Abstract

Computational physicists are commonly faced with the task of resolving discrepancies between the predictions of a complex, integrated multiphysics numerical simulation, and corresponding experimental datasets. Such efforts commonly require a slow iterative procedure. However, a different approach is available in casesx where the multiphysics system of interest admits closed-form analytic solutions. In this situation, the ambiguity is conveniently broken into separate consideration of theory–simulation comparisons (issues of verification) and theory–data comparisons (issues of validation). We demonstrate this methodology via application to the specific example of a fluid-instability-based ejecta source model under development at Los Alamos National Laboratory and implemented in flag, a Los Alamos continuum mechanics code. The formalism is conducted in the forward sense (i.e., from source to measurement) and enables us to compute, purely analytically, time-dependent piezoelectric ejecta mass measurements for a specific class of explosively driven metal coupon experiments. We incorporate published measurement uncertainties on relevant experimental parameters to estimate a time-dependent uncertainty on these analytic predictions. This motivates the introduction of a “compatibility score” metric, our primary tool for quantitative analysis of the RMI + SSVD model. Finally, we derive a modification to the model, based on boundary condition considerations, that substantially improves its predictions.

1 Introduction: The Value of Analytic Solutions

A common challenge for physics verification and validation efforts is to resolve discrepancies between the predictions of a complex multiphysics numerical simulation and corresponding experimental datasets. Distinguishing issues of model validation (e.g., Does the physics model reflect reality?) from those of code (e.g., Is the model implementation free of errors?) and solution verification (e.g., Is the implementation numerically accurate?) can be challenging, particularly when the underlying physics models and/or their implementations are themselves under evaluation and thus subject to frequent alteration. Definitively separating these issues often requires a slow, and therefore costly, iterative procedure [1, Fig. 4]. Occasionally, however, the system of interest may admit closed-form analytic solutions (here abbreviated as “theory”). This breaks the ambiguity by distinguishing between theory–simulation comparisons (i.e., issues of verification) and theory–data comparisons (i.e., issues of validation) as illustrated in Fig. 1.

Fig. 1
Relationship between analytic solutions, experimental data, and numerical simulations
Fig. 1
Relationship between analytic solutions, experimental data, and numerical simulations
Close modal

This work applies this analytic solution approach to the combination of Los Alamos's Richtmyer–Meshkov instability (RMI)-based ejecta source model [25] with a prescription for a self-similar velocity distribution (SSVD) [6,7], here abbreviated as the “RMI + SSVD” source model. We begin by presenting an analytic solution [810] for piezoelectric ejecta mass measurements relevant to a particular class of high explosive-driven experiments [11,12]. Next, to aid quantitative analysis, we define the concept of “model compatibility scores.” With these formalisms in place, we proceed to assess the RMI + SSVD source model in two ways. First, we use the analytic solutions to verify the quality of the model implementation in the continuum mechanics code flag; second, we use the analytic solutions to validate the accuracy of the model predictions. Then, motivated by the validation assessment, we derive from first principles a modification to the SSVD—based on boundary condition requirements inherent in the raw data—that substantially improves the model predictions.

2 Predictions of the Richtmyer-Meshkov Instability + Self-Similar Velocity Distribution (RMI + SSVD) Source Model

It has been known for over 60 years [13] that the passage of a shock through a free metal surface tends to generate a fine spray of microscopic particulates (ejecta). Perhaps one of the most fundamental questions arising from the study of this phenomenon pertains to the amount of material produced: more concretely, the aggregate mass of the ejecta. Several diagnostic techniques exist to tackle this question. One method involves fielding piezoelectric sensors in the path of the expanding ejecta cloud. Under certain conditions, the time-dependent piezoelectric voltage measured by this sensor [V(t)] is straighforwardly related to the accumulation of ejecta mass per unit area of the collecting surface [m(t)] [14,15]. Here, we refer to the latter quantity as the piezoelectrically inferred mass.

In order for an ejecta source model to be predictive, it must accurately calculate the dynamics of the ejecta cloud, including its time-dependent mass deposition. Piezoelectrically inferred masses are therefore a crucial touchstone for the development of theoretical and numerical ejecta models. This work focuses on measurements of ejecta launched into vacuum by a singly shocked planar metal coupon with a constant free-surface velocity. This scenario describes several experiments using high explosive-driven tin coupons [11,12], and the complete piezoelectric dataset from one such experiment has been provided to us by the lead experimenter (Buttler). (This corresponds to the experimental data in Fig. 1.) Further details of that experiment are comprehensively documented in the associated publication [11].

