a analytic pseudo-spectral method for 3- and 5-sided surface patches
DESCRIPTION
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 PresentationTRANSCRIPT
![Page 1: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/1.jpg)
![Page 2: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/2.jpg)
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
![Page 3: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/3.jpg)
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)
![Page 4: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/4.jpg)
1u
0u
![Page 5: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/5.jpg)
Often periodic boundary conditions are used:
1u
0u physical space
),0( vX
),1( vX
),0( vX u
),1( vX u
v
u
2
10
![Page 6: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/6.jpg)
1u
0u
![Page 7: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/7.jpg)
![Page 8: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/8.jpg)
Marine Propeller (each blade a single patch)
![Page 9: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/9.jpg)
Wine Glass (three patches)
![Page 10: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/10.jpg)
![Page 11: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/11.jpg)
Lipid Membranes in Two Component Systems (Doubly Periodic Lawson Surface)
![Page 12: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/12.jpg)
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)(
![Page 13: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/13.jpg)
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
![Page 14: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/14.jpg)
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’.
![Page 15: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/15.jpg)
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.
![Page 16: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/16.jpg)
![Page 17: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/17.jpg)
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
![Page 18: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/18.jpg)
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
![Page 19: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/19.jpg)
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.
![Page 20: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/20.jpg)
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
![Page 21: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/21.jpg)
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
![Page 22: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/22.jpg)
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
![Page 23: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/23.jpg)
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
![Page 24: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/24.jpg)
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
![Page 25: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/25.jpg)
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
![Page 26: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/26.jpg)
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
![Page 27: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/27.jpg)
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)
![Page 28: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/28.jpg)
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),(),(
![Page 29: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/29.jpg)
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
![Page 30: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/30.jpg)
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
![Page 31: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/31.jpg)
EXAMPLE:
Section of blend between circular cylinder and a
flat plane at to the cylinder axis4
7N510),(),( vuXvuX exact
![Page 32: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/32.jpg)
Second example of approximation to 4-sided PDE surface patch
Corresponding polynomial corner solution ),( vuXc
![Page 33: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/33.jpg)
Third example of approximation to 4-sided PDE surface patch
![Page 34: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/34.jpg)
![Page 35: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/35.jpg)
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
![Page 36: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/36.jpg)
)(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
![Page 37: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/37.jpg)
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
![Page 38: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/38.jpg)
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
![Page 39: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/39.jpg)
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
![Page 40: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/40.jpg)
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
![Page 41: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/41.jpg)
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
![Page 42: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/42.jpg)
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)
![Page 43: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/43.jpg)
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
![Page 44: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/44.jpg)
EXAMPLE:
5 -sided patch with remainder term.
7N
![Page 45: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/45.jpg)
5 -sided patch without remainder term.
7N
![Page 46: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/46.jpg)
Polynomial Corner Solution
![Page 47: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/47.jpg)
Singularity Solution
![Page 48: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/48.jpg)
![Page 49: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/49.jpg)
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
![Page 50: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/50.jpg)
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
![Page 51: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/51.jpg)
EXAMPLE:
3 -sided patch with remainder term.
7N
![Page 52: A Analytic Pseudo-Spectral Method for 3- and 5-sided Surface Patches](https://reader036.vdocument.in/reader036/viewer/2022062517/5681332c550346895d9a25b2/html5/thumbnails/52.jpg)