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A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches. M.I.G. BLOOR, M.J.WILSON Department of Applied Mathematics Leeds University. The PDE Method. Parameter space. Physical space. is a partial differential operator (usually elliptic) - PowerPoint PPT Presentation
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u x
y
z
Parameter space Physical space
),( vuX
),()),((, vuFvuXLm vu
is a partial differential operator (usually elliptic) of order m in the independent variables u and v.
mvuL ,
)),(),,(),,((),( vuzvuyvuxvuX
10
1
0),(2
2
22
2
2
vuXv
au
Usual partial differential equation:
a = ‘smoothing parameter’Solve over finite region subject to boundary conditions on function and parametric derivatives:
u
v
),0(
),0(
vX
vX
u
)1,(),1,( uXuX v
),1(
),1(
vX
vX
u
)0,(),0,( uXuX v 1
1
(1)
1u
0u
Often periodic boundary conditions are used:
1u
0u physical space
),0( vX
),1( vX
),0( vX u
),1( vX u
v
u
2
10
1u
0u
Marine Propeller (each blade a single patch)
Wine Glass (three patches)
Lipid Membranes in Two Component Systems (Doubly Periodic Lawson Surface)
When the solution is periodic, we can, in principle, express it in the form:
)sin()()cos()()(),(1
0 nvuBnvuAuAvuX n
N
nn
where 2
01
0000 )( uauauaauA nv
nnv
nnv
nnv
nn ueaueaueaeauA 4321)(
nvn
nvn
nvn
nvnn uebuebuebebuB 4321)(
IN THIS SITUATION WE CAN FIND AN ANALYTICAPPROXIMATION TO THE SURFACE, AND THIS RESULTSIN A VERY FAST METHOD FOR GENERATING AND REGENERATING A PDE.
HOWEVER, WHEN WE NEED A FOUR-SIDED PATCH,WE ARE BASICALLY FACED WITH SOLVING THE BIHARMONIC EQUATION IN A RECTANGULAR DOMAIN
024
4
22
4
4
4
v
x
vu
x
u
x
),(ufx
),(vgx
),(uFv
x
),(vGu
x
,, buav
,, avbu
THE SOLUTION OF TWO-DIMENSIONAL BIHARMONICEQUATION IS A CLASSICAL PROBLEM, WITH MANY APPLICATIONS IN MECHANICS,
E.G.
• CREEPING, VISCOUS FLOW IN A RECTANGULAR CAVITY
• EQUILIBRIUM OF ELASTIC MEMBRANE
• BENDING OF CLAMPED THIN ELASTIC PLATE SUBJECT TO A NORMAL LOAD.
ACCORDING TO MELESHKO (1998) ‘IT REPRESENTS ABENCHMARK PROBLEM FOR VARIOUS ANALYTICAL AND NUMERICAL METHODS’.
WE COULD USE A STANDARD NUMERICAL METHODSUCH A FINITE-DIFFENCE OR FINITE-ELEMENT.
BUT THE SURFACE PATCH WOULD BE REPRESENTEDDISCRETELY.
WE ARE SEEKING A FAST METHOD OF SOLUTION THAT PRODUCES A CONTINUOUS, ANALYTICALAPPROXIMATION TO THE SURFACE.
