mathematical modeling of uncertainty in computational mechanics

Mathematical modeling of uncertainty

in computational mechanics

Andrzej PownukSilesian University of Technology

Polandandrzej@pownuk.com

http://andrzej.pownuk.com

Schedule• Different kind of uncertainty• Design of structures with uncertain parameters• Equations with uncertain parameters• Overview of FEM method• Optimization methods• Sensitivity analysis method• Equations with different kind of uncertainty

in parameters• Future plans• Conclusions

],[3 PPP

],[2 PPP

],[1 PPP

%5.20 PP

[kN] 100 P

[m] 1L1P 2P

3P

1 2 34

6

5

7 89

1 0

111 2 1 3

1 4

1 5

1 61 7 1 8

1 9

2 0

2 12 2 2 3

2 4

2 5

2 62 7 2 8

2 9

3 0

3 1

3 23 3

3 4

3 5

3 6

3 7

3 8

3 9

4 0

4 1

4 2

4 3

4 4

4 5

4 6

4 7

4 84 9

5 0

5 1

5 2

5 3 5 4

5 5

5 6

5 7

5 85 9

6 0

6 1

6 2

6 3 6 4

6 5

6 6

6 7

6 86 9

No Error % No Error % No Error % No Error %

1 107.057 % 21 42.4163 % 41 109.399 % 61 31.4828 %

2 78.991 % 22 34.0332 % 42 20.0367 % 62 95.903 %

3 38.2972 % 23 9.30111 % 43 109.399 % 63 48.1069 %

4 52.0345 % 24 100.427 % 44 0.833207 % 64 68.1526 %

5 22.7834 % 25 0.833207 % 45 116.216 % 65 22.7834 %

6 68.1526 % 26 116.216 % 46 100.427 % 66 52.0345 %

7 95.903 % 27 20.0367 % 47 34.0332 % 67 78.991 %

8 48.1069 % 28 116.216 % 48 9.30111 % 68 38.2972 %

9 31.4828 % 29 48.1793 % 49 42.4163 % 69 107.057 %

10 22.6152 % 30 116.216 % 50 15.3219 %

11 38.0037 % 31 8.87714 % 51 33.6609 %

12 91.4489 % 32 19.1787 % 52 119.633 %

13 45.9704 % 33 35.7319 % 53 47.414 %

14 11.913 % 34 18.7494 % 54 9.99375 %

15 24.3824 % 35 6.38613 % 55 24.3824 %

16 9.99375 % 36 18.7494 % 56 11.913 %

17 119.633 % 37 19.1787 % 57 91.4489 %

18 56.7357 % 38 35.7319 % 58 45.9704 %

19 33.6609 % 39 8.87714 % 59 38.0037 %

20 15.3219 % 40 48.1793 % 60 22.6152 %

1:09 5/153

Rod under tension

21 ,0

0

uLuuu

xndx

xduxAxE

dx

d

Lu(x)

x

E,An

1A

2A

xA

Differential form of equilibrium equation

E – Young modulus.A – area of cress-section.n – distributed load parallel to the rod,u – displacement

1:09 6/153

Different kind of uncertainty

1:09 7/153

Floating-point and real numbers

Rh 0

h0h

0h - parameter

20 he.g.

Floating-point numbers emh 100

1:09 8/153

Uncertain parameters Taking into account uncertainty

using deterministic corrections.

Control problems

Gregorian and Julian calendar vs astronomical year (common years and leap years)

hhh 0

steering wheel is necessary

1:09 9/153

Uncertain parameters Semi-probabilistic methods

0hh

N ...21

- safety factor

i - partial safety factor

This method is currently used in practical

civil engineering applications(worst case analysis)

Some people believe that probability doesn't exist.

Law constraints

1:09 10/153

Uncertain parameters

Random parameters

],,[:],[ hhhPhhP

Rhh :

Using probability theoryone can say that buildings are usually safe ...

1:09 11/153

Uncertain parameters

Bayesian probability

BP

APABPBAP

||

Cox's theorem - "logical" interpretation of probability

1:09 12/153

Uncertain parameters Interval

parameters

],[ˆ hhhh

Interval parameter is not equivalent to

uniformly distributed random variable

1:09 13/153

Uncertain parameters

Set valued random variable

Upper and lower probability

nRhh :

AhPAPl :

AhPABel :

1:09 14/153

Uncertain parameters Nested family of random sets

Nhhh ...21

}:{ hxPxF x

xF

1h

2h

3h

1F

2F

3F

0x

0xF

1:09 15/153

Uncertain parameters

Fuzzy sets

0F

0F

1F

1F

1

0

xF

F

F

x

xy Fxfyx

Ff

:sup

Extension principle

1:09 16/153

Uncertain parameters

Fuzzy random variables

Random variables with fuzzy parameters

RFhh :

RFphpRFh ,,:

Etc.

