Linear shaped-charge jet optimization using machine learning methods Free

Shaped charges are used in a variety of applications for creating deep incisions or perforations in hard, dense materials such as stone and metal. Depending on the intended use, the geometry of a shaped charge can take several forms including a radially symmetric (e.g., a conical shape) or linear configuration (i.e., with linear planar symmetry). Linear shaped charges are used to create deep linear cuts and have many commercial applications in industries including mining, oil and gas extraction/transmission, structural demolition, and rocket systems.^1–3 As a specific example, a linear shaped charge can be used to quickly sever oil or natural gas pipelines, which may be necessary during an emergency scenario.⁴ Linear shaped charges are also used for rapid stage separation in multistage rocket systems for spacecraft launch vehicles.^5,6

Most linear shaped-charge designs involve encasing some type of high explosives (HEs) in a shell of metal or other materials. The casing is open on one side with a concave cavity cut into the HE material. This cavity is usually lined with a thin material that we, subsequently, refer to as the liner. A detonator (or detonators) is used to ignite the HE, which produces a detonation wave that travels through the HE to the liner. When this detonation wave reaches and interacts with the liner, it produces a Richtmyer–Meshkov instability that manifests as the liner material in the cavity begins to invert and is pulled into a narrow high-velocity jet that forms at the central apex of the liner cavity.^7–9 This phenomenon is also sometimes referred to as the Munroe or Neumann effect.¹⁰ Much of the hydrodynamic theory describing the process of liner jet formation was laid out by Birkhoff et al.¹⁰ and Pugh et al.,¹¹ whose work provides an analytical basis for this phenomenon and the penetrative properties of the jet. For many applications, it is desirable for this jet to strike a target (i.e., the material in which an incision is to be made) and penetrate as deeply as possible into this target, which is located at some specified distance from the linear shaped charge.

Work conducted by a number of researchers^12–17 has demonstrated that detonation wave properties can be modified and controlled to alter jetting and penetration properties for various types of shaped charges. Our work investigates several linear shaped charge design parameterizations using two-dimensional hydrodynamics simulations and design optimization methods. We use these design parameterizations to modify properties of the HE detonation waves prior to interaction with the liner material with the goal of altering jet characteristics to increase the penetration depth into a given material target. We run numerous simulations in parallel on a high performance computer (HPC) to generate large datasets. These datasets are then used to train neural network models for use as a surrogate to aid in iteratively optimizing the penetrative properties of the jet. In our analysis, we generally leave the liner angle, thickness, and material unchanged, as these properties have been somewhat extensively studied in previous works.^18–23 

Our parameterizations include altering the positions of multiple detonators and the addition of metal inclusions within the HE. We provide justification for the selection of these design parameterizations using analytical expressions that describe the dynamics of detonation wave interactions, such as the pressure increase from the collision of detonation waves.^13,24,25 Due to the high computational time cost of simulating the penetration of a jet into a target, we also investigate potential optimization metrics that correlate to penetration depth^10,26 but do not require us to fully simulate the penetration of the jet into a target. This is particularly useful for our optimization analysis that involves conducting numerous hydrodynamics simulations. From our analysis, we produce several optimized linear shaped-charge designs with a higher penetration depth than the baseline case and, in some cases, accomplish this using less HE.

The configuration, including dimensions, of our two-dimensional hydrodynamics simulation for the baseline case that we compare to our optimized designs is shown in Fig. 1. Our two-dimensional simulation configuration has linear planar symmetry boundary conditions and extends infinitely in the $z$-direction out of the page. The linear shaped charge in this configuration consists of HE surrounded by a stainless steel casing. On the open side of the HE, an angled cavity protrudes into the HE and is lined with copper. The liner angle that we use for most of our simulations is $α= 60 °$⁠. In front of the liner, we use air at ambient conditions (i.e., atmospheric pressure and room temperature). A single detonator is placed near the back of the HE region, which is used to initiate the detonation wave that strikes the copper liner. Note that the air region extends further than what is shown in Fig. 1 and the total length of the simulation in the $x$-direction is equal to 10 cm.

