IV European Conference of Computational Mechanics
description
Transcript of IV European Conference of Computational Mechanics
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IV European Conference ofIV European Conference ofComputational Mechanics Computational Mechanics
Hrvoje GotovacHrvoje Gotovac, Veljko Srzić, Tonći Radelja, Vedrana Kozulić, Veljko Srzić, Tonći Radelja, Vedrana Kozulić University of Split, Department of Civil and Architectural Engineering, CroatiaUniversity of Split, Department of Civil and Architectural Engineering, Croatia
Explicit Adaptive Fup Collocation Method Explicit Adaptive Fup Collocation Method (EAFCM) for solving the parabolic problems(EAFCM) for solving the parabolic problems
PresentationPresentation
ECCM, 21 May 2010, Paris, France.ECCM, 21 May 2010, Paris, France.
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Presentation outlinePresentation outline
1.1. General conceptGeneral concept2.2. Fup basis functionsFup basis functions3.3. Fup collocation transform (FCT) - space Fup collocation transform (FCT) - space
approximationapproximation4.4. Explicit time integration for parabolic stiff Explicit time integration for parabolic stiff
problemsproblems5.5. Numerical examplesNumerical examples6.6. ConclusionsConclusions7.7. Future directionsFuture directions
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1. General concept1. General concept
Developing adaptive numerical method Developing adaptive numerical method which can deal with parabolic flow and which can deal with parabolic flow and transport stiff problems having wide range transport stiff problems having wide range of space and temporal scalesof space and temporal scales
Ability to handle multiple heterogeneity Ability to handle multiple heterogeneity scalesscales
Application target: unsaturated and Application target: unsaturated and multiphase flow, reactive transport and multiphase flow, reactive transport and density driven flow in porous media, as density driven flow in porous media, as well as structural mechanics problemswell as structural mechanics problems
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Saturated – unsaturated flowSaturated – unsaturated flow
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Interaction between surface and Interaction between surface and subsurface flowsubsurface flow
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Geothermal convective processes Geothermal convective processes in porous mediain porous media
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Typical physical and numerical Typical physical and numerical problemsproblems
Description of wide range of space and temporal Description of wide range of space and temporal scalesscales
Sharp gradients, fronts and narrow transition Sharp gradients, fronts and narrow transition zones (‘fingering‘ and ‘layering’)zones (‘fingering‘ and ‘layering’)
Artificial oscillations and numerical dispersion – Artificial oscillations and numerical dispersion – advection dominated problemsadvection dominated problems
Description of heterogeneity structureDescription of heterogeneity structure Strong nonlinear and coupled system of Strong nonlinear and coupled system of
equationsequations
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Motivation for EAFCMMotivation for EAFCM
Multi-resolution and meshless approachMulti-resolution and meshless approach Continuous representation of variables Continuous representation of variables
and all its derivatives (fluxes)and all its derivatives (fluxes) Adaptive strategyAdaptive strategy Method of lines (MOL)Method of lines (MOL) Explicit formulation (no system of Explicit formulation (no system of
equations!!!)equations!!!) Perfectly suited for parallel processingPerfectly suited for parallel processing
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Flow chart of EAFCMFlow chart of EAFCMSTART
EFFECTIVE GRID
T = t0
2. CALCULATION SPACEDERIVATIVES AND WRITE
EQUATIONS IN THEGENERAL FORM
ADDITIONALPOINTS
CALCULATE BASICGRID VIA FCT
TOTAL GRID
INITIAL CONDITIONu(0,x)
