The hypersingular boundary element method revisited

Transcription

1 Boudary Elemets ad Other Mesh Reducto Methods XXXVII 7 The hypersgular boudary elemet method revsted N. A. Dumot Cvl Egeerg Departmet, Potfcal Catholc Uversty of Ro de Jaero, Brazl Abstract Ths s the sequel of a paper preseted at the BEM/MRM Coferece four years ago, whch the covetoal, collocato boudary elemet method was reformulated by proposg a smple, however cosstet dervato o the bass of the weghted-resduals statemet. It was show that the sgle-layer potetal matrx G should be geeral rectagular ad satsfy some spectral propertes (orthogoalty to the space of ubalaced boudary tracto forces) the same way as the double-layer potetal matrx H s orthogoal to rgd-body dsplacemets, whe modellg a fte elastc body. Moreover, a subtle mprovemet was proposed for the terpolato of tracto forces, the case of curved boudares, whch was meat to just smplfy the umercal mplemetato. I the preset paper, t s cocluded that the proposed mprovemet s fact a ecessary oe f strct cosstecy of the formulato s requred ad more emphatcally f a cosstet hypersgular formulato s to be mplemeted. It s also show that the correct hypersgular formulato requres that the dscotuous parts (two free terms) of the matrx H be obtaed depedetly from the matrx G. Motvato of the preset developmets was the applcato of the hybrd boudary elemet method to stra gradet elastcty, whch oly maes use of the matrx H together wth ts hypersgular couterpart. Although ths paper s of a rather theoretcal ature, a smple umercal example s show to llustrate the ecessty of the proposed mprovemets. Keywords: boudary elemets, weghted-resdual methods, hypersgular formulato, curved elemets. ISSN X (o-le) WIT Trasactos o Modellg ad Smulato, Vol 57, 014 WIT Press do:10.495/be370031

2 8 Boudary Elemets ad Other Mesh Reducto Methods XXXVII 1 Itroducto The hypersgular formulato of the collocato boudary elemet method seems to have bee cosoldated almost two decades ago wth the semal wors doe maly by Gugga ad some co-worers (as revewed Referece [1]) ad Matc ad Pars [], as well as tag to accout the coceptual remars made by Muherjee [3], to meto just oe complemetary cotrbuto. Sce the, several umercal mplemetatos reported the lterature apparetly attest the approprateess of the proposed developmets. However, there stll are some coceptual ssues to be addressed as a result of the theoretcal qures carred out startg from varatoal prcples ad cosstecy checs of the classcal, collocato boudary elemet method [4 6]. Motvato of the preset developmets s a varatoally-based applcato of the boudary elemet method the stra gradet elastcty [7, 8], from whch t turs out that the terpolato fucto proposed Referece [5] for the evaluato of the sgle-layer potetal matrx s ot just a subtle mprovemet of the boudary elemet method, but a decsve requremet whe dealg wth hypersgularty ad curved boudares. Ths s partcularly ecessary f cosstecy of the formulato regardg rgdbody rotato s to be checed, a two-dmesoal (D) mplemetato (sce a soparametrc 3D mplemetato ca oly be proved fully cosstet for a restrcted class of boudary elemets). The most mportat coceptual cotrbuto of the preset paper s probably the demostrato that the hypersgularty features of the double-layer potetal matrx should ad fact ca be dealt wth depedetly from the correspodg term of the sgle-layer potetal matrx. Ths sheds lght o some htherto elusve f ot uduly gored features of ths formulato. 1.1 Problem formulato A smple, cosstet dervato of the BEM was preseted Referece [5] ad shall ot be repeated here. However, t should be restated that oe s dealg wth the statc aalyss of a elastc body submtted to tracto forces t o part of the boudary ad to body forces b the doma. Dsplacemets u are ow o the complemetary part u of. Oe s loog for a adequate approxmato of the stress feld that satsfes equlbrum the doma, b 0, (1) j j also satsfyg the boudary equlbrum ad compatblty equatos, t alog, j j u u o u, () where j s the outward ut ormal to. Idces, j, (also, l) may assume values 1, or 3, as they refer to the coordate drectos x, y or z, respectvely, for a geeral 3D aalyss. Sum s dcated by repeated dces. Partcularzato to D aalyss as well as to potetal problems s straghtforward. WIT Trasactos o Modellg ad Smulato, Vol 57, 014 WIT Press ISSN X (o-le)