1:09 17/153

Design of structures with

uncertain parameters

1:09 18/153

Design of structures Safety condition

0A

P

PAE,

P – load,A – area of cross-sectionσ – stress

1:09 19/153

Safe area

A

PAP 0

Safe area

0A

P

1:09 20/153

Design of structures with interval parameters

A

P AP 0

Safe area

],[ 000

1:09 21/153

Design of structures with interval parameters

A

P

AP 0

],[ 000

0P

0A

0P],[ 00

PPP

}],,[],,[:{ 000000 APPPPA

1:09 22/153

More complicated cases

P11AE 22AE

2L1L

PPEEEEPEEAAAA ,,,,,,,:, 2211212121

PEEAA ,,,, 2121 - design constraints

1:09 23/153

Design constraints

Pu

u

L

AE

L

AEL

AE

L

AE

L

AE0

2

1

2

22

2

22

2

22

2

22

1

11

011 E 022 E

2211 ,, EEEEPP

1

0

111

11

u

u

LL

2

1

222

11

u

u

LL

1:09 24/153

Geometrical safety conditions

maxmin uuu

inu

maxu

1:09 25/153

Applications of united solution set In general solution set of the design

process is very complicated.

In applications usually only extreme values are needed.

hhhuuhu ,,:

hhhuuhu ,,:

1:09 26/153

Different solution sets

United Solution Set

Controllable Solution Set

Tolerable Solution Set

BAXBBAAXBA ,,:,

BAXBBAAXBA ,,:,

BAXBBAAXBA ,,:,

1:09 27/153

Example

]6,2[],2,1[,: BABXAXX

United Solution Set 6,12,1

]6,2[X

Tolerable Solution Set

]3,2[X

Controllable solution set

X

]6,2[2,1: XXX

]6,2[2,1: XXX

1:09 28/153

Example United solution set

Tolerable solution set

Controllable solution set

]4,2[4,1: XXX

4,

2

1

4,1

]4,2[X

]2,1[X ]4,2[4,1: XXX

X

]4,2[,4,1,: BABAXX

1:09 29/153

][ 00

Safety of the structures

00 P

AP

AP

0

0A

P

][PP

0A

P- true but not safe

- unacceptable solution

PAE,

1:09 30/153

Safety of the structures

000 ,,:

A

PPPA - Definition

of safe cross-section

000 ,,:

A

PPPA - Definition

of safe cross-section

or

1:09 31/153

More complicated safety conditions

lim it s ta te

uncerta in lim it s tate

1

2

crisp sta te

uncerta in sta te

1:09 32/153

It is possible to check safety of the structure using united solution sets

trueYXYYXX ,,:

falseYXYYXX ,,:

1:09 33/153

Equations with uncertain parameters

1:09 34/153

Equations with uncertain parameters

Let’s assume that u(x,h) is a solution of some equation.

huhxuu x ,

How to transform the vector of uncertain parameters

through the function uin the point x?

1:09 35/153

Transformation of uncertain parameters through the function ux

h

uhuu x

0h

00 huu x

1:09 36/153

Transformation of interval parameters

],[:)(],[ 00,0,0 hhhhuuu xxx

huu x

],[ 00 hh

],[ ,0,0xx uu

h

1:09 37/153

Transformation of random parameters

dhdu

uhf

du

dhuhfuf h

hu

Transformation of probability density functions.

hfh - the PDF of the uncertain parameter h is known.

PDF of the results

1:09 38/153

Transformation of random parameters

1:09 39/153

Main problem

The solution ux(h) is known implicitly and sometimes it is very difficult to calculate the explicit description of the function u=ux(h).

0,...,,,,2

jii xx

u

x

uuhx

1:09 40/153

Analytical solution

In a very few cases it is possible to calculate solution analytically. After that it is possible to predict behavior of the uncertain solution ux(h) explicitly.

Numerical solutions have greater practical significance than analytical one.