Figure 2 depicts the problem geometry for this scenario. We begin by defining the source areal mass function (AMF) as a moment of the ejecta particle distribution at creation
mc(w,tc)1Amfc(m,w,tc)dm
(1)
Fig. 2
Simplified problem geometry. Dashed line: initial (solid) coupon free surface for times t≤t0. Blue line: postshock (liquid) coupon free surface at time tc>t0, after impulsive acceleration to lab-frame velocity ufs. The shocked surface launches ejecta particles of relative velocity w toward a piezoelectric sensor situated at initial distance h with collecting area A. Ejecta particle creation may be instantaneous at t0, or it may be sustained over an unknown interval [t0,tcf]. The current treatment sets t0=0 for simplicity with no loss of generality. The problem is defined such that all quantities are non-negative.
Fig. 2
Simplified problem geometry. Dashed line: initial (solid) coupon free surface for times t≤t0. Blue line: postshock (liquid) coupon free surface at time tc>t0, after impulsive acceleration to lab-frame velocity ufs. The shocked surface launches ejecta particles of relative velocity w toward a piezoelectric sensor situated at initial distance h with collecting area A. Ejecta particle creation may be instantaneous at t0, or it may be sustained over an unknown interval [t0,tcf]. The current treatment sets t0=0 for simplicity with no loss of generality. The problem is defined such that all quantities are non-negative.
Close modal
This function exists even if the underlying distribution function fc is unknown. It represents the ejecta mass generated per unit area of the free surface from particles of relative velocity w at creation time tc, and therefore carries units of mass per unit volume. Similarly, the sensor AMF ma(u,t) represents the ejecta mass collected per unit area of the sensor from particles of lab-frame velocity u at measurement time t. Causality and conservation of mass together require
ma(u,t)=(uuufs)mc[w=uufs,tc=uthuufs]
(2)

where here the arguments of mc embody the shift from the source model's reference frame, which is moving with the free surface, to the lab frame of the motionless sensor.

If the ejecta cloud mass per unit area is conserved during transport to the sensor (as is assumed by the standard piezoelectric data analysis procedure), then it follows directly that the true mass accumulation on a motionless sensor prior to arrival of the free surface (i.e., t<tfs(h/ufs)) is given by
mt(t)=htufsdw0tc(w,t)mc(w,tc)dtc
(3)
where the lower limit on the velocity integration comes from the causality requirement, and
tc(w,t)(w+ufs)thw
(4)

is the creation time required for a particle with relative velocity w to reach the sensor at measurement time t.

As explained in Ref. [15], the procedure for obtaining the piezoelectrically inferred cumulative ejecta mass, mi(t), begins by equating pressure on the sensor with the ram pressure of the ejecta cloud, P(t)=ρ(t)v2(t), where ρ is the average mass density of the cloud and v is the velocity of particles arriving at time t. A further assumption is that the ejecta particles are created instantaneously; in this case, particles arriving at time t must have a velocity determined by the time-of-flight: v=(h/t). Taken together, this yields a simple expression for the inferred mass per unit area
mi(t)=0tρ(t)v(t)dt=0tP(t)v(t)dt=1h0tP(t)tdt
(5)
Experimentally, in a regime where the piezoelectric sensor has a linear response function, the pressure is simply related to the voltage trace [15]
P(t)=1ARS0tV(t)dt
(6)
(This relationship is useful for computing variances on mi, as described below.) Analytically, however, the pressure on the sensor is calculated directly as the momentum flux of arriving particles
P(t)=htufsma(u,t)udu=htufs(u2uufs)mc[uufs,uthuufs]du
(7)

When combined with Eq. (6), this yields a prediction for the time-dependent piezoelectric voltage trace.

Combining Eqs. (5) and (7) yields the analytic expression for the piezoelectrically inferred cumulative mass
mi(t)=mt(t)+1hhtufswdw0tc(w,t)tcmc(w,tc)dtcmt(t)+Δm(t)
(8)

For any given source model described by a source AMF mc, the analytic predictions provided by Eq. (3) and (8) (i.e., “theory” in Fig. 1) are directly comparable to the piezoelectric experimental data. Previous work [9,16] has found that the epistemic uncertainty represented by mi(t)mt(t) is generally quite small.

The RMI source model [25] posits that imperfections on the free surface seed the growth of a Richtmyer–Meshkov fluid instability, and that the ejecta originate from the disintegration of RMI spike features. This provides an ejecta mass production rate. Experimental observations suggest the spike features evolve self-similarly, thereby yielding a velocity distribution (SSVD) for the particles. The combined RMI + SSVD model predicts a cumulative ejecta mass produced up to time tc in particles with relative velocity in [w,η˙s] [6,7]
m(t)=23m0ln(1+tcβτ)·wη˙sg(w)dw0η˙sg(w)dw
(9)
where g(w)((ξe(ξw/η˙s)+1)/(ξ+1)). When the RMI spike-tip velocity (η˙s, the maximum relative velocity in the spike) is constant and equal to its asymptotic value, the physical quantity described by Eq. (9) arises from a source AMF of the form
mc(w,tc)=23m0·1tc+βτ·ξeξwη˙s+1(2eξ)η˙s·Π0tcf(tc)
(10)

where the boxcar function terminates ejecta production at a specific shutoff time, tcf. The asymptotic-velocity approximation is used to keep the analytic derivations tractable. Calculations using the RMI ejecta package in flag, which solves numerically for the temporal evolution of η˙s [17], find this quantity reaches its asymptotic value in approximately 100 ns. This timescale is much shorter than the tcf values required for the RMI source model to produce the measured ejecta masses in relevant tin coupon experiments [11,12,18,19]. The spike tip velocity is therefore approximately constant for most of the RMI ejecta production interval for the shots under consideration here.

When the ejecta arrival (measurement) times at the sensor exceed the production interval at the source (i.e., t>tcf) as for the experiments considered here, calculations must account separately for t<t* and t>t*, where t*((h+η˙stcf)/(η˙s+ufs)) represents the final arrival (depletion) time for the fastest ejecta (i.e., particles with w=η˙s). This caveat is necessary to ensure that all particles have wη˙s: t>tcf implies, via causality, that w((hufst)/(ttcf)), yet the latter quantity exceeds η˙s when t<t*.

