AUTOMATIC. STRUCTURED ... generation is closely coupled to both the design and analysis software and because ..... to Dr. Donald. Kinsey and Lt. John Seo.
N95- 28751
AUTOMATIC
STRUCTURED (SOME John
GRID
GENERATION
RESTRICTIONS
R. Chawner
and
USING
GRIDGEN
APPLY)
John
P. Steinbrenner
Pointwise, Inc. Bedford, Texas
OVERVIEW
The
authors
of structured
have
noticed
grid generation.
in the recent
The
motivation
grid generation behind
literature
an emphasis
such work is clear;
despised task in the grid - analyze - visualize triad of computational analysis (CA). generation is closely coupled to both the design and analysis software and because of grid quality
are lacking,
"push
button"
grid generation
usually
results
on the
grid generation
automation
is easily
the most
However, because 1) grid 2) quantitative measures
in a compromise
between
speed,
control, and quality. Overt emphasis on automation obscures the substantive issues of providing users with flexible tools for generating and modifying high quality grids in a design environment. In support of this paper's tongue - in - cheek title, many features of the Gridgen software are described. Gridgen no stretch of the imagination an automatic grid generator. Despite this fact, the code does utilize automation techniques that permit interesting regenerative features.
is by many
INTRODUCTION
Gridgen is comprised
[1] is a software
of two codes:
system
Gridgen
for generation
(see Figure
of 3D, multiple
1), an interactive
block,
program
structured
that
provides
grids.
The
capabilities
system ranging
from geometry model import through volume grid initialization and analysis software pre-processing; and Gridgen3D, a batch program for volume grid refinement. Gridgen is a monolithic program in the sense that it consists of a single process and a single graphics window. Gridgen's window, however, is repositionable and resizable. Gridgen also manages its own non-overlapping windows inside the main window. User interaction with Gridgen is directed through text - based button menus activated via the mouse or keyboard "hot keys". A 3D, graphical image of the grid is always present and may be easily panned, zoomed, rotated, and otherwise
customized
by the user.
The
manner in which grids are constructed using Gridgen is is based on a hierarchical paradigm, beginning with the geometry model, and proceeding to curve, surface, and volume elements as shown in Figure 2. The geometry model, provided as input to Gridgen, serves as the foundation for the hierarchy. The user constructs
curves
which
are in turn
used to build
the topological
The grid for each of the hierarchical components is an implicit result of maintaining a distinction between the geometry model, allows
changes
design
(CAD)
to be rapidly
Users operate assigns variety
system.
Gridgen After
propagated in much importing
either
forward
model
throughout
as a product (database),
a number of points to each connector, and distributes those of distribution functions. The connectors are then selected
(domains). Transfinite interpolation (TFI), elliptic partial differential PDE methods are available for controlling the distribution of grid points 463
and volume
components.
but separate part of the component. The the hierarchical components, and the grid
or backward
the same manner the geometry
surface
designer
the grid system. operates
the user draws
a computer curves
aided
(connectors),
points along the connector using a as the boundaries of surface grids equation (PDE), and hyperbolic on the domains. The domains are
then selectedas the boundariesof methods
are available
and exports
formatted
for grid point data
volume
grids
control
for the chosen
(blocks).
within analysis
Again,
the blocks.
TFI, Finally,
the grid generation minutia and bookkeeping as possible and grid quality. This is made possible through Gridgen's the inter-relationships between the 1D, 2D, and 3D grid
in Figure
shapes
(Reference
is a user as much of
automatically.
blocking
Some
GRID
in the context of grid generation for an external [2] and illustrated in Figures 3, 4, and 5, was
that studied its wake structure relative to the base slant for an evaluation of computational fluid dynamics (CFD)
[3]). The
PDE
conditions
so that the user may concentrate on topology data hierarchy. The data hierarchy maintains components (connectors, domains, and blocks,
Gridgen's automation features will be demonstrated automotive shape. This bluff body, described in Reference
automative
and hyperbolic
grid generation using Gridgen design has been to automate
may be propagated up or clown the hierarchy in the following sections.
