grid generation ts sir

7/18/2019 Grid Generation TS Sir

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AN INTRODUCTION TO

GRID GENERATION

BY

T.SUNDARARAJAN

DEPT. OF MECHANICAL ENGG.

PROFESSOR, IIT MADRAS


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STEPS INVOLVED IN MODELING

PRE-PROCESSING (CREATION GEOMETRY &MESH AND APPLICATION OF BOUNDARYCONDITIONS)

ANALYSIS (SOLUTION OF GOVERNINGEQUATIONS --> M!!, M"#$%'# & E%$*+%$)

POST-PROCESSING (ESTIMATION OF

DESIRED QUANTITIES- SAY, HEATTRANSFER FROM PREDICTION OFTEMPERATURE, ETC.)


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PRE-PROCESSING

CREATION OF GEOMETRY USINGSOLID MODELING OR SURFACEMODELING TECHNIQUES

DISCRETISATION OF THE GEOMETRYINTO MESH BY AUTOMATIC GRIDGENERATION METHODS

APPLICATION OF BOUNDARYCONDITIONS AND SPECIFICATION OFINPUTS


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AUTOMATIC GRID GENERATION

DISCRETISATION OF COMPLEGEOMETRIES INTO A MESH /ITH

DESIRED PROPERTIES GENERATION OF STRUCTURED MESHFOR FINITE DIFFERENCE0 FINITEVOLUME METHOD APPLICATIONS

GENERATION OF UNSTRUCTRED MESHFOR FINITE ELEMENT METHOD

ADAPTIVE MESH FOR REGIONS OFSHARP GRADIENTS IN SOLUTION


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AIRCRAFT MODEL


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DISCRETISATION ERROR

)x(0x

TT

!3

x

dx

Td

!2

x

dx

Td

x

TT

dx

dT

i1i

2

i

3

3

i

2

2

i1i

i

+=

=

+

+

)(0

!3!2

1

2

3

3

2

2

1

xxTT

x

dx

Tdx

dx

Td

x

TT

dx

dT

ii

ii

ii

i

+=

+

=

)x(Ox2

TT

dx

dT 21i1i

i

+

=

+


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GRID ADAPTION

SINCE DISCRETISATION ERROR DEPENDS ONGRADIENTS, /HEREVER GRADIENTS ARE HIGH,STEP SI1E SHOULD BE SMALL.

/E CAN PROPOSE THAT DT0D 2, /HERE 2 ISTHE STEP SI1E. OR, DT0D.2 3 CONSTANT. REFINING THE GRID AT LOCATIONS OF HIGH

GRADIENTS AND DEREFINING AT LOCATIONS OF

LO/ GRADIENTS /ILL LEAD TO OPTIMAL USEOF GRID POINTS. HENCE ONE CAN GETACCURATE SOLUTIONS /ITH LESS NUMBER OFGRID POINTS.


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LAUNCH VEHICLE


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STRUCTURED GRID GENERATION

INVOLVES MAPPING OF COMPLE GEOMETRYINTO A SIMPLE RECTANGULAR SHAPE

BODY FITTING COORDINATES ARE EMPLOYED

SIMPLIFICATION GEOMETRY RESULTS INCOMPLE FORM OF GOVERNING EQUATIONS

LENGTHS, AREAS, VOLUMES AND VECTORS(VELOCITIES ETC.) NEED TO BE TRANSFORMED

MAPPING TRANSFORMATION SHOULD BESMOOTH, PREFERABLY CONFORMAL ANDPROVIDE CONTROL OVER GRID SPACING


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STRUCTURED GRIDS

MAPPING BY BODY- FITTING COORDINATE

TRANSFORMATIONS

ISOPARAMETRIC MAPPING OF SUB-DOMAINS

AND CREATION OF MULTI- BLOC4 GRIDS THE SEQUENCE OF MAPPING DETERMINES

/HETHER THE FINAL MESH /ILL BE A PSEUDO-

RECTANGULAR, O-TYPE, C-TYPE OR H-TYPE

MESH


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PSEUDO- RECTANGULAR MESH


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ISOPARAMETRIC MAPPING

x N x and y N yii

m

i i

i

m

i= == =

1 1

( , ) ( , )


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MULTI- BLOC4 STRUCTURED GRIDS


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MAPPING CONFIGURATIONS


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TRANSFORMATION

RELATIONS

Coordinatetransformation implies 3(x, y)and = (x, y).