Combining Eqs. (3) and (10) yields the following expression for the true ejecta mass accumulation on the sensor
mt(t)={2m0ξ3η˙s(2eξ)htufshufstttcfeξwη˙sln[p(t)+q(t)w]dw+2m03η˙s(2eξ)htufshufstttcfln[p(t)+q(t)w]dw+2m0ξ3η˙s(2eξ)ln(1+tcfβτ)hufstttcfη˙seξwη˙sdw+2m03η˙s(2eξ)ln(1+tcfβτ)hufstttcfη˙sdw(t>t*)2m0ξ3η˙s(2eξ)htufsη˙seξwη˙sln[p(t)+q(t)w]dw+2m03η˙s(2eξ)htufsη˙sln[p(t)+q(t)w]dw(tt*)
(11)
where p(t)1+(t/βτ) and q(t)(ufsth/βτ). Similarly, Eqs. (8) and (10) together yield
Δm(t)={2m0βτξ3ηs˙h·[p(t)1]2eξhtufshufstttcfweξwηs˙dw+2m0βτξ3ηs˙h·q(t)2eξhtufshufstttcfeξwηs˙dw2m0βτξ3ηs˙h(2eξ)htufshufstttcfweξwηs˙ln[p(t)+q(t)w]dw+2m0βτ3ηs˙h·[p(t)1]2eξhtufshufstttcfwdw+2m0βτ3ηs˙h·q(t)2eξhtufshufstttcfdw2m0βτ3ηs˙h(2eξ)htufshufstttcfwln[p(t)+q(t)w]dw+2m0ξ3ηs˙h(2eξ)·[tcfβτln(1+tcfβτ)]hufstttcfηs˙weξwηs˙dw+2m03ηs˙h(2eξ)·[tcfβτln(1+tcfβτ)]hufstttcfηs˙wdw(t>t*)2m0βτξ3ηs˙h·[p(t)1]2eξhtufsηs˙weξwηs˙dw+2m0βτξ3ηs˙h·q(t)2eξhtufsηs˙eξwηs˙dw2m0βτξ3ηs˙h(2eξ)htufsηs˙weξwηs˙ln[p(t)+q(t)w]dw+2m0βτ3ηs˙h·[p(t)1]2eξhtufsηs˙wdw+2m0βτ3ηs˙h·q(t)2eξhtufsηs˙dw2m0βτ3ηs˙h(2eξ)htufsηs˙wln[p(t)+q(t)w]dw(tt*).
(12)

Certain terms in mi(t) and mt(t) must be evaluated using multivalued special functions (i.e., they contain a branch cut in the complex plane). However, only unphysical (acausal) scenarios lead to complex or multivalued solutions. Strict enforcement of causality, such that all evaluations pertain to particles arriving at the downstream sensor after they are created at the source, is sufficient to keep these evaluations strictly real-valued.

2.1 Uncertainty Bounds on the Analytic Predictions.

The analytic calculations take as input a set of experiment parameters, and these carry their own measurement uncertainties. Table 1 lists the uncertainty values applied to these parameters and the source from which each value was derived. As these measurements are uncorrelated, we apply the standard variance formula [20] to these values in order to estimate 1σ uncertainty bands for the analytically calculated mt(t) and mi(t) (See Sec. 3 for a discussion of the uncertainty levels used in this study.). The results shown here may underestimate the true uncertainty as they do not incorporate the measurement uncertainty on the postshock free surface velocity, ufs. The uncertainty contribution owing to the assumption of instantaneous creation (a necessary piece of the piezoelectric analysis [15] and therefore also of the mi derivation) is negligible compared to the other factors incorporated here [9,16].

Table 1

Experimental uncertainties used to assign uncertainty bands to analytic predictions

UncertaintySource
σh= 25 μmVogan et al. [11], Table I caption
σΩ= 0.5 OhmsLeCroy LT374 data sheet [21]
σA= 0.1 mm2Estimated pin radius uncertainty σha
σS= 1.55 pC/NVogan et al. [11], Sec. Ib
σt0= 0.216 μsVogan et al. [11], Sec. III. Ac
UncertaintySource
σh= 25 μmVogan et al. [11], Table I caption
σΩ= 0.5 OhmsLeCroy LT374 data sheet [21]
σA= 0.1 mm2Estimated pin radius uncertainty σha
σS= 1.55 pC/NVogan et al. [11], Sec. Ib
σt0= 0.216 μsVogan et al. [11], Sec. III. Ac
a

This is a blind estimate: σh is the published uncertainty on the pin distance measurement, whereas the true uncertainty on the radius of the piezoelectric sensor collecting surface should be determined by an unrelated measurement or the manufacturer's specifications.

b

As per Ref. [11], literature and manufacturer values for the piezoelectric sensitivity of LN sensors vary between 20.9 and 24 pC/N (crudely, 22.45±1.55 pC/N).

c

The uncertainty in the on-axis shock breakout time is 0.116 μs, compounded with an overall timing jitter of 100 ns.

3 Compatibility Scores

In order to quantify comparisons with the analytic predictions, we define a new metric. The “compatibility score” represents the fraction of the physical quantity of interest (in this case, time-dependent ejecta mass delivered to the sensor) over which two methods for obtaining that quantity exhibit overlapping uncertainty bands.