1:09 41/153

Newton method

01,, hxhxu

0, uhx 0,, uhx or

01,,

hhxhhxu

Etc.

1:09 42/153

Continuation method

Continuation methods are used to compute solution manifolds of nonlinear systems. (For example predictor-corrector continuation method).

1:09 43/153

Many methods need the solution

of the system of equations with interval parameters

hhhuFuhu ,0,:

x

y

hhhuFuhu ,0,:

hu

1:09 44/153

Interval solution of the equations with interval parameters

hu - smallest interval which contain the exact solution set.

hhhuFuhu ,0,:

1:09 45/153

Methods based on interval arithmetic

Muhanna’s method Neumaier’s method Skalna’s method Popova’s method Interval Gauss elimination method Interval Gauss-Seidel method etc.

1:09 46/153

Methods based on interval arithmetic

These methods generate the results with guaranteed accuracy

Except some very special cases it is very difficult to apply them to some real engineering problems

1:09 47/153

Overview of FEM method

1:09 48/153

Finite Element Method (FEM)

1:09 49/153

Real world truss structures

1:09 50/153

Truss structure

1:09 51/153

Boundary value problem

E – Young modulusA – area of cross-sectionu – displacementn – distributed load in x-direction

21,0

0

uLuuu

ndx

duEA

dx

d

1:09 52/153

Potential energy

LLL

Nunudxdxdx

duEAuI

000

2

2

1

N – axial forceL – length

0, uuI

1:09 53/153

Finite element method

QKuVufuL ,

i

iih uxNxuxu )()(

1:09 54/153

Truss element 1D

u1u E,A1 2

Lu(x)

x

E,An

1A

2A

xA

u ,1 xu ,2

E,A

1:09 55/153

Truss element 2D

x

yxLu ,,1

xLu ,,2

yLu ,,2

yLu ,,1

xu ,1

xu ,2

yu ,2

yu ,1

1

2

1:09 56/153

Truss element 3D

1:09 57/153

Variational equations

0, uuI

Frechet derivative

0,1

lim0

uuIuIuuIuu

00

00

LLL

uNdxundxdx

ud

dx

duEA

1:09 58/153

Variational equations

L

dxdx

ud

dx

duEAuua

0

),(

LL

uNudxnul0

0

)(

)(),( , uluuaVu

1:09 59/153

Galerkin’s method

i

iih uxNxuxu )()(

(v)v),( ,v luaV hh

QKu

1:09 60/153

Ritz’s method

),...,()( 1 Nh uuIuI

)(),( 2

1)( uluuauI

0),...,( 1

i

N

u

uuI

QKu

1:09 61/153

Parameter dependent system of equations

hhhQuhKuhu ,:

1:09 62/153

Optimization methods

1:09 63/153

hh

hfhuL

hh

hfhuL )(),(

)(),(

i

i

i

i

umax

u

umin

u

hh

hQuhK

hh

hQuhK )()(

,)()(

i

i

i

i

umax

u

umin

u

1:09 64/153

These methods can be applied to the very wide intervals

h

The function

)(huu

doesn't have to be monotone.

1:09 65/153

Numerical example

02

3

,0)0(

,02

3 ,0

2

),(

2

2

2

2

2

2

2

2

dx

Lud

dx

udLu

Lu

xqdx

udEJ

dx

d

q

L

2L

1:09 66/153

Numerical data

2

3,

2dla

12848248

9

24

1

EJ

1

20,dla

1284824

11

)(433

4

434

LL

xqL

xqLL

xqLqx

Lx

qlx

qlqx

EJxu

Analytical solution

1:09 67/153

0 5. 15.1

0 037.

0 022.

y x( )

x

q

L

2L

Interval global optimization method

1:09 68/153

Other optimization methods

DONLP2 and AMPL

Till today the results in some cases are promising however sometimes

they are very inaccurate and time-consuming.

