finite-element simulation code for high-power

8/6/2019 Finite-Element Simulation Code for High-power

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4 0 6 5. Hum phries, Jr., and C. EkdahlAxial current Target

J Magnetic fieldMagnetic pressure

Metal linerFIGURE 1. Implod ing liner experimental g eom etry. A pulsed axial current creates a toroidal m agneticfield that drives the liner radially inward. In some experiments, the liner collides with a target to creatematerial states of high temperature and pressure.

The finite-element approach exhibits good stability for collisions between objects andshock convergence on axis. Crunch can be used autonomously by experimental scientists. The physical basis of theprogram and the command set are fully documented. The program and supporting data are highly portable. The standard version runs on IBM-standard personal computers with support for common graphics and hardcopy devices.

Two principles guided development to ensure that code results would be independent of thecode developer. All material data are access ible to the user. Information on the equation of state, electricalconductivity, and material strength is stored in external ASC II libraries. Material mo delscan be corrected or updated without changing the program. Crunch has no adjustable internal parameters for material models. Results are absoluteconsequences of the information in the external libraries. Any extrapolation beyond therange of input data terminates a run.The following sections document mathematical methods applied in Crunch and data re-sources we have assembled for personal comp uters. Section 2 covers the hydrodyn amic dif-ference equ ations with emph asis on cylindrical sy stems. The relationships follow from integralexpressions of conservation of mass, momentum, and energy over elements. Here, elementsare volumes w ith a unique m aterial identity that may move through the solution s pace. Section3 describes equation-of-state information to complete the hydrodynamic equations. We haveconverted the Sesame library to a language- and machine-independent form. Strength modelsfor elastic m aterials are discussed in Section 4. Section 5 covers numerical m ethods to calcu-late distributions of pulsed magnetic fields in a moving medium with temperature-dependentelectrical conductivity. Section 6 reviews the conductivity library we assembled to cover thetemperature range of interest for imploding liner simulations.In magnetic simulations with a single drive current Crunch includes a coupled circuit torepresent pulsed-pow er drivers. Section 7 describes the versatile model where all circuit com-ponents m ay have user-specified time variations. Section 8 reviews organizational aspects ofCrunch, including user interfaces and options for multiple drive currents. Section 9 illustrates


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Finite-element MHD simulation code 407some capab ilities of the code with two exam ples. The first, the Sedov blast wave problem , is astandard benchmark test for hydrodynamic codes. The second, an exploding wire simulation,illustrates a coupled calculation with nonlinear magnetic diffusion.2. Integral Hydrodynamic Equations

Num erical hydrodynam ic equations are difference representations of conservation of mass,momentum, and energy. The Crunch equations are referenced to elements. Figure 2 showselement divisions for a ID simulation. Depend ing on symm etry, the slices represent thin plates,cylindrical shells, or spherical shells. The two sets of indices apply to elements and elementboun daries. We shall denote boundary q uantities with uppercase letters and element quan titieswith lowercase. For example, element i with average radius r, has boundaries at Ri-i and /?,.Table 1 lists symbo ls used in the paper. Elem ents retain their material identity as they move andchange size during the calculation. The method is similar to Lagrangian finite-difference cal-culations [see, e.g., Potter (1973)]. An inherent limitation of the approach is the difficulty ofmodeling processes like mixing. On the other hand, the element-centered approach has twoadvantages:

Automatic zone refinement for compressional phenomena like shocks. Ability to model explosive processes where the solution volume size may change byorders of magnitude.In this paper we shall concentrate on cylindrical system s. The extension to planar and spher-ical systems is straightforward. Conservation of mass implies that element masses do notchange during the simulation. Consider an element with initial boundaries Roi-\ and Roi andinitial density poi. The mass is given by

In this paper factors of n are included for clarity. Redundant floating p oint calculations havebeen eliminated from the program. The boundaries move in response to forces. The density atany time is related to the boundary positions byPi = m, (2)

Element iBoundary i

FIGURE 2. Index conventions for IDfinite-elementsimulations. In cylindrical geometry, the elementsare cylindrical shells.


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408 5. Hum phries, Jr., and C. EkdahlTABLE 1. Symbols and units

Symbolr?m,-P?P?w?JP?


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Finite-element MHD simulation code 409TABLE 1. Continued

Other quantitiesSymbol Quantity UnitsLc Liner inductance HL, Inductance of power connections Hh Liner height mA, B Circuit model compon ent matrices

Note that equations (1) and (2) do not incorporate approximations based on small elementwidth. This feature avoids numerical problems when elements com press to cylindrical or spher-ical axes. Furtherm ore, the model allows the use of large elements. The average element radiuscorresponds to the center-of-mass coordinate. Assuming a uniform density, the average radiusof a cylindrical element is related to the boundary radii by

f + Rf-i2 (3)

We express conservation of momentum as an equation of motion for element boundaries.The object is to find the boundary velocitiesdRiV,-T. (4)

The time rate of change of m omentum at boundary i equals the time derivative of velocity timeshalf the masses of adjacent elements,

The force on the boundary is the sum of forces from adjacent elements. Summing pressure andmagnetic forces gives the equation of motion( ^ ^ ( r i 2 + 1 - r , 2 ) . (6)

The new elem ent qua ntities in equation (6) are the pressure/?,-, artificial v iscosity force w,, andcurrent density j , . Current density flows in the axial direction for the geometry of figure 1. Asexplained in Section 5, the toroidal magnetic field B t is a boundary quantity. The magneticforce equals the integral of the force density over half the volumes of adjacent elements. Thepressure force equals the difference in pressure in the adjacent elements multiplied by thecylindrical area at the boundary. The artificial viscosity force term damps no nphysical os cil-lations at shock fronts. The physical rationale for artificial viscosity and its inclusion in thehydrodynamic equations are covered in Potter (1973). We used an adaptation of the vonNeumann-Richtmyer form (Neumann & Richtmyer 1950; Richtmyer & Morton 1967) forfinite-difference solutions,

w = -Cp& 2 dva? (7)In equation (7) C is an adjustable parameter to spread the shock over several mesh d ivisions,A is the mesh scale length, and dv/dx is the spatial derivative of velocity.


