MESH-FREE METHODS FOR HEAT TRANSFER AND FLUID …fluid.ippt.gov.pl/sarler/MESHFREE_AMAS.pdf · ppcpπ = =∑ rrr F= (){}1, , , , , , , , ,22 2 π ppppppppppppp= x y zxy yz xzx y z

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MESH-FREE METHODS FOR HEAT TRANSFER AND FLUID FLOW

Boidar arlerLaboratory for Multiphase Processes

Nova Gorica Polytechnic, Nova Gorica, Slovenia

Institut Podstawowych Problemov TechnikiPolska Akademia NaukCentrum Doskonalosci

Nowoczesne Materialy i KonstrukcjeWarszawa, Polska, kwiecien 14, 2003


MESH METHOD MESH-FREE METHOD

MOTIVATION

BASIC LITERATURE

S.N. Atluri, S. Shen, The Meshless Method, Tech Science Press, Los Angeles, 2002.

G.R. Liu, Mesh Free Methods, CRC Press, Boca Raton, 2003.

NEW COMPUTATIONAL DISCIPLINE

GENERAL TRANSPORT EQUATION

( ) ( ) ( )( ) ( )v D Stρ ρ Φ Φ∂ϒ Φ +∇ ⋅ ϒ Φ = ∇ ⋅ ∇Φ +

∂r

Boundary conditionsRND Γ∪Γ∪Γ=Γ

;D DpΦ = Φ ∈Γ

r

;N Np

nΓ

∂Φ = Φ ∈Γ

∂r

( );R R Rref p

nΓ

∂Φ = Φ Φ −Φ ∈Γ

∂r

Dirichlet

Neumann

Robin


Initial conditions

( ) 00, ;p t pΦ = Φ ∈Γ∪Ωr r

Solution

( )0, ;p t t pΦ +∆ ∈Γ∪Ωr r

DIFFUSION TENSOR

11 12 13

21 22 23

31 32 33

d d dD d d d

d d d

Φ

=

1 0 00 1 00 0 1

I =

'D D I DΦ Φ Φ= +

11 12 13

21 22 23

31 32 33

'd D d d

D d d D dd d d D

Φ

Φ Φ

Φ

−

= − −

Splitting

POISSON REFORMULATION

Θ⋅∇+=Φ∇ θ2

( ) /S Dt

θ ρ Φ Φ∂ = ϒ Φ − ∂

( ) /v D DρΦ Φ = ϒ Φ − ∇Φ

'rΘ

( ) ( )0 0 /S Dt

ρ ρθ Φ Φ ϒ Φ − ϒ Φ≈ − ∆

Approximate expression for scalar inhomogenous part

Scalar inhomogenous part

Vector inhomogenous part

TAYLOR EXPANSION

( )Φ−Φ+≈ Φ,θθθ

( ) ( )Φ−Φ⋅∇+⋅∇+Φ−Φ+=⋅∇+=Φ∇ ΦΦ ,,2 ΘΘΘ θθθ

( )Φ−Φ+≈ Φ,ΘΘΘ

Final Poisson form

ITERATION STRATEGIES

Nonlinear

Highly non-linear

2 1j j jθ+∇ Φ = +∇ ⋅Θ

( ) ( )2, ,

1 1 1j jj jj j jθ θ Φ+ + +

Φ∇ = + −Φ +∇Φ Φ Φ⋅ +∇⋅ −ΦΘ Θ

( )1 1j j j jrelcθ θ θ θ− −= + − ( )1 1j j j j

relc− −∇ ⋅ = ∇ ⋅ + ∇ ⋅ −∇ ⋅Θ Θ Θ Θ

( ) ( ) ( )p pc T vc T k Ttρ ρ

∂+∇ ⋅ = ∇ ⋅ ∇

∂r

Tcp≡Φ ( )ϒ Φ ≡ Φ

p

kDc

Φ ≡ ' 0D ςξΦ ≡ 0SΦ ≡

0Φ∆

−Φ∆

≡tkc

tkc pp ρρ

θtkcp

∆≡Φ

ρθ,

x xvρΘ ≡ Φ Φ≡Θ yy vρ

xx vρ≡Θ Φ, yy vρ≡Θ Φ,

EXAMPLE SCALAR (HEAT) EQUATION

( ) ( ) ( ) ( )1 refP T Ttρ ρ µ ρ β

∂ +∇ ⋅ = −∇ +∇ ⋅ ∇ + − − ∂v vv v a

x yv vΦ ≡ Φ ≡

( ) ( )ϒ Φ = Φ ϒ Φ = Φ

D Dµ µΦ Φ≡ ≡

EXAMPLE VECTOR (MOMENTUM) EQUATION

' '0 0D Dςξ ςξΦ Φ≡ ≡

EXAMPLE VECTOR (MOMENTUM) EQUATION

( ) ( ), ,1 1x x ref y y refS P a T T S P a T Tρ β ρ β ≡ − + − − ≡ − + − −

0 0S S

t t t tρ ρ ρ ρ

θ θµ µ µ µ µ µ

≡ Φ − Φ − ≡ Φ − Φ −∆ ∆ ∆ ∆

, ,t tρ ρ

θ θµ µΦ Φ≡ ≡

∆ ∆

2x x xvρ ρ

µ µΘ ≡ Φ Θ ≡ Φ

2y y yvρ ρ

µ µΘ ≡ Φ Θ ≡ Φ

, ,2x x xvρ ρµ µΦ ΦΘ ≡ Φ Θ ≡

, , 2y y yvρ ρµ µΦ ΦΘ ≡ Θ ≡ Φ

METHOD OF PARTICULAR SOLUTIONS

θΦParticular solution

0Φ+Φ=Φ θ

0ΦHomogenous solution

Solution

PARTICULAR SOLUTION θΦParticular solution satisfies the Poisson equation but

not necessarily the boundary conditions

( ) ( )2, ,

θ θ θ Φ Φ∇ Φ = + Φ −Φ +∇⋅ +∇ ⋅ Φ −ΦΘ Θ

2 θ θ∇ Φ = +∇⋅Θ

HOMOGENOUS SOLUTION

0ΦHomogenous solution satisfies the Laplace equation

with the modified boundary conditions

002 =Φ∇

0 ;N Np

n nθ

Γ Γ

∂ ∂Φ = Φ − Φ ∈Γ

∂ ∂r

( )0 ;R R R

ref pn n

θ

Γ Γ

∂ ∂Φ = Φ Φ −Φ − Φ ∈Γ

∂ ∂r

0 ;D DpθΦ = Φ −Φ ∈Γ

r

AN INDUSTRIAL EXAMPLE OF OUR DRBEM RESEARCHflow simulation in D.C. casting of aluminum alloys

