Nonsmooth Optimization Algorithms in Some Problems of Fracture Dynamics

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1 Iellge Iformao Maageme, 2, 2, do:.4236/m Publshed Ole November 2 (hp:// Nosmooh Opmzao Algorhms Some Problems of Fraure Dyams Absra V. V. Zozulya Cero de Ivesgao Cefa de Yuaa A.C, Coloa Chuburá de Hdalgo, Yuaá, Méxo E-mal: zozulya@y.mx Reeved Oober 6, 29; revsed July, 2; aeped Sepember 5, 2 Mahemaal saeme of elasodyam oa problem for raed body wh osderg ulaeral resros ad fro of he ra faes s doe lassal ad wea forms. Dffere varaoal formulaos of ulaeral oa problems wh fro based o boudary varaoal prple are osdered. Nosmooh opmzao algorhms of Udzawa s ype for soluo of ulaeral oa problem wh fro have bee developed. Covergee of he proposed algorhms has bee suded umerally. Keywords: Ulaeral Coa, Fro, Cra, Varaoal Prples, oudary Varaoal Fuoal, Nosmooh Opmzao Algorhm. Iroduo The mahemaal formulao of he elasodyam problem for a raed body, ha aes o aou he possbly of ra edge oa erao ad he formao of areas wh lose oa, adheso ad sldg, was preseed frs []. The algorhm for he soluo of hs problem was elaboraed [2] ad s based o a heory of subdffereoal fuoals ad he fdg of her saddle pos. May examples of he ra faes oa erao ad fro fluee o he fraure mehas reros have bee osdered he boo by Guz ad Zozulya [3] ad revew papers [4-6]. I hese ases he oa area s a pror uow ad he ulaeral odos have o be mposed o he relave dsplaemes ad he muual raos. The ulaeral oa resro wh fro a be wre as a equaly for he dsplaeme ad rao veors. As a resul a omplee se of boudary odos a ra faes s wre as a sysem of equaos ad equales. The presee of equaly ype boudary odos mples he boudary problems o be olear, whh requres he vesgao of orrespodg boudary value problems. Mahemaal formulao of he problem of ra faes oa erao he dyam ase has bee doe [3-5,7]. Se he osras oer boudary varables oly, s aural o loo for a umeral soluo by meas of boudary egral equao (IE) mehod. Approah s based o use of fudameal soluos. I [8] IE formulao va eergy mehod s based o boudary m-max prple,.e., a prple expressed erms of he boudary uows. Frs me boudary varaoal formulao of elasodyam oa a problem wh frao was proposed [2] ad he was exeded ad appled o elasodyam problems for bodes wh ras wh osderg ulaeral froal oa of he ra faes Guz ad Zozulya [3,5]. There are may algorhms for ulaeral oa problems wh fro. eause of oleary of he problem mos of hem lude dsrezao ad erave proedure o sasfy ulaeral osras. Ierave algorhms of suh ype [9,] are amed Uzawa s ype algorhms. For he frs me Uzawa s ype algorhm was proposed for soluo elasodyam froal oa problem for body wh ra []. The he approah was developed ad more algorhms were proposed [2,-3]. The algorhms are based o saddle po fdg ad projeo o he se of ulaeral resros ad fro respevely. I [4] was show ha proposed algorhms are overge ad was sudes rae of overgee. Some mahemaal problems relaed exsee ad uqueess of he problem of ulaeral oa problem wh fro were suded [7,5,6]. More formao relaed o mahemaal for-

