Download - Smoothed particle hydrodynamics for fluid dynamics...2 Mathematics of smoothed particle hydrodynamics some facts and basic mathematics kernel and particle approximations of a function

Smoothed particle hydrodynamics for fluid dynamics

Xin Bian and George Em Karniadakis

Division of Applied Mathematics, Brown University, USA

class APMA 2580on Multiscale Computational Fluid Dynamics

April 5, 2016

1 / 50

Homework: you should start

Hw # 4: Finite difference method + MPI for Helmholtz equations

If you need a multi-core machine: apply for a CCV accounthttps://www.ccv.brown.edu/

2 / 50

Outline

1 Backgroundhydrodynamic equationsnumerical methods

2 Mathematics of smoothed particle hydrodynamicssome facts and basic mathematicskernel and particle approximations of a functionfirst and second derivatives

3 Particles for hydrodynamicscontinuity and pressure forceviscous force

4 Classical mechanics for particles ⇒ hydrodynamicsdensity estimateequations of motion

5 Numerical errors

6 Research challenges

7 A short excursion to other particle methods

3 / 50

Outline





5 Numerical errors



4 / 50

https://www.ccv.brown.edu/

Outline





5 Numerical errors



5 / 50

Conservation law of mass: continuity equation

Total mass flows out of the volume V (per unit time) by surface integral∮ρv · df, (1)

where ρ density, v velocity and df is along the outward normal.The decrease of the mass in the volume (per unit time)

− ∂

∂t

∫ρdV . (2)

For a mass conservation, we have an equality

− ∂

∂t

∫ρdV =

∮ρv · df =

∫∇ · (ρv)dV , (3)

which is valid for an arbitrary V . The continuity equation reads

∂ρ

∂t+∇ · (ρv) = 0. (4)

6 / 50

Conservation law of momentum: Euler equations

Total pressure force acting on the volume (surface to volume integral)

−∮

pdf = −∫∇pdV . (5)

For a unit of volume, momentum equations in Lagrangian form read

ρdv

dt= −∇p or

dv

dt= −∇p

ρ. (6)

Considering the particle derivative is related to the partial derivatives as

d

dt=

∂

∂t+ v · ∇, (7)

the Euler equations in Eulerian form read

∂v

∂t+ v · ∇v = −∇p

ρ. (8)

7 / 50

Conservation law of momentum: Navier-Stokes equations

For real fluids, we need to add in viscous stress due to irreversible processand assume that the viscous stress depends only linearly on derivatives ofvelocity.Without derivation, the Navier-Stokes equations read

∂v

∂t+ v · ∇v = −1

ρ[∇p + η4 v + (ζ + η/3)∇∇ · v] (9)

For an incompressible fluid ρ = const. and ∇ · v = 0.Therefore, the momentum equations simplify to

∂v

∂t+ v · ∇v = −1

ρ(∇p + η4 v) . (10)

For a compressible fluid, an equation of state is called for

p = p(ρ,T = T0). (11)

8 / 50

Outline





5 Numerical errors



9 / 50

Mesh-based discretizations

Prof. Karniadakis will cover the mesh-based methods in other lectures

finite difference method

spectral h/p element method

... ...

10 / 50

Mesh-free discretizations: mesh-free = mess-free?

Some particle methods

smoothed particle hydrodynamics (SPH)moving least square methods (MLS)vortex methodVoronoi tesselation... ...

mesh-free ≈ mess-free

no mesh generationLagrangian, no v · ∇vcomplex moving boundaryincorporation of new physics... ...

11 / 50

Outline





5 Numerical errors



12 / 50

Outline





5 Numerical errors



13 / 50

Integral representation of a function

It was invented in 1970s (author?) [12, 8].

