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