EXAMPLE
the subject of a wind tunnel test vehicle was also used as the basis
PDE,
the user sets boundary
software.
From the description above it should be apparent that in - the - loop task. Therefore, one of the goals of the software
respectively) so that the user's changes of these automation tools are described
elliptic
system
and surface
grids
created
using
angle. This software for
Gridgen
are shown
6. DATABASE
Database
is Gridgen's
is to be generated. curve and surface
term
Gridgen's geometry.
for the
geometry
those
listed
of the object
on and
database capability is based mathematically The database also includes relational data
etc. Since Gridgen is dedicated to the generation used to create the database. Gridgen can import including
model
around
which
the grid
on nth degree, rational, such as grouping, color,
Bezier names,
of hexahedral grids, the user's CAD software may be the database from several industry standard file types,
below.
• PLOT3D (bilinear surfaces) • Patran V2.5 Neutral 1 • IGES 2
The
bilinear
database
surfaces
for the
bluff-body
grid
in Reference [2]. The surfaces in this particular database surfaces for the fore- and mid-bodies, and one containing
were created
from
the drawings
and
projection
of grid components the database entity 1Packet 2Entities
types 000,
of surface
grids.
More importantly,
tabular
were stored in two files: one containing the four aft-body surfaces.
Gridgen offers several methods for placing grid points on the database be drawn directly on the surfaces ("Line on DB" and "Curve on DB" segment grids,
and
Gridgen
automatically
including curves types), projection maintains
data the five
that may of curve
the adherence
to the database. This relationship between the grid and the database is maintained via name. Each entity in Gridgen's database has a unique name and every grid component
32 and 33 100, 102, 104,
106,
110, 116,
124,
126,
128, 212,
464
308, 314,
402, 406,
408
that
adheres
The
utility
to the
The
Gridgen entire list
of this
database feature
saves
the
name
is explained
provides of database
of the
in the
entity
following
and
the
database
coordinates
(see
Figure
7).
paragraphs.
several commands that allow commands is listed below.
changes
• Import
• Export
• Examine
• Name
• Group • Delete
• Ungroup • Translate
• Reduce • Scale
* Copy • Rotate
to be made
interactively
to the
database.
• Intersect
When of the
grid
the
also
modified
off"
mode.
wish
to create
have
to the a grid
via the
entity
constructs
the
name
type
that
files
can When
cp aft40.net
the
steps
baseline
is saved
can
Rotate) by
are
the
ahead
baseline
data
form
already
associates
grid
type
applied
by the
stored
user,
in Figure
of the
base
and
the
slant
all
7) are
made
IGES
components
files
and with
20 ° base this
it's
contain
in a "hands
created
slant it's
simple: entity
that
we
database case
and
important
Gridgen names,
uses
Gridgen
where
was
imported,
psurface
for a parametric
surface),
file
angle
will
aftxx.net
generated
be
been
for the
don't
entity
(e.g,
in the
named
is first
For
which
also
to accomplish
basename-type-number
of entity entity
a database In order
files,
which
of the
can
has
entities.
Patran
file from
of time
40 ° case
Gridgen
database
and
for the
a variation
that
database
slant
will be to create
of the
of the number
be created the
PLOT3D string
of the
40 ° base
40 ° configuration.
for naming
For
require
new
where
we copy
the
the
several
xx is the
40 ° database
aft base
databases, slant
the
angle,
file to a generic
ie
file
Aft.net
grid
will
be generated
to a Gridgen
the
cp
20 ° database
aft20.net
2.
Import
into
3.
Import
the
database
and
determined
file.
with
In order
references
to database
to generate
grids
for the
surfaces other
called base
Aft-psurface-n
slant
angles
case,
ie.
and
the
following
be used.
