Modern introduction to_grid-generation

COSMASE Shortcourse Notes
EPF Lausanne 1996

Modern Introduction to
Grid Generation

Jochem H¨ userc
a

Yang Xia

Department of Parallel Computing
Center of Logistics and Expert Systems
Salzgitter, Deutschland

c Jochem H¨ user
a
All Rights Reserved
Notes prepared for COSMASE Shortcourse in Grid Generation
EPF Lausanne, 23-27 September 1996

Contents

1.0 Overview and Status of Modern Grid Generation . . . . . . . . . . . . . . . 1

2.0 Methods of Differential Geometry in
Numerical Grid Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3.0 Introduction and Overview of Numerical
Grid Generation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.0.1 General Remarks on Grid Generation
Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.0.2 A Short History on Grid Generation . . . . . . . . . . . . . . . . . . . 9

3.0.3 What Is a Good Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.0.4 Aspects of Multiblock Grid Generation . . . . . . . . . . . . . . . . . . 14

3.1 Equations of Numerical Grid Generation . . . . . . . . . . . . . . . . . . . . . 16

3.1.1 Elliptic Equations for 2D Grid Generation . . . . . . . . . . . . . . . . 16

3.1.2 Elliptic Equations for Surface Grid Generation . . . . . . . . . . . . . 19

3.1.3 Elliptic Equations for 3D Grid Generation . . . . . . . . . . . . . . . . 23

3.2 Grid Generation Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2.1 Computational Aspects of Multiblock Grids . . . . . . . . . . . . . . . 25

3.2.2 Description of the Standard-Cube . . . . . . . . . . . . . . . . . . . . . 26

3.2.3 The Grid Grid Generation Toolbox . . . . . . . . . . . . . . . . . . . 30

3.2.4 Input for Grid Generation in 2D and 3D . . . . . . . . . . . . . . . . . 31

i

3.2.4.1 Rectangle Grid Example . . . . . . . . . . . . . . . . . . . . 32

3.2.4.2 Diamond Shape 6 Block Grid Example . . . . . . . . . . . . 33

3.3 Local Grid Clustering Using Clamp Technique . . . . . . . . . . . . . . . . . . 41

3.3.1 Hyperboloid Flare in F4 Windtunnel Grid . . . . . . . . . . . . . . . . 43

4.0 Grid Adaptation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.0.1 Adaptation by Controlfunctions . . . . . . . . . . . . . . . . . . . . . 48

4.0.2 Algebraic Adaptation Algorithms . . . . . . . . . . . . . . . . . . . . . 52

5.0 A Grid Generation Meta Language . . . . . . . . . . . . . . . . . . . . . . . . 59

5.0.1 Topology Input Language . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.0.2 TIL Code for the Cassini-Huygens
ConÞguration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.0.2.1 TIL Code for 6 Block Cassini–Huygens Grid . . . . . . . . . 71

5.0.2.2 TIL Code for 24 Block Cassini–Huygens Grid . . . . . . . . 73

5.0.3 TIL Topology and Grid Quality . . . . . . . . . . . . . . . . . . . . . . 74

6.0 Tools for Automatic Topology DeÞnition . . . . . . . . . . . . . . . . . . . . . 79

6.0.1 Example Topology 1: A Circle in a Box . . . . . . . . . . . . . . . . . . 80

6.0.2 Example Topology 2: Two Circles in a Box . . . . . . . . . . . . . . . . 93

6.0.3 Example Topology 3: Cassini–Huygens Space Probe . . . . . . . . . . 99

7.0 Parallelization Strategy for Complex Geometries . . . . . . . . . . . . . . . . 102

7.0.1 The Abstract Parallel Machine . . . . . . . . . . . . . . . . . . . . . . 102

7.0.2 General Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7.0.3 Encapsulation of Message Passing . . . . . . . . . . . . . . . . . . . . 103

7.0.4 Aspects of Software Engineering: C versus Fortran . . . . . . . . . . . 105

7.0.5 Numerical Solution Strategy: A Tangled Web . . . . . . . . . . . . . . 106

ii

List of Figures

2.1 Curvilinear nonorthogonal Coordinate System. The contravariant base vectors e i
ˆ
point in the tangential directions but are not unit vectors. Vectors ei are Cartesian
unit vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3.1 In this chart the complete modelization process starting from the surface descrip-
tion of the vehicle, coming from the design ofÞce, to the visualized results of a 3D
ßow solution is depicted. The necessary data Þles along with the corresponding
software modules are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2 Standard cube in CP (computational plane). Each cube has its own local coordinate
system. The grid is uniform in the CP. . . . . . . . . . . . . . . . . . . . . . . . 26

3.3 Mapping of a block from SD to CP. Arrows indicate orientation of faces, which
are numbered in the following way: 1 bottom, 2 front, 3 left, 4 right, 5 back, 6 top.
The rule is that plane ζ 1 corresponds to 1, plane η 1 to 2 and plane ξ 1 to 3. 27

3.4 Orientation of faces. Coordinates ξ, η, ζ are numbered 1, 2, 3 where coordinates
with lower numbers are stored Þrst. . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.5 Determination of orientation of faces between neighboring blocks. . . . . . . . . 28

3.6 The Þgure shows the overlap of two neighboring blocks. . . . . . . . . . . . . . 29

3.7 The 8 possible orientations of neighboring faces are shown. Cases 1 to 4 are ob-
tained by successive rotations e.g. 0, 1 π, π and 3 π. The same situation holds for
2 2
cases 5 to 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.8 A 6 block grid for diamond shaped body. This type of grid line conÞguration
cannot be obtained by a mono-block grid. . . . . . . . . . . . . . . . . . . . . . 35

3.9 Grid line can also be clustered to match the physics of the ßow; e.g. resolving a
boundary layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.10 Control information for the 6 block diamond grid. This command Þle is also used
by the parallel ßow solver. ÒÞle diamond contains the actual coordinate values. . 36

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3.11 Coordinate values of the Þxed (physical) boundaries of the 6 block diamond grid. 37

3.12 So called clamp technique to localize grid line distribution. The real power of this
technique is demonstrated in the Hyperboloid Flare in F4 Windtunnel grid. . . . 41

3.13 Clamp 1 at hyperboloid ßare. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.14 Clamp 2 at hyperboloid ßare. . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.15 A conventional topology for a hyperboloid ßare in a windtunnel. Although the
topology allows a reÞnement of the grid from an Euler to a N–S grid, this re-
Þnement extends into the far Þeld and thus causes a substantial computational
overhead. Numbers denote block numbers, doted lines are block boundaries, solid
lines are grid lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.16 Topology of 36 block hyperboloid ßare. This topology is one part of the topology
of 284 block grid, see Fig. 3.18. . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.17 36 block grid for hyperboloid ßare. . . . . . . . . . . . . . . . . . . . . . . . . 45

3.18 Topology of 284 block grid for hyperboloid ßare in windtunnel. . . . . . . . . . 45

3.19 284 block grid for hyperboloid ßare in windtunnel. . . . . . . . . . . . . . . . . 46

4.1 1 block adptive grid for supersonic inlet, adapted by control functions. . . . . . . 50

4.2 3 block adaptive grid for forward facing step, adapted by control functions. . . . 51

4.3 One–dimenisional monitor surface (MS). The initial grid in the physical SD is
uniform. Lifting up the grid points produces the grid on the monitor surface. The
variable s denotes arc lehgth on the MS . . . . . . . . . . . . . . . . . . . . . . 52

4.4 Repostioned grid on the monitor surface. Grid points are uniformly distributed on
the monitor surface so that arc length spacing is constant. When projected down
back to the physical solution domain, this results in a clustering according to the
gradient of the monitor surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.5 Repositioning of grid points on the monitor surface according to the magnitude of
curvature, resulting in a clustering of grid points to regions where curvature κ 0. 53

4.6 Adaptation of a grid using the monitor surface technique to capture a moving
shock together with a strong circular gradient (vortex). The initial grid has uniform
grid spacing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.1 Navier-Stokes grid for a four-element airfoil, comprising 79 blocks.The Þrst layer

of grid points off the airfoil contour is spaced on the order of 10 6 based on chord
length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

iv

5.2 The Þgure shows the block structure of the four element airfoil generated by
GridPro. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.3 Grid for a T joint that has numerous industrial applications. . . . . . . . . . . . . 61

5.4 Sphere in a torus. The input for this grid is presented in the following tables. . . . 62

5.5 Complete 3D grid for a generic aircraft with ßaps, constructed from analytical
surfaces. However, the topology is exactly the same as for a real aircraft. . . . . . 62

5.6 The picture shows a blowup of the engine region of the generic aircraft.Future
TSTO or SSTO vehicles will exhibit a similar complex geometry, necessitating
both the modelization of internal and external ßows. . . . . . . . . . . . . . . . . 63

5.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.9 Topological design for the Huygens space probe grid. In this design all elements,
such as SSP, GCMS and sensors are ignored. . . . . . . . . . . . . . . . . . . . 72

5.10 The 6 block Huygens grid is depicted. It is bounded by a large spherical far Þeld,
in which the Huygens space probe is embedded. The ratio of the far Þeld radius
and the Huygens radius is about 20. . . . . . . . . . . . . . . . . . . . . . . . . 75

5.11 Illustration of the block topology for the 24 block Huygens grid. Left: Topology
top view. The far Þeld, the Huygens body and GCMS are clamped by quads. Right:
Topology side view. Again clamps are used for the 3D topological design. . . . . 75

5.12 24 block grid for Huygens space probe including SSP. . . . . . . . . . . . . . . 76

5.13 Topological description for a two–loop hyperquad is depicted. Some attention has
to be given to the placement of the wireframe points. In both cases, the topology
remains unchanged while wireframe corners are different. It is obvious that in the
second case an overlap is encountered and grid lines will be folded. . . . . . . . 77

5.14 The wireframe should reßect the actual geometry. Although wireframe coordinates
are not Þxed, a better initial solution will lead to faster convergence and inproved
grid quality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.15 The grid quality is inßuenced by the initial coordinate values of the wireframe
corners A distorted wireframe may cause grid skewness and folded lines. . . . . 78

6.1 In this Þrst example the grid for a circle in a box has to be generated. First the
user has to construct the geometry (physical boundaries) as shown in the Þgure.
Second, the blocking toplogy has to be speciÞed. . . . . . . . . . . . . . . . . . 81

v

6.2 To start with the example, invoke the automatic blocking manager with command
az. The AZ–manager window appears. Switch to 2–dim on the menubar, as shown
in Þgure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.3 AZ–manager has mini CAD capability that is used to construct the boundary of
the solution domain. To this end click surf on the menubar and select load:–plane
from this menu. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6.4 new text needed AZ–manager has mini CAD capability that is used to construct
the boundary of the solution domain. To this end click surf on the menubar and
select load:–plane from this menu. . . . . . . . . . . . . . . . . . . . . . . . . 82

6.5 To construct the outer box, input contour data as indicated. The Þrst surface to be
generatedd, is the west side of the box. It is of type textbfplane that actually is a
line in 2D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.6 Input plane information as shown. The next surface is the south side of the box.
Note that surfaces are consecutively numbered (surf id). . . . . . . . . . . . . . 83

6.7 Input plane information as shown. This surface is the east side of the box. Note
that surface orientation has to be reversed by specifying –side. . . . . . . . . . . 84

6.8 Input plane information as shown. This plane is the north side of the box. Note
that surface orientation has to be reversed by specifying –side. Press unzoom to
make a box Þt on screen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.9 Select load:–ellip from the surf on the menubar. Input ellip information as shown
to construct the circle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.10 A surface may not have the desired size. In order to resize it, deselect cut (if it is on,
i.e. the radio button is pressed) of the SHOW option on the Command Panel. Then
activate hand from the CUT–P option on the command panel. Next, activate the
proper surface (blue color) – in this example the proper edge of the box. Position
the mouse curser over the handle and drag it in the indicated direction. . . . . . 86

6.11 Activate the other box surfaces in the same way and drag them as shown in Þgure
a. The dragging shoul be done such that the box contour becomes visible. The
circle should be visible as well. Now wireframe points can be placed. Wireframe
point don’t have to ly on a surface, but they should be close to the surface to which
they are going to be assigned. Wireframe points are created by pressing ”c” on
the keyboard and clicking at the respective positions with LB. Make sure that
Corner Creation Mode appears in the upper left corner of the window. Finish
point positioning as depicted in Þgure b. . . . . . . . . . . . . . . . . . . . . . . 87

6.12 Link the corners by pressing ”e” on the keyboard and click each pair of points with
LB. Make sure that Link Creation Mode appears in the upper left corner of the
window. After completing the linkage process, a wireframe model should like as
depicted in the Þgure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

vi

6.13 Activate surface 0. Press ”s” on the keyboard and click the two points, marked
by ”X”. If points are selected, their color turns from yellow to white. That is the
corners are assigned to surface 0. . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.14 Repeat this action for surface 1, shown in Þgure. . . . . . . . . . . . . . . . . . 89



6.17 Activate surface 4 and assign the remaining 4 points to the circle. . . . . . . . . 90

6.18 Since GridPro is 3D internally, it is necessary to explicitly remove two blocks.
The Þrst block to be revoned is formed by the 4 points which build diagnoals, as
shown in Þgure. Press ”f” and click two points, marked by triangles. Perform the
same action for points marked by crosses. A red line between the corresponding
diagnoals indicates the succesful performance. . . . . . . . . . . . . . . . . . . 91

6.19 Save the topology Þle (TIL code) by selecting topo and clicking TIL save. Select
Ggrid start to generate the grid. . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.20 Select topo from the menubar and click Ggrid stop. The generation process is
now suspended. Select grid from the menubar and click load new, the grid Þle
”blk.tmp” can be read. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.21 Select panel=T from the menubar and click Grid. The grid appears as wireframe
model. At the right side of the menu window, select STYLE and click lines under
stop. Click shell by Make Shell at the right low side of the menu window. The
grid appears. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.22 Constrct geometry in the same way as described in example 1. . . . . . . . . . . 93

6.23 It is suggested to place the wirefrace points in a row by row fashion. Other toplo-
gies, of course, would be possible. . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.24 Press TIL read under item topo of the menubar to read in the geometry data. . . 94

6.25 A row by row pattern results in a wireframe point numbering as shown. . . . . . 95

6.26 Assignment of wireframe points to surfaces is exactly as in the Þrst example. . . 95

6.27 Activate surface 1 and assign wireframe points. . . . . . . . . . . . . . . . . . . 96



vii




6.33 Start Ggrid by selecting Ggrid start. . . . . . . . . . . . . . . . . . . . . . . . 98

6.34 The topology to be generated for Huygens is of type box in a box. The outer 8
wireframe points should be placed on a sphere with radius 5000, the inner 8 points
are located on the Huygens surface. . . . . . . . . . . . . . . . . . . . . . . . . 99

6.35 Construct wireframe model as in Þgure. . . . . . . . . . . . . . . . . . . . . . . 100

6.36 Assign surface 2 as in Þgure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.37 Assign surface 1 as in Þgure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

7.1 Flow variables are needed along the diagonals to compute mixed second deriva-
tives for viscous terms. A total of 26 messages would be needed to update values
along diagonals. This would lead to an unacceptable large number of messages.
Instead, only block faces are updated (maximal 6 messages) and values along di-
agonals are approximated by a Þnite difference stencil. . . . . . . . . . . . . . . 107

7.2 The Þgure shows the computational stencil. Points marked by a cross are used
for inviscid ßux computation. Diagonal points (circles) are needed to compute the
mixed derivatives in the viscous ßuxes. Care has to be taken when a face vanishes
and 3 lines intersect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

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Concepts of Grid Generation 1

1.0 Overview and Status of Modern Grid Generation

These lecture notes are the less mathematical version and an excerpt from a forthcoming book
being currently written by the Þrst author, entitled Grid Generation, Computational Fluid Dy-
namics, and Parallel Computing, envisaged for the second half of 1997.

These notes are the continuation of courses in grid generation that were given at Mississippi
State University (1996), for the International Space Course at the Technical University of Stuttgart
1995), at the Istituto per Applicacione del Calculo (IAC), Rome (1994), the International Space
Course at the Technical University of Munich (1993), and at the Von Karman Institute, Brussels
(1992). Material that has been prepared for presentations at the California Institute of Technology
and the NASA Ames Research Center has also been partly included. However, in cooperation with
P.R. Eiseman, PDC, White Pains, New York, substantial progess has been achieved in automatic
blocking (GridPro package). This material is presented here for the Þrst time. To keep the number
of pages for this article at a manageable size, the introduction to parallel computing and the section
on interfacing multiblock grids to ßow solvers had to be omitted. The reader interested in these
aspects should contact the authors. Some recent publications can be found on the World Wide Web
under ... In addition, a web address www.cle.de is being built where further information will be
available. Furthermore, a CD-Rom containing a collection of 2D and 3D grids can be obtained
(handling fee) from the authors.

With the advent of parallel computers, much more complex problems can be solved, pro-
vided computational grids of sufÞcient quality can be generated. The recent Workshops hely by
the Aerothermodynamics Section of the European Space Agency’s Technology Center (ESTEC)
(1994, 1995) has proved that grid generation is one of the pacing items in CFD.

Although the emphasis is on grid generation in Aerodynamics, the codes presented can be
used for any application that can be modeled by block-structured grids.

In these notes Numerical Grid Generation is presented in a way that is understandable to the
scientist and engineer who is more a programmer, rather than a mathematician and whose main
interest is in applications. The emphasis is on how to use and write a grid generation package.
The best way to understand theory is to develop tools and to generate examples. Much emphasis
is given to algorithms and data structures, which are presented from a real working code, namely
the Grid and GridPro [44] package, built by the authors. No claim is made that the packages
presented here are the only or the best implementation of the concepts presented. Hence, a certain
amount of low level information is presented. However, this is critical to understanding a general
2D and 3D multi-block grid generator, and is missing from other presentations on the subject.

The multi-block concept gets an additional motivation with the advent of parallel and dis-
tributed computing systems, which offer the promise of a quantum leap in computing power for
Computational Fluid Dynamics (CFD). Multi-blocks may well be the key to achieve that quantum
leap, as will be outlined in the chapter on grid generation on parallel computers, presenting the key
issues of the parallelizing strategy, i.e. domain decomposition, scalability, and communication. A
discussion of the dependence of convergence speed of an implicit ßow solver on blocknumber for
parallel systems wil also be given.

2

Since grid generation is a means to solve problems in CFD and related Þelds, a chapter on in-
terfacing the Þnal grid to the Euler or Navier-Stokes solver is provided. Although, in general, grids
generated are slope continuous, higher order solvers need overlaps of 2 points in each direction.
The software to generate this overlap for multi-block grids is also discussed.

First, the grid construction process is explained and demonstrated for several 2D examples.
The object–oriented grid generation approach is demonstrated in 2D with a 284 block grid, viz. the
”Hyperboloid Flare in the F4 windtunnel”, using both the Grid and GridPro packages. In 3D, we
start with a simple mono-block grid for the double ellipsoid, explaining the 3D mapping procedure.
After that, a multi-block grid for the double ellipsoid is generated to demonstrate the removal of
singularities. Next, surface grid generation is demonstrated for the Hermes Space Plane, and a
mono and a 7-block grid are generated. The last vehicle that is presented is the Space Shuttle
Orbiter where surface grid generation is given for a 4-block grid, modeling a somewhat simpliÞed
Orbiter geometry, namely a Shuttle without body ßap. In the next step, the surface grid and volume
grid for a 94-block Euler mesh of a Shuttle with body ßap are presented. The rationale for the
choice of that topology is carefully explained. In the last step, it is shown how the 94-block Euler
grid is converted to a 147-block Navier-Stokes grid, using the tools provided by Grid . In 3D,
object–oriented grid generation in exemplied by the Electre body in the F4 windtunnel. Along
with the examples, the questions of grid quality and grid adaptation are addressed.

Concepts of Differential Geometry 3

2.0 Methods of Differential Geometry in
Numerical Grid Generation

In the following a derivation of the most widely used formulas for nonorthogonal curvilinear
coordinate systems ( Fig. 2.1) is given as needed in numerical grid generation. Variables x1 xn
denote Cartesian coordinates and variables ξ 1 ξ m are arbitrary curvilinear coordinates. The

approach taken follows [9]. It is assumed that a one-to-one mapping A M exists with
¼ x1´ξ1 ξm µ
½
x´ξ1 ξm µ : (2.1)
xn ´ ξ1 ξm µ
As an example, we consider the surface of a sphere where Rm R2 and ξ1 θ π 2 θ
π 2; ξ2 ψ 0 ψ 2π. The coordinates x1 x2 and x3 correspond to the usual Cartesian coordi-
nates x, y and z. In the physical space, we have the surface of a sphere, whereas in the transformed
space θ and ψ form a rectangle.

e3
e2
e1

e3

e1
e2

Figure 2.1: Curvilinear nonorthogonal Coordinate System. The contravariant base vectors e point
i
ˆ
in the tangential directions but are not unit vectors. Vectors ei are Cartesian unit vectors.

