WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

TOPOLOGY OPTIMIZATION WITH PADÉ APPROXIMANTS 1611
k k k k k k k k
fcosΩt
m m m m m m m m
u 1
Figure 1. A simple mass–spring chain with a harmonic load.
related Krylov subspace projection method since the bi coefficients are not directly computed with
this method. To determine if the Krylov method is suitable for a design optimization procedure
is left for future studies. The problems with ill-conditioning can be significantly reduced if the
computations are carried out with higher precision numerics (e.g. quad precision as available on
some 64-bit platforms). Apart from the simple test case analysed in Section 2.3, the use of higher
precision numerics is also left for future work.
To conclude the analysis the individual steps involved in constructing the PA are listed:
• Solve the original equation at the chosen expansion frequency ( = 0) by a factorization
method (Equation (17)) and compute N + 1 gradients (w.r.t. at = 0) by forward-/back
substitutions (Equations (18)–(19)).
• Compute the bi coefficients by solving the over-determined system of equations in
Equation (25).
• Compute the coefficients ai with Equations (22)–(23).
2.3. Analytical example: forced vibrations of a mass–spring chain
In this example the accuracy of the numerical scheme is demonstrated and the issue of numerical
precision is addressed. The simple mass–spring chain in Figure 1 is considered which allows for
analytical computation of the PA.
The steady-state vibration amplitudes of the masses are governed by a set of equations of the
form Equations (2)–(3), in which the load vector is
f ={0000000 f } T
if the harmonic load of magnitude f acts on the rightmost mass.
In order to compare with the numerical scheme, the de-tuning parameter = − 0
(cf. Equation (4)) is introduced. With introduced, S can be written as
S = (− 2 0 M + i0C + K) + (−20M + iC) + (−M) 2
in which simple mass-proportional damping C = cM is assumed.
The solution is written as
in which H = S −1 . The amplitude of the rightmost mass is
u 8
(29)
(30)
u = Hf (31)
u8 = H88 f (32)
Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 72:1605–1630
DOI: 10.1002/nme