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410 S. Humph ries, Jr., and C. EkdahlCrunch advances hydrodynamic quantities using the standard time-centered leapfrogmeth od. The boundary velocities V; are defined at half time steps, and all other quantitie sapply at integral steps. Throughout this paper the superscript n denotes the time step, so that

jn+i/2 _ {n + frt/2. Replac ing time deriva tives in equation (6) with time-centered differenceoperators gives an equation to advance the boundary velocity,/ Afi/n+l/2 _ i/n-l/2 _ 'Vj Yf

X [(p?+ , + < + , - Pi - 0477/?? + B?(yA-i +7")fl-( '1/+2| + ' "2)] - (8)Equation (7) has the following difference representation,

Wj = C P / | * | M1 l ( * l W1 /> \9)

where C is a constant in the range 1-10. Given the modified velocities, the next step is toadvance the boundary radii to the next integral time step,R? +l =R? + Vr+U 2At. (10)

New element densities and average radii can be determined from R"+l using equations (2)and (3).The internal energy M, is an element p roperty eq ual to the total energy of element i dividedby m,. At present Crunch does not model chan ges of , resulting from convection or radiationtranspo rt. Although simple to code, we omitted electron thermal condu ctivity contributions forthree reasons: Thermal cond uction in solids and liquids is negligible compared to energy transport by

shocks. Thermal transp ort coefficients are not well known at high temperature and pressure. Energy transport in gases and plasmas is probably dom inated by convection.Under the limiting assumptions ch anges of internal energy in hydrodyn amic calculations resultfrom work performed by pressure, artificial viscosity force, and elastic stress. In magneticproblems ohmic heating also contributes. The work performed by pressure and artificial vis-cosity force on element i in a time step is -(/? ,+w ,)A V ;, where AVJ is the change in elem entvolum e. The time rate of change of energy from resistive heating in an element with electricalconductivity tr, equals;2/cr , times the element volum e. With these contributions, the equationto advance internal energy is

The first term in brackets is a time-centered expression involving the advanced value of pres-sure. The p ressure is estimated by the two-step proc ess described in the next paragraph. It is notnecessary to apply more complex time-centered expressions for the second term. In cases ofinterest the time scale for resistive heating is much longer than that for shock heating.To close the set of equations we must find the new element pressures p" + ' corresponding tomodified values of density and internal energy, p " + 1 and u"+x . The values are determined fromthe Sesame equ ation-of-state library discussed in the next section. The library con sists of 2Dtables of pressure and internal energy as functions of density p and temperature r : p{p,r) and

M( p , r ) . With known values of density and internal energy the temperature r can be determinedwith an inverse interpolation. We use a modified two-step m ethod (Potter, 1973) to advance the


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Finite-element MHD simulation code 411pressure and preserve time centering in equation (11). The advanced internal energy " + " isfirst estimated from equation (11) using only/?". Sesam e interpolations give estimates of the ad-vanced temperature T " + " and pressure p"+ ". T he quantity (p"+p"+u)/2 is then used in equ a-tion (11) to yield an improved value u" + '. The interpolations are repeated to find T/ '+ ' a nd /? " + ' .3. Equation-of-state information

The Sesame Library maintained by LANL contains computed values of thermodynamicproperties at high temperature and pressure for 131 materials. The standard format of thelibrary is not convenient for personal computers; the data are contained in a single large binaryfile. It is necessary to apply a set of FORTR AN subroutines supplied by the laboratory to accessinformation (Abdallah et al. 1980). We converted the Sesam e data to a portable format that wasboth language and machine independent. We wrote programs to dissect the library and to createindividual AS CII files for each materia l. The da ta are accessible w ith an editor, and it is easy tocreate new tab les. With data comp ression the library fits on three 1.4-MB floppy disks.The files in the PC Sesame Library have descriptive names of the form ALUM3713.SES.The first four characters of the prefix denote the type of material, while the next four give theLANL table numbers. Each material data file contains a standard header and four tables. Theheader provides information to load values into spreadsheets or program s. The first table containsthe discrete values of density for tabulation of the pressure and internal energy: p , , p 2 , . . . ,p(Np).The second table lists temperature values: TU T 2 ) . . . ,TJ V T . The third table gives correspond-ing pressure values with density indices as the outer loop: p(l,l), p(\,2),...,p(l,NT),p(2,l),...,p(i,j),...,p(Np,NT - 1), p(N p,N T). The ordering differs from the standardSesame arrangem ent and allows more efficient interpolation. The fourth table contains v aluesof internal energy: W(1,1) ,K(1,2 ) , . . . , u(lNT),u(2,l),...,u(iJ),...,u{Np,N T-l),u(Np,N T).For backward compatibility, we m aintained the practical units of the Los Alamo s tables: p ing/cm 3 (1000 kg/ m 3 ), r in K, p in GPa (10 9 Pa), and u in MJ/kg (106 J/kg) .

We also developed a structured FORTRAN software unit, OPEN_SES, that handles opera-tions on the Sesam e tables. All storage and m anipulation of data are internal to the unit, withcommunication via subroutine and function calls. The functionality of OPEN_SES does notrequire data structure definition in the calling program. For exam ple, the statementCALL SesLoad(MatFile)

transfers material data from a Sesame file to memory storage controlled by the software unit.Here, the quantity M a t F i l e is the 8-character PC Sesame file prefix. A subsequent call to theinteger function NCo de ( ) returns the operation status. Related functions include N T a b l e ( )(num ber of tables successfully loaded in mem ory) and NMem ( ) (available memory bytes fortable storage). The subroutine

S e sC onvF a c to r s ( R hoC onv ,Te m pC onv ,P r e s sC onv ,UC onv ,Ve lC onv)sets up unit conversions for all input and output d ata. For exam ple, the statement

CALL S e s C o n v F a c t o r s ( 1 . 0 E 3 , 1 . 0 , 1 . 0 E 9 , 1 . 0 E 6 . 1 . 0 )changes hydrodynamic quantities to SI units. OPEN_SES assigns an integer material numberM atN o to tables in the order they are loaded. The function to return the stored pressure valuefor density p/ and temperature TJ has the name and pass parameters P I J (M atN o, I , J ) .

