WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...
WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...
WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...
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1194 J. Opt. Soc. Am. B/Vol. 22, No. 6/June 2005 J. S. Jensen and O. Sigmund<br />
Hˆ e = rmin − distk,e, e Ddistk,e rmin, k =1, ..., N D . 16<br />
Thus the filtered sensitivities is a modification of the<br />
original sensitivities based on a weighted average of the<br />
sensitivities in neighboring elements within a fixed range<br />
(specified by r min). In the examples shown, we have used<br />
r min=2.5 times the element size.<br />
C. Continuation Method and Penalization<br />
The solution of the optimization problem turns out to be<br />
strongly nonunique, which results in multiple local<br />
maxima. Some of these are based on local resonance effects<br />
that lead to a poor performance away from the target<br />
frequency and to strong out-of-plane scattering and,<br />
consequently, to poor transmission for a real threedimensional<br />
structure. A way to reduce the chance of ending<br />
up in such a maximum is to apply a continuation<br />
method in which we attempt to convexify the object function<br />
by changing the original problem into a smoother<br />
one. After a converged design is obtained for the smooth<br />
problem, we gradually change the problem back into its<br />
original form during a continued optimization procedure.<br />
As shown for phononic bandgap structures, 18 the presence<br />
of strong damping smoothes the dynamic response<br />
considerably, and in Ref. 13 this was used with advantage<br />
to avoid undesired local extrema in the optimization of<br />
these structures.<br />
Similar to the case of phononics (elastic waves), we<br />
here add an artificial damping term C art to the model:<br />
C = C abs + C art, 17<br />
where C abs is the damping matrix stemming from the absorbing<br />
boundaries. We let the extra damping matrix be<br />
proportional to the mass matrix, such that<br />
C art = M, 18<br />
where is a real and positive constant. In a typical optimization<br />
procedure we start out with strong damping,<br />
say, =0.1, and then gradually decrease until the performance<br />
of the structure no longer can be improved.<br />
Even when avoiding the pitfall of resonance-based local<br />
maxima, we may still end up with an unfeasible design<br />
with elements that have nondiscrete design variables x e,<br />
i.e., values other than strictly 0 or 1. For other applications<br />
of topology optimization, various penalization methods<br />
have been developed to avoid this problem. For<br />
bandgap-type problems, the need for penalization appears<br />
to be small 13 in that the highest possible material contrast<br />
is favorable for maximum wave reflection. However,<br />
if the problem is not strictly of the reflection type, we may<br />
end up with some gray elements.<br />
Here we propose a new scheme inspired by an explicit<br />
penalization scheme introduced in Ref. 19. Instead of adding<br />
a penalization term directly to the objective, we use<br />
an implicit variant by introducing the penalty as an extra<br />
damping term for elements in D:<br />
e<br />
Cpen =4xe1−xeMe , e D, 19<br />
thereby causing elements with 0x e1 to induce an energy<br />
loss and hence be expensive for the objective function.<br />
The penalization scheme is typically employed when<br />
a converged design is obtained and gray elements have<br />
appeared. Then is gradually increased from, e.g., <br />
=0.01 until the gray elements vanish. However, if a lot of<br />
gray elements appear in the design, it can be beneficial to<br />
have a nonzero throughout the optimization procedure.<br />
In the following we call the pamping coefficient as an acronym<br />
for penalization damping.<br />
D. Algorithm<br />
The optimization algorithm is outlined in the following:<br />
1. Set up a finite-element model and choose an appropriate<br />
design domain D and target frequency .<br />
2. Choose an initial design, i.e., distribution of xe in D, and initial values of artificial damping, e.g., =0.1.<br />
3. Compute the elementwise material coefficients Ae and Be. 4. Solve system equations and compute objective func-<br />
ˆ<br />
tion J and filtered design sensitivities dJ/dxe.<br />
5. Update design with the mathematical programming<br />
tool, the Method of Moving Asymptotes. 17<br />
6. Repeat steps (3)–(5) until design convergence is<br />
maxx new−x e, where is a small positive constant.<br />
7. Decrease artificial damping, e.g., new=/2.<br />
8. Repeat steps (3)–(7) until the performance is no<br />
longer improved.<br />
9. If gray elements appear in the design, repeat steps<br />
(3)–(5) with increasing values of , starting from, e.g., <br />
=0.01. Alternatively, carry out the whole optimization<br />
procedure with a nonzero .<br />
3. S<strong>IN</strong>GLE-FREQUENCY OPTIMIZATION<br />
We now use the basic algorithm to design the T junction<br />
for maximum transmission at three separate frequencies.<br />
Figure 2 shows the design domain. A domain consisting of<br />
ten unit cells is chosen, but the choice is arbitrary, and it<br />
Fig. 2. Design domain D and initial material distribution.