 1 2 - n 1 " 1 Ln r 1

(6.31)

For the computation of these expressions in the original space the flexibility ma.trix s is needed

(6.TZ)

1 A.E. Green: A not.e on stress syst-ems iu aelotropic materials. Phd. AflW 34, 416 418 (1943).

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70 CH/tPTE.'H 6. t.'LilSTlCLTY

The constants o:~., k = 1. 2 are determined by

2s12 + s.1.3 Ut + 0:2 = ----

822

Additionally we need (k = 1, 2)

and sn

Ut0'2 = -. 822

'l'z + 1 ,r2. ··t - •• 2, (lk

arctan [ ;_ . ] . v nk:r1

ln Fourier space the fundamental traction i is obtained by applying the traction operator

A' ; _ ... [ Cm·t·:f:tl/t + ~C1212:r21/~ Cu22:r21/f + iC1212it1/~ ] L- z c , , 1c· A , c-· A , 1c·' - ., ··nzz:rtl/:1 + 2 ·1212:r 21/1. ·2222:r 21/:j + 2 / tnzJ:tfli (6.33)

to t.he fundamental solution {!, by simply multiplying the t.wo matrices .A; and ( r . .'Jote that Al, is not synnnetric. The lengthy expressions for the orig­inal space are omitted here, they can be found for example in [GP97].

The Galerkin BIE for orthotropic elasticity in JR2 is finally obtained by in­setting the cleri vecl Fourier fundamental solLLt.ion {! and the corresponding traction operator A~ in (6.16) and (6.17).

The clamped spruce board As cited in [CS8:3] the behaviour of a spruce board wiLh its grain parallel

to the :r :~ axis may be modelled by the model proposed in this section. The inversion of the flexi bilit.y matrix

~ ] x w-1

11.500 [kPa.- 1

].

lea.ds to the elast.icity coefficient.s, cf. (6.32) ,

[

Cnn C:m2 0 l [ 0.065:-3 O.!B67 C1

0m Czm 0 = 0.0367 1.7242

o ~c1212 o o lnserted in (6.:30) the Fourier fundamental solution is obtained easily. As sample Dirichlet problem the square [0. 1] X [0, 1] is taken charged with a single Dirac-load in the direction of :1:2 at. the center :1: 1 = :r 2 = 1/ 2. The resulting displacements in the interior are given in Fig.6.2 aml reflect dearly the ani~otropic stiffne~s. The singularity ofv.2 at. the load point :r = (1/2, 1/ 2) is correctly represented (the value i~ missing in the graphic representation).

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6.2. ;\:'~TSOTROPfC C>\SF: 71

tlisplacerrwnt ?J 1

Figure (L2: u1 , n 2 in the interior for the Diri.ehlet problem of t.mnsvcr~e isor.ropie elasticity

6.2.4 Transverse isotropic elasticity m R:s

The fundamental solution T n three dirnensions, a.n analytic.: funda.menta.l solution in the original space

can ouly be found {()r transw:>rse isotropie elast.ieity, d. :.\Jis88]. lf tlte plaue of isotropy lies in the (x1 , ;r;!)-plane the relevant values of the elasticity tensor Cklnm an~ [GZ601

Cnll (\ 1:.!2 c111:1 0 0 0 Cuu Cu;;:; 0 0 0

C:n:n 0 0 0 C1:11:1 0 0

(6.:H)

C:m3 0 lC 2(•1111 - Cu:.~2)

wtu~rc the t.nmsv<!rsc ism.wpy irnpli<!;;

\Ve usn the COlll!llOll ahbrevia.l.ious (\ 1 C'1111: C:1:1 = C:l:\:1:1, C1 :1 =

Cn.33: C,H - Cm3, C;;u - Cm2 and derive the transformed differential ope­ra.t.or from (6.24)

i-'(i.) = C:t:/:1:1;:!

Ct>f/:.! + (\4:1:~ + C1 x~ (}'lJ:;I:/;:l

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72 CHl1V.l'Lll 6. .EL'lSTlCIT'r.

with C1 = C11 - C61;, C~ = Cn + C44 and f = J:i:f + :1:~. Hence the Fourier fundamental solutions is

~ishimura. [~is88] has derived an analytical expression in t.he original space, compare also Kroner [Kro5:3],

\Ve need r.- 1 = C 1.!/C66 and K·2 ,3 as the two distinct roots of

and rk = J K~; (:ri + J:~) + :r~, rk = rk + l:r:ll · Finally the following constants have to be decermincd

The values for [{3• L:3 , M3 can be obtained by interchanging r"2 and tc:3•

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Chapter 7 ~.·.··· ~-- .. ,. ; ·. -~! ' ,_ \ . '!f.·,, ·.

'-."· '

l Plates

Oustav Hobert Kirchhoff

7.1 The Thin Plate

In thi~ ehaptcr Fourier HE:VI b applied to plate problems. In the firr:;t part, the model for t.he bending of plat.es l::; analysed which is due w Kirchhof[1

and whieh is valid f()r thin plates where shear deformations can be negleeted. For t.his t.!wory, r:olloea.l.iou a.pproadtes a.re well est.a.hlishod for t.he original space, e.g. lAP92, I3es9lj. The Galerkin I3IE were only recently developed in );H98:. In the ;w<:ond part, W<~ cmt;;ider pla.t.es on \Yin k l<~r fonmhtt.iom; for which a collocation approach can be found in }ah98]. Finally, lhe refined plat<~ thwry of R<~issncr is discnss<~d, d. [i\ P92, J-k~91].

i\ II these approaeher:; were de vel oped for isot.ropic plates in the original Hpace. Henf'e, t.he Galerkiu .!:HE {()r the auisotropie case is newly derived here in t.he original aH well as in the transf(wmecl r:;paee.

7.1.1 Isotropic case

The differential equation and its fundamental solution Tlw rnidsurfa<x~ of t.ho plate is situa-t-ed in the (:r 1, :r-2) plmw. u:(:r); :r =

:r1 , :t2 denotes lhe ouL-of-plane bending displacemenl. We deline lhe unit outward nornutl ~~- (tt1, llz)T and the unit tangent T- (Tl, T2)r- ( -l'z, l'lf· The momeut mkl aud the shr.ar components (}f. a.ro const,rueted from the sl.ross

1 Kin:hhoff. C.: lber das Clei<:hgewic.:ht und die lkwegung ciner dat;ti~dwu Scheibe . ./. Htim· und Angewandk A1athtmatik 40 (1850), 51 88.

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74 CHilPTl'.'H 7. PL/UES

tensor CTkt by

j·h/ '2

'Ink/= okl(:r, :r:J):r:J d:r3 , 1>/ 2

They can be related to w by

'{!Ik( -J(klmnOrnn'W

IJk a1mkt = -Dakllw

:F H /ilk/ :F IIA H

f{klmn:i:mXnii';

ix(rhkt = iD:i:ki'1:i:(tiJ; t.'h;J

1J - -:---::-12(1- D2 )

(7.1)

(7.2)

where sununation has to be applied for repeated indices k, l, m, n = 1, 2. E, D, D are the '(oung's modulus ibrsticit y, the P oisson'scoefficient, and the flexural rigidity of the plate, respectively . rlkt is Kronecker's symbol, and h is the thickness of the plate.

Figure 7.1: Plate jor'Cf!.S and moments

The differential equation for t.he bending of t.hr isotropic Kirchhoff plate is

D D.D.w = f D(:i:i + :i:~fti! = f. (7.3)

with f as transversal load per unit area. Thus the ditl'erential opera.tor is

P(a) = DD.D. :F H

The in verseof P(x) is the transformed fundamental solution

lxl2 IV(x) =- (ln l:z:l -ln lrol)

8JrD 4

(7.4)

(7.fi)

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7.1. THE THIN PL/l'.l'J:: 75

with l:rl = )xi+ :L1I:i:l = J:i:f + ::i:~ . For the original space, the fundamental solution is taken from the literature, e.g. [FB98]. The constant. r0 is arbitrary and often chosen such that ln r~ = 1 or just as To = 12 .

The boundary differential operators The boundary quantit.ies >vhich are relevant in the context of BE).-1 are the

deflection w, the normal slope c;v, the normal bending moment m,n and the equivalent Kirchhoff shear q,/ = 1/kqk + dmT / ds (we assume that the normal vector // is piecewise constant)

'W :F H w;

'f,/ IJk(hw 4 (/;,/ i iJ1 .. ::i:ku'; (7.6) rn,/ 1/ki/(TnH 4 rh,/ 1/A: IJrlhu;

q,/ 1/kqk + TkfAn![ml/fTm 4 (],/ 1/A.ib: + iTkikrhlm.f/lTm,

where the twisting moment. for the Kirchhoff shear is defined as mT = ntkWkTt, awl the differentiation with respect to the arc Lengths of the boun­dar:y is d/ ds = TkOk· At the i-th comer, there is a particular comer force J: which is related to a jmnp of the t\visting 1nornent rn.T around the corner

.,.,.i · ( ·r - ·r') r1··- ('l' - .,.i ) T ''- ··c - "T ·.-"-c. · (7.7)

This corner force is for arbitrary angles;;

(7.8)

2 The mathematical meaning of thi~ arbitrarine~" i~ that one can add arbitrary homo­geneous solutions to the fundamental solution. FDr a.ll To we have

The fundamental solution is the convolutional inverse of r.he differeutial operator. This is independeut of the choice of r0 . The following is valid for all r 0

P(8)W = J

ln the transformed space, this arbitrariness lies in the freedom to choose a particular regularization procedure for the singularity at [i[ = 0, cf. [Dud97b].

3 These particular comer tenus can also be derived by a rigorous distributional reasoning <b di~cu~sed iu Chapter 3.2.

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76 CHilPTl'.'H 7. PL/UES

with the differential operator (see Fig.7.2 for the definition of JF' , JF' )

A~.

:F A' f-t c

+

+

(1- D)D (z;i'vi'- viillz;)(D11 - D22)

2D(l/t'/)i' - '4',4'- /)1'1/1i + 1/zil/ii)al'l·

-(1- o)D(l/t'~4i- l/1'1/zi)(xi- :£~ ) 2D(I/tiuti - ~4'1/t' - 1/~i/)~i + 1/ziuii):£,1:2.

Figure 7.2: Dcfinit.ion of !.he normal vectors 11- i , 11-; for !.he corner term

Thus, the three boundar.)r ditl'erential operators A:b,m,q for a boundary element with the normal 1) and the corner diH'erentiaJ operator A;. are defined as

for A' IJ~Ok :F A' il/k:l·k; ·'_p,_, = f-t = 'P 'P

for Ai - l(klnmviu{Dmn :F Ai J(klmnl1£vfim.i·n: m.,) = +-+ = m m

for A' = - D l/iaw- :F A' = iDuf}:k:i't i l+ (7.9) q.., q +-+ q

K . , ' ~'a + iK Hrnn T1:1J/, T/im:i;n;[;l'; - klmn Tpl/k 11 rnnp for .fe A' (7.8) :F A' (7.8). c see +-+ c see

T he boundary quantity and its transform are obtained vm A~w or A~tii (k = o.p,rn,q,c) .

For any well-posed problem, half of t.hc boundary data must be giv cn.For each pair of dual v ariablet> (dual in the sent>e that the pairt> lead to work t erms) either one of thet>e two variables or the relation b etween both (a Robin type boundary condition) must be prescribed. There are four different

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7.1. THE THIN PL/l'.l'J::

t.ypes of boundary condition :

w(x) = wr(x) for X E fw; m 1,(.r) = Tnvr(:r) for x E fm;

77

In addition, we have to prescribe for each corner poinL x~ either Lhe jump of Lhe twisLing moment. f~(:r~) or the displacement w(x~) at this poinL

For establishing the I3IE, the following derivatives of the fundamental solu­t.ion lF are required: The fundament.al slope <f>v (with lnr6 = 1)

u' :r I A ·i T·F k' ·k l I I <> = t ' = -- n x

v ., 411D

the fundamental normal moment, cf. [Bes91] for the original space,

Af.., = A:)Y = _ __1:__ [2(1 +D) ln lxl + (:) + D)(11; · Vl:rl)2

8r.

and the fundamental Kirchhoff shear

The fundamental c:orner fon:e is

.r +--+

+(1 + 31/)(Ti . Vj:rl) 2]

f:,.. = A i l·l' c ,

(7.12)

(7.13)

(7.14)

The evaluation of the derivatives of the fundamental solution in Fourier space is very easy compared to that. in the original space.

The symmetric Galerkin BIE The Somigliana identity in analogy to (2. 7) as a weak form equivalent to

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78 CHilPTl'.'H 7. PL/UES

(7.3) is, cf. [FB98, .Jah98],

h:(x)w(x) ~· .f(y)H'(:r- y) drl (7.15) .!

+ /' q,/(y)Tl' (:r- y) dfy- /' m,/(y)ci>,(:r- y) dfy . r . r

+ { :rAy)M,_,(x- y) drv- { w(y)CJ,;(:r- y) dfv lr lr + L JJuDTl' (:r- y~)- L ·w(u~)Fc(:~:- '!ID·

According to [Bes91], the free term is'"= 6q/(27i), i.e. the percentage of the total angle 211. The Galerkin version is obtained by additional weighting with test function rp~ (q is t.he dual variable of w). It is in distributional notation

( q)~l' f * H') + ( ~!' IJv *' IV ) - ( 0?~1 , mv * 1>,/) + (q)/I' :p" * M")- (qrf1, ·w * Q") (7.16)

+ L (~1, fc(Y~)!V(:J;- y~J)- L (\frb, w(y~)J~(:1;- /J~J).

Once more, the boundary facwr r" is obtained implicit.ely a.s shown in the paradigmatic example for the heat equation by Wx = x(:r)u:(:z:) as defined in (3.4). \Ve introduce now the discretizations for all boundary quantities. Because of the high order of t.he differential operator (order four) we have to respec.t certain continuit;y requirements for the trial functions. The approxi­mation of the deflection w must be C1 at the nodes (the tangential derivative nmst. be c.:ontinuous) and the trial functions for the slope should be C 0

, i.e. continuous. Therefore, the boundary values are approximated by Hermite polynomials [FB98], (.)' = d(.) / ds = T~Dk(.) is t.he tangential derivative,

(7.17)

The functions (p:,., c~iP arc construc:lcd from t.hc trial funct.ions for t.hc reference clement. (L~ is the Length of t.hc i-Lh clement.)

, ;,0 - { \U\\r -

rY~ = { . .,.

xi(3- 2:r l) (:r1 - 1 )~ (1 + 2:r 1)

0

:z:I(x1 - 1)L~- l

x1 (:r,- 1) 2 L~ 0

for t.he (i- 1)-th element for t.he i-th element. otherwise

for Lhe ('i - 1 )- Lh clement for t.he i-th element. otherwise

(7.18)

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7.1. THE THIN PL/l'.l'J::

These functions have the interpolation properties

G0° (x· - .. ,./) - X / w • - ,, - <Jk( d _ ,!,o (x = J:' / ) = 0 I 'r'w

($

d •0 ( i --_1-Q,_~ :r =X ) = Okl· ( 8 y

79

(7.19)

The ot.her quant.ities of the boundary. i.e. m,/ and qf/ , arc approximat.cd by piecewise linear polynomials

for the (i- 1)-th element for the i-th element otherwise

These trial functions are in distributional notation:

:d(::>- 2:rt)H(:rl)H(1- xt)6(:r2) -12 + 6ii1 + ii}~-ii:r + 6ii1c-iir + 12c-ii: r

Yf (x1 - 1)2 (1 + 2:rt)H(:r1)H(l- x1)6(:r2) 12 - i.i:i - Gii·1 - Gi.l\ e-ih - 12.!:1 e-i:h

:J:t

For the slope trial functions we get

:ri(xl- 1)l1 1H(rt)H(1- :1·t)6(:r2 )

. 6 + x' 2e ;,f! - 4' ;.r'· e i.h- Ge i;t,- 2; ·>· L''- 1 ·1 · ' · 1 -· · , _,_ 1

e xi :r1(x1- l?t;H(.Tt)H(l- :r1)6(x2)

6- xi - 4i:i:t - 2iilc-ii, - 6c- i;i:J lj ' 1

XI

The Lransformations of the linear Lrial functions arc

·0 ~Dm~q x 1H(x t)H(l- xt)J(:r2 )

F 70 -1 + li:l e-u~ + e-·•iJ

H ~~Jm,q :Z:i

•0 9m,q (1 - xt)H(x t)H(l - :~: 1 )6(:r2 )

:F ~0 1- eii' -),;J;I

H (/!m,q ' 2 XI

(7.20)

(7.21)

(7.22)

(7.23)

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80 CHilPTl'.'H 7. PL/UES

\Vith the exception of the deflection itself, all boundary quantities are depen­dent on t.hr normal vector, t.hry arr discontinuous at. corner point.s. Honer we drfinr t.wo different. nodal values at. thrsr eornrrs.

