WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

R
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
q1=1
q2=1
q1=1
q2=3
q1=3
q2=3
0
0 0.1 0.2 0.3 0.4 0.5
design variable
0.6 0.7 0.8 0.9 1
R
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
q1=0.25
q2=0.25
q1=1
q2=0.25
q1=1
q2=1
0
0 0.1 0.2 0.3 0.4 0.5
design variable
0.6 0.7 0.8 0.9 1
Figure 2 - Reflection, R, versus the design variable ̺, left: polynomial interpolation (Eq. (6)),
right: inverse polynomial interpolation (Eq. (7)), for different values of the parameters q1 and
q2. Parameters: ρ2/ρ1 = κ2/κ1 = 500.
Fig. 2(right) shows reflection versus ̺ for different parameter values. The curves are
clearly more well-behaved with a non-vanishing slope for ̺ → 1. The different parameter
combinations have been used in the optimization algorithm and the simplest
choice of q1 = q2 = 1 yields good results with well-defined air and solid regions in the
optimized designs. Naturally, a simple 1D model cannot fully account for all necessary
properties, and more work should be put into fully understanding the effect of the
choice of interpolation functions.
DISCRETIZATION AND SENSITIVITY ANALYSIS
To solve the model equation (with the appropriate boundary conditions) and the optimization
problem, we discretize the complex amplitude field p and the design field ̺
using finite elements:
N
Nd
p = φipi, ̺ = ψi̺i, (8)
i=1
where φi and ψi are basis functions, and the vectors p = {p1 p2 ...pN} T and ̺ =
{̺1 ̺2 ...̺Nd }T contain the nodal values of the two fields. We use the commercial
FEM package FEMLAB on top of MATLAB for assembling and solving the discretized
equation:
S(̺)p = f(̺). (9)
Throughout this work we use a triangular element mesh, and a quadratic approximation
for the pressure field and a linear approximation for the design field.
We now derive the sensitivities of the objective functional Φ(̺,p) with respect
to a single design variable. We use the adjoint method and obtain the adjoint equation
as:
S T λ = −1 ∂Φ
( − i 2 ∂pR
∂Φ
)
∂pI
T , (10)
4
i=1