Traditional Discretisation Methods. Finite Difference Finite Element Finite Volume

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1 Introdcton to element based comptng --- fnte volme and fnte element methods. Mesh generaton Joanna Szmelter Loghborogh Unversty, UK

2 Tradtonal Dscretsaton Methods Fnte Dfference Fnte Element Fnte Volme

3 Fnte Dfference Method -1,,1-1,1, 1,1 1, x x 2,, 1 1, x 0 2, 2, 0 1, 1 1, 1, = = y v v x t y v x t y 2 O x central x x,, 1, forward x O

4 Fnte Volme Method 0 0 = = d y v d x d t y v x t 0 0 = Γ = Γ Γ Γ y x y x S v S V t d n v d n d t From Gass Dvergence Theorem: Γ n x n y

5 Fnte Element Method Element Nodes = = d y v d x d t y v x t Weghted resdal analyss: = = v W d y N W d x N N W d t W d y N v W d x N N W d t N N N N N =

6 Fnte Element Method Weghted resdal analyss: If W s chosen to be the same as N the method s called Galerkn method. 0 = v W d y N W d x N N W d t 0 = Γ Γ Γ Γ y x v d n N N d n N N v d y N N d x N N d N N t For easy mplementaton of bondary condtons ths s ntegrated by parts. 0 0 = = Yelem Xelem elem v t v d N y N d N x N d N N t B B M = 0 Yelem Xelem elem v t B B M

7 Fnte Element Method 1 Dvdes comptatonal space nto elements t M elem B Xelem B Yelem v = 0 For every element constrcts M, B, B ;,, v elem Xelem Yelem elem elem elem 2 Reconnects elements at nodes Agglomerates for the whole doman M = = e e M elem elem, B X, = = e e B elem Xelem, B, v = Y = e e v B elem Yelem 3 As a reslt a set of algebrac eqatons s formed and ts solton follows M B X BY v = 0 t

8 SHAPE FUNCTIONS The tetrahedron famly of elements N And when dervatves are of nterest: d dn dx dx We know fnctons N, they are freqently polynomals -obtanng ther dervatves s easy. After Zenkewcz et al FEM 2000

9 SHAPE FUNCTIONS Example: a lnear nterpolaton of a scalar T n a trangle. The vale of T n an arbtrary pont alpha s approxmated by: 1N1 2N2 3N3 α N A / 1 = 1 A element where A s an area 3 α A 1 2 1

10 SHAPE FUNCTIONS Crvlnear elements can be formed sng transformatons Isoparametrc elements

11 Typcal Examples of Data Strctres Strctred Pont based --- I,J,K ndexng Set of coordnates and connectvtes Natrally map nto the elements of a matrx,,

12 Typcal Examples of Data Strctres Strctred Bnary trees

13 Typcal Examples of Data Strctres Unstrctred Element based connectvty Element Element Element nformaton related to shape fnctons

14 Typcal Examples of Data Strctres Unstrctred Edge geometrcal nformaton

15 Edge based data

Tranglaton Strctred Meshes Cartesan grds wth mappng

16 Selected Mesh Generaton Technqes Unstrctred Meshes Drect tranglaton Advancng Front Technqe Delanay Tranglaton Strctred Meshes Cartesan grds wth mappng and/or mmersed bondares Varants of cosahedral meshes Others Others

17 An example of the drect tranglaton Redced Gassan Grd Rossby-Harwtz Wave Prmary mesh 5 days Dal mesh 14 days

18 Advancng Front Technqe Smltaneos mesh pont generaton and connectvty

19 Delanay Tranglaton How to connect a gven set of ponts?

20 Delanay Tranglaton Create Vorono polygons,.e. The constrct that assgns to each pont the area of the plane closer to that pont than to any other pont n the set. A sde of a Vorono polygon mst be mdway between the two ponts whch t separates

21 Delanay Tranglaton If all pont pars of whch have some bondary n common are ont by straght lnes, the reslt s a tranglaton of the convex hll of the ponts.

22 Delanay Tranglaton mesh constrcted from the redced Gassan grd ponts

23 Flxes can be constrcted for the srfaces whch ct cells sold Geometry conformng meshes

24 Meshng technqes for mesh adaptvty -for lower order elements are: pont enrchment h-refnement, and atomatc regeneraton mesh movement Dynamc regeneratons for Meshes tlsng a mappng are partclarly effcent -for hgher order elements p-refnement ncreases order of shape fncton that s the order of polynomal sed for elemental nterpolaton

25 The Edge Based Fnte Volme Dscretsaton Edges Medan dal comptatonal mesh Fnte volmes 25

26 Geosphercal framework Szmelter & Smolarkewcz, J. Compt. Phys. 2010

27 A stratfed 3D mesoscale flow past an solated hll Redced planets Wed & Smolarkewcz, QJR

28 Stratfed mesoscale flow past an solated hll on a redced planet 4 hors Fr=2 Fr=1 Fr=0.5 Hnt & Snyder J. Fld Mech. 1980; Smolar. & Rotnno, J. Atmos. Sc ; Wed & Smolar., QJR 2009

