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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 dl i x d jy d kz d = + +
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
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