UC-34

Issued: November 1976

Accuracy of the Conventional Lagrangian SchemeI

for One-Dimensional Hydrodynamics

by

Wildon Fickett

!I

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Notation.

,

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At - time stepAy, y = x, p, u, or p - calculated minus exact solution at the given time at the test particle

or leading shockN - total number of computation cellsp - pressureq - artificial viscosityp - densityt - timeu - particle velocityU - shock velocityx - distance2- thicknessSub-zero - initial

Glossary

Body - the body whose motion is calculatedCalculation - a calculation with the PAD codeCell - a computation cell of the finite-difference methodCell size - the cell width, proportional to N-1Dial - a parameter of the numerical methodExtrapolation - to N = UJHE = high explosiveNumber of cells - the total number of computation cells NPiston - a left boundary whose motion is a specified function of timeProblem - a test problemResidual error - a non-zero extrapolated value of AyShell - a section of the body consisting of a single material

‘-_*-

apshot - a profile at fixed time,

Standard calculation - a calculation with the standard dial settings for the particular. ~ ~ problem‘~ ~ , Test particle - the particle whose motion is studiedi!je l-- -“

ZLn :_.

—b .d——ml ..s—

..-

E=OI Units

?;s:~ “’- “3-~=% The units are those of the coherent S1 set based on mm, mg, and ALS,which give p in GPa, p~~~1

9“~ ~ ~~n_ Mg/mS, u in kmls, and e in MJlkg.

- In1.

Significant Figures and Exponents

Numerical values are rounded to three significant figures. Positive and negative exponentsare indicated by P and N. Thus 3N4 = 3 x 10-4.

...111

ACCURACY OF THE CONVENTIONAL LAGRANGIAN SCHEME

FOR ONE-DIMENSIONAL HYDRODYNAMICS

by

Wildon Fickett

ABSTRACT

The accuracy of the conventional Lagrangian scheme for one-dimensionalhydrodynamics is assessed by detailed comparison with a variety of testproblems having exact solutions. The choice of test problems is made in the

context of wave motions in the pressure range of detonations in condensed

explosives.For each test problem the calculation is made for 16,32, 64, 128,and 256

computation cells. The results and the exact solution are compared along

particle and shock paths by superposing a plot of the exact solution on a plotof the calculated result. Also, the integrated errors over a typical time spanare plotted against the reciprocal number of cells.

The effect of changing such parameters of the computational method asthe form and amount of artificial viscosity is investigated.

For these relatively simple geometries, accuracy better than that found fortypical experimental geometries can be achieved with 100or 200cells and aCDC 7600computer time of a minute or two.

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1. INTRODUCTION

The “standard” Lagrangian numerical method for one-dimensional inviscid hydrodynamicsdescribed in 1967 by Richtmyer and Morton’ has remained a de facto standard, with onlyminor variations, and continues to be used extensively for a variety of problems. Some tests ofits accuracy have been made,2 but we needed a detailed study in the field of shock anddetonation waves in the pressure range associated with condensed explosives.

The first part of this work required a set of test problems that admit exact solutions, andthese have been assembled.’ Here we report on the testing of the standard numerical method,as embodied in the PAD code’ running on the Control Data Corp. 7600 computer, againstsome of the test problems. Throughout this report, the unmodified word “calculation” meansa calculation made with PAD.

The method uses a number of “dials” or parameters of the numerical method, such as theform and amount of artificial viscosity, the number and distribution of the computation cells,and the size of the time step. Typical “standard” dial settings are chosen for each problem

(see Sec. II) and, by varying only the number of computation cells, we can display the resultsof the calculation and its comparison with the exact solution. Section III contains com-parisons made with different dial settings. Results are summarized in Sec. IV.

The finite-difference method places p and v at the cell center and x and u at its right boun-dary. (These locations become important primarily when one is looking at large-gradientregions. ) The method does not treat shocks explicitly, so the leading shock is located and itsstate determined by a separate procedure which differs slightly from that described in themanual.’ This and other minor changes in the code are described in Appendix A. The codekeeps a “least-active-cell” marker just ahead of the leading shock, and saves time by notcalculating for the inactive cells ahead of it. Only the active cells appear in the snapshots.The left and right boundaries are represented by dummy cells, and the left dummy cell alsoappears in the snapshots. Its state is set equal to that of the first interior cell, except for x andu, which are calculated from the boundary prescription. The left dummy cell is not includedin the count for the number of computation cells.

The calculations are set up so that the principal wave moves through the body from left toright. In most cases the wave is produced or supported by a prescribed left-boundary motion.We call this left boundary the piston, and prescribe its velocity as a function of time. If thebody is of a single material, we use a uniform cell size throughout. If there are two slabs ofmaterial, or shells, a uniform cell size is used within each shell, but a different cell size is usedbetween shells. Throughout the calculation, we use a fixed time step set at one-quarter toone-t bird of the Courant value for the cell requiring the smallest value. For each problem, thecalculation is made with 16, 32, 64, 128, and 256 cells. For the standard calculations, an op-timum artificial viscosity was chosen for each problem by examining the shock’s anatomyand making the choice for sharpness and critical damping at a typical pressure. Results ofthis preliminary study are described in Appendix B. The optimum viscosities are markedwith a superscript c in Table B-II.

The calculated results are given for each problem in turn. The problem numbers are thoseof Ref. 3. For each calculation, there are three types of plot: (1) snapshots—profiles at fixedtime, (2) particle histories—the history of the position and state of a test particle (fluid ele-ment), approximated as the history of a particular computation cell, and (3) the lead-shockhistory-the history of the position and state of the leading shock, determined as describedabove. We make the comparison with the exact solution in one oft wo ways: (1) by superpos-ing the exact solution as a dashed line on the plot of the calculated particle or shock history,or (2) by displaying the difference Ay between the calculated and exact solutions:

Ay = (calculated solution) – (exact solution) ,

where y is x, p, u, or p, at each time of the histories. The standard scale labels of the code areused in both cases, but the difference plots are labeled by the word “difference” in the title atthe top of the page (and are also distinguishable by their appearance and small range). Thecode chooses the scales of each plot to span the range of the data plus a small margin. Thisgives maximum resolution, thereby making life tolerable for the designer of the calculation,but does give a different scale for each plot. The histories show every time step and thesnapshots show every cell. The points are connected by straight lines, and each point ismarked with a dot where this does not make too dense a line. The slight imprecision inplotting resolution is not caused by hardware limitations but by old software, whose resolu-t ion is 1024 points per page width (four plots per page for all plots). The finite resolution canoccasionally be discerned in the form of small steps in a nearly horizontal portion of a curve.

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In contrast to the rough snapshots, the particle histories are always smooth, with approx-imately sinusoidal noise. This can be understood by regarding the set of computation cells asa system of coupled oscillators. Typically, the passage of a shock excites in each cell a smallspurious ringing. Adjacent cells show similar ringing patterns, but snapshots will show arandom-looking noise because the adjacent cells are caught at different phases. The ringing iswell resolved in the histories because the time step is so small (about one-quarter of the spacestep). The shock histories show a random-looking but still periodic noise, due partly to thenature of the locating recipe. The minor period is the transit time of the shock through a cell,and the major period is the time required to approximately repeat the same phase for thetime of shock entry into a cell relative to the beginning of a time step.

The results for each problem are displayed in a set of plots selected from the standardsequence given in Table I. Not all types of plots are presented for all problems. For some typesof plots there may be more than one page, showing different values of N.

The particle histories are for a particular test particle specified for each problem. Thesnapshots are at a test time (also specified for each problem), usually the time when the testparticle is about halfway through the wave. The particle history over a truncated time rangeis included (item 3 of Table I) because, in those problems with a leading shock, the passage ofthe shock over the particle generates such large errors that later errors are hard to see on thesame scale. The truncation consists of removing the initial time segment containing the largeshock-passage errors. The artificial viscosity q has no counterpart in the exact solutions, all ofwhich have discontinuous shock transitions. The value of q displayed in some of the shockhistories is the maximum (quadratically) interpolated value in the profile at each time.

The extrapolation plots present for each problem the mean absolute value of Ay over thepertinent time range vs N-’. The truncated time range is used for the particle, and the fulltime range for the shock, except in problem 7, where the truncated range is used for both. Themean absolute value is

where i numbers the times of the calculation steps covering the truncated time range (t,+, = t,+ At), and y, is the value of y at time i. In most cases these plots become more or less linear as

they approach N-* = O and extrapolate to ]@I = Othere, indicating agreement with the ex-act solution in the limit of zero cell size. Where this is not the case, we refer to the ex-trapolated value of I~[ as the residual error.

There is probably no single measure that is suitably representative of the overall error in allcases. In most cases, the error does not change sign anywhere in the time range studied, sothat the mean absolute value is identical to the arithmetic mean and is quite suitable. Whenthe error changes sign only once or twice in the range, the absolute value is still the most ap-propriate, for it gives the mean size of the error, whereas the arithmetic mean could give amisleadingly small value through cancellation. When the oscillations are much larger thanthe mean error, the absolute value is less appropriate, for it then measures mainly the am-plitude of oscillations. For some of these cases we have also supplied estimates of thearithmetic mean.

II

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A. Results

●Problem 1, Rarefaction. Test problem 1, Fig. 1, is a centered, simple rarefact ion wavepropagating into a semi-infinite slab of aluminum, initially at rest and uniformly isother-mally precompressed to p = 30. The left boundary is a piston initially at x = O; the rarefactionis generated by instantaneously decreasing its velocity at t = Ofrom u = Oto U = –1.542,

the exact value being chosen to make the final pressure exactly zero. The calculationthickness is k = 10 and the coarsest calculation has At = 0.04.The test particle is initially at x= 5 (cell N/2).

The calculation rounds off the wave head, with motion beginning early, then lags and un-dershoots at the tail, producing a small spurious tension there. The artificial viscosity turnson after the minimum velocity point of the undershoot. Its peak value is approximatelyproportional to cell size. Typically, the pattern of differences from the exact solution isqualitatively the same, regardless of cell size. For this problem there is no shock and thus noneed to truncate the time range. The cell-size extrapolation is quite smooth and nearly linear.

●Problem 2, Shock Deceleration. Problem 2, Fig. 2, looks at the deceleration of a shock bya following rarefaction wave. The left edge of a semi-infinite slab of aluminum is a piston in-itially at x =0. At t = —0.1304, its velocity is increased instantaneously from u = O to u =1.743, generating a flat-topped shock of p = 37.3. At t = O, its velocity is instantaneouslyreduced to zero again, generating a centered rare faction wave which, with the special equa-tion of state used, brings the pressure down to exactly zero. The rare faction head overtakesthe shock at x = 10, t = 1 (the initial piston velocity and time having been chosen to givethese even numbers). The calculation thickness is i = 20and the coarsest calculation has At= 0.04. The test particle is at x = 10 (cell N/2).

The calculated particle motion is similar to that in problem 1. Because of the finite shockwidth, the test particle begins to move early, first overtaking the exact trajectory, then lagg-ing behind it in the undershoot at the tail of the rarefaction wave. The passage of the particlethrough the shock generates some quickly damped ringing, most easily seen in the p-u plot.The number of cycles of ringing appears to be approximately proport ional to N; the frequencyis, of course, proportional to N-i, as is the shock thickness.

The calculated shock is well below the true value, and” the first hint of the slope discon-tinuity at the overtake appears at 128 cells. The magnitude of the shock-position error in-creases slightly with time, with the calculated position lagging behind the true value. Bothextrapolations are reasonably smooth.

● Problem 3, Free-Surface Deceleration. Problem 3, Fig. 3, shows the decelerat ion of a freesurface (previously accelerated by a shock) by a rarefaction wave. The left edge of analuminum slab is a piston initially at x = –1.5; the right edge is a free surface at x = 8.5. At t= – 1, the piston velocity is increased instantaneously from u = O to u = 1.5, therebygenerating a flat-topped shock with p = 29.4. At t = Othe piston reaches x = Oand its velocityis instantaneously reduced to zero, generating a centered rarefaction wave. The shock reachesthe free surface at t = 0.421, accelerating it to u = 3.02; the rare faction head reaches it at t =1.35, initiating the deceleration; and the rarefaction tail reaches it at t = 2.36, at which pointits velocity has been reduced to u = 0.0189. The coarsest calculation has At = 0.04. The testparticle is at the free-surface (cell N).

The particle motion is again similar to that of problem 1, with the same early motion but aless pronounced undershoot at the tail. The initial acceleration of the free surface by theshock (not shown here) leaves the density high by - 1%. This art ifact (exaggerated density

r.

4

error at an edge or interface) is characteristic of the calculational method and is not reducedby decreasing the cell size, because the artificial shock width is reduced proportionately. Itcan have either sign, depending on the type and magnitude of the artificial viscosity. Passageof the entire rarefaction wave through the free surface leaves this edge-cell density error vir-tually unchanged. Note that a small artificial viscosity, approximately proportional to cellsize, is present throughout the deceleration, because the interaction region of the forward-facing rare faction wave with the free surface has a small positive velocity gradient.

.Problem 7, Decelerating Shock from HE. In problem 7, Fig. 4, a slab of HE is driving ashock into a magnesium slab. This problem looks at the motion of the interface between thetwo materials and at the deceleration of the shock by the transmitted Taylor rare faction waveof the detonation. The left boundary of the explosive is a piston initially at x = O, the ex-plosive/magnesium interface is initially at x = 10, and the magnesium slab is considered to besemi-infinite. At t = O an instantaneous-reaction, steady, unsupported Chapman -Jouguetdetonation is initiated at the piston, whose velocity drops instantaneously from u = Oto u =– 1, producing a Taylor wave of sufficient extent that its tail characteristic does not affect themotion in the time range of interest. The shock impedance of the magnesium is sufficientlylow that the wave reflected into the explosive is a (weak) rarefaction. The calculation beginswhen the detonation front reaches the interface at t = 1.18. The complete detonation wavehas been precalculated and entered as an initial condition, and the result is a pronounced in-itial discontinuity at the interface. The explosive thickness is 2 = 10 and the magnesiumthickness is a nominal i = 12. The coarsest calculation has At = 0.04. Half of the computa-tion cells are in each material, so that the magnesium cells are 1.2 times larger than the ex-plosive cells. The test particle is at the interface (last explosive cell) on the explosive side.

The calculated particle motion shows a strongly damped oscillatory transient coming fromthe initial discontinuity and with a time period inversely proportional to cell size. The posi-tion error increases with time, with the calculation running ahead of the true solution. Thetransient peak value of q (not shown) increases slightly as the cell size decreases, presumablybecause the interface cell samples less of the Taylor wave. Again, the shock errors are ap-preciable. The early shock history is, of course, affected by the starting transient. Thesnapshots show some ringing behind the shock and suggestions of the typical undershoots atthe tails of both the Taylor and reflected rarefaction waves. The typical interface densityerror, mostly in the inert, is evident in the 128-cell snapshot. The particle-density extrapola-tion shows what may be a small residual error. We expect some residual density error becausethe particle is at an interface.

.Problem 8, Reverberation. Problem 8, Fig. 5, examines the free-surface motion in analuminum plate driven by a supported (flat-topped) Chapman-Jouguet detonation in a semi-infinite slab of explosive. The detonation can be assumed to initiate at the left boundary ofthe explosive, and is treated in the same approximation as in problem 7. In problem 8,however, the piston velocity is raised to the Chapman-Jouguet particle velocity to producethe desired flat-topped wave. The detonation drives a p = 29.0 shock into the plate, which ac-celerates by the usual sequence of reverberations.

The calculation gives the explosive a nominal thickness of* = 30 (large enough that thepresence of the left boundary does not affect the plate motion in the time range of interest)and places the explosive/aluminum interface initially at x = 20. The calculation begins attime t = 2.50, with the detonation front at the interface, and with the detonation wave en-tered as an initial condition as in problem 7. The coarsest calculation has At = 0.04. Half of

the computation cells are placed in each material, su that the explosive cells are three timeslarger than the aluminum cells. The test particle is at the plate’s free surface (cell N).

The exact calculation of the complete free-surface motion is quite complicated, and wehave not reproduced all of it here. Instead, we use only the second constant-state plateau, ex-tending from t = 7.27 to t = 10.12. The full time range is t = 6 to 12, including the precedingand following reverberations. The truncated time range is t = 8.4 to 9.4, approximately thecent ral t bird of the plateau.

As in problem 3, the calculation shows an initial high density left over from the initialshock acceleration of the free surface. At the beginning of the plateau, the pressure and den-sit y show rapidly damped ringing. The velocity overshoots, but rings less. Most oft he ringinghas disappeared from the truncated time range at 256 cells. In addition to the expectedresidual density error, both the position and velocity show small residual errors which are dis-cussed in the next section.

●Problem 9, Blast Wave. Problem 9, Fig. 6, is a spherical blast wave in a -y = 7 polytropicgas. The initial pressure is zero (to achieve the required infinite shock strength) and the in-itial density is unity. The center is at x = O.The problem is designed so that at t = 1 the shockpasses through x = 10 with p = 25, U = 10, u = 2.5. The wave is generated by a pistonprogrammed to follow a particle path of the exact solution. The piston is initially at x = 14.4.At t = 1/4 its velocity increases instantaneously from u = O to u = 5.74, generating a shockwith p = 132.0, U = 23.0, u = 5.74. Thus there is a fivefold decrease in shock strength over thetime of the calculation. The calculation extends from x = 14.4 to x = 35.6, the outer boundarybeing chosen so as to place the test particle at x = 25 on the right boundary of cell N/2. Thecoarsest calculation has At = 0.01, a small value at the observation time required by thestrong initial shock.

The particle starts moving too soon and remains a nearly constant distance ahead of the ex-act solution. A feature not previously seen is the very slow damping of the ringing behind theshock. The 128-cell snapshot implies that the ringing terminates abruptly at t > 1.75. Wehave verified this by an additional calculation in which the particle history extends to time 2.The 128-cell truncated-time particle history from this calculation is included for comparison.Another instance of this sudden termination of ringing will be presented in the problem-2variation in the next section. We are unable to explain this abrupt termination.

The shock also runs ahead of the exact solution in a similar way. The shock noise patterndiffers in appearance from the previous problems because of the small time step—at the ob-servation time the shock requires about 13 time steps to cross a cell.

●Problem 10, Steady Detonation. Problem 10, Fig. 7, calculates the reaction zone of asteady, supported Chapman-Jouguet detonation in Composition B represented by apolytropic gas equation of state with -y = 3. As in problem 9, the desired flow is generated byprogramming the piston velocity to be that of a particle passing through the steady solution.The reaction rate is a function only of A, the degree of reaction. The velocity of the piston, in-itially at x = O, increases instantaneously from u = O to the spike value u = 4.25, thendecreases following the steady solution until the reaction is finished, and finally is left at theconstant Chapman-Jouguet value of u = 2.125. The reaction-rate multiplier merely sets thetime scale. It is chosen for unit reaction time. The corresponding reaction-zone lengt h is 5.31.The calculation thickness is i? = 20. The test particle is initially at x = 10 (cell NE). The COar-

sest calculation has At = 0.025. The simple reaction rate used makes p and u linear functionsof time; this gives zero truncation error in the calculation’s integration of u for the piston posi-t ion, which is thus calculated exactly (excluding roundoff error).

t

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For the smaller values of N, the gross features of the particle motion are like those ofproblem 9: the particle begins to move too soon and continues ahead of the exact solution,and oscillations persist through the reaction zone. With increasing N, the oscillation is essen-tially undamped. Its period increases with time. The undershoot at the end of the reactionzone is qualitatively similar to that at the end of a rarefaction wave, but is smaller and ismasked by the oscillations. The artificial viscosity, of course, always turns on at this point.With increasing N, a nonzero q first appears within the reaction zone at N = 128, anddecreases in size with increasing N. The position error shows a curious change at N = 256: forsmaller N it is always positive when the particle emerges from the shock, and then itdecreases. For N = 256, the error is initially negative, and then increases. We have not madecalculations to examine in detail the final state following the reaction zone, but it appears tohave positive Ax and Ap, a negative Ap, and a Au whose sign is not certain. The final-stateerrors in p, u, and p are considerably smaller than the amplitude of the oscillations within thereaction zone.

The shock also shows some curious features. Trends in the errors with time first becomeevident at N = 64, with Ax increasing and Ap and Au decreasing. At N = 128 and N = 256(not shown), the trend in Ax is reversed, and that in Au has become more pronounced. For allN, the (peak value of) q oscillates between 16 and 24.

III. PARAME7FER VARIATIONS

In this section we consider the effects of varying the numerical computation dials. We choseproblem 7 as the main testing ground for the fairly extensive and systematic set of variationslisted in Table II. In addition to this set for problem 7, we have investigated the effect ofomitting the energy advance in problem 1, and the effect on a shock of switching to aquadratic (only) viscosity in problem 2. We have rerun problem 8 with the quadratic viscosityto reduce as far as possible the braking effect of the artificial viscosity on the compressionwaves. We have rerun problem 10 with the quadratic viscosity to see if this would reduce thesmall residual error in the mesh-size extrapolation. To save space and make the results easierto correlate we have settled on a small standard set of plots for most of the variations. Thetabular summary of the next section is also pertinent.

To check the effect of roundoff (not listed in Table II), we used the hardware option of theCDC 7600 which offers a choice of truncation or rounding in the floating-point operations.The standard option, used for all of our calculations, is truncation. We completely recompiledthe code, specifying the rounding option, and then repeated the standard calculation forproblem 2. The recompilation changed all arithmetical operations except those in the ex-ponential routine, which came from the system library. The 250-cell calculation had 1.6 x 10’“points” (the advance of one cell through one time step). The differences between truncationand rounding ranged from 5 parts in 1014for x to 1.4 parts in 1011for u. The results for calcula-tions with fewer cells showed that the error was roughly proportional to ~.

A. Results

Variations on Problem 1. In problem 1, we eliminated the energy by replacing the fullequation of state

p(m) = w(dp. – 1) + (-Y – l)pe

~ = 3, a = 0.264, PO = 2.79

with the corresponding isentrope function r

p(p) = kp~, k = 0.01254,

calculated to pass through the initial state. The exact solution is unchanged because the flowis isentropic. The calculation is different because it no longer depends on the energy advance.However, the changes were very small: the differences from the exact solution changed by lessthan 1% (oft he differences). In the full equation of state, the first term, which depends on thedensity only, is dominant, being about four times the second term in the initial compressedstate.

Variations on Problem 2, Fig. 8. The next variat ion is the quadrat ic viscosity q = O, 1.414for problem 2 (see Appendix B for the viscosity specificat ion). With no linear term, dampingis very slow, and the viscosity is turned on at some points behind the shock. Oscillation stopsabruptly about halfway through the wave, at a time that is almost independent of N. Some ofthe errors are larger than for the standard case; for example, the shock posit ion error is almosttwice as large.

The results illustrate the effect of an underdamped shock produced by a quadratic q or asmall linear q. For a flat-topped shock, a simple shock-state recipe like ours will have apositive residual error: the oscillations produce a peak pressure above the true value, and thismaximum is picked up by the state recipe. The same will be true for sufficiently large N whena rarefaction wave follows the shock, for the effect of the fixed physical gradient over the firstoscillation period decreases with N because the period goes as N-1. Another int crestingproperty may be inferred from the results: different particles have sufficiently differenthistories that some exceed the peak pressure of the exact solution and some do not. The par-ticular particle displayed does not exceed the peak pressure, but the shock history shows thatsome do.

Variations on Problem 7

● q = 0.3, 0 - Underdamped Shock, Fig. 9. The snapshots show that the large oscilla-tions at the front are rapidly damped; even so, the amplitude of the later oscillation is about80% greater than for the standard case. Velocity and density errors also are appreciably largerthan for the standard case.

● q = 0.5~ O- Critically Damped Shock, Fig. 10. The behavior here is about the same asin the standard case. The oscillation amplitude and particle errors are about the same, exceptfor an appreciably larger density error. The shock history noise is greater than in the standardcase.

● q = 0.7, 0- Overdamped Shock, Fig. 11. With overdamped shock, the oscillation am-plit ude is less than in the standard case, and shows some irregularity. It is curious that withthis reduced amplitude of oscillation, the snapshots are rougher than those for the standardcase. The particle errors are comparable to those for the standard case, with Ax a littlesmaller and Ap a little larger. The shock history noise is about the same as for the standardcase.

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•~ = O, 1.414 - Quadratic, Fig. 12. As expected, the oscillation amplitude here islarge—about three times larger than that for the standard—but the particle errors are com-parable. An interesting feature of the pressure snapshot is the smooth portion centered on theinterface. Unlike problem 2 with this q (Fig. 8) in which the oscillations stopped abruptly,they here continue; presumably the smooth portion of the snapshot indicates that all oscilla-tions are in phase. The shock histories show the same qualitative behavior as in problem 2with this q, and the remarks there are applicable here.