2 638 mulao ad umeral soluo of he ulaeral oa problems a foud [7,8]. The am of hs paper s o prese varaoal formulao of he elasodyam problem for body wh ra wh osderg possbly for ulaeral ra faes oa erao wh fro. Varaoal formulao of he problem, whh s based boudary varaoal prple s preseed. Nosmooh fuoals ha orrespod o ulareral froal oa odos are osrued. The ase of he ra fe elas meda s osdered more deals. I order o sudy overgee of he proposed algorhms, wo problems relaed o he ra faes oa erao uder ao of he harmo eso-ompresso ad shear waves have bee solved umerally usg IE mehod. 2. Classal Formulao of he Problem Le us osder dyamal loadg of ra a fe homogeeous, leally elas body. The ra s desrbed by a orrespodg oreed mddle surfae se we suppose ha oly small deformaos our. We assume ha dsplaemes of body pos ad her grades are small. I hs ase R \ ( 2,3) he dffereal equaos of equlbrum dsplaeme may be preseed he form 2 A u b u, x R \, j j [, ] () q ( x, ) F ( q ( x, )) The operaor A j for a sorop body has he form A ( ), (2) j j j ad are Lame osas, ad, j s a Kroeer s symbol, x deoes he paral dervaves wh respe o spae, deoes he paral dervaves wh respe o me. Throughou hs paper we use he Ese summao oveo. If he problem s defed o a fe rego, he he soluo of Equao () s uquely deermed by assgg dsplaemes ad veloy veors he al sa of me. The he al odos are u ( x, ) u ( x), u ( x, ) v ( x), x V (3) Addoal odos a he fy mus be sasfed u x O r 2, j O( r ) for r (, ) ( ) (, ) (l( )) 3-D ase (4) u x O r, j O( r ) for r 2-D ase (5) Here r s he dsae he 3-D ad 2-D Eulda spaes respevely. The dffereal operaor Pj : uj p s alled sress operaor. I rasforms he dsplaemes o he raos. For homogeeous sorop elas medum has he forms j j P (6) Here are ompoes of he ouward u ormal veor, s a dervave dreo of he veor x ( ) ormal o he surfae V. The oa fores qx (, ) q( x, ) e whh arse o he ras edges durg he erao are deoed by qx (, ) σ( x, ) ( x ) (7) σ( x, ) ( x, ) e e s he sra esor; j j x ( ) ( xe ), ( x) ( x) ( x ) ; ( x ) ad ( x ) are he ormal u veors dreed o he posve sde of he oppose ras edges. The dsplaeme dsouy veor haraerzes muual dsplaemes of he ras edges u ( x, ) u ( x, ) u, (8) u ad u (, ) x are dsplaemes of oppose ras edges. Furhermore, we mpose he followg Sgor osras u ho, q,( u ho) q, x (9) ad Coulomb s fro law: q q u, q q u q, x () wh λτ u( x) q for x ; he Coulomb s fro oeffe s here assumed o be osa. Here he ormal ad ageal ompoes of he dsplaeme dsouy o are deoed by u ( x, ) u ( x, ), u ( x, ) u( x, ) u ( x, ) ( x ), () ad he ormal ad ageal ompoes of he oa fores o are deoed by q( x, ) q( x, ) ( x ), q ( x, ) q( x, ) q ( x, ) ( x ). (2) Classal formulao of he elasodyam problem for body wh ra wh osderg oppose ra sdes erao osss soluo of he al boud-

