Penn State University College of Earth and Mineral Science

Department of Energy and Mineral Engineering

Smooth Particle Hydrodynamics (SPH)

Presented By Ahmed Nasreldeen

Long Fan Keru Liu

Group Presentation EGEE 520 Course

Spring 2017

Ø  HistoricalPerspective

Ø  GeneralPrinciples.

Ø  GoverningEquations

Presentation Outlines

Ø  Hand-CalculationExample

Ø  NumericalExample

Ø  ExampleApplications

Ø  Introduction

Smoothed Particle Hydrodynamics (SPH)

FluiddynamicsThefluidisrepresentedbyaparticlesystem

Someparticlepropertiesaredeterminedbytakinganaverage

overneighboringparticles

The Smoothed Particle Hydrodynamics(SPH) method works by dividing the fluidintoasetofdiscreteelements, referred toas particles. These particles have a spatialdistance (known as the "smoothinglength", typically represented in equationsby h), over which their properties are"smoothed"byakernelfunction

It is a computational method used for simulating the dynamics of continuummedia,suchassolidmechanicsandfluidflows.Ithasbeenusedinmanyfieldsofresearch,including astrophysics, ballistics, volcanology, and oceanography. It is a mesh-freeLagrangianmethod(wherethecoordinatesmovewiththefluid)

Smoothed Particle Hydrodynamics

Basic Concepts of Computational Hydrodynamics

Grid Based Methods

Particle Based Methods

Ø  Eulerianapproach(FDM)

Ø  Lagrangianapproach(FEM)

Ø  Meshfreemethodscanusebothapproaches

Ø  SmoothedParticleHydrodynamics

Ø  DissipativeParticleDynamics

Ø  BrownianDynamics

Eulerian vs Lagrangian descriptions

Eulerian method Concernedwithfluidproperties

(velocity,density,pressure,temperature)ataspecificspace-

timepoint(x,y,z,t).

Controlvolumeapproach

NamedafterSwissmathematicianLeonhardEuler

(1707-1783).

Lagrangian method

Concernedwithaparticularparticleoffluidasitmovesthroughspacetoreflectthebehavioroftherestparticles.

Controlmassapproach

NamedafterItalianmathematicianJosephLagrange

(1736-1813).

Eulerian vs Lagrangian descriptions

LagrangianMethods Euler ianMethods

GridAttachingonthemoving

material Fixedinthespace

TrackMovementofanypointon

materials

Mass,momentum,andenergyfluxacrossgridnodesandmesh

cellboundary

TimeHistoryEasytoobtaintime-historydataatapointattachedon

materials

Difficulttoobtaintime-historydataatapointattachedon

materialsMovingBoundaryandInterference

Easytotrack Difficulttotrack

IrregularGeometry Easytomodel Difficulttomodelwithgood

accuracy

LargeDeformation Difficulttohandle Easytohandle

Meshfree and Meshfree Particle Methods

Ø  Thebasicideaofthemeshfreemethodsistodiscretizethecontinuumthroughaset of nodes without the connective mesh in order to follow the deformationexperienced by thematerial and avoid the degradation of the numerical resultmaintainingasuitablecomputationaleffort.

Ø  Whenthenodesassumesaphysicalmeaning,i.e.theyrepresentmaterialparticlescarrying physical properties, the method is said to be meshfree particle andfollows,ingeneral,aLagrangianapproach.

Ø  AccurateandstablenumericalsolutionsforintegralequationsorPDEswithallkindofboundaryconditions

Limitations of Grid Based Methods

Not Suitable for problems involving: Ø  Largedisplacements.

Ø  Largeinhomogeinities.

Ø  Movingmaterialinterface.

Ø  Deformableboundaries.

Ø  Freesurfaces.