We construct these simulations using the MARBL hydrodynamics code.^27,28 MARBL is an arbitrary Lagrangian–Eulerian code that uses a high-order finite element method to solve the conservation of mass, momentum, and energy equations.²⁹ Our simulation mesh has approximately 10 600 elements, with higher mesh refinement in the HE and liner region and along the path of the jet. These elements are quadratic for kinematic fields and piecewise-discontinuous linear for thermodynamic fields. Our simulations also output a density and a velocity field image on a fixed Cartesian grid at equally spaced time steps throughout the simulation using the Ascent software library.³⁰ We use these Ascent images to calculate the fitness metric for our optimization analysis, which is described further in Sec. II C.

The materials in our simulations are modeled using the Livermore Equation of State (LEOS) library. The material equations of state for copper, stainless steel, and air, come from LEOS tables 290,³¹ 3010, and 2260, respectively. They are based on the quotidian equation of state with extension formalism.^32,33 For the copper and stainless steel in our simulations, we use the Steinberg–Guinan strength model.³⁴ For the copper strength model, we use a shear modulus of 0.477 Mbar, initial yield stress of $1.2× 10 − 3$ Mbar, and a maximum work hardening stress of $6.4× 10 − 3$ Mbar. Similarly, for the stainless steel strength model, we use a shear modulus of 0.770 Mbar, initial yield stress of $3.3× 10 − 3$ Mbar, and a maximum work hardening stress of $2.5× 10 − 3$ Mbar. To simulate the detonation and combustion dynamics of the HE, we use a reactive flow model,^35–37 which has the ability to account for a broad range of detonation and shock phenomena not captured by programmed burn models.^38,39 We neglect the strength of the unreacted HE in our simulations.

While the analysis of Birkhoff et al. provides an excellent basis for understanding shaped-charge jet formation, it does overlook some important aspects of this process. For instance, the work of Pugh et al.¹¹ provides additional analysis on shaped-charge jet formation, such as accounting for variable collapse velocity $v 0$⁠, to explain the large discrepancies between the Birkhoff et al. formulation and the jet length observed in experiments. The variable collapse velocity $v 0$ proposed by Pugh et al. implies that the collapse angle $β$ (and consequently the angle $δ$⁠) does not remain constant as the detonation wave propagates from the liner apex to the liner base (i.e., the points where the liner contacts the outer casing). According to Pugh et al., even a small gradient or variation in the collapse velocity $v 0$ can produce a large gradient in the jet velocity $v j$⁠. This suggests that an optimal detonation wave angle $ω$ may also change over time as the detonation wave sweeps along the liner from the apex to the base. Figure 5 shows an example of how $β$ may vary as the detonation wave moves along the liner from apex to base.

This analytical theory provides an incentive for investigating different detonation wave front profiles where the angle $ω$ can be varied along the detonation wave front, which allows us to find an optimal wave front for optimizing a penetration metric. While these analytical relations provide a reasonable starting point for understanding shaped-charge jet formation, hydrodynamics simulations are often needed to fully capture the flow dynamics present in the formation of the jet and slug. In the following sections, we describe our methodology for altering and shaping the detonation wave front with the objective of optimizing a fitness metric related to jet penetration depth into a target.

The ultimate goal of our optimization analysis is to increase the penetrative properties of the jet. While we can directly simulate the penetration of the shaped-charge jet into a target, this is computationally inefficient when running numerous simulations, such as for our optimization analysis. Therefore, a correlation between penetration depth and a fitness metric based on more easily computable jet characteristics (e.g., jet velocity and density) is of great value. A fitness metric based on jet characteristics also allows us to use optimization methods to investigate how different designs influence these characteristics. A well-known analytical relation for predicting the penetration depth of the RMI jet was proposed by Birkhoff et al.¹⁰ Several other empirical and analytical relations for predicting penetration depth, as well as other damage metrics, have also been proposed.^26,41,42 We start by considering the penetration depth relation proposed by Birkhoff et al.¹⁰ 