The treatment of the corner force .fc and the corner deflection We in Fourier space is enabled by defining particular comer ! rial functions <D~. = rl (:1; - :r~)

fc = f D(:z:- :r~)

We = ttl' rl(:z:- :r~)

The contributions of t.he corner terms to (7.Hi) can be written as

These discretizations result finally in the discretized Galerkin BIE

~'\~q ,'\-'m

(7.24)

(<7),. fx * H') + L qi (11~, (p~ * W)- Lm' (~, q):n "'<!>,/)

(7.26)

J.V,. J.V1:

+ Lt'(dl~,o~dt · >- Lttli((b-~,qj~ "'t~> -

with ( (f>{t' 1.1.'\) = ( (p~, X~J; p{1 is the polynomial defined for the test func­tion (/I~ . The kernels of this BIE can be highly singular, cf. [FB98] for the regularization. A more formal presentation of t.his Galerkin BIE is

l\iq /\ 'm

(~,fx*lV)+ Lq;(riJ~ ,O~ *lV) - Lm; (o~,o~ *A~W)

/V:p J.Vw

+ L p' (?:_!, (<0 * A;!lH')- L ttl ' (1~1' rp:, "'A:llV) (7.27)

i.Vc iV~

+ L t' ( <D-/1, qj~"' l'V) - L m' ( (jj;1, 0~~ * A~ll'),

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7.1. THE THIN PL/l'.l'J:: 81

which finds its Fourier equivalent in (after cancelling (271)-2)

(7.28)

For a symmetric Galerkin method, additional BIEs are required which will be given in t.hc following for Fourier space. If needed t.hey can be transferred easily t.o t.hr original space. Thr I3IE for t.hr normal slope is obtained by applying the adjoint. of AI,, = -A~ on (7.28) and by choosing t.he dual t.cst function. IIere we need a moment test. function O?n· The I3IE is then

(7.29)

!Vq I'v·m

-" q" (cY (-:r) cbi .~E1i: ) + "mi (Jr! ( -:i:) o' AJ'""~i, li:) L..J .·Jn l.·q 1'? L.....J .rn '.rn '~" .,j i i

;\''1' .'\'\' ..

- L Pi ( q~n( - :r), (;>4J,A:"lt·'J + L 111 ' ( ¢>{n( -:i-), q;~,.A~A;1 l·t') i ·i

iVf.. ~'\:-c

- L f' ( ¢~" (-i), J;A~l-t:)- L tu ' ( J~,J -:i:) , «~:.A~A~.l't). ·i i

The other I3IE are obtained analoguously. I3y using the operator notation introduced in (7.9), the following scheme can be established for the total system

with

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82 CHilPTl'.'H 7. PL/UES

and

-AAJ c

B = (I -A~

I is the identity operator sm:h that IT"f' = TV The transformed terms in the operator matrix A of (7.:30) are obtained by simple multiplication. The scalar product on the left-hand side of (7.29) ca.n be used for regularization purposes in the original as well as in the transformed space. The differentiation Bwx. = B{w(.r) x(x)} should be done carefully taking into account. that the product of the defiect.ion wit.h t.he eutoff-dist.ribution of the domain has to be evaluated. A de! ailed discussion of this regulariz:a! ion is not. within the scope intended herc. lL will be published separately.

The free terms on the right-hand side As an example, \Ve investigate

\ &!,, (.xz;UAw + wz;~fAx)) \(;r!,, xcp,/) + (#,, w5(1p)z;1,Dk1/J J ,

(7.31)

with x = H(1;~·) a:,; cutoff-distribution of t.he domain n. The Local support of t.he test. function Q~n is part of the boundary, i.e. part of that which is described by 5(~;), Therefore. we have for the last scalar product locally something of the character

with some cx•-function p(:r) and constants C 1,2 . This apparently divergent int.egral is of character 1i(O). The same is t.rue for all diffcrcnt.ial operat.ors A~,,m,q,c · According to t.he author's experience, all t.hese singularities cancel with the singularitiets of the right-hand side. The proofs for this are beyond the scope of this thesis and will be publitshed later.

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7.1. THE THIN PL/l'.l'J:: 83

Sample integrations / isotropic case In section 5.1.2, we defined linear Lest. and t.rial funet.ions, for example

An exemplary analytical integration of one entry of the matrix leads in the single Layer case to (we have chosen To = ve)

The corresponding int.egral in Fourier space is computed as

.Jsgn.i2e- i:"

4:i:8

The kernel in Fourier spaee is hypersingula.r and is regularized as shown in t.he example for t.he isotropic heat equation.

Example : Fourier BEM for the clamped square plate For a damped t;quare plate with n = [0, 1] X [0 , 1] and with two elernent.s

at each side, the t;ystenl (7.:JO) can be reduced because of w = 0, 'Pv = 0 on

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84 CHilPTl'.'H 7. PL/UES

the total boundary and due to We = 0 for all corner point.s. The right-hand side of t.hc Fourier DIE is

(7.:3:3)

For Lhe volume forces f we get

(7.:34)

The free term on the Left-hand side is zero because of w = ·Pu = 0 along the boundary. Linear trial and test functions are chosen for the normal moment and the Kirchhoff shear forces at the boundary. For the corner forces we have the four trial functions

(p~ = rl(J'I )0(:1:2)

(p~ = ri'(::r1- 1)r5(:rL)

(p~ = J(::r1 - 1)J(:rL- 1)

(p~ = J(::ri)J(:z;2- 1)

:F H

:F H

4 4

(;2 = 1: .. c

~Z~ == c-iX~_; (;3 = e i (.h ··h). ,' c 1

(;l = e ii: , .. c.

Therefore. we get the following system of :)6 equations (Nq 16, Nc = 4):

(7.:35)

j = 1 ... Nq: 0 ( <{ ( T),/, TV) + (~( ( -±), ~ ~~lt') (7.:16]

. 1 "' 0 J = ... iVc:

( ¢~J -x), ~ ?:n.A~)i') + ( c;~( - :/·), ~ ?;:l~t')

· · (.;;,I -i ),f,A~ li-") - ( ~k'), ~ ¢~A;, 1-V)

+ ( biu(-i·), t 6~.A~.A~rt)- (bin( -x) , ~ 6;.A~ll')

- (\'~(-:r),j"1V)- (i~(-x),~0~1T·t') I Nm ) I Nc ) + \ ~;~ ( -;[;): l,= ~;;ll.A~)l' - \ ~ ( -:i: ), l,= 0;~ 1-l' .

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0.00241

-0.06036

-0.06036

-0.06036

-0.06036

0.00241 0.00241

0.002410.00087

0.52196

0.00087

0.000870.00087

0.52196

0.52196

0.52196

Kirchhoff shear forces normal moments

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x1xx x

xxxx 1 1

12

2

2 2

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7.1. THE THIN PL/l'.l'J:: 87

The symmetric Galerkin BIE for orthotropic plates To establish the Fourier DIE, the following boundary differential operators

are derived from (7.39) and (7.40) together with (7.6)

Ai 'P (7.!13)

A' Tf\ u~~~; (:rf + 1712i~) D11 + 4u\J;~i\:r2Dt2 + ~~~ ~~~(.i:~ + D12if)Dn; Ai

lj

+ +

A ' c

+

. i (·' i i A ; i A ) ( ' 2 - ' 2)D WI .r1 + IJ2l/2.T1- l/1/J2.T2 .l'1 + I/12.T2 11

4i(u~v~~~~.i\ + 11ivi1{i·z)i'tX2Dt2 it1~(i2 + l/i//i:z:2- t/1/.{i:t)(i~ + 1712:i:i)D22;

-(T;i'v;;-i- l/1;1/2') Ji:f + l/t2:i:DDu- (:i:~ + vu:i:f)Dn]

2(1/t;I/ti- 1/;!iuii- 1/1;1/1; + I/1'I/l.'i)i:li2DI2·

Introducing (7.12) and (7Jt:)) in (7.::m) Leads to the Ga.lerkin BIE for or­t.hot.ropic thin plates.

7 .1.3 General anisotropic case

The differential equation and its fundamental solution The strain-stress relations for general anisotropic plates a.re with the fle­

xibilities akl :

a·12crn + a22cr22 + azocr12; al6/2cru + a26/2cr22 + fLGG/2crl2.

They arc linked Lo the corresponding rigidities 1Jk1 by

C1 (a22U.66 - U.~G); Ct (au a6r;- u.io);

(7.44)

r:1 (atr;U.:w- a12115r;); c1 (a12a2o - a22a1G);

lJn 1}66

D2a r:t(auu.n -ai2); (7./li)) c1 (a12a1r, - aua2G);

with

1 12 I (J II (1.12 aw l f3 det l a12 fL22 a2a

J Ct I aw fL2G aaa

The bending and twisting moments are

m11 rnn

rn22

-(1Ju8u + lJ12822 + 21Jto0t2)w; - (DtG011 + D2r,022 + '2Dr,Gau)w; -(Dt2011 + Dnon + '2D2Gau)w.

(7.46)

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88 CHilPTl'.'H 7. PL/UES

and the shear forces are

qt = -[DnDm + 3Dt&Dm + (D12 + 2D&G)Dm + D26D222]; qz = -[Dt&Dm + (Dtz + 2Doo)Dm + 3Dz6Dm + D22D222].

(7.47)

The differential operator for general anisotropic thin plates is, cf. [LekGS],

P(D) [DnDuu + '1DtGOmz + 2(Dt2 + 2Dr,r,)Dun +4DzoD1222 + DzzD2ml

[Dni:7 + 4Dt6:{/i:2 + 2(D12 + 2D66)i:T.i~ +4D A ,'.:l + D ' 4] 26XtX2 22X2 ·

\Vhich leads to t.he Fourier fundaruental solut.ion

The boundary operators The relevant boundary operators are in Fourier space

A' 'P

A;n + +

-4;1 +

+ +

A' r.

+

(7.48)

(7.50)

Once more, the Galerkin I3IE can be obtained by inserting these opera­tors and the anisot.ropic fundament.al solution into t.he Ga.lerkin BIE of the isotropic case.

7.1.4 The thin plate on a Winkler foundation

The fundamental solution for the isotropic case The behavior of an elastically on continuous springs supported plate

(Fig. 7.5) is described by

(D!:::.!:::.+k)w=f (7.51)

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7.1. THE THIN PL/l'.l'J:: 89

It is in v ertedb y the fundamental solution, cf. [.J ah98] for the original space,

H' ---ke1 -a2 • ( l:rl)

211 D a a

2 ~·= tlJ (tl:rl) = ---4

. 4

.Jo - dt; 211 D . 0 Dt + ka a.

F B

1

1J(ii + :!:~)2 + k . (..., F.2) I • • )

with a = ( D / k) 111. kei i~ the Kelvin function of order zero and J 0 is the Bessel

function of order zero. Due to the fact that the differential operator ha~ the

Figure 7.i): P late on a Flinkler fo·undat ion

same cleri vat.ions as in t.he case of t.he unbeddcd plate ( k = 0), the boundary different-ial operat.ors can be taken from (7.9). The BlE can therefore be adopted frorn (7.30).

The fundamental solutions for bedded anisotropic plates The fundamental ~olutions for the orthotropic and for the general

ani~otropic case can be taken from the unbedded plate by adding in the denominaOJr the ~tiffness k. \Ve get for orthotropic plates

A 1 ll' = ' ,·1 '.2 ' 2 AA • ' D 11 .t 1 + 2D:JJ 1 x~ + D 221 2 + k

and for general anisotropic plates

lT7 = [Dt d~ + 4DtG:i'{iz + 2(Dt2 + 2DGG) :rf:i"~ + 11D2Gi l i :1 + D22i~ + k] 1

(7.5:3)

(7.54)

7.1.5 Combined bending and stressing of thin plates

The isotropic thin plate The differential equation for thin plate:; subj ected to compressive in-plane

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90 CHilPTl'.'H 7. PL/UES

forces iVA-1 is

4

The Fourier fundamental t;olution for arbitrary i.Vkt it;

(7.56)

For Lhe isotropic ease iVkt = JV, [Bcs91] gives as fundamental soluLion in the original space :

for tensile N : TV 1

2;r(2 [ln[::r[ + Ko(([.rl)]; (7.S7)

for compressive N : vV 1

- · ~ -2 [2ln[:r[ +i;rHG(([:r[)]. (7.58) ·L1c,

The anisotropic thin plate The Fourier fundamental solutions for orthotropic and general anisotropic

platet; under membrane forc:es J\lk1 and on elastic: foundations with stiffness k are

orthotropie :

anisotropic :

1 D11 :J·i + 2D:l:i:i:f:~ + Dn:i:l + k - Nkt:i:k:i·t'

[D11:i·t + 4Dir,:i·{i·z + 2(DI2 + 2Dr,r,) :ri:r·~

+lDzGid::~ + D22ij + k- 2\T;,ti:kiL] 1.

(7.S9)

(7.60)

To the author's knowledge, expressions for the original space are not avail­able.

7.2 Refined Plate Theories

7.2.1 The thick plate

To refine Kirchhoffs plate theory R.cissner proposed a model where the influ­ence of transverse shearing strains and the effects of the stress o-:;:J perpendi­cular to the plate are not neglec:ted'1. The presentation in the original space is essentially based on [AP92]. In the following, we distinguish between Greek indices CY, .S = 1, 2 and Latin indices k, l, rn = 1, 2, 3.

4 Reb~rter, E-.: On bendinp; of elastk plat.e~. Qum"t. Appl. i'vfath. 5, (1947).

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7.2. HHFlNE.'D PLA:n; THEOlUHS 91

The differential equation and its fundamental solution As in t.he Kirehhoff model, we examine a plat.c with eonst.anl. thickness

h which is lying in the (:r 1, :r-2)-plane. :r:J = 0 is t.hc mean surface which is subjected t.o a transverse load p(J:),J· = :r 1,J:2 and t.o volume forces fk· As generalized displa<.:en1ent:; u., we ta.ke the rotationt:~ u1 = :p1 , ·u2 = cp2 and the defie<.:tion u.3 = w in the direction of :r3 . The bending curvatures "-a.8 and the transverse rotations of the normals of the plate ~;01 are

and (7.61)

These values are relat.ed via the const-itutive equat-ion to the generalized internal forces

iJ - -2 +--_(>,a.\ p; 1- {I

(7.62)

D = (Eh:3)/(12(1- D2)) is the same rigidity as in the Kirchhoff model, cf. (7.2). The characteristic quantity .\2 = 10/h2 related t-o Reissner's model follows frun1 the assumption of parabolic dit:~tribution of the stress cr33 along X.J, e.g. [AP92],

:r-3 ( 4:r~) 0';;:; = p(:r:1, :!:~) :-'- :3 - -; ·

2h h-(7.63)

It should be noted that the generalized moment iho/1 has a term related Lo t.he transversal load p. \Ve define ij,. = q" and

thus:

The equilibrium equations are

1.1 _., fllu/J + --_6,/J A ~p,

' 1 - v '

D( 1 - D) "-a8 + --_ 6aaK·U · . [ 1.1 ]

1- ll

. D - 2, -ba = -fa---_ A Oap:

1 - J.I

- b:J = -h- p.

\Ve get finally the part-ial differemia.l equat-ion

(7.64)

(7.65)

(7.66)

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92 CHilPTl'.'H 7. PL/UES

and the differential operator

D(1 - P) [ (D. -.A?) + c1Dn C1U12 ->..2()1

l P(a) 2

c1D21 (6.- >..~) + c1D22 -)..'2()2 (7.67) )..'2()1 )..2()2 )..26_

[ (1 "12 2) .-2 Ct:hJ:'2 i)..2:f·l

l DO!- 1) :r +>.. +ct.D1 P(:r)

2 Ct:E'2Xt (l:i:l2 + >..2) + Ct:E~ i)..2:J·'2

-iA2i'1 -i>..2i·z >..21:rl'2

with Ct = (1 + P) /(1- D) and 6. =Ott +On, I:W = i:f + 1-~. The right-hand side is with c:-2 = D / (1 - P)

(7.68)

The inverse leads Lo Lhc Fourier ftmdamcnLal solution :

The version in the original space it; given for example in [Ant88] and repro­duced here

1 [(sD (z) 2 ) _ (SA(z) , ) I I I I] -D --_ - ln Z + 1 OaJ - --_ + 2 00 :r 03 X ; 8;r 1 - ll 1 - 1/

-l.J:~" = 1D lxlonlxl (ln z:~ - 1); (7.70)

8;r

z --- lnz-2z . 1 [( 2 8 ) 2 2) 16r,1J.\2 1 - j) '

with l:rl = J:rf + :d, .z = >..l:rl and

2 ( 1) A(z) = Ko(z ) + :;; K1 (z) - :;; and B(z) = Ko(z) + ~ ( Kt(z)- ~) .

