Table Of ContentTHE FUNDAMENTALS OF
GRID GENERATION
Patri
k M. Knupp and Stanly Steinberg
(cid:13)
1992by Patri
kM. Knupp
Printed September 8, 2002
Grid for the Continental Shelf of the Gulf of Mexi
o
(cid:13)
1992 by P.M. Knupp, September 8, 2002 i
To a Sense of Spa
e, Montana.
Contents
1 Preliminaries 1
1.1 The Goals of Grid Generation . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Tools of Grid Generation . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Mappings and Invertibility . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Spe
ial Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Trans(cid:12)nite Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Appli
ation to Hosted Equations 23
2.1 Transformationof the 1D Hosted Equation . . . . . . . . . . . . . . . 24
2.2 Dis
retization of the 1D Transformed Equation . . . . . . . . . . . . . 26
2.3 Transformationof the 2D Hosted Equation . . . . . . . . . . . . . . . 30
2.4 Dis
retization of the 2D Transformed Equation . . . . . . . . . . . . . 33
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3 Grid Generation on the Line 41
3.1 Introdu
tion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 Generatorsthat Control Grid Spa
ing . . . . . . . . . . . . . . . . . . 42
3.3 1-D PoissonGrid Generators . . . . . . . . . . . . . . . . . . . . . . . 46
3.4 Numeri
al Implementation. . . . . . . . . . . . . . . . . . . . . . . . . 48
3.5 Minimization Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.6 The Cal
ulus of Variations. . . . . . . . . . . . . . . . . . . . . . . . . 53
3.7 A Fourth-Order Grid Generation Equation . . . . . . . . . . . . . . . 54
3.8 Feature-Adaptive Moving Grids . . . . . . . . . . . . . . . . . . . . . 56
3.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4 Ve
tor Cal
ulus and Di(cid:11)erential Geometry 59
4.1 Ve
tor Cal
ulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.1.1 Ve
tor Geometry of Curves . . . . . . . . . . . . . . . . . . . . 61
4.1.2 Ve
tor Geometry of Surfa
es . . . . . . . . . . . . . . . . . . . 62
4.1.3 Normals and Gradients on Impli
it Surfa
es and Curves . . . . 65
4.2 General Coordinate Relationships . . . . . . . . . . . . . . . . . . . . 65
4.2.1 Coordinate Relationships In One Dimension . . . . . . . . . . . 66
4.2.2 Coordinate Relationships In Two Dimensions . . . . . . . . . . 67
4.2.3 Coordinate Relationships In Three Dimensions . . . . . . . . . 70
4.2.4 Coordinate Relationships On Curves and Surfa
es . . . . . . . 74
4.3 Elementary Di(cid:11)erential Geometry . . . . . . . . . . . . . . . . . . . . 74
4.4 Ve
tor Cal
ulus and Di(cid:11)erential Geometry in the Plane . . . . . . . . 77
4.4.1 First-Order Relationships . . . . . . . . . . . . . . . . . . . . . 77
4.4.2 Se
ond-Order Relationships . . . . . . . . . . . . . . . . . . . . 79
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1992 by P.M. Knupp, September 8, 2002 iii
5 Classi
al Planar Grid Generation 82
5.1 Introdu
tion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.2 The Planar Grid-Generation Problem . . . . . . . . . . . . . . . . . . 83
5.3 Non-Ellipti
Grid Generators . . . . . . . . . . . . . . . . . . . . . . . 84
5.3.1 Algebrai
Grid Generation. . . . . . . . . . . . . . . . . . . . . 84
5.3.2 Conformal and Quasi-ConformalMapping Te
hniques . . . . . 88
5.3.3 Orthogonal Grid Generation . . . . . . . . . . . . . . . . . . . 89
5.3.4 Hyperboli
and Paraboli
Grid Generation . . . . . . . . . . . 90
5.3.5 Biharmoni
Grid Generation . . . . . . . . . . . . . . . . . . . 93
5.4 Ellipti
Grid Generation . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.4.1 The Simplest Ellipti
Generator (Length) . . . . . . . . . . . . 94
5.4.2 The Winslow or Smoothness Grid Generator . . . . . . . . . . 95
5.4.3 Trun
ation Errorin Grid Generation: A Case Study . . . . . . 97
5.5 The Inhomogeneous TTM Grid Generator . . . . . . . . . . . . . . . . 98
5.6 Controlling the Grid Near the Boundary . . . . . . . . . . . . . . . . . 102
5.6.1 The Neumann Boundary Condition. . . . . . . . . . . . . . . . 102
5.6.2 The Steger-SorensonApproa
h . . . . . . . . . . . . . . . . . . 104
5.7 Solution-Adaptive Algorithms . . . . . . . . . . . . . . . . . . . . . . . 106
5.7.1 Grid Adaption with InhomogeneousTTM . . . . . . . . . . . 106
5.7.2 The Deformation Method . . . . . . . . . . . . . . . . . . . . . 107
6 Variational Planar Grid Generation 109
6.1 Introdu
