Mathematical modeling of uncertainty in computational mechanics
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Transcript of Mathematical modeling of uncertainty in computational mechanics
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Mathematical modeling of uncertainty
in computational mechanics
Andrzej PownukSilesian University of Technology
http://andrzej.pownuk.com
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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
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],[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
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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
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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
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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
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Uncertain parameters
Random parameters
],,[:],[ hhhPhhP
Rhh :
Using probability theoryone can say that buildings are usually safe ...
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Uncertain parameters
Bayesian probability
BP
APABPBAP
||
Cox's theorem - "logical" interpretation of probability
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Uncertain parameters Interval
parameters
],[ˆ hhhh
Interval parameter is not equivalent to
uniformly distributed random variable
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Uncertain parameters
Set valued random variable
Upper and lower probability
nRhh :
AhPAPl :
AhPABel :
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Uncertain parameters Nested family of random sets
Nhhh ...21
}:{ hxPxF x
xF
1h
2h
3h
1F
2F
3F
0x
0xF
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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
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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
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Design of structures with interval parameters
A
P AP 0
Safe area
],[ 000
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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
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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
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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
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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,
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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
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More complicated safety conditions
lim it s ta te
uncerta in lim it s tate
1
2
crisp sta te
uncerta in sta te
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It is possible to check safety of the structure using united solution sets
trueYXYYXX ,,:
falseYXYYXX ,,:
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Equations with uncertain parameters
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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
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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
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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 )()(
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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
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Galerkin’s method
i
iih uxNxuxu )()(
(v)v),( ,v luaV hh
QKu
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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 ,:
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Optimization methods
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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
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These methods can be applied to the very wide intervals
h
The function
)(huu
doesn't have to be monotone.
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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
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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
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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
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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[
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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
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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
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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
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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
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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.
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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
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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
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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
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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