The tangent vectors or base vectors at a point P ¾ M are deÞned by

∂x
ek : ; k 1´1µm (2.2)
∂ξk

The tangent vector ek points in the direction of the respective coordinate line. These base
vectors are called covariant base vectors. A second set of base vectors is deÞned by

ei ¡ e j δij (2.3)

The ei are called contravariant base vectors and are orthogonal to the respective covariant
vectors for i j. Covariant and contravariant vectors are related by the metric coefÞcients (see

4

below).
For the above example the two tangent vectors eθ and eψ are obtained by differentiating each of
the functions x´θ ψµ y´θ ψµ and z´θ ψµ with respect to either θ or ψ.
A physical vector can either be represented by contravariant or covariant components

v vi ei v je j (2.4)

where the summation convention is employed. In two dimensions, contravariant components
of a vector are found by parallel projections onto the axes, whereas covariant components ar ob-
tained by orthogonal projection.
According to its transformation behavior, a vector is called contravariant or covariant. Let vibe the
components of a vector in the coordinate system described by the x an let vi be the components in
i

the system ξ i

(α) A vector is a contravariant vector if its components transform in the same way as the coordi-
nate differentials:
∂xi
dxi ∂ξi dξ j
∂xi
(2.5)
vi ∂ξ j v j

(β) A vector is covariant if it transforms in the same way as the gradient of a scalar function:
φ´x1 xn µ φ´ξ1 ξm µ
∂φ ∂φ ∂ξ j ∂ξ j ∂φ
∂xi ∂ξ j ∂xi ∂xi ∂ξ j (2.6)
∂ξ j
vi ∂xi v j

In order to measure the distance between neighboring points, the Þrst fundamental form is
introduced:

gi j : ei ¡ e j (2.7)

The gi j are also called components of the metric tensor. The components of the inverse matrix
are found from

gi j gik δk
j (2.8)

The distance ds between two neighboring points is given by

Ô
ds ´gi j dξi dξ j µ (2.9)

In Cartesian coordinates there is no difference between covariant and contravariant compo-
nents since there is no difference between covariant and contravariant base vectors. Therefore the


matrix gi j (the ˆ denotes the Cartesian system) is the unit matrix.
ˆ
The components of gi j in any other coordinate system can be directly calculated using the chain
rule:

∂xk ∂xl
ˆ ˆ
gi j ˆ
gkl (2.10)
∂xi ∂x j

In order to Þnd the transformation rules of derivatives of scalars, vectors and tensors, the
Christoffel symbols of the Þrst and second kinds are introduced. Suppose that coordinate ξj is
changed by an amount of dξ j . This changes the base vector ei by dei . Since dei is a vector, it
can be represented by the system of base vectors ek . Further, dei is proportional to dξ j . One can
therefore write

dei Γkj dξ j ek
i (2.11)

where the symbols Γkj are only coefÞcients of proportionality. These symbols are also called
i
k
Christoffels symbols of the second kind. Taking the scalar product with e , one obtains directly
from equation 2.11

Γkj
i ei j ¡ ek (2.12)

where a comma denotes partial differentiation. The Christoffels symbols of the Þrst kind are
deÞned as

Γi jk : gil Γljk (2.13)

If the base vectors are independent of position, the Christoffels symbols vanish. They are,
however, not tensor components, which follows directly from their transformation behaviour.
The relationship between the Γkj and the glm is found in the following way. Insertion of the deÞni-
i
tion of the metric components glm in equation (2.7), into equation (2.12) and interchanging indices
leads to

∂g jl
Γijk
1 ij
2
g
∂ξk
· ∂gklj ∂gklj
∂ξ ∂ξ
(2.14)

If we use equation (2.2), the deÞnition of the tangent vector, along with equation (2.12), one
obtains the computational useful form

∂ξi ∂2 xl
Γijk (2.15)
∂xl ∂ξ j ∂ξk

6

The knowledge of the grid point distribution, then, allows the numerical calculation of the
Christoffels symbols. If we contract the Christoffels symbols, i.e. upper and lower indices are the
same and are summed over, equation (2.14) yields

Ô
Γiji
1 im ∂gmi
2
g
∂ξ j
Ô
1 ∂ g
g ∂ξ j
´lnÔgµ j (2.16)

Ô
where g is the determinant of the metric tensor, that is g is the Jacobian of the transformation.
In two dimensions with curvilinear coordinates ξ η and Cartesian coordinates x y, one Þnds

yη xη
ξx ξy
ηx ηy J 1 yξ xξ
Ô (2.17)
g J ´xξyη yξxηµ
In curved space, partial differentiation is replaced by covariant differentiation which takes
into account the fact that base vectors themselves have non-vanishing spatial derivatives. In the
following, the nabla operator

∂
∇: ei (2.18)
∂ξi

is used. For the calculation of the cross product the Levi-Civita tensor is introduced.

´1 for i 1 j 2 k 3 and all even permutations
ˆ
εi jk : 1 for all odd permutations (2.19)
0 if any two indices are the same

where ˆ again denotes the Cartesian coordinate system. From equation.(2.0) we know the trans-
formation law for covariant components, and hence

∂xl ∂xm ∂xn
ˆ ˆ ˆ
εi jk ˆ
εlmn ˆ
J εi jk (2.20)
∂xi ∂x j ∂xk

where J is the Jacobian of the transformations. Forming the determinant from the components
of equation (2.10), results in g J2 , and therefore

Ô
εi jk gεiˆjk (2.21)

Raising the indices in equation (2.21) with the metric, one obtains

Ô i jk
εi jk ˆ
gε (2.22)


The cross product of two vectors a and b in any coordinate frame is then written as

Ô
a¢b ai b j ei ¢ e j ai b j εi jk ek gεi jk ai b j ek
ˆ (2.23)

It should be noted that equations (2.21) and (2.22) can also be derived by starting from the
well-known relation in Cartesian coordinates

ˆ ˆ
ei ¢ e j ˆ ˆ
εi jk ek (2.24)

Insertion of

∂xl ∂xm ∂xk n
ˆ
ei ˆ
el ; e j ˆ
em ; ek e ; (2.25)
∂xi
ˆ ∂x j
ˆ ∂xn
ˆ

into equation (2.24) and multiplication of the resulting left and right sides by

∂xi ∂x j
ˆ ˆ
(2.26)
∂x p ∂xr

with summation over i and j Þnally leads to

Ô
e p ¢ er gε prn en
ˆ (2.27)

With the above equations it is now possible to derive the transformation rules (i)-(viii) needed
for the Euler or SWEs and for the grid generation equations themselves:

(i) Gradient of a scalar function h:
∂
∇h ei h ei h i gil h i el (2.28)
∂ξi

(ii) Gradient of a vector Þeld u
∂ j ∂u j
∇u ei
∂ξi
´u e j µ ∂ξi
· umΓim
j
e je j (2.29)

where equation (2.11) was used.
(iii) Divergence of a vector Þeld u
∇¡u ek ¡ ∂ξk ´ui ei µ
∂ ∂u j
∂ξi
· ek ¡ emui Γm
ik
(2.30)
∂ui
∂ξi
· ui Γm
im uii · 1
2g g j u
i Ôg ´Ôgu j µ j
1

where equation (2.16) was used.

8

(iv) Cross product of two vectors (here base vectors ei , e j since they are most often needed)
Ô
ei ¢ e j εi jk ek gεi jk ek
ˆ J εi jk ek
ˆ (2.31)
k
It should be noted that this formula dirctly gives an expression for the area dA and the
volume dV .
dAk : dsi ¢ ds j ; d : dsk ¡ ´dsi ¢ ds j µ (2.32)
where i j k are all different. The vector dsi is of the form dsi : ei dξi (no summation over
i). Thus we obtain
dAk Jdξi dξ j ; dV Jdξi dξ j dξk (2.33)

(v) Delta operator applied to a scalar function ψ:

∂ ∂ψ 1 Ô
∆ψ ∇ ¡ ´∇ψµ ek ¡ el gil gil ψ i l · ψ i Ô ´ ggik µk (2.34)
∂ξk ∂ξi g

where the components of the gradient of equation (2.28) were inserted into equation (2.30).

(vi) For transient solution areas, i.e. the solution area moves in the physical plane but is Þxed in
the computational domain, time derivatives in the two planes are related by

∂f ∂f ∂f ∂xi
∂t ∂t
· ∂xi ∂t
(2.35)
ξ x t ξ

where ´∂xi ∂t µξ are the corresponding grid speeds, which can be calculated from two con-
secutive grid point distributions.

(vii) Relation between covariant and contravariant components of the metric tensor:

∆ik
gik (2.36)
g

where ∆ik denotes cofactor ´i kµ of the matrix of the metric tensor. In two dimensions we
have
g11 g 1 g22 ; g12 g21 g 1 g12 ; g22 g 1 g11 ; (2.37)

(viii) Relationship between grid point distribution and Christoffels symbols, eqn.(16):

∂ξi ∂2 xl
Γijk (2.38)
∂xl ∂ξ j ξk

Numerical Grid Generation 9

3.0 Introduction and Overview of Numerical
Grid Generation Techniques

3.0.1 General Remarks on Grid Generation
Techniques

CFD of today is marked by the simulation of ßows past complex geometries and/or utilizing com-
plex physics. A comparison between the impact of the numerical technique and the computational
grid used, reveals that in many cases grid effects are the dominant factor on the accuracy of the
ßow solution. Despite this fact, the number of researchers working in grid generation is at least
an order of magnitude smaller than the number of scientists active in CFD. The reason for this is
most likely that publishing a paper in CFD is much more rewarding. First, a CFD paper is easier
to publish since, e.g. a modiÞcation of a numerical scheme in 1D is sufÞcient to justify a new
paper. Second, publishing a grid generation paper involves a large amount of programming (time
consuming) and algorithm development as well as computer graphics to visualize the grid. In the
past, algorithms and programming have been considered to be outside the engineering domain.
Hence, the majority of the engineering software is still in Fortran. Data structures and Object-
Oriented-Programming (OOP) are not taught in engineering courses and thus their importance is
not recognized in this Þeld. As a consequence, many codes in industry are not state of the art.
Strange enough, the latest hardware is used together with the software concepts of the late Þfties.

With the advent of massively parallel systems and high end graphics workstations a new level
of performance in aerodynamic simulation can be achieved by the development of an integrated
package, coined PAW (Parallel Aerospace Workbench). PAW would provide the aerospace engi-
neer with a comprehensive, partly interactive package that, accepting the CAD geometry from the
design ofÞce, provides modules for surface repair, followed by quasi-automatic grid generation
(via a special language) and Þnally produces and visualizes the ßow solution. Software devel-
opment has to be based on standards like UNIX, ANSI C X11, Tcl/Tk, and Motif as well as on
Open GL. For parallelization PVM (Parallel Virtual Machine) or MPI (Message Passing Interface)
should be used (or a similar package), generating a hardware independent parallel ßow solver.

3.0.2 A Short History on Grid Generation

Numerical grid generation has the dual distinction of being the youngest science in the area of
numerical simulation and one of the most interesting Þelds of numerical research. Although con-
formal mappings were used in aerospace for airfoils, a general methodology for irregular 2D grids
was Þrst presented in the work of A. Winslow [1]. In 1974, a paper entitled ”Automatic numer-
ical generation of body Þtted curvilinear coordinate systems for Þelds containing any number of
arbitrary two-dimensional bodies” appeared in the J. of Computational Physics [2], authored by
Thompson, Thames, and Mastin. This paper can be considered a landmark paper, originating the
Þeld of boundary conformed grids, and making it possible to use the efÞcient techniques of Þnite
differences and Þnite volumes for complex geometries.

The advent of high-speed computers has made ßow computations past 3D aircraft and space-

10

craft as well as automobiles a reality. No longer are scientists and engineers restricted to wind
tunnel and free ßight experiments. On the numerical side, the development of high-speed comput-
ers as well as graphics workstations has allowed aerodynamicists to perform intricate calculations
that were unthinkable a few decades ago. A clear understanding of many of the aspects of subsonic
to hypersonic ßows, not attainable by wind tunnel experiments, is one of the important results of
these calculations. In addition, with the advent of massively parallel systems, a new era of CFD
(Computational Fluid Dynamics) has started, which will allow realistic simulations of turbulent
ßows with real gas effects (high temperature) past complex 3D conÞgurations. The purpose of
these notes is to present a clear, vivid, comprehensive treatment of modern numerical grid gener-
ation, based on boundary conforming grids, with emphasis on the actual grid generation process
as applied to complete geometries. Special emphasis is placed on algorithms and examples will be
presented along with a discussion of data structures and advanced programming techniques, topics
nearly almost always neglected in the scientiÞc and engineering Þeld. With the advent of modern
programming languages like C and C++, a revolution in scientiÞc programming is taking place.
Engineers, used to Fortran since the late Þfties, Þnally begin to realize that a new way of thinking
is needed in software engineering, characterized by the concept of object-oriented-programming.
Only with this approach it can be hoped to reduce the size and complexity of source code, to fully
use available hardware, to write and maintain very large software packages and to implement them
on arbitrary computer architectures. Grid generation packages like GridPro, and Grid code [20]
are based exclusively on C, X-Windows, and Open Gl, allowing full portability. Although nearly
two decades have past since the work of Thompson et al., the development of generally applicable
and easy to use 3D grid generation codes still presents a major challenge. In the case of structured
grids it was soon recognized that branch cuts and slits or slabs did not offer sufÞcient geometrical
ßexibility. Therefore, the majority of structured grid generation codes now employs the multiblock
concept, which is unstructured on the block level but results in a structured grid within a block.
To determine the block topology of a complex SD (solution domain) requires a certain effort. The
recent AZ–manager code (see Sec.??) is a major step forward in automatic topology deÞnition.

Grid generation has to embedded in the overall solution process that is depicted in Fig. 3.1.
Most important are tools for data conversion to directly use CAD data and to automatically inter-
face a grid to the ßow solver.

In the chart of Fig.3.1 the complete modelization process to obtain a ßow solution is shown. It
is assumed that a surface description of the vehicle is available, generally coming from the design
ofÞce in a CAD format, e.g. CATIA. This information has to be processed to convert it into a
format that can be read by the modules of Grid . Various conversion modules are available. The
chart then shows the grid generation process, where each module generates output that can be read
by the following modules, producing some kind of pipeline. The interfacing to the ßow solver
is done automatically and results can be visualized directly by the Plot3D package from NASA
Ames or by any other package that supports this format, e.g. the widely known Tecplot software
from AMTEC Engineering.

In practice, problems may be encountered with the grid generated and the ßow solution may
not be obtained, either because there is no convergence of the ßow solution, or after a few iterations
the solution may blow up. This depends very much on the ßow physics and to a large extent on
the experience of the user. These codes do not yet have reached a stage where they can be used as
a black box. A case with Ma of 25 and thermo-chemical nonequilibrium Navier-Stokes equations
is of course much harder to solve than an Euler problem at Ma 2. The stiffness of the equations


PAW-Diagram
Parallel Aerodynamics Workbench
surface/grid
visualization
modification
triangular grid
surface plot3d
plot3d
mbview
NASA CFD−
standard
CAD−data

Tripla GridPro
IGES Halis PDC Halis p3d2cmd
PDC−CLE

TIL
code

topology TIL code wireframe
builder file
cmd
wireframe file
visualization
modification
Visual3 flow ParNSS
MIT solution PDC−CLE
project
file

Figure 3.1: In this chart the complete modelization process starting from the surface description
of the vehicle, coming from the design ofÞce, to the visualized results of a 3D ßow solution is
depicted. The necessary data Þles along with the corresponding software modules are shown.

12

depends on the physics and also on the grid. A Navier-Stokes grid for a 3D vehicle with a cell
aspect ratio of 106 in the boundary layer is not only much more difÞcult to generate but it is also
much harder for the numerical scheme to converge to the physical solution in an efÞcient way.
Although the emphasis of these lecture notes is on grid generation, some results for Euler and N-S
calculations are presented.


3.0.3 What Is a Good Grid

First, the discussion will be restricted to multi–block grids. It is felt that any kind of geometry can
be gridded by this approach, resulting in a superior grid quality when compared with unstructured
grids, in particular for viscous ßows. However, it should be kept in mind that grids of tens of thou-
sands of blocks have to be generated (automatically). On the block level the grid is completely
unstructured. The large number of blocks is needed because of complex topology, for instance,
there may be hundreds of bodies in the ßowÞeld, or, because the solver is run on a MIMD (Mul-
tiple Instruction Multiple Data) massively parallel system that may comprise several thousand
processors.

Second, the answer is straightforward, namely that it is not possible to deciding by simply
looking at a grid whether it is good or not. However, there exist several criteria that tell when
grid quality is not good.

º high degree of skewness

º abrupt changes in grid spacing

º insufÞcient grid line continuity (C0 , C1 , or C2 )

º nonalignment of the grid with the ßow

º insufÞcient resolution to resolve proper physical length scales

º grid topology not well suited to sufÞciently cover the ßow physics

º grid lacks special features needed by physical submodels

º grid is not singularity free (e.g. at stagnation point)

The Þrst three points are mostly independent of the ßow physics. Their effect is that they
reduce the order of the numerical scheme, and hence the accuracy of the results may deteriorate.
However, the magnitude of these effects will depend on the underlying ßow physics.

The other topics are strongly coupled with the ßow physics. It is known from numerical error
investigations that locally 1D ßow (proper grid alignmnet) will give an improved numerical accu-
racy, e.g. grid lines conforming to the boundary of a body will produce better accuracy. Length
scales have to be properly resolved (e.g. boundary layer), otherwise the physics cannot be ade-
quately represented, resulting in, for example, the wrong drag or skin friction. Grid topology also
has an inßuence on the physical results. An O-type topology for a 4 element airfoil may be ade-
quate for Euler solutions, but may be insufÞcient to resolve the wake resulting from viscous ßow,
necessitating a C-H type topology. Physical submodels may, for example, need grid line orthogo-
nality at Þxed boundaries. This is the case for some turbulence models to determine the distance
from the wall. Grids having a singularity, e.g. at the nose part of an aircraft, may lead to substan-
tially increased computing time because of a slowdown in convergence, since no information can
cross the singularity.

14

3.0.4 Aspects of Multiblock Grid Generation

Structured grids use general curvilinear coordinates to produce a body Þtted mesh. This has the
advantage that boundaries can be exactly described and hence BCs can be accurately modeled.
In early grid generation it was attempted to always map the physical solution domain (SD) to a
single rectangle or a single box in the computational domain (CD). For multiply connected SDs,
branch cuts had to be introduced, a procedure well known in complex function theory and analytic
mapping of 2D domains, e.g. the Joukowski airfoil proÞle. However, it became soon obvious
that certain grid line conÞgurations could not be obtained. If one considers for example the 2D
ßow past an inÞnitely long cylinder, with a small enough Re number, it would be advantageous
if the grid line distribution would be similar to the streamline pattern. A 2D grid around a circle
which is mapped on a single rectangle necessarily has O-type topology, unless additional slits
(double valued line or surface) or slabs (blocks are cut out of the SD) are introduced. The main
advantage of the structured approach, namely that one only has to deal with a rectangle or a box,
that is a code is needing only 2 or 3 ”for” loops (C language), has been lost. The conclusion is
that this amount of structuredness is too rigid, and some degree of unstructuredness has to be
introduced. From differential geometry the concept of an atlas consisting of a number of charts is
known. The set of charts covers the atlas where charts may be overlapping. Each chart is mapped
onto a single rectangle. In addition, now the connectivity of the charts has to be determined. This
directly leads to the multiblock concept, which provides the necessary geometrical ßexibility and
the computational efÞciency for the Þnite volume or Þnite difference techniques.

For a vehicle like the Space Shuttle a variety of grids can be constructed (see Sec. ??). One
can start with a simple monoblock grid that wraps around the vehicle in an O-type fashion. This
always leads to a singular line, which normally occurs in the nose region. This line needs special
treatment in the ßow solution. It has been observed that convergence rate is reduced, although
special numerical schemes have been devised to alleviate this problem. Furthermore, a singularity
invariably leads to a clustering of grid points in an area where they are not needed. Hence, com-
puting time may be increased substantially. In addition, with a monoblock mesh gridline topol-
ogy is Þxed. Additional requirements with regard to grid uniformity and orthogonality cannot be
matched. The fourblock grid shown in Sec. ?? removes the singularity but otherwise retains the O-
type structure. The number of gridpoints is reduced substantially. The grid is smooth across block
boundaries. Therefore, no special discretization in the ßow solver across boundaries is needed. A
94 block Euler grid for the Shuttle has been generated, which includes the body ßap. The larger
number of blocks is needed to get a special grid line topology, and to better represent the ßow
physics (see discussion in Sec.??).

