IV European Conference of
Computational Mechanics
Hrvoje Gotovac, Veljko Srzić, Tonći Radelja, Vedrana Kozulić
University of Split, Department of Civil and Architectural Engineering, Croatia
Explicit Adaptive Fup Collocation Method
(EAFCM) for solving the parabolic problems
Presentation
ECCM, 21 May 2010, Paris, France.
1
Presentation outline
1.
2.
3.
4.
5.
6.
7.
General concept
Fup basis functions
Fup collocation transform (FCT) - space
approximation
Explicit time integration for parabolic stiff
problems
Numerical examples
Conclusions
Future directions
2
1. General concept
 Developing
adaptive numerical method
which can deal with parabolic flow and
transport stiff problems having wide range
of space and temporal scales
 Ability to handle multiple heterogeneity
scales
 Application target: unsaturated and
multiphase flow, reactive transport and
density driven flow in porous media, as
well as structural mechanics problems
3
Saturated – unsaturated flow
4
Interaction between surface and
subsurface flow
5
Geothermal convective processes
in porous media
6
Typical physical and numerical
problems





Description of wide range of space and temporal
scales
Sharp gradients, fronts and narrow transition
zones (‘fingering‘ and ‘layering’)
Artificial oscillations and numerical dispersion –
advection dominated problems
Description of heterogeneity structure
Strong nonlinear and coupled system of
equations
7
Motivation for EAFCM
 Multi-resolution
and meshless approach
 Continuous representation of variables
and all its derivatives (fluxes)
 Adaptive strategy
 Method of lines (MOL)
 Explicit formulation (no system of
equations!!!)
 Perfectly suited for parallel processing
8
Flow chart of EAFCM
START
INITIAL CONDITION
u(0,x)
T = t0
1. GRID ADAPTATION
CALCULATE BASIC
GRID VIA FCT
ADDITIONAL
POINTS
TOTAL GRID
EFFECTIVE GRID
2. CALCULATION SPACE
DERIVATIVES AND WRITE
EQUATIONS IN THE
GENERAL FORM
FIND ADAPTIVE
TIME STEP - dt
9
CONTINUE
3. PERFORM TEMPORAL
NUMERICAL INTEGRATION
GET NEW VECTOR
u(t,x) AT TIME - T + dt
T = T + dt
YES
T < TMAX
NO
END
10
2. Fup basis functions

Atomic or Rbf class of
functions
 Function up(x)
up'( x)  2 up (2x  1)  2 up (2x  1)

Fourier transform of
up(x) function


up( t )  
sin t 2 j
j
t2
j 1

up( x) 
1
2

e
itx

up( t ) dt

Function Fupn(x)

k n2

Fupn ( x )   Ck ( n ) up x  1  n  n 1 
2
2


k 0
11
Function Fup2(x)
Fup2(x)
0.0
5/9
26/9
5/9
0.0
-0 .5 0
-0 .2 5
0 .0 0
0 .2 5
0 .5 0
-8.0
0.0
0 .2 5
0 .5 0
x
Fup2'(x)
0.0
8.0
-0 .5 0
-0 .2 5
0 .0 0
x
Fup2''(x)
0.0
64.0
-0 .5 0
64.0
-0 .2 5
0 .0 0
0.0
0 .2 5
0 .5 0
x
-128.0
Fup2'''(x)
1536
512
0.0
0.0
0.0
0.0
0.0
-0 .5 0
-0 .2 5
0 .0 0
0 .2 5
0 .5 0
x
0.50
x
-512
-1536
3  2
14
IV
Fup2 ( x)
2
14
2
2
-0.50
14
2
-0.25
3  2
0.00
14
0.25
3  2
14
14
14
12
Function Fup2(x,y)
13
Function Fupn(x)






Compact support
Linear combination of n+2 functions Fupn(x)
exactly presents polynomial of order n
Good approximation properties
Universal vector space UP(x)
Vertexes of basis functions are suitable for
collocation points
Fupn(x,y) is Cartesian product of Fupn(x) and
Fupn(y)
14
3. Fup collocation transform (FCT)

Any function u(x) is presented by linear combination of
Fup basis functions:
u( x ) 
J  ( 2 jmin  j  n 2 )

j 0

k n 2
d kj  kj x 
j - level (from zero to maximum level J)
jmin - resolution at the zero level
k - location index in the current level
dkj - Fup coefficients
kj - Fup basis functions
n - order of the Fup basis function
15
6
1.2
a)
5
b)
1
u (x)
f(x)
0.5
j
0
f(X), U (X)
ABS (f(X) - U 0(X))
0
4
3
2
k=0
k=1
k=2
k=3
k=4
0
-0.5
1
0
0.5
1
1.5
0
2
0.5
X
0.8
0.7
0.6
0.5
0.4
0.3
1
1.5
0
2
0
0.5
1
1.5
2
1.5
2
1.5
2
X
X
1
6
0.9
1
5
u (x)
f(x)
k=0 k=1 k=2
k=3 k=4 k=5
k=6 k=7 k=8
2
f(X), U 1(X)
0.5
4
ABS (f(X) - U 1(X))
1
j
0.9
0.1
0
0
-0.5
0.8
0.7
0.6
0.5
0.4
0.3
0.2
1
-1
0.1
0
0
0.5
1
1.5
2
0
0.5
X
1
1.5
0
2
0
0.5
1
X
X
1
6
0.9
1
5
u (x)
f(x)
0.5
f(X), U 5(X)
4
3
2
ABS (f(X) - U 5(X))
5
j
1
0.2
-1
3
c)
1.1
0
-0.5
0.8
0.7
0.6
0.5
0.4
0.3
0.2
1
-1
0.1
0
0
0.5
1
X
1.5
2
0
0.5
1
X
1.5
2
0
0
0.5
1
X
16
Spatial derivatives
u ( x) 
J  ( 2 jmin  j  n 2 )
j
k
j 0
k n 2

