*4.2. Numerical Results*

Axisymmetric ALE and Eulerian numerical methods incorporating the spatial and temporal profiles of the laser beam pressure pulse were implemented in the simulation of the LIW process. In the axisymmetric ALE simulation, the effect of the standoff distance between the foils on the impact angles, nodal velocities, springback, and overall shape of the deformed flyer foil were investigated. Using the Eulerian technique, the jetting phenomenon and the interlocking of the foils along the weld interface were simulated. In addition, the hypothesis that simulation results assuming a constant initial flyer velocity and shape are not as accurate is tested.

The axisymmetric ALE simulation result for the same sample case (laser fluence of 31.08 J/cm2, and standoff distance of 0.26 mm) is compared to the experimental result, as shown in Figure 13. The diameter of the welded region obtained from experimental and numerical results is 2.5 mm and 2.2 mm, respectively (12% difference). The overall shape of the deformed foils was successfully simulated, showing good agreemen<sup>t</sup> with experimental results as seen in Figure 13. Since the effect of the double-sided tape (discussed in experimental results) was neglected, the annular gap observed in the experiments was not fully captured. Therefore, an investigation into the effect of double-sided tape can be pursued in the numerical simulations of the LIW process as part of future research.

**Figure 13.** Axisymmetric simulation results: deformed shapes and comparison to experimental results; Contour plots include S: von Mises stress (Pa), V: velocity (m/s), and PEEQ: equivalent plastic strain.

To investigate the importance of the temporal profile of the laser beam pressure pulse, an axisymmetric ALE simulation for a case with a larger standoff distance (0.40 mm) and higher laser fluence (36.06 J/cm2) was created. The simulation results for this case at different times during the second step of the LIW process are shown in Figures 14 and 15. As seen in these figures, a significant amount of springback is observed. This is due to the fact that the plasma pressure pulse duration (~300 ns, see Figure 8) is significantly shorter than the collision start time (366 ns, see Figures 14 and 15). This means that the pressure load application ends before the flyer collides with the target. Since there is no force to resist the bouncing of the foils, the springback occurs due to the very high flyer velocities (~1550 m/s, see Figure 15) and low impact angles in the center of the impact region.

**Figure 14.** Axisymmetric simulation results: von Mises stress (Pa) contour plot and deformed shape at different times during the collision.

**Figure 15.** Axisymmetric simulation results: velocity (m/s) contour plot and deformed shape at different times during the collision.

A closer look at the velocities obtained in the collision region (over 5600 m/s, see Figure 16) reveals that jetting velocities were reached in the ALE domain. However, the ALE method is not capable of fully capturing the jetting phenomenon. Note that jetting is one of the known requirements for a successful weld [24,31,32] and happens when metal particles from the foils reach very high velocities upon collision and are ejected from the surface. The resulting 3D images of the deformed foils at 1 μs for this sample case are shown in Figure 17.

**Figure 16.** Axisymmetric simulation results at 400 ns: (**a**) Jetting velocities (m/s); (**b**) The resultant velocity vectors in the collision region.

One of the most crucial parameters in the successful execution of LIW is the amount of initial gap between the two foils [25]. If this standoff distance is too small, then the flyer will not have enough time to sufficiently accelerate and/or might not reach the required minimum impact angle before making contact with the base. Conversely, if this gap is too large, the flyer might not reach the base plate or might exceed the maximum required impact angle before the collision with the base. Therefore, it is very important to assess the effect of standoff distance on the impact angles and velocities in different regions of the flyer. The axisymmetric ALE simulation results for sample cases with a laser fluence of 31.08 J/cm<sup>2</sup> and standoff distances from 0.1 mm to 0.4 mm are provided in Table 6.

**Figure 17.** Resultant 3D images of the deformed foils and von Mises stress (Pa) contour plots (the axysimmetric simulation at 1 μs): (**a**) Overview of the geometry; (**b**) Closer view of the springback.


