Automatic structured grid generation using Gridgen (some restrictions ...

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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