SLIDE 1
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 1/138
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 2/138
partial differential equation and its application in computational - - PowerPoint PPT Presentation
partial differential equation and its application in computational - - PowerPoint PPT Presentation
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 1/138
SLIDE 2
SLIDE 3
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 3/138
Plane stress problem in theory of elasticity
, , 1,2 , , ) 1 ( 2 ) 1 ( 2
* * , ,
= = = = + − + + x t n x u u f u E u E
u
- mass density, E, - material constant,
- mass force.
f
= x u u ,
SLIDE 4
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 4/138
] , [ ˆ
+ − =
h h h
} ) ( : { ˆ =
h h h
F
Triangular fuzzy number
) (h
F
F
1
−
h
+
h
+ − = 1 1
h h
−
h
+
h
SLIDE 5
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 5/138
L L L L L L L
L n
L m
q 1
E
2
E
SLIDE 6
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 6/138
Data
. 1 , ] [ ], 10 2 , 210 [ ˆ 0, , ] [ ], 231 , 189 [ ˆ , 1 , ] [ ], 10 2 , 210 [ ˆ 0, , ] [ , ] 231 , 189 [ ˆ
2 1 2 1 1 1
= = = = = = = = GPa E GPa E GPa E GPa E
3 . =
, 1 = m kN q
] [ 1 m L =
SLIDE 7
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 7/138
n m DOF Elements Time 5 5 72 50 00:00:01 10 10 242 200 00:00:09 20 20 882 800 00:03:50 30 40 2542 2400 01:27:52
Time of calculation
Processor: AMD Duron 750 MHz RAM: 256 MB
SLIDE 8
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 8/138
Numerical example
Shell structure with fuzzy material properties
SLIDE 9
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 9/138
Equilibrium equations
- f shell structures
= + = + = + + = + −
x p n M ds d n M x p n M b n T b M b T b M b T , ) ( | , | | |
3 3
where
1,2 , , , , , | = = + =
x u u u u u
SLIDE 10
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 10/138
Numerical data (=0)
], [ ] 10 2 . 2 , 10 . 2 [
5 5
MPa E
,
3 . , 2 .
L=0.263 [m], r=0.126 [m], F=444.8 [N], t=
] [ 10 38 . 2
3 m −
SLIDE 11
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 11/138
Numerical results (fuzzy displacement)
=0: =1: u = -0.04102 [m].
] [ 03748 . , 043514 . m u − −
Using this method we can obtain the fuzzy solution in one point. The solution was calculated by using the ANSYS FEM program.
SLIDE 12
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 12/138
The main goal of this presentation is to describe methods of solution
- f partial differential equations
with fuzzy parameters.
SLIDE 13
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 13/138
Basic properties
- f fuzzy sets
SLIDE 14
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 14/138
Fuzzy sets
R x x R
F F
→ ] 1 , [ ) ( :
x
) (x
F
1
)} ( ), ( { ) ( x x min x
B A B A
=
)) ( ), ( ( ) ( x x T x
B A B A
=
)} ( ), ( { ) ( x x max x
B A B A
=
)) ( ), ( ( ) ( x x S x
B A B A
=
SLIDE 15
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 15/138
Extension principle
)} ( ),..., ( ), ( { ) (
2 1 ) ,..., , ( ) (
2 1
n F F F x x x f y F f
x x x min max y
n
=
=
) ,..., , (
2 1 n
x x x f y =
), ( ) (
) ( ) (
x
x F f y F f
max y =
=
), (
n
R F F ). ( ) ( R F F f , : R R f
n →
SLIDE 16
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 16/138
Fuzzy equations
SLIDE 17
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 17/138
Fuzzy algebraic equations
h y H = ) , (
) ( h y y =
, :
n m n
R R R → H
) (
m
R F F
] 1 , [ ) ( : → h h
F m F
R
) ( ) (
) , ( : ) (
h y
h y H h F F H
max =
=
SLIDE 18
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 18/138
Fuzzy differential equation (example)
) ( , (0) , R F F h y y x h dx dy = =
2
2 ) , ( y hx h x y + =
) ( )) ( | (
2
2 :
h max x y
F y x h h F
=
+ =
