,' . IDNA-TR-88-286

AD-A211 728

SOME NEW ALGORITHMS FOR BOUNDARY VALUE PROBLEMSIN SMOOTH PARTICLE HYDRODYNAMICS

P. M. CampbellMission Research Corporation1720 Randolph Road, SEAlbuquerque, NM 87106-4245

6 June 1989

Technical Report

CONTRACT No. DNA 001-88-C-0079

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~ A r ede r ivation of the basic equat i ons of smoothed particle hydrodynamics i s g iven tha t removes some o( the intui t'i ve e lement s in ea t·lier work. New algor i thms .1 1· e ob t ai ne d for hydrodynami cs lvith hea t flow tn-:1t-..:include bounda r y t e rms . These a l gorithms pe rmit appl ication of the method to sys t e ms driven rv e xterna l pressures or heat sources . Hydrodynami · work terms a r e J iffer ent in the ne\v algor ithms and these terms may e limina te some difficulties encounte r ed with expressions (ounJ i n the 1 ite r a tor-e . The met hod developed for thermal diffusion y ields the fl uxes direc tly a nd may be ext e nded to a f lux limi ted dif f usion formulation of r adiation t rans port. /( rl,l}. (-~, .. , '

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PREFACE

The author thanks Robert F. Stellingwerf for introducing him to the SIImethod and for many enlightening discussions. This work was supported under Con-tract DNA 001-88-C-0079.

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TABLE OF CONTENTS

Sect ion Page

PREFACE iii

CONVERSION TABLE iv

1 INTRODUCTION 1

2 KERNEL ESTIMATES NEAR BOUNDARIES 2

3 MOMENTUM EQUATION WITH BOUNDARY PRESSURE 5

4 EQUIVALENT FINITE DIFFERENCE FORMS 10

5 WORK TERMS IN THE ENERGY EQUATION 15

6 HEAT CONDUCTION TERMS IN THE ENERGYEQUATION 19

7 CRITIQUE OF BROOKSHAW'S METHOD OF CALCU-

LATING THERMAL DIFFUSION 22

8 CRITIQUE OF SOME MANIPULATIONS IN THE

LITERATURE 26

9 EXTENSION OF HIGHER DIMENSIONS 28

10 SUMMARY OF NEW SPH ALGORITHMS 34

11 LIST OF REFERENCES 36

SECTION 1

INTRODUCTION

Although particle methods in hydrodynamics have been remarkably suc-

cessful in many problems, 1- 4 current methods suffer from an inadequate treatmentof boundary conditions. This is particularly evident when one tries to include heattransport in the energy equation.5 -6 Without accurate values of the flux at the bound-aries, it is not possible to calculate the net exchange of energy between a fluid and itsenvironment.

Another class of problems requiring accurate treatment of boundary condi-tions is illustrated by the shock tube problem where the shock is driven by a piston.

In this case the particle method must be able to accommodate an externally-applied (boundary) pressure. The method should also be able to handle conductionheating of the gas by a hot wall or piston; and, finally, boundaries must be treate(laccurately to include radiative heat exchange, in particular, radiative cooling.

In this paper, some new algorithms are developed for smooth particle hy-(Iro(lynlamics in problems where external boundary conditions are imposed. Althouigihthermal and radiative diffusion are included, radiation transport is neglected in thepresernt paper.

U WIII nllllll m l

SECTION 2

KERNEL ESTIMATES NEAR BOUNDARIES

The standard kernel estimate of a function is given by 3

ff,(x) = f(x')W(x - x', h)dx', (I

where the kernel is normalized by

XJim W(x- x',h) dx' 1. (2)h -,o f

In these expressions the domain of definition is 0 < x < X.

As long as h -* 0 one would not expect normalization problems near theboundary, but ini practical applications kernels with finite h are used. One would thenexpect difficulties within x - h of the boundary because the integrals are truncated bythe boundary. For example

lir W(x - x')dx' (Z- X f 2

which is obvious from Fig. 1.

