Numerical integration

Transcription

1 Numeicl itegtio Alyticl itegtio = ( ( t)) ( t) Dt : Result ( s) s [0, t] : ( t) st ode odiy diffeetil equtio Alyticl solutio ot lwys vilble d( ) q( ) = σ = ( d ) : t 0 t = Numeicl itegtio 0 t t 2. t. t t N = T = N [0, T ] = [ t, t ] = [ t, t ] Time itevl/step "" Δ t = t t Time icemet ( t ) = g(,,,...) Appoimte solutio (depedig o 2 the vlues t pevious discete times)

2 Numeicl itegtio Alph method = f ( t ( ), t) Defiig : = ( t) : = ( t ) : = t ( Δt) Appoimtio: Equtio to solve ( ) Itegtio Δt = = f (, t ) = 0.5 = = 0 = = = t t Δ t t Δ t Fowd Eule (eplicit) scheme 2 Ck-Nicholso (mid poit ule) scheme Bckwd Eule (implicit) scheme

3 Numeicl itegtio = ( ( t)) ( t)) Dt Ukows Itegtio t= t 0 = 0,, ( ), ( ) q d σ = = Δt = ( ) Dt = ( )

4 Itegtio of the costitutive equtio: Idetifictio of the cuet stte THEOREM : ( equivlece) Poof: ) = = 0 = = Δt = 0 = = ( ) = b) t t t COROLLARY: > > ( equivlece) Poof: ) > > b) > >

5 ))Itegtio of the costitutive equtio: Idetifictio of the cuet stte Stte idetified t time i the cuet time itevl [ t, t ] Ulodig/NeutlodigTheoem = = 0 = = Δt g= 0 Coolly = 0 > > > = = ( = ) > 0 Δt Δt [ ( )] = ( [( ) ] ) = g = 0 g Lodi 2t Δt Δt Δt

6 Iviscid poblem Iviscid cse d implicit itegtio 0 Iviscid cse = [ ( )] Implicit itegtio = Δt Δt Δt = Numeicl itegtio iheits the popeties of the model d ecoves the solutio of the te idepedet poblem fo the iviscid cse d implicit itegtio

7 Numeicl itegtio: stbility lysis f ( ( t), ( t)) = 0 = t = 0 0 = b eos do ot popgte (stble itegtio) [ ( )] [ Δt( )] Δt Δt Δt = = Δ t Δ t Δt [ Δt( )] = Δt [ Δt( )] Δt b [ Δt( )] Δt 0 Cck-Nicholso (mid-poit ule) [ Δt( )] Δt 2 2 Bckwd-Eule

8 Numeicl itegtio: ccucy lysis Accucy () t = () t () t Alph method ( ) () t = () t () t ( ) Fist ode ccucy/cosistecy: Numeicl solutio fulfills this equtio s = ( ) ( ) ( t ) = ( t ) = = = ; = = = t t t t ( ) = Δt Secod ode ccucy: Numeicl solutio fulfills this equtio s Δt 0 = = ; = = = t t t 0

9 Numeicl itegtio: ccucy lysis Alph method = = ( ) Δt Δt t t ( t ) = ( ) = [ ( t ) ( t )] Tkig time deivtives: Δt t t = ( ) [ ] t t = [ ] [ ] [ ] Δt

10 Numeicl itegtio: ccucy lysis () t = () t () t ; () t () t () t = ( ) ( ) Tkig the limit Δt 0 ( t = t t ) Δt = ( ) ( ) = ( ) holds 2 Δt = 2 = ( ) ( ) ( ) holds oly fo = 2 Alph method: Summy of stbility/itegtio lysis esults Stble: = [,] Fist ode ccute: = [0,] Secod ode ccute: 2 = 2

11 Cosistet (lgoithmic) tget opeto σ () tg Alytic tget opeto = = ( d) lg, t g ( ) ( ) σ lim lg, = t g ( ) 0 lg, : Algoithmic tget opeto Δ t = σ = [ d ] : q ( ) d ( ) = σ = Σ (elstic/ulodig) = [ Δt( ) ] Δt ( ) (lodig) Δ t Δt ( ) δσ = lg, ( ): δ