) ( ) ( R F x yF
SLIDE 19
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 19/138
Definition of the solution
- f fuzzy differential equation
) ( , (0) ), , , ( R F F h y y h y x f dx dy = =
) ( )) ( | (
) ( ), , ( ), ( :
h max x y
F y y h x y dx dy x,h y ξ h F
=
= = =
) ( ) ( R F x yF
SLIDE 20
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 20/138
Fuzzy partial differential equations
) ( , , ) , ,..., , , , (
2 2 m k k
R F F V = h u h x u x u x u u x H
) ( )) ( | (
, ) , ,...., , , ( ), , ( :
h x u ξ
u h x u x u h u, x H h x u ξ h F V F
k k
max =
= =
) ( ) (
n F
R F x u
SLIDE 21
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 21/138
Algebraic solution set
] 8 , 2 [ ] , [ ] 2 , 1 [ =
+ − x
x ] 4 , 2 [ ˆ = x
United solution set
] 4 , 1 [ } ], 4 , 2 [ ], 2 , 1 [ : { ˆ = = = b x a b a x x ] 8 , 2 [ ] 2 , 1 [ = x
Controllable solution set
} ], 8 , 2 [ ], 2 , 1 [ : { ˆ b x a b a x x = = = = ]} 8 , 2 [ ] 2 , 1 [ : { ˆ x x x
Tolerable solution set
} ], 8 , 2 [ ], 2 , 1 [ : { ˆ b x a b a x x = = ] 4 , 2 [ ]} 8 , 2 [ ] 2 , 1 [ : { ˆ = = x x x
SLIDE 22
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 22/138
Remarks
} ) ( : { ) (
) (
=
y y cl x F
x F
). ( ) ( x F sup x F
+
=
), ( ) ( x F inf x F
−
=
dx x dF dx x dF dx x dF x F x F dx d x F dx d ) ( ) ( , ) ( )] ( ), ( [ ) (
+ − + −
= = =
Buckley J.J., Feuring T., Fuzzy differential equations. Fuzzy Sets and System, Vol.110, 2000, 43-54
SLIDE 23
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 23/138
This derivative leads to another definition
- f the solution of the fuzzy differential equation.
- Goetschel-Voxman derivative,
- Seikkala derivative,
- Dubois-Prade derivative,
- Puri-Ralescu derivative,
- Kandel-Friedman-Ming derivative,
- etc.
SLIDE 24
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 24/138
Applications of fuzzy equations in computational mechanics Physical interpretations
- f fuzzy sets
SLIDE 25
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 25/138
Equilibrium equations of isotropic linear elastic materials
,
2 2
t u X x
i i j ij
= +
,
kl ijkl ij
C =
, 2 1 + =
i j j i ij
x u x u
, ,
* u i i
x u u =
, ,
*
= x t n
i j ij
. ), ( ) , (
*
=
=
x x u t x u
t
SLIDE 26
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 26/138
Uncertain parameters
- Fuzzy loads,
- Fuzzy geometry,
- Fuzzy material properties,
- Fuzzy boundary conditions e.t.c.
SLIDE 27
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 27/138
Modeling of uncertainty
Probabilistic methods
, : R X →
). (x f 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
- f uncertainty.
SLIDE 28
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 28/138
Random sets interpretation
- f fuzzy sets
) ( ) ( ˆ : ˆ R I H H →
} ) ( ˆ : { ) ( =
A H P A Pl
) ( ˆ ... ) ( ˆ ) ( ˆ
2 1 n
H H H
)} ( ˆ : { }) ({ ) ( = =
H h P h Pl h
F
SLIDE 29
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 29/138
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
- f 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
SLIDE 30
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 30/138
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
SLIDE 31
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 31/138
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
SLIDE 32
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 32/138
Random sets interpretation
- f fuzzy sets
P
) (P
F
) ( ˆ
1
H ) ( ˆ
2
H ) ( ˆ
3
H ) ( ˆ
4
H
2
P
= ) (
2
P
F
1
= ) (
1
P
F
1
P
0.5
2 1 } { } { ) (
2 1 1
= + =
P P P
F
} {
4
P
1 } { =
i i
P
} {
3
P } {
2
P } { 1
P
SLIDE 33
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 33/138
This is not a probability density function
- r a conditional probability
and cannot be converted to them.
