Tutorial 4: FUNDAMENTAL SOLUTIONS: I-SIMPLE AND COMPOUND OPERATORS

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1 Boary Elemet Commicatios 00 Ttorial 4: FNDAMENTAL SOLTIONS: I-SIMPLE AND COMPOND OPERATORS YOSSEF F. RASHED Dept. o Strctral Egieerig Cairo iversity iza Egypt yosse@eg.c.e.eg Smmary a objectives I the ttorial we presete other eamples o the erivatio o the boary itegral eqatio i the irect orm. Maily elasticity a plate i beig problems were iscsse. I this ttorial we will iscss the eiitios a the methos o erivatio o ametal soltios. The se o sch soltio withi the boary elemet metho was iscsse i the ormer ttorial. A table presets the commoly se orms o ametal soltio is give. Also a metho base o simple aalogy to the algebraic partial ractio is iscsse to ecompose compo ieretial operators. I the et ttorial we will cotie iscssig how to set p the ametal soltios or comple matri operators. Deiitios The ametal soltio ca be eie i the most simple way as the respose e to it sorce i a iiite problem. For eample whe we say ij is the ametal soltio or isplacemets i elasticity problems that meas: ij is the isplacemet at poit i the irectio j e to it poit loa applie at i the i irectio. It ca be see that ij is a erel betwee two-poits. From Betti reciprocal theory it is clear that ij = ji. Mathematically the ametal soltio o a problem is the soltio o the goverig ieretial eqatio whe the Dirac elta is actig as a orcig term appears o the right ha sie []. It has to be ote that o boary coitios is orce to simlate the iiite atre o the problem. I other wors it is the particlar soltio o the problem correspoig to the Dirac elta istribtio. Provie that the Dirac elta posses the siglar atre the ametal soltio is also siglar. The ame ametal came rom the act that it is the soltio o the most ametal problem i mechaics which eals with a it sorce i a iiite boy. It is calle also the pricipal soltio; the siglar soltio or the ree-space ree s ctio. From this eiitios the ametal soltio ca be eie as ollows: L = δ where L is a scalar ieretial operator a δ is the Pal Dirac elta i which is the sorce poit a the is a iel poit. I L is a matri operator eqatio ca be re-writte as ollows:

2 L * ij j = δ δi It has to be ote that o boary coitios are eorce i both eqatios a. I this ttorial we will iscss the erivatio o the ametal soltio i eqatio whereas the erivatio o ametal soltios or matri operators as that o eqatio will be iscsse i the et ttorial. sel properties o the Pal Dirac elta The ollowig properties o the Pal Dirac elta col be se i the erivatio o the ametal soltio []: whe = δ = 0 whe lim 0 δ = δ F = F 5 i which is a arbitrary omai. For simplicity it will be chose as either a circle or sphere or two- a three-imesioal problems respectively []. It has to be ote that we alreay have mae se o the thir property eqatio 5 i the ormer two ttorials. I this ttorial the seco property eqatio 4 will help i the steps o the ametal soltio erivatio. Methos o erivatio Derivatio o ametal soltios is a legthy tas or iiclt operators. However i may cases this col be a systematic procere. The geeral techiqe or erivig the ametal soltio is to se itegral trasorms sch as Forier Laplace or Hael trasorms []. Sch a techiqe ivolves complicate mathematics a has very sophisticate proceres. Thereore it will be covere i latter ttorial as a avace topic. I the et sectio we will emostrate the erivatio o the ametal soltio or well-ow simple operators sch as the Laplacia. The proceres will be escribe or both two- a three-imesioal problems. I sectio 5 a table will be presete to smmarize the ametal soltios or commoly se simple operators. The i sectio 6 a simple aalogy to the algebraic partial ractio techiqe will be se to ecompose compo operators to the simple orms presete i sectio 5. 4

3 4 Fametal soltios sig itegratio I this sectio we will emostrate a techiqe base o irect itegratio a maig se o the properties o the Dirac elta to costrct the ametal soltio. The startig poit o this techiqe is to solve the ollowig homogeeos eqatio: L = 0 6 This eqatio ca be solve sig ay simple techiqe i the calcls sch as irect itegratio i polar cooriate separatio o variables variatio o parameters or sig comple variable trasormatio or the case o two imesioal problems oly. Some costats will appear e to the itegratio proceres. I orer to obtai sch costats we ca mae se o the Dirac Delta property i eqatio 4 ater combiig it with eqatio to give: lim 0 L = As the omai ca be chose arbitrary oe ca preset it as a small circle sphere or the threeimesioal case o a rais ε or simplicity. The the limit i eqatio 7 will be perorme as ε 0. It has to be ote that the itegratio i eqatio 7 col be perorme easily by trasormig it to the boary o the circle sig the itegratio by parts proceres presete i ttorial. More etails abot this metho is presete by Rahma i Re. []. The ollowig eamples will emostrate that iea. 4. Laplace operator i two-imesio Cosier the Laplace eqatio i two-imesio: = δ 8 The irst step to costrct the ametal soltio is to solve the ollowig homogeeos eqatio: = 0 9 Or i polar cooriate it col be epresse as ollows: r = 0 0 r r r By itegratio oe ca obtai: = a b l r I orer to obtai the costats a a b i eqatio we will mae se o eqatio 7 to give: 7

