WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

which leads to the expressions
TOPOLOGY OPTIMIZATION WITH PADÉ APPROXIMANTS 1619
c 1 c
= + i
ū 2 uR
c
=
uI
1
c
u Di = 1
c
ū Ēi = 1
N
N k=1
N
N k=1
N
N k=1
ū T LDi
u T LĒi
u T L (61)
Large savings in computational time and memory use are obtained if Di and Ei are not explicitly
computed. Instead, the formulas for Di and Ēi are inserted into Equations (62)–(63)
c
u D0 = 1
N
N k=1
c
u DN+1 =− 1
N
N k=1
(62)
(63)
ū T L (64)
1
B
N
(ū
j=1
T Lũ j)u + j
in which B = 1 + N i=1 bi i is the denominator of the PA and
c
u Di = 1
N
N k=1
c
ū Ēi =− 1
N
N k=1
1
B ( ˜bi(ū T L) + u +
1
¯B
N
j=1
N
N+1−i uN+1
j=1
¯Q j,N+1−i(ū T Lũ j)¯h T
for i = 1, N. In Equation (66)–(67) the following symbols have been introduced
h T = u T N
N+1 −
˜Q T j
= N
Q j,N+1−lu
l=1
∗ l
u
l=1
+ l uN+1u T N+1−l
(65)
(ū T Lũ j) ˜Q T j ) (66)
which are independent of the frequency and must be computed only once for each optimization
iteration. This is also the case for the scalar u +
N+1−i uN+1. The following quantities that appear in
Equations (64)–(67) should be computed for each frequency evaluation point k:
˜bi = N
b j−i
j=i
j
ũ j = N
ul− j l − u j
l= j
Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 72:1605–1630
DOI: 10.1002/nme
(67)
(68)
(69)
(70)
(71)