(Lucy,1977),(Gingold&Monaghan,1977)

(Miyama,Hayashi&Narita,1984)

(Benz,Slattery&Cameron,1986)

(Springel,2005)

(Monaghan,2005)

(Monaghan,1992)

Historical Perspective

Simulatingastrophysicsproblems

Fragmentationincollapsingmolecularclouds

Collisionproblems

Appliedtosolidandfluidmechanics

Cosmologicalsimulations

Computergraphics

Advantages of SPH Method

Ø  Itcanobtainthetimehistoryofthematerialparticles.Theadvectionandtransportofthesystemcanthusbecalculated.

Ø  The free surfaces, material interfaces, and moving boundaries can all be tracednaturallyintheprocessofsimulationregardlessthecomplicityofthemovementofthe particles, which have been very challenging to many Eulerian methods.Therefore, SPH is an ideal choice for modeling free surface and interfacial flowproblems.

Ø  The distinct meshfree feature of the SPH method allows a straightforwardhandlingofvery largedeformations, sincetheconnectivitybetweenparticlesaregeneratedaspartofthecomputationandcanchangewithtime.

Ø  SPH is suitable for problems where the object under consideration is not acontinuum.Thisisespeciallytrueinbio-andnano-engineeringatmicroandnanoscale,andastrophysicsatastronomicscale

Ø  SPHiscomparativelyeasierinnumericalimplementation,anditismorenaturaltodevelopthree-dimensionalnumericalmodelsthangridbasedmethods.

Ø  Pureadvection istreatedexactly.Forexample, if theparticlesaregivenacolour,andthevelocityisspecified,thetransportofcolourbytheparticlesystemisexact

Ø  Withmorethanonematerial,eachdescribedbyitsownsetofparticles,interfaceproblemsareoftentrivialforSPHbutdifficultforfinitedifferenceschemes.

Ø  Particlemethodsbridge thegapbetween thecontinuumand fragmentation inanaturalway.

Ø  The resolution can be made to depend on position and time, which makes themethodveryattractiveformostastrophysicalandmanygeophysicalproblems.

Ø  SPH has the computational advantage, particularly in problems involvingfragments,dropsorstarsthatthecomputationisonlywherethematteris,withaconsequentreductioninstorageandcalculation.

Disadvantages of SPH Method

Ø  ThemaindisadvantageofSPHisitslimitedaccuracyinmulti-dimensionalflowsduetonoise.Thisnoiseseriouslymessesuptheaccuracythatcanbereachedwiththetechnique,especiallyforsubsonicflow,andalsoleadstoaslowconvergencerate.

Ø  ParticularlyproblematicinSPHarefluidinstabilitiesacrosscontactdiscontinuities,suchasKelvin-Helmholtzinstabilities.Theseareusuallyfoundtobesuppressedintheirgrowth.

Ø  Anothergenericproblem is that the artificial viscosity is operating at some levelalso outside of shocks, giving the numerical model a relatively high numericalviscosity,whichlimitstheReynoldsnumbersthatcanbeeasilyreachedwithSPH.

Industrial applications of SPH Method

Ø  Aerospaceindustries

Ø  Carindustries

Ø  Energyproductionindustries

Ø  Industrialprocessing

Ø Marineandcoastalengineering

Ø  Environmentalandgeophysicalproblems

Ø  Fluid-structureinteraction Ø Otherapplications

(Astrophysics,Bioengineering,Hydrodynamics)

15

The basic step of the method (domaindiscretization,fieldfunctionapproximationandnumericalsolution):Ø  Thecontinuum:

Asetofarbitrarilydistributedparticleswithnoconnectivity(meshfree);Ø  Fieldfunctionapproximation:Theintegralrepresentationmethod

Ø  Convertingintegralrepresentationintofinitesummation:Particleapproximation

16

Integral representation of a function:

ThecontinuumAsetofarbitrarilyparticles𝑓(𝑥)=∫Ω↑▒𝑓(𝑥 )𝑊(𝑥− 𝑥 ,ℎ)𝑑𝑥  

17

Integral representation of a function: Ø  Theinterpolationisbasedonthetheoryofintegralinterpolantsusingkernels

thatapproximateadeltafunction

Ø  Theintegralinterpolantofanyquantityfunction,A(r)

Ø  where:risanypointindomain(Ω),Wisasmoothingkernelwithhaswidth.

