Copyright c 2006 Tech Sciece Press CMES, vol12, o2, pp109-119, 2006 Aalysis of Circular Torsio ar with Circular Holes Usig Null-field Approach Jeg-Tzog Che 1, We-Cheg She 2, Po-Yua Che 2 Abstract: I this paper, we derive the ull-field itegral equatio for a circular bar weakeed by circular cavities with arbitrary radii ad positios uder torque To fully capture the circular geometries, separate forms of fudametal solutio i the polar coordiate ad Fourier series for boudary desities are adopted The solutio is formulated i a maer of a semi-aalytical form sice error purely attributes to the trucatio of Fourier series Torsio problems are revisited to demostrate the validity of our method Torsioal rigidities for differet umber of holes are also discussed keyword: Torsio, Null-field itegral equatio, Fourier series, Circular hole, Torsioal rigidity 1 Itroductio oudary value problems always ivolve several holes or more tha oe importat referece poit It is coveiet to be able to expad the solutios i alterative ways, each way referrig to differet specific coordiate set describig the same solutio Accordig to the idea, we develop a systematic approach icludig the adaptive observer system ad degeerate kerel for fudametal solutio i the polar coordiate ad employ Fourier series to approximate the boudary data I the past, multiply coected problems have bee carried out either by coformal mappig or by special techiques Lig [Lig C (1947)] solved the torsio problem of a circular tube with several holes Muskhelishvili [Muskhelishvili N I (1953)] solved the problem of a circular bar reiforced by a eccetric circular iclusio Che ad Weg [Che T; Weg I S (2001)] have itroduced coformal mappig with a Lauret series expasio to aalyze the Sait-Veat torsio problem They cocered with a eccetric bar of differet ma- 1 Distiguished Professor, Departmet of Harbor ad iver Egieerig, Natioal Taiwa Ocea Uiversity, Keelig, Taiwa Email: jtche@mailtouedutw 2 Graduate studet, Departmet of Harbor ad iver Egieerig, Natioal Taiwa Ocea Uiversity, Keelig, Taiwa terials with a imperfect iterface uder torque ased o the CVEM (complex variable boudary elemet method), Shams-Ahmadi ad Chou [Shams-Ahmadi M; Chou S I (1997)] have ivestigated the torsio problem of composite shafts with ay umber of iclusios of differet materials ecetly, Ag ad Kag [Ag W T; Kag I (2000)] developed a geeral formulatio for solvig the secod-order elliptic partial differetial equatio for a multiply-coected regio i a differet versio of CVEM To avoid mesh geeratio for fiite elemet or boudary elemet, meshless formulatio is a promisig directio [Ji (2004), Sladek V; Sladek J; Taaka M (2005), Wordelma C J; Aluru N ; avaioli U (2000)] The preset formulatio ca be see as oe kid of meshless methods, sice it belogs to boudary collocatio methods ecause the coformal mappig is limited to the doubly coected regio, a icreasig umber of researchers have paid more attetios o special solutios However, the extesio to multiple circular holes may ecouter difficulty It is ot trivial to develop a systematic method for solvig the torsio problems with several holes Crouch ad Mogilevskaya [Crouch S L; Mogilevskaya S G (2003)] utilized Somigliaa s formula ad Fourier series for elasticity problems with circular boudaries Mogilevskaya ad Crouch [Mogilevskaya S G; Crouch S L (2001)] have solved the problem of a ifiite plae cotaiig arbitrary umber of circular iclusios based o the complex sigular itegral equatio I their aalysis procedure, the ukow tractios are approximated by usig the complex Fourier series However, for calculatig a itegral over a circular boudary, they did t express the fudametal solutio usig the local polar coordiate y movig the ull-field poit to the boudary, the boudary itegral ca be easily determied usig series sums i our formulatio due to the itroductio of degeerate kerels Mogilevskaya ad Crouch [Mogilevskaya S G; Crouch S L (2001)] have used the Galerki method to approach boudary desity istead of collocatio approach Our approach ca be exteded to the Galerki formulatio oly for the circular ad a-