. q = O, 0, 2- PIC Form, Fig. 13. The main difference from the standard is the ‘40~o

reduction in the oscillation amplitude. The particle errors are about the same or slightlysmaller, and the shock histories are very similar. The snapshots are slightly rougher.

● At Halved, Fig. 14. Figure 4 for the standard case shows that the time resolution is

good—about 20 steps per oscillation cycle. Reduction of At would therefore be expected tomake little difference; this was found to be the case, with the particle oscillation about thesame and the errors in p, u, and p slightly reduced.

To investigate the increased noise in the shock history, we made additional runs showingevery particle at every time step over a small region. The switchover from one cell to the nextin the shock finder is the main source of noise. In brief, the explanation is that the smallertime step resolves this switchover process better and larger fluctuations result. The velocity isnoisiest. Its rise through the shock lags behind the pressure rise. At the switchover, thevelocity of the maximum-pressure cell still has a way to go to reach the top, resulting in alarge dip in the velocity history recorded by our simple recipe.

. At Doubled, Fig. 15. Surprising y, doubling At produces little difference from the stan-dard case. The oscillation amplitude is slightly reduced, and the errors are almost the same.The shock history noise is less regular and the snapshots are rougher.

● Changed Cell-Size Ratio, Fig. 16. Instead of having 8 cells each in the explosive andinert material, we change to 6 in the explosive and 10 in the inert, changing the inert/ex-plosive cell-size ratio from 1.2 to 0.72. Compared to the standard, the oscillation amplitude isincreased about 400A, the position error is significantly reduced, and the other errors arechanged a little. The shock histories differ slightly, showing smaller errors and a morepronounced long-period oscillatory component with the changed cell-size ratio.

Variations on Problem 8, Fig. 17. The next variation is on problem 8 with the quadraticviscosity q = O, 1.414. Again, the most striking feature is the characteristic lack of dampingassociated with removal of the linear viscosity. The oscillation amplitude is muchlarger—sufficiently so that the original errors in p and u, of fixed sign with small oscillationssuperimposed, are converted into errors of alternating sign. The mean velocity error here iscomparable to that in the standard calculation, but the oscillation amplitude is now an orderof magnitude larger than the mean error.

The residual errors are still with us: –0.0076 and –0.001 for Ax and Au, compared with–0.0068 and –0.0018 in the standard case. These values for the present case have beenarrived at not by looking at the extrapolation plots, whose main component is the large os-cillations, but by extrapolating averages over the truncated time range. We had hoped to findthe averages greatly reduced by the change to the quadratic viscosity, which should reducethe frictional braking effect of the Viscosity on the finite-width compression waves of thereverberation. Such is not the case. We are unable to offer any other explanation of theresidual errors.

Variations on Problem 10,Fig. 18. Finally, we have rerun problem 10 with the quadratic-only viscosity, q = O, 1.414. The oscillations are several times larger than in the standardcase, but do decay with time. The mean particle errors are appreciably larger for x and p, butsmaller for p and u. The position error has the same form and sign for all N. The viscositywithin the reaction zone is several times larger than in the standard case and regularly dis-tributed in time. The small trends with time in the shock history are different, with Ax hereincreasing with time for N = 128, and decreasing at late times for N = 256. We have includedthe snapshots for N = 256 because they show an interesting pattern of “beats,” presumablycaused by changes in phase of the individual-cell oscillations.

The residual mean particle errors are reduced. By examining the graphs we have estimatedmean errors, for x over the truncated time range, and for p, u, and p in the neighborhood oft= 1.8. The estimated residual mean errors for the standard and for the present calculation are

Standard q = o,1.414

Ax –0.013 –0.009Ap –0.09 +0.02Ap +0.007 ‘“o.Au +0.008 +0.006

Thus switching to the quadratic q reduces the residual error but does not remove it entirely.Apart from remarking on the persistence of the oscillations, we have no explanation of theresidual errors or of the small trends in the shock history.

IV. SUMMARY AND CONCLUSIONS

The 128-cell errors for each calculation are given in Table III. All are the absolute meanvalue as defined in Sec. II. In a few cases where the oscillations are much larger than themean error, we have added in parentheses graphical estimates of the mean error over severalcycles in the middle of the time range.

In most cases, the values given are representative in the sense that the linear form inferredfrom them for the function Ay(N-’) would be reasonably good. However, there can be ap-preciable curvature of either sign; in these cases the inferred linear form would be lesssatisfactory.

Our results can be summarized as follows:

. Ringing is caused by a shock followed by a wave that decreases the pressure. Typically, asmall residual ringing persists for a long time but may vanish abruptly.

● Errors due to smearing of discontinuities have the expected sign.

. Shock errors (as defined by our recipe for locating the shock) are significantly larger thanparticle errors, but the difference varies from a factor to 2 to 3 in problem 9, to 10 or more inproblem 2.

r.

a.

● Recognized sources of residual error—overheating at an interface and overshoot at a noisy

shock—appear as expected and should be remembered in drawing conclusions.

10

..

. Small, but clear-cut residual errors in problems 8 and 10 remain unexplained.

● The sensitivity of the results to variations in the parameters of the finite-difference methodis not as great as we had thought. The effects of such variations are sufficiently diverse that itis difficult to come up with general recommendations.

APPENDIX A

CODE CHANGES

The shock state is determined in a slightly different way than that described in Ref. 2. It islocated as described there—at each time, its position is taken as that of maximum q asdefined by a quadratic interpolation. But its state is taken to be that of the maximum-p cell,typically 3 or 4 cells behind that of maximum q. The value of u is that at the center of thiscell, taken as the mean of the left- and right-edge values.

The code provides for centering the energy advance in time by iterative simultaneous solu-tion of the energy advance and the equation of state. For an equation of state linear in theenergy

p(v,e) = f(v) + g(v)e ,

the iteration is avoided by explicit solution. The current version of the code also allows thetime-saving explicit solution through local linearization of the equation of state, taking

f(v+) = p – (t3p/~e)ve

g(v+) = (tip/i3e)v ,

with e the old energy and v+ the new volume, and with p and (tip/ae). evaluated at (v+ ,e).The problems affected here are problems 2 and 7, which use the Walsh equation of state. Ifchemical reaction is present, as in problem 10, iteration is required to center the advances ofboth X and e. The standard number of passes n is 2, and we use this value. A trial calculation

with n = 4 showed little difference.

11

APPENDIX B

ANATOMY OF THE SHOCK

We used three forms of the artificial viscosity: Landshoff (linear), Richtmyer-vonNeumann (quadratic), and PIC (linear). Written in linear combination, these are

q = &JC(AX)U. + B*P(AX)21U, [UX + Cplul (Ax)uX . (B-1)

We specify the viscosity by the values of the coefficients A, B, and C. (If the value of C isomitted from the specification, C is zero. ) For each of the three forms, we have chosen coef-ficients ranging from too small (underdamped) to too large (overdamped). Linear combina-tions oft he Landshoff and Richtmyer-von Neumann forms are popular, and we have includeda number of these. We obtain an infinite-strength shock for the polytropic gases (~ =3 and -y= 7) by taking zero initial pressure. For these systems, a pure Landshoff viscosity would neverproduce a finite pressure because it has a sound-speed multiplier and the initial sound speedis zero. So for these two systems we have added a small quadratic term to those calculationsfor which we would otherwise have chosen a pure Landshoff form. The value of B used inthese cases is too small to have a significant effect on the shock shape.

The calculation is a shock driven by a constant-pressure rear-boundary condition. Theshock is displayed after it has traversed 50 cells. We used an arbitrary choice of AxO = 10throughout. The time steps relative to the Courant value At *( = c/Ax) in the shocked state aregiven in Table B-I. The ratios vary somewhat because we used time steps of the power of 2next larger than 0.25 At*. In several cases we also tried time-step values twice as large andhalf as large as the standard. Cutting the time step to half the standard value usually makesno significant difference. In some cases, doubling it does make a difference, and we have in-cluded some such cases in the results.

The results are given in Figs. B-1 and B-2 and Table B-II. From them we have chosen anopt imum viscosity for each material, the criterion being that the damping be crit ital. These

optimum values are marked with a superscript c in Table B-II. The viscosity variationsdescribed in Sec. III are marked with a superscript b.

Figure B-1 shows (1) snapshots for aluminum, (2) the history of p, u, p, and q fort he fiftiethcell as it passes through the shock, for each of the materials at the optimum viscosity, and (3)snapshots and profiles for the quadratic viscosity used in several of the variations given inSec. III. The shock crosses a cell in about four time steps, so the histories have about fourtimes more points through the shock than do the profiles, and are smoother and more infor-mative. Figure B-2 shows the velocity history for each entry in Table B-I.

Table B-I gives estimates of the shock thickness 6 measured in number of cells. Wecalculate 6 from

d = (U/AxO)~ , (B-2)

where ~ is the time width of the shock as illustrated in the first velocity-history frame of Fig.

B-I. We definer as the time width of the shock determined by the straight line tangent to thepoint of maximum slope in the particle velocity history. (This definition of course becomesmeaningless for very large or very small viscosities.) The range of the time scales throughoutFig. B-2 is set inversely proportional to the shock velocity U so that 6 is proportional to ~, as

F.

.

s

12

.

r.

measured by a ruler laid on the page. Table B-I also gives an approximate value of the peak-to-peak particle-velocity amplitude of the first peak in units of the particle-velocity jumpthrough the shock, for underdamped shocks which show such oscillation. This measure isrecorded as zero whenever the rise is monotone or slightly bumpy.

As expected, the shocks go from under-to overdamped as the viscosity is increased, and theLandshoff form with polytropic gas shows more cotnplex oscillations. Away from the op-timum viscosity, doubling of the time step makes the history rougher and increases the os-cillation. When the viscosity is large enough (q = (5, 0.125, O) for -y = 7 polytropic gas), onesees the beginning of instability.

REFERENCES

1. R. D. Richtmyer and K. W. Morton, Difference Methods for Initial-Value Problems, 2nded. (Interscience, New York, 1967), p. 295.

2. P. J. Roache, Computational Fluid Dynamics (Hermosa Publishers, Albuquerque, NM,1972), p. 326.

3. W. Fickett and W. C. Rivard, “Test Problems for Hydrocodes, ” Los Alamos ScientificLaboratory report LA-5479 (August 1974).

4. W. Fickett, “PAD-A One-Dimensional Lagrangian Hydrocode, ” Los Alamos ScientificLaboratory report LA-591O-MS (April 1975).

13

TABLE I

DISPLAY SEQUENCE

1. Snapshots

2. Test-Particle History

A. Exact solution superimposed on first page

B. A (calculated – exact), all quantities, both pages

C. Same as B with truncated time range

3. Leading-Shock History

A and B as above

4. l~j vs N-’

..

p,u, p,qvsx

first page - x, p, u, p vs t

second page - p-u, p-p, q-t

same as 2

x,p, u,pvs N-’

TABLE II

PARAMETER VARIATIONS

Figure Problem Variation

------

89

101112131415161718

122777777778

10

Remove energy dependenceRoundoffq = o, 1.414q =0.3, oq =0.5, oq = 0.7,0q = o, 1.414q=o, o,2At halvedAt doubled6/10 cellsq = o, 1.414q = o, 1.414

14

TABLE III

SUMMARY OF 128-CELL ERRORS

Problem Variation

1

p

3

‘1

8

9

10

~

7

7

7

7

7

7

7

7

8

10

Standard

““

n

“

,,

“

u

q = o, 1.414

q = 0.3,0

q = 0.5,0

q = 0.7,0

q = o, 1,414

q=o, o,2

At halved

At doubled

6/10 cells

q = o, 1.414

q = o, 1.414

—

Path”

P

Ps

P

Ps

P

Ps

Ps

Ps

Ps

Ps

Ps

Ps

Ps

Ps

Ps

Ps

P

Ps

Ax

0.0031

0.00360.042

0.0052

0.0120.030

0.037

0.0400.15

0.0130.019

0.00240.053

0.0120.035

0.0120.026

0.0100.038

0.0120.036

0.0110.033

0.0120.032

0.0120.030

0.00900.025

0.032

0.0330.081

Ap

0.24

0.182.6

0.17

0.120.59

3.2N4

0.2020.353

0.31 (–0.35)0.99

0.450.98

0.121.45

0.120.67

0.120.86

0.120.48

0.120.85

0.120.45

0.120.59

0.170.45

0.091 (-o)

0.64 (–0.32)1.2

Au

0.013

0.0140.11

0.014

0.00680.049

0.0014

0.0110.023

0.11 (0.0025)0.092

0.0120.058

0.00760.15

0.00690.083

0.00640.075

0.C0760.066

0.00700.075

0.00490.062

0.00680.049

0.00590.043

0.0086 (–0.0019)

0.033 (o *0.01)0.12

Ap

0.062

0.00430.047

0.015

0.00740.019

0.029

0.00660.0098

0.0062 (–0.0068)0.054

0.0120.017

0.0180.042

0.0140.021

0.00990.028

0.00660.013

0.00700.027

0.00810.014

0.00740.019

0.00870.015

0.020

0.015 ( –0.014)0.032

‘P = particle, S = shock

15

..

Substance

TABLE B-I

SHOCK STATES

1. Aluminum (Walsh)p. =

u=

2. PMMA (Walsh)p. =

u=

:3.I+]lytropic Gas-y=

p. =

-i. Polytropic Gas-y=

p“ =

2.795.33 + 1.34 u

1.192.57 + 1.54 U

7O,po=l

3O,pO = 1.6

P—

30

15

15

60

u, p—-

1.473.49

2.151.87

1.941.33

4.333.20

c, u At/At”

7.417.30

5.835.88

8.877.75

7.508.66

0.23

0.26

0.15

0.19

16

TABLE B-II

CALCULATED SHOCKS

1.A1

2. PMMA (Plexiglas)

u-OscillationaViscosity (%)

0.1,0,00.2,0,00.3,0,00.4,0,0bo,1.414,00,2,0‘0.3,1,00,0,2

bo,3,o,0bo.5,o,0bo.7,o,0bo,1.414,0CO.3,1,00,0,1bo,o,20,0,3

3. PG (Polytropic gas), -y = 71,0.125,01,5,0,125,0same, 2At3,0.125,05,0.125,00,2,0same, 2 At0.3, 1,00.5, 1,0same, 2 At‘0.2, 2,0same, 2At0,0,20,0,3

451500

15600

3330

100

2700

6727

11352

107

205

1921

285

4. PG,7=31,0.125,0 292,0.125,0 04,0.125,0 00,2,0 4‘0.3, 1,0 00,0,2 0

Thickness(No. cells)

1.01.61.92.91.82.62.61.7

1.01.32.01.91.61.01.41.8

0.60.80.30.91.81.91.91.01.11.02.02.00.91.1

0.7

2.02.91.51.5

‘Peak-to-peak amplitude (first peak to first trough) in u-oscillation, expressed as percentage of shock u,‘Variations for effect of viscosity on accuracy (Sec. III).‘Optimum viscosity used for most calculations.

17

PA03.3F6 - 7!5.M14 IwNz LAC2A ●1. RARE. *

TFICKIZTIGY. .0E5 SEC CY4 RU4. 1.855 SEC ON J3B

>

I 1 1 I

30 -

25 :

20 -

15 :

10 :

5 :

0 -t 1 1 I

-5 0 5 10 15 20

POSiti Oll X (mm)

I 1 1 I I I I

.20

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c’~-5 0 5 10 15 Zo

POSiti Orl X (mm)

Fig.

Problem 1. Rarefaction.

-1

-1

ERU4 1. 16 CELLS 07/?1/75

1. RARE. / TIME= Z.0000E+OO

1 1 1 1 ~

o :

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3.6

3.4

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3.0

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Position x (mm)

I I 1 1 I

i:J,,,,-5 0

Standard calculation.

5 10 15 Zo

POS., t Ofl x (mm)

:

:t

.●

PAD3.3f6 - 75.M14 RUN 2 LACi?A ‘1. RARE. *TFICKETIGY. 3.438 SEC ON RLJN, 5.209 SEC ON .KIB

25

20 t

1 1 I 1

30 -

L

15 -

10 -

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0 5 10

Pos;t,orl )( (mm)

15

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2.8[

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1. RARE. / PARTICLES

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9.5 -

9.0 -

8.5 -

8.0 -

t

t..l.l..l .l . . ..l . . . . . ..l1.0 1.5 2.0 2!.5 3.0 3.5

Time t (Ps)

3.6 ....,.,,,,, I J

3.4 :

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3.0 :

2.8 :

1.0 1.5 .?.0 2.5 3.0 3.5

a

3

30

25

.20

15

10

5

0

~“ ”’’’’’’’’’”’’’’’’”’’’’”’”:

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1.0 1.5 2.0 2.5 3.0 3.5

Time t (Ps)

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\

-1.2 - \

\

-1.6 -1 I I 1 !

1.0 1*5 2.0 2.5 3.0 3.5

Time t (Ps)

20

Time t (Ps)

FA03.3F6 - 75 JLJ -14 RUN 2 LAC2A ‘1. RARE. ”1~1 CKf TIGY. k.8\3 sEC W RUN. 6.614 SEC ON JOB

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9.0

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8.0

3.6

3.4

3.2

3.0

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T,me t (Ps)

1 1 1 1 1

1 1 1 ! 1

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T,me t (PS)

30

25

c?c1 20

aa 15LJMWal 10

c+5

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VI\E

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-.4

-.8

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-1.6

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1. RARE. / PARTICLES

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t. .lc. .% l!, cc, lcc, l... ,,,.,1.0 1.5 2.0 2.5 3.0

Time t (tISI

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T,me t (PS)

21

PA03. W6 - 75J.AI* Rw? 1.AC2A .1. RARC. -TFICKETIGY, .!x?e SEc ON RUN, ?.296 SEC ON JOB

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Velocity u (km/s)

PA03.3F6 - 75.iu14 RW 3 LAC.?A ‘1. RARE. oTFICKETIGY. .3B0 SEC IM Rw. 16.351 SEC m m

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Veloc, ty u (km/s)

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muN1. 16 CELLS 07/2! /75

1. RARE. / PARTICLES

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1. RARE. / DIFFERENCE 1 PARTICLES

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●✌✎

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22

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1. RARE. / DIFFERENCE Z PARTICLES

L I 1 1 1 1 1

.

1 [ 1 1 1

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02 -

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a

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Time t (Ps)Time t (tIs)

1 1 1 1 1.05 -

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-.05 -

-.10 -

-.15 -

1 1 # I 1 [ 1

m\

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3

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Time t (PS)Time t (Ps)

Fig. 1 (cent)

23

PA03.3F6 - 75.U14 RUN3 LAC2A ● I . RARE. ●

TFICKETIGY, 4.232 SEC EN RW, ?0.203 SEC ~ .JC8

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.004 -

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o

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:c1

0.

J

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0

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o

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1. RARE. / DIFFERENCE Z PARTICLES

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1.0 1.5 Z.o Z*5 3.0

T!me t (Ps)

1 I 1 1 I 1

1 1 1 1 I -1

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Time t (PS)

..

Fig. 1 (cent)

24

:

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l/(NO. OF CEL.LS)

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16 :’ 1 1 r 1

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4 -

0 :

, 1 1 1 1

Lo 5 10 15 Zo

POS, t,Orl X (mm)

[’”’”’’”’’’’”’’’’’’’”11

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POS, t,Ofl X (mm)

.20

.8

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o.

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> I I I I 1

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POS, t,Orl X (mm)

I I 1 1 1

3.Z -

3.1 -

3.0 -

Z.9 -

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N--=w

Fig. 2.Problem2. Shock deceleration. Standard

5 10 15 Zo

Pos,t,on x (mm)

calculation.

4

..

i

26

.

PAD3. S6 - 7S.JJ-14 RW5 LAC2A ‘.?. SHOCK DECEL*TFICKETIGY. 6.358 SEC CN RI-N. 36. 51t2 SEC CN JW

.

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..

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POSit, on x (mm)

-1

0-0 4 8 12

POSiti Orl X (mm)

ERUN q. 1.?13CELLS 071?1175

2. SHOCK DECEL / TIME= 1.4000E+O0

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3.4

3.2

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o 4 8 12

POSitiOfl X (mm)

f i 1 1 1 1 1 1 1

o~4 8 lZ

Pos,tlon x (mm)

PA03. X6 - 75u14 RUNS LAC.?A ..? . SHCCK OECEL “TFICKETIGY. .719 SEC CN RUN. 30.903 SEC CN JOB

I 1 1 1 1 1 1

10.6 -

10.5 -

E

x

10*Z -

10.1 -

! 1 I 1.5 1.0 1.5 2.0 2.5

T,me t (Ps)

3.4

3.21

\

\

\

P1 . . ..l . . . . . . . . . . . . ..i.5 1.0 1.5 2.0 Z.5

T,me t (Ps)

aJL>VIMal

k

J

x.+J.

ERw I, 16 CELLS 07121/75

Z. SHOCK DECEL / PARTICLES

[ 1 1 t 1/. 1 1

1.6

1.2?

.8

.4

0

.5 1.0 1.5 2.0 2.5

T,me t (Ps)

t’”r’’’’ ”””’’ ””’’ ”””~

\

\

\

.5 1.0 1.5 2.0 2.5

T,me t (Ps)

I

28

..

..

.

PAD3. 3F6 - 75J& 1~ RUN 5 LAC2A ~? SHOCK OECEL*TFICKETIGY. 9.495 SEC ON RW, 39.679 SEC ON JOE

10.6

10.5

10.4

10.3

10.2

10.1

10.0

.5 1.0 1.s 2.0 2.5

T,me t (Ps)

L 1 1 I 1 1

tl 1 I 1 [ i

.5 1.0 1.5 2.0 2.5

Time t (Ps)

ERw 9, 12s3 CELLS 07/21 /75

Z. SHOCK OECEL / PARTICLES

L 1 i 1 I 1 J

.5 1.0 1.5 2.0 2.5

T,me t (Ps)

l-’ 1 1 I 1 -14

>Iii. .l##. l. ##.,.,,,! 1.5 1.0 1.5 zoo Z.5

T,me t (PS)

Fig. 2 (cent)

29

PAD3.3F6 - 75.M14 171m5 LAC2A .2. SHOCK OECEL -TFICKEIIGY. .95* SEc CN Ru’A. 31.138 SEC CW JOB

~“ ~

I-l. l 1 ! I 1 <

0 ..2 .4 .6 .8

Velo C,t Y u (kmzs)

PA03.3F6 - 75JA14 RLN5 LAC2A ‘?. SHOCK OECELOTFICKETIGY. 9.82? SEC CN RLW. ~o. oo6 SEc m m

&

w

30

25

20

15

10

5

0

1 1 I 1 1 4 1 1 1

1 1 I 1 I 1 1 I I

o .4 .8 1.2 1.6

Vel.Ocit Y u (km/s)

ERu4 1. !6 CELLS 07/?1/75

2. SHOCK DECEL Z PARTICLES

1 1 f # 1

16 -

12 :

8 -

4 -

0. -

# I t 1 12.7 Z.8 Z.9 3.0 3.1 3.2

Emm 4. 128 CELLS 07/.?I /75

Z. SHOCK DECEL i PARTICLES

/

t { 1 1 1 —r—-.l--l--

30 -

/;

Z5 -

Zo :

15 :

10. -

5-

0-1 ! I 1 1 1. 1.,., 1 J

Z.8 3.0 3.2 3*4

.’

Dens,ty ~ (Mq/mJ)

30

---- —-.F’AU3.3F6 - 75JU14 RUN 6 I-AC2A ●I?. SHIXK IXCEL*TFICKETIGY. 1.460 SEC ON RUN, 58.705 SEC C&’ JW

●

.

1 1 1 1 ,

.04 -

. Oz :

0 .

-.0.2 -1 1 1 1

.5 1.0 1.5 2.0 2.5

Cl/

T,me t (tIs)

1 1 r [.4 :

.2 :

0,

.2:

! ! 1 1 1 1

ERUN 2, 32 CELLS 07/.? 1/75

Z. SHOCK DECEL / DIFFERENCE Z PARTICLES

I 1 1 I 115 -

4

10

5

0 -J-5:

-10:

-15:r

.5 1.0 1.5 2.() 2.5

Time t (ps)

.5 1.0 1.5 2.0 2.5

T,me t (tIs) Time t (Fs)

31

PA03.3F6 - 75wJ-lh RW 6 LAC2A ●2. SI+XK OECELO ERLN II. 1.?8 CELLS 071.?1 /75

SHOCK DECEL / DIFFERENCE / PARTICLESlFICKIIIGY, 8.511 SEC w Ru4. 65.756 SEC CN WE

z.