The critical routines for hydrod ynamic codes are those that interpolate and invert quantities.The function P (M at N o, Rh o , Te mp ) returns the pressure in the current units for materialMatNo at density Rho and temperature Temp. OPEN_SES routines use polynomial interpola-tion with user-specified accuracy up to sixth order. Interpolation routines return an error codethrough NC od e ( ) for out-of-range values or interpolation errors. The inversion functionTem p (M at N o, Rh o , P r e s s ) gives temperature as a function of density and pressure using


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412 5. Humphries, Jr., and C. Ekdahlthe following method. The routine identifies the pressure table row with density p, closest toRho and scans temperature columns for pressure entries that bracket P r e s s . A ID interpola-tion gives the temperature T(/O/,Press). The procedure is repeated on nearby rows to build atable of values: ...,r(p/_i,Press), T(p /;Press), T ( P / + 1 , P r e s s ) , . . . . The number of entries de-pends on the interpolation order. A final interpolation gives temperature at the target pressure.The routine is fast enough so that it is unnecessary to build inverted tables T( p,p) in memory.Other OPEN_SES routines calculate sound speed and points on the shock Hugoniot curve [see,e.g., Courant & Friedrichs (1991)].4. Material strength models

This section reviews elastic material models in Crunch. The normal strain in a medium is adimensionless quantity that equals the change in the length of an element divided by its originallength, while normal stress is the force per unit area that must be applied to a solid body toproduce a strain.Shear stresses that give angular distortions of the element do not appear in 1Dcalculations. In cylindrical simulations there are two possible normal strains, err and eee, causedby the two stresses srr and sss. Figure 3 illustrates the relationship between strains and stressesin a cylindrical system. An element moves only in the r direction but may have stresses andstrains in 9 because of radial compression or expansion.

We can identify two limiting cases where the system is unconstrained in z (plane stress) orclamped {plane strain). Following Fung (1965) and Huddlestone (1961), the stress-strainrelationships for the two cases are

1err = (Srr ~ VS ee),

= ~g (~vsrr + see) (12)

or

= -[-v(l + v)srr + (l-v2)see]. (13)

'00

rr

FIGURE 3. Stress on an element in a cylindrical system with applied force in the r directional only,showing the origin of the hoop stress.


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Finite-element MHD simulation code 413In equations (12) and (13), the quantity E is Young's modulus and v is Poisson's ratio (whichgenerally has a value near 0.25).

Concentrating on the constrained case, the relationships in equation (13) give the stresses as"(1 - v)err

(14)verr + {\-v)eee(1 + 0( 1- 21 ')Radial motion of an element results in an azimuthal strain. Noting that the strain equals thechange in length along 6 divided by the original length and assuming continuity of the medium,the azimuthal strain in element i at time t" is

r!> - r?eSe i= Jr-. (15)

where the quantity r" is the average element radius [equation (3)] and the superscript o refersto quantities at the initial time. The radial strain is given by the expression

{ 'The quantities on the right-hand side of equation (16) are boundary radii. Inspection of figure 3shows that azimuthal stress exerts a component of radial force, the hoop stress. For a givenvalue of sggj, the radial force per unit area at the center of mass of element i from materialexpansion or compression is given by the expression

SggjjR? ~ R"-1 ] . .r?

It is relatively easy to incorporate elastic forces in planar Crunch simulations. The stressesappear as extra contributions in the boundary equation of motion. For example, in a planarsimulation the force per unit area acting at boundary i is

p?-pUi + < ~ *W+, ~ d + *,+ ,- (18)Equation (18) holds for positive values of strain (extension) because the fluid equation of statecontributes no force. Equivalently, pressure entries in the Sesame tables are always positive.On the other hand, the tables contribute elastic forces in elements with negative strain (com-pression). In this case there is a redundancy if stresses are included. The following prescriptionholds for a planar numerical model: use (p" + w") for the force per area in an elastic elementwhen e" < 0 and use (p" + w" s") when e" < 0 if the element is unbroken. An element breaksif a positive value of st exceeds the yield stress or if the temperature reaches the melt value. Theadjusted force is used in both the equation of motion and internal energy equation.

Material strength contributions to cylindrical problems are more involved. Consider radialexpansion of a cylindrical element. The azimuthal strain and stress are positive, resulting in aninward radial hoop force. Conservation of volume gives a radial compression (negative strain);therefore, the direct radial force is given by equation-of-state contributions. The radial averageof this force over the elements of a cylindrical shell is approximately zero, while the hoop stressgives a cumulative inward force. Next, suppose the cylindrical shell is pushed inward. There iscompression in the azimuthal direction and expansion in the radial direction. The elastic forcesresist the radial expansion, giving a net outward force associated with the azimuthal strain.An element breaks if the radial stress exceeds the yield value. Following these considerations,the following conventions are applied in Crunch. In unbroken elements only positive radial


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414 S. Humph ries, Jr., and C. Ekdahlstresses are included in the equation of motion while the hoop force is included for positive andnegative values of the azimuthal stress. Elastic forces are represented by an extra term inequation (6),

rM -srri + srri+, . (19)The radial stress terms are included only if they have a positive value.We used the Steinberg (1996) parametric model in Crunch for the elastic properties of ma-terials. The following equation gives Yo ung's modulus as a function of density, pressure, andtemperature:

E(p,p,r) = Eo\ 1 + A ^ - B ( T - 300)1. (20)In equation (20) Eo (PA) is the value of You ng's m odulus at ambient co nditions, p (PA) is thepressure, T (K) is the temperature, and 77 is the compression,

1 7 = - . ( 2 1 )PoThe yield stress is proportional to the relative value of Young 's modulus multiplied by a func-tion of the element strain that represents work hardening:

Y=YJ(e). (22)

The function is given byYofie) = ^ [ 1 + /3( + *,) "] =s Ym. (23)

The sym bols in equations (22) and (23) represent the following quantities: Yo (Pa) is the yieldstrength at the Hugoniot elastic limit, yma x (Pa) is the work hardening maxim um, and /? and nare work hardening p arameters. Steinberg (1996) also gives an expression for the melt point ofa material in compression:

L( I^ (24)In equation (24) rmo is the melt temperature at ambient conditions, y o and a are model param-eters, and 17 is given by equation (21). We have created an ASCII library that contains thematerial parameters in SI units.5. Magnetic diffusion

This section reviews num erical methods in Crunch to treat magnetic diffusion in cylindricalsystem s like that of figure 1. The solution volume may contain layers of different mate rialsincluding conductors and insulators. All layers participate in the hydrodynamic solution, butonly the condu ctors participate in the magn etic field solution. To begin, supp ose that there is asingle specified drive current I(t) and that the conducting layers are contiguo us. We shall callthe set of conductors the magnetically active region. The goal is to find the toroidal m agneticfield distribution o ver the region as a function of time. Figure 4 shows index conv entions forthe magnetic field ca lculation. Element quan tities include r" J" (axial current density in A /m 2)and or" (electrical conductivity in mho/m). The temperature-dependent element conductivitymay change with time. Boundary quantities include R" and V? +U 2. The quantity Rmin is the


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Finite-element MHD simulation code 415

FIGURE 4. Index conventions for quantities in the cylindrical magnetic diffusion calculation.

radius of the inner edge of the magnetically active region, and Rmax is the radius of the outeredge. Because the element quantity jz is proportional to the spatial derivative of toroidal mag-netic field, we take B" (T) as a boundary quantity. This association is consistent with the fieldboundary conditions:

B(R min,t) = O,B(R ma x,t) = 2nR n (25)

E(r,t) = E(ro,t) - - \ B(r',t)dr'dB(r',lr dB(r',t) dro dr= E{ro,t) - dr1 - B(rB,t) + B(r,t) -Jr at at dt

The Crunch difference equations follow from the differential equation of magnetic diffusionin a moving medium [see, e.g., Humphries (1997)]. Farad ay's law gives the difference in axialelectric field at radii ro and r in the active region as

(26)

(27)

(28)

(29)

The other defining relationships are Ohm's law,j(r,t) = *(r,t)E(r,t),

and Ampere's law,2irr' dr'j(r',t).

