WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

Response, ln(u 8 u - 8 )
TOPOLOGY OPTIMIZATION WITH PADÉ APPROXIMANTS 1613
10
8
6
4
2
0
-2
-4
-6
exact
N=7 (quad)
N=7
-8
0 0.5 1 1.5 2
Frequency, Ω
Figure 2. Comparison between exact response and Padé approximant with
different numbers of expansion terms N.
log(error)
0
-10
-20
-30
-40
N=5
N=3
N=7
N=5
N=3
-50
0 10 20 30 40 50
Numerical precision (# of significant digits)
Figure 3. Error of the bi coefficients versus the numerical precision in the computations.
approximation with N = 7 is generally superior to lower-order approximations but it is noted that
the improvement when going from N = 5 to 7 is limited.
The error introduced by insufficient numerical precision is illustrated in Figure 3. The
ill-conditioning of the P matrix leads to inaccurate computations of the bi coefficients. This
Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 72:1605–1630
DOI: 10.1002/nme