B.arler & J.Perko (2000)

challenges ?

motivation ?

massmomentumenergyspecies

nonlinear b.c.nonlinear m.p.phase-change


GLOBAL WEAK FORMS

( )2 ( ) ;p S p p∇ Φ = ∈Ωr r r

( )( ) ; DDp p pΦ = Φ ∈Γ

r r r

( )( ) ; NNp p p

nΓ

∂Φ = Φ ∈Γ

∂r r r

Dirichlet (or essential) boundary conditions

Neumann (or natural) boundary conditions

Poisson equation

( )( ) ;p p pΦ ≈ Φ ∈Ω∪Γr r r%

DOMAIN ERROR

BOUNDARY ERROR

( ) ( )2 ( ) 0;p p S p pΩ = ∇ Φ − ≠ ∈Ωr r r r%R

APPROXIMATE SOLUTION

( ) ( )( ) 0;DD

Dp p p pΓ

= Φ −Φ ≠ ∈Γr r r r%R

( ) ( )( ) 0;NN

Np p p pnΓΓ

∂= Φ −Φ ≠ ∈Γ∂

r r r r%R

CONCEPT OF TRIAL FUNCTION

BEST SOLUTION

1( ) ( )

N

n nn

p c pψ=

Φ =∑r r%

Nullification of errors ΩR ΓRand in some fashion

CONCEPT OF ADMISSIBLE FUNCTION

Trial function should satisfy minimum requirements

0C 1C continuous, for example

( )n pψr

CONCEPT OF ADMISSIBLE FUNCTION

0C

1C( )n pψr

xp

POINT COLLOCATION METHODWe consider the errors vanish in certain points

; 1,2,...,jp j NΩ=r

; , 1, 1,...,jp j N N N NΩ Ω Ω Γ= + +r

domain points

boundary points

POINT COLLOCATION METHODWe introduce boundary conditions indicators

( ) 1;0;

DjD

j Dj

pp

pχ

∈Γ= ∉Γ

r ( ) 1;0;

NjN

j Nj

pp

pχ

∈Γ= ∉Γ

r

Equations for determination of the coefficients

( ) 0jpΩ =rR

( ) ( ) ( ) ( ) 0D ND N

j j j jp p p pχ χΓ Γ

+ =r r r rR R

( ) ( )2

1

( )N

n nn

p c p S pψΩ=

= ∇ −∑r r rR

( ) ( )1

( )D

N

n n Dn

p c p pψΓ

=

= −Φ∑r r rR

( ) ( )1

( )N

N

n n Nn

p c p pnψ

Γ= Γ

∂= −Φ

∂∑r r rR

CASE 1: Number of trial functions equal to the numberof constraining points

N N NΩ Γ= +

CASE 2: Number of trial functions smaller than the numberof constraining points

N N NΩ Γ< +

We seek an approximate solution*j

c

( )j jp εΩ =rR

( ) ( ) ( ) ( )D ND N

j j j j jp p p pχ χ εΓ Γ

+ =r r r rR R

such that the summed point-wise square error is minimised

* 0j jicε ε

∂=

∂

Limiting case where the number of collocation points tendsto infinity

( ) ( ) ( )2 2 2 2D D N N

D N

p d p d p dυ υ υ εΩ Ω Γ Γ Γ ΓΩ Γ Γ

Ω+ Γ + Γ =∫ ∫ ∫r r rR R R

υΩ ≡ penalty parameters

WEIGHTED INTEGRAL SQUARE ERROR

CONCEPT OF CONTIGUOUS SUBDOMAINS

CONCEPT OF NON-CONTIGUOUS SUBDOMAINS

SUBDOMAIN/AVERAGE ERROR METHODContiguous (non-overlapping) subdomains

k kΩ Γ

We may set

( ) 0jp dΩΩ

Ω =∫r

k

R

( ) ( ) ( ) ( ) 0D ND N

j j j jp p p p dχ χΓ Γ

Γ

+ Γ = ∫r r r r

k

R R

THE FINITE VOLUME METHOD

( ) ( )2 0n n j jc p d S p dψΩ Ω

∇ Ω− Ω =∫ ∫r r

k k

( ) ( ) 0n n j jc p d S p dψΓ Ω

∇ ⋅ Γ − Ω =∫ ∫rr r

k k

THE WEIGHTED RESIDUAL METHOD

( ) ( ) 0jp p dνΩΩ

Ω =∫r r

#

( )jpνr test function

The domain integral should be enforced for all test functions !

The test function is in genaral expressed as

( ) ( )j n n jp b pν ν=r r

SPECIAL CASES OF THE TEST FUNCTIONS

( )1;0;

k

k

pp

pν

∈Ω= ∉Ω

rr

r The Finite Volume Method

( ) ( )jp p pν δ= −r r r

The Point Collocation Method

( ) ( )p pν Ω=r r

# The Least Square Error Method

THE GLOBAL UNSYMMETRIC WEAK FORM (GUWF-I)

( ) ( ) ( )2 0j jp S p p dνΩ

∇ Φ − Ω = ∫r r r%

Requirements on

Requirements on

( ) 1: jp CΦ Φ ∈r% %

:ν none

THE GLOBAL SYMMETRIC WEAK FORM (GSWF)

( ) ( ) ( )2 0j jp S p p dνΩ

∇ Φ − Ω = ∫r r r%

Requirements on

Requirements on( ) 0: jp CΦ Φ ∈r% %

( ) 0: jp Cν ν ∈r

( ) ( ) ( ) ( ) ( ) ( ) 0j j j jp p d p p S p p dν ν νΓ Ω

∇Φ ⋅ Γ − ∇Φ ⋅∇ + Ω = ∫ ∫rr r r r r r% %

Galerkin method:

Petrov-Galerkin method:

( ) ( )n np pνΦ =r r%

( ) ( )n np pνΦ ≠r r%

THE GLOBAL UNSYMMETRIC WEAK FORM (GUWF-II)

