WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
Int. J. Numer. Meth. Engng 2007; 72:1605–1630
Published online 10 April 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.2065
Topology optimization of dynamics problems
with Padé approximants
Jakob S. Jensen ∗,†
Department of Mechanical Engineering, Solid Mechanics, Technical University of Denmark,
Nils Koppels Allé, Building 404, Denmark
SUMMARY
An efficient procedure for topology optimization of dynamics problems is proposed. The method is based
on frequency responses represented by Padé approximants and analytical sensitivity analysis derived using
the adjoint method. This gives an accurate approximation of the frequency response over wide frequency
ranges and a formulation that allows for design sensitivities to be computed at low computational cost
also for a large number of design variables. Two examples that deal with optimization of forced vibrations
are included. Copyright q 2007 John Wiley & Sons, Ltd.
Received 30 October 2006; Revised 1 February 2007; Accepted 2 March 2007
KEY WORDS: Padé approximants; topology optimization; forced vibrations
1. INTRODUCTION
The solution to a discretized linear vibration problem with harmonic excitation can be expressed
in the exact form
u =
2Nd−2 i=0
ãi( − 0) i
2Nd i=0 ˜bi( − 0) i
in which Nd is the number of degrees of freedom in the model, ãi and ˜bi are expansion coefficients,
is the excitation frequency, and 0 is an arbitrary expansion frequency. This representation of
the solution is rarely used except for systems with only few degrees of freedom for which the
corresponding small number of expansion coefficients can be found analytically. Solutions for larger
problems are usually obtained numerically with a factorization method or an iterative method.
∗ Correspondence to: Jakob S. Jensen, Department of Mechanical Engineering, Solid Mechanics, Technical University
of Denmark, Nils Koppels Allé, Building 404, Denmark.
† E-mail: jsj@mek.dtu.dk
Copyright q 2007 John Wiley & Sons, Ltd.
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