In the verification sector (Fig. 1), the two methods are the analytic source model prediction and the associated numerical simulation; in this case, the compatibility score seeks to answer the question, “How faithfully does the numerical code implementation represent the intended physics?” In the model validation sector, the methods are analytic prediction and experimentation; here, the compatibility score addresses the question, “How well does the physics hypothesis accommodate the experimental data?”

It is beneficial to compute separate “verification” and “validation” scores in this manner, as this makes it straightforward to identify a variety of scenarios. For instance, if the model prediction and numerical simulation agree yet are both inaccurate representations of the data owing to a shared constant bias, this will manifest as a high verification score but a low validation score. Alternatively, if the model accurately accommodates the data but the numerical simulation deviates from the model prediction, this will manifest as a low verification score but a high validation score.

For the purposes of the current discussion, we assume the uncertainty bands or error bars are sufficiently well-defined such that the overlap criterion is a reasonable definition of compatibility. (Of course, when the error bars represent true statistical variances, it is possible for a model prediction and a measurement to be compatible even when the associated error bars are not contiguous [22].)

The compatibility score calculation is straightforward. Given a pair of time-dependent ejecta mass accumulation curves—with or without associated uncertainty bands—we construct a function that is unity everywhere the curves overlap and zero elsewhere. Next, we convert this function of measurement time into a function of the fraction (percentage) of total ejecta mass delivered to the sensor. The integral of this function is the compatibility score. Because the score is guaranteed to land in the range [0, 100] (with 0 indicating total incompatibility and 100 indicating perfect compatibility), it constitutes a simple, intuitive, and easily calculated metric related, under certain conditions, to other quantitative assessment methodologies [23].

Score values calculated this way depend upon the criteria used to establish compatibility. The appropriate uncertainty level for quantitative assessments in a particular application may be determined by several criteria: stakeholder requirements, model use tolerances, etc. [22]. The compatibility score metric, computed with 1σ uncertainties, was sufficient to answer the questions articulated earlier in this section.

4 Verification: Theory–Simulation Comparisons

In flag simulations of an ejecta experiment, the piezoelectric sensor is represented by a tally surface. The mass accumulation on this tally surface is derived from the passage of computational macroparticles, which the flag ejecta package uses to represent the physical ejecta particles [17]. The code parameter numperptcl determines the number of physical ejecta particles represented by each simulation macroparticle; lower values yield better statistics at increased computational expense. The tally surface calculation is a proxy for the mass accumulation that would be inferred from a piezoelectric voltage trace. Statistical noise in the tally surface calculation varies inversely with the number of macroparticles; the number of macroparticles generated in each computational time-step scales inversely with the RMI timescale βτ and numperptcl. The tally noise is additionally sensitive to ϑ, the ratio of ejecta cloud transit time to the ejecta production interval: larger ϑ values mean the cloud becomes more distended during transit (owing to the distribution of particle velocities), thereby reducing the number of macroparticles arriving on the tally surface per unit time. Shot noise therefore becomes more prominent with increasing ϑ.

In the limits βτ0 and ϑ0, the fully time-dependent source model calculation in flag exhibits excellent agreement with analytic predictions from the time-independent approximation to RMI + SSVD source model, as shown in Figs. 3 and 4. There we also see that, as expected, reducing numperptcl improves simulation/theory agreement, albeit slowly. These findings are quantified via the associated compatibility scores, which are listed in Table 2. Because these calculations do not at present account for the full envelope of statistical variability that would arise from an ensemble of flag calculations, the tabulated values should be taken as lower bounds on the compatibility scores.

Fig. 3
Predicted and simulated ejecta mass accumulation at the sensor for Vogan et al. [11] shots 6 and 8. Blue: analytic predictions [mt(t)] from the stationary velocity distribution approximation. Red, orange: flag tally surface results. The analytic predictions were constructed to match the first-arrival times in the flag calculations, and both the analytic and numerical calculations for each shot used a tuned tcf (see below). Note shot 8 is outside the domain of validity of the RMI source model owing to the large kh0 value, so a data comparison would not be appropriate for this shot, though theory/simulation comparison remains valid. The flag tallies saturate at late times owing to radial expansion of the simulated ejecta cloud; the standard piezo analysis, reflected in the analytic calculations, inherently assumes [∇·u]r=0. All numerical and analytic calculations used ξ=7.2.
Fig. 3
Predicted and simulated ejecta mass accumulation at the sensor for Vogan et al. [11] shots 6 and 8. Blue: analytic predictions [mt(t)] from the stationary velocity distribution approximation. Red, orange: flag tally surface results. The analytic predictions were constructed to match the first-arrival times in the flag calculations, and both the analytic and numerical calculations for each shot used a tuned tcf (see below). Note shot 8 is outside the domain of validity of the RMI source model owing to the large kh0 value, so a data comparison would not be appropriate for this shot, though theory/simulation comparison remains valid. The flag tallies saturate at late times owing to radial expansion of the simulated ejecta cloud; the standard piezo analysis, reflected in the analytic calculations, inherently assumes [∇·u]r=0. All numerical and analytic calculations used ξ=7.2.
Close modal
Fig. 4
As Fig. 3, for Vogan et al. [11] shots 10 and 12
Fig. 4
As Fig. 3, for Vogan et al. [11] shots 10 and 12
Close modal
Table 2

Theory/simulation compatibility scores for the four Vogan et al. [11] shots in Figs. 3 and 4 (1σ variance on the theoretical predictions, 0 variance on the simulation results).

