sci.math.num-analysis SUNA04, Five Point Star ======================= 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 2-----1-----3 | 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 * E-mail: Han.deBruijn@RC.TUDelft.NL --| Fax: +31 15 78 37 87 ----* Blood