3 Boudary Elemets ad Other Mesh Reducto Methods XXXVII 9 1. The fudametal soluto The covetoal, collocato boudary elemet method may be derved for fudametal solutos ad gve as j u p, (3) m u ( u u C ) p, (4) j m r r r where u, for s 1, are rgd-body dsplacemets that are multpled by s prcple arbtrary costats C sm, ad p m are arbtrary (vrtual) force parameters, wth m characterzg both locato ad drecto of applcato of p m. The, ad u jm m are fuctos wth global support of the coordates ad drectos of p m referred to by m (the source pot), as well as of the coordates ad drectos referred to by (the feld pot), where the effects of are measured. p m 1.3 Numercal boudary dscretzato of dsplacemets ad forces The dsplacemets u ad the tracto forces t are the problem s uows alog ad u, respectvely. They are approxmated alog as jm r s sm m u ud, (5) t tt J J at ut, (6) d where d, for 1, s a vector of d odal dsplacemets ad u (, ) u are terpolato fuctos wth local support, usually pecewse polyomals chose as fuctos of the parametrc varables (, ), for 3D problems, such a way that, at the odal pots,. Sce the tracto forces t are surface attrbutes, the t parameters t are also surface attrbutes that deped o the outward ormal of the boudary pot at whch t s physcally t d attached. Geerally,, as the boudary may ot be etrely smooth, wth more tha oe ormal at some pots. The terpolato fuctos t t (, ) also have local support ad should be cosstetly defed as eq (7), where u are gve pecewse as the same polyomals u for the dsplacemets of eq (6), J s the Jacoba of the coordates trasformato alog the boudary segmet ad J at s the value of J at the odal pot, so that t smlarly as for the dsplacemets. u WIT Trasactos o Modellg ad Smulato, Vol 57, 014 WIT Press ISSN X (o-le)

4 30 Boudary Elemets ad Other Mesh Reducto Methods XXXVII 1.4 Isoparametrc formulato ad a frst cosstecy chec The boudary geometry s approxmated from the odal attrbutes usg the same terpolato fuctos u of eq (10), whch cossts a soparametrc represetato of the problem. Ths s expressed as x u x, (7) where x are odal Cartesa coordates ad x x(, ) are the Cartesa coordates evaluated for the parametrc varables (, ). For a geerally curved boudary segmet expressed accordg to eq (7), a geeral lear dsplacemet feld gve odally as d eq (5) s exactly reproduced alog (, ). Ths the l bass of a C 0 formulato of the fte elemet method a hgher order dsplacemet feld ca be oly approxmately represeted. For a D problem, the outward ut ormal to a boudary segmet s gve terms of a sgle parametrc varable by x 1 y, wth J x y y J x. (8) Sce, from eq (7), x u x u x, eq (8) may also be wrtte dcal otato terms of the odal values of the outward ut ormal as where For a 3D problem, J J u at (D problems). (9) x y, z, y, z, 1 y x, z, x, z,, (10) J x y x y z,,,,,,,,,,,,,,,, J y z y z x z x z x y x y. (11) Sce eq (10) volves products of dervatves, t s obtaed stead of eq (9) J J u at (3D problems, geeral). (1) It may be checed that ths latter equato s oly exactly satsfed for lear (threeode) ad quadratc (sx-ode) tragles as well as for the lear (four-ode) quadrlateral elemet. As a result, a costat stress feld ca be exactly reproduced accordg to eq (6) for ay curved D boudary elemet compare wth (9) whereas ths s oly codtoally true the 3D case. WIT Trasactos o Modellg ad Smulato, Vol 57, 014 WIT Press ISSN X (o-le)