1. Copy
Since
as the
term
knows
grid
therefore,
basename
the
Recall
goal,
a text
sequence
slant.
for the
same
is a descriptive
aft40.net.
the
from
is the
Then,
Scale,
(as
variations
a grid
the
file.
number
database
entities
Our
convention
is the
user
that
20 ° base
it's
in the
basename
Since
(Translate,
modified
on parametric
assume
name. that
Gridgen's stored
based
let's for the
think
to understand names
commands
to the
grid
As before,
Gridgen
the
modification
adhering
automatically.
Changes
entities
shape
elements
Gridgen entity
Gridgen
called
same
generic
name
files
Body.net
used
for the
baseline
Aft.net
Gridgen
doesn't
file to the
the
database
file that
notice
was
created
a difference
Aft-psurface-1
has
for the
and
40 ° case.
between
the
been
imported 465
Aft.net.
current and
and there
previous are
grid
database references
configuration to that
entity)
(ie,
a
the
grid is automatically regenerated in terms files. Therefore, with a little forethought,
of the new database through the simple act of importing the the user can parametrically change the shape of the database
without
An example
requiring
significant
40 ° grid was changed
grid rework.
into the 20 ° grid simply
of this is shown
by importing
in Figure
a new database
8 which
shows
how the
file.
CONNECTORS
consists
The one dimensional grid of three sets of information:
and parameters The
describing
shape
element in Gridgen's the 3D curve shape,
the distribution
is defined
interactively
up of one or more segments
where
data hierarchy is the connector. Each connector the number of points on the connector (dimension),
of grid points
along
by how the user
a segment
is a primitive
the connector.
draws
the connector.
Each
curve
type including
those
connector listed
is made
below.
• Line (a 3D polyline) • Curve (a 3D cubic Bezier curve) • Akima Curve (a 3D cubic Bezier
curve
• Line on DB (same
constrained
• • • •
as a Line but
with slopes
the segments
are represented
via Akima's
to lie on a database
Curve on DB (same as a Curve but constrained Akima Curve on DB (same as an Akima Curve Circular Arc Conic Section
Mathematically,
defined
surface)
to the database) but constrained
by cubic,
rational,
method)
to the database)
Bezier
curves.
This underlying
similarity
allows the segments to be converted with a single Gridgen command from a Line, to a Curve, to an Akima Curve, and even to a generalized Bezier segment as shown in Figure 9. The Bezier segment affords the user that
the opportunity to edit the the Gridgen user can create
Gridgen's
connector
slope point of the Bezier control polygon. This flexibility is provided so connectors with the desired shapes. Editing control points is only one of
modification
commands
• Add Segment • Translate
• Insert Segment • Stretch
• Split
• Join
which
are listed
below.
• Erase Segment • Rotate
• Edit
Control
• Project
Points
onto DB
The user assigns a number of grid points (called the dimension) to each connector a numerical value or by graphically picking a series of existing connectors and using dimensions
for the current
The distribution are shown in Figure
by either typing the sum of their
connector. of grid points
10. Breakpoints
along the connector are locations
along
is defined
by several
the connector
at which
parameters,
some of which
a grid point
will lie and at
which the grid point spacing will be explicitly defined by the user. Breakpoints also divide the connector into subconnectors. In each subconnector the user controls the number of grid points (subject to the total connector
dimension),
clustering
at each breakpoint,
and distribution
function
from this list: • hyperbolic • geometric
tangent progression
• monotonic • copied
rational
from other 466
quadratic connector(s)
spline
type which
may be selected
Although separately, two
they
will
grid
are
is the
database
attributes
coupled. the
most
This
allows
This
whole
of a connector
connector
intensive
task
described
in the
are
user
propagated
(shape,
to change
of the
which
(as
connectors
the
coupling
labor
is modified
to the
three
automatically.
reworking
connectors
changes blocks.
of the
all
be updated
without
the
each
can
dimension, any
one
connector's make
in Gridgen.
