WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

in which
TOPOLOGY OPTIMIZATION WITH PADÉ APPROXIMANTS 1627
A = u0 + N
B = 1 + N
ai
i=1
i
bi
i=1
i
Equation (A2) is differentiated w.r.t. a design variable
and by differentiating the expressions for A and B
in which the notation u ′ 0 = a′ 0
Bu ′ = N
(A3)
(A4)
Bu ′ = A ′ − B ′ u (A5)
i=0
is used for simplicity.
From Equations (22)–(23) a ′ i can be expressed as
a ′ i
= i
a ′ ii
N
− b
i=1
′ ii
u (A6)
b ju
j=0
′ i− j
+ i
b
j=1
′ jui− j
and this expression is inserted into Equation (A6) to yield
Bu ′ = N
i
b ju
i=0 j=0
′ i
i− j + b
j=1
′ jui−
j i
N
− b
i=1
′ ii
u (A8)
which after reorganizing terms is given as
Bu ′ = N
i
i + N
i
with
i=0
b ju
j=0
′ i− j
i=1
b
j=1
′ jui− j
After some index manipulation we can write this expression as
Bu ′ = N
˜biu
i=0
′ i
˜bi = N
+ N
b j−i
j=i
j
ũib
i=1
′ i
ũi = N
u j−i j − u i
j=i
− b ′ i u
i
(A7)
(A9)
(A10)
(A11)
(A12)
Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 72:1605–1630
DOI: 10.1002/nme