Finite Volume Methods (1D-2D)

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1 e Volume Mehods (D-D) Adaped from Noes o Trase lows by Aruro eo ad Shallow-Waer equaos by Adrew Slegh Aruro eo, Orego Sae versy (all 03) /6/03 Aruro S. eo, OS

2 e Volume Shock-Capurg Mehods Ably o hadle ereme flows Trasos bewee sub / supercrcal flows are easly hadled Oher echques have problems wh ras-crcal flows Seep wave fros ca be accuraely smulaed /6/03 Aruro S. eo, OS

3 Dam break Source: A usrucured ode-ceered fe volume scheme for shallow waer flows wh we/dry fros over comple opography, Nkolos ad Dels 3 /6/03 Aruro S. eo, OS

4 Dam break (amao) hp:// 4 /6/03 Aruro S. eo, OS

5 Illos Trase Model amao (Aruro eo, OS) hp:// 5 /6/03 Aruro S. eo, OS

6 Coservave Equaos (D) S Q A gi A Q Q ga S o S f gi S 0 /6/03 Aruro S. eo, OS 6

7 I ad I Trapezodal chael Base wdh B, Sde slope S = Y/Z I h B h S 3 I h db d h 3 ds d ecagular, S = 0 I h B A B A B I db d 7 Source erm S f Q Q A (4/3) /6/03 Aruro S. eo, OS

8 ecagular Prsmac h hu hu hu gh S gh 0 S S o f 8 V d d S d V /6/03 Aruro S. eo, OS

9 -dmesos I -d we have a era erm: frco S G y hv hu h huv gh hu hu gh hv huv hu G y y f o f o S S gh S S gh S 0 (/3) v u u h S f /6/03 Aruro S. eo, OS 9

10 e volume formulao or homogeeous form.e. whou source erms recagular corol volume - space + V d d 0 -/ +/ 0 -/ +/ /6/03 Aruro S. eo, OS

11 e Volume ormulao (Co.) Defg as egral averages e volume formulao becomes / /, d / /, d / /, d / /, d / / /6/03 Aruro S. eo, OS

12 e Volume ormulao So far o appromao was made The soluo ow depeds o how he egral averages are esmaed I parcular, he er-cell flues +/ ad -/ eed o be esmaed. /6/03 Aruro S. eo, OS

13 e Volume -D If odes ad sdes are labelled as : A A 3 V A 3 4 Soluo s V Where s s ormal flu for sde ec. A 4 s s s3 3 s4 4 3 /6/03 Aruro S. eo, OS

14 V -D ecagular Grd y -/,j+/ G j+/ +/,j+/ -/, j +/ -/,j-/ Soluo s G j-/ +/,j-/ 4 y G G /, j /, j, j / /6/03 Aruro S. eo, OS, j /

15 Goduov mehod for flu compu. ses formao from he wave srucure Assume pecewse lear daa saes + -/ +/ - - Cells Daa saes (0) -/ (0) +/ + 5 lu calculao s soluo of local ema problem /6/03 Aruro S. eo, OS

16 ema Problem The ema problem s a al value problem defed by 0, f f / / The soluo of hs problem ca be used o esmae he flu a +/ 6 /6/03 Aruro S. eo, OS

17 ema Problem (Co.) The ema problem s a geeralsao of he dam break problem Dam wall Deep waer a res Shallow waer a res 7 /6/03 Aruro S. eo, OS

18 Dam Break Soluo h Waer levels a me =* Evoluo of soluo v Velocy a me =* arefaco Wave srucure Shock * 8 /6/03 Aruro S. eo, OS

19 Appromae ema Solvers No eed o use eac soluo Eac soluos requre eraos ad are compuaoally epesve or some problems, eac soluos may o es May ema solvers are avalable (oe s ad H are mos popular) 9 /6/03 Aruro S. eo, OS

20 Appromae ema Solvers oe s Solver Goverg equaos are appromaed as: A ~ A A Where s obaed by oe averagg ~ h h h ~ u~ u h h u h h c~ c c 0 /6/03 Aruro S. eo, OS

21 oe s Solver (Co.) Properes of mar Ege values gh Ege vecors Wave sreghs lu s gve by c u ~ ~ ~ c u ~ ~ ~ c u ~ ~ ~ ) ( c u ~ ~ ~ ) ( u c h h ~ ~ ~ u c h h ~ ~ ~ h h h / ~ ~ ~ j j j j /6/03 Aruro S. eo, OS

22 H Solver S S Hare, a, Va eer * Assume wave speed Cosruc volume * Iegrae 0 * * S S S S / Or, * S S S S / S S /6/03 Aruro S. eo, OS

23 H Solver Wha wave speeds o use? Oe opo: S S or dry bed (rgh) m u c, u m u c, u S u T T c c c T T Smple, bu robus S u c 3 /6/03 Aruro S. eo, OS

24 Hgher Order Space Cosruc ema problem usg surroudg cells May creae oscllaos Pecewse recosruco Need o use lmers /6/03 Aruro S. eo, OS

25 mers Oba a grade for varable cell, Grade obaed from mer fucos Provde grades a cell faces mer =G(a,b) u u a / / / u u b /6/03 Aruro S. eo, OS 5

26 mers (Co.) A geeral lmer G a, b ma 0,m m 0,ma a, b,ma, b a, b,maa, b for a for a 0 0 β= gve MINMOD, β= gve SPEBEE Va eer G a, b a b a a b b 6 /6/03 Aruro S. eo, OS

27 Hgher order me 7 Needs o advace half me sep MSC-Hacock Prmve varable m varable W Evolve he cell face values /: W, W, W 0 pdae as ormal solvg he ema problem usg evolved W, W A W W W A W / / /6/03 Aruro S. eo, OS

28 Alerave me advace Aleravely, evolve he cell values usg Solve as ormal Procedure s d order space ad me W W / / / /6/03 Aruro S. eo, OS 8

29 We / Dry os We / Dry fros are dffcul Source of error Source of sably oud : llg of sorm-waer ad combed sewer sysems loodg - udao 9 /6/03 Aruro S. eo, OS

30 Dry fro speed * u c uc S u c We bed Dry bed c 0 S u c * Dry fro s fas Ca cause problem wh me-sep / Coura umber 30 /6/03 Aruro S. eo, OS

31 Soluos for we/dry fros Mos popular way s o arfcally we bed Provde a small waer deph wh zero velocy Ca drascally affec fro speed Need o be very careful abou dvdg by zero 3 /6/03 Aruro S. eo, OS

32 Illos Trase Model amao hp:// 3 /6/03 Aruro S. eo, OS

Contents. Radial Basis Function Networks. introduction. In MLP. Radial Basis Function Network. In RBFN

Contents. Radial Basis Function Networks. introduction. In MLP. Radial Basis Function Network. In RBFN Radal Bass Fuco Neorks Coes Iroduco Archecure Desgg Learg sraeges MLP vs RBFN roduco roduco I MLP Compleely dffere approach by veg he desg of a eural eork as a curve-fg appromao problem hgh-dmesoal space