Taking derivatives and combining equations (24)-(28) givesdBHt

1 1 a SB av-(rB)-v-r-BT.afio r or or or


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416 S. Humphries, Jr., and C. EkdahlThe difference equations follow from equation (29). For spatial differencing we preserve theelement viewp oint by treating material properties as element quantities. For time differencingwe apply the method of Dufort and Frankiel (1953), which exhibits num erical stability for anychoice of time step. Substituting space- and time-centered difference operators gives an equa-tion to advance the field at a boundary point over a time interval f"+1 - f""1 = 2Af.

_where

B"+\R"+\(r"+\ r")a = j ^; ,^ _ B f_,Jg?_,(rfl.,-r/ ')

(R?-RU)r?trtn '


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Finite-element MHD simulation code 417collapses. Section 9 shows an example of a nonlinear magnetic diffusion calculation with avoid.6. Tem perature-depen dent e lec tr ica l conduct iv i ty

Imploding liner simulations require data on electrical conductivity over a broad range of ma-terial conditions, from the solid to plasma states. Following the method of Lindem uth (1985),we assembled material from several sources to create a conductivity library for common metals(aluminum, copper, gold, iron, lead, molybdenum , silver, titanium, and tungsten) over the tem-perature range 300 to 10 8 K. The present library is limited to materials near solid density, ad-equate for models of the acceleration phase of imploding liners. The low-tem perature range (300to 1000 K) is covered by standard references. We adopted data from exploding wire experi-ments (Tucker & Toth 1975) for the range from 1000 K through vaporization (-20,000 K). Theo-retical results (Lee & Mo re 1984; Rinker 1988) were used above 100,000 K. In the intermediaterange of vaporized metal we connected the data with smooth interpolations. The main limita-tion of the present tables is that they do not include pressure depend ence.Tucker and Toth (1975) reported the conductivity of metal wires subject to pulsed currentsas a function of volume-averaged action. (Action is the time-integral of the square of thecurrent density.) The data cover the solid-l iqu id-v apo r transition range that is poorly describedby theory. Inference of the volume resistivity depends on the assumption that the wire crosssection does not change significantly during heating. The data are therefore invalid after thewire burst point. In the experiments changes of internal energy resulted almost entirely fromohmic heating. The changes of internal energy inferred from the action in Tucker and Toth(1975) were in good agreement with tabulated values for liquification and vaporization. Wedetermined the temperature from the internal energy by Sesame table interpolations at soliddensity. Figure 5 shows resulting values of volume resistivity as a function of temperature foraluminum. The results are close to handbook values in the low-temperature range. The resis-tivity rises at melt and rapidly increases following vaporization. In the wire experiment withcurrent constrained by inductance the ohmic heating rate escalated at the vaporization pointrapidly leading to wire explosion.We used the electron transport tables of Rinker (1988) maintained at LANL for high-temperature conductivity. Tabulated values for partially ionized material in the range T >10 5 K appeared reaso nable. Values converged to those of Spitzer and Harm (1953) at the highend of the temperature range. The tables were invalid at lower temperatures, as shown infigure 6a. We took a pragmatic approach to fill in the missing data in the range 10 4 to 105 K.The interpolation shown in figure 6b starts from the exploding wire burst point and follows asmooth transition to the high-temperature Rinker data. Data for the other metals followedsimilar behavior. Results were collected in an ASCII conductivity library.

7. Circuit model for pulsed-power driversCrunch includes a circuit model for self-consistent calculations of drive current in linersimulations [see, e.g., Struve et al. (1998)]. Figure 7 shows the generic pulsed-power circuitsuggested by R. Reinovsky of LAN L. Com ponents are divided into three sets: generator, fuse,

and load. With the exception of the generator capacitance Cg, all components may have as-signed time variations. With this feature, the model can address an array of pulsed-powerdevices. For example, a time-dependent value for Rf enables simulations of capacitor bankswith fuses. An inductive storage device is represented by assigning an infinite value to C s,initiating a seed current in the first loop, and setting time-dependent values for Lg, Rg, and Rf.This v ersatility requires that the numerical solution method give physically reasonab le resultsfor large time-scale d ifference in different sections of the circuit. In this section, w e shall first


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418 S. Humphries, Jr., and C. Ekdahl400

300

I& 200y.2806

100Solid

1 MVaporizi

elt J*

ition }

Burst pcu

/ '/

\

5000 10000 15000Temperature (K)

FIGURE 5. Volume resistivity as a function of temperature. Results derived from the exploding wire datao fTucke r&Toth (1975) .

cover a robust method to advance voltages and currents and then discuss circuit contributionsfrom the magnetohydrodynamic solution.Figure 7 show s seven circuit compo nents. We can write a time-centered difference equationfor each com ponen t. The resulting set advances the seven values of voltage and cu rrent over aninterval Af from time t" to f"+1.yn + yn+l _ yn _ yn+\ + ; n +l_ + _ Rn+l/2 = 0 >

^4" + V 4 n + 1 - V " - V , " + rn+l/2 _

V,": n + l _ :n _ :n+l i :n

A,3n + V 3 n + 1

-L nf+U 2 = 0,

/? r l / 2 = o,(33)

V4n + V 4 " + ' - V3" - V3" It L", +U 1 = 0 ,


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Finite-element MHD simulation code 4 1 9

10

5o110

1"S 106

Aluminum (2.7 gm/cm )

\ Tucker-Toth

Rinker _,y

10J 10* 103T e m p e r a t u r e ( K )

10

Aluminum - Interpolated conductivity

10

I 107

1rj 106

10J

andbook

\\

acker-Toth

InterpcAlation //

f Rinker

10J 10" 10J 10Tempera ture (K)

10' 10

FIGURE 6. Aluminum electrical conductivity as a function of temperature at solid density, (a) Data sources.Open squares: results derived from exploding wire data of Tucker & Toth (1975). Filled circles: Resultsfrom the tables of Rinker (1988). (b) Final values used in the conductivity library including an interpola-tion at intermediate temperature.