To assess these fitness metrics using our simulations, we include a tungsten target at a distance of 5.0 cm from the liner and use a parameterization in an array of simulations to vary the jet properties such that we obtain a variety of penetration depths. We use a tungsten target over other lower-density materials (e.g., steel) to reduce the time required for the jet to fully penetrate into the target and consequently reduce the time required to run these simulations. We then compute $Φ 1$ and $Φ 2$ for each simulation at a time just before the jet strikes the target. We calculate these fitness metrics using images providing velocity and density fields that are output from our simulations using the Ascent software library,³⁰ which are output at equally spaced time steps. The correlation between penetration depth into the target and the fitness metrics $Φ 1$ and $Φ 2$ are plotted in Figs. 7(a) and 7(b), respectively. As can be seen in Fig. 7, the fitness metric $Φ 2$ correlates much more strongly with penetration depth. The fitness metric $Φ 1$ may simply not be accurate enough for the shaped charge scenario that we consider. Therefore, for our optimization analysis, we compute $Φ 2$ as our fitness metric in place of simulating the penetration of the jet into a target. We also use the maximum value of the fitness metric over all time steps. While the stand-off distance between the liner and the target is an important parameter for determining penetration depth of a given jet, we are more concerned with finding the most penetrative jet at any stand-off distance occurring at any time throughout our simulation for our optimization analysis.

While there exists a great variety of gradient-free optimization methods, we employ a surrogate model based approach where the surrogate fitness metric is denoted by $Φ ^( z)$⁠. This approach is data-driven, needing no information other than values of $Φ( z)$ evaluated at selected sample points in $Z$⁠, but it results in a continuous approximate model that is useful for sensitivity analysis. Some excellent review articles on gradient-free optimization include those of Vu,⁴³ Jones,⁴⁴ Koziel,⁴⁵ and Stork.⁴⁶ In order of increasing complexity, surrogate-based optimization may be based on polynomials, radial basis function interpolation, Gaussian process regression (a.k.a. Kriging), or artificial neural networks (ANNs).⁴⁷ The ANN approach is applied here, specifically a two hidden layer fully connected network using Gaussian activation functions.⁴⁸ The training can be done iteratively and the $i$th iteration of the algorithm proceeds through the following general steps:

1. Define the sample domain $Z i$⁠.

2. Generate $k i$ sample points in $Z i$⁠.

3. Perform $k i$ hydrodynamics simulations.

4. Compute $k i$ fitness metrics.

5. Train ANN to learn the fitness metric as a function of the design variables to create surrogate model $( z , Φ ^ ( z ) )$⁠.

6. Use limited-memory Broyden–Fletcher–Goldfarb–Shanno (LBFGS) repeatedly from random starting points to find all the local maxima of $Φ ^( z)$⁠.

It should be noted that in the literature many gradient-free optimization methods employ a combination of exploration (or diversification) and exploitation (or intensification). The approach described above is heavily weighted toward exploitation meaning that most random exploration is performed during the initial sampling and little subsequent random sampling is performed. As an example, for three design variables, the initial exploration may consist of $3 5=243$ initial samples (we generally round this up to $300$ samples), which is presumed to be a high enough sample density to have one sample in the neighborhood of the global optimum. Subsequent iterations then refine only the most promising regions of design space.

We perform the sampling in Step (1) using a traditional Latin hypercube. A traditional Latin hypercube is an $n$-dimensional sampling with two key properties: (1) a quasi-uniform space filling (no unusually large gaps between the samples); (2) non-collapsed, meaning that no coordinate is replicated. Latin hypercube samples are not unique, and finding the “best” set of samples can be a complicated optimization problem in itself. However, a traditional Latin hypercube gives us a reasonable sampling spread throughout much of the sample domain.

