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Numerical solutions of fuzzy
partial differential equation
and its application
in computational mechanics
Andrzej Pownuk
Char of Theoretical Mechanics
Silesian University of Technology
Andrzej Pownuk
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Numerical example
Plane stress problem
in theory of elasticity
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Plane stress problem
in theory of elasticity
E
E
u, 
u,   f   0, ,   1,2
2(1  )
2(1  )
u  u* , x   u
 n  t* , x   
 - mass density,
E, - material constant,
f  - mass force.
u,
u

x
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Triangular fuzzy number
1
F
h0
 F (h )
h
h1  h1
h
h0
hˆ  {h :  F (h)  }
hˆ  [h , h ]
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q
L
E2
L
n L
E1
L
L
L
L
L
m L
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Data
Eˆ 01  [189, 231], [GPa],   0,
Eˆ 1  [210, 210], [GPa],   1,
1
Eˆ 02
Eˆ12
 [189, 231], [GPa],   0,
 [210, 210], [GPa],   1.
  0.3
 kN 
q  1  , L  1 [m ]
m
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Time of calculation
n
5
10
20
30
m
5
10
20
40
DOF
72
242
882
2542
Elements
50
200
800
2400
Time
00:00:01
00:00:09
00:03:50
01:27:52
Processor: AMD Duron 750 MHz
RAM: 256 MB
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Numerical example
Shell structure with
fuzzy material properties
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Equilibrium equations
of shell structures
T  | b M  | b   0
T  b  M  | b 3  0
T  n  b M  n  p  , x  
M  | n 
d
( M   n )  p 3 , x  
ds
where
 
u |  u ,   u ,
 


u 
u ,   , ,   1,2
x

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Numerical data (=0)
E [2.0 105 , 2.2 105 ] [ MPa],
  0.2, 0.3,
L=0.263 [m], r=0.126 [m], F=444.8 [N], t= 2.38103 [m]
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Numerical results (fuzzy displacement)
=0: u   0.043514,0.03748 [m]
=1: u = -0.04102 [m].
Using this method we can obtain
the fuzzy solution in one point.
The solution was calculated
by using the ANSYS FEM program.
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The main goal of this presentation
is to describe methods of solution
of partial differential equations
with fuzzy parameters.
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Basic properties
of fuzzy sets
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Fuzzy sets
 F : R  x   F ( x )  [0,1]  R
 F (x )
1
x
 AB ( x )  min{ A ( x ),  B ( x)}
 AB ( x )  T ( A ( x ),  B ( x ))
 AB ( x )  max{ A ( x ),  B ( x )}
 AB ( x )  S ( A ( x ),  B ( x ))
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Extension principle
y  f ( x1 , x2 ,..., xn )
 f ( F ) ( y) 
max
y  f ( x1 , x2 ,...,xn )
min{ F ( x1 ),  F ( x2 ),...,  F ( xn )}
 f ( F ) ( y )  max  F (x),
y f (x)
f : R n  R,
F  F ( R n ),
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f ( F )  F ( R ).
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Fuzzy equations
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Fuzzy algebraic equations
H( y , h )  0

H: R R  R ,
n
m
n
y  y ( h)
F  F (R )
m
 F : Rm  h   F (h) [0, 1]
 H ( F ) (y) 
max  F (h)
h:H( y ,h ) 0
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Fuzzy differential equation
(example)
dy
 h  x,
dx
y(0)  y0 ,
h  F  F ( R)
hx2
y( x, h) 
 y0
2
(  | y F ( x )) 
max2
h:  
h x
 y0
2
 F (h)
y F ( x)  F ( R)
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Definition of the solution
of fuzzy differential equation
dy
 f ( x, y, h),
dx
( | yF ( x)) 
y(0)  y0 ,
h  F  F ( R)
max
h: ξ  y ( x,h),
dy
 y ( x ,h ), y ( 0) y0
dx
 F (h)
y F ( x)  F ( R)
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Fuzzy partial differential equations
u  2 u
k u
H( x, u, , 2 ,..., k , h)  0,
x x
x
(ξ | u F (x )) 
u V ,
max
h: ξ  u ( x ,h ), H ( x ,u, h,
h  F  F (Rm )
u
 u
,...., k ,h ) 0, uV
x
x
k
 F ( h)
u F (x)  F ( R n )
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[1, 2]  x  [2, 8]
Algebraic solution set
[1, 2]  [ x  , x  ]  [2, 8]
xˆ  [2, 4]
United solution set
xˆ  {x : a  [1,2], b  [2,4], a  x  b}  [1,4]
Controllable solution set
xˆ  {x : a  [1,2], b  [2,8], a  x  b}
xˆ  {x : [1,2]  x  [2,8]}  
Tolerable solution set
xˆ  {x : a  [1,2], b  [2,8], a  x  b}
xˆ  {x : [1,2]  x  [2,8]}  [2,4]
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Remarks
Buckley J.J., Feuring T., Fuzzy differential equations.
Fuzzy Sets and System, Vol.110, 2000, 43-54
F ( x)  cl{ y :  F ( x ) ( y )  }
F ( x)  inf F ( x),
F ( x)  sup F ( x).