Given a scalar function f (r) of spatial coordinate r , its integralrepresentation reads

f (r) =

∫f (r ′)δ(r − r ′)dr ′, (12)

where the Dirac delta function reads

δ(r − r ′)

{∞, r = r ′

0, r 6= r ′(13)

and the constraint of normalization is∫ ∞∞

δ(r)dr = 1. (14)

14 / 50

Outline





5 Numerical errors



15 / 50

SPH 1st step: smoothing or kernel approximation

(author?) [8]; (author?) [12]

Replace δ with another smoothly weighting function w :

f (r) ≈ fk(r) =

∫f (r ′)w(r − r ′, h)dr ′, (15)

where kernel w has properties

1 smoothness

2 compact with h as parameter

3 limh→0 w(r − r ′, h) = δ(r − r ′)

4∫

w(r − r ′, h)dr ′ = 1

5 symmetric

6 ... ...

compact: B-splines, Wendland functions ...(author?) [15, 16, 19]

Gaussian: 1a√π

e−r2/a2

16 / 50

Compact kernel and its normalization

a cubic function as reads (h = 1 for simplicity)

w(r) =

{CD(1− r)3, r < 1;0, r ≥ 1.

(16)

If we require the constraint of normalization in two dimension∫ 2π

0

∫ 1

0C2(1− r)3rdrdθ = 1⇐⇒ C2 =

10

π(17)

a piecewise quintic function reads

w(r) = CD

(3− s)5 − 6(2− s)5 + 15(1− s)5, 0 ≤ s < 1(3− s)5 − 6(2− s)5, 1 ≤ s < 2(3− s)5, 2 ≤ s < 30, s ≥ 3,

(18)

where s = 3r/h and C2 = 7/(478πh2) and C3 = 1/(120πh3).

f (r) = fk(r) + error(h)

17 / 50

SPH 2nd step: summation or particle approximation

Integral ⇐⇒ summation

fk(r) =

∫f (r ′)w(r − r ′, h)dr ′(19)

fs(r) =

Nneigh∑i

fiw(r − ri , h)Vi , (20)

where Vi is a distance, area and volumein 1D, 2D, and 3D, respectively.Therefore,

fk(r) ≈ fs(r) (21)

fk(r) = fs(r) + error(∆r , h)

summation within compact support

18 / 50

An example: evaluation of density and arbitrary function

∀i of particle index, mass mi , density ρi , and Vi = mi/ρi .

fs(r) =

Nneigh∑i

fiw(r − ri , h)Vi =

Nneigh∑i

mi

ρifiw(r − ri , h). (22)

What is the density at an arbitrary position r?

ρs(r) =

Nneigh∑i

mi

ρiρiw(r − ri , h) =

Nneigh∑i

miw(r − ri , h). (23)

What is the density of a particle at rj?

ρs(rj) =

Nneigh∑i

miw(rj − ri , h). (24)

What is the value of a function of a particle at rj?

fs(rj) =

Nneigh∑i

fimi

ρiw(rj − ri , h) =

Nneigh∑i

fiWji = fj (25)

19 / 50

Outline





5 Numerical errors



20 / 50

Gradient of a function

f (r) ≈ fk(r) =

∫f (r ′)w(r − r ′, h)dr ′ (26)

⇓∇r f (r) ≈ ∇r fk(r) = ∇r

∫f (r ′)w(r − r ′, h)dr ′ (27)

⇓

∇r fk(r) =

∫∇r f (r ′)w(r − r ′, h)dr ′ +

∫f (r ′)∇rw(r − r ′, h)dr ′ (28)

⇓∇r fk(r) =

∫f (r ′)∇rw(r − r ′, h)dr ′ (29)

⇓

∇r f (r) ≈ ∇r fk(r) ≈ ∇r fs(r) =

Nneigh∑i

mi

ρif (r ′)∇rw(r − r ′, h) (30)

21 / 50

Other first derivatives

∇r · f (r) ≈ ∇r · fk(r) ≈ ∇r · fs(r) =

Nneigh∑i

mi

ρif (r ′) · ∇rw(r − r ′, h) (31)