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420 5. Humphries, Jr., and C. Ekdahl

I J

mFIGURE 7. Circuit model for pulsed-power drivers.

The time-dependent circuit component values are set at the intermediate time step. We canexpress equation (33) succinctly in terms of a circuit vector:

C" =v,"

(34)

After moving all quantities at tn+l to the left-hand side and quantities at t" on the right-handside, equation (33) becomes

where the coefficient matrices are(35)

n+1/2 _

2L"g+xn/Mpn+l/2

-2L}+U 2/M001

00

R}+x n2LfmlL t

_R?+U2-2L",+m /A t

0

1- 100000

001

- 1000

00001

- 10

0101010

- 100000

2CJM

(36)


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Finite-element MHD simulation code 421and

gn+1/2 _

~Rg2L ng+U 2/A t

Rn+W2-2LfxnlA t

00

- 1

00

-R'}+U 22L" f+U 2/A t

Rn+U2-2L" l + U2 /A t

0

- 1100000

00

- 11000

0000

- 110

0- 10

- 10

- 10

- 100000

2C O /A f

(37)

We can rewrite equation (35) in a form to advance the circuit vector,C " + 1 = I n v ( A " + 1 / 2 ) B ' I + 1 / 2 C " (38)

The first step in Crunch circuit calculation is to set initial values for the circuit vector [equa-tion (34)]. For a simple capacitor bank, the loop currents are zero and the charge voltageapplies at V] and V5. If circuit component values do not change with time, the matrix prod-uc t I nv ( A" + 1 / 2 ) B n + l / 2 is evaluated only once if any of the components vary, the matricesmust be updated at each time step.The time variations of generator and fuse components follow user-supplied tables. Thesetables are ASCII files with multiple data lines containing a time and a comp onent value. Thefollowing consideration applies in preparing tables. The voltage across a series combination of

resistor and inductor isd ( dLV= iR+ (Li) = i\R+ dt \ dt

di.dt(39)

Equation (39) is cons istent w ith the form of equation (33) if the value of the resistor is taken asthe effective value (R + dL/dt). The third column in figure 7 represents the pulsed-pow er load,all comp onents past the fuse. In Crunch the load indu ctance is divided into two parts, L t = L,+Lc. The constant quantity L, represents the inductance of transmission lines and the experi-mental chamber volume outside the initial liner radius, /Cax- The quantity Lc is the extrainductance resulting from the liner imp losion. For a cylindrical liner of height h, the inductanceequals

" m a x(40)

where R^ is the present liner outer radius. Note that equation (33) does not give the comp o-nent value at the intermediate time step. This has little effect on the calculation because thetime scale for changes in the drive current is much longer than the hydrodynamic time thatdetermines At. Following equation (39), the load resistance has two components, R, = Rc +dLjdt. The time derivative of liner inductance is related to the velocity of the outer boundaryvelocity by

dh cIt (41)The liner resistance (which includes the effects of magnetic field penetration) equals the volt-age on the outside of the liner divided by the drive current (f2 in figure 7):


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422 5. Hum phries, Jr., and C. Ekdahl

In equation (42), j " and a" are the current density and electrical conductivity in the outerelement.8. Crunch organization

This section discusses how the methods discussed in the previous sections are implementedin Crunch. Although the CGraph postprocessor has a full graphical user interface, the mainCrunch code is controlled by script files. The text files consist of a series of lines with standardcomm ands and param eters. There are several motivations to preserve this venerable technique. It is easier to modify an existing script with an editor than to walk throu gh a complex menu

structure. Script preparation automatically docu ments a run. Scripts can easily be incorporated in batch file procedures for extended run seq uences.To make the process as painless as possible, Crunch has a free-form line parser with detailederror checking. The parser accepts flexible number formats, comment lines, blank lines, andindentations to create readable scripts. Com mands may appear in any order; Crunch collects allinput data initiating a run. Table 2 shows an exam ple of a script file for an imploding liner run.Crunch hand les ID calculations in three geometries: planar, cylindrical, and spherical. Thereare three operation modes: hydrodynamic, magnetic diffusion, and coupled magnetohydro-dynam ic. The magnetic modes are only definable for planar and cylindrical calculations. In thepresent version of the code, there are three ways to drive materials:

Applied p ressure on the inner or outer boundary w ith a user-specified waveform. Assignm ent of initial velocities to elemen ts. Coupled magnetic acceleration.Setting up a system geometry is simply a matter of defining element boundaries and materialidentifies. In Crunch, elements are grouped into layers, contiguous sets with common proper-ties. Layers are defined by the command

LAYER 1 1 0.0450 0.0465 50The exam ple sets up Layer 1, associates its properties with M aterial 1, assigns inner and ou terbound aries of 0.0450 and 0.0465 m, and divides it into 50 equal elements. The program handlesup to 5000 elements in 200 layers. Layers may represent different materials, or several layerscould be used for variable m esh resolution in a single object. Single-element layers can be usedfor objects that are approximately hom ogeneou s over the problem time scale. An exam ple is acompressed gas inside an imploding liner.Material num bers are associated with stored tables for equation-of-state information, elec-trical conductivity, and strength param eters. Layers with the same material number share tablesbut may have different characteristics. Adjustable layer properties include the initial thermo -dynam ic state (density and temp erature with pressure and internal energy set from the Sesam etables), artificial viscosity force coefficients, inclusion in the magnetically active region, ap-plication of elastic stresses, and initial velocity. There are two options for setting velocity incylindrical and spherical prob lems: uniform or radially w eighted. In the latter case, velocitiesare assigned so that there is no initial com pression {dp/dt \0 = 0.0) for all elements of the layer.