( ) ( ) ( ) ( )j jp p p p dν νΓ

∇Φ −Φ ∇ ⋅ Γ + ∫rr r r r% %

Requirements on

Requirements on

( ) 1: jp Cν ν ∈r

:Φr

none

( ) ( ) ( )2 0j jp S p p dνΩ

+ ∇ − Φ Ω = ∫r r r%

CONCEPT OF CONTIGUOUS SUBDOMAINS

CONCEPT OF NON-CONTIGUOUS SUBDOMAINS

MESH-FREE INTERPOLATION OF TRIAL AND TEST FUNCTIONS

Shepard Functions InterpolationShepard (1968)Reproducing Kernel Particle InterpolationLiu, Chen, Uras & Chang (1996)Partition of Unity InterpolationBabuka & Melenk (1997)Moving Least Squares InterpolationLancaster & Salkauskas (1981)Point Interpolation MethodLiu, (1995)Radial Basis Function InterpolationWendland (1995)Method of Fundamental SolutionsChen, Golberg (1998)

CONCEPT OF NON-CONTIGUOUS TRIAL AND ERROR FUNCTIONS

EXAMPLE:Test function: Moving Least SquaresTrial function: Gaussian

SHEPARD INTERPOLATION

( ) ( ) ( )1

N

n nn

p p pξ=

=∑r r rF = F

( ) ( )

( )1

nn N

nn

w pp

w pξ

=

=

∑

rr

r

( ) | | ; 2n nw p p p µ µ−= − =r r r

1968

RADIAL BASIS FUNCTIONS INERPOLATION

( )

5 2 3 4

2 3 41 8 40 48 25 5 ;

0 ;

n n n n nn

n

n

r r r r r r rp r r r r r

r r

ψ

− + + + + ≤ = >

r SS S S S S

S

=

( )

6 2 3 4 5

2 3 4 51 6 36 82 72 30 5 ;

0 ;

n n n n n nn

n

n

r r r r r r r rp r r r r r r

r r

ψ

− + + + + + ≤ = >

r SS S S S S S

S

=

1995

THE MOVING LEAST SQUARES APPROXIMATION

( ) ( ) ( )1

N

n nn

p p c pπ=

=∑r r rF =

( ) 2 2 21, , , , , , , , ,x y z x y y z x z x y zp p p p p p p p p p p p pπ =r r

( ) 1, , ,x y zp p p pπ =r r

Linear Basis

Quadratic Basis


EVOLUTION

1994-2000

FSB-INTRBF

1987-1993

FSB-INTD-INT

2001-

FSRBF


MESHLESS METHODS EVOLVING FROMFUNDAMENTAL SOLUTION BASED NUMERICAL METHODS

BDIM : use of fundamental solution boundary polygonisationdomain polygonisationdomain integrationbounday integration

( ) ( ) ( )( ) ( )v D Stρ ρ Φ Φ

∂ϒ Φ +∇ ⋅ ϒ Φ = ∇ ⋅ ∇Φ +

∂r

RESEARCH DIRECTION: BDIM-DRBEM

DRBEM: use of fundamental solutionboundary integration boundary polygonisationuse of radial basis functionsdomain pointisation

BDIM : use of fundamental solution boundary polygonisationbounday integration domain polygonisationdomain integration

Semi mesh-free !

RESEARCH DIRECTION: DRBEM-DRMFS

DRBEM: use of fundamental solutionboundary integration boundary polygonisationuse of radial basis functions domain pointisation

DRMFS: use of fundamental solution boundary pointisationfictitious boundaryuse of radial basis functionsdomain pointisation Polygon-free !

RESEARCH DIRECTION: DRMFS-RBFCM

DRMFS: use of fundamental solutionboundary pointisationfictitious boundaryuse of radial basis functions domain pointisation

RBFCM: boundary pointisationuse of radial basis functionsdomain pointisation

Polygon-free !

BDIM

DRBEM

DRMFS

RBFCM


Motivation

RBF interpolation

RBF solution of PDE

Advanced solution strategies

Natural convection in porous media - Darcy model

Thermal design of hollow bricks

Natural convection in solid-liquid system

Stefan problem

SCOPE

( ) ( )n np p sψ ψ= −r r

where is the Euclidean norm•

RADIAL BASIS FUNCTIONS

: is a continuous function R Rψ + →

(0) 0ψ ≥

is a continuous function ψ

; , ,p p i x y zξ ξ ξ= =rr

RADIAL BASIS FUNCTION APPROXIMATION

( ) ( )n np pψ ς≈r rF

1n nm mς −= Ψ F( ) ( )m m n m n mn np pψ ς ψ ς≡ = ≡r rF F

RBF Approximation

Calculation of Coefficients (Collocation, Least Squares)

Cartesian Coordinate System

( ) ( )n np p sψ ψ= −r r r

RADIAL BASIS FUNCTIONS - THEORY

1982 R. Franke: interpolation of scattered data, comparisonof different RBFs

1984 S. Stead: calculation of partial derivatives

1990 E.J. Kansa: approximation of surfaces1990 E.J. Kansa: parabolic, hyperbolic, elliptic PDE

1996 G.E. Fasshauer: symmetric solution of PDE1992 Z. Wu: Hermite-Birkhoff interpolation

nr3

nr2 2 where is a shape parameter.nr c c+

2

2 1

log , k 1, in 2D

, k 1, in 3D

kn n

kn

r r

r −

≥

≥22

nc re−

RADIAL BASIS FUNCTIONSGLOBALLY SUPPORTED

Conical radial:Cubic radial:

Multiquadric

Polyharmonic splines

Gaussians

EXAMPLES OF RADIAL BASIS FUNCTIONS

Polyharmonic splines:

( )2 2( )r r log rβϕ −=

Radials:

2 1( )r r βϕ −=

Compactly supported:3

1: 1- 3 1( )

1: 0

r r rc c crrc

ϕ

< + = >

ϕ = +r c2 21 rϕ = +

GLOBALLY SUPPORTED RBFs

(1 ) , 0 1(1 )

0, 1

nn r r

rr+

− ≤ ≤− =

>For d=1: ϕ

ϕ

ϕ

= − ∈

= − + ∈

= − + + ∈

+

+

+

( )( ) ( )( ) (8 )

11 3 11 5 1

0

3 2

5 2 4

r Cr r Cr r r C

For d=2, 3: ϕ

ϕ

ϕ

ϕ

= − ∈

= − + ∈

= − + + ∈

= − + + + ∈

+

+

+

+

( )( ) ( )( ) ( )( ) ( )

11 4 11 35 18 31 32 25 8 1

2 0

4 2

6 2 4

8 3 2 6

r Cr r Cr r r Cr r r r C

Wendlands CS-RBFs

RADIAL BASIS FUNCTIONSCOMPACTLY SUPPORTED

ϕ = − +( )1 2r ϕ = − ++( ) ( )1 4 14r r

COMPACTLY SUPPORTED RBFs


PARTICULAR SOLUTION ONCE AGAIN

θΦParticular solution satisfies the Poisson equation but not necessarily the boundary conditions

