Smoothed Particle Hydrodynamics · 2014. 4. 16. · Smoothed Particle Hydrodynamics S.P. Korzilius Promotor: W.H.A. Schilders Supervisor: M.J.H. Anthonissen April 9th 2014, Eindhoven

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Smoothed ParticleHydrodynamicsS.P. Korzilius

Promotor:W.H.A. Schilders

Supervisor:M.J.H. Anthonissen

April 9th 2014, Eindhoven

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/ department of mathematics and computer science

Outline

• SPH

• Particle clustering

• Remedies

• Results

• Future work

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SPH

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Basic idea of SPH

Observe that:

f(x) =

∫Ω

f(y)δ(x− y) dy.

Replacing δ by a smooth function W gives:

f(x) ≈∫

Ω

f(y)Wh(x− y) dy.

Representing the domain Ω by a set of particlesleads to the discrete approximation:

〈fi〉 :=∑j∈Si

fjWh(xi − xj)Vj

where Si represents the setof particles in the supportdomain of particle i.

The kernel function should:

• have the delta function property

• satisfy the unity condition

• have compact support

• be radially symmetric

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Derivatives in SPH

From the kernel approximation we can derive that:

f(x) ≈∫

Ω

f(y)Wh(x− y) dy =⇒ f′(x) ≈

∫Ω

f′(y)Wh(x− y) dy.

Partial integration then gives:

f′(x) ≈ f(y)Wh(x− y)

∣∣∣∣∂Ω

+

∫Ω

f(y)W′h(x− y) dy

.=

∫Ω

f(y)W′h(x− y) dy,

where the derivative is with respect to x. Representing the domain by a set ofparticles gives:

⟨f′i

⟩=∑j∈Si

fjW′h(xi − xj)Vj

In practice, the following expression is often used:

⟨f′i

⟩=∑j∈Si

(fj − fi)W ′h(xi − xj)Vj

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Application to fluid flow

SPH is usually applied to the compressible N-S equations. With the volumeof a particle defined as Vj =

mjρj

, we have

• Conservation of mass:Dρ

Dt= −ρ∇ · v −→

⟨Dρi

Dt

⟩= ρi

∑j∈Si

mj

ρj(vi − vj) · ∇Wh(xi − xj)

−→ 〈ρi〉 =∑j∈Si

mjWh(xi − xj).

• Equation of state:

∆p =ρ0c

20

γ

[(ρ

ρ0

)γ− 1

]−→ 〈∆pi〉 =

ρ0c20

γ

[(ρi

ρ0

)γ− 1

].

• Conservation of momentum:

ρDv

Dt= −∇p+∇ ·T + ρg −→

⟨Dvi

Dt

⟩=∑j∈Si

mj (Pij + Πij)∇Wij + g.

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Particle clustering

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Particle clustering (1)

Problem:• For small particle distances

the kernel derivative vanishes.• Via the momentum equation this

leads to a diminishing repulsiveforces between approachingparticles:

|〈Fij〉| = |mimjPij∇Wij | .

Price (2012)

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Particle clustering (2)

• Accuracy error associated withparticle clustering is small.

• Waist of computational effort.

• Leads to questionable results.

Different kernel functions have been used:

W∗(R) = α

(√2−√R),

W∗∗

(R) = α

(1

4R

2 − R + 1

).

∗Schüssler and Schmitt (1981), ∗∗ Johnson and Beissel (1996).

These gave unsatisfactory results.

Remedies:• Better performing kernel with

nonzero derivative at origin.• Particle collisions.

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Remedies

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Convex kernel (1)

We look for a kernel Wnew that has nonzeroderivative at the origin and still satisfies(most of) the kernel requirements.

We use the following algorithm:

• Start with regular-shaped kernel function:

Worg(R) = αorg(2− |R|)3

(3

2|R|+ 1

)for 0 ≤ |R| ≤ 2,

where R := x−yh .

• Define

W′new(R) =

+Worg(R) forR < 0,

−Worg(R) forR > 0.

• Integrate W ′new(R) to find Wnew(R).

• Normalise Wnew(R) to satisfy unity condition.

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Convex kernel (2)

For the given kernel the algorithm leads to the following convex kernel:

Wnew(R) = αnew(2− |R|)4

(3

4|R|+ 1

)for 0 ≤ |R| ≤ 2.

Consequences:• Different weight distribution: more weight is “lost”, which has

a negative effect on the accuracy.• Better particle distribution has a positive effect on accuracy.

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Particle collisions (1)

We can use “real” collisions to keep particles apart. In a fully elastic collisionmomentum and energy are conserved:

miui +mjuj = mivi +mjvj

12mi|ui|

2 + 12mj |uj |

2 = 12mi|vi|

2 + 12mj |vj |

2

=⇒ v

elastici =

(mi −mj)ui + 2mjuj

mi +mj.

In a fully inelastic collision only momentum is conserved, but we now thevelocities are equal afterwards:

miui +mjuj = mivi +mjvj

vi = vj

=⇒ v

inelastici =

miui +mjuj

mi +mj.

where u is the velocity before and v the velocity after the collision.

Combining both gives

vi = Ecvelastici + (1− Ec)v

inelastici =⇒ vi = ui −

mj

mi +mj(1 + Ec)(ui − uj).

with elasticity parameter 0 ≤ Ec ≤ 1.

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Particle collisions (2)

Use this one-collision expression in the following way:

1. Consider only velocity in inter-particle direction,2. Apply to all particles closer than δc to each other.

This gives: δc

vi = ui −∑j∈Ci

mj

mi +mj(1 + Ec)

(uij · rij)rijd2ij

. support domain

Order of computations:Compute new densitiesCompute new pressuresCompute new accelerationsCompute new velocities

← Apply collisions; update velocitiesCompute new positions

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Results

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Rotating shaft

Flow of air (ρ = 1) around a shaft with speed 1000 rpm:

Wendland kernel Particle collisions Convex kernel

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Hydrostatic pressure

Errors in numerical pressure gradients, compared to−ρg,for a column of water (ρ = 1000) with initial zero-pressure:

Method h = 1.5d h = 2.0dWendland kernel 1.32 % -Particle collisions 1.32 % -Convex kernel 2.76 % 1.20 %

(Smoothing length can be increased for convexkernel without leading to particle clustering.)

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Future work

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Future work

• Write paper on “convex kernel” and “particle collisions” andpresent them at SPHERIC 2014.

• Write paper on two-dimensional Laplacian estimate,

(Presented one-dimensional estimate at SPHERIC 2013.)

• Work on the problem of “tensile instability”.

Tensile instability occurs if an equation of state allows for negativepressures; the net force between particle pairs becomes attractive,causing a numerical instability.

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Questions?

Documents

Smoothed Particle Hydrodynamics · 2014. 4. 16. · Smoothed Particle Hydrodynamics S.P. Korzilius Promotor: W.H.A. Schilders Supervisor: M.J.H. Anthonissen April 9th 2014, Eindhoven