K0 , K1 arc the modified Bessel functions.

The Galerkin boundary integral equations The U defined in (7.69) are the fundamental quantities in the field. The

corresponding boundary terms (normal slope, tangential :;lope, and displace­Inent) f~' = (&,_, &.,-, Tl' ) at the i-th part of the boundary with the normal

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7.2. HHFlNE.'D PLA:n; THEOlUHS 93

//' = ( tJi, //2) are obtained by

(7. 71)

wit.h: //~ () l v~ 0 . 0 1

The transform of the boundary differential operator A~ to obtain the miss­ing fundamental quantities \~j = (Al,~,_,, i1i 7 , t};J (the bending moment, the twisting moment, and the real shearing force) is. cf. [AHFEZ99],

A'i 1:.•i l.ll- l'

ilht/ik 1 1/ . - i ~ - 2-. UJ/JkXA·

0

with Ti = (rf, r~) as the i-t.h tangential veet.or.

(7.72)

Because the ditl'erential operator is of order two and not of order four as in the Kirchhoff' model, linear shape functions ¢~>,T.w are sufficient for the boundary "displacements'' :.Pv, zp7 , w, whereas constant functions ¢~1,t.q can be taken for t.he deri va.ti ves nlw, m 117 , q,1 . The discretization;; are :

.!.V,/

'·' ~ """"nio·i (···) '1'1-~ ,.....,_, .f.......t : f/ .t, ~

I\~m ."f\{t

rnw::::::: ~ m'<t>:n(:r), rn,n::::::: ~ t'¢~(:r) ,

The boundary· integral equation is

with

I

-A' II

l''v'w

w::::::: ~tu' 0:Jr),

Nq

CJv ::::::: ~ q'(XJ:r).

7.2.2 Thick plates on Winkler foundations

The fundamental solution

(7.73)

The substitution p ---+ p - k1D in (7.65) leads to the equilibrium equation

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94 CHilPTl'.'H 7. PL/UES

for a t.hick plate on a \Vinkler foundation (k is t.he modulus of subgrade react.ion). The transformed differential operator is

with c~ = 2kD/(1J(l - i/)2 >.2) and C;J = 2k/(1J(l - D)). The fundamental solu! ion is t.he inverse

(7.75)

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Chapter 8

Waves

8.1 Transient Problems Lonl Rfl.ylt:·i!Jh

8.1.1 Fourier BIE for transient problems

ln l.lw nmt,ext. of t,ra.nsient. dynamic: pmhll~m:-; t.lw symbol (':) is n;.;ed to dl~now a, spatial and temporal Fourier tra,nsformed quantity, i.e. we define in analogy to ('1.1) :

(8.1}

l, ;;;A:: k = 1, 2,:1 are t.he t.irne aud the spal.ial eoordiual.es. The corresponding differentimion~ arc denoted by i.lt, U,.. The quantities ;,:, J,. are the cireuhtr frequl~ney and l.lw wa.vl~ nnrnbm-s. For t.lw dynarnir Fourier BE.\1 dw seala.r produet and Lhe eonvolution have to be redefined, cf. Eqs. (:J.2,)) and c:.Lw),

sealar product: (a, b) .l.l .. a(x, t} b(x,t} dx dt; (8.2}

eonvohttion: r ;· a(y, T} b(x- Y: t- T} dy dT. (8.:J} Jn?... J;.,r:.

The l:onvolntion r.lworem combined wir.lt Paneva.l's equality lm1.d t.o t.he fun­damemal equaliLy for Lhe dynamic Fourier OEM

(a(:~:,t),b(x, t) * c(;r, t)) = (271:11+1 (r£(-;r, -;,:),b(X,:.;) c(i,w)). (8.4)

Litl:mt.Hw for r.lw t.ra.llit.iomtl H f<:IVI l:mt lw fo11ud in [Am.88, l·ks87, Hcs97, I3onmf.

95

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96 CHl1Y.lEll 8. VV>l \'1'.'5

8.1.2 The elastic bar

The differential equation and its fundamental solution To illustrate the general principle of Fourier BEivl for transient problems

the example of an elastic bar is taken as prototype. lt is a two-dimensional problem in ~1 x JR 1 •

The differential equation and its Fourier t.ransfonn deseribing t.he dyrmrnie behavior of t.he bar are

.- 2 '2 (-E.4U1 +pot) n(xr,t)

(E4xi- pw2) u(x1 ,w)

j('J·r, t)

/(ir,w).

(8.5)

with EA., p a.s the longitudinal rigidity and the mass density, respectively. u is the longitudinal displacement and f is the volmne force in :r 1 direction. The traction q at t.he boundary with normal vi i:}

:F +--+ (8.6)

The boundary consists only oft.he two end points oft-he bar. The fundamental

solution and it.s derivative arc obtained by sct.t.iug J = 8(:rl)t5(t) 4 / = 1 :

u 1 4sgn (t) [H( -:z:r- c:Pt) - H( -:z;1 + cpt)] (8.7)

u 1 EAi:T- pw2 '

with Cp = j E.4./ p a.:-; wa.ve velocity.

Causal fundamental solution Obviously, the fundamental solution is not. causal, a wave is traveling in

direet.ion of negative time. As proposed in [Dud97b], pp 58, a simple resid­ual calculus, based on the theorem of Paley-\'Vicner, renders for t.he causal fundamental solution

u (8.8)

(;

1 Compared t o the static examples the notation for t-he traction t is clw.uged here to r1 t.o avoid confmiou \Vith t as time.

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8.1. llLL\;'SlnNT PlW1JLJ:;i\1S 97

Dynamic Galerkin BIE for the bar The boundary quantities n, IJ arc approximated bj· spatial q'J~1~u (:r 1) and

t.crnporal trial functions 'P;?,a (t)

Temporal and spatial test functions ~~ (x) , t.p~2 (t) lead to the displacement Galerkin BIE, d. [Bar99] for mal.hcmat.ical proofs,

( c~yh.~h 1. * c' + ~ qi1 i2 1 c~;,,~h 6,1 ·~'2 , u) ! q Yq ' X I I L \! q Yq , ! q '1-'q ' ' (8.10)

ii :·i ~

Once more, the free term " is hidden in t.he dist.rlbut.ional notation. All quantities are formally extended frurn their hounded :mpport.s to JR1 X JR1 a~ ~hown in Chapter :J. The equivalent DIE i~ in Fourier space

I (;.;I (-5: ) .V2 (-· ·) fi.) \.'I l 'f"q "' ' X ( (b/i(-5: I)j;fl2 (-~·),fJ') (8.11)

+ 2.:: q' I;., ((~I (-1:1) :Pf/ ( -UJ) ' ¢~1 $~2 C)

1- 1 ,'1-:l

where the factor (2;r)-:1 wa:; cancelled. Its analogue, the traction DIE is

(8.12)

+ L u' 1i 2

( ~;~~ ( - i t) .;"{12

( -UJ), <?;;,1 :P:i A{A~O) , ll ,tz

For thit; example, the formal presentation is exaggerated but facilita.teb the trant;fer to problem::; which are rnore complex.

Page 99: Fourier BEM: Generalization of Boundary Element Methods by Fourier Transform

EA, ρ

L = 2

L/4

f(x,t) 1

-1/2-1

t

f(x,t)

Page 100: Fourier BEM: Generalization of Boundary Element Methods by Fourier Transform

u(L,t)

q(0,t)

Page 101: Fourier BEM: Generalization of Boundary Element Methods by Fourier Transform

100

The causality in Fourier space is achieved as discussed in Section 8.1.2 :

for the~~ : (8.17)

:F +-t 0

for the ~:l u

F [r +-t

In Fourier space, the Galerkin BIE equivalent. to (8.14) is, compare (8.11) and (8.12),

( (;?,' ( -i ):P{12

( -w), fxD J (8.18)

+ L qiJi> ( ¢?i ( -:r)#,' ( -w ), c)~iz/J:,'(I)

and for the Lract.ion DIE we get.

(8.19)

+ LH;,;, (0;~' (-:i:)~L2 ( -w),0;~1 zP~2A{A;0) . l J ,Z·J

8.1.4 Waves in isotropic elastic media

The fundamental solution and the Galerkin BIE The dynamic behaviour of the isotropic, homogeneous ela.<;t.ic continuum

is described by Lame's equat.ion

tt Dun" + (A+ tt) Dkrur - p Dttnk

4 -~I iri.:l'ilk - (A+ p.) :i·,,i:(ii.t + p:;}·uk

k, l = 1,2,:3;

(8.20)

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8.1. llLL\;'SlnNT PlW1JLJ:;i\1S 101

which is an extension of Navier's equation (G.1) by the inert.ial term pDuuk

(pis the mass density). In analog;-,· to (6.4) the dynarnie differential operator is obtained as

[

.. , .. \ 1[) [) c;D. + c1 11 - tt

P(D) = cfD~~ ciD:u

Its transform is

cf 012

c;D. + cfD~2 - Du ciD:~~

2 A ...

c1.Yt X2

c:; [2'[2 + cfi~ - c.v·2 ci:h:i:2

and hence its Fourier fundamental solution

(8.21)

( 8.22)

For two dimensional problems we have to take the upper left part of this matrix. The traction boundary operator A:

1 is identical to that of the statie

c:ase A't, see (6.15). In for example [ES75] the three-dimensional fundamental solution was derived for the orii!,inal spac:e

And for two dimension:,; we have, e.g. [Kyt9G],

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102

The Galerkin BIE The vcet.orial Galer kin I3IE for general transient dynamic problems is ( cf.

[~IDC89] for t.he original space) in Fourier space

I 7.h( .··)·r;J·'l-( . ·) .- ) _ I __ ;jl( .-.), ;..i'2( , ·) f. r"r) \ qr;t -.I. 't"rt -w ' llx - \ '!'(! -.I. 'i"<t -JJ ' X .J (8.2o)

+ '"""' q ;1,2 I d.~ 1 (-i:) · ~i2 (- ' ·) 0./i1 ;.,i2l_r) ~ \' q • 't"q "' ' 'q 't'q '

- ( 0~' ( -.i') p~" ( -w ), At' ftx) = - ( ~~ ( -.f·):p~·j ( -cv'), fxA~' (; ) (8.27)

_ L qitiz ( ~~ ( - .i')j.,iu2 ( -w ), 3~~0~2 A{/(')

+ ~ u1·1i2 ( 3{,1

( -i')pt" (-w ), 6:t<P~2 A~1 A~1 LJ). tL,Z2

8.1.5 Initial conditions

ln (8 .26) and (8.27) the initial conditions were neglect.ed. To t.ake them into account the Galerkin BlE (8.26) has to be extended on the right-hand side by, cf. [Bon95a],

( <;/1' ( -:i:)f>// (-iN'), 11~ (!) + ( <;/1' (-:i; )f>/t" ( -cv·), h~D) . (8.28)

The additional terms for the second BIE (8.27) are

( (Y,,; ( -x);f{,~ ( -w), 1t~A{1fr) + ( ~~ ( -i )j;-~2 ( -w ), D~O). (8.29)

with ii~, i;~ as the transformed initial values for u and v = Ut,U in the domain n a.t timet = 0. F = DtU 4 V = iwO is the t.ime derivative of t.he fundamental solution U. These volume integrals might be evaluat.ed directly, i.e. by a discretization of the domain wit.h volume cells as explained in Chapt.er 10 for nonlinear problems. Alternatively, several authors have proposed special routines to transfer the volume integrals by a second Green's identity to the boundaries (double or nmltiple reciprocity I3E~I) .

8.1.6 Transient plate problems

Thin plates Tu obtain the dynamic analogue of the theory pre:;ented in Chapter 7.1,

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8.1. llLL\;'SlnNT PlW1JLJ:;i\1S 103

an inertia term and eventually a damping term has t.o be added to (7.:3) :

(8.:)0)

with p, cas the mass density and the viscous damping coefficient, respectively. To the authors knowledge, there are no transient fumlamental solutions for the original space. The Fourier fundamental solution is

ll-"(i. [<;) = ( ') ''F 1 I :t. 1J :i:i + :i;~ + icw- lpw (8.:31)

The boundary differential operators of t.l1e stat.ic problem are also valid for t.l1e dynamic case. Hence the Galerkin BIE is in analogy to (7.30)

(8.32)

+ I:ui''" (1Y'(-:i;)f>'" (-w),d/19'"ATll). "1.!,·1.'2

8.1. 7 Dynamic analysis for bended and stressed plates

The consideration of in-plane forces leads to the differential equation

(8.:3:3)

where Nkt are the known constant in-plane compressive forces. Hence. t.he fundamental solution for dynamic stability analysis is

' 1 H' = '' A' ' ' • - A A • D(J:f + x~F + u~·- hpc.v•~- i\1k rJ:kxr

(8.34)

By setting Lhe righL-hand side Lo zero, Lhe dynamic stability of Lhin plaLes can be invesLigated.

For the particular case of isoLropic in-plane forces Nkt = ±N ( compres­sive I tensile) and vanishing dam ping, [I3es91] cites a fundamental solution in the frequency domain :

TV=- :t 1

:t !_ [H6(ai:~:l)- H6(i;l:~:l)], n·+~i ·1

with ( = N/ lJ, (J1 = h[Y.-J2 I lJ, and

02 - ~ [ J ('1 + 4,81 ± (2] : {2 -~ [J(1+4.31 =t=(].

(8.35)

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104 CHl1Y.lEll 8. VV>l\'1'.'5

8.1.8 Dynamic poroelasticity

The differential equation and its fundamental solution Thr linear rlasLie model presented in Section 8.1.4 ean br extended Lo

describe the behaviour of a porous, fluid-saturated mrdium2. In addition to the Lame coefficients ,\, p, two poroelastic coefficients have to be dehned [Kyt96]:

2 2 2 (1 - 2D)(l + Du? H = -~, 1J , - . - -

9 (1- 21Ju)(l/u- 1!) 2 1 + i}u

Q = -1n·1J - H, :3'" 1 1- 217u ·

where /', ilu, B denot.e t.he porosit.y, the undrained Poisson's ratio, and the Skcmpton pore pressure coefficient\ rcspectivcl,y. Then, the stress-strain re­lations arc

(8.37)

The superscripts s, f refer to the elastic solid and fiuid parts. The fiuid pore pressure is related to thr fluid stress by

1 r p = --a. ~~

(8.38)

By assuming linear compatibility (small displacements of the solid u" and of the fiuid 1/)

the differential equations for the solid part

c\ + 11 + ~l2

) og. u( + ttDrru'k + CJork uf - F ( Dru'k - Dtu.D

- Pn Dtt uf. - P12 Dtt u~ = - fJ:,

and for the fluid part

(8.39)

(8.40)

2 ::VI.A. iliut: Theory of propagation of ela~tic wave~ in a fiu.id-~aturated porou~ ~olid, Pa.rt 1: Low-frequency range , J. Licoust. Soc. Amc1·. 28 (1956) . 168- 178; Part 11. ibid 28 (1956), 179- 191.