COCONUT Projecthttp://www.mat.univie.ac.at/~neum/glopt/coconut/

Main problems: time of calculations, accuracy

1:09 69/153

Sensitivity analysis method

1:09 70/153

Monotone functions

1x 2x

)( 1xf

)( 2xf

0)(

dx

xd f

)(}ˆ:)(sup{ xfxxxfy

)(}ˆ:)(inf{ xfxxxfy

1:09 71/153

Sensitivity analysis

If 0)( 0

x

xf, then )(),( xyyxyy

If 0)( 0

x

xf, then )(),( xyyxyy

),(xfy ].,[ xxx

]3,1[,2 xxy

,2)(

xdx

xdy ,422

)2(

dx

dy ,1)( xyy 9)( xyy

]9,1[ˆ y

1:09 72/153

Truss structure example

1:09 73/153

Accuracy of sensitivity analysis method (5% uncertainty)

Accuracy in %

0 1,04E-02

0 0,00E+00

0,003855 0,00E+00

0 0,00E+00

0 0,00E+00

0 0,00E+00

0 1,89E-03

0 5,64E-01

0,026326 0,00E+00

0 4,87E-03

0 1,21E-03

0 0

18 – interval parameters

1:09 74/153

Extreme value of monotone functions

),...,,( 21 nxxxfy

nn xxxxxx ˆ,...,ˆ,ˆ 2211

nxxx ˆ...ˆˆˆ 21 x

)}ˆ(:)(min{ xxx Verticesyy

)}ˆ(:)(max{ xxx Verticesyy

n2 - calculations of y(x)

1:09 75/153

Complexity of the algorithm, which is based on sensitivity analysis

),(xfy .xx

,1x

f

,2x

f

nx

f

… - n derivatives

),,...,,( 21 nxxxfy .,...,, 21

nxxxfy

We have to calculate the value of n+3 functions.

,......, ixf

00 ,..., ni xxf 1

n

,,1 ,..., n

ni xxfy 2

1:09 76/153

Vector-valued functions

nxxxyy ,...,, 2111

nxxxyy ,...,, 2122

nmm xxxyy ,...,, 21

…

In this case we have to repeat previous algorithm m times.We have to calculate the value of m*(n+2) functions.

1:09 77/153

Implicit function

)()( xQyxA

)()()( 1 xQxAxy

yxAxQy

xAkkk xxx

)()(

)(

1:09 78/153

Sensitivity matrix

n

mmm

n

n

x

y

x

y

x

y

x

y

x

y

x

yx

y

x

y

x

y

...

............

...

...

21

2

2

2

1

2

1

2

1

1

1

x

yx 2y

2xy

1:09 79/153

Sign vector matrix

mn

mm

n

n

SSS

SSS

SSS

sign

...

............

...

...

21

222

21

112

11

x

y 2S

1:09 80/153

Independent sign vectors

,ji SS .)1( ji SS

jijiji S *****

** )1(,, SSSSSS

Number of independent sign vectors:

],1[ m

1:09 81/153

Complexity of the whole algorithm.

2*p – solutions (p times upper and lower bound).

],1[ mp

.21,12121 mnnpn

)()( xQyxA 1 - solution

n - derivatives .ixy

yxAxQy

xAkkk xxx

)()(

)(

)(xy

1:09 82/153

All sensitivity vector can be calculated in one system of equations

yxAxQy

xAkkk xxx

)()(

)(

yAQ

RHSkk

k xx

],...,[)( 1 nkx

RHSRHSy

xA

Complexity of the algorithm:

.22,12222 mp

kkx

RHSy

xA

)(

1:09 83/153

Sensitivity analysis method give us the extreme combination of the parameters

We know which combination of upper bound or lower bound will generate the exact solution.

We can use these values in the design process.