**Table 6.** The effect of standoff distance based on axisymmetric LIW simulation.

As mentioned earlier, due to the Gaussian profile of the laser beam, the applied pressure load on the flyer is axisymmetric. Therefore, the resulting impact velocities are maximum in the center of the laser spot and decrease in the radial direction away from the center. Impact angles and velocities at the point of collision for the sample case are shown in Figure 15 (at t = 366 ns). The drastic velocity gradients between different regions of the flyer show the importance of incorporating the spatial profile of the laser beam. As seen in Table 5, the maximum velocity is reached at a distance between 0.3 and 0.4 mm away from the initial flyer position. This means that if the standoff distance is large enough, at a certain time the flyer starts to decelerate, showing the importance of considering the temporal profile of the laser beam. If a constant velocity had been assumed, not only would the changes in accelerations and velocities be ignored over time, but so would the gradients between the different regions of the flyer have been neglected. At the point of collision, starting from the center and moving away radially, the impact angle starts at zero degrees, gradually increases to a maximum, and then slowly decreases back to zero close to the edges of the flyer. This shows that assuming an initial shape or a constant impact angle does not give an accurate representation of the LIW process. To support this claim, two di fferent LIW configurations were simulated using an Eulerian technique applying a constant downward initial flyer velocity of 800 m/s. One assumed an initial flyer impact angle and the other assumed a curved initial flyer shape. These additional simulation results are provided for the sake of comparison with the primary work here in which measured spatial and temporal profiles of the laser beam pulse have been incorporated. As seen in Figure 18, the application of a constant initial velocity results in excessive rebounding of the foils upon collision and is therefore not realistic. Moreover, despite using an Eulerian grid, the jetting phenomenon is not observed at any time during the corresponding LIW simulations.

**Figure 18.** *Cont*.

**Figure 18.** Eulerian volume fraction (EVF) of the aluminum foil in angled and curved LIW orientations at different times in Eulerian simulations assuming an initial flyer shape and constant velocity.

Implementing an Eulerian simulation, the jetting phenomenon and the interlocking of the foils along the weld interface were mimicked. The Eulerian simulation results for a sample case with a 37.30 J/cm<sup>2</sup> laser fluence and a 0.4 mm standoff distance are shown in Figure 19. These results are the volume fraction plots of the aluminum flyer foil. It can be seen that in contrast to some of the reports in the literature [30–32], most of the jet consists of steel target particles. This is despite the fact that aluminum is the softer material compared to steel. Therefore, an experimental investigation into the composition of the jet in the LIW process can be pursued as part of future research.

**Figure 19.** *Cont*.

**Figure 19.** Jetting phenomenon at different times and interlocking of the foils along the weld interface in Eulerian simulation of the LIW process.

As seen in Figure 19, the interlocked foils bounce off in the center of the weld region. This could be attributed to the high fluence of the laser pulse (37.30 J/cm2) and modeling of the fixed metal specimen as a rigid body, while in reality the metal specimen is made of aluminum. Therefore, the effect of the fixed metal specimen material on rebounding of the foils can also be further studied as part of future research.

### **5. Conclusions and Future Work**


**Author Contributions:** Conceptualization, S.S.; methodology, S.S., G.H.G., M.I.H., S.F.S. and H.Y.; software, S.S., G.H.G., M.I.H., S.F.S, and H.Y.; validation, S.S.; formal analysis, S.S., G.H.G., S.F.S. and H.Y.; investigation, S.S., G.H.G. and M.I.H.; resources, A.S.M., D.Q.; data curation, S.S.; writing—original draft preparation, S.S.; writing—review and editing, S.S., A.S.M., S.F.S., G.H.G., and H.Y.; visualization, S.S.; supervision, A.S.M.; project administration, A.S.M.; funding acquisition, A.S.M., D.Q.

**Funding:** This research was funded by The University of Texas at Dallas.

**Conflicts of Interest:** The authors declare no conflict of interest.