110 Copyright c 2006 Tech Sciece Press CMES, vol12, o2, pp109-119, 2006 ular cases However, it may ecouter difficulty for the eccetric case Two requiremets are eeded: degeerate kerel expasio must be available ad distictio of iterior ad exterior expressio must be separated Therefore, the collocatio agle of φ is ot i the rage 0 to 2π i our adaptive observer system This is the reaso why we ca ot formulate i terms of Galerki formulatio usig orthogoal properties twice Free of worryig how to choose the collocatio poits, uiform collocatio alog the circular boudary yields a well-posed matrix O the other had, ird ad Steele [ird M D; Steele C (1991)] have also used separated solutio procedure for bedig of circular plates with circular holes i a similar way of the Trefftz method ad additio theorem I this paper, the ull-field itegral equatio is utilized to solve the Sait-Veat torsio problem of a circular shaft weakeed by circular holes The mathematical formulatio is derived by usig degeerate kerels for fudametal solutio ad Fourier series for boudary desity i the ull-field itegral equatio The, it reduces to a liear algebraic equatio After determiig the ukow Fourier coefficiets, series solutios for the warpig fuctio ad torsioal rigidity are obtaied Numerical examples are give to show the validity ad efficiecy of our formulatio 2 Formulatio of the problem What is give i Figure 1 is a circular bar weakeed by N circular holes placed o a cocetric rig of radius b The radii of the outer circle ad the ier holes are ad a, respectively The circular bar twisted by couples applied at the eds is take ito cosideratio Followig the theory of Sait-Veat torsio [Timosheko S P; Goodier J N (1970)], we assume the displacemet field to be u = αyz, v = αxz, w = αϕ(x,y), (1) where α is the agle of twist per uit legth alog the z directio ad ϕ is the warpig fuctio Accordig to the displacemet field i Eq (1), the strai compoets are ε x = ε y = ε z = γ xy = 0, (2) γ xz = w x + u z = α( ϕ x y), γ yz = w y + v z = α( ϕ y +x), ad their correspodig compoets of stress are (3a) (3b) σ x = σ y = σ z = σ xy = 0, (4) σ xz = µα( ϕ x y), σ yz = µα( ϕ +x), (5) y where µ is the shear modulus There is o distortio i the plaes of cross sectios sice ε x = ε y = ε z = γ xy = 0 We have the state of pure shear at each poit defied by the stress compoets σ xz ad σ yz The warpig fuctio ϕ must satisfy the equilibrium equatio 2 ϕ x 2 + 2 ϕ = 0 id, (6) y2 where the body force is eglected ad D is the domai Sice there are o exteral forces o the cylidrical surface, we have t x = t y = t z = 0 y substitutig the ormal vector, the oly zero t z becomes t z = σ xz x +σ yz y = 0 o (7) Figure 1 : Cross sectio of bar weakeed by N (N = 3) equal circular holes y substitutig (5) ito (7) ad rearragig, the boudary coditio is ϕ x x + ϕ y y = y x x y = ϕ = ϕ o, (8)
Aalysis of Circular Torsio ar 111 where is the boudary I Figure 1, we itroduce the expressios for the positio vector (x k,y k ) of the boudary poit o the kth circular hole x k = acosθ k +bcos( 2πk ), k = 1, 2,, N, N 0 < θ k < 2π, (9) y k = asiθ k +bsi( 2πk ), k = 1, 2,, N, N 0 < θ k < 2π, (10) ad the uit outward ormal vector =( x, y )= ( cosθ, siθ) for the ier circular boudaries, we have ϕ = bcos(2πk N )siθ k bsi( 2πk N )cosθ k o k, (11) where k (k = 1, 2,, N) is the kth boudary of the ier hole, θ k is the polar agle with respect to the origi of the kth hole For the outer boudary, the