SLIDE 34
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 34/138
) ( ) ( ˆ : ˆ R I X X →
R X X →
) ( :
]} , [ ) ( : { ]) , ([ b a X P b a PX =
} ] , [ ) ( ˆ : { ]) , ([ =
b a X P b a Pl
]) , ([ ]) , ([ b a Pl b a PX
) ( ˆ ) ( ,
X X
SLIDE 35
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 35/138
Random sets
) ( ) ( ˆ : ˆ R I H H →
Probabilistic methods
) ( ) ( ) ( = =
− +
H H H
Fuzzy methods
) ( ˆ ... ) ( ˆ ) ( ˆ
2 1 n
H H H
Semi-probabilistic methods (interval methods)
) ( ˆ ... ) ( ˆ ) ( ˆ
2 1 n
H H H = = =
- r
another procedures.
SLIDE 36
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 36/138
Design of structures with fuzzy parameters
} ) ( {
f f
P g Pl P = h
) (
) ( :
h sup P
F h g h f
=
SLIDE 37
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 37/138
Equation with fuzzy and random parameters
, ) ( : R X X →
), ( ) ( ˆ : ˆ R I H H →
)}. ( ˆ : { ) ( =
H h P h
F
} )) ( ˆ ), ( ( : ) , {( =
+
H X g P Pf
SLIDE 38
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 38/138
) ( ) (
) , ( : ) (
h sup x
F h x g h F x
=
=
+ x F x f
x x P P ) ( } {
) (
)) ( ( ) ( ) (
) (
x E x dP x P
F F x f
= =
− +
} )) ( ˆ ), ( ( : ) , {( =
+
H X g P Pf
SLIDE 39
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 39/138
General algorithm
), ( ) ( ˆ : ˆ R I →
H H
)}. ( ˆ : { ) ( =
H h h P
F
V = u f(h) h u, L , ) ( ), ( ) ( h Q u h K = ) (h g y = ) ( ) (
) ( : ) ( :
y sup sup P
F g y y F g f
= =
+
h
h h
) ( ) (
) ( : ) (
h
h h F g y F g
sup y =
=
SLIDE 40
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 40/138
Other methods of modeling of uncertainty:
- TBM model (Philip Smith).
- imprecise probability
(Imprecise Probability Project, Buckley, Thomas etc.).
- etc.
SLIDE 41
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 41/138
Numerical methods of solution
- f partial differential equations
SLIDE 42
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 42/138
Numerical methods of solution
- f partial differential equations
- finite element method (FEM)
- boundary element method (BEM)
- finite difference method (FDM)
1) Boundary value problem. 3) System of algebraic equations. 4) Approximate solution. 2) Discretization.
SLIDE 43
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 43/138
Finite element method
Using FEM we can solve very complicated problems. These problems have thousands degree of freedom.
Curtusy to ADINA R & D, Inc.
SLIDE 44
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 44/138
Algorithm
= = + − x u x f y u x u , ,
2 2 2 2
= + − fvd vd y u x u
2 2 2 2
= + fvd d x v y u x v x u
SLIDE 45
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 45/138
+ = d x v y u x v x u v u a ) , (
= fvd l ) (
) ( ) ( , v l u,v a V v =
SLIDE 46
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 46/138
1
2
n
i i
=
V Vh
, ) ( ) (
=
i i i h
x u x u
=
i i i h
x v x v ) ( ) (
ij
) ( =
j i x
- shape functions
SLIDE 47
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 47/138
), , (
j i ij
a K =
) (
i i
l Q =
) ( ) ( ,
h h h h h
v l ,v u a V v =
Q Ku =
System of linear algebraic equations
SLIDE 48
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 48/138
Approximate solution
, ) ( ) (
=
i i i h
x u x u
Q K u
1 −
= ) ( ) ( x u x uh
SLIDE 49
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 49/138
Numerical methods of solution
- f fuzzy
partial differential equations
SLIDE 50
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 50/138
Application of finite element method to solution
- f fuzzy partial differential equations.
