Jun 3, 1990 - exact conservation laws are found to exist for all physically relevant ... that SOH conserves angular momentum centered at the zone, but not at ... In many fonnulations, space is also discrctized into ... upon a physical principle, such as conservationof angular momentum,would be preferable to one based on.
VCRL- JC-105926 PREPRINT
CONSERVATION OF ENERGY, MOMENTUM, AND ANGULAR MOMENTUM IN LAGRANGIAN STAGGERED-GRID HYDRODYNAMICS
Donald E. Burton
This paper was prepared for submittal to THE NUCLEAR EXPLOSIVES CODE DEVELOPERS' CONFERENCE Monterey, California NoveDlber6-9,199O NoveDlber 16, 1990
This is ~ preprint of ~ p~per intended for public~tion in ~ joum~1 or proceedings. Since ch~nges nuy be m~de before publication, this preprint is m~de ~vail~ble with the undersunding th~t it will not be cited or reproduced without the permission of the author.
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DISCLAIMER This documenl wa.. prepared a.. an accounl or work "po_'red by an allency or Ihe lJniled Seal"" Gowernmenl. Neilher Ihe Uniled Sial"" Gowernmenl nor Ihe UniwersilY or Calirornia nor any or Iheir employees. makes any warranly. e!(pres.s or implied. or usumes an~' lelealliabililY or r""ponsihilil)' ror Ihe accuracy. compleleness, or uwrul. n"".. or any inrormalion. apparatu... produci. or proces.'! disclosed. or represenls Ihal il.. uw uld nol inrrinICe privalely owned riIChlS. Rererence herein 10 any specilic commercial prodUCIs, prllCe!Os.or ""nice by Irade name. Irademark. manuraclurer. or olherwi"". does nol nec""sarily conslilule or imply it.. endorsemenl. recommendalion. or raworinll hy Ihe Uniled Siaies Gowernmenl or Ihe lIniwersilY or Calirornia. The vie"'" and opinion.'! or aulhors e"'pre..sed herein do nol nec"""arily slale or feneel Ihow or Ihe Uniled Sial"" Gowernmenl Of Ihe UniwersilY or Calirornia. and shall not he used ror adwerlisinll or product endorwmenl purposes.
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CONSERVATION OF ENERGY, MOMENTUM, AND ANGULAR MOMENTUM IN LAGRANGIAN STAGGERED-GRID HYDRODYNAMICS
DonaldE. Bunon LawrenceLivennoreNationalLaboratory Livermore,California ABSTRACT The conservation properties for general formulations of Lagrangian ,staggered-grid hydrodynamics (SOH) are investigated in the small timestep limiL Fonn-preserving deftnitions and exact conservation laws are found to exist for all physically relevant quantities (mass, linear momentum, total energy, and angular momentum). The analysis is multi-dimensional and applies to arbitrarily connected grids. The momentum equation is found to lead to an exactly conserved energy current between internal and kinetic energy ftelds. The energy current is not properly calculated by SOH methods in which the wode is differenced independently of the differenced momentum equation. This leads to the commonly observed lack of energy conservation in SOH methods. It is also shown that SOH conserves angular momentum centered at the zone, but not at the node. These angular momentum results apply not only to SGH but to any method using a lumped mass approximation in the acceleration equation. This conservation accounts for the empirical observation in SOH that the motion of the zone centers is free from the spurious vorticity seen in the nodes. Consequently, vorticity can be calculated directly from the zone velocities and need not be separately integrated. This leads to an improved method for removing the spurious vorticity called the "Damped Excess Vorticity" (DEV) method.