This
previous
section).
automatically
the
user
grid
are
attributes allows
much
As will
the
of the
attributes
connector
up
distribution)
more
specified
and
the
user
efficient
the
other
to edit
since
the
creating
updating
process
occurs
whenever
be shown
in the
following
sections,
component
hierarchy
to domains
and
DOMAINS
Surface
grids
domain's
perimeter.
arranged
into
must will
map
four
into
from when the case,
edges
such
that
detects
opposite
TFI
coordinates grid point TFI
all of the
uses
user's have
grid
points
by
the
known
grid
same
the
an
point
data
the
ensures
that
the
surface
grid
each
will
u and
point. same
the
be
domain Gridgen
transfinite (obtained
Furthermore, database
database
v are
the
called
the
must
defined
connector
grid
of the
(ie,
been
lie on the
space
comprise they
points
method
interior
ff = [ u v ]T where
points
but
has
grid
domain
parametric
of the database surface (see Figure 7). This which are then evaluated in terms of the
of grid
perimeter
algebraic
used form
that
perimeter
g = [ x y z ]T on each F at
to bound
connectors
the
number
domain
applying
in the
of the
on
used
to interpolate
connectors data
of the
be
the
Once
automatically
boundary
selection
may
edges
distribution)
be applied
the
rectangle).
uses
point
that will
by
of connectors
surface
method
grid
algorithm
Parametric
the
This
connector's
parametric
space) interior
number
create
Gridgen TFI
created
Any
(TFI).
each
are
3
an I x J computational
automatically
interpolation
(domains)
surface
surface.
parametric
In this (domain
interpolation yields (u, v) coordinates at each database surface resulting in F = [ x y z ]T.
adhere
to
the
database
without
any
need
for
projections. This themselves
same type of auto-TFI are modified. Domain
• Translate
• Stretch
• Split
• Join
The
new
whatever
TFI
changes
propagate Surface
Gridgen's wise, selected variety
connector
method
grid
elliptic
successive control
over
3Gridgen
quality
change
data
The
some
can also create
is used to
(SOR) elliptic are
to
re-TFI
the
domain
the
user.
Figure
by applying
2) is called
orthogonality) solves
with
method
listed
by
Gridgen
algorithm
belong shown
DB
clustering,
PDE
that those
onto
domain (see
Specifically,
of which
a domain
the
hierarchy
(smoothness,
relaxation
connectors include
• Project
applied
techniques.
functions.
of attributes,
last
Gridgen's
PDE
* Rotate
shape
was
up
logic is applied when modification commands
surface This
can
and
467
PDE technique
domains
using
modification,
where
editing. improved
to a connector.
by
using
factors allows
below.
a hyperbolic
or the
automatically
of
equation
relaxation
flexible
be
grid
type
forward
Poisson's
optimal
is quite
to domains below.
the
application
an explicit,
([4])
subject
user
to apply
of point-
to
user
a wide
• Control Functions - LaPlace(smoothness) - Fixed Grid (smoothness) - Thomas-Middlecoff (clustering)[5] - Sorenson(orthogonality)[6] - nilgenstock-White(orthogonality)[7] [8] • BoundaryConditions - Fixed - Float - Ortho
• -
SurfaceShape free fixed database
• NumericalSolution relaxationfactor
Oneof the most important elliptic PDE attributes is the surfaceshape.This attribute in combination with the controlfunctionsallowsthe grid pointsto redistributethemselvesin order to obtain the desiredgrid qualitieswhile alsoadheringto the desiredshape.Of course,the mostimportant shapethat usersneedto maintainis the shapeof surfacegridsconstrainedto the database.Therearetwo techniques with which Gridgencan makegrid points adhereto the database.As was the casefor TFI, if Gridgen determinesthat the surfacegrid pointslieon the database,the elliptic PDEswill be solvedin the parametric space(u, v) of the database and the model space coordinates will be obtained from the known surface shape
_"= f(u,
a simple
v) (parametric
projection
the
database
entities
to
by
Gridgen).