Ø  Thewidth,orcoreradius, isascalingfactorthatcontrolsthesmoothnessorroughnessofthekernel.

𝐴↓𝐼 (𝑟)=∫Ω↑▒𝐴(𝑟 )𝑊(𝑟− 𝑟 ,ℎ)𝑑𝑟  

18

Integral representation into finite summation

Ø  Numericalequivalent

Ø  where j is iterated over all particles, Vj is the volume attributed

implicitly to particle j, rj the position, and Aj is the value of anyquantityAatrj

Ø  ThebasisformulationoftheSPH

𝐴↓𝑆 (𝑟)=∑𝑗↑▒𝐴↓𝑗 𝑉↓𝑗 𝑊(𝑟− 𝑟↓𝑗 ,ℎ) 

𝑉= 𝑚/𝜌 

𝐴↓𝑆 (𝑟)=∑𝑗↑▒𝐴↓𝑗 𝑚↓𝑗 /𝜌↓𝑗   𝑊(𝑟− 𝑟↓𝑗 ,ℎ) 

𝐴↓𝐼 (𝑟)=∫Ω↑▒𝐴(𝑟 )𝑊(𝑟− 𝑟 ,ℎ)𝑑𝑟  

Ø  Up tonow, various kernel functions havebeendeveloped andused inthe SPH method, among which the most widely used are the cubicsplinekernelfunctionandtheWendlandkernel.

−𝑊(𝑟↓𝑖𝑗 ,ℎ)=𝐶↓ℎ  █�{█(2−𝑞)↑3 −4 (1−𝑞)↑3  𝑓𝑜𝑟 0≤𝑞≤1� (2−𝑞)↑3  𝑓𝑜𝑟 1≤𝑞≤2�0 𝑓𝑜𝑟 𝑞>2  � 𝐶↓ℎ =15/(4𝜋ℎ↑2 )� 𝑊(𝑟↓𝑖𝑗 ,ℎ)=𝐶↓ℎ  █�{█(2−𝑞)↑4 (1+2𝑞)−4 (1−𝑞)↑3  𝑓𝑜𝑟 0≤𝑞≤2��0 𝑓𝑜𝑟 𝑞>2  � 𝐶↓ℎ =7/(64𝜋ℎ↑2 ) 

19

q= 𝑟↓𝑖𝑗 /ℎ 

Eachparticleisspecifiedbyastatelist:mass,velocity,position,force,density,pressure

),,,,,( iiiiii prm ρFvParticle i

Governing Equations

Theaccelerationofaparticleis

gravityi

einteractivi

viscosityi

pressurei

i aaaadtdv

+++=

Governing Equations

Assumewehavethesamemassforallparticles,mi=m

The mass m is calculated by

( ) ( )particlesofnumberTotal

volumeTotalfluidofDensitym ⋅=

Particle mass

Let us now go back to the weighted averages…

Howtodeterminethedensityofaparticle?

ijr

∑=j

ijji rWm )(ρ

Weight function or Kernel function

Surface tracking

WecanfindthesurfacebymonitoringthedensityIfthedensityataparticledeviatestoomuchcomparedtoexpecteddensitywetagitasasurfaceparticle

Pressure

Wegetthepressurefromtherelation:

( )02 ρρ −= isi cp

whereCsisthespeedofsoundandρ0isthefluidreferencedensity

),,,,,( iiiiii prm ρFv

Ø  Thenextpropertywefocusonistheforce

Ø  Velocitiesandpositionsarecalculatedfromtheforcesinawaysimilartoanordinaryparticlesystem

Ø  Butbeforewegointothatweneedtolearnmoreabouttakingaverages…

In SPH, the average is formally defined as follows:

∫ −=V

drrrWrArA ')'()'()(

Fornumericalsolutions,thesummationisusedinsteadofintegration.