3 A.\V. Skempton: The pore pressure coefficieuts A and B. Geotcchrmtue 4 (1954), 14:3 147.

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8.2. Sil(110i\i"\U'r. PlWHLHAlS' 105

are obtained. F is the viscous force generated by the relative movement between the solid and the fluid and represents viscous damping. p1 1, p22 arc the effective mass densities of the solid and the fluid, rcspeetivcly, and !ht =

p21 is a dynamic coupling parameter. The transformed syslcrn of differential equations is

[7\ +7\J (:i: ,t.0·)u = -l (8.42)

with the transformed differential operators ( C\ = ). + p + (J2 / H)

-C\i1i1 -C1i1i2 -C1:h:i::; -(.J:i:l:i:l -Q:h:i::.!. -(Ji:l:i::; -C\i2i 1 -C1i:2i2 -C1:i::2 :i::; -Q:i::.!.i:l -Q:i::.!. :i::.!. -(Ji::.!.:i::;

Pt -CI::i::; i: t -C~:J-3:i:2 -C\:i::J:i::J -Q:i::Ji t -Q:i:;j;/;2 -(Ji:)':;j -Qi:t:i:l -Q;/; L;/;2 -(J:i: li:l -R:r1 :rt -Ri·t:i:2 -R:r1:r:l

-Qi:'l:i: l -Q;/;2;/;2 -(J:i:2:i:;j -Ri:2:J·t -Ri·2:i:2 -R:r2:r:l

-Q:'i:3:i: l -Qi3i2 -Qi:3i:3 -R:r3:r1 -Ri·3i:2 -Ri:.3i:.3

A.l 0 0 A:.!. 0 0 0 .. 11 0 0 .12 0

At -p,[:r[2 iFw PttW2; - +

. 11 Jh p2 0 0 0 0 !b +iFw Ptzcu2; "12 0 0 AJ 0 0 +

A) -iFw 2

D A2 () () A3 () + P22w ,

D () Az () () A . .J

and with the displacements and volmue forces

''I' ( iii iii ·u' iii ii~ 'f ); u 3 uJ

p· ( 11 f' n 'r Af ' r ) . 2 fl f2 fJ Ilence the Fourier fundamental solution is

D = [Pt +P2r 1• (8.43)

For a 13Etvl in the original space, sec \Vic be [\Vic93].

8.2 Stationary Problems

8.2.1 The stationarity assumption

Stationary problems arc obtained by assuming t.hat. all quant.it.ics of the dif­ferential equation vary with c-i«Jot, where l.0'o is a fixed value. Hence vvr have for a quant-ity a

a(x, t) = a(:r, wo) e a(:r, w) = h(i: ,wo)6(w - l.0'o)- (SAl)

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106

The time depemleney is separated; the quasi-statie problem for a(:r , w0 ) 4 a( i, t.v'o) can be solved first before reintroducing t.he sinusoidal time behaviour. Thus in t.he following the t.ime dependency is omitt.ed in the presentation.

8.2.2 Scalar waves in isotropic media

The fundamental solution and the Galerkin BIE The stationary scalar wave equation is obtained by replacing Ott -+ -w;}

and l.v' -+ w0 in (8.14) which results in the Helmholtz equation

(.6. + k~) u(:r) =-J(:r) F H (k~- l:i;n Ft(:/;) = -hn. (8.45)

with the given wave number kp = l.v'~ /c~ . The fundament.a.l solutions are. cf. [Bon95a] for the original space,

for JR.2 . l {; (:r) = -Hi>(kplxl)

4 4 . 1

(8.46) U(i·) = '2 1·'-1"2: kl'- .L

1 F A 1 for R3 U(x) = -- e'·kp i :~:l H U(i) = k2 - 1·'-12. (8.47)

4r.l:~:l .. 'p .L

H6 is the zero order Hankel function of the first kind. The boundary diffe­rential operator is identical to that for the Poisson equation

A' = -v' · \7 q F

+-+ A~ = -iv'· · .i:. ( 8.48)

The Galerkin I3IE is similar to that of the Poisson equation, compare (5.:32),

(i:U9)

,l\;~q 1Yu

2:: qi ( ~J - x ). 6:/J) + 2:: ni ( ~( -.t-). 6:,.A~C) ; 1.

( &,, ( -:! ). ixA//I) (8.50)

I'v·q ;\"LI

+ L q; ( (),( - i:), ¢;1.4~0) - L u; ( 2Yu (-:I-) . (p~A{1A;/f).

The interior Dirichlet problem The interior Dirichlet problem for the Helmholtz equation is

(.6. + k;,) ·u(:r) =- f(:r), :r E r2: (8.51)

u = ur = 0, X E f.

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8.2. S''L\"1'/0Y.\/Ct' 1'/lOHI./-.'MS 107

Fig)Ll showt- t.he amplitude of t.he veloeit_v pot.ential a for :.J..' = 10 [rad/s] caused by a single soLtrce in the center of the sqLtare ll = [0, lj x _0, lj. The

Figure 8.:~: The 'twlocily polenliu1 u due lo a sinyle r:enlered source ·wilh :.Jo =

10 fmdjsj

boundaries are discretized by four constant trial functions for each side which i~:; not really sufficient for the aecurm.e fulti llment of the imposed boundary nmdil.i(ms.

Sonunerfeld's radiation condition For exterior problems (e.g. lwles iu an iuiiuit.e domain) au additional cou­

dition, 8ommerfeld 's n-1dim.ion condition, is required t.o get unique results. Tt. distinguishes between incoming and outgoing waves. \Ve have for the lR" DLOO

incoming :

oul.goi ng :

0(.) is t.he Lamla.u symbol.

8.2.3 Elastic waves in isotropic media

The stationary fundamental solution 13y substituting"'-' = :.;;0 in tho transient. funda.ment.al sol1nion (8.2:3) we

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108

obtain

cfil ;/;~ r-: 5;2 + r:~ :i;~ - wa

cf:i:3i;2 (8.5:3)

The corresponding expression for the fundamental solution in the original space can be found. for example in [Bon95a.]. The st.a.tiona.ry problems are not. elaborated further, the derivation of the stationary fundament.al solutions from t.hc transient fundarncnt.al solution is always trivial in Fourier space.

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Chapter 9

Thermoelasticity

.I.M. C:. Duhamel

9.1 Coupled Thermoelasticity

9.1.1 Coupled anisotropic thermoelasticity

The differential equation and itH fundamental ~;olution 'Jhr l.hrory of l.hrnnodasLicil .. v l.ab's int.o accouul. i.lw coupliug lJrl.wrcu

elastic stresses and t.emperat.cue, d. [DSSH9, :'\ow7ii, K .. ogO}I. \\'e present. first the fully coupled theory where the interaction between tempera t.ure and mechanical forces is taken into ac:eount. The equat.ions are linearized and the h!'at condn<:t.iiJn is moddkd by FonricT's la.1N. Th!'u, sPvc·nd simplifil'd rnodds

under additional assumptions are in v est.ig·at.ed.F or all these problems, the Fourier fundamenal3olution and the DIE are deri vee\. Fundamemal solutions in t.lce original space are onl:y available for "-r:y special caoes.

The two fuudameut.al axioms for thenuoelasl.ieity are the firsL

(9.1)

and r.lw sc·(~OJHI Ia w of t.hcTmcHiynarnics

(0.2)

1 Thc firs~ work on a coupled dwory lbr stresses eau;:cd by l'orecs and dwnua.l ;:~rcs;.c;. is lTim3:3 j : .J .1\1. C. Duhamel: IVH•muir ~ur le cakul de~ action~ mulkula.ire~ dev ~ lumket< pm lr~~ ehangr•nwnt., dn t.empi~rarnrr~ dam lr~s r:orp' solidr~~.lvfP.m. (},rod . . 1d . .'in.'I'(J,nt.l itrn.ngm-8

5 (1838), 440--498.

109

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110 CK\PTJ::H 9. THt;UAlOHLASTlClT'r.

c, k are the internal and the kinetic energy per unit volume and D1w is the heat. source density. IJk, 8, T denote the cornponents of the heat flux veet.or, the ent.ropy per unit volume, and t.he t.mnperature, respectivcL:y. The simplest. (and Linearized) model for the heat conduct.ion is Fourier's Law

(9.:3) where K;.1 is the symmetric thennal conductivity tensor. According to the Duhamel-\-eurnann relation, the stn:>.'-iS tensor o is depending on the strain tensor E and the temperature differences () = T- T0 (with T0 as the temper­ature at the natural state)

(9.4)

The entropy s is related to E and () by

·j r:c B 8 = _i;~;:tEkt +To . (9.5)

cklrnn is the elasticity tensor measured at constant temperature and c,. de­notes the speeific heat measured at eonstant strain. 8kt aecounts for the interaction between strain a.nd temperature. The dynamic equilibiurn is

(9.6)

p is the mass density and u, f denote the displacements and volurue forces, respectively. The final system of different.ial equations describing the cou­pled thermoelasticit.y consists of the linearized dynamic equilibrium and the linearized heat conduction equation

Cklmn()n(Um - piJtt'Uh, - ./1kl()l(j

K ktakt() - c,Dt() - To.3klatatuk

The transform of (9.7) leads t.o

- h; -at w·

P=

and

Cw-n:i':nXt c'2l~n:1:n:i:l - ~·2

C:mn:i:n:i;/ -Tu(htwi:l

f,

Cu;~.,}-n:i':t

C~t:~ni-n:i:t C:}{:;n:i:rJt - (YvJ

2

-T[)3:JlWXf

f1' = ( /L h h - iww ) .

(9.7)

(9.8)

The Fourier fundamental solution for coupled thermoelasticity in arbitrary anisotopic media is obtained by inversion

['; - p' -1 ( ··' · ") ; - J. , w. (9.9)

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9.1. COUPLt.'D THt.'H!VlOJ::Li'lSTlCITY 111

The boundary operator The first quantity needed for the I3Eivi algorit.lun is t.hc t.raetion t at t.he

i-t.h boundary panel wit.h t.he normal 1)

As second t.crm we need the normal heat. flux

(9.11)

These relations can be converted into operator notation, d. [BSS89],

4 (9.12)

with u from (9.8) and the boundary operator

A' t [ Clllnv fDn C ll2n z;f Dn C 11:Jn z;j Dn - ;'ill iJ{

l C2nnv fDn C212nz1{un C21:Jn z;f Dn - ;3211/i (9.13)

C3nnvfDn C :JI2n z;f Un C :JI:Jn z;f Un - .3:]11!{ 0 0 0 -J{kf i!:Jlz

[ iCwn l/f':L·" iC 112" 11f i n iClf3n1J{Xn - ;'J1wt

l 1C 2llnvf':'i:n iC212n v[ i:, "iCzrJn IJ{ .in - /1211/i 1C:Jllnvf':l:n iC;Jl2n v[ i:, "iC313nP{ .in - /'i:JIIJl

0 0 0 -K~;;wi,il

and

(ti)l' ( t' ti t' i ) . = ' l '2 :~ q,l 4

The Galerkin BIE The Galrl·kin 131E ca n be adopted from cla.stodynamics, i. e. from (8.26)

and (8 .27) , by inserting the fundamental solution (9.9) and t.he boundary op erat.or (9.13). This leads to t.he displacement. Galerkin 131E

( rP{' ( -:L· )tjJ~" ( - w), Uy) = ( rP{ ' ( - i ) 9~" ( - w·),JJr) (9.14)

+ 2:: f;' ;" ( J>i t (-i·)VJ{2 ( - w·), i>~' cp~z rj)

- L u''h ( rPi' (- :[:):p{z ( - w·) , ¢:i cp~J ~'lf) : Z t 11-2

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112 CK\PTJ::H 9. THt;UAlOHLASTlClT'r.

and the traction Galerkin l::HE :

- ( rb{/ ( -i )0(/ ( -w), A{' fry) = - ( \;{/ ( -i )¢(,2 ( -u) ), fxA{' {;) (9.15)

_"' titi 3 /(],j, (-'I··) ·;.,i"(- · •) (i)]-i, ~i3 _,1j,{j'; ) ~ \ i"It · · 't'u '-" ' , t 'Yt ~

+ L u''i' ( c;;,' ( -:i:):P;/ ( -w ), ~;,' \;;,' .4{' A~'(;). tt,"l2

9.1.2 Coupled isotropic thermoelasticity

The differential equation and its fundamental solution For isotropic media the elasticity components are

and the isotropic thermal constants are, d. [BSS89J,

where ·y0 depends on the coefficient. of linear thermal expansion 0:1

The thermal diff'usivit.y 1\o is

Ao 1\'·o = C- E ·

The diH'erentiaJ equation (9. 7) is then simplified to, d. [BSS89J,

pauuk + (A+ Jl)Dkrut - pDuu.A - ''r'oO~cfJ 1

akk(J - - OtB - TJoOtOr.Vr, /{, [)

(9.1G)

(9.17)

(9.18)

(9.19)

with 1/o = ~:oT0/ /\o and Q = n) c, . The Fourier fundamental solut.ion is hence

[., = p' l(·lA' 0 •)• " ... , w' (9.20)

where the transfonned ditl'erential operator is defined as

[

-Cti:dl -At -Ct i :d2 A A A. "-

-cl:•r2:l't -c1X2X2 - ih -Ct:i:J :[:l -c1 i :3:i:2

7]o.i'tev' l]o:hec;

-CtXtX.3

-Ct X2X.3

-cti.J:r-3 - At T]o:'i:3!.;J

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9.2. SllvlPLlFlJ:;D THt.:lUv101'.'L1STlC MODt:LS 113

9.2 Simplified Thermoelastic Models

9. 2.1 Thermal stresses

In eases where the deformation is mainly caused by thermal sources than by external mechanical forces, it can be assumed that the coupling tenn T0 3krUtU(Hk is negligible compared to ccDt&, cf. [KiigOO]. This means that the thermoelastic dissipation has no significant influence on the thermal field. The seeond equation in (9.7) is then decoupled from the first

(9.21)

and solved seperately. In a. second step, the obtained t.emperature field H is regarded as "body force'' for t.he elastic fields in the first equation of (9.7). The fundament.al solution is then

C112ni:ni:r Ctun:l:,:/:1 - pw2

C:mn ;[; n:t: r 0

C ll:1n):,,i:l c2l3n:l:,:/;/

C:J/:Jn:t:n:l:r - {JWZ 0

9.2.2 Coupled quasi-static thermoelasticity

If Lhe exLernal sources f. w vary slowly in Lime compared with the c:harac:­Lerisllc: frequency cjL the inertia Lerms can be omiUed (cis the speed of Lhe clasLle waves and L Lhe charac:LerisLic: lengLh of Lhe body). The coupled system of quasi-static: differential equations is then

cklmn()nlU.m - 3k/(}l(J

KktOktB - c,8tB - T0.3kl8t8l'uk

The Fourier fundamental solution is

C u:ln:i:.,,:i:, C'lnrJn:i;l

C :JC:>n:i:n :i;l -T0.3:;rw5:1

-h; -OtW.

i.Bllil VJ21 :l:r i.8:Jt :l:r

Kkti:ki-1 + i u.)C,

(9.22)

r

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114 CK\PTJ::H 9. THt;UAlOHLASTlClT'r.

9.2.3 Uncoupled quasi-static thermoelasticity

Combining the two approximations, i.e. neglecting the coupling term To.BkzDtDruk and the inertia term -pDttak, the uncoupled quasi-static model is obtained. The differential equation is then

Ck rmnOn(Um - /'>k rEJJJ K ktiAtfJ - c, DtfJ

and as fundamental solution we get

9.2.4 Stationary thermoelasticity

-.fk;

-O(W,

(9.24)

In the stationary state, all physical variables are independent of time, leading to t,he Poisson equaLion for heat, conduction and to _\Javier 's equation for elasticity with a body source Lerm caused by the temperature

Cktrnrl)n(Urn - /'>kraJ) K kcDkt(}

The corresponding fundamental solution is

-.fk;

-O(W.

i(h t:h i.32li:l i./1:3/i: l

J(kl·iki·l

(9.25)

r (9.26)

For some fundamental solutions in the original space see Balas et a.l. [BSS89].

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Chapter 10

Non-linearity

10.1 Physical Non-linearity

10.1.1 Inelastic problems

The differential equation for inelastic materials In the f()Jlowing, all physical terms are given in raLe tenns. Assuming linear

strain-displacement. relationship, the total strain rate can be decomposed imo a linear ,:"t, and a non-linear part s~1 • cf. lDal9:3j,

(1 0.1)

Thr Liurar part is rdatr:d by Hooke's law t,o tht' r:la.sl.ic sl.n·ss ra.l.r:s Lhrongh, compare (6.:2~),

(10.2)

with Cuwn = A J~,/5mn - '211 Jh11 51u in the isotropic case. The subst,iJ.ut,iou of ~L· by i - '=~N yield::;

(UU)

The Pquilibrium cqna.l.ion i~

(10.1)

which lead~ after introduction of (10.1) to t.hc diffcrcnt.ial equation

(1 0.3)

llfi

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116

The left-hand side of (10.5) is identical to that obtained in the linear case that means all non-linear contributions can he regarded a.s fictitious volume forces. Similarly, we get fictitious tractions Oil t.he boundary

(10.6)

IIence, the non-liilear problem is reformulated as a linear one where the IlOn-linearity is put Oil the right-haild side aild treated as initial stress and bounda.ry tractions. The final solution is obtained. by a st.a.ndard iteration process, e.g. [Dal9:3]. Each step consist.s of a totally linear problem and is solved as discussed in the previous chapters.