The hydrodynamics simulations performed in Step (3) can be performed in parallel. A scalable asynchronous task scheduler Merlin⁴⁹ is used to manage the workflow. This workflow scheduler runs all of the physics simulations, including the separate tasks of mesh generation, hydrodynamics, and the fitness metric post processing of Step (4). The machine learning in Step (5) utilizes the Pytorch library.⁵⁰ We use a fully connected two hidden layer network with Gaussian activation functions and 50 to 100 nodes for each hidden layer.

In Step (5) of the above algorithm, the well-known LBFGS algorithm is used to find all of the local maxima of the ANN model. The LBFGS algorithm is a quasi-Newton method, requiring gradients but not Hessians. The gradient of the model is easily computed using the built-in automatic differentiation capability of Pytorch. This method of course converges to a local maximum, and our goal is to find the global optimum. It is a simple matter to execute the LBFGS algorithm over and over again with $O( 10 5)$ different starting points in the domain $Z$⁠. It must be noted that while a single hydrodynamics simulation may require $8$ CPU hours, evaluation of the ANN model only requires approximately $0.1$ seconds, so this is not an expensive proposition. While no statistical proof is provided, it was observed that this approach found between 3 and 10 distinct local maximum thus the ANN model does not suffer from extreme oscillations and the global optimum of the ANN model is indeed found.

The above algorithm can be repeated multiple times where new samples are generated in a refined (or narrowed) sample domain within some fraction of the original domain centered at the global optimum found in the previous iteration (e.g., within $±5%$⁠). However, we found that a single iteration followed by further simulations run within a refined sample domain (within $±5%$⁠) was generally sufficient for our purposes. For a study with three design parameters, we typically use 300 hydrodynamics simulations in the first iteration and 100 further simulations in the refined sample domain. Each of our two-dimensional hydrodynamics simulations can be run in less than an hour on a single GPU (NVIDIA V100 on the Lassen HPC⁵¹). This constitutes the optimization methodology that we use. We apply this methodology to the design parameterizations described in the following section.

For our analysis, we investigate many different design parameterizations. However, for brevity, we will only focus on a few of the best performing parameterizations. The design parameterizations that we consider include altering the position of one or more detonators within the HE region, altering the time delay between the ignition of multiple detonators, and the addition of a metal inclusion (we use Cu for this purpose) within the HE region. When the HE detonation wave strikes the inclusion, it produces a shock wave that travels through the inclusion at a slower shock velocity $u s$ than the detonation wave velocity $u d$⁠. These parameterizations allow us to tune the detonation wave front to some degree and even split the detonation wave into multiple portions to induce detonation wave collisions.

The diagrams in Fig. 8 show several sample designs that we investigate in our study. The parameterization in Fig. 8(a) involves using three detonators, where two detonators are symmetrically placed about the central axis in the HE region and a third detonator is varied along the central axis in the $x$-direction. Figure 8(b) shows a parameterization that also involves three detonators, but introduces a time delay between the detonator on the central axis and the symmetric detonators (represented by different detonation wave radii in the diagram). The two symmetric detonators can be varied only along the stainless steel casing and the central detonator fixed near the edge of the casing. In this parameterization, either the two symmetric detonators or the central detonator can detonate first. The parameterization of Fig. 8(c) involves using a copper inclusion in a triangular or chevron shape. As shown in the diagram, the dimensions of the inclusion are varied, but symmetry about the central axis is maintained.

As previously mentioned, the design parameterizations that we have chosen allow us to modify the properties and shape of the detonation wave (or waves) that strike the copper liner. One phenomenon of particular interest is the localized increase in pressure that occurs due to collisions between detonation waves and whether this localized pressure increase is able to significantly affect properties of the resulting jet. The work of Zhang et al.²⁵ demonstrates that the locations of detonation wave collisions can be used to predict where fractures form along a steel plate coupled to HE with an array of detonators. Liu et al.¹³ have also investigated the collision of detonation waves in relation to shaped charges applications. The maximum pressure in the collision region $P 2$ (i.e., the region where the detonation waves collide and overlap) relative to the original pressure of the individual detonation waves $P 1$ can be correlated to the incidence angle $ϕ 0$ between two symmetric converging detonation waves. An analytical expression for the ratio $P 2/ P 1$ is derived using the Rankine–Hugoniot jump conditions in the  Appendix. In the  Appendix, we also plot $P 2/ P 1$ vs incidence angle $ϕ 0$ for a range of heat capacity ratios $γ$ for different types of HE. We hypothesize that we may be able to enhance jetting by taking advantage of this phenomenon.