The
advantage
(no
additional
of the (those
database much near
the
seam the
the
from bluff
one body
parametrically
important
the
forcing
the
the
connector on
move
subject
by a new
shape
The
elliptic
defined
by
may
also
may
then
be used
region
of the
define
region
to which
elliptic sections)
PDE the
attributes. user
may
grid.
elliptic two
refined
grid invoke
thus
has
set-up
automated lie
making grid
on
the
points
points
may
been
created
as to which
to
all of a grid
PDEs)
those
grid
that
grid
solution. in the
the point's
the
same
calculation
that
lie on or
migrate
grid
TFI
and
across
on the
surface
points
are
may
should the
PDE
require method
468
connector
connector's
the of
solved
float
shape
or
is done
the
original
connector
of this
feature
is shown
the
name
methods
subdomain
subsets effects
resetting
of the
PDE
attributes.
the
isn't than with the grid they
is replaced
in Figure
the fit over
the
points;
shape
are
its own
allows
treats
domain
implies,
that
stuck
that
to restrict
saves
re-application without
2) being
condition
interior
known
shape Rather
condition
boundary as
is not
when
is required.
a boundary
of as templates
Each
Even
manner
PDE
be thought
be made.
modifications
as
connector)
same
example
which, elliptic
(the
domains.
the
The
solver An
across the
PDE
points.
two
provide
domains
sub-domains the
lines
fine-tuning
the
perimeter
between
techniques
When
Subdomains
When
only
notations
grid's
shared of grid
PDE
solution.
with
grid
with
of a surface
variation
to the
changes
simply
shape
1) manually
defined
a surface
elliptic
is
maintains
easy
has
that
database
automatically
much
Gridgen
technique,
surfaces, Of course,
surfaces
connectors
elliptic
PDE
user
to a small the
for
between the
11 shows
precise
between
subject
parametric
projection.
it's
detects
of the
the solver
is computed
is that
Gridgen
on
PDE
conventionally.
true
shared
the
database
forebody
for smooth
to float
to the
Subdomains
the
Gridgen's
a connector
apply
(which
solution
points
elliptic
Fortunately,
when
for
grid
in the
technique
projections).
multiple require
run
orientation PDE
Specifically, differences
Figure
that
to choose
shape
points
will
solved
case
inadequate,
finite
two
is especially
(no
for keeping
being
elliptic
faster
span
are
requirement
user
something
the
technique
domain
projection
technique.
surfaces
spanning
This
the
automatically
to another.
which
and
domains
surface
and
user.
will
the
second Each
parametric
much
in
database
It is often by the
used
When
between
it's
surface
code
efficient.
seam
and
points
surface more
onto, of the
database
The
technique).
project
attributes)
application neighbors
technique).
(conventional
12.
of a domain. of the domain
unique methods
methods grid
and
combination (see
previous
of
BLOCKS
Volume
grids
(blocks)
are created
perimeter. 4 Any number of domains into six faces such that opposite faces
by the
user's
selection
of the domains
an I x J × K computational parallelepiped). Once the block automatically create the volume grid points by applying TFI. Block modifications, of changes
down
including
Gridgen's
data
and maintains both used in two blocks. transferred
exported
Gridgen the
entire
scale, and rotate, editing)
has
been
will result
to the domains
one of the most cumbersome As a result of Gridgen's data
that
use that domain.
for the chosen
analysis
also offers an interactive
blocking
(backward
perimeter
the block's
defined
Gridgen
in automatic
and
will
propagation
connectors.
aspects of multiple block grid generation: hierarchy, the code automatically detects
Changes
to a domain
hierarchy and cause automatic re-TFI of the are exported for use with the analysis software
specifically
comprise
full- and partial-face interblock connections simply by noting that the same domain is Changes to a domain's grid points by the PDE solver, for example, are automatically
to both blocks
up Gridgen's data boundary conditions
copy, translate,
hierarchy
Gridgen has also automated interblock connectivity detection.
that
may be used on the block's perimeter but they must be arranged have the same number of grid points (ie, the block must map into
system.
perimeter
volume grid points. the connectivity data
will propagate Also, when the is formatted and
software.
tool for re-dimensioning
This feature
on a block's
allows
the
user
(changing
to select
any
the total number connector
in the
of grid points) blocking
and change its number of grid points. That change is propagated automatically throughout maintain dimensional consistency (ie, maintain I × d domains and I x J × K blocks).
system
the system
to
MISCELLANEOUS
One
final
automation
feature
deserves
mention.
Gridgen
can
import
"raw"
grid
points
(from
a
PLOT3D grid file, for example). When this is done, Gridgen automatically creates connectors, domains, and blocks, from the raw grid data. Therefore, the user may then avail him/herself of all of Gridgen's commands, including analysis boundary condition setting and file export. Conversely, the Gridgen user may export
the grid points
from any connector,
domain(s),
or block(s)
to a PLOT3D
file for use in other
applications.
SUMMARY
Various
features
within
Gridgen
have been described
and their
unique
automatic
aspects
have been
emphasized. These features contrast sharply with the fact that Gridgen is not an automatic grid generator, s It is not the intent of this paper to diminish the many published and unpublished works on automatic grid generation. Indeed, the successful configuration - limited procedures that this paper
and
Gridgen
4Gridgen can also create SWe define an automatic user
itself
have
been
application of CFD in a design environment often depends on may be applied rapidly and easily by the CFD novice. Instead, motivated
by the desire
to emphasize
a block by applying a hyperbolic PDE technique to a domain. grid generator as one which will create a suitable grid given
input.
469
grid quality
a geometry
model
and software
and
little
or no
flexibility. Theseemphaseshaveled to the developmentof a visually based,3D, interactiveapplication environmentthat automates much of the low level grid maintenance tasks, freeing the user to apply his/her judgement as to grid suitability. Gridgen's ability to automate much of the process is directly to its data hierarchy in which the curve, surface, and volume grids are maintained as secondary of connector, domain, and block topology components.
attributable components
ACKNOWLEDGEMENTS
Gridgen is a product of Pointwise, Inc. GRIDGEN Version 9 was sponsored by NASA Ames Research Center through Computer Sciences Corp. Thanks to Mr. Matthew Blake of NASA and Dr. Jin Chou of CSC. GRIDGEN Version 8 was sponsored by NASA Langley Research Center through Computer Sciences Corp. Thanks to Dr. Robert Smith of NASA and Dr. Jamshid Abolhassani of CSC. GRIDGEN Version 6 was sponsored by the United States Air Force Wright Research and Development Center. Thanks to Dr. Donald Kinsey and Lt. John Seo. All trademarks used herein are the property of their respective owner.
REFERENCES
[1] Steinbrenner, J.P., and Chawner, wise, Inc., Jan. 1995.
J.R.,
"The
Gridgen
User Manual:
Version
[2] Ahmed, S.R., Ramm, G., and Faltin, G., "Some Salient Features of the Vehicle Wake," SAE Technical Paper Series No. 840300, SAE International Detroit,
MI, March
L.W.,
R.L,
"The
3DGRAPE
S., et al, Pineridge
[8] White, J.A., "Elliptic Number of Boundaries",
Journal
of Computational
Physics,
J.F., "Direct Control of the Grid Point Distribution Journal, Vol. 18, pp. 652-656, 1979.
[7] Hilgenstock, A., "A Fast Method Boundary Control", Numerical by Sengupta,
Point-
Time - Averaged Ground Congress and Exposition,
Aerodynamics Workshop," Second International Industry, Seville, Spain, October 1988.
