WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

5.1 Padé approximants 45
h
2h
A
f cosΩt
Response (log-scale)
a) b)
c) d) e)
2
0
-2
-4
-6
-8
-10
-12
fig. c)
fig. d)
-14
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Frequency, Ω
Figure 5.1 a) An unsupported 2D elastic body (plane stress) subjected to a harmonic
load, b) Response in point A for the three optimized structures in c,d,e). Dashed lines
indicate the optimization frequency intervals with the structure in c) optimized for the
single frequency Ω = 1 only. From paper [18].
resolution of the frequency range corresponding to 100 points. Fig. 5.1b displays the
response of the structures. A difference in the performance for the three structures
is seen; a very low response can be obtained for a single frequency but the response
is very sensitive to the excitation frequency and just a small frequency de-tuning
will result in a significantly higher response. With frequency range optimization a
much more robust performance is obtained. Furthermore, it also results in much
more well defined structures with fewer design variables with intermediate values
between 0 and 1. The reason for this is, however, not fully understood.
Numerical problems arise if the number of expansion functions is too high. The
coefficients are computed on the basis of derivatives at the expansion frequency
and more functions require higher order derivatives. This puts a large demand
on the numerical precision. This is illustrated in Fig. 5.2 which is based on a 1D
model. The error in the bi-coefficients is here plotted versus the precision used in
the numerical scheme. Double precision, which is the standard in most numerical
libraries (corresponding to 16 significant digits), yields unacceptable high errors if
e.g. N = 7 is chosen (for this particular example). Quad-precision numerics (32
significant digits) would, in this case, yield acceptable accuracy.
fig. e)