tractio-free coditio is specified Thus, the problem of torsio is reduced to fid the warpig fuctio ϕ which satisfies Laplace equatio of Eq (6) ad the Neuma boudary coditios of Eq (11) for the ier boudary ad zero tractio o the outer boudary 3 Method of solutio 31 The dual boudary itegral equatios ad ullfield itegral equatios We apply the Fourier series expasios to approximate the potetial u ad its ormal derivative o the boudary u(s k )=a k 0 + =1 (a k cosθ k +b k siθ k ), s k k, k = 1, 2,, N, (12) t(s k )=p k 0 + =1 (p k cosθ k +q k siθ k ), s k k, k = 1, 2,, N, (13) where t(s k )= u(s k )/ s, a k, b k, p k ad q k ( = 0, 1, 2, ) are the Fourier coefficiets ad θ k is the polar agle The itegral equatio for the domai poit ca be derived from the third Gree s idetity [Che, J T; Hog, H -K (1999)], we have 2πu(x)= T (s, x)u(s)d(s) U(s, x)t(s)d(s), x D, (14) 2π u(x) = M(s, x)u(s)d(s) L(s, x)t(s)d(s), x x D, (15) where s ad x are the source ad field poits, respectively, D is the domai of iterest, s ad x deote the outward ormal vectors at the source poit s ad field poit x, respectively, ad the kerel fuctio U(s, x)=lr, (r x s ), is the fudametal solutio which satisfies 2 U(s,x)=2πδ(x s), (16) i which δ(x s) deotes the Dirac-delta fuctio The other kerel fuctios, T (s,x), L(s,x) ad M(s,x), are defied by T (s,x) U(s,x), L(s,x) U(s,x), s x M(s,x) 2 U(s,x) s x, (17) y collocatig x outside the domai (x D c ), we obtai the dual ull-field itegral equatios as show below 0 = T (s, x)u(s)d(s) U(s, x)t(s)d(s), x D c, (18) 0 = M(s, x)u(s)d(s) L(s, x)t(s)d(s), x D c, (19) where D c is the complemetary domai ased o the separable property, the kerel fuctio U(s,x) ca be expaded ito degeerate form by separatig the source poits ad field poits i the polar coordiate [Che J T; Chiu Y P (2002)]: U i (,θ;ρ,φ)=l 1 m ( ρ )m cosm(θ φ), ρ U(s,x)= U e (20) (,θ;ρ,φ)=lρ 1 m ( ρ )m cosm(θ φ), ρ >
112 Copyright c 2006 Tech Sciece Press CMES, vol12, o2, pp109-119, 2006 where the superscripts i ad e deote the iterior ( > ρ) ad exterior (ρ > ) cases, respectively I Eq (20), the origi of the observer system for the degeerate kerel is (0,0) for simplicity It is oted that degeerate kerel for the fudametal solutio is equivalet to the additio theorem which was similarly used by ird ad Steele [ird M D; Steele C (1992)] Figure 2 shows the graph of separate expressios of fudametal solutios where source poit s located at = 100, θ = π/3 are show below L(s,x)= L i (,θ;ρ,φ)= ( ρm 1 )cosm(θ φ), > ρ m L e (,θ;ρ,φ)= 1 ρ + ( m )cosm(θ φ), ρ > ρ m+1, (22) M(s,x)= M i (,θ;ρ,φ)= ( mρm 1 )cosm(θ φ), ρ m+1 M e (,θ;ρ,φ)= ( mm 1 )cosm(θ φ), ρ > ρ m+1 (23) Sice the potetial resulted from T (s,x) ad L(s,x) kerels are discotiuous across the boudary, the potetials of T (s,x) for ρ + ad ρ are differet This is the reaso why = ρ is ot icluded i expressios of degeerate kerels for T (s,x) ad L(s,x) i Eqs (21) ad (22) Figure 2 : Graph of the separate form of fudametal solutio (s =(10,π/3)) y settig the origi at o for the observer system, a circle with radius from the origi o to the source poit s is plotted If the field poit x is situated iside the circular regio, the degeerate kerel belogs to the iterior case U i ; otherwise, it is the exterior case After takig the ormal derivative with respect to Eq (20), the T (s,x) kerel ca be derived as T (s,x)= T i (,θ;ρ,φ)= 1 + ( ρm )cosm(θ φ), > ρ m+1 T e (,θ;ρ,φ)= ( m 1 ρ )cosm(θ φ), ρ > m, (21) ad the higher-order kerel fuctios, L(s,x) ad M(s,x), 32 Adaptive observer system After movig the poit of Eq (18) to the boudary, the boudary itegrals through all the circular cotours are required Sice the boudary itegral equatios are frame idifferet, ie objectivity rule, the observer system is adaptively to locate the origi at the