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.