sci.math.numanalysis
SUNA04, Five Point Star
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Consider the well known five point star. Function values (f1,f2,f3,f4,f5) are
defined at its nodal points. These values can be interpolated
5 by a (Finite Element) polynomial, which is defined by:

 f = a1 + a2.x + a3.x.x + a4.y + a5.y.y
213
 Specify for the nodes: (0,0), (1,0), (+1,0), (0,1), (0,+1).
 This results in 5 equations for the unknowns (a1,a2,a3,a4,a5):
4
f1 = a1 ; f2 = a1  a2 + a3 ; f3 = a1 + a2 + a3
f4 = a1  a4 + a5 ; f5 = a1 + a4 + a5
From which it follows that: a1 = f1 ; a2 = (f3  f2)/2 ; a4 = (f5  f4)/2
a3 = (f3  2.f1 + f2)/2 ; a5 = (f5  2.f1 + f4)/2
These are well known finite difference schemes for the zero'th, first and second
partial derivatives on the five node star.
By substitution of the a's, the function f is expressed as follows:
f = f1 + (f3  f2)/2.x + (f3  2.f1 + f2)/2.x.x +
+ (f5  f4)/2.y + (f5  2.f1 + f4)/2.y.y
Rewrite this:
f = (1  x.x  y.y).f1 + 1/2.( x + x.x).f2 + 1/2.(+ x + x.x).f3
+ 1/2.( y + y.y).f4 + 1/2.(+ y + y.y).f5
Sic! Here we find the (Finite Element) Shape Functions of the five point star.
They are: N1(x,y) = 1  x.x  y.y
N2(x,y) = 1/2.( x + x.x) ; N3(x,y) = 1/2.(+ x + x.x)
N4(x,y) = 1/2.( y + y.y) ; N5(x,y) = 1/2.(+ y + y.y)
It is necessary to introduce now local coordinates (h,k) instead of (x,y).
Then, a (Finite Element) Isoparametric Mapping can be defined as follows:
x = N1(h,k).x1 + N2(h,k).x2 + N3(h,k).x3 + N4(h,k).x4 + N5(h,k).x5
y = N1(h,k).y1 + N2(h,k).y2 + N3(h,k).y3 + N4(h,k).y4 + N5(h,k).y5
The partial derivatives dx/dh, dx/dk, dy/dh, dy/dk are calculated:
dx/dh = (1/2  h).(x1  x2) + (1/2 + h).(x3  x1)
dy/dh = (1/2  h).(y1  y2) + (1/2 + h).(y3  y1)
dx/dk = (1/2  k).(x1  x4) + (1/2 + k).(x5  x1)
dy/dk = (1/2  k).(y1  y4) + (1/2 + k).(y5  y1)
Jacobian determinant: J = (1/2  h)(1/2  k).J1 + (1/2 + h)(1/2  k).J2
+ (1/2  h)(1/2 + k).J3 + (1/2 + h)(1/2 + k).J4
Where: J1 = (x2  x1).(y4  y1)  (x4 x1).(y2  y1)
J2 = (x4  x1).(y3  y1)  (x3 x1).(y4  y1)
J3 = (x5  x1).(y2  y1)  (x2 x1).(y5  y1)
J4 = (x3  x1).(y5  y1)  (x5 x1).(y3  y1)
The shape functions of the five point star's Jacobian are therefore identical
to those of a (Finite Element) quadrilateral: see for example O.C. Zienkiewicz'
"The Finite Element Method" or the previous poster on "Convex quadrilaterals".
This quadrilateral defines an area inside the star as indicated in the figure:
5 These area's "happen" to be ADJACENT when the F.D.
 stars are combined into a (curvilinear) grid.
J3________J4
   The determinants J1,J2,J3,J4 denote the area's of
2________1________3 triangles which are spanned by the star's "arms".
   For the *inverse* of an isoparametric mapping to
________ exist, it is required that the Jacobian J has a
J1  J2 uniform (positive) sign. Since J is in general a
 bilinear interpolation of the Jk's (k = 1,2,3,4),
4 it is necessary and sufficient that the latter are
positive: J1 > 0 , J2 > 0 , J3 > 0 , J4 > 0 .
The geometrical meaning of this is that a five point star must not be distorted
in such a rigourous way that its arms "overcross" each other.
Performing the isoparametrics for an arbitrary function f (partial derivatives):
df/dx = dh/dx.df/dh + dk/dx.df/dk = (dy/dk.df/dh  dx/dk.df/dk) / J
df/dy = dh/dy.df/dh + dk/dy.df/dk = ( dy/dh.df/dh + dy/dk.df/dk) / J
The derivatives d(x,y)/d(h,k) and J were calculated above.
If these formulas are specified for the special case that (h,k) = (+1/2,+1/2)
or any other of the "inscribed" quadrilateral nodes, then they reduce to well
known (Finite Element) schemes for taking derivatives at a triangular element:
Re: SUNA, Triangle Isoparametrics.
This means that a devising a special Finite Element Method for five point stars
in a curvilinear grid is rather pointless: such a method would NOT behave very
differently from an conventional F.E. method on ordinary triangles! I feel that
the above is already a nice demonstration of the Unification Principle at work
(Re: SUNA, The Manifesto).
Disclaimer: the above stuff was published earlier in this group. I nevertheless
decided to post it, in order to preserve continuity with past & future.
To be continued ...

* Han de Bruijn; Applications&Graphics  "A little bit of Physics * No
* TUD Computing Centre; P.O. Box 354  would be NO idleness in * Oil
* 2600 AJ Delft; The Netherlands.  Mathematics" (HdB). * for
* Email: Han.deBruijn@RC.TUDelft.NL  Fax: +31 15 78 37 87 * Blood