WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

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