WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

706 J.S. Jensen / Comput. Methods Appl. Mech. Engrg. 198 (2009) 705–715
material properties. Means for realization of time varying properties
(such as the dielectric constant) is the use of non-linear optic
material or with the use of ferroelectric/ferromagnetic materials.
In order to keep the derivations simple and at the same time
consider structures that are realizable as mechanical systems, it
is chosen to consider the case of time varying stiffness and a constant
material density. The starting point is a dynamic FE model:
M€u þ C _u þ KðtÞu ¼ fðtÞ; ð1Þ
in which M and C are constant mass and damping matrices and KðtÞ
is the time dependent stiffness matrix. The vector u contain the nodal
displacements and fðtÞ is the time dependent load.
The paper is organized as follows. Section 2 introduces the
space–time design variables and the basic notation is defined. In
Section 3, the time integration algorithm is described and basic
simulation results are presented and compared to exact results
from literature. Section 4 covers sensitivity analysis w.r.t. space–
time design variables. In Section 5, an example of an optimized dynamic
structure is given. The space–time distribution of stiffness in
a 1D elastic rod is optimized in order to minimize wave transmission;
a so-called dynamic bandgap structure. In Section 6, the results
are summarized and conclusions are given.
2. Space–time design variables for a one-dimensional problem
In traditional topology optimization with the density approach
[1], a single design variable is introduced for each element in the
finite element model. This is illustrated in Fig. 1 for a single spatial
dimension.
The design variables are denoted x1; x2; ...; xN where N is the
number of elements in the model. The design variables are collected
in the vector x. The value of xi governs the material properties
of the corresponding element according to a specified material
interpolation model [3].
In the proposed formulation, we extend the traditional approach
by allowing the material properties in a spatial element
to change in time. This is facilitated by introducing a two-dimensional
design element grid (for one spatial dimension), see Fig. 2.
An array of design variable vectors is introduced:
X ¼fx1; x2; ...; xMg; ð2Þ
x = { x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9
x N } T
Fig. 1. Traditional design variable concept for topology optimization with the
density approach for one spatial dimension.
M xM = { x1 x2 x3 x4 x5 x6 x7 x8 x9 x N } T
M M M M M M M M M
j xj = { x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9
x N } T
j j j j j j j j j
x1 = { x1 1
1 x2 x3 x4 x5 x6 x7 x8 x9 x N } T
1 1 1 1 1 1 1 1
T1+MΔT T1+(M-1)ΔT T1+jΔT T1+(j-1)ΔT T1+ΔT T1 Fig. 2. Extended topology optimization approach with space–time design variables
for one spatial dimension.
t
in which each design vector in the array contains the element-wise
design variables for a specific time interval. For time interval j, the
design vector components are specified as
xj ¼fx j
1
; xj
2 ; ...; xj
NgT : ð3Þ
In (2) M is the number of time intervals in which the material properties
can attain different values. The temporal design starts at
t ¼ T1 and continues to t ¼ T2 ¼ T1 þ MDT. For simplicity, uniform
intervals DT are specified but non-uniform intervals can readily
be used with the presented formulation.
The choice of DT should depend on the specific problem considered,
such as the frequency of the wave, but should also take
into account the temporal discretization used in the time integration
algorithm. In the present work a central-difference explicit
scheme with a fixed time step Dt is applied. In this case it is necessary
that Dt DT in order for the numerical results to be accurate.
Another and perhaps more natural choice, could be to use a
space–time finite element formulation, e.g. [15]. This will be subject
for future work.
3. Numerical simulation of dynamic structures
A standard central-difference explicit scheme is used to solve
(1) (see, e.g. [12]). Based on the displacement vector un at the current
time step and at the previous time step un 1, the displacement
vector at the next time step unþ1 is approximated as
1
ðDtÞ 2 Munþ1 fn Knun þ
2
M 2
ðDtÞ
1
Dt C
!
un
1
M 2
ðDtÞ
1
Dt C
!
un 1; ð4Þ
in which fn and Kn is the load vector and stiffness matrix at the current
time step. The scheme in (4) is efficient, especially if M is diagonal
(this simplification is made throughout the examples in this
paper). However, one has to ensure that the time step Dt is sufficiently
small (CFL-condition [12]) to ensure stability of the scheme.
In the following sections, (4) is used to simulate the behavior of
structures with a time dependent stiffness matrix. The examples
will serve both as an illustration of typical behavior observed when
the material properties vary in time, as well as documentation for
the applicability (and limitations) of the proposed time integration
scheme.
3.1. Instant change of material properties
The propagation of a wave in a medium that experiences an instant
change of the material parameters was studied theoretically
in [36], in which it is shown that a forward travelling wave splits
up in a ‘‘transmitted” forward travelling wave and a ‘‘reflected”
backward travelling wave, and that they both retain the shape of
the original wave. Fig. 3 shows a 1D elastic rod in which the material
properties for t < t0 are q ¼ E ¼ 1 and for t ¼ t0 the stiffness is
changed to E ¼ E0 whereas the density is unchanged. Shown also
are snapshots of the simulated wave motion.
Weekes [36] predicts the amplitudes of the forward travelling
wave, T, and the backward travelling wave, R:
ffiffiffiffiffi
pffiffiffiffiffi
p
E0 þ 1
T ¼
2 ffiffiffiffiffi p ; R ¼
E0
E0 1
2 ffiffiffiffiffi p : ð5Þ
E0
In Fig. 4 the amplitude of the forward and backward travelling
waves are estimated based on simulation of the wave propagation
with (4). The amplitudes are given relative to the input amplitude
and are computed for different values of E0. The results are compared
to the exact values in (5) and the agreement is seen to be very