In the following paragraphs, we present optimal results for several of our design parameterizations obtained using the optimization methodology described in Sec. II D. Figure 9 shows a density plot for an early time “initial” state for each of the design parameterizations that we present. Figure 9(a) shows the baseline case with a single detonator centered at the left of the HE region. Figures 9(b)–9(d), show the optimal initial states for the design parameterizations described in Figs. 8(a)–8(c), respectively. Density plots at a time just prior to the detonation wave striking the liner are shown in Fig. 10 for the same parameterizations as in Fig. 9. Velocity vectors where the size of the vector corresponds to magnitude are overlaid on the plots in Fig. 10. Pressure plots for the baseline case and for each of the design parameterizations are shown in Fig. 11.

Ultimately, the optimal designs from several of our parameterizations produce very similar detonation wave fronts. This appears to be a case of “convergent evolution” where our vastly different parameterizations produce the same outcome when optimized. This wave front can be seen for three different parameterizations in Figs. 11(b)–11(d), where the front is slightly concave as opposed to the convex wave front seen in the baseline case in Fig. 11(a). Localized high-pressure regions along the wave front can also be observed in Figs. 11(b)–11(d). These high-pressure regions along the wave front correspond to collisions between distinct detonation waves. These distinct detonation waves either emanate from the detonators or for the case in Fig. 11(d) have been “split” into distinct waves by the metal inclusion. However, these high-pressure regions appeared to have a fairly minor effect on jetting properties. This is likely due to the fact that these high pressure regions are very localized and do not affect liner collapse velocity outside of a relatively small area. Further discussion on the effect of detonation wave collisions is provided later in this section and in the  Appendix.

Density plots of jets produced by the baseline case and for an optimal case are shown in Figs. 12(a) and 12(b), respectively. The optimal jet in Fig. 12(b) has a longer length, higher tip velocity, and more of the mass is concentrated toward the front of the jet than the baseline jet in Fig. 12(a). All of these properties contribute to increasing the fitness metric $Φ 2$ for the optimal jet and consequently the penetration depth.

We also found that this optimal detonation wave front can be reproduced in a simpler manner by arraying three or more detonators along a specific function, such as a semicircle or parabola. For a semicircular arrangement, we refer to the center of the full circle as the focal point $F$⁠, which corresponds to the semicircle radius (see Fig. 13). Using the Merlin workflow software,⁴⁹ we ran several sets of simulations using $N d=3$⁠, $5$⁠, $7$⁠, and $60$ detonators arrayed along a semicircle to investigate the effect of modifying $F$⁠. For $N d=60$⁠, the detonator array essentially produces a continuous detonation wave (see Fig. 14). Figure 15 plots the fitness metric $Φ 2$ [see Eq. (16)] vs $F$ for several sets of simulations involving different values of $N d$⁠. Figures 15(a)–15(c) are for cases where the liner angle $α= 60 °$⁠, $45 °$⁠, and $30 °$⁠, respectively. Note that $F<0$ implies that the array of detonators is convex and produces a convex detonation wave (similar to the baseline case) and $F>0$ implies that the array of detonators is concave and produces a concave detonation wave. As the magnitude of $F$ increases, the detonation wave becomes flatter.

