WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

1616 J. S. JENSEN
The derivative of c w.r.t. a single design variable xi is then
c ′ = c
xi
+ c
u
uR
′ c
R + u
uI
′ I
in which () ′ = d/dxi and the scalar c/xi and the row vectors c/uR, c/uI can be found
directly for a given objective function.
To facilitate the computation of u ′ R and u′ I it is exploited that u′ can be expressed in terms of
the derivative of the Taylor expansion coefficients ui in the following form:
u ′ = N+1
i=0
(38)
(Diu ′ i + Eiū ′ i ) (39)
The derivation of the matrices Di and Ei is lengthy and given in the Appendix.
Equation (39) is written out in terms of the real and imaginary parts u ′ R and u′ I
u ′ 1
R =
2
u ′ I =−i
2
N+1
i=0
N+1
i=0
((Di + Ēi)u ′ i + ( ¯Di + Ei)ū ′
i
and Equations (40)–(41) are inserted into Equation (38):
c ′ = c
xi
+ c
u
N+1
i=0
(40)
((Di − Ēi)u ′ i − ( ¯Di − Ei)ū ′ i ) (41)
(Diu ′ i + ¯Diū ′ i
) + c
ū
N+1
in which the derivative-like terms c/u and c/ū are defined as
c
c 1
=
u 2
uR
c 1 c
=
ū 2 uR
i=0
− i c
uI
+ i c
uI
(Ēiu ′ i + Eiū ′ i ) (42)
The adjoint approach [18] is now used to replace the terms u ′ i and ū′ i in Equation (42) with
terms that are easier to compute. The expression for c ′ is augmented with two series of additional
terms
c ′ 0 = c′ + N+1
k
i=0
T i R′ i
N+1
+
i=0
¯k T
i ¯R ′ i
where k T i are vectors of Lagrangian multipliers and Ri are the residuals defined from Equations
(17)–(19) as
(43)
(44)
(45)
R0 = P0u0 − f (46)
R1 = P0u1 + P1u0
(47)
Ri = P0ui + P1ui−1 + P2ui−2, i = 2, N + 1 (48)
Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 72:1605–1630
DOI: 10.1002/nme