3. OVERVIEW OF NUMERICAL METHODS

Transcription

1 3 OVERVIEW OF NUMERICAL METHODS 3 Inroducory remarks Ths chaper summarzes hose numercal echnques whose knowledge s ndspensable for he undersandng of he dfferen dscree elemen mehods: he Newon-Raphson-mehod, a powerful echnque o numercally solve nonlnear equaons; he Gauss-Sedel-mehod, a relaxaon echnque whch produces gradually mprovng approxmaons for he soluon of lnear sysems of equaons; he Euler-mehod or frs-order Runge-Kua-mehod, o numercally solve nal value problems of frs-order dfferenal equaons; he mehod of cenral dfferences, whch can be consdered as an mproved verson of he Euler-mehod; Newmark s -mehod, for he me negraon of second-order dfferenal equaons lke he equaons of moon We shall no deal n deal wh he deeper problems lke convergence analyss, marx nverbly The neresed reader s advsed o consul he wde leraure of numercal mahemacs; an excellen sarng pon s Belyschko e al (000) 3 The Newon-Raphson mehod The am of hs mehod s o deermne a vecor û whch sasfes he gven nonlnear equaons f ( u, u,, u ) 0 f ( u, u,, u ) 0 n n or shorly f ( u, u,, u ) 0 n n f( u) 0, where f(u) s a connuously dfferenable funcon of u I s assumed ha we have an nal esmae u 0 for û, and also assumed ha for any arbrary u, we are able o deermne f(u) as well as s Jacoban marx, K(u): dfu ( ) Ku ( ) du or, n a more dealed form: df p( u, u,, un) K pq du Noe ha he explc form of f(u) and K(u) s no necessary o know; bu we mus be able o calculae hem for any arbrary u q 4

2 Kaaln Bag: Secon 3 Fundamens of The Dscree Elemen Mehod Overvew of Numercal Mehods f Ku ( 0) f( u0) û u0 u Fgure The Newon-Raphson-mehod for a lnear equaon Imagne frs ha f s a lnear funcon of u In hs case K s consan (he same for any u) Assume, n addon, ha K s no sngular, e s nverable In hs case he equaons are easy o solve n a sngle sep: uˆ u K( u ) f( u ) (Fgure llusraes hs for he very specal case n = ) However, for a non-lnear f he above calculaon leads o a nex approxmaon: u u K( u ) f( u ) f Ku ( 0) f( u0) û u u u0 u Fgure The Newon-Raphson-mehod for a nonlnear equaon whch can be used o produce one more approxmaon of he roo: u u K( u ) f( u) (Insead of nverng K, an alernave s o solve he lnear equaons 5

3 Kaaln Bag: Secon 3 Fundamens of The Dscree Elemen Mehod Overvew of Numercal Mehods for u + ) K ( u ) u u f ( u ) Ths has o be repeaed unl he magnude of f(u k ) (eg he Eucldean norm, T f f f ) becomes smaller han a pre-defned error lm In addon o he ermnaon creron f [a prescrbed value], he value of u u s also ofen racked, whch s parcularly helpful f he funcon f(u) s raher fla, e K s nearly sngular around he roo Usually he mehod quckly converges o a roo of he equaon, assumng ha he nal guess u 0 was close enough However, here are many suaons when he mehod fals or badly converges, for nsance: Bad sarng pon: eg K s sngular or nearly sngular a he sarng pon; or he sarng pon s no n an nerval from where he mehod converges The sarng pon may also ener an nfne cycle, jumpng back-and-forh beween wo (or perhaps more) pons whou a convergence The dervave K does no exs a he roo, or s dsconnuous near he roo Among he dfferen dscree elemen echnques hs mehod s appled for nsance n he DDA mehods and n he Bag-Bojar-mehod 33 Relaxaon: The Gauss-Sedel mehod Quas-sac DEM models are based on he soluon of a sysem of lnear equaons As wll be explaned laer, hey apply he well-known dsplacemen mehod of srucural analyss, so he equlbrum equaons correspondng o each degree of freedom gve he sysem of equaons o solve, whose unknowns are he characersc dsplacemens of he model In hese sysems he number of equaons equals he number of unknowns The problem o solve can be formulaed as follows: Deermne he unknown vecor u: u u u u n so ha would sasfy he equaons where K and f conss of gven consans: K u f 0 6