5 Boudary Elemets ad Other Mesh Reducto Methods XXXVII Basc equato of the collocato boudary elemet method The collocato boudary elemet method, as gve the lterature, s stated as jmu j d m d umt dt umbd ext, (13) where, for brevty ad sce t s of o cocer for the preset developmets, a error term that multples the costats C sm of eq (4) s dropped, as compared wth the cosstet dervato of Refereces [4, 5]. Equato (1) may be wrtte matrx format as Hd Gt b, (14) d d where H H m R s a ematc trasformato matrx, G d s a flexblty-type matrx ad b R G m R d t b m s a vector of odal dsplacemets equvalet to the appled body forces. The product p T Hd has the meag of vrtual wor, where p pm are the vrtual forces troduced eqs (3) ad (4) [5]. The frst motvato for troducg the defto of t as eq (6) was the smplfcato obtaed the evaluato of G, for curved boudares, sce the Jacoba of d cacels out. It ca be ow see, accordg to the prevous Secto, that, for a D problem wth a sotropc materal, eq (13) holds, as stated, exactly for a geeral lear dsplacemet feld, whch s ot the case of the htherto proposed boudary elemet mplemetatos [9]. For a 3D problem, eq (6) provdes oly codtoally a exact statemet for a lear dsplacemet feld, but s evertheless a smpler ad more accurate formulato. The sgle-layer potetal matrx G s geeral rectagular, as stated eq (13), sce the tracto forces deped o the outward ormal to the boudary. The fact that G s rectagular should ot hder the correct proposto of a umercal model ad the ultmate evaluato of all relevat quattes of a mechacal problem [9]. The use of dfferet tracto forces at a boudary corer leads to locally more accurate results tha as terms of the equvalet odal forces of the fte elemet method. The double-layer ad sgle-layer potetal matrces H ad G comprse ther defto sgular ad mproper tegrals, respectvely, whe source (dex m) ad feld (dex ether or ) refer to the same odal pots. The, specal care must be tae the umercal tegratos. Equato (13) follows the most commo developmets of the lterature, whch the boudary ext ecloses the sgularty, or source, pot characterzed by the dex m. Three mathematcally equvalet forms of the matrx H H are [6] m H u d u d m jm j jm j m ext u d u d fp jm j jm j ds, (15) WIT Trasactos o Modellg ad Smulato, Vol 57, 014 WIT Press ISSN X (o-le)

6 3 Boudary Elemets ad Other Mesh Reducto Methods XXXVII usg as the boudary that leaves the source pot outsde the doma (meag that the fudametal soluto s actually aalytcal ), as llustrated o the left of Fgure 1 for the D case. The expresso used eq (13) s the secod oe both eq (15) ad the Fgure, wth the tegrato ultmately carred out, as requred a drect umercal evaluato, for the boudary splt to a fte part ad a dscotuous part, as llustrated o the rght of both eq (15) ad Fgure 1. The procedures for the umercal evaluato of the fte ad dscotuous parts of the tegral eq (15), for ether D or 3D problems, s well documeted the lterature ad shall be ot repeated here. Fgure 1: Illustrato of the three mathematcally equvalet forms of eq (15) for a bouded doma [5]. Hypersgular mplemetato of the matrx H.1 Prelmary developmets The complete theoretcal developmet of the hypersgular formulato of the collocato boudary elemet method was prmarly formulated by Gugga ad co-worers, as revewed Referece [1], who also credts valuable depedet achevemets to Matc ad Pars []. Ther le of reasog, whch tertwes the matrces H ad G the evaluato of boudary dsplacemet dervatves the Cartesa coordate drectos, shall ot be revewed at preset. The relevat fact to be brought to lght s that, the varatoal, hybrd boudary elemet formulato, use s made oly of the matrx H, ad a dfferet le of reasog s ecessary whe dealg wth problems of gradet elastcty, for stace [8]. The followg developmet s proposed for the frst tme, to the author s best owledge. Although ot show before, let y explctly characterze the Cartesa coordates of the source pot m (to whch the ut pot forces p of eqs (3) ad (4) are appled) ad x be the feld pot, where the effect of m p m s evaluated ad alog whch tegrato s ultmately carred out o the boudary eq (15) ths seems to be a commo otato the lterature, as gve by Gugga [1], for stace, although Muherjee [3] uses exactly the oppose. The frst expresso of the double-layer potetal matrx H eq (15) s derved pror to the dscretzato of the boudary dsplacemets u as eq (5) wth respect to the ormal drecto η of the boudary pot at whch the source s appled: WIT Trasactos o Modellg ad Smulato, Vol 57, 014 WIT Press ISSN X (o-le)