Shotkh0βτ (ns)ϑnumperptclScore
081.275.791.7810691.4
060.199.9841.4110666.6
060.199.9841.4110478.1
120.2577.634.4610611.9
120.2577.634.4610433.8
100.08165.24.9510613.2
Shotkh0βτ (ns)ϑnumperptclScore
081.275.791.7810691.4
060.199.9841.4110666.6
060.199.9841.4110478.1
120.2577.634.4610611.9
120.2577.634.4610433.8
100.08165.24.9510613.2

The flag calculations solved numerically for the temporal evolution of the spike-tip velocity, η˙s, while the analytic theory calculations used a time-independent approximation. The scores listed here should be taken as lower bounds.

The RMI + SSVD source model is implemented in a subgrid fashion [17]. Furthermore, in these calculations, the computational macroparticles travel ballistically from the simulated coupon surface to the tally surface without a background mesh. Thus, these results are independent of the simulation resolution (i.e., the concept of asymptotic mesh-resolution regimes may be inapplicable here).

This analysis quantitatively demonstrates that in the limit of low noise, the time-independent approximation to the RMI + SSVD source model is a good representation of the fully time-dependent source model. The excellent agreement in the limits βτ0 and ϑ0 constitutes a major step toward verifying the RMI + SSVD ejecta source model implementation in flag. Furthermore, because the analytic predictions involve independently derived equations, which are not coded into the flag ejecta package, this agreement also serves as an independent check on the analytic derivations.

However, as βτ increases, the simulation results rapidly diverge from the analytic model prediction. Additional work may be required to reduce the strong dependence upon the RMI timescale. A closer examination of noise in the flag calculations is ongoing.

5 Validation: Theory–Data Comparisons

When comparing model predictions to the experimental data, we restrict our attention to low-kh0 shots, as these lie within the domain of applicability of the RMI + SSVD source model.

5.1 Thresholding.

The piezoelectric ejecta mass data from planar tin coupon experiments [11,12] (and many others) commonly indicate the presence of a very low mass, early arriving “tail” to the distribution of ejecta particles. This tail contains roughly 1% of the total measured ejecta mass, and may be governed by physics beyond the scope of the RMI + SSVD ejecta source model. The model itself specifically targets 99% of the ejecta mass [24], namely, the material not included in this tail. Thus when computing analytic model predictions for a validation study, we use input parameters extracted from piezoelectric datasets thresholded at 99% of the total measured mass. In particular, the analytic calculations are constructed to match the (empirical) thresholded data-onset time, t99. Investigation finds t99 corresponds—within the applicable uncertainty bands—to the first-arrival time, ta0, predicted by the source model for approximately half of the kh0<0.5 shots in the Vogan et al. [11] dataset. Pinning the analytic calculations to t99 is therefore an idealization, but one that simplifies theory/data comparison.

5.2 Ejecta Production Interval (tcf).

The RMI + SSVD source model does not specify the ejecta production interval. To assess the sensitivity of this study to this unknown parameter, we investigated two methods for setting it:

  • Tuning: Here, tcf was set separately for each shot such that at the end of the production interval, the source model had produced exactly 99% of the published total ejecta mass for that shot. This ideal production interval establishes a baseline for comparison, by forcing the model prediction and the thresholded data to agree at the final measurement time.

  • Prescribing: Here, tcf=aλ/ufs for each shot (a=30,40,50). The value a = 40 produces empirical agreement with the measured time-integrated masses [5]. We vary a to assess the impact of deviating from this nominal value.

Owing to the logarithmic time dependence of the RMI source model, changing the prescribed production interval by ±25% (a=40±10) changes the total predicted mass for these shots by only ±10%8%. This is within the published 1σ10% measurement uncertainty on integrated ejecta masses obtained from lithium niobate (LN) piezoelectric sensors [12,14,25]. Thus, for these particular shots, distinct values of a within this range are indistinguishable in the piezoelectric ejecta mass data.

5.3 Scores.

Given a specification for the time-dependent piezoelectric measurement uncertainty, a compatibility score can be computed from the piezoelectric data and the RMI + SSVD model prediction (which, as explained above, carries an approximate 1σ variance derived from uncertainties on the input parameters). This metric reflects the percentage of mass for which the model prediction, with its associated uncertainty, is compatible with the specified uncertainty bounds on the experimental data.

The published 1σ measurement uncertainty on the cumulative ejecta mass per unit area at the final measurement time, obtained from LN piezoeletric sensor measurements such as those under consideration here, is approximately ±10% [12,14,25]. Based on this uncertainty at a single time, we construct two hypothetical models, which are intended to represent simplistic bounding cases on the true but unknown time-dependent piezoelectric ejecta mass measurement uncertainty:

  • Constant: the uncertainty is ±10% at all times.

  • Linear: the uncertainty is 100% at the time of first arrival at the sensor (i.e., the maximum possible uncertainty at the data-onset time), followed by a linear decline to ±10% at the final measurement time.

The constant case represents the most straightforward extrapolation from the known uncertainty at the final measurement time to uncertainties at all measurement times. The linear case reflects the fact that the nonzero uncertainty in the true arrival time of the ejecta cloud's leading edge at the sensor translates into a 100% uncertainty in the detected mass at that time (i.e., has the first particle arrived or not?). The higher compatibility scores obtained from the linear model provide a useful upper bound on the data's compatibility with a 1σ uncertainty on the model prediction.

