Hidrológiai Közlöny 1982 (62. évfolyam)

7. szám - Gáspár Csaba: Perem-integrálegyenlet módszer alkalmazása szivárgási problémákra

Gáspár Cs.: Perem-integrálegyenlet módszer Hidrológiai Közlöny 1982. 7. sz. 323 1. táblázat. A perem-integrálegyenlet módszer és a véges differencia, ill. véges elemmódszer főbb numerikus jellem­zőinek összehasonlítása TaőA. 1. CpaeHenue enaenux umßpoeux noKa3amesieü Memo­doe uwneppaAbHOio ypaeHenun no Konmypy, Konennux 3Ae­Menmoe u Koneimix pa3Hocmeü Table 1. Comparison of the principal numerical charac­teristics of the boundary integral equation-, the finite difference- and finite element methods Jellemzők megnevezése VDM, VEM PIEM N NX N N 2 Csomópontok, rács­pontok száma N­A mátrix mérete N'-XN* A mátrix nemnulla elemeinek száma N 2 A mátrix struktúrája ritka, sávos tele A mátrix egyéb spéci- szimmetrikus, nemszimmetri­ális tulajdonsága pozitív definit kus A megoldás művelet­• ' AM \7G AT 3 igenye ISI ^ sorszámozásának problémája. Ezen felül a külön­féle típusú peremfeltételek egységesen és egyszerű en kezelhetők. u HopMajibnoe npüH3B0flH0e OT nero; L, K, R — HHTer­panbHbie oriepaTopbi jiorapn(])MimecKHx, AByxcJioiÍHbix H MOHOCJioiÍHbix noTeHunajiOB. ABTOP flOKa3i,iBaeT, HTO nepexoA K ypaBHemno no KOHTypy (7) jiBJiflercji KoppeiCT­HbiM, T. e. BCHKoe peuieHne (1) — H TOJibKO OHH — Mory r 6biTb noJiyieHbi nyTeM peuieHiiji ypaBHCHUH (7). Bo BTOpOÍÍ 'laCTH Iipe«CTaBJI>ieTC5I B03M0>KH0CTb ripn­MeHeHHíi Meroaa K peiueHino HEKOTOPBIX RANHMHBIX 3a/;a<I ycTaHOBHBiiieiíoi 4>HJibTpauHH. (12) ecTb HHTerpajibHoe ypaBHCHiie no KOHTypy (JmjibTpauHii nepe3 oflHopo /myio ii n30TponHyio cpeay, a (15) TO >i<e «JIJI CJTOHCTOH HSO­Tponnoií cpe^bi. riocjie 3Toro aBrop paccMarpuBaeT pemeHiin npoöjieivi (J)HJ7bTpaUHH CO CBOÖOflHÓft nOBepXHOCTblO [4], [7] MeTO­aaMH iiHTerpajibHbix ypaBnennii no KOHTypy. B rpeTbeii MacTH paccMarpbiBacTca mfCJieiiHoe peme­HHC HHTerpajibHbix ypaBHeHHií no KOHTypy. flpuMeHe­HHeM MeTOÄOB KOJiJiOKauHH HHTerpajibHoe ypaBneHiie npuBOflincí! K ciicTeMe jiHHeiÍHbix ypaBHeHHií (18)—(19) — (20), K03(|)(J)HUHGHTbI KOTOpOÍÍ paCMHTblBaiOTCH 3KCHJIH­IJHTHblM nyTeM. B TaöJi. 1. npHBOÄHTCil cpaBiiemie Mero/ia nmeipajib­Hbix YPABHEHHIÍ no KOHTypy c METOAAMH KOHCMHUX 3Jie­MCHTOB H KOHeMHblX paSHOCTeíí. ßeMOHCTpiipyeTCJl, MTO KOJiHMecTBO BbiMHCJieHiiii npH peilieHHII MCTOAOM HHTer­pajibHbix ypaBHeHHií no KOHrypy na nopjiflOK HHJKC, neM B flpyrnx cjiynaíix. IRODALOM [1] C. Baiocchi—V. Comincioli—E. Magenc.r- -G. A. Pozzi: Free boundary problems in the theory of fluid flow through porous media? existence and uniquess theorems Ann. Mat. Pura Appl. 96(1973) 1—82. [2] C. A. Brebbia: Recent Advances in Boundary Ele­ment Methods Pentech Press, London, 1978. [3] T. A. Cruse — F. J. Rizzo: Boundary-Integral Equation Method: Computational Applications in Applied Mechanics AMD, ASME, New York , 1975. [4] Haszpra Ottó: Háromdimenziós szabadfelszínfl szi­várgás fokozatos közelítő meghatározása elektroli­tos analóg modellel, Hidrológiai Közlöny, 1978. 