Many High-Energy-Density Physics Program experiments involve abrupt collisions be-


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Finite-element MHD simulation code 423TABLE 2. Crunch script file example

ATLAS BENCHMARK* Liner parameters:* Aluminum* Height: 0.04 m* Inner radius: 0.0450 m* Outer radius: 0.0465 m* - GENERAL CONTROL -

SETMODE: LINERGEOMETRY: CYLINTMAX: 6.5E-6DT: 2.0E-9STEPCHANGE: 5.8E-6 0.1E-9STEPCHANGE: 6.0E-6 0.5E-9INTERPTYPE POLY

* - MATERIAL PROPERTIES -SESAME 1 ALUM3715MATCOND 1 TABLE ALUM

* - LAYER PROPERTIES -LAYER 1 1 0.0450 0.0465 50SETINIT 1 2.700E3 298.0VISCOSITY 1 10.0

* - MAGNETIC PROPERTIES -CGEN: 1250.OE-6LGEN: 10.0E-9RGEN: 2.0E-3LFUSE: 0.0RFUSE: 80.0LTLINE: 0.0INIT 0.0 0.0 240.0E3 0.0 0.0 0.0 240.0E3SETB 50 1CIRCFILE 2

* DIAGNOSTICS HISTORY 50DTIME: 1.0E-6

ENDFILE

tween layers that are initially disconnected. For simulations of these experiments, Crunchemploys a special single-element void layer to preserve continuity of elements. The void m odelhas a single parameter DBounce that can be set by the user. In the absence of a definition, thecode uses a default value equal to the smallest initial element width. Voids have zero pressurewhen their width exceeds DBounce: R" /?"_, > DB oun ce. In this case, the void has no effecton the dynam ics of adjacent elemen ts. If the width drops to DB ounce, the void is squashed. Asquashed void is incompressible (R f = /?"_i + DBounce) and has a pressure equal to theaverage of pressures in adjacent elements. For short time steps adjacent elements can adjust tothe sudden chang e of forces at the void bound aries preserving num erical stability.

Crunch can make tim e-step adjustments to preserve num erical accuracy and stability un dernormal circumstances. In magnetohydrodynamic runs, the program uses the smaller of the


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424 S. Humph ries, Jr., and C. Ekdahlhydrodynamic or magnetic time. To find the hydrodynamic time, Crunch searches for theminimum element width. This width is divided by either the element initial velocity or a ge-neric sound speed of 1000 m/s. The time step equals the result divided by a hydrodynamicsafety factor (default of 10). To find a magnetic time at the beginning of a calculation, theprogram searches the magnetically active region for the minimum of the quantity

WiiRi-Ri-i)2- (43)The magn etic time step equa ls the value divided by the magnetic safety factor (default of 2). Atlater times when magnetic field values are available to estimate second spatial derivatives,Crunch searches for a minimum of the quantity

(n+1 n) [ ri+1 a i+ , (R i+1 - R/) - r, ar, (fl, - /?,_,)]"Bi F ~ ^ ITR ( 4 4 )Autom atic time-step selection often fails in extreme circumstances like high-speed layer col-lisions or shock convergence on axis. In this case, the user can fine-tune the time step to passthrough the danger zone. A typical run takes less than one minute on a personal computer, andtwo or three iterations are usually sufficient to achieve a successful solution. To help in thisprocess the program records run variables and makes a graceful exit in the event of errors inintrinsic functions or floating point operations.Crunch makes extensive use of external data tables in ASCII format. Previous sectionscovered com plex tables that are generally not prepared by the user: the Sesame equation-of-state tables and the libraries STRENGTH.LIB and CONDUCT.LIB. These material tables aretreated as absolute data; there is no way to adjust their values within the code. A second classof tables prescribes time-depend ent qu antities. These tables are generally prepared by the userto model specific experiments. Examples include waveforms for applied pressure, drive cur-rent, or pulsed-power circuit com ponen ts. Tables consist of several data lines, each con tainingtwo real numbers (time and tabulated q uantity). Nonuniform time intervals and unsorted entrylines are allowed. The tables are processed with a free-form parser and may contain comm entlines, indentations, and a variety of real-number formats. The command to load a waveformtable may include the two parameters TMult and VMult to scale entries as they are loaded. Thisfeature makes it possible to perform parameteric scans without creating a large number oftables or to maintain a library of normalized waveforms.

To begin a run, Crunch reads com mands from the control script and stores param eters. Theprogram then checks the consistency of input data, opens output files, and defines elementdimensions. The next step is to set initial thermodynamic conditions for the elements. Theprogram then initializes strength properties of elastic elements. The default is to include apositive strain that produces a stress that counteracts the initial element pressure force. Thisfeature prevents spurious layer expansion during initial accelerations or drifts. For magneticcalculations, Crunch sets temperature-dependent element conductivities and determines thelimits of the magnetically active region. The program assigns initial current densities if there isan initial bias current. For a calculation with a drive circuit Crunch finds values of the com-ponents at t = 0 and sets up the advancing matrices. To complete initialization, the programcalculates a safe time step if there is no user specification.

In the main time loop, the first step is to check whether elastic elements have broken ormelted and to remove them from the material strength calculation. The program then sumsrelevant element forces to advance the boundary velocities according to equation (6). The nextsteps are to advance bo undary po sitions [equation (10)], calculate new element densities [equa-tion (2)], and update artificial viscosity contributions [equation (9)]. Crunch finds changes of


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Finite-element MHD simulation code 425internal energy and new values for element pressure and temperature following the methoddescribed in Section 2. The following sequence of operations occurs for magnetic calculations.Temperature-dependent electrical conductivities are updated. If there is an external drive cir-cuit Crunch updates the coefficient matrices A and B [equations (36) and (37)], calculatesthe product Inv(A)B with an LU decomposition, and advances the circuit vector followingequation (38) to determine a new value for the drive current. Alternatively, the code inter-polates user-supplied tables to find one or more drive currents at the half time step. The mag-netic field values are advanced with equation (30), and then field arrays are shifted, preservingold values for the Dufort-Frank iel algorithm. Finally, new values of the element current den-sity are calculated.

Crunch advances the time and checks whether an automatic step adjustment is required.The final activity in a cycle is to record information. The program supports several typesof diagnostics. A data dump is a record of the properties of all elements at a specifiedtime. Depending on the calculation mode, the dump may contain lists of hydrodynamic,elastic, and magnetic quantities. Crunch has a flexible system to set dump times. The crite-ria, which act in combination, are as follows: a uniform number of elapsed steps, a uniformtime interval, or up to 25 user-specified times. Dump information can be recorded intwo formats: a plot file in binary format passed to the CGraph program or an ASCII listingfile. The listing file also contains results of several global calculations. Hydrodynamic list-ings include total kinetic energy, momentum, and change of internal energy. Magnetic list-ings include total stored magnetic energy and the time integral of ohmic power dissipation.In runs with a drive circuit, Crunch makes files of Rc(t) and Lc{t) that can be ported tocircuit simulation software. The program can make up to 10 history files for specified ele-ments. These files contain detailed records of the time-dependent hydrodynamic and mag-netic quantities.