min,min,1 ,..., n

ni xxfy max,max,1 ,..., n

ni xxfy

1:09 84/153

Example

,

1111

1111

1111

1111

4

3

2

1

4

3

2

1

Q

Q

Q

Q

y

y

y

y

],2,1[ix

,

222

3

3222

444

4321

4

4321

4321

4

3

2

1

xxxx

x

xxxx

xxxx

Q

Q

Q

Q

1:09 85/153

Analytical solution

4321

321

4321

4321

4

3

2

1

xxxx

xxx

xxxxxxxx

y

y

y

y

]2,1[ix

]5 ,1[

]6 ,3[

]8 ,4[

]8 ,4[

1:09 86/153

Sensitivity matrix

1111

0111

1111

1111

4

4

3

4

4

3

3

3

4

2

3

2

4

1

3

1

2

4

1

4

2

3

1

3

2

2

1

2

2

1

1

1

x

y

x

yx

y

x

yx

y

x

yx

y

x

y

x

y

x

yx

y

x

yx

y

x

yx

y

x

y

x

y

x 1y

1:09 87/153

Sign vectors

4

4

3

4

4

3

3

3

4

2

3

2

4

1

3

1

2

4

1

4

2

3

1

3

2

2

1

2

2

1

1

1

x

y

x

yx

y

x

yx

y

x

yx

y

x

y

x

y

x

yx

y

x

yx

y

x

yx

y

x

y

signsignx

yS

4

3

2

1

1111

1111

1111

1111

1111

0111

1111

1111

S

S

S

S

S sign

1:09 88/153

Independent sign vectors

11

11

11

112*

1**

S

SS

1111

1111

1111

1111

4

3

2

1

S

S

S

S

1:09 89/153

Lower bound- first sign vector

1

1

1

1

)(

4

3

2

1

1*

x

x

x

x

Sx

2

3

4

4

))((

))((

))((

))((

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

1*4

1*3

1*2

1*1

1*

11*

1*

Sx

Sx

Sx

Sx

SxQSxASxy

y

y

y

y

))(())(())(( 1*

1*

1* SxQSxySxA

]1,1,1,1[1* S]2,1[ix

1:09 90/153

Upper bound- first sign vector

2

2

2

2

)(

4

3

2

1

1*

x

x

x

x

Sx

4

6

8

8

))((

))((

))((

))((

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

1*4

1*3

1*2

1*1

1*

11*

1*

Sx

Sx

Sx

Sx

SxQSxASxy

y

y

y

y

]1,1,1,1[1* S

]2,1[ix

1:09 91/153

Lower bound – second sign vector

]1,1,1,1[2* S

2

1

1

1

)(

4

3

2

1

2*

x

x

x

x

Sx

1

5

5

5

))((

))((

))((

))((

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

2*4

2*3

2*2

2*1

2*

12*

2*

Sx

Sx

Sx

Sx

SxQSxASxy

y

y

y

y

]2,1[ix

1:09 92/153

Upper bound – second sign vector

1

2

2

2

)(

4

3

2

1

2*

x

x

x

x

Sx

5

6

7

7

))((

))((

))((

))((

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

2*4

2*3

2*2

2*1

2*

12*

2*

Sx

Sx

Sx

Sx

SxQSxASxy

y

y

y

y

]1,1,1,1[2* S

]2,1[ix

1:09 93/153

Interval solution

1

3

4

4

))}(()),(()),(()),((min{ 2*

2*

1*

1* SxySxySxySxyy

5

6

8

8

))}(()),(()),(()),((max{ 2*

2*

1*

1* SxySxySxySxyy

]5 ,1[

]6 ,3[

]8 ,4[

]8 ,4[

yThe solution is exact

1:09 94/153

Taylor expansion method

m

iii

i

iii hh

h

uuu

10,

00

hhh

m

iii

i

iii hh

h

uuu

10,

00

hh

m

iii

i

iii hh

h

uuu

10,

00

hh

1:09 95/153

The function u=u(h) is usually nonlinear

0h h

u

huu

000 hh

dh

hduhuhuL

1:09 96/153

Accuracy of two methods of calculation (20% uncertainty)

1:09 97/153

Accuracy of two methods of calculation (50% uncertainty)

1:09 98/153

Comparison 50% uncertainty

 Sensitivity method [%] Taylor method [%]    Comparison [%]

-0,03 -1,19 43,01 -48,34 143466,7 3962,185

-37,1 -0,39 -11,27 -46,95 69,62264 11938,46

-1,53 -0,24 28,41 -44,04 1956,863 18250

-0,25 -4,3 -41,91 21,75 16664 605,814

-0,29 -0,28 43,11 -47,35 14965,52 16810,71

-0,33 -0,04 -45,43 38,26 13666,67 95750

0 -1,97 31,88 -45,78 inf 2223,858

-13,59 -15,68 -32,33 -30,86 137,8955 96,81122

Si

SiTi

du

dudu

,

,, %100

Si

SiTi

du

dudu

,

,, %100Tidu ,

Sidu ,

Tidu ,

Sidu ,

1:09 99/153

Time of calculation(endpoints combination method)

1:09 100/153

Time of calculation(First order sensitivity analysis)

1:09 101/153

Time of calculation(First order Taylor expansion)

1:09 102/153

Comparison

Number of interval

parameters Sensitivity Taylor %

105 2 0,02 9900

410 452 1,22 36949

915 15 208 16,64 91294

Time in seconds

1:09 103/153

APDL description

N 1 0 0 N 2 1 0

MP 1 EX 210E9 F 3 FX 1000 R 1 0.0025

(description of the nodes)