~-, 1 1

-1

.9Zo

10

0

-lo

T 1 1 I f “-r-l

.012

.008

.004

I I 1 1 1 1

&

o

-.004 J 1 1 1 1 1

.5 1.0 1.s 2.0 2.5

Time t (PS)

.5 1.0 1.5 2.0 2.5

T,me t (PS)

LI 1 I r .81, I 1 1 1 11

.4 -

.2 :

0 :

-.2 -

L

.4 -

() :~

-.4 -

-.8 -

I 4 1 t t 1.5 1.0 1.5 2.0 2.5

LN-

tl t 1 1 1

.5 1.0 1.5 Z.o Z.5

Time t (t’s) Time t (PS)

:

Fig. 2(cont)..:

32

.PA03.3F6 - 75JW14 Ru4 7 L AC2A 02. SHOCK OECEL.TFICKEIIGY. 6.663 SEC ON RUN. 89.832 SEC ON JOE

ERUN 4. 1?8 CELLS 07/,21/75

●

.

2. SHOCK OECEL .’ DIFFERENCE. TRUNCATED TIME / PARTICLES

1 I 1 1 r 1 , f

.008 -

.006 -

.004 -

.002 -

0 -

.002-

-.004 1 1 i 1 ! 1 #

.0

.0

5

0

E .005\:

0C/

; -.005

-.015

1.2 1.6 2.0 Z.4

T,me t [t’s)

1 I 1 1 ( 1 [ ,

1 I ! 1 1 ! 1

1*Z 1.6 Z.o Z.4

T,me t (tLs)

A

t-l~

. 1.6 Z.o 2.4

.04

.02

0

T,me t (PS)

1 1 i 1 1 1 1

I 1 1 1 ) 1 , 1 I

l.z- 1.6 Z.o Z*4

Time t (ps)

..

.-. . .--. ,. --..-., ,-,=PA03.3F6 - 75W14 RUN 7 LAC.?A “.?. SHOCK OECEL .TFICKETIGY, 6.S62 SEC ON RUN, 90.030 SEC CN m

.2. SHOCK DECEL 1

&

-.02 0 . Oz .04

Velocity u (km/s)

0.

01

Ck

>

UKSl Y, ICU LLLL> Ul*Cir12

DIFFERENCE r TRUNCATEO TIME 1 PARTICLES

“4~

o

-.4

-.8

J

1 1 1 1 1 ! J-.015-.O1G.005 o .005 .010 .015

Oens,tY e (Mq/mJ)

“o=~

.Ozo -

.015 -

.010 -

.005 -

0 - 1 I 1 1 1 1 t

●✎

✎

..

;.

1.Z 1.6 Z*O Z*4

T,me t (PS)

34

t

*. PAD3.3F6 - 75.ML14 RW 5 LAC2A .?. SHOCK OECEL”

TFICKCTIGY. .371 SEC ON RUN. 30.55w SEC ChN JOB

m.

*

EE

x

+J

18

16

14

12

10

8

6

, 1 1 1 1 1 1 1 1

/

t I t I t I 1 1 1

.4 .8 1.2 1.6 2.0

T,me t (Vs.)

I [ ! I 1 I

\

3.20 i

t., .l!. ii!. l. !.l. ll. i.l.,. l.!. 1,.l.i.4 .8 1.2 1.6 2.0

T,me t [Ps)

ERUN 1. 16 CELLS 07/21/75

Z. SHOCK DECEL / LEAD SHOCK

1 I I 1 1 I [ I I

—— -

\

35 -\

\

30 - \\

\

25 -\

\\

L

.4 .8 1.2 1.6 Z*O

Time t (Ps)

1.8 I 1 1 1 I 1 1 i [—— —

\“ i

1.6 -

1.4 -

1..2 -,

\

\\

\

.4 .8 1.2 1.6 2.0

T,me t (PS)

..

Fig. 2(cont)..’

35

PA03.3F6 - ?W14 RW5 LAC.?A 0?. SHOCK OECEL alFICKETIGY. 8.9E5 SEc w w. 39.169 SEc m m

EE

18

16

14

lC

I I I t 1 I 4 1 1

, t t 1 I 1 1 1 I6

.4 .8 1.2 1.6 2.0

T,me t (Ps)

3.60

3.55

3.50

3.45

3.40

3.35

1 I 1 1 1 1 I I I

—— —

3,3(3~.4 .8 1.2 1.6 2.0

T,me t (Psi

2c1

a

36

32

28

24

Zo

fRw4, I?E CELLS 07/21/75

2. SHOCK DECEL / LEAD SHOCK

1 1 I 1 1 I 1, IJ1 I I—

t [ 1 1 1 1 1 1 1 -1.4 .8 1.2 1.6 2.0

1.7

1.6

1.5

1.4

1.3

1.2

1.1

T,me t (PS)

I 1 1 I I b I I 1—

t.c., .o. t.8.1. t.l... *.l.8. fift... a<,.a<.4 .8 1.2 1.6 2.0

Tcme t (PS)

..

*

.

..

.PAC13.3~6 - 75JUlq. RUN 7 LAC.?A .2. SHOCK DECEL -TFICKETIGY. 22.405 SEC ON RW4.

CRUN 5,105.573 SEC ON JOS

256 CELLS 07/?1 /75

Z. SHOCK DECEL / DIFFERENCE. TRUNCATE: TIME / PA-RT[CLE

,.020

EF

x .015

co

w .010m

:

.005

0

I l“’’I’’ ’’1’ ’’’1 ’’’’ 1’’”1’ “’/”

/

I 1,, ,! 1,,,,1,,,,1,,,,1,,,,1,7

o .01 .02 .03 .04 .05 .06

l/lNO. OF CELLS)

I l“’’I’’ ’’1’ ’’’1 ’’’’ 1’ ’’’1’” ‘l”_

.025 –

.020 –E\

;.015 –

@J

; .010 –

i

ko .005 –

o –1 1 I 111,, ,, [,, ,, 1,, ,, 1,, ,,1, ,,{1,,-

0 .01 .02 .03 .04 .05 .06

t1.2 —1

1 11 Il“’’I’’ ’’1’ ’’’; ’’’’ 1’’’ ’1’1

1.0 –

.8 –

.6 –

.4 —

.2 –

o – l, 11, l,, l, l,, ,, l,, ,,1, ,,,l,To .01 . Oz .03 .04 .“05 .06

I/(NO. OF CELLS)

I

. 12 –

.10 –

.08 –

.06 –

.04 –

.02 –

o –1 1 11 1 1,, ,, 1,, ,, 1,, ,, 1,, ,, {,,,,1,,-o .01 .02 .03 .04 .0S .06

I/(NO. OF CELLS) I/(NO. OF CELLS)

Fig. 2(cont)

37

PA03.3F6 - 75w-lq KW6 L&C?& ●2. SHOCK OECEL*TFICKETIGY. 25.233 SEC W RW, E2. q79 SEC ~ JW

I l“’’I’’ ’’1’ ’’’1 ’’’’1’ ’’’1’” ‘l”-

.30 –

.25 –

.20 :

.15 ~

.10 ~

.05 –

o –II, ,, 11, ,111111 ill, ,111, ,llr

l/[NO. O

0 .01 .02 .03 .04 .05 .06

IF CELLS)

“’”~ ‘ ‘.Z5

.Zo

.- 1

I 1 I 1 I{lollltillltti

.10

.05

0 ~~,,,,l,,l,,,l,,,,,lj

0 .01 . Oz .03 .04 .05 .06

0.

ER~ 5. ?% CELLS 07121 !75

Z. SHOCK DECEL / DIFFERENCE / SHOCK

16 1l“’’I’’ ’’1’ ’’’1 ’’’’ ;’ ’’’1’’” l“-

lZ –

8 –

4 —

0 –1 Ill, 11, 11, 11, llllllllllllllr

0 .01 .02 .03 .04 .05 .06

l/(NO. OF CELLS)

1

.6 –

.5 –

.4 —

.3 :

●Z :

.1 –

o –

l/(NO. OF CELLS)

38

0 .01 .OZ .03 .04 .05 .06

I/(NO. OF CELLS)

..

.. PA03. 3F6 - 75JUL 14 RuN 9 LAC? 03. FREE-SURFACE OCCEL .

7FICKETIGY. .064 SEC CF4 RUN. 106.671 SEC 04 JOBERUN I. 16 CELLS 07/?1 /75

-4 -

-8 -

-12 -

-16 -

r

I 1 1 1 1 r

~z 4 6 8 10 lz

POS,tiOrl X (mm)

.20, , 1 1 1 1 1 1

. 15 -

.10 -

.05 -

0-

1

!

I ( 1 t ! { 1 1

0246

Poslt,on

Problem

x

3.

8 10 12

(mm)

3. FREE-SURFACE DECEL. / TIME= 1.8000E+O0

Free-surface deceleration.

L“’’’’” ”’’’ ’’’’ ”’r’”l “’I’dL

1.6 -

1*Z -

.8 -

.4 -

,-

2.6

2.4

2.2

2.0

024

Pos

6 8 10 12

tiOll X (mm)

( 1 1 I 1 1 J

1 1 ! 1 I 1 i

O z 4 6 8 10 lz

Posltlorl x (mm)

Standard calculation.

PAL13. X6 - 75,1L14 IWN9 LAC2 .3. FREE -SLWACC DECEL .0 ERIN 4. 128 CELLS O-)/i?l fziTf lChTTIGY. ~.581 ZC CN RLN, 111. 188sEccNm

1 I I 1 1 , I 1

i) -

-4 -

-8 -

-12 -

k-16 I

~ 1“I ! 1

0 2 4 6“8 10 12

Pos, t,on x (mm]

I 1 1 I , 1 3

.04C

.03C

.0.20

.010

0—!—J—L—~

O Z 4 6 8 10 12

<. FREE-SURFACE DECEL. / TIME: 1.8000~’00

1.6

1.2

.8

.4

01 1 1 1 1 # 1

0 Z 4 6 8 10 I.Z

POS4t(Orl X (mm)

.Z.8~

2.6 :

2.4 i’

2.2. -

2.0, -

v..

.

I

POS, t,Orl X (mm)

Fig. 3(cont)

Posltlorl x (“m)

..

;

40

.. PAD3.3F6 - ;5w4.14 RUN 9 L AC? .3. FREE-SURFACE OCC:L, .

TFICIWTIGY,

.?

12.5

12.0

1

1

.5

.0

10.5

2.78

2.76

2.74

2.72

2.70

.?58 SCC ON RuN. 106. f365 SfC ~ JCf3

—~.———r

r

J I 1 t 1

1.0 1.5 2.0 ‘2.5 3.0

T,me t (PS)

1 1 1 1-1

1 1 I

1.0 1.5 2.0

T,me t (PS

—

~1-L

2.5 3.0

ERLW 1. 16 CELLS 07/?1/75

3. FREE-SURFACE DECELO / PARTICLES

1 1 ,

‘i

--~1~

o -——————— -

-.5

-1.0

-1.5

-2.0

-2.5-

~. 1.5 2.0 2.5 3.()

3.(

20S

2.C

1*S

1.0

.5

0

T,me t (PS)

1 1 1 1 1

\

\

~1.0 . 2.5 3.0

T,me t (PSI

..

>

rAL-13.3F6 - -3.Al14 KW49 LAC2 ‘3. FREE -S4RFACC OECEL. olfl WETIG>. 5.*W SEc m FwN. 11.?.531 Scc m JOE

EE

1

1

12.0 -

.s -

.0 -

10. s -

t , , 1 4

.085

.080

.075

.070

.065

1.0 1.5 2.0 2.5 3-.0

T,me t (Ps)

7 1 I # I-1

1.0 1.5 2.0 2.5 3.0

T~me t (Ps)

20

a

a1.J

WMw

Ct

m\E

i!

0

-.10

-.ZO

-.30

-.40

El?l!N L!. 1?8 CELLS 07 J.?IJ75

3. FREE-SURFACE f3ECEL. / F’ARTICLES

3.0

Z.5

2.0

1.5

1.0

.5

1.0 I*S Z.o .2.s 3.0

T,me t (Ps)

1 1 I I I1

J 1 1 ‘ .~o

1.0 1.5 Z*O Z*5 3.0

T,me t (PSI

.

42

*. PAD3.3F6 - 75 JUL14 RUN 9 LAC2 .3. FREE -SUWACE OEW. .

TFICKf TIGY, 6.2?5 SCC ON RUN. 11?.83? SEC ON JOB

*

o

-.10

-.20

-.30

-.40

1 1 1 I 1 1 [

j

A .5! 1 ! 1 1 !

1.0 1.5 2.0 2.5 3.0

VeloC, ty u (km/s)

ERUN + , 128 aLL5 07/21/75

3. FREE -SUR”FACE DECEL. / PARTICLES

L’r’’’h’l’r’l’l 1 1 I 1

a

alL

iIAu

Ck

-.qo~.076 .080 .084

r

Density ~ (MqzmJ)-z.700

—[[’’’’’’’’’’’’’”’””’ ‘i

.006 :

.004. ~

.002: -

0. 1 1 I 1 11.0 1.5 2.0 2.5 3.0

T,me t (PS)

PAD3.3J-6 - 75J.Ll~ MJNlo LAC2 .3. FREE-SURFACE DECEL . . EFNN 1, 16 CELLS 07/.?1 175

TFICKCTIGY. .I15SECLB4RIJ4. 1?5.587 sEC CN .JX3

3. FREE-SURFACE OECEL. / DIFFERENCE / PARTICLES

1 1 t i I

.08 -

.06 -

.04 -

.02 -

0 -

-.0.2 -0 I I , I

.02

0

-.02

-*O4

-.06

-.08

1.0 1.5 2.0 2.5 3.0

T,me t (Ps)

, 8 1 1

I ! 1 1 1

1.0 1.5 2.0 2.5 3.0

T,~e t (IJS)

-1

-1

-z

-z

o

.5

.0

.5

.0

.5

.3

.2

.1

0

-.1

-.Z

1 1 1 I / 1

1 I I ..~--l-

1.0 1.5 2.0 Z.5 3.0

Time t (PsI

1 1 [ i I

J I 1 I L

1.0 1.5 Z*O 2.5 3.()

T,~e t (IJS)

..

:.

44

.. PAD3.3F6 - 75.NL14 Rwlo LAC2 ●3. FREE-SURFACE OECEL. O

TFICKETIGY, 5.23o SEC CN RWJ,ERLJ4 %, 12s Culs 07/?1/75

130.702 SEC CU4 JCQ

3. FREE-SURFACE DECEL. / DIFFERENCE / PARTICLES

..

i:

0

1 I 1 1 I I

.012 -

.008 -

.004 -

0 -

~. . 2.5 3.0

T,me t (Ps)

L 1 1 1 1

.020

.016 ;1

.012 -

tl 1 1

1.0 1*5 2.0

Time t (tIs

1 t i

2.5 3.0

a

alLJ

MmaJ

:

J

1 1 f 1 1

0 : 1

-.10. -

-.20:

-.30;

-.40- , I t ! 1

.04

0

-.04

-.08

1.0 1.5 2.0 2.5 3.()

Time t (PS)

1 [ 1 1 1

I ! I 1 1

1*O 1.5 2.0 2.5 3.0

Time t (tIs)

45

P4D3.3F6 - 75w14 FlR410 LAC.? ‘3. FREC-SURFACE DECEL .-TFICKEIIGY. 17.55E Scc m I?lm. 1%3.030 SEC CN .x9

3. FREE

o .01 .Oz .03 .04 .05 .06

I/( NO. OF CELLS)

E\

:

C/

.030

.Ozo

.010

0

1

I

o .01 .02 .03 .04 .05 .06

I/( NO. OF CELLS)

Em 5. 256 CELLS 07/.?1 /75

SURFACE OECEL. / DIFFERENCE / PARTICLE

1.Z

.8

.4

0

1 l“’’I’’ ’’1’ ’’’1 ’’’’ ;’ ’’’1’’” l’~

I 1,, ,, [,,,01,,,,1,,,,1,,,,1,,, 11,,-

0 .01 . Oz .03 .04 .05 .06

I/(NO. OF CELLS)

1

.10 –’1 I J 1 I“’’l ’’’’ l’’’ ’l ’’’’ l’’’ ’l’l_

.08 –

.06 --

.04 –

. Oz –

o –1,, , ,1,,,,1,,,,1,,,,1,,,,1,7

o .01 .Oz .03 .04 .05 .06

l/(NO. OF CELLS)

..

Fig. 3 (cent).\.

46

.. PA03. 3F6 - 75JULIW RUN12 LAC? .7. CfCEL . SHCKK FROM HE”

lFICKETIGY. .054 S~C ~ RUN. lW .423 SEC ON JCX3ERuN I , 16 CELLS 07/.?1 /75

7. L)ECEL. SHOCK FROM HE / TIME= Z.0000E+OO

[“’’”’”’’’””“’””72.0

1.51

,..,’. . . . .,1,,.,(,

L

1.0 -

.5 -

0 -

-.5 -

wLJMwal --+c+

~o 5 10 15 20 0 5 10 15 20

Posltlon x (mm)Position x (mm)

5 -1 1 1 1 1

4 -

:0--,

0-3 -

+2VI 2 -0

;

>1 -

0 1 1 1 I 1 !

1.$

1.6

1.4

1.2

0 5 10 15 20

Position x (mm) F’osltlorl x (mm)

Fig. 4.Problem 7. Decelerating shock from HE. Standard calculation.

47

.PAlJ3.3f6 - 75 MI* FaNl? LAC2 .7. OECEL. SHOCK FROM We ERLN4.

IrlcKErl GY. 8.59? SEc CN w. 152.961 SEC ON .JW

128 CELLS 07/21/75

7. DECEL. SHOCK FROM HE / TIME= .Z.0000E+OO

1 I 1 1 1 1 I l., 1 1d,

Z*O -

1.5 -

1.0 -

.5 -

0 -

-.5 -

-1.0 :I ! ! I 1 1 1 I 1

Iz -

8 -

/

4

0

.

:0

a

alLJ

mI/la

k

o 4 8 lZ 16

POSit, Ofl X (mm)

o 4 8 lZ 16

POSit, Otl X (mm)

1,,.l,!,,,l.,,t,,[,,. 1,,,1.,,1,,,1LI&-1’118s I’8.18”1 “1’ 1’’- 1’1’1’1”4”4

1.84.0

3.0

Z.o :

1.0

1.6

1.4

1.2

-i-d J., .1. I.l. I.l. I I t 1 1 t 1.

0 4 8 lZ 16

Pos,t,on x (mm)

o 4 8 lZ 16

Posit,orl X (mm)

:

Fig. 4(cont).\

48

8PA@3.3F6 - 75.AK14 FWN1.? LAC2 ●7. CCCEL. SHOCK ~ROM HE.

1.+18 SEC CN RUN, 1+5.787 SEC ON JCd3lfl CKETIGY,

12.5

12.0

11.5,

11.0

10.5

10.0

—

2.0

1.8

106

1.4

I 1 1 1 1 i T---T

1.2 1.4 1.6 1.8 2.0 2.2 2.4

l,~e t (b’s)

1.2 1.4 1.6 1.8 2.0 2.2 2.4

l,~e t (Psi

ERUN 1. 16 CELLS 07/21/75 -

7. DECEL. SHOCK FROM HE / PARTICLES.-

1 1 1 ) I I I

25 -

\Zo -

15 -

\

10 - \

5 I I 1 I I ! !

1.2 1.4 1.6 1.8 2.0 2.2 2.4

2.8

Z.4

2.0

1.6

1.Z

T,me t (PS)

—1 1 { 1 { 1 r

\

\

\ 1\ \i I I ! ! ! ! 1

1.2 1.4 1.6 1.8 2.0 Z.Z Z.4

T,me t (Ps)

..

49

PAL13.3F6 75W.1% RUN!.? LAC.2 ,7. CECCL. SHWK FRON W“lfl ChTIIGY. 10.636 SIC m RW. 155.005 SEc m m

1 1 , , 1 1 1 ~

EE

‘?.0

1.5

1.0

()*5

10.0

2.0

1.8

1.6

1.4

I 1 1 1 I ! ( I

1.2 1*4 1.6 1.8 2.0 2.2 2.4

T,~e t (Ps)

I 1 1 1 I 1 1

, I 1 1 1 1 I

I.z 1.4 1.6 1.8 2.0 2.2 2.4

T,me t (Fs)

U?LNq. IZE CELLS 07/21/75

7. OECEL. SHOCK FROM HE / PARTICLES

30 1 I ) t I I

Z5 -

\\

20 -

1s -

10 -

51’’’’’’’’’’’’”’’’”” ‘“’’”

3.0

Z.5

2.0

1.5

1.2 1.4 1.6 1.8 2.0 2.2 2.4

T,me t (PsI

! 1 1 1 1 t r

1 1 t I 1 ! L1.2 1.4 1.6 1.8 2.0 2.2 2.4

T,me t (Ps)

9.

.’

.. PAD3.3F6 - 75wlk RuN 13 LACZ’

rFICKETIGY.●7. CCCEL. SI+3CK FRCWI HE*

3.61? SEC ON RUN, 167.840 SEC ON JOEERUN 4, 128 CELLS 07/.? 1/75

3C-J-,x

EE

x

Ko+J

IA

2

140

130

120

110

100

7. DECEL. SHOCK FROM HE / DIFFERENCE, TRUNCATE-D TIME / PARTICLES

I r 1 1 1 1

3~

. 1.8 2.0 Z.Z z.4

T,~e t (PS)

‘.0051 I I I I 1 r 1

-.006

E_\ .007

:

(!/ -.008

+5

$-.009

2-.010

-.011

1.4 1.6 1.8 Z.O Z.Z ,z.4

Time t (PS)

0.

al

-.08

I

L 1 I ! 1 1 I 1

-*1Z :

fh

-.16 -

-.ZO : u

-.Z4 -,i

I 1 1 ! 11.4 1.6 1.8 2.0 Z.z z’.4

Time t (Ps)

.012

.010

.008

.006

.004

1.4 1.6 1.8 Z.o z.z ze4

T,me t (tIs)

Fig. 4 (cent)

PAD3.3F6 - 75.AL14 Rwli2 LAC2 .7. CECEL SHOCK fFKM K-IF ICKEIIGY, 1! .536 SEC CN RUN. 155.907 sEC CN JOB

a

1 1 1 I

~~1.5 2.0 2.5 3.0

Veloc, tx “ (km/s)

PA03,3F6 - 75.AA14 R(R4I3 LACZ ,7. DECEL. SHOCK FRCX4 W“rrlcKcf16y. 3.E36 XC C’N RbT4. 168.065 SEC CN JOS

7. DECEL. SHOCK FROM HE-/

a

al

-..24 1.. 1,.,.1.,..1....!....!. . ..1....1....1.. !

&

ERM %. 12B CELLS 07/?1 /75

7. DECEL. SHOCK FROM HE 1 pAf?TICLES

30 1. 1.,.,1 1. I t.. . 1 1.. .-F

25 -

20 -

15 -

10 -

L5~J.u.b-. L

}.4 1.6 1.8 2.0

Oerrs, ty ~ (Mq/m J)

IRW *. I?B CELLS 07/2i~75

DIFFERENCE . TRUNCATED TIME 1 PARTICLES

1 I 1 1 I 1

-.08 :

-.1.2 :

-.16 :

-*ZO :

-.24 -,, ,,, ,,, .,,,,,,,,,,,,, ... ,...,:

.004 .006 .008 .010 .01.2 -.011-.010-.009-.008-.007- .006-.005

Veloc,ty u (km/s) Oews,ty c (Mq/mJ]

Fig. 4(cont)

52

a.

PA03.3F6 - 75JLA.14 ml? LAC2 ●7. OEC!ZL. SHOCK FROM HE*TFICKETIGY, .359 SEC IW Rw, 144.72S SEC CN &@

w.

x

i 1 1 118 :

I 1 1

16 :

14 :

12.-

10 -,:/

1 1 1 I 1 1-1.2 1.4 1.6 1.8 2.0 2.2 2.4

T,,ne t (Ps)

Z*O5 1 8 1 1 I 1 t

Z*OO -

mE 1.95 -

\

\ \

:\

\- 1.90 -v

\\

\$ 1.85 - \ .

i

j 1.80 -

1.75-

! ! 1 1 f 1 [1.Z 1.4 1.6 1.8 Z.O Z.Z Z.4

T,me t (tIs)

7.