ceter of circle i the boudary itegral Adaptive observer system is chose to fully employ the property of degeerate kerels Figures 3 ad 4 show the boudary itegratio for the circular boudaries i the adaptive observer system It is oted that the origi of the observer system is located o the ceter of the correspodig circle uder itegratio to etirely utilize the geometry of circular boudary for the expasio of degeerate kerels ad boudary desities The dummy variable i the circular itegratio is agle (θ) istead of radial coordiate () 33 Liear algebraic system y movig the ull-field poit x k to the kth circular boudary i the sese of limit for Eq (18) i Fig 3,
Aalysis of Circular Torsio ar 113 If the domai is ubouded, the outer boudary 0 is a ull set ad N C = N It is oted that the itegratio path is couterclockwise for the outer circle Otherwise, it is clockwise For the k itegral of the circular boudary, the kerels of U(s,x) ad T (s,x) are respectively expressed i terms of degeerate kerels of Eqs (20) ad (21), ad u(s) ad t(s) are substituted by usig the Fourier series of Eqs (12) ad (13), respectively I the k itegral, we set the origi of the observer system to collocate at the ceter c k to fully utilize the degeerate kerels ad Fourier series y collocatig the ull-field poit o the boudary, a liear algebraic system is obtaied Figure 3 : Sketch of the ull-field itegral equatio i cojuctio with the adaptive observer system Figure 4 : Sketch of the boudary itegral equatio for domai poit icojuctio with the adaptive observer system we have N C 0 = T (s k,x j )u(s k )d k (s) k=1 k N C U(s k,x j )t(s k )d k (s), k=1 k x D c, (24) where N C is the umber of circles icludig the outer boudary ad the ier circular holes I the real computatio, we select the collocatio poit o the boudary [U]{t} =[T]{u}, (25) where [U] ad [T] are the ifluece matrices with a dimesio of N C (2M +1) by N C (2M +1), {u} ad {t} deote the colum vectors of Fourier coefficiets with a dimesio of N C (2M + 1) by1iwhich[u], [T], {u} ad {t} ca be defied as follows: U 00 U 01 U 0N U 10 U 11 U 1N [U]=, U N0 U N1 U NN T 00 T 01 T 0N T 10 T 11 T 1N [T]=, (26) T N0 T N1 T NN {u} = u 0 u 1 u 2 u N, {t} = t 0 t 1 t 2 t N, (27) where the vectors {u k } ad {t k } are i the { form of a k 0 a k 1 b k 1 a k } Tad M bk M { p k 0 p k 1 q k 1 p k } T, M qk M respectively; the first subscript j (j = 0, 1, 2,, N) i [ ] [ ] U jk ad T jk deotes the idex of the jth circle where the collocatio poit is located ad the secod subscript k (k = 0, 1, 2,, N) deotes the idex of the kth circle where boudary data {u k } or {t k } are specified, N is the umber of circular holes i the domai ad
114 Copyright c 2006 Tech Sciece Press CMES, vol12, o2, pp109-119, 2006 Preset method Table 1 : Comparisos of the preset method ad covetioal EM oudary desity discretizatio Auxiliary system Formulatio Observer system Sigularity Fourier series Degeerate Null-field Adaptive observer No pricipal value kerel itegral system equatio Covetioal EM Costat elemet Fudametal solutio oudary itegral equatio Fixed observer system Pricipal value (CPV, PV ad HPV) Table 2 : Torsioal rigidity of circular cylider with a eccetric hole (a/ = 1/3) Table 3 : Torsioal rigidity of a circular cylider with a rig of N holes (a/ = 1/4,b/ = 1/2) M idicates the trucated terms of Fourier series The ifluece coefficiet matrix of the liear algebraic system is partitioed ito blocks, ad each off-diagoal block correspods to the ifluece matrices betwee two differet circular holes The diagoal blocks are the ifluece matrices due to itself i each idividual