∑≈j

ijjj

ji

rWAm

A )(ρ

∑ ∇≈∇j

ijjj

ji

rWAm

A )(ρ

∑ ∇≈∇j

ijjj

j

irWA

mA )(22

ρ

∑

∑

≈

≈

jijj

jijj

j

ji

rWm

rWm

)(

)(ρρ

ρ

Meshlessmethod!!

Velocities and Forces

Motionequationinelasticity:

extFdtdv

ρσ

ρ11

+⋅∇=

where: εσ C=εµσ !+Ι−= p

AllthistogetherproducesthefollowingfluidequationcalledNavier-Stokesequation

gvv++∇⋅∇+∇−= extFp

dtd

ρρµ

ρ11

Convert each term on the RHS in Navier-Stokes to SPH-averages

Ø  Firstterm(pressure)becomes:

∑ ∇≈∇−j

ijiji

rWPp )(1ρ

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+−= 22

j

j

i

i

j

jij

ppmP

ρρρ

where

30

∑ ∇≈∇−j

ijiji

rWPp )(1ρ

The second term (viscosity):

∑ ∇≈∇⋅∇−j

ijiji

rW )(2Vvρµ

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+−= 22

j

j

i

i

j

jij

mρρρ

µvvV

where

Summary

Ø  Theaccelerationofaparticlecannowbewritten:

gravityi

externali

viscosityi

pressurei

i aaaadtdv

+++=

∑ ∇≈j

ijijpressurei rWPa )(

externali

i

externali Fa

ρ1

≈

),0,0( ga gravityi −≈

∑ ∇≈j

ijijviscosityi rWVa )(

Hand-Calculation Example

33

Problem Description

34

𝑋↓𝑗 =(4,3)

𝑉↓𝑖 =(1,1)

𝑉↓𝑗 =(0.5,1)

-1

7

1 52 3 4 6 7

3

6

5

4

2

1

0-1

0

ℎ

Mass/m

Density/ρ

Pressure/P

Viscosity/μ

Velocity/v

Location/(x,y) h ∆t

Particlei 1 1 1 1 (1,1) (2,1) 2.5 1

Particlej 1 1 1 1 (0.5,1) (1,2) 2.5 1

𝑋↓𝑖 =(2,2)𝑟↓𝑖,𝑗 

35

Calculation of 𝛻W↓𝑖𝑗 

Usingthenumericalmethodtosolve𝛻W↓𝑖,𝑗 𝜕𝑊↓𝑖𝑗 /𝜕𝑥  = (− 45∗((2−0.001)↑2 + 1↑2 )↑3/2  /4π∗ ℎ↑5  + 45∗((2−0.001)↑2 + 1↑2 )/2π∗ ℎ↑4  − 15/π∗ℎ↑2  )−(− 45∗5↑3/2  /4π∗ ℎ↑5  + 45∗5/2π∗ ℎ↑4  − 15/π∗ℎ↑2  )/0.001 =−2.43∗10↑−1 𝜕𝑊↓𝑖𝑗 /𝜕𝑦  = (− 45∗(2↑2 + (1−0.001)↑2 )↑3/2  /4π∗ ℎ↑5  + 45∗(2↑2 + (1−0.001)↑2 )/2π∗ ℎ↑4  − 15/π∗ℎ↑2  )−(− 45∗5↑3/2  /4π∗ ℎ↑5  + 45∗5/2π∗ ℎ↑4  − 15/π∗ℎ↑2  )/0.001 =−1.21∗10↑−1 