tion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.2 The Cal
ulus of Variations. . . . . . . . . . . . . . . . . . . . . . . . . 110
6.2.1 Minimization Theory. . . . . . . . . . . . . . . . . . . . . . . . 113
6.2.2 Ellipti
ity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.3 Variational Grid Generation in the Plane . . . . . . . . . . . . . . . . 117
6.3.1 The Length Fun
tional. . . . . . . . . . . . . . . . . . . . . . . 119
6.3.2 The Area Fun
tional . . . . . . . . . . . . . . . . . . . . . . . . 121
6.3.3 The OrthogonalityFun
tional . . . . . . . . . . . . . . . . . . . 122
6.3.4 Combinations of Fun
tionals . . . . . . . . . . . . . . . . . . . 123
6.3.5 The AO Fun
tional. . . . . . . . . . . . . . . . . . . . . . . . . 124
6.3.6 The Referen
e Grid and the Repli
ation Idea . . . . . . . . . . 125
6.4 Numeri
al Algorithms for Variational Generators . . . . . . . . . . . . 127
6.5 The Dire
t Optimization Method . . . . . . . . . . . . . . . . . . . . 128
7 Tensor Analysis and Transformation Relationships 131
7.1 Introdu
tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.2 Tensor Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.2.1 The Tra
e of a Tensor . . . . . . . . . . . . . . . . . . . . . . . 132
7.2.2 The Gradient of a Ve
tor . . . . . . . . . . . . . . . . . . . . . 132
7.2.3 The Divergen
e of a Tensor . . . . . . . . . . . . . . . . . . . . 133
7.2.4 The Gradient of a Tensor . . . . . . . . . . . . . . . . . . . . . 134
7.2.5 Curl and Lapla
ian . . . . . . . . . . . . . . . . . . . . . . . . 135
7.2.6 The Tensor Produ
t . . . . . . . . . . . . . . . . . . . . . . . . 135
7.3 TransformationRelations . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.3.1 Gradient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.3.2 Divergen
e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7.3.3 Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7.3.4 Lapla
ian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
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1992 by P.M. Knupp, September 8, 2002 iv
7.3.5 Derivation of the Winslow Equations . . . . . . . . . . . . . . . 141
7.3.6 Type Invarian
e of the Hosted Equation . . . . . . . . . . . . . 142
7.3.7 Transformation of the Time Derivative . . . . . . . . . . . . . 142
8 Advan
ed Planar Variational Grid Generation 145
8.1 Introdu
tion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
8.2 Variational Prin
iples . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
8.2.1 Fun
tionals of the Metri
Tensor . . . . . . . . . . . . . . . . . 146
8.2.2 Rigid Body Transformationsof the Domain . . . . . . . . . . . 149
8.3 The Euler-LagrangeEquations . . . . . . . . . . . . . . . . . . . . . . 150
8.3.1 The General Planar Euler-LagrangeEquations . . . . . . . . . 150
8.3.2 The Euler-LagrangeEquations for the Prin
iple I4 . . . . . . . 152
8.3.3 Numeri
al Implementation . . . . . . . . . . . . . . . . . . . . 154
8.3.4 The Covariant Proje
tions . . . . . . . . . . . . . . . . . . . . 156
8.3.5 Tensor Form of the Euler-LagrangeEquations . . . . . . . . . 157
8.3.6 Logi
al Spa
e Weighting . . . . . . . . . . . . . . . . . . . . . 160
8.4 The Se
ond Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
8.5 Inverse Mapping Approa
h . . . . . . . . . . . . . . . . . . . . . . . . 162
9 Grid Generation in Three Dimensions 165
9.1 Volume Di(cid:11)erential Geometry . . . . . . . . . . . . . . . . . . . . . . 166
9.2 Approa
hes to Three-dimensional Grid Generation . . . . . . . . . . . 168
9.2.1 The Volume Grid-Generation Problem . . . . . . . . . . . . . . 168
9.2.2 3D Trans(cid:12)nite Interpolation . . . . . . . . . . . . . . . . . . . 170
9.2.3 3D Thompson-Thames-Mastin . . . . . . . . . . . . . . . . . . 171
9.3 The Variational Method . . . . . . . . . . . . . . . . . . . . . . . . . . 172
9.3.1 3D Variational Prin
iples . . . . . . . . . . . . . . . . . . . . . 172
9.3.2 The 3D Euler-LagrangeEquations . . . . . . . . . . . . . . . . 173
9.3.3 A Variational Approa
h to the Steger-SorensonAlgorithm . . 176
10 Variational Grid Generation on Curves and Surfa
es 177
10.1 Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
10.1.1 Di(cid:11)erential Geometry of Curves . . . . . . . . . . . . . . . . . 177
10.1.2 Transformation Relations on a Curve . . . . . . . . . . . . . . 179
10.1.3 Parametri
Approa
h to Variational Curve Grid Generation . 181
10.1.4 Lagrange Multiplier Approa
