mathematical modeling of uncertainty in computational mechanics

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Mathematical modeling of uncertainty in computational mechanics Andrzej Pownuk Silesian University of Technology Poland [email protected] http:// andrzej.pownuk.com

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Mathematical modeling of uncertainty in computational mechanics. Andrzej Pownuk Silesian University of Technology Poland [email protected] http://andrzej.pownuk.com. Schedule. Different kind of uncertainty Design of structures with uncertain parameters - PowerPoint PPT Presentation

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Page 1: Mathematical modeling  of uncertainty  in computational mechanics

Mathematical modeling of uncertainty

in computational mechanics

Andrzej PownukSilesian University of Technology

[email protected]

http://andrzej.pownuk.com

Page 2: Mathematical modeling  of uncertainty  in computational mechanics

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

Page 3: Mathematical modeling  of uncertainty  in computational mechanics

],[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

Page 4: Mathematical modeling  of uncertainty  in computational mechanics

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 %

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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

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Different kind of uncertainty

Page 7: Mathematical modeling  of uncertainty  in computational mechanics

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Floating-point and real numbers

Rh 0

h0h

0h - parameter

20 he.g.

Floating-point numbers emh 100

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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

Page 9: Mathematical modeling  of uncertainty  in computational mechanics

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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

Page 10: Mathematical modeling  of uncertainty  in computational mechanics

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Uncertain parameters

Random parameters

],,[:],[ hhhPhhP

Rhh :

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

Page 11: Mathematical modeling  of uncertainty  in computational mechanics

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Uncertain parameters

Bayesian probability

BP

APABPBAP

||

Cox's theorem - "logical" interpretation of probability

Page 12: Mathematical modeling  of uncertainty  in computational mechanics

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Uncertain parameters Interval

parameters

],[ˆ hhhh

Interval parameter is not equivalent to

uniformly distributed random variable

Page 13: Mathematical modeling  of uncertainty  in computational mechanics

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Uncertain parameters

Set valued random variable

Upper and lower probability

nRhh :

AhPAPl :

AhPABel :

Page 14: Mathematical modeling  of uncertainty  in computational mechanics

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Uncertain parameters Nested family of random sets

Nhhh ...21

}:{ hxPxF x

xF

1h

2h

3h

1F

2F

3F

0x

0xF

Page 15: Mathematical modeling  of uncertainty  in computational mechanics

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Uncertain parameters

Fuzzy sets

0F

0F

1F

1F

1

0

xF

F

F

x

xy Fxfyx

Ff

:sup

Extension principle

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Uncertain parameters

Fuzzy random variables

Random variables with fuzzy parameters

RFhh :

RFphpRFh ,,:

Etc.

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Design of structures with

uncertain parameters

Page 18: Mathematical modeling  of uncertainty  in computational mechanics

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Design of structures Safety condition

0A

P

PAE,

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

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Safe area

A

PAP 0

Safe area

0A

P

Page 20: Mathematical modeling  of uncertainty  in computational mechanics

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Design of structures with interval parameters

A

P AP 0

Safe area

],[ 000

Page 21: Mathematical modeling  of uncertainty  in computational mechanics

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Design of structures with interval parameters

A

P

AP 0

],[ 000

0P

0A

0P],[ 00

PPP

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

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More complicated cases

P11AE 22AE

2L1L

PPEEEEPEEAAAA ,,,,,,,:, 2211212121

PEEAA ,,,, 2121 - design constraints

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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

Page 24: Mathematical modeling  of uncertainty  in computational mechanics

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Geometrical safety conditions

maxmin uuu

inu

maxu

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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 ,,:

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Different solution sets

United Solution Set

Controllable Solution Set

Tolerable Solution Set

BAXBBAAXBA ,,:,

BAXBBAAXBA ,,:,

BAXBBAAXBA ,,:,

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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

Page 28: Mathematical modeling  of uncertainty  in computational mechanics

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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

Page 29: Mathematical modeling  of uncertainty  in computational mechanics

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][ 00

Safety of the structures

00 P

AP

AP

0

0A

P

][PP

0A

P- true but not safe

- unacceptable solution

PAE,

Page 30: Mathematical modeling  of uncertainty  in computational mechanics

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Safety of the structures

000 ,,:

A

PPPA - Definition

of safe cross-section

000 ,,:

A

PPPA - Definition

of safe cross-section

or

Page 31: Mathematical modeling  of uncertainty  in computational mechanics

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

Page 32: Mathematical modeling  of uncertainty  in computational mechanics

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

trueYXYYXX ,,:

falseYXYYXX ,,:

Page 33: Mathematical modeling  of uncertainty  in computational mechanics

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Equations with uncertain parameters

Page 34: Mathematical modeling  of uncertainty  in computational mechanics

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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?

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Transformation of uncertain parameters through the function ux

h

uhuu x

0h

00 huu x

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Transformation of interval parameters

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

huu x

],[ 00 hh

],[ ,0,0xx uu

h

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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

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Transformation of random parameters

Page 39: Mathematical modeling  of uncertainty  in computational mechanics

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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

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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.

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Newton method

01,, hxhxu

0, uhx 0,, uhx or

01,,

hhxhhxu

Etc.

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Continuation method

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

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Many methods need the solution

of the system of equations with interval parameters

hhhuFuhu ,0,:

x

y

hhhuFuhu ,0,:

hu

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Interval solution of the equations with interval parameters

hu - smallest interval which contain the exact solution set.

hhhuFuhu ,0,:

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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.