∇r×f (r) ≈ ∇r×fk(r) ≈ ∇r×fs(r) =

Nneigh∑i

mi

ρif (r ′)×∇rw(r−r ′, h) (32)

22 / 50

Second derivatives

Note that we have following identity (author?) [4]∫dr′[f (r′)− f (r)

] ∂w(|r′ − r|)∂r ′

1

rijeij

[5

(r′ − r)α(r′ − r)β

(r′ − r)2− δαβ

]= ∇α∇βf (r) +O(∇4fh2). (33)

Therefore,

1

ρi

(∇2v

)= −2

Nneigh∑j

mj

ρiρj

∂wij

∂rvij (34)

1

ρi(∇∇ · v) = −

Nneigh∑j

mj

ρiρj

∂wij

∂r(5eij · vijeij − vij) (35)

(36)

23 / 50

Outline





5 Numerical errors



24 / 50

Outline





5 Numerical errors



25 / 50

Particles for hydrodynamics

continuity equation is accounted for by

ρi =

Nneigh∑j

mjWij , ri = vi (37)

pressure force: −∇p/ρ

FCi =

Nneigh∑j

FCij =

Nneigh∑j

−mj

(pj

ρ2j

)∂w

∂rijeij , (38)

bad: not antisymmetric by swapping i and jrecognize −∇p/ρ = − p

ρ2∇ρ−∇ pρ

FCij = −mj

(pi

ρ2i

+pj

ρ2j

)∂w

∂rijeij , (39)

26 / 50

Outline





5 Numerical errors



27 / 50

viscous force

In general

FDij =

mj

ρiρj rij

∂w

∂rij

[(5η

3− ζ)vij +

(5ζ +

5η

3

)eij · vijeij

](40)

For inompression flows ∇ · v = 0, therefore,

N∑j

5

ρiρj rij

∂wij

rijeij · vijeij ≈

N∑j

1

ρiρj rij

∂wij

∂rijeij . (41)

FDij = 2η

mj

ρiρj rij

∂wij

∂rijvij ≈ 10η

mj

ρiρj rij

∂wij

∂rijeij · vijeij . (42)

Either choice is fine, but they are different.

28 / 50

Summary of SPH for isothermal Navier-Stokes equations

continuity equation:

ρi =

Nneigh∑j

mjWij , ri = vi (43)

momentum equations: (author?) [4, 11]

vi =

Nneigh∑j 6=i

(FCij + FD

ij

), (44)

FCij = −mj

(pi

ρ2i

+pj

ρ2j

)∂w

∂rijeij , (45)

FDij =

mj

ρiρj rij

∂w

∂rij

[(5η

3− ζ)vij +

(5ζ +

5η

3

)eij · vijeij

](46)

weakly compressible: (author?) [2, 13]

p = c2Tρ, or p = p0

[(ρ

ρr

)γ− 1

](47)

p0 relates to an artificial sound speed cT29 / 50

Outline





5 Numerical errors



30 / 50

Outline





5 Numerical errors



31 / 50

Density estimate or sampling

Given a set of point particles with mass mi , what is the density estimatefor a position at r .

ρs(r) =

Nneigh∑i

miw(r − ri , h). (48)

where kernel w has properties1 smoothness2 compact with h as parameter3∫

w(r − r ′, h)dr ′ = 14 symmetric5 ... ...

Eq. (48) is more fundamental than the summation form presented early

fs(r) =

Nneigh∑i

fimi

ρiw(r − ri , h). (49)

32 / 50

Outline





5 Numerical errors



33 / 50

Least action

Define the Lagrangian L as

L = T − U, (50)

where T and U are kinetic and potential energies, respectively. For a setof particles

L =N∑i

mi

(1

2v 2i + ui (ρi , s)

)(51)

Define the action as

S =

∫Ldt. (52)

Minimizing S such that δS =∫δLdt = 0, where δ is a variation with

respect to particle coordinate δr. We have (author?) [17]

