WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

TOPOLOGY OPTIMIZATION WITH PADÉ APPROXIMANTS 1607
variables. The present paper is related to these works [15, 16] but focuses on optimization problems
with a large number of design variables such as found in topology optimization.
In Section 2 it is shown how the PA is computed and the validity of the approximation is demonstrated
with simple examples along with a discussion of the importance of numerical precision.
In Section 3 the sensitivity analysis of a PA response function with respect to a set of design
variables is treated in detail. The approach is based on adjoint analysis which makes the
method suitable for the case of many design variables. Two examples of topology optimization of
forced vibration problems with PAs are presented in Section 4. Finally, conclusions are given in
Section 5.
2. FREQUENCY RESPONSES WITH PADÉ APPROXIMANTS
The basis of the proposed method is the computation of the frequency response using a PA. The
proposed procedure differs slightly from what appears to be the standard implementation of method
(see e.g. [5]). The difference will be pointed out in the following.
Time-harmonic motion of a linear damped dynamic system is governed by the discretized set
of equations
in which the system matrix S is defined as
Su = f (2)
S =− 2 M + iC + K (3)
M is the mass matrix, C the damping matrix, K the stiffness matrix, and f is the loading vector.
The frequency of excitation is denoted and u is a vector containing the discretized nodal values
of the complex amplitude function.
An approximate solution for u is now sought in the vicinity of some expansion frequency 0.
The de-tuning parameter:
= − 0
is used to define the closeness of the excitation frequency to this expansion frequency.
Equation (2) is rewritten as
in which the new matrices are defined as
(P0 + P1 1 + P2 2 )u = f (5)
P0 = S(0) =− 2 0 M + i0C + K (6)
P1 = S
(0) =−20M + iC (7)
P2 = 1
2
2 S
2 (0) =−M (8)
Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 72:1605–1630
DOI: 10.1002/nme
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