The CGraph program is a powerful analysis tool to create screen and hardcopy plots fromCrunch output. Figure 8 illustrates the graphical user interface, w hich accesses several menus.The functions of the Main Menu are to review data files, load runs, and transfer to submen us.The Spatial Menu controls plots of spatial variations of any hydrodynam ic or magnetic q uan-tity for a given data du mp . The plots feature autom atic grid selection, zoom capabilities, op-tional material or layer symbols, and superposition capability for any number of plots. Thefeatures apply to both screen and hardcopy plots. The General Menu creates plots similar tospatial plots except that any hydrodynamic or magnetic quantity can be assigned as the inde-pendent and dependent variables. The Time Menu creates plots of quantities in any element asa function of time. Here, CGraph scans the plot file extracting information on the specifiedelement and quantity for all stored data d ump s. The resolution is therefore determined by thenumber of dumps. These plots can be superimposed to show variations of hydrodynamic ormagnetic quantity in two or mo re elements. The History Menu opens any probe file associatedwith the run and displays any of the stored quantities as a function of time in a specifiedinterval. Finally, the Movie Menu controls routines that display spatial plots for different datadumps sequentially to simulate time animations.

9. ExamplesCrunch is used extensively by LANL researchers and undergoes continual testing. Thissection illustrates results with two simple calculations. More complex runs with liner acceler-ation, target collision, and shock conv ergence are described in Hump hries and Ekdahl (1996 ).The first exam ple is a standard hydrody namic test (Clover 1993), the Sedov blast wave prob-lem. A total energy U is deposited instantaneously in a small spherical volume at the center of


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426 S. Humphries, Jr., and C. Ekdahl

CGRAPH 2.6 - Hydrodynanics AnalysisField Precision, Copyright 1997J \\V

\

SPATIAL PLOT INFORMATION

Quantity: DENSITYUnit: (gn/c*3)Tine: 6.06883E-66Dump 6 out of 6X AxisXHin: 4.58816E-S2XHax: 6.29116E-01XTick: l.eeeeeE-eiy Axiswin: e.eweE*eo

Vltax: Z.6249E*eiVTick: s.eeeooE*eeRETURN to continue ...

File: at bench.CPLVariable: SENSITVCurrent dunp: 6Current elenent: 1No hardcopy pending

FIGURE 8. Mo nochro me repro duction of the CG raph screen showing a spatial plot (elemen t density as afunction of position). Status window at bottom right.

a uniform gas distribution with zero temperature and density po. The properties of th e resultingshock can be determined for an ideal gas described by gamma law. Here, the pressure andinternal energy are related to density and temperature byu = CVT,P = (y- \)PCJ. (45)

In equation (45), C v is the specific heat at constant volum e. We constructed a new Sesame table(IGAS0001.SES) taking y = 5/3 and Cv = 8640 J/kg K (hydrogen). Zel'dovich and Raizer(1966) give the position of the shock front as a function of time after energy deposition asr ( 0 = 1-15 )\Po)

{/\ 1/5 2/5 (46)The density at the shock front equals ps = po (y + l) /( y - 1) = 4p o . For po - 0.1 kg/m 3 andU = 617 J, equation (46) predicts a shock front radius of 0.05 m at 5 p.s. Our simulation used90 spherical-shell elements d istributed to a radius of 0.06 m. The initial energy was depositedin a sphere of radius 0.002 m com posed of five elem ents. The spherical m ass of 3.351 X 10~9kG and initial temperature of 2.135 X 10 7 K give an internal energy of u = 1.84 X 1 0 " J/kG .Table 3 lists the input file for this problem. The interesting control statements are DT andSTEPCHANGE. DT sets the short initial time step of 0.25 ns to resolve expansion of the small


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Finite-element MH D simulation code AllTABLE 3. Crunch script for the Sedov blast wave problem

SEDOV BLAST WAVE* - RUN CONTROL -SETMODE: Hydro

GEOMETRY: SphereINTERPTYPE PolyTMAX: 5.01E-6DT: 0.25E-9STEPCHANGE 0.5E-6 5.OE-9

* - MATERIAL PROPERTIES -SESAME 1 IGAS0099

* - LAYERS -LAYER 1 1 0.0000 0.002 5SETINIT 1 0.1000 2.135E7VISCOSITY 1 2.50LAYER 2 1 0.0020 0.060 90SETINIT 2 0.1000 0.00VISCOSITY 2 2.50

* - DIAGNOSTICS -MAKEDIAG 5.0E-6NHIST 1HISTORY 70*

ENDFILE

heated sphere. The STEPCHANGE function raises A/ to 5.0 ns at 0.5 /AS, after the shock hasexpanded significantly. The LAYER commands set up two layers for the heated sphere and thequiescent gas, both associated with Material Table 1. The DIAGN OSTIC com mands call for asingle spatial dump at 5.0 /AS and a detailed history at Element 70 (initial radius r = 0.0436 m)with a record at each time step. The plot of p{r) at 5.0 /AS in figure 9 is direct output fromCGraph. The symbols show the location of element centroids. The advantage of the element-based approach is apparentelements are concentrated near the shock front. The Crunch re-sult is in excellent agreement with theory for the position of shock front and the magnitude ofthe density peak.

The second exam ple illustrates a nonlinear magnetic diffusion problem w ith multiple cur-rent segments. An exploding copper wire of radius 0.5 mm is located inside a copper returncylinder. The wire is defined as Layer 1. Layer 3 is the return conductor, a coaxial shell withinner diameter 1 mm and thickness 1 mm. The conductivities of the wire and return conductorare given by a table from CONDUCT. LI B . Both layers have 25 elem ents. Layer 2 is the single-element void separating the conductors. The following comm ands control the magnetic solution.Current 1 CDAMP 1.0E-6 0.5E6Current 2 CDAMP 1.0E-6 -0.5E6SetB 25 1SetB 26 1SetB 51 2

The first command defines Current Table 1. He re CDAMP. CUR is a normalized critically dam pedpulse. The two adjustment factors give a peak current of 500 kA at 1.0 /AS. The second com-


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4284.WQE-04

5. Hum phries, Jr., and C. EkdahlCCRAPH (Field Precision, 1997) File: sedov.CPL

.8

1 J

rf/

//

1

r (cm) I.0OO0E4OQTine: 5JB82E-IDump No: 1

Y- Ax i s in t erv a l : 5 .C8 8 8 E- 8 5 ( g n /cn3 )X-Axis in te rv al : 1 .8888E+8B cm

FIGURE 9. Simu lation of the Sedov blast wave problem . CGraph plot of density as a function of positionat 5 /JLS. Ideal gas with y = 5/3. Initial density: 10~4 g/cm 3. Initial energy of 617 J deposited in a sphereof radius 0.2 cm . 90 elements. Open circles: heated elements. Open triangles: quiescent elem ents.

mand defines Current Table 2 as the negative of Table 1. The third command associates Table 1with Boundary 25, the outer surface of the wire. The fourth comm and sets the same current atBoundary 26, the inside edge of the return conductor. These specifications remove the voidfrom the magnetic calculations by ensuring that both surfaces have the same enclosed current.The final statement associates Table 2 with Boundary 51, the outer radius of the return con-ductor. The com bined effect of statem ents 2 and 5 is to constrain the net system cu rrent to zero.Figure 10 shows distributions of current density (a) and temperature (b) at 0.8 /is. The distri-bution ofjz in the outer conducto r differs from the exponentia l variation exp ected at low currentbecause of reduced conductivity in the heated region. Figure 10b shows that the temperature onthe inner surface is 1200 K. The diffusion process in the wire is highly nonlinear. The currentdensity is confined to a thin region at an inwardly propagating front by heating and reducedconductivity in the material behind. When the front reaches the axis, the simulation showsrunaway heating and wire explosion.

