Smoothed Particle Hydrodynamics (SPH): corrections and … · 2013-11-19 · For further reading...

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Smoothed Particle Hydrodynamics (SPH):corrections and applications

Q.Z. Hou

April 7 2010

Outline

1 Motivation

2 SPH and corrective schemesSmoothed particle hydrodynamicsCorrective schemes

3 Test problemsWater hammerHeat conductionStress wave propagationShock tube

4 Summary and future workBibliography

Outline

1 Motivation

Physical problem

Laboratory set-up

Bozkus and Wiggert (1997)

Practical problem

Industrial pipelines

Oil pipeline

Steam pipeline

Difficulties in traditional CFD methods

Two fundamental frameworks:Eulerian description: mesh fixed in space, objects moveacross it; calculation of mass, momentum and energydistributionLagrangian description: mesh attached to the material,cells deform; no mass flux, flux of momentum and energyat cell boundaries

Difficulties:Eulerian grid: irregular/complex geometry, locations ofinhomogeneities, free surfaces, moving interfaces,deformable boundariesLagrangian grid: mesh generation, large deformationboth: explosion, high-velocity impact, discrete particles

Outline

1 Motivation

What is SPH?

SPH: smoothed particle hydrodynamics

1. movies3D simulation of dam breakpouring water in a glass (like commercial advertisement forbeer).Hollywood, Pixar Animation Studio, Industry Light andMagic, fluid simulations in Ice Age and 2012

2. key ideafollow particles’ movement, information attached to them isused as weighted average in numerical approximation

Fundamentals of SPH

Integral representation

f (x) =

f (ξ)δ(x − ξ,h) dV

Function reconstruction

f (x) ≈∫

Ωf (ξ)W (x− ξ,h) dV

Representation of the derivative

∇f (x) ≈ −∫

Ωf (ξ)∇W (x − ξ,h)dV

with smoothing function W (x − ξ,h)and smoothing length h

Smoothing function

Requirements:

unity:∫

Ω W (x − ξ,h) dξ = 1;compact support: W (x − ξ,h) = 0, |x − ξ| > κh;positivity: W (x − ξ,h) ≥ 0, |x − ξ| ≤ κh;symmetry: W (x − ξ,h) = W (ξ − x ,h);δ function consistency:

limh→0

W (x − ξ,h) = δ(x − ξ);

monotonically decreasing with respect to |x − ξ|smooth

Cubic spline – smoothing function W

Cubic spline and its derivatives

Discretisation (particle approximation)

Ω is divided into N particles, with volume ∆Vj , (j = 1, · · · ,N)

f (x) ≈∫

Ωf (ξ)W (x − ξ,h) dV

⇒ f (x i) ≈∑

f (ξj)W (x i − ξj ,h)∆Vj

movement of particlesgive each particle a fixed mass mj and replace the volume ∆Vjby

∆Vj =mj

position evolutionxn+1

i = xni + ∆t vn

Discretisation for derivative

∇f (x) ≈ −∫

Ωf (ξ)∇W (x − ξ,h)dV

⇒ ∇f (x i) ≈ −∑

f (ξj)∇W (x i − ξj ,h)∆Vj

introduce the following notation

Wij := W (x i − ξj ,h),

Wij,α :=(∂Wij∂xα

)i, where α indicates a spatial direction

Particle distribution

1D example: f (x) = x , x ∈ [0, 1],N = 5, κh = 12

f1 ≈∑5

j=1 fjW1j∆Vj = f1W11∆V4 + f2W12∆V2 = 16 f2 = 1

24 6= 0

function reconstruction derivative approximation

Corrective schemes (1D)

Expanding Taylor series for f (ξ) around point x yields

f (ξ) ≈ f (x) +∂f (x)

∂x(ξ − x)

Multiplying with W and integrating over Ω∫Ω

f (ξ)WdV ≈ f (x)

WdV +∂f (x)