Consider what this does to the kernel estimate of the density. Assume a denseconstant spacing of particles in the vicinity of the boundary X. Far from the wall, thekernel estimate gives

pW) jp (x') W(x - x', h) dx'

Srn W(x- - ,h)

P I

2

x x

2h 2h 2h h -_

Figure 1. A typical smoothing kernel, showing how the

symmetrical form is truncated as the particle ap-proaches a boundary.

where pC is the constant value. However, following Eq. 3, as x -* X, the kernel estirate

becornes

p(X) lim p(x')W(x - x',h)dx'

- m W(X-xh)

1PC (5)

So within -- . of the boundary the interpolation feature of the kernel estiniate giv's

spurious resuls. Also, presumably, one finds

P(X) lirn P(x')V, (x - x')dx'

2

for the case of a uniform pressure in the neighborliood of the boundary as in Fig. 2.

3

-PC

1/2 PC

h X

Figure 2. The falloff in a uniform pressure distribution asgiven by a kernel estimate in the neighborhoodof the boundary.

In summary, we have seen that because of the truncation of the normalizationintegral within -- h of a boundary, interpolation implied by the kernel estimate rTnaynot be accurate. Renormalization is a possibility but does not appear to work. Also.the boundary value method presented below seems to require the apparent failuremnanifested in Eq. 3.

4

SECTION 3

MOMENTUM EQUATION WITH BOUNDARYPRESSURE

All treatments of SPH of which the author is aware require integration byparts when deriving kernel estimates but simply drop the boundary terms. (See Section8 below.) This is presumably because the kernel is supposed to mimic a 6-function

which goes to zero sufficiently far from the particles representing the edge of the system.

However, if we are to consider problems where the particles interact with aboundary condition, such as an externally-applied pressure, we must specifically includethe boundary terms. In this section the two procedures developed by Monaghan forthe momentum equation 2' are modified to include boundary pressure.

In the first case, write the momentum equation as

dv 1 9Pdt p dx

-4( a. (7)

The kernel estimate of this equation is formed by changing the independent variableto x', multiplying by W(x - x')dx' and integrating over the domain 0 < x < X. Oneo)t.ains

dv, fX a !'j( z)x

dt , (' P)

-- (P) f , W-(x - x')d,()

where the subscrip,. . ienotes "kernel estimate." The second integral has been lin-carized by eval,iving (P/p") by its kernel estimate and taking it outside the integralsign. This is justified since W(x -- x') acts like 6(x - x').

5

After integrating by parts, one finds

dv, - [PW ( j ' x (P) aW(x- 'ddt p 10 P a

())lWxo '- ( P) jX paW (xX') dX', (9)P oP

where we have used dW/drzx -OW/ox' assuming a symmetric kernel.

If we now evaluate the integrals by the particle method, replacing

f f(x')W(x - z') dz' fj W (x - . , (10)i Pi

we obtain an expression for the momentum equation with external boundary pressures,

dto -+ p W (x - o)

2 2 ± W- X)

Thus, particle i feels the effect of the boundary pressure if it is within range of theboundary and W(xi - o) ) 0. The boundary pressure enters in a term similar to thatfor any other particle j.

Does this expression conserve momentum? If we multiply by m, and sum.we find

6

d5 re, v, Po m, mI ) + il W(x, - o)

-/P /"\ dm, +~PX Mi X + ilWX

-ZZ j 2~m 2 (9 Xj (12)• j (Pj (Pil 0x

The last term in the equation vanishes because OW/ax, is antisymmetric in i andj. To get a form where P, and Pj appear symmetrically was the main point in themanipulation of P and p in Eq. 7. This insures the vanishing of the last term in Eq. 12which is necessary for conservation.