The simulation data plotted in Fig. 15 demonstrates that there is generally a distinct optimal value of $F$⁠. As expected, the optimal value of $F$ changes depending on the liner angle $α$ that is used. For instance, for the case where $α= 30 °$ and $N d=3$⁠, the optimal value is $F≈3.6$ cm, whereas the optimal value is $F≈2.5$ cm for the case where $α= 60 °$ and $N d=3$⁠. This optimal value of $F$ also varies slightly as $N d$ is increased. For the optimal value, $F>0$⁠, which indicates that the optimal detonation wave front is slightly concave. This is consistent with the optimal detonation wave fronts produced from our other design parameterizations. Figure 14 shows the detonation wave prior to striking the target for the case where $F=2.5$ cm, $N d=60$⁠, and $α= 60 °$⁠, which is clearly similar to the detonation wave fronts produced by our other optimal designs shown in Figs. 10 and 11.

The natural question arises as to why these particular detonation wave fronts tend to optimize jetting. A partial answer to this question can be understood by considering the diagrams in Figs. 4 and 5. For a given liner collapse angle $β ^$⁠, we assume that there is a corresponding optimal collapse velocity $v 0$ that will produce values of $v j$ and $m j$ that correspond to an optimal fitness metric [e.g., the metric $Φ 2$ in Eq. (16)]. Since $β ^$ is not constant and varies as the detonation wave sweeps across the liner, the optimal collapse velocity $v 0$ will also vary over time. By tuning the incidence angle of the detonation wave $ω$⁠, which corresponds to the shape of the detonation wave front, we can ideally ensure that we obtain the optimal collapse velocity $v 0$ at all times because $v 0$ is directly dependent on $ω$ via Eq. (5). We note that altering the type of HE used or making minor modifications to the liner geometry could alter the values of the optimal parameters needed to achieve an optimal detonation wave front. For instance, using a different type of HE would affect the detonation velocity and would likely require modifying the detonator locations to reproduce the optimal detonation wave front.

The plots in Fig. 15 also appear to demonstrate that the pressure increase from detonation wave collisions (⁠ $P 2/ P 1$ as described in the  Appendix) are having a relatively minor effect on our fitness metric $Φ 2$ and consequently the penetrative properties of the jet. As the number of detonators $N d$ along the semicircular array is increased, the point detonators tend to blend into a single continuous line and there are essentially no detonation wave collisions induced between the detonation waves emanating from the individual point detonators. Figure 15 demonstrates that, for $N d=60$ where detonation wave collisions are negligible, there is very little change in $Φ 2$ from the simulations that use $N d=5$ where detonation wave collisions are induced. This is a surprising result and appears to indicate that these detonation wave collisions have a less significant effect on the jetting properties than the angle $ω$ at which the detonation wave strikes the liner.

Our optimization study demonstrates that there is an optimal detonation wave front profile that increases the penetration depth of the shaped-charge jet into a given material. We were able to reproduce this detonation wave front profile using several different parameterizations that involve modifying the locations of multiple detonators, the time delay between detonators, and using a metal inclusion within the HE region. The fact that we are able to arrive at very similar optimal detonation wave fronts indicates a “convergent evolution” effect and implies that we have obtained a fairly optimal point in the parameter search spaces that we considered.

The collision between detonation waves is another phenomenon that we investigated in our study. Using our design parameterizations, we are able to instigate collisions between detonation waves, which increases the pressure in the collision region between the waves. We had originally hypothesized that the localized pressure increases due to detonation wave collisions may help to increase jet velocity and penetration depth. However, we found that the shape of the detonation wave profile appeared to be a much more significant factor in optimizing the linear shaped charge jetting than these localized pressure increases from detonation wave collisions.

Future work may involve applying some of the lessons learned in this study to other linear shaped-charge geometries. This includes modifying liner properties to observe the effect that this has on the optimal detonation wave front. Experimental studies may also provide some additional insights and provide a means of validating the optimal designs that we developed in this study.

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344 and was supported by the LLNL-LDRD Program under Project No. 21-SI-006. We also thank J. Cassell, A. B. Kostinski, and W. J. Schill for helpful discussions and guidance related to the research presented in this paper. LLNL IM: LLNL-JRNL-847005.

