cae grid geometry

8/13/2019 CAE Grid Geometry

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CFD Lecture 6

Dr. Thomas J. Barber



Numerical ConceptsNumerical Properties: Time-Accurate vs. Time-archin!

" Time-marchin!: stea#$-state solution %orm unstea#$e&uations

' (nterme#iate solution has no meanin!

" Time-accurate: time-#epen#ent) vali# at an$ time step



Numerical ConceptsNumerical Properties o% *cheme

" Consistenc$

' (mplies numerical e&uations !o to PD+ as ∆t) ∆, !o to . ' (mplies #iscretiation can be reverse# /usin! Ta$lor0 to !et PD+

" *tabilit$

' Ten#enc$ o% error in solution o% al!ebraic e&uations to #eca$

' (mplies numerical solution !oes to e,act solution o% #iscretie# e&uations

" Conver!ence

' *olution o% appro,imate e&uations approaches e,act set o% al!ebraic e&ns.

' *olutions o% al!ebraic e&ns. approaches e,act solution o% P.D.+.1s as ∆ x ∆ t ⇒ 0

+,act *olution

2

3overnin!

P.D.+.1sL/20

*$stem o% Al!ebraic +&uations

Appro,imate *olution

u

Discretization

Consistency

Convergence

as ∆ x ∆ t ⇒ 0



4o5 !oo# are the results

" Assess the calculation %or ' 3ri# in#epen#ence

' Conver!ence /mathematical0: resi#uals as measure o% ho55ell the %inite #i%%erence e&uation is satis%ie#.

7 Loo8 %or location o% ma,imum errors

7 Loo8 %or non-monotonicit$

( )

( )

2

2

,

max

. . . 0

. . . ( , ) 01

. . . ( )

,

n n

i j ij

n

ij

i j

n

ij i j

P D E Lu

F D E Lu x y

R M S error L norm N

Max error at x y

ε

ε

ε

=

= ≠

=

=

∑



Preprocessin!

3eometr$ 9

3ri# 3eneration (ntro;



<h$ #oes preprocessin! ta8e so lon!

" 4i!h-%i#elit$ anal$sis process time /historicall$0

" ulti-%i#elit$ anal$sis approach is t$picall$ use#) but

' Do the$ start 5ith lo5 %i#elit$ results

' Are the hi!h %i#elit$ anal$sis results too late to have an e%%ect

Geometry & Grid

(60%)

Solution

(20%)

Analysis

(20%)

0D !eanline

"D

2D Streamline

#D



Co#e Preprocessin!3eometr$

" CAD !eometr$ must be a#apte# = cleane# up %or !ri# !enerationan# %lo5 solution

' eliminate bolts) %illets) real !eometr$ e%%ects

' eliminate casin! !aps or treat 5ith a boun#ar$ con#ition

" *o%t5are not available to maintain common #atabase

' CAD ⇒ CFD anal$sis) 4eat Trans%er anal$sis) >

' CFD ⇒ ?D anal$ses ⇔ @D anal$ses ⇔ D anal$ses

" 3eometrical topolo!ies are %re&uentl$ not suite# to simple!ri##in! concepts



Co#e Preprocessin!3eometr$: 3as Turbine Combustor

$lo rom

'ig Comressor

$lo into

'ig *ur+ine

$uel

,n-ector



Co#e Preprocessin!3eometr$: Combustor Liner

Dilution .et



Co#e Preprocesin!3ri# 3eneration ptions

" *tructure# 3ri# =*tructures Data

" 2nstructure#Data = *tructure#

3ri#

i,j+1

i-1,j i,j

i,j-1

i+1,j

X , i

Y, jUi,j

61

35 36

11

37

X

Y

U3660

10 12

62



Co#e Preprocessin!3ri# 3eneration

" Attributes o% a ba# !ri#

' some points insi#e bo#$

' %e5 points on or near sur%ace




*ransormation to a ne coordinate system *ransormation to a stretced grid



Co#e Preprocessin!3ri# 3eneration - 3eneric Topolo!ies

/locstructured

1 '