BEFORE DEALING WITH 3 & 5 SIDED PATCHES,
LET US CONSIDER A 4 SIDED PATCH:
LET BE A REGULAR 4-SIDED PATCHBOUNDED BY 4 REGULAR SPACE CURVES SUCH THAT:
),( vuX
)(4),(2),(3),(1 ufufvfvf
)(4)1,(
)(3),1(
)(2)0,(
)(1),0(
ufuX
vfvX
ufuX
vfvX
u
v
),( vuX)(2 vf
)(4 vf
)(3 vf
)(1 uf 1
1
APPROACH: WE SEEK AN ANALYTIC APPROXIMATION OF THE FORM:
),(),(),(),( vuXrvuXcvuXpvuX
WHERE
),( vuXpREPRESENTS THE SUM OF SEPARABLE EIGENSOLUTIONS OF THE 4-ORDER OPERATOR OF EQ (1)
)()exp( uv
),( vuXcREPRESENTS A POLYNOMIAL SOLUTION OF EQ (1) THAT TO ENSURE THAT CORNER CONDITIONS ARE SATISFIED
),( vuXr (SMALL) REMAINDER TERM TO ENSURE CONTINUITY AT PATCH BOUNDARIES
BOUNDARY CONDITIONS:
Positional continuity at the corners implies:
)1(1)0(4
)1(4)1(3
)0(3)1(2
)0(2)0(1
ff
ff
ff
ff
Boundary conditions on normal derivatives:
)(4)1,(
)(3),1(
)(2)0,(
)(1),0(
ufvuX
vfuvX
ufvuX
vfuvX
v
u
v
u
Note: functions on RHS may be chosen to ensure tangent-plane continuity with adjacent patches.
Regularity at corners implies :
)0(4)1(1
)1(3)1(4
)1(2)0(3
)0(1)0(2
)1(1)0(4
)1(4)1(3
)0(3)1(2
)0(2)0(1
u
v
u
v
v
u
v
u
ffu
ffv
ffu
ffv
ffv
ffu
ffv
ffu
)(1 ufu)(2 vf
u
)0,0( u
v
Continuity of twist vectors at corners implies :
)1(1)0(4
)1(4)1(3
)0(3)1(2
)0(2)0(1
vu
uv
vu
uv
fufv
fvfu
fufv
fvfu
WE NOW SEEK A POLYNOMIAL ‘CORNER’ SOLUTION OF THE FORM:
)(6
0 0),( inn
n
n
i kc vuAvuX
WHICH SATISFIES THE 12 CORNER CONDITIONS:
)1(2)0,1(
)0(3)0,1(
)1(2)0,1(
)0(2)0,0(
)0(1)0,0(
)0(1)0,0(
fvXc
fuXc
fXc
fvXc
fuXc
fXc
v
u
v
u
)0(4)1,0(
)1(1)1,0(
)0(4)1,0(
)1(4)1,1(
)1(3)1,1(
)1(3)1,1(
fvXc
fuXc
fXc
fvXc
fuXc
fXc
v
u
v
u
(Note 28 vector constants to be determined)KA
AND WHICH MATCHES THE 4 TWIST VECTORS:
)1(1)1,0(
)1(3)1,1(
)0(3)0,1(
)0(1)0,0(
vuv
vuv
vuv
vuv
fuXc
fuXc
fuXc
fuXc
AND WHICH IS A SOLUTION OF EQ (1):
0),(2
2
22
2
2
vuXcv
au
THIS GIVES 22 CONDITIONS WITH WHICH TOFIND THE 28 KA
THE REMAINING 6 ARE OBTAINED FROM THECONDITIONS:
)0(2)0,0(
)0(1)0,0(
)0(4)1,0(
)1(4)1,1(
)1(2)0,1(
)0(2)0,0(
uuuuuu
vvvv
uuuu
uuuu
uuuu
uuuu
fXc
fXc
fXc
fXc
fXc
fXc
FINDING THE EIGENSOLUTION
The eigensolution is defined by
),( vuXp
),( vuXp
),(),(),( vuXcvuXvuXp
and satisfies Eq (1) and also the modified (homogenous) boundary conditions:
)1,()(4)1,(
),1()(3),1(
)0,()(2)0,(
),0()(1),0(
uXcufuXp
vXcvfvXp
uXcufuXp
vXcvfvXp
)1,()(4)1,(
),1()(3),1(
)0,()(2)0,(
),0()(1),0(
uXcufvuXp
vXcvfuvXp