O, 5 =5(, )and y = y(, )

Here, (x,y) are physial oordinates and (,)are !ody"fittin# oordinates.


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$%T&'C %&'*T'%+

d

d

dx dy

dx dy

dx

dy

x y

x x

x y

x y

=+

+

=

dxdy

x xy y

dd

=

x y

x y

x x

y y J

y x

y x

=

=

11

x y z

x y z

x y z

x x x

y y y

z z z

=

1


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TRANSFORMATION OF

DERIVATIVES

T

xT

T Tx x x= = +

T

y TT T

y y y= = +

( ) ( ) ( )

( ) ( )

= + = + + +

+ + + +

2 2 2 2 2

2

2

2 2

2

2

2

2

T T TT T T

T T

xx yy x y

x y x x y y


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TRANSFORMATION OF

COORDINATES

( )

CT

t kT

x

T

y Q x y C T T T

k

Jx y

Tx y

Ty y x x

TQ

p p t t +

= + +

+ + + +

=

2

2

2

2

2

2 2

2

2

2 2

2

2

2

2 0

( , )

( ) ( ) ( , )


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LENGTHS, AREAS, VOLUMES

kdA e dl e dl =

e dl i dx jdy i x d jy d = + = +

e dl i dx jdy i x d jy d = + = +

( ) kdA e dl e dl k x y y x d d k J d d = = =

( )( )dv e dl e dl e dl J d d d = = .


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VECTORS (COVARIANT,

CONTRAVARIANT)

e dl i x d jy d kz d = + +



e

i j kx y z

x y z

=

=+ +

+ + 2 2 2

e

i j kx y z

x y z

=

= + +

+ + 2 2 2

e

i j kx y z

x y z

=

= + +

+ + 2 2 2


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METHODS OF STRUCTUREDMESH GENERATION


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STRUCTURED GRID METHODS

ALGEBRAIC MAPPING (ANALYTICAL &

TRANSFINITE INTERPOLATION)

CONFORMAL MAPPING

USE OF ELLIPTIC, PARABOLIC &

HYPERBOLIC PARTIAL DIFFERENTIAL

EQUATIONS


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ALGEBRAIC MAPPING

* -

C

=0 =1

=0

=1

* -

C

=0

=1

=0 =1

(x,y) (,)


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ALGEBRAIC MAPPING

= f x y f x y f x y1 1 3( , ) . / ( , ) ( , )

= f x y f x y f x y 2( , ) . / ( , ) ( , )

Consider the simply onneted domain shon in 3i#4re

hose sidesAB, BC, CD and * are #i5en !y the e64ationsf1(x,y) = 0, f

2(x, y) = 0,f

(x, y) = 0 andf

!(x, y) = 0 respeti5ely.

7itho4t loss of #enerality8 one an map the 4r5es *- and

C onto the lines = 0and = 1.0, 4sin# a transformation of the

form

+imilarly, it an !e ass4med that


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TRANSFINITE

INTERPOLATION9 *pply 4nidiretional interpolation in

"diretion (or "diretion) !eteen the !o4ndary #rid data #i5en on the 4r5es = 0and = 1 (or = 0 and = 1)and o!tain the oordinates x:

py

p: for

e5ery interior and !o4ndary point.9 Cal4late the mismath !eteen the interpolated and the at4al

oordinates on the = 0and = ' (or = 0 and = 1) !o4ndaries.

9 ;inearly interpolate the differene in

the !o4ndary point

oordinates in (or )diretion and find the orretion to !e appliedto the oordinates of e5ery interior point.