(material characteristics)

(forces)

(other parameters – cross section)

1:09 104/153

Interval extension of APDL language

MP EX 1 5 F 3 FX 5 R 1 10

(material characteristics)

(forces)

(other parameters – cross section)

Uncertainty in percent

1:09 105/153

Web applications

http://andrzej.pownuk.com/interval_web_applications.htm

Endpoint combination method

Sensitivity analysis method

Taylor method

1:09 106/153

Automatic generation of examples

1:09 107/153

The APDL and IAPDL code

1:09 108/153

The results

1:09 109/153

Calculation of the solutionbetween the nodal points

12

3

eeu 1

eu 2eu 3

eu 4

eu 5

eu 6

1x2x

3x

0x

eee uxNxu )()(

1:09 110/153

)(),(),( huhxNhxu eee

Extreme solution inside the elementcannot be calculated using only the nodal solutions u.(because of the unknown dependency of the parameters)

Extreme solution can be calculated using sensitivity analysis

m

eee

h

usign

h

usign

),( , ... ,

),( 00

1

00 hxhxS

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Calculation of extreme solutions between the nodal points.

1) Calculate sensitivity of the solution.(this procedure use existing results of the calculations)

m

eee

h

usign

h

usign

),( , ... ,

),( 00

1

00 hxhxS

2) If this sensitivity vector is new then calculatethe new interval solution.

The extreme solution can be calculated using this solution.

3) If sensitivity vector isn’t new then calculatethe extreme solution using existing data.

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Use of existing commercial FEM software

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Use of existing commercial FEM software

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Papers related to sensitivity analysis method

Pownuk A., Numerical solutions of fuzzy partial differential equation and its application in computational mechanics,

Fuzzy Partial Differential Equations and Relational Equations: Reservoir Characterization and Modeling (M. Nikravesh, L. Zadeh and V. Korotkikh, eds.), Studies in Fuzziness and Soft Computing,

Physica-Verlag, 2004, pp. 308-347

Neumaier A. and Pownuk, A. Linear systems with large uncertainties,

with applications to truss structures(submitted for publication).

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Monotonicity tests

Taylor expansion of derivative

Interval methods

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Monotonicity tests

m

jjj

jiii

hhhh

u

h

u

h

u

1

002

0 )()()()( hhh

m

jjj

jiii

hhhh

u

h

u

h

u

1

002

0 )ˆ()()()ˆ(ˆ

0hhh

If

then function

)(huu

is monotone.

1:09 117/153

High order monotonicity tests

...))(()(

2

1)(

)()()(

1

0002

002

0

m

j

m

j

m

kkkjj

jijj

jiii

hhhhhh

uhh

hh

u

h

u

h

u hhhh

...)ˆ()()()ˆ(ˆ

01

002

0

m

jjj

jiii

hhhh

u

h

u

h

u hhh

If

then function

)(huu

is monotone.

1:09 118/153

j

i

h

u )(0 h

hhhQuhK ),()(

)()()(

)(

huhKhQu

hK

iii hhh

)()( hQuhK

Exact monotonicity tests based on the interval arithmetic

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Finite difference method

x

xxfxxf

x

f

dx

df

2

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Slightly compressible flow- 2D case

t

p

B

cVqy

y

p

B

kA

yx

x

p

B

kA

x oc

bsc

yycxxc

)(1 o

o

ppc

BB

),,(),( * txptxpp

),,(),( * txq

n

txp

q

.),(),( 00 xxptxp

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Example

Injection well

Production well

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Interval solution (time step 1)p_upper(t) - p_lower(t)

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“Single-region problems”

xx 1 xx 2 xx 3

]2,1[x,321 xxxy

xxxx 321

xxxxxy 2321

2,

4

1]}2,1[:{ 2 xxx

x

y

1 2-1

1

2

xxy 2

4

1

1:09 124/153

Multi-region problems

1x 2x 3x

5,4]}2,1[,,:{ 321321 xxxxxx

Solution of single-region

problem

Solution of multi-region

problem

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More constraints – less uncertainty

,321 xxxy

321 xxx constraints:

Result with constraints(single-region)

Results without constraints(multi-region)

2,

4

1 5,4

].2,1[ix

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Multi-region case

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Data filealpha_c 5.614583 /* volume conversion factor */beta_c 1.127 /* transmissibility conversion factor */