ERUN 1, 16 CELLS 07/21/75

DECEL. SHOCK FROM HE / LEAD SHOCK

1 1 1 I 1 1 1

Z4

,~\

\

20 -\\

\

16 \l\

l.z-

V ~~

rl.l. 1 ! 1 1 1 Ii1.Z 1.4 1.6 1.8 Z.O Z.Z Z.4

Ttme t (Ps)

i T I 1 1 1 t\

\

Z.4 F\

\

‘“0 w1.6

17 N I‘-c~

1.4 1.6 1.8 Z.O Z.Z Z.4

Ttme t (PS)

Fig. 4 (cent)

53

PAD3.3F6 - 75JJL14 RIJ412 L AC? 07. MCCL. SHOCK FROM 1+-IF ICKEIIGY. 9.4%9 SEC W RUN. 153.818 Scc m m

18[. I , 1 1 1 1 I J

1 ..2 1.4 1.6 1.8 2.0 Z..Z Z.4

T,me t (Ps)

@/

1 1 # 1 1 1 1

Z.lo -

Z.05 -

Zoo -\ .

\1.95 - \

i1.90 -

1.85 -r

I I ! 1 1 , I

I*Z 1.4 1.6 1.8 Z.O Z.Z Z04

T,me t (Ps)

ERW II. I?fl CELLS 07/I?l~75

7. DECEL. SHOCK FROM HE / LEAD SHOCK

t.!cs.l.t,c,l.,.,.*.!t.ti1.Z 1.4 1.6 1.8 2.0 2.2 2.4

T,me t (PS)

I 1 1 [ 1 # 1

\

Z.6 : \

Z.4 :

Z..z:

Z.o :

cL.,..l.t.t.,.l.t.l...t .I.fi1.Z 1.4 1.6 1.8 2.0 2.2 2.4

Time t (Ps)

Fig. 4 (cent)

54

% PA03.3F6 - 75JW19 RUN13 LAC2 ●7 . WCEL SHCCK FROM liE*TFl CKE71GY. 9.891 SEC ON RUN. 174.1?0 SEC ON JOE

.08

.06

.04

.0.2

0

7. DECEL. SHOCK FROM HE /

1, !! ,1,,,,1,,,,1,,,,1,,,,1,,

o .01 .02 .03 .04 .05 .06

I/(NO. OF CELLS)

}’I’’’’l’’’’l’’’’l”d’’l’’’’l’’’’l”dn

.

o .01 .02 .03 .04 .05 .06

I/(NO. OF CELLS)

ERUN 5, 256 CELLS 07/21 /75

DIFFERENCE, TRUNCATED TIME / PARTICLE

I-,

1.Z

1“67/

/

7“

o~<f,,,,,,,,,,,,,o .o1- .OZ .03 .04 .05 .06

l/(NO. OF CELLS)

1 l“’’I’’ ’’1’ ’’’1 ’’’’ 1’’’ ’1’’”1 “-

.lZ –

.08 –

.04 –

0 –1 I 11 1 1,, ,, 1,, ,, !,, , ,1,,,,1,,,,1,,-0 .01 .Oz .03 .04 .05 .06

I/(NO. OF CELLS)

:

-b.

Fig. 4 (cent)

55

PAD3.3f6 75ml~ RUN 3 LAC2 .7. CECEL. WCK FRCM WI- Cw 5. .?56 CELLS 081 12)75

IF IIXEIICU. 13.356 SEC ON I?U’J, 3~.E56 SEc w m

-1. DECEL. SHOCK FROM HE / C! INFERENCE. TRUNCATED TIME / SHOCK

, l“’’I’’’’ 1’’’’ 1’’’’ 1’’’’ !’’”1’ ‘-I I

, 1 I I I I

/’: - : “’’’’’’’’”7:

t..-.- !3 —

Z.o

-() –.._EE

:c-l 1<

.10

.05

c!-~,,,,,,,,,,,,,)1{

0 .01 .0.2 .03 .04 .05 .96

i.

.

/.0 –

5 ~

o : 1, !,, 1, 1111111 [111111111 !111

o .01 .Oz .03 .04 .05 .06

I/(NO. OF CELLS) l/(NO. OF CELLSI

1

●lo –

.08 –

.06 –

.04 –

.Oz –

o – I, t,, lttlll! ltlllllll!tlllft

o .01 . Oz .03 .04 .05 .06

IA\E

x

J

.30 I

.25 –

.Zo –

.15 –

.10 –

.05 –

o –1 I111Ilv, ,11, llltlol llllllllllltlio .01 .0.2 .03 .04 .05 .06

4

l/(NO. OF CELLS)

Fig. 4 (cent)

I/(NO. OF CELLSI

56

PAD3.3F6 - 75w14 RuFl15 LAC2 ‘S. REVERBERATION*TFICKETIGY, .150 S&C ~ Rw, 175.631 SEC ON JOB

CRUN I, 16 CELL’+ 07/.? 1175

8. REVERBERATION / TIME= 8.5000E+O0 “-

..

30 -T

25 -

20 -

15 -

10 -

5 -

0 -

[ 1 I 1 1 I I1 1 1 1 1 I 1 [,

4*.

/=

.

4.5

4.0

3.5

3.0

Z.5

z.oloL_____JZo 30 40 50

tl....l....J....l... !....l...,l,,,.l,,,.l10 Zo 30 40 50

Posltlofl x (mm) Pos, tion X (mm)

l“’’’’’”’”’’’’’’’’’’’’”’’’’’’’’’’’’1’’”1 1 1 , 1 1 [ 1 1

Z.8 -

Z.6 -

Z.4 -

Z*Z -

Z.o -

1.8 -

1.6 -

1 1 1 1 ! 1 1 I10 Zo 30 40 50

.05 -

.04 -

.03 -

●OZ -

.01 -

I‘~10 Zo 30 40 50

Pos, t,on x (mm) POSit,Orl x (mm)

Problem8. Reverberation. Standard calculation.

57

PA03.3F6 - 75w14 PJM15 LAC.? ●E. REvEREERATIONSTFICKETIGY, 9.975 SEc m RW, 185.456 SEc CYi JOE

11”’’’ ”8”””81” ”’I”8”’1’’”8” ““1’’’’ {’””1

ERLN +, 1?6 CELLS 0711? I 175

8. REVERBERATION / TIME= 8.5000E+O0.

I f, I 1,..,1.,,, 1 r., I 1

4.5 -/’-

4.0 -

3.5 -

3.0 -

Z*5 -

r

L

5 :

0 -

5 -

‘~10 Zo 30 40 50

L zeo~10 20 30 40 50

.

POS, t,Orl X (mm) Pos, t,on X (mm)

. 101,.,..,....! . . . . . . . . . . . . . . . . . . . { . . . . . . . . . # 1 # 1 l.. 1 I 1

Z.8 -

Z.4 -

Z.o -

1.6 -

I 1 I t 1, t I !,,

.08

\

VI.04

0

0I 1 I 1 1 ! 4 1 I

10 Zo 30 40 50

Position x (mm)

10 Zo 30 40 50

POSlt, Orl X (mm],

“.. PAD3.3F6 - 75AfLlh RUN15 LAC2 .S. REVERBERATION~

TFi~K~TiGY. .<.3 SEC CN RUN. 175.905 SEC CN JOE

--

EE

x

+$

z.

z.

z.

z.

1 I ! 1 I I 1

65 -

60 -

55 -

50 -

45 -

40 -,[L!.. !ii . . ..l. ..l . . ..l. .il . . ..l6 7 8 9 10 11 lZ

Time t (Ps)

1 1 1 1 1 1 r

j

8Z ;

80 ;

78 )

2

76 ;

—— ——[ 1 I I 1 [ I6 7 8 9 10 11 lZ

al

ERWi 1

8. REVERBERA-

Z.o

1.5

1.0

.5

0

-.5

5.5

I/l 5.0\

3 4.5

3.5

16 CELLS 07121/75

ION / PARTICLES

t.l.#..l. #..ll... !.!. .! . . ..! . . ..l.6 7 8 9 10 11 lZ

Time t (Ps)

, , . . ., . . . . , . . . . . . . . , 1 .,,,1..,-P

I 1 ! I ( I 16 7 8 9 10 11 lz

T,me t (Ps)

Fig. 5 (cent)

Time t (tIs)

PA03.3F6 - 75.JJ-111 Rlnl15 LAC.? ●E. REVEREERATION*TFICKETIGY, 13. ~12 SEc m m, lEE. Wl SC CN J~

65

60

55

50

45

40

1 I I I 1 [ 1 J

1 1 I I I I 1

6 7 8 9 10 11 12

T,me t (PS)

,

: ~J/’’.A:1 I 1 # 1 I I

.070

.060

.050I

— —— —E1...l..l...l..,3i.l.,36 7 8 9 10 11 12

T,me t (Ps)

Em h, I.?E CELLS 071.?1/75

8. REVERBERATION / PARTICLES

I t 1 1 I I I

●4

.3

.2

.1

0 ;,,t,,,,~Av

5.Z

4.8

4.4

4.0

3.6

6 7 8 9 10 11 12

T,me t (Ps)

I 1 1 1 1 1 1;J

~8 9 10 11 Iz

T,me t (tIs)

,“

,

$.

Fig. 5(cont)

-.‘

PAD3.3F6 - 75&JL14 RUN15 LAC? .8. REVERBERATIC+J*TFICKETIGY, 13.906 SEC CN RW. 1S9.367 sEC ON JOB

a

Lo’’’’’’’’ I” IT IO I’I” IT Io’I III< I’

L1.l.l! !.l.4.1. !jl. t,l,,,,,,,, ,,, ,,,,, ,,3.6 4.o 4.4 4.8 5.2

velocity u (km/s)

LRUN I+. 1?S CELLS 07121175

8. REVERBERATION / PARTICLES

1 1 1 I I 1 1 , 1

.4

.3

.Z

.1

0

1 1 1 1 1 [ ! 1.07Z .076 .080 .084 .088

Density e (Mq/mJl-z.700

1 1 1 1 1 I I

.012

~‘; !

.010 -,

.008

.006

.004

“OO? ,,,,,,,,,4

6 7 8 9 10 11 Iz

Time t (PS)

Fig. 5(cont)

..

61

. . . . ..- ----- “---- . . .

PAD3.3F6 - 7S.k.Ll~ RW16 LAC2 ●B. REVEREERATlfJi”TFICXETIGY. 8.954 SEC ~ RUN. 229.296 SEC CN JOB

8. REVERBERANT ION /

1 I 1 1 # I 1

380

376

372

368

S16

514

512

510

J 1 1 I 1 1

UYo

x

2c)

&

alL

:Wal

Lk

8.4 8.6 8.8 9.0 9.2 9.4

T,me t (Ps)

L 1 I 1 1 I 1 1

Ino

8.4 8.6 8.8 9.0 9.2 9.4

T,me t (tIs)

x

LRW Y, la LLLL3 Ulr Cl ,13

DIFFERENCE, TRUNCATED TIME / PARTICLES

lC

5

c

-5

-lC

-1 1 1 1 t [ 4

J ! # f I 1 1

8.4 8.6 8.8 9.0 9.2 9.4

Time t (Ps)

F\/ \

vA

t, I I I I I 18.4 8.6 8.8 9.0 9..? 9.4

T,me t (Ps)

..,

.t

.

a

..

62

“4.

“-.

PAD3.3F6 - 75J(JL14 RuN16 LAC2 ●E. REVERBERATION*TFICX2TIGY, 30.137M SEc GN RUN, 251.818 SEC c+4 J3B

156 -

15Z -

148 -

r

8. REVERBERATION /

t 1 1 1 1 1

160 -

tl.1 1 1 1 1 !8.4 8.6 8.8 9.0 9*Z 9.4

T,me t (PS)

1 1 1 1 [ I

510 !

50(-):

490 .

i

480 ;

470 :I 1 ! 1 1

8.4 , 8.6 8.8 9.0 9.Z 9.4

T,me t (Ps)

ERUN 5. 256 CELLS 07/,?1/75

DIFFERENCE, TRUNCATEO TIME / PARTICLES

1 1 I 1 I 1

-z -

-4 -

-6 -

-8 -

-lo -

-lZ -

-14 -, I 1 1 1 18.4 8.6 8.8 9.0 9.Z 9.4

T,me t (PS)

1 I 1 1 I 1-146 -

-148 :

-150 -

-15Z -

1 1 1 1 I I8.4 8.6 8.8 9.0 9.2 9.4

Time t (Fs)

63

-- . . . .---, - “.. .-. .-

PA03. 3F6 - 75M 14 RuN16 LAC2 ●8. REWRBERAT [W.TFICKETIGY. 9.17S SEC CN RUN, 230. 1.?? SEC CN J~

8. REVERBERATION .’

l“’’I’”’’ 1’’’’ 1’’’’ 1”’” 1 “’’1’’”

10 -

5 -

0 -

-5 -

-lo -l.. I I I 1 i 1

-140 -138 -136 -134

ty u (km/s) X105Veloc

Lnox

U(UI W. IcI! LLLL3 Ut~Cl~13

DIFFERENCE. TRUNCATED TIME / PARTICLES

10

5

0

-5

-lo

1 I ! i 1 1.,..1= 1

1 # [ 1 I I I d

510 512 514 S16

Oens,tx t’ (Mq/mJ) X106-Z8000

1 I 1 1 # t

16 :\

12 :

8 :

4 -

0 :1 1 1 1 t

8.4 8.6 8.8 9.0 9.2 9.4

T,me t (PS)

.b

64

PAD3.3F6 - 75.XJLl~ RUN16 LAC2 .8. REVCRBCRATION+ LRuN 5.TFICI(ETIGY. 31.3(?9 SEC EN RUN.

256 CCLLS 07/?1 (7525?. ?72 SEC ON JOEl

8. REvERBERATION / DIFFERENCE, TRUNCATED TIME / PARTICLE

EE

*30

.20

●lo

o

1 I“’’t ’’’’ l’’’ ’l ’’’’l’’”

-1 ll!:!lfi tltl, llll##!llt4[llli-

o .01 .02 .03 .04 .05 .06

l/(NO. OF CELLS)

.030 JI’’’’l ’’’l l’’’ ’l ’’’’l’” ‘l ’’’’ l’J_

.025 –

: .020 –

:

@/ .015 –

x+J

m .010 –K

2.005 –

P0(!)

1 i 1 1 t 1, 11111, 1, 111111, 11,11, ,,11(

o .01 .02 .03 .04 .05 .06

l/(NO. OF CELLS)

.25

z .20c)

a. 15

Ci.05

VI

0>

0

.030

.0.20

.010

0

I

0-+

1 1 i I 1 Il!l!ll 1111{1, lllli, llilllll-

o .01 .02 .03 .04 .05 .06

l/(NO. OF CELLS)

1 1“’’1’’’’1’’’’1’’’’1’’’’/’” ‘1’~

I 1,, !, 1,, ,, 1,, ,, 1,, ,, [,,,,1,,-

o .01 .02 .03 .04 .0s .06

I/(NO. OF CELLS)

Fig. 5 (cent)

65

PAD3.3F6 - 75.AJ-14 RWle LAC2 ●9. BLAST 14AVE0

TFICXETIGY, .130 SEC CN RW. 253.+71 SEc m m

L“’I’’” I’’’’-’’’’’’{”’’’’’”’”” “’’’’”~

12 -

8 -

4 -

0 -

16 20 24 28 32 36

POS,t,O~ X (mm)

L.I-l.I.l.ll”l’l”l’l”-I “’1’’’!”1””1~l~

4.0 :

3.0

2.0 ;

1*O

1o~16 20 24 28 32 36

POS,t!Ofl X (mm)

Fig.Problem 9. Blust wave.

2.0

1.5

1.0

.5

0

ERlR4 1, 16 CELLS 07121/75

9. BLAST WAVE / TIME= 1.5000E+O0

1 8 t ! I I I I I I 11

J. I 1 1 1 I 1 ! [. t I I

16 20 24 28 32 36

Post t,on x (mm)

1.2 -

1.1 -

1.0 -

.9 -

.8 -

.7tr.t,lc,.l.,.l.,.9cf.1,.o.l.t.{ .1.,..16 20 24 28 32 36

POSiti Orl X (mm)

Standard calculation.

.

2

i

66

-..

“.

PA03.3F6 - 75JW14 RUN I Ei LAC2 .9. 8LAST WAVE-TfICKETIGY. 7.ohl SIC m RUN. 260.381 SEC ON X@

0.

0-

>

12 -

8 -

4 -

0 -

18 20 22 24 26 28 30

POS, t,Ori X (mm)

5!.,.!, I I I t I

4 -

3 -

2 -

1 -

0 J1 ! I 4 1 1

18 20 22 24 26 28 30

POS, t,Orl X (mm)

1.0

t

ERUN 4, 1.?8 CELLS 07/,? 1/75

9. BLAST WAVE / TIME= 1.5000E+O0

I 1 1 1 1 1 I

2.0 -

1.5 -

I

“:=18 20 22” 24 Z6 Z8 30

POS,t,Ofl X (mm)

L“ \ I I I 1 I 1

w

LLbLk&- -18 20 Z2 Z4

POSitlOll X

1 ! {

Z6 28 30

(mm)

;

*●

PA03.3J6 - 75.RJ-1~ WNle LAC2 ●9. ELAST UAVEQ

TFICKETIGY, .741 SEC m RW, 25+ . 08? SEC CN JC@

l’I’’’’ [’’”’ 8’”’’ 1’’”’ 1”’’’ 1’””1 “’’1 ’’’’’’’’ ”’”4

Z6.4 -

E

: Z6.O -x

co

? Z5.6 :m:

Z5.Z ~

.

t., .. ..l... U.l....k....l ....l. t..l!j.8 1.0 1.Z 1.4 1.6

Time t (Ps)

L’l”’r’ I’’’’ ”r’’””l ””ll’’’”l””l ““’8 ”’’’ 1””~”~

1.00 — —1 . ...1.. 1 . . . . 1 . ...1....1 1.. .<1.

.8 1.0 1*Z i.4 1.6

T,me t (Ps)

:c)

0.

m\E

M

J

ERWI, 16 CELLS 071.?1/75

9. BLAST WAVE / PARTICLES

Z5 -’I 1,...1,.,,1, ],,,,.1,,..~

\

\

Zo -

15 -

10 -

5 -

0 -,— ,— 1,. A.. .L. . .. L.1. ..! I!. .4J-I-.

.8 1.0 1*Z 1.4 1.6

Z*5-r,,,l..,r-

Z.o -

1.5

1.0

.5

f

o -— ,-,. 1,, , 1..,.1-.1...,1....1, ... !.

.8 1.0 1.2 104 1.6

T,me t (PS)

Fig. 6(cont)

68

..● PA03.3F6 - 75 WLl Ii RUN 113 LAC2 ●9. BLAST !4AVE.

lFICKETIGY$

.

26.4

26.0

25.6

25.2

1.3C

1.20

1.10

1.00

8.73! SEC ON RUN. 262.071 SEC ON tiB-

1 1 r 1, , 1 1 1 I !1

t.l..!... !.,,.1,,.,,,,.,,,,,,,,,..,,,,,,.,.,,,;

.8 1.0 1.2 1.4 1.6

T,me t (PS)

, 1 , 1, 1 8 1 I T

1 ...1.... 1,, ,,1 1 1,. , 1.. ,,! .,,,1

.8 1.0 1.Z 1.4 1.6

z<

2(

Is

lC

5

0

1 1

-1

ERUN I+, 128 CELLS 07/21/75

9e BLAST WAVE / PARTICLES

I i,., 1,., 1, 1, 1, 1, ! 1

.8 1.0 1.2 1.4 1.6

T,me t (PS)

2.0 -

1.5 -

1.0 -

.5-

0- -l~

1.2 1.4 1.6

T,me t (PS)

Fig. 6(cont)

Time t (PS)

PA03.3J-6 - 75U14 RW18 LAC? 09. BLAST WAKOTFICKETIGY. 9.139 SEC C$4 RLN, .?6?.479 SEc ON .Xx3

c?c)

0.

aL3

IAMal

&

o .5 1.0 1.5 2.0 2.5

Veloclty u (km/s)

CRUN ~. 1.?S CELLS 07)?1?75

9. BLAST WAVE / PARTICLES

Z5

Zo

15

10

5

0

1.OO1. 051.101.151. ZO1. Z5 1.30

Dens, ty ~ (MqzmJ)

1,,.,[..,.l.,,, I..,,I,,..T..T

8

6 ;

4 ~

z :

0 ; J1 1.,.,1,.,.1,.,,1,..,8. .,~

.8 1.0 1.Z 1.4 1.6

T,me t (Psi

...

.

.

Fig. 6(cont)

-a-

70

..b PAD3.3F6 - 75JlLlk RUN20 LAC2 .~. FE A%l UAVF.

TFICKFTIGY. 5.*9O SE[ ON RLN.. .

3ol.730-iEc liN :&

9. BLAST WAVE ,/-.

I I [ 1 1 1 1 t4.20-

410 :

400 -

390 :

tl. .. l. .. l...,,.,,,,,,.,,,, ,,, ,,.,,.{1.0 1.2 1.4 1.6

T,me t (Ps)

L

E\ -.006 -

:

w

:-.oo8 -

ic

:

-.010. -

r

I 1 1 1 1 1 I 1-.004

Ii

1.0 1.2

T

~~1.4 1.6

me t (Ps)

ERUN +. 128 CELLS 07/.? 1/75

DIFFERENCE, TRuNCATED TIME / PARTICLES

o

-.Z

-.4

-.6

-.,l~d. 1.Z 1.4 1.6

T,me t (I-IS)

lo’’’’’’’’l”’’’’’’”’’’’’’’’’’’’’’’’”lL

.03 -

.Oz -

.01 -

0 -

-.01 -

J

i

-“oz~1.0 1.Z 1.4 le6

Time t (PS)

Fig. 6(cont)

71

PA03.3F6 - ?5An 14 FIJN,? LAC2 .9. BLAST WAVE-TflcKETIDv.

EKW4. 128 CELLS L18i 131757,?76 sEC ~ RIN, 8.418 SEC m .x9

9. BLAST WAVE / DIFFERENCE. TRUNCATED TIME / PARTICLES

EE

1.0 1.2 1.4 1.6 1.8 2.0

T,me t (Ps)

1 I

-.0041 1 t 1 1

#

ll. l! l. t. l.!. l.l. l.l.l!1.0 1..2 1.4 1.6 1.8 2.0

T,me t (PS)

IA\E

J/

(tot=2)

o

\

-ef3~1.0 1.2 1.4 1.6 1.8 2.0

T,me t (Ps)

~“’~

.03 -

● 02 -

.01 -

0 -

-.01 -

-.OZL-[ I I I ! 1 1 i

1.0 1.Z 1.4 1.6 1.8 z.o

T,me t (Ps)

. ..

.-

Fig. 6 (cent)

72

s

J.

PA03.3F6 - 75JIJLI< RL!wo LAC2T~l CI(~TIGY.

“9. BLAST WAVIO5.763 SEC IX RN. CR(JNLI,

302.003 SEC CN WE133 CELLS 07/? I /75

9. BLAST WAVE z DIFFERENCE, TRUNCATEI) TIME / PARTICLES

-.02 -.01 0 .01 .Oz .03

Veloc, ty u (km/s)

1

0.

aJL3

vlmc1

L+

-.8~-.008 -.006 -.004

Dens,tx e (Mq/mJ)

.lz, ,! .. l . . . .. !.. rr, r!, ,rr,r ,,, ,,, ,,,,,,,

0-

>

.10 -

.08 -

.06 -

.04 -

.oz-

0 ,,,,,l;1.0 I.z 1.4 1.6

—

T,me t (PS)

Fig. 6 (cent)

73

ERw I. 16 CELLS 07/?1175

9. BLAST WAVE / LEAD SHOCK

x

Ko+-J

IA

2

3.2 !“’’”’’’’ ”’--’’’’’’’’’’’’’”’’’’’’’’’”’”’’’”:

30 :

28 !

Z6 :

1 1 ,. , I 1....1,,..1. 1 1.8 1.0 1.2 1.4 1.6

T,me t (tIs)

1—————————

1.3Z 1

r.t . . ..l . . ..l.. c.l . . ..l . . ..! . . . . . . . . . . . . ..l . . ..l.J.8 1.0 1.Z 1.4 1.6

30

Z5

Zo

15

10

Z.8

Z.6

Z.4

Z.z

Z*O

1.8

~,.Is,,.I,T..I,s..I=,.. lr,..t,,.~.[....[.”

“/

\

l., 1 l,, ..l, . ..lc. ..t. .,. t, . ..1 U. LA,

.8 1.0 1*Z 1.4 1.6

Time t (PS)

l“’’’””’ or”’’’’’’’ ”r””n’’””’’’” ”’’”’’””n:\

J

1,, ,.r, ,,, [,. ,.l

.8 1.0 1*Z I*4 1.6

T,me t (PS)T,me t (Ps)

74

“.I

-.PAD3.3F6 - 75.AJI- lq RUN 113 LAC2 ‘9. BLAST HAVE-TFICKITIGY. 7.907 SEC CN RUN. ?61. ?+7 SEC CN JOE

-..