d
d
dF
(
x
)
dF
( x)
F ( x)  [ F ( x ), F ( x )]   
, 
dx
dx
dx
 dx
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 dF ( x )

dx

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- Goetschel-Voxman derivative,
- Seikkala derivative,
- Dubois-Prade derivative,
- Puri-Ralescu derivative,
- Kandel-Friedman-Ming derivative,
- etc.
This derivative leads to another definition
of the solution of the fuzzy differential equation.
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Applications of fuzzy equations
in computational mechanics
Physical interpretations
of fuzzy sets
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Equilibrium equations of
isotropic linear elastic materials
ij
 2ui
 Xi   2 ,
x j
t
ij  Cijklkl ,
1  ui u j
ij 


2  x j xi
ui  ui* ,
x  u ,

,


*
ijn j  ti ,
x   ,
u( x, t ) t 0  u* ( x), x  .
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Uncertain parameters
- Fuzzy loads,
- Fuzzy geometry,
- Fuzzy material properties,
- Fuzzy boundary conditions e.t.c.
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Modeling of uncertainty
Probabilistic methods
X  :   R,
f X (x ).
Semi-probabilistic methods
x

x
Usually we don’t have enough information to calculate
probabilistic characteristics of the structure.
We need another methods of modeling
of uncertainty.
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Random sets interpretation
of fuzzy sets
Hˆ  :     Hˆ  ()  I ( R)
Pl( A)  P{ : Hˆ  ()  A  }
Hˆ  (1 )  Hˆ  (2 )  ...  Hˆ  (n )
 F (h)  Pl({h})  P{ : h  Hˆ  ()}
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Dubois D., Prade H.,
Random sets and fuzzy interval analysis.
Fuzzy Sets and System, Vol. 38, pp.309-312, 1991
Goodman I.R., Fuzzy sets as a equivalence class
of random sets. Fuzzy Sets and Possibility Theory.
R. Yager ed., pp.327-343, 1982
Kawamura H., Kuwamato Y.,
A combined probability-possibility evaluation theory
for structural reliability.
In Shuller G.I., Shinusuka G.I., Yao M. e.d.,
Structural Safety and Reliability,
Rotterdam, pp.1519-1523, 1994
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Bilgic T., Turksen I.B., Measurement of membership function
theoretical and empirical work.
Chapter 3 in Dubois D., Prade H., ed.,
Handbook of fuzzy sets and systems, vol.1
Fundamentals of fuzzy sets, Kluwer, pp.195-232, 1999
Philippe SMETS, Gert DE COOMAN,
Imprecise Probability Project,
etc.
Nguyen H.T., On random sets and belief function,
J. Math. Anal. Applic., 65, pp.531-542, 1978
Clif Joslyn, Possibilistic measurement and sets statistics. 1992
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Ferrari P., Savoia M., Fuzzy number theory to obtain
conservative results with respect to probability,
Computer methods in applied mechanics and engineering,
Vol. 160, pp. 205-222, 1998
Tonon F., Bernardini A., A random set approach
to the optimization of uncertain structures,
Computers and Structures, Vol. 68, pp.583-600, 1998
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Random sets interpretation
of fuzzy sets
 F ( P2 )  1
 F ( P1 )  0.5
 F (P )
1
 F ( P1 )  P{1}  P{2 } 
2
Hˆ  (4 )
 P {4 }
Hˆ  (3 )
 P {3}
 P {2 }
Hˆ  (2 )
 P {1}
Hˆ  (1 )
P1
P2
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
i 
P{i }  1
P
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This is not
a probability density function
or a conditional probability
and
cannot be converted
to them.
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Xˆ  :     Xˆ  ()  I ( R)
X  :     X  ()  R
 , X  ()  Xˆ  ()
P ([a, b])  P{ : X  () [a, b]}
X

Pl([a, b])  P{ : Xˆ  ()  [a, b]  }
P ([a, b])  Pl([a, b])
X

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Random sets
Hˆ  :     Hˆ  ()  I ( R)
Probabilistic
methods
H  ()  H  ()  H  ()
Hˆ  (1 )  Hˆ  (2 )  ...  Hˆ  (n )
Fuzzy
methods
Semi-probabilistic
methods
(interval methods)
Hˆ  (1 )  Hˆ  (2 )  ...  Hˆ  (n )
or
another procedures.
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Design of structures
with fuzzy parameters
Pf  Pl{g (h)  0} 
0
Pf
Pf  sup  F ( h )
h: g ( h ) 0
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Equation with fuzzy
and random parameters
X  :     X  ()  R,
Hˆ  :     Hˆ  ( )  I ( R),
 F (h)  P{ : h  Hˆ  (  )}.
Pf  P{(, ) : g( X  (), Hˆ  ( ))  0}
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
Pf
 P{(, ) : g( X  (), Hˆ  ( ))  0}
 x( F ) ( x) 
Pf 
sup  F (h )
h:g ( x ,h )0
 P {x} 

x ( F ) ( x)
x

Pf 

x ( F ) ( x )dP ( x )
 E ( F ( x))