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CONFORMAL MAPPING

THIS USES MAPPING TRANSFORMATION OFTHE FORM / 3 F(1) OR 1 3 G(/) /HERE 1 3 6 7 Y AND / 3 U 6 7 V

DEPENDING ON THE TRANSFORMATIONFUNCTION USED, SOME GEOMETRY IN (,Y)DOMAIN CAN BE MAPPED INTO A SIMPLERSHAPE IN (U,V) DOMAIN

MAPPING IS CONFORMAL, ECEPT AT A FE/SINGULAR POINTS

CAN BE USED FOR 8-D DOMAINS


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CONFORMAL MAPPING

9 3 ' 6 7: 3 ; (


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SCH/ART1 CHRISTOFFEL

TRANSFORMATION

z C " # " # " # d" $k k nkn= +( ) ( ) ...( )1 21 2

/2$$ @73 1

I 7! $!* " !2"9 2 , = =n

i ik1 2


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PDE BASED METHODS

ELLIPTIC AND HYPERBOLIC PDES ARECOMMONLY USED

ELLIPTIC PDE TRANSFORMATION- ..+ YY+ 11= P(, , ) ..+YY+11=Q(,,)

..+ YY+ 11= R(, , )

HERE, P, Q, R ARE CONTROL FUNCTIONSFOR OBTAINING DESIRED STEP SI1EVARIATION


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STRUCTURED MESH


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STRUCTURED MESH

GENERATION BY FEM

% & dxdyi =/ 2 0

% dxdyi =/ 2 0


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UNSTRUCTURED MESH

GENERATION INVOLVES ALGORITHMIC DIVISION OF

GIVEN GEOMETRY USING DESIRED

ELEMENT SHAPES SPECIFIC REGIONS OF THE GEOMETRY

COULD BE REFINED (INTRODUCTION

OF NE/ NODES & ELEMENTS) ORDEREFINED (REMOVAL OF NODES0

ELEMENTS)


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UNSTRUCTURED MESH


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UNSTRUCTURED MESH

GENERATION

TRIANGULATION METHODS

- DIVISION OF LARGE TRIANGLES

- DELAUNAY TRIANGULATION - ADVANCING FRONT TECHNIQUE

QUADTREE0 OCTREE METHODS

S/APPING0 SMOOTHING OFADJACENT ELEMENTS MAY BE

REQUIRED


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THREE DIMENSIONAL GRID

GENERATION


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ADAPTIVE MESH GENERATION

MESH IS REFINED0 DEREFINEDDEPENDING ON SOME MEASURE OFERROR

ERROR MEASURES CAN DEPEND ONGRADIENTS, FLU MISMATCH, FINITEELEMENT RESIDUE ETC.

DEPENDING ON THE LEVEL OFERROR, THE ETENT OF ERROR CANBE DECIDED


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STRUCTURED MESH


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STRUCTURED MESH

ADAPTION

%a ' dT dx d T dx k d T dx

n n= + + +1

2 2 ... /

x = 7

T2$!$ $ $? ! $" $'7-?7!7+'7"% #$2"?!


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MODELLING OF PREMIED

COMBUSTION IN A

PROPAGATING FLAME

= D PREMIED FLAME


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=-D PREMIED FLAME

PROPAGATION

)exp(2

2

TkQx

T

t

T=

).exp(2

2

TkQxt =

0=

=

xx

T

0=x

*t x = 08

*t x=18 T = To,x= 0 x = 1


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ERROR BASED GRID ADAPTION

=j

jj TxNT )( =j

jj xN )(

0)).exp((2

2

=

dxTkQxT

t

TNi

dxtx)p

*

ne

n

),(12)(

2

2

2

%2

n(t) =

n

n t+ )(2%2(t) =


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ERROR ON UNIFORM MESHES


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PREDICTION ON UNIFORM MESHES


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SOLUTION /ITH ADAPTIVE MESH


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H-VERSION & MOVING MESHES


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ERROR ON ADAPTIVE MESHES


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CONCLUSIONS

AUTOMATIC GRID GENERATION HELPS IN

REDUCING DRUDGERY OF USER INPUT

ADAPTIVE GRID GENERATION IMPROVES

THE ACCURACY OF PREDICTIONS STRUCTURED OR UNSTRUCTURED

MESHES MAY BE GENERATED DEPENDING

ON THE DEMAND OF THE APPLICATION

grid generation ts sir

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