(ξ − x)WdV

f (ξ)WdV ≈ f (x)

Corrective scheme for derivative

Taylor series expansion

f (ξ) ≈ f (x) +∂f (x)

∂x(ξ − x)

Multiplying with ∂W∂ξ and integrating over Ω∫

Ωf (ξ)

∂W∂ξ

dV ≈ f (x)

∂W∂ξ

dV +∂f (x)

(ξ − x)∂W∂ξ

CSPM: corrective smoothed particle method

Corrective scheme in discretized form

function approximation∫Ω

f (ξ)WdV ≈ f (x)

⇒ f (x) ≈∫

Ω f (ξ)WdV∫Ω WdV

⇒ f (xi) ≈∑

f (xj)Wij∆Vj∑Wij∆Vj

derivative approximation∫Ω

f (ξ)∂W∂ξ

dV ≈ f (x)

∂W∂ξ

dV +∂f (x)

(ξ − x)∂W∂ξ

⇒(∂f (x)

)i≈∑

[f (xj)− f (xi)]Wij,x ∆Vj∑(xj − xi)Wij,x ∆Vj

Same 1D example

function reconstruction derivative approximation

Same 1D example: f = (f1, f2, f3, f4, f5)T

f ≈ Af , ∂f/∂x ≈ Bf

Interpolation matrix A2/3 1/61/6 2/3 1/6

1/6 2/3 1/61/6 2/3 1/6

1/6 2/3

4/5 1/51/6 2/3 1/6

1/6 2/3 1/61/6 2/3 1/6

1/5 4/5

Difference matrix B

2 0 12

−12 0 1

2 0 12

−12 0

−1 1−1

2 0 12

−12 0 1

2 0 12

−1 1

2D: f (x , y) = sin πx sin πy , (x , y) ∈ [0,1]× [0,1]

Reconstruction of function f and particle approximation of thefirst derivative fx

Outline

1 Motivation

Water hammer

Typical reservoir-pipe-valve (RPV) system

Governing equations

∂v∂t

∂p∂x

+∂φ

∂x, (momentum equation)

∂p∂t

= K∂v∂x

(continuity equation)

linear artificial viscosity φ =

Cv,x , v,x < 00, v,x > 0

Solving two first order equations by CSPM

CSPM:(∂f∂x

∑[f (xj )−f (xi )]Wij,x ∆Vj∑

(xj−xi )Wij,x ∆Vjfor f = v and f = p

IC: v0i = v0; p0

i = p0

Algorithm for single time stepstep 1: apply BC vn+1

N = 0step 2: calculate pressure

pn+1i = pn

i + ∆t[K(∂v∂x

]step 3: apply BC pn+1

1 = p0step 4: calculate velocity

vn+1i = vn

i + ∆t[

(∂p∂x

(∂φ

Results

instantaneous valve closing linear valve closing

Pressure history at valve

Heat conduction in a bar

Temperature distribution at different times, left - energy balance(two first order eqs.), right - heat conduction (one second order

Wave propagation in an elastic bar

Snapshot at t = 10 µs

Shock tube (moving particles)

Shock tube problem

Outline

1 Motivation

Summary

difficulties in traditional methodsfundamentals of SPHcorrections to SPH1D test problems of CSPM

Future work

new corrective schemesirregular particlesmoving particles2D / 3D problems

For further reading

Bozkus Z and Wiggert D.C (1997)Liquid slug motion in a voided line. Journal of Fluids andStructures, 11, 947-963

Liu G.R and Liu M.B (2003)Smoothed Particle Hydrodynamics: A Meshfree ParticleMethod.

Chen J.K., Beraun J.E., Carney T.C (1999)A corrective smoothed particle method for boundary valueproblems in heat conduction. International Journal forNumerical Methods in Engineering, 46: 231-252

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Smoothed Particle Hydrodynamics (SPH): corrections and … · 2013-11-19 · For further reading...

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