Maria Bayard Dühring - Solid Mechanics

3.3 The optimization problem 15
3.3 The optimization problem
The purpose of the topology optimization is to optimize the objective function Φ. For
the three wave propagation problems considered Φ has a similar form and depends
on the field variables uj or their derivatives with respect to xj, which in both cases
will be denoted wj. The functions wj are in general complex valued so in order to
get a positive and real valued objective function it is the squared absolute value of
wj or a sum of these that is optimized. The generic formulation of the optimization
problem where the response has to be minimized takes the form
min
ξ
subject to
Φ =
Ωo j=1
J
|wj(r, ξ(r))| 2 dr, objective function (3.3)
1
Ωd dr
ξ(r)dr − β ≤ 0,
Ωd
volume constraint (3.4)
0 ≤ ξ(r) ≤ 1 ∀ r ∈ Ωd, design variable bounds (3.5)
J
Skjuj − fk = 0, governing equations (3.6)
j=1
The only constraint that is employed in the optimization problems considered is
the volume constraint (3.4). It can be included in the formulation to put a limit
on the amount of material distributed in the design domain Ωd. The parameter β
indicates the fraction of the volume that is allowed to contain solid material and
it takes values between 0 and 1, where β = 1 corresponds to no limit. When the
problem considered has several local minima the optimized designs will be strongly
dependent on the choice of β.
With the objective function (3.3) the problem is optimized for one driving frequency.
But for wave propagation problems it is often relevant to optimize for a
frequency interval in order to get a broadband solution. The objective function can
therefore be changed such that the objective is to minimize the sum of responses
for a number of target frequencies ωi in the interval considered, as explained in
[53]. The chosen interval is divided into M equally sized subintervals. The target
frequency in each subinterval, which results in the highest value of Φ, is determined
by approximating Φ as function of the frequency using Padé expansions, see [64].
The problem is then optimized for the new objective function Ψ, which is the sum
of Φ evaluated at the determined target frequencies and divided by the number of
intervals M to get the average value
min
ξ
Ψ =
ω1,...,ωM
maxωi∈Ii (Φ(ωi))
, I1 = [ω1; ω2[, . . . , IM =]ωM; ωM+1]. (3.7)
M
Here ωM+1 − ω1 is the entire frequency interval and Ii are the equally sized subintervals.
By this procedure, Φ is minimized at all the target frequencies, which are
updated at regular intervals during the optimization.