WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

1620 J. S. JENSEN
Based on the expressions given in Equations (64)–(67) the adjoint fields can be computed from
Equations (54)–(56) and finally the design sensitivities from Equation (57).
4.1. Optimized reinforcement of a 2-D elastic body
As the first example, the dynamic response of the 2-D elastic body in Figure 4 (with material
properties E = 1, = 1, = 0.3) is optimized by distributing a maximum amount of 25%
reinforcement material. The material properties of the reinforcement material are:
E = 2, = 2, = 0.3
and the plane stress model with unit thickness is considered. The structure is discretized using
80 × 40 bi-linear quadratic elements and the load is applied in three nodes in the middle of the
lower boundary.
A standard Matlab implementation of the topology optimization algorithm is used. Specific
details are not included here and the reader is referred to [23] for implementation details and
to [19] for a general presentation of the method. Design updates are found with the method of
moving asymptotes (MMA) [24]. As material interpolation the RAMP-model [25] is implemented
with penalization parameter p = 5. A sensitivity filter is not used and the symmetry of the designs
is explicitly enforced by averaging the left–right sensitivities. In the initial design the 25% reinforcement
material is uniformly distributed in the entire body and 100–150 iterations have been
used in the examples.
The damping coefficients = 0.5 and = 0.005 are reused from the analysis example in
Section 2.4 with the important note that the damping is kept independent of the design by being
proportional only to the constant part of the mass- and stiffness-matrix.
The first objective is to minimize the response in point A (Figure 4), averaged over a finite
frequency range. Equation (58) becomes
c1 = 1
N
N k=1
u Aū A
by specifying L to have one unit entry in the diagonal that corresponds to the vertical amplitude
in A.
Figure 6 shows four examples of optimized material distribution. Figure 6(a) shows the distribution
when the response is minimized for a single frequency = 1, whereas the designs in
Figure 6(b)–(d) are obtained when the response is minimized for larger frequency ranges. The
number of PA expansion terms is N = 7 and the number of frequency evaluation points for the
objective function is N = 100. For the smallest optimization interval ( = 0.9−1.1) asinglePAis
used but in order to obtain good approximations for the larger intervals the number of patched PA
expansions is increased to two and three for the intervals = 0.8–1.2and = 0.7–1.3, respectively.
Figure 7 shows the PA for the initial structure and the PA for the optimized body corresponding to
the optimization interval = 0.9–1.1 (Figure 6(b)). A low response is seen in the entire optimization
interval, which is accomplished by the creation of several closely spaced anti-resonances. The
frequency responses for the initial and the optimized designs are also computed using a standard
direct approach for a high number of frequencies (thin solid lines) to illustrate the validity of the
PA approximations.
Figure 8 shows the response for the other optimized designs in Figure 6. The structures and their
corresponding responses are very dependent on the optimization interval. The structure optimized
Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 72:1605–1630
DOI: 10.1002/nme
(72)