Maria Bayard Dühring - Solid Mechanics

efractive index component n11 changes and the change depends on the different strain components -
the more the strain components change the more n11 changes. The optimization problem is only stated
for the piezoelectric model and the aim is to maximize an expression, which is dependent on one of the
normal strain components. The objective function Φ is thus chosen as the squared absolute value of the
normal strain in the vertical direction in the output domain Ωo and the formulation of the optimization
problem takes the form
max
ξ
log(Φ) = log
Ωop
|S2(r, ξ(r))| 2 dr
, objective function, (23)
subject to 0 ≤ ξ(r) ≤ 1 ∀ r ∈ Ωd, design variable bounds. (24)
To obtain better numerical scaling the logarithm of the objective function is used. To check that the
acousto-optical interaction has indeed improved by the optimization the optical model is solved for both
the initial design and the optimized design for the case where a wave crest is at the waveguide and half a
SAW phase later where a wave trough is at the waveguide. For these two cases the difference in effective
refractive index for the fundamental mode ∆neff,1 will take the biggest value possible for the initial design
as shown in [6]. ∆neff,1 is normalized with the squared applied electric power P and this is then defined
as the measure for the acousto-optical interaction, see [6] for the expression for P . So the final goal of
the optimization is to increase this measure.
4.5. Discretization and sensitivity analysis
The mathematical model of the acousto-optical problem is solved by finite element analysis. For the
piezoelectric model the displacement-electric potential vector is defined as w = [w1, w2, w3] = [u1, u2, V ].
The complex fields wi and the design variable field ξ are discretized using sets of finite element basis
functions {φi,n(r)}
wi(r) =
N
Nd
wi,nφi,n(r), ξ(r) = ξnφ4,n(r). (25)
n=1
The degrees of freedom corresponding to the four fields are assembled in the vectors wi = {wi,1, wi,2, ...wi,N } T
and ξ = {ξ1, ξ2, ...ξNd }T . A triangular element mesh is employed and second order Lagrange elements
are used for the complex fields wi to obtain high accuracy in the solution and for the design variable ξ
zero order Lagrange elements are utilized. The commercial program Comsol Multiphysics with Matlab
is employed for the finite element analysis. This results in the discretized equations
n=1
3
Kjiwi = fj, (26)
i=1
where Kji is the system matrix and fj is the load vector, both being complex valued due to the PMLs.
To update the design variables in the optimization algorithm the derivatives with respect to the design
variables of the objective function must be evaluated. This is possible as the design variable is introduced
as a separate field. The objective function depends on the complex field vector w2, which is via Eq.(26)
an implicit function of the design variables and can be written as w2(ξ) = w R 2 (ξ) + iw I 2(ξ), where w R 2
and w I 2 denote the real and the imaginary part of w2, respectively. Thus the derivative of the objective
function Φ = Φ(w R 2 (ξ), w I 2(ξ), ξ) is given by the following expression found by the chain rule
dΦ
dξ
= ∂Φ
∂ξ
+ ∂Φ
∂w R 2
∂w R 2
∂ξ
+ ∂Φ
∂w I 2
∂wI 2
. (27)
∂ξ
As w2 is an implicit function of ξ the derivatives ∂wR 2 /∂ξ and ∂wI 2/∂ξ are not known directly. The
sensitivity analysis is therefore done by employing an adjoint method where the unknown derivatives
are eliminated at the expense of determining an adjoint and complex variable field λ from the adjoint
equation
∂Φ
Kj2λ = −
∂wR 2
5
− i ∂Φ
∂wI
, (28)
2