Now what about the boundary terms? From Eq. 3, we have

1

m,W(x, - o) = I , (13)i2

so that Eq. 12 becomes

d 1E- MiV, -Po+0 Oop0 M 2W (x1 ,-o)

dt 2 P

1P - Px Mi f )W(xt-X). (14)~ ~ P P -, \/-

Here P, and Px are the given boundary values of the external pressure. The otherterms are the interpolated values of the pressure at the boundary given by the internalcalculation. If this calculation is accurate (apart from the factor of 1/2), it must yield

1 ( ;j )0i i

and

7

Px - px 1: m, W(x, - X), (16)i (Pi

giving for Eq. 12

d mivi Po - Px (17)dt

A similar analysis is possible for Monaghan's other conservative form of themomentum equation. First write

dv 1 OP p11 2 O9p/2- 2 (18)dt p ax p ax

Then the kernel estimate of the momentum equation becomes

d = -2 p) 10 ax' W(x - x')dx'

= -2 p1/ -)p12W x

-2 (p) / 2-- -- (X - x')dx' . (19)

Evaluating the integral by the particle method gives

dv, - 2 (F)2 [p 0 '/'W(Xi - o) -pXk'2W(Xi _ X)]

- 2W(Xi Xi) (20)) PiP, Oxi

where, again, we note the fortuitous appearance of the factor of 2 in the bounldaryterms which will compensate the falloff represented by Eq. 3.

8

After multiplying by m and summing, we obtain

d E-nivj = 2P,/ 2 m pm, W(X, - o)

ii P )

- 2 E E mimjP ,' 2 W (x-x) (21)y PiPj 09xi

where because of the symmetry in i and j, the last term is zero; and since

1 p,/2 - i Em, /2 f~ W(X, -o0)2 P 1i

2m, ("p1W(X, - X) , (22)

we get the conservation of momentum,

d-E Mv = P - Px (23)

It is curious that in both of these forms the boundary terms enter in such away that the factor of 1/2 from the normalization anomaly at the boundary is requiredfor a conservative momentum equation. This is not true of all forms, however. Forexample, the first expression for the pressure gradient in Eq. 7 does not produce aboundary term with the factor of 2 present.

9

SECTION 4

EQUIVALENT FINITE DIFFERENCE FORMS

If particles are assumed equally spaced, it is possible to derive the equivalentfinite difference form which the kernel estimate represents.6 It is instructive to look atthese forms, particularly in the neighborhood of the boundary, to get a feeling for theaccuracy of the particle method.

For the purpose of illustration, consider the kernel shown together with itsderivative in Fig. 3,26

1(2 u2 I 0 < lul <

W(x-' (2- u) 3 < u 2 (21)6h( Iul

0 2 < Jul

where u (x - x')/h, and

w = law (25)

Ox h Ou

Now, consider a series of equally spaced equal mass particles far from anyboundary as shown in Fig. 4. The momentum equation in the form of Eq. 11 is

10

W

h 2h

aw/a x

Figure 3. The smoothing kernel given by Eq. 24 and its(continuous) first derivative.

1l

dv, ME +_________ -xj

l rn[( )+( )] (2l ) (l6)

R P2i+1 + il

since OV ' 1/Dx, is zero -xcept for j =I - 1 and j i + I as can be seen from Fig.3.From Eqs. 24-25 we have

-1

19W (x, - Xj) 0 (27)

Oxi

which leads to Eq. 26 above. This simplifies to the central difference formula

dt 2h i+l i-

where we have used rn/p = h for equally spaced particles.

Let us now consider the case where the boundary falls within h to the right ofparticle i. In this case, the j = i + 1 term is missing but the boundary term is present.Specifically, let us assume the boundary X occurs -- 0.6h to the right of particle i whereW(x, - X) - 1/2h. Eq. 11 then becomes

dv[dt P xK (L)x + (M il Px()

M m [(p 2h12 (As (29)

12

i-2 i-I i i+1 i+2

(a)

Figure 4(a). A series of equally-spaced particles far from a boundary.

p ._i Pi P --

2h

i-I i _

(b)

Figure 4(b). Equivalent difference form for the pressure gradient in theneighborhood of the boundary.