−𝑊(𝑟,ℎ)=𝐶↓ℎ  █�{█(2−𝑞)↑3 −4(1−𝑞)↑3  𝑓𝑜𝑟 0≤𝑞≤1� (2−𝑞)↑3  𝑓𝑜𝑟 1≤𝑞≤2�0 𝑓𝑜𝑟 𝑞>2  � 𝐶↓ℎ =15/(4𝜋ℎ↑2 )� 

q= 𝑟↓𝑗,𝑖 /ℎ = √�5 /2.5 <1

Particlej

Particlei

𝑊↓𝑖𝑗 = 45𝑟↓𝑖𝑗↑3 /4𝜋ℎ↑5  − 45𝑟↓𝑖𝑗↑2 /2𝜋ℎ↑4  + 15/𝜋ℎ↑2  

𝛻W↓i,j = 𝜕W↓ij /𝜕x , 𝜕W↓ij /𝜕y =(−2.43∗10↑−1 ,−1.21∗10↑−1 );

Calculation of 𝛁𝑾↓𝐣𝐢 

36

Usingthenumericalmethodtosolve𝛻W↓𝑗𝑖 𝜕𝑊↓𝑗𝑖 /𝜕𝑥  = (− 45∗((−2−0.001)↑2 + 1↑2 )↑3/2  /4π∗ ℎ↑5  + 45∗((−2−0.001)↑2 + 1↑2 )/2π∗ ℎ↑4  − 15/π∗ℎ↑2  )−(− 45∗5↑3/2  /4π∗ ℎ↑5  + 45∗5/2π∗ ℎ↑4  − 15/π∗ℎ↑2  )/0.001 =−2.43∗10↑−1 𝜕𝑊↓𝑗𝑖 /𝜕𝑦  = (− 45∗(2↑2 + (−1−0.001)↑2 )↑3/2  /4π∗ ℎ↑5  + 45∗(2↑2 + (−1−0.001)↑2 )/2π∗ ℎ↑4  − 15/π∗ℎ↑2  )−(− 45∗5↑3/2  /4π∗ ℎ↑5  + 45∗5/2π∗ ℎ↑4  − 15/π∗ℎ↑2  )/0.001 =−1.21∗10↑−1 

−𝑊(𝑟,ℎ)=𝐶↓ℎ  █�{█(2−𝑞)↑3 −4(1−𝑞)↑3  𝑓𝑜𝑟 0≤𝑞≤1� (2−𝑞)↑3  𝑓𝑜𝑟 1≤𝑞≤2�0 𝑓𝑜𝑟 𝑞>2  � 𝐶↓ℎ =15/(4𝜋ℎ↑2 )� 

q= 𝑟↓𝑗,𝑖 /ℎ = √�5 /2.5 <1

Particlej

Particlei

𝛻W↓𝑗𝑖 = 𝜕W↓𝑗𝑖 /𝜕x , 𝜕W↓𝑗𝑖 /𝜕y =(−2.43∗10↑−1 ,−1.21∗10↑−1 );

𝑊↓𝑗𝑖 = 45𝑟↓𝑗𝑖↑3 /4𝜋ℎ↑5  − 45𝑟↓𝑗𝑖↑2 /2𝜋ℎ↑4  + 15/𝜋ℎ↑2  

Calculation of 𝛁2𝑾↓i,j 

37

Usingthenumericalmethodtosolve𝛻2W↓𝑖,𝑗 𝛻↑2 𝑊↓𝑖𝑗 |↓𝑥 = 𝑊↓𝑖+0.001,𝑗 −2𝑊↓𝑖𝑗 + 𝑊↓𝑖−0.001,𝑗 /0.001↑2  = −0.257075−2∗(−0.257317)−0.257558/0.001^2 =5.1∗10↑−1  𝛻↑2 𝑊↓𝑖𝑗 |↓𝑦 = 𝑊↓𝑖,𝑗+0.001 −2𝑊↓𝑖𝑗 + 𝑊↓𝑖,𝑗−0.001 /0.001↑2  = −0.257196−2∗(−0.257317)−0.257438/0.01↑2  −2.36∗10↑−1 

𝛻2W↓i,j = 𝛻↑2 𝑊↓𝑖𝑗 |↓𝑥 , 𝛻↑2 𝑊↓𝑖𝑗 |↓𝑦 =(−5.1∗10↑−1 ,−2.36∗10↑−1 );