3 639 ary value problem ()-(5) wh osderg Sgor oa odos (9) wh fro (). I lassal elasodyams he equaos of moo () ad al odos (2) mus be sasfed exaly (see [9]). Ths meas ha he ompoes of he dsplaeme veor should be fuos of he lass 2.2. l, C ( R ) C ( ). Here C ( R ) s a fuoal spae of fuos, wh smooh dervaves wh respe o he spae oordaes ad l smooh dervaves wh respe o he me. I order o sasfy all he equaos of elasodyams he lassal sese, he ompoes of he sress-sra sae should belog o he followg fuoal spaes 2,2, u C ( R ) C ( ), j, (3),, j C ( R ), p C ( ) These requremes of lassal elasodyams are very srge. Therefore may mpora physs ad egeerg problems, parular problems wh ulaeral resros ad fro, have o lassal soluo. For hs reaso s eessary o osder weaeed formulaos o elasodyam problems. Wh suh a approah s o eessary o fulfll all he elasodyams equaos he lassal sese. 3. Varaoal Formulao of he Problem whou Coa Codos I order o formulae a elasosa oa problem for body wh ra wee form we wll osder he boudary varaoal prple rodued [2,5,,2]. I [2,5] have bee show ha he ase f body wh ra ouped fe rego he boudary varaoal fuoal may be preseed he form ( up, ) u ( y, ) u ( x, ) F ( xy,, ) dsd 2 j j p ( y, ) u ( y, ) dsd (4) px (, ) p( x, ) e p( x, ) qx (, ), p s a veor of gve loadg appled o he ra edges, qx (, ) s a veor of oa fores,. The boudary varaoal fuoal (4) s smooh ad Gaeaux-dffereable, herefore he followg odo of fuoal mma ae plae ( ) u (5) ad he problem of fdg mma s equvale o he followg egral equao uj( x, ) Fj( x, y, ) dsd p ( y, ) (6) We a represe boudary varaoal fuoal (4) he form ( up, ) Fu, u p, u (7) 2 F s marx egral operaor defed (6). The varaoal formulao of a elasodyam problem for raed body whou ulaeral osras (7) ad fro (8) s as follows: Fd up, K ( up, ) suh ha (8) ( up, ) m { [ u, p]} u, p K ( u, p) K ( u, p) uh ph x /2, /2. { ( ), ( ), } 4. Nosmooh Fuoals for Ulaeral Coa Codos wh Fro (9) I order o formulae boudary odos form of equales (9) ad () wee form le us osder a maxmal moooe operaors : u p. For eah maxmal moooe operaor may be defed wh auray up o a osa ompoe ovex sem-ouous from below fuoal j suh, ha j. Here s deoed he subdffereal of he osmooh fuoal (see [8] for deals). 4.. Sgor oudary Codos Fuoal Spae Le /2, u H ( ) ad q /2, H ( ) sasfy followg odos u, h q. q,( u h ), Here, deoes he dualy parg bewee he fuoal spaes H /2, ( ) ad H /2, ( ). The orrespodg fuoal has he form, f u h ( u) (2), oherwse The ojugae fuoal has he form, f q ( q) (2), oherwse 4.2. oudary Codos wh Coulomb Fro Les /2, 2 u ( H ( )) ad q ( H ( )) /2, 2

4 64 sasfy followg odos f q q he u, f q q he u q ad also ( q q ), u. Here, deoes he dualy parg bewee he fuoal spaes /2, 2 /2, 2 ( H ( )) ad ( H ( )). The orrespodg fuoal has he form ( u ) q, u (22) The ojugae fuoal has he form, f q q ( q ) (23), oherwse 4.3. Sgor oudary Codos wh Fro These boudary odos may be osdered as ombao of he prevously osdered boudary odos. Really les u /2, H ( ) ad H /2, ( ) sasfy followg odos q u h, q, q,( u h ) ad also /2, 2 /2, 2 u ( H ( )) ad q ( H ( )) sasfy followg odos f q q he u, f q q he u q ad also ( q q ) u. We osder fuoals suh ha, ( u) ( u ) ( u ) ad, ( ) ( ) q q ( q ) (24) These fuoals have he form q, u, f u h, ( u ) (25), oherwse, f q, q q ( q ) (26), oherwse, 4.4. Ses of Admssble Dsplaemes ad Trao for Sgor oudary Codos wh Fro For