δS =

∫ (∂L

∂v· δv +

∂L

∂r· δr)

= 0 (53)

34 / 50

Euler-Lagrangian equations

δS =

∫ (∂L

∂v· δv +

∂L

∂r· δr)

= 0 (54)

consider δv = d(δr)/dt and d/dt = ∂/∂t + v · ∇

⇓

δS =

∫ {[− d

dt

(∂L

∂v

)+∂L

∂r

]· δr}

dt +

[∂L

∂v· δr]tt0

= 0 (55)

assume variation vanishes at start and end times and furthermore, δr isarbitrary. Therefore, we have the Euler-Lagrangian equations

d

dt

(∂L

∂vi

)− ∂L

∂ri= 0. (56)

35 / 50

Equation of motions for particles

From the Lagrangian L =∑N

i mi

(12 v 2

i + ui

)we know

∂L

∂vi= mivi ,

∂L

∂ri= −

Nneigh∑j

mj∂uj

∂ρj

∂ρj∂rj

(57)

Some basic thermodynamics: dU = TdS − PdVSince V = m/ρ, so dV = −mdρ/ρ2. For per unit mass we have

du = Tds − P

ρ2dρ. (58)

For a reversible process ds = 0, therefore ∂ui/∂ρi = p/ρ2. Put everythingknown into the Euler-Lagrangian equations, we get

vi =

Nneigh∑j 6=i

−mj

(pi

ρ2i

+pj

ρ2j

)∂w

∂rijeij . (59)

36 / 50

Conservation laws

Euler hydrodynamics

total mass M =∑N

i mi .

total linear momentum

d

dt

N∑i

mivi =N∑i

midvidt

=N∑i

N∑j

−mimj

(pi

ρ2i

+pj

ρ2j

)∂w

∂rijeij = 0.

(60)

total angular momentum

d

dt

N∑i

ri ×mivi =N∑i

mi

(ri ×

dvidt

)(61)

=N∑i

N∑j

−mimj

(pi

ρ2i

+pj

ρ2j

)∂w

∂rij(ri × eij) = 0. (62)

Similarly for the viscous forces.37 / 50

Outline





5 Numerical errors



38 / 50

Errors in density estimate: kernel error

Recall the kernel approximation

ρk(r) =

∫ρ(r′)w(r − r′, h)dr′, (63)

Expanding ρ(r ′) by Taylor series around r

ρk(r) = ρ(r)

∫w(r − r′, h)dr′ +∇ρ(r) ·

∫(r′ − r)w(r − r′, h)dr′

+∇α∇βρ(r)

∫δr′αδrβw(r − r′, h)dr′ + O(h3). (64)

Recall∫

w(r − r′, h)dr′ = 1 and odd terms vanish due to symmetric w ,

ρ(r) = ρk(r) + O(h2). (65)

39 / 50

Errors for a function f : kernel error and summation error

Similarly as for density estimate:

f (r) = fk(r) + O(h2). (66)

Recall the particle approximation or summation form

fk(ri ) ≈ fs(ri ) =

Nneigh∑j

mj

ρjfjw(ri − rj , h). (67)

Let us do Taylor series on f (rj) around ri

fs(ri ) = fi

Nneigh∑j

mj

ρjw(rij , h) +∇fi ·

Nneigh∑j

rjimj

ρjw(rij , h) + O(h2). (68)

To have error of O(h2), we need

N∑j

mj

ρiw(rij) = 1,

N∑j

rjimj

ρiw(rij) = 0, (69)

which is not guranteed in practice (depends on configurations, ∆x , and h).40 / 50

Outline





5 Numerical errors



41 / 50

Challenges

error analysis due to particle configurations

consistency and conservation at the same time

convergence for a practical purpose

coarse-graining from molecular dynamics

42 / 50

Outline





5 Numerical errors



43 / 50

Algorithmic similarity: pairwise forces within short range rc

in a nutshell, ∀ particle i in SPH, SDPD, DPD, or MD, the EoM:

vi =∑j 6=i

(FCij + FD

ij + FRij

)(70)

options for different components

weighting kernel or potential gradient in MDequation of statedensity fieldthermal fluctuationscanonical ensemble / NVT: thermostat... ...