−𝑊(𝑟,ℎ)=𝐶↓ℎ  █�{█(2−𝑞)↑3 −4(1−𝑞)↑3  𝑓𝑜𝑟 0≤𝑞≤1� (2−𝑞)↑3  𝑓𝑜𝑟 1≤𝑞≤2�0 𝑓𝑜𝑟 𝑞>2  � 𝐶↓ℎ =15/(4𝜋ℎ↑2 )� 

q= 𝑟↓𝑗,𝑖 /ℎ = √�5 /2.5 <1

𝑊↓𝑗𝑖 = 45𝑟↓𝑗𝑖↑3 /4𝜋ℎ↑5  − 45𝑟↓𝑗𝑖↑2 /2𝜋ℎ↑4  + 15/𝜋ℎ↑2  

Calculation of 𝛁𝟐𝑾↓𝐣,𝐢 

38

Usingthenumericalmethodtosolve𝛻2W↓𝑗,𝑖 𝛻↑2 𝑊↓𝑗𝑖 |↓𝑥 = 𝑊↓𝑗+0.001,𝑖 −2𝑊↓𝑗𝑖 + 𝑊↓𝑗−0.001,𝑖 /0.001↑2  = −0.257558−2∗(−0.257317)−0.257075/0.001↑2  =−5.1∗10↑−1  𝛻↑2 𝑊↓𝑗𝑖 |↓𝑦 = 𝑊↓𝑗,𝑖+0.001 −2𝑊↓𝑗𝑖 + 𝑊↓𝑗,𝑖−0.001 /0.001↑2  = −0.257438−2∗(−0.257317)−0.257196/0.001↑2  =−2.36∗10↑−1 

𝛻2W↓𝑗𝑖 = 𝛻↑2 𝑊↓𝑗𝑖 |↓𝑥 , 𝛻↑2 𝑊↓𝑗𝑖 |↓𝑦 =(−5.1∗10↑−1 ,−2.36∗10↑−1 );

−𝑊(𝑟,ℎ)=𝐶↓ℎ  █�{█(2−𝑞)↑3 −4(1−𝑞)↑3  𝑓𝑜𝑟 0≤𝑞≤1� (2−𝑞)↑3  𝑓𝑜𝑟 1≤𝑞≤2�0 𝑓𝑜𝑟 𝑞>2  � 𝐶↓ℎ =15/(4𝜋ℎ↑2 )� 

q= 𝑟↓𝑗,𝑖 /ℎ = √�5 /2.5 <1

𝑊↓𝑗𝑖 = 45𝑟↓𝑗𝑖↑3 /4𝜋ℎ↑5  − 45𝑟↓𝑗𝑖↑2 /2𝜋ℎ↑4  + 15/𝜋ℎ↑2  

Calculation of Acceleration, Velocity and Location

39

𝑽↓𝒊 𝒏𝒆𝒘 = 𝑽↓𝒊 + 𝒂↓𝒊 ∗∆𝒕 =(𝟏,𝟏)+(𝟏.𝟓𝟒𝟐,𝟎.𝟕𝟎𝟓)=(𝟐.𝟓𝟒𝟐,𝟏.𝟕𝟎𝟓); 𝑿↓𝒊 𝒏𝒆𝒘 = 𝑿↓𝒊 + 𝑽↓𝒊 𝒏𝒆𝒘 ∗∆𝒕 =(2,2)+(𝟐.𝟓𝟒𝟐,𝟏.𝟕𝟎𝟓)=(𝟒.𝟓𝟒𝟐,𝟑.𝟕𝟎𝟓);