1. GRID ADAPTATION
FIND ADAPTIVETIME STEP - dt
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GET NEW VECTORu(t,x) AT TIME - T + dt
T = T + dt
3. PERFORM TEMPORALNUMERICAL INTEGRATION
END
T < TMAXYES
CONTINUE
NO
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2. Fup basis functions2. Fup basis functions Atomic or RAtomic or Rbfbf class of class of
functions functions Function up(x)Function up(x)
Fourier transform of Fourier transform of up(x) functionup(x) function
Function FupFunction Fupnn(x)(x)
up x up x up x'( ) ( ) ( ) 2 2 1 2 2 1
up tt
t
j
jj
( )sin 2
21up x e up t dtitx( ) ( )
12
0k
1nnkn 22n
2k1xup)n(C)x(Fup
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Function FupFunction Fup22(x)(x)
3 2 1 4 3 2 1 4
3 21 4
- 0 . 5 0 - 0 . 2 5 0 . 0 0 0 . 2 5 0 . 5 0
- 0 . 5 0 - 0 . 2 5 0 . 0 0 0 . 2 5 0 . 5 0
- 0 . 5 0 - 0 . 2 5 0 . 0 0 0 . 2 5 0 . 5 0
- 0 . 5 0 - 0 . 2 5 0 . 0 0 0 . 2 5 0 . 5 0
- 0 . 5 0 - 0 . 2 5 0 . 0 0 0 . 2 5 0 . 5 0
x
x
x
x
x
F u p 2 ( x )
F u p 2 ' ( x )
F u p 2 ' ' ( x )
F u p 2 ' ' ' ( x )
F u p xI V2 ( )
5 / 90 . 0 2 6 / 9 5 / 9 0 . 0
0 . 0 8 . 0 - 8 . 0 0 . 0
0 . 0 6 4 . 00 . 0
- 1 2 8 . 0
6 4 . 0 0 . 0
0 . 0
5 1 2
0 . 0
- 5 1 2- 1 5 3 6
1 5 3 6
0 . 0 0 . 0 0 . 0
2 1 4 2 1 4
2 1 4 2 1 4
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Function FupFunction Fup22(x,y)(x,y)X Y
Z c)
X Y
Z
a)
X Y
Z
b)
X Y
Z e)
X Y
Z
d)
X Y
Z
f)
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Function FupFunction Fupnn(x)(x)
Compact supportCompact support Linear combination of n+2 functions FupLinear combination of n+2 functions Fupnn(x) (x)
exactly presents polynomial of order nexactly presents polynomial of order n Good approximation propertiesGood approximation properties Universal vector space UP(x)Universal vector space UP(x) Vertexes of basis functions are suitable for Vertexes of basis functions are suitable for
collocation pointscollocation points FupFupnn(x,y) is Cartesian product of Fup(x,y) is Cartesian product of Fupnn(x) and (x) and
FupFupnn(y) (y)
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Any function u(x) is presented by linear combination of Fup basis functions:Any function u(x) is presented by linear combination of Fup basis functions:
jj - level (from zero to maxim - level (from zero to maximumum level level J)J)
jjminmin - resolution at the zero level - resolution at the zero level
kk - location index in the current level - location index in the current level
- Fup coefficients- Fup coefficients
- Fup basis functions - Fup basis functions
nn - order of the Fup basis function - order of the Fup basis function
xd)x(uJ
0j
)2n2(
2nk
jk
jk
jminj
jkd
jk
3. Fup collocation transform (FCT)3. Fup collocation transform (FCT)
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X
j
0 0.5 1 1.5 20
1
2
3
4
5
6
k=1 k=3 k=5 k=7 k=8k=0 k=2 k=4 k=6
X
j
0 0.5 1 1.5 20
1
2
3
4
5
6
a)
k=0 k=1 k=2 k=3 k=4
X
f(X),
U0 (X
)
0 0.5 1 1.5 2
-1
-0.5
0
0.5
1
u0(x)f(x)
b)
X
ABS
(f(X
)-U
0 (X))
0 0.5 1 1.5 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
c)
X
f(X),
U1 (X
)
0 0.5 1 1.5 2
-1
-0.5
0
0.5
1
u1(x)f(x)
X
AB
S(f(
X)-
U1 (X
))
0 0.5 1 1.5 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
X
j
0 0.5 1 1.5 20
1
2
3
4
5
6
X
f(X),
U5 (X
)
0 0.5 1 1.5 2
-1
-0.5
0
0.5
1
u5(x)f(x)
XAB
S(f(
X)-
U5 (X
))0 0.5 1 1.5 2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
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Spatial derivativesSpatial derivatives
J
j
n
nkp
jk
pjkp
pJ
j
n
nk
jk
jk
jjjj
xdxdd
xdxudxdxu
0
)22(
20
)22(
2
minmin )()()(
jk
jk
jk
jk
uuu
d
1
1
54651441
Nlbuxdxud jj
k
pjlk
jkp
jl
p jj
min
min
2...,,0;)( 2
0
,,
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4. Time numerical integration4. Time numerical integration Reduces to system of Reduces to system of OOrdinary rdinary DDifferential ifferential
EEquations (ODE) for adaptive grid and quations (ODE) for adaptive grid and every time step (t – t+dt):every time step (t – t+dt):
With appropriate initial conditions:With appropriate initial conditions:
1,...,1;),,,,()( )2()1( NiuuuxtFtdtud
iiiii