7 Boudary Elemets ad Other Mesh Reducto Methods XXXVII 33 ud ( y, x) ( x) ( y) u ( x)d ( x) jm, j jm, j fp dsc ( y, x) ( x) ( y) u ( x)d ( x) jm, j ( y, x) ( x) ( y) u ( x)d ( x ). jm, j (16) (The Gree letter has already bee used to characterze oe of the boudary parametrc varables for a 3D mplemetato, as eqs (10) ad (11). However, ths latter use,, s dexed, whch helps to eep varables apart.) The left-had sde of the above equato uses the compact otato of the paper, whereas the explct depedece o source ad feld varables s show o the rght. The tegral s also show as splt to a fte ad a dscotuous part, followg the otato of eq (15). I alluso to Gugga s developmets, ad correspod exactly to lm 0 ad e fp dsc lm 0, respectvely, also usg the s followg a sphere of radus to characterze the splt domas llustrated Fgure 1. Gugga [1] does ot seem to le resourcg to a fte-part tegral, whereas Muherjee [3] maes some ot so clear dstcto betwee a (Hadamard) fte-part tegral ad a Cauchy prcpal value. However, as proposed Referece [10], the fte part of a hypersgular tegral ca be evaluated etrely o mathematcal terms ad depedetly from the cocept of a Cauchy prcpal value. Next, let u ( x ) eq (16) be expaded about u y ( 0) as a seres of : du y( 0) u ( ) ( 0) ( ) ( x u y O u y q y O ). (17) d I ths equato, y y ( 0) s explctly wrtte to emphasze that u y as well as t dervatves are sgle-valued fuctos. The ormal dervatve qy duy d s troduced to smplfy otato, but subject to posteror terpretato. Addg ad subtractg terms to the tegrals of eq (16), t results ud ( yx, ) ( x) ( y) u ( x)d ( x) jm, j jm, j fp jm, dsc j ( y) ( y) jm, ( y, x) j( x)d ( x) dsc ( y) jm, ( y, x) j ( x) ( y) d ( x ). dsc ( yx, ) ( x) ( y) u ( x) u ( y) q y d ( x) u (18) q The uderled tegral terms are the same oes of eq (16). The terms bracets the secod row correspod to the expaso of the dsplacemet u ( x ) show eq (17). It s justfed to place u y ad ts dervatve, as sgle-valued fuctos at y y ( 0), outsde the tegrals of the thrd ad fourth rows. Sce the sgularty of the product 1, ( yx, )d ( x ) s of order O( ) for ether a D jm WIT Trasactos o Modellg ad Smulato, Vol 57, 014 WIT Press ISSN X (o-le)

8 34 Boudary Elemets ad Other Mesh Reducto Methods XXXVII or a 3D problem ad the terms bracets are of order O( ), accordg to eq (17), the whole tegral the secod row of the above equato vashes whe 0. The dcated fte-part tegral the frst row of ths equato meas that oly the boudary dsplacemet u ( x ) s to be terpolated, the umercal mplemetato stage, accordg to eq (5). The tegrals of the thrd ad fourth rows of ths equato are dealt wth separately the followg Sectos. However, t s worth already otcg the subtle dfferece allocatg ( y ) ether outsde or sde the tegrals. Ths wll be elucdated later o the frame of a mechacal terpretato. It ca be advaced that, the thrd row, the odal dsplacemet u y s depedet from the local boudary geometry gve by dsc, ad ( y ) just deotes the drecto of a fxed, appled vrtual double force. O the other had, q ( y ) the fourth row deotes a dsplacemet gradet that s ormal to the boudary, characterzed by ( y ), whch must be cosdered as varyg alog dsc.. Evaluato of the tegral the fourth row of eq (18) for a D problem I the followg, t s show how to evaluate the term the fourth row of eq (18) for the D problem, wth results that ca be extrapolated to the 3D case, as carred out the expaded verso of the preset paper. I matrx format, jm, j (1 ) r r r r 1 0 xy xy 4 (1 ) r J r r c cs r xc xsyc 4 4 (1 ) r cs s xsyc ys (19) wth the otato x r c ad y r s, for smplcty. The term jm, j d eq (18) has the meag of gradet double dsc tractos that perform vrtual gradet wor o the gradet dsplacemets q ( y ), whch are by defto located at the source pot y. The, must be alged wth q, that s, pot outward alog the crcular excluso boudary. Ths justfes * wrtg jm, j pm, polar coordates for tegrato over a arch of crcle of arbtrarly small radus, cetered o the sgularty pot ad spag from to, terms of the boudary coordate that substtutes for. Sce d d, r r 1, r 0, x x cos c, y y s s, the expresso of j, j becomes, polar coordates, WIT Trasactos o Modellg ad Smulato, Vol 57, 014 WIT Press ISSN X (o-le)