4 Kaaln Bag: Secon 3 Fundamens of The Dscree Elemen Mehod Overvew of Numercal Mehods k k k k k k k k k n n K and n n nn f f f f n There exs several mehods o solve large sysems of lnear equaons Drec solvers (eg Gauss-Jordan elmnaon, LU-facorzaon) am o produce he exac (up o numercal roundng errors) soluon, eg by nverng he coeffcen marx K, or by oher echnques: hey oban he soluon n a fne number of calculaon seps Relaxaon mehods, on he oher hand, produce a sequence of approxmaons of he soluon, hopefully geng closer and closer o he exac soluon The Gauss-Sedel-mehod wll be parcularly mporan for us I works n he followng way: Sarng from an nal u 0 esmaon, he mehod prepares a u, u, u, u +, seres of approxmaons, convergng o he exac soluon Denoe he j-h scalar componen of he soluon by u j, and s -h esmaon by u j, From he -h esmaon, u, he +-h esmaon, u +, s calculaed as follows Calculae he r resdual vecor belongng o he -h approxmaon: r K u f If u were equal o he exac soluon, hen he componens of r were all zeros, and he calculaons should be ermnaed However, f hs s no he case, hen he scalar componens of r are no all zeros A general, say j-h componen s: r k u k u k u f j, j, j, jn n, j Fnd ha scalar n r whch has he larges absolue value, say hs s he p-h componen of r : r max r p, j, ( j) (We can also say ha he p-h equaon s sasfed wors by he -h approxmaon of he soluon) Calculae now wha value should be sand a he p-h poson of u so ha hs wors value n he resdual would become zero In oher words, deermne u p, o replace u p, so ha p r k u k u k u f 0 p, ph h, pp p, ph h, p h h p Afer some rearrangemen, he soluon for u p, s: p n u p, : k phuh, k phuh, f p k pp h h p The res of he componens of u + reman equal o he correspondng componens of u : uj, : uj, for all j p Then le := +, and a nex approxmaon can be prepared n he same way n 7

5 Kaaln Bag: Secon 3 Fundamens of The Dscree Elemen Mehod Overvew of Numercal Mehods Furher and furher eraon seps have o be done, unl he componens of he resdual vecor r become suffcenly small (eg he Eucldean norm of r can be he used for a ermnang creron) Mehod of Cross, wdely appled n he 0eh cenury n manual srucural analyss, s based on hs relaxaon echnque Mehod of Cross s for frames whose nodes have roaonal degrees of freedom The momen balance equaons of he nodes are a sysem of lnear equaons, and he unknowns are he roaons of he nodes whch lead o such a poson where he momen balance s sasfed for all nodes In any sep, he node wh he larges equlbrum error s seleced, and whle keepng all ohers fxed, he consdered node s allowed o roae no he equlbrum poson The mehod of Kshno, a quas-sac DEM echnque, apples hs ype of relaxaon 34 Soluon mehods for nal value problems 34 Inroducory remarks Tme-seppng DEM algorhms nend o deermne (or, a leas, approxmae wh suffcen accuracy) he me-dependen dsplacemens u() and veloces du() v() d of he model a he dscree me pons,,,, +,, sarng from a known u( 0 ) and v( 0 ) whch belong o he nal me 0 The me-seppng algorhms calculae a seres of u( ), v( ); u( ), v( ); u( ), v( ), u( ), v ( ); approxmaons for whch he equaons of moon, e he equaons d u() M f (, u( ), v( )) d are suffcenly accuraely sasfed a he,,,, +, me nsans There are a vas number of numercal echnques o numercally solve nal value problems of dfferenal equaons Par of hese echnques are explc: as explaned already n Secon, means ha when consderng a me nerval, hose u and v values (generalzed dsplacemens and veloces) belongng o he endpon + are deermned n such a way ha he equaons of moon are compled a me pon and he values a + are predced from he approxmaed u and v values belongng o The mplc me negraon echnques are more relable In hese mehods he u and v values belongng o he endpon + are calculaed n such a way ha he equaons of moon should be sasfed a he endpon of he me nerval, whch s done wh he help of a gradually mprovng eraon scheme: he approxmaed values of u and v a + are checked and modfed agan and agan, unl a suffcenly exac mach s reached; and hese values are hen used as he sarng daa for he nex mesep There are hree mehods whch wll be parcularly mporan n dscree elemen modellng: () he frs order Runge-Kua-mehod (Euler-mehod); () he mehod of cenral dfferences; () Newmark s -mehod 8

