Mesh Free Methods for Fluid Dynamics Problems - TU/e · PDF fileDe. v. CONTINUITY EQ. ... 2003 Mesh free methods for fluid dynamics problems 1111. Updating of smoothing length h

Mesh Free Methods for Mesh Free Methods for Fluid Dynamics ProblemsFluid Dynamics Problems

RemoRemo MineroMinero

17 December, 200317 December, 2003

17 Dec, 200317 Dec, 2003 Mesh free methods for fluid dynamics problemsMesh free methods for fluid dynamics problems 22

OutlineOutline

IntroductionIntroduction

Smoothed Particle Hydrodynamics (SPH)Smoothed Particle Hydrodynamics (SPH)

MeshlessMeshless PetrovPetrov--GalerkinGalerkin Method (MPGM)Method (MPGM)

Local Radial Point Interpolation Method (LRPIM)Local Radial Point Interpolation Method (LRPIM)

ConclusionsConclusions


Why Why MfreeMfree methods in CFD?methods in CFD?

Standard techniques (FDM, FVM, FEM) have limitations when Standard techniques (FDM, FVM, FEM) have limitations when dealing with:dealing with: large distortionslarge distortions free surfacesfree surfaces deformable boundariesdeformable boundaries

Remedies: reRemedies: re--meshing techniquesmeshing techniques can be expensive and really complicated (especially in 3D)can be expensive and really complicated (especially in 3D)

MFreeMFree methodsmethods Only displacement of points within the domain: easy insertion orOnly displacement of points within the domain: easy insertion or

deletion of pointsdeletion of points


MfreeMfree methods for methods for PDEsPDEs: general idea: general ideaDomain representationDomain representation node generationnode generation

Representation of u (and Representation of u (and its derivatives) in a support its derivatives) in a support domain through shape domain through shape functionsfunctions

support domain definitionsupport domain definition shape function creationshape function creation

Plug in the representation of f Plug in the representation of f into the equations, get a into the equations, get a discrete equivalent, solve it discrete equivalent, solve it

( ) ( ) ( )∑=

−Φ=n

1iii uu xxxx


Smoothed Particle HydrodynamicsSmoothed Particle HydrodynamicsAuthors:Authors: Lucy (1977); Lucy (1977); GingoldGingold and Monaghan (1977, 1982); Monaghan and Monaghan (1977, 1982); Monaghan

(1987, 1988) (1987, 1988)

IdeaIdea

Definitions:Definitions: kernel approximation and particle approximationkernel approximation and particle approximation W = weight or smooth or kernel functionW = weight or smooth or kernel function h = smoothing lengthh = smoothing length ΩΩ = support domain= support domain

( ) ( ) ( ) ξξxξx duu −δ= ∫+∞

∞−

( ) ii

i Vh,Wu ∆−≅ ∑ ixx( ) ( ) ( )∫Ω −= ξξxξx dh,Wuuh

Exact integral representation of u(x)

SPH


Choice of the Weight FunctionChoice of the Weight Function

hd ixx −

=

exponential weight functionW3 =

quartic splineW2 =

cubic spline weight function (defined piecewise)W1 =

h variable from point to point ! method very flexible


Properties of weight functionsProperties of weight functions

PositivityPositivity: : W W > 0, in > 0, in ΩΩ

Compactness: Compactness: W = 0, outside W = 0, outside ΩΩ

Unity:Unity: ∫∫ W( W( xx -- ξξ ,h) d,h) dξξ = 1 = 1

because of Cbecause of C00 consistencyconsistency

Monotonically decreasing Monotonically decreasing away from 0

Delta function behavior in the limit

Cp consistency if ∫∫ xxpp W( W( xx -- ξξ ,h) d,h) dξξ = 0= 0

No C1 consistency close to boundary: Reproducing Kernel Particle Method (RKPM) in order to guarantee it

away from 0


NavierNavier--Stokes (NS) equationsStokes (NS) equations

v⋅∇ρ−=ρDtD

αβ

αβα

+∂σ∂

ρ= Fv

x1

DtD

αβαβεερµ+⋅∇

ρ−=

2p

DtDe v

CONTINUITY EQ.

MOMENTUM EQ.

ENERGY EQ.

