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