WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

Space–time topology optimization for one-dimensional wave propagation
Jakob S. Jensen *
Department of Mechanical Engineering, Solid Mechanics, Nils Koppels Allé, Building 404, Technical University of Denmark, 2800 Lyngby, Denmark
article info
Article history:
Received 11 March 2008
Received in revised form 30 September 2008
Accepted 3 October 2008
Available online 29 October 2008
Keywords:
Topology optimization
Wave propagation
Transient loading
Dynamic materials
1. Introduction
abstract
Topology or material layout optimization has gained popularity
as a design tool in academics and industry. This is mainly due to
the large design freedom and the use of adjoint sensitivity analysis
that allows for efficient handling of the many element design
variables usually appearing in the discretized models. Topology
optimization was originally introduced to design stiff lightweight
mechanical structures [2] and has since been extended to a variety
of different settings. A fairly recent overview of applications can be
found in [4] and new applications appear regularly, e.g. the recent
works on optimization of ferromagnetic materials [9], incompressible
materials [7], electrostatically actuated devices [38], acoustostructural
interaction [37], and damage detection [20].
A number of papers have considered optimization of structures
based on its transient response, see, e.g. [17] for a review, and recently
there has been an increasing interest in using topological
design variables for these transient problems, e.g. in structural
dynamics [27,34], for thermal problems [33,21], crashworthiness
[29], electromagnetics [10,28], and structural wave propagation
[13]. In this paper, the topology optimization concept for transient
loads is extended with design variables in the temporal domain, in
order to allow for the optimized material properties to vary in
time. The extended method is demonstrated for wave propagation
in a one-dimensional (1D) model of an elastic rod with time dependent
Young’s modulus. The present study is closely related to
recent works [25,26] that present a solid mathematical foundation
* Tel.: +45 45 25 42 50; fax: +45 45 93 14 75.
E-mail address: jsj@mek.dtu.dk
0045-7825/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.cma.2008.10.008
Comput. Methods Appl. Mech. Engrg. 198 (2009) 705–715
Contents lists available at ScienceDirect
Comput. Methods Appl. Mech. Engrg.
journal homepage: www.elsevier.com/locate/cma
A space–time extension of the topology optimization method is presented. The formulation, with design
variables in both the spatial and temporal domains, is used to create structures with an optimized distribution
of material properties that can vary in time. The method is outlined for one-dimensional transient
wave propagation in an elastic rod with time dependent Young’s modulus. By two simulation examples it
is demonstrated how dynamic structures can display rich dynamic behavior such as wavenumber/frequency
shifts and lack of energy conservation. The optimization method’s potential for creating
structures with novel dynamic behavior is illustrated by a simple example; it is shown that an elastic
rod in which the optimized stiffness distribution is allowed to vary in time can be much more efficient
in prohibiting wave propagation compared to a static bandgap structure. Optimized designs in form of
spatio-temporal laminates and checkerboards are generated and discussed. The example lays the foundation
for creating designs with more advanced functionalities in future work.
Ó 2008 Elsevier B.V. All rights reserved.
for spatio-temporal design problems for the 1D and 2D wave
equation.
By including design variables in the time domain the optimization
problem becomes equivalent to the classical problem of
optimal control [18]. Holtz and Arora [14] solved an optimal control
problem by using adjoint sensitivity analysis and mathematical
programming and exemplified the method with obtaining
optimized trajectories for a scalar control force. The contribution
of the present work can be viewed as an extension to [14] by considering
the material layout problem with topological design
variables.
The dynamics of structures with time varying material parameters
have been studied in a number of works. Clark [11] demonstrated
numerically the possibility for vibration control by using
stiffness switching induced by piezoelectric materials, Ramaratnam
and Jalili [30] considered a bi-stiffness spring setting and
demonstrated vibration control theoretically and experimentally,
and Issa et al. [16] considered a similar problem with stiffness
switching induced by a controllable hinge and showed vibration
attenuation numerically and experimentally. Other examples of
geometric stiffness control have been demonstrated in [19,31]
and alternatively magneto- or electro-rheological fluids can be
used to obtain a similar control of the stiffness in the temporal domain.
Recently reported is the possibility for using a combination
of non-linear materials and external high-frequency excitation that
results in an effective change of the material stiffness [6].
General properties of space–time varying materials (denoted
‘‘dynamic materials”) has been studied, e.g. in [22,5]. The recent
monograph [24] gives an extensive coverage of the subject with
a special emphasis on space–time variation of electromagnetic