Aalysis of Circular Torsio ar 115 hole After uiformly collocatig the poit alog the kth circular boudary, the submatrix ca be writte as [ ] U jk = [ ] T jk = U 0c U 0c U 0c U 0c jk (φ 2M) U 1c U 1c U 1c U 1c jk (φ 2M) U 1s U 1s U 1s U 0c U 1c U Mc U Ms U Mc U Ms U Mc U Ms U Mc jk (φ 2M) U Ms U Mc T 0c T 0c T 0c Tjk 0c(φ 2M) T 0c T 1c T 1c T 1c Tjk 1c(φ 2M) T 1c U 1s jk (φ 2M) U 1s jk (φ 2M) U Ms T 1s T 1s T 1s Tjk Mc(φ 1) Tjk Ms(φ 1) Tjk Mc(φ 2) Tjk Ms(φ 2) Tjk Mc(φ 3) Tjk Ms(φ 3) Tjk Mc(φ 2M) Tjk Ms(φ 2M) Tjk 1s(φ 2M) Tjk 1s(φ 2M+1) T Mc T Ms, (28) (29) Although the matrices i Eqs (28) ad (29) are ot sparse, they are diagoally domiat It is foud that the ifluece coefficiet for the higher-order harmoics is smaller It is oted that the superscript 0s ieqs (28) [ ] ad (29) [ disappears ] sice siθ = 0 The elemet of U jk ad T jk are defied respectively as U c jk (φ m)= U(s k,x m ) cos(θ k ) k dθ k, k = 0, 1, 2,, M, m = 1, 2,, 2M +1, (30) U s jk (φ m)= U(s k,x m ) si(θ k ) k dθ k, k = 1, 2,, M, m = 1, 2,, 2M +1, (31) Tjk s(φ m)= T (s k,x m ) cos(θ k ) k dθ k, k = 0, 1, 2,, M, m = 1, 2,, 2M +1, (32) Tjk s(φ m)= T (s k,x m ) si(θ k ) k dθ k, k = 1, 2,, M, m = 1, 2,, 2M +1, (33) where k is o sum ad φ m is the polar agle of the collocatig poits x m alog the boudary The explicit forms of all the boudary itegrals for U,T,L ad M kerels are listed i the Appedix esides, the limitig case across the boudary ( < ρ < + ) is also addressed The cotiuous ad jump behavior across the boudary is also described y rearragig the kow ad ukow sets, the ukow Fourier coefficiets are determied Equatio (18) ca be calculated by employig the relatios of trigoometric fuctio ad the orthogoal property i the real computatio Oly the fiite M terms are used i the summatio of Eqs (12) ad (13) After obtaiig the ukow Fourier coefficiets, the origi of observer system is set to c k i the k itegratio as show i Fig 4 to obtai the iterior potetial by employig Eq (14) The differeces betwee the preset formulatio ad the covetioal EM are listed i Table 1 4 Illustrative examples ad discussios I this sectio, we deal with the torsio problems which have bee solved by Caulk i 1983 [Caulk D A (1983)] The cotours of the axial displacemet are plotted i three cases The torsioal rigidity of each example is calculated after determiig the ukow Fourier coefficiets Case 1: A circular bar with a eccetric hole A circularbar ofradius with a eccetric circular holes removed is uder torque T at the ed The torsioal rigidity G of cross sectio ca be expressed by G µ D = r 2 dd N k=1 k ϕ ϕ d k, (34) The results of torsioal rigidity for each case are show i Table 2 The exact solutio derived by Muskhelishvili is listed i Table 2 for compariso Our solutio is better tha that of Caulk obtaied by IE whe the hole is closely spaced
116 Copyright c 2006 Tech Sciece Press CMES, vol12, o2, pp109-119, 2006 2 15 1 05 0-05 -1-15 -2 2 15 1 05 0-05 -1-15 -2 Figure 5 : Axial displacemet for the circular bar weakeed by two holes 2 Figure 6 : Caulk s data (Solid lies idicate results from the first-order solutio ad dashed lies from the umerical solutio of the exact boudary itegral equatio) [Caulk D A (1983)] 15 1 05 0-05 -1-15 -2-2 -15-1 -05 0 05 1 15 2 Figure 7 : Axial displacemet for the circular bar weakeed by three holes Figure 8 : Caulk s data (Solid lies idicate results from the first-order solutio ad dashed lies from the umerical solutio of the exact boudary itegral equatio) [Caulk D A (1983)] Case 2: A circular bar with two circular holes A circular bar of radius b with two equal circular holes removed is uder torque T at the ed The boudary curve of kth ier cavity is described by usig the parametric form of (x k,y k ) i Eqs (9) ad (10) What is brought out is the problem subject to zero tractio o the outer boudary ad Neuma boudary coditio defied i Eq (11) o all the ier circles Figures 5 ad 6 show the results usig the preset method ad those from the first-order approximatio solutio ad the exact boudary itegral equatio derived by Caulk [Caulk D A (1983)] Twety-oe collocatig poits are selected o all the cir-