h to Variational Curve Grid Gen-
eration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
10.2 Surfa
es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
10.2.1 Di(cid:11)erential Geometry of Surfa
es . . . . . . . . . . . . . . . . 189
10.2.2 Transformation Relations on a Surfa
e . . . . . . . . . . . . . 191
10.2.3 Parametri
Approa
h to Variational Surfa
e Grid Generation . 194
10.2.4 LagrangeMultiplierApproa
htoVariationalSurfa
eGridGen-
eration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
10.2.5 The Surfa
e Covariant Proje
tions . . . . . . . . . . . . . . . . 199
11 Contravariant Fun
tionals: Alignment, Diagonalization, and Rota-
tion 201
11.1 Fun
tionals Based on the Ellipti
Norm . . . . . . . . . . . . . . . . . 202
11.1.1 The Alignment Fun
tional . . . . . . . . . . . . . . . . . . . . 203
11.1.2 The Diagonalization Fun
tionals . . . . . . . . . . . . . . . . . 204
11.2 Non-Symmetri
Form of the Diagonalization Equations . . . . . . . . 206
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1992 by P.M. Knupp, September 8, 2002 v
11.3 Winslow's Variable Di(cid:11)usion Generator . . . . . . . . . . . . . . . . . 207
11.4 Lo
al Condition with Non-Symmetri
Matri
es . . . . . . . . . . . . 208
11.5 Alignment with an Ellipti
Generator . . . . . . . . . . . . . . . . . . 212
A Tensor CoeÆ
ients 230
A.1 PlanarTensor CoeÆ
ients . . . . . . . . . . . . . . . . . . . . . . . . . 230
A.1.1 Covariant Form . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
A.1.2 ContravariantForm . . . . . . . . . . . . . . . . . . . . . . . . 231
A.2 Volume Tensor CoeÆ
ients . . . . . . . . . . . . . . . . . . . . . . . . 232
A.2.1 Mixed Covariant and ContravariantForm . . . . . . . . . . . . 232
B Fortran Code Dire
tory 234
B.1 Analyti
Grid Generators . . . . . . . . . . . . . . . . . . . . . . . . . 234
B.1.1 Programlinear . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
B.1.2 Programoned. . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
B.1.3 Programbilinear . . . . . . . . . . . . . . . . . . . . . . . . . . 235
B.1.4 Programtwod . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
B.2 Trans(cid:12)nite Interpolation (TFI) . . . . . . . . . . . . . . . . . . . . . . 235
B.3 Hosted Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
B.3.1 One-D-Programs . . . . . . . . . . . . . . . . . . . . . . . . . . 236
B.3.2 Two-D Programs . . . . . . . . . . . . . . . . . . . . . . . . . . 236
B.4 LINE GENERATORS . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
B.5 Hyperboli
Grid Generator . . . . . . . . . . . . . . . . . . . . . . . . 237
B.6 TTM Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
B.7 Variational Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
B.7.1 Length Fun
tional . . . . . . . . . . . . . . . . . . . . . . . . . 237
B.7.2 Weighted Combination Fun
tional . . . . . . . . . . . . . . . . 238
B.7.3 Metri
Elements Fun
tional . . . . . . . . . . . . . . . . . . . . 238
C A Rogue's Gallery of Grids 239
List of Figures
1.1 Transformation, map, or
oordinate system . . . . . . . . . . . . . . . 2
1.2 A grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Boundary topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 A bilinear map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 Polar
oordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.6 Paraboli
ylinder
oordinates . . . . . . . . . . . . . . . . . . . . . . 14
1.7 Ellipti
ylinder
oordinates . . . . . . . . . . . . . . . . . . . . . . . 14
1.8 Horseshoe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.9 Modi(cid:12)ed horseshoe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.10 Bipolar
oordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.11 Spheri
al
oordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.12 Boundaries of planar regions . . . . . . . . . . . . . . . . . . . . . . . 18
1.13 TFI Grid on the horseshoe . . . . . . . . . . . . . . . . . . . . . . . . 20
1.14 TFI Grid on the Swan . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.15 TFI Grid on the Chevron . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1 Solution of the BVP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Central di(cid:11)eren
es and averages . . . . . . . . . . . . . . . . . . . . . 27
2.3 One-dimensional dis
retization . . . . . . . . . . . . . . . . . . . . . . 28
2.4 Grid points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.5 Two-dimensional sten
il . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1 One-dimensional grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2 Nonlinear iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.1 Parametri
urve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2 Parametri
surfa
e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.3 Impli
it
urve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.4 Coordinate lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.5 Coordinate
urves and surfa
e . . . . . . . . . . . . . . . . . . . . . . . 71
4.6 Parallel Tangents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.1 Algebrai
grid generation using multiple
urves . . . . . . . . . . . . . 85
5.2 Riemann Mapping Theorem . . . . . . . . . . . . . . . . . . . . . . . . 89
5.3 Hyperboli
grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.4 Horseshoe region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.5 Inhomogeneous-TTM grids. . . . . . . . . . . . . . . . . . . . . . . . . 101
6.1 Convex fun
tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.2 Referen
e spa
e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
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1992 by P.M. Knupp, September 8, 2002 vii
8.1 Lo
ation of the entries of B . . . . . . . . . . . . . . . . . . . . . . . . 154
9.1 The volume grid-generation problem . . . . . . . . . . . . . . . . . . . 169
10.1 Curve parameterization . . . . . . . . . . . . . . . . . . . . . . . . . . 182
10.2 Geometry of
urve-grid trun
ation errors . . . . . . . . . . . . . . . . 187
10.3 Surfa
e parameter spa
e . . . . . . . . . . . . . . . . . . . . . . . . . . 195
C.1 Unit square grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
C.2 Trapezoid grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
C.3 Annulus grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
C.4 Horseshoe grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
C.5 Swan grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
C.6 Chevron grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
C.7 Airfoil grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
C.8 Dome grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
C.9 Valley grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
C.10Ba
kstep grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
C.11Plow grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
C.12C grids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
List of Tables
1.1 Logi
al spa
e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Coordinate maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Consisten
y
onditions . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1 Grid points (i; j (cid:21)0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.1 Coordinate
urves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2 Coordinate surfa
es . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3 Coordinate
urves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.4 Coordinate surfa
es . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.5 Planar inner produ
t relationships . . . . . . . . . . . . . . . . . . . . 78
7.1 Summary of an operator's domain and range . . . . . . . . . . . . . . 132
7.2 Covariant tangent tensor produ
t properties . . . . . . . . . . . . . . . 136
8.1 Unweighted Variational Prin
iples . . . . . . . . . . . . . . . . . . . . 147
8.2 First partial derivatives of the variational prin
iples . . . . . . . . . . 153
8.3 Covariant proje
tions of the planar Euler-Lagrange equations . . . . . 157
9.1 3D Variational Prin
iples . . . . . . . . . . . . . . . . . . . . . . . . . 173
10.1 Properties of the Ja
obian matrix and its generalized inverse . . . . . 192
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(cid:13)
1992 by P.M. Knupp, September 8, 2002 ix
PREFACE
This book has two parts, re(cid:13)e
ting the fa
t that the authors have two major
goals. The (cid:12)rst part of this book, Chapters One through Five, is intended to be
used as a textbook for a
ourse
overing the mathemati
al fundamentals, the basi
algorithms, and some appli
ations of stru
tured grid generation. The se
ond part,
Chapters Six through Eleven, presents results from the authors' original resear
h in
grid generation. The two parts of the book are written in the same style and the
transition from textbook to monograph is gradual.