INTRODUCTION In 1950, von Neumann and Richtmyer1 published a spatially and temporally staggered differencing scheme (VNR) for the hydrodynamics equations. Their method remains widely used today, and there are many variations on the VNR method which we will generically term staggered-grid hydrodynamics (SOH)2-6. In SOH, space is discretized into zones fonned by lines drawn between nodes. In many fonnulations, space is also discrctized into nodal control volumes surrounding nodes, fonning a median mesh. Typically, the median mesh is constructed from line segments drawn between the zone centers and the centers of adjacent zone sides as shown dashed in Figure 1. In SOH, the spatial centering of coordinates and velocity are at nodes while mass, internal energy, and pressure are within zones; variables are usually centered temporally at the full time step except for velocity which is at the half. Traditionally, SOH fonnulations are fonnulated to exactly conserve only linear momentum, and the conservation of other quantities is only approximate. We will show that SOH can be made to exactly conserve several additional quantities. Trolio and Trigger7.8pointed out in 1961 that the VNR method was not exactly energy-conserving and proposed conservative implicit and explicit methods for the one-dimensional equations. These methods retained the spatial staggering of VNR but relinquished the temporal staggering. Extensions of the TroHo and Trigger methods to higher dimensions do not appear to have been widely disseminated. It is also well known that SOH suffers from spurious vorticity on the scale of the mesh size because of degrees of freedom unconstrained by the difference equations which are perturbed by discretization approximations. There exist several ad hoc schemes successful in reducing "hourglass" distortion (see Figure 2) without affecting physical "shear" modes2.9. These schemes are not effective against spurious Figure 1. Zone (solid). "chevron" distortion because they consider only defonnation of single zones. node (dashed),and comer Burton10 devised a method which is effective against both hourglass and chevron
(shaded)controlvolumes.
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modes without simultaneouslydamping physical "bending" modes. This method made use of the empirical observationthat the motionof the zonecentersis free fromthe spuriousvorticityseenin the nodes. A schemebased upon a physical principle, such as conservation of angular momentum,would be preferable to one based on empiricism. Angularmomentumhasbeenlargelyignoredin publisheddiscussionsof SOHdifferencing. APPROACH
J
We have reexamined conservation laws in Lagrangian SGH by using a control volume approach in which the difference equations are expressed in conservation form. In this fonn, the time evolution of a numerical quantity Q is equal to the spatial integral of flux q about some control volume boundary. The collection of control volumes must span all space. but different quantities can have different sets of control volumes. When discretized, the integral becomes a sum of currents J =S. q,
EPEfj
ES8~ Figure 2. Modes of deformation: (a) hourglass, (b) chevron, (c) bending, (d) shear. Spurious modes are (a) and (b).
where S is a surface area. For the form to be conservative, the currents entering the control volume must have opposite sign to those exiting an adjacent control volume, so that the net current at the boundary sums to zero. In interpreting SGH, we have found it necessary to extend these notions to encompass some nonlocality in the boundary current, permitting conservative interactions with physically nonadjacent control volumes. We will refer to this as a nonloeal conservation form. Nonlocal conservation has a larger volume of interaction and therefore a lower spatial order of integIjltion than a local form, but is still fully conservative. In SOH, the control volumes and currents are determined by the traditional differencing. Instead of postulating Q and checking it against the conservation equation, our procedure will be to define Q to be that quantity which appears in the conservation equation. If we can then show that the boundary currents sum to zero, the numerical quantity Q is exactly conserved by the scheme. Fmally, we must establish whether the numerical quantity Q and its numerical time integral Q bear any relation to the physical quantities of interest This is by no means guaranteed. If the forms of these numerical quantities correspond to our physical intuition, they are good numerical analogs of the physical system. A numerical quantity is further said to beform-preserving7 when its functional dependence matches that of the physical variable. The concept of fonn preservation is especially important because many physical quantities are functionally related: velocity, linear momentum, kinetic energy, and angular momentum. However, they can be numerically evaluated from independent integrals which do not automatically preserve the functional relationships. For form-preserving quantities, separate numerical integration is unnecessary, but may still be useful as an independent check of the conservation properties of the numerical scheme. Let us introduce the notion of a corner which is that portion of space linking a zone with a node. typically the intersection of the zone and nodal control volumes as shown in Figure 1. Comers are characterized by the set of indices {c, z, n} which identify respectively the comer, zone, and node. Once an extensive quantity is defined at one location, it can be conservatively mapped to the comers of the control volume and then to other control volumes, so that the comer can be made to play the role of a conservative accounting device. Although it is not particularly important that the comer quantities themselves be fonn-preserving or satisfy conservative evolution equations, most do. As we are concerned primarily with spatial differencing, the timestep will be assumed to be infinitesimal For finite timestep, energy conservation is especially sensitive to the details of the time centering. For a discussion of the consequences of finite timestep as well as special circumstances in axisymmetric systems, see Reference 11.