Aalysis of Circular Torsio ar 117 2 15 1 05 0-05 -1-15 -2-2 -15-1 -05 0 05 1 15 2 Figure 9 : Axial displacemet for the circular bar weakeed by four holes holes of equal radii is regarded as the third example I a similar way, the cotour plot of the axial displacemet is show i Figure 7 Good agreemet is made after comparig with the Caulk s data i Figure 8 Case 4: A circular bar with four circular holes The fourth problem is a circular bar weakeed by four equal circular holes uder torque I Figure 9, our result of axial displacemet agrees well with the values i the dashed lie of Figure 10 which are solved by usig the boudary itegral equatio esults obtaied by usig the preset method for Case 2, Case 3 ad Case 4 are listed i Table 3 After compariso, our results agree well with Caulk s data obtaied by IE formulatio Case 5: Lig s examples [Lig C (1947) ] Table 4 shows a compariso of the torsioal rigidities of three cases with differet geometries of circular holes computed from the preset method, IE formulatio [Caulk D A (1983)] ad first-order approach [Caulk D A (1983)] We have ot oly calculated the torsioal rigidity but also tested the rate of covergece of Fourier terms of the case with seve cavities as show i Fig 11 Test of Parseval s sum for boudary desities was also implemeted to esure the covergece The preset solutios are a improvemet over Lig s results i every case The large differece i the secod example i Table 4 may ascribe to Lig s legthy calculatio i error as poited out by Caulk [Caulk D A (1983)] 5 Coclusios Figure 10 : Caulk s data (Solid lies idicate results from the first-order solutio ad dashed lies from the umerical solutio of the exact boudary itegral equatio) [Caulk D A (1983)] cular boudaries i the umerical implemetatio After beig compared with the results of Figure 6, the umerical results are cosistet with those of the boudary itegral equatio Case 3: A circular bar with three circular holes Ulike Case 2, a circular bar weakeed by three circular The torsio problems of circular shaft weakeed by several holes have bee successfully solved by usig the preset formulatio Our solutios match well with the exact solutio ad other solutios by usig the boudary itegral equatio for the three Caulk s cases There are oly 41 collocatio poits uiformly distributed o each boudary for more accurate results of torsioal rigidity with error less tha 1 % after comparig with the kow exact solutio egardless of the umber of circles, the proposed method has great accuracy ad geerality Through the solutio for several problems, our method was successfully applied to cases of multiple holes Furthermore, our method preseted here ca be used to problems which satisfy the Laplace operator
118 Copyright c 2006 Tech Sciece Press CMES, vol12, o2, pp109-119, 2006 Table 4 : Torsioal rigidity of Lig s [Lig C (1947) ] examples torsioal rigidity 0729 0728 0727 0726 0725 0724 0 10 20 30 40 Fourier terms Figure 11 : Torsio rigidity versus the umber of Fourier terms Ackowledgemet: Fiacial support from the Natioal Sciece Coucil uder Grat No NSC91-2211- E-019-009 for Taiwa Ocea Uiversity is gratefully ackowledged efereces Ag W T; Kag I (2000): A complex variable boudary elemet method for elliptic partial differetial equatios i a multiply-coected regio, Iteratioal Joural of Computer Mathematics, 75, 515-525 ird M D; Steele C (1992): A solutio procedure for Laplace s equatio o multiplecoected circular domais, ASME Joural of Applied Mechaics, 59, 398-404 ird M D; Steele C (1991): Separated solutio procedure for bedig of circular plates with circular holes, ASME