9 Boudary Elemets ad Other Mesh Reducto Methods XXXVII 35 (1 ) 1 0 c cs, jm j dsc 4 (1 ) 0 1, (0) 4 (1 ) cs s ad t follows that a correspodg matrx of free terms C * m pm, d C m s expressed as cs c (1 ) c cs, (1) whch s bouded regardless of the value of. As t turs out, the tegral the fourth row of eq (18) leads to the same matrx of free terms as the case of the classcal elastcty a result that ca be extrapolated for the 3D problem..3 Evaluato of the tegral the thrd row of eq (18) for a D problem The tegrad the thrd row of eq (18) s also evaluated polar coordates for tegrato over a arch of crcle of arbtrarly small radus, cetered o the sgularty pot ad spag from to, terms of the boudary coordate that substtutes for, accordg to the prevous Secto, although eepg ( y ) depedet from. As proposed depedetly by Gugga [1] ad Matc ad Pars [] (although the subject s clearer ad smpler Gugga s papers), the tegral ( ) lm f b d lm a, () 0 0 where f ( ) s bouded the tegrato terval ad ot supposed to vash as 0, ca be formulated as the dcated sum of two terms. The frst of them, b, teds to fty wth 0 but must cacel wth a smlarly behavg term the tegral o the rght the frst row of eq (18), such that, ud jm j be ultmately bouded. I fact, the tegral o the rght the frst row of eq (18) s evaluated terms of fte parts, whch meas that ts ubouded part s dsregarded oly because of ts cacellg wth b. The, a s the oly term of actual terest eq (). Its evaluato s carred out by expadg a geerc tegrato lmt about 0 a Taylor seres alog the boudary, expressed as d ( ) (0) O( ). (3) d 0 Sce these developmets do ot preset ay coceptual dfferece to Gugga's proposto, oe may sp further detals ad summarze that, usg d xy yx d 0, (4) 3 0 J 0 WIT Trasactos o Modellg ad Smulato, Vol 57, 014 WIT Press ISSN X (o-le)

10 36 Boudary Elemets ad Other Mesh Reducto Methods XXXVII where s the sged curvature of the plae curve ( ), the evaluato of the fte-part tegral (that s, wthout the fte term lmb ) the thrd row of 0 eq (18) ca be expressed as the matrx a m of free terms: where fm ( ) fm ( ) am ( y) fp jm, ( y, x) j ( x)d ( x ), (5) dsc (1 ) fm( ) x scy cx sy 4 (1 ) (6) c cs xc xsyc 3cx sy 4 (1 ) cs s xsyc ys. As observed by Gugga [1], f the boudary s straght, the 0 ; f the boudary s curved, but smooth at the sgularty pot, the, ad f ( ) f ( ). I ether case, a 0. m 3 Hypersgular mplemetato of the matrx G m The sgle-layer potetal matrx G, preseted eqs (13) ad (14), has ts ormal gradet expresso pror to the dscretzato of the boudary tracto forces as eq (6) gve prcple by the fte ad dscotuous parts u td u ( yx, ) ( y) t ( x)d ( x) ( y, x) ( y) ( x)d ( x) ( y, x) ( y) ( x)d ( x ). (7) m, m, um, t um, fp dsc t m Ths s htherto a developmet smlar to the oe of eq (16), for the double-layer potetal matrx H. However, whereas the dsplacemet u ( x ) the defto of H s a sgle-valued fucto that ca be expaded as eq (17), the tracto forces t ( x ) are surface attrbutes that deped o the outward ormal ( x ) to ( x ) ad, the umercal dscretzato to be adopted, dscotuously defed from boudary segmet to boudary segmet. Ths justfes smply ascrbg t ( x ) 0 alog dsc, (8) wth the result that oly the fte part of eq (7) s to be evaluated, although t 1 volves a r or r sgularty for D or 3D problems, respectvely. WIT Trasactos o Modellg ad Smulato, Vol 57, 014 WIT Press ISSN X (o-le)