" !ore comlicated grids can +e constructed +y com+ining te +asic grid

toologies cylinder in a duct

1verset or Cimera

Cartesian 3olar

/ot tae advantage o natural symmetries o te geometry



Co#e Preprocessin!3ri# 3eneration - 3eneric Topolo!ies

Cartesiansteise

" !ore comlicated grids can +e constructed taing advantage o simle elements

4nstructuredy+rid

Dimension 4nstructured Structured

@D trian!ular &ua#rilateral D tetrahe#ra he,ahe#ra



3ri# Topolo!$ De%initions = ptions

" an$ #i%%erent cell=element an# !ri# t$pes are available." Choice #epen#s on the problem an# the solver capabilities." Cell or element t$pes:



node

face

cell

facecell

node

edge

2D computational grid

3D computational grid

cell

center

Terminolo!$

" Cell control volume into5hich #omain is bro8en up.

" No#e !ri# point.

" Cell center center o% a cell." +#!e boun#ar$ o% a %ace.

" Face boun#ar$ o% a cell.

" one !roupin! o% no#es)%aces) an# cells:

' <all boun#ar$ one. ' Flui# cell one.

" Domain !roup o% no#e)%ace an# cell ones.



3ri# t$pes: structure# !ri#

" *in!le-bloc8) structure# !ri#. ' i)E)8 in#e,in! to locate nei!hborin! cells.

' 3ri# lines must pass all throu!h #omain.

" bviousl$ can1t be use# %or ver$ complicate#

!eometries.



" Di%%erent t$pes o% he,ahe#ral !ri#s.

" *in!le-bloc8. ' The mesh has to be represente# in a sin!le bloc8.

' Connectivit$ in%ormation /i#enti%$in! cell nei!hbors0 %or entire mesh isaccesse# b$ three in#e, variables: i) E) 8.

*in!le-bloc8 !eometr$ Lo!ical representation.

" *in!le-bloc8 meshes ma$ inclu#e ? #e!ree corners.

+ +

+ +

Face meshin!: structure# !ri#s



3ri# t$pes: multibloc8

" ulti-bloc8) structure# !ri#.

' 2ses i)E)8 in#e,in! 5ithineach mesh bloc8.

' The !ri# can be ma#e upo% /some5hat0 arbitraril$-connecte# bloc8s.

" ore %le,ible than sin!lebloc8) but still limite#.

Source: www.cfdreview.com



" Di%%erent t$pes o% he,ahe#ral !ri#s.

' ulti-bloc8. ' The mesh can be represente# in multiple bloc8s.

ulti-bloc8 !eometr$ Lo!icalrepresentation.

' This structure !ives %ull control o% the mesh !ra#in!)usin! e#!e meshin!) 5ith hi!h-&ualit$ elements.

' anual creation o% multi-bloc8 structures is usuall$more time-consumin! compare# to unstructure#meshes.

Face meshin!: multibloc8



3ri# t$pes: unstructure#

" 2nstructure# !ri#.

' The cells are arran!e# in an arbitrar$ %ashion.

' No i)E)8 !ri# in#e,) no constraints on cell la$out.

" There is some memor$ an# CP2 overhea# %or unstructure#re%erencin!.

Unstructured mesh on a dinosaur



2nstructure# 3ri#

Face meshin!: unstructure# !ri#s

" Di%%erent t$pes o% he,ahe#ral !ri#s. ' 2nstructure#.

' The mesh has no lo!ical representation.



Face meshin!: Gua# e,amples

" Gua#: ap.

" Gua#: *ubmap.

" Gua#: Tri-Primitive.

" Gua#: Pave an# Tri-Pave.



3ri# t$pes: h$bri#

" 4$bri# !ri#. ' 2se the most appropriate cell t$pe in an$ combination.

