Page 242 - Basic Structured Grid Generation
P. 242
Index 231
intrinsic definition, 30 Thomas Algorithm, 125–127, 132–134
intrinsic derivatives, 36 Torus, 62–3
metric tensor, 38–40 Transfinite interpolation (TFI):
principal normal, 31 bilinear transformation, 93
radius of curvature, 31 Boolean sum of projectors in three
Serret–Frenet formulas, 34, 38 dimensions, 97
tangent vector, 31 Boolean sum of projectors in two
torsion, 33–5 dimensions, 94
Stretching transformations, 98, 109, 120–1 in three dimensions, 96
Eriksson function, 101 in two dimensions, 94
hyperbolic sin functions, 100 trilinear transformation, 97
hyperbolic tangent functions, 109 Transformation law for:
Summation convention, 3–4 Christoffel symbols, 16–17
Surface of revolution, 48, 52, 55–6, 61, 66 contravariant second-order tensors, 11
Surfaces: contravariant vectors, 11
angle between co-ordinate curves, 49 covariant second-order tensors, 11
Beltrami operator, 73, 141 covariant vectors, 10–11
Christoffel symbols, 51–2 mixed second-order tensors, 12
Codazzi equations, 69 third-order tensors, 13
covariant derivatives, 52 TTM equations, 119
elliptic, hyperbolic, parabolic points,
62, 65 Under-relaxation, 129
Euler’s Theorem, 66
first fundamental form, 47
fundamental existence theorem, 70 Variational methods:
Gaussian curvature, 65 for one-dimensional grids, 157
Gauss’s equation, 68 for plane two-dimensional grids, 164
Gauss’s formula, 67, for space-curves, 161
geodesic curves, 54–58 for surface grids, 171, 174, 175
grid generation, 141, 171, 174, 175 harmonic maps, 172
lines of curvature, 66 Vectors:
mean curvature, 64 magnitude, 9
metric tensors, 47 scalar product, 9,11
Meusnier’s Theorem, 63 vector product, 14, 50
non-singular points, 43 Volume element, 8
normal curvature, 60, 63 Voronoi polygon, 191, 199
Riemann–Christoffel tensor, 53,
68–9 Weight functions:
second fundamental form, 61 one-dimensional grids, 137–9, 150, 157, 162
surface Frenet equations, 58 two-dimensional grids, 166–8
tangent plane, 43 Weighted area functional, 167
umbilics, 66 Weighted L-functional, 166–7
Weingarten’s equations, 67 Winslow equations, 118