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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

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Overview of FEM method

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Finite Element Method (FEM)

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Real world truss structures

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Truss structure

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Boundary value problem

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

21,0

0

uLuuu

ndx

duEA

dx

d

Page 52: Mathematical modeling  of uncertainty  in computational mechanics

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Potential energy

LLL

Nunudxdxdx

duEAuI

000

2

2

1

N – axial forceL – length

0, uuI

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Finite element method

QKuVufuL ,

i

iih uxNxuxu )()(

Page 54: Mathematical modeling  of uncertainty  in computational mechanics

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Truss element 1D

u1u E,A1 2

Lu(x)

x

E,An

1A

2A

xA

u ,1 xu ,2

E,A

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Truss element 2D

x

yxLu ,,1

xLu ,,2

yLu ,,2

yLu ,,1

xu ,1

xu ,2

yu ,2

yu ,1

1

2

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Truss element 3D

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Variational equations

0, uuI

Frechet derivative

0,1

lim0

uuIuIuuIuu

00

00

LLL

uNdxundxdx

ud

dx

duEA

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Variational equations

L

dxdx

ud

dx

duEAuua

0

),(

LL

uNudxnul0

0

)(

)(),( , uluuaVu

Page 59: Mathematical modeling  of uncertainty  in computational mechanics

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Galerkin’s method

i

iih uxNxuxu )()(

(v)v),( ,v luaV hh

QKu

Page 60: Mathematical modeling  of uncertainty  in computational mechanics

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Ritz’s method

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

)(),( 2

1)( uluuauI

0),...,( 1

i

N

u

uuI

QKu

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Parameter dependent system of equations

hhhQuhKuhu ,:

Page 62: Mathematical modeling  of uncertainty  in computational mechanics

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Optimization methods

Page 63: Mathematical modeling  of uncertainty  in computational mechanics

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hh

hfhuL

hh

hfhuL )(),(

)(),(

i

i

i

i

umax

u

umin

u

hh

hQuhK

hh

hQuhK )()(

,)()(

i

i

i

i

umax

u

umin

u

Page 64: Mathematical modeling  of uncertainty  in computational mechanics

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These methods can be applied to the very wide intervals

h

The function

)(huu

doesn't have to be monotone.

Page 65: Mathematical modeling  of uncertainty  in computational mechanics

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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

Page 66: Mathematical modeling  of uncertainty  in computational mechanics

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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

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0 5. 15.1

0 037.

0 022.

y x( )

x

q

L

2L

Interval global optimization method

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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

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Sensitivity analysis method

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Monotone functions

1x 2x

)( 1xf

)( 2xf

0)(

dx

xd f

)(}ˆ:)(sup{ xfxxxfy

)(}ˆ:)(inf{ xfxxxfy

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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

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Truss structure example

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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

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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)

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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

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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.

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Implicit function

)()( xQyxA

)()()( 1 xQxAxy

yxAxQy

xAkkk xxx

)()(

)(

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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

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Sign vector matrix

mn

mm

n

n

SSS

SSS

SSS

sign

...

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

...

...

21

222

21

112

11

x

y 2S

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Independent sign vectors

,ji SS .)1( ji SS

jijiji S *****

** )1(,, SSSSSS

Number of independent sign vectors:

],1[ m

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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

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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

)(

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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

Page 84: Mathematical modeling  of uncertainty  in computational mechanics

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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

Page 85: Mathematical modeling  of uncertainty  in computational mechanics

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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[

Page 86: Mathematical modeling  of uncertainty  in computational mechanics

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

Page 87: Mathematical modeling  of uncertainty  in computational mechanics

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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

Page 88: Mathematical modeling  of uncertainty  in computational mechanics

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Independent sign vectors

11

11

11

112*

1**

S

SS

1111

1111

1111

1111

4

3

2

1

S

S

S

S

Page 89: Mathematical modeling  of uncertainty  in computational mechanics

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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

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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

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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

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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

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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

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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

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The function u=u(h) is usually nonlinear

0h h

u

huu

000 hh

dh

hduhuhuL

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Accuracy of two methods of calculation (20% uncertainty)

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Accuracy of two methods of calculation (50% uncertainty)

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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 ,

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Time of calculation(endpoints combination method)

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Time of calculation(First order sensitivity analysis)

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Time of calculation(First order Taylor expansion)

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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

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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)

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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

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Web applications

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

Endpoint combination method

Sensitivity analysis method

Taylor method

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Automatic generation of examples

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The APDL and IAPDL code

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The results

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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 )()(

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)(),(),( 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

Page 114: Mathematical modeling  of uncertainty  in computational mechanics

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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

Page 116: Mathematical modeling  of uncertainty  in computational mechanics

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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.

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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.

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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

Page 120: Mathematical modeling  of uncertainty  in computational mechanics

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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

Page 121: Mathematical modeling  of uncertainty  in computational mechanics

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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

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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

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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

Page 136: Mathematical modeling  of uncertainty  in computational mechanics

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RShh R RShh R

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

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

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

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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

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Combination of random and fuzzy parameters

hruu ,r – random parameterh – fuzzy parameter

][:,: hhAhruPAPl

A

u hhduhufAPAP ][:,][

or

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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

,:

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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