7 Trian!les an# &ua#rilaterals in @D.

7 Tetrahe#ra) prisms an# p$rami#s in D.

' Can be non-con%ormal: !ri#s lines #on1t nee# to match atbloc8 boun#aries.

trian!ular sur%ace mesh

on car bo#$ is &uic8 an#

eas$ to create

prism la$er

e%%icientl$ resolves

boun#ar$ la$er

tetrahe#ral

volume mesh

is !enerate#

automaticall$

non-conformal

interface



omplex !eometrie"

Surface me"# for a grid

containing onl$ tetra#edra

Tetrahe#ral mesh

" *tart %rom D boun#ar$mesh containin! onl$trian!ular %aces.

" 3enerate mesh consistin!

o% tetrahe#ra.



" Flo5 ali!nment 5ell#e%ine# in speci%ic re!ions.

" *tart %rom D boun#ar$an# volume mesh:

' Trian!ular an# &ua#rilateral%aces.

' 4e,ahe#ral cells.

" 3enerate onal h$bri#mesh) usin!:

' Tetrahe#ra. ' +,istin! he,ahe#ra.

' Transition elements:p$rami#s.

Surface me"# for a grid containing

#exa#edra, p$ramid", and tetra#edra

(and pri"m")

onal h$bri# mesh



" Parametric stu#$ o% comple,!eometries.

" Noncon%ormal capabilit$allo5s $ou to replace portion

o% mesh bein! chan!e#." *tart %rom D boun#ar$ mesh

or volume mesh.

" A## or replace certain partso% mesh.

" Hemesh volume i% necessar$.

%onconformal me"#

for a valve port

nonconformalinterface

Noncon%ormal mesh



esh namin! conventions - topolo!$

" *tructure# mesh: the mesh %ollo5s a structure#i)E)8 convention.

" 2nstructure# mesh: no re!ularit$ to the mesh.

" ultibloc8: the mesh consists o% multiple bloc8s)each o% 5hich can be either structure# orunstructure#.



esh namin! conventions ' cell t$pe

" Tri mesh: mesh consistin! entirel$ o% trian!ular elements.

" Gua# mesh: consists entirel$ o% &ua#rilateral elements.

" 4e, mesh: consists entirel$ o% he,ahe#ral elements.

" Tet mesh: mesh 5ith onl$ tetrahe#ral elements." 4$bri# mesh: mesh 5ith one o% the %ollo5in!:7 Trian!les an# &ua#rilaterals in @D.

7 An$ combination o% tetrahe#ra) prisms) p$rami#s in D.

7 Boun#ar$ la$er mesh: prims at 5alls an# tetrahe#raever$5here else.

7 4e,core: he,ahe#ra in center an# other cell t$pes at 5alls." Pol$he#ral mesh: consists o% arbitrar$ pol$he#ra.

" Noncon%ormal mesh: mesh in 5hich !ri# no#es #o notmatch up alon! an inter%ace.



esh &ualit$

" For the same cell count) he,ahe#ral meshes 5ill!ive more accurate solutions) especiall$ i% the!ri# lines are ali!ne# 5ith the %lo5.

" The mesh #ensit$ shoul# be hi!h enou!h tocapture all relevant %lo5 %eatures.

" The mesh a#Eacent to the 5all shoul# be %ineenou!h to resolve the boun#ar$ la$er %lo5. (nboun#ar$ la$ers) &ua#) he,) an# prism=5e#!ecells are pre%erre# over tri1s) tets) or p$rami#s.

" Three measures o% &ualit$: ' *8e5ness.

' *moothness /chan!e in sie0.

' Aspect ratio.



" T5o metho#s %or #eterminin! s8e5ness:

?. Base# on the e&uilateral volume:7 *8e5ness

7 Applies onl$ to trian!les an# tetrahe#ra.

7 De%ault metho# %or tris an# tets.

@. Base# on the #eviation %rom a normalie# e&uilateralan!le:

7 *8e5ness /%or a &ua#0

7 Applies to all cell an# %ace shapes.

7 Al5a$s use# %or prisms an# p$rami#s.

max max minθ θ − −

&0

&0

&0

&0,

optimal (e'uilateral) cell

actual cell

circumcircle

esh &ualit$: s8e5ness



"Common measure o% &ualit$ is base# on e&uian!le s8e5.