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MASS AND MASS FRACTIONS In SOH, mass is fundamentally defined in terms of the initial zone volume and density. Mass fractions are
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weightsused to derme comer and nodalmassesin teons of the zonemass. The mass fraction«Pc==Me/Mz is also usedto conservativelydisbibuteextensivequantitiesfromzone to comer,and 'I' e ==Me/Mf1' from node to comer. The fractions can be defined in a number of ways providing the foUowingconditions are met: Z
f1
I,«Pe= I, 'l'e= 1 , e e Z
f1
The notation Land e
Le
means to sum over those comers c contained in zone z and surrounding node n
respectively. In terms of the mass fractions, the various masses are
=«PeMz = 'I' , Mf1 a LMe =L«PeMz , e e Me
eMf1
f1
z Mz
= LMe e
f1
z
= L'I'eMf1 e
.
Since the volume of each comer is independently specified by the definition of the median mesh, the density of material can in principle vary between comers in a zone. We will limit the discussion to fully Lagrangian systems. Since the mass fractions are assumed constant, aU mass will be conserved providing the zone masses are held constant,
FORCE AND MOMENTUM In the case of momenb.1m,the conserved currents are forces J =Se · (J z = fe acting on nodes due to the zonal stress tensor (Jz as shown in Figure 1. Although the forces can be applied anywhere along the nodal control volume boundary, we assume them to be applied at the zone center, which is a point on the boundary to be defined later. The surface area Se has a contribution from two segments of the median mesh. Without violating our fundamental assumptions, a higher-order scheme can be easily constructed by calculating separate forces for each segment of the median mesh. In the absence of body forces, we note that Newton's third law is satisfied by forces within the zone, so that the currents passing through the zone from one nodal control volume to another are conserved z
Le fe =0 . The evolution of nodal momentum is calculated from Newton's second law with the total force on a node due to surrounding zones calculated as follows. By conservatively disbibuting the force currents by nodal mass fraction, the second law is also satisfied for comers and zones,
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n
Fc =='I'cFn='I'cLrc" c' z Fz a LFc
c
z
z
.
n
=L'I'cFn =L'I'cLrc" c c c'
Pz a Fz
z
z.
z.
=L'I'cFn=L'I'cPn= LPc . e e e
In the zone equations, the forces re' are actually applied at the surface of an extended zone formed by the median meshpassingthroughthe centersof adjacentzonesas shownin Figure 3. The effect of the 'lie is to conservatively divide the nonlocal comer currents amongst the zones with which they interacL Because forces rc sum to zero at the interfaces of the nodal control volumes, the equation for the nodal momentum is in conservation form. Those for the zone are in nonlocal conservation form because the cwrents sum to zero at the extended zone boundary instead of the true zone boundary. The acceleration equations for the zone are of lower spatial order than those for the node.
S
ACCELERATION AND INTEGRALS The evolution equations for velocity are calculated from the respective masses and momenta.
r- ---- _Ir ,.
I I
I.
. I
'I'c
. Rz
I
L
.
I
I 8----- - .I The time integrated velocity, position, and momenta are given as follows, are always expressible in terms of the motion of the nodal coordinates,and are form-preserving. At this point. we will define the point'z to be the zonecenJer.
I , Idt lic = z
I
'n ==': + dt Un
Un == dt lin Uc ==
Un
I
Uz == dt liz
'c ==': +
,
= L