( ) ( )Φ−Φ⋅∇+⋅∇+Φ−Φ+=Φ∇ ΦΦ ,,2 ΘΘθθ

2n nψ ς∇ Φ =

2 n nψ ψ∇ =

n nψ ςΦ =

4

2

41 116 32 ( )

( ) l

ln

nr r

r

r

r r rψ

ψ =

= −

( )32 2 2 2 2 2 21 1

2 2

9 ( ) ln +c 4

( ) +c

c

+c

r c r r

r r

c c r

ψ

ψ = − + +

=

+

( ) ( )21 3

2 2 2 202 2

2rr c r cψ

− −= + + +

AXISYMMETRIC MULTIQUADRICS

( ) ( )21 1

2 2 2 202 2

4rr c r cψ

+ −= + − +

( )1

2 2 212

r cψ+

= +

( )3

2 2 2112

r cψ+

= +

20

3

2 ( ) ( )ArK k D k

l lψ

π π= +


1 ( )A lE kψπ

=

3

( )18Al C kψπ

=

20( ) ( )

2ArlE k K k

lψ

π= −

( )1/ 2/ 2

2 2

0

( ) 1 sinK k k dπ

ϕ ϕ−

= −∫


( )1/ 2/ 2

2 2

0

( ) 1 sinE k k dπ

ϕ ϕ+

= −∫2 2( ) 2(2 ) ( ) (1 ) ( )C k k E k k K k= − − −

2

1( ) ( )1

D k E kk

=−

2 2 2( ) ( )r r z zl p s p s c= − + − + 24 r rp skl

=

RADIAL BASIS FUNCTION APPROXIMATION

( ) ( ) ( ) 1n n nmn mp p pψ ψς −≈ = Ψ

r r rF F

( ) ( ) ( ) 1nn n nm mp p p

p p pξ ξ ξ

ςψ ψ −∂ ∂ ∂≈ =

∂ ∂Ψ

∂r r r FF

( ) ( ) ( )2 2 2

2 2 21

n nn nm mp p pp p pξ ξ ξ

ςψ ψ −∂ ∂ ∂≈ =

∂ ∂Ψ

∂r r r FF

Evaluation of Function Values

Evaluation of First and Second Partial Derivatives

RADIAL BASIS FUNCTION COLLOCATION METHOD

;;

D d pB b p

= ∈Ω= ∈Γ

FF

m mψ ς≈F

Partial differential equation (boundary value problem)

RBF approximation of solution


;;

D d pB b p

= ∈Ω= ∈Γ

FF

;;

m m

m m

D d pB b pψ ςψ ς

≈ ∈Ω≈ ∈Γ

1

1

;

;nm m

nm

n

mn

D d p

B b p

ψ

ψ

−

−

≈ ∈Ω

≈ ∈

Ψ

Ψ Γ

F

F

,1,2,... 1, ,...,n NN NΓ ΩΓ +=


Ax a=r r

i

i

ba

d

=

ri m

i m

BA

Dψψ

=

1

1i n n m

i n n m

BA

Dψψ

−

−

Ψ= Ψ

[ ]mx ς=r

[ ]mx =r F i

i

ba

d

=

r

Calculation of Coefficients

Calculation of Physical Variables

System of Collocation Equations

( ) ( )N N N NΩ Γ Ω Γ+ × +


1 1

( ) ( )N N

s n n s n nn n N

B Dψ ς ψ ςΓ Ω

Γ= = +

≈ +∑ ∑F

Symmetric approximation

,s sB D adjoint operators acting on sr


Ax a=r r

i

i

da

b

=

r

i m i m

i m i m

B B B DA

D B D Dψ ψψ ψ

=

[ ]mx ς=r


Hermite System of Collocation Equations

Hermite interpolation scheme: improves accuracy !

( ) ( )N N N NΩ Γ Ω Γ+ × +


1 1

augN NN

n n n nn n N

Pψ ς ςΩΩ

Ω

+

= = +

≈ +∑ ∑F

Polynomial augmentation

Physically consistent interpolation

Improves condition number

1,x,y,xx,yy,...