αβαβαβ τ+δ−=σ p

αβαβ µε=τ

( ) αβα

β

β

ααβ δ⋅∇−

∂∂+

∂∂=ε v

xv

xv

32

being:

Newtonian fluid

The superscripts α, β, and γ denote the coordinate directions


Representation of variablesRepresentation of variables

Two steps in SPH:Two steps in SPH: Kernel approximation + Particle approximationKernel approximation + Particle approximation

For a PDE, we express also the derivatives of fFor a PDE, we express also the derivatives of f

ij

ij

ij

ij

ij

ij

ij

jiiji r

Wrr

Wr

W∂

∂=

∂∂−

=∇xxx

∑=

ρ=

n

1jijj

j

ji Wf

mf( ) ( ) ( ) ξξxξx dh,Wff −= ∫

( ) ( ) ( )h,rWh,Wh,WW ijjijiij =−=−= xxxx

ijij

n

1j j

ji Wf

mf ∇

ρ=∇ ∑

=

being:

being:


Discrete representation of NS equationsDiscrete representation of NS equations

∑=

=ρn

1jijji

Wm( )∑=

∇⋅−=ρ n

1jijijij

i

WvvmDtD

α

=β

αβαβ

=α

α

+∂∂

ρρεµ+εµ

+∂∂

ρρ+

−= ∑∑ Fxx

v n

1j i

ij

ji

jjiij

n

1j i

ij

ji

jij

i

Wm

Wppm

DtD

αβ

= =β

α

=α

βαβ δ

∇⋅

ρ−

∂∂

ρ+

∂∂

ρ=ε ∑ ∑∑

n

1j

n

1jijiji

j

j

i

ijji

j

jn

1j i

ijji

j

j

iW

m32

xW

vm

xW

vm

v

αβαβ

=

εερµ+∇⋅

ρρ+

= ∑ iii

iij

n

1jij

ji

jij

i 2W

ppm

21

DtDe v

or

with:

1 2 21 vs

! mass conservationbut more CPU time and some edge effect

2


Updating of smoothing length hUpdating of smoothing length h

Key role of h:Key role of h: h too small, n too small, results no accurateh too small, n too small, results no accurate h too big, local information smoothed outh too big, local information smoothed out

h treated as a variable:h treated as a variable: known at the beginning:known at the beginning: updated solving:updated solving:

ok for slow varying density, more complicated procedure for fastok for slow varying density, more complicated procedure for fastexpansion/contraction (e.g. in gases)expansion/contraction (e.g. in gases)

( ) 0i

30i

N

1jj h

34m

0i

ρπ=∑=

( )∑=

∇⋅−ρ

−=ρρ

−=niN

1jij

ni

nj

nij

Dni

ni

ni

Dni

ni

ni Wm

nh

DtD

nh

DtDh vv nD = number of

dimensions


Problems close to the boundary due to one side summation: introduction of virtual points virtual points outside the boundary with outside the boundary with ρρii, p, piiand and vvii for solid boundariesfor solid boundaries

Efficient search of points within the support domain

Code StructureCode Structure

Numerical simulation of shocks: avoids unphysical particle penetration


Lid driven cavity problemLid driven cavity problem

x=0.5

÷

y≈0.8

Re = 10 and 41x41 particles

Comparisons with FDM having the same grid


Free Surface FlowFree Surface Flow

S

H

Subscripts:

exp experiments

m Monaghan

p SPH


MeshlessMeshless PetrovPetrov--GalerkinGalerkin (MLPG) method(MLPG) method

Authors:Authors: Lin and Lin and AtluriAtluri (2001); G.R. Liu and (2001); G.R. Liu and YanYan (2001)(2001)

Procedure:Procedure: local residual formulation integrated over a local local residual formulation integrated over a local

(simple) (simple) quadraturequadrature domaindomain field variables approximated at any point using field variables approximated at any point using

Moving Least Squares (MLS)Moving Least Squares (MLS)


Moving Least Squares (MLS)Moving Least Squares (MLS)

ΩΩ

Approximation of u:Approximation of u:

pp((xx) = vector of polynomials (basis functions)) = vector of polynomials (basis functions) m = number of polynomialsm = number of polynomials aaTT((xx) = a) = a00(x) a(x) a11(x) a(x) amm(x) = vector to be (x) = vector to be

determineddetermined

aa((xx) depends on u() depends on u(xxii), being x), being xii in in Ω

( ) ( ) ( ) ( ) ( )xaxpxxx Tj

m

1jj

h apu ==∑=

n points here

Ω


Determination of Determination of aa((xx))

( ) ( ) ( )[ ]∑=

−−=n

1i

2ii

Ti uWJ xaxpxx

0J =∂∂a

( ) ( ) ( ) ( )( ) ( )∑∑∑== =

− Φ==n

1iii

n

1i

m

1jiji

1j

h uupxu xxBxAx

Existence of A-1 if n>>m

weight function shape function


Properties of MLSProperties of MLS

CCkk consistency, if k the complete order of monomials in consistency, if k the complete order of monomials in pp easy to set the desired level of consistencyeasy to set the desired level of consistency