Since multiblock grids are unstructured on the block level, information about block connec-
tivity is needed along with the of each block. For reasons of geometrical ßexibility it is mandatory
that each block has its own local coordinate system. Hence blocks can be rotated with respect to
each other (Sec. 3.2.3). Slope continuity across neighboring block boundaries is achieved by over-
lapping edges (2D) or faces (3D). For grid generation an overlap of exactly 1 row or 1 column is
needed (2D). The ßow solver ParNSS needs an overlap of 2 faces.

The solution domain is subdivided into a set of blocks or segments (in the following the words
block and segment are used interchangeably). The overlap feature facilitates the construction of
the ßow solver substantially and allows the direct parallelization of the code on massively parallel


systems using message passing.

Each (curvilinear) block in the physical plane is mapped onto a Cartesian block in the com-
putational plane (CP). The actual solution domain on which the governing physical equations are
solved is therefore a set of connected, regular blocks in the computational plane. However, this
does not mean that the solution domain in the computational plane has a regular structure, rather
it may look fairly fragmented. For the parallelization the important point is that there is no near-
est neighbor relation for the blocks. Therefore communication among blocks follows an irregular
pattern. A parallel architecture that is based on nearest neighbor communication, e.g. lattice gauge
theory problems, will not perform well for complex aerodynamic applications, simply because of
the communication overhead, caused by random communication.

The grid point distribution within each block is generated by the solution of a set of three
Poisson equations, one for each coordinate direction. The right hand side of the Poisson equations
is used for grid point control and is determined from the speciÞed grid point distribution on the
surfaces. That means that Þrst the right hand side is determined for each grid point on a surface and
then the values of the control functions are interpolated into the interior of the solution domain,
with some additional smoothing. In this context a grid point is denoted as boundary point if it
lies on one of the six faces of a block in the CP. However, one has to discern between physical
boundary points on Þxed surfaces that can be used for computation of the right hand side of
the Poisson equation and matching boundary points on overlap surfaces connecting neighboring
blocks. The positions of the latter ones are not known a priori but are determined in the solution
process (see above). All grid point positions on the faces of a block must be known before the
Poisson equations for this block can be solved to determine the grid point positions of the interior
points.

A coordinate transformation of the governing physical equations and their respective boundary
conditions from the physical plane to the computational plane is also required.

16

3.1 Equations of Numerical Grid Generation

3.1.1 Elliptic Equations for 2D Grid Generation

In the following the elliptic PDEs used in the grid generation process will be derived, using the
mathematical tools of the previous section. First, we wish to explain the motives leading to this
approach. In the past, Þnite differences and Þnite elements have been used extensively to solve
computational problems. The latter have been mainly used in structural mechanics. About two
decades ago, Thompson et al.( see e.g. [2]) introduced the concept of a boundary Þtted grid (BFG)
or structureg grid (SG), that is a grid whose grid lines are aligned with the contours of the body.
Clearly, such a grid has to have coordinate lines, i.e. it cannot be completely unstructured. In
computational ßuid dynamics (CFD) in general and in high speed ßows in particular many ßow
situations are encountered where the ßow in the vicinity of the body is aligned with the surface, i.e.
there is a prevailing ßow direction. This is especially true in the case of hypersonic ßow because
of the high kinetic energy. The use of a SG, allows the alignment of the grid in that direction,
resulting in locally quasi 1D ßow. Hence, numerical diffusion can be reduced, i.e. better accuracy is
achieved. A BFG exactly matches curved boundaries, and for complex SDs generally will consist
by a set of so called blocks, which are connected. In the approach taken by the authors, a SD may
be covered by a set of hundreds or even thousands of blocks. Second, SGs can be made orthogonal
at boundaries, facilitating the implementation of BCs (Boundary Condition) and also increasing
the numerical accuracy at boundaries. Furthermore, orthogonality will increase the accuracy when
algebraic turbulence models are employed. In the solution of the N-S (Navier-Stokes) equations,
the BL (Boundary Layer) must be resolved. This demands that the grid is closely wrapped around
the body to describe the physics of the BL (some 32 layers are used in general for SGs or UGs).
Here some type of SG is indispensable. In addition, to describe the surface of the body a structured
approach is better suited.

The resolution of the BL leads to large anisotropies in the length scales in the directions along
and off the body. Since the time-step size in an explicit scheme is governed by the smallest length
scale or, in the case of chemical reacting ßow, by the magnitude of the chemical production terms,
extremely small time steps will be necessary. This behavior is not demanded by accuracy but to
retain the stability of the scheme. Thus, implicit schemes will be of advantage. In order to invert the
implicit operator, factorization is generally used, resulting in two factors if LU decomposition (that
is, factoring in the direction of the plus and minus eigenvalues of the Jacobians) is employed, or in
three factors if the coordinate directions are used. For the unstructured approach there is no direct
way to perform this type of factorization. Moreover, the use of the so-called thin layer approach,
that is retaining the viscous terms only in the direction off the body, reduces the computer time by
about 30 %. Since there are no coordinate lines in the UG, this simpliÞcation is not possible. A
fairly complex procedure would be needed to artiÞcially construct these lines. Moreover, the ßow
solver based on the UG is substantially slower than for the SG. This is due to the more complicated
data structure needed for UGs. Factors of 3, and by some authors of up to 10, have been given in
the literature.

In order to calculate heat loads for vehicles ßying below Ma 8, turbulence models have to
be used, for example, the difference in surface temperature for S¨ nger at cruising speed (around
a
Ma 5) is about 500 K depending on laminar or turbulent ßow calculations. Depending on the real


surface temperature encountered in ßight a totally different type of vehicle has to be designed
since a cooled Titanium wall cannot withstand a temperature of 1300 K for a long period (about
20 minutes). Therefore turbulent calculations are of high importance. Only a SG can provide the
alignment along with the orthogonality at the boundary to accurately perform these calculations.

In general, SG provide sophisticated means both for clustering and adaptation using redistri-
bution or local enrichment techniques. A comparison of these two approaches is given by Dannen-
hoffer [2] where local enrichment gives somewhat better results. However, it is much more costly
to use. In many cases of practical interest, for example Þne resolution of a bow shock or a canopy
shock and in situations where shocks are reßected, the alignment of the grid can result in a more
accurate solution than randomly Þlling the space with an enormously large number of smaller and
smaller tetrahedrons or hexahedrons.

The highest degree of freedom of course is obtained in UGs. The majority of physical phe-
nomena encountered in external and also in internal ßows exhibit certain well ordered structures,
such as a bow shock or system of reßected shocks or some type of shock-shock interaction, which
can be perfectly matched by adaptive grid alignment, coupled to the solution process.It is therefore
felt, however, that mesh redistribution and alignment is totally adequate for the major part of the
ßow situations encountered in CFD, especially in aerodynamics.

Only if a very complex wave pattern evolves due to special physical phenomena, for example,
generating dozens of shock waves coming from an explosion, the UG seems to be advantageous.

In addition, the coupling of SGs with UGs is possible as has been shown by Weatherill and
Shaw et al. [40]. Such a grid is called a hybrid grid.

A mixture between the boundary Þtted grid approach and the completely unstructured ap-
proach is the use of multiblock grids, which has been employed in the Grid [27] and in the
GridPro[35] packages. On the block level the grid is unstructured, that is the blocks itself can
be considered as Þnite elements, but within each block the grid is structured and slope continuity
is provided across block boundaries (on the coarsest level). If a block is reÞned locally, this fea-
ture cannot be maintained. Recently a 94 block grid (Euler) and a 147 block grid (N-S) for the
Space Shuttle have been generated, demonstrating the variability of this approach. Moreover, the
automatic zoning feature of GridProhas been used to generate a grid of several thousand blocks
(see Sec. ??).

For the derivation of the transformed grid generation equations one starts from the original
Poisson equations and then the role of the dependent and independent variables is interchanged.

∆ξ P; ∆η Q (3.1)

Using (2.34), the transformation equation of the Laplacian, and the fact that ξ and η are coor-
dinates themselves, we Þnd

∆ξ gik ´ξ i k Γik ξ
j
j µ gik ei k ¡ e1 (3.2)

18

1
∆ξ Ô yη ´g11 xξξ · 2g12 xξη · g22 xηη µ xη ´g11 yξξ · 2g12 yξη · g22 yηη µ P (3.3)
g

In a similar way we obtain for the η coordinate

∆η
·1
Ô yξ ´g11 xξξ · 2g12 xξη · g22 xηη µ xξ ´g11 yξξ · 2g12 yξη · g22 yηη µ Q (3.4)
g

By means of (2.37), the gi j can be expressed in terms of gi j . For stability reasons in the nu-
merical iterative solution of the above system of equations, the above equations are rewritten in
the form

g22 xξξ 2g12 xξη · g11 xηη · g´xξ P · xη Qµ 0
g22 yξξ 2g12 yξη · g11 yηη · g´yξ P · yη Qµ
(3.5)
0

These equations are quasilinear, where the non-linearity appears in the expression for the met-
ric coefÞcients.

For 2D, using the original functions P and Q and coordinates x, y and ξ, η, one obtains:

g22 yξξ 2g12 yξη · g11 yηη · g´yξ P · yη Qµ
(3.6)
0

where

g11 x2 · y2
ξ ξ g12 g21 xξ xη · yξ yη g22 x2 · y2
η η
g11 g22 g g12 g21 gg12 gg21 g22 gg11 (3.7)
g ´xξ yη xξ yη µ2

is used. The numerical solution of Eqs.(3.6) along with speciÞed control functions P, Q as
well as proper BCs is straightforward, and a large number of schemes is available. Here a very
simple approach is taken, namely the solution by Successive-Over-Relaxation (SOR). The second
derivatives are described in the form

´xξξ µi j xi·1 j 2xi j · xi 1 j
´xηη µi j xi j·1 2xi j · xi j 1 (3.8)
´xξη µi j 1 4´xi·1 j·1 xi 1 j·1 xi·1 j 1 · xi 1 j 1µ

Solving the Þrst of Eqs.(3.6) for xi j yields the following scheme


xi j 1 2´αi j · γi j µ 1 αi j ´xi·1 j xi 1 j µ
βi j ´xi·1 j·1 xi 1 j·1 xi·1 j 1 · xi 1 j 1µ·
γi j ´xi j·1 xi j 1 µ· (3.9)
1 2Ji2 j Pi j ´xi·1 j xi 1 j µ·
Ó
1 2Ji2 j Qi j ´xi j·1 xi j 1 µ

where the notation J2 g, αi j g22 , 2βi j g12 , and γi j g11 was used. Overrelaxation is
achieved by computing the new values from

xnew xold · ω´x xold µ 1 ω 2 (3.10)

The solution process for a multiblock grid is achieved by updating the boundaries of each
block, i.e. by receiving the proper data from neighboring blocks and by sending overlapping data
to neighboring blocks. Then one iteration step is performed using these boundary data, iterating on
all interior points. After that, those newly iterated points which are part of the overlap are used to
update the boundary points of neighboring blocks and the whole cycle starts again, until a certain
number of iterations has been performed or until the change in the solution of two successive
iterations is smaller than a speciÞed bound.

3.1.2 Elliptic Equations for Surface Grid Generation

x
A curved surface can be deÞned by the triple y ú vµ,
z
where u and v parametrize the surface. DeÞning a new coordinate system on such a surface only
means another way of parametrization of the surface, i.e. a transformation from ú vµ to ´ξ ηµ,
which is a mapping from R2 R3 . To obtain the new coordinates Poisson equations are used. The
derivation for the surface grid generation equations is similar as in the 2D or 3D case. Using the ∇
operator of equation 2.18, the surface grid generation equations take the form

j
∆ξ gik ξ i k Γik ξ j P
j
(3.11)
∆η gik η i k Γik η j Q

In this case the derivatives are with respect to u and v, and not with respect to ξ and η. There-
fore, the metric is given by the transformation of the parameter space ú vµ to physical space
´x y zµ, i.e ei ∂i ´x y zµ with ∂1 ∂u and ∂2 ∂v . To obtain the familiar form of the equations,
i.e. ú vµ values are the boundary points in the ´ξ ηµ grids in the computational plane (CP), the
dependent and independent variables in Eqs. 3.11 need to be interchanged. Introducing the matrix
M of the contravariant row vectors from the transformation ú vµ ´ξ ηµ,

ξu ηu
M (3.12)
ξv ηv

20

and the inverse matrix, M 1 , formed by the covariant columnvectors,

uξ uη
M 1 vξ vη (3.13)

along with the deÞnition of S : ´ξ ηµT the surface grid equations Eqs. 3.11 take the following
form:

gik Γ1 P
gik S i k M ik
Q (3.14)
gik Γ2
ik

From this folllows

gik ξ i k gik Γ1
M 1 M 1
P ik
Q (3.15)
gik η i k gik Γ2
ik

The surface metric induces, in comparison with the plain 2D case, additional terms that are
independent of ξ η and act as some kind of control functions for the grid line distribution. To
determine the Þnal form of the equations the role of the independent and the dependent variables
in the second derivatives ξ i k η i k have to be interchanged. This is done as follows. The matrix of
covariant row vectors M can be written in the form

ξu ξv 1 vη uη
M ηu ηv vη uξ (3.16)
J

Hence, ξuu ´ξu µu
v ¡η
where ∂u ξu ∂ξ · ηu ∂η . This results in the following second
J u
derivatives

1 Ò Ó
ξuu ξu v2 uξξ 2vξ vη uξη · v2 uηη ξv v2 vξξ 2vξ vη vξη · v2 vηη
η ξ η ξ (3.17)
J2

1
ξu u2 uξξ 2uξ uη uξη · u2 uηη
ξvv J2 η ξ Ó (3.18)
ξv u2 vξξ 2uξ uη vξη · u2 vηη µ
η ξ

¡
ξu uη vη uξξ uη vξ · vη uξ uξη · uξ vξ uηη
¡
ξuv 1
J2 ¡
ξv uη vη vξξ uη vξ · vη uξ vξη · uξ vξ vηη
¡· (3.19)

ηuu 1
J2
ηu v2 uξξ 2vξ vη uξη · v2 uηη
η ξ
(3.20)
ηv v2 vξξ 2vξ vη vξη · v2 vηη
η ξ


The quantities ηvv and ηuv are transformed similar. With the deÞnition of S stated above the
formulas can be summarized as:

1 v2 uξξ 2vξ vη uξη · v2 uηη
η ξ 1
Suu M : Ma (3.21)
J2 v2 vξξ 2vξ vη vξη · v2 vηη
η ξ J2

¡
uη vη uξξ uη vξ · vη uξ uξη · uξ vξ uηη
Suv
1
M ¡
uη vη vξξ uη vξ · vη uξ vξη · uξ vξ vηη
:
1
Mb (3.22)
J2 J2

1 u2 uξξ 2uξ uη uξη · u2 uηη 1
M´ 2 µ
η ξ
Svv : Mc (3.23)
J2 uη vξξ 2uξ uη vξη · u2 vηη
ξ J2

Sustituting the above formulas for S i j into Eq. 3.15 yields

1 11
g a · 2g12 b · g22 c
¡ · M 1 P

gik Γ1
ik (3.24)
J2 Q gik Γ2
ik

Sorting the vectors a b c with respect to the second derivatives of r ú vµ, then one obtains

1
αrξξ 2βrξη · γrηη
¡ · M 1 P

gik Γ1
ik (3.25)
J2 Q gik Γ2
ik

where

α g11 v2 2g12 uη vη · g22 u2
η η
β g11 vξ vη g12 úξ vη · vξ uη µ· g22 uξ uη (3.26)
γ g11 v2 2g12 uξ vξ · g22 u2
ξ ξ

M 1 has been deÞned previously. In the numerical solution of Eq. 3.26 that uses simple SOR,
only terms rξξ and rηη contain values ui j and vi j for which the surface equations have to be solved
for. One has to note that the metric is also a function of variables u v. In the case of planes the
equations reduce to the well known 2-dimensional grid generation equations.

For the numerical calculation of the surface Christoffel symbols the following formulas are
used. Given a function f ú vµ, derivatives fu , fv , fuv , fuu and fvv are needed. For surfaces in 3D
space one has

e1 ´xu yu zv µ
e2 ´xv yv zv µ (3.27)

22

The numerical procedure for the calculation of the metric is as follows. First, all derivatives
xu xv xuu xuv xvv etc. have to be calculated. After that, the 5 quantities gi j and gi j Γ1j have to be
i
stored at each grid point. These quantities and not the coordinate values ´x y zµ should be interpo-
lated in the generation of the surface grid.

Using the relations of Sec. 2.0

g11 x2 · y2 · z2
xu xv · yu yv · zu zv
u u u
g12 (3.28)
g22 x2 · y2 · z2
v v v

the determinant of the metric tensor is

g g11 g22 g2
12 (3.29)

and the contravariant metric coefÞcients are :

g11 g22 g
g12 g12 g (3.30)
g22 g11 g

From this the Christoffel symbols can be determined.

Γ1 e1 ¡ ei k
ik (3.31)
Γ2
ik e2 ¡ ei k

e1 g1k ek g11 e1 · g12 e2 ´·g22e1 g12e2 µ g
(3.32)
e2 g2k ek g21 e1 · g22 e2 ´ g12e1 · g11e2 µ g

Γ1
11 e1 ¡ e1 1 e1 xuu · e1 yuu · e1 zuu
x y z
Γ1
12 e1 ¡ e1 2 e1 xuv · e1 yuv · e1 zuv
x y z (3.33)
Γ1
22 e1 ¡ e2 2 e1 xvv · e1 yvv · e1 zvv
x y z

Resulting in

e1 g11 xuu · 2g12 xuv · g22 xvv
¡·
gi j Γ1j
i x
e1 g11 yuu · 2g12 yuv · g22 yvv
¡·
y
e1 g11 zuu · 2g12 zuv · g22 zvv
¡ (3.34)
z

An analog result follows for gi j Γ2j . All terms have a common factor 1 g.
i


3.1.3 Elliptic Equations for 3D Grid Generation

The following general coordinate transformation from the cartesian coordinate system, denoted
by coordinates ´x y zµ, to the CD, denoted by coordinates (ξ, η, ζ), is considered. The one to one
transformation is given by (except for a Þnite (small) number of singularities):

x x´ξ η ζµ; ξ ξ´x y zµ
y y´ξ η ζµ; η η´x y zµ (3.35)
z z´ξ η ζµ; ζ ζ´x y zµ

Since there is a one-to-one correspondence between grid points of the SD and CD (except for
singular points or singular lines), indices i, j, and k can be used to indicate the grid point position.
The grid in the CP is uniform with grid spacings ∆ξ = ∆η = ∆ζ = 1. Similar as in 2D, a set of
Poisson equations is used to determine the positions of (ξi , η j , ζk ) in the SD as functions of x y z.
In addition, proper Boundary Conditions (BC) have to be speciÞed. Normally, Dirichlet BCs are
used, prescribing the points on the surface, but for adaptation purposes von Neumann BCs are
sometimes used, allowing the points to move on the surface, in order to produce an orthogonal
grid in the Þrst layer of grid points off the surface. The Poisson equations for the grid generation
read:

ξxx · ξyy · ξzz P
ηxx · ηyy · ηzz Q (3.36)
ζxx · ζyy · ζzz R

where P Q R are so-called control functions that depend on ξ, η and ζ.

However, this set of equations is not solved on the complex SD, instead it is transformed to the
CD. The elliptic type of Eqs.(3.36) is not altered, but since ξ, η and ζ are coordinates themselves,
the equations become nonlinear. In more compact notation the elliptic generation equations read.

∆ξ P; ∆η Q; ∆ζ R (3.37)

where planes of constant ξ, η or ζ form the boundaries and x, y, z are now the dependent
variables. Equations 3.37 are solved in the computational plane using Eq.2.18. The x, y, z coor-
dinate values describing the surfaces which form the boundary of the physical solution domain,
are now used as boundary conditions to solve the transformed equations below. To perform the
transformation the ∆-operator in general curvilinear coordinates is used:

j
∆Ψ gik Ψ i k Γik Ψ j (3.38)

where Ψ is a scalar function. The determinant of the metric tensor, g, is given by the
transformation from (x,y,z) to ´ξ η ζµ. The co -and contravariant base vectors are given by
ei ∂i ´x y zµ ; ei ∂i ´ξ η ζµ with ∂1 ∂ξ ∂1 ∂x etc.