Figures 5 and 6 show analytic predictions from the time-independent approximation to the RMI + SSVD source model, computed using tuned tcf values, coplotted against thresholded piezoelectric data with a constant ±10% uncertainty. Table 3 lists compatibility scores for the full gamut of scenario combinations considered here.

Fig. 5
Predicted and piezoelectrically measured ejecta mass accumulation at the sensor for Vogan et al. [11] shots 6 and 10. Blue: analytic predictions [mi(t)] from the stationary velocity distribution approximation. Black: pieozoelectric data (courtesy of Buttler), thresholded at 99%. The analytic predictions shown here were computed using a tuned tcf value and were constructed to match the data-onset time for the thresholded data (t99). A constant model for the time-dependent uncertainty was applied to the thresholded data. Compare column 2 of Table 3.
Fig. 5
Predicted and piezoelectrically measured ejecta mass accumulation at the sensor for Vogan et al. [11] shots 6 and 10. Blue: analytic predictions [mi(t)] from the stationary velocity distribution approximation. Black: pieozoelectric data (courtesy of Buttler), thresholded at 99%. The analytic predictions shown here were computed using a tuned tcf value and were constructed to match the data-onset time for the thresholded data (t99). A constant model for the time-dependent uncertainty was applied to the thresholded data. Compare column 2 of Table 3.
Close modal
Fig. 6
As Fig. 5, for Vogan et al. [11] shots 11 and 12
Fig. 6
As Fig. 5, for Vogan et al. [11] shots 11 and 12
Close modal
Table 3

Theory/thresholded data compatibility scores computed for low-kh0 Vogan et al. [11] shots

ShotTuneda =30a =40a =50
Constant measurement uncertainty
655.638.1≈0≈0
1047.749.634.614.1
1127.133.328.420.4
1260.958.349.136.8
Linear measurement uncertainty
664.852.436.40
1058.860.747.832.9
1143.151.844.834.9
1279.878.073.369.7
ShotTuneda =30a =40a =50
Constant measurement uncertainty
655.638.1≈0≈0
1047.749.634.614.1
1127.133.328.420.4
1260.958.349.136.8
Linear measurement uncertainty
664.852.436.40
1058.860.747.832.9
1143.151.844.834.9
1279.878.073.369.7

The piezoelectric data are courtesy of Buttler. Results for both tuned and prescribed tcf values are listed. The range of prescribed values corresponds to a range of total ejecta masses smaller than the published measurement uncertainty [12,14,25]. The analytic predictions from the time-independent approximation to the RMI + SSVD source model were constructed to match the data-onset time for the thresholded data (t99), which may slightly artificially increase the scores, and were given 1σ uncertainties as described above.

For the particular shots and scenarios considered here, the degree of congruence between predictions of the RMI + SSVD ejecta source model and the thresholded (99%) piezoelectric data ranges from ≈0 to ≈80%. In some cases, a scenario that yields a high score for one shot yields a very low score for a different shot: using a prescribed tcf value with a =50 and a linear model for the hypothetical measurement uncertainty yields a score of 70% for shot 12 but 0% for shot 6. Of the eight scenarios considered here, only one (prescribed tcf with a =30 and a linear measurement uncertainty model) yields scores greater than 50% for all shots.

In addition, model/data congruence can be highly sensitive to the ejecta production interval, tcf: variations in tcf over a range undetectable within the uncertainty on the total ejecta mass (see above) can significantly alter the level of agreement in these hypothetical scenarios (e.g., by 3.5 × for shot 10 under the constant measurement uncertainty scenario).

More generally, and as demonstrated in Figs. 5 and 6, the RMI + SSVD source model tends to overpredict the mass stored in the earliest-arriving ejecta, relative to the thresholded data. Compatibility between the analytic predictions of the source model and the thresholded piezoelectric data tends to occur within the later-arriving material.

6 Boundary Conditions for Ejecta Source Models

Guided by the model validation assessment in Sec. 5, we naturally contemplate how, or whether, alterations to the source model might improve the congruence between its analytic predictions and the data.

A valuable clue to the behavior of the model predictions can be obtained by considering general characteristics of the raw piezoelectric voltage measurements (see Sec. 2) in this class of experiments [11,12]. In particular, such traces (1) rise smoothly from the baseline, and (2) are continuous everywhere, to within the resolution of the measurement (e.g., Ref. [11, Fig. 4]). These properties encode at least three mathematical requirements on the analytically predicted voltage trace:

  1. V(ta0)=0

  2. V(ta0)=0

  3. limtt+V(t)=limttV(t)t

where ta0 is the ejecta first-arrival time at the sensor.

We consider the general class of ejecta source models featuring a stationary velocity distribution, or which are well-approximated as such. Such models are described by source AMFs of the form
mc(w,tc)=m0g(tc)f(w)Πt0tcf(tc)
(13)
where the boxcar function is used to ensure termination of ejecta production at t=tcf. For instance, the stationary approximation of the RMI + SSVD ejecta source model (Eq. (10)) has
g(tc)=231tc+βτf(w)=ξeξwŵ+1(2eξ)ŵ
(14)

We may apply the analytic voltage calculation described in Sec. 2 to the general source AMF of Eq. (13) to obtain mathematical expressions for the initial voltage, initial voltage slope, and magnitude of any discontinuities in the voltage trace.

