By adjusting the N, F, and G parameters in the continuous models, we produced results similar to those observed for the DTM with thin weak layers (N=4.2, F=2.2, G=2.2) and thick weak layers (N=21.4, F=1.0, G=5.0). The depth and radius of the crater produced in the continuous target were within 1% and 6% of the craters produced in the discrete targets with thin- and thick-weaker layers, respectively. The maximum and final volume show small discrepancies between the DTMs and CTMs: for the thin-layer best-fit model, the maximum and final volume differ by 3.3% and 5.1%, respectively, while differences of 10% and 6%, respectively, are observed for the thick layer best fit model. We can also compare the runtimes for the CTMs and the DTMs, although it should be noted that these were simulations were run on a cluster with core clock speeds that ranged from 2.0 GHz to 2.53 GHz, so significant deviations in runtime are observed within each set of models; average runtimes and the size of the high-resolution zone for each target geometry are listed in Table 3Error! Reference source not found.. Since the anisotropic strength model results in higher runtimes (due to increased time spent during the advection step), the 15 cppr CTMs are comparable to the 20 cppr DTMs in terms of computational efficiency. We note that the resolution of the CTMs can be set arbitrarily; the parameters in the anisotropic strength model are set independent of the resolution, allowing us to simulate thin layers without the need for higher resolutions and consequently higher runtimes. For example, the CTM can be kept at a resolution of 15 cppr and the anisotropic strength parameters can be set to approximate layers of 100 m thickness; a target geometry with a minimum layer thickness of 100 m for a projectile with a radius of 750 m would require a resolution of 30 cppr in the DTMs, allowing us to save hundreds of computational hours.
Table 3: Comparisons of average simulation runtimes for the CTM and DTM target geometries. Also included is the size of the high-resolution zone in the computational mesh. Simulation Set Resolution Average Runtime (hours) High- Resolution Mesh Size Minimum Layer Thickness (1 cell; 4 cells) CTM 15 cppr 92 400x255 DTM 15 cppr 51 400x255 50; 200 m 20 cppr 88 533x340 37.5; 150 m 25 cppr 235 666x425 30; 120 m 30 cppr 408 800x510 25; 100 m
The most significant differences between the models are observed when comparing the uplift of the base layer and the thickness of material on the crater floor. Examining the plot of BLU and TAB for the discrete models (Fig. 2.7) and the continuous models (Fig. 2.13), the relationships between the uplift and inward collapse of material for the two models are distinct. For the DTM, the BLU increases to a maximum and then decreases (Fig. 2.7a), while an increase in N leads to a gradual decrease in BLU (Fig. 2.13a). For our TAB measurements, the DTM shows little change until weak layers reach a thickness of 2 km, after which a steady increase is observed (Fig. 2.7b). For the continuous models, the TAB increases nearly linearly as N is increased (Fig. 2.13b).
Furthermore, there are also regions in the continuous target models (primarily in the suite of simulations with increasing N) in which subsurface layers undergo noticeable deformation, as visualized using tracer particles (Figs. 2.15a–b). This deformation is not present in the isotropic models, nor is it present in the DTMs. The deformed region in the continuous target models appears to form due to the inward collapse of material initially transported just outside the transient cavity rim or displaced in the wall of the transient cavity, which counteracts the outward collapse of the temporary central peak, resulting in the apparent faulting observed in the crater floor region. It is notable that bedding planes become vertically oriented at a similar radial distance from the crater centre at the Haughton impact structure (Osinski and Spray 2005). These zones of deformation have also been observed in numerical models of Haughton (Collins et al. 2008). Our results thus support the hypothesis that the area of most significant deformation at Haughton and other similar-sized craters – i.e., the zone of vertically oriented bedding planes– is due to the inward collapse of the transient cavity walls and rim interacting with the outward collapse of the central peak (in agreement with field observations from Osinski and Spray (2005)).
The continuous target models do a reasonable job at replicating the inclusion of weak layers in the target (as evidenced by the reproduction of final depth and radius, maximum and final volume, and the BLU and TAB; see Table 2); although there are some shortcomings (e.g., in the deformation of layers observed at ~10 km when the anisotropy parameters are set to relatively large values, discussed in the preceding paragraph). The depth-diameter trend observed when increasing the parameters used in the Tsai-Hill criterion (primarily
the N parameter), as well as when increasing the thickness of the interbedded weak layers, are nearly identical (Fig. 2.11). Aside from subtle differences in target deformation (Figs. 2.14, 2.15) and transient cavity excavation (which is more noticeable when weak layers are set to a lower friction coefficient; Fig. 2.9), the adjustments to the yield criterion correctly reproduce subsurface impacts into targets with uniform subsurface layering. Furthermore, although the results of impacts into the discrete layered targets appear to be resolution independent, there is a minimum resolvable thickness of those layers. To halve the minimum thickness of the weak layers in the DTMs (say, for example, from 50 m to 25 m), the spatial resolution needs to be doubled (in our case, from 15 cppr to 30 cppr). This is the main advantage of the continuous target models; the thickness of included ‘layers’ are not specified, and therefore the resolution can be set independent of target structure. By adjusting the Tsai-Hill parameters appropriately, the crater morphology generated by an impact into layered targets can be approximated without the need for increasing spatial resolution or generating a more complex target geometry.
Setting these parameters also presents the greatest challenge in using the anisotropic strength model: settling on an appropriate choice for the range of parameters in question to model a given target geometry. For instance, using these anisotropic parameters to find a best-fit model for Haughton, which possesses a very complex target stratigraphy with many thin, weak layers interbedded with thicker, stronger layers (e.g., Osinski and Spray 2005) is a non-trivial task. We provide a range of parameters (N ~ 3 – 22; F, G ~ 1 – 5) that allow us to fully describe the simplified geometries explored in the DTMs. We suggest that these parameters be set freely within this range to replicate the target geometry, noting that increasing the N parameter most closely correlates with an increase in the ratio of weaker layers to stronger layers within the target (i.e., the total weak layer thickness in our analyses); F and G can be subsequently adjusted to fine-tune the desired fit. For instance, in our best fit models, N=4.2 provided a good fit to a DTM geometry in which the weaker layers made up ~14% of the target sequence, while N=21.4 fit well when the composition of the sedimentary sequence was ~86% weaker material. However, the target geometries explored in the DTM are idealized: the weak layers are distributed evenly throughout the target sequence and are all of equal thickness. Although we do not consider more complex target geometries in this work (i.e., unevenly distributed weaker layers, targets containing
weaker layers of varying thickness, etc.), researchers looking to adapt this strength model to their simulations may use our results and best-fit parameter selections as a starting point.
It should be noted that numerous studies have analyzed stress-strain and failure stress data from triaxial compression tests of inherently anisotropic geological materials. From these data, the Tsai-Hill yield parameters may be computed. For instance, Young and Silvestri (1979) utilized the Tsai-Hill yield criterion to describe the anisotropic behavior of St. Louis clay. They determined that the parameters used in the yield criterion are F=1.5, G=H=0.5, and N=4.91. The yield envelope fit to uniaxial and triaxial compression and extension data of Four-mile gneiss provide temperature-dependent parameters in the modified Tsai-Hill criterion: the constants varied from F=4.03–5.72, G=2.41–3.25, H=1.76–1.49, and N=6.23–10.0 between room temperature and 700℃ (Gottschalk et al. 1990). Although the goal of this study was to explore the range of parameters suitable to replicate the effect of thin, weak layers evenly distributed within the target sequence on complex crater formation, we note that the anisotropic strength model could be further validated by replicating small scale triaxial experiments within iSALE.