F V = h u h x, f h u, x, L , ), ( ) (
Parameter dependent boundary value problem.
F = h h Q u h K ), ( ) (
F = h h u u ), (
) (
n F
R F u
SLIDE 51
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 51/138
-level cut method
} ) ( : { ˆ =
h h h
F
} ˆ : { ) (
) (
= u u u
u
sup
F The same algorithm can be apply with BEM or FDM.
} ˆ ), ( ) ( : { ˆ
= = h h h Q u h K u u
SLIDE 52
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 52/138
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
} ˆ ), ( ) ( : { ˆ
= = h h h Q u h K u u
We can solve these equation
- nly in special cases.
SLIDE 53
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 53/138
Solution set of system of linear interval equations is very complicated.
= [1,2] [-1,1] [1,2] [2,4] [2,4] [1,2]
2 1
x x
1 2
3 3 3 3
) ( B A, hull
) ( B A,
SLIDE 54
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 54/138
Monotone functions
−
h
+
h
) (
− − =
h u u
) (
+ + =
h u u
h
) (h u u =
), (
− − =
h u u ). (
+ + =
h u u
SLIDE 55
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 55/138
} ˆ ), ( ) ( : { ˆ
= = h h h Q u h K u u
m
R
h ˆ
m
2
system equations have to be solved.
Sensitivity analysis
If
h u
, then
) ( ), (
+ + − −
= = h u u h u u
If
h u
, then
) ( ), (
− + + −
= = h u u h u u
1+2n system of equation (in the worst case) have to be solved.
SLIDE 56
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 56/138
Multidimensional algorithm
= h h h Q u h K ˆ ), ( ) (
) ˆ ( 1,..., ), ( ) ( ) ( ) ( ) (
= = − = h h h u h K h Q h u h K mid m i h h h
i i i
n i h u sign h u sign
m i i i
,..., 1 , ,...,
1
= =
S
SLIDE 57
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 57/138
Calculate unique sign vectors . ,..., 1 ,
*
k q
q
=
S ) 1 (
− =
j i
S S If , then
.
j i
S S
Calculate unique interval solutions )] ) 1 ( , ˆ ( ), , ˆ ( [ ˆ
* * *
− =
i i i
S h u S h u u Calculate all interval solutions
*
ˆ ˆ }, ,..., 1 { }, ,..., 1 {
=
j i
k j n i u u
SLIDE 58
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 58/138
Computational complexity
1+2n system of equation (in the worst case) have to be solved.
= ... ... ... ... ... ... ... ... ...
1 1 1 1 1 m n n m i i m
h u h u h u h u h u h u h u =
... ...
1 n i
S S S S
All sign vectors
... ...
* * * 1 *
=
k q
S S S S
Unique sign vectors
i
s
SLIDE 59
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 59/138
This method can be applied
- nly when
the relation between the solution and uncertain parameters is monotone.
) (h u u =
SLIDE 60
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 60/138
According to my experience (and many numerical results which was published) in problems of computational mechanics the intervals are usually narrow and the relation u=u(h) is monotone.
h ˆ
SLIDE 61
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 61/138
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
SLIDE 62
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 62/138
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
- f 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
SLIDE 63
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 63/138
Particular case - system of linear interval equations
=
n n nn n n
Q Q X X K K K K ˆ ... ˆ ... ˆ ... ˆ ... ... ... ˆ ... ˆ
1 1 1 1 11
=
F n F n F nn F n F n F
Q Q X X K K K K ... ... ... ... ