Automatic topology generation and generalised B spline mapping

465

AUTfXATIC TOPOLOGY GENERATION ARD GENERALISED B SPLINE MAPPING A ROBERTS British

Aerospace

SUMMARY

Completion anomalies be

defined

with

differential through

for

have

very

equations

topological

accelerated spline

strategies

been

few

singularities and

regular

the

the

of

mapping

finite

can then of

for

the

modelling

schemes provide

Relaxation

for

topological

realisation

Such B spline

accuracy.

than those

of

structures

The

B spline

field.

band’

removal grid

statements.

in

‘wide

for

complicated

a generalisation

can then be more efficient

topologically

of

explicit

uees

convergence

programmed

Synthesis

in a cube cluster.

sequences

difference

with

B

methods even

grids.

INTRODUCTION The Multi-Volume for

linking

Data

a range

of

Structure

industrial

I Pig

1.

The

use

The

automatic

aircraft be

and

augmented

The

grid

with

default is

3.4.5

Data Bases

units

partitioned

are

field

cryptic

coordinates

for

pod.

are

generated

a

number of

partitioned

into meeting

along

To

fields

such

Each unit

volumes.

language.

or 6 blocks

system

1,2

with

mapping

problems.

‘starters’

an engine

the adjacent

topology

CPM methods.

Representation

common organisational

such as

a

purpose of

to Methods

techniques

topological

using

a general to a range

)-Clebsch

presents

‘details’

component and

configuration

face

multi-grid

in the field

combination

can

of

is

bases

--

The MVDS System Links

singularities

(MVDS) data

as

The library permit

a

refers

display

wing-body to both an

of

such units

of

the

grid

automatically. cubic

junction

blocks

lines

connected

in the field.

face

to

466

Fig. 2.

A Field Partitioned with Singularities in the Field.

Each volume has a different

grid structure at each multi-grid

resolution

level:-

3 Volume

Fig. 3.

Level 0

Level 1

Level 2

Multi-grid structures in a volume

At every level the partition boundary is half a parametric interval from the nearest grid surface.

Topological

singularities

then occur between grid

points:-

Fig. 4. Mapping Singularities at Node Points. Power -1

Power 0

Power +l

Power +2

467 The grid

is

said

to

be regular

In

every

junction

lines

have no numerical

line.

axes

may be

such

shock

across

surfaces B spline

are not applicable.

regions

regions

significance

discontinuous

singularities,

in

the

where four

partition

boundaries

although

the orientation

partition

boundaries.

and vortex

sheets

techniques

volumes

and junction

of

the parametric

Near

the usual

meet along

topological

operator

can then be used for

expansions

modelling

through

such anomalies. Tesselation

of Surfaces

A tesselation sided

cells.

suitable field,

of

for

by

required

is

a decomposition

operations

cube clusters.

definition

information

a surface

Tesselation

are

combined

is

with

not

a general

decomposition

of a 3-D field

in

principle

but

unsuitable

progressive

synthesis

using

grid

operations

that to

with

via

into

due

to

a tetrahedron

cluster

We are

preserve

in the

the

Automatic

structures.

four

strategies

many singularities

problems.

that

surface

illustrate

practicable

conceptual

of

of

here

For cube clusters

enumeration

produces

used

mass

is

possible

then left

the

of

conversion

cube

with

cluster

conventions. We consider surface. point cell

of to

tesselating old

a general

We can select that the

cell.

polygonal

A median line

median

point

of

operation

is

then

decomposition

Old

5.

Tesselation

Operations

point

of a simply

of a line

is

then a line

of

the

boundary line

New

or a cell

joining

enclosing

the reclassification

Median Lines

Deleted

Fig.

one

as an additional

E

decomposition

any interior

cell of

of

connected

closed

to be the median

the median point boundary

every

of a

lines.

median line

The of

an

the new decomposition.

Median Lines

String

466 Repeated suitable

use of

for

The median endless cell

lines

strings

segments

of

strings,

connected

one

consisting

at

of

a

The general cell

of

a

lines

forming

boundary

line

string

then always where it

bend points

or,

if

a strategy

are

a set

join

of

of

up to

cells

string.

closed

left

form

where each

We may note any complete

surface

with

the

this

of

connected

closed

of

boundary

concave

the

second left

that

string

we can delete

cell

minimum tesselation

sequence

we can synthesise

surface separate

from

the

capture

definition

left

turn

turn.

enclosing

by enumeration

respect

string

any closed

strings

with

minimum

is

except

for is

string

the

of

new cell

of the boundary possible

to a ‘captured

The string

if

zone’.

the nominated then defined

the The bend

by the

by one segment of the string.

are no bend points,

the new cell

of progressive

to

can be defined

the first

there

then

A more cryptic

always

follows

is of

string

string. is

follows

The reallsation

surface

is

deletion

connected

By reversal

a pair

line

that

points

by the

we

operation

into

string

tesselatlons

at a time.

The boundary

segments

until simply

synthesis

boundary

string.

of

surface.

a simply

two cells.

one string

a family

a closed

string

median line

preserved

closed

of

produces

of

A cell

a particular

time,

just

tesselation

tesselation

tesselation

are

For any tesselation

any

the

median lines.

conventions

a simply

operation

technique.

of

of

contains

tesselation for

the tesselating

a multi-grid

following

based on just

this

convention

two elementary

can follow

operations:-

+ # Capture Fig.

6. These

operating First completed Second strings.

Delete

One Cell

Elementary

Operations

operations

can

at two priority priority

controlled

by

autonomous

propagation

sequences

levels.

propagation

or a corner priority

be

Two Cells

on a Surface

is

propagation

along

is

propagation

through

strings

until

a loop

is

encountered. propagation

an area

enclosed

by new

469

Bend Points Fig.

7.

Basic

Captured

Capture

Cube Cluster

The direct

Strings

Operations

analogue

of

a tesselated

which the minimum cube cluster ‘external

Captured

Phases

cube’

consisting

have median surfaces

of

closed

consists

of cubes

all

surface

of a finite

space

not

is

cubic

included

in

bounded by a median line

a cube

cluster

volume enclosed the

of

finite

in by an

cube.

each of

four

We

volume

interfaces:-

--_lall m m

Fig.

8. The Three Median Faces of a Cube.

Median lines faces

of

then

join

which volume

up to

each

an interface

is

contains the

intersection

can be regarded

of median faces

form edgeless

volume

string

the triple

of

one cube to a pair

sheets. facets

intersection

A volume

of of

as connecting

of an adjacent sheet

a particular a pair

of volume sheets

of

is

a pair

cube. a set

median

and of volume strings.

of

surface

volume sheets.

of median

Median faces volumes sheet.

in A

Each volume is

470 The cube cluster conventions are conserved by the deletion of any complete volume sheet. deletion

of

sequence.

Any cube cluster can be reduced to the minimum cube cluster by volume

sheets

one

at

a

time.

Sheets

may

be deleted

in any

By reversal of such sequences any cube cluster can be synthesised

one sheet at a time using one of many different sequences. We have the pair of elementary operations:-

Capture One Volume

Delete Two Volumes

Elementary Operations in a Cube Cluster

Fig. 9.

We can then have first priority propagation along strings, second priority propagation propagation

throught

sheet

segments

bounded

by

through a captured volume cluster.

bend lines meet.

strings

and

third

priority

Corners are points where three

The specification must then enumerate explicitly:-

all corners with 1,2,4 or 8 inserted into one old volume

for any bend line not terminating at either end by a corner one sample volume referring to 1,2 or 4 bend lines.

for any surface not bounded anywhere

by bend lines one sample volume

referring to 1 or 2 new surfaces. All other instructions are implicit. After grid relaxation each median surface will form a surface for which the unit normal displays

of

is continuous the

grid

and

almost for

everywhere.

contours

of

Such surfaces are useful for field

functions.

Where

median

surfaces form closed sheets the Euler characteristic3 can be used to relate the

471 number

of

h

handles3

to

the

of

sum

the

topological

powers

s where

the

power of a singularity is four less than the number of volumes

topological

meeting along a singularity line.

For such tesselated surfaces we have

s I 8(h-1) This is a useful rule when selecting a suitable topology. Solid and Null Volumes At the lowest level of the internal logic the two elementary operations are However,

used and the topology is always a properly connected cube cluster.

the required grid has sheets that can be bounded by an envelope or aircraft surfaces,

regarded

designate

volumes

as

intrusions

of

in a cube cluster

permitted through solid volumes.

the envelope as

into the

We

field.

can

No propagation

solid volumes.

is

A null volume is a volume of zero thickness

with one great side connected to some old volume or some solid volume and five faces connected to new volumes that may be null volumes.

Solid Null Volume

Fig. 10. The

null

volumes There

conventions. propagate

zones

bridge

six

apparent

propagating

anomalies

forms as

1, 2 and 4 bend lines respectively

along strings. segment.

are

Apparent Propagation Forms

Fig. 11.

thus

shown.

the

cube

Forms

cluster 3,

7,

8

as first priority propagation

Forms 2 and 6 propagate 1 or 2 sheets through some median sheet

Form 1 propagates type distinctions.

with

in

the

same

sheet

numbers

as

the

The shaded zones represent core original

volume.

With

the

qualification SOLID all core zones are designated as solid in the insertion of a sheet.

The other volumes are volumes of the new sheet.

introduce 1, 2, IIor 8 corner points of the new sheet.

Forms 5, 6, 9, 10

472

The control of propagationIs based on the examinationin turn of each of the six volumes that enclose the core volume. Each volume either is or is'not a null volume. The great side can be adjacent to a solid volume,a field null volume, or an old field volume. If it is adjacentto an old field volume, that volume may appear already at the same or higher priority level or at a lower level or not appear in the pendingoperationslists. Table 1

NULL VOLUME TESTS AND ACTIONS

TEST VOLUME IS NOT NULL ADJ. VOL IS SOLID ADJ. VOL IS NULL VOLUME ADJ. VOL ALREADY IN SAME OR HIGHER LEVEL ADJ. VOL ALREADY IN LOWER LEVEL ADJ. VOL NOT ALREADY RECORDED

ACTION IF TRUE RETURN DESIGNATENULL VOLUME AS SOLID DELETE PAIR OF NULL VOLUMES RETURN DELETE LOWER LEVEL REF. AND RECORD RECORD PENDINGOPERATION

In practice few sheets require explicit identificationof more than two volumes. If all sheets in a 'detail'intersectone sheet then all instructions can refer to that sheet. These conventions provide a cryptic language for the introduction of additional'starters'and 'details'. TopologicalDisplays In order to define the topologyof a library unit in the cryptic language the user must draw a series of sketches.

A set of sketches in fig. 12

represent the developmentof a simple wing-body configuration. In this case all explicit statementsrefer to the horizontalplane of symmetry. Each one line statementspecifieswhich volume (denotedby l), which operationand, if required, what is the fractional width of the new sheet.

If the set of

statements is accepted a full set of displays of solid surfaces and field sheets can be requested. With no further data solid surfacesgeneratedfrom the sketches in fig. 12 are displayed in the form shown in fig. 13.

Other

examples of display generated from the cryptic topology are shown in fig 14. These show a pitot intake, a side intake and a fan-in wing with adjacentfield sheets.

*

Next Volume

r= =

Solid Volume Wake

Fig. 12 User's Sketchesfor a Wing Body Combination.

Fig 13a. Displayof ResultingSolid Surfaces.

474

Sheet A View Looking Aft

Sheet B

View from Side

Sheet C View from Side

Sheet D View from Side Fig 13(b)

Median Surfaces after Relaxation

475

Fig

14.

Other

Displays

Generated

from

the

Topology

Definition.

476 Basis

Function

The

B

alternative spline

gapping

Spline

schemes

topologically required

to

integer

finite

turn

model

relevant

The basis

in

regular

although is

and

of are

through

near

of

integer the

equations

of

the

flat

function

The

B spline

within

here

function function

format analysis

are

not

equations

are

defined

follows:51 , E2 9ES

parametric

fi(E’,c2,63)

field

xiJ

scalar

At all

coordinates

functions.

X1, X2, X3

Cartesian

ej(S1,E2,E3)

basis

functions

field

function

coefficients

*

coordinates

c1 , E2 , c3 we have

+ 1

Iej(E1,S2,E3)

fi(S1,E2,E3) =

1 ej(E1,E2,S3,)Aij

.I axi

ad =

at2

axk

-_

6i

k

Near a mapping singularity

J, = A +

a*

D

BP

first

the general

linear

field

BP Xp

and we want the numerical

axp

we consider

,

,ijas -aci

formulae

a-4 acJ

to satisfy

= Bk Bk

,

at all

Jl,ij

points

= 0

in

generalisation

expansions

tensor

as

The integer conventions

basis

The basis

operator

space

regarded

basis

remains the

be

conventions.

are violated.

where and

will

spllne

grid.

singularities

conventions

singularities

function

the

reallsatlons

regions

spline

schemes

difference

interpretations

the equations

valid. as

477 These

conditions

provided

are

satisfied

= fi

Xi

and

A4j

for

all

points

near

any

mapping

singularity

i = 1,2,3

depends on

1

The use of the same basis

2

Derivation matrix

JI = fq

3 A + 1 BP Xpj p=l

=

This conservation

3

at

that

of

the

inversion

The use of the

functions

inverse

for

Xp and $.

derivations

from

the

explicit

derivative

by

is not valid

near

at each point. ‘flat

space’

formula

for

k ( ij )

The conservation

of intrinsic

mapping singularities 1 2 3

A mixture

of analytic,

Interpolated

values

The general

derivatives

of linear

functions

using:-

tensor

B spllne

and finite

of the inverse formula

for

difference

derivatives

derivatives

k { ij 1

Subject

to

such

basis

In the numerical

between the for

use

of

basis

a plane

Integer

unit

of

we require

functions

of

basis

of

functions

computed errors

normal

function surface

Spline

Let y(u)

of quadratic

a common set

the accuracy

computed

Using

treatment

conventions

to

improve

the accuracy

the form:-

A + BPXp + Cp2XpX2

J,= With the

function

near

a mapping

conventions and small

the

for

there

in the field singularity

error

of

a spherical

is

a close

equations

connection

and accuracy

on a spherical

the

unit

normal

should

surface.

Mapping

be the integer

spline

Yipqr

be the spline

weights

generating

function

cl A

be the order

of the polynomical

segments

be the order

of derivatives

continuous

E

be the order

of operand

which a spline

T

be the number of non-zero

gi

be a tripolynomical

for

segments

of y

operand of order

of y

between segments

Q

fit

is

exact

of y

of

surface. be zero

470 The tripolynomical operand is defined by E

E

E

a:0

b=O c-0

The integer spline system is applicable to a simple Cartesian grid with unit grid interval in parametric space 5~,52,53. For such a grid the basis functions and their coefficients are defined by ej(~1,~2,~3) = y(51-p)y(s2-q)y(s3-r) j = j(p,q,r) Xij = Wipqr The generating function y consists of T non-zero segments where each segment is a polynomial

of order Q extending

over a unit interval.

The function y is

symmetric about the origin and derivatives of order R are continuous between segments. For an operand of some order E the fit is exact and all derivatives are exact at all points.

For a general operand the functions fi are tripolynomial

functions of order Q within each mapping cell where the mapping cell boundaries are ss I i

for T even

for all integer i

5s = i+l/2 for T odd Regarding

a

function as the derivative

of order

zero of that function

the

general explicit derivative within a mapping cell is expanded in the form

with

continuous across both 5~ = i and 5~ = i + I/~

Comparison of Generating Functions Let Y,(u) = if -l/2 < u < l/2 then 1 else 0 +1/2 and YT+,(u) = J YT(u/v) dv - l/2 i.e. YT is the Tth power of convolution of the unit square pulse Let Z(u) be a function with 4 non-zero segments fitted to the following values at the integer values of u:-

479

TABLE 2 FUNCTION "

-2

Z(u)

0

-1

0

-1

+2 -1

dZ(u)/du

00000

d2Z(u)/du2

00000

+l

+2 0

Fig 16. Form of the Z Function

Fig 15. Form of the Y Functions

We will compare four generatingfunctionsthat providethe basis for the three and five point finite differenceschemes (5PTFD and 5PTFD) and the cubic and quintic B spline schemes (CBS and QBS).

For one dimensionalcalculationsof

the point values, first derivatives,and second derivativesat the grid points, the correspondingoperatorsP, F, S, applied to the weights can be expandedin standard finite differencenotation for each of the generatingfunctions. We then have:TABLE 3

CHARACTERISTICS OF FOUR DISCRETISATION SCHEMES

Q

R

E

T

Y4 + z/6

5

2

2

4

Y4

3

2

3

4

5

2

4

6

5

4

5

6

Scheme

3PTFD CBS

Y

SPTFD QBS

Y6

P

1 1+62/c 1

S

F

!J6

62

!J&

62

(l-62/&

(b62112)62

400 For 3-D applications the operators P, F, S applied parallel to the parametricaxis 6' are denoted by P,, F,, S, respectively. The formulae for typical derivativesare then expanded:TABLE 4 THREE DIMENSIONALDERIVATIVES Derivative

B Spllne Form A

Finite Difference

B Spline Form B

Pl P2 P3 Bi

fi

Fi fi

Pl P2 P3 Bi

F1 Pi' fi

a2fi/a(E1j2

Sl fi

St PP P3 Bi

s, Pi' fi

a2f /a61162 i

F2 F3 fi

Pl F2 F3 Bl

s2 P;' s3 Pj' fi

fi

fi

af,ml

In a relaxationsequence the weights are updated using and the relaxation

factor

the

residual

errors

ri

s:-

Win + 1 = Win + s ri For example in the relaxation

CS,P;'Fi

:

coverging

to the solution

of

0

S

the relaxation fin+l

sequence

-fim

and this

can be expanded as

+ ssl S, P,,,

expression

form

Ps+2 (din

can be further

= ei(w,sl

- fi-

)

expanded using tri-harmonic

operands

of

the

+ u2 C2 + w3 C3)

gi For harmonic

operands

the

ratio

of

the

computed

values

of

and SIP;’

gi and

F,

PT’j2gi

17.

to

the exact

analytic

derivative

over

the

W, range

as shown in fig

First

Second Derivative

Fig

17

Derivatives

of Cos (wx) as Percentage

These equations The CBS scheme derivatives can produce

may be interpreted

uses

tricubic

at all

points

to

CBS: the

approximation

Deriv.

computing

a tricubic

replacement

times

for

First ___

Deriv. - - - YPTFD

of the Exact Derivatives

as follous:-

mapping cells

for

of

to

achieve

operand. of

1 + d2/6

the calculation

precise

values

The 3PTFD system by 1 in

the

of derivatives

of

is

expansion reduced

all

a fast of

P

relative

to CBS by up to 70%. Conditions

especially

potential

flow

numerical

first

achieved Errors reduction

of

Relaxation Since

by

errors

of

(~61)

based on first

the same accuracy

derivatives for

other is

terms

order

an operator

been

recovered is

order

derivatives first for

6 4 by the

require

is

derivatives

use

of

transonio

from CBS to

of

the

of

the accuracy

second

order

derivatives.

