Smoothed particle hydrodynamics for fluid dynamics...2 Mathematics of smoothed particle...

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

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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/

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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

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

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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)

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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)

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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)

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Mesh-based discretizations

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

finite difference method

spectral h/p element method

... ...

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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... ...

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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)

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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

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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)

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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

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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)

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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)

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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)

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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)

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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)

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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.

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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)

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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)

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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)

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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)

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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)

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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)

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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)

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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

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Challenges

error analysis due to particle configurations

consistency and conservation at the same time

convergence for a practical purpose

coarse-graining from molecular dynamics

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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]

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academic toy: several interesting elements

∼ 12,000 SDPD particles (author?) [3]

hypnotized?

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academic toy: several interesting elements

∼ 12,000 SDPD particles (author?) [3]

hypnotized?

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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.

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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.

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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.

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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.

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