The incremental boundary integral equations Equation (10.5) can be solved by a BEivl based on the linear fundamental

solution U. Transferring the collocation approach proposed by Dallner [Dal93] to a Galerkin method, the following I3IE cail he defined in Fourier space

l ~j( ")' \ l ~i( ·" ) f': f·; l ~j( ' ) oNA· [~\ \(Pt. -.T , n:x I = \ G'\ -.r ' xu + \(Pt. -:r ' E " /I i iVu

(10.7)

+ L e ( (){( - :i·), (?~ 0) - L u' ( ¢{ ( - :i:), ~;~A: O): N, i

The only difference to the linear case lies in the integral containing the non­liilear st.raiil i~'. U is t.hc linear fundamental solution and A{ is the corres­poilding bouildary opera! or. A IT is the stress differential operator

A. '(' . u == Z -'k lrnn:Ln, (10.9)

which generates thr, fuildameilt.al stress tr.nsnr 'I'.,pkl from the fundamental solution Upm :

(Hl.lO)

The discretization of the locally plastifying zone The discretization of the boundary quanti tie:; in (10. 7) aml (Hl.8) can be

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10.1. PH'fSlCltL NON-LL\'.K·1R1TY 117

r

....,____ _____...

....,____ _____...

....,____ n _____... ....,____ _____...

....,____ _____...

....,____ _____...

....,__~------------~~~~~~------------~_____..

Figure 10.1: The dom.ain discr·etiz ation of a locally plastifying r·egion esti­tnated a priori, cf. [Dal93}

taken directly from the linear case. Required is ::;till a partition of the pla.s­tifying zone o[\· in to cellt;, see Fig. 10. 1. The non-linear ::;train rate i t\ for the next increment in the iteration proce:;s hw; to be approximated in the inner cells by local volume trial functions (or alternat.ively by a dual reciprocity concept. which uses global t.rial funet.ions like radial basis functions. wavelets, etc:., e.g. [Pol96]). This zone is a priori not. known but can oft.en be estimated roughly. The rest. of the domain n remains undisc:retiz;cd as long a<> volume forces f arc z;cro.

The non-linear :;train rate i t is approximated here by i'Vc standard polyno­mial trial functions

4

<!>;, = (1 - :r1 )(1- :1·2)X~

"';:J - (.1 r )x· x0 ' ''e - --·1 2. ·"

(10.11)

Figure 10.2: Linear trial functions f or rectangular / triangular elements (lR~)

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118

The cutoff· dist.ribut.ions x~ for rectangular reference cells are for the !Rn

n

x~(:r) = IT H(xk)H (1 - :rk) k·-1

and for triangular reference cells in R2 and in R3 , cf. [Dud97b],

for JR2 :

4 X~ (:r)

{·~ (i) ~'-c

X~ (:r)

.~~ (:i.·)

H(:r 1 )H(J:2 )H(1- :r 1 - :r2 ) (10.13)

1 [1 . 1 . ] .·. .. ~ (1- e '·''')--::---- (1- e IX") ; Xt- X~ Xt X£

H (:rt)H(:r2)H (x3)H(1- :r·1- :r2- x3) i(i:1i:2 + i:1i:1- :i·f)(e-;"2 + e-iia)

i:1.i:2i"3(.i\- .i2)(i:1- .f·l) "ie- 'h + iei(h -:i:z - :i:3) i + ie-i.(:i:2+i:3)

:!:t(i"t- :i:£)(:i:t- :i::l) + :h:!:2:i::l

Arbitrary trial functions c)~ of arbitrary polynomial degree defined on trans­lated, rotated and dilated cells are obtained by the application of (::3.19) and (:3.20). Fig.l0.2 shows the rectangular and t-riangular reference elements with linear trial funct.ions for t.he R2 . Fig.10.3 gives t.he same for three dimensions.

:r3 .T2

~x,

oZ, = X1(1- .r.2)(1- :T;J)X0

Q>~ = (1- x1):1:2(l- :r·3)x0

o; = (1- :rl)(l- J::~ ):r:d)

"'.· 2. = (1 - x· • )·r· ·r ·.vu ""<' 1 . ·2· ·3 ~. c)4 = T X (1 - :r· ).·.v0

'" ·12 3 ~

Figure 10.3: Linear trial fnnction.s for rectangular·/ triangular elements {!R3 )

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10.2. Gt;OM.t:TWCi'lL f\iON-LlNK'lWT'r-

The additional BIE for the interior degrees of freedom The discretized versions of (10.7) and (10.8) arc now

119

(10.14)

-i Nu

+ L:e(<1~(-:i:),(3~0) - L:u'(ri>U-:r).r3~~0) ; ''"' i

- ( 0;;,( -x), .AU'x) =- ( \;!J - :I-),] x.A~C) - f= ei ( 0;U -:i:), 3~~-(/r) i

.i.Vt 1\:LL

- L ti ( ¢~( -:£) . ri>tA{u) +Lui ( ~( -:i-) , cp~,A{AtC') . .z i

For the additional degrees of freedom , the coefficients e; of the interior strain rates E:~1 , t.hc following J31Es arc required, cf. [Dal93],

~ ;_V~

( J;i.( -i) , A,~) = ( J;i.( -i) , ./ xA/i) + L e' ( \b-;,( - i:), ¢~A,A,J') (10.15) ;

i 1Vu

+ L t' ( 1;u- :!: ) ' 3~ A, D) - L ui (<llc (-:!: ) ' 3~ A,~ D) ' N,, .

with

4

The algorithm for the ~olution is not pre~ented here, it can be taken for ex­mnple from Dallner [Dal9:3]. A~ for the linear ca~e, the Fourier BEl\-1 approach enables the solution of non-Linear prohlen1s ·where the fundamental solution is not. known .

10.2 Geometrical Non-linearity

10.2.1 Large deflection of thin elastic plates

The von Karman equation A~ an example of a I3E.\:I for geometrical non-linear problen1s, the mode­

rately large deflection of thin ela~tic plate~ is comidered. IIere, it is a,:,;surned

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120

that the strains in the mid-plane (:z;1 , :.r'l) of the plate are given b,v, cf. [Kyt.9G, \V.JTOO],

(10.17)

wit.h

This means that all non-linear tenus in the non-linear compatibility equation of elastic cont.inua

(10.18)

a.re neglected except those due to the deflection w in direction :r 3 . The in­plane displacements u1 , ·u2 are assumed t.o be small. The membrane forces ,vkl are then obtained by an integration across t.he thickness h

with \TL

' kl

(10.19)

h is the constant t.hickness of t.he plate and Ckfrnn is the elasticity tensor. The bending and twist.ing moments arc as in t.he linear theory, compare (7.2),

(10.20)

whereas there is an additional non-linear tenn for qk :

qt + q~; (Hl.21)

with iJ1rnkt,

The equilibrium equat.ion i.Jkqk + f = 0 is expressed in w

(10.22)

with t.hc isot.ropic plate rigidity lJ = £h2 /[12(1- 1:J2 )] as in the linear case. f is the transversal load per unit area. The in-plane equilibrium is

(Hl.2:3)

\Ve get finally after introducing a. scalar Airy type stress function F with

(HLH)

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10.2. Gt;OM.t:TWCi'lL f\iON-LlNK'lWT'r- 121

t.he von Ktirrruin equations for the large defiection of isotropic thin plates

6.6.F 6.6.w

.A- (a12wih~w- Ou'WOn'W) (10.25)

J I lJ + hj lJ ( Dnt'Du w - 2D12t'Dnw + Du FD22w) .

The two left-hand sides of these equations arc linear and lead to Lhc DIE presented in Chapter 7. The boundary operators and Lhc funr:lamcnLal solu­tions for both equations can be taken from the I3E:\I for linear thin plates (Kirchhoff) .The non-linear terms are shifted to the right-hand side and r:an be treated similar w volume forces. They lead to volume integrals for each increment of the iterat.ive solution.

The domain terms in the original space According to \Yang et al. [\V.JTOO], the problem stated above can be solved

by a dual reeiprocit.y BEI\L They propose to transfer the volume integrals of the right-hand side, which a.re linked to the non-linearity and to the vo­Lume forces, to boundary int.egrals. The st.arting point is the following series expansion in global domain interpolation funct.ions 6h

NIL

Lt (w, w) ;::::: L a'(?L(:r); i= l

f I lJ + L2(w, F) ;:::::

with the operators

L 1 (w,w)

Lz(w, F)

i= l

E (Dt2w D12w - Du·wanw);

hi D ( DzzF Dn·tv - '2a12F D12·tv + DnFiJ22w) .

(10.26)

N, L are the boundary and dmnain collocation points. Crucial for the duaJ reciprocity approach is the knowledge of t/!{1 with

6.6. «;i'z = c)~.

\Vang et al. [\V.JTOO] propm;e the following radial basi:; function:;

<!J~ = c1 + czl :rl + c.1lxl 2 + c.1l:rr>

(10.27)

(10.28)

which arc Lranslated ! o each collocation point. A second reciprocity trans­forms the domain int.egra.b to boundary integralb. Thus, \Ve get for the volume t.enn of the displar:ement I3IE

1 1 N+L . . ' D [! + £ 2(w, F)]8 TV = D L b'6.6.~/;~(y) 8 TV (10.29)

i=l

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122

the following expressions for each collocation point at :r = :r', compare (7.15),

D..D..IJ}tl * ~v ~;/ (:J.: )1J'tlr) (10.:30)

{ A<tqj'1(y)H'(.r.- y) df9 - { Arnl/'~(u)A)V(.r- y) dfy Jr Jr + { Arl;!·~·t(y)AmlY(.r.- y) df9 - { ~·11 (y)AqlV(:r- y) dfy Jr Jr + L Ac·t;'!~Jy)lV(x -1/)- L 't/.,~1 (y)A)V(x- y1

'').

k

The solution is then computed iteratively together with the linear BEI\1. For the details see [\VJTOO]. The main problems are :

• how to find the qih for which t'Jh in (10.27) arc known,

• how to compute the derivat.ives of ·~;~ 1 at the boundary panels,

• and how to choose the collocation points in the interior domain.

\Vang et al. [\VJTOO] proposed for the second point a ~econd series approx­imation. To the author~ knowledge, there is no Galerkin dual reciprocity met.hod published yet which might simpli(y t.he quest.ions mentioned above. \Vavelets might be a good choice for 9~ 1 , but this is beyond t.he scope of this thesis.

10.2.2 Dual reciprocity methods in Fourier space

The evaluation of interpolation functions in Fourier space In contrast t.o the t.raditiona.l approach, t.he first problem, the evaluation

of ~:;l in (10.27), is very simple in Fourier space. \Ve have for arbitrary global int.erpolation functions 6h

(10.31)

The second problem, t.he comput.at.ion of (;0 (or it.s derivatives) at the bound­ary in (10.30) is also simple if t.he interpolation functions qih arc polynorni­a.ls, trigonometric or exponential functions becam;e the transformations of these functionb lead to derivatives of Dimc-di~tribution~. If x0 (:r) denotes the cutoff di~tribution of a boundary panel defined in (:3.17) or (:3.18), and

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10.2. Gt;OM.t:TWCi'lL f\iON-LlNK'lWT'r- 123

6h = L~;: 2..:1 Cktxi: is an arbitrary n-dimensional polynomial. compare ( 4.17), t.he values of q'Jh at the boundary panel can be computed in Fourier space by

1 "' . ,o ~ ~ -l --.J - o -( -. -Q!) "' X = Ckt t u;.x . 211)" .. " . . ~

k I

Transferring this idea to the dual rec:iproc:ity functions qf1 and their deriva­tives pi(D) t/'h leads to the evaluation of the following c-onvolution

'i ( · .!i · O ~~-1 :r u' · JJk-o( · ·· )·· ' ( 1 j• p'i ( ') J." 'll( A ) -(.. )" P :r)~· :n *X = LJ LJ 1. c ,d (··A2 _. 2 ) 2 X :1: - !J dy, 10.32)

271 k l ft" y I + Y2

which is trivial for all suggested. interpolat.ion funct.ions and. all straight boundary panels due to the Dirac-distributions in the kernels. The third dif­ficulty, t.he choice of collocation points, might be less severe \Vhen a. Galer kin method is chosen, i.e. by choosing oh as additional test functions for the interior.

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Chapter 11

Wavelets illberto c(}_./der·6n

11.1 Fundamentals of Wavelet Theory

11.1.1 Data compression by wavelets

,\it. hough thC' first. mathematical paper on wavelet. transform was published by Cnlrkr6n 1 it. t.ook l.wenl.y yra.rs till Goupillaud, Grossma.n11 aml 1\lorlrl. applied it iu rw eugiueeriug coutext,2

Soon aft.erwards, wavelets became one of the most vivid research areas in mathematic:-; and crnoput.ational engineering·. ·Thoy have: found their way rapidly iut.o i ndm;t.rial applieat.iom because of t.heir rather promisiug pro­pert.ies as meaus of a very efficient,, hierarchical data represent.ation. They are mainly applied in t.he lields of image processing· and sip;nal analysis, al­though, in the author's opinion, they might be of advantage wherever a large mnonnl. nf dal.11 is hiindlr~d. Tlw n~a.rlPr ini.Pn~si.Pd in th<' largf'r r:oui.Pxt is rdcrrr~d l.o l.lw wPhpagr•:-; ht.l.p:// ww1v. wa.vdds.urg r:dil.r~d by \Vim Swddr:us.

As the Fourier U'ansform, the wavelet t.ransform is an integral transform technique which has a cominLLOLLS and a discrete version. The combination with a rnult.ircsolution algorithrn leads to a fast wavelet t.ransforrn which mig;hl. [)(' Ui'iC~d ii.i'i an a.ll.r·mativc~ lo l.lw FFT ('.g. Lieb rLir·971. 1-'init.(~ dPmcnl. mel.lwcb were developed t.o rAprA<'ent. local chara.(·.tAriSLics, spectral met.hods

1 A.P. Cakkr6n: luccnaodink spaces nud iwcrpolmioa, lhc COiuplex method. St"Udia .tvlath. 24 ll961), 113 190.

~P. Goupillaml., A. Grossmann anti .L\.forld.: Cydr~-(kt.a,,·(~ and rda.t.cd t.ranHfrmm in S('isrnic ~ip;nal urmlysis. Geocxplomlion. 2:1, El~cvicr Science Pub!. (1984/8'-i), 8'-i-102.

12fi

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126 CH:\PTP:R II. W;\VFLE'TS

were used for modelling global properties. The wavelet transform is designed for r.he optimal. combi.rmt.ion of l.ocal and global. infonrmr.imt.

In 1 !)!)1, the fir~t paper [HCIWl] on wavelet.;, in t.he context of HEM a.ppeared. The subsequent. publieat.ions, e.g. [DPS!n, I3V!H, Ra.t!Wa, Ra.t96h, Ra.t.97, LS~l7: PSS!:l7, PS~l7, Wan97b, SeMll)j, eoncentrat.ing on wavelet discretization;, for boundary eh~rrwul md.hods, suc<:l~edl~d in wdu<:iug dw work a.llll sl.ora.gl~ of lhe discreti~alion scheme from O(;Yz) w O(N(lnN)") (wiLh a either z.ero or 1.1 ~mall positive integer). Tlw l~f<~!Vl nmtriee~, whieh an: fully populated, are approximaLed by sparse maLrices with a "finger ~tructure" (Fig.ll.l). 13ecause of t.he oseilhtting ehamcter of t.he wavelets (vanishing moments), some non-essr.nt,ia.l oli-dia~ona.l t.enns r·a.n be uc~lef't,cd on eaf'h diseretizal.iou level of the hierarchical dat.a representation wit.hout affeeting the eon vergence

Figure 11.1: The "fi11ger st.ructure" of the a.pproximated nE\;f-mat.rix ob­lairwd hy a wavelet discrcl.izat.ion Beh98]

UntorLunately. this compre~slon algorilhm requires a nested sequence of meshes on the boundary r and therefore a suffieiently regular geomet.ry of the problem. This is not. realist.ic for prar:l.ieal problems whif'h haw.• in general highly complex geomet.ries. To circumvent. this problem, Sduuidlin mul Sc~hwa.b [SSOO]Imw proposed a wavdl:t. Gllll:rkin BBM on unst.mc:t.un:d. meshes which i::. based on an agglomerated wavelet basis. The algorithm st.ruc:t.un~!-l the da.t.11 in dn!->t.crs gathering da.t.a from pllrt.!-l of t.lw gl:mru:t.ry which are not. necessarily eon11er:l.ed. :\o hierarehiealmesh is rec1uired.

