4.5 Numerical examples
4.5.2 Chain of elastic beads
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500 1000 1500 2000
Taget Ball
Frequency
Time Step [μsec]
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0 500 1000 1500 2000
Incomming Ball
Frequency
Time Step [μsec]
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Contact Constraint
Frequency
Time Step [μsec]
Figure 4.16: Head-on collision of two soft elastic balls of radius 0.025 m. Histogram of time steps selected by asynchronous energy-stepping forℎ= 5⋅10−5 J.
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Taget Ball
Frequency
Time Step [μsec]
0 5 10 15 20
0 1000 2000 3000 4000 5000 6000
Incomming Ball
Frequency
Time Step [μsec]
0 5 10 15 20
0 1000 2000 3000 4000 5000 6000
Contact Constraint
Frequency
Time Step [μsec]
Figure 4.17: Head-on collision of two soft elastic balls of radius 0.025 m and 0.050 m. Histogram of time steps selected by asynchronous energy-stepping forℎ= 5⋅10−5 J.
The values of the material constants correspond to steel and are equal to 𝜆0 = 9.70⋅1010 Pa, 𝜇0= 8.26⋅1010 Pa, the density is𝜌= 7780 kg/m3.
Front view Front view
Figure 4.18: Setup of the chain of ten elastic beads with diameter 3/8” and impact velocity of 0.44 m/s. On the right hand side, it is shown a detail of the finite-element mesh employed in the calculations.
The time histories of the total energy, potential energy and kinetic energy of the asynchronous energy-stepping solution corresponding to ℎ = 10−7 J are shown in Figure 4.19. The localized nature of the dynamics is clearly observed in the time histories of the local—as opposed to global—
energies depicted in Figure 4.20 for the first four beads of the chain. It is also worth noting that, before and after the solitary wave travels along a bead, the local energy is exactly preserved by asynchronous energy-stepping. Similarly, all global momentum maps of the system and all local momentum fluxes across bead boundaries are exactly preserved by asynchronous energy-stepping.
In particular, Figure 4.21 shows the exact conservation of total linear momentum and the change in shape of the local contributions𝐽(𝑠) as the solitary wave propagates along the chain.
As we have discussed previously, there are two unique properties of asynchronous energy-stepping:
i) automatic and asynchronous selection of the time step size in each subdomain, and ii) exact conservation of all local and global momentum maps of the systems. These features allow for a confident and straightforward analysis of the local behavior of a Lagrangian system. For example, the formation and propagation of solitary waves are of particular interest to the dynamics of a one- dimensional chain of elastic beads. Numerous experimental and theoretical investigations have been conducted with the aim of understanding the shape and the speed of these solitary waves (see, for
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0.5 1 1.5 2 2.5 3 3.5x 10−4
Total Energy
Kinetic Energy
Potential Energy
Time [μsec]
Energy [J]
Figure 4.19: Chain of ten elastic beads. Asynchronous energy-stepping solution with ℎ= 10−7 J.
Time history of total, kinetic and potential energies.
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0 1 2 3
x 10−4
Striker
Time [sec]
Energy [J]
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0 1 2 3
x 10−4
Bead # 2
Time [sec]
Energy [J]
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0 1 2 3
x 10−4
Bead # 3
Time [sec]
Energy [J]
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0 1 2 3
x 10−4
Bead # 4
Time [sec]
Energy [J]
Figure 4.20: Chain of ten elastic beads. Asynchronous energy-stepping solution with ℎ= 10−7 J.
Time history of total (black curve), kinetic (dashed blue curve) and potential (blue curve) local energies.
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−16
−14
−12
−10
−8
−6
−4
−2 0 2x 10−4
Total Momentum Striker
Bead # 2
Bead # 3
Bead # 4
Time [μsec]
Linear Momentum [Nsec]
Figure 4.21: Chain of ten elastic beads. Asynchronous energy-stepping solution with ℎ= 10−7 J.
Conservation of total linear momentum and time history of individual 𝐽(𝑠)—only the component align with the chain of beads is shown.
example, [9] and references therein). However, the amount of vibrational kinetic energy retained in each bead during and after collision has never been accounted for. This is clearly a local behavior of the system that can be readily studied with asynchronous energy-stepping. The vibrational kinetic energy of each bead is defined as
VKE(𝑠)=
𝑁𝑠
∑
𝑎=1
1
2𝑚𝑎∥𝑞˙𝑎− ⟨𝑞⟩˙ (𝑠)∥2 (4.45)
where ⟨𝑞⟩˙ (𝑠) is the center of mass velocity of the 𝑠-th bead. Figure 4.22 illustrates the amount of vibrational kinetic energy in the system as the solitary wave travels along the chain and it also shows the individual contribution of each bead. It is worth noting that, for this particular finite-element mesh and initial conditions, the vibrational kinetic energy trapped in the system is not negligible and naturally accumulates as the solitary wave travels along the chain.