varaoal formulao of he ulaeral oa problems wh fro also are used ses of admssble dsplaemes K,,( u, p) K ( u, p) K ( u) K( u ), K, ( u, p) K( up, ) K( u ), K, ( u, p) K( u, p) K( u) K( u ), /2, K( u) { uh ( ), u h }, x, } (27) /2, K( u) { uh ( ), u for q q ad u q for q q, x} ad rao K,,( σ) KT( σ) K ( σ) K( σ ), K, ( σ) KT( σ) K ( σ ), K, ( σ) KT( σ) K( σ ) /2, ( ) { ( ), q, } K σ σ H x (28) K σ σ H q p x /2, ( ) { ( ),, } 5. Varaoal Formulao of he Problem wh Coa ad Fro I order o formulae a elasodyam oa problem for body wh ra varaoal form we osder fuoal (7) o he se of admssble dsplaemes (27) or admssble rao (28). I he frs ase he problem s formulaed he form Fd u ad psuh ha up u p (29) (, ) sup f { * * (, )} p K (, ),,(, ) u p u K u p I he seod ase he problem s formulaed he form Fd u ad psuh ha up u p (3) (, ) f sup { (, )} * u K ( u, p) p * K,, ( u, p ) I order o formulae a elasodyam oa problem for body wh ra wee form usg osmooh fuoals (23) ad (24) we wll osder he boudary varaoal prple he form Fd u ad psuh ha ( up, ) f sup { ( u, p )} (3),,,, u, p K ( u, p) ( up, ) ( up, ) ( u ) (32),,, I he same way we a osder he omplemeary fuoal ( up, ) ( up, ) ( q ). (33),,, I hs ase he problem well be Fd u ad psuh ha ( up, ) supf { ( u, p )} (34),,,, u, p K ( u, p) Fuoal (29) ad (3) has a smple form, bu he

5 64 ses of resros (27) ad (28) are omplae, hey oas ulaeral osras (9) ad (). Fuoals (3) ad (3) are more omplae ad osmooh, bu he se of resros (9) s smple, does o oa ulaeral osras (9) ad (). Whh for s more s preferable deped o algorhm used for umeral soluo of he problem. I s eessary o meo ha he boudary varaoal prples are usually used wh EM. 6. Dual Varaoal Formulao ad Uzawa s Opmzao Algorhm We a reformulae above varaoal problems usg dualy feaure. O hese dual formulaos are based Uzawa s osmooh opmzao algorhms. Le us osder dual formulaos ad orrespodg Uzawa s algorhms for he problems uder osderao. 6.. ouadry Varaoal Prples I Le us rodue fuoal L (35) ( upq,, ) ( u, p ) q, ( u ) whh s osdered o he followg ses of resros up, K ( u ), q K, ( σ ) (36) Dual o (3) varaoal formulao of he oa problem wh fro for elas body wh ra has he form * L ( upq,, ) f sup sup L ( u, p, q ) (37) * u, p K( u, p) q K, ( σ) The Uzawa s algorhm ludes he followg seps: ) spefy a al value q K, ( σ ), 2) solve he mmzao problem for ow q ad deerme he uow quay u, p K ( u, p ) L ( u, p, q ) fsup { L ( u, p, q )} up, K ( up, ) up, K ( up, ) f sup ( up, ) q, ( u) 3) orre he quay K, σ q o sasfy he osras (38) q P [ q ( u )] (39) ( ) P s he operaor of projeo K, ( σ) /2, H ( ) o K, ( σ ) ad oeffe s seleed so as o provde he bes overgee of he algorhm, 4) proeed o he ex sep of erao ouadry Varaoal Prples II. Le us Irodue Fuoal L (4) * * ( up,, u) ( u, p ) u, ( q ) Whh s osdered o he followg ses of resros up, K ( u ), uk, ( u ) (4) Dual o (34) varaoal formulao of he oa problem wh fro for elas body wh ra has he form * L ( up,, u) fsup sup L ( u, p, u ) (42) * u, p K( u, p) qk, ( σ) The Uzawa s algorhm ludes he followg seps: ) spefy a al