The metltrHI prescnt.cd in [SSOO] is based on the lifting 8cherw~, a "seeoud generation wavelet algorithm" developed by \Vim Sweldens and uses ideas from other uu:r.hod;; for the~ n~dudion of cmnplc~xir.y in HE\l a.lgort.ilnns, i.e. ti·om multipole expansion [Rok80] and panel clu::.tering [11::'\18!.1].

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11.1. FUNDiL\1t;Nl'i1LS OF \\-:1\1 t;LK.l' THt;OHY- 127

In this thesis, it is not possible to present. a wavelet based BE.\1 in full detail. The intention of the following sections is t.o give a short inLroduc.Lion Lo wavelet theory and t.o show how t.he compression of BE:VI matrices can also be obtained for the Fourier BE\1.

11.1.2 The wavelet transform

The windowed Fourier transform The representation of a function f in the original space gives infonna­

tion about its Local character. The corresponding Fourier image .f reflects its global (spectral) properties. The first at.tempt to combine these t.wo repre­sentat.ions has let to t.he Gabor transform (or windowed Fourier t.ransformYJ

;

+oo Q,.{J}(b, i ) :=

00

f(x )g,.(:r- b) e ixi: dx, (11.1)

where Yet is the window function often chosen w; (with a fixed CY > 0)

(11.2)

Q,. {!} gives the local and the global infonnation of f. The standard deviation

1 8(} = --. llg,, 112

(1L3)

is the measure for the width L(ga) = 2fo of the :r:-i: window. The Fourier transform of g,, also a Gaussian function, generates the spectral window. The height of the :r-i window is L(g,.) = 1/(4/(:t) . lienee, the areil of the window has the following eonstant value (Fig.ll.2) :

(11.4)

Heisenberg's Uncertainty Principle for an arbitrary window function g

A 1 L(g)L(g) 2: 2" (lL'"J)

states that (11.4) is the minimal window area. Therefore, the choice of g, in ( 11.2) is optimal. The Gabor transform analyses f h;y windows with constant

3 D. Ga,bor: Theory of Cornrnunica,tion, J. IEE (London) 93 (1946), 429 457.

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128

:c i

------~ ' i

J:2 ·-·-·-·-·- -·i·-·- -·-·-@ -·-·-·-·-·-·i·-·-·-·-·-EJ ' ' . ~

Figure 11.2: The :r-:i: window for ! he Gabor Lransform (lcf!) and for Lhe wavelet t.ran~form (right) [Chu92]

area which is it.s main di~advantage. High and low o~cilla.t.ing functions cannot be represent.ed adequately. An additional dilat.ional (or scale) parameter can be introduced if g = \!) satisfies [Chu92]

l:-x:· 1j•(x) d:r = D (11.6)

which leads to the continuou~ wavelet t.ransform.

The continuous wavelet transform The continuous wavelet t.ransfonn as a t.ransfonn which ada.pt.s its window

size according to the local and global information density of .f is defined by

,;·l oo ·(T -b) Wdf}(a, b) := lat 2 -ex; f(x)~; ~ cl:z:, a, bE lit (11.7)

(":") denotes t.he complex conjugat.e. ~; is called mother wavelet from which the wavelets -~;a , b are const.nu.:t.ed by translation and dilation

(11.8)

1!J has to satis(y the admissibility condition [Chu92]

' ;·+x l·tp(:!:W , C 10 := . -

1

,-

1

-d.r < X· . . - co X

(11.9)

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11.1. FUNDiL\1t;Nl'i1LS OF \\-:1\1 t;LK.l' THt;OHY-

The x-i window is given by

[b + a:rm- a:r,., b + a;rrn + a;r,] X [

' ' A ' ] Xm - x,., :rm + x,. , a a.

where Xm, im. denote the cent.cr of the window

1 li·:X'

' 2 Xrn := ll«'ll~ -x :z:l~;(:r)l dx,

and :rr, :'i·r the corre~ponding radii (11-112 i~ the L£-Norrn) :

The rcc.onst.ruc:t.ion formula for the wavelet transform is

with t>a,f1 as the dual wavclCL.

The discrete wavelet transform The discrcl c wavelet. transform wiLh a scale facLor (} = 2 is

w,i.{.f}(ai, bk,J) ·- (!, 't),,j) = /~ .f(:t) 4'>k,j(:r) dx;

with discrete values for a. , b

-j (1) - (}

k bk· · - --:bu. ,) 21 ' j, k E Z.

129

(11.10)

(11.11)

(11.12)

(11.13)

110 > 0 is I he sampling rate which is often chosen as b0 = 1. For Lhc recon­struction, the ·wavelet t;· has to satisfy the stabilit.Y condition, i.e. t;· has to

generate a fmme4 , cf. [Chu92],

•ill! II~ :-::; l: I(!, 4'k ,}) 12 :-::; n 1!11~ (11.14)

·lA frame is a colleCLion of vectors rLi of a Hilbert space X so that no clement of X is orthogoual to all a, [Bla98]. In contmst. to bases it is uot r·equired that the a, be linearly independent. Hence, frames are redundant, ba~e~ not.

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130

for some positive const.ant.s 0 <A~< B < oc (A, Bare called frame bounds). The reconstruction formula is then

J(:r) = 2:::: cl,,1tZ·k,J, h,jc:'L

Ck,J = (f, 1/'I,,J) . (11.15)

ck,.i are the coefficients of !he wavelet expansion. In the case of orthogonal wavelets, the dual wavelet '~'>k,j is Linked to 't/>r,,.i by the biorthogonality relation ( 6iJ is the Kronecker symbol)

k, ,j,l,rn E Z. (11.16)

The dual is obtained in Fourier space by [Chu92]

' .. -··(~) v3(:i:) l'' T ·= -----:-''---'------./ . . . "'ex! I ~~·(:i; + 2Ttj I) 12

L...-]1=-x-

(11.17)

Alternatively to (11.15), f can be reconstructed by the dual wavelets

f(:z:J = 2:::: dk,;~·k,j, dk,j = (r lf;k,j) . (11.18) k,JE;L

11.1.3 Multiresolution analysis

Decomposition into a sequence of hierarchical structured details Around Hl86, t.he rrrultir·esol'Ution analysis (:\IRA), a new method for per­

forming discrete wavelet analysis, was developed [~vial89, ~vfey9D]. The coef­ficients c;,k (or d1,k) on the finest Level (smallest value a1) are computed ex­plicitely. From these, all coefficients on all coarser levels can be constructed by a recursive algorithm. Thus, the total computational effort is reduced rcmarka.bly; the wavelet. analysis based on a rnult.iresolut.ion algorithm is also called fast wavelet tmnsfonn (F\VT).

A function .fa is sampled on the finest level j = 0 at regular intervals 6.J: = ~.

Then this sequence is split into a coarser version f_ 1 at the coarser ~cale with 6.:r = 2,; and into the complementary detaih; d_1 at scale 6:r = .;. The repe­tition of this process leads to a sequence fo , f_1 , f _2 , . . . of coarser and coarser approximations \Vit.h complementary sequence~ d0 , d_1 , d_2 , . .. of dew.ils re­moved at every scale. Each detail dj is then described by a superposition of wavelets ~/;k ,j , k E Z with a special mother wavelet. determined recursively. The original function .fo can be recovered after N iterations by summing all details and adding the coarsest approximation f _ N

fo = f N + d tv· 1 1 + d N 2 + · · · + do. (11.19)

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11.1. FUNDiL\1t;Nl'i1LS OF \\-:1\1 t;LK.l' THt;OHY- 131

Nested sequence of function spaces for the approximations J1 The l'VIRA constructs a nested sequence of veet.or spaecs

(11.20)

satisfying the following properties

n v; = {0}, dos U Fi = L2 (R). (11.21) j EZ JE"i.

where clos denotes the elosure of a vector space. This means that the spaees lj, j E Z have only the trivial function q> = 0 in l:Ornrnon and t.hat. all elements of L~ can be represented by fund.ions of these spal:es. The spal:e l~i-l is obtained from l~i by scaling

rb(x) E l-~ {::} rf>(2.r) E l·}-1, (11.22)

and the spaces 1j are dosed with respect to translation

ri>(x) E \j {:} 0'/(:r- k) E ~j, k E Z. (11.23)

The space \!j is spanned by the scaling func:tions (Dk,j(:z:), k E Z which arc used to approximate fi m1 the j-th level of coarseness5 .

Nested sequence of function spaces for the details d1 The details dj on level j are the complement of the coarse approximation fj.

Hence. we define function spaces Hi i for the details which are complementary t.o l·r

7• They sat.isfy

(11.2·1)

·which implies that v; ean be expressed by the sum of the spaees of details lVr and Vv on t.he coarsest level

(11.2i))

The spaces lVj are spanned by wavelets ~;k,j· k E Z, i .e. the det.ails dJ are expressed. by a sum of wavelets ~:k,j, k E Z \Vhile the coarsest approximation .fN off is described by d>k ,N·

''The scaling functiom; c,!Jk,J (:t), k E Z are t.lw R zesz busts of the vector space lj [U>IR94].

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132

Mother wavelets 'ljJ constructed from mother scaling functions ¢ The !'villA starts by defining a m other· scaling f unction 00 which has to

satisfy a.<.; a necessary and sufficient condition the scaling relation [Chu92]

o(.r) = ~ ~k<i>(2x - k) . (11.26) k--00

The s c:aling functions 6k,i E l' j are obtained by translating and dilating ¢ :

(11.27)

They are used as a Riesz basis for Vj. The dual sealing functions 0;/i:,j defined by the biorthogonality relation on level j

(11.28)

lead to the dual :\IRA. The scaling functions are orthonormal at. every scale but not. from one scale to the other. The Fourier transform of the scaling equation (11.26) is

with (11.29)

The rmmerical realizations of a :\IRA algorithm are based on the coefficients ~k from equation (11.26). Therefore, fi is called genemting function of the l'viR.A. The mother wavelet is then construeted from the mother scaling func­ticm in Fourier space by, see for example Bla.tt.er [Bla98] for the just.ificat.ion,

(11.30)

The relation in the original space is given by Blatter as

t~{z:) = ~ g,q.i(2:z:- k), with k= - oo

The algorithm for the fast wavelet transform The two fundamental recursive relations for the fast wavelet algorithm

't) .,,j = ~ 9t 1>zk l ,j 1

l

(11.:32)

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. . .c c

d d

c

d

c

d

0 1 2

21 N-1 N

NN-1c

start ck,0 = < f, k,0ϕ >

G G G G

HHHH

finer level coarser level

coefficients on the level jdetails

coefficients on the level japproximations

for all k

recursion

number of levelsN

for all j = 1..N

for all j = 1..Nj - 1

j - 1d j = G c

jc = H c

coefficients on the finest level

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134

with the adjoint operators defined by H*cj = :Z::IEZ ~k-~1c1 and G*c:j = 2...:/E:?: 9k- '2(C[.

11.2 Wavelet Galerkin Discretization

The Galerkin l3El\l Leads to symmet.ric:, but. full;y populated matrices which becomes relevant. for large problems. Then, the computational effort. is not determined by the computation of the matrix entries hut by the inversion of the matrix. As recent research ha:; ~hown, there are ~eveml possihilitie~ for cornpressing the llEI\l matrices. In our context, we present first ideas on the compression by· wavelets for the Fourier BEI'vi which are closely related to the work done by Schneider [Sch98], Schwab [LS97. SSOO] and coworkers.

It is far beyond the scope intended here to give a cmnplete present-ation of the compression algorit.hms. \Ve only want. to shmv that t.he existing approaches can be t-ransferred easily to the Fourier BE:\·L The explicit realization might be t.hc subject of an independent. thesis.

11.2.1 Matrix compression in the original space

Wavelets as trial and test functions \Vith t.he exception of a very recent prepublication [SSOOJ, all wavelet based

BE:\Is found in the literature require a sufficient.ly regular and structured ge­mnetry. They assume t.hat. the boundary can be decomposed int.o N0 curvi­linear, either quadrilateral or triangular surface pieces ri. The corresponding reference domain rj i~ either the unit triangle or the unit ~quare, e.g. [PS97]. The wavelet analyt>is is then constructed on thit> reference domain : r 0 it> divided into a sequence of 111

, l = l..L :;ubsquares or subtriangles {f;,,1} by successively halving the sides l times. On these, (jd discont.inuous pieee\vise polynomials of degree d are defined.

This ends up in j = l..JV0 surface pieces, l = l..L levels of subdivisions. k = 1..41 su belements on each Level l, and m = l..Od sealing functions for each subelement.. ¢r denotes the sealing function with the multiindex 1 = (j, k, l, m.) and t~'.J is the complementary wavelet. -"'ot.c t.hat. each sealing function leads t.o 3 wavelets, i.e. we have for the wavelets m = 1..311,1 and .) = (j, /;:,!,3m). The wavelets arc piecewise polynomials in local coordinates [PS97]. The left part of Fig.llo'l shows an example for a linear wavelet on a. triangular panel which is given by [PSS97]. The right part shows the real and imaginary part of the Fourier transformed wavelet.

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5

7

4

6

41

-5

-3

-2-5

0

-12

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136

The matrix entries are computed from

G~," = (~;.!', ~:.r * AU) , Hy, 1 = (V.l', 1/!.J * U) ,

An estimate fur the ent.ries is given by [PS97]h

I GIL I < C'd-2(d. 2)2. (d+2)(1H') 7.1'.! - ~ ·y.r

which depends on the distance between two wavclct.s defined by

dy.r = dist(S1 , Sp ), with S.r = {x E::: l"; ~i:.r(.r) # 0}.

(11.39)

(11.H)

(11.42)

Cis a constant independent of the levels l'.l. If the two wavelets are seperated by a distance d.J'.J larger than a certain truncation parameter til' .1 (depending on the level l and Ievell') the entries can be neglected without affecting the a."iymptot.ic convergence rat.e [PS97, Sch98]. Hence, as Long a.'l ! he geometry is sufficiently strud ured, most. of the entries of ! he st.iffness matrix can be a priori replaced with zero. The minimal value for the t.runcat.ion parameter is determined by [PSS97]

(11.43)

with some number a 2: 1. The compressed Galcrkin scheme is stable if a is sufficiently large. The values a, n' arc given by [PS97]

s+d+1 (\: 2: 2(d + 1) ,

with s, s' E [D, d + 1).

, , s'+d+l 0: ? - -:-------;--

- 2(d + 1) ,

The presentation given here is only a rough sketeh, for a realization it is essential to study the literature cited above.

11.2.2 Matrix compression in Fourier space

Transform of the trial and test wavelets As long as the wavelets are defined on straight elements, the wavelet dis­

cretization presented in the last section can be transferred to the Fourier

('This is essentially clue to the vanishing moment properties of the wavelets [Sch98]

/ x"~'!J dx = 0, for [n[ ::; d. (11.10)

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11.2. H'!l\!ELET Gl!.LEHKlN DlSCRK.fl:tilTlOIV 137

BE\1. The truncation criterion (11.4:3) and the distances d.l',.J between the wavclct.s remain valid. VVc only have to find t.he transformed wavelets and t.he transfonna!.ions for !.he scaling func!.ions on t.hc coarsest. level. They arc defined as piecewise polynomials on triangular or qua.driiate:·al refcrcnee cl­ement:-;. lienee, the tramJormed trial and te:-;t wavelets 1/'J, 1/-'J' are obtained by a combination of ('1.15) to (11.17). The real and imaginary part of the transformed sample \vavelet are visualized on the right-hand side of Fig.ll A .

Transform of the multiscale Galerkin BIE The remaining non-zero matrix entries can be computed by the transform

of (11.:39)

L 1 ~ ~ ' '" ) c.~ . .~= (2")" \ u.,(-i), u.~Au , Hf,J = -r· 1) lt3pr -:i:) , v~.~D). (11.44) 271 n \

The vanishing moment. condition (11.6)

as fundamental condition for wavelets leads to an annihilation of the singu­larities of the fundamental solution at i = 0 (in the case of elliptic operators). We have for example for the Laplacian in two dimensions LJ = -(:i:i + i:D 1

and hence for the kerncl

(11A5)

It. is onl;y a!. t.he coarsest. level that the singularity remains at. i = 0, because t.he scalar product. with t.wo scaling functions is not. vanishing at. i = 0 :

lim (/>N ( - :[· )(/>N(; -j. 0. :i:I--+U

(11A6)

These ideas remain to be developed. Here, they are intended a,<, a preview of the future development. oft he Fourier BE\I.