63

5PTFD is

of

3PTFD is

so

not

that

the

necessary.

methods.

eight

use

less

of

of

for

of

SPTFD.

first

samples for

points

with conformol

the accuracy

as

that

as many grid

CBS.

of

non-orthogonality

Vj.

remote

than

needed to match CBS accuracy

flow For

same order

times

derivatives

for

mapping

are known exactly.

the treatment

values

the

by the

of

relaxation

3-D potential

for

non-zero of

case

accuracy

term

and block

solution

Sl 00s (&cl)

critical

held

of

as methods based on second hand,

with

uses

accuracy

the

the In this

The change

‘upwinding’

the

where the most critical On the

grid.

critical.

in the

:

3PTFD are

not

can be based on line

F, Fl cos

methods

is

reduction

dominated

to

an orthogonal

derivatives

without are

favourable

in 2-D using

first

derivatives

However, A seven over

point

the grid

the

accuracy

and the w range

derivatives

numerical of

of

over operator

the harmonic

first and

CBS scheme 5PTFD has which

this

would be spectrum.

482

Compared at constant band width the reduction in computing time for the finite difference system can be less than 25%.

In a relaxation sequence the critical

condition for the selection of E is wl = w

=w 2 3 = IIfor the finite difference system and wl = II , w2 = w3 = 0 for the B spline system due to the replacement

of 1's by ,!Ss P,+l P,+2. * 8

This can imply a value of E for the B spline system

three times that for the finite difference system. For a solution of the Euler equations by time stepping no numerical second differences

are required and no upwinding errors are necessary.

The correct

selection of latent shock surface requires accurate modelling of the wave front with zero rate of propagation.

The three point operators of CBS match the w

band width for the second derivatives of five point finite difference scheme and for the first derivatives of a seven point finite difference scheme. five point operators of the QBS system provide an even wider w band width. suitable application

of errors

to weights

the maximum

The By

Courant number can be

raised by a factor of ~'3. Thus 2-D potential flow using conformal mapping is especially favourable to the 3PTFD

scheme.

In most

other cases the wide band accuracy of B spline

schemes and their effect on rates of convergence are significant.

Experience

with grid relaxation demonstrates that the predicted relaxation factors can be achieved in the final convergence phase.

Topological Singularities In

a

topologically

differentiation

regular

grid,

numerical

inter,olation

and

operators which produce results within a patch require values

within a work zone that extends some distance H beyond the patch boundaries in all directions.

TABLE 5

The distance H is the halo width given by:-

THE HALO WIDTH

grid point operators T Even

T/2

-1

T Odd

(T-1)/2

interpolation operators T/2 (T-1)/2

483

Fig 18. A Work Zone in an Infinite Grid

Fig 19. Copy Core Elements into Halo

Zones

The numerical operators are always applied to a work zone as if the halo zone

represented

grid.

grid points in adjacent volumes

in a topologically regular

All grid points in the core zone of the work zone represent grid points

within the patch and all values associated with such grid points are regarded as independent variables.

The edges of a core zone consist of overlapping zones of width H within the core zones. zones.

The corners of the core zone are the intersections of the edge

Edge zones values are always copied into edge zones of the halo of an

adjacent carpet.

With T even this is sufficient to ensure analytic continuity

across a partition line except for parametric distance (T-1)/2 from the node point at each patch corner.

At all regular node points the values in a corner

of a core zone are copied into a corner of the halo of the diagonally opposite carpet.

For T odd this is sufficient to ensure analytic continuity up to the

node points. Where

N carpets meet at a node point with N i 4 the set of N adjacent

corner zones of the halo of the adjacent carpets are paired collectively with the set of N adjacent corner zones of the core zones of the adjacent carpets.