Joural of Applied Mechaics, 59, 398-404 Caulk D A (1983): Aalysis of elastic torsio i a bar with circular holes by a special boudary itegral method, ASME Joural of Applied Mechaics, 50, 101-108 Che J T; Chiu Y P (2002): O the pseudodifferetial operators i the dual boudary itegral equatios usig degeerate kerels ad circulats, Egieerig Aalysis with oudary Elemets, 26, 41-53 Che J T; Hog H K (1999): eview of dual boudary elemet methods with emphasis o hypersigular itegrals ad diverget series, ASME Applied Mechaics eviews, 52, pp 17-33 Che T; Weg I S (2001): Torsio of a circular compoud bar with imperfect iterface, Joural of Applied Mechaics, ASME, 68, 955-958 Crouch S L; Mogilevskaya S G (2003): O the use of Somigliaa s formula ad Fourier series for elasticity problems with circular boudaries, Iteratioal Joural for Numerical Methods i Egieerig, 58, 537-578 Ji (2004): A meshless method for the Laplace ad biharmoic equatios subjected to oisy boudary data, CMES: Computer Modelig i Egieerig & Scieces, 6No3, 253-262 Lig C (1947): Torsio of a circular tube with logitudial circular holes, Quarterly of Applied Mathematics, 5, 168-181 Mogilevskaya S G; Crouch S L (2001): A Galerki boudary itegral method for multiple circular elastic iclusios, Iteratioal Joural for Numerical Methods
Aalysis of Circular Torsio ar 119 i Egieerig, 52, 1069-1106 Muskhelishvili N I (1953): Some basic problems of the mathematical theory of elasticity, Noordhoff, Groige Shams-Ahmadi M; Chou S I (1997): Complex variable boudary elemet method for torsio of composite shafts, Iteratioal Joural for Numerical Methods i Egieerig, 40, 1165-1179 Sladek V; Sladek J; Taaka M (2005): Local itegral equatios ad two meshless polyomial iterpolatios with applicatio to potetial problems i ohomogeeous media CMES: Computer Modelig i Egieerig & Scieces, 7No1, 69-84 Timosheko S P; Goodier J N (1970): Theory of Elasticity, McGraw-Hill,NewYork Wordelma C J; Aluru N ; avaioli U (2000): A meshless method for the umerical solutio of the 2- ad 3-D semicoductor Poisso equatio, CMES: Computer Modelig i Egieerig & Scieces, 1No1, 121-126 Appedix A: Appedix Aalytical evaluatio of the itegrals ad their limits The degeerate kerels are described i Eqs (20), (21), (22) ad (23), ad orthogoal process is show below:
120 Copyright c 2006 Tech Sciece Press CMES, vol12, o2, pp109-119, 2006 Orthogoal process Limit ρ Orthogoal process Limit ρ Orthogoal process Limit ρ Orthogoal process Limit ρ U(s,x) ad U (s,x)t (s)d(s) 2π 0 U i cos(θ)dθ = π 1 ρ cos(φ), ρ 1 2π 0 U i si(θ)dθ = π 1 ρ si(φ), ρ 1 2π 0 U e cos(θ)dθ = π 1 +1 ρ cos(φ), < ρ 2π 0 U e si(θ)dθ = π 1 +1 ρ si(φ), < ρ π 1 ρ cos(φ)=π 1 1 cos(φ), ρ π 1 ρ si(φ) = π 1 1 si(φ), ρ π 1 +1 ρ cos(φ) = π 1 cos(φ), < ρ (Cotiuous for < ρ < + ) π 1 +1 ρ si(φ) = π 1 si(φ), < ρ T (s,x) ad T (s,x)u(s)d(s) 2π 0 T i cos(θ)dθ = π ( ρ ) cos(φ), > ρ 2π 0 T i si(θ)dθ = π ( ρ ) si(φ), > ρ ) 2π 0 T e cos(θ)dθ = π( ρ cos(φ), < ρ ) 2π 0 T e si(θ)dθ = π( ρ si(φ), < ρ π ( ρ ) cos(φ) = πcos(φ), > ρ π ( ρ ) ) si(φ) = πsi(φ), > ρ π( ρ cos(φ) =-πcos(φ), < ρ (jump for < ρ < + ) ) π( ρ si(φ) =-πsi(φ), < ρ L(s,x) ad L(s,x)t (s)d(s) 2π 0 L i cos(θ)dθ = π ( ρ 1 ) cos(φ), > ρ 2π 0 L i si(θ)dθ = π ( ρ 1 ) si(φ), > ρ ) +1 2π 0 L e cos(θ)dθ = π( ρ cos(φ), < ρ ) +1 2π 0 L e si(θ)dθ = π( ρ si(φ), < ρ π ( ρ 1 ) cos(φ) =-πcos(φ), > ρ π ( ρ 1 ) si(φ) = πsi(φ), > ρ ) +1 π( (jump for < ρ < + ) ρ cos(φ) = πcos(φ), < ρ ) +1 si(φ) = πsi(φ), < ρ π( ρ M(s,x) ad M (s,x)u(s)d(s) 2π 0 M i cos(θ)dθ = π ρ 1 cos(φ), ρ 2π 0 M i si(θ)dθ = π ρ 1 si(φ), ρ 2π 0 M e cos(θ)dθ = π cos(φ), < ρ ρ +1 2π 0 M e si(θ)dθ = π si(φ), < ρ ρ +1 π ρ 1 cos(φ) = π 1 cos(φ), ρ π ρ 1 si(φ) = π 1 si(φ), ρ π cos(φ) = π 1 ρ +1 cos(φ), < ρ (Cotiuous for < ρ < + ) π si(φ) = π 1 ρ +1 cos(φ), < ρ