WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

J. S. Jensen and O. Sigmund Vol. 22, No. 6/June 2005/J. Opt. Soc. Am. B 1193
cell. Thus the entire computational domain consists of
1515+391019 2 =178,695 elements. On a 2.66
-GHz computer with 1-Gbyte RAM, a solution for a single
frequency of Eq. (4) takes approximately 10 s. This is sufficiently
fast for the model to be used in our iterative optimization
algorithm in which several hundred solutions
need to be computed in a single optimization procedure.
A. Design Variables and Material Interpolation
The design domain D is a subdomain of . We assign one
design variable x e to each finite element within D:
0 x e 1, e D. 6
It is impossible to solve the discrete combinatorial
problem for more than, e.g., 100 design variables. Instead,
we use continuous design variables so that we can apply a
gradient-based algorithm. We let the design variable govern
the distribution of air and dielectric in each element
in D by letting the material coefficients A e and B e vary
linearly with x e:
A e = A 1 + x eA 2 − A 1,
B e = B 1 + x eB 2 − B 1, 7
such that, for x e=0, the material in the element will be
material 1 (air) and, with x e=1, it will be material 2 (dielectric).
It is now the task for the optimization algorithm
to identify the values of x e in D that optimize our chosen
objective function and, additionally, to ensure only discrete
values, 0 and 1, of the design variables in the final
design in order to obtain a well-defined structure.
B. Objective Function and Sensitivity Analysis
The goal is to maximize the power transmission through
the waveguide, thereby minimizing the reflection at the T
junction. The power transmission is found from the timeaveraged
Poynting vector:
Px = PxPyT = 1
ARiū u. 8
2
We consider the vertical component of the power transmission,
P y, averaged in two separate domains J1 and
J2 near the two output waveguide ports [see Fig. 1]:
J1 = 1
e Py, Ny eJ1 J 2 =− 1
N y
e Py, 9
eJ2 where N y are the number of finite elements in the y direction
in J1 and J2 . The vertical component of the timeaveraged
Poynting vector in each element is given as
e 1
Py =
2 AeRiue T e e Qyū , 10
where ūe denotes the complex conjugate of the element
e
nodal values. Qy is defined as
e
Qy = NT
y NdS y=0. 11
We want to maximize the total power transmitted
through the output ports for a given frequency , which
leads to the optimization problem:
max J,
xe J = J 1 + J 2, 12
in which we additionally enforce a simple symmetry condition
to ensure a symmetrical design with J 1=J 2. The
optimization problem in expressions (12) is solved with
the mathematical programming tool, the Method of Moving
Asymptotes, 17 in combination with the computed sensitivities
of the objective function with respect to the design
variables.
Analytical sensitivity analysis is an essential part of a
fast optimization method and can be performed, e.g., with
the adjoint method, 12 leading to the expression
dJ
dx e
=2RTd− 2M + iC + K
u, 13
dxe where it has been assumed for simplicity that the design
domain D neither includes the wave input boundary inp
nor overlaps J and where is the solution to the adjoint
problem
− 2M + iC + KT = i
4 e e T T e AeQy − Qy ū ,
eJ 14
where the summation should be understood in the normal
finite-element sense. Note that Eq. (14) can be solved essentially
without computational effort if the original problem
is solved by a direct factorization method. This means
that one can obtain all the sensitivities [Eq. (13)] simply
by solving the original problem [Eq. (4)] with an extra
right-hand side (load case) and inserting the result into
Eq. (13).
In structural mechanics the problem of mesh dependency
arises when the standard formulation is used. This
implies that if the finite-element mesh is refined, finer details
will appear in the optimized structure. One can
avoid this by introducing a heuristic mesh-independency
filter. 12 In optics this dependency is not inherent in the
problem, but the filter can nevertheless be applied with
advantage to avoid too fine details in the structure. The
variant used here is based on filtering the computed sensitivities
in the following way:
dJ ˆ
=
dxe ˆ
H e
dJ
Hˆ e
dxe , 15
where the summation is performed over all elements in
the design domain and where Hˆ i is a convolution operator
defined as