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General algorithm
ˆ :  H
ˆ ( )  I ( R),
H


ˆ ( )}.
 F (h)  P{ : h  H
L( u, h)  f (h), u V
K (h)u  Q(h), y  g (h)
 g ( F ) ( y )  sup  F ( h)
h: y  g ( h )
Pf  sup  F (h)  sup  g ( F ) ( y )
h:g ( h )0
y: y 0
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Other methods of modeling of uncertainty:
- TBM model (Philip Smith).
- imprecise probability
(Imprecise Probability Project,
Buckley, Thomas etc.).
- etc.
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Numerical methods of solution
of partial differential equations
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Numerical methods of solution
of partial differential equations
- finite element method (FEM)
- boundary element method (BEM)
- finite difference method (FDM)
1) Boundary value problem.
2) Discretization.
3) System of algebraic equations.
4) Approximate solution.
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Finite element method
Using FEM
we can solve very
complicated problems.
These problems
have thousands
degree of freedom.
Curtusy to ADINA R & D, Inc.
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Algorithm
  2u  2u 
  2  2   f , x  
y 
 x
u  0, x  
  2u  2u 
   2  2 vd   fvd
x
y 


 u v u v 
  x x  y x d   fvd
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 u v u v 
a(u, v)   

d
x x y x 

l ( )   fvd 

v V , a(u,v)  l (v )
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
Vh  V
i
i
n
2
uh ( x )   ui  i ( x ),
1
vh ( x )   vi  i ( x )
i
i
i ( x j )  ij - shape functions
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vh Vh , a(uh ,vh )  l (vh )
System of linear algebraic equations
Ku  Q
Kij  a(i ,  j ),
Qi  l (i )
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Approximate solution
1
uK Q
uh ( x )   ui  i ( x ),
i
uh ( x)  u( x)
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Numerical methods of solution
of fuzzy
partial differential equations
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Application of finite element method
to solution
of fuzzy partial differential equations.
Parameter dependent boundary value problem.
L(x, u, h)  f (x, h), u V , h  F
K (h)u  Q(h), h  F
u  u( h), h  F
uF  F ( Rn )
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-level cut method
hˆ   {h :  F (h)  }
uˆ   {u : K(h)u  Q(h), h  hˆ }
u( F ) (u)  sup{ : u  uˆ  }
The same algorithm can be apply
with BEM or FDM.
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http://zeus.polsl.gliwice.pl/~pownuk
51/138
uˆ   {u : K(h)u  Q(h), h  hˆ }
Computing accurate
solution is NP-Hard.
Kreinovich V., Lakeyev A., Rohn J., Kahl P., 1998,
Computational Complexity Feasibility of Data Processing
and Interval Computations.
Kluwer Academic Publishers, Dordrecht
We can solve these equation
only in special cases.
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
52/138
Solution set of system of linear
interval equations is very complicated.
[1,2] [2,4]  x1  [-1,1]
[2,4] [1,2]   x    [1,2] 

  2 

2
3
3

( A ,B )
hull
3

( A ,B )
1
3
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
53/138
Monotone functions
u   u(h  )
u  u(h )
u   u(h  )
h
u   u(h  ),
h
h
u   u(h  ).
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
54/138
uˆ   {u : K(h)u  Q(h), h  hˆ }
hˆ   Rm
2 m system equations have to be solved.
Sensitivity analysis
u
 0 , then
h
u
If
 0 , then
h
If
u   u(h  ),
u   u(h  )
u   u(h  ),
u   u(h  )
1+2n system of equation
(in the worst case)
have to be solved.
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
55/138
Multidimensional algorithm
K(h)u  Q(h), h  hˆ 




u
(
h
)

Q
(
h
)

K
(
h


0
0
0)
K (h0 )


u( h0 ), i  1,...,m
hi
hi
hi
h0  m id( hˆ  )

 ui 
 ui 
, i  1,...,n
 ,..., sign
S   sign
 h1 
 hm 


i
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
56/138
Calculate unique sign vectors
*
q
S , q  1,...,k.

i

j
If S  (1)  S , then S  S .

i

j
Calculate unique interval solutions
*
*
ˆ
ˆ
uˆ  [u(h , Si ), u(h , (1)  Si )]
*
i
Calculate all interval solutions

i
*
j
i {1,...,n}, j {1,...,k}, uˆ  uˆ
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
57/138
Computational complexity
si
 u1
 h
 1
 ...
u  ui

h  h1
 ...
 un

 h1
1+2n system of equation (in the worst case)
have to be solved.
...
...
...
...
...
u1 
hm 

... 
ui 

hm 
... 
un 

hm 
All sign
vectors
S1 
 
 ... 
S   S i  
 
 ... 
S n 
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
Unique
sign vectors
S1* 
 