value u K, ( u ), 2) solve he mmzao problem for ow u ad deerme he uow quay u, p K ( u, p ) L ( u, p, u ) fsup { L ( u, p, u )} up, K ( up, ) up, K ( up, ) f sup ( up, ) u, ( q) 3) orre he quay (43) u o sasfy he osras u P [ u ( q )] (44) K, ( u) P s he operaor of projeo K, ( u) H /2, ( ) o K, ( u ) ad oeffe s seleed so as o provde he bes overgee of he algorhm, 4) proeed o he ex sep of erao. Nex we wll show how hese algorhm appled o some problems of fraure dyams. More applao oe a fd he boo [3] ad revew papers [4-6]. 7. Harmo Loadg of he Cra Ife Elas Rego Le a load, whh hages harmoally me * p( x, ) Re{ p ( x ) e } be appled o ra edges. Moreover, we suppose ha o he ra edges he ulaeral resros (9) ad fro () should be sasfed. I [4,5] was show ha hs ase he oa erao veor s o harmo ad a o be preseed he form q( x, ) Re{ q ( x ) e }. Therefore * ompoes of he oa fores ad dsplaemes dsouy veors have o be expaded o Fourer seres, whh deped o he loadg parameer, ( x, ) F ( ( x, )) Re{ ( x, ) }, u x F u x u x e q q q e (45) (, ) ( (, )) Re{ (, ),

6 642 T q ( x, ) F( q ( x, )) q ( x, ) e d, 2 T u ( x, ) F( u ( x, )) u ( x, ) e d. 2 Here, F ad (46) F are dre ad verse dsree Fourer rasforms. Fourer oeffes of he ompoes of he oa fores ad dsplaemes dsouy are relaed by he followg egral equaos uj( x, ) Fj( x, y, ) ds p ( y, ) (47) eause of he ulaeral resros (9) ad fro () he problem beomes a osruvely olear oe. Ths meas ha he fuoals (2) - (26) defe ulaeral oa odos wh fro po by po ad fuoal spaes a o be rewre frequey doma beause of her oleary. As resul of all above varaoal formulao of he problem a o be formulaed frequey doma. Therefore we wll use vaaoal formulaos (32) ad (33) spae-me doma ad adap algorhms -4 for soluo of he problem he ase of harmo loadg wh osderg ulaeral resros (9) ad fro (). Algorhm ludes he followg seps: ) spefy a al value u ( x, ) K, ( u ), 2) alulae Fourer oeffes u ( x, ) F ( u ( x, )) (48) 3) alulae q ( x, ) subsug ow u ( x, ) he erals equao (, ) j(, ) j(,, ) (, ) q y u x F x y ds p y (49) 4) alulae q usg ow q ( y, ) q F q (5) ( x, ) ( ( x, )) 5) orre he quay u o sasfy he osras u ( x, ) P [ u ( x, ) q ] (5) K, ( u) P s he operaor of projeo o he ses K, ( u ) u h ad u q ad oeffe s seleed so as o provde he bes overgee of he algorhm, 6) proeed o he ex sep of erao. Algorhm 2 ludes he followg seps: ) spefy a al value q ( x, ) K ( q ),, 2) alulae Fourer oeffes q ( x, ) F ( q ( x, )) (52) 3) alulae u ( x, ) solvg erals equao for ow q ( x, ) p ( y, ) q ( y, ) u ( x, ) F ( x, y, ) ds (53) j j 4) alulae u usg ow u ( x, ) u F u (54) ( x, ) ( ( x, )) 5) orre he quay q o sasfy he osras q P [ q u ] (55) K, ( q) K, ( q) P s he operaor of projeo o he ses q ad q q, ad oeffe s seleed so as o provde he bes overgee of he algorhm, 6) proeed o he ex sep of erao. Algorhm 3 ludes he followg seps: ) spefy a al value u ( x, ) K, ( u ), 2) alulae Fourer oeffes u ( x, ) F ( u ( x, )) (56) 3) alulae q ( x, ) subsug ow u ( x, ) he erals equao q y u x F x y ds p y (57) (, ) j(, ) j(,, ) (, ) q x usg ow q ( y, ) 4) alulae (, ) q F q (58) ( x, ) ( ( x, )) 5) orre he quay q o sasfy he osras K, ( q) q ( x, ) P [ q ] (59) K, ( q ) P s he operaor of projeo o he ses q ad q q, 6) alulae Fourer oeffes q ( x, ) F ( q ( x, )) (6) 7) alulae u ( x, ) solvg erals equao for ow q ( x, ) p ( y, ) q ( y, ) u ( x, ) F ( x, y, ) ds j j 8) alulae u usg ow u ( x, ) (6)