13

since W(x1 - X) = 1/2h and the j = i+1 term is missing. Using h = m/p, this reduces

to

dv, (310[(

dt 2h x (0i-1

which is a fairly good approximation. As shown in Fig. 4 it gives a central differenceforin based on Px situated 2h from Pi-1 , although it was assumed that the boundarywas -- 1.6h from xi-1. All finite difference algorithms are uncertain to within Ax = h,so these boundary terms are probably as good as one would expect of most differencemethods.

14

SECTION 5

WORK TERMS IN THE ENERGY EQUATION

We now consider the work terms which appear in the energy equation in thepresence of boundary pressures. In order to insure conservation, this term must beexpressed in a form that is complimentary to the right-hand side of the momentumequation so that the kinetic energy appears correctly.

We write the work terms as

d P Ov

dt p ax

- a + V a a

where c is the internal energy per unit mass. Taking the kernel estimate of Eq. 31yields

dc ~ ~ W~ -xa P x')dx'

+ Vj~ ~ W(x - x')dx' ,(32)

where the second term has been linearized as before. This term is now rewritten usingthe kernel estimate for the momentum equation,

dt, Xa I W (x x)dx - ( P),] f X -a 'V (X - X')(dir' . (33)dt (),1

15

If we multiply by v, we get

vKj~ ~ W(x -x')dx'=-(m,)fX ,W(x - x')dx', (3)

which gives for the energy equation

- () x°w(- x)dx' (37)

Next, integrating by parts we obtain

& , dv,, [ W[ -

dt + Rt - p 0o P ax

-- (PV X)p-a (x - x')dx' (36)

When this expression is evaluated by the particle method, we get the energy

equation expressed in terms of total energy,

di + di -12+ = o)V- [ ) 2 + 2 ),] xW(, - X)

- Vm + (k)] aw(x, -x,) (37)

16

We can verify that Eq. 37 insures conservation if we multiply by 7n, and sum,

d12) 1 (PV" _____-

I I P(iPi Ps

(Pv)x- PxE miP2 S i POi

-F ZPm)m, a VV W(X-~ (38)

The left-hand side is the time rate of change of total energy, internal plus kinetic. Thedouble sum on the right is zero because of the antisymmetry of OW,/Ox,. And asbefore, we have

Po mi -PW - 0Pi Pi 2

Px _m, (Pv W X -1 (Pv)x , (1 (,))

giving

d 7(f1 = (PV) - (Pv)x()

dt 2vi

The energy Fq. 37 is expressed in terms of the total energy. After the parliclesare imoved using the momentum equation, the change in kinetic energy is known. ThenEq. 37 can be solved f - r the internal energy (,. In sone cases, it may be desirable tohave the energy equation expressed in terms of the internal energy alone.

17

The explicit appearance of the kinetic energy term can be eliminated usingthe momentum Eq. 11,

v( v + poW(xi - o)

- ( v ( PXW(X - X)x P

-Zm, + 8W,+ 1-ox,)()j () j Vii P ' wx j

After subtraction, this leads to a form of the energy equation where the kinetic energydoes not appear explicitly,

[V0

dt [VX - (j W(X - X)

- > m, [Vi - d,[ OX, (42)

In summary, we have developed a conservative expression, Eq. 37, for thework done on a fluid by pressure forces that is complementary to the momentum Eq. IIin the presence of externally-applied (boundary) pressures. These two equations insureIhe, conservation of momentum and energy.

18

SECTION 6

HEAT CONDUCTION TERMS IN THE ENERGYEQUATION

Let us now consider the addition of thermal heat conduction terms in theenergy equation. This will be especially important for ICF applications. Neglectingother terms for the moment, the energy equation becomes

dc 1a ( _

dt pax (KaxIlaFp ax(,a

where K is the thermal conductivity, T the temperature and F the heat flux. Weshall approach this equation in the same way as the momentum equation and obtain acompletely parallel formulation.