𝑎↓𝑖 = 𝑎↓𝑖↑𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 + 𝑎↓𝑖↑𝑣𝑖𝑠𝑐𝑜𝑠𝑖𝑡𝑦 + 𝑎↓𝑖↑𝑒𝑥𝑡𝑒𝑟𝑛𝑎𝑙 + 𝑎↓𝑖↑𝑔𝑟𝑎𝑣𝑖𝑡𝑦 =∑𝑗↑▒𝑃↓𝑖𝑗 𝛻𝑊(𝑟↓𝑖𝑗 )+  ∑𝑗↑▒𝑉↓𝑖𝑗 𝛻↑2 𝑊(𝑟↓𝑖𝑗 )+ (1/𝜌↓𝑖 ) 𝐹↓𝑖↑𝑒𝑥𝑡𝑒𝑟𝑛𝑎𝑙 +(0,−𝑔)=(−2.43∗10↑−1 ,−1.21∗10↑−1 )−3.5(−5.1∗10↑−1 ,−2.36∗10↑−1 )+(0,0)+(0,0)=(1.542,0.705)

Calculation of Acceleration, Velocity and Location

40

𝑽↓𝒋 𝒏𝒆𝒘 = 𝑽↓𝒋 +𝒂𝒋∗∆𝒕 =(𝟎.𝟓,𝟏)+(−𝟏.𝟓𝟒𝟐,−𝟎.𝟕𝟎𝟓)=(−𝟏.𝟎𝟒𝟐,𝟎.𝟐𝟗𝟓); 𝑿↓𝒋 𝒏𝒆𝒘 = 𝑿↓𝒋 + 𝑽↓𝒋 𝒏𝒆𝒘 ∗∆𝒕 =(4,3)+(−𝟏.𝟎𝟒𝟐,𝟎.𝟐𝟗𝟓)=(𝟐.𝟗𝟓𝟖,𝟑.𝟐𝟗𝟓);

𝑎↓𝑗 = 𝑎↓𝑗↑𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 + 𝑎↓𝑗↑𝑣𝑖𝑠𝑐𝑜𝑠𝑖𝑡𝑦 + 𝑎↓𝑗↑𝑒𝑥𝑡𝑒𝑟𝑛𝑎𝑙 + 𝑎↓𝑗↑𝑔𝑟𝑎𝑣𝑖𝑡𝑦 =∑𝑖↑▒𝑃↓𝑗𝑖 𝛻𝑊(𝑟↓𝑗𝑖 )+  ∑𝑖↑▒𝑉↓𝑗𝑖 𝛻↑2 𝑊(𝑟↓𝑗𝑖 )+ (1/𝜌↓𝑗 ) 𝐹↓𝑗↑𝑒𝑥𝑡𝑒𝑟𝑛𝑎𝑙 +(0,−𝑔)=−(−2.43∗10↑−1 ,−1.21∗10↑−1 )+3.5(−5.1∗10↑−1 ,−2.36∗10↑−1 )+(0,0)+(0,0)=(−1.542,−0.705)

Calculation Results

-1

7

1 52 3 4 6 7

3

6

5

4

2

1

0-1

0

ℎ𝑉↓𝑗𝑝𝑟𝑒 

𝑉↓𝑖𝑛𝑒𝑤 𝑉↓𝑗𝑛𝑒𝑤 

𝑉↓𝑖𝑝𝑟𝑒 

Mass/m

Density/ρ

Pressure/P

Viscosity/μ

Velocity/v

Location/(x,y) h ∆t

Particlei 1 1 1 1 (2.5,1.7) (2,1) 2.5 1

Particlej 1 1 1 1 (-1.0,0.3) (1,2) 2.5 1

For each time step: Ø  DiscretefromintegrationtosummationØ  NumericalapproximationforKernelFunctionØ  CalculatedensityforeachparticleØ  CalculatepressureforeachparticleØ  Calculatealltypeofaccelerationsforeachparticle,andsumitupØ  Findnewvelocitiesandpositionsbyusingthesamesummationmethodas

before…

Numerical Example

43

Simulation Example

Ø  SoftwareIntroduction

Ø  Softwarepreparation

Ø  NumericalComputation

Ø  ParameterSetup

Ø  Coding

Ø  Visualizationofresult

Software Introduction

Ø  Codesource:Opensource.