7 643 u F u (62) ( x, ) ( ( x, )) 9) orre he quay u o sasfy he osras K, ( u ) u ( x, ) P [ u ] (63) K, ( u) P s he operaor of projeo o he ses u h ad u q, ) proeed o he ex sep of erao. Algorhm 4 ludes he followg seps: ) spefy a al value q ( x, ) K, ( q ), 2) alulae Fourer oeffes q ( x, ) F ( q ( x, )) (64) 3) alulae u ( x, ) solvg erals equao for ow q ( x, ) p ( y, ) q ( y, ) u ( x, ) F ( x, y, ) ds (65) j j 4) alulae u usg ow u ( x, ) u F u (66) ( x, ) ( ( x, )) ) orre he quay u o sasfy he osras u ( x, ) P [ u ] (67) K, ( u ) K, ( u) P s he operaor of projeo o he ses u h ad u q, 2) alulae Fourer oeffes u ( x, ) F ( u ( x, )) (68) 3) alulae q ( x, ) subsug ow u ( x, ) he erals equao (, ) j(, ) j(,, ) (, ) q y u x F x y ds p y (69) 4) alulae q usg ow q ( y, ) q F q (7) ( x, ) ( ( x, )) 5) orre he quay q o sasfy he osras K, ( q) q ( x, ) P [ q ] (7) K, ( q ) P s he operaor of projeo o he ses q ad q q, 6) proeed o he ex sep of erao. 8. Numeral Sudy of he Algorhms Covergee Covergee ad omparso aalyses of he above four algorhms were doe for wo es problems wh he followg parameers. The raed maeral has he followg mehaal haraerss: elas modulus E 2 GPa, Posso s rao.25, spef desy 78 g / m. The fe ra s loaed he 3 plae R 2 x : x2 ad s surfae s desrbed by he Caresa oordaes x : l x l, x, x (72) Teso-Compresso Wave Le harmo eso-ompresso P-wave wh mulple frequees propagaes ormally o he ra surfae. The de wave s defed by he poeal fuo (, ) ( x2) x 2e (73) 2 s he amplude of he de wave, / s he wave umber, 2 s he veloy of he P-wave, 2 /T s he frequey, T s he perod of wave propagao, ad are he Lame osa, ad s he desy of he maeral. Followg [3-5] we osder wo separae problems: he problem for de waves ad he problem for refleo waves. Obvously, he ase uder osderao he problem for de wave s rval. Therefore we wll pay aeo o soluo of he problem for he refleed waves. The load o he ra s edges aused by he de waves has he form p Re{ p ( x ) e }, * 2 2 p (74) * I hs ase he ra surfaes are subjeed o he boudary odos p p for x (75) p p for x, as s show he Fgure. Wh osderg oa erao a he ra edges, he load veor o he ra edges has he form s p2( x, ) (76) p2( x, ) q2( x, ) x, ; q2, xe e s a rego of lose oa, whh s vared durg me. The fore of oa erao a he ra edges q ad dsplaeme dsouy (ra opeg) 2