Following Section 3, we write

d a (aF\ FOpdt ax p : plx

Which has the following kernel estimate

d(k a ( W(x x')dx'

dt Z, a X' \p

( f x ap W(x - x')d x' (.15)

After integration by parts, one obtains

19

-f(x - x')dx'dt - 10 ki a

(F [pW(Xx,

(F) LxpaW(x-x)dx, (1I6)

which in particle form becomes

- [(D + ()] pxW(x, -d--F : + (F]PeW(xi _ )

o 2

- _m (F ) + ( F )] OW(xi ) (47

P i P (x

It is clear that energy conservation follows in the same way as demonstrated above.

Thus, in Eq. 47 we have the energy equation with heat conduction in. aform completely analogous to the momentum equation, Eq. 11. The boundary valuesfor the fluxes, F, and Fx appear in a similar way to the boundary pressures. Andsince experience has proven Eq. 11 to be a successful particle form of the momentumequation, one would expect similar results for Eq. 47.

There remains only to calculate the fluxes for use in Eq. 47. We write F inthe form

F 10T

0- (T) K-T apo- .=- K~ (-8)

which, by a now-familiar procedure, leads to the particle equation,

20

(~i) K, + poW(X, - o)

- [i (T\x ('T)]\xiX

- K m + W(x - X)(49)

S 2 i P2axi

So the temperatures define the fluxes through Eq. 49 and the fluxes determine theenergy transport through Eq. 47.

It is possible to develop an equivalent treatment of heat conduction basedon the product of square root approach used in Eq. 18, but this is unlikely to be anybetter than the method already given. Other approaches have been proposed for heatconduction (which neglect boundary terms, however). The method of Brookshaw isdiscussed in the following section, and a variation of Brookshaw's method by Monaghangives similar results in test calculations.5

21

SECTION 7

CRITIQUE OF BROOKSHAW'S METHOD OFCALCULATING THERMAL DIFFUSION

In a 1985 paper,6 Brookshaw discusses several approaches to adding radiativeheat diffusion to SPII. He describes Lucy's early attempt 7 at solving the energy equation

dS aFdt ax

F 4acT' i9T3Kp ax' (50)

by using the particle method

( T d S ) . = _ m F j ' W x idt ,j P 0x

F = _( 4acT 3\i (T) aW(x -x,) (13Kp )mj- Px (51)X

It is stated that since W(x, - xj) --+ 0 at the boundary of the fluid, T - 0,and one has an insulating boundary condition. The statement is made, "...so there isno reason to assume that the SPH equation will automatically take into account thecorrect boundary condition."

It is easy to see where Lucy and Brookshaw go wrong here. In order to get theparticle equations, Eq. 51, they had to integrate by parts and drop the very boundaryterms they are so concerned about. For example

IT -I -W(x -x')dx'dt j a x'

, x, aW

-FW(x - x')Io - 10 F dx (x - x')dx', (52)

22

which becomes in particle form

(dS) FpT- FW(x,-o)-FxW(x, -X)

-m (F OW x- Xj) (5)P i x

Even though boundary terms are present here, this is not a good form touse, since it does not guarantee conservation. It is better to use the method developedin Section 6. Test calculations showed Eq. 51 was inaccurate near the boundary andalso the method was sensitive to particle positions. Brookshaw presented a differentapproach to calculating heat diffusion based on a difference algorithm that he claimedgives much smoother results than Eq. 51. The method given in Section 6 should beaccurate near the boundaries, but it may be that Brookshaw's approach would giveadded smoothness. For this reason, we shall attempt to derive Brookshaw's algorithmwith boundary terms included.

Write the energy equation as

dc a a) 154)P dt -~ a(K -O9x]

and form the kernel estimate

PKd K-: Jx 'IK 9}IWx -x'dxI, (55)

where curly brackets are used to indicate a special treatment for that particular term.Integrating by parts yields

P, K- w(x - X') + f ( (- X')dx'. (56)

23

Now, Brookshaw appeals to a Taylor series expansion to write a differenceform for the term in curly brackets.