p
d u ( x)
j
 d k x   d x p 
J  ( 2 jmin  j  n 2 )

j 0

k n 2
p j
d
 k ( x)
j
dk
d xp
ukj1 
1
 5 46  5  ukj 
d kj 
144
ukj1 


p
j
d u ( xl )

p
dx
2 jmin  j
jmin  j
j
j, p
u
b
;
l

0
,
...,
2
N
 k k ,l
k 0
17
4. Time numerical integration
 Reduces
to system of Ordinary Differential
Equations (ODE) for adaptive grid and
every time step (t – t+dt):
d ui (t )
 F (t , xi , ui , ui(1) , ui( 2) ) ; i  1,...,N  1
dt
u0  U 0 (t ) or
 With
u 0
 D0 (t ) u N  U N (t ) or
x
u N
 D N (t )
x
appropriate initial conditions:
ui  u(t , xi )
18
Stabilized second-order Explicit
Runge-Kutta method (SERK2)
 Recently
developed by Vaquero and
Janssen (2009)
 Extended stability domains along the
negative real axis
 Suitable for very large stiff parabolic ODE
 Second – order method up to 320 stages
 Public domain Fortran routine SERK2
19
5. Numerical examples
 1-D
density driven flow problem
 2-D
Henry salwater intrusion problem
20
Mathematical model
 Pressure-concentration
 Fluid
formulation
mass balance:
p
0 C
S
 n c
    q  QP  QR
t
t
p
0
 Salt
n
mass balance:
C
  q   C     n DH   C   (C   C ) QR
t
21
t = 0.0
t = 0.02
t = 0.02
7
1
6
0.75
5
*
4
j
C
0.5
3
2
0.25
1
0
0
0
0.25
0.5
x /
t = 0.5
t = 0.16
0.75
1
0
0.25
0.5
x
0.75
1
0.75
1
0.75
1
t = 0.16
7
1
6
0.75
5
*
4
j
C
0.5
3
2
0.25
1
0
0
0
0.25
0.5
x
0.75
1
0
t =t 0.980
= 0.30
1
1
x
t = 0.30
6
5
0.6
4
Cc
*
0.5
7
0.8
0.75
j
0.5
0.4
3
0.25
0.2
2
0
0
0.25
1
0
0
0.25
0.5
0.5
1
x
0.75
1.5
2
1
0
0.25
t = 0.46
0.5
x
t = 0.46
7
1
6
0.75
5
j
C
*
4
0.5
3
2
0.25
1
0
0
0
0.25
0.5
x
0.75
1
0
0.25
0.5
x
0.75
1
22
t = 200 (s)
t = 200 (s)
0.8
0.8
0.6
0.6
Y
1
Y
1
0.4
0.4
0.2
0
0.2
0.7
0.1 0.5
0.3
0.9
0
0.5
1
1.5
0
2
0
0.5
X
1
1.5
2
1.5
2
X
t = 600 (s)
t = 600 (s)
0.8
0.8
0.6
0.6
Y
1
Y
1
0.4
0.4
0.3
0.2
0.5
0.1
0.2
0.9
0.7
0
0
0.5
1
X
1.5
2
0
0
0.5
1
X
23
t = 3 600 (s)
t = 3 600 (s)
0.8
0.8
0.6
0.6
Y
1
Y
1
0.4
0.4
0.1
0.3
0.5
7
0.
0.2
0
0
0.5
1
0.2
0.9
1.5
0
2
0
0.5
X
1
1.5
2
1.5
2
X
t = 12 000 (s)
t = 12 000 (s)
0.8
0.8
0.6
0.6
Y
1
Y
1
0.4
0.4
0.1
0.3
0.5
0.7
0.2
0
0
0.5
1
X
1.5
0.9
0.2
2
0
0
0.5
1
X
24
Number of collocation points
and compression coefficient
CR 
N non adaptive  2 500000
N non adaptive
N adaptive
N adaptive
1000
 5 000
6000
5000
800
4000
N
CR
600
3000
400
2000
200
0
1000
0
3000
6000
t (s)
9000
12000
0
0
3000
6000
t (s)
9000
12000
25
6. Conclusions
 Development
of mesh-free adaptive
collocation algorithm that enables efficient
modeling of all space and time scales
 Main feature of the method is the space
adaptation strategy and explicit time
integration
 No discretization and solving of huge system
of equations
 Continuous approximation of fluxes
26
7. Future directions
 Multiresolution
description of heterogeneity
 Development of 3-D parallel EAFCM
 Time subdomain integration
 Description of complex domain with using
other families of atomic basis functions
 Further application to mentioned processes in
porous media and other (multiphysics)
problems
27