12. [5] Haszpra Ottó: A dunakiliti tározó szivárgócsator­náinak hatékonyságvizsgálata. Összefoglaló jelen­tés. VITUKI, Budapest, 1980. [6] Sz. G. Mihlin: Linyejnüje uravnyenyije v csasztnüh proizvodnüh. Moszkva, 1977. [7] R. L. Taylor —C. B. Brown : Darcy flow solution with a free surface. Proc. ASCE, J. Hydr. Div. Hy2, 1967. [8] V. Sz. Vlagyimirov: Bevezetés a parciális differen­ciálegyenletek elméletébe. Műszaki Könyvkiadó, Budapest, 1979. npHMeHeHHe MeTOfla KOHTypHbix HHTerpajibHbix ypaBHeHHií K peuieHHto npoöjieiw (JM.nbTpamin Taiunap, lI. MeTOfl KOHTypHbix HHTerpajibHbix ypaBHeHHií oÖJierMHT HJIH Booßme «EJIAET BOSMOJKHMM peineHeH CJIOHÍHMX ÄByiwepHbix 3A«aM (JiHJibTpaqHH B TOM cjiynae, ecjw 3Ha­MeHHji napaiweTpoB nac HHTepecyioT TOJibKO Ha rpaHHue HCCJIE^YEMOFI oöjiac™ HJIH B ORPAHIMEHHOM KOJIHQECTBE TOMeK BHyTpH 3T0H 06jiaCTH. CymHOCTb MeTO^a B TOM, MTO ÄH({)(j)epeHUHajTbHbie ypaBHeHHií ([>H.ribTpauHH npn noMOiUH anriapaTa TeopHH noTeHmiajioB npeo6pa3yeTC5i B HHTerpajibHoe ypaBneime, Koropoe onpe^ejieHO TOJibKO rpaHHue oöjiacTH. TaKHM 0öpa30M jjByxMepHyio 3aaaiy MO>KHO CBECTH K OÄHOMEPHOH. TpeöyeMoe KÓJlHHeCTBO BblMHCJieHHH TaK>Ke CHH>KaeTCJI. ECJIH MaTeMaTimecKyio M0«ejib HccjieAyeMoii npoő­jieMbi MO)KHO 3anncaTb B BHAe ypaBHeHHH üyaccoHa (1), TÓ cooTBeTCTByiomee KOHTypnoe HHTerpajibHoe ypaBHe­HHe noJiyMaeTCji corjiacHO BbipaweHHio (7), rae u h v oöo3naMaioT peiuemie ypaBiiemiíi (1) Ha rpamme oöjiacTH Application of boundary integral equation!) for the solution of seepage problems by Gáspár, Cs. The solution of a number of complicated two-dimen­siol seepage problems is facilitated, or made at all possible by the method of boundary integral equations, provided that the values of certain parameters are needed only along the boundary of, and/or at a few internal points only within, the domain under consi­deration. The method eonsits essentially of reducing with the help of the potential theory, the differential equation describing seepage flow to an integral equation­mwhich is defined along the boundary of the domain considered only,^ so that substantially it is sufficient to solve one-dimensional a linear problem instead of the original twodimensional problem. In this manner the computations needed for a solution are materially reduced in volume. If the mathematical model of the problem under consideration is Poisson's equation (1), then the corres­ponding boundary integral equation is obtained as Eq. (7), where u and v are the value, which the solution Eq. (1) assumes at the boundary of the domain and the normal derivative, thereof, respectively, whereas L, K and R are the integral operators of the logarithmic-, the double-layer- and single -layer potentials, respec­tively. It is demonstrated that the changeover to the boundary integral equation (7) is a correct one, in that all solutions of Eq. (1) — and only those — can be derived from the solutions of Eq. (7). In Section 2 the applicability of the method is illustra­ted for some tvpical problems in steady seepage flow. Eq. (12) is the boundary integral equation of seepage in uniform, isotropic média, while Eq. (15) applies to layered, isotropic soils. Hereafter the successive­approximation solution [4], m of unconfined seepage problems is connected with the method of boundary integrals. In Section 3 the numerical solution of theboundary integral equations is considered. Using the collocation method the integral equation is reduced to the set of linear equations (18)—(19)—(20), the coefficients of which are found explicitly. The method of boundary integral equations is finally compared for the numerical values obtained with the finite-difference- and finite­element methods (Table I), demonstrating that the boundary integral method required less computation effort by at least one order of megnitude than the latter.

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