WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

708 J.S. Jensen / Comput. Methods Appl. Mech. Engrg. 198 (2009) 705–715
does not depend on the material properties of the second material
(i.e. E0).
Fig. 6 shows the computed wavenumber shift compared to the
exact value (8). A good accuracy of the numerical results is noted
for values of the interface speed up to about V ¼ 0:5, but for higher
values the numerical simulations overestimate the shift. This is
due to an insufficient number of elements per wavelength resulting
from the increase in wavenumber (decrease in wavelength).
The presented results have been computed with 80 elements per
wavelength in material 1 (for V ¼ 0). For V ¼ 0:6 this has decreased
to only 20 elements per wavelength thereby leading to
inaccurate results. The shift in wavenumber is seen to approach
infinity when V ! 1. This corresponds to the case when the interface
speed approaches the wave speed.
Similarly to the previous example, the energy in the wave is not
conserved and the increase in energy can be computed based on
the results in [35]. The exact results are shown in Fig. 8 (the
lengthy but explicit formula is not given) and compared to the
numerical results for four different values of E0. As seen in Fig. 8
the results deviate significantly even for relative small values of
E0 (such as E0 ¼ 1:25) also for V 0:5. Thus, the reason for this discrepancy
is not insufficient discretization of the wave.
Wavenumber shift, ln Δ γ /γ 0
Energy
100
10
1
0.1
0.01
0 0.2 0.4 0.6 0.8 1
Interface speed, V
Fig. 6. Shift in the wavenumber of the reflected wave versus the interface speed V.
Comparison between exact results (solid line) and numerical simulation (discrete
markers).
1200
1000
800
600
400
200
0
total mechanical
energy
potential
energy
kinetic
energy
0.2 0.3 0.4 0.5 0.6
Time, t
0.7 0.8 0.9 1
Fig. 7. Total mechanical, potential, and kinetic energy versus time. Instant change
of stiffness to E0 ¼ 3att = 0.85 s.
ln(ΔE mek /E 0 )
10
1
0.1
0.01
0.001
0.0001
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Interface speed, V
Fig. 8. The relative jump in the total mechanical energy versus the interface speed V
for E0 ¼ 1:5, E0 ¼ 1:25, E0 ¼ 1:11, E0 ¼ 1:05, E0 ¼ 1:02 (top-down). Comparison
between exact results (lines) and numerical simulation (markers).
The reason for the overestimation of the energy increase is the
time integration scheme and the origin of the problem is illustrated
in Fig. 9. The plots show the computed wave velocities for
Velocity
Velocity
150
100
50
0
-50
-100
-150
0 0.2 0.4 0.6 0.8 1
Axial position
150
100
50
0
-50
-100
-150
0 0.2 0.4 0.6 0.8 1
Axial position
Fig. 9. Wave velocities for V ¼ 0:3 for two different values of E0, left: E0 ¼ 1:05 and
right: E0 ¼ 1:5.