Since

K(x') K(x) - (X- X') OKax

aT I12T =x) x - (x- x')- + (X- x -)'- + ..,(57)

we can write

K - - [K(x) + K(x')] T(x) - T(x')axT OK T (

2K(x) aT _(X - XI) K(x)--a2T + aK aTJ + (58)

The kernel estimate, Eq. 56, becomes

dE, {K cI} W(X -X')]X-~

fX T(x) - T(X') _(+ f tK(x) + K(z')] x' W(x - x')dx' (59)x ax

where Brookshaw only discusses the integral and assumes everything goes to zero atthe boundaries.

It is interesting to see something like 2 k under the integral sigl,. To seehow the difference form gives the intended heat conduction term, substitute Eq. 58into Eq. 59

dc T a a7') 1

'd-- - (X - X) - yK ) + ... W(X - X')

fX~ I _ (x-_ x)~ -a -(9T) + _._(+o 2KI1X.. -(-') K + . W(x - x') dx' .((0)

24

When the integral is performed, term-by-term cancellation gives

K dE- + O(h 2) (61)

which is the result obtained by Brookshaw. Including the boundary terms in theintegration by parts does not change the result given by series expansion.

Brookshaw's scheme with boundary terms can now be written as

df, T(x T( )(x - ))

P" dt [{K(x) + K(x')} T(x)- - 'dt X - X1

+ 1 [K(x) + K(x')] T(x) - T(x') a W(x - x') d' ,(62)oX - X ax ~ 62

which gives in particle form

__ 1 T, -T 1 T,_-Tx_

S K o) [ W(X-)+ K, + KxI - - X W (Xi -x)dt Pi xi - o Xi - X

+ m Y- (KS. + Ki) T" - T7 a W (Xi - xj) .(63)j PiPj xi - xj axi

It is clear that Eq. 63 gives conservation since the summation is antisymmetricin i and j and the necessary factors of 2 are present in the boundary terms. Whetherthis approach to heat conduction is better than that represented by Eqs. 47-49 can onlybe determined by numerical testing. Eq. 63 certainly solves the problem of an insulatingboundary condition, since any particle within 2h of the boundary (which presumablyis on the order of one diffusion length) will exchange energy with the environmenticprcscnted by the temperatures T, and Tx.

25

SECTION 8

CRITIQUE OF SOME MANIPULATIONS IN THELITERATURE

In the foregoing, it has been shown that boundary terms neglected in otherderivations of SPH algorithms2 - 7 are essential for many problems and particularlyfor energy transport problems. The question arises as to how these terms were soconsistently neglected over the last decade.

In Gingold and Monaghan's paper, 4 they approach the particle method asfollows:

Kernel Estimate P,,(x) = f P(x')W(x - x')dx',

Particle Approx. P, = E m, -P 2

The Gradient (= m 'WPxi - x( ) a (64)Ox, P a91

However, if one calculates the pressure gradient directly from the kernel estimate onefinds:

Kernel Esti~mate f~ W W(x - x') dx' ,(X, + / ' W x-x)xIntegration PWaP ( - x')dx'(1P

Particle Approx. 49P PxIV(x, - X) - PoW(x, - o)09x,

+ Zm,( P) aw(x,-x,) (65)j j 26,

26

This shows clearly that one must be careful of taking derivatives throughparticle approximations to kernel estimates. In Monaghan's 1982 paper,3 he mentionsintegration by parts but says he is assuming W, the function, or both go to zero onthe boundary. This is better than what one sees in References 4 and 6, because it ismathematically defined even though unnecessarily restrictive.

The point to be made here is that in dealing with SPH equations, it is saferto carry everything through to final form as an integral equation and only then replacethe integrals with sums over particles.