No No KroneckerKronecker δδ function property:function property: special attention for essential BCspecial attention for essential BC

Tasks for weight function W:Tasks for weight function W: give smaller weights to residuals give smaller weights to residuals

far from xfar from x ensure smoothness when x movesensure smoothness when x moves in practice: same W as SPHin practice: same W as SPH

( ) ( )∑=

≠Φ=n

1ijijij

h uuxxu


NavierNavier--Stokes equations (Stokes equations (ΨΨ--ωω form)form)

∇=∇⋅∂∂−ω∇=ψ∇⋅

ω=ψ∇

TTxTRaPrPr

2

2

2

u

u

yu

xv

∂∂−

∂∂=ω

xv

yu

∂Ψ∂−=

∂Ψ∂=

( )

( )0d

0dTTWxTRaPrPrW

0dW

i

i

i

2i

2i

2i

=Ω

=Ω∇−∇⋅

∂∂+ω∇−ψ∇⋅

=Ωω−ψ∇

∫∫

∫

Ω

Ω

Ω

u

u

local residual formulation

( ) ( )

( ) ( )

( ) ( )∑

∑

∑

=

=

=

Φ=

ωΦ=ω

ΨΦ=Ψ

n

1iii

n

1iii

n

1iii

TT xx

xx

xx

derivatives to be expressed too ! complicated

W = weight function;used the same as SPH

strong form

Ψ = stream function

ω = vorticity

Pr = Prandtl number

Ra = Railegh number


Ψ

Ψ

Ψ

T=cost

T=cost

T=cost

T

T

T

Ra = 103

Ra = 104

Ra = 105

Natural Convection Natural Convection in a Square Cavityin a Square Cavity

special treatment for ω on the boundary

driver is ∆T

Simulations with different distribution of points

256 x 256 257 x 257


Comparison with benchmark solutionComparison with benchmark solution

Nu = local Nusselt number


Local Radial Point Interpolation Method Local Radial Point Interpolation Method (LRPIM)(LRPIM)

Authors:Authors: G.R. Liu and G.R. Liu and YanYan (2001)(2001)

Idea:Idea: want to have the want to have the KroneckerKronecker delta function property in delta function property in

order to better deal with BCorder to better deal with BC still have field variables (and their derivatives) still have field variables (and their derivatives)

expressed in terms of shape functionexpressed in terms of shape function


Point Interpolation MethodPoint Interpolation Method

( ) ( ) ( )axpxx Tn

1iii

h apu ==∑=

Polynomial based PIM; a to be determined

( ) n,,1iu iT

i …=∀= axp a determined imposing the Kroneckerdelta function property

P = momentum matrix

Does P-1 exist?PaU = UP 1a −=

( ) ( ) ( ) i

n

1ii

1Th u)(u ∑=

− Φ=== xUxΦUPxpx u expressed in terms of shape function


Characteristics of PIMCharacteristics of PIM

P-1 does not exist with this configuration of points Remedy 1: a small shift


Radial Point Interpolation Method (RPIM)Radial Point Interpolation Method (RPIM)

( ) ( ) ( )axRx Ti

n

1ii

h aRxu ==∑=

( ) ( )rRR ii =x ( ) ( )2i

2i yyxxr −+−=

Ri radial basis functions

being (in 2D):

( ) ( )∑Φ=i

iih uu xx

Momentum matrix R symmetric and invertible for any distribution of points


Solve equations + resultsSolve equations + results

Plug in f(x) = Plug in f(x) = ΣΦΣΦiiffii (and the derivatives of f)(and the derivatives of f) in local in local residual form of residual form of NavierNavier--Stokes equationsStokes equations


ConclusionsConclusions

MfreeMfree methods interesting for CFD problems to methods interesting for CFD problems to overcome some limitations of traditional methods overcome some limitations of traditional methods

Several Several MFreeMFree methods applied in CFD:methods applied in CFD: they differ for shape constructionthey differ for shape construction they show good accuracy in some test problemsthey show good accuracy in some test problems

CriticsCriticsSome details missing, e.g. how to deal with BCSome details missing, e.g. how to deal with BC

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Mesh Free Methods for Fluid Dynamics Problems - TU/e · PDF fileDe. v. CONTINUITY EQ. ... 2003 Mesh free methods for fluid dynamics problems 1111. Updating of smoothing length h