Ax a=r r

0

i

i

dba =

r

0

i m i m

i m i m

Ti m

B B PD D PAP

ψψ

=


Polynomial augmentation

Polynomial augmentation improves condition number

( ) ( )aug augN N N N N NΩ Γ Ω Γ+ + × + +

[ ]mx ς=r


1 ;mn nmB b pψ − ≈ ΓΨ ∈F 1 ;mn nmD d pψ − ≈ ΩΨ ∈F

PARADOX SITUATION

1,2,...,n NΓ= , 1,...,n N N NΓ Γ Ω= +


D d=F=

RBF approximation

* *D d ddΩ Ω

Ω = Ω∫ ∫F=F F

** * *s sc dd d d

n nΓ ΓΩ Γ Γ

∂ ∂= Ω+ Γ − Γ

∂ ∂∫ ∫ ∫F FF F F= F

Green formula


RBF approximation*

* * *( )sc d d dn n

dΓ ΓΩ Γ Γ

∂ ∂= Ω+ Γ − Γ

∂ ∂∫ ∫ ∫F FF F F= F

1 1

( ) ( )N N N N N

s n n s n nn n N N

D Bψ ς ψ ςΓ Ω Ω Γ Γ

Ω Γ

+ + +

= = + +

≈ +∑ ∑F


Ax a=r r

i

i

da

b

=

r

i m i m

i m i m

B D B BA

D D D Bψ ψψ ψ

=

[ ]mx ς=r


Green System of Collocation Equations

Green interpolation scheme: improves consistency

( ) ( )N N N N N NΩ Γ Γ Ω Γ Γ+ + × + +


RADIAL BASIS FUNCTIONS - TRANSPORT PHENOMENA

1998 M. Zerroukat, H. Power, C.S. Chen: heat conduction

2000 M. Zerroukat, K. Djidjeli, A. Charafi: advection-diffusion

2001 B. arler, J. Perko, C.S. Chen: Navier-Stokes

2001 J. Perko, B. arler: Solid-liquid phase change (FBP)

2002 H. Power, V. Barraco: unsymmetric versus symmetric

2002 W. Chen, Green interpolation scheme - proposal2003 B. arler, R. Vertnik, Green interpolation scheme - application

1996 G. E. Fasshauer, Hermite interpolation scheme

2003 I. Kovačević, B. arler, Stefan problem (MBP)

DΓ

NΓ

SATUTURATED DARCY POROUS MEDIA

P v Tr

pc k Kµ β ε

0 p v fKµ

= −∇ − +rrDarcy Momentum Conservation

0v∇ ⋅ =rMass Conservation

1 ( )reff a T Tρ β = − − r r

Boussinesq Body Force

Energy Conservation 2( )pc vT k Tρ ∇ ⋅ = ∇r

GOVERNING EQUATIONS

PREVIOUS APPROACHES

1970 FDM: B.K.C. Chan, C.M. Ivey, J.M. Barry

1981 FEM: G.E. Hickox, C.M. Gartling

1984 FVM: V. Prasad, F.A. Kulacki

2001 BDIM: R. Jecl, L. kerget, E. Petriin

2000 DRBEM: B. arler, D. Gobin, B. Goyeau, J. Perko, H. Power

DΓ

NΓ

RBFCM SOLUTION STRATEGY

P v Tr

•

•

••

•

••

•••

•

•••

•

•

••

•

•

•

•

•

( ) ( )n np p sψ ψ= −r r r

nprpc k Kµ β εinterpolation

solution

( ) ( )n np pψ ςΦ ≈r r

0vnΓ

∂=

∂r

r

Darcy Momentum Conservation

Energy Conservation

( ) ; DT p T pΓ= ∈Γr r

BOUNDARY CONDITIONS

( ) ; NT p F pn ΓΓ

∂= ∈Γ

∂r r

r

non-permeable slip boundary conditions

12 ( );jj jv f pK

P µ+∇ = ∇ ⋅ − + ∈Ωrr r

1 ( ) ; j jjP n v f n pKµ

Γ Γ+∇ ⋅ = − + ⋅ ∈Γ

rr r r r

1 ; jj f n pn

P ΓΓ

+∂= ⋅ ∈Γ

∂

r r rr

SOLUTION PROCEDURE FOR PRESSURE

Pressure Poisson Equation

Neumann Pressure Boundary Conditions

1 ; ref ej

r fP P p p+ = =r r

1 1 ( ); ,j j jv P f x yKξ ξµ

ξξ

+ +∂= − + =

∂

r

11 1( ) 0jj jv v v+ ++∇ ⋅ = ∇ ⋅ + =r%r r

SOLUTION PROCEDURE FOR VELOCITY

1 11 jj jv v v +++ = +r%r r

Does not obey mass conservation in general !

1 1 1 1( )j j j jP v v vK Kµ µ+ + + +∇ = − = − −r r r%%

2 1 1j jP vKµ+ +∇ =+ ∇ ⋅

r%

1 0; jP pn

+

Γ

∂= ∈Γ

∂r%r

SOLUTION PROCEDURE FOR PRESSURE CORRECTION

Neumann Pressure Correction Boundary Conditions

Pressure Correction Poisson Equation

1 0; jrefP p p+ = =

r r%

SOLUTION PROCEDURE FOR CORRECTED VELOCITY

BODY FORCE UPDATING

1 1 1 j P jre

jl

Kc Pv vpξξξµ

+ + +∂= −

∂%

( )( )1 1j j f jrel

jTf f c f f++ = + −

SOLUTION PROCEDURE FOR TEMPERATURE

1jT +

Pressure

Velocity

Pressure Correction

Temperature, Body Force

1 1

1

j j

j j

v v v

T T T

ε

ε

+ +

+

− ≤

− ≤

r r

EndNO

ITERATION STRATEGY

Velocity Correction

PRESSURE POISSON EQUATIONALGEBRAIC SYSTEM OF COLLOCATION EQUATIONS

1 1 ji n i nm

jm i in f n

pPξ ξ ξ

ξ

ψ Γ+−

Γ∂

Ψ =∂

Boundary Nodes

Domain Nodes

2 21 1

2 21 j

in in nm in nmx

jm

y

fp p p

P ξξ

ψ ψ ψ− −+ ∂ ∂ ∂

+ Ψ = Ψ ∂ ∂ ∂ 1

ref

jN m m refP Pδ + =

PRESSURE CORRECTION POISSON EQUATIONALGEBRAIC SYSTEM OF COLLOCATION EQUATIONS

1 1 0i n i nj

mmnp

Pξξ

ψ −Γ

+∂Ψ =

∂%

Boundary Nodes

Domain Nodes

2 21 1 1

21

2 j

in in nm n nx y

m mj

i vPp p K p ξ

ξ

µψ ψ ψ− + − +

∂ ∂ ∂+ Ψ = Ψ ∂ ∂ ∂

%

1 0ref m

jN mPδ + =%

ENERGY EQUATIONALGEBRAIC SYSTEM OF COLLOCATION EQUATIONS

( ) ( )1 1

,0 0

D ND N

D N

p pp p

p pχ χ

∈Γ ∈Γ= =

∉Γ ∉Γ

r rr r

r r

Boundary Nodes

Domain Nodes

1 1 1( )D N D Ni i n nm i i n i

jnm i i i imn T F

pTξ

ξ

χ ψ χ ψ χ χ− −Γ Γ Γ

+∂Ψ + Ψ = +

∂

2 21 1 1

2 2 0p i n nm m in in nmx y

jmc v k

p p pTξ

ξ

ρ ψ ψ ψ− − + ∂ ∂ ∂

Ψ − + Ψ = ∂ ∂ ∂

Boundary Condition Indicators

Generalized Thin Plate Splines of Order k

( ) 2 2 log ,kn n np r rψ −=

r

Generalized Multiquadrics of Order k

( ) ( )1

2 2 2k

n np r cψ = +r

( ) ( ) ( )( )1

22 2n x nx y nyr p p p p p= − + −

r

Free parameter c in multiquadrics !

