WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

TOPOLOGY OPTIMIZATION WITH PADÉ APPROXIMANTS 1609
A set of Taylor expansion coefficients are defined: ui = (1/i!)u (′ i)
0 , so that Equation (14) can be
written as
u = u0 + Nt
ui i
(15)
i=1
In order to find ui, Equation (15) is inserted into Equation (5)
(P0 + P1 1 + P2 2
) u0 + Nt
ui i
= f (16)
and the terms are matched by the order of the de-tuning parameter . This gives the following set
of equations:
i=1
P0u0 = f (17)
P0u1 =−P1u0
(18)
P0ui =−P1ui−1 − P2ui−2, i = 2, Nt (19)
The zeroth order equation (Equation (17)) is equivalent to the original Equation (5) for = 0. It is
assumed that this equation is solved by a factorization method so that P0 is available in factorized
form. The benefit of this is evident from Equations (18)–(19) since the coefficients u1, u2 and so
forth can be computed simply by forward-/back substitutions. The factorized P0 will be used also
to compute design sensitivities.
2.2. Coefficients ai and bi
The combination of Equations (15) and (13) can now be utilized to find the unknown PA coefficients:
u0 + Nt
i=1
u0 + Nt
ui
i=1
i
ui i = u0 + N i=1 ai i
1 + N i=1 bi i
Both sides of Equation (20) are multiplied with the denominator
1 + N
= u0 + N
and terms are matched by the order of i
bi
i=1
i
ai
i=1
i
(20)
(21)
a1 = u1 + b1u0
(22)
ai = ui + i−1
b jui− j + biu0, i = 2,...,N (23)
j=1
from which the ai coefficients can be computed if the bi coefficients are known. Equating the
terms of order N+1 gives
0 = uN+1 + N
(24)
in which the bi coefficients are the only unknowns.
b juN+1− j
j=1
Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 72:1605–1630
DOI: 10.1002/nme