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Chapter 12

Conclusions

12.1 Results

This thesis is the first principle work on the generalization of the bound­ary element. method by spatial and temporal Fourier transform. The tra­ditional Galerkin boundary· integral equations are reformulated by means of t.he convolution Lheorcm and Parseval's idcntit.y. In cont.rast l.o t.he t.radil.ional method, all quantities, !.he !.rial and Lest functions as well a.<; the fundamental solution a.nd its derivatives, are only required in Fourier space. The matrices a.re evaluated directly, no inverse tra.nsform is necessary.

For complicated physical problems (e.g. anisotropic elasticity ) , the funda­mental solution is only known explicitel.Y in Fourier spa.ce. Hence, approa.ches via traditional BE:\I encounter a lot of difficulties which are avoided in the method presented here. Therefore, the field of applications of the BE:\{ is enormously enlarged.

The new approach is valid for all linear problems where the material pa­rameters do not depend on location or time. The examples presented were taken from stationary heat conduction, elasticity, thin and thick plates with or wil hout. \Vinkler foundation. Tta.nsicnt. and stat.ionary dynamic problems ·were discussed for elas t.iciLy, porodasLiciLy, plat.cs, and I hennoclastir.it.y.

Geometrical and ph;ysical non-linearities can be modelled by iterat.ive proce­dures where the non-linear contributions arc shifted Lo the right-hand side of the equatioru; and treated as initial volume terms for each increment.

The Galer kin BE~vl leads Lo symmet.ric, but fully populal cd mal rices. It was shown t.hat. recent, methods based on Lhe faBI wavelet, transform t.o sparsify

139

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140 CH/tPTJ::H 12. COi\iCLUSlONS

these matrices can be transferred t.o Fourier BEJ\l.

For the realization of Fourier BE:VI, a rigorous dist.ribut.ional representation of t.he boundary element. method \Vas developed. It enables t.he correct treat­ment of the jumps and singularit.ies oceuring in t.he boundary int.egral equa­tions of traditional BEM or Fourier BEM. The distributional concept is nec­essary for the differentiation of boundary integral equations required for the symmetric Galerkin I3EivL It results in demonstrating that all singularities (which are often non-integrable) caneel in the original space if the integrals are combined adequately.

12.2 Open Questions

There are three n1ain open questiom;: Firstly, all examples were computed by int.egration routines which are not really optimized for the oscillat.ing kernels occurring in the integrals of the Fourier BE:-1. First ideas may be taken from a paper published by Averbueh et al. [AJJC+oo] on t.he computation of oscillatory int.egrals by wavelets.

And secondly, the Fourier BEivi is still rest.ricted to cases where the material parameters are constant with respect to time and space. The character of the differential equations changes totally if this assumption is abandoned. A way of establishing a llEI\l for t.his kind of problems may be read in the work of Pomp [Pom98].

Finally, the approach presented here is only valid for straight elements. For the implementation of curved elements, the approach of Ancira [AS97] might be useful. The curved elements are approximated locally by tangential straight. clement.s of t.he same proportions. The kernel and t.he Jacobian are expanded in Taylor series.

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Acknowledgements

The author would like to express his gratitude for the grants which have supported this thesis. The grant from the French Alfred-Kast.ler foundation, which provided support for a. stay a.t t.he EcoLe Polyt.echnique, Laboratoire de :VIecanique des Solides (L:\.1S), in Paris, gave me the opportunity to de­velop the initial ideas. l am particularly indebted to Prof. .\tare Bonnet for his criticism and comments and the benefit. of his experience in the field of boundary element methods. They provided the real incentive to my work.

Furt.her encouraged by the fruitful and open discussions and by the kindness of all members of t.he 1.\'IS, I went back to l\Iunich. Prof. Harry Grund mann, who had already supervised my PhD thesis, hosted me during the following two years. Once more l am deeply grateful for his personal and professional support. Better working conditions cannot be found. The financial support, a grant especially for the postdoetoral thesis, from the German Research Foundation (DFG) is gratefully acknowledged.

I am also indebted to many other people working in the field of boundary clement methods. First of all to Andreas Pomp who helped me to unders tand some deeper mathematical items related to my work. Prof. .Joachim Gwin­ner from the l :niversitiit der llundeswehr in .\Iunich was always open to my problems. The constructive criticism of Prof. Friedel Hartmann helped me to complete rny work. Prof. Kuhn in Erla.ngen, Prof. \Vendland in Stuttgart, and Prof. rvia.ier in l'viilan gave me the opportunity to present my work. \Vithout. these discussions I \Vould not have progressed as fast as I did. Prof. Guig­giani in Pisa and Attilio Frangi in rviilan helped me a lot with their questions and comments. I would like to thank Prof. Ernst Rank for pushing me more towards engineering applications. I was sometimes lost in self-generated ar­tificial problems.

I would like to thank my family, my wife and my children, for t.heir fiexibilit.Y and patience, and my father for his critical reading.

141

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Appendix A

Glossary

A. I Distribution theory

support: The support of a function c.?(:z:)

supp{ 6} = {:r E JRn: 6(:r) # 0}

is defined as the closure of the set. of all point.s J: E JR" where (j) is not. vanishing. ~ingle points :r with ¢(:r) = 0 are included in this set.

em-continuous: A function ¢ is called C:"'-cont.inuous if it is continuous up to its nr-t.h derivative.

differentiation: Dk<i> denotes the k-t.h differentiation of a function q) with respect to :rk. i.J" with the mult.iindex cr standing for

n

lnl = Ltr.k, nk E No . k=l

Holder-continuous: A function 6 E em). is Hiilder continuous of order .\if it. satisfies for two points .T = 6. 6 up to its lo:l-th derivativ-e (l(xl S: m) t.hc condition ( C is an arbit.rary constant.)

0<.\S:L

Lipschitz-continuous: A function rj> is called Lipschitz continuous if it. is IIi.ilder-c..:ont.inuou:; with .\ = 1.

143

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144 LtPPl:.'i\;'DlX .-1. GLOSSL1RY

C 1 boundary: A botmdary is a C1 boundary if for every boundary point :Do E on one can find a C1 function 'U' in a neighbourhood of :ro with

1p (x0) = 0, dq(xo) -1- 0.

LP-spaces: The Banach spaces LP, p ::;:. 1 are defined as the space of func­tions 0~ which arc t.o the p-t.h power integrable. They arc endowed with t.hc norm

[j. ] 1/ p lk>IIP = __ [<i>(.r)[P d.r < ex).

H_n

For p = 2, the scalar product of two functions ¢ 162 can be defined

(rf>t,rh) = _i., 0)l(x) rP2(x) d:r.

The L 2 is a Hilbert space1 .

Sobolev spaces: Sobolev spaces rvm··1' are spaces of functions ¢ which are up to their m-t.h derivative in LP

lV"'•P = { (;); ()" E LP(fl); for all [a[ :S: rn }.

If p = 2 the spaces n :m,z are notated by Hm.

trace operator: For a C111-regular dmnain, the trace operator~/ of orders :S: rn _yields the boumlary vahw1-; of a function ¢ up to the k-th derivative (k as an int-eger srnaller than s- 1/2). VVe have

with the traces as the normal derivatives

n,o - _' - ()J(b I

u . - [)vi im' for all :j = 0, .. . , k.

generalized functions: The :;et of generalized functions n include all linear and continuous functionals. They are defined via smne test functions q)

Hence t.he properties of t.he test functions define t.he set of generali~:ed

functions. In the following , the four main types are given with the corresponding sets of tes t functions .

1 In the context of the preseut. thesis, t.he scalar 1n·oduct is defined without the couju!!;at.e mrnplex. The funct.i0m; are real-valued.

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1\.1. msTn.muno:v THF:ORY 145

distdbutions:l: The t.est functions <:'.{r) E D(H) - qj"'(n) are bounded, po!->~css a ~~ornpm:r. support, and Hn~ infinit.oly oft<m ~~ont.iTmonsly dif­ferentiable. They and all lheir derivalives vanish at the boundary. To d1~monstr<J.t.c t.ha.t. r.his s<~t. is not ~~rnpty, Hiirrnand<~r givns tlu~ follow­ill~ example : For all dosed balls with the eeuter :r0 c n c_ H." and the radius r > 0 whieh are t.otally in the domain U, there exi~;ts an exponential test, funetion {.F'i~.A.l wit.h :r0 = 0, r = 1).

1:1.·1 < ,. l:rl > r.

A distribution u E= V'(H) is defined by the sealar produet with the te8t. funct.ion c/J

u( 6) = (u. <;)) = { u(x)o(:c) d:r < x. ln?.t~

Figmo A.l: 1'wo-dim.1~nsional tr~st fnndion <i.{r:) E Cgo(n)

tempered distribution: The set of tesl fLmctions can be enlarged to all funct.ions cj> E S(R.n) whit'h dedinc exponentia.lly including all their 1lcriva.tiw~s (t.lw Ga.u::.sia.n i~ a.n nxa.rnple). The eorrespoll(ling (dua.l) distributions u. E S'(IR") are ealled tempered distributions. I3ecmu;e we l..ta.vc weakened t.lw eondit,ions ou t.!w t.cst. funetions, t.lw set of tmnpered distributions is smaller than that of distributions. The set S'(JR") is closc~d with respect. t.o t.he Fourier transform whid1 w<Js t.lw hist.ori1~al.

reason tor their definition.

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146 LtPPl:.'i\;'DlX .-1. GLOSSL1RY

distribution with compact support: If the condition at infinity is aban­doned one can define t.est. functions 0) E ex(~") = £ (~n). To avoid difficulties at. infinit.y, t.he corresponding distributions 'n E £' (~n) have to have COIIl]JaCt support. Their Fourier t.ransforrn is a ex-function without any local singularities.

ultradistributions: The Fourier transform of a distribution leads not evi­dentJy to a distribution. For example, the Fourier transform of el' :.r I E

~ yields r\'(:£ 1 - i), i = A which is not. a dist.ribut.ion. Because the Fourier t.ransform of generalized functions is defined via t.he Fourier transform of the test. fum:tions, we get the test furu:tions 0? E Z(C") for the ultradistributions u E Z'(C11

) by Fourier transform of the test funct.ions 9 E D(O) of the distributions.

generalized differentiation: The differentiation of generalized funct.ions is defined via partial intcgrat.ion, i.e. the differential operator is shift.ed to the test functions.

(iAn, 9) = - (u, fA¢) , 9 E C~.

Dec:ause of the definition of the test functions rp, the distributions are infinitely often different-iable. Jumps and singularities can be differen­tiated.

differentiation of a product: lf the multiplication of two distribut.ions u1 , u2 is defined, we have for the differentiation of a product (Leilmiz)

iA (u1u2)

convolution of distributions: The convolution

llt*Uz:= r 11t(X-y)u2(y)dy; Jan :r,y E R".

of two distributions n1 and n2 is defined as long as either u.1 or u.2 has compact support [H6r90]. \Ve ha.ve for n1 E D'(~2) and. n2 , U.:J E E' (R")

commutation association

'Ill * 'U2

u1 * (u2 * u 3)

u2 * u1

(u1 "' u 2 ) * u3

differentiation of a convolution product: \Ve have for the differentia­tion of a eonvolutional product

which rnearu; that we can exchange differentiation and the integration related to convolution [IIi.ir90].

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iLl. DlSTIUlJUTlON THHOHY 147

Cauchy principal value: Some singular integrals can be regularized by Laking into account. the fac!. that the singularities cancel. If for ex­ample, the integral is singular at. :r0 E [u, II] then the principal value is obtained from

v.p . . l' q1(:r) dx = r!i;6~ {.[xo-c q1(:r) dx + 1:." rp(1:) d:r} .

The two integrals are infinite, but the limit of the sum is finite.

Finite-part integral: The finite part (Hada.ma.rds principal value) as an extension of the concept of Cauchy's principal va.lue is defined as the regular part of a singular integral. If an integraLlon leads to a function

.'v- b g(E) = ~~6L ~,:1 + bN-tln c: + bN+2 + 0(1)

J-l

then the term b,_~ is called the finite part. of g (E), [McLOO],

p.f. g(t:) = b,vl2·

This means that all singular contributions are neglect.ed.

Gauss-Green formula: If a domain n is described by a cutoff distribution x we have for t.he product of x with a distribution u, [Hrir90]

df is the Euclidian surface measure on an and I/ t.he exterior unit normal. The integration weighted with o = 1 of this equality yields the theorem of Gauss

{ 1· o1 (uy) dO=- { (ux)D1 {1} dn = 0 = 1 i31ud0- { 'UI/ 1 df . .In .Jn n Jr Fourier transform of tempered distributions: For tempered distribu­

tions u E S', the Fourier transform ii is defined by the transform of the test function rp

qi E S.

Parseval's identity: The invariance of the scalar product concerning the Fourier transform is called Parscval's idenitit.y

1 ( A A \ (27i)n U.t, u~f;

1 j' A (A)' (A) l ' -(. )" U. J .:r n2 x r :1: .

2Jr _ltll

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148 i'tPP.ENDJX A. GLOSS/illY

Alternatively- for real valued distributions, we have

Fourier transform of distributions: The transform il E Z'(Cn) of a. dis­tribution u E 'D'(n) is defined via. Parseval's identit.y

This means that it is constructed from the Fourier transform ilz E

Z(C") of t,he test. funct,ion \bD E V(n). Hence we have also defined the Fourier transform of ultradistrihutions.

theorem of Paley-Wiener: lf the support of a distribution u is cont.ained in a convex and compact domain n ~ &." with a cutoff distribution x then the transform of u has no local singularities [Hor90]. Consequently, every distribution wit.h compact. support. has a Fourier transform which is an cnt.ire analytic function in en .

jumps and singularities: A distribution has singularities of degree s E lR if it.s transform behaves at. infinity like O(lil"). The Dirac-distribution is of degree s = 0 and jumps arc singulari! ies of degree s = -1.

A.2 Boundary Element Method

boundary integral equations: The boundary integral equations arc ob­tained by a Smnigliana identity \Vhere the source points are put on the boundary itself.

direct and indirect BEM: In t.hc dirce!. l3E_\l, the real physical bound­ary quantities are used. The indirect TIE:\-1 defines fictitious sources of unknown density on the boundary.

fundamental solution: The fundamental solut.ion U is Lhc response of the infinite medium to a single source .f = 5

P(D)U = J.

It is the inverse of the differential operator P(D).

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149

Green's function: A solution U ofP(D)U = b is called Green's function if its takes into account particular boundary conditions.

reciprocity relation: If the virtual work is computed for a first state of equilibrium by choosing a second state as virtual state, and if a second virtual work is formulated by exchanging the two states, the two virtual works will be equal. This equalit.y is called reciprocity relation and 1s also known as Betti's t.heorem.