27

SECTION 9

EXTENSION TO HIGHER DIMENSIONS

Consider first the kernel estimate for the momentum equation, Eq. 7. Invector notation, this becomes

- _f V1 W(iF- /'I,h)d 3 rI

- f)J V'pW(I#F- i'1,h)d3r', (66)

where we assume the kernel is spherically summetric, extendiig over a range R - 2habout the point . The volume of integration V is therefore a sphere of radius I?centered on Fas in Fig. 5. An exception to this is when Fis within range of a boundaryas in Fig. 6. Then integration by parts yields a surface term, where the boundarypressure at the surface influences the particle at point .

Integrating by parts in Eq. 66 gives

dt7 IB (P) W(IF- F'l,h)d§' - f P(~ VW(jF- F, h)d r'

- (w('-F ',h)d§'- (- pVW(IF - F'Kh)d3 r', (6 7)

where the surface integrals are over that area of the boundary intercepted by the non-zero range of W. The particle approximation to the kernel estimate becomes

28

r

0

Figure 5. Range of influence of kernel W(IF- F'I) about thepoint F.

B

(IA

r

FBB 0

Figure 6. Range of influence of W intercepts boundaryplane B over area AS.

29

a<

Figure 7. The area on a plane boundary intercepted by thesphere is a disk of radius a.

d 6 - ) + 4) PBW (lr, - F ,h g

-(4 + (4)] VW(1t - Fjj,h), (68)

where PB and PB are the density and pressure at the boundary averaged over AS,, r13is the normal point at the boundary closest to , and i is the unit vector normal tothe boundary at ASi, all as shown in Fig. 6.

The calculation of ASi is fairly easy if one neglects curvature of the boundary.which is justified in most cases since h is supposed to be small. In three dimensions.

the area intercepted by a sphere of radius R located a distance d = Jr- rBI along thenormal to a plane boundary, as illustrated in Fig. 7, is

AS, = (R -I- )• (69)

In 2D rectangular geometry (x, y) one considers a unit length in z. Hence, the rangeof influence of W is a cylinder of radius R and unit length. In this case, which isillustrated in Fig. 8, the area intercepted by W on a plane boundary is

As, = 2 (R2 - i- frB) /. (70)

30

x

Figure 8. The area on a plane boundary intercepted by the

cylinder of unit length is a rectangle of height 2a.

'his formula does not apply to the situation in 2D cylindrical geometry where, be-cause of the symmetry, a particle is toroidal in shape. There does not appear to be

a simple expression that covers all boundary configurations in this geometry. In theone-dimensional cases discussed earlier, one considers a unit length in each of the two

transverse directions giving AS = 1. Eq. 68 then reduces to the ID expression derivedearlier. The area AS, in these three cases can be written generally as

AS, C(a)(R' - - B12) - ' , (71)

where a =z 1,2,3 is the geometrical order, and C(a) = (1,2, 7r), respectively. It should

be rioted that an exception to Eq. 71 may occur at the corner of intersecting boundaries.

The normalization anomaly discussed in Section 2 is not altered in highergeometries. In this case, one has

lim f W(Ir- 'i, h)d 3 r' (72)F-FB ,- 2 ,(2

and for a uniform distribution of particles in the neighborhood of the boundary,

imW(lB - FI,h) I p, (7:3)

31

Likewise, the boundary value of the pressure or any similar function in the particle approximation would be expressed as

(74)

In order to demonstrate conservation of momentum from Eq. 68, one obtains the total component of momentum in any direction x,

~ d~ A

~m'di ·:t -I

-ps L Fni(Ps~S.. x)W(Iii- fsl, h)

-ps }:m, (~~§. · x) W(lii- fsj,h), • p,

(75)

where the double sum vanishes because of the antisymmetry in i and i. The first term is the particle approximation to one-half of the boundary value of the net force on the system of particles,

1~ .... - L.J(P ~s, · z)s , 2 •

(76)

and by the same arguments presented earlier in Section 3, the second term should be a good approximation to this expression also. Conservation is then expressed as