11 Boudary Elemets ad Other Mesh Reducto Methods XXXVII 37 4 Coclusos As outled, the double-layer potetal matrx G s rectagular, a cosstet umercal mplemetato, whch also maes use of a mproved represetato of the tracto forces alog the boudary, as surface attrbutes tmately related to the boudary outward ormal through the Jacoba J. The proposed hypersgular developmets treat the sgle-layer ad doublelayer potetal matrces G ad H depedetly from each other. Ths s proposed as coceptually superor to the developmets the techcal lterature. The ecessty of such a proposto comes from the varatoal, hybrd boudary elemet method, whch does ot maes use of the matrx G, ad thus eeds to deal wth the matrx H depedetly. Moreover, developmets the stra gradet theory of elastcty have led to the preset developmets. To be emphatc, developmets other tha the preset oes do ot lead to verfably cosstet results gradet elastcty [8]. The sgular ad hypersgular mplemetatos of the collocato boudary elemet method ca be compactly represeted as fp dsc H H 0 d d d G ut b u fp dsc dsc fp t, (9) R R R H H H d d q R R q Gu t bu where the frst row of equatos s eq (14) ad the matrx H s represeted by ts fte ad dscotuous parts. The secod row of equatos shows o the left-had sde H fp correspodg to the term o the rght the frst row of eq (18), dsc R H d R d correspodg to the term o the thrd row of eq (18) ad developed dsc Secto.3, ad H correspodg to the term o the fourth row of eq (18) ad R q developed Secto.. The ormal dsplacemet gradets q were troduced eq (17) ad have the same boudary dstrbuto ad oretato of the tracto fp forces t. The matrx G R correspods to the fte part represetato of the u t gradet of G gve eq (7), eepg md, accordg to eq (8), that ts dscotuous couterpart s vod. The smplest cocevable umercal verfcato of eq (9) s proposed for the example of Fgure, whch smulates a sotropc, elastc body whose strogly curved boudary s gve by two quadratc elemets [8]. It s checed that the expaded matrx H of eq (9) s orthogoal to rgd body traslatos ad rotatos. Ths equato s also exactly checed wth the umercal capacty of represetato for a appled lear dsplacemet feld, to whch dsplacemet gradets q ad tracto forces t preset a strog varato alog the curved boudares. Ths d of umercal assessmet s more covcg that carryg out a covergece aalyss that may just masquerade local errors related to the evaluato of H ad G for a large umber of degrees of freedom. WIT Trasactos o Modellg ad Smulato, Vol 57, 014 WIT Press ISSN X (o-le)

12 38 Boudary Elemets ad Other Mesh Reducto Methods XXXVII Fgure : Very smple example wth oly two quadratc curved elemets [8]. Acowledgemet Ths wor was supported by the Brazla ageces CAPES, CNPq ad FAPERJ. Refereces [1] Gugga, M., Formulato ad Numercal Treatmet of Boudary Itegral Equatos wth Hypersgular Kerels, eds. V. Slade ad J. Slade, Sgular Itegrals Boudary Elemet Methods, Chapter 3, , Computatoal Mechacs Publcatos, Southampto, [] Matc, V. & Pars, F., Exstece ad evaluato of the two free terms the hypersgular boudary tegral equatos, Egeerg Aalyss wth Boudary Elemets, 16, 53-60, [3] Muherjee, S., CPV ad HPF tegrals ad ther applcatos the boudary elemet method, Iteratoal Joural of Solds ad Structures, 37, , 000. [4] Dumot, N.A., A Assessmet of the Spectral Propertes of the Matrx G used the Boudary Elemet Methods, Computatoal Mechacs (1), 3-41, [5] Dumot, N.A., The boudary elemet method revsted, Boudary Elemets ad Other Mesh Reducto Methods XXXII, ed. C.A. Brebba, 7-38, WIT Press, Southampto, 010. [6] Dumot, N.A., Assessmet of the spectral propertes of the double-layer potetal matrx H, Boudary Elemets ad Other Mesh Reducto Methods XXXV, eds. C.A. Brebba ad A.H-D. Cheg, 09-19, WIT Trasactos o Modellg ad Smulato, vol 54, WIT Press, Southampto, 013. WIT Trasactos o Modellg ad Smulato, Vol 57, 014 WIT Press ISSN X (o-le)

13 Boudary Elemets ad Other Mesh Reducto Methods XXXVII 39 [7] Dumot, N.A. & Huamá, D., A varatoal boudary elemet approach for stra gradet elastcty, ICSECM 013 The 4th Iteratoal Coferece o Structural Egeerg ad Costructo Maagemet 013, Proceedgs of the Specal Sesso o Advaced Materals, vol 01, 75-89, Kady, Sr Laa, 013. [8] Dumot, N.A. & Huamá, D.: A hybrd varatoal formulato for stra gradet elastcty Part II: boudary elemet mplemetato, submtted CMES Computer Modelg Egeerg & Sceces, 014. [9] Brebba, C.A., Telles, J.C.F. & Wrobel, L.C., Boudary Elemet Techques, Sprger-Verlag: Berl ad New Yor, [10] Dumot, N.A., Smplfed assessmet ad evaluato procedure of ftepart hypersgular tegrals, accepted BETeq Advaces Boudary Elemet Techques XV, Florece, Italy, 014. WIT Trasactos o Modellg ad Smulato, Vol 57, 014 WIT Press ISSN X (o-le)