The C runch code is in its second version. We anticipate future expansions to cover researchrequirements of the High-Energy-Density Physics Program. The next three areas of develop-ment are detonation m odels, detection of conditions for spallation, and radiation transp ort. Theadvantages of the element approach to shock hydrodynamics become more apparent in 2Dmodels. We have applied the conservation equations of Section 2 to conformal triangular m eshesto create the Vogon code. This program handles planar and cylindrical 2D systems. The hy-drodynamic package for Vogon that uses Sesame table equation-of-state data has been com-pleted and extensively tested. We are currently working on coupled m agnetic acceleration for2D liner simulations.


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Finite-element MHD simulation code 429a3.8998Et88

-5.192CE+87

CCRAPB (Field Precision, 1997) File: benchl2.CPL

1/ J_

\

r (cm) 2J08BE-81Time: 7.9999E-07Dump No: 4

r-A xi s in te rv al : 5.BBBBE+B7 (A/cm2)X-Axis in te rv a l: 2.B8BBE-B2 cm

9.3951E+03

.OSeBE+80

CGRAPH (Field Precision, 1997j i T

f

i/J

,

r -\

r |

> File: benchl2.CPL

r

1 i _

r -

r ~ J -*

r (cm) 2.0000E-81Time: 7.9999E-B7Dump No: 4

Y-Axis in te rv al : lJBB8EtB3 (deg-K)X-Axis in te rv al : 2J8BBE-B2 cm

FIGURE 10. Simulation of an exploding wire, CGraph plots. Open circles: elements of the copper wirewith initial radius 0.05 cm (Layer 1). Open squares: elem ents of the coppe r return conductor with initialinner radius 0.10 cm and outer radius 0.20 cm (Layer 3). The void element (Layer 2) is not plotted, (a)Current density as a function of radius at 0.8 /JLS. (b) Temperature as a function of radius at 0.8 (is.


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430 5. Humphries, Jr., and C. EkdahlAcknowledgments

We would like to thank Bard Bennett for preparing an ASCII version of the Sesame tablesand Robert Reinovsky for suggesting and testing the pulsed-power circuit model. This workwas supported under LANL Subcontract C50430017-35.

R E F E R E N C E SABDALLAH, J. et al. 1980 Los Alamos National Laboratory Report LA-8209.ALIKHANOV, S.G. & KONKASHBAEV, I.K. 1973 Nucl. Fusion 14, 3.BENNET, B.I. et al. 1978 Los Alamos National Laboratory Report LA-7130.CHERNYCHEV, V.K. et al. 1987 Megagauss Technology and Pulsed Power Applications, C M . Fowler

et al., eds. (Plenum Press, New York).CLOVER, M.R. 1993 Los Alamos National Laboratory Report X6:MRC-93-126 (unpublished).COURANT, R. & FRIEDRICHS, K.O. 1991 Supersonic Flow and Shock Waves (Springer-Verlag, NewYork), p. 138.DUFORT, E.C. & FRANKIEL, S.P. 1953 Math. Tables and Other Aids to Comp. 1, 135.EKDAHL, C. & HUMPHRIES S. 1998 In Seventh Int. Conf. on Megagauss Magnetic Field Generation andRelated Topics (Sarov, Russia) (in press).FUNG, C. 1965 Foundations of Solid Mechanics (Prentice-Hall, Englewood Cliffs, NJ).HOCKADAY, M.P. et al. 1998 In Proc. of 10th IEEE Int. Pulsed Power Conf. (in press).HUDDLESTON, J.V. 1961 Introduction to Engineering Mechanics (Addison-Wesley, Reading, MA).HUMPHRIES, S. 1997 Field Solutions on Computers (CRC Press, Boca Raton, FL), Chap. 12.HUMPHRIES, S. & E K D A H L , C. 1996 IEEE Trans. Plasma Sci. 24, 1334.LEE, Y.T. & M O R E , R.M. 1984 Phys. Fluids 27, 1273.LINDEMUTH, LA. 1985 J. Appl. Phys. 57, 4447.LYON, S.P. & JOHNSON, J.D. (eds.) 1992 Los Alamos National Laboratory Report LA-UR-92-3407.N E U M A N N , J. & RICHTMYER, R.D. 1950 J. Appl. Phys. 21, 232.PARKER J. 1993 A Primer on Liner Implosions (Los Alamos National Laboratory, Los Alamos, NM)(unpublished).PARSONS, W.M. et al. 1997 IEEE Trans. Plasma Sci. 25, 205.POTTER, D. 1973 Computational Physics (Wiley, New York), Chap. 9.RICHTMYER, R.D. & M O R T O N , K.W. 1967 Difference Methods for Initial-Value Problems, 2nd ed.(Interscience, New Y ork).RINKER, G.A. 1988 Phys. Rev. A 37, 1284.SHERWOOD, A.R. et al. 1980 Megagauss Physics and Technology, P.J. Turchi, ed. (Plenum Press, NewYork), p. 391.SPITZER, L. & H A R M , R. 1953 Phys. Rev. 89, 977.STEINBERG, D.J. 1996 Lawrence Livermore Laboratory Report UCRL-MA-106439.STRUVE, K.W. et al. 1998 In Proc. of 1997 Pulsed Power Conf. (in press).TUCKER, T.J. & TOTH, R.P. 1975 Sandia National Laboratories Report SAND-75-0042.TURCHI, P.J. et al. 1980 Megagauss Physics and Technology, P.J. Turch i, ed. (Plenu m Press, New Y ork),p. 375.ZEL'DOVICH, YA . B. & RAIZER, YU. P. 1966 Physics of Shock Waves and High-Temperature Hydro-dynamic Phenomena (Academic Press, New York), p. 93 .

finite-element simulation code for high-power

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