uXcufvuXp
vXcvfuvXp
vv
uu
vv
uu
Important to note that
are all zero at the 4 corners of the patch
Xpu
Xpv
Xpuv
Xp
LOOK FOR A SEPARABLE SOLUTION OF THE FORM
)()exp( uv
THAT SATISFIES THE ABOVE HOMOGENEOUS BOUNDARY CONDITIONS
IT TURNS OUT THAT IS OF THE FORM )(u
)cos(sin)sin()sincos()sin(sin)( uuuuuu
A SO-CALLED PAPKOVICH-FADLE FUNCTION
WHERE SATIFIES THE EIGENVALUE EQUATION
22sin
(WITH COMPLEX ROOTS)
THUS IS OF THE FORM),( vuXp
n nnnnnn vCvBuvuXp )exp()exp(),(Re),(
m mmmmmm uEuDv )exp()exp(),(Re
WHERE ARE CONSTANTS
DETERMINED FROM THE BOUNDARY CONDITIONSBY A LEAST-SQUARES FIT
nB nC nD nE
Note that in practice we truncate the above series so that
N
n
tr etcvuXpvuXp Re),(),(
NOW OUR APPROXIMATE SOLUTION IS GIVEN BY
),(),(),( vuXcvuXpvuX tr
WHICH IS APPROXIMATE IN THE SENSE THAT BOUNDARY CONDITIONS ARE NOT EXACTLY SATISIFIED AT ALL POINTS ON BOUNDARIES.
TO ENSURE GEOMETRIC CONTINUITY ADD INA REMAINDER TERM THUS
),(),(),(),( vuXrvuXcvuXpvuX tr
TO MAKE SURE THAT SATISFIES THEBOUNDARY CONDITIONS.
),( vuX
),( vuXp
NOTE THAT IN THIS WORK IT IS CONVENIENT TO CHOOSE TO BE A COON’S PATCH.),( vuXr
NOTE THAT AS THE NUMBER OF TERMS N INCLUDED
IN THE SERIES FOR INCREASES, THEN
GENERALLY DECREASES.
),( vuXp tr
),( vuXr
NOTE THAT WE HAVE AN ANALYTIC EXPRESSIONFOR EVERYWHERE.),( vuX
EXAMPLE:
Section of blend between circular cylinder and a
flat plane at to the cylinder axis4
7N510),(),( vuXvuX exact
Second example of approximation to 4-sided PDE surface patch
Corresponding polynomial corner solution ),( vuXc
Third example of approximation to 4-sided PDE surface patch
PROCEED BY ASSUMING THAT PATCH IS PRODUCEDBY MAPPING FROM RECTANGULAR REGION OF PARAMETER SPACE AS BEFORE.
AND THAT 4 OF THE 5 VERTICES COINCIDE WITH THE CORNERS OF
WITHOUT LOSS OF GENERALITY CHOOSE THE FIFTHVERTEX TO LIE ALONG U=1, THUS:
v
)(2 vf
)(4 vf
)(3 vf
)(1 uf 1
1
),1( vs
u
)(4)1,(
)(3),1(
)(2)0,(
)(1),0(
ufuX
vfvX
ufuX
vfvX
Positional boundary conditions along edges as before:
where is continuous in v but may have a discontinuous derivative at singularity. Derivative conditions as before:
)(3 vf
)(4)1,(
)(3),1(
)(2)0,(
)(1),0(
ufvuX
vfuvX
ufvuX
vfuvX
v
u
v
u
Where could be discontinuous at singularity. )(3 vfu
ASSUME THAT ALL CONDITIONS ON THE FUNCTION
AND ITS DERIVATIVES AT CORNERS OF
HOLD AS FOR THE 4-SIDED PATCH
NOW LOOK FOR A SINGULARITY SOLUTION