4 lim 0 or = a bl r lim 0 = Applyig the itegratio by parts or ree s seco ietity it gives: a bl r lim = 4 ε 0 se the polar cooriate otatio where = rθ oe ca obtai: θ= π lim a blε = 5 ε 0 θ= 0 ε Notig that or a circle = the we ca obtai: a is a arbitrary costat a b =. The π ial orm o the ametal soltio ca be writte as: = a l r 6 π It has to be ote that i carryig ot the itegratio i the ormer eample the ollowig alterative trasormatio col be se []: z = i a z = i 7 a it is easy to prove that: z z = a z z = 8 i 4. Laplace operator i the three-imesio Similar to the two-imesioal case Eqatio 9 ca be re-writte as: r r r = 0 9 r By itegratio oe ca obtai:

5 b = a 0 r By satisyig the coitio i eqatio 7 oe ca obtai: lim ε 0 γ= π γ= 0 θ= π θ= 0 b a ε θγ = ε ε Notig that or a sphere = a = r θγ the we ca easy obtai: a is arbitrary costat a b =. The the ial orm o the ametal soltio ca be writte as: 4 π = a 4πr 5 Commo ametal soltio The most commoly ametal soltios are se as basics or may problems i comptatioal mechaics are presete i Table or oe- two- a three-imesioal case [4]. Table : Fametal soltios or most commoly se operators. Eqatio Oe-imesioal Two-imesioal Three-imesioal Laplace = δ = = l r = π 4πr Helmholtz iλr = si λ = H λr = e λ = δ λ 4i 4πr Moiie Helmholtz = si iλ = K0 r λ = δ λ π λ λr = e 4πr Bi-harmoic 4 = r l r = δ 8π Where H a K 0 are Hael a Bessel ctios respectively. 6 Partial ractio aalogy I the ormer sectio we have emostrate simple proceres to erive the ametal soltio or simple operators sch as the Laplacia. I this sectio we will preset a techiqe base o a aalogy to the algebraic partial ractio to ecompose compo operator to simple operators o

6 the orms give i Table. I orer to emostrate this metho we will cosier the ollowig eamples: 6. Eample : Cosier i we wat to costrct the ametal soltio o the ollowig compo operator: a b = δ From Table we ca obtai the ollowig ametal soltios a where: a = δ 4 a b = δ 5 I orer to obtai the ametal soltio i terms o a we try to mae a aalogy to the partial ractio theory as ollows: δ = a b δ = a b a b a b b a 6. Eample : δ a b 6 7 = 8 The ollowig eample will be cosiere to mae the iea more clear. Cosier that we wat to costrct the ametal soltio o the ollowig operator: a = δ 9 From Table we ca obtai ametal soltios a where: = δ 0 a a = δ

7 Followig the same aalogy o the previos eample oe ca write: = δ a δ δ = a a a a a = 4 As it ca be see that the partial ractio aalogy helps i ecomposig compo scalar operators to well-ow simple operators. 7 Coclsios I this ttorial we have emostrate simple proceres or erivig the ametal soltios or well ow a commoly se ieretial operators i comptatioal mechaics. We also covere a techiqe base o a aalogy to the simple cocept o algebraic partial ractios to obtai the ametal soltio or compo operators cosist o proct o the well-ow simple operators. I the et ttorial we will cover the se o Hörmaer metho to ecople comple operators. We will give may eamples to show the step-by-step erivatio o the ametal soltio erels. 8 Eercise: - sig the metho escribe i sectio 4 obtai the ametal soltio or the moiie Helmholtz eqatio. - se the partial ractio aalogy i the ametal soltio or the ollowig operator: 4 a = δ 5 9 Soltio o the eercise i ttorial I orer to erive the irect boary itegral eqatio or the shear-eormable plate beig restig o the two-parameter Pastera oatio moel the ollowig ietity ca be writte: or [ Q Q q K ] 0 Mβ β = 6 [ Q Q q ] K 0 Mβ β = 7 The irst itegral will lea to the same reslts as escribe i Ttorial. Now we will cosier the seco itegral:

8 K = K 8 = K The irst itegral o the right ha sie will remais as it is whereas the seco itegral ca be ecompose sig the itegratio by parts proceres ree s seco ietity twice i similar maer as we se beore or Laplace eqatio to give: = 9 The the ial itegral eqatio ca be writte as: P p i i i i β β = 0 Q Q M 40 A the ametal soltio cam be obtaie rom: β β δ = δ Q M 4 δ = δ Q 4 Maig se o the properties o the Dirac Delta oe ca write: i i i i P = i i i p 4 Which is the reqire boary itegral eqatio. For more etails o the erivatio the reaer ca coslt Re. [5].

9 Reereces a Frther Reaig [] Kythe P.K. Fametal Soltios or Dieretial Operators a Applicatios Birhäser Press Berli ermay 996. [] Rahma M. Mathematical Methos with Applicatios WIT Press Sothampto K Bosto SA 000. [] Katsiaelis J.T. The Aalysis o Plates o Elastic Foatio by the Boary Itegral Eqatio Metho Ph.D. Thesis Polytechic Istitte o New Yor 98. [4] Brebbia C.A. Fametals o Boary Elemets I New Developmets i Boary Elemet Methos Proceeig o the Iteratioal Semiar iversity o Sothampto CMP Press Sothampto K 980. [5] Rashe Y.F. Boary Elemets Formlatios or Thic Plates WIT Press Sothampto K 000.