WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

Response, ln(u A u - A )
TOPOLOGY OPTIMIZATION WITH PADÉ APPROXIMANTS 1615
2
0
-2
-4
-6
-8
N=3
-10
0 0.2
direct solution
N=6 (7 PA expansions)
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Frequency, Ω
Taylor (N=10)
Figure 5. The response of point A in Figure 4 using direct computation of the frequency response (solid
line) and PAs with different numbers of expansion coefficients.
to deteriorate, again due to an ill-conditioned P matrix. To demonstrate the accuracy of the PA a
standard Taylor-expansion with 10 terms has been included in the figure for comparison.
As mentioned in Section 2.3 the problem with ill-conditioning can be overcome with higher
precision numerics. Another possibility is to construct the frequency response with several PAs
each with a different expansion frequency. This is illustrated in Figure 5 in which seven separate PA
expansions (with N = 6) have been computed with equidistant frequency spacing and then patched
together. This multiplies the computational time accordingly, but offers an implicit control of the
accuracy of the PA expansions by requiring that two patches should coincide at their midpoint
frequency.
N=9
N=6
3. DESIGN SENSITIVITY ANALYSIS
The core of this work is to use the PA directly for optimization of the dynamic response. The first
step is to compute the sensitivity of a chosen objective function, denoted c, w.r.t. a set of design
variables. Analytical expressions for the sensitivities are obtained using the adjoint method.
A real-valued objective function with the general form
c = c(u, x) (36)
is considered. The function depends on the PA (u) and may also explicitly depend on the set of
design variables defined in the vector x ={x1, x2,...,xNx }T .
In the general case with non-zero damping u is complex and conveniently split up in the real
and imaginary parts
c = c(uR, uI, x) (37)
Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 72:1605–1630
DOI: 10.1002/nme