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