/* size of the block */

dx 100dy 100h 100

/* time steps */time_step 15number_of_timesteps 10

reservoir_size 20 20

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Interval solution (time step 5)

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Comparison Single region - Multi-region

[0,55] [psi] [0, 390] [psi]

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Sensitivity in time-dependent problems

),(),( 1 hpQphpA ttt

11

),(),(),(

tt

k

t

kk

tt

hhhphpAhpQ

phpA

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Sensitivity

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Calculation of total rate and total oil production

PN

wi

wsciT tqtq

1

)()(

NTS

iiiTP ttqN

1

)(

wf

w

e

cwfscsc pp

sr

rB

khppqq

2

1ln

20

1:09 133/153

Interval total rate

PN

wwi

wsciT ptqtq

1

),()(

PN

wwi

wsciT ptqtq

1

),()(

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Interval total oil production

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Exact value of total rate

PN

wi

wsciT tqtq

1

)()(

PP N

w k

iwsc

N

wi

wsc

kiT

k h

p

p

tqtq

htq

h 11

)()()(

)( iTR tqsign

hS

1:09 136/153

RShh R RShh R

),()( RiRi tt hpp ),()( RiRi tt hpp

))(,()( RiiTiT ttqtq p))(,()( RiiTiT ttqtq p

)](),([)]([ iTiTiT tqtqtq

1:09 137/153

Equations with different kind of uncertainty

in parameters

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Combination of random and interval parameters

hruu ,r – random parameterh – interval parameter

][:,: hhAhruPAPl

A

u hhduhufAPAP ][:,

or

1:09 139/153

Combination of random and fuzzy parameters

hruu ,r – random parameterh – fuzzy parameter

][:,: hhAhruPAPl

A

u hhduhufAPAP ][:,][

or

1:09 140/153

Combination of random and random sets parameters

hruu ,

r – random parameterh – random set parameter (set valued random variable)

AhruPAPl ,:,

etc.

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Calculation risk of cost using Monte Carlo method

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Interval web applications

http://andrzej.pownuk.com/interval_web_applications.htm

NodeNumberOfNode 0NumberOfChildren 2Children 1 2 IntervalProbability 0.05xMinMin 1xMinMax 1.1xMidMin 2.0xMidMax 2.0xMaxMin 6xMaxMax 6.11NumberOfGrid 1ProbabilityGrids 2DistributionType 3End

Node NumberOfNode 1NumberOfChildren 1Children 2 xMinMin 1xMinMax 1.1xMidMin 3xMidMax 3.11xMaxMin 6xMaxMax 6.11NumberOfGrid 2DistributionType 2End

Node NumberOfNode 2xMinMin 1xMinMax 1.1xMidMin 3xMidMax 3.11xMaxMin 6xMaxMax 6.11NumberOfGrid 3DistributionType 1 End

ResultsXmin 0Xmax 10NumberOfSimulations 2000NumberOfClasses 10NumberOfGrid 2DistributionType 2End

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Future plans

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Future plans -uncertain functions

Equivalent of random fields

xEE

x

E

P

L

Different kind of dependences – not only interval or random constraints.Time series with interval, fuzzy, random sets parameters.

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Future plans - software (web applications)http://andrzej.pownuk.com/download.htm

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Future plans - design and optimization under uncertainty

hfhxf opt

xx

),(min

}:{][ hhhfhf optopt

hh

dx

hxdfxhx ,0

,:

1:09 147/153

Taking into account economical constraints

C

0C 0CCP

00 RCCP

0R

- real cost

- assumed cost

- investment risk

- acceptable risk level

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Cooperation with commercial companies

ChevronTexaco http://www.chevrontexaco.com/

Commercial FEM companies

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Conclusions

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Conclusions In cases where data is limited and

pdfs for uncertain variables are unavailable, it is better to use imprecise probability rather than pure probabilistic methods.

Using interval methods we can create mathematical model which is based on very uncertain information.

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Presented algorithms are efficient when compared to other methods which model uncertainty, and can be applied to nonlinear problems of computational mechanics.

Sensitivity analysis method gives very accurate results.

Taylor expansion method is more efficient than sensitivity analysis method but less accurate.

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Conclusions

It is possible to include presented algorithms in the existing FEM code.

In calculations it is possible to use different kind of uncertainty (crisp numbers, intervals, random variables, random sets, fuzzy sets, fuzzy random variables etc.)

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Thank you