Ø  Dimension:2-D&3-D.

Ø  Numericallycomputation.

Ø  VisualizationroutineswithParaView.

Software preparation

Ø  Windows:IntelVisualFortranSilverfrostFTN95GNUgfortrancompileronCygwin

Ø  Mac:GNUgfortran

Ø  Linux:GNUgfortrancompileronCygwin ParaView5.3.0(Visualization)

Numerical Computation

Numericallycomputefree-surfaceelevation:

1)Foragiven(X,Y)position.

2)ComputenumericalMASSatdifferentZpositionsusingMASSvaluesofneighbouringfluidparticles:

3)Wewillchooseaswaveelevation:theZvalueforwhichm=0.5*m(reference)

Numerical Computation

Numericallycomputevelocity:1).Foragivenlocation.2).Computenumeicalvelocityusingvelocityvaluesofneighbouringfluidparticles:

Numerical Computation

Numericallycomputepressure:1).Foragivenlocation.2).ComputenumericalPRESSUREusingPRESSUREvaluesofneighbouringfluidparticles:

Numerical Computation Numericallycomputeforces:1)Forarangeofboundaryparticles2)WecomputenumericalACCELERATIONofthoseboundaryparticlessolvingtheparticleinteractionswithfluidneighbouringparticles:3)SummationofACCELERATIONvaluesofthoseboundaryparticles:

Parameter Setup

KernelFunction:Ø  Gaussian

Ø  Quadratic

Ø  Cubic

Ø  Wendland

Parameter Setup

Time-steppingalgorithm:Ø  Predictor-Corrector

Ø  Verlet

Ø  Symplectic

Ø  Beeman

Parameter Setup

Viscositytreatment:Ø  Artificialviscosity

Ø  Laminar

Ø  Laminarviscosity+Sub-ParticleScale

Parameter Setup

DensityFilter:Ø  ZerothOrder–ShepardFilter

Ø  FirstOrder–MovingLeastSquares(MLS)

Other choices

Ø  Kernelcorrection(None)

Ø  Equationofstate(Tait’sequation)

Ø  Boundaryconditions(Dalrymple)

Ø  Geometryofthezone(Box)

Ø  Initialfluidparticlestructure(BCC)

Coding

InCygwin,withvirtualLinuxplatform:

Ø  CompilesSPHysicsgen_2DbySPHysicsgen.make.

Ø  RunSPHysicsgen_2Dwithinputfile(Case1.txt),thenobtainaoutputfile(Case1.out).

Ø  CompileandgenerateSPHysics_2DbySPHysics.make,placestheSPHysics_2Dexecutable.

Ø  ExecuteSPHysics_2D.

Ø  VisualizebyParaView.

Bug:‘Xilink’CommandinSPHYSICS.makfilecannotberecognize

bynewversionofCygwin.

Solution:RemodifythePHYSICS.makfileasfollows:

Coding-bug and solution

Visualization of result

Visualization of result (t=0, 15,30,45)

Visualization of result (t=60, 75,90,105)

Visualization of result (t=120, 135,149)

Example Applications

62

Uses in astrophysics Smoothed-particlehydrodynamics'sadaptiveresolution,numericalconservationofphysicallyconservedquantities,andabilitytosimulatephenomenacoveringmanyordersofmagnitudemakeitidealforcomputationsintheoreticalastrophysics.

Uses in fluid simulation

First,SPHguaranteesconservationofmasswithoutextracomputation.Second,SPHcomputespressurefromweightedcontributionsofneighboringparticles.Finally,SPHcreatesafreesurfacefortwo-phaseinteractingfluidsdirectly.ForthesereasonsitispossibletosimulatefluidmotionusingSPHinrealtime.

Uses in solid mechanics ThisfeaturehasbeenexploitedinmanyapplicationsinSolidMechanics:metalforming,impact,crackgrowth,fracture,fragmentation,etc.

Questions

66