~ d~ A ~(P"S .... A) L.Jm•y·x = L.J u ,.x B· i t i

(77)

32

By a similar series of steps the work terms in the energy equation of Section 5

are written

de + d ( ' 2 A ,[, (( -i;B1,hdt dt 2VKJ - + (SpW rj 2 Bh

- mi -() + (O) . VW(! - r'I,h), (78)

and conservation of energy follows in a like manner,

d 12 = (PA. ' - (79)

2 i

The heat flux terms in Section 6 are easily generalized to the form.

with the fluxes given by

-- K, + >r (w4) (4)] W ih)(8

(Ti

33

I

SECTION 10

SUMMARY OF NEW SPH ALGORITHMS

A suggested approach to SPH is summarized here that allows boundary valuesfor pressure, temperature and heat flux to be specified. This allows problems to besolved where the fluid is driven by an externally-applied pressure, temperature or both.It also allows the fluid to lose energy to a cooler environment. Although the diffusiontreatment is derived for thermal conduction, radiation which is in equilibrium with thematerial temperature can be handled in the same way.

The momentum equation for particle i is written

di B- + (P)i] PBW(I- rBj,h)ASjdt P

- P) + "

- jZ ( + Mh) VW(ji-F1 1,h) (82)

In this equation PB is the external scalar pressure imposed on the fluid, specified asproblem input. The quantity PB is the boundary value of the density given by

PB = 2 mjW(IF - FBI,h) . (83)

The factor of 2 appears as in Eq. 74 because of the normalization anomaly at theboundary.

From Eq. 82 we see that any particle i within range of the boundary, i.e.,Iri - FBI< 2h, begins to be influenced by the external pressure. This pressure terInenters in a form similar to that of any neighboring particle except W appears in theboundary term rather than VW. This equation is, therefore, different than if onuassumed that a fixed particle represented the boundary with a pressure defined there.

34

The work term which appears in the energy equation is written

dti +dr I -2- F+ 1, "A B(~- h)- dt 2' + (P6)] - ",PBW(Ir,

-( m g + (k)] VW(I - Fj, h). (84)

After the particles are moved using Eq. 82, the change in kinetic energy is known andEq. 84 is used to obtain the change in internal energy. From the new values of p, andci, the temperatures and pressures can be calculated.

If thermal or equilibrium radiation diffusion is present, the following heat fluxterms must be added to the work terms in the energy equation

dt Work Terms - J iPBW(I - FB ,h)

-m, +.VW(JF - F , h). (85)

The heat fluxes F, are defined at each particle position just as the pressures and tem-peratures are. The fluxes are obtained from the temperatures by

B -K, + - A PW(IF - B , h)

-K, m ( + VW(Ii - ?-jh). (86)P i P

It is not clear how best to implement the diffusion equation in the finite dif-ference approximation. Some numerical experimentation may be necessary. However,the above set of SPH equations allows for boundary conditions on P, f and T andconserves energy, momentum, and mass.

35

SECTION 11

LIST OF REFERENCES

1. Stellingwerf, R. F., "Smooth Particle Hydrodynamics in Three Dimensions,"MRC Report MRC/ABQ-N-384 (1988).

2. Monaghan, J. and Lattanzio, J., "A Refined Particle Method for AstrophysicalProblems," Astron. Astrophys 149, 135 (1985).

3. Monaghan, J., "Why Particle Methods Work," Siam J. Sci. Stat. Comput. 3,422 (1982).

4. Gingold, R. and Monaghan J., "Kernel Estimates as a General Particle Methodin Hydrodynamics," J. Comput. Phys. 46, 429 (1982).

5. Monaghon, J. "SPH Meets the Shocks of Noh," Monash University reprint (1987).

6. Brookshaw, L., "A Method of Calculating Radiative Heat Diffusion in ParticleSimulations," Proc. ASA 6, 207 (1985).

7. Lucy, L., Astron. J. 82, 1013 (1977).

36

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