Fractal basis functions and the CVBEM

Engineering Analysis with Boundary Elements 20 (1997) 337-339 © 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0955-7997/97/$17.00

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Research Note Fractal basis functions and the CVBEM T.V. Hromadka II a'* & R.J. Whitley b aDe~artment of Mathematics, California State University, Fullerton, California 92634, U.S.A. Department of Mathematics, University of California, lrvine, California 92616, U.S.A. (Received 21 March 1997; accepted 6 June 1997)

The Complex Variable Boundary Element Method or CVBEM has recently been applied to the use of new fractal basis functions for defining a global trial function on the problem domain. In that recent advance, a topic for future research was the need for development of an algorithm to construct a global trial function that converges to the true problem boundary conditions, assumed to be continuous on the boundary. In this paper, such an algorithm for constructing a sequence of fractal basis functions is presented. © 1998 Elsevier Science Ltd. All rights reserved. 1 INTRODUCTION

where f(z) is a complex function defined over a domain fl containing F, A~'y= ~'i - ~'J- i; and q0 is a point in I' in the arc from ~'j_ l to ~':. For an integrable function f, for example f continuous on F, the following limit exists:

The Complex Variable Boundary Element Method (CVBEM) is a complex variable function approximation method that develops an approximator, &(z), that is composed of a pair of two-dimensional functions, go(z) = dp(z) + i(b(z) where ~(z) and ¢(z) are both harmonic conjugate functions and hence exactly satisfy the Laplace equation. Several papers and books (e.g. Ref. l) develop the CVBEM in detail. The focus of this note is the development of a mathematical equation that links a function, o~(z), analytic in a simply connected domain, fl, with a simple closed boundary contour, F, to a series expansion of ~0(z) using fractal basis functions. The series expansion includes a direct representation of a discretization algorithm which is more explicit than that of Hromadka and Whitley. 2

/(z) dz

(2)

independent of the choice of points ~'j and 7j, where k is the maximum value of the lengths IIA~'j[I,j = 1,2 .... , n. If F is smooth, being composed of m smooth curves I ' t , r 2 ..... I'm, joined end to end (i.e. a piecewise smooth curve), then

le,z, (3)

1.1 Complex function integrals 1.2 Fractal basis functions and bisection partitioning of boundary I'

The boundary curve F, of length L, with parametric equation ~"= ~'(t), 0 --< t ~ 2, will be assumed to be smooth with g'(t) having a continuous nonzero derivative ~"(t) at every point 0
Let Zr and zs be two successive points, in the usual positive direction, on r . Then a linear fractal basis function is defined, on the arc Crs, of F, bounded by zr and zs, by

n

Sn = Z f ( ~ j ) A ~ j

(1)

o;

t~c,~

( ~ - ~).

t E c.

rs

j=l

t-----~)-~" (~'~~ Cts i s - it'

*Corresponding author. 337

(4)

T. V. Hromadka, R. J. Whitley

338

The extension of a linear fractal basis function to a constant, parabolic, cubic, or other basis function is similar. A sequence of successive bisection of arcs in 17 can be used to add partition points on 17, and thereby define a set of fractal basis functions. First, a reference point, ~'j, is located on F. The arc C~1 is all of 17, beginning and ending at point ~'b transversed in the usual counterclockwise direction. The first fractal basis function is denoted by 2

(~-)

I I

where point ~'~ serves as both the beginning point and endpoint of arc Cib with midpoint ~'2 ~ I' (the midpoint of arc C~ is located a distance fi~/2 from ~'~, as measured along C~ ~, where f ll is the arc length of C~ 1). The second pass in partitioning I" results in locating two more points ~') and ~'~ in 17 where ~'3 is the midpoint of arc C~2, and ~'4 is the midpoint of arc C2~. The corresponding fractal basis functions are

For the bisection partitioning algorithm, the sequence of successive partitioning passes result in the global fractal function. Given the initial point ~'1 E l " with value ('01 = CO(~'I )' zc 2~ I G(~') =COl d- E E k=lu=l

t ~trs ~ (~')

(6)

where k is the bisection algorithm partition pass number; for this k-th pass, 2 k-I points are added, namely all the points halfway between the points already in the partition; dr, ds, dt are the arc distances, along 17, in the positive direction, from initial point ~'l @ F to partition points ~'r, ~'s, ~'~, given by dr = (u - I)L/2 ~- I, ds = uL/2 k - 1, dt = ( 2 u - I)L/2 k, respectively; and 6tr.~ is the weighting defined by 6'r~= co(~',) - ¢°(~'r) + (~'0 2

(7)

After k0 passes the remainder of the series, R~,(~'), is given by

3

A (~-) 12

zc

2k I

E

Y

t

(8)

k=/%+lu=l

and

with the truncated global fractal function

4

zx (~-), 21

ko 2~

respectively. The third pass in partitioning r results in locating four more points, ~'5, ~'6, ~'7, ~'8 at the midpoints of arcs C~3, C3:, C24, Cab respectively. The above method of partitioning 17 results in a set of partition points, each adjacent pair being at the same arclength distance from each other, for each iteration of the algorithm. Practically speaking, it is more convenient to partition [0,1 ] by 0 = to < t~ < tN < 1 and use ~'j = f(tj) as partition points on 17.