WHICH WILL GIVE THE FORM OF THE SOLUTION
IN THE NEIGHBOURHOOD OF THE SINGULARITY
),( vuXs
USING LOCAL POLAR COORDINATES
cos
sin1
rvsv
ru
EQUATION (1) SATISFIED BY BECOMES),( vuXs
0),(11
2
2
2
22
2
vuXsrrrr
r ),1( vs
DENOTING COORDINATE(S) WITH SINGULARITY
LOOK FOR SOLUTION OF THE FORM
)( fr
EXPAND BOUNDARY CONDITIONS ABOUT (1,vs)FOR SMALL VALUES OF ALONG ANDr 0
THE VALUE OF DETERMINED FROM DEPENDENCE OF BOUNDARY CONDITIONS, AND THE 4 ARBITRARY CONSTANTS IN CAN BE FIXED FROM THE 4 BOUNDARY CONDITIONS
r
)(f
REPEAT FOR ALL COORDINATES WITH A SINGULARITY
TO FIND THE COMPLETE LOCAL SOLUTION ),( vuXs
NOW INTRODUCE A SOLUTION DEFINED BY),( vuXm
),(),(),( vuXsvuXvuXm
WHICH SATISFIES MODIFIED BOUNDARY CONDITIONS
)1,()(4)1,(
),1()(3),1(
)0,()(2)0,(
),0()(1),0(
uXsufuXm
vXsvfvXm
uXsufuXm
vXsvfvXm
)1,()(4)1,(
),1()(3),1(
)0,()(2)0,(
),0()(1),0(
uXsufvuXm
vXsvfuvXm
uXsufvuXm
vXsvfuvXm
vv
uu
vv
uu
NOTE THAT SATISFIES EQ (1) AND IS REGULAR.
THUS CAN BE FOUND BY WRITING
AND USING THE METHOD FOR THE 4-SIDED PATCH, I.E.
),( vuXm
),( vuXm
),(),(),(),(),( vuXrvuXcvuXpvuXsvuX tr
),(),(),(),( vuXrvuXcvuXpvuXm tr
EXAMPLE:
)4/)cos(3(,0,0()(3
15.0)0),sin(),(cos(0)(1
5.00))cos(5.01(),sin(),(cos()(3
)),sin(),(cos()(3
)0),sin(),(cos(2)(1
vhvfu
vvvsvfu
vvhvvvf
hvvvf
vvvf
)(4),(2),(4),(2 ufvufvufuf are cubics
chosen so that consistency conditions are satisfied at corners
Singularity at (1,0.5)
FOLLOWING METHOD OUTLINED ABOVE, AND
IDENTIFYING WITH Z COORDINATE,
THE FOLLOWING BOUNDARY CONDITIONS ON APPLY:
hr
hr
hr
rhhr
1
01
),(2
)0,(
A solution for can be found
WhereHence the solution can be found
)( rf)cos()sin()cos()sin()( DCBAf
),( vuXm
EXAMPLE:
5 -sided patch with remainder term.
7N
5 -sided patch without remainder term.
7N
Polynomial Corner Solution
Singularity Solution
PROCEED AS BEFORE USING PARAMETRICMAPPING FROM 4-SIDED DOMAIN
)(4)1,(
)(3),1(
)(2)0,(
)(1),0(
ufuX
vfvX
ufuX
vfvX
u
v
)(2 vf
)(4 vf
)(3 vf
)(1 uf 1
1
CONSTANT
Boundary conditions on normal derivatives as before:
)(4)1,(
)(3),1(
)(2)0,(
)(1),0(
ufvuX
vfuvX
ufvuX
vfuvX
v
u
v
u
But note not readily available from adjacent patches and so must be chosen with care to satisfy regularity conditions on parametric derivatives.
)(3 vfu
OTHERWISE PROCEED AS BEFORE TO SEEK SOLUTIONOF THE FORM
),(),(),(),( vuXrvuXcvuXpvuX tr
EXAMPLE:
3 -sided patch with remainder term.
7N