 ... 
S *  S q * 
 
 ... 
S k * 
58/138
This method can be applied
only when
the relation between the solution
and uncertain parameters
u  u(h)
is monotone.
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
59/138
According to my experience
(and many numerical results
which was published)
in problems of computational mechanics
the intervals hˆ  are usually narrow
and the relation u=u(h)
is monotone.
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
60/138
Akpan U.O., Koko T.S., Orisamolu I.R., Gallant B.K.,
Practical fuzzy finite element analysis of structures,
Finite Elements in Analysis and Design, 38 (2000) 93-111
McWilliam S., Anti-optimization of uncertain structures
using interval analysis,
Computers and Structures, 79 (2000) 421-430
Noor A.K., Starnes J.H., Peters J.M.,
Uncertainty analysis of composite structures,
Computer methods in applied mechanics and engineering,
79 (2000) 413-232
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
61/138
Valliappan S., Pham T.D., Elasto-Plastic Finite Element Analysis
with Fuzzy Parameters, International Journal
for Numerical Methods in Engineering, 38 (1995) 531-548
Valliappan S., Pham T.D., Fuzzy Finite Analysis
of a Foundation on Elastic Soil Medium.
International Journal for Numerical Methods and Engineering,
17 (1993) 771-789
Maglaras G., Nikolaidids E., Haftka R.T., Cudney H.H.,
Analytical-experimental comparison of probabilistic methods
and fuzzy set based methods for designing under uncertainty.
Structural Optimization, 13 (1997) 69-80
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
62/138
Particular case system of linear interval equations
F
 K11
... K1Fn   X 1  Q1F 

   
 ... ... ...   ...    ... 
 K F ... K F   X  Q F 
nn   n   n 
 n1

 Kˆ 11
... Kˆ 1n   X 1  Qˆ1 

   
 ... ... ...   ...    ... 
 Kˆ  ... Kˆ    X  Qˆ  
nn   n 
 n1
 n
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
63/138
K 0u0  Q0

 u 0
K0
hi
Si


Q0
hi


K 0
hi

u0

 ui 
 ui 

,..., sign
 sign

 h1 
 hm 
Si  Sj* , where j  Ci
i*
i*
i*
ˆ
ˆ
ˆ
X   [X(h , S ), X(h , (1)  S )]
j*
ˆ
ˆ
X i  X i , where j  Ci
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
64/138
Computational complexity
of this algorithm
1+2p - system of equations.
i*
p - number of independent sign vectors S  .
p  [1, n ]
[1  2, 1  2 n ]
- system of equations
n - number of degree of freedom.
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
65/138
Calculation of the solution
between the nodal points
u 6e
3
x3
u5e
x0

u 2e
1
x1
u e ( x )  N e ( x )u e
e
u 4e
u1e
2
u 3e
x2
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
66/138
ue (x, h)  Ne (x, h)ue (h)
Extreme solution inside the element
cannot be calculated using only the nodal solutions u.
(because of the unknown dependency of the parameters)
Extreme solution can be calculated
using sensitivity analysis
e
e





u
(
x
,
h
)

u
( x 0 , h 0 ) 
e
0
0
, ... , sign

S  sign




h

h
1
m





Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
67/138
Calculation of extreme solutions between
the nodal points.
1) Calculate sensitivity of the solution.
(this procedure use existing results of the calculations)
e
e





u
(
x
,
h
)

u
( x 0 , h 0 ) 
e
0
0
, ... , sign

S  sign




h

h
1
m





2) If this sensitivity vector is new then calculate
the new interval solution.
The extreme solution can be calculated using this solution.
3) If sensitivity vector isn’t new then calculate
the extreme solution using existing data.
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
68/138
L
Numerical
example
q
4
E,
L
3
Plane stress problem
in theory of elasticity
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
2
L
1
69/138
Plane stress problem
in theory of elasticity
E
E
u, 
u,   f   0, ,   1,2
2(1  )
2(1  )
u  u* , x   u
 n  t* , x   
 - mass density,
E, - material constant,
f  - mass force.
u,
u

x
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
70/138
Finite element method
Ku=Q
K
B
T
DBd,

Q  N fd 
T

e
u( x )  N( x )u,
N
T
tdS,
 
ui ( x)  N ij ( x)u j .
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
71/138
u 6e
x3
3
 x1 
x   ,
 x2 
u5e
e
u 2e
1
x1
u 4e
u1e
2
 u1 
u 
u2 
u( x )  N ( x ) u
u 3e
x2
 u1 
u 
 2
0
N 2 (x)
0
N 3 (x)
0   u3 
 u1 ( x )   N1 (x )
u( x )  

 


N1 ( x )
0
N 2 (x)
0
N 3 ( x )  u 4 
u 2 ( x )   0
u5 
 
u6 
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
72/138
 x13 
x3   3 
 x2 
 x11 
x1   1 
 x2 
e
u6
3
x3

u 2e
1
x1
u5e
 x12 
x2   2 
 x2 
e
u 4e
u1e
2
u 3e
x2
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
73/138
N i (x)  ai  bi x1  ci x2
N i (x j )  ij
N1 (x) 
2 3
x1 x2
3 2
 x1 x2
2
 ( x2
3
3
 x2 ) x1  ( x1
2
 x1 ) x2