Gk,(~')=co, + X

fractal function,

(9)

k=lu=l

Suppose that co satisfies a Lipschitz condition on 17: Ico(zj) - co(z2)l -< MIz~ - z21

(10)

for some constant M and all z l and z2 on 17. This will be true, for example, if the derivative w' exists and is continuous, and therefore bounded on I'. In this case the sum 2~ I

1.3 Global

t

X 6'r, ~ (~').

t

s= uZ= l 'rs rs

G(~-)

Consider a continuous function, co(z), defined on a domain f~ that contains a piecewise smooth curve, 17. For any partition of F, by n points ~'l, ~'e..... ~', placed successively in the positive direction along I', a global trial function, G(~'), can be defined for ~"E 17 by a sum of fractal basis functions weighted by point values of co(z):

(ll)

has only two non-zero terms; the factor t

,a (~-) rs

is bounded in absolute value because the curve is smooth I

I ~(~-)1. _B G(~') =

Ik- ~'+ l (~')1cok

(5)

where wk = W(fk), with coo = c0,. This fractal basis function is analogous to the usual linear polynomial basis function approximation, but in the usual approach, all points ~'k-i, ~'k, ~'k+l are defined initially. Another approach is to use an algorithm that describes a sequence of successively finer partitioning of F, where the fractal basis function is weighted by the difference between the newly added partition point value, co(~'0, and, for the linear basis function, the linear approximation (co(~'0 + (co(~'~))/2.

(12)

and 16tr~l ~

CO(~t)~CO(~'r) +

I

co(~t)~co(~s) _< M 2k - I

(13)

hence

BM ISI ~ _k---~T

(14)

and so the series for Gko(~') converges absolutely and uniformly on r .

339

Fractal basis functions and the CVBEM

given by eqn (1) is

1.4 C o m p l e x V a r i a b l e B o u n d a r y E l e m e n t M e t h o d (CVBEM)

The CVBEM approximator, &(z), defined inside the problem domain, fl, with boundary, F, is 1 f G(~')d~" ~(z) = 2~iJr ~---~' z E fl

(15)

If o~(z) is analytic on fl U r, then for z E fl, 1 o~(z)=~

r

Gk°(~')d~" ~- 1 S Rko(~')d~', z E ~2

~'-z

~

r

g'-z

~ ~ c~s

(f--~'r). ( ~ ' , - ~'r)'

~. E Crt

~______)). (~s-

~"~ c,s

( ~ ' s - ~'t)'

where Crs = Ca O Cts, and C~t N Ct, = ~'t. Consequently, for z ~ fl, (16)

The typical CVBEM application involves a problem domain where increased accuracy can be obtained by irregular placement of nodes, an aspect which we will now neglect. The CVBEM approximation can be expanded for z ~ f~ by

= ~o~ Sr Gk.(f)d(f) if f- z 2TiJr f - z

1 ~/

)k ( ~ ' ) : rs

0;

t

c,, ~

z = \ ~'t - g'L] (ln(~'t - z) - ln(~'r - z)) Z - - ~'s

-- ( ~--S~__ ~t) (ln(~'s -- z) -- ln(~'t -- z)) where the logarithm used must have its branch cut prescribed as described in Ref. 2.

!

2 CONCLUSIONS

k__l.=l -~ ~'--Z

~r/ r

t

+ ~ . ~.. ~r~Jc~ ~-S_z k=lu=l

(17)

due to t

A(~') = 0 for ~'~ Cr~ The first integral in the eqn (17) expansion is 2~ri. To integrate

A new algorithm for constructing a global trial function by a sequence of fractal basis functions is presented. The series converges to the true problem boundary conditions, under mild conditions of continuity. Because of the series representation structure, error evaluation of the approximation is readily achieved by using standard series evaluation techniques, given that the solution to the boundary value problem satisfies a Lipschitz condition on the problem boundary (see eqns (8) and (14)).

t

REFERENCES

consider the typical situation that C~ is a line segment connecting ~'~ and ~'~.Then, ~'t = (~'r + ~'s)/2, and l

~(~')

1. Hromadka II, T.V., The Best Approximation Method in Computational Mechanics. Springer-Verlag, 1993. 2. Hromadka II, T.V. and Whitley, R.J. Expansion of the CVBEM into a series using fractals. Engineering Analysis and Boundary Elements, 1993, 12, 17-20