The following results were obtained by converting the generic source AMF of Eq. (13) to the sensor-frame AMF via Eq. (2); then deriving a generic expression for the pressure on the sensor via Eq. (7); and, finally, differentiating that expression to obtain the analytic voltage via Eq. (6).

6.1 Initial Voltage: V(ta0).

The procedure described above yields [26]
V(ta0)=ARSm0hû4ŵg(0)f(ŵ)
(15)

It immediately follows that any ejecta source model in this class will produce V(ta0)=0only if g(0)=0,f(ŵ)=0, or both. Equation (14) shows the RMI + SSVD source model satisfies neither condition; it is therefore guaranteed to predict a voltage trace characterized by a finite value at the first arrival time, i.e., V(ta0)0.

6.2 Initial Voltage: V(ta0).

Similarly, a second differentiation of the expression for the pressure on the sensor yields [26]
V(ta0)=ARS{m0h2[û6ŵ24û5ŵ]g(0)f(ŵ)+m0hû5ŵ2g(0)f(ŵ)m0h2û6ŵg(0)f(ŵ)}
(16)

It immediately follows that any stationary-velocity-distribution source model will produce V(ta0)=0only if g(0)=g(0)=0, g(0)=f(ŵ)=0,f(ŵ)=f(ŵ)=0, or some combination thereof. Equation (14) shows the RMI + SSVD source model satisfies only f(ŵ)0 for sufficiently large ξ (such as the value ξ=7.2 used in all the analytic and numerical calculations of this study). The source model is therefore guaranteed to predict a voltage trace characterized by a finite slope at the first arrival time, i.e., V(ta0)0.

6.3 Continuity: limtt+V(t)=limttV(t)t.

Clearly, from the derivations summarized in Sec. 2, the mathematical functions derived for t<t* and t>t* are continuous functions. The only measurement time at which the pressure derivative (and hence the voltage) might become discontinuous is t=t*. Applying the derivation of the generic expression for the voltage prediction to both branches of the calculation, and evaluating these at t*, yields [26]
ΔV(t*)=ARSm0hufstcfû4ŵg(tcf)f(ŵ)
(17)

from which it immediately follows that the predicted voltage trace will be continuous at t=t* only if g(tcf)=0,f(ŵ)=0, or both. The RMI + SSVD source model satisfies neither condition; it is therefore guaranteed to predict a discontinuous voltage trace, i.e., ΔV(t*)0.

6.4 A Modification to the Source Model.

Combinations of boundary and initial conditions that simultaneously satisfy all three voltage requirements include
g(0)=f(ŵ)=0f(ŵ)=f(ŵ)=0g(0)=g(0)=g(tcf)=0

The third scenario suggests a model incorporating a smooth “start-up” and a natural termination at tcf. This, combined with the variability of compatibility scores with production interval (Sec. 5), argues strongly for the need to augment the source model with additional physics to accommodate a natural termination. This is a topic of ongoing research.

Meanwhile, the second scenario can be satisfied, or approximately so, with a very simple modification to the SSVD. Given f(ŵ)0 for the values of ξ under consideration here, we need only construct a new velocity distribution, f̃(w)κ[f(w)f(ŵ)] where the normalization factor κ ensures mass conservation via 0f̃(w)dw=1. Then, f̃(ŵ)=0 by construction and f̃(ŵ)=f(ŵ)0. As shown in Fig. 7, this modification improves the voltage predictions exactly as expected.

Fig. 7
Predicted and measured piezoelectric voltage traces for Vogan et al. [11] shots 6 and 10. Black: data (courtesy of Buttler). Blue: analytic prediction from the original SSVD. Green: analytic prediction computed from the modified SSVD. The original model generates discrepant values of V(ta0), V′(ta0), and ΔV(t*).
Fig. 7
Predicted and measured piezoelectric voltage traces for Vogan et al. [11] shots 6 and 10. Black: data (courtesy of Buttler). Blue: analytic prediction from the original SSVD. Green: analytic prediction computed from the modified SSVD. The original model generates discrepant values of V(ta0), V′(ta0), and ΔV(t*).
Close modal

One can also show, via a similar analysis [26], that ensuring f(ŵ)=f(ŵ)=0 has the additional benefit of enforcing a very smooth rise to the analytically predicted cumulative ejecta masses: mt,i(ta0)=m˙t,i(ta0)=m¨t,i(ta0)=mt,i(ta0)=0. This alteration to the model prediction is so beneficial as to obviate the need for data thresholding, as illustrated by the theory–data comparisons in Figs. 8 and 9, as well as by the associated compatibility scores listed in Table 4.