1 x11
  1 x12
x12
x22
1 x13
x23
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
Etc.
74/138
K

B T DBd,
e
 N1

 x1
B 0

 N
 1
 x1
0
N1
x2
N1
x2
N 2
x1
0
N 2
x2
0
N 2
x2
N 2
x1
N 3
x1
0
N 3
x2

0 

N 3 
x2 
N 3 

x1 


1 
0 

E
 1
D
0 
2
1  
1 
0 0

2 

Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
75/138
Geometry of the problem
L
Fuzzy parameters:
q
E1, E2 , E3 , E4
Real parameters:
4
E,
L
3
q , , L
2
L
1
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
76/138
Numerical data
L=1 [m],
 kN 
q  1  ,
m
  0.3.
=0
[189, 231] [GPa]
=1
210 [GPa]

[189, 231] [GPa]
210 [GPa]
Eˆ 3
Eˆ 4
[189, 231] [GPa]
210 [GPa]
[189, 231] [GPa]
210 [GPa]

Eˆ 1
Eˆ 2

Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
77/138
Numerical results
Fuzzy stress
ˆ 1y
=0
[0.96749, 0.974493] [kPa]
=1
0.971063 [kPa]
ˆ 2y
[1.02833, 1.02955] [kPa]
1.02894 [kPa]
ˆ 3y
[0.98086, 1.01719] [kPa]
0.999086 [kPa]
ˆ 4y
[0.982807, 1.01914] [kPa]
1.00091 [kPa]

Fuzzy displacement
Nr
uˆ i ,   0 [m]
Nr
uˆ i ,   0 [m]
Nr
uˆ i ,   0 [m]
1
2
3
4
[0, 0]
[0, 0]
[0, 0]
[0, 0]
5
6
7
8
[3.2517e-14,7.49058e-13]
[3.81132e-12, 4.692e-12]
[-1.5243e-12,-4.9879e-13]
[ 4.4199e-12, 5.4275e-12 ]
9
10
11
12
[-1.5134e-12,1.0498e-12]
[8.1381e-12,9.9465e-12]
[-3.1758e-12,-1.7949e-13]
[8.7620e-12,1.0709e-11]
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
78/138
Numerical example
Truss structure
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
79/138
Numerical example
(truss structure)
d 
du 
  EA   n  0
dx 
 dx 
 Boundaryconditions

L
L

du dv
a(u, v)  EA
dx,
dx dx

l (v)  nvdx  ...,
0
0
v V , a (u,v)  l ( v )
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
80/138
P3
P1
P2
P=10 [kN]
Young’s modules
the same like
in previous example.
  0.3
L=1 [m]
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
81/138
Interval solution:
axial force [N]
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
[ 3145.34, 4393.45 ]
[ 1482.48, 1914.16 ]
[ -172.138, -221.845 ]
[ 164.454, 279.737 ]
[ -958.619, -936.417 ]
[ 2459.35, 2536.53 ]
[ 1527.83, 1546.14 ]
[ -343.544, -357.966 ]
[ 1708.72, 1617.27 ]
[ -840.883, -841.035 ]
[ 1132.62, 1189.25 ]
[ 1532.73, 1547.37 ]
[ -338.641, -356.736 ]
[ 3028.51, 2962.81 ]
[ -932.071, -929.76 ]
[ -278.358, -245.009 ]
[ 1656.79, 1671.62 ]
[ -214.586, -232.489 ]
[ 4264.06, 4221.36 ]
[ -169.222, -168.335 ]
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
-751.05, -742.133 ]
453.902, 470.55 ]
-1417.47, -1433.55 ]
6437.89, 6417.04 ]
-7444.75, -7432.58 ]
-200.408, -202.065 ]
-2196.2, -2197.33 ]
283.42, 285.763 ]
4020.01, 4013.59 ]
-200.408, -202.065 ]
-9461.8, -9431.91 ]
3589.87, 3583.79 ]
-3488.96, -3478.74 ]
713.715, 704.035 ]
4929.89, 4924.37 ]
720.439, 696.638 ]
3580.36, 3594.25 ]
-3482.95, -3485.36 ]
-9466.06, -9427.23 ]
4010.55, 4024 ]
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
-194.644, -208.406 ]
-2188.83, -2205.43 ]
275.268, 294.73 ]
-7448.38, -7428.59 ]
-194.644, -208.406 ]
6417.52, 6439.45 ]
451.658, 473.02 ]
-1419.72, -1431.08 ]
-738.486, -755.954 ]
-166.773, -171.028 ]
4242.96, 4244.56 ]
1655.57, 1672.95 ]
-215.805, -231.149 ]
-266.518, -258.031 ]
-930.146, -931.887 ]
3007.62, 2985.78 ]
1531.23, 1549.04 ]
-340.144, -355.068 ]
1144.66, 1176 ]
-839.969, -841.95 ]
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
61
62
63
64
65
66
67
68
69
[
[
[
[
[
[
[
[
[
1686.62, 1641.68 ]
1528.04, 1545.77 ]
-343.334, -358.339 ]
2470.18, 2524.72 ]
-947.416, -949.597 ]
253.654, 185.319 ]
1683.18, 1701.27 ]
-188.192, -202.832 ]
3683.74, 3761.16 ]
82/138
Truss structure
(Second example)
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
83/138
n L
L
L
L
L
P
P
Eˆ i , A
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
84/138
Data
Eˆ   [189, 231] [GPa],   0,
Eˆ   [210, 210] [GPa],   1,
L  1 [m ],
A  0.0001[m 2 ],
Pˆ  [9, 11] [kN],   0,
Pˆ  [10, 10] [kN],   1.

Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
85/138
Time of calculation
n
200
300
400
500
DOF
804
1204
1604
2004
Elements
1000
1500
2000
2500
Time
00:02:38
00:08:56
00:20:46
00:39:45
Processor: AMD Duron 750 MHz
RAM: 256 MB
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
86/138
Monotonicity
tests
(point tests)
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
87/138
Monotone solutions.
(Special case)
Ku  Q(h)   α j h j
j
Qi (h)   ijh j , ij  R
j
1 j 
Q  
  ...   α j  const
h j
 nj 
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
88/138
K
0
h j

u
K
1  Q
K

 h j h j
h j


q   K 1α j  const


u  u(h) - linear function.
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
89/138
Natural interval extension
f ( x )  x  x,
2
ˆf ( xˆ )  xˆ 2  xˆ
fˆ ([1, 2])  [1, 2]  [1, 2]  [1, 2] 
 [2,4]  [2,1]  [4,5]
 1 
f ([1, 2])   ,2
 4 
f ( xˆ )  fˆ ( xˆ )
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
90/138
Monotonicity tests
u(h) u(h0 )


hi
hi
m
 2u( h 0 )
(h j  h 0j )
hi h j
j 1

If
uˆ(hˆ  ) u(h0 )
0


hi
hi
m
 2 u( h 0 ) ˆ
(h j  h 0j )
hi h j
j 1

then function
u  u (h )
is monotone.
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
91/138
High order monotonicity tests
u(h) u(h0 )


hi
hi
m
 2 u( h 0 )
1
(h j  h 0j ) 
hi h j
2
j 1

m
m
j
k

 2 u( h 0 )
(h j  h 0j )(hk  hk0 )  ...
hi h j
If
uˆ(hˆ  ) u(h0 )
0


hi
hi
m
 2 u( h 0 ) ˆ
(h j  h 0j )  ...
hi h j
j 1

then function
u  u (h )
is monotone.
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
92/138
Numerical example
(Reinforced Concrete Beam)
Data
Concrete
E  1.3,1.5  104 MPa
ct  0 MPa
Steel
E  2.0,2.2  105 MPa
  0.2,0.3
Geometry
a  0.127 m
b  0.152 m
A  0.019 m2
0
Numerical result
=0: u2 x  0.182,0.200104 m
=1: u2 x  0.190,0.190104 m
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
93/138
In this example
commercial FEM program ANSYS
was applied.
Point monotonicity test can be applied
to results which were generated
by the existing engineering software.
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
94/138
Taylor model
m
u(h)
0
 u( h  ) 

i 1
0
u(h )
hi
h  h ,
i
0
i
0
h
 mid(hˆ  )
0
du
(
h
0
)
u( h )  u( h ) 
( h  h0 )
dh
u
h0
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
u(h )
h
95/138
Approximate interval solution
ˆu  uˆ (hˆ  )  u(h0 ) 
m

0
u(h  )
i 1
hi
hˆ
0
i  hi
,
uˆ  u(hˆ  ).
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
96/138
Computational complexity
u( h 0 )
0
u ( h  )
hi
- 1 solution of K
1
- the same matrix K
1
1 - point solution
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
97/138
Akapan U.O., Koko T.S., Orisamolu I.R., Gallant B.K.,
Practical fuzzy finite element analysis of structures.
Finite Element in Analysis and Design, Vol. 38, 2001, pp. 93-111
u L ( h)  u ( h 0 ) 

i
u(h 0 )
1
0
(hi  hi ) 
hi
2

i
j
u(h 0 )
(hi  hi0 )(h j  h 0j )
hi h j
u L ( h)  u ( h)
u  u L (h )
u L ( h )
u  u(h )
u(h )
h0
h
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
h
98/138
Finite difference method
0
du(h )
dx
d
2
0
u(h )
2
dx


0
u(h
 h)
2h
0
u( h
du( h )

dx
0
 h)  u(h
0
0
 h )  2  u(h )  u(h
2
h 
0
du( h )
dx

d
2
0
u ( h )
2
dx
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
 h )
0
( h  h )
99/138
Monotonicity test based
on finite difference method (1D)
du( h )

dx
0
du( h )
dx

d
2
0
u( h )
2
dx
0
( h  h )
0
du( h0 )
0
0
u
(
h


h
)

u
(
h
0
0
dx

  h ) h
h  h  2 0  h 
d u( h )
u( h0  h )  2  u( h0 )  u( h0  h )
dx2


If h  hˆ function is monotone.
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
100/138
Monotonicity test based on
finite differences
and interval extension (1D)
(1)
uˆ
0
2
0
ˆ
du( h ) du( h ) d u ( h ) ˆ
0



