11

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.

22

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

33

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

44

Saturated – unsaturated flowSaturated – unsaturated flow

55

Interaction between surface and Interaction between surface and subsurface flowsubsurface flow

66

Geothermal convective processes Geothermal convective processes in porous mediain porous media

77

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

88

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

99

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

1010

GET NEW VECTORu(t,x) AT TIME - T + dt

T = T + dt

3. PERFORM TEMPORALNUMERICAL INTEGRATION

END

T < TMAXYES

CONTINUE

NO

1111

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

1212

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

1313

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)

1414

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)

1515

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)

1616

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

1717

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

,,

1818

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

1919

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

2020

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

2121

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

2323

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)

2424

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)

2525

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

2626

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

2727

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