6 Kaaln Bag: Secon 3 Fundamens of The Dscree Elemen Mehod Overvew of Numercal Mehods The Euler-mehod and he mehod of cenral dfferences s suable for he calculaon of frsorder dfferenal equaons (e when he dfferenal equaons conan only he frs dervave of he unknown funcon) The equaons of moon are, on he oher hand, secondorder equaons Forunaely, general hgher-order dfferenal equaons can easly be ransformed no frs-order equaons n he followng way: Consder he k-h order dfferenal equaon: k u ( ) ( ) ( ) ( ), ( ), u, u k d d d d f u, u k d d d d Inroduce he followng noaons: y ( ) u( ) du() y() d d u() y3() d k d u() yk () k d and usng hem, he orgnal dfferenal equaons can be wren n he form: dyk () f (, y( ), y( ), y k ( )) d The relaons beween he funcons y (), y (),, y k () are gven by he (k-) dfferenal equaons dy() () y() d dy() () y3() d dy (k-) () k yk () d Le he k-h equaon be he orgnal dfferenal equaon by usng he new noaons, as seen above: dyk () (k) f(, y( ), y( ), yk ( )) d Summarze hese k equaons no he followng sysem: dy( ) fˆ(, y ( )) d where he noaons are: 9

7 Kaaln Bag: Secon 3 Fundamens of The Dscree Elemen Mehod Overvew of Numercal Mehods y() y() y() y3() y( ) ; fˆ (, y( )) yk () yk () yk () f(, y( )) A frs-order (bu of course much larger) sysem of dfferenal equaons s receved hs way The equaons of moon are of second order, so he applcaon of he above ransformaon s sraghforward Inroduce he followng noaons: u( ) ˆ v( ) y( ) : ; f (, u( ), v( )) : M f(, u( ), v( )) ; f(, u( ), v( )) : v( ) f (, u( ), v( )) or shorly: dy( ) fˆ(, y ( )) d The soluon of such a dfferenal equaon means o fnd he funcon y() f s frs dervave s known, and he nal value, e he y( 0 ) whch belongs o he gven 0 me nsan s specfed (Hence he name nal vaule problem ) 34 Frs order Runge-Kua-mehod (Euler-mehod) Le he nal value of he funcon y() be y 0 : y( ) y, 0 0 and he funcon y() sasfes he followng dfferenal equaon n whch f ˆ(, y ( )) s gven, no necessarly explcly, bu a leas n he sense ha can be calculaed for any and y(): d y( ) f ˆ(, y ( )) d Our am s o deermne hose y, y,, y, y +, values whch belong o he me nsans,,,, +, (The ncreasng ndces unlke n he prevous cases where hey ndcaed beer and beer approxmaons of he soluon now denoe ha me proceeds) Accordng o he Euler-mehod, from value y whch belongs o he value y + belongng o + s calculaed n he way llusraed n Fgure 3: Calculae ˆf a (, y ) ( means o approxmae he frs dervave of he unknown funcon y() a ): h f ˆ(, y ) Assumng ha hs frs dervave remans consan on he (, + ) nerval, he approxmaon of y + a + (denong he lengh of he nerval by, so ): y : y h 30