Somigliana's identity: If one state in a reeiprocit,y relation is chosen as Lhe state related to the fundamental solution, the resulting equaliL.Y is called Somlgliana's identity.

trial and test functions: The knuwn and unknown quantities arc approx­imated by trial functions which are located on the houmlary. The equa­tions are then weighted (or tested) by test functions which are either identical to the trial function::; (Galerkin approach) or which are Dirac­dist.ributions (collocation approach). In engineering mechanics, dte test functions are often called virtual functions.

single layer potential: The single layer potential is defined as the convo­lution of a boundary quantity t with the fundamental solution U

l-,.{t} = ;· t(y)U(:r- y) ely. 1'

F is the operator of the single layer potentiaL

double layer potential: The double layer operator J( yields a convolution with the normal derivative (with respect toy) of the fundamental so­Lution

}( {u} = f u(y) ,~0 U (:r.: - y) ely. • I' Ul/y

Its adjoint operator is obtained by taking the normal derivative at the point :r

K'{t} = { t(y)~i) . U(x-y)dy. Jr uiJ,,

hypersingular potential: The hypersingular operator H" results from a double differentiation

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150 LtPPl:.'i\;'DlX .-1. GLOSSL1RY

Due to its singularity, this integral cannot be dift"erentiated directly under t.he int.egral sign. Taking into aceoun! the differentiation of a eonvolut.ion product (sec above) we have

TV{u} = u(:r) * [-v, · v(l/y · vU(:r))].

where the support of "If is restricted t.o the boundar}'·

free term: Because of the sifting property of the Dirac-distribution, the Somigliana identity relluees one volume integral to a single value which is r:alled free ternL For the dired BE!\I, this single tenn takes into account the shape and the regulaxity of the boundary.

source and field point: In the case of the collocation BEvi, the boundary integral equations are solved for discrete source points of the boundary by setting the Dirac-distribution (the "source" for the fundamental solution) on these points. The integration is computed with respect to the field (observation) point.

singularities (2D): \Ve distinguish the following types of singularities:

weak singular strong singular hypcrsingular

JR2 O(ln !:rl) O(!:r!- 1) O(!:r!-2

)

JR:l O(!xl-1) O(!:rl-2) O(!:r!-:J)

\Veak singular functions can be integrated without regularization through a. coordinate transformat-ion and reg·ular Gaussian quadra­ture. Strong singularities require a Cauchy Principal Value regular­ization (direct. approach [ GG90]) or regularization by rigid body mo­tion technique (indirect approach [Bon95a]). For hypersingular terms, Hadamarcls concept of finite parts (direct. technique [Gui94]) or indirect methods ([GIVII90]) ean be used.

regularization: For the numerical treatment of singular integrals, a regu­larization procedure is required. The singularities of the DIE are can­celling in total, therefore it is advit;able to regroup the integral terrm;

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151

such that no non-int.egrable singularities occur. The perhaps simplest method for this is obtained by converting the free t.crm to a rigid mode integral (regularization by rigid motion). For a more detailed study, sec the review paper of Tanaka cl. al. [TSS94].

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Appendix B

Special Distributions

Definition B.l The Dirac-distribution Then-dimensional Dirac-dist.ribLLt.ion o(:z:) = rr~- 1 o(:r:k) is defined by

!. 8(:r) d:r . J;l~

o(x)

1.

0

:r E IR";

for all l:rl # 0.

(I3.1)

This means that. b"(x) is concentrated on a single point. x = 0 with infinite density 6(0) --t X· but. finite measure J 6 d:r = 1. It. is the identity object concerning convolution

u = u"' 6 = .h" u(y)6(1:- y) dy , :z: , y E IR" . (8.2)

and its Fourier transform is

6(:z;) 1, XE R". (13.3)

The Fourier t.ransfonn of the translctted Dirac-distribution 6 is even defined for wrnplex y E en, d [Cha87],

1Y(x - y) C- -i<y/i:>

' :z;, y E <C". (B.4)

All differentiatioru; can be expressed by a convolution \Vith 6" = i1"'6

u * (0''6) = .L. u(y)a" o(:r _ y) dy = a"u(:z;), :r,y E !Rn. (13.5)

153

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154 APPJ:;NDIX 13. SPE.'CHL DlSTlUHUTIONS

The product with another distribution u can be simplified by the following rclat.ion1

A:

u(.r.) D~6(.r- y) = 2:= ( k ~I ) (-1) 1D1u(y) D~-16(x- y) . (B.G) { _ ()

If a boundary· DO is described by 'U>(J:) = 0 with a function 'U' E c= then the integration of a distribution ·u along this boumlary can be described by the scalar product

( 13. 7)

The gradient. v~·~ of Lhc hypcrsurfac:c ~~~ = 0 should not vanish, i.e., V(b f::- 0 for ·~·:, = 0 and the outer unit normal of this hypersurface is

v'li> () = ___ '_ lvt.'·l· (B .8)

The chain rule is still valid, we get

(B.9)

Distributions can be represented by series which is often useful for practical computations. For the Dirac-distribution, there are many ways to construct. a sequence of functions or regular distributions <i>k{.r), k = 1, 2, .. . which converges to t.hc Dirac-distribution. The 0?k have t.o sat.isfy t.hc following two properties, cf. [GS64], vol.l, pp.34,

1. For any Af > 0 and for lal S Af and lbl S JVI, the quant.iLics

(B .lO)

musL be bounded by a eonsLanL independent of a, b, or k (in oLhcr words, depending only on Al ).

2. For any fixed nonva.nishing a and b, we must have

l' / ·& , (C) JC _ { 0 for a < b < 0 and 0 < a < b, A:_!!~ cpk " c " - 1 for u < 0 < b . . a

(ll.ll)

1 If u. is siu!iular at. the same point as 6, see the theory of uou-liuear dist.ribut.ions [Col84, Col85].

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155

One-dimensional examples for these sequences are, cf. [Kyt9G, LigGG, Boi9G],

l { ,.2 f (x'- ,·' ) - e-,. ,. ' ;:k 0.

k 1

11 1 + k 2:r'2'

k -k2 x 2 -e ft 1 sin2 b:

[;:r[ < E ,

[;r[ > E: (B.l2)

\Ve need these sequences for the multiplication of distributions, e.g. for H(:r)r5(:r) = r5(:r)/2 and the evaluation of the free terms in BIEs.

Definition B.2 The Heaviside-distribution The Hea.viside-distribution H(1:) is obtained by the integration of the Dira.c­

d istri bution

H(:r) = r o(y) ely= { 1 ... J -oo 0 , "

x>O ]: < 0.

(B. 1:3)

In the literature, there are several definitions for the value at :r = 0. It is in our context, i.e. for linear distributions, determined by ([Boi96])

(I3.14)

For the multi-dimensional Heaviside-distribution, we can define the cutoff distribution for a domain n c JR." by (compare Chapter 3)

x(x) •~ { ~(x) which can be expressed by

.r E 0

.r E iJO

.r ~ C2 = n u an. (B.15)

(I3.16)

with a function t'J E C:xo(JR.") as defined for (B.7). The intcgrat.ion of a dis­t.ribut.ion a over t.hc domain II can be described by

(II(4''), u) = { u(x) d:r. J ~·?_0

(I3.17)

The value K" ( x) on the boundary DO is uniquely defined, a~; shown in the following example.

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156 APPJ:;NDIX 13. SPE.'CHL DlSTlUHUTIONS

a) b)

Figure J3.1: 1:-'xamples for the edge and corner terms 1r(x), :r E 811.

Example B.l The edge value K

The value K(:r) of the cutoff distribution x of the semicircle

x (x) = II(:rl)II(l- X~- :r~). (D.l8)

can be determined by exploiting the sifting property of the Dirac-distribution

K(:r) = ;· x(y)S(:r- y) (ly, • E 2

:1: E an . (D.l9)

This is for the origin x1 = :~: 2 = 0 (Fig.B.la)

t.:(O, 0) = .l2

II(y!)II(l - :yf - y~)S(yt)r5(y2) d:y1 dy2

L2 H(yl)r5(,1/l)J(y2) dyl dy2.

The int,roduct,ion of a Dirac-sequence yields

k2 1 ., ., ., 1 rc (O, 0) = lim - . HUll) e- k-(v;+v:;) dy1 dy2 = :-·

k-+ ·X> 7r :R' 2

This is the standard factor r;(:r) = 1/2 for smooth parts of a boundary2

(D.20)

For the corner at x1 = 0, :~·2 = 1 (Fig.B.lb), x(u) is approximated loea.lly by !he tangential cutoff distribution H(yi)H(l - /)2). The value K(O, 1) is !hen obtained from

2 This value can be found iu t.he literature of non-linea.r geueralized functions [Boi96]. p. 48, example 2.32, a.nd [St.e95], p.47, example 2.

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157

which is precisely the value t.:(:r) known for a rectangular corner. For arbitrary angles () (Fig. I3. 2), we get in ~2

( ) k2 /' ( ) ( (} ) k"(y" I y") l B(:r) 1"1.· x = lim - II y2 II y1 tan - y2 e 1 · 2 ( y = --. k--+oo 7r , Jt2 271

(13.21)

IJ = (0, -lf

Figure B.2: Comer· part of the cutojj' distribution for arbitrary angles ()

The gradient. of the cutoff disLribuLion leads to a definition of the normal vect.or of the boundary even for non-smoot.h an.

Example B.2 The gradient of the cutoff distribution The normal vector u = - \hp = 2(xL :r2)T of a unit circle is obtained fron1

V'x = V'H(l - x1- .r.~) = J(l - .r7 - :r~) ( =~:~ ) ° C- ~:~~~ :r~) 21~1 ( =~;~~ ) .

For non-smooth boundaries, the cutoff distribution x is defined by a product (or a sum of products) of Hcavlslr:le-dlstributlons Tik H('t!JA} Its gradient V'x leads Lo a normal vector I/ defined as for smool h boundaries.

Example B.3 The normal vector for a rectangular corner The normal vector u for a rectangular corner in ~~ (Fig. B.:)) is defined by

the gradient. of the cutoff distribution x = H(xl)H(::r2 ) :

n ( r5(:rl) II(:r2 ) ) '( )H( ) ( 1 ) -( )H( ) ( 0 ) v X = Il(xdJ(xi) = u X1 :1"2 O + 6 :1·2 X1 1 .

Thus, the normal vectors on the edges are 11 = ( - 1, 0 yr for ::~· 1 = 0 and I/ = (0, -1)1' for :r2 = 0. l\ ote that for Lhe first dlffercntlat.ion, there is no special t.erm at. t.he corner x 1 = :r~ = 0.

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LiS APPENDIX B. SPECIAL DISTR.IBUTWNS

1/ = (-l,O)T I/ = (0, -l)T

Figure 8.3: Cv.tojj' distribution of rectangular' corner and corre.sponding nor­mal vectors 11 defined by V' x

Example B.4 The normal vector for corners with arbitrary angles An arbitrary angle () in R2 is described by t.hr cutoff distribution X =

H(J:2)H(:~: 1 tan()- :r2 ). The gradient. leads t.o

Thus, the normal vector is I/ = (0, -l)T for the edge :1:2 = 0 and I/

(-tanH, 1) for ::r2 = ::~· 1 tan H.

Example B.5 The gradient of x for the semicircle The gradient of the semicircle (d. Example B.l) is

Vx V {H(:rl)H(l- x7 - :d)

( 6(x1)H(l - .d -. ~~) + H.(.T1)6.(1- xi. - .r~)( -2.rl) ) .

H(:r!)o( l - :rf- ::~:D( - 2xz)

The normal veet.or is hence 1/ = ( -1, 0)1 for t.he straight. part of the boundary :r 1 = 0 and I/ = (- 2:~: 1, - 2:r2)T for the curved part.

For differential operators of higher degree (higher than two), additional derivatives of the cutoff' distribution x are required. They lead to additional corner terms which is shown in the next example.

Example B.6 Higher derivatives of the cutoff distribution The second derivatives of t.hr rectangular corner of Example 8.3 with X = H (J: 1 )H (:~:2 ) arc

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159

Thus, we get for the mixed derivative a12x particular corner terms b(x1 )6(:z;2).

For an arbitrary angle wit.h x: = H(:r2)H(:r 1 tan(} - :r.2), we get. as second differentiations

Dux H(x2)6'(:r1 tan (J- :r2 ) tan2 e, D12X 6(:r2)6(x1 tan (J- :r2) tane- H(x2)8'(:r1 tan (J- :1·2) tane,

D2L\ Dl2X, 022X 6'(:r2)H(:r1 t.an g- :r2)- 26 (:rz)<)(:rl tanO- :r:z) +

+ H(:r2)1\' (:r 1 t.an g- :r2) .

Once more, we obtained additional corner terms.

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Appendix C

Integration of BEM matrices

C .1 Analytical integrations

C.l.l Singular integrals by Fourier transform

A Large number of the one- and t\vo-dimensional integrations can be com­puted analytically by means of the symbolic computation program IvlAPLE [\Vat97]. For this, some additional tools were developed based on the rou­t.ine for t.he analytical Fourier transformation. The main problem with this was the treatment of the singularities. The integration tools implemented in .\lAPLE arc in general not able to handle strong and hypcrsingular integrals. The Fourier transform is based on distribution theory and therefore by far better suited for singular integrals.

The n-dirnensional integrations in Fourier space of the kernels .fare computed by successively applying the following formula for ik? k = 1 ... n

r J(i-~;:) d:ik = .! J¥;1

C.1.2 Additional regular Fourier pairs

In the .\lAPLE version used for this thesis, t.hc analytical Fourier t.ransforrn is ba:,;ed on a table of predefined pairs of Fourier trant;form. Because they are in general not sufficient for the integrations required here, additional pairs have to be defined which can be easily done by the :'lifAPLE routine addtable.

161

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162 J.PPJ::NlJlX C. 1NT£.'01UlT10N OF 13£M MATIUC£.'5

For example, the Fourier transform of the exponential function is added to the Fourier table with

:F H

The complex Dirac-distribution is a ult.ra.distribution. which is not. imple­mented originally. Similarly, the following pairs arc added (sec Lighthill [Lig66] for the justification) :

me: N.

csgn is t.he complex signum function.

C.1.3 Additional singular Fourier pairs

Light hill gives as transform of sgn :rk/x~:', with an integer m > 0 the following result :

sgn :t'k :F ' ( -ii:k)m-1 (1 I . I ""') -- H -2 n Xk + C .

:1:/_'' (rn - 1)! (C.2)

The constant C occurs because the functions sgn :rk/:z:'k' arc determined only up to an arbitrary multiple of the Dirac-distribution J(m l) at the singular point :rk = 0, see Lighthill [Lig66], p.57. This constant C has to be defined for each fundamental solution which is shown for the exarnple of the Poisson equation in the following.

Example C.l.l Poisson equation The integmtion of the Fourier fundam.ental solution U should yield the value of the fundamental solution U(O) at :r = 0 in the original :;pace. Hre have for· the Laplacian in ~2 :

' r - 1 L -- ~ 2 ~ 2 ·

:I:l + :I:2

The flr·.'it integm.tion O'IHT :1:2 = - C)O ... :J0 yiddii

1 ;· 1 , 1 S)Sn :h - (2r.F 1<:1 if+ i:~ d:r-2 = - :zr. 2i:l .

llaiied on (C.2) with m = 1, we get for· the irdegrntion with respect to :/; 1 the following divcrye11t e:rprcs,o;ion

-- ~ di-1 = -- Lim (ln l:r1l +C) . 1 h. sgnf 1

211 ' JRl 2J"l 2r. ,., 40

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C.l. ilN/I.LYTlC/I.L L\Tt;GRATlONS 163

The known fundamental solution in the original space is already given in equatiott ( 2. 5)

The limit process :r~,; -+ 0, k = 1, 2 yields

Thus it can be .seen that the two constant.s C and Co are the sa·rne. It is well known that the constant Co in the original space can be chosen arbitra­rily. it e1;presses additional homogeneous solutions which can be added to the fundamental solution without changing the solution. In genem.l it is chosen to be zem C = C0 = 0. lienee we have e8tabli8hed the following Fourier pair for the lvfAPLE table

The table entr·ies fm· rn > 1 an; obtained in a ;;irnilo.r- uwnnr.or (Tab. C.1.1 ). For the Frmr·ier pendant to e"-"'k sgn :rkj1·k, c1 E C the table Tab. C.l.l can be

original space :F ++ Fourier space

sgn :z:~,;jxk 4 - 2(ln [:z:k I) sgn T 2 jx2 L • • 'k ' A:

:F ++ 2iik(ln [ik[- 1)

sgnxUx~ :F ++ 2.rU2!(ln[ikl- 3/2)

sgnxkjxt, :F ++ -2ii:l,/3!(ln hi- 11/ 6)

c I ~ sgn x'j, x A. :F ++ -2it./4!(ln [:i\[- 25/12)

Table C.l: Fourier pairs for sgn :z: ~J x'['

·used by substitut ing :'rk by xk + ic1 . ]\lute that then lnfi:.-[ should be replaced by the complex logarithmic function

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164 J.PPJ::NlJlX C. 1NT£.'01UlT10N OF 13£M MATIUC£.'5

C.2 Numerical integrations

As an example for the numerical integrations, the computation of the fol­lowing matrix emries for the stationary heat conduction equation are inves­tigated (see (f_J.:3)

All integrals have inherited their singularity frorn the fundamental solution (r. \Ve compute first the off-diagonal entries by subtracting from them the diagonal term fl.ii = Hii - ;\ji HJ.i for all j -1- i

).Ji = L~/L~ is the rat.io between the length oft.he i-th element and the Length of element j. These integral are regular and can be computed numerically.

The diagonal kernels are simpler than the off-diagonal kernels because they are not composed by a trial ami test function which have different orientation. They are computed analytically as shown in the previous section. The final matrices are obtained by

Page 161: Fourier BEM: Generalization of Boundary Element Methods by Fourier Transform

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