24

gi j ei ¡ e j ; Γijk ei ¡ e j k ´¡ denotes scalarproductµ (3.39)

If ξ, η or ζ is inserted for Ψ, the second derivatives are always zero and Ψ i 1 only for j=1
if variable ξ is considered etc. Eqs. 3.37 then become

gik Γ1
ik P ; gik Γ2
ik Q ; gik Γ3
ik R (3.40)

or in vector notation

¼ g11x · 2g12x · ½
ξξ ξη P
E ¡ g yξξ · 2g yξη ·
11 12 Q (3.41)
g11 zξξ · 2g12 zξη · R

¼ ½
e1
where E e2 is the matrix consisting of the contravariant row vectors. The inverse
e3
matrix is F ´e1 e2 e3 µ consisting of the covariant column vectors. Thus E 1 F and Þnally the
3d grid generations are in the form

¼ g11x · ½
ξξ P
g yξξ ·
22
F ¡ Q (3.42)
g33 zξξ · R


3.2 Grid Generation Concepts

3.2.1 Computational Aspects of Multiblock Grids

As has been discussed previously, Boundary Fitted Grids (BFG) have to have coordinate lines, i.e.
they cannot be completely unstructured. In CFD in general, and in high speed ßows in particular,
many situations are encountered for which the ßow in the vicinity of the body is aligned with the
surface, i.e. there is a prevailing ßow direction. This is especially true in the case of hypersonic ßow
because of the high kinetic energy. The use of a structured grid (SG), allows the alignment of the
grid, resulting in locally 1D ßow. Hence, numerical diffusion can be reduced, i.e. better accuracy
is achieved. A BFG exactly matches curved boundaries, and for complex SDs will consist of a
set of blocks. In the present approach, a SD may be covered by a set of hundreds or thousands of
blocks. Second, SGs can be made orthogonal at boundaries and almost orthogonal within the SD,
facilitating the implementation of BCs (Boundary Condition) and also increasing the numerical.
Numerical accuracy will increase further when algebraic turbulence models are employed using an
almost orthogonal mesh. In the solution of the N-S (Navier-Stokes) equations, the BL (Boundary
Layer) must be resolved. This demands that the grid is closely wrapped around the body to describe
the physics of the BL (some 32 layers are used in general for SGs). Here some type of SG is
indispensable.

In addition, to describe the surface of the body a structured approach is better suited. The
resolution of the BL leads to large anisotropies in the length scales in the directions along and
off the body. Since the time-step size in an explicit scheme is governed by the smallest length
scale or, in the case of chemical reacting ßow, by the magnitude of the chemical production terms,
extremely small time steps will be necessary. This behavior is not demanded by accuracy, but to
retain the stability of the scheme. Thus, implicit schemes will be of advantage. In order to invert the
implicit operator, a SG produces a regular matrix, and thus makes it easier to use a sophisticated
impicit scheme. Moreover, the use of the so-called thin layer approach, that is retaining the viscous
terms only in the direction off the body, reduces computer time by about 30 %. Since there are no
coordinate lines in the UG, this simpliÞcation is not directly possible. A fairly complex procedure
would be needed to artiÞcially construct these lines. Moreover, the ßow solver based on the UG
approach is substantially slower than for SGs. This is due to the more complicated data structure
needed for UGs. Factors of 3, and by some authors of up to 10, have been given in the literature.

An important point for the accuracy of the solution is the capability of grid point clustering and
solution adaptation. In general, SGs provide sophisticated means both for clustering and adaptation
using redistribution or local enrichment techniques. A comparison of these two approaches is
given by Dannenhoffer [2] where local enrichment gives somewhat better results. However, it is
much more costly to use. In many cases of practical interest, for example Þne resolution of a bow
shock or a canopy shock and in situations where shocks are reßected, the alignment of the grid
can result in a more accurate solution than randomly Þlling the space with an enormously large
number of smaller and smaller tetrahedrons or hexahedrons. The highest degree of freedom of
course is obtained in UGs. It is therefore felt, however, that mesh redistribution and alignment is
totally adequate for the major part of the ßow situations encountered in external ßows, especially
in aerodynamics.

26

The large majority of physical phenomena encountered in external and internal ßows exhibit
certain well ordered structures, such as a bow shock, or system of reßected shocks or some type
of shock-shock interaction, which can be perfectly matched by adaptive grid alignment, coupled
to the ßow solution.

Only if a very complex wave patterns evolve due to special physical phenomena, for example,
generating dozens of shock waves, the UG has advantages.

In addition, the coupling of SGs with UGs is possible as has been shown by Shaw et al. [].
Such a grid is called a hybrid grid. A mixture between the boundary Þtted grid approach and
the unstructured approach is the use of multiblock grids, which is followed in Grid []. On the
block level the grid is completely unstructured but in each block the grid is structured and slope
continuity is provided across block boundaries.

3.2.2 Description of the Standard-Cube

The following description is used for both the grid generator and the ßow solver, that is the same
command Þle is used. All computations are done on a standard–cube in the computational plane
(CP) as shown in Fig. 3.2. The coordinate directions in the CP are denoted by ξ, η and ζ and block
dimensions are given by I, J, and K, respectively.

ζ

6

3
2

5
4
η
ξ
1

Figure 3.2: Standard cube in CP (computational plane). Each cube has its own local coordinate
system. The grid is uniform in the CP.

In the CP, each each cube has its own right-handed coordinate system ´ξ η ζµ, where the
ξ direction goes from left to right, the η direction from front to back and the ζ direction from
bottom to top. The origin of this coordinate system in the lower left front–corner (Fig. 3.2). The
coordinate values itself are given by the proper grid point indices i, j, k in the ξ, η and ζ directions,
respectively. That means that the values reach from 1 to I in the ξ direction, from 1 to J in the η


direction, and from 1 to K in the ζ direction. Each grid point represents an integer coordinate value
in the computational plane.

A standard box has six faces, which will be numbered in the same way as a dye. Face 1 is the
ξ η plane with a ζ–value of 0, face 6 is the ξ η plane with a ζ–value of K 1. In Fig.3.2 this
corresponds to the bottom and top faces. Face 2 is the front plane and face 5 the back plane, which
means face 2 is deÞned as the ξ ζ plane with a η value of 0 and face 5 is the ξ ζ plane with a
η value J 1. Faces 3 and 4 then denote the left and right right boundaries of the standard box in
Fig.3.2. Both are η ζ planes, having ξ–values 0 and I 1, respectively.

A simple notation of planes within a block can be achieved by specifying the normal direction
along with the proper coordinate value in that direction. For example, face 2 will be uniquely
deÞned by describing it as a J (η) plane with a j value 1, i.e. by the pair (J, 1) where the Þrst
value is the direction of the normal vector and the second value is the plane index . Thus, face 4 is
deÞned by the pair ´I I 1µ. This notation is also required in the visualization module.

Grid points are stored in such away that the I direction is treated Þrst, followed by the J and
K directions, respectively. This implies that K planes are stored in sequence. In the following
the matching of blocks is outlined. First, it is shown how the orientation of a face of a block is
determined. Second, rules are given how to describe the matching of faces between neighboring
blocks. This means the determination of the proper orientation values between the two neighboring
faces. The input description for the Grid control (topology) Þle as used in Grid is shown.

Y

Z

6
4
2

X

physical domai computational domain

Figure 3.3: Mapping of a block from SD to CP. Arrows indicate orientation of faces, which are
numbered in the following way: 1 bottom, 2 front, 3 left, 4 right, 5 back, 6 top. The rule is that
plane ζ 1 corresponds to 1, plane η 1 to 2 and plane ξ 1 to 3.

To determine the orientation of a face, arrows are drawn in the direction of increasing coordi-
nate values. The rule is that the lower coordinate varies Þrst and thereby the orientation is uniquely
determined. The orientation of faces between neighboring blocks is determined as follows, see Fig.
3.5. Suppose blocks 1 and 2 are oriented as shown. Each block has its own coordinate system (right
handed). For example, orientation of block 2 is obtained by rotation of 3 2 about the η–axis – ro-
tations are positive in a counter clockwise sense – and a subsequent rotation of 3 2π about the new
ζ–axis. Thus faces 4 and 6 are matching with the orientations as shown, determined from the rules

28

ζ ζ

η
6

5
3 4
2 4

1
ξ ξ

Figure 3.4: Orientation of faces. Coordinates ξ, η, ζ are numbered 1, 2, 3 where coordinates with
lower numbers are stored Þrst.

180 z
z

h
6 6
2
5 4 4 3
3 5
2

1 x 1
x
h cut
cut

4 map
4

Figure 3.5: Determination of orientation of faces between neighboring blocks.


overlap

overlap

Figure 3.6: The Þgure shows the overlap of two neighboring blocks.

1 2 3 4

5 6 7 8

standard plane

Figure 3.7: The 8 possible orientations of neighboring faces are shown. Cases 1 to 4 are obtained
by successive rotations e.g. 0, 1 π, π and 3 π. The same situation holds for cases 5 to 8.
2 2

30

shown in the previous Þgure. All cases in group 1 can be obtained by rotating about an angle of 0,
1 2π or 3 2π. The same is valid for elements in group 2. The code automatically recognizes if the
orientation between two faces needs a mirroring. Thus cases 1 and case 7 in Fig. 3.7 are obtained
by rotating case 1 by π 2 and then do the mirroring. Thus, we have determined the input needed
for Fig. 3.5, namely how to match face 4 of block 1 to face 6 of block 2. Rotations are denoted
by integer 0, 1, 2, and 3 marking the corresponding orientations. However, if control faces are
speciÞed, that is, the point distribution on the face of a certain block, that will be used to calculate
the control functions for the current face of the block under consideration, values 1 to 8 have to be
provided. This is clear, because the code cannot anticipate in which orientation the distribution is
going to be used. The control information for a 3D case therefore contains one of the 8 orientation
numbers.

3.2.3 The Grid Grid Generation Toolbox

In the following the two codes developed by the authors Grid and GridProwill be presented. In
addition, all grids presented have been generated by these tools. Other codes can be found in the
references.

Grid is a general grid generation package for multiblock 2D, surface, and 3D solution do-
mains that may be of arbitrary shape. Grid is a collection of ANSI Croutines and is modeled
following the Unix toolbox concept. Modules for surface grid generation, 3D grid generation as
well as for adaptation are provided, including grid enrichment. An algebraic-elliptic generation
technique is used with convergence acceleration. Its name is derived from using the star as a
wildcard, which stands for the collection of the modules that make up Grid . Grid visualization
is based on X-Windows and Motif. All major workstations are supported, including Unix PCs.
Therefore, the package can be implemented on nearly any type of computer, ranging from a PC to
a Cray. Grid generation is very fast and efÞcient, which is shown by the fact that several large have
been generated and visualized on a PC. The package comprises interactive tools for surface grid
generation and for grid enrichment, i.e. there are very efÞcient means for grid point clustering,
e.g., to obtain a Navier-Stokes grid from an Euler grid. For the Hermes grid generation an Euler
grid of 150,000 points was converted into a Navier-Stokes grid of 300,000 points in a few minutes
on a PC. Grid can be used in every area where structured grids are needed, and provides special
features, such as grid generation on surfaces.

Grid is a collection of C routines, fairly small in size, e.g. the 3D multiblock grid generation
module, Grid , consists of about 1500 lines in C. Modules are build from reusable subroutines
and can be used in arbitrary order, each designed for a special task.

The grid generation modules Grid , Grid , and Grid use a combination of algebraic and el-
liptic grid generation methods, where elliptic techniques are mainly used for smoothing purposes.
The grid visualization module Xivis is based on X11, allowing this system to be run on virtually
any type of computer, from a PC under Linux, Solaris, or Unix station to an SGI Power Chal-
lenge etc. To speed up convergence, 3D blocks can be interactively cut out of the SD. Grid point
distribution on the block faces is frozen, while points in the interior are iterated. This technique
was successfully used when Grid generated the 150,000 points Euler grid for the Hermes space
plane: 20 iterations with SOR were used for the complete spacecraft, while a block of about 8,000


points was cut out around the winglets and iterated 100 times to improve grid quality and then was
inserted back. In Grid dynamic storage allocation is used, so the user never has to specify any
array dimensions. Special data types are used, which allow a much more compact programming.
All graphics functions are running under X-Windows so the code is fully portable.

Grid input consists of a control Þle, describing the connectivity and orientation with respect to
neighboring blocks. Control functions for grid adaptation can also be speciÞd. Each block has its
own local coordinate system. All grids are slope continuous across block boundaries. Hence block
boundaries are not visible. Grid description is described by a set of keywords used in the control
Þles and the coordinates Þles. The widely used Plot3d (NASA Ames) grid Þle format is supported.

3.2.4 Input for Grid Generation in 2D and 3D

The Grid package expects geometrical objects as input. In order to identify the object type, the
actual data must be proceeded by an identiÞer. The following object types can be speciÞed in
Grid input- and output Þles:
Òline2d, Òline3d, Òplane2d, Òplane3d, Òvol3d, Òcntrl2d, Òcntrl3d.
File formats can be either in ASCII or binary, which is automatically detected by the input routines.
One additional command is ÒÞle Þlename, which redirects the input to Þle Þlename until an end-
of-Þle is encountered. After that, reading is resumed by the calling Þle, which in general is the
control (command) Þle. The ÒÞle command can be nested. Furthermore, there are three internal
object types, that means, object types, which are used only inside a module, namely Òerror, Òend,
Òdigit.

The meaning and usage of these objects is as follows. Lines (curves) are needed as boundaries
of planes and surfaces, which are embedded in 2- or 3-dimensional space. That is, each point
is given by a pair (x,y) or by a triple (x,y,z). The same is true for surfaces. In case of a plane,
object Òplane2d, two coordinate values are needed while for a surface in 3D, object Òplane3d),
three coordinate values are required. A volume is part of 3-dimensional space, therefore the object
name Òvol3d. For the description of the matching of neighboring blocks in 2D or 3D control
structures Òcntrl2d or Òcntrl3d are needed. This concerns the command Þles of modules Grid ,
Grid , and Grid . The above mentioned ÒÞle command is a feature which makes the input more
readable, especially for the large input Þles needed in Grid . Instead of placing an object of type
Òplane3d or Òvol3d directly after the control information of a block, which in general comprises a
large number of coordinate values to describe the geometry of the boundary surface or the volume
grid, the user can reference to a Þle containing that object. It is recommended to write the control
information in one Þle and to create as many separate objects of type Òplane2d, Òplane3d or
Òvol3d as there are blocks. Then a Þle reference is made to the respective Þle containing the plane
or volume data. Volume data are used in a restart case. If a multiblock grid with a larger number
of blocks (see Sec. ??) is to be generated, the use of a separate Þle for the boundary data of each
block is the only possible way. For the parallel grid generator this feature has to be used, because
it would be completely impractical if each node read the entire input.

32

3.2.4.1 Rectangle Grid Example

In this section we perform the simple task of generating a monoblock grid for a rectangle. How-
ever, this simple example allows us to introduce most of the concepts needed for advanced grid
generation. In Sec. 3.2.4.2 we generate a 6 block grid for a diamond shaped body. In order to
generate a grid two types of informa tion have to be provided, namely the block connectivity
information and the grid point distribution on the physical boundary of the solution domain (SD).

First the so called control Þle or command Þle (extenion .cmd) has to be speciÞed. The com-
mand Þle describes the blocking of the solution domain. It also contains the references to those
Þles that contain the surface grid. Since this is the Þrst example, a 2D problem is considered only.
Therefore, the surface grid simply consists of the grid point distribution on the four edges (sides) of
the rectangle. The grid point distribution can be generated interactively, using the tools (see Chap-
ter Grid Generation Tools) Poli and Spline2D. In this example it is assumed that the command
(connectivity) information is stored in the Þle rect.cmd and the geometry data (edge description)
is stored in rect.lin. The Þrst line of the command Þle contains the keyword cntrl2D that simply
denotes a 2D SD. The 1 in next line denotes the block number, followed by the dimensions of
the block (ξ η directions)11 10. It should be noted that two different coordinate systems are used.
In Physical Space (also denoted as the Physical Plane) coordinates are described by (x y z). For
complex geometries, curvilinear coordinates (ξ η ζ) are needed whose directions are conforming
to the shape of the conÞguration. The elliptic grid generation equations as well as the ßow equa-
tions are solved in the transformed or Computational Space (Plane) (see Sec ...). In this plane
coordinate directions are denoted by I, J, and K. In Grid the following rule applies (respectively
in three dimensions):

Rule: In specifying the dimensions of a block the Þrst number denotes the number of grid
points in the ξ ´I µ direction (number of columns), the second one gives the number of points
in the η ´J µ direction (number of rows).

A second input rule determines the orientation of the edges with respect to the axes of the
coordinate system. If a grid comprises several blocks, each block can have its own local coordinate
system and these systems can be rotated with respect to each other. In 2D the rotation is determined
automatically by the code. The four edges of a block are numbered 1, 2, 3, and 4. However, to
enable the code to perform the automatic detection of rotated blocks, the following rule must be
obeyed (for three dimensions see later chapter):

Rule: With respect to the coordinate axes of the block the numbering of the edges has to be
such that: edge 1 is east, edge 2 is north, edge 3 is west, and edge 4 points south.

In what follows, a description of the simple format of the command Þle is given, sufÞcient for
the rectangle example. The complete command Þle format is described in section ??. The next
four lines describe the edges of the rectangle.

The Þrst integer gives the number of the edge, followed by the type of the side. A type 0
or 1 describes a Þxed boundary, a type 2 or 3 indicates a matching boundary, i.e. an interblock
boundary. In this example, there are only Þxed edges, hence sides are either of type 0 or 1. A side
of type 1 is used for interpolating an initial grid. For each point on this edge the corresponding


point on the opposite edge is known. If this edge is of type matching, the corresponding opposite
side in the neighboring block is investigated. If this edge is Þxed, a linear interpolation between
the two corresponding Þxed edges is performed. If the edge is matching, the algorithm continues
through all blocks until a Þxed side is encountered. Since the interpolation is linear, the quality
of the initial grid depends on the choice of the edge where the interpolation process starts from.
To obtain a good initial grid, several edges can be used. Moreover, an interpolated grid may be
partly overwritten in this process, allowing complete freedom. The interpolated grid speeds up the
grid generation process by a factor of 5 - 10, making it possible to generate grids with hundreds
of thousands of points on a workstation. The next two integers are needed to describe the block
connectivity, specifying the block and edge number of the neighboring block. If an edge is not
neighbored, these numbers are 0. Integers 5 and 6 describe the so called control functions (see next
Chapter) used for grid point clustering. The last line, Þle rect.lin is a reference to the geometry Þle
that contains the coordinate values of the 4 edges of the rectangle in exactly the same order as the
edges occur in the command Þle. For example, it would have been possible to describe edge 2 Þrst
and the edge 1 in the command Þle. In that case, however, the sequence of the edge coordinates in
Þle Þle rect.lin must be consistent with the order of the edges in Þle rect.cmd.

Ò cntrl2d
1
11 10
110000
200000
300000
400000
Þle rect.lin

In Þle Þle rect.lin the Þrst line indicates that data are of type line 2d, that is are part of a
2D curve. The next value gives the number of coordinate pairs (x, y) that form this line. It should
be noted that all physiacl coordinates are given with respect to a Cartesian coordinate system that
holds for all blocks.

3.2.4.2 Diamond Shape 6 Block Grid Example

To illustrate the multiblock concept, a simple 6 block grid for a diamond shape is constructed, see
Fig. 3.8. The control Þle information for the grid of Fig. 3.8 is shown in Fig. 3.10. The meaning
of the control information is explained below. Since the example is 2D, the Þrst line of this Þle
starts with Òcntrl2d. In Fig. 3.11 the corresponding coordinate values are speciÞed for all Þxed
(physical) boundaries. The proper type of these boundaries is therefore Òline2d.

In the following a complete description of the topology of multiblock grids is given. Having
read this information, the reader should refer back to the example in Fig. 3.10.

Each data Þle begins with a line indicating the object type. In 3D, this line has the form object
type
for example

34

Òline2d Òline2d
10 10
10 0 00
10 1 01
10 2 02
10 3 03
10 4 04
10 5 05
10 6 06
10 7 07
10 9 09
10 10 0 10
Òline2d Òline2d
11 11
0 10 00
1 10 10
2 10 20
3 10 30
4 10 40
5 10 50
6 10 60
7 10 70
8 10 80
9 10 90
10 10 10 0


Figure 3.8: A 6 block grid for diamond shaped body. This type of grid line conÞguration cannot
be obtained by a mono-block grid.

Figure 3.9: Grid line can also be clustered to match the physics of the ßow; e.g. resolving a bound-
ary layer.

36

Òcntrl2d
1
8 11
1 2 3 3 0 0
2 3 2 4 0 0
3 1 0 0 0 0
4 1 0 0 0 0
2
8 11
1 3 4 3 0 0
2 0 0 0 0 0
3 1 0 0 0 0
4 2 1 2 0 0
3
7 11
1 2 5 3 0 0
2 0 0 0 0 0
3 3 1 1 0 0
4 1 0 0 0 0
4
7 11
1 2 6 3 0 0
2 0 0 0 0 0
3 2 2 1 0 0
4 1 0 0 0 0
5
8 11
1 0 0 0 0 0
2 2 6 4 0 0
3 3 3 1 0 0
4 1 0 0 0 0
6
8 11
1 0 0 0 0 0
2 0 0 0 0 0
3 3 4 1 0 0
4 3 5 2 0 0
ÒÞle diamond.lin

Figure 3.10: Control information for the 6 block diamond grid. This command Þle is also used by
the parallel ßow solver. ÒÞle diamond contains the actual coordinate values.