),( ii xtuu
)()( 00
00 tDxu
ortUu
)()( tDxu
ortUu NN
NN
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SStabilized second-order tabilized second-order EExplicit xplicit RRunge-unge-KKutta method (utta method (SERK2SERK2))
Recently developed by Vaquero and Recently developed by Vaquero and Janssen (2009)Janssen (2009)
Extended stability domains along the Extended stability domains along the negative real axisnegative real axis
Suitable for very large stiff parabolic ODESuitable for very large stiff parabolic ODE Second – order method up to 320 stagesSecond – order method up to 320 stages Public domain Fortran routine SERK2Public domain Fortran routine SERK2
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5. Numerical examples5. Numerical examples
1-D density driven flow problem1-D density driven flow problem
2-D Henry salwater intrusion problem2-D Henry salwater intrusion problem
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Mathematical modelMathematical model
Pressure-concentration formulationPressure-concentration formulation Fluid mass balance:Fluid mass balance:
Salt mass balance:Salt mass balance:
RPcp QQ
tCn
tpS
q 0
0
RQCCCnCtCn )(
HDq
2222
x
c
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1t = 0.980
x
C*
0 0.25 0.5 0.75 1
0
0.25
0.5
0.75
1
t = 0.0t = 0.0t = 0.02
x
C*
0 0.25 0.5 0.75 1
0
0.25
0.5
0.75
1
t = 0.5 /t = 0.16
x
C*
0 0.25 0.5 0.75 1
0
0.25
0.5
0.75
1
t = 0.30
x
C*
0 0.25 0.5 0.75 1
0
0.25
0.5
0.75
1
t = 0.46
x
j
0 0.25 0.5 0.75 10
1
2
3
4
5
6
7
t = 0.02
x
j
0 0.25 0.5 0.75 10
1
2
3
4
5
6
7
t = 0.16
x
j
0 0.25 0.5 0.75 10
1
2
3
4
5
6
7
t = 0.30
x
j
0 0.25 0.5 0.75 10
1
2
3
4
5
6
7
t = 0.46
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0.1
0.90.3
0.50.7
X
Y
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1t = 200 (s)
X
Y
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1t = 200 (s)
0.1
0.90.3
0.7
X
Y
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1t = 600 (s)
X
Y
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1t = 600 (s)
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X
Y
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1t = 3 600 (s)
X
Y
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1t = 12 000 (s)
0.10.3
0.5 0.70.9
X
Y
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1t = 12 000 (s)
0.10.3
0.50.9
X
Y
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1t = 3 600 (s)
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Number of collocation points Number of collocation points and compression coefficientand compression coefficient
t (s)
CR
0 3000 6000 9000 120000
200
400
600
800
1000
t (s)
N
0 3000 6000 9000 120000
1000
2000
3000
4000
5000
6000
adaptive
adaptivenonR N
NC
0005
0005002
adaptive
adaptivenon
N
N
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6. Conclusions6. Conclusions Development of mesh-free adaptive Development of mesh-free adaptive
collocation algorithm that enables efficient collocation algorithm that enables efficient modeling of all space and time scalesmodeling of all space and time scales
Main feature of the method is the space Main feature of the method is the space adaptation strategy and explicit time adaptation strategy and explicit time integration integration
No discretization and solving of huge system No discretization and solving of huge system of equations of equations
Continuous approximation of fluxesContinuous approximation of fluxes
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7. Future directions7. Future directions
Multiresolution description of heterogeneityMultiresolution description of heterogeneity Development of 3-D parallel EAFCMDevelopment of 3-D parallel EAFCM Time subdomain integration Time subdomain integration Description of complex domain with using Description of complex domain with using
other families of atomic basis functionsother families of atomic basis functions Further application to mentioned processes in Further application to mentioned processes in
porous media and other (multiphysics) porous media and other (multiphysics) problemsproblems