" De%inition o% e&uian!le s8e5:

5here: θ

ma, lar!est an!le in %ace or cell.

θmin smallest an!le in %ace or cell.

θe an!le %or e&uian!ular %ace or cell.7 e.!.) 6 %or trian!le) I %or s&uare.

" Han!e o% s8e5ness:

−

−

−

e

mine

e

emax ,1(0

maxθ

θ θ

θ

θ θ

θ min

θ max

0 1

e"t wor"t

+&uian!le s8e5ness



" Chan!e in sie shoul# be !ra#ual /smooth0.

" Aspect ratio is ratio o% lon!est e#!e len!th to shortest e#!elen!th. +&ual to ? /i#eal0 %or an e&uilateral trian!le or a s&uare.

"moot# c#ange large *ump in

in cell "i+e cell "i+e

a"pect ratio 1 #ig#-a"pect-ratio 'uad

a"pect ratio 1 #ig#-a"pect-ratio triangle

esh &ualit$: smoothness an# aspect ratio



*trivin! %or &ualit$

" A poor &ualit$ !ri# 5ill cause inaccurate solutions

an#=or slo5 conver!ence." inimie e&uian!le s8e5:

' 4e, an# &ua# cells: s8e5ness shoul# not e,cee# ..

' Tri1s: s8e5ness shoul# not e,cee# ..

' Tets: s8e5ness shoul# not e,cee# .I.

" inimie local variations in cell sie: ' +.!. a#Eacent cells shoul# not have Ksie ratio1 !reater than @.

" (% such violations e,ist: #elete mesh) per%ormnecessar$ #ecomposition an#=or pre-mesh e#!es an#

%aces) an# remesh.



inade'uate etter flow

/

3ri# #esi!n !ui#elines: resolution" Pertinent %lo5 %eatures shoul# be a#e&uatel$ resolve#.

" Cell aspect ratio /5i#th=hei!ht0 shoul# be near one5here %lo5 is multi-#imensional.

" Gua#=he, cells can be stretche# 5here %lo5 is %ull$-#evelope# an# essentiall$ one-#imensional.

low Direction



3ri# #esi!n !ui#elines: smoothness

" Chan!e in cell=element sie shoul# be !ra#ual /smooth0.

" (#eall$) the ma,imum chan!e in !ri# spacin! shoul# beM@:

"moot# c#ange

in cell "i+e"udden c#ange

in cell "i+e 4D/

5 5 5

∆xi ∆xi61

2.1x

x

i

1i ≤∆

∆ +



3ri# #esi!n !ui#elines: total cell count

" ore cells can !ive hi!her accurac$. The #o5nsi#eis increase# memor$ an# CP2 time.

" To 8eep cell count #o5n: ' 2se a non-uni%orm !ri# to cluster cells onl$ 5here the$ are

nee#e#.

' 2se solution a#aption to %urther re%ine onl$ selecte# areas." Cell counts o% the or#er:

' ?+ are relativel$ small problems.

' ?+ are interme#iate sie problems.

' ?+6 are lar!e. *uch problems can be e%%icientl$ run usin!

multiple CP2s) but mesh !eneration an# post-processin! ma$become slo5.

' ?+O are hu!e an# shoul# be avoi#e# i% possible. 4o5ever) the$are common in aerospace an# automotive applications.

' ?+ an# more are #epartment o% #e%ense st$le applications.



*olution a#aption

" 4o5 #o $ou ensure a#e&uate !ri# resolution) 5hen $ou#on1t necessaril$ 8no5 the %lo5 %eatures *olution-base#!ri# a#aption

" The !ri# can be re%ine# or coarsene# b$ the solver base#on the #evelopin! %lo5:

' *olution values. ' 3ra#ients.

' Alon! a boun#ar$.

' (nsi#e a certain re!ion.



A#aption e,ample: %inal !ri# an# solution

2D planar "#ell - contour" of pre""ure

final grid

2D planar "#ell - final grid

cae grid geometry

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