8 Kaaln Bag: Secon 3 Fundamens of The Dscree Elemen Mehod Overvew of Numercal Mehods y h y y Fgure 3 The man sep of he Euler-mehod Then he analyss of he nex nerval follows Fgure 4 llusraes ha snce a he endpon of he nerval s no checked wheher he equaons of moon are sasfed whn he necessary accuracy, he errors of he consecuve approxmaons accumulae, and he esmaed y, y,, y, y +, values may devae from he exac y() more and more as ncreases y h h y y y Fgure 4 Euler-mehod: Increasng devaons from he exac soluon The Euler-mehod can smply be appled o he DEM equaons of moon Inroduce he noaon du() d v() dv( ) f (, u( ), v( )) d (here he f (, u( ), v( )) funcon s known ), and he nal values belongng o 0 are he gven u 0 and v 0 : 3

9 Kaaln Bag: Secon 3 Fundamens of The Dscree Elemen Mehod Overvew of Numercal Mehods u( 0) u0 v( ) v 0 0 From he already known (u, v ) approxmaons whch belong o, hose (u +, v + ) belongng o a + can be calculaed by usng he dervave, h, belongng o : h v f (, u, v ) whch yelds u u u v h v v v f (, u, v ) The man dsadvanage of he Euler-mehod s s explc naure (he accumulang errors of he approxmaons), whch can be avoded by usng he mplc verson of he mehod Is man dea s formulaed as y : y h, where h f ˆ(, y ) Snce y + s unknown, f ˆ(, ) y canno be calculaed However, by usng an erave scheme, beer and beer approxmaons can be produced for y +, and he eraon s sopped when he consdered measure of he error of he equaons becomes suffcenly small Usng he noaons appled n he equaons of moon of DEM, he fundamenal sep of he mplc Euler-mehod s u u v v v f (, u, v ) Ths mehod s appled n he Conac Dynamcs mehods of dscree elemen modellng 343 The mehod of cenral dfferences Ths mehod can be consdered as an mproved verson of he explc Euler-mehod For he sake of smplcy, whou dealng wh he general noaons wh (y(), h()), he applcaon for he equaons of moon wll be nroduced here only Based on he u approxmaon of he funcon u() a, and on he approxmaon of he funcon v() n he mddle pon of he nerval (, ), denoed by v /, he approxmaons belongng o he me nsans laer by are calculaed as and v : v f (, u, v ) / / / 3

10 Kaaln Bag: Secon 3 Fundamens of The Dscree Elemen Mehod Overvew of Numercal Mehods u : u v / and hen he nex me nerval can follow (A he begnnng of he whole analyss, e a he frs nerval when = 0, v / can be approxmaed by he gven nal value v 0 ) Due o s smplcy and effecveness, hs mehod s very wdely appled: can be found n several commercal sofwares lke PFC, UDEC, EDEM ec 344 Newmark s -mehod As every mplc echnque, Newmark s -mehod analyses he (, + ) nerval o fnd he unknowns belongng o + so ha he equaons of moon would be sasfed a he endpon, e a + The mehod apples for second-order dfferenal equaons lke he equaons of moon Le us search for he soluon of he equaon Ma( ) f (, u( ), v( )) n whch u( ) u( ) v( ) d, a( ) d d d and gven he u( 0) u0, v( 0) v0 nal values For hose u(), v(), a() funcons whch exacly sasfy he equaons of moon, he resdual funcon r(, u( ), v( ), a( )) f(, u( ), v( )) M a( ) would be consanly zero for every ; hese u(), v(), a() funcons would ndeed descrbe he real hsory of he analysed sysem Assume now ha he u, v and a numercal soluons whch belong o ndeed sasfy he equaons; and our am s o fnd he u +, v + and a + belongng o so ha r(, u, v, a ) 0 Accordng o Newmark s -mehod he veloces and acceleraons a + are approxmaed n erms of he unknown u + wh he help of he parameers and whch conrol he behavour of he me negraon: a : u u v ( ) a v : v ( ) a a These formulas are based on he followng bass: The velocy a + s approxmaed wh he help of an average acceleraon, a : v : v a where 33