Òline2d Òline2d Òline2d
11 7 11
-1 -1.0 -0.3 0 1 -1
-1 -0.9 -0.2 -0.1 1 -0.9
-1 -0.8 -0.1 -0.2 1 -0.8
-1 -0.7 0.0 -0.2 1 -0.7
-1 -0.6 0.1 -0.2 1 -0.6
-1 -0.5 0.2 -0.1 1 -0.5
-1 -0.4 0.3 0 1 -0.4
-1 -0.3 Òline2d 1 -0.3
-1 -0.2 7 1 -0.2
-1 -0.1 -0.3 -1 1 -0.1
-1 0.0 -0.2 -1 1 0
Òline2d -0.1 -1 Òline2d
8 0 -1 8
-1.0 -1 0.1 -1 0.3 -1
-0.9 -1 0.2 -1 0.4 -1
-0.8 -1 0.3 -1 0.5 -1
-0.7 -1 Òline2d 0.6 -1
-0.6 -1 7 0.7 -1
-0.5 -1 -0.3 1 0.8 -1
-0.4 -1 -0.2 1 0.9 -1
-0.3 -1 -0.1 1 1 -1
Òline2d 0 1 Òline2d
8 0.1 1 11
-1.0 1 0.2 1 10
-0.9 1 0.3 1 1 0.1
-0.8 1 Òline2d 1 0.2
-0.7 1 7 1 0.3
-0.6 1 -0.3 0 1 0.4
-0.5 1 -0.2 0.1 1 0.5
-0.4 1 -0.1 0.2 1 0.6
-0.3 1 0.0 0.2 1 0.7
Òline2d 0.1 0.2 1 0.8
11 0.2 0.1 1 0.9
-1 0 0.3 0 11
-1 0.1 Òline2d
-1 0.2 8
-1 0.3 0.3 1
-1 0.4 0.4 1
-1 0.5 0.5 1
-1 0.6 0.6 1
-1 0.7 0.7 1
-1 0.8 0.8 1
-1 0.9 0.9 1
-1 1.0 1 1

Figure 3.11: Coordinate values of the Þxed (physical) boundaries of the 6 block diamond grid.

38

Òcntrl2d

After this control line, object speciÞc information is expected. One can write any number of
objects (in this example Òcntrl3d) after this control line as needed. The object speciÞcation is
valid until the next control line is encountered or if the end of the current input Þle is read. All
control lines, which can not be identiÞed are converted to the internal object type error. In control
Þles, command information and geometrical information of objects may be mixed, .e.g., one can
start with the control information of type Òcntrl3d to specify the block. After that, a line may be
given

plane3d

to tell Grid that surface information for this block follows or a line

Þle Þlename

could be given to indicate that surface data for the next block are stored in Þle with name
Þlename.

To be more speciÞc, the input format for the different object types will now be explained in
detail.

In short notation an object of type Òline2dhas the following form:

line2d
I
x(1) y(1)
..
..
..
x(I) y(I)

Object Òline2d indicates the boundary line of a 2D SD. The value of I speciÞes the number of
data points, followed by I pairs of x and y values, denoting the boundary points.

Objects plane2d and plane3d are of the following form.

plane2d
IJ
x(1,1) y(1,1)
..
..
x(I,1) y(I,1)
x(1,2) y(1,2)
..
..
x(I,J) y(I,J)

plane3d
IJ
x(1,1) y(1,1) z(1,1)
..
..
..
x(I,J) y(I,J) z(I,J)


In Òvol3d the I-index is running fastest, followed by the J and K indices, so the K planes (I x
J coordinates each) are stored consecutively.

vol3d
IJK
x(1,1,1) y(1,1,1) z(1,1,1)
..
..
x(I,1,1) y(I,1,1) z(I,1,1)
x(1,2,1) y(1,2,1) z(1,2,1)
..
..
x(I,J,1) y(I,J,1) z(I,J,1)
x(1,1,2) y(1,1,2) z(1,1,2)
..
..
x(I,J,K) y(I,J,K) z(I,J,K)

The control information for 2d reads

cntrl2d
nos
IJ
s1 st nb ns cb cs
s2 st nb ns cb cs
s3 st nb ns cb cs
s4 st nb ns cb cs

where nos is the block number and I, J are the number of grid points in the respective direc-
tions. The next four lines describe the four edges of the block. s1 to s4 denote the side-number
where 1 is east, 2 north, 3 west, and 4 south. st is the side-type. 0 means Þxed side, 1 is a Þxed
side which is used for computing the initial algebraic grid, before the Poisson equations are used.
A side of type 2 is a matching side (overlap). In this case, the corresponding values for nb and
ns have to be given where nb is the number of the neighboring block and ns the number of the
matching side of this block. If st is 0 or 1, these values should be set to zero. cb and ct denote
the control block and control side from which the distribution of boundary points is to be taken to
calculate the controlfunctions for the right hand side of the Poisson equations. Hence, these values
should reference a Þxed side, which must have the same number of grid points as the current side.
It is not necessary to set s1 to 1 etc. The edge control information can be in any order. The only
restriction is that the same order is used when the boundary data are read. The 2D code allows
for orthogonal grids at a Þxed boundary (type 5). The distance of the Þrst layer of grid points of a
Þxed boundary (type 6), and a combination of distance control and orthogonality (type 7) can also
be chosen. The prescribed distance values needed for side type 6 and 7 are speciÞed at the end of
the input control Þle. A similar format is used for the control information in 3D.

cntrl3d
nos
IJK
s1 st nb ns nr cb cs cr

Again, nos denotes the block number, I, J and K are the dimensions in x,y and z-direction, re-

40

spectively. Each block has 6 sides, so for each side there is one line with side-speciÞc information.
s1 to s6 are the side numbers as described for the standard block. st again is the side type, where
a 1 denotes the side used for initialization. In addition to the neighboring block nb and the neigh-
boring side ns, rotation value nr is necessary. The same is true for the control side information cb,
cs, and cr. If no clustering is desired, these values are 0. Variables must be in the following range.
Only integer values can be assumed.

¯ s1 s6: [1,6] side number

¯ st: [0,3] side type

¯ nb: [1,N] (N is total number of blocks) neighboring block

¯ ns: [1,6] neighboring side of block nb

¯ nr: [0,3] rotation needed to orient current side to neighboring side

¯ cb: [1,N] block number for control information

¯ cs: [1,6] side number for control information

¯ cr: [1,8] orientation for control information

For a Þxed side control information can be used from other sides (faces) or from other blocks
and the orientation of these sides (faces) has to be speciÞed (cr). In the 3D case a value between
1 and 8 must be given. In the 2D case, there are only 2 possibilities, namely the sides are either
parallel (+1) or antiparallel (-1), which is detected by the code. If control functions are speciÞed
only on one side, and no control functions are given at the opposite Þxed side, the control values
at the opposite boundary are assumed to be zero. The control values within a block are determined
by linear interpolation.


Figure 3.12: So called clamp technique to localize grid line distribution. The real power of this
technique is demonstrated in the Hyperboloid Flare in F4 Windtunnel grid.

Figure 3.13: Clamp 1 at hyperboloid ßare.

3.3 Local Grid Clustering Using Clamp Technique

Along Þxed walls a large number of grid lines is required in order to simulate boundary layers.
In the remaining solution domain this requirement only causes memory overhead and reduces the
convergence speed. It is therefore mandatory to localize the grid line distribution. To this end the
clamp clip technique is used, see Fig. 3.12. The principle of it is to build a closed block system
connected to the physical boundary. The number of grid lines can be controlled within the block. If
the grid is reÞned along the physical boundary, a reÞnement at the outer boundary is also obtained.
Using clamp clips, the grid lines can be closed in clamp blocks. The local grid reÞnement can be
achieved without inßuencing the far Þeld grid, see Figures 3.13 and 3.14.

42

Figure 3.14: Clamp 2 at hyperboloid ßare.


Figure 3.15: A conventional topology for a hyperboloid ßare in a windtunnel. Although the topol-
ogy allows a reÞnement of the grid from an Euler to a N–S grid, this reÞnement extends into the
far Þeld and thus causes a substantial computational overhead. Numbers denote block numbers,
doted lines are block boundaries, solid lines are grid lines.

3.3.1 Hyperboloid Flare in F4 Windtunnel Grid

In Fig. 3.15 a conventional topology is shown to capture the boundary layers both for the windtun-
nel walls and the hyperboloid ßare. The Þrst disadvantage is that all gridlines are extending into
the farÞeld, generating a huge amount of Þnite volumes in areas that don’t need a Þne resolution.
Second, identiÞcation of single objects is not possible.

As a more complex example of the above strategy a ”hyperboloid ßare in the F4 windtunnel”
grid is presented. One of the major constraints in the generation of this grid was to contain the
high degree of grid line clustering around the hyperboloid ßare close to the body. To this end the
”clamp” technique described in Fig. 3.12 was used. Introducing clamp clips to locations near phys-
ical boundaries keeps the grid lines local. Fig. 3.16 shows the local topology of the hyperboloid
ßare.

44

Figure 3.16: Topology of 36 block hyperboloid ßare. This topology is one part of the topology of
284 block grid, see Fig. 3.18.


Figure 3.17: 36 block grid for hyperboloid ßare.

Figure 3.18: Topology of 284 block grid for hyperboloid ßare in windtunnel.

46

Figure 3.19: 284 block grid for hyperboloid ßare in windtunnel.

Grid Adaptation 47

4.0 Grid Adaptation Techniques

For the adaptation of a grid, knowledge about the ßow solution is needed to obtain the desired grid
resolution in all regions that exhibit large variations. Adaptive grid methods monitor the solution
and adjust the grid dynamically, concentrating grid points in areas of larger solution variation.
In many ßows of interest, these regions have a regular structure so that SGs are well suited for
adaptation. The following discussion is restricted to grid point movement.

Common to all adaptation techniques is the speciÞcation of a weight function, obtained from
the solution features of the ßow, denoted by M1 M2 Mn . The Mi are nonnegative functions.
Generally, the weight function w is a linear combination of the Mi

w 1 · c1 M1 · c2 M2 · · cnMn (4.1)

For example the following expressions could be chosen as weight functions:
w 1 · β ∂Φ , using the gradient or w 1 · β κ where κ denotes the curvature of the Φ.
∂ξ
Φ is the so called monitor surface (explained below) and could be constructed, e.g. from the Ma
number distribution.

The basic formula for the redistribution of grid points is the so called equidistribution statement
for the weight function w that, written as Þrst order differential equation, takes the form

w sξ c (4.2)

or in differential form

w ds cdξ (4.3)

where where c constant or c c´ξµ, ds is arc length, and ξ denotes the coordinate of the
computational domain. Assuming a constant c, the Þrst order differential equation of Eq. 4.2 can
be differentiated again, leading to the second order differential equation

´w sξµξ 0 (4.4)

Eqs. 4.2 and 4.4 can be interpreted in several ways, giving rise to various adaptation tech-
niques. The Þrst adaptation technique presented is concerned with the usage of PDEs. When em-
ploying elliptic PDEs for grid generation, the weight function can be used to calculate the so called
control functions P´ξ ηµ and Q´ξ ηµ (2D) that form the rhs of the elliptic PDEs, which are then
called Poisson equations. The effect of P and Q can be described using an example from electro-
statics. If there are no charges in the SD, the electrostatic potential is obtained by solving Laplace’s
equation. Introducing charges into the SD, that is the rhs of the Laplace equation is different from

48

zero (Poisson equation), concentrates the equipotential lines in the vicinity of the charges (nega-
tive) or repels the potential lines (positive charge). If the areas where to cluster the grid are known
a priori, the boundary point distribution could be speciÞed accordingly and P and Q could be com-
puted from this distribution, in order to achieve the same clustering in the interior of the SD as on
the physical boundaries. Without controlfunctions, the Laplace operator would equidistribute the
grid points in the interior.

The second technique which is widely used is of algebraic nature, obtained by interpreting
Eq. 4.2 in a discrete way. A monitor surface (MS) (monitor curve in 1D) is positioned above the
physical SD, e.g., if the SD is a 2D region in the ´x yµ plane, the MS, denoted by Φ´x yµ, is a
curved surface in 3D space that is directly above the 2D region. The MS is a piecewise linear
(smooth) surface. It is constructed by the user; for example, it can be the linear combination of
one or more of the variables that determine the physics of the problem. To capture a shock, Ma
number could be used, combined with the pressure, p, to cluster points in the boundary layer. The
amount of clustering is given by the weight function, which may depend on the gradient or the
curvature of the monitor surface. Examples of monitor surfaces will be given in the next section.
Adaptation is performed by operating on a surface grid lying on the MS. The basic features of the
adaptation algorithm are given below where it is assumed that a grid already exists in the physical
SD, and a ßow solution is available.

Regardless of the dimension, algebraic adaptation is always done on a curve by curve basis.

º the initial surface grid is obtained by projecting (lifting) the grid points of the physical
domain up to the monitor surface.

º surface grid points are subsequently repositioned on the MS by equally distributing the
weight function, w, based on the arc length of the coordinate curves.

º after that, the new surface grid on the MS is projected back down to the physical domain
to yield an improved grid that more accurately represents the ßow physics and thus reduces
the numerical error.

The two techniques mentioned above are derived from the same principle, namely the equilib-
rium statement (see below) that can be interpreted as a differential equation, leading to a Poisson
equation or an integral form, resulting in an algebraic approach. The two techniques are described
in more detail in the subsequent sections.

4.0.1 Adaptation by Controlfunctions

Let us reconsider the grid generation equations

∆ξ P; ∆η Q (4.5)

Interchanging the role of dependent and independent variables leads to

Grid Adaptation 49

g22 yξξ 2g12 yξη · g11 yηη · g´yξ P · yη Qµ 0 (4.6)

These equations are quasilinear, where the non-linearity appears in the expression for the met-
ric coefÞcients.

If we set the control functions P and Q to zero, we obtain the transformed Laplace equation.
The control functions can be used to adapt the grid lines to better represent certain solution features
or to match the distribution of the boundary points in the interior. The Þrst process is a dynamic
one. The latter is performed only once, before the computation is started. In principle, however,
there is no difference between these two approaches. To visualize the distribution of the grid lines
generated by Laplace’s equation, one can assume that each grid point is coupled to its four nearest
neighbors by springs having equal spring constants. The equilibrium of such a spring system gives
the distribution of the coordinate points. If the distribution of boundary points is equidistant, one
observes the following fact: A convex boundary repels grid lines, whereas a concave boundary
attracts grid lines. Very often, however, the opposite behavior is desirable. For example, when
a storm surge model is used for the calculation of the water level in a bay (convex area), high
resolution near the shore is required or, in aerodynamics, the BL for a convex surface might have
to be modeled.

One of the simplest ways to achieve the desired clustering could be to change the distribution
of the boundary points representing the physics, i.e. to choose a dense distribution in the direction
off the convex surface to capture the BL. Since the Laplacian produces an equidistribution in the
interior, the effects of the boundary grid point distribution would be smoothed out within a few
grid points and therefore would not be felt in the interior. For that reason, a Poisson equation is
needed and control functions P and Q have to be determined.

Since a boundary is either a ξ- or an η-line (in 2D), all partial derivatives in the other direction
vanish. Let us assume that ξ only depends on x and η only depends on y. If the boundary is not a
straight line, x and y must be replaced by arc length s. If we determine P from a boundary formed
by an η-line (i.e. ξ varies) and Q from a boundary which is formed from a ξ-line, we Þnd from
equations

g11 xξξ xξ P; g22 yηη yη Q; (4.7)

Also, since xξξ , xξ as well as yηη , yη can be calculated from the boundary point distribution,
equations can be solved for P and Q, yielding

xξξ yηη
P: g11 ; Q: g22 (4.8)
xξ yη

Expanding now Eq. 4.4 and dividing by w, yields
wξ
sξξ s
w ξ

50

Supersonic Inlet: Euler Computation Stationary Solution
19x129 Adaptive Grid
200

150

100

50

0
0 50 100 150 200 250 300 350 400 450 500 550

Physical Parameters:
Mach Number Contour
Ma=3
(computation with restart file) o
AoA=-0.5
Numerical Parameters:
CFL=10 (Implicit Solution)
-4
S-W part=10

1.67494 1.63224
1.58953
2.27274 2.23004

2.91325

Figure 4.1: 1 block adptive grid for supersonic inlet, adapted by control functions.

Comparison with the Þrst equation in Eqs. 4.7 leads to

wξ wη
P g11 ;Q g22 (4.9)
w w

or when resolved for w
P
w c1 exp dξ
g11

where c1 is the constant of integration. Using the relation for P (same holds for Q), the weight
function w can be directly incorporated in the grid generation equations.

The use of these control functions guarantees that the speciÞed boundary point distribution is
obtained in the interior of the SD, i.e. this procedure yields the same grid point distribution along
ξ- and η-lines as on the boundary. The numerical values of P and Q in the interior are determined
by linear interpoloation between two opposite Þxed sides. For curvilinear boundaries, arc length s
replaces x and y.

Figs. 4.1 and 4.2 illustrate the grid adaptation by using control functions. It can be seen that
the adapted grids possess a high degree of smoothness [45].

Grid Adaptation 51

Forward Facing Step: Solution Adaptive Grid

Figure 4.2: 3 block adaptive grid for forward facing step, adapted by control functions.

52

ψ

χ

Figure 4.3: One–dimenisional monitor surface (MS). The initial grid in the physical SD is uniform.
Lifting up the grid points produces the grid on the monitor surface. The variable s denotes arc
lehgth on the MS

ψ

dsi

χ

Figure 4.4: Repostioned grid on the monitor surface. Grid points are uniformly distributed on the
monitor surface so that arc length spacing is constant. When projected down back to the physical
solution domain, this results in a clustering according to the gradient of the monitor surface.

4.0.2 Algebraic Adaptation Algorithms

For the algebraic adaptation it is assumed that a 1D, or 2D grid has already been generated. The
algorithm adapts this grid according to the speciÞed set of weight functions that are generally
obtained from the physical solution. Two algorithms will be presented that are extremely fast, so
that they can be used for time dependent problems, too. The basic idea of these algorithms is taken
from the papers [32], [33], and [31]. First, consider the one-dimensional problem of grid point
clustering. Examples of monitor surfaces are given in the following three Þgures.

The following notations are used in one dimension. Their extension to higher - dimensions is
straightforward.

Grid Adaptation 53

ψ

χ

Figure 4.5: Repositioning of grid points on the monitor surface according to the magnitude of
curvature, resulting in a clustering of grid points to regions where curvature κ 0.

º the continuous variables in the physical SD, on the MS, and in the computational SD are
given by x s, and ξ, respectively.

º there are two sets of grid points: the initial grid points denoted by x , sI and the repositioned
I
i i
grid points which are given by xi si where i 2´1µN 1, and N is the total number of grid
points (or Þnite volumes).
I
º the initial and repostioned points on the monitor surface are also denoted as P and Pi .
i

º grid points in the computational domain are denoted by ξi . In general, ∆ξi 1 is chosen.

º the weight function depends on arc length, i.e., wi w´si µ. Of course, x could also be chosen
as the independent variable, because there is a one-to-one mapping between the grid points
of the physical SD and those on the MS.

º half point values (cell faces) are averaged, e.g. wi·1 :
2
1
2 ´wi · wi·1µ

Large w result in small ∆si , i.e. there is a clustering of grid points where wi is large.

We Þrst present the redistribution approach as described in [32] and [33], setting c constant.
After that, the algorithm of Hsu et al. [31] is outlined that uses the equidistribution statement with
a variable c´ξµ. In both algorithms the original arc length of the MS and curve remain constant.
The one-dimensional version of the algorithms employs the equidistribution statement in the form:
si·1
wds c´ξi·1 ξi µ
si

where c is determined from the requirement of constant arc length.

1. MS Determine the monitor surface from the ßow solution, e.g. use the Ma number distribution.

2. Lift up Lift the initial grid ´xI µ from the physical SD up to the MS, resulting in a grid ´sI µ. If,
i i
for example, Ma is used, one could directly set ΦI : Ma´xI µ -Þltering of Ma values may
i i

54

be needed - resulting in a table ´xI ΦI sI µ i 2´1µN 1 where arc length is computed by
i i i
piecewise linear approximation, using the Euclidean norm.