EE

x

w.

1 1, [Tr.,i., 1,,,1,.,,1,, 1 1 1

30 !

Z8 :

.2’6 f

i

Z4 :

t.l....l . . . ..! . . ..%! ..l . . ..! . . ..l... %!..,.,,,,,,,].8 1.0 1.2 1.4 1.6

T,me t (Ps)

P“’”’’”’ ”’’”r”’’ ”’’’’ ”’’’’’’’’’’ ””’’’’’ ”’’””3[—_ —______jo0 .032m i. L

(

.0Z8 -.E\

~

.0.24 :@/

$

.020 -:

:.

.016 ‘1 1“ ‘1.,, ,1, .,, ,,, ,,, ,. ..,, ,,, ,, ...,, 1,

.8 1.0 1.Z 1.4 1.6

&

aLJ

1A

M

u

m\E

A

ERUN +. 133 CELLS 07/,? [/75

9. BLAST WAVE / LEAD SHOCK

~lrr,l,r,lrl,,,.i. ,S,l,,,l....r....,.

30 -

Z5 -

Zo -

15 -

l%,,,!, !,, ,, {,, , ,1 .,, .,,,,.,. ,, !,, , 1,, 1,

Z.8

Z.6

Z.4

.Z.Z

2.0

1.8

.8 1.0 1.2 1.4 1.6

T,me t (Ps)

\'''' ''''' ''''' '''"' '"''' '''"" 'r''' 'o''rr''r'`''oJ

l,. ., [,.,,,,,,,,,,,,,,%, ,,, , .,,,,,,,,,,,,,. 1,

.8 1.0 1.Z 1.4 1.6

Time t (FS) Time t (Ps)

Fig. 6(cont)

75

PA03.3F6 - 75wlq 13JN19 LACZ .9. 6tAST WAVESTFICKETIGY.

ERU!I, 16 CELLS 071.?1 /75.3m SEC m Rt.N. .?75 .*77 SEc m m

9. BLAST WAVE / DIFFERENCE / LEAD SHOCK

l-f , i t { ! [ I I 1 t 1 1 r—

-2. 0 ;

-3.0 ~

-4.0 -

-5.0 :

, , .L

-Z.o :

:2 c)

i -3.0 )0.

alLJIAW -4.0 -a

k

-5.0:

1 1 1-.14 -.iz -lo -.(38 -.06 -.04

1 I 1-.10 -.09 -.08 -.07 -.06

Veloc,tx u (kmzs) Dens,ty r? (Mqzmj)

-1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tr*l---r-=*r-

-.2.0-

2c)

w -Z.5 -x+J

W0

-3.0 -:

>

-3.5 -

1. . . . 1., . ., .,, ,,, ,, .,, ... ,., .,#. ,,. ,,, ,,, .,L

.8 1.0 1.Z 1.4 1.6

T,me t (PSI

.

..

.

Fig. 6(cont)

76

:

--●

--. PAD3.3F6 - 75JU14

lrl CKEll GY.RUN.?O LAC2 “9. BLAST NAVE.

16,775 SEC CN RUN.ERuN 5. ?56 CELLS 07/21/7.

313.015 SEC EN JOB

‘3. BLAST WAVE / DIFFERENCE, TRuNCATED TIME / PARTICLE

%.

EE

x

Ko

LA

:n

r“’’’”7i.10

.05

1off’

I 1 1,, , ,1,,,,1,,,,1,,,,1,,,,1,,

.040

.030

.020

.010

0

0 .01 .02 .03 .04 .05 .06

I/(NO. OF CELLS)

l“’’I’’ ’’1’ ’’’1 ’’’’ 1’’”1’ “’[”-

!

0 .01 .02 .03 .04 .05 .06

l/(NO. OF CELLS)

1 l“’’I’’ ’’1’ ’’’1 ’’’’ ;’’’ ’1’’”1 ‘(

1.2 —

.8 –

.4 —

o –1 1 I 1 ! 1, !, ,1,,,,1,,,,1,,,,1,,,,1,,-

o .01 .02 .03 .04 .05 .06

l/(NO. OF CELLS)

1 !“’’I’’ ’’1’ ’’’1 ’’’’ 1’’’ ’1’11 rlll-

.08 F

.06 :

.04 :

.02 :

1,, , ,1,,,,1,,,,1,,,,1,,,,1,;

o .01 .02 .03 .04 .05 .06

l/(NO. OF CELLS)

Fig. 6(cont)

77

PAC13. K6 - 75M14 Fwi19 LAC,? .9. SIAST LIAVE -TFIChZTIGY. ?O. *5+ SEc CN m. ?95.5~7 SEC CN JOB

.8 l“’’1’’’’1’’” 11I! 1II1 tI1t 1

.6 : ‘ ‘/~

.4 —

;,/,,,,l,,,,,,,,,,,,,,lj

/

..2

0

0 .01 .Oz .03 .04 .05 .06

I/(NO. OF CELLS)

rl“’’I’’’’ 1’’’’ 1’’’’ 1’’’’ 1’’”1’ _

.06

.04 ~

● Oz ~

o :1,, ,, 1,, ,, 1! 111 !!l,lll, ,11!1+

o .01 .02 .03 .04 .05 .06

I/(NO. OF CELLS)

ERIJ4 5. Z% CCLLS 07/?1 /75

9. BLAST WAVE / DIFFERENCE / SHOCK

3.0 1l“’’I’’ ’’1’ ’’’1 ’’’’ ;’ ’’’1’’” l“-’

Z.5 –

Z.o –

1.5 –

1.0 –

.5 –

o –! I1 tl,t#!llt!l[tl!lltl!l18111111-o .01 .OZ .03 .04 .05 .06

l/(NO. OF CELLS)

1 l“’’I’’ ’’1’ ’[ ’1 ’’’’ 1’ ’’’[’” ‘[”-

.06 –

.05 –

.04 :

.03 :

.Oz :

.01 –

0 –1,, , ,1,,,,1,,,,1,,,,1,,,,1,,-

0 .01 .OZ .03 .04 .05 .06

l/(NO. OF CELLS)’

.-.

e.

.

Fig. 6(cont)

.

78

:

●✎

P,VJ3.3F6 - 75~Llh RLNi?? LAC2 ‘IO. STEADY DETONATION - ERUN 1. 16 CELLS 07/?1 /75-.. lFICKETIGY. .?02 SEC ON RUN. 314.293 S~C CN JOB

10. STEADY DETONATION / TIME= 1.7000E+O0

4.0 :“’’’’’’’’’’’’” 1 1 I )

3.0 ;

2.0. -

1.0:

0:1 1 [ 1 I 1 1

%.50

t

40

3011

Zo

10

IL

4 8 lz 16

Pos,t,orl x (mm)

4 8 12 16

Posltlon x (mm)

!“”’’’’’’’’’’’’’’’’’’’’’’’’”ll

Zo -

15 -

10 -

5 -

0 -1 1 ! ! ! 1 i

Z.4

2.01

.-IAE

2

1“6~4 8 lZ 16

4 8 lZ 16

Pos, t,on x (mm) Posltloi-1 x (mm)

Fig. 7.Problem 10. Steady detonation. Standard calculation.

79

PA03.3F6 - 75UI% Ku?@? LAC? ● 10. STEADY DETONATION ERUN9. 12E CELLS 07JZ’I /75TFICUETIGY, 9.666 SEC Cti RUN, 323.757 SEc CN .#3f3

60 r i I r { 1 1

a3f3;~

aL;IA 20 -alk

10 -

0 -I I I I 1 I

4

20

> 5

0

6 8 10 12 14

Pos, t,on x (mm)

4 6 8 10 12 14

Pos, tlon x (mm)

10. STEADY DETONATION / TIME= 1.7000E+O0

4.C

3.0

Z’. c

1.0

0I I I I 1

4 6 8 10 12 14

POS, t,Orl X (mm)

2.8 -

2.4 -

.2.0

1.6L4 6 8 10 12 14

on x (mm]

.-‘

Fig. 7(cont)

80

:

-.

PAL13.3F6 - 75AL14.-. .. —- Kut+?? LAC.5 ,10. STEAOYEILICINATION. ERuN I . [6 CELLS 07/?1175

-.. 1! [LkX1l GY. .339 SEC CW RUN, 314. ~30 SEc m JOB

10. STEADY DETONATION / TIME= 1.7000E+O0

1.01, 1 1 1 I 1 I 2.0 1 1 I 1 1 1 i

1.5 -

1.0 -

.5-

0- ~ ,,,,,~j1 1 1 t

‘4 8 12 16

-9.

.8 -

.6 -

.4 -

.2 -

0-

..

u0

I I , , I 1 1 I I4 8 12 16

F’osltlof-l x (mm) POSit,On x (mm)

PA03. 3F6 75JA I* RUQ? LAC2 .10. slcmy DEToNATIoN.lflcKCTIGY. 9.825 SEC ON RUN. 3?3.916 SLC ON JC9

ERUN4. 1?8 CELLS 07/2’1 /75

10. STEAOY DETONATION / TIME= 1.7000E+O0

Z.or’’’’’’’’ r’ro”r’

1 [

1.0 -1 1 i

.8 -

.6 -

.4 -

.Z -

0-1 ! 1 [ I

1M 1.5 -

1.0 -

.5 -

Kax

w Q

‘u

:IY

1

‘~4 6 8 10 Iz 14

4 6 8 10 [z 14..

-..

Poslt!orl x (mm)Posltlorl x (mm)

Fig. 7 (cent)

81

PA03. W6 - 75M14 mRi22 LAC.2 .10. STEAOY OETONAll CNOlFICKETIGY,

14.0

13.0

12.0

11.G

Io.o

.873 SEC ON m. 319.%4 SEc CN w

r , r-’--l I 1 1 1 1 4

a

aJ1-

:mal

L+

4 1 1 t 1 I 1 , 1

1.2 1.6 2.0 Z.4

T,me t (tIs)

t r r 1 1 1 1 1 I 4

\

-!

1.2 1.6 2.0 Z.4.—

T,me t (PS)

mw 1. 16 CELLS 07/al~75

O. STEADY DETONATION / PARTICLES

t

501

:c1

40I

m

30 -

Zo -

10 -

~~,,,,,,,,,,,,,,,,,o—

1.Z 1.6 Z.o Z.4

T,me t (tIs)

\

:+2 1.0

~/,,,,,,,,,t,,,,,,,<o—

1*Z 1.6 Z.o Z*4

Time t (PsI

.:.

r.

Fig. 7(cont)

.“

.

PAD3.3F6 - 75-J&l% Rtt&Z2 LAC2 ●lo. sTEAOY DETWATIW*TFICKETIGY, 13..?05 SEC C?4 RLN, 327.2% SEC CN JOE

14*O I I 1 1 1 ! I 1

13.0 :

12.0 :

11.0 :

10.0 . I 1 I 1 1 ! I !1.2 106 2.0 2.4

T,me t (PS)

-1

.~1.2 1.6 2.0 2.4

Time t (Ps)

EFWN ~, 128 CELLS 07121/75

10. STEADY DETONATION / PARTICLES

501

40 -

30 -

20 -

10 -

0 -—1 1 1 1 1 t I 1 ~

4.0

3.0

2.0

1.C

c

.I

-I

1.2 1.6 2.0 2.4

Time t (VS)

1 1 I , 1 1 1 1 4

1.2 1.6 2.0 2.4

Time t (Ps)

:

-..

Fig. 7(cont)

PAD3.3F6 - 75Jl14 KWN?F’ LAC.? .10. SIEAOY OCTCNAll ON- El?.m IlrlcKc TIGY. 1.0% SEC CM RLN. 315. lE17 sEC ON .J3E

16 CELLS 07/?1175

ION / PARTICLES10. STEADY DETONA

-1....,,., 1 1 a 1 r r 1

50 -

40 -

30 -

.20-

10 -

0;/

1 1 ! 1 ! !o 1.0 2.0 3.0 4.0

so

4020

c1 30a(.3IAIA

Zoill

c+10

0

1 I I I 1 T

I 1 ! 1 1 1

,.8 2.0 .2.2 2.4 2.6 .2.8 3.0

Velocity u (km/s) oens, t. c (Mq/m J)

PA03.3F6 - 75.M_l~ RLF&?? LAC2 ‘IO. STEACIY ~TONATICW. ER!J44, I?E CELLS 07 J?IJ75

lrl CKETIGY. 13.530 SEC 04 RU4. 327.621 SEC GN .X63

10. STEADY DETONATION / PARTICLES

60,

50

& 30a

k10

0

60

50

a 30

01.. , 1 1 1. [. I I 1 i

o 1.0 2.0 3.0 4.0

Veloc, ty u (km/s)

1 , t 1 I I i 1

t. t. o.1. o.l. t. l. t. l. o.l. ,.l. o.!.* t1.6 Z.o Z.4 Z.8 3.Z

Density r? (Mqzmj)

“-

.

,.

.

.

:

-.Fig. 7(cont)

-..

-..

PA03.3F6 - 75JJ-lk MJN2h LAC? .10. STEAOY OETONATION” ERIJN h.TFICKCTIGY. 10.385 SEC LT4 RUN.

12E CELLS 07(.? 1/75398.924 SEC ON .MB

10. STEADY DETONATION .’ OIFFERENCEr TRUNCATED TIME / PARTICLES

140! { 1 1 1 1 1\ i

t[. $. l. o.l. 8. {. $., .8. ,.t. lc.j1.2 1.6 2.0 2.4

T,me t (tIs)

.005[, .,., 1 [ 1 , 1 1

‘ -.005:

w -.010

-.020

1)4!.1.(!.!.(. .!. !.!11.2 1.6 Z*O

1 ,me t (Ps

2.4

.

.

_*

.

0 -

z -

4 -

6 -

8 -

F!’.!.l. l.!. !.l. !l!.l. l.l.,. j$.Z 1.6 Z.o Z.4

Time t (PSI

1“’’’’’’’’’’’’’’’’’”’’” ““J.03\

.Oz -

.01 -

0 -

-.01 -

-*OZ -

(-li. j. l. !, l#!. ,,, ,1, ,., ,,, l,,,j1.2 1.6 Z.o .2.4

T,me t (Psi

..

-..

Fig. 7(cont)

85

PA03.31_6 - 75.AIL14 Kl!N.?+ LAC.2’ .10. STEADY DETONATION” Em 5.

lFICKETIGY. ?.2. 161 SEC CN m.

256 CELLS 07121/75

4?0.700 SEC CN .nE

10. STEADY DETONATION .’ DIFFERENCE. TRUNCATEO TIME / PARTICLES

“.

.

1 1 r I 1 1 1

15 -

10 -

5 -

0 -

-5 -

-lo -I f ! ! 1 I !

I I I I 1 ! I I .I,..

0 -

-.1 -

-.Z -

-*3 -

-.4 -

,

EE

0.

OJL

?IAa

k

x

-.5 - 1’ J

1 1 1 I 1 t [1.Z 1.6 Z.o Z.41*Z 1.6 Z.o 2.4

T,me t (Ps)Time t (tIs)

i r I 1 I t I

.006 1. 4

.015 -

.010 -

.005 -

J

.004

.Ooz

o

.00Z

.004

0’

.005

I

-.OIO; I’ Ii

1:. !.l.,.,.,.,.,,. Il.z 1.6 Z.o Z-4

I! hi!. l. l. l. t. l.l. l., .l.,.l1.Z 1.6 Z*O 2.4

T,me t (Ps)T,me t (Ps)

Fig. 7(cont)..

--.

86

-..

-..

.

PAD3.3F6 - 75 JJL14 RuN24 LAC? .10. STEADY DETONATION- ERUN q, 128 CELLS 07/.21/75TFICKETIGY. 10.618 SEC C+l RuN, 399.157 SEC ON Jus

10. STEADY DETONATION ,’ DIFFERENCE, TRUNCATED TIME / PARTICLES

1“’1 1 [ 1 1 1 1 r 1 I I 1 I 7*L

o

-..2

-.4

-.6

-.8

1 , I I 1 I 1 1

-.0.2 -.01 0 .01 .02 .03

Vetoc,tx u (km/s)

256 Cells1 I 1 I I 1 1

.06 -

.os~

.04 :

.03 ;

.02 :

.01 -

0 - I1 1 1 ! ! 1 1

. .

1..2 1.6 2.0 2.4

T,me t (Ps)

Fig. 7(cotzt)

.2 -

0 -

-.2 -

-.4 -

-.6 -

-.8 -

1. .. (, ..1 . . ..! ..1... 1.1

-.020 -.015 -.010-.00s o .005

Derlsity t’ (Mg/mJ)

L“’’’’’’’’’’’’’’’’’’’” ““’”JL

.06 -

.05 -

.04 -

.03 :

.02 -

.01 -

r

J

o1 1 1 1 1 I 1 I

1.2 1.6 2.0 2.4

Time t (PS)

a.

87

PAD3.3f6 - 75-14 Rw.?z LAC? ●IO. STEAOY CC TDNATION”lFICKETIGY, .521 SEC m RUN. 311t.61? SEC ON JDB

r 1 1 1 1 I I I

18 -

16 -

14 -

12 -

10 -

8 1 4 ! ! I 1

1.0 1.2 1.4 1.6 1.8 2.0

Time t (Ps)

t # T 1 1 I r

3.20 :—————————

3.10 :

.2.90

2.80

1.0 1.2 1.4 1.6 1.8 2.0

—. .— ——Time t (Ps)

al

EM_Nl. 16 CELLS 07/.? 1)75

10. STEADY DETONATION / LEAD SHOCK

58

56

54

52

1.0 1.2 1.4 1.6 1.8 2.0

4.Z

4.1

4.0

3.9

3.8

T,me t (Ps)

1 I 1 i I 8 r

— — —— —— —— .

[1 .! .l. l. !. l.!. l.!. l.l.i1.0 1.Z 1.4 1.6 1.8 Z.O

-- . ——Ki-e t (1’s)— ‘–-

-..

..

.

.

:

a.

-.. PAD3.3F6 -

TFICKETIGY

,.=. . ,,. “, ,., -. ,. -.., “-. . . ,. Vm. a . . . . . . . . . r-, ,., . c, .,-, , t. “-, ,-, .-, C,3.An ,. nimc> Lm. c -IU. 31 LMUT ur_lwwIIu Iv- LKUN 3, O’1 LLLI-> UI, C1, 12

4.500 SEC CN RuN, 356.4.25 SCC ON J3B

10. STEADY DETONATION / DIFFERENCE / LEAD SHOCK

[’ 1 1 t I I “-lI 1 1 1 i 1 1 1

z1 -1.2E.

-.

L

-1.4 -

-1.6 -

-1.8 -

-2.0 -

-2.2 -

.08 -

.06 -

.04 -

r

x

co

a

I

I 1 ! 1 [ 1 1 I1.0 1.2 1.4 1.6 1.8 2.0

T,me t (Ps)

-.090~ , I I 1 ! 1 ~

1.0 1.2 1.4 1.6 1.8 2.0

T,me t (tIs)

r 1 1 1 1 r r

-.14. .

-.-.15

-.16

-.17

-.18

-.19.-

1111

I 1 ! 1 1 I 1

1.0 1.2 1.4 1.6 1.8 2.01.0 1.2 1.4 1.6 1.8 2.0

T,me t (PS) T,me t (PS)

Fig. 7(cont)

89

PA03.3F6 - 75U14 fiw23 LAC2 O1O. STEAOY MTCFJATl IX40 U?L?i 4.TFICKETIGY. 11.5EOSECGNRW.

1.33 CELLS 07/.?1 /75363.50% SEc m .x@

EE

x

+J

VI

:

10*

[’ 1 1 1 1 1 1

.04 -

.03 -

● Oz -

.01 -

0 -

-.01}, 1 I 1 I I i

1.0 1.2 1.4 1.6 1.8 2.0

T,me t (FS)

11’’ ”l ”’’’’” 1 I I 1

-.045

t

--6050 -.E\

:-.055 -

bA -.060.-+JW;-. o65 -0

,

-.070

I_.075~

1.0 1.2 1.4 1.6 1.8 2.0

T,me t [Fs)

STEADY DETONATION / DIFFERENCE / LEAD SHOCK

I 1 I 1 1 1 I 1-.6 -

-.8 -

:0

-1.0 -aalL3 -1.2 -WMak -1.4 -

-1.6 -r

.088

.092

.096

I I 1 1 1 I I1.0 1.2 1.4 1.6 1.8 2.0

T,me t (tIs)

L.I’1 1 1 I r

1

-.loo~,1 ‘1 ‘

I 1 t I1.0 1.2 1.4 1.6 1.8 2.0

T,me t (Ps)

.-.

...

Fig. 7(cont)

90

.-.

PAD3.3F6 - 75 JuLlq RuN24 LAC2 .10. STEAOY OE1ONATION.TFICKCTIGY, 33.008 SEC ON RbT4.

muN 5, 256 CELLS 07/211754.?1 .547 sEC CN J3B

10. STEADY DETONATION / DIFFERENCE, TRUNCATED TIME / PARTICLE

..

EE

x

1 l“’’I’’ ’’1’ ’’’1 ’’’’ 1’ ’’’1’’” l“

.16 :

.12 ~

.08 :j

.04 :

0 :1,,,,1,,,,1,,,,1,,,,1,,,,1,:

o .01 .02 .03 .04 .05 .06

I/(NO. OF CELLS)

r

.08 f

.06 ~

.04 :

.0.2F

o :

0 .01 .0.2 .03 .04 .05 .06

I/(NO. OF CELLS)

1 l“’’I’’ ’’1’ ’’’1 ’’’’ ;’’’ ’1’’”1 ‘~1.6 –

1.2 —

.8 –

.4 —

0.–

o .01 .02 .03 .04 .05 .06

I/(NO. OF CELLS)

PI’’’’l ’’’’ ’’’’ ’l ’’’’ l’’’ ’l””l ‘1

.040 >

.030 :

n?n.ULU

> .010 y

1,,,,11,,,1,,,,1,,,,[,,,,/,,o .01 .02 .03 .04 .05 .06

I/(NO. OF CELLS)

Fig. 7(cont)

91

PA03.3F6 - 75.AA-14 RUN23 LAC? ‘IO. STEAOY OETE!NAIICNOrr]~TIGy.

ERLN 5.35.909 SEc m m. 387. s33 SEC CN .mB

?56 CELLS 07121175

-..

10. STEADY DETONATION / DIFFERENCE / SHOCK -..

.40

.30

.20

.10

0

● 30

.25

.20

.15

●lo

.05

0

I l“’’l ’’{l l’l[’l ’’’’[’’”

r

~

! 1, !l,l, ,,,1, ,,,1, ,,,l!, ,,1,,

o .01 .Oz .03 .04 .05 .06

I/(NO. OF CELLS)

1 l“’’I’’ ’’1’ ’’’1 ’’’’ 1’’’’1” “[”-

1 1, !,, (,,,,1,,,,1,,,,1,,,,1,,1

0 .01 .Oz .03 .04 .05 .06

alLJ

M1A

m\E

u

I l“’’I’’ ’’[’ ’’’1 ’’’’ ;’’’’[” “l”-

5 –

4 —

3 –

z –

1 –

0 –1 !1!11,, , ,1,,,,1,,,,1,,,,1,,,,1,7

0 .01 . Oz .03 .04 .05 .06

l/(NO. OF CELLS)

.5

.4

.3

.Z

.1

0

1 l“’’I’’ ’’1’ ’’’[ ’’’’ 1’ ’’’1’” ‘[’J-

1 1,, , ,1,,,,1,,,,1,,,,1,,,,1,7

0 .01 .Oz .03 .04 .05 .06

l/(NO. OF CELLS)

Fig. 7(cont)

I/(NO. OF CELLS)

92

... PAD3.3F6 - 75JJLlk Rui19 LAC3 ‘SHOCK DECEL. . Q = O, 1.411+.

TFICKETIAN, 6.231 SEC CN R(X4, lE~. 166 SEC CN ~ERUN 4, 1?S CELLS 07/,?2/75

Z. SHOCK DECEL., Q = 0. 1.414 / TIME= 1.4000E+O0

1.6 L. r,, .,. ,., ,r>,,r ,., .,,,,30 t

25 :

20 -

15 -

10 -

5 :

L

1.2 -

.8 :

.4 -

-1

I

4

o~,d,,,,,; d1 1 [ ! t [ 1

0 4 8 12

POS, t,Orl X (mm)

o 4 8 12

POS,t,Orl X (mm)

3.4

3.Z

3.0

Z.8

[ 1 1 1 1 ! 1 1 10 4 8 lZ

Posit,on x (mm)

o 4 8 lZ

F’os,t,on x (-mm)

Fig. 8.Pro blem2. q = O, 1.414, quadratic.

