WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

TOPOLOGY OPTIMIZATION WITH PADÉ APPROXIMANTS 1617
that all vanish at equilibrium so that c ′ 0 = c′ . Equation (45) is rewritten with Equations (46)–(48)
inserted
c ′ 0 = c′ + N+1
+ N+1
i=0
i=0
(k T i P0 + k T i+1 P1 + k T i+2 P2)u ′ i
(¯k T
i ¯P0 + ¯k T
i+1 ¯P1 + ¯k T
i+2 ¯P2)ū ′ i
N+1
+
N+1
+
i=0
(k
i=0
T i P′ 0 + kT i+1P′ 1 + kT i+2P′ 2 )ui
(¯k T
i ¯P ′ 0 + ¯k T
i+1 ¯P ′ 1 + ¯k T
i+2 ¯P ′ 2 )ūi
where kN+2 = kN+3 = 0 have been introduced for simplicity and it has been assumed that the load
f is independent of the design.
Inserting Equation (42) into Equation (49) yields
c ′ c
0 =
xi
+ N+1
i=0
+ N+1
i=0
+ c
u
N+1
i=0
(Diu ′ i + ¯Diū ′ i
) + c
ū
(k T i P0 + k T i+1 P1 + k T i+2 P2)u ′ i
(¯k T
i ¯P0 + ¯k T
i+1 ¯P1 + ¯k T
i+2 ¯P2)ū ′ i
N+1
i=0
(Ēiu ′ i + Eiū ′ i )
N+1
+
(k
i=0
T i P′ 0 + kT i+1P′ 1 + kT i+2P′ 2 )ui
N+1
+
i=0
(¯k T
i ¯P ′ 0 + ¯k T
i+1 ¯P ′ 1 + ¯k T
i+2 ¯P ′ 2 )ūi
The Lagrangian multipliers are now chosen so that the terms that involve u ′ i and ū′ i vanish. This
condition is fulfilled if
c
u
N+1
Diu
i=0
′ i
+ c
ū
N+1
Ēiu
i=0
′ i
N+1
+
i=0
(49)
(50)
(k T i P0 + k T i+1P1 + k T i+2P2)u ′ i = 0 (51)
which assures that the complex conjugate of this equation is also fulfilled and consequently that
all terms in Equation (50) with u ′ i and ū′ i disappear. The terms in Equation (51) are collected
N+1
i=0
c
u Di + c
ū Ēi + k T i P0 + k T i+1P1 + k T i+2P2
u ′ i = 0 (52)
which holds if the following conditions for ki are satisfied:
k T i P0 + k T i+1 P1 + k T i+2 P2 =− c
u Di − c
ū Ēi, i = 0, N + 1 (53)
Equation (53) is conveniently solved for ki by starting with i = N + 1
P T 0 kN+1
c
=−
u DN+1
T
c
−
ū ĒN+1
T
Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 72:1605–1630
DOI: 10.1002/nme
(54)