3. Integrate Determine the weight function ( e.g. magnitude of gradient of Ma number) and in-
tegrate the equidistribution statement from (all values are taken from the initial grid; super-
script I has been ommitted for notational simplicity) s1 to s on the MS. c is obtained by
integrating from s1 to sN
F ´sN µ
c
ξN ξ1
where
s
F ´sµ : wdl
s1

The values of F ´sµ and F ´sN µ are related by the formula

ξ ξ1
F ´sµ F ´sN µ (4.10)
ξN ξ1

Note: In practice the integration is done numerically using the trapezoidal quadrature rule.

4. Build table From step 3 a table of values ´sI F ´sI µµ i 1´1µN is obtained. It should be noted
i i
that the grid sI is not distributed as required by the equidistribution statement and also would
i
not lead to a uniform grid in the compuatational domain when projected back and therefore
has to be repositioned.

5. Find F ´s j µ Determine the F values for the repositioning of grid points. Since the left and right
boundary values, denoted as sL and sR , are known where s1 sI sL , sN sI
1 N sR and
requiring that the ξ j values have uniform grid spacing in the computational domain (as
usual) with ∆ξi 1, one obtains from Eq. 4.11 the implicit condition for the repositioning
of the s j values, namely

j 1
F ´s j µ F ´sN µ j 2´1µN 1 (4.11)
N 1

It is evident that in order to obtain the new sj values, F ´s j µ has to be inverted.

6. Inversion The value F ´s j µ is used to search the table from step 4, returning an index m´ jµ for
which
F ´sI µ F ´s j µ F ´sI ·1 µ j 2´1µN 1
m m

Linear interpolation within interval ´m m · 1µ is utilized and yields

F ´s j µ F ´sI ·1 µ
m
αj (4.12)
F ´sm·1 µ F ´sI µ
I
m

Insertion of α j directly results in

sj sI · α j ´sI ·1 sI µ j
m m m 2´1µN 1 (4.13)

Grid Adaptation 55

7. Repositioning The new values in the physical domain are found by projecting back down
xj xI · α j ´xI ·1 xI µ j
m m m 2´1µN 1

Using a more compact notation Pj : ´x j y j µ results in
Pj PI · α j ´PI ·1 PI µ j 2´1µN 1
m m m

8. Smooth Relax the repositioned grid. Elliptic grid generation is used for a few smoothing itera-
tions (3–4).

It should be noted that the equilibrium statement in step 3 was integrated such that the resulting
integral could be solved for ξ and, as a consequence, F ´sµ had to be inverted to obtain the new arc
length values. Another possibility is to divide Eq. 4.3 by w so that the integration directly gives the
values for s. In that case, however, weight function w has to be determined as a function of ξ and
not of arc length s. Hence, w´s´ξµµ is needed, that is s´ξµ has to be inverted to provide ξ ξ´sµ. In
practical computations, the algorithm is used as presented. In Fig. 4.6 the result of an adaptation
is depicted. The grid was adapted to capture a moving shock and a strong circular gradient (the
details are given in [33]).

If this approach is to be applied in clustering points along a curve instead along the x axis, step
7 of the above algorithm has to be replaced by steps 5 to 7 as described in the algorithm of Hsu
below.

The MS algorithm can be extended to a 2D physical SD, requiring a MS in 3D space. To each
gridline in the physical SD there is a corresponding curve in the MS. The curvature for this curve
can be splitted into two parts, namely into normal and geodesic curvature, denoted by κ and κg .
n
The normal curvature is a measure of the rate at which the direction of the tangent leaves the MS
while the geodesic curvature is the curvature of the curve in the MS. Since κ represents changes
n
of the MS itself, it is used in the weight function, i.e. w 1· κn .

The algorithm of Hsu does not use a monitor surface, but works directly on the curve s s´ξµ.
Adaptation in 3D is performed by independenty looping over all gridlines in directions ξ, η and ζ,
that is there is no direct coupling of the points in a plane or within a block.

The algorithm of Hsu uses c c´ξµ. The clustering of grid points is along a given curve,
denoted by points Pi .

∆si wi λci (4.14)

The constant λ has been introduced to avoid an additional constraint among the c that follows
i
from the assumption of constant arc length. Therefore, λ is chosen to satisfy this constraint. It
should be noted that s is now the arc length of the curve along which the points are to be redis-
trituted.

Values for ci are not yet known. They are determined from the initial grid point distribution.
ci : ∆sI ; i 2´1µN 1
i

56

Figure 4.6: Adaptation of a grid using the monitor surface technique to capture a moving shock
together with a strong circular gradient (vortex). The initial grid has uniform grid spacing.

Grid Adaptation 57

For wi 1 this choice of the ci ensures that the grid point distribution remains unchanged,
provided λ is determined such that total arc length remains constant. From Eq. 4.14 one obtains

∑ ∆sI
λ: c
∑ wii
i
(4.15)

Constant arc length also requires
∑ ∆si ∑ ∆sIi
The implementation of an adaptation algorithm based on this approach, comprises the following
steps :

1. Determine ci ∆sI obtained from the initial grid.
i

2. Compute λ ∑ ∆sI
c
i
; this choice of λ guarantees that total arc length of curve s´ξµ remains
∑ wii
constant.
c
3. Determine ∆si λ wii

4. Arc length Calculate the arc length up to point i: si ∑i
k 1 ∆sk

5. Find index Find index m´ jµ ε N such that: sI
m si sI ·1
m

6. Interpolate Use linear interpolation between points ´xm ym µ and ´xm·1 ym·1 µ, to determine
the repositioned coordinates. This is done by using y am x · bm with the parameters

yI ·1 yI
m m
am : I
xm·1 xIm
yI xI ·1 yI ·1 xI
m m m m
bm :
xI ·1 xI
m m

7. Reposition Next ∆si ´xi·1 · yi·1 µ the repositioned coordinates ´xi·1 yi·1 µ have to be de-
termined. Starting from a boundary point, the repositioned coordinate values ´x yi µ are
i
assumed to be known.

∆si ´xi·1 xi µ2 ·´yi·1 yiµ2
´xi·1 xiµ2 ·´amxi·1 · bm yiµ2
This equation has to be solved for xi·1 . The equation has two solutions. Step 5 requires:
xI
m xi·1 xI ·1 and yI
m m yi·1 yI ·1 with yi·1 am xi·1 · bm
m

8. Smooth Relax the repositioned grid. Elliptic grid generation is used for a few smoothing itera-
tions (3-4).

Notes to the algorithm:

58

¯ If the straight line in 7 has a slope of more than π 4, the role of x and y is interchanged. This
can be easily checked by comparing yI ·1 yI with xI ·1 xI .
m m m m

¯ In order to have a smooth grid point distribution, the weight functions w should be Þltered, e.g.
i
averaged. Otherwise kinks in the repositioned grid may occur.

¯ Adaptation works on a curve by curve basis. For 3D a loop over all coordinate directions is
needed.

¯ The request that total arc length remains Þxed, may be changed. Arc length has no physical
meaning.

¯ The adaptation is extremely fast. It can be used for moving meshes.

¯ Adaptation can be performed in a step by step fashion. For example, one can adapt the grid to
an oblique shock, while preserving the Þne grid in the BL. Suppose that grid lines in the
BL are ξ lines (η = constant). For all of these lines wi 1 is chosen. Then the Þne grid in
the BL remains unchanged. For adaptation of the oblique shock η lines (ξ = constant) are
adapted the using pressure gradient or Ma number gradient. The Þnal grid depicts a very
sharp oblique shock and also retains the Þne BL grid. No points from the BL are moved to
the oblique shock region.
On colorplate E a solution adapted grid for a monoblock cone, constructed in that way, is
shown. The grid is adapted to capture both the oblique shock and the BL.

Grid Generation Language 59

5.0 A Grid Generation Meta Language

5.0.1 Topology Input Language

With the parallel computers of today substantially more complex ßuid ßow problems can be tack-
led. Complete aircraft conÞgurations, complex turbine geometries, or ßows including combustion
have been computed in industry. Consequently, geometries of high complexity are now of inter-
est as well as very large meshes, for instance, computations of up to 30 million grid points have
been performed. Clearly, grid generation codes have to be capable to handle this new class of
application.

Convential grid generation techniques derived from CAD systems that interactively work on
the CAD data to generate Þrst the surface grid and then the volume grid are not useful for these
large and complex grids. The user has to perform tens of thousands of mouse clicks with no or
little reusability of his input. Moreover, a separation of topology and geometry is not possible. An
aircraft, for example, has a certain topology, but different geometry data describe different aircraft
types. The topology deÞnition consumes a certain amount of work, since it strongly inßuences
the resulting grid line conÞguration. Once the topology has been described, it can be reused for
a whole class of applications. One step further would be the deÞnition of objects that can be
translated, rotated and multiplied. These features could be used to build an application speciÞc
data base that can be used by the design engineer to quickly generate the grids needed.

In the following a methodology, which comes close to the ideal situation described above,
is presented. To this end a completely different grid generation approach will be presented. A
compiler type grid generation language has been built, based on the ANSI–C syntax that allows
the construction of objects. This grid generation process can be termed hands off grid generation.
The user provides a (small) input Þle that describes the so called TIL code to built the wireframe
model, see below, and speciÞes the Þlenames used for the geometry description of the conÞgura-
tion to be gridded. A variety of surface deÞnitions can be used. The surface can be described as
a set of patches (quadrilaterals) or can be given in triangular form. These surface deÞnitions are
the interface to the CAD data. In general, a preprocessor is used that accepts surface deÞnitions
following the NASA IGES CFD standard [?] and converts all surfaces into triangular surfaces. That
is, internally only triangular surfaces are used. In addition, the Gridpro code allows the deÞnition
of analytic surfaces that are built in or can be described by the user in a C function type style.
The user does not have to input any surface grids, that is, surface and volume grids are generated
in the same run without any user interaction. This approaoch has the major advantage that it is
fully reusable, portable, and that highly complex grids can be built in a step by step fashion from
the bottom up, generating a hierarchy of increasingly complex grid objects, for instance see Sec.
5.0.2.

TIL allows the construction of complex grids by combining predeÞned objects along with op-
erators for placement of these objects. For example, the grid around an engine could be an object
(also referred to as component). Since an aircraft or spacecraft generally has more than one engine
located at different positions of its structure, the basic engine object would have to be duplicated
and positioned accordingly. In addition, the language should be hierarchical, allowing the con-
struction of objects composed of other objects where, in turn, these objects may be composed of

60

more basic objects etc. In this way a library can be built for different technical areas, e.g. a turbo-
machinery library, an aircraft library or a library for automobiles. The Topology Input Language
[35] has been devised with these features in mind. It denotes a major deviation from the current
interactive blocking approach and offers substantial advantages in both the complexity of the grids
that can be generated and the human effort needed to obtain a high quality complex grid. No
claims are made that TIL is the only (or the best) implementation of the concepts discussed, but is
believed that it is a major step toward a new level of performance in grid generation, in particular
when used for parallel computing.

TIL has been used to generate the grids thar are presented below. The surfaces used are de-
Þned analytically or are given by a CAD system. The surface deÞnition Þles are speciÞed outside
of the GridPro environment. The grid generator needs three different Þles as input. The TIL Þles
(extension .fra) contain the topology deÞnitions. The toplogy of a complex grid can be organized
into components or objects that may be grouped. A schedule Þle is needed (extension .sch) to
specify directions for the numerical scheme of the grid generator, i.e. to obtain a certain conver-
gence level (the user should keep in mind, however, that there is nothing like a converged grid).
The GridPro [35] package reduces the amount of information needed by an order of magnitude.
The following Þgures show 2D and 3D grids that have been generated with the automatic zoning
(blocking) approach.

The versatility and relative ease of use (comparable in difÞculty in mastering LATEX) will be
demonstrated by presenting several examples along with their TIL code. The examples a presented
demonstrate the versatility of the approach and show the high quality of the grids generated.

Figure 5.1: Navier-Stokes grid for a four-element airfoil, comprising 79 blocks.The Þrst layer of

grid points off the airfoil contour is spaced on the order of 10 6 based on chord length.

In order to generate a grid The following steps have to be followed in the grid generation process.


Figure 5.2: The Þgure shows the block structure of the four element airfoil generated by GridPro.

Figure 5.3: Grid for a T joint that has numerous industrial applications.

62

Figure 5.4: Sphere in a torus. The input for this grid is presented in the following tables.

Figure 5.5: Complete 3D grid for a generic aircraft with ßaps, constructed from analytical surfaces.
However, the topology is exactly the same as for a real aircraft.


Figure 5.6: The picture shows a blowup of the engine region of the generic aircraft.Future TSTO
or SSTO vehicles will exhibit a similar complex geometry, necessitating both the modelization of
internal and external ßows.

Figure 5.7:


Step 0 In a preprocessing step, all surfaces have to be prepared, providing a certain smoothness.
This procedure is called carpeting, see for example [37] and is not in GridPro. For the
proper surface format, see below.
Step 1 In the Þrst step, all surfaces (curves in 2D) that form the physical boundaries of the solution
domain have to be labeled, that is labels for the corners (or vertices) have to be provided.
next, a list of all neighboring vertices has to be speciÞed. The user has complete freedom in
the deÞnition of the surfaces.
Step 2 In the second step, the user designs the block topology. This is a creative process that can-
not be entirely automated. The blocking depends on the ßow physics and, unless speciÞc
rules are built into the grid generator, the expertise of the aerospace engineer is needed. The
choice of the topology itself leads to a clustering of grid points. Therefore, selecting the
proper topology may be more effective regarding grid adaptation than the use of additional
control functions. The major advantage is that TIL can be used to generate simple compo-
nents Þrst that can the be assembled to form complex grids. Thus, a component library can
be built up.

In the following some general explanations about TIL are given. The concepts function and com-
ponent are used interchangeably. Again, it is stressed, as a general rule, that the user has to label
the surfaces (curves in 2D) that form the boundary of the solution domain (SD). How the SD is
subdivided into surfaces is up to the user. Any surface is identiÞed by its corners (or vertices),
which are labeled in increasing order. Approximate corner coordinates should be speciÞed, that
is, the vertex need not be directly on the surface, but should not be positioned too far from it. The
code will do the necessary projections and will also construct a grid on each surface. The labeling
only serves to determine the grid topology. In addition, the user has to specify a list of neighboring
(or connecting) corners that may be on other surfaces. In the following some of the TIL rules are
presented that will be needed to understand the torus-sphere example, presented below.

º Any grid generated by GridPro is comprised by components.
º Components itself may comprise more basic components.

º The validity of component names (or numbers) is restricted to the component in which they
have been deÞned. This also holds for variable names.
º Components in TIL are similar to functions in ANSI C or C++. A realization of a component
is achieved by using a function. In the deÞnition of a function its name is followed by the
formal list of parameters, enclosed in parentheses (see e.g. Table 5.2). The component is
called using the values of the actual parameter list.
º Each function contains a body in which the necessary deÞnitions and statements occur (see,
e.g. Table [5.2]).
º TIL provides special data types, namely cIN, sIN, cOUT, and sOUT. In the formal param-
eter list, variables of that data type can be declared, as can be seen from Tables [5.2 - 5.4].
The letters c and s stand for corner and surface, respectively while IN and OUT specify data
to be imported or exported. In other words, IN indicates that variables are of input type and
OUT denotes output variables. TIL is more speciÞc than a computer language since it also
speciÞes the direction of the data transfer.

66

º If a negative integer value is speciÞed for the In or OUT data types, the corresponding IN
or OUT array is initialized with 0 (for example see Table 5.3).

º In referencing corners there exist several possibilities. For example, if one speciÞes cOUT(2
4..6) an array cOUT[1] to cOUT[4] is allocat ed, containing the coordinates of corners
labeled 2, 4, 5, and 6 in the respective component. The range parameter is given by . The
array is allocated in the calling function (or parent function). The elements of this array can
be accessed by, e.g. 1:1 or 1:3. The Þrst integer gives the component number of the child
function, the second one references the array index.

º In connecting corners i on component I and j on component J, it sufÞces to either specify
the connectivity in I or in J, i.e. only once.

º Arrays begin with index 1. This is in contrast to ANSI C where arrays start with index 0.

º Variable names are restricted to the block in which they are deÞned.

The input data for the sphere in a torus example, presented in Fig. 5.4, are given in the following
tables. The cross section of the torus has a radius of 0.5. The central circle of the torus is in the y-z
plane (x-axis points up, y-axis goes to the right, z-axis points away from the reader) with its center
at the origin and a radius of 1.5. The sphere has a radius of 0.25 and its center is at (0, 0, 1.5).

In Table 5.1 the components comprising the torus-sphere grid are read in. The bINt() function
is the parent function to all other functions (components). The INPUT statement is used to read
in predeÞned components, follwed by the component number and the component name with the
associated list of the actual arguments. The Þrst line after BEGIN of function main deÞnes the
torus surface analytically; its deÞntion is contained in the Þle surf/torus.h. The torus surface is
labeled surface 1. The following INPUT statement reads in component torus, which consists of
4 cross sections, each deÞned as a component of type tsec (see below). In the argument list it is
speciÞed that the analaytic torus surface is an input surface to component torus, and that corners
1 4 of component 1 (these are the 4 inner points of a cross section forming the square that can
be seen in Fig. 5.4) as well as corners 1 4 of component 2 (again a square) of component torus
are made available to function main via the cOUT statement. It is clear from Fig. 5.4 that these
corner points have to be known in component ball since there is a connection between these cross
section corners and the corresponding corners on the ball surface. In addition, the same corner
point on the ball surface is connected to its 3 neighboring corner points on the cube, forming a 4D
hypercube topology. The next INPUT statement reads in component ball, labeled surface 2, and
shifts its origin by (0 0 1.5). In this context the £ does not denote a multiplication. The argument
list of ball contains a cIN data type for 8 corners. Thus, the 8 corners having been made available
by the cOUT statement in the torus argument list serve as input for the ball component. It should
be noted that 1:1..8 used in cIN refers to the numbering in function main(), while the cOUT data
type uses the numbering of component torus. This distinction is important in order to address the
correct variables.

In Table 5.2 the topology input for the sphere is depicted. The Þrst line after BEGIN speciÞes a
surface, numbered surface 1 of type ellipsoid (ellipse in 2D) with 3 numbers denoting the lengths
of the major axes in the x,y, and z coordinate directions. The topology of the ball component
is equivalent to a cube. Hence, the coordinates of the 8 corner points (or vertices) of that cube
are connected to corners c 1 - c 8 that belong to two cross sections of the torus component.


# Begin wire frame file #

COMPONENT bINt ()
# Define component bINt (ball_in_torus)

BEGIN
S1 -implic "surf/torus.h";
INPUT 1 torus (sIN (1),cOUT (1:1 . . 4 2:1 . . 4));
# Input a torus and export 8 corners from it.

INPUT 2 (0 0 1.5) *ball (cIN (1:1 . . 8));
# Input a ball with translation (0 0 1.5) and
# import 8 corners.

END

Table 5.1: Input of the toplogy Þles of the torus and ball (sphere) components.

COMPONENT ball(cIN c[1..8]) #define component ball
BEGIN

s 1 -ellip (4 4 4); #define surface by id
c 1 -0.25 0.25 -0.25 -s 1 -L c:1;
c 2 -0.25 0.25 0.25 -s 1 -L c:2 1;
c 3 0.25 0.25 0.25 -s 1 -L c:3 2;
c 4 0.25 0.25 -0.25 -s 1 -L c:4 3 1;
c 5 -0.25 -0.25 -0.25 -s 1 -L c:5 1;
c 6 -0.25 -0.25 0.25 -s 1 -L c:6 2 5;
c 7 0.25 -0.25 0.25 -s 1 -L c:7 3 6;
c 8 0.25 -0.25 -0.25 -s 1 -L c:8 4 7 5;
x b c:1 c:7 1 7;

# define corners by id, initional position, surfaces
# that the corner is on and the links (edges) to other
# defined corners that the corner has.
# exclude 2 blocks defined by corner pairs.
END

Table 5.2: Toplogy Þle for the ball component.

68

COMPONENT torus()
BEGIN
s 1 -analyt "surf/torus.h"; # Define surface
INPUT 1 (1 0 0 0 0.707 0 0 0.707 0)*
tsec(sIN (1),cIN (-8),(-8),cOUT (1..8));
INPUT 2 (1 0 0 0 -0.707 0 0 0.707 0)*
tsec(sIN (1),cIN (1:1..8),(-8),cOUT (1..8));
INPUT 3 (1 0 0 0 -0.707 0 0 -0.707 0)*
tsec(sIN (1),cIN (2:1..8),(-8),cOUT (1..8));
INPUT 4 (1 0 0 0 0.707 0 0 -0.707 0)*
tsec(sIN (1),cIN (3:1..8),(1:1..8),cOUT (1..8));
x f 1:1 3:1 1:2 3:2 1:3 3:3 1:4 3:4;
x f 1:5 3:5 1:6 3:6 1:7 3:7 1:8 3:8;
# Input tsec(torus_section) with transformation
# ((1 0 0) (0.707 0) (0 0.707 0)).
END

Table 5.3: Topology Þle for the torus component.