8 644 Fgure. Reagular ra uder ormal loadg. u u should sasfy he oa osras he form 2 2 u, q, u q x, (77) Loadg p2( x, ) o he ra edges ad her opeg u2( x, ) may be expaded o Fourer seres p2( x, ) F ( p2( x, )), u2( x, ) F ( u2( x, )) (78) p2( x, ) F ( p2( x, )), u2( x, ) F ( u2( x, )) (79) Fourer seres expasos of he dsplaeme dsouy ( ) u2 x ad he rao p 2 ( x ) are relaed by he IE of he form p x ) F. P. F ( x y, ) u y ) d, 2 ( 22 2 (,, 2,,, x (8) The erels F (, ) 22 x y may be obaed from fudameal soluos for he 2-D seady-sae wave equaos of elasodyams, whh s well ow ad may be fd [3,4,2]. Ths problem was solved usg above four algorhms. Depedee of he algorhms overgee rae o wave umber s preseed Fgure 2. Aalyss of hese daa shows ha all algorhms are overge ad obaed resuls ode for all algorhms, bu overgee rae s dffere. A aalyss of resuls Fgure 2 reveals ha Algorhm 3 ad Algorhm 4 have sgfaly faser overgee for all wave umbers Shear H-Wave Le harmo shear H-wave wh mulple frequees propagaes ormally o he ra surfae. The de wave s defed by he poeal fuo Fgure 2. Covergee of he algorhms for dffere wave umbers. (, ) x ( 2 2) x e, (8) s he amplude of he de wave, 2 / 2 s he wave umber, 2 s he veloy of he H-wave, 2 /T s he frequey. Followg [3,4] we osder wo separae problems: he problem for de waves ad he problem for refleo waves. Obvously, he ase uder osderao he problem for de wave s rval. Therefore we wll pay aeo o soluo of he problem for he refleed waves. The load o he ra s edges aused by he de waves has he form * p( ) Re{ p( x) e * 2 x, }, p 2 (82) I hs ase he ra surfaes are subjeed o he boudary odos p p for x, (83) p p for x, as s show he Fgure 3. Wh osderg oa erao a he ra edges, he load veor o he ra edges has he form s p( x, ) p( x, ) (84) q( x, ) x, ; q, xe e s a rego of lose oa, whh s vared durg me. The fore of oa erao a he ra edges q ad dsplaeme dsouy (ra opeg) u u u should sasfy he oa osras he form q q2 u, q q2 u q x, (85) s a fro rao; u q s a oeffe ha deped o qualy f oa faes, q 2 s

9 645 Fgure 3. Reagular ra uder shear loadg. a ormal oa fore, he problem uder osderao s ow before. Loadg p( x, ) o he ra edges ad her opeg u may be expaded o Fourer seres F p ( p ( x, )) u( x, ) F ( u( x, )) (86) p( x, ) F ( p( x, )), u( x, ) F ( u( x, )) (87) I Guz ad Zozulya 2, 22 was show ha Fourer seres expasos of he dsplaeme dso- uy ( ) u x ad he rao p ( x ) are relaed by he IE of he form p ( x ) F. P. F ( x y, ) u ( y ) d,,, 2,,, x (88) The erels F ( x y, ) may be obaed from fudameal soluos for he 2-D seady-sae wave equaos of elasodyams, whh s well ow ad may be fd [3,4,2]. Ths problem was solved usg above four algorhms. Depedee of he algorhms overgee rae o wave umber s preseed Fgure 4. Aalyss of hese daa shows ha all algorhms are overge ad obaed resuls ode for all algorhms ad overgee rae s o dfferg sgfaly. Our alulaos show ha all four above algorhms are overge elasodyam problems wh oa ad wh fro for fe raed body. I s mpora o meo ha Algorhm 3 ad Algorhm 4 have sgfaly faser overgee boh ases froless oa problem ad problem wh fro. 9. Colusos Ths paper prese varous varaoal formulaos of elasodyam problem for body wh ra wh osderg possbly for ulaeral ra faes oa erao ad fro. Varaoal formuaos s based o boudary varaoal prple ad o fudameal solu- Fgure 4. Covergee of he algorhms for dffere wave umbers. os. Nosmooh fuoals ha orrespod o ulareral froal oa odos are osrued. Ierave algorhms of he Uzawa s ype ha are based o projeo