SPH: (author?) [14]

SDPD: (author?) [4]

DPD: (author?) [10]; (author?) [5]; (author?) [9]

MD: (author?) [1]; (author?) [7]; (author?) [6]; (author?) [18]

44 / 50

academic toy: several interesting elements

∼ 12,000 SDPD particles (author?) [3]

hypnotized?

45 / 50

academic toy: several interesting elements

∼ 12,000 SDPD particles (author?) [3]

hypnotized?

46 / 50

References I

[1] M. P. Allen and D. J. Tildesley. Computer simulation of liquids.Clarendon Press, Oxford, Oct. 1989.

[2] G. K. Batchelor. An introduction to fluid dynamics. CambridgeUniversity Press, Cambridge, 1967.

[3] X. Bian, S. Litvinov, R. Qian, M. Ellero, and N. A. Adams.Multiscale modeling of particle in suspension with smootheddissipative particle dynamics. Phys. Fluids, 24(1), 2012.

[4] P. Espanol and M. Revenga. Smoothed dissipative particle dynamics.Phys. Rev. E, 67(2):026705, 2003.

[5] P. Espanol and P. Warren. Statistical mechanics of dissipative particledynamics. Europhys. Lett., 30(4):191–196, May 1995.

[6] D. J. Evans and G. Morriss. Statistical Mechanics of nonequilibriumliquids. Cambridge University Press, second edition, 2008.

47 / 50

References II

[7] D. Frenkel and B. Smit. Understanding molecular simulation: fromalgorithms to Applications. Academic Press, a division of Harcourt,Inc., 2002.

[8] R. A. Gingold and J. J. Monaghan. Smoothed particlehydrodynamics: theory and application to non-spherical stars. Mon.Not. R. Astron. Soc., 181:375–389, 1977.

[9] R. D. Groot and P. B. Warren. Dissipative particle dynamics:Bridging the gap between atomistic and mesoscopic simulation. J.Chem. Phys., 107(11):4423–4435, 1997.

[10] P. J. Hoogerbrugge and J. M. V. A. Koelman. Simulating microsopichydrodynamics phenomena with dissipative particle dynamics.Europhys. Lett., 19(3):155–160, 1992.

[11] X. Y. Hu and N. A. Adams. A multi-phase SPH method formacroscopic and mesoscopic flows. J. Comput. Phys.,213(2):844–861, 2006.

48 / 50

References III

[12] L. B. Lucy. A numerical approach to the testing of the fissionhypothesis. Astron. J., 82:1013–1024, 1977.

[13] J. J. Monaghan. Simulating free surface flows with SPH. J. Comput.Phys., 110:399–406, 1994.

[14] J. J. Monaghan. Smoothed particle hydrodynamics. Rep. Prog.Phys., 68(8):1703 – 1759, 2005.

[15] J. J. Monaghan and J. C. Lattanzio. A refined particle method forastrophysical problems. Astron. Astrophys., 149:135–143, 1985.

[16] J. P. Morris, P. J. Fox, and Y. Zhu. Modeling low Reynolds numberincompressible flows using SPH. J. Comput. Phys., 136(1):214 – 226,1997.

[17] D. J. Price. Smoothed particle hydrodynamics andmagnetohydrodynamics. J. Comput. Phys., 231:759 – 794, FEB 12012.

49 / 50

References IV

[18] Mark E. Tuckerman. statistical mechanics: theory and molecularsimulation. Oxford University Press, 2010.

[19] Holger Wendland. Piecewise polynomial, positive definite andcompactly supported radial functions of minimal degree. Advances inComputational Mathematics, 4(1):389–396, 1995.

50 / 50