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0.5 1 1.5
2x 10−5
Vibrational Kinetic Energy
#2 #3 #4 #5
Time [μsec]
Energy [J]
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0 1 2 3
x 10−4
Total Energy TE−VKE
Kinetic Energy
Potential Energy
Time [μsec]
Energy [J]
Figure 4.22: Chain of ten elastic beads. Asynchronous energy-stepping solution with ℎ= 10−7 J.
The top figure illustrates the amount of vibrational kinetic energy (VKE) in the system as the solitary wave travels along the chain. The bottom figure shows the time history of individual components VKE(𝑠).
Finally, we investigate the computational efficiency of asynchronous energy-stepping over energy- stepping and explicit Newmark for the setup depicted in Figure 4.18. We compare the execution time of asynchronous energy-stepping forℎ= 10−6J and the mesh partitioned into 10 subdomains (𝑇AES-10,10−6), with execution times of energy-stepping forℎ= 10−6 J (𝑇ES,10−6) and explicit New- mark for a small time step Δ𝑡= 10−4 𝜇sec (𝑇NM,10−4)—this time step is 10 times smaller than the average time step resulting from the energy-stepping calculations. As each numerical time integra- tor provides a different approximation of the exact solution of the problem, it is not only important to compute the timing of each simulation but also to quantify the accuracy of the numerical ap- proximation. To this end we additionally compute the asynchronous energy-stepping solution for ℎ= 10−7J and we compare the time history of energies for each numerical solution as an indication of accuracy. Figure 4.23 shows that the explicit Newmark solution quickly loses stability and blows up, even though the time step employed is small and therefore the execution time is 25 times larger than𝑇AES-10,10−6. Figure 4.24 shows that asynchronous energy-stepping solutions withℎ= 10−6J and with ℎ= 10−7 J are in reasonable agreement. In contrast, Figure 4.25 shows that the energy-
stepping solution is less accurate than its asynchronous counterpart for the same energy step. The execution time for these different cases is summarized in Table 4.1 in terms of speedup relative to 𝑇AES-10,10−6(see Section 4.4.2). It bears emphasis that the asynchronous energy-stepping solution is more computationally efficient and more accurate than its synchronous counterpart. These results suggest that, for this particular finite-element mesh and initial conditions, a speedup of the form
𝜎= 𝑇ES,10−7
𝑇AES-10,10−6 ∼5.2
is more appropriate than𝜎=𝑇ES,10−6/𝑇AES-10,10−6 ∼2.5, as shown in Figure 4.26. Nonetheless, a speedup of∼2.5 is consistent with the fact that, for this example, the solitary wave has a support of∼4 beads and the chain comprises 10 beads.
𝑇AES-10,10−6 𝑇ES,10−6 𝑇NM,10−4 𝑇AES-10,10−7 𝑇ES,10−7
Speedup 1 ∼2.5 ∼25.0 ∼1.8 ∼5.2
Table 4.1: Chain of ten elastic beads. Speedup is defined as𝑇 /𝑇AES-10,10−6.
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0 1 2 3 4 5
x 10−4
Total Energy
Kinetic Energy
Potential Energy
Time [μsec]
Energy [J]
Figure 4.23: Chain of ten elastic beads. Explicit Newmark solution with Δ𝑡 = 10−4 𝜇sec. Time history of total, kinetic and potential energies— grey curves correspond to the asynchronous energy- stepping solution withℎ= 10−7 J.
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0.5 1 1.5 2 2.5 3 3.5x 10−4
Total Energy Kinetic Energy
Potential Energy
Time [μsec]
Energy [J]
Figure 4.24: Chain of ten elastic beads. Asynchronous energy-stepping solution with ℎ= 10−6 J.
Time history of total, kinetic and potential energies— grey curves correspond to the asynchronous energy-stepping solution withℎ= 10−7 J.
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0 0.5 1 1.5 2 2.5 3 3.5x 10−4
Total Energy Kinetic Energy
Potential Energy
Time [μsec]
Energy [J]
Figure 4.25: Chain of ten elastic beads. Energy-stepping solution withℎ= 10−6J. Time history of total, kinetic and potential energies— grey curves correspond to the asynchronous energy-stepping solution withℎ= 10−7 J.
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0 0.5 1 1.5 2 2.5 3 3.5x 10−4
Total Energy Kinetic Energy
Potential Energy
Time [μsec]
Energy [J]
Figure 4.26: Chain of ten elastic beads. Asynchronous energy-stepping solution with ℎ= 10−6 J.
Time history of total, kinetic and potential energies— grey curves correspond to the energy-stepping solution withℎ= 10−7 J.