EXPLODEDVIEW Fig 20.

In the exploded a,b,c

PHYSICAL SPACE

Work Zones near a Singularity

--

are

carpets. The defined

view of

each

copied

We then require basic

function

in matrix

the N work zones into

the

edge

to determine conventions

the grid

zones

of

the values

then

permit

point the

values

halos

of

in the N halo the

halo

in the zones two adjacent

corners

corner

p,q,r--

values

to

be

format

Then

A

[

1x[:I+[Blx

Then

[;]

The

square

continuity

matrices conditions

=

with

[

A-‘xP]

dimensions

modified

[!]

x

NH’,

to ensure

that

NHL represent A is

column matrices have dimensions NH’,M where M is The first

set

O

[iI=

of continuity

conditions

is:-

a

selected

not a singular

the number of

fi

set

matrix. functions.

of The

405 CONTINUITY RANGES

TABLE 6

Parametric distance from node

From

Continuity along grid lines

T-+4

0

T=i6

l/2

fi a2fi / a(&')2

T=+8

3/z

a4fi / a(El)4

consideration

of

symmetry

and

antisymmetry

for

any

P,F,S, there are always H redundant equations in this system.

definition

of

To satisfy basis

functions conventions we require that when all elements of a,b,c are unity then all elements of p,q,r are also unity.

The condition that the sum of the radial

tangential vectors at the node point is zero is virtually mandatory. violations

very near the node points with little effect elsewhere. auxiliary

Minor

of this constraint lead to violent excursions in double curvature There are then H-l

solutions and an empirical parameter is used to control tuning for

double curvature effects. An alternative

system of contraints drops one order on the continuity at

half a parametric interval and introduces the constraint that a common limit of Jn is approached at the node point for every carpet. empirical

This system requires

parameter to tune for double curvature and another to

one

control J.

This form has the merit that surfaces can be transferred directly to AD2000 and other panel systems.

In field solutions the expansion of J,about the singularity can be expanded in the form JI -

A+BiXi+CijXiXj + DijhXiXjXk + +

For any values of A and Bi modelling throught the singularity should be exact when all the other coefficients are zero. possible

to neutralise

With the CBS scheme it should be

the total source strength for unit length due to the

contribution from Cij, for the region near a singularity line independently for N = 3,5,6. The total doublet, quadrupole and higher order violations of conservation should

have effects

that subside at least as rapidly as the inverse second

power of distance from the singularity line.

466

Fig 21

Grid Lines and Direction Cosine Contours.

Concluding Remarks

The MVDS data base is the most versatile convention compatible with generalised B

spline mapping.

versatility

The present

cryptic

language does not realise

of the data base but it appears

to be adequate

the full

for a range of

practical problems. As a grid

point

bench mark cases.

technique the B spline system is slow for the standard

However, for 3-D relaxation and Euler solutions the superior

harmonic band width, numerical stability and accuracy for first derivatives are relevant. is

Used with topologically complicated computing grids the alternative

the finite

difference

volume

technique

which

is no more

accurate

than

the finite

system away from mapping singularities and has no special treatment

of mapping singularities. The MVDS system therefore, is suitable as a replacement for the existing facilities

which

locate

mapping

singularities

at

critical

regions

of

the

aircraft surfaces. ACKNOWLEDGEMENTS The programming in Aerospace, Weybridge

the

work

described

was

done

by

Clive

Stops,

British

List of References 1.

Detyna, E. (1981) Variational Principles and Gauge Theory. An Application to continuous Media. Department of Mathematics, University of Reading.

2.

Roberts, A. (1980) The treatment of Shocks in Fast Solving Methods. In Numerical Methods in Applied Fluid Dynamics, Hunt, B. ed., Academic Press

3. Lipschuts, M. (1969) Schaum's Outline Series Differential Geometry MCGRAW HILL Book Company.