RADIAL BASIS FUNCTIONS

2 2 2(1.25)c = l Franke (1975)

Kansa (1990)2c κ∝ Cheng (2002)

( ) ( )01M x xp p pψ + = −

r

( ) ( )02M y yp p pψ + = −

r

( )3 1M pψ + =r

3 Augmentation Polynomials

Scaling Constants

( )0 12x x xp p p+ −= − ( )0 1

2y y yp p p+ −= −

AUGMENTATION POLYNOMIALS

( ) ( )201M x xp p pψ + = −

r

( ) ( )( )0 02M x x y yp p p p pψ + = − −

r

( ) ( )203M y yp p pψ + = −

r

( ) ( )04M x xp p pψ + = −

r

( ) ( )05M y yp p pψ + = −

r

( )6 1M pψ + =r

6 Augmentation Polynomials

AUGMENTATION POLYNOMIALS

( , ) 0x yv p pnΓ

∂=

∂r

r

W E

N

S

( , )x yT p p T− +=

( , ) 0x yT p pn

+

Γ

∂=

∂r

NUMERICAL EXAMPLE

( , )x yT p p T+ −=

( , ) 0x yT p pn

−

Γ

∂=

∂r

NUMERICAL EXAMPLE-DIMENSIONLESS FORM0p p

pp

ξ ξξ

ξ

−=

∆

y

x

pA

p∆

=∆

p p pξ ξ ξ+ −∆ = −

2* p yc Ka p T

Rak

ρ β

µ

∆ ∆=

refT TT

T−

=∆

p yc pv v

kρ ∆

= p yc K pP v

kρ

µ

∆=

T T T+ −∆ = −

NUMERICAL EXAMPLE-DIMENSIONLESS FORM

0v∇ ⋅ =r

*0 yP v Ra T i= −∇ ⋅ − +r r

( ) 2v T T∇ ⋅ = ∇r

Dimensionless mass conservation

Dimensionless Momentum Conservation

Dimensionless Energy Conservation

( , ) 0x yv p pnΓ

∂=

∂

rr

W E

N

S

1 1( , )2 2yT A p− = +

1( , ) 02xT p

nΓ

∂+ =

∂r

NUMERICAL EXAMPLE-DIMENSIONLESS FORM

1 1( , )2 2yT A p+ = −

1( , ) 02xT p

nΓ

∂− =

∂r

CALCULATION OF STREAMFUNCTION

y

y

p

x yp

v dpψ−

′ = + ∫x

x

p

y xp

v dpψ−

′ = − ∫

1y

y

p

n y nm xmp

dp vψ ψ−

−′ = + ⋅Ψ∫ 1x

x

p

n x nm ymp

dp vψ ψ−

−′ = − ⋅Ψ∫

CALCULATION OF LOCAL NUSSELT NUMBER

( , )( )

x yx

y

T p ppNu p

A T

−∂∂

=∆

1( , )( )

n x y nm mx

y

p p TpNu p

A T

ψ − −∂Ψ

∂=

∆

CALCULATION OF GLOBAL NUSSELT NUMBER

( , )y

y

p

x y yxp

T p p dpp

NuA T

+

−

−∂∂

=∆

∫

1( , )y

y

p

n x y y nm mxp

p p dp Tp

NuA T

ψ

+

−

− −∂ ⋅Ψ∂

=∆

∫

CALCULATION PARAMETERS

0.5Prelc =

0.001frelc =

0.0001vε =

0.0001Tε =

grid A Ra* ∆Nu ∆ψ20x20 1.0 25 +0.0163 -0.004230x30 1.0 25 +0.0085 +0.000420x20 1.0 50 +0.0031 -0.006730x30 1.0 50 +0.0446 -0.014520x20 1.0 100 +0.0621 -0.039530x30 1.0 100 +0.0315 +0.006840x20 0.5 50 +0.0306 +0.003320x40 2.0 50 +0.0575 +0.0049

ref

ref

NuN

NuNuu−

∆ = min

m

min

in

ref

ref

ψψ

ψψ

−∆ =

NUMERICAL RESULTS

D.Gobin, B.Goyeau, 2000 reference solution (40000 FVM)

TEMPERATURE AND VELOCITY FIELDS

10 x 10 20 x 20

TEMPERATURE AND VELOCITY FIELDS

30 x 30 30 x 30 nonuniformH.Sadat, S.Couturier, 2000 nonuniformity

STREAMFUNCTION

A=0.5 A=2

40 20× 20 40×

MID-PLANE x VELOCITY COMPONENT

10 x 10 20 x 20

MID-PLANE x VELOCITY COMPONENT

30 x 30 30 x 30 nonuniform

MID-PLANE y VELOCITY COMPONENT

10 x 10 20 x 20

MID-PLANE y VELOCITY COMPONENT


SURFACE AND MID-PLANE TEMPERATURES

10 x 10 20 x 20

SURFACE AND MID-PLANE TEMPERATURES


HOT SIDE NUSSELT NUMBER

10 x 10 20 x 20

HOT SIDE NUSSELT NUMBER


0.0372Nu∆ = − 0.0040ψ∆ = −DRBEM 30x30

0.0315Nu∆ = − 0.0068ψ∆ = +

COMPARISON DRBEM-RBFCM

B.arler, D.Gobin, B.Goyeau, J.Perko, H.Power, Int.J.Numer.Methods Fluids, 2000, pp. 279-312.

RBFCM 30x30

*

1100

ARa=

=

Good News: Results of Comparable Accuracy !

HIGHER ORDER THIN PLATE SPLINES

Results Diverge !

*

1100

ARa=

=

6

8

10

logloglog

r rr rr r

POLYNOMIAL AUGMENTATION

Improves the Condition Number (left and right Nusselts numbers)

*

1100

ARa=

=

Degrades the Accuracy if with higher polynomial augmentation

A=1

Ra=100

Da=10e-3

( ) 22

1 p v fK

v vvρ µε

µ= −∇ − + +∇ ⋅ ∇

rr r rr

DARCY-BRINKMAN MODEL

MESHFREE RBF COLLOCATION METHODJ.Perko, B.arler, C.S.Chen (2001)

natural convection in a single phase system natural convection in a solid-liquid system


MODULAR BRICK TB39

TB39

TB39 T10


The equipment for experiment, Prof.Poredo, Laboratory for Refrigeration, University of Ljubljana

86

620

188

59

72

230

43

115

Ø16

86

Dimensions of the cross section of the experimental vessel. Small shaded region is modelled.

ICE FORMATION SIMULATIONS

Water initially at 10.0C

Water initially at 0.5C

Tube surface at -5.0C

ICE FORMATION SIMULATIONS

One-domain temperature formulation

Voller-Swaminathan scheme

Most simple RBFCM using multiquadrics

B

A4.3

Grid Type I. Grid Type I includes 307 gridpoints, 76 on the boundaries and 231 in the domain.

A

B

2.15

Grid Type II. Grid Type II includes 1199 gridpoints, 150 on the boundaries and 1049 in the domain.

Temperature history of gridpoint A during the first hour of solidification calculated with grid Types I and Types II.

