WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

1618 J. S. JENSEN
which is solved for kN+1, and then carrying on to solve the remaining equations
P T 0 kN
c
=−
u DN
T
c
−
ū ĒN
T
− P1kN+1
(55)
P T 0 ki
c
=−
u Di
T
c
−
ū Ēi
T
− P1ki+1 − P2ki+2, i = N − 1, 0 (56)
It is noted that with P0 already factorized all Lagrangian multipliers can be computed by forward-
/back-substitutions.
With ki given from Equations (54)–(56) the final expression for c ′ becomes
c ′ = c ′ c
0 =
xi
+ 2Re
N+1
(k
i=0
T i P′ 0 + kT i+1P′ 1 + kT i+2P′ 2 )ui
4. TOPOLOGY OPTIMIZATION OF FORCED VIBRATIONS
A formulation has been developed that allows for the design sensitivities to be computed without
overwhelming computational effort regardless of the number of design variables. This makes the
present method well suited for optimization problems with many design variables such as in
topology optimization [19]. In a standard topology optimization implementation with the density
approach, a single design variable is used for each finite element in the model, typically leading
to thousands of variables. Therefore, a fast sensitivity analysis procedure is essential for practical
applications.
Two forced vibration problems are considered as application examples. Topology optimization
of forced vibrations has been studied previously, e.g. by [1, 19–22]. All studies considered the
dynamic performance at single or multiple excitation frequencies. With the proposed method the
performance of the structure can be optimized in entire frequency ranges at little extra cost.
The average response in the frequency range from 1 to 2 is considered:
c =
1
2 − 1
2
u T Lū d ≈ 1
1
N
N k=1
(57)
u T Lū (58)
in which L is a diagonal matrix that is used to indicate the degrees of freedom that are considered in
the optimization. The integral is well approximated by the sum if a sufficient number of evaluation
points N is used (50–100 points are typically used in the examples). The objective function is
written in terms of real and imaginary parts:
c = 1
N
N k=1
so that the complex derivatives can be computed
c 1 c
= − i
u 2 uR
c
uI
(u T R LuR + u T I LuI) (59)
= 1
N
N k=1
ū T L (60)
Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 72:1605–1630
DOI: 10.1002/nme