Chapters One through Five provide a uniform mathemati
al treatment of funda-
mental grid generation algorithms. The mathemati
al orientation of the book is a
result of our desire to provide the (cid:12)rst systemati
treatment of this material in text-
book form. Although mathemati
ally oriented, the book proves few theorems so as
to keep the book a
essible to a large audien
e. Chapters One through Five should
bene(cid:12)t engineers and other users of grid generation software by providing a simple
explanation of
urrent algorithms. It should also be of interest to mathemati
ians,
who will be qui
k to note gaps in the underlying mathemati
al theory.
The se
ond part of this book (Chapters Six through Eleven) turns to more
theoreti
al aspe
ts of grid generation, primarily of interest to those who do resear
h
on the algorithms of grid generation. These later
hapters summarize the authors'
original resear
h results. This part is not
onsidered de(cid:12)nitive sin
e a
tive resear
h
intoboththepra
ti
alandtheoreti
alaspe
tsofgridgenerationis
urrentlyunderway.
Certainapproa
hesandalgorithmsareemphasizedoverothers,re(cid:13)e
tingtheauthors'
orientation. Sin
e grid generationis an open resear
htopi
, it is understandable that
othershavingdi(cid:11)erentobje
tivesmaydisagreewiththeemphasesandinterpretations
given. Forexample, generationofgridsassolutionstopartialdi(cid:11)erentialequationsis
emphasizedmorethan algebrai
methods eventhough the latter
ertainly
onstitutes
an e(cid:11)e
tive approa
h to grid generation. Similarly, variational grid generation is
emphasized over more
lassi
al partial di(cid:11)erential equation te
hniques. It is hoped
thatinthelongrun,themathemati
alapproa
htothesubje
twillhelpidentifytruly
robust methods that will serve as the basis of a (cid:12)nal solution to the problem of grid
generation on
ompli
ated obje
ts.
The book does not introdu
e readers to the pra
ti
al details of generating grids
on
ompli
ated obje
ts requiring pat
hing of multiple blo
ks, unstru
tured grids,
intera
tive grid generation, et
. Adequate dis
ussion of these important topi
s would
double the size of this book and
arry us far a(cid:12)eld from the basi
ideas we wish to
present; a separate book on these pra
ti
al matters would be a useful
ompanion to
the present volume.
In re
ent times, there has been
onsiderable interest in unstru
tured grids (see
Carey, 1993, [24℄). For very
omplex geometries, satisfa
tory unstru
tured grids
are easier to generate than stru
tured grids. On the other hand, the unstru
tured
approa
h has limitations. For example, there is a nontrivial
omputational
ost in
using unstru
tured grids. The EÆ
ien
y Note on page 44 (in Se
tion 5.1.3) of the
Thinking Ma
hines FORTRAN manual (Thinking, 1992, [203℄) makes
lear that,
for the CM5, programs using stru
tured grids are more eÆ
ient than ones using
unstru
tured grids. For problems involving relatively simple geometries, stru
tured
grid algorithms may be preferable. There has also been skepti
ism
on
erning the
appli
ability of unstru
tured meshes to problems of vis
ous (cid:13)ow (Mavriplis, 1992,
[134℄). To be
onsistent with the goal of being an introdu
tion to grid generation,
thisbookisrestri
tedtoregularquadrilateralmeshesandsimply
onne
teddomains.