The following 3 numbers specify the initial ccordinates of the corner. These coordinate values
only serve to provide an initial solution for the grid generation process. The Þnal coordinates
of the corners are computed by the program. The -s option followed by an integer indicates the
surface that it belongs to. The link option, -L, lists all corner points to which the current corner
is connected. The x command excludes pairs of corner points that otherwise would be falsely
connected.

Table 5.3 gives the TIl input for the torus component. It is important to note that the torus
component itself comprises 4 components, named tsec (Table 5.4). The topology of the torus is
obtained by cutting it in the cross section plane. This cut results in 2 cross sections (see Fig. 5.4),
while the other 2 are the left and right cross sections connecting to the sphere surfaces. Component
tsec represents such a cross section of the torus and thus is a more basic object (again see Fig. 5.4).
Each cross section contains 8 corners and its corners are either connected to 2 other cross sections
( 8 corner points for each cross section), which is the case for the cross sections at the cut, or
to another cross section (8 corner points), one of the sphere surfaces (4 corner points), and to the
corner points above and below the sphere of another cross section. Again, reference is made to Fig.
5.4 for visualization. Therefore, each tsec component needs an input of 16 corner points, which
is reßected in its argument list (see Table 5.4. The 9 numers after the INPUT statement form a
matrix, used for the transformation of the original cross section coordinates. The argument list in

The command g (see Table 5.4) speciÞes the number of grid points to be distributed along an
edge. The Þrst two variables specify the corners (vertices) of the edge. Vertices can be identiÞed
by integer values or by variable names. The third value is the number of grid points.

The surface of a torus is generated by rotating a circle about an axis that is in the plane of the
circle without intersecting it. A point on the surface is determined by angles Θ and Φ that are in
the y-z plane and in the cross section plane, respectively. The torus surface is given by the formula


COMPONENT tsec(sIN t,cIN c[1..8],C[1..8])
# Define component tsec(torus_section)
BEGIN
c 1 -0.35 1.5 0 -L c:1 C:1;
c 2 -0.35 2.7 0 -L c:2 C:2 1;
c 3 0.35 2.7 0 -L c:3 C:3 2;
c 4 0.35 1.5 0 -L c:4 C:4 3 1;
c 5 -0.5 1.0 0 -s t -L c:5 C:5 1;
c 6 -0.5 3.0 0 -s t -L c:6 C:6 2 5;
c 7 0.5 3.0 0 -s t -L c:7 C:7 3 6;
c 8 0.5 1.0 0 -s t -L c:8 C:8 4 7 5;
x f 5 7;
g 1 c:1 24 1 C:1 24;
END

Table 5.4: Topology Þle for the tsec component.

# Begin schedule file #
step 1: -w 100 -R sing 0.4 -r 4 43 -R adj 107 -S 100
step 100: -g 8:0 24 16:8 24 24:16 24 24:0 24
-R sing 0.4 -R sharp 0 - S
write -a -D 0 -f dump.tmp
write -a -D 3 -f blk.tmp
# End schedule file #

Table 5.5: Schedule Þle to specify output format.

70

#define FUNCU ya=sqrt(y*y+z*z)-1.5,1.0 - (ya*ya + x*x)*4
/* Define torus surface */
#define Ulen (-1.0 ) /* -1.0 for no period */
#define FUNCV 0.0 /* Do not care in this case */
#define Vlen (-1.0)
#define FUNCW 0.0
#define Wlen ( -1.0 )
#define FUNCZ 0.0
#define FUNCY 0.0
#define FUNCX 0.0

Table 5.6: Surface deÞnition Þle for the sphere surface.

#define FUNCU (x*x + y*y + z*z)*16 - 1.0
/* Define ball surface */
#define Ulen (-1.0 ) /* -1.0 for no period */
#define FUNCV 0.0 /* Do not care in this case */
#define Vlen (-1.0)
#define FUNCW 0.0
#define Wlen ( -1.0 )
#define FUNCZ 0.0
#define FUNCY 0.0
#define FUNCX 0.0

Table 5.7: Surface deÞnition Þle for the torus surface.

(R 1 5 rc 0 25)

r´Θ Φµ ´ R · rccosΦ cosΘ R · rc cosΦ sinΘ rc sinΦµ
FUNCU, FUNCV, and FUNCW are the transformed coordinates that would be denoted as u v w
in a mathematical context, being functions of the Cartesian variables x y z. Ulen, Vlen, Wlen spec-
ify the interval for which the transformed variables are deÞned. FUNCX, FUNCY, and FUNCZ
describe the inverse transformation.


5.0.2 TIL Code for the Cassini-Huygens
ConÞguration

In the following we present the TIL code to generate a 3D grid for the Cassini–Huygens space
probe. Cassini–Huygens is a joint NASA–ESA project and will be launched in 1997. After a ßight
time of 8 years, the planet Saturn will be reached and the Huygens probe will separate from the
Cassini orbiter and enter the Titan atmosphere, Saturn’s largest moon [?]. Titan is the only moon
in the solar system possessing an atmosphere (mainly nitrogen). During the two hour descent
measurements of the composition of the atomsphere will be performed by several sensors located
at the windward side of tyhe space probe. In order to ensure that the laser sensor will function
properly dust particles must not be convected to any of the lens surfaces of the optical instruments
[46]. Therefore, extensive numerical simulations have been performed to investigate this problem.

In order to compute the microaerodynamics caused by the sensors the proper grid has to be
generated. A sequence of grids of increasing geometrical complexity has been generated. The
simplest version, comprising 6 blocks, does not contain the small sensors that are on the windward
side of the probe. With increasing comlexity the number of blocks increases as well. The Þnal grid,
modeling the SSP and GCMS sensors, comprises 462 blocks. However, it is important to note that
each of the more complex grids is generated by modifying the TIL code of its predecessor.

5.0.2.1 TIL Code for 6 Block Cassini–Huygens Grid

The general approch for constructing the Cassini–Huygens grids of increasing complexity was
to Þrst produce an initial mesh for the plain space probe without any instruments which is of
relatively simple shape equivalent to a box. Thus the Þrst topology is a grid that corresponds to a
box in a box. The reÞnement of the grid is achieved by adding other elements designed as different
objects. The combination of all objects results in the Þnal grid. As the Þrst step in generating the
Huygens grid, a topology for the Huygens body (without any additional elements such as SSP,
GCMS and sensors) is designed, shown in Fig. 5.9. This topology describes the spherical far Þeld
and the body to be simulated. To generate this grid the following TIL code has been written. A 6
block grid is obtained, shown in Figs. 5.10 and ??. This grid has a box–in–box structure: the outer
box illustrates the far Þeld and the interior one is the Huygens body. The corner connection to be
initialized results in the wire frame topology, shown in Fig. ??.

72

Far-Field

block 5 block 4
Huygens Surface block 6

block 3

block 1 block 2

Figure 5.9: Topological design for the Huygens space probe grid. In this design all elements, such
as SSP, GCMS and sensors are ignored.

TIL Code for 6 Block Huygens Space Probe Grid:
SET GRIDDEN 47
COMPONENT Huygens()
BEGIN

s 1 -tube "HuygensProfil.lin";
s 2 -ellip(0.00015625 0.00015625 0.00015625) -o;

INPUT 1 (650 0 0 0 650 0 0 0 650)*
loop(sIN (1),cIN (-4),(-4),cOUT (1..4));

INPUT 2 (0 0 -4525)*(4525 0 0 0 4525 0 0 0 4525)*
loop(sIN (2),cIN (1:1..4),(-4),cOUT (1..4));

INPUT 3 (0 0 325)*(650 0 0 0 650 0 0 0 650)*
loop(sIN (1),cIN (1:1..4),(-4),cOUT (1..4));

INPUT 4 (0 0 4525)*(4525 0 0 0 4525 0 0 0 4525)*
loop(sIN (2),cIN (3:1..4),(2:1..4),cOUT (1..4));

END

COMPONENT loop(sIN s, cIN c[1..4], C[1..4])
BEGIN

c 1 -1 - 1 + 0 -s s -L c:1 C:1;


c 2 1 - 1 + 0 -s s -L c:2 C:2 1;
c 3 1 + 1 + 0 -s s -L c:3 C:3 2;
c 4 -1 + 1 + 0 -s s -L c:4 C:4 1 3;
END

5.0.2.2 TIL Code for 24 Block Cassini–Huygens Grid

By modifying the previous topology the SSP object is generated. The SSP is of cylindrical form
which can be described by introducing eight corner points, shown in Fig. 5.11. The wire frame
topology has different conÞguration than design topology. The wire frame topology has different
conÞguration than design topology. The relationship beween corners reaim unchanged, shown in
Fig. ??. This kind of topology deÞnition allows the subdivision of a more complex object into a
set of simple ones. Adding objects into the principal body, a TIL code will be generally extended.
This principle can be seen in the following TIL code.

TIL code for 24 block Huygens
COMPONENT HuygensSSP()
BEGIN

s 1 -plane(0 0 -1 162.4);
s 2 -quad "../Data/HuygensData/HuygSSP.dat";
s 3 -ellip(0.000078125 0.000078125 0.000078125) -o -t 0 0 300 -c
1000;

INPUT 1 (0 248 162.4)*(80 0 0 0 80 0 0 0 80)*
holeSSP(sIN (1),(2),cIN (-8),(-8),cOUT (1..8));

INPUT 2 (0 248 0)*(80 0 0 0 80 0 0 0 80)*
loop(sIN (-1),(2),cIN (1:5..8),(-4),cOUT (1..4));

INPUT 3 (0 248 -50)*(40 0 0 0 40 0 0 0 40)*
loop(sIN (-1),(-1),cIN (1:1..4),(2:1..4),cOUT (1..4));

INPUT 4 (0 248 -12500)*(100 0 0 0 100 0 0 0 100)*
loop(sIN (3),(-1),cIN (3:1..4),(-4),cOUT (1..4));

INPUT 5 (0 0 -12500)*(12800 0 0 0 12800 0 0 0 12800)*
loop(sIN (3),(-1),cIN (4:1..4),(-4),cOUT (1..4));

INPUT 6 (0 0 -50)*(690 0 0 0 690 0 0 0 690)*
loop(sIN (-1),(-1),cIN (5:1..4),(3:1..4),cOUT (1..4));

INPUT 7 (650 0 0 0 650 0 0 0 650)*
loop(sIN (2),(-1),cIN (6:1..4),(2:1..4),cOUT (1..4));

74

INPUT 8 (0 0 650)*(650 0 0 0 650 0 0 0 650)*
loop(sIN (2),(-1),cIN (7:1..4),(-4),cOUT (1..4));

INPUT 9 (0 0 700)*(690 0 0 0 690 0 0 0 690)*
loop(sIN (-1),(-1),cIN (8:1..4),(6:1..4),cOUT (1..4));

INPUT 10 (0 0 13100)*(12800 0 0 0 12800 0 0 0 12800)*
loop(sIN (3),(-1),cIN (9:1..4),(5:1..4),cOUT (1..4));

x b 7:1 8:3;
x f 1:5 1:7 2:1 2:3 5:1 5:3 6:1 6:3 7:1 7:3;
g 2:1 1:5 30;
END

COMPONENT holeSSP(sIN s,s1,cIN c[1..8],C[1..8])
BEGIN
c 1 -0.5 -0.5 0 -s s -L c:1 C:1;
c 2 0.5 -0.5 0 -s s -L c:2 C:2 1;
c 3 0.5 0.5 0 -s s -L c:3 C:3 2;
c 4 -0.5 0.5 0 -s s -L c:4 C:4 1 3;
c 5 -1.0 -1.0 0 -s s s1 -L c:5 C:5 1;
c 6 1.0 -1.0 0 -s s s1 -L c:6 C:6 2 5;
c 7 1.0 1.0 0 -s s s1 -L c:7 C:7 3 6;
c 8 -1.0 1.0 0 -s s s1 -L c:8 C:8 4 5 7;
END

COMPONENT loop(sIN s,s1, cIN c[1..4], C[1..4])
BEGIN
c 1 -1 -1 0 -s s s1 -L c:1 C:1;
c 2 1 -1 0 -s s s1 -L c:2 C:2 1;
c 3 1 1 0 -s s s1 -L c:3 C:3 2;
c 4 -1 1 0 -s s s1 -L c:4 C:4 1 3;
END

5.0.3 TIL Topology and Grid Quality

The code, however, is not completely foolprooff and situations may occur when inconsist toplogi-
cal data is speciÞed. GridPro tries to do some repair when information is missing or is inconsistent.
In these cases, error messages are issued that quickly lead to the detection of the error. The most
common blunders are described below and are depicted in Figs. 5.13 to 5.15.

Grid topology describes the connectivity among the blocks that comprise the solution domain.
Instead of describing the block topology, a 3D wireframe is given, from which the block topolgy
can be derived. The wireframe is speciÞed by the user, see examples below. The wireframe itself
is a collection of points and edges whose relations are to be speciÞed. This information allows the


Figure 5.10: The 6 block Huygens grid is depicted. It is bounded by a large spherical far Þeld, in
which the Huygens space probe is embedded. The ratio of the far Þeld radius and the Huygens
radius is about 20.

Farfield
Farfield

GCMS

GCMS
Huygens

Huygens

Figure 5.11: Illustration of the block topology for the 24 block Huygens grid. Left: Topology top
view. The far Þeld, the Huygens body and GCMS are clamped by quads. Right: Topology side
view. Again clamps are used for the 3D topological design.

76

Figure 5.12: 24 block grid for Huygens space probe including SSP.

ordering of points which represent the vertices of the wireframe structure. The positions of these
points must be given as spatial coordinates, and these positions are used as initial conditions for
the grid generator. The accuracy of coordinate information plays an important role for the grid
quality. For instance, one may generate an intersecting hyperquad with the same topology, but
different initial coordinates, shown in Fig. 5.13. In Figs. 5.14 and 5.15 an example is given to
demonstrate the inßuence of the initial wireframe coordinates on the grid quality. In the Þrst case,
there is no skewness in the conÞguration. Based on this initial condition, a grid without folded
lines is generated, shown in Fig. 5.14. In the second case, the grid is skewed and has folded lines,
shown in Fig. 5.15.


A B A B

a b a b

d c d c

D C D C

Figure 5.13: Topological description for a two–loop hyperquad is depicted. Some attention has
to be given to the placement of the wireframe points. In both cases, the topology remains un-
changed while wireframe corners are different. It is obvious that in the second case an overlap is
encountered and grid lines will be folded.

Figure 5.14: The wireframe should reßect the actual geometry. Although wireframe coordinates
are not Þxed, a better initial solution will lead to faster convergence and inproved grid quality.

78

Figure 5.15: The grid quality is inßuenced by the initial coordinate values of the wireframe corners
A distorted wireframe may cause grid skewness and folded lines.

Automatic Blocking 79

6.0 Tools for Automatic Topology DeÞnition

The next major step forward in automatic blocking is to provide a tool that by taking graphi-
cal input automatically produces the TIL code described in Sec. 5.0.1. AZ–manager that is now
described, is such an interactive topology generator. In this context, topology deÞnition means
constructing a wireframe model by placing wireframe points, linking these points, and assigning
them to Þxed surfaces. Once the topology deÞnition has been done, all grids are produced au-
tomatically starting GridPro from within AZ–manager and visualizing the grid while it is being
computed. AZ–manager has miniCAD capabilities allowing the user to built his own geometry.
This feature is particularly useful for external ßows wherethe outer boundary is not of Þxed shape.

The topology generation process comprises the following stages.

Topology The topology deÞnition process comprises four stages:

º Generating a geometry description or using existing CAD data.
º Constructing the set of wireframe points.
º Linking corresponding wireframe points.
º Assigning wireframe points to surfaces.

In the following general rules for AZ–manager usage will be outlined, concerning mouse usage,
rotating a 3D objects, and activating a surface as well as performing actions on it. It is also de-
scribed how ro read in CAD data as the general means for surface description of 3D bodies. The
reader should refer to this section when actively using AZ–manager, because only then the full
meaning of the rules will become clear.

Mouse Usage All three mouse buttons are used.
LB: Drag to translate the object across the screen.
MB: Rotation (see description below).
RM: Drag the box that is appearing to desired size and then release mouse button. The area in the
box will be enlarged.

Rotation (Orientation of Coordinate System) There are two different modes of rotation. On
the Command Panel under topic Rotation two view options world and eye can be selected. Under
world view, a rotation is about the active coordinate axis. To activate an axis, move the mouse
cursor over the axis such that the rotation symbol occurs and press LB. The axis becomes a thin
line, indicating that it is now the active axis, i.e. the axis of rotation. Yellow denotes the x-axis,
green the y-axis, and purple the z-axis. Under eye view, the axis of rotation lies always in the
view (screen) plane. Moving the mouse in a certain direction in the view plane, automatically and
uniquely deÞnes the axis of rotation being perpendicular to the direction of motion. It should be
noted that this axis of rotation is invisible.

Activating a Surface Surface deÞnition: AZ–manager has some CAD capabililty. The user
can deÞne surfaces in 2D (lines) as well as in 3D (surfaces). Surfaces are numbered consecutively,

80

starting from 0. Only one surface can active at a time. Only the active surface is affected by user
operations.

A body or conÞguartion comprises a set of surfaces. The user needs to have the possibility to
scroll through these surfaces and to perform certain actions on them. The active surface, on which
actions are performed, is selceted by pressing surf on the command panel (the command panel
is the right side of the screen, containing a multitude of commands). By using or scrolling
is done and the corresponding surface is selected. Pressing the hand button, the active surface is
marked by blue handles. Dragging these handles resizes the active surface.

Setting the Display Style of Objects There are various possibilties of displaying objects on
the srceen, using the button stop of the STY LE option on the command panel, one selects
the current static display style, while button move governs the display style during the mouse
movement. For the 2D examples below, one should select lines. One should experiment with the
options available to fully understand their meaning.

Cutting Plane (CP) To place wireframe points in 3D space, a plane has to be deÞned, termed
the cutting plane. All wireframe points generated are automatically positioned on this plane. The
CP can be moved by the user, i.e. translated and rotated to any desired position. Its usage is
described in Example 3.

Surface Geometry Description In example 3 the surface of the Cassini–Huygens Space Probe
is described by a set of 6 rectangular patches, given in pointwise form, that have been extracted
from CAD data. Several other IGES formats are also supported. The Huygens surface data is in
the Þle HuygensSurf.dat. A second Þle with extension conn is needed by GridPro, describing
the connectivity of the patches. Using a conversion program, the Þle HuygensSurf.dat.conn is
automatically generated. The format of this Þle is of no concern to the user, he just has to remember
that it is needed by GridPro.

Fit Objects on Scree When surfaces are read in or constructed they may not Þt on the screen,
that is, they may not be visible, because of a distorted scaling. Press unzoom as many times as
necessary to Þt objects on screen. When surfaces are generated, they might have to be scaled.
Turn hand on and make the surface active. The surface turns blue and then resize by dragging the
surface handles.

6.0.1 Example Topology 1: A Circle in a Box

The following example constructs a 4 block grid as a 2D projection of a rod (cylinder) in a box, i.e.
a doubly connected region. Gridlines wrap around the rod in an O-type fashion. However, since
the outer boundary is formed by a box, special care has to be taken by the grid generator to have
grid lines close to the box corners. A clustering algorithm is needed to achieve that. The user does
not have to specify any parameters, the clustering is performed automatically in GridPro based
on boundary curvature. The following set of pictures serve as a step by step introduction on how
to actually generate the grid. Stepping through the pictures, the reader should make sure that his
results are similar to those depicted in the Þgures. Since wireframe points are interactively placed,
results will not be identical with the pictures.


160,180

s4

s1 s3

s5
Y 50,50
r=20

s2

0,0 X

Figure 6.1: In this Þrst example the grid for a circle in a box has to be generated. First the user
has to construct the geometry (physical boundaries) as shown in the Þgure. Second, the blocking
toplogy has to be speciÞed.

general topo surf grid 3-dim
2-dim
...

Figure 6.2: To start with the example, invoke the automatic blocking manager with command az.
The AZ–manager window appears. Switch to 2–dim on the menubar, as shown in Þgure.

82

general topo surf grid
load: file
load:-plane

...

Figure 6.3: AZ–manager has mini CAD capability that is used to construct the boundary of the
solution domain. To this end click surf on the menubar and select load:–plane from this menu.

surf id:

type:

center:

normal:

orient:

n-space:

cancel ok

Figure 6.4: new text needed AZ–manager has mini CAD capability that is used to construct the
boundary of the solution domain. To this end click surf on the menubar and select load:–plane
from this menu.


surf id: 0

type: -plane

center: 0 0 0

normal: 1 0 0

orient: +side

n-space:

cancel ok

Figure 6.5: To construct the outer box, input contour data as indicated. The Þrst surface to be
generatedd, is the west side of the box. It is of type textbfplane that actually is a line in 2D.

surf id: 1

type: -plane

center: 0 0 0

normal: 0 1 0

orient: +side

n-space:

cancel ok

Figure 6.6: Input plane information as shown. The next surface is the south side of the box. Note
that surfaces are consecutively numbered (surf id).