"An Ad Hoc SOR Method",
[5] Thomas, P.D., and Middlecoff, by Elliptic Equations", AIAA [6] Sorenson, 1989.
from
1984.
[3] Misegades, K.P., "1988 Automobile Supercomputing in the Automotive [4] Ehrlich, 1981.
10", available
Book:
Theory,
Users'
For The Elliptic Grid Generation Press
Ltd.,
Manual,
on
Vol. 44, pp. 31-45,
in Meshes
NASA
Generated
TM-102224,
Generation of Three-Dimensional Grids in Computational Fluid Mechanics
Swansea,
July
With Full '88, ed.
UK, 1988, pp. 137-146.
Grid Generation With Orthogonality and AIAA paper no. 90-1568, June 1990.
470
Examples",
Conference
Spacing
Control
On An Arbitrary
_r
y
:
t Ei._
I onl
I_ull
t I _
l"llt4
-1,1_'_le6
Figure
1:
blocks volumes 3D
The
Gridgen
screen.
o f r
_
W a
d surfaces domains b
2D
a c
k W connectors
a
ID
curves
_--_,.,-_
r
d
geometry database
model
Figure
2
/
/
(;ridgen's
dala
471
hierarchy.
/
1044
222(1-cos(40))
'
__1
222cos(40)tan
_
I I I I I i_mum_oa
I ! I I
__88
,
Figure 3: Schematic of a bluff-body, external automotive tomation features. All dimensions in millimeters. Adapted
shape from
II
used for demonstration Reference [2].
of Gridgen's
au-
Y
0
288.0
""¥ ...........
5
273.121521
........
10 258.013519
2
12.5
250.298187
15
242.432068
20
226.102539
"""
25
208.698853
"'",
30
189.814743
", 35
168.921402
", 40
145.301163
288 ',,"
, •
""'" ""i'"',
II
222
Figure 4: Detailed schematic of the various millimeters. Adapted from Reference [2].
base
angles
472
for the bluff body
in Figure
3. All dimensions
in
5800
t
1044
3000 530
280
tv F
I-+
1.35.1044
F
2.43.1044
V----" _I
1
3O00
3000
I I I
! H'H I
I...ml_QIB._.,
I I
I
I I ! !
Figure 5: Schematic of the bluff body Adapted from Reference [2].
in Figure
3 installed
473
in the wind tunnel.
All dimensions
in millimeters.
Figure on the
6: A 15 block grid containing 520,000 surface of the body is also shown.
Body-psurface-
grid
points
was created
for the
1
bluff
body.
A close-up
of grids
Body-psurface-5
x=O
_
y =188
x=822 Y = 288 z = 194.5
z=0
Figure
u=0.0
u=0.7
v=0.667
v = 1.0
7: Gridgen
to be easily
maps
grid
elements
to the
database
via
changed. 474
the
entity
name
allowing
the
underlying
shape
Figure8: The grid for the 40° baseslant angle(left) waschangedto the 20° baseslant (right) simplyby importinga differentdatabasefile.
°Q:i
J Linear
Segment:
Cubic •
Figure another.
9: Gridgen's Note
that
control
unified the
point
geometric
control
Segment: O
foundation
points
Bezier
for all three
allows
slope
point
segments
segments
are
Segment:
to be easily the
converted
from
same.
/
A []
control point
Figure
10:
Terminology
•
break point used
in the
• distribution 475
grid point of grid
points
O along
cursor
a connector.
one
type
to
r
; ; ,+;.:+';,
/ / 1tINl!
Figure
11: The
beyond onto
the the
Figure
grid
surfaces.
12:
The
surface are
the
grid
on the
boundaries
All other
connector
points
between
upper
forebody
of the are
surfaces).
maintained
two
surface
spans
two
The
filled
on the
grids
solution. 476
database circles
surfaces
may
change
surfaces denote
(the grid
vertical points
lines
that
extending
are projected
parametrically.
shape
according
to
the
elliptic
PDE