12 Cosistet (lgoithmic) tget opeto Stess diffeetitio δσ = ( d ) : δ d ( ) δ : δ = : : δ = s : δ = : : δσ ( d ) : δ (elstic/ulodig) s = Δt = lg, : δ [( d ) d' σ σ ]: δ (lodig) Δt ( d ) (elstic/ulodig) Δt H q( ) = ( d ) d' s s d ' = ) lg, (lodig) ( 2 Δt ( ) tg, s Additiol tem O( Δt) ; δ 0 (elstic/ulodig) = Δt δ ) (lodig) Δt

13 Cosistet (lgoithmic) tget opeto lg, ( d) (elstic/ulodig) Δt = ( d ) d' s s Δt tg, Additiol tem O( Δt) Remk If: = 0 lg, = tg, d If: Δ t = 0 lg, = Remk 2 If: 0 = ( d ) d' σ σ lg, (lodig) Algoithmic d lyticl tget opetos mtch Algoithmic viscous tget opeto mtch (i the iviscid limit) the oe fo the te idepedet dmge model

14 Itegtio of the costitutive equtio Numeicl lgoithm: INPUT DATA[ t, t Δt = t ], t = : : σ = : Step Compute = : : = s : = ( ), Step 2 If Elstic Ulodig = d = d = q( ) ; ; = ( d ) lg, s = ( d ) s EXIT

15 Itegtio of the costitutive equtio Numeicl lgoithm: Step 3 If > [ ] Δt( ) Δt q( ) = ; d = Δ t Δt σ = ( d ) σ EXIT = ( d ) lg, Δt H q( ) ( σ σ ) Δt (Lodig) 2 ( ) OUTPUT DATA [ t, t Δt = t ],, σ lg, Fo = 0d = the iviscid model is ecoveed!!!!

16 Sti-te Viscosity depedecies The sti te d the viscosity hve simil effects o the stistess eltio σ σ Effects of sti te = 0 s = 0 s 9 = 0 s 2 Effects of viscosity = 0 MP s 2 = 0 MP s 3 = 0 MP s

17 Fomultio i Voigt s Nottio Tkig ito ccout the symmety of the stess d sti tesos, theey c be witte i vecto fom: REMARK The double cotctio ( σ:) is equivlet to the scl (dot) poduct σ : ({ } { }) { } { } σ:= σ σ ij ij = σ i i 2 d ode tesos vectos γ y γ z 2 2 y z ot. = y y yz γ y y γ = yz 2 2 z yz z γ z γ yz z 2 2 σ y z σ y σ y yz z yz σ z VOIGT S NOTATION y z = R γ y γ z γ yz σ σ y def σ y z yz def 6 {} z 6 { σ} = R

18 Hooke s Lw Hooke s lw i tems of the stess d sti vectos: σ = : = λ 2μ Whee D is the mti of elstic costts: { σ} = D { } σi = Dij j i {,...6} ν ν ν ν ν ν ν ν ν ν E ( ν ) ν ν D = 2ν ( ν)( 2ν) ( ν ) 2ν ( ν ) 2ν ( ν )

19 Ivese elstic costitutive equtio = : σ ν ν = E 2E Whee is the elstic complice mti: D D { } = D { σ} j = ( ) ijσi i {,...,6} ν ν E E E ν ν E E E ν ν E E E E = ; G = 2( ν ) G G G D

20 Algoithm computtios i Voigt s ottio The epessios used i the lgoithms of the model ed: { } { } {}{ D}{} { } { D } { } = : : = σ: = σ = = σ σ { σ} = ( d ( )) { σ} { σ} = D { } {} D {} { } D { } { } { } = = σ σ = σ { σ} Dlg, = = ( d ) D {} Δt Δt H q( ) ({ } { } 2 σ σ ) ( ) 6 6

21 2D Fomultio i Voigt s Nottio Hooke s lw i tems of the stess d sti vectos: def y γ y γ ot y y 2 = y = y γ y y 2 3 {} = REMARK The double cotctio ( σ:) is equivlet to the scl (dot) poduct σ : ({ } { }) { } { } σ:= σ σ ij ij = σ i i 2 d ode tesos Whee is the mti of elstic costts: vectos σ σ y y σ y VOIGT NOTATION def σ y y 3 { σ} = σ

22 2D Fomultio i Voigt s Nottio Hooke s lw i tems of the stess d sti vectos: σ = : = λ 2μ { σ} = D { } σi = Dij j i {, 2, 3} Whee D is the 2-D mti of elstic costts: ν 0 E D = ν 0 2 ν ν Ple stess ν 0 E D = ν 0 2 ν ν Ple sti beig: ν ν = ν E E = 2 ν

23 END OF LECTURE 5