o he se of ulaeral resros ad fro are proposed. I was show ha he ase f varaoal formulao s based o prples formulaed oly for boudary he IE mehod may be used. The ase of he ra fe elas meda s osdered more deals ad four ew algorhms are proposed. To sudy overgee of he proposed algorhms wo problems relaed o he ra faes oa erao uder ao of he harmo eso-ompresso ad shear waves have bee solved umerally usg IE mehod.. Referees [] V. V. Zozulya, O Solvably of he Dyam Problems Theory of Cras wh Coa, Fro ad Sldg Domas, Dolady Aadem Nau Urasoy SSR, Vol. 3, 99, pp , Russa. [2] V. V. Zozulya, Mehod of oudary Fuoals Coa Problems of Dyams of odes wh Cras, Dolady Aadem Nau Urae, Vol. 2, 992, pp , Russa. [3] A. N. Guz ad V. V. Zozulya, rle Fraure of Cosruve Maerals uder Dyam Loadg, Nauova Duma, Kev, 993, Russa. [4] A. N. Guz ad V. V. Zozulya, Fraure Dyams wh Allowae for a Cra Edges Coa Ierao, Ieraoal Joural of Nolear Sees ad Numeral Smulao, Vol. 2, No. 3, 2, pp [5] A. N. Guz ad V. V. Zozulya, Elasodyam Ulaeral Coa Problems wh Fro for odes wh Cras, Ieraoal Appled Mehas, Vol. 38, No. 8, 22, pp [6] V. V. Zozulya ad P. I. Gozalez-Ch, Dyam Fraure Mehas wh Cra Edges Coa Ierao, Egeerg Aalyss wh oudary Elemes, Vol. 24, No. 9, 2, pp

10 646 [7] V. V. Zozulya, Mahemaal Ivesgao of Nosmooh Opmzao Algorhm Elasodyam Coa Problems wh Fro for odes wh Cras, Ieraoal Joural of Nolear Sees ad Numeral Smulao, Vol. 4, No. 4, 23, pp [8] C. Polzzoo, A oudary M Max Prple as a Tool for oudary Eleme Formulaos, Egeerg Aalyss wh oudary Elemes, Vol. 8, No. 2, 99, pp [9] J. Cea, Opmzao, Teore e Algorhms, Duod, Pars, 97, Freh. [] I. Eelad ad R. Temam, Covex Aalyss ad Varaoal Problems, Norh-Hollad, 975. [] V. V. Zozulya, Varaoal Prples ad Algorhms Coa Problem wh Fro, I: N. Masoras, V. Mladeov,. Suer ad L. J. Wag, Eds., Advaes Sef Compug, Compuaoal Iellgee ad Applaos, WSES Press, Davers, 2(a), pp [2] V. V. Zozulya, Varaoal Prples ad Algorhms Elasodyam Coa Problem wh Fro, I: S. N. Alur, M. Nshoa ad M. Kuh, Eds., Advaes Compuaoal ad Egeerg Sees, Tehology See Press, Puero Vallara, Mexo, 2(b). [3] V. V. Zozulya ad O. V. Meshyov, Use of he Cosraed Opmzao Algorhms Some Problems of Fraure Mehas, Opmzao ad Egeerg, Vol. 4, No. 4, 23, pp [4] V. V. Zozulya ad M. V. Meshyova Sudy of Ierave Algorhms for Soluo of Dyam Coa Problems for Elas Craed odes, Ieraoal Appled Mehas, Vol. 38, No. 5, 22, pp [5] V. V. Zozulya, Fraure Dyams wh Allowae for Cra Edge Coa Ierao, I: C. Cosada, P. Shavoe ad A. Moduhows, Iegral Mehods See ad Egeerg, rhauser, oso, 22, pp [6] V. V. Zozulya ad P. Rvera, oudary Iegral Equaos ad Problem of Exsee Coa Problems wh Fro, Joural of he Chese Isue of Egeers, Vol. 3, No. 3, 2, pp [7] N. Kuh ad J. T. Ode, Coa Problems Elasy, SIAM Publaos, Phladepha, 987. [8] P. D. Paagoopoulos, Iequaly Problems Mehas ad Applaos, Covex ad No Covex Eergy Fuos, rhauser, Sugar, 985. [9] A. C. Erge ad E. S. Suhub, Elasodyams, Vol. 2. Lear Theory, Aadem Press, New Yor, 975. [2] J. Domguez, oudary Elemes Dyams, Compuaoal ad Mehaal Publshg, Souhempo, 993.