Maria Bayard Dühring - Solid Mechanics

5.2 Results 29
the sum of the squared magnetic field amplitudes in the x1- and x2-direction in the
core area, Ωc. Hence, the optimization problem is given as
max
ξ
log(Φ) = log
Ωc j=1
2
|Hj(r, ξ(r))| 2 dr, objective function (5.3)
subject to 0 ≤ ξ(r) ≤ 1 ∀ r ∈ Ωd, design variable bounds (5.4)
The morphology-based filter described in section 3.5 is applied to avoid meshdependent
solutions and to enforce 0-1 designs. To prevent convergence to a local
optima a continuation method is applied where αs has a big value from the beginning
of the optimization such that the problem is modified to a smoother problem.
During the optimization αs is gradually reduced to its more realistic value. Vector
elements are used for H1 and H2 in order to avoid spurious modes. Lagrange
elements of second order are used for H3 and of zero order for ξ.
5.2 Results
An example of the optimization is shown in figure 5.2(a) and 5.2(b) where the initial
and the optimized designs are plotted, respectively. The distribution of the energy
measure from the objective function normalized with its maximum value for the
initial design, Φmax, is indicated with the contour lines. The objective function
is increased 3.75 times for the optimized design. All parts are connected in the
optimized geometry and it is close to be a 0-1 design. The symmetry around the
vertical axis is kept in the optimized design, but the 60 degree symmetry from the
initial design has vanished. The objective function in the optimized design has been
redistributed such that the modes shape is extending more in the vertical direction
Figure 5.2 The center region of the holey fiber where black indicates silica and white is
air. The contour lines show the distribution of the energy measure from the normalized
objective function Φ/Φmax. (a): Initial design. (b): Optimized design.