Temperature history of gridpoint B during the first two hours ofsolidification calculated with grid Type I and Type II.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11

0.18

0.19

0.2

0.21

0.22

0.23

0.24

Gridpoints of grid Type IGridpoints of grid Type IIInterphas e boundaries grid Type IInterphas e boundaries grid Type IIS urface of tube

Interphase boundaries after one and two hours of solidification calculated with grids Type I and II.

0 300 600 900 1200 1500 1800 2100 2400 2700 3-6

-4

-2

0

2

4

6

8

Time (s )

Tem

pera

ture

(C)

Gridpoint A

Temperature histories of gridpoint A calculated with three time-steps

Grid Type III for numerical computation of ice thickness. Grid Type III includes 580 gridpoints,92 on the boundaries and 488 in the domain.

0

2

4

6

8

10

12

14

16

18

0 10 20 30 40 50 60 70 80Time (min)

Thic

knes

s of

ice

(mm

)

measured thichnesses of ice computed by grid Type I computed by grid Type III

A comparison of measured and computed results of ice thickness using the RBFCM


Promissing development and use of RBFCM for different transport phenomena

Method is very simple for numerical implementation

No need for polygonisation of the domain and boundary

2D and 3D numerical implementation is almost identical

Full matrices

Matrices might be ill conditioned

CONCLUSIONS

Fasshauers symmetric formulation

double consideration of boundary points in recirculating flow situations

compactly supported RBF

multilevel RBF

automatic domain decomposition

p - adaptivity through free parameters in RBF

r - adaptivity (elliptic mesh generator)

PRESENT RESEARCH


HOMOGENOUS SOLUTION BOUNDARY ELEMENT METHODS

Phase Change with Convection - Modelling and Validation, Udine, Italy, September 2-6, 2002.

0 ** 0 * 0( ) ( , ) ( ) ( , ) ( ) ( ) 0cp p s d p p s s s

n nψ

ψ ψΓ ΓΓ Γ

∂Φ ∂Γ − Φ − Φ =

∂ ∂∫ ∫rr r r r r r r r

* 01( , ) log( )2 | |

rp sp s

ψπ

=−

r rr r

* 1 1( , )4 | |

p sp s

ψπ

=−

r rr r

2D

3D

Drawback: boundary polygonisation, complicated integrals

HOMOGENOUS SOLUTIONMETHOD OF FUNDAMENTAL SOLUTIONS

Phase Change with Convection - Modelling and Validation, Udine, Italy, September 2-6, 2002.

0 *( , )n np sψ ζ ∗Φ =r r

1 3 1colN Nψ ψ

Γ Γ

∗ ∗+ −= =

02 2colN N x xp pψ ψ

Γ Γ

∗ ∗+ −= = −

03 1colN N y yp pψ ψ

Γ Γ

∗ ∗+ −= = −

04colN N z zp pψ ψ

Γ Γ

∗ ∗+ = = −

2 *( , ) 0n np sψ ζ ∗∇ =r r

MFS SOLUTION PROCEDURE

• i, m = 1, 2, • • •, NΓcol

• j, n = 1, 2, • • •, NΓ

• ξ = x, y if not stated otherwise

( ) ( ) ; 1, 2, ,n np p s n Nψ ζ∗ ∗ΓΦ ≈ − = ⋅⋅⋅

r r r

Key Point in Solution Procedure

( ) ( ) ; 1, 2, ,n np p n Nψ ζ∗ ∗ΓΦ ≈ = ⋅⋅⋅

r r

GOVERNING EQUATIONSGeometry

Non-Linear Laplace Equation

Boundary Conditions

• domain Ω• boundary Γ; RND ΓΓΓ=Γ UU

( ) 0)( =∇⋅∇ TTk

DDTT Γ∈= p;

NN

kTT

nΓ∈−=

∂∂

Γ

p;

( ) NRref

R

TTk

TTn

Γ∈−−=∂∂

Γ

p;

DΓ

NΓ

RΓ

TRANSFORMED GOVERNING EQUATIONSKirchhoffs Transform, Inverse Kirchoffs Transform

Transformed Non-Linear Laplace Equation

Transformed Boundary Conditions

( ) DDT Γ∈Φ=Φ p;

NN

kT

nΓ∈−=Φ

∂∂

Γ

p;0

( )[ ] ( ) ( )( ) RRref

RR

ref

R

TddTT

kTTT

kT

nΓ∈

−Φ−ΦΦ

Φ+Φ−=−Φ−=Φ

∂∂

Γ

p;00

( ) ∫=ΦT

T

dTkkT

0

,0

( ) ( )ΦΦ

ΦddTT ,

020 =Φ∇k


Boundary Condition Indicators

( ) ( ) ( )1 1 1, ,

0 0 0

D N RD N R

D N R

p p pp p p

p p pχ χ χ

∈Γ ∈Γ ∈Γ= = =

∉Γ ∉Γ ∉Γ

r r rr r r

r r r

jnjn b=∗ *ζψ

Algebraic System of Equations


first NΓ equations

( )ΦΦ∂

∂+

∂∂

+= ∗

Γ

∗

Γ

∗∗

ddT

kT

nn

R

inRiin

Niin

Diin

0

ψχψχψχψ

( ) ( ) ( )

−ΦΦ

Φ−Φ

−+

−+Φ= R

ref

RRi

NNi

DDii T

ddTT

kT

kTTb

00

χχχ

( )in nψ ψ p∗ ∗=r

remaining NΓ - NΓcol augmentation equations

( ) 1 2; , , ,jn nj j col colψ ψ p j N N N∗ ∗ ∗Γ + Γ + Γ= = = ⋅⋅⋅

rψ

,,,;0 21 Γ+Γ+Γ== NNNjb colcolj

( ) 1 20; , , , .n col colψ p j N N N∗Γ + Γ + Γ= = ⋅⋅⋅

r


( ) ( ) ( )

−ΦΦ

Φ−Φ

−+

−+Φ= −∗∗ R

ref

RRi

NNi

DDinin T

ddTT

kT

kTT

00

1 χχχζ ψ

In Case of Robin Boundary Conditions and Non-Linear Material Properties

ε≤Φ−Φ ii

For all Collocation Points pi

Explicit Expression for the Coefficients

TWO-DIMENSIONAL THERMAL TEST No.15PROPOSED BY NAFEMS

NAFEMS No.15 BENCHMARK TEST

grid 2x60+2x100

point (0.6,0.2)value 18.3xxxx value 18.25xxx

2-10 typical grid distances

absolute accuracy 0.00001


TEMPERATURES SUM

0

2

4

6

8

10

12

14

16

18

20

0 2 4 6 8 10 12 14 16

Distance [/]

Σ T

* 10

5 [°C

]