84

surf id: 2

type: −plane

center: 160 0 0

normal: 1 0 0

orient: −side

n−space:

cancel ok

Figure 6.7: Input plane information as shown. This surface is the east side of the box. Note that
surface orientation has to be reversed by specifying –side.

surf id: 3

type: −plane

center: 0 180 0

normal: 0 1 0

orient: −side

n−space:

cancel ok

Figure 6.8: Input plane information as shown. This plane is the north side of the box. Note that
surface orientation has to be reversed by specifying –side. Press unzoom to make a box Þt on
screen.


surf id: 4

type: -ellip

center: 50 50 0

semi-u: 20 0 0

semi-v: 0 20 0

semi-w: 0 0 1

orient: +side

cancel ok

Figure 6.9: Select load:–ellip from the surf on the menubar. Input ellip information as shown to
construct the circle.

86

general topo surf

SHOW: axis surf cut
y

1
"hand" CUT-P:
hand

2
x 3
CURRENT:
surf

Figure 6.10: A surface may not have the desired size. In order to resize it, deselect cut (if it is on,
i.e. the radio button is pressed) of the SHOW option on the Command Panel. Then activate hand
from the CUT–P option on the command panel. Next, activate the proper surface (blue color) – in
this example the proper edge of the box. Position the mouse curser over the handle and drag it in
the indicated direction.


8 7

4 3

1 2

5 6
a b

Figure 6.11: Activate the other box surfaces in the same way and drag them as shown in Þgure a.
The dragging shoul be done such that the box contour becomes visible. The circle should be visible
as well. Now wireframe points can be placed. Wireframe point don’t have to ly on a surface, but
they should be close to the surface to which they are going to be assigned. Wireframe points are
created by pressing ”c” on the keyboard and clicking at the respective positions with LB. Make
sure that Corner Creation Mode appears in the upper left corner of the window. Finish point
positioning as depicted in Þgure b.

88

Figure 6.12: Link the corners by pressing ”e” on the keyboard and click each pair of points with
LB. Make sure that Link Creation Mode appears in the upper left corner of the window. After
completing the linkage process, a wireframe model should like as depicted in the Þgure.

CURRENT
surf
0(-1)

Figure 6.13: Activate surface 0. Press ”s” on the keyboard and click the two points, marked by
”X”. If points are selected, their color turns from yellow to white. That is the corners are assigned
to surface 0.


CURRENT
surf
1(-1)

Figure 6.14: Repeat this action for surface 1, shown in Þgure.

CURRENT
surf
2(-1)


90

CURRENT
surf
3(-1)


CURRENT
surf
4(-1)

Figure 6.17: Activate surface 4 and assign the remaining 4 points to the circle.


Figure 6.18: Since GridPro is 3D internally, it is necessary to explicitly remove two blocks. The
Þrst block to be revoned is formed by the 4 points which build diagnoals, as shown in Þgure.
Press ”f” and click two points, marked by triangles. Perform the same action for points marked by
crosses. A red line between the corresponding diagnoals indicates the succesful performance.

general topo
TIL read
TIL save
Ggrid start
Confirm

Current Topology Is Complete.
Launch Ggrid ?

ok cancel

Figure 6.19: Save the topology Þle (TIL code) by selecting topo and clicking TIL save. Select
Ggrid start to generate the grid.

92

general topo
TIL read
TIL save
Ggrid start

Ggrid stop

Figure 6.20: Select topo from the menubar and click Ggrid stop. The generation process is now
suspended. Select grid from the menubar and click load new, the grid Þle ”blk.tmp” can be read.

grid panel=T
load new Topolog
Grid
Surface

Figure 6.21: Select panel=T from the menubar and click Grid. The grid appears as wireframe
model. At the right side of the menu window, select STYLE and click lines under stop. Click shell
by Make Shell at the right low side of the menu window. The grid appears.


6.0.2 Example Topology 2: Two Circles in a Box

160,180
s3

110,130
r=20

s5
s0

s2
s4

Y 50,50
r=20

s1
0,0 X

Figure 6.22: Constrct geometry in the same way as described in example 1.

Figure 6.23: It is suggested to place the wirefrace points in a row by row fashion. Other toplogies,
of course, would be possible.

94

SET DIMENSION 2
SET GRIDEN 8

COMPONENT DoubleCircle ()
BEGIN

s 1 -plane (1 0 0 0);
s 2 -plane (0 1 0 0);
s 3 -plane (-1 0 0 160);
s 4 -plane (0 -1 0 180);
s 5 -ellip (0.05 0.05 0) -t 50 50 0;
s 6 -ellip (0.05 0.05 0) -t 110 130 0;

END

Table 6.1: It is als possible to read the geometry in a TIL Þle. This TIL Þle only contains the
geometry of the solution domain. In practice, CAD data will be available that will be used for
geometry description.

general topo
TIL read

Ggrid start

Figure 6.24: Press TIL read under item topo of the menubar to read in the geometry data.


31 32 33 34 35 36

25 26 28 29 30
27

19 20 21 22
24
23

13 14 15 16 17 18

7 8 9 10 11 12

1 2 3 4 5 6

Figure 6.25: A row by row pattern results in a wireframe point numbering as shown.

CURRENT

surf
0(0)

Figure 6.26: Assignment of wireframe points to surfaces is exactly as in the Þrst example.

96

CURRENT

surf
1(1)

Figure 6.27: Activate surface 1 and assign wireframe points.

CURRENT

surf
2(2)



CURRENT

surf
3(3)


CURRENT

surf
4(4)


98

CURRENT

surf
5(5)



general topo

Ggrid start

Figure 6.33: Start Ggrid by selecting Ggrid start.


SET DIMENSION 2
SET GRIDEN 10

COMPONENT Huygens ()
BEGIN

s 1 -linear "HuygensSurf.dat";

END

Table 6.2: This TIL code is read in using topo TIL read as in former examples). The Þle Huy-
gensSurf.dat contains the Huygens surface data as described in ??. The Huygens surface is dis-
placed on the srceen and can be rotated as described in ??. Select surf and click load:–ellip.
Construct a circle with radius 5000.

6.0.3 Example Topology 3: Cassini–Huygens Space Probe

In this example we will do our Þrst 3D problem and, at the same time, will use surface description
data coming from a CAD system (see above).

Y
z=-5000

inner box

X

Z
z=5000

Figure 6.34: The topology to be generated for Huygens is of type box in a box. The outer 8
wireframe points should be placed on a sphere with radius 5000, the inner 8 points are located on
the Huygens surface.

100

z=-5000 z=5000

(((
((( (((
(((
((( (((

z=-500 z=500
(a) (b)

Figure 6.35: Construct wireframe model as in Þgure.

(((
(((
CURRENT
(((
surf
1(1)

Figure 6.36: Assign surface 2 as in Þgure.


(((
(((
(((
CURRENT
surf
0(0)

Figure 6.37: Assign surface 1 as in Þgure.

102

7.0 Parallelization Strategy for Complex Geometries

7.0.1 The Abstract Parallel Machine

The following philosphy is basic to the parallelizing approach of multiblock codes. In writing
cost effective parallel software whose quality can be ensured, one has to separate the methods of
the application from the details of implementation on a particular parallel architecture, with its
particular processors and perhaps a vendor-speciÞc parallel Þle system. This can be achieved by
writing code for an abstract machine, then implementing the abstract machine on speciÞc hardware
platforms.

The abstract machine has a communication system with these methods:

º Sending locally blocking tagged messages,

º Blocking receives of messages chosen by tag or by source processor,
º A system message buffer of given Þxed size,
º A collective operation, called ”reduce” in MPI for sums over all the processors, and other
such operations.

In addition, if needed by a CFD code, the abstract machine has a processor with these BLAS1
vector methods:

º Adding vectors and multiplying by a scalar (saxpy/daxpy operations).
º Gather/Scatter operations on index vectors.

The abstract machine has the following Þle system methods:

º Collective synchronized Open of a Þle,

º Multiple atomic seek-and-read operations with a shared Þle pointer,
º Multiple write operations with a single shared Þle pointer.

Some parallel processing systems support these operations very efÞciently. For example the vector
operations on a Cray or IBM RS/6000 processor, also implemented as pipeline operations on RISC
processors such as the i860. When a processor does not directly support one of the methods, it can
be implemented in software.

The abstract machine speciÞed above is well suited for unstructured or structured mesh com-
putations, being portable to any software or hardware environment, yet taking advantage of the
speciÞc machine conÞguration to maximum advantage.

Parallel Grid Generation 103

7.0.2 General Strategy

Parallelization is achieved by domain decomposition. Do-loop level parallelization will neither be
economical nor effective.

Since for structured grids the grid topology (that is the number and size of blocks needed
to obtain a certain gridline conÞguration) is dictated by the physics, load balancing is achieved
by mapping any arbitrary number of blocks to a single processor. This demands a slightly more
sophisticated message passing scheme, but leads to a much more ßexible approach. One might
surmise that for unstructured grids a domain decomposition algorithm will result in a load balanced
application, making the mapping of several subdomains (blocks) to a single processor unnecessary.
However, this may not be the case for very large grids, because a lenghty optimization stage might
be required, for instance, to use simulated annealing to minimize the cost function.

The decompositions will be performed statically via pre-processing of the grids rather than
dynamically during the ßow solutions. This is adequate for the current problems However, future
developments may include unsteady calculations on moving and/or solution adaptive grids and
thus any modiÞcations made will keep these future developments in mind while not allowing
them to dominate the initial parallelization effort.

In order to minimize code maintenance only a parallel version of the code is kept. A single
processor machine is simply considered as a one processor parallel machine.

All codes are implemented under PVM or MPI, but the communication libraries of Intel and
Ncube are directly supported as well as the IBM parallel environment.

7.0.3 Encapsulation of Message Passing

Message passing routines are called indirectly, using a set of encapsulated routines that in turn call
PVM routines.

EfÞcient coding: In parallel programming a number of operations, such as global reduction
operations (”global sum”) are required quite frequently. Since PVM and MPI lack this type of
operations they are realized by the application itself. These pieces of code are assembled in a
communication library.

The parallelization of I/O can be very different with respect to the programming models
(SPMD, host-node - not recommended) and I/O modes (host-only I/O, node local I/O, fast parallel
I/O hardware, etc.) supported by the parallel machines. The differences can also be hidden in the
interface library.

Portability: Encapsulation of message passing routines helps to reduce the effort of porting a
parallel application to different message passing environments.

Source code: Encapsulation allows to keep only one source code both for sequential and par-

104

allel machines.

Maintenance and further development: Encapsulation keeps message passing routines local.
Thus, software maintenance and further development will be facilitated.

Common message passing subset: The table below lists the basic message passing operations
which are available in most of the message passing environments. These routines are functionally
equivalent although the semantics of the arguments and return valves may differ in detail. Porta-
bility can be highly increased by restricting oneself to use only operations included in the common
subset for implementing the interface routines.

º Since each processor of the parallel machine takes one or more blocks, there may not be
enough blocks to run the problem on a massively parallel machine. We have tools to auto-
matically split the blocks to allow the utilization of more processors.

º The blocks may have very different sizes, so that the blocks must be distributed to the
processors to produce a reasonable load balance. We have tools to solve this bin-packing
problem by an algorithm discussed below.

An extremely simple message-passing model is ismplemented, consisting of only send and
receive. The simplicity of this model implies easy portability.

For an elementary Laplace solver on a square grid, each gridpoint takes the average of its four
neighbors, requiring 5 ßops, and communicates 1 ßoating-point number for each gridpoint on the
boundary. The Grid grid generator is a more sophisticted elliptic solver, where about 75 ßops
occur per internal grid point, while grid coordinates are exchanged across boundaries. Ourßow
solver,ParNSS, in contrast does a great deal of calculation per grid point, while the amount of
communication is still rather smalli.

Thus we may expect any implicit ßow solverto be highly efÞcient in terms of communication,
as shown by the timing results below. When the complexity of the physics increases, with turbu-
lence models and chemistry, we expect the efÞciency to get even better. This is why a ßow solveris
a viable parallel program even when running on a workstation cluster with slow communication
(Ethernet).

Parallel Machines and CFD

For the kinds of applications that we are considering in this report, we have identiÞed
four major issues concerning parallelism, whether on workstation clusters or massively
parallel machines.

Load balancing As discussed above, the number of blocks in the grid must be equal
to or larger than the number of processors. We wish to distribute the blocks to processors
to produce an almost equal number of gridpoints per processor; this is equivalent to min-
imizing the maximum number of gridpoints per processor. We have used the following
simple algorithm to achieve this.


The largest unassigned block is assigned to the processor with the smallest total num-
ber of gridpoints already assigned to it, then the next largest block, and so on until all
blocks have been assigned.

Given the distribution of blocks to processors, there is a maximum achievable parallel
efÞciency, since the calculation proceeds at the pace of the slowest processor, i.e. the one
with the maximum number of gridpoints to process. This peak efÞciency is the ratio of
the average to the maximum of the number of gridpoints per processor, which directly
proceeds from the standard deÞnition of parallel efÞciency.

Convergence For convergence acceleration a block-implicit solution scheme is used,
so that with a monoblock grid, the solution process is completely implicit, and when
blocks are small, distant points become decoupled. Increasing the number of processors
means that the number of blocks must increase, which in turn may affect the convergence
properties of the solver. It should be noted [?] that any physical ßuid has a Þnite infor-
mation propagation speed, so that a fully implicit scheme may be neither necessary nor
desirable.

Performance It is important to establish the maximum achievable performance of
the code on the current generation of supercomputers. Work is in progress with the Intel
Paragon machine and a Cray YMP.

Scalability Massively parallel processing is only useful for large problems. For a ßow
solver, we wish to determine how many processors may be effectively utilized for a given
problem size – since we may not always run problems of 10 000 000 gridpoints.

7.0.4 Aspects of Software Engineering: C versus Fortran

Only a few remarks shall serve to highlight the discussion. Fortran has been the program-
ming language of science and engineering since its inception in the late Þfties. It has been
updated in 1977 and recently new versions called Fortran90 (or HPF) have been devised.
On the other hand, ANSI-C is a quasi de-facto standard that is available on virtually any
hardware platform. In the following, the most important features of a modern high-level
language are listed, mandatory to write portable, safe, robust, efÞcient, and reusable code
that can also be maintained.

º Special Parallel Languages not needed for CFD
º Desirable features of a high level language for CFD:
– wide availability and standardized
– portability (sequential and parallel machines)
– rapid prototyping (high productivity)
– compact code

106

– versatile loop and conditional commands
– dynamic storage allocation
– recursion
º Advanced features
– pointers
– user deÞned data types (structures)
– classes (information hiding)
– conditional compilation (implementation of drivers for different parallel ma-
chines)

The conclusion drawn is the following:

º F77: does not possess modern constructs
º HP Fortran and F90: some modern features available
º C: all features available, except classes ; dynamic storage allocation limited
º C++: all features available Recommendation : C++ language of choice for the 90’s,
but not all of its advanced features are needed for CFD

7.0.5 Numerical Solution Strategy: A Tangled Web

It is important to note that the succesful solution of the parallel ßow equations can only
be performed by a Triad numerical solution procedure. Numerical Triad is the concept
of using grid generation, domain decomposition, and the numerical solution scheme
itself. Each of the three Triad elements has its own unique contribution in the numeri-
cal solution process. However, in the past, these topics were considered mainly separately
and their close interrelationship has not been fully recognized. In fact, it is not clear which
of the three topics will have the major contribution to the accurate and efÞcient solution
of the ßow equations. While it is generally accepted that grid quality has an inßuence on
the overall accuracy of the solution, the solution dynamic adaptation process leads to an
intimate coupling of numerical scheme and adaptation process, i.e. the solution scheme is
modiÞed by this coupling as well as the grid generation process. When domain decom-
position is used, it may produce a large number of independent blocks (or subdomains).
Within each subdomain a block-implicit solution technique is used, leading to a decou-
pling of grid points. Each domain can be considered to be completely independent of its
neighboring domains, parallelism simply being achieved by introducing a new bound-
ary condition, denoted as inter-block or inter-domain boundary condition. Updating these
boundary points is done by message–passing. It should be noted that exactly the same ap-
proach is used when the code is run in serial mode, except that no messages have to be
sent to other processors. Instead, the updating is performed by simply linking the receive


h

Figure 7.1: Flow variables are needed along the diagonals to compute mixed second derivatives
for viscous terms. A total of 26 messages would be needed to update values along diagonals. This
would lead to an unacceptable large number of messages. Instead, only block faces are updated
(maximal 6 messages) and values along diagonals are approximated by a Þnite difference stencil.

buffer of a block to its corresponding neighboring send buffer. Hence, parallelizing a
multi-block code neither demands rewriting the code nor changing its structure.

A major question arises in how the decomposition process affects the convergence rate
of the implicit scheme. First, it should be noted that the N-S equations are not elliptic, un-
less the time derivative is omitted and inertia terms are neglected (Stokes equations). This
only occurs in the boundary layer when a steady state has been reached or has almost been
reached. However, in this case the Newton method will converge quadratically, since the
initial solution is close to the Þnal solution. The update process via boundaries therefore
should be sufÞcient. In all other cases, the N-S equations can be considered hyperbolic.
Hence, a full coupling of all points in the solution domain would be unphysical, because
of the Þnite propagation speed, and is therefore not desirable and not needed. To retain
the numerical order across block (domain) boundaries, an overlap of two points in each
coordinate direction has been implemented. This guarantees that the numerical solution
is independent of the block topology. The only restriction comes from the computation of
ßow variables along the diagonals on a face of a block (see Fig. 7.1), needed to compute
the mixed derivatives the viscous terms.

It would be uneconomical to send these diagonal values by message passing. Imagine
a set of 27 cubes with edge length h 3 assembled into a large cube of edge length h. The
small cube in the middle is surrounded by 26 blocks that share a face, an edge, or a point
with it. Thus, 26 messages would have to be sent (instead of 6 for updating the faces) to
fully update the boundaries of this block. Instead, the missing information is constructed
by Þnite difference formulas that have the same order of truncation error, but may have
larger error coefÞcients.

To continue the discussion of convergence speed it should be remembered that for
steady state computations implicit techniques converge faster than fully explicit schemes.

108

3
3 2

4 1
4 1

Figure 7.2: The Þgure shows the computational stencil. Points marked by a cross are used for
inviscid ßux computation. Diagonal points (circles) are needed to compute the mixed derivatives
in the viscous ßuxes. Care has to be taken when a face vanishes and 3 lines intersect.


The former are generally more computationally efÞcient, in particular for meshes with
large variations in grid spacing. However, since a full coupling is not required by the
physics, decomposing the solution domain should result in a convergence speed up, since
the inversion of a set of small matrices is faster than the inversion of the single large
matrix, although boundary values are dynamically updated. On the other hand, if the
decomposition leads to a blocksize of 1 point per block, the scheme is fully explicit and
hence computationally less efÞcient than the fully implicit scheme. Therefore, an optimal
decomposition topology must exist that most likely depends on the ßow physics and the
type of implict solution process. So far, no theory has been developed.

However, a number of numerical experiments has been performed with the ParNSS
code [?], clearly demonstrating the convergence speed up. Block numbers have been var-
ied from 5 to 140, using an otherwise identical grid.

Second, domain decomposition may have a direct inßuence on the convergence speed
of the numerical scheme. In this paper, the basis of the numerical solution technique is the
Newton method, combined with a Conjugate-Gradient technique for convergence accel-
eration within a Newton-Iteration. In the preconditioning process used for the Conjugate-
Gradient technique, domain decomposition may be used to decrease the condition number
(ratio of largest to smallest eigenvalues) of the matrix forming the left hand side, derived
from the discretized N-S equations. In other words, the eigenvalue spectrum may be com-
pressed, because the resulting matrices are smaller. It is shown in [?] that this ratio is a
measure of the convergence speed for generalized conjugate residual algorithms. Having
smaller matrices the condition number should not increase; using physical reasoning it is
concluded that in general the condition number should decrease.

From these remarks, it should be evident that only a combination of grid generation
scheme, numerical solution procedure, and domain decomposition approach will result in
an effective, general numerical solution strategy for the parallel N-S equations on complex
geometries. Because of their mutual interaction these approaches must not be separated.
Thus, the concept of numerical solution procedure is much more general than devising a
single numerical scheme for discretizing the N-S equations. Only the implementation of
this interconnectedness in a parallel solver will lead to the optimal design tool.

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