( h  h )
2
dx
dx
dx
If , )1( u
ˆ 0
then function u  u (h ) is monotone.
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
101/138
Monotonicity test based
on finite difference method
(multidimensional case)
u(h0 )
hi

m

 2u(h0 ) *

hj  h0j  0,

h

h
i
j
j 1

i  1,..., m

*
hi
 hˆi , i  1,..., m
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
102/138
We can check how reliable this method is.
h*
hˆ 
( hˆ  , h* )
*
ˆ
(h , h )  (hˆ  )  sup h2  h1
h1 ,h2hˆ 
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
103/138
Monotonicity test based on
finite differences
and interval extension
(multidimensional case)
uˆ(1i)
u(hˆ  ) u(h0 )



hi
hi
(1)
0  uˆi ,
m

 2u(h0 ) ˆ
hj  h0j

h

h
i
j
j 1


i  1,..., m
In this procedure we don’t have to solve any equation.
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
104/138
More reliable monotonicity test
y
du L ( h )
dh
y 
duˆ L ( hˆ )
dh
du(h )
dh
~ˆ
duˆ L (h )
dh
h
h
h0
~ˆ
ˆ )
ˆ
ˆ
~ˆ
d
u
(
h
)
d
u
(
h
h  hˆ  L   L 
dh
dh
h
h
h
~ˆ
duˆ L ( h )
0
dh
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
105/138
duˆ L (hˆ )
0
dh
Subdivision
hˆ1  hˆ2  hˆ
duˆ L (hˆ1 )
0
dh
duˆ L (hˆ2 )
0
dh
11
12
1
ˆ
ˆ
ˆ
h  h  h
duˆ L (hˆ11 )
ˆL (hˆ12 )
d
u
0
0
dh
dh
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
hˆ121  hˆ122  hˆ12
106/138
If width of the interval hˆ i.e.


ˆ
w(h )  h  h



is sufficiently small,
then extreme values of the function u
can be approximated by using
the endpoints of given interval hˆ.
u  min{u(h ), u(h )},

u


 min{u(h ), u(h )}.
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
107/138
Exact monotonicity tests
based on
the interval arithmetic
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
108/138
K(h)u  Q(h), h  hˆ 
K(hˆ  )u  Q(hˆ  )
ˆ (hˆ ) K (hˆ )

u

Q

 ˆ ˆ
ˆ (hˆ )
K


u(h  )

hi
hi
hi
uˆi ( hˆ  )
0
h j
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
109/138
Numerical example

1 d  dT(r) 
R

r

R
:
 rλ
Q 0
2
 1
r dr 
dr 


dT r 
: -λ
 α T r   Tb 
r  R1
dr

: T r   Tt
 r = R2





ˆ   Tˆ 
Tˆn 
T
1

,...,


 

ˆ 

T
 hull 









T
 
ˆ  K
ˆ 



Q
ˆ ,
ˆ  
K

T

 

λ

λ



Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
110/138
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
111/138
Sometimes system of algebraic equations
is nonlinear.
K ( h, u)u  Q( h)
In this case we can apply
interval Jacobean matrices.
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
112/138
F(u, h)  0
F(u, h) u F(u, h)

 0, i  1,..., m
u hi
hi
ui

h j
 F1
 u
 1
 ...
 Fn
 u11

...
...
...
F1
ui 1
...
Fn
ui 1
F1
h j
...
Fn
h j
F1
ui 1
...
Fn
ui 1
...
...
...
F1 
un 

... 
Fn 
un 

F
u
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
113/138
ui

h j
F
(u1 ,..., ui 1 , h j , ui 1 ,..., un )
F
u

 F 
F

  const, sign
sign
 (u1,..., ui 1, h j , ui 1,..., un )
 u 

 ui
sign
 h j


  const



  const


Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
114/138
ˆ (x, u(hˆ ), hˆ )

F
(
x
,
u
(
h
),
h
)

F


h  hˆ  ,

u
u
Regular interval matrix
ˆ,
A  A
A  0.
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
115/138
F(u, h)  0, h  hˆ 

ˆ uˆ , hˆ
F
 
h
Fˆ uˆ , hˆ





(u1 ,..., ui 1 , h j , u j 1 ,..., un )
It can be shown that if the following interval Jacobean matrices
are regular,
then solutions of parameter dependent system of equations
are monotone.
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
116/138
Numerical example
L
q8
q11
q9
q12
q7
q2
q1
P
P
q10
q5
q3
P
P
H
q6
q4
H
Uncertain parameters: E,A,J.
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
117/138
Equilibrium equations
of rod structures
d2 
d 2u 
 EJ
  q( x)
2 
2 
dx 
dx 
L

2
2
L

d ud v
a(u, v)  EJ 2 2 dx, l (v )  qvdx  ...
dx dx
0
0
v V , a (u,v)  l ( v )
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
118/138
 0.054 0.0554 
4
J 
,
[
m
],