Grid 2x60 + 2x100 = 320IMSL solver, double precision

14,0

15,0

16,0

17,0

18,0

19,0

20,0

21,0

22,0

23,0

24,0

Grid size

T [C

]

0.5 grid distance 1 grid distance 5 grid distance 6 grid distance

2x2 3x5 6x10 9x15 12x20 15x25 30x50 45x75 60x100 90x150 120x200 180x300 240x400


SOLUTON AT DIFFERENT GRID DENSITY AND SOURCE DISTANCE

0,000

0,001

0,010

0,100

1,000

10,000

Grid size

log ∆

T [C

]

0.5 grid distance 1 grid distance 5 grid distance 6 grid distance

2x2 3x5 6x10 9x15 12x20 15x25 30x50 45x75 60x100 90x150 120x200 180x300 240x400

NAFEMS No.15 BENCHMARK TEST- ABSOLUTE ERROR

ACCURACY AT DIFFERENT GRID DENSITY AND SOURCE DISTANCE

NUMERICAL EXAMPLES

WHERE IS THE PROPER POSITION OF SOURCE POINTS ?

WHERE POSITION CHANGE DOES NOT INFLUENCE THE RESULTS !

NUMERICAL EXAMPLES

CASE WITH CONSTANT THERMAL CONDUCTIVITY

0.6 WkmK

=

( )[ 0.6 0.005 ]

0.5 0.7

o

o

T C WkC m K

W Wkm K m K

= −

≤ ≤

CASE WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY

conservative 30%

Brick MB/2S, constant thermal conductivity

Brick MB/2S, variable thermal conductivity

20 oT C= + 20 oT C= −

20 oT C= + 20 oT C= −

Brick MB/2H, constant thermal conductivity

Brick MB/2H, variable thermal conductivity

20 oT C= +

20 oT C= +

20 oT C= −

20 oT C= −

BRICK TB39 - NUMERICAL RESULTS

number of collocation nodes = 4552, computing time = 5min (PIII, 933 MHz), COMPAQ FORTRAN+IMSL

BRICK TB39 - ALTERNATIVE, NUMERICAL RESULTS

INITIAL BUBBLE SHAPE

Initial non-deformed bubble shape. The 64 boundary collocation nodes are represented by•, the 64 fundamental solution poles are represented by +. The upper isopotential linerepresents Φ = +0.01 [m2/s], the lower Φ = -0.01 [m2/s]. The lines are equipotentiallyspaced by 0.001 [m2/s].

RIGID BUBBLE PRESSURE

Initial pressure arround non-deformed bubble as a function of interphase boundary length. The length isscaled with the total length. Symbols coincide with the boundary collocation points. ↑ represents the partof the pressure γκ−refP and ↓ represents the part of the pressure Pref.

RIGID BUBBLE POTENTIAL FLOW COMPONENTS

Initial velocity arround non-deformed bubble as a function of interphase boundary length. The velocity isscaled with the reference velocity and length with the total length. Symbols coincide with the boundarycollocation points. • represents the scaled absolute value of velocity 0/ vv , 0/ vvx→ and ./ 0vvy↑

FINAL BUBBLE SHAPE

Final deformed bubble shape. The positions of the initial boundary collocation nodes are represented by ο.

FINAL BUBBLE PRESSURE

(Almost) final pressure arround deformed bubble as a function of interphase boundary length.

FINAL POTENTIAL FLOW COMPONENTS

Final velocity arround deformed bubble as a function of interphase boundary length.

SOLUTION PROCEDURE

Description of the Bubble Shape

Γ⋅⋅⋅=oo

Nkk ,,2,1;p

Γ−− ⋅⋅⋅=

−+

−=

ooooo

l Nkpppp ykykxkxkk ,,2,1;2/12

1

2

1

The Total Euclidean Length Γl Equals to

ΓΓ ⋅⋅⋅==o

ll Nkkkk ,,2,1;δ

( ) ( ) Γ⋅⋅⋅==oo

ll Nlp ll ,,2,1;ξξ ζψ

SOLUTION PROCEDURE

Cubic Splines

( ) ΓΝ⋅⋅⋅=−=o

lll ,,2,1;3 kkkψ

( ) ( ) 121

==−Γ+Γ Νll oo ψψ

N

( ) ( ) lll oo ==−Γ+Γ Ν 12

ψψN

( ) ( ) 3

3

lll oo ==Γ+Γ Ν

ψψN

3 Augmented Functions

SOLUTION PROCEDURECompatibility Conditions

( ) ( ) ,0 ξξ ζψζψ llll =Γl ( ) ( ) ,0 ξξ ζψζψ llll dd

dd

ll

l=Γ ( ) ( ) ξξ ζψζψ llll d

ddd 02

2

2

2Γ

ΓΓ =

ll

l

ΓΝ⋅⋅⋅==o

,,2,1,; ljb jljlξξ ζψ

( ) ΓΓ Ν⋅⋅⋅=⋅⋅⋅==oo

l ,,2,1,,,2,1; lNkklkl ψξψ

Γ⋅⋅⋅==oo

Nlpb ll ,,2,1;ξξ

jljl pξξξζ

o1−= ψ

( ) ( ),02 ll ψψξ −= ΓΝ −Γ

lψ 02=

−ΓΝb

( ) ( ),01 ll ψψξ

ll

l ∂∂

−∂∂

= ΓΝ −Γψ 0

1=

−ΓΝb

( ) ( ),02

2

2

2

ll ψψξ

ll

l ∂∂

−∂∂

= ΓΝΓψ 0

1=

−ΓΝb

SOLUTION PROCEDURE

Components of the Bubble Normal

,

2/122−

+

+=

lll

ooo

dpd

dpd

dpd

n yxyx

2/122−

+

−=

lll

ooo

dpd

dpd

dpdn yxx

y

Bubble Curvature2/322

2

2

2

2

−

+

−=

llllll

oooooo

dpd

dpd

dpd

dpd

dpd

dpd yxxyyxκ

The First and the Second Derivatives

( ) ( ) ,ξξ ζψ nndd

dpd

ll

ll

o

= ( ) ( ) ξξ ζψ nndd

dpd

ll

ll

o

2

2

2

2

=


Documents

MESH-FREE METHODS FOR HEAT TRANSFER AND FLUID …fluid.ippt.gov.pl/sarler/MESHFREE_AMAS.pdf · ppcpπ = =∑ rrr F= (){}1, , , , , , , , ,22 2 π ppppppppppppp= x y zxy yz xzx y z