11 Kaaln Bag: Secon 3 Fundamens of The Dscree Elemen Mehod Overvew of Numercal Mehods a : ( ) a a, he dsplacemen a + s calculaed from anoher average acceleraon, a : where : u u v a a : ( ) a a Afer some rearrangemen, he above wo formulas are receved (Noe ha for specal values of and he mehod becomes equvalen o oher, prevously shown mehods For = = 0, for example, he explc Euler-mehod s gven: he veloces and acceleraons are assumed o be he same along he whole nerval as a s begnnng pon, and no eraon s needed for he calculaon For = 0 and = /, he mehod of cenral dfferences, also explc, s gven For oher values of and oher me negraon mehods are receved, beng mosly mplc) If choosng an approxmaon for u +, he wo formulas provde an approxmaon o v + and a + These can be nsered no he equaon r(, u, v, a ) 0, where he unknowns are now only he componens of u + Ths (usually nonlnear) equaon can hen be numercally solved, eg wh he Newon-Raphson-mehod, and afer fndng u + suffcenly exacly, from he wo formulas a + and v + are also receved Ths mehod s he fundamen of he DDA models 345 Remark: Sably of he soluon An mporan aspec of he me negraon of he equaons of moon (and of dfferenal equaons n general) s sably Sably can loosely be defned as he propery of an negraon mehod o keep he errors resulng n he negraon process of a gven equaon bounded a subsequen me seps An unsable mehod wll make he negraon errors ncrease exponenally, and an arhmec overflow can be expeced even afer jus a few me seps Snce sably depends no only on he gven mehod bu also on he ype of problem, for a one-dmensonal case he es equaon y =y, where s a complex valued consan, s used o characerze he sably properes of a gven mehod Ths characerzaon s performed by defnng he se of values of and for whch he correspondng mehod s sable Algorhms ha are sable for some resrced range of values (,) are called condonally sable When usng such mehods, he me sep should be chosen dependng on he characerscs of he problem as defned by (or a se of ) In he case of a nonlnear problem for whch he value of changes wh me, he algorhm may be sable for some par of he negraon and unsable for anoher Consequenly, s very mporan when usng condonally sable algorhms o know n advance he range of values (,) for whch he mehod s sable and o compare wh he possble range of values of he gven problem For hs purpose he regon of absolue sably of a mehod s defned as ha se of values (,) for whch a perurbaon n he soluon y wll produce a change n subsequen values whch does no ncrease from sep o sep The regon of absolue sably s an nrnsc 34

12 Kaaln Bag: Secon 3 Fundamens of The Dscree Elemen Mehod Overvew of Numercal Mehods characersc of he mehod whch should be consdered pror o he use of condonally sable algorhms As an example, Euler s mehod descrbed above s condonally sable and mus be less han / o assure sably An algorhm s sad o be A-sable or uncondonally sable f he soluon o y =y ends o zero as when he Re()<0, whch means ha he numercal soluon decays o zero whenever he correspondng exac soluon decays o zero An A-sable algorhm may be also defned as an algorhm whose regon of absolue sably s he complee lef half complex plane ncludng he magnary axs The mos mporan consequence of he A- sably propery s ha here s no lmaon on he sze of for he sably of he negraon process: hs s why A-sable algorhms are also called uncondonally sable Obvously, hs propery s very mporan and generally desred n he negraon of mulbody and oher engneerng sysems, snce he analys would only have o be concerned wh he sep sze for accuracy purposes and no for sably The uncondonal sably of Newmark s mehod s guaraneed for / The mehod of cenral dfferences s, on he oher hand, only condonally sable: he maxmal allowed lengh of he mesep depends on he larges egenfrequency of he sysem Snce he mehod of cenral dfferences s mos wdely appled n DEM, he reader wll mee he problem of allowable mesep lengh eg a he BALL-ype models and a UDEC Quesons 3 Inroduce he Newon-Raphson-mehod! 3 Inroduce he Gauss-Sedel-mehod! 33 Inroduce he Euler-mehod! 34 Inroduce he mehod of cenral dfferences! 35 Inroduce Newmark s -mehod! 35