12 
 12
E  [210, 220] [GPa],
A[0.052 , 0.0552 ] [m2 ], L=H=1 [m], P=1 [kN].
q1 [m]
q2 [m]
qi
0.035716
0.000008
-0.011230
0.035716
-0.000021
-0.011230
qi
0.037414
0.000009
-0.010718
0.037414
-0.000017
-0.010718
q7 [m]
q8 [m]
q10 [m]
q11 [m]
qi
0.082163
0.00009
-0.007494
0.082163
-0.000033
-0.007494
qi
0.086067
0.000010
-0.007151
0.086067
-0.000026
-0.007151
q3
q9
q4 [m]
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
q5
[m]
q6
q12
119/138
Optimization methods
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
120/138
min ui


ui  L(u, h)  f (h)

ˆ
h

h


min ui


ui  K (h)u  Q(h),

ˆ
h

h


max ui


ui  L(u, h)  f (h)

ˆ
h

h


max ui


ui  K (h)u  Q(h)

ˆ
h

h


Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
121/138
These methods can be applied
to the very wide intervals
hˆ  .
Function
u  u(h)
doesn't have to be monotone.
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
122/138
Numerical example
L
2
L
q
 d2 
d 2u 
 2  EJ 2   q( x ),
dx 
 dx 

d 2u(0)
d 2  3L 
  L
 3L 
 0,
u
0
u 2   0, u 2   0,
2
2 
dx
dx  2 
 
  
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
123/138
Numerical data
Analytical solution
 1  1 4 ql3
ql 4 
 L


qx

x

dla
x

0, 




48
128
 2
 EJ  24
u( x )  
3

L  qL3
qL4 
1  1 4 9

 L 3L 
 EJ  24 qx  48 qL x  2   48 x  128  dla x   2 , 2 




 

Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
124/138
L
2
L
q
x
0.5
1
15
.
0.022
0.037
y(x)
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
125/138
Other methods
and applications
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
126/138
Iterative methods
Popova, E. D., On the Solution of Parametrised Linear Systems.
In: W. Kraemer, J. Wolff von Gudenberg (Eds.):
Scientific Computing, Validated Numerics,
Interval Methods. Kluwer Acad. Publishers, 2001, pp. 127-138.
Muhanna L.R., Mullen L.R., Uncertainty in Mechanics.
Problems - Interval Based - Approach. Journal of Engineering
Mechanics, Vol. 127, No.6, 2002, pp.557-566
K(h)u  Q(h), h  hˆ 
Andrzej Pownuk
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127/138
Kij (h)
k
 Cij hk
Q j (h)  C kj hk
Inner solution
Outer solution
i)
(i )
ˆu(INNER
ˆ
ˆ
 u  uOUT
(i )
uˆ OUT
 uˆ
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
128/138
Valliappan S., Pham T.D., 1993,
Fuzzy Finite Element Analysis
of a Foundation on Elastic Soil Medium.
International Journal for Numerical
and Analytical Methods in Geomechanics, Vol.17, s.771-789
In some cases we can prove,
that the solution can be calculated using only
endpoints of given intervals.
The authors were solved some special
fuzzy partial differential equations
using only endpoints of given intervals.
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
129/138
Load combinations in civil engineering
Many existing civil engineering programs
can calculate extreme solutions
of partial differential equations
with interval parameters (only loads) e.g:
- ROBOT (http://www.robobat.com.pl/),
- CivilFEM (www.ingeciber.com).
These programs calculate
all possible combinations
and then calculate the extreme solutions
(some forces exclude each other).
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
130/138
Fuzzy eigenvalue problem
detM(h)  K (h)   0
(i )

 { : det(M(h)  K(h)), h  hˆ }
(i )
(i )
( |  F )

(i )
sup{ :    }
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
131/138
Upper probability
of the stability
Pl{Re( )  0} 
(i )
(i )
sup ( |  F )
:0
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
132/138
Random set Monte Carlo simulations
Pl(u0  u( Hˆ ))  P{ : u  u( Hˆ  ())}
In some cases we cannot apply
fuzzy sets theory to solution of this problem.
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
133/138
Conclusions
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
134/138
Conclusions
1) Calculation of the solutions
of fuzzy partial differential equations
is in general very difficult (NP-hard).
2) In engineering applications the relation
between the solution and uncertain parameters
is usually monotone.
3) Using methods which are based on sensitivity
analysis we can solve very complicated problems
of computational mechanics.
(thousands degree of freedom)
Andrzej Pownuk
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135/138
4) If we apply the point monotonicity tests
we can use results which was generated
by the existing engineering software.
5) Reliable methods of solution
of fuzzy partial differential equations
are based on the interval arithmetic.
These methods have
high computational complexity.
6) In some cases
(e.g. if we know analytical solution)
optimization method can be applied.
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
136/138
7) In some special cases we can predict
the solution of fuzzy partial differential equations.
8) Fuzzy partial differential equation can be applied
to modeling of mechanical systems (structures)
with uncertain parameters.
Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk
137/138