29 Smth, Advances n Geophys 1979; Hnt, Olafsson & Bogealt, QJR 2001

30 Nonhydrostatc Bossnesq montan wave Szmelter & Smolarkewcz, Comp. Flds, 2011 NL/Uo = 2.4 Comparson wth the EULAG s strctred mesh reslts --- very close wth the lnear theores Smth 1979, Drran 2003: over 7 wavelenghts : 3% n wavelength; 8% n propagaton angle; wave ampltde loss 7%

31 Gravty wave breakng n an sothermal stratosphere Lpps & Hemler Drran Prsa et al JAS 1996, Smolarkewcz & Margoln, Atmos. Ocean 1997 Klen, Ann. Rev. Fld Dyn., 2010, Smolarkewcz et al Acta Geoph

32 Smolarkewcz et al Acta Geoph 2011 Gravty wave breakng n an sothermal stratosphere Nonhydrstatc Edge-Based NFT EULAG CV/GRID Drran Lpps & Hemler 32

33 Statc mesh adaptvty wth MPDATA based error ndcator Schär Mon. Wea. Rev Recommended mesh ca10000 ponts Coarse ntal mesh 80x45 =3600 ponts and solton Adapted mesh 8662 ponts and solton 33

34 Cartesan dx= Dstorted prsms dx= tetra dx= Stratfed flow past a steep solated hll 34

35 Stratfed flow past a steep solated hll Hnt & Snyder JFM 1980 Smolarkewcz & Rotno JAS

36 Hanazak, JFM 1989 Low Frode Nmber Fr=0.3 Flow Past a Sphere and Hem-Sphere Hybrd mesh 36

37 Hanazak, JFM 1989

38 Convectve Planetary Bondary Layer Schmdt & Schmann JFM 1989 Smolarkewcz et al JCP 2013 Edge-based T Edge-based C EULAG SS LES Observaton 64x64x51 38

39 What nherent constrants does a flow solver choce pose on a mesh? CONVERGENCE OF FINITE-VOLUME MPDATA ON UNSTRUCTURED MESH

40 Selected references for frther detals Smolarkewcz, PK, Szmelter, J, Wyszogrodzk, AA 2013 An nstrctred-mesh atmospherc model for nonhydrostatc dynamcs, Jornal of Comptatonal Physcs, 254,pp Smolarkewcz, PK and Szmelter, J 2011 A Nonhydrostatc Unstrctred-Mesh Sondproof Model for Smlaton of Internal Gravty Waves, Acta Geophysca, 596,pp Szmelter, J and Smolarkewcz, PK 2011 An edge-based nstrctred mesh framework for atmospherc flows, Compters and Flds, 461, pp Szmelter, J and Smolarkewcz, PK 2010 An edge-based nstrctred mesh dscretsaton n geosphercal framework, Jornal of Comptatonal Physcs, 229, pp Smolarkewcz, PK and Szmelter, J 2009 Iterated pwnd schemes for gas dynamcs, Jornal of Comptatonal Physcs, 2281, pp Smolarkewcz, PK and Szmelter, J 2008 An MPDATA-based solver for compressble flows, Internatonal Jornal for Nmercal Methods n Flds, 568, pp Szmelter, J and Smolarkewcz, PK 2006 MPDATA error estmator for mesh adaptvty, Internatonal Jornal for Nmercal Methods n Flds, 5010, pp Smolarkewcz, PK and Szmelter, J 2005 MPDATA: An Edge-Based Unstrctred-Grd Formlaton, Jornal of Comptatonal Physcs, 2062, pp Zenkewcz OC, Taylor RL, The Fnte Element Method,5 th Ed, Btterworth-Heneman, 2000 Szmelter, J and Pagano, A 1995 Vscos flow modellng sng nstrctred meshes for aeronatcal applcatons, Lectre Notes n Physcs, Sprnger 453, pp Evans, A, Marchant, MJ, Weatherll, NP, Szmelter, J, Natksmah, DK 1992 A Pont Enrchment Strategy for Polygonal Fnte Volme Meshes, Appled Mathematcal Modellng, 16, pp Szmelter, J, Marchant, MJ, Evans, A, Weatherll, NP 1992 Two-dmensonal Naver-Stokes Eqatons - Adaptvty on Strctred Meshes, Comp. Meth. Appl. Mech. Eng, 101, pp Evans, A, Marchant, MJ, Szmelter, J, Weatherll, NP 1991 Adaptvty for Compressble Flow Comptatons Usng Pont Embeddng on 2-D Strctred Mltblock Meshes, Internatonal Jornal for Nmercal Methods n Engneerng, 32, pp Zenkewcz, OC, Szmelter, J, Perare, J 1990 Compressble and Incompressble Flow: An Algorthm for All Seasons, Comp. Meth. Appl. Mech. Eng, 78, pp W, J, Zh, JZ, Szmelter, J, Zenkewcz, OC 1990 Error Estmaton and Adaptvty n Naver-Stokes Incompressble Flows, Comptatonal Mechancs, 6, pp

An Improved Isogeometric Analysis Using the Lagrange Multiplier Method

An Improved Isogeometric Analysis Using the Lagrange Multiplier Method An Improved Isogeometrc Analyss Usng the Lagrange Mltpler Method N. Valzadeh 1, S. Sh. Ghorash 2, S. Mohammad 3, S. Shojaee 1, H. Ghasemzadeh 2 1 Department of Cvl Engneerng, Unversty of Kerman, Kerman,