Fig. 8
Predicted and piezoelectrically measured ejecta mass accumulation at the sensor for Vogan et al. [11] shots 6 and 10. Black: unthresholded pieozoelectric data (courtesy of Buttler). Blue, Green: analytic predictions [mi(t)] computed using the original and modified (boundary aware) forms of the SSVD, respectively. The analytic predictions were computed using a tuned tcf value and were constructed to match the data-onset time for the data (ta0). A constant model for the time-dependent uncertainty was applied to the data. These plots show approximate 1σ uncertainty bands.
Fig. 8
Predicted and piezoelectrically measured ejecta mass accumulation at the sensor for Vogan et al. [11] shots 6 and 10. Black: unthresholded pieozoelectric data (courtesy of Buttler). Blue, Green: analytic predictions [mi(t)] computed using the original and modified (boundary aware) forms of the SSVD, respectively. The analytic predictions were computed using a tuned tcf value and were constructed to match the data-onset time for the data (ta0). A constant model for the time-dependent uncertainty was applied to the data. These plots show approximate 1σ uncertainty bands.
Close modal
Fig. 9
As Fig. 8, for Vogan et al. [11] shots 11 and 12. Notice how applying a slightly larger uncertainty band on the analytic predictions (e.g., 1.5σ) would significantly improve theory/data compatibility for the modified SSVD case (green).
Fig. 9
As Fig. 8, for Vogan et al. [11] shots 11 and 12. Notice how applying a slightly larger uncertainty band on the analytic predictions (e.g., 1.5σ) would significantly improve theory/data compatibility for the modified SSVD case (green).
Close modal
Table 4

Theory/unthresholded data compatibility scores computed for low-kh0 Vogan et al. [11] shots, using both the original and modified forms of the SSVD.

1σ2σ
ShotOriginalModifiedOriginalModified
634.888.652.097.2
1032.276.450.983.1
1121.746.840.289.8
1240.453.157.585.5
1σ2σ
ShotOriginalModifiedOriginalModified
634.888.652.097.2
1032.276.450.983.1
1121.746.840.289.8
1240.453.157.585.5

The piezoelectric data are courtesy of Buttler. These results were computed for both 1σ and 2σ uncertainties on the data and model, using a constant measurement uncertainty on the data and tuned tcf values in the model.

7 Conclusions

The existence of a closed-form analytic solution for the time-dependent mass accumulation upon a motionless piezoeletric sensor (specifically in the case of planar, single-shock ejection into vacuum from a constant-velocity free surface) provides a powerful tool for conducting both verification and validation studies of Los Alamos's RMI + SSVD ejecta source model. To quantify these studies, we have introduced a “compatibility score” metric.

The scores are context-dependent. Nevertheless, we have found that the simple intuitive nature of the metric, coupled with its ease of calculation, makes it a valuable tool for the verification and validation studies conducted here. Furthermore, under certain conditions, the compatibility score may be viewed as a limiting case of other metrics [23].

Comparison of the analytic predictions to numerically computed tally surface data from the continuum mechanics code flag has verified the implementation of the RMI + SSVD source model in that code, particularly in the limit of low statistical noise. Deviations between simulation and theory grow quickly as a function of the RMI timescale βτ, which impacts the number of computational macroparticles generated in the calculation.

Comparison of the analytic predictions to thresholded piezoelectric data finds the RMI + SSVD ejecta source model tends to overpredict the amount of mass stored in the earliest-arriving (highest-velocity) material. The level of agreement between the model predictions and the thresholded data varies widely across shots and uncertainty scenarios, and is sensitive to the production interval, which is unspecified in the current source model.

The analytic formalism is an extremely powerful tool for exploring modifications of the source model. The voltage and mass predictions are both significantly improved by adjusting the SSVD to bring its properties in line with requirements imposed by global properties of the piezoelectric voltage data in this class of experiments [11,12].

Acknowledgment

The authors thank Fred Wysocki for many enjoyable and enlightening conversations, Alan Harrison for insightful suggestions and feedback, and William Buttler for providing piezoelectric ejecta mass data. This work was supported by the U.S. Department of Energy through the Los Alamos National Laboratory. Los Alamos National Laboratory is operated by Triad National Security, LLC, for the National Nuclear Security Administration of U.S. Department of Energy (Contract No. 89233218CNA000001).

Funding Data

  • U.S. Department of Energy through the Los Alamos National Laboratory (Funder ID: 10.13039/100000015).

Nomenclature

     
  • a =

    coefficient on prescribed tcf value

  •  
  • A =

    collecting area of sensor

  •  
  • fc =

    ejecta particle distribution function at creation

  •  
  • h =

    initial distance from free surface to sensor

  •  
  • mc =

    source areal mass function

  •  
  • mi =

    piezoelectrically inferred mass accumulation

  •  
  • mt =

    true mass accumulation

  •  
  • m0 =

    areal density normalization

  •  
  • R =

    terminating impedance of the piezoelectric circuit

  •  
  • S =

    piezoelectric sensitivity

  •  
  • t =

    particle measurement (arrival) time at the sensor

  •  
  • ta0 =

    ejecta first-arrival time at the sensor

  •  
  • tc =

    particle creation time at the free surface

  •  
  • tcf =

    ejecta production termination time

  •  
  • tfs =

    free-surface arrival time at the sensor

  •  
  • t0 =

    shock breakout time at the free surface (=0 by fiat)

  •  
  • t99 =

    time at which 1% of the total ejecta mass has arrived

  •  
  • t* =

    final arrival time for particles with w=η˙s

  •  
  • u =

    lab-frame particle velocity

  •  
  • ufs =

    lab-frame free-surface velocity

  •  
  • w =

    particle velocity relative to the free surface

  •  
  • ŵ,û =

    maximum relative and lab-frame ejecta velocities

  •  
  • βτ =

    RMI timescale

  •  
  • η˙s =

    RMI spike-tip velocity

  •  
  • λ =

    initial seed perturbation wavelength

  •  
  • ξ =

    SSVD scaling factor (=7.2 for all calculations here)

  •  
  • ϑ =

    ratio of ejecta transit time to tcf

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