The authors have no conflicts to disclose.

Dane M. Sterbentz: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). Charles F. Jekel: Conceptualization (equal); Methodology (equal); Software (equal); Writing – review & editing (equal). Daniel A. White: Conceptualization (equal); Investigation (equal); Methodology (equal); Software (equal); Writing – original draft (equal); Writing – review & editing (equal). Robert N. Rieben: Resources (equal); Software (equal); Writing – review & editing (equal). Jonathan L. Belof: Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Resources (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Most shape charge designs involve a single detonator or a single line of detonators that create a single detonation wave that interacts with the liner material. However, the collision between multiple detonation waves initiated from separate detonators can increase pressure in the collision region between the detonation waves above the pressure of the individual waves, which we hypothesized could be used to enhance the kinetic energy of the jet. In the following paragraphs we use the Rankine–Hugoniot equations to derive the bounds on this pressure increase and its relational dependence on the angle at which the detonation waves collide (i.e., the incidence angle). We begin by considering the symmetric collision of two detonation waves in two dimensions as shown in Fig. 16.

When two detonation waves collide, the detonation wave front behind the collision point is converted to a shock wave that traverses the products that have been reacted by the opposing detonation wave. This causes a refractive effect as the shock wave is not self-sustaining and is moving through a new medium (i.e., the reacted products rather than the reactants through which the detonation waves move). The collision of two detonation waves creates three distinct regions, which are the undetonated region ahead of the detonation wave (Region 0), the region behind each detonation wave (Region 1, which excludes the collision region), and the collision region (Region 2) between the two detonation waves [see Figs. 16(b) and 17]. We denote properties of these regions using the subscripts $0$⁠, $1$⁠, and $2$⁠, respectively.

Following the methodology described by Zhang et al.,²⁵ Bdzil and Short,⁵² Zhang,²⁴ and Liu et al.,¹³ we can approximate the behavior within the three regions of the box shown in Fig. 16(b) (i.e., the three regions near the collision point) as being equivalent to the stationary reflection of an oblique detonation wave at a rigid wall where the reflected wave is a shock wave. Under this approximation, we can describe the behavior in the three regions using the diagram of Fig. 17. Regions 0 and 1 are separated by the incident oblique detonation wave with angle $ϕ 0$ and Regions 1 and 2 are separated by the reflected shock wave with angle $ϕ 2$⁠. Due to the refractive effect, the angle of incidence $ϕ 0$ is generally not equal to the angle of reflection $ϕ 2$⁠. Note that the velocity at which mass enters perpendicularly to the stationary oblique detonation wave is equal to the detonation wave velocity $u d$ in this approximation. In the following paragraphs, we develop a relation for the ratio of the pressure near the collision point in the collision region to the pressure of the individual detonation waves (i.e., $P 2/ P 1$⁠) as a function of the incidence angle $ϕ 0$⁠.

By simultaneously solving the implicit relation in Eq. (A25) and the expressions in Eqs. (A12), (A15), and (A19), we can compute the pressure ratio $P 2/ P 1$ as a function of only the incidence angle $ϕ 0$ and the ratio of specific heats $γ$ (which is a material property of the HE through which the detonation waves are propagating). Figure 18(a) provides a plot of $P 2/ P 1$ vs the incidence angle $ϕ 0$ for several values of $γ∈[2,3]$⁠. Similarly, Fig. 18(b) provides a plot of $θ$ vs the incidence angle $ϕ 0$ for several $γ$ values. According to Bdzil and Short,⁵² there is a critical incidence angle $ϕ 0 , c$ (generally between $40 °$ and $50 °$⁠) above which the reflected shock wave becomes irregular. At this point, the assumption that the velocity $u 2$ is parallel to the rigid wall breaks down and the formulation described above is no longer valid. For $ϕ 0> ϕ 0 , c$⁠, the collision point tracks along a line rather than a single point. The plots in Fig. 18 are only plotted up to this critical incidence angle $ϕ 0 , c$⁠.