93

PA03.3F6 - 75~19 llN19 LAC3 ●-K DECEL. . Q - 0. l.~l~e ERl14 *, 12E CELLS 07/.??/75.-.

Tilclcriw. 9.359 SEc m RLN, 187. ?9+ SEc CN JOE

2. SHOCK DECEL. , Q = 0. 10414 / PARTICLES

I I 1 1 #

10.6 -

10. s -

10.4 -

10.3 :

10..Z -

10.1 -

10.0 -1 1 I 1 t

.5 1.0 1.5 Z*O Z.5

T,me t (Ps)

-1

.5 1.0 1.5 200 2.5

T,me t (Ps)

30

Zo

10

0

I I I { 1

11 . . .. l. ..l. .i. l....l i.5 1.0 1.s Z.o Z.5

T,me t (Ps)

l-’ 1 1 I 1 -1

FI . . .. I . . . . . . . ..I . . . . I-I.5 1.0 1.5 Z.o 2.5

T,me t (PS)

..*

Fig. 8(cont)

94

. ..

●✎✎

PAD3.3F6 - 75.lA14 KuJ21 LAC3 ●SHOCK DECEL. . O = O, l.q 14*TFICKSTIAU. 6.%1 SEC M i?w,

ERLN 9.?37.513 SEC ~ .MB

12S CELLS 07/22/75

2. SHOCK DECEL.. Q = 0. 1.414 / DIFFERENCE, TRUNCATED TIME / PARTICLES

.006[, .,.,.,.,.,.,.,.,., ., .,.,.,.,,, L, I T 1 1 1 1 [ J

.004

EE .Ooz

:1

x

K o0

.-(/l -. Ooz

:

-.004

-.006

,

;d,.o,,,,,,;1.Z 1.6 Z.o Z.4

1.0

01L

; -1.0IAal

k-Z*O

1.Z 1.6 Z*O Z.4

Time t (tAs) Time t (PS)

t

4

-1

-.04

L .!..! .!!..1,,,4-.08

1 1 1 1 1 1 1I.z 1.6 Z*O

jZ*4 1*Z 1.6 Z*O .2.4

T,me t (PS) Time t (PS.)

Fig. 8 (cent)

95

PA03. S6 - 75.AA19 KIJN?l LAC3 ●SHOCK OECEL. . O = O. 1.41~0 Em 5. 256 CELLs 0712?175

TFICKETIAN. ?l .700 SEC CN MN, 252.652 SEC CN .X9

2. SHOCK DECEL. . Q = Or 1.414 / DIFFERENCE. TRUNCATED TIME / PARTICLES

# 1 1 1 1 1 t

.003 -

.Ooz -

EE

.001 -x

Ko 0 -+J

:-.001

-.002 - >1

-ooo3~1.Z 1.6 Z.o Z.4

T,me t (tIs)

I I ( I I 1 I 1 1

. Oz -

.01 -

0 -

-.01 -

-*OZ -

-.03 I1 # I I 4 ! 1 1

1.Z 1.6 Z.o 2.4

a

IA\.3

1 I I 1 I I I 1 11.0

.5

0

-.5

-1.0

-1.5

t8. t. i. 1. l. 1., .1 .,. l.*. l.t.t.t .11.Z 1.6 Z*O Z.4

Ttme t (Ps)

L’ ”’”’’’’’’’”’’’’’’’”’”’’’’” “J.06 -

.04 -

.Oz -

0 -

-*OZ -

-.04 -

-.06 ,1, 1 I I 1 I I -11.2 1.6 Z.o 2.4

.*6

...

Time t (Ps) T,me t (Ps)

Fig. 8 (cent)c

.7.

96

=..

...

PAD3.F6 - WIq RUU!ITFICUETIAH.

LAc3 ‘SMOCKIXCEL. , Q = O, l. I+Itt*6.76o SEC 04 Ru4, 237.711 SEc CN Jet?

ERU4 9, 1?E CELLS 07/?.?/75

2. SHOCK DECEL. , Q = 0. 1.414 / DIFFERENCE. TRUNCATED TIME / PARTICLES

a.

1.0

0

–1.0

-2.0

_~o .04 .08

Velocity “ (km/s,)

@

I.,

-Z*O v7

Density I? (Mqzm3)

.-WI0

:.->

Time t (tLs)

Fig. 8(cont)

97

pm3.3r6 - 7W. A14 KW19 LAC3 ● SHOCKOECEL. . 0 =TFICKETIAU. 8.651 SEC ~ RUN. IES.786 SEC 04 JIX

Lo’’’”’’’’’ ”’”’ ””6’’’’’’’”’”’’””6 ‘ ‘--l

18

1 /;

,[/t 1 1 1 I I 1 1

.4 .8 1.2 1.6 2.0

lime t (tIs)

l“’’ ”’”’”’”’”’”’’’’’’’”’’’’’”’” ““”l

3.65

t

L

t3*35~

.4 .8 1.2 1.6 .

T,me t (PsI

0, l.ql~” ERUN9, 12S CELLS 07/2.??75

2. SHOCK OECEL., Q ❑ Or 1.414 / LEAD SHOCK

a

40

36

32

Z8

Z4

~.4 .8 I.z . .

Time t (Ps)

~J.4 .8 1.Z 1.6 2.0

-..

.-.

.

Fig. 8(cont)

-..

98

●✎✎ PA03.3F6 - 75J.Alq IWN.2 LAC2 ●7, CCCEL. SHOCKFROMHE, Q - 0.3, O* ERUN 4,

TFIcKETIAU, 8.5.23 SEC CN RUN,126 CELLS 07/22/75

10. ~73 SEC ON JOE

7. DECEL. SHOCK FROM HE. Q = 0.3, 0 / TIME= Z.0000E+OO

[“’’’”’’’’”’’’’’’’’’’’’’’’’’’’r”~[“1’’’ !’’’ [’’’ 1’’” 1’’’’”’’’’’’’’” “’’”4

12 -

8 -

41W0 t -1.0

1, 1 1 ! 1 , 1 1 1 1 1t.1.8.1.8.1. t.!. 8.1.1 .t.8.1. #.t. *.!.4t. i0 4 8 12 16

Pos,tiofl X (mm)

o 4 8 12 16

POSitlOfl X (mm)

1“8TY“’’’’’’’”’’’’’’’’’’’’’’’”

1.6I

1.4

1.2

~l___Jyj,,,,0 1 1 1 ! 1 1 1 1 1 l!i

o 4 8 12 16 0 4

Posit

underdarnped.

8 12 16

on x (.nm)POS,tiOfl X (mm)

Fig. 9.

Problem 7. q = 0.3, 0, linear

99

PA03.3F6 - 75.M14 Iiui? LACZ ●7. OECEL. 2J+WK FRCWHE. O - 0.3. O“ ERLN 4, 128 CELLS 07/22/75

TFICKETIAU, 10.51tl SEC CN RU4. 12.1191 SEC 04 JW7+ DECEL.

1 1 t I I 1 I J

12.0

11.5

1.0

0.s

10. C

2.0

1.8

1.6

1*C

1 1 1 1 I 1 I

1.2 1.4 1.6 1.8 2.0 2.2 2.4

Time t (Ps)

8 i 1 I 1 I I J

\

1 1 1 ! ! ! 13

1.2 1.4 1.6 1.8 2.0 .2.2 Z.4

T,me t (Ps)

SHOCK FROM HE. Q = 0.3, 0 / PARTICLES

30 ( 1 I I I r 1 1

25 -

\Zo -

15 - d

10 -

5 1 I 1 1 1 I I

1.Z 1.4 1.6 1.8 Z.O Z.2 2.4

Time t (Ps)

L“’’’]’’’’’’”’”’”’’’’” ‘“

3.5 -

3.0 -

2.5 -

Z.o -

1.5 -

t.,.,.S..l.*.l.,.#B,.L1.Z 1.4 1.6 1.8 Z.O Z.Z 2.4

Time t iPs)

. .

.-.

.

Fig. 9 (cent)$

●.

-.. PA03.3F6 - 75.UI* RU43 LAC.? ●7. OECEL. SHOCKFROMHE, O = 0.3, 0.

TFIcxETIAu, 5.706 SEC 04 RW, 19.999 SEc CN .xlEERUN4, 1.?8 CELLS 07/?? /75

7. DECEL. SHOCK FROM HE. Q = 0.3, 0 / DIFFERENCE, TRUNCATEO TIME / PARTICLES...

.

EE

x

Ko.-+.-VI

:

I I 1 1 1 1

.014 -

.013 -

.012 -

.011 -

.010 -

tl. l.!. !.l, ,,, ,,, ,., .,~1.4 1.6 1.8 2.0 2.2 z.4

T,me t (Ps)

-.016 -/.E

;z

-.020 -b

$.-IJl

&.oz4 -

-.0281 ‘ “ 1 1 1 1 ! 11.4 1.6 1.8 Z.o 2.2 2.4

a

alL

i1AOJ

k

VI\E

J!

.-

-.05

-.10

-.15

-.20

-.25

-.30

-.35

~. 1.6 1.8 2.0 .2.2 z.4

T,me t (Ps)

1 T 1 1 1 1

.030 )

.020 : J

.010. -

[o

f1 1 1 ! ! ! 1

1.4 1.6 1.8 2.0 Z.Z .7.4

Time t (PS) T,me t (PS)

101

PA03.3F6 - 75.M-1* Ru4.? LACZ ●7. CCCEL. SI+2CKFR~ ~. Q - 0.3. 0“ UWr4 4, l= CELLS 07f22175

TFICKETIAH. 9.W SEc CN RUN. 11.316 SECWX@7. DECEL. SHOCK FROM HE, Q = 0.3. 0 / LEAD SHOCK

18 1 1 1 1 i 1 !

16 :

14 :

12 :

10:1 I 1 1 1 ! 1

1.2 1.4 1.6 1.8 2.0 2.2 204

T,me t (Ps)

.2..20 1 I 1 1 I 1 1

I 1

1.80P, ,.,. 1 1 I 1 1 I

1.2 1.4 1.6 1.8 2.0 2.2 2.4

T,me t (Ps)

r I 1 I I I 1 I

30

.25

~1

\

20

1s}

t

t.li...,,~’~1..2 1.4 1.6 $.8 2.0 2.2 2.4

T,me t (tIs)

.2.6

2.4

2.0

, .3

~es~1.2 1.4 1.6 1.8 2.0 2.2 2.4

T,me t (Ps)

.-.

...

.

r

Fig. 9 (cent)

-.

102

-.. PAD3.3F6 - 75JJL 14 RUN 4 LAC2

lFICXETIAU,●7 GfCEL SHOCKFRC+lHE.

8.634 SEC ON RLN, 29.856 SEC ON JCSa - 0.5, 0. ERw.JI+, 1?8 CELLS 07/.?? /75

FROM HE. Q = 0.5, 0 / TIME= Z.0000E+OO7. OECEL. SHOCK

“..

12 -

8 -

4 -

0 -1 1 1 ! t # 1 1 t I 1

2.

1.

1.

.

/

.

-1.

1’

0 4 8 12 16

POSit,On X (mm)

o 4 8 12 16

POSit,On X (mm)

5 -’I 1 I I 1 I I 1 1

4 -

3 -

z -

1 -

0 f [ ! ! 4 1

/t

1.6;

1.4}

/

IJ—1—L. t.l.t.t.n.l. t.l. ,.l. t.l .,.,.$.,%,, ,., .,. jo 4 8 12 16 0 4 8 12 16

POSit(On X (mm) POSit,Ofl X (mm)

:

.’

Fig. 10.Problem 7. q = 0.5, 0, linear critically damped.

103

PA03.3F6 - 75.Ml~ RuN 4 LACI? ‘7. OECEL. SH3CKFROMHE. OTFICKETIAU. 10.67? SEC ON RU4. 31.8% SEC ON .J2S

7. DECEL.

12*O

11.5

11.0

10.5

10.0

2.0

1.8

1.6

1.4

. 8 1 1 I ( .T-’=---

1 t ! 1 t 1

1.2 1.4 1.6 1.8 2.0 2.2 2.4

T,me t (Fs)

L I 1 I I I 1 I

\

1.2 1.4 1.6 1.8 2.0 2.2 2.4

T,me t (Ps)

= 0.5, 0. EFK!N4. 12E CELLS 07/?2/75

SHOCK FROM HE. Q = 0.5. 0 / PARTICLES

30 I I I 1 1 1 I

25 -

\20 -

15 -

10 -

c 1 1 t 1 I 1 1

J 1.2 1.4 1.6 1.8 2.0 2.2 2.4

3.5

3.0

2.5

2.0

1.5

T,me t (Ps)

1 1 [ 1 1 I r

I I 1 1 [ t 1 I

1.2 1.4 1.6 1.8 2.0 2.2 2.4

Time t (Ps)

. .

.

.“.

f

*.

--.

ERUN4, 128 CELLS 07/2? /75

-..

PA03.3F6 - 75JJL14 RUN 5 LAC2 .7. OEcEL. StiOCKFROMw. o - 0.5, 0.TFICKETIAN. 5.873 SEC W GUN, 39.571 SEC CN WE

7. DECEL. SHOCK FROM HE, Q = 0.5. 0 / DIFFERENCE, TRUNCATED TIME / PARTICLES

[130 :

1.20 !

110: -

loo~

140

1.4 1.6 1.8 Z.O Z..Z 2.4

T,me t (tIs)

1 I { 1 1 [

-.012 :

r“E\

:–.014 :

Q

A+$--.016: -q

?

2

-.018: -

1t, 1 I 1 1 ! 11.4 1.6 1.8 Z.o z.z z.4

T,me t (Ps)

1A\E

x

J

“01>

-.05, 1 1 1 1 1 1 I

-.10 -

-.15 -

h-.20 -

-.z5-

d1 1 1 1 1 I

1.4 1.6 1.8 2.0 z.z 2.4

T,me t (tIs)

.016

.012

.008

.004

1.4 1.6 1.8 2.0 .2.2 2.4

Time t (FS)

105

PA03. ZF6 - 75JJ-14 KwJk LAC2 ●7. CCCEL. SHOCKFl?@l FE, O = 0.5. 0.9.W5 SEc CN Rw, 30.707 SEc w .JcnTFICKETIAU.

18

16

EE

lC

.Z.1O

2.05

2.00

1.95

1.90

1.85

Em ~, 12E CELLS 07/?2/75

7. DECEL. SHOCK FROM HE. Q = 0.5, 0 / LEAD SHOCK

r. 0 I 1 I ! 1 111.2 1.4 1.6 1.8 2.0 2.2 2.4

T,me t (Ps)

1“’’’’”’’’’’’’’’”’’’” ““J

I. 1. !. l. !. !. t. l.!. l.!. l.!., i1.2 1.4 1.6 1.8 2.0 2.2 2.4

.28 -I 1 1 { I I I

24 -

\

20 -

16 -

I 1 I I [ I I

2.6

2.4

Z.Z

Z.o

1.Z 1.4 1.6 1.8 2.0 Z.Z Z.4

Time t (Ps)

I t 1 1 1 1 I

\L

!. !. !. l. !l!. l.l. (l!.!,! .,11.2 1.4 1.6 1.8 2.0 Z.Z Z.4

-..

...

Tjme t (tIs)

Fig. 10(cont)

T,me t (PS)

r

*.

106

. ..

PA03. ZF6 - 75JW Ih RUN6 LACI? ●7. CECEL. SJ+CCKFFWI Iif,TFICKETIAW. 8.697 SEC CN RuN, ~9.411 SEC CN JOB

7. DECEL. SHOCK

o = 0.7. O* ERUNI+, I?B CELLS 071?? /75

FROM HE, Q D 0.7, 0 / TIME. Z.0000E+OO

Lo”’’’’’’’ ”I” I’’’ ”’’’ I’’’ I’’’ I’” I’r’l”l

-..l“’”’’’’’’’’’’’’’’’”’’’’’’’’’’[’’’’’’’’”-

2.0 -

1.5 -

1*O :

.5 :

0 -

-.5 -

/’//’4 -

0 -1 I 1 1 I 1 1 ! 1 1

0 4 8 12 16

POS,t,Ofl X (mm)

o 4 8 12 16

Position x (mm)

5 -’1 T 1 1 1 1 1 T 1

4 f-

>

-1,,, 1 1 1 I I 1 I J 1

1.8

1.6

1.4

L

3 -

2 -

1 -

-1

1.2

0 I I 1 1 I I 1 1 !H-J J. I. I.ll,l. I, I,!. ,,!, [. I ., .1,,,1,,,1,,

0 4 8 12 16

Pos, t,on x (mm)

o 4 8 12 16

Pos, t,on x (mm)

Fig. 11.

Problem 7. q = 0.7, 0, linear overdamped,.

PA03.3F6 - 75.LIL14 RW6 LACI? ●7. CCCEL. WCC)( FROMHE, OTFICKETIAW. 10.7*6 sEC CN Rw.J, 51. ~60 SEC CN WE

7. DECEL.

I I I 1 [ 1 8

12.0 -

11.5 -

11.0 -

10.5 -

1o.o- 1 1 [ t t t I

1.2 1.4 1.6 1.8 2.0 2.2 2.4

T,me t (PS)

1’ 1 1 [ T 1 1 I

2.0 i

\

1.8 :

1.6 :

1.4 :

1 1 1 I ! ! 1 11.2 1.4 1.6 1.8 2.0 2.2 2.4

Time t (IJs)

= 0.7, 0. Eiiui9. 12S CELLS 0712.2/75

SHOCK FROM HE. Q ❑ 0.7. 0 / PARTICLES

30 I 1 1 1 I , I 1

25 -

\20 -

15 -

10 -

5 t I I 1 t I I1.2 1.4 1.6 1.8 2.0 2.2 2.4

T,me t (tIs)

1 [ 1 1

)I I I

3.0 -

\

2.5 -

2.0 -

1.5-

t I I 1 1 t I1.2 1.4 1.6 1.8 2.0 2.2 Z.4

T,me t (Ps)

.-

.

,..

Fig. 11 (cent)

108

r

●.

PAD3.3F6 - 75JUL14 RUN7 LAC? ●7. DECEL. SI+JCKFROMHE, 0 = 0.7, 0“ ERUN~,TFICKETIAU,

1.?8 CELLS 071.??/755.931 SEC W Rw, 59.195 SEc m WE

7. DECEL. SHOCK FROM HE, Q = 0.7, 0 / DIFFERENCE. TRUNCATEO TIME / PARTICLES

-..

*o

x

EE

x

co

+

IA

:

1 T 1 1 1 I120 :

110 ;

[100 ;

90 :

80-’ ! 1 1 1 1

1.4 106 1.8 2.0 2.2 2.4

T,me t (Ps)

1.4 1.6 1.8 2.0 2.2 Z*4

T,me t (PS)

-.008 -

; -.009 -E

;E--- 010 -

G

!-.011 -mE

~-*OIZ -

L“”’

-.dL-————1.4 1.6 1.8 Z.O Z.2 Z.4

T,me t (Fs)

in\

:.Olz -

J

A?.-u

.008 -:-

>

.004 -

tl!!!!l l.l. l.!! !!lj1.4 1.6 1.8 Z.O Z.Z Z.4

T,me t (PSI

?

.a

Fig. 11 (cent)

109

PM3.3F6 - 75.AJLI% W6 LAC? 07. CXXEL. SlfCXKFROMHE, O = 0.7. 0° Em *,TFICKETIAU. 9.561 SEC ON Rw,

I.?E CELLS 07/22175SO.275 SEC ON m

7. DECEL. SHOCK FROM HE, O = 0.7, 0 / LEAD SHOCK

EE

x

18

16

14

12

10

1 1 t 8 1 1 1\

t

L ! I ( I 1 ! I1.2 1.4 1.6 1.8 2.0 2.2 2.4

T,me t (PS)

L’ T I I I ! # 112.10 -

2.05 -

2.00 -

1.9s -

1.90 -

1.85 -

\ ,\

\\

t. 1. !. l. !. l!. l.!.{!.!.! .111.2 1.4 1.6 1.8 2.0 2.2 2.4

.28 -1 i i I 1

\

1 I

.24 -

\

20 - \

16 -

I t 1 ! ! 1 I

2.6

2.4

2’.2

.2.0

1.2 1.4 1.6 1.8 2.0 2.2 2.4

Time t (Ps)

# , I 1 [ I I 1

\

\

r

I I I I ! I I1.2 1.4 1.6 1.8 2.0 2.Z 2.4

T,me t (Ps)

-..

.“e

T,me t (IIs)

Fig. 11 (cent)

110

Y

PA03.3F6 - 75J&I% RIN8 LAC2TFICKETIAIJ.

●7. OECEL. SJ+XK FROMHE, O = 0, I. 1+11+.8.563 SIC CN RUN, 69.104 SEc m .J3a

ERw 4. !?8 CELLS 07/22/75

7. DECEL. SHOCK FROM HE. Q = 0. 1.414 / TIME= Z.0000E+OO

a

v

1 ( 1 1 1 1 ( 1 1 116 -

12 -

/ ~

8 -

4

i‘< -

o~l qf ! 1 I 1 ! Io 4 8 12 16

Pos, t(orl x (mm)

F“’”’’’’’’’’’’’’’’’’’’’’’’r’’’r’’’’”J

0;/ I 1 1 ! 1 I ! ! 1 io 4 8 12 16

POS, tiOll X (mm)

3

al>

.

1“’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’’”1

2.0 -

1.5 -

1.0 -

.5 -

0. -

-.5-

l.Ort.[. t!n?n[. #,, ,t, #,,,,,,,,.,,,,,,,,, ,,. {

o 4 8 12 16

Pos it Ion x (mm)

L’ I 1 1 1 1 1 1 1 1j

1.8 :

1.6. -

1.4: -

j

1.2:

1 1 1 1 f 1 1 1 1 10 4 8 12 16

Posit,cxl x (mm)

Fig. 12.Problem 7. q = O, 1.414, quadratic.

111

PA03.3F6 - 75.,W14 RiNe LAC2 ●7. OECEL. SHOCKFROMHE, O = O, 1.41+$TFICKETIW. 10.582 SEC CN RU+4. 71.123 SEC CN WE

ERw 9, 12E CELLS 07/2.? /79

EE

x

12.0

11.5

11.0

10.5

10.0

2.0

1.8

1.6

1.4

— —7. DECEL. SHOCK FROM HE, Q = 0, 1.414 / PARTICLES

f i I 1 1 [ r

I ! # 1 1 1 11.2 1.4 1.6 1.8 2.0 2.2 2.4

T,me t (PS)

I 1 1 1 I 1 I

\

1.2 1.4 1.6 1.8 2.0 2.2 2.4

T,me t (PS)

25 -

20 -

15 -

10 -

I 1 ! 1 1 15 L

1.2 1.4 1.6 1.8 2.0 2.2 2.4

2.8

.2.4

2.0

1.6

1.2

Time t (Ps)

I 1 1 i

i

T 1 1

\

j

1 1 I 9 I 1 t1.2 1.4 1.6 1.8 2.0 2.2 Z.4

Time t (Ps)

.*.

.-.

r

112

. ..

.*.

PA03. F6 - 75WL14 KUN 9 LAC2 ●7. DECEL. SHOCKFROMHE, Cl = O, 1.~14* ERUN4,TFICKETIAM, 5.722 SEC W RUJ,

1?S CELLS 07/2? (7578.655 SEC C+i JC12

=ro

x

EE

x

Eo

*,-ul

2

7. DECEL. SHOCK FROM HE, Q z 0. 1.414 / DIFFERENCE, TRUNCATED TIME / PARTICLES

1.4 1.6 1.8 2.0 2.2 .2.4

Time t (Ps)

0

-.10

-.ZO

-.30

11 ! 1 1 1 I 1

II ! ! ! ! ! ~1.4 1.6 1.8 Z.O 2.z z.4

Time t (PS)

I 1 1 1 1 1 i 1 L f 1 [ 1 1 1 J

1.4 1.6 1.8 Z.O Z.Z Z.4

T,me t (tIs)

--*OOZ -

-.004 -.

~-.006 -z

w -.008 -

$.- -.010 -m

:0–.012 -

-.0141

m\

:

.Ozo

.010

//

0

-“010~1.4 1.6 1.8 2.0 .2.2 2’.4

T,me t (tIs)

..

113

PA03.3F6 - 75JJ-14 RUN 8 LAC2 07. OECEL. SHOCKFROMFE. O = O. 1.414” ERIN~. 129 CELLS 07/22175TFICICETIAU, 9.393 SEC ~ RUN. 69.935 SEC W .KIE

7. DECEL. SHOCK FROM HE, Q ❑ 0. 1.414 / LEAD SHoCK

16 :

14 :

12 :

10 :1 1 ! I I 4 I

1.2 1.4 1.6 1.8 2.0 2.2 2.4

T,me t (Ps.)

1“ [ 1 8 T 1 1 r

2.10

.2.05

Z*OO

1.95

1.90

1.85

I 1 1 t I { I 1

1.2 1.4 1.6 1.8 .2.0 2.2 .2.4

Time t (Ps)

28

Z4

Zo

16

Lr [’’’’’’’’”’’’”’””” ‘“’”-l

~Lil1.Z 1.4 1.6 1.8 Z.O 2.Z Z.4

T,me t (Ps)

1 i I 1 1 I 1\

\

t. [ 1 1 1 1 1 11

1.Z 1.4 1.6 1.8 Z.O Z.Z Z.4

Time t (PS)

-..

...

.

9

4

.

“..

PA03.3F6 - 75JW14 FRmlo LAC2 .7. CECEL. SHOCKFRC+lHE, Q = O, 0. .?* ERUN 4,TFICKETIAW, 8.515 SCC ON RUN, BB.3?6 SEC ON .J3B

1.?9 CELLS 07/.22175

7. DECEL. SHOCK FROM HE, Q : 0, 0. 2 / TIME= 2.0000E+OO

“*●

L“’r ’” ’’’’ r” ’’’’’’’’’r’””” “’’’r”

12 -

8 -

4 -

0 -1 1 t I I 1 1 ! 1 i

[“’1 ’1 ’’”’’’’’’’’’’’’’’’’’’’’ 1”’”” ““1

/

-’eo~0 4 8 12 160 4 8 12 16

POS,t,Orl X (mm) POSlt,On X (mm)

1 1 1 1 I 1 1 ml 15

4 -

3 -

z -

1 -

0 -

1.4

I

1.2

,&,,,,,,jt.l.lll!l. jil. L.l. !.l. !.!.l. t.l.ll.l. i

o 4 8 12 16

POSit,On X (mm)

o 4 8 12 16

Pos, t,on x (mm)

Problem 7. q = O, 0, 2, PIC...

115

PA03.3F6 - 75.,kLlh lwNlo !-AC.? ●7. OECIL. SHOCKFROJlHE,TFICKETIAW. 10.517 SEC CN RUi. 90.3?8 SEC ~ JSS

7. DECEL.

[“”’’’’’’’’’’’’’’’’’’’” “1

0=0. 0,2S ERw 4, 12S CELLS 07/22175

SHOCK FROM HE. Q = 0. 0, Z / PARTICLES

30 ~

. ..

.2.0

1.8

1.6

1.4

1*Z 1.4 1.6 1.8 Z.O Z.Z Z.4

T,me t (Ps)

I 1 1 I 1 1 14

I 1 1 I 1 1 I il.Z 1.4 1.6 1.8 Z.O Z.Z Z.4

T,me t (PS)

I.z 1.4 1.6 1.8 Z.O Z.Z Z*4

Time t (Psi

.

~~1.Z 1.4 1.6 1.8 Z.O Z.Z Z.4

Time t (Ps)

[ 1 1 1 1 i I r

3.0

Z.5

Z.o

1.5

1 .1.0.1.,.,.1.,.,.1., .!.,.!

Fig. 13(cont)

116

a

..

“-G

“..

1

PA03.3F6 - 75.AJL14 RUNI 1 LAC2 ●7.TFICKETIAU.

CCCEL. SHOCKFROFlHE,5.747 sEC @l RUN. 97.882 SEC ON J~

7. DECEL. SHOCK FROM HE, Q = 0, 0, .2 /

1

130

120 :

110 :

100 ?

90 :8

1.4 1.6 1.8 2.0 2.2 2.4

a

0=0,0,2* ERLN 9, Ii% CELLS 07f2~75

DIFFERENCE, TRUNCATED TIME / PARTICLES

.

.

.

●

✎

L 1 1 1 1 1 1

08 :

12 :

16 :

20 :

24 ;

t I I 1 1 1

104 1.6 1+8 Z*O Z.z Z.4

T,me t (Ps) Time t (Ps)

I I { I { i 1 I I 1 1 1 1 t 1 1L

-.006 -

E\ -.007 -

:

w –.oo8 -

$.

; -.009 -

~

-.010~J

t,.,.l.,1,.j.,.f..fl1 .005

I11 .11 .l!l .111.111 1v

1.4 1.6 1.8 2.0 Z*Z Z.4 1.4 1.6 1.8 Z.O Z*Z Z.4

Time t (tIs) T,me t (Fs)

Fig. 13(cont)

..

117

PA03.3F6 - 75uIq KvNIo LAC2 ‘7. CECEL. SHGtKFRCfl WZ, O = O, 0, 2*TFICKETIAU.

Em 4.9.32S SEC w W, 89.139 SEC CN J~

[28 CELLS 0712.?175

7. DECEL. SHOCK FROM HE, Q = O, 0, 2 / LEAD SHOCK

EE

x

18. . T 1 1 I I 1 1

16 :

14 :

12 ;

101 1 [ I 1 ! I

1.2 1.4 1.6 1.8 2.0 .2.2 2’.4

T,me t (Ps)

_ Z.05 -

E

$ Z*OO -

v 1.95 -:

: 1.90 -

~

1.85 -

,.80~1.Z 1.4 1.6 1.8 Z.O Z.Z Z.4

Z8 -1 1 1 1 [ 1 1

Z4 -

\

Zo -

16 -

1 I I 1 8 , t 11.2 1.4 1.6 1.8 Z.O 2.Z Z.4

Time t (tIs)

Z.4

Z*Z

I

‘-oL1.Z 1.4 1.6 1.8 Z.O Z.Z Z.4

.“d

.‘.

.

T,me t (PS) T,me t (PS)

“-. PAD3.3F6 - 75Jwlq RuNl? LAC2 ●7. DECEL. SHCCKFROMHE, HALF DEL T*

TFICXETIAU, 13.590 SEC C(4 RUN.ERUN4.

11?.633 SEC ON JW128 CELLS 07/2.? 175

7. DECEL. SHOCK FROM HE, HALF DEL T / TIME= 2.0000E+OO

“*.

.

0.

o-

>

16 1 1 , 1 1 1 I 1 I

12 -

8 -

4 -

0 -1 I 1 1 1 t 1 1 10 4 8 12 16

Pos, tiorl x (mm)

1 1 1 i I 1 I ( 1

4.0

3.0 :

EZ.o ;

1.0 )

o : 1 1 1, 1 1 1 d—1—l—Lo 4 8 12 16

M\

.$J

/

1 1 1 1 r I I 1 1 1

.2.0 -

1.5 - >

1.0 -

.5. -

0 -

-.5-

-1.o-1 [ 1 1 1 1 1 1 ! I i

0 4 8 12 16

POS,tlOrl X (mm)

1 1 I 1 r I 1 1 I

1.8 :

1.6 :

1.4. -

1.2:

1 I 1 1 I 1 1 t 10 4 8 12 16

POSitiOn x (mm) POSitiOn x (mm)

Fig. 14.

Problem 7. At halved.

119

PA03. 3F6 - 75.M. 14 ml? LAC.? .7. ~CEL . SHUCKFRCtl HE. HALF OEL .T-TFICKETIAU. 17.8?4 SEc m m, 116.868 SEC ON m

ERtJN*, 1.?6 CELLS 07/.?.?175

7. DECEL. SHOCK FROM HE, HALF DEL T / PARTICLES

12.0

11.5

11.0

10.5

10.0

I 1 I I I 1

1 t 1 I 1 1 1

I.Z 1.4 1.6 1.8 2.0 2.2 .2.4

Time t (vs)

l“’’’”’’’’’’’’’”’”’’”2.0

}

1.2 1.4 1.6 1.8 2.0 2.2 2.4

T,me t (Fs)

30

Z5

Zo

15

10

5

3.C

Ze$

Z.c

1.5

1 1 t I 1 1 111.Z 1.4 1.6 1.8 Z.O Z.Z Z.4

T,me t (Ps)

1 0 1 1 1 1

1. !. 0., .8 .1.8 . [.,.1.,.! . . . .

I.Z 1.4 1.6 1.8 Z.O Z.Z Z.4

Time t (Ps)

--.

..*

.

Fig. 14(cont)r

;’

120

“.. HALF OEL 1- ERuN 4. 1?S CELLS 07/.?2/75

DIFFERENCE. TRUNCATED TIME / PARTICLES

PAD3.3F6 - 75wL14 lwN13 LAC2 ‘J. CECEL. SHOCKFRC+lHE.TFICKETIA!4. 17.684 SEC ON RW. 136.946 SEC ~ J~

7. DECEL. SHOCK FROM HE. HALF DEL T /

.*. [ 1 1 1 , 1 ,,,, ,.,, ,,,,

M@’30

Zc

Lla

-.15

-*ZO

-.25

L

~11 f ! 1 1- ll. t. lj!, .ill!. !.l

1.4 1.6 1.8 2.0 Z*Z 2.41.4 1.6 1.8 2.0 Z.z 2.4

T,me t (PS) T,me t [Ps)

.-

.006

.008

0

z

.Olz

.008

.004

0-.0

1.4 1.6 108 .?.0 Z.Z 2.4

T,me t (Ps)

1.4 1.6 1.8 Z.” Z.Z Z.4

T,me t (PS)

.s

121

PA03.3F6 - 75.Ll-lq Ruil? LAC2 .7. OECEL. SHOCKFR~ HE, HALF DEL T- GwN q,TFICKETIAU. 16.147 *C CN W,

12E CELLS 07/2? /75115.190 SEC ~ .)CM

7. DECEL. SHOCK FROM HE; HALF DEL T / LEAD SHOCK

EE

x

181. t 1 r i 1 [ 1

16 :

14 :

12 ;

10 :1 1 1 1 I 1 !

1.2 1.4 1.6 1.8 2.0 2.2 2.4

T,me t (Ps)

l“’’’’’ ’’’’ ’” ’’’’l”’” ““4

2.10 :

.2.05 -

2.00 -

1.95 -

1.90 -

1.85 -

,osc)~1.2 1.4 1.6 1.8 2.0 2.2 2.4

Time t (Ps)

28

24

20

16

t. i. !. l. l. l. l. l.l. l.l. lit.ll1.2 1.4 1.6 1.8 2.0 2.2 2.4

Time t (Ps)

l\ I 1 1 1 I I 1

2.6

2.4

2.2

2.0

I I 1 I 1 I 8

\

1

I. 1. c. t., .l. n.t. t.l.a.t .I.li1.2 1.4 1.6 108 2.0 2.2 2.4

T,me t (Ps)

Fig. 14(cont)

122

r

;

“..

“..

PA03.3F6 - 75.NL14 RIJN14 LAC2” ‘7. OECEL. SHOCKrROM K. JUICE OEL 1“TFICKETIAU. 8.625 SEC CN RUN. 1~6.941 SEC CN JOB

ERUN 4. 1.?8 CELLS 07/.?? /75

7. DECEL. SHOCK FROM HE, TWICE DEL T / TIME= .Z.0000E+OO

1.2 -

8 -

r

/’ /

t1.0 I

/.5

0

-.5

I

4

0 ‘p,,,,,,,,, -1.0I t ! 1 1 1 1 1 1 1 1

0 4 8 12 16

POSltlOn X (mm)

o 4 8 12 16

POSit,Ofl X (mm)

L.! I I ! 1 1. 1 I 1 I J Ei,,,,,,,$.!,l,,.1,1.1,,,, ,ISI,I,I-

4.0

3.0

1

1.8

1.6

1.4

1.2

Z.o

1.0

1o~0 4 8 lZ 16

t 1, 1 1 I 1 1 1 1 1 10 4

Pos 1t

8 12 16

Otl X (mm)POS,t,Orl X (mm)

? Fig.

Problem 7. At doubled.

;

123

PAD3. 3F6 - 75J.J- 14 RUN14 LAC2 .7. ECEL. SHOCKFRC+lWE,TFICKETIAU. 10.672 SEC CN RUN, IW.9EE SEC ON JC@

7. DECEL.

2.0

1.s

11.0

10.5

10.0

i I 1 I [ I

1 t 1 1 1 1 11.2 1.4 1.6 1.8 2.0 2.2 2.4

T,me t (PS)

\

rWICE DEL 1. ERUN 4. 128 CELLS 07122f75

SHOCK FF!OM HE, TWICE DEL T / PARTICLES

3C

25

Zo

15

10

5

3.0

.2.5

Z.o

1.5

1 1 1 1 1 I r

! I I ! ! ! 1

1.2 1.4 1.6 1.8 Z.O Z.Z 2.4

Time t (P.s.)

1 1 I 1 I 1 r

t 1 1 11.Z 1.4

1 1 11.6 1.8 Z.O .Z.Z Z.4 1.2 1.4 1.6 1.8 Z.O Z.2 2.4

T,tie t (Ps) Time t (Ps)

.“.

.

Fig. 15 (cent)

124

r

PA03.3F6 - 75JJL14 R(R4I5 LAC2 .7. OECEL. SHOCKFROfl HE TNICE DEL T*TFICKETIAW. 5.657 SEC CN R~, 156.640 SEC CN JOE

ERUN %. 12S CELLS 07/2.2/75

*ox

EE

x

Ko.-+.-

7. DECEL. SHOCK FROM HE. TWICE DEL

140. , 1 I 1 1 1

130 :

120: -

110: -

100:

L~

. 2.0 2.2 2.4

T,me t (VS)

-.005

-.006

.E -.007;z

@/-.008

2

“$ -.009

~-.010

-.011

1.4 1.6 1.8 .2.0 z.z Z*4

Time t (P S.)

T/

J

DIFFERENCE. TRUNCATED TIME / PARTICLES

1 1 1 1 1 1

-.08 :

-.lZ. -

P A

r-.16T

v

-.ZO:

-.Z4 - ,/

1 t 1 1 11.4 1.6 1.8 z.o z.z z.4

Time t (IJS)

.Olz

.010

.008

.006

.004

1.4 1.6 1.8 Z.o Z.z 2.4

T,me t (1-LS)

Fig. 15(cont)

125

PA03. ~6 - 75WII! KW14TFICKETIAU.

LAC? ●7. CECEL. SWCK FRW HE, TuIcE CEL T*9.477 SEc CN Rw. 147.7% *C m Jm

ERU’44, 128 CELLS 071.?2175

7. DECEL. SHDCK FROM HE, TWICE DEL T / LEAD SHOCK

181 .,.,.,.,.,.,.,.,., .,. 1 1J

16 :

14 :

~

12 :

10. -1 1 I I i ! I

Z*IO

2.05

.?.00

1.95

1.90

1.85

1.2 1.4 1.6 1.8 2.0 2.2 2.4

Time t (tIs)

1 1 I 1 1 I 11

t I I 1 1 I I i1.2 1.4 1.6 1.8 2.0 2.2 2.4

T,me t (Ps)

c?c)

0.

alL

IiIAal

k

28 ~1 I 1 1

\

1 I i

.24 -

\

20 -

16-

1 t 1 I 1 1 t1.2 1.4 1.6 1.8 2.0 2.2 2.4

Time t (Ps)

f 1 1 I # I I\L

2.6 i

2.4 :

Z.z F

Z*O. -

\

J 1 1 , I ! i t i1.Z 1.4 106 1.8 Z.O Z.Z z.4

Time t (tIs)

.“. I

.-.

Fig. 15(cont)

126

r

i

“..

“*.

.

PA03. 3F6 - 75JLl 19 RUN16 LAC.? ●7. OECEL. SHOCKFROM!-E, 6I1O CELLS,TFICKETIAL.A, 8.595 SEC CN RLN. 166.395 SEC ON JOB

ERUN 4, 128 CELLS 07/22/75

7. DECEL. SHGCK FROM HE, 6/10 CELLS / TIME= Z.0000E+OO

12 -

8 -

o 4 8 12 16

POSit,Orl x (mm)

4 -

3 -

z -

1 -

0 I I 1 ! 1 1 1 1 1 *o 4 8 12 16

w

.-wl

:0

/

[ 1 1 1 1 1 1 r 8

.2.0 -

1.5 -

1.0 -

.5 -

0-

-*5-

-1.01 ! ! ! t 1 1 1 10 4 8 12 16

Position x (mm)

I 1 1 1 1 1 1 1 1 I

1.8 :

1.6 :

1.4: -

1.2:

1 1 t 1 t ! ! ! I t ! 10 4 8 12 16

Position x (mm)Pos,t,on x (mm)

Fig. 16.Problem 7. 6/10 cells in HE/inert.

127

PA03.3F6 - 75.Mlq Ku.116 LAC2TFICKETIAU,

07. i)ECEL. SHOCKFRCfl I-E,10.63~ SEC w FUN, 166.43% SEC Ch’ JW

7. DECEL

1

E

:1

x

1Ao

al

1

T,me t (Ps)

VIc

:

2.0

1.8

1.6

1.4

1 1 1 I 1 1 1

]

\

I t 1 1 I I 11.2 1.4 1.6 1.8 2.0 2.2 2.4

6/10 CELLS* Em %. 1.28 CELLS 07/.??2175

. SHOCK FROM HE. 6/10 CELLS / PARTICLES

a

T,me t (Ps)

IA

3

I [ 1 I t 1 II

2.8 -

\

2.4 :

2.0 :

1.6. -

1.2 1.4 1.6 1.8 2.0 2.2 2.4

T,me t (Fs) T,me t (Ps)

.-.

Fig. 16(cont)

128

f

i

-..

.

.“

.

PAD3.3F6 - 75.lJLlh liw17 LAC2 ‘7. DECEL. SHOCKFROMHE. 6)10 CELLS-TFICKETIAU,

ERLJN+.5.817 SEC CN RUN.

128 CELLS 07/?2/75176.092 SEC ON WB

7. DECEL. SHOCK FROM HE, 6/10 CELLS / DIFFERENCE, TRUNCATED TIME / PARTICLES

105

100

95

90

85

80

1.4 1.6 1.8 2.0 2.2 2.4

T,me t (PS)

-.006

I I 1 1 1 I 41.4 1.6 1.8 2.0 2.2 2.4

Time t (Ps)

L?c1

0.

r 1 r 1 , 1 1

-*lo A d

-.15 F

.020

m\ .015

ii

J

k .010+.-

:+

; .005

[ I ! 1 1 1 I 11.4 1.6 1.8 2.0 2.2 2.4

T,me t (WS)

1 1 1 1 I I 1

OL’ 1 1 ! 1 ! I1.4 1.6 1.8 2.0 2.2 z.4

T,me t (lJs)

*

i

Fig. 16(cont)

129

PA03.3F6 - 75JU14 IWN16 LAC2 ●7. OECEL. sWCK FROMHE,TFICKETIAU. 9.438 SEC CN R~. 167.239 SEC CN JG9

7. DECEL.

EE

x

Ko

16 :

14 :

12 :

10 :1 I 1 1 I I I

1.2 1.4 1.6 1.8 2.0 2.2 2.4

T,me t (Fs)

1 1 1 1 1 ! I

2.10 -

.2.05 -

E\

g .2 .00 -\

& \1.95 -

>

IA; 1.90 -n

1.85 -

! I I 4 f 1 !

1.2 1.4 1.6 1.8 2.0 2.2 2.4

Time t (Ps)

I5/10 CELLS* ERW 4, 128 CELLS 07J?I?175

SHOCK FROM HE. 6/10 CELLS / LEAD SHOCK

1 I I.28 -

I 1 I 1

Z4 -

\

Zo -

16 -

Z.6

2.4

Z*Z

Z*O

t.1.11.11..l.l,l”lI.Z 1.4 1.6 1.8 Z.O 2.2 2.4

T,me t (Ps)

r 1 I 1 1 I I

\

\

I 1 1 I I ._LA—Ll—l

1.Z 1.4 1.6 1.8 2.0 Z.Z Z.4

T{me t (Ps)

.:

.“.

.

-.

“..

.

PA03.3F6 - 75&l_14 w? LAC? .S. F$WEREfRATICN, O - 0. I .414*TFICXETllW. 10.016 SEC CN Iiw, 11.6&w sEc m&n

ERu4 4, 12E CELLS 07/i3/75

8. REVERBERATION, Q = 0, 1.414 / TIME= 8.5000E+O0

30

25

20 IL

1s -

10 :

5 -

0 -I 1 1 1 9 1 t i

10 20 30 40 50

Pos it iOn x (mm)

L

T 1 1 1 1 1 1 1 1.12

.10

.08

.06

.04

.02

0 ~L .-#/L++_4dJ]10 20 30 40 50

Position x (mm)

4.5

4.0

3.5

3.0

.2.5

1 1 1 1 1 I 1 1

Z*O I 1 1 1 I 1 1 I10 Zo 30 40 50

POSiti Ofl X (mm)

LL“”’’’’’”’11,’rI,..rI .

2.8 -

Z*4 -

Z.o -

1.6-

1 1 1 1 1 1 1 110 Zo 30 40 50

POSition x (mm)

Fig. 17.Problem 8. q =0, 1.414, quadratic.

131

PK)3.3S6 - 75.M-14 KU42 LAC2 ●8. F$XWERATICN, O - 0. l.%1~”TFICXETIU(.

ERW II, 12E CELLS 0712%17513.514 SEC m m. 15.121 SEC W .X9

8. REVERBERATION. Q = O. 1.414 / PARTICLES

EE

x

Ko.-W.-vl

2

00P.

N

.E

>x

1 I 1 T 1 I T 165

60

55

5C

45

t.i.. .l . . ..l . . ..l . . ..l . . ..l . . ..l. l6 7 8 9 10 11 12

Time t (Ps)

L“l ’” ”” 1’’” ’l ’”””l”” 1“’’1’””1

t————

6 7 8 9 10 11 12

Time t (PS)

r. r 1 1 1 1 1 1 1

.4 -

.Z -

0 -

-.2 -

4

I II-.4 -

1 1 1 1 I ! I

6 7 8 9 10 11 12

Time t (Ps)

5.5

5.0

4.5

4.0

3.5

I 1 1 1 1 I 1

I 1 I I 1 1 16 7 8 9 10 11 12

Time t (tIs)

.

Fig. 17(cont)

..-

-..

PAD3.X6 - 75.AL14 liui3 LAG! “8. REVEK=RATI~, Q - 0. 1.%14*TFICKETIW, 8.753 SEc 04 RN.

mm *,5s.8s0 SEc CN .X9

1.?ECELLS 07/,?l+z75

8. REVERBERATION, Q = O, 1.414 / DIFFERENCE. TRUNCATEO TIME / PARTICLES

330 1 r 1 1 t T .20

*o 325 -x

.1020

: 3Z0 -b

x al oK L

o.. 315 - ;* vl.- alM k -.102

310 -

[ 1 1 1 1 18.4 8.6 8.8 9.0 9.2 9.4

-.208.4 8.6 8.8 9.0 9*Z 9.4

Time t (tIs) Time t (PS)

I-’’’”’’’’’”” , 1 1

8.4 8.6 8*8 9.0 9.Z 9.4

.010

0

.010

. Ozo

~8.6 8*8 9.0 9..2 9.4

*

1!

Time t (PS)

Fig. 17 (cent)

Time t (tIs)

133

PM3.3F6 - 75.AL19 FitN3 LAC2 ●S. REvEREcRATIcN, 0 - 0, 1.414.TFICKETIU(. 30.709 SSc CN RtN,

ERm 5,n.835 sc m x43

.?s6 CELLs 07/.2%/75

8. REVERBERATION. Q = 0. 1.414 / DIFFERENCE, TRUNCATED TIME / PARTICLES

uox

EE

x

Ko

1 1 1 1 1 1

128 :

124 :

120 :

116 -

112 ;1 t 1 1 1

8.4 8.6 8.8 9.0 9.2 9.4

.022 :

.020 :

.018 -

&

Time t (tIs)

1 # 1 1 I 4

J

.016~, :, J I 1 1 t -18.4 8.6 8.8 9*O 9.Z 9.4

Time t (PS)

I 1 I 1 1 1 I 1.10

.05

0

.05~,

.10

I t 1 1 1 I I 18.4 8.6 8.8 9.0 9.2 9*4

Time t (Ps)

“010~

.005

0I.005

.010

/

i

[ 1.”!.!.!.,.,.,.,,, 1 18.4 8.6 8.8 9.0 9.Z 9.4

Time t (IAS)

Fig. 17(cont)

134

. .. PAD3.~6 - ~lv MN. ? LAG? ●8. REVZRKRATIU4, Q - 0, 1.%14*TFICIGITII.J(. lW.005 SEc CN RU4. 15.613 SEC CN JIX

ERU4 %, 12E CELLS 07/~/75

“..

.

I I I i

.6 -I

.4 -

.2 -

0 -

-.2 -

-.4 -1 1 1 1 1

8. REVERBERATION. Q = 0, 1.414 / PARTICLES

a

k

3.5 4.0 4.5 5.0 5.5

Velocity u (km/s)

eox

v

.>

.4 -

.2 -

0 -

-.2 -

-.4 “1 I t t 1 [ 1

.055 .060.065.070.075.080 .085

Density ‘? (Mqzm3 )-2.700

40

30

20

10

0

6 7 8 9 10 11 12

Time t (Ps)

Fig. 17 (cent)

135

PAD3.3F6 - _EW1-19 FW3 LA(2 ●8. iWEFSZiUTIC?+, Cl = O, 1.414.TFIC%ETIW, 31.15esEcm Rw,

m 5,78.2s5 Sfc CN Jc9

Z% CfLLS 0712%llS

8. REVERBERATION, Q = 0, 1.414 / DIFFERENCE, TRUNCATED TIME / PARTICLE

EE

x

co.-*.-Vl

c?

1[“ ’’1’ ’’’1 ’’’’ 1’ ’’’[’’’’1’” ‘1’~

.30 :

.20 :

.10 :

0 :l, 1111, ,1111, ,, l,, ,,1, ,,,l,r

O .01 .02 .03 .04 .05 .06

l/(NO. OF CELLS)

I l“’’I’’ ’’1’ ’’’1 ’’’’ 1’ ’’’1’” ‘

.020 –

.

:.015 –

~

w

k .010 –w..VIc

: .005 –

o y I I 1 1 [,, , ,1,,,,1,,,,1,,,,1,,,,1,7

o .01 ●O2 .03 .04 .05 .06

I/(NO. OF CELLS)

a

M\E

x

J

.6 ~l’’’’l ’’’’ ”’’’ ’l ’’’’ l’’”-l ““1’~

.5 –

.4 —

●3 –

.2 –

.1 –

o –1 1 t I 1 1!, , ,1,,,,1,,,,1,,,,1,,,,1,,-

o .01 .02 .03 .04 .05 .06

l/(NO. OF CELLS)

1l“’’I’’ ’’[’ ’’’1 ’’’’ 1’ ’’’1’’” l“-.06 –

.05 –

● 04 :

.03 :

. 0.? –

●O1 –

o- –1,, , ,1,,,,1,,,,1,,,,1,,,,1,;

0 .01 .02 .03 .04 .05 .06

l/(NO. OF CELLS)

-.

.‘.

.

Fig. 17(cont)

136

-.

.-.

PAD3.~6 - 75.-UI4 I?U45 LAC2 ●IO. STEADYDETCNATICN, Q = 0, 1.%14* EIWN 4,TFICKTIW, 9.s+3 SEC CN Ku4,

I.ZWCELLS 071Z++f1569.239 SEc Ct4 Jc9

10. STEADY IJETONATION. Q = 0. 1.414 / TIME= 1.7000E+O0

0.

>

60

50

40

30

20

la

c

1 1 ! 1 1

I [ [ 1 I 14 6 8 10 12 14

POSiti Oll X (mm)

20, 1 1 1 1 [ r

5 -

0 -

5 -

0 -( 1 1 1 1 I L4 6 8 10 12 14

POS it iOn X (mm)

3.

2.

z.

2.

1.

4 6 8 10 12 14

POSitiOfl X (mm)

[ 8 1 1 I

z - 4

8 -

4 -

0 -

6 -L 1 1 1 1 t4 6 8 10 12 14

Fig. 18.

Problem 10. q = O, 1.414, qu.udmtic.

POSitiOfl X (rIIWI)

PAD3. X6 - 7!5JJ-14 KU45 LAC2 ●1O. STEADYCCTWTICN. Q . 0. 1.419*TFICICETIW.

Ef?LN5,.z6. @t3 SEC m RiN,

2S6 CELLS 07/I??tf75105.639 SEC N JIX

10. STEADY DETONATION, Q = 0. 1.414 / TIME= 1.7000E+O0

0.

al

60

30 ;

20 -

10 -

0 -I 1 I

L1 1

4 6 8 10 12 14

POSitiOll X (mm)

1“ I 1 I 1 I 1

-’

4 6 8 10 12 14

POSitiOrl X (mm)

W

L I 1 4 1 I J

4.0

3.0

IL2.0 1

1.0 :

0 1 ! 1 1 t i4 6 8 10 12 14

POSitiOn x (mm)

I 1 8 1 13*Z :

2.8 -

2.4 -

rZ.o -

I

1.6 - L1 1 1 I [ \

4 6 8 10 lZ 14

Position x (mm)

<

.

?

7

Fig. 18(cont)

138

. .

●

.“

PAD3.S6 - WI* IWJ5 LAC.?●1O. STEAOYIXTCNATICN. Q - 0. 1.414” ERu4 ~,TFICKTIW, 13. ?61 SEc 04 Mm,

li?E CELLS 07/?+ /7592.756 SC m .mB

10. STEADY DETONATION. Q = 0. 1.414 / PARTICLES

14.0 \“’”’”’’’”’”’”’”’”’”’””””’1

_ 13.0EE

x

K 12.00 1/.-+.-VI

: 11.0

I /“1.2 1.6 2.0 Z.4

Time t (Ps)

---1 r 1 I 1 i I 13.2 :

2.8 :

Z*4 :

2.0 -

1.6 - A1 [ I 1 1 I 1 1

1.2 1.6 2.0 2.4

Time t (Ps)

1 T 1 I f [ 1 r i60 -

50 -

40 -

30 -

20 -

10 -

0 ;J1 1 1 1 [ 1 1

1.2 1.6 2.0 2.4

Time t (Ps)

L. 1 1 1 1 1 1 1 1 A

4.0

3.0 :

2.0 i

1.0

o— ,.l..J2.0 2.41.2 1.6

Time t IJs)

?

i

139

PAD3.X6 - 75.ALIW W7 LAC2 ●IO. STEAOYCCTCNATICW,Q - 0. 1.+1** ER5J4.TFICKETIW, 10. W3 SEc m RUN.

128 CELLS 071.?+175165.554 SEC CN JC@

10. STEADY DETONATION. Q = 0, 1.414 / DIFFERENCE, TRUNCATED TIME / PARTICLES

.038~ [“’’’’’’’”’’’”’’’”’”’’’’” ‘“l

EE

x

co.-+

I.036 b

.034 :

.032 :

.030 :1 I 1 1 1 1 [

0.

z -

1 -

0 -

-1 -

-2 -

-3 -,, ,,, ,,, ,,,.,,, ,,,.,,j

1.2 1.6 2.0 2.4 1.2 1.6 2.0 .2.4

Time t (Ps) Time t (PS)

[“’”’’’’’’’’ ”’’’’’ ”’”’’’’”1 l“’’’’’ ’’’’ ’’’’ ”’” ’’’” ’’””~.021-1, 4

0 -

-.02 -

-.04 -

-.06 -

-.081’’’’’’’’’”’’’’’”” “’’’”’”’”11.2 1.6 Z*O Z.4

J

.10 -

.05 -

0 -

-.05 -

-.10 -

ll. t. !. l. l. !. !. i. l.!. l.!. l.,,i1.Z 1.6 Z*O Z.4

Time t (tIs)

Fig. 18 (cent)

Time t (PS)

140

●

PA03.3f6 - 7%lJ-1~ KWJ7 LAC.2 ●IO. STEADYDETCNATIN. Q - 0. I.’+ I*” ERw 5,

TFICKETILW.

256 CELLS 07124175

3-?.751 s-EC CN RUi. 187.762 SEc CN m

10. STEADY DETONATION, O = O, 1.414 / DIFFERENCE, TRUNCATED TIME / PARTICLES

.* 145Z.o

*o 140

x 1.0

:

1350

EE 0. 0

x a

130L

c 30 M. 1A

w a -1.0.-111 125 k

:-Z. o

120

1.2 1.6 2.0 2.4 1.2 1.6 2.0 .2.4

Time t (Fs) Time t (Ps)

1 I 1 I 1 1 r

L’1”’’’’’ ”’”’’’’’”’’’’’”’’’”’”< . 10 I i

li.’.t.l.l...l.,lt’ I 11 .0 .! .0.1.,.,.,.,.,., ., .1.,.1

1.2 1.6 .2.0 2.4 1.2 1.6 2.0 2.4

Time t (Ps) lime t (Ps)

Fig. 18(cont)

141

PAo3. ~6 - -15.Al-Ih KU47 LAC2 ●IO. STEADYOETCNATICN, Cl - 0. 1.411!*TFIWETIIW. 10.7s1 SC CN m. Elw4 w. li?E CELLS 071.?+175

165.793 SEC ~ JC910. STEADY DETONATION, Q = 0. 1.414 / DIFFERENCE. TRUNCATED TIME / PARTICLES

a

[’ 1 1 I 1 I

2 [

1 -

0 -

-1 -

-2 -

-3 - 1 1 1 1 1-.10 -.05 0 .05 .10

Velocity “ (km/s)

l“” ’” ’’’’ ’’’” ’’8””8 “1

-.08 -.06 -.04 -.02 0 .02

.20

.15

.10

. 0s

Fig. 18(cont)

DensitY C’ (Mq/m3)

I [ 1 1 # I 1

0[

1.2

i

I

1

tJn

1 t t I !1.6 2.0 2.4

Time t (PS)

.-.

,*.

?

‘1

142

. PAD3.~6 - WI* iW16 LAC2 ●IO. STEAOY~T@lATICN. Q = O, 1.414s ERW *,. TFICKETIW, 11. -mosEcmlw4,

1.?8 CELLS 07/?9/751.29.6’55 SEC ~ .K9

10. STEADY DETONATION. Q = 0. 1.414 / DIFFERENCE / LEAD SHOCK

EE

x

Kc.-W.-

vl

:

E\

0/

.x+J.-

ulr

i 1 , ! t 1 1 i

.060 Ill’1 1 1 1 { 1 {

1.0 1.2 1.4 1.6 1.8 2.0

Time t (tIs)

r 1 1 1 I 1 I I}

o -

-.02 -

-.04 -

-.06 -

‘:;L1.0 1.2 1.4 1.6 108 Z*O

&

[ 1 I 1 1 T 1 1

-l=o~1.0 1.Z 1.4 1.6 1.8 Z*O

Time t (PS)

I 1 1 1 1 1 I 1L

-.08 -

M -.10 -\E

1-.lZ -

J

A* -.14 -.-

:+ -.16 -

:

-.18 -

,-.zot,

,,1 1 1 1 1 4

1.0 1*Z 1.4 1.6 1.8 2.0

Time t (PS) Time t (tIs)

Fig. 18(cont)

143

PAD3.3f6 - W14 lWi6 LAC2 ●IO. STEADYUZTCNATICW, Q - 0. 1.414.TFIcKETIW,

EFKN 5.3t.697 SEc CN Rm,

?56 CELLS 07/.3/75162.613 SEC CN .X9

10. STEADY DETONATION, 0 = 0. 1.414 / DIFFERENCE / LEAD SHOCK

EE

x

co

‘P.-

w

11 I 1 1 1 I J

.032 -

.028 -

.024 -

.020r ‘ 1 I t I 1 1

1.0 1.2 1.4 106 1.8 2.0

Time t (tIs)

1 8 I 1 1 1 1 J

ll. !. l. l. l.t. l.!. lo!.t .tj1.0 1.2 1.4 1.6 1.8 2.0

Time t (IJS)

2.5, I 1 1 1 1 I 1

2.0 -

1.5 -

1.0 -

.5 -

0 1 I I 1 1 ! d1.0 1.2 1.4 1.6 1.8 2.0

Time t (tIs)

-.04

-.06

-.08

-.10

-.12

-.14

Lo’’’’’’’ ””’’ ”’’’ ”’’’”d

-.lb~1.0 1.2 1.4 1.6 1.8 2.0

Time t (PS)

.“.,

.

Fig. 18(cont)

144

“-

>

PAD3..2 - -iwEP23 RW 8 LACI .1 SH3CK. ●AL. P=30* WL -i% -?TFICKETITN.

02/20/75

.SEE SEC Ct4 RWJ. 1?.927 SEc m m

AL, P:30 / u : 0.3. 1, 0 / TIME: 6.8 SOOE+01

}

I .5 ~oo~o~oooo~oo~ooo ooooooooooooo

o

1.0 -

0

.5 -

0

0 - 000I 1 k I 1

.350 300 350 Voo 450 500 550

L30 Boooooooooooooooo ooooooooooooo

25 -0

?0 -

015 -

10 -

5 -0

0

0 - 00

! ! I { I

250 300 350 400 *5O 500 550

m\E

x

a

uL3

WVIOJ

c+

POSitiOn X (mm) Position X (mm)

10 c 1 1 1 ,

8 -

6 -

500000000000000000000000000000E

; 3.+ -0

z

b

o: 3.2 -.-IAL

63.0 -

0

0

.?.0 - 0 ()< ,1 , c t 1

250 300 350 *OO 450 500 550

0-

‘t0

0

0

00[ . ...1....1....1....!.. Al’

250 300 350 400 h50 500 550

VIou

2

I

o

00 0000000000000000000000000000

m.

>

POSitiOn X (mm)

Fig.

Pos; t;on x (mm)

B-1.

Snapshots and particle histories for all optimum and one quadratic viscosity.

145

PA03 .2 - 7WEP,?3 R(.N 8 LACI. I 51+XK. ‘AL, P=30*TFICKETITN. 1.504 SEc CN lam, 13.563 SEc W Xe

:(3

a

alL

;IA01

c+

:0

w

:.-vl0:. .>

1 1 1 1

40 -

35 -

30 -

25 -

?0 -

15 -

10 -

5 -

01 1 I !

60 65 70 75 80 85

Time t (Pz)

10 1 1 # 1

e -

6 -

4 -

.

2 -

0I 1 I 1 1

60 65 70 75 80

Time t (Ps)

.-

WL-79-’? 02/20/75

AL. P,30 / Q : C).3. 1, 0 / F’ARTICLES

2“0~I .5 -

.0 -

.5 -

0

60 65 70 75 80 e5

Time t (PS)

3.8 1 1 r #

3.6 -

3.4 -

3.2 -

3.0 -

2.8[. I t 1 I

60 65 70 75 80 i

“.

.%

.

Time t (IJS)

..-

>

.

Pm3.2 - 74$EI%23 ml% LACI.2 SIKCX. .PfrlA, P=15*TFICKETITN, 1.014 SEC CNRU4. 23. IEO SEc m X9

PMMA ,I 1 I I 1 I

15

:b

0000000000000 0000000000000000000 00000

:0

i

a

OJt.

10 -

iIAOJ oc+

5 -

I . ..l.!

o0 - 000

250 300 350 400 450 500

POSitiOn X !mm)

0-

10 I 1 1 1 I

8 -

6 -

4 -0

.2 -

0

0 ~0 ooo-

1 1 1 t t

250 300 350 400 450 500 550

WL-71t-2 0.?/?0/75

P=ls / o = 0. 1.414. 0 / TIME= 8.5000E+01

30~

.?.5 -

2.0 - 0

I .5 -

1.0 -

0

.5 -

0 -0

!ooo-

1 1 1 1.z-50 300 350 900 450 500 550

POS; tiOrl X (mm)

2.2. 1 1 1 1 I

?.0 -

0000000000(~dwooo”ooooooo 0000000001.B-

I .6 - 0

1.4 -

~.250 300 350 900 450 500 550

POSit;Orl X (mm)

Fig. B-1

Position x (mm)

(cent)

147

pm3. 2 - 7162P.23 RU414 LACI .2 SJ+CCK. ●W. P=15* WL-7+-2 0.?/.?0/75

TFICKETITN, 1.766 SEC CN RN, .?3.932 SC m .Jcm

1

15 -

10 -

5 -

0I 1 , 1 I 1

so 85 90 95 100 I 05

Time t (Pz)

1 1 # 1 1 1

1 1 1 t 1 1

80 85 90 % I 00 I 05

Time t (Ps)

PMMA , P=15 / o = o? 1.414, 0 / PARTICLES

3“0~2.5 -

W

: 2.0 -

.Y

J1.5 -

hw.-

:

.5 -

m 85 Sm 95 100 105

2.2

.2.0

..

I .4

1 .Z

Time t (Ps)

T I I 1 , 1

I 1 9 1 t 1

80 85 90 95 I 00 105

Time t (Ps)

.-.

.9.

.

PA03.2 - 7%EP23 RLN15 LACI ..2 SMCK. ●IWW, P=15. WL-71+-2 02/20/75TFICKETITN, 1.79? SEC C$4 Ru4, 25.928 SEC CN .KB

l“’’’’’’’’ ”’’’’’’’’’””’”’”””’

>

#

>.

20

t15 -

10 -

5 -

0I 1 1 # 1 I

so as 90 E 100 105

Time t (PS)

1 I 1 1 1 I

--J,j~80 85 90 95 ! 00 105

Time t (tIs)

PMMA, P:IS / Q : ().3, 1, 0 / PARTICL

3.o~

2.0 -

.5 -

.0 -

.5 -

0 - 1 I I 1 I 1m 85 90 95 I 00 I 05

Time t (PS)

s

2.2. 1 1 1 1 I [

2.0 -

I .8 -

I .6 -

1.4 -

-il.. I t I 1 ! 1eo 135 90 95 I 00 I 05

Time t (P S.)

Fig. B-1 (cent)

149

PA03 .2 - 7WZP23 RU430 LACI.3 SFWK. ●PG. G-7. P-15. WL-71+-2 02/20/75TFICKETITN, 1.135 SEC CNRW. 46.12? SC CM .Jc9

I 1 1 1

20 -

15 -

10 -

5 -

01 I t 1

60 65 70 75 80

Time t (tIz)

~

Time t (tIs)

PG. G=7, P=15 / Q= 0.2. 2. 0 / PARTICL

3“o~.2.5 -

W

; ?.0 -

-v

J1.5 -

$.-

2

.5 -

0 1 I 1 I60 65 70 75 m

Time t (I-IS)

1. 1 , 1

1.4 -

1.3 -

1.2 -

1.1-

1.01 I t I

60 65 70 75 6U

Time t (Ps)

‘s

.

k

c.

..- PAD3 .2 - 7WEP23 RU46 LAC1.4 S!+XK. *PG. G=3, P=60*

TFICKETINI . 1.799 sEC CN RIJ4. 11.$06 SEC W.JW

PG ,

●

v

.-ul0

;.->

T 1 1 I

80 -

70 -

60 -

50 -

*O -

30 -

20 -

10 -

01 1 I 1

55 60 65 70 75

Time t (Ps)

40 1 1 # 1 1

35 -

30 -

25 -r

20 -

15 -

10 -

5 -

0I ! 1 I

55 w 65 70 75

Time t (t’s)

l+L-~-2 03/13/75

G=3. P=60 / O ❑ 0.3. 1. 0 / PARTICLES

6 -

5 -

Q -

3 -

2 -

1 -

0

I 1 1 1

5 60 65 70 75

Time t [Ps)

1 1 1 I

3.5 -

3.0 -

.?.5 -

2.0 -

,.51-&___& t I

65 70 75

Time t (Ps)

Fig. B-1 (cent)

151

PAD3.3 - 75UAYI RW 2TFICKETIPU, 3.5S9 SEC ~ Ru+, 3.599 SEc m JM

.2.

1.

1.

.

5 -

0 -

5 -

0 t { I I

60 65 70 75 80 85

T,me t (Ps)

1.5 -

1.0 -

.5 -

0

60 65 70 75 80 85

T,me t (PS)

06/06/75

Q = 0.1. 0. 0 / PARTICLES

1.5 -

1.0 -

.5 -

0 1 1 I 1

60 65 70 75 80 85

Time t (Ps)

Z.o

1.5

1.0

.5

0

I 1 I 1

1 I 1 I

60 65 70 75 80 85

T,me t (PS)

Fig. B-2.Particle velocity histories for all cases in Table B-II. The frames are arranged in the same or-der as the table. The label at the upper right gives the viscosity value for the topmost leftframe, and the frames are in clockwise order–upper left, upper right, lower right, lower left.

.-.

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152

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,“or-n1.5 -

1.0 -

.5 -

0 1 1 1 [

60 65 70 75 80 85

Time t (Ps)

L >

1.5 -

1.0 -

.5 -

0 -I [ 1 1

60 65 70 75 80 85

Time t (Ps)

06/06/75

Q = 0. 1.414. 0 / PARTICLES

60 65 70 75 80 85

Time t (Ps)

z.

1.

1.

.

60 65 70 75 80 85

Time t (~s)

x Fig. B-2(cont)

153

PM3.3 - 7Y’IAY1 RU41 ITFICKETIPU. 16.157 SC ON Rwi. 16.157 SE[ W JOE

3.0 -1 I [ 1 1 1

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2.0 -

1.5 -

1.0 -

.5 -

0 1 I I 1 ! 1

80 85 90 95 100 105

T,me t (Ps)

::~

.?.0 -

1.5 -

1.0 -

.5 -

0 I I 1 t I

80 85 90 95 100 105

Time t (Ps)

3.0

.?.5

2.0

1.5

1.0

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0

3.0

2.5

2.0

1.5

1.0

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0

06/06/75

Q : 0.3, 0, 0 / PARTICLES

I I 1 8 I 1

I I 1 1 I 1

80 85 90 95 100 105

T,me t (Ps)

1 1 1 I 1 t

80 85 90 95 100 105

T,me t (PS)

*

Fig. B-2(cont)

154

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06406 {75

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0

80 85 90 95 100 105

Time t (PS)

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rt

I 1 1 I 1

80 85 90 95 100 105

Time t (tIs)

3.0

Z.5

2.0

1.5

1.0

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3.0

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1 1 I 1 I 1

80 85 90 95 100 105

Time t (Ps)

1 1 I 1 I [

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80 85 90 95 100 105

Time t (I-IS)

b Fig. B-2(cont)

155

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Time t (Ps)

Z*O -

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0 I 1 t 1

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T,me t (Ps)

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1.0 -

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0 ),1 1 1

60 65 70 75 80

Time t (Ps)

‘:yZ.o -

1.5 -

1.0 -I i

L

60 65 70 75 80

Time t (Ps)

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Fig. B-2(cont)

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60 65 70 75 80

T,me t (tIs)

3.0 -1 I T 1

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2.0 -

1.5 -

1.0 -

.5 -

0 1 1 1 [60 65 70 75 80

Time t (PS)

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0 I 1 t I60 65 70 75 80

Time t (PS)

3.0 - I 1 r 1

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1.0 -

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0 1 1 1 160 65 70 75 80

Time t (IJs)

Fig. B-2(cont)

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3.0 -1 1 1 r

2.5 -

Z.o -

1.5 -

1.0 -

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0 I 1 1 t

60 65 70 75 80

T,me t (tIs)

3.0

Z*5

Z*O

1.5

1.0

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0

f # 1 1

1 I ! I

60 65 70 75 80

Time t (PS)

06106175

0= 0.5? 1? O / PARTICLES

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Z*O -

1.5 -

1*O -

.5 -

0 I I 1 t60 65 70 75 80

Time t (PS)

3.0 -1 1 1 1 -1

Z*5 -

Z*O - -1

1.5 -

1.0 -

.5 -

0 ! 1 I 160 65 70 75 80

Time t (~s)

:f

1,

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2.5 I.2.0-

1.5 -

1.0 -

.5 -

0 1 1 1 I60 65 70 75 80

T,me t (PS)

6 -’ 1 1 i

5 -

4 -

3 -

2 -

1 -

0I 1 1 1

55 60 65 70 75

Time t (PS)

06/06175

Q= “~ 0. Z / PARTICLES

2.0 -

1.5 -

1.0 -

.5 -

0 1 1 I [60 65 70 75 80

Time t (FS)

6 -’ [ ( 1

5 -

4 -

3 -

2 -

1-

01 1 I 1

55 60 65 70 75

Time t (Ps)

Fig. B-2(cont)

159

PA03.3 - 75MAY1 RWJ37TFICKETIW, 47.519 SEC CN Ru.J, L+7.51E SEC CN .AX3

06/ 06/ 75

m\

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A‘P

L

5 -

4 -

3 -

2 -

1 -

0[ I 1 1

55 60 65 70 75

Time t (PS)

6 -’ I 1 1

5 -

4 -

3 -

z -

1 -

01 1 t 1

55 60 65 70 75

Time t (tIs)

J

Fig. B-2(cont)

Q = 4. 0.125, 0 / PARTICLES

6 -’1 1 1

5 -

4 -

3 -

z -

1-

01 [ 1 1

55 60 65 70 75

6

5

0

Time t (tIs)

T I I I

J 1 I #

55 60 65 70 75

Time t (tIs)

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