WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

1606 J. S. JENSEN
To compute a frequency response with a fine frequency resolution can be computationally costly
since a numerical solution must be obtained for each considered frequency. Furthermore, if the
analysis is the basis for an iterative optimization procedure this approach will in most cases be
unfeasible. In this work, an efficient approach for obtaining the frequency response based on the
solution form in Equation (1) is used and applied in a topology optimization procedure. Other
methods can be used to reduce computational costs, e.g. the modal expansion technique which is
especially applicable to lower frequency ranges where only a few modes influence the response [1].
However, optimization of eigenvalues and eigenvectors often leads to non-smooth problems that
might pose difficulties for optimization algorithms. Thus, the investigation of alternatives seems
justified.
In the proposed procedure, the expression in Equation (1) is used with fewer expansion terms
in the numerator and denominator so that a good approximation for u is obtained only in a
neighbourhood of 0. It will be demonstrated how the coefficients ãi and ˜bi can be computed by
performing only a single factorizing of the system matrix S =− 2 M + iC + K for a chosen
expansion frequency = 0, followed by relatively inexpensive forward-/back substitution in order
to obtain gradients at 0. The rational polynomial expansion in Equation (1) can be recognized as
the Padé approximant (PA) associated with an infinite power series of the form ∞ i=0 ci( − 0) i
[2]. This paper will not go into the theory behind the PA. But due to the connection between the
solution form in Equation (1) and a PA, the term PA is used throughout this work to describe the
approximation and to characterize the proposed method.
PA approximants have been popular as means for computing the frequency response of different
dynamic systems at a low computational cost. They are the backbone of the so-called asymptotic
waveform evaluation (AWE) methods [3], used for circuit analysis and for electromagnetic wave
scattering problems, e.g. [4, 5]. In[6] the application to electromagnetic problems was extended
to treat problems in bounded and open domains and in [7] PAs were used to obtain the response
of vibro-acoustic systems over wide ranges of frequencies.
The pure acoustic problem was considered by Djellouli et al. [8]. They used the PA to compute
scattering of acoustic fields and compared the results to those obtained by a standard Taylor
expansion. The PA algorithm was shown to outperform the Taylor expansion and give a good
approximation over wide frequency ranges. The acoustic problem was treated also by Malhotra
and Pinsky [9] and the work was extended in [10] by using a Krylov subspace projection method.
In fact, the Krylov method has a strong link to matrix-valued PAs [11], also referred to as the
Padé-via-Lanczos connection (PVL) [12].
The possibility for using PA approximants to compute frequency responses with a high-frequency
resolution was utilized in a previous optimization study by the author [13]. In that work, a photonic
crystal waveguide was designed with a topology optimization approach. The PAs were
used to accurately pinpoint a number of the most critical frequencies in the operational frequency
range at a low computational cost. The optimization considered the performance for these
critical frequencies which were repeatedly updated during the optimization process. Another optimization
study was reported in [14], in which a PA was used to approximate the dynamic
response for large variations of a design variable and used as a basis for finite element-based size
optimization
Only few works have considered direct optimization of the frequency response based on its
PA. In [15] a procedure was developed for sensitivity analysis of PAs for 3-D microwave device
applications. The formulation was suitable for shape/size optimization with a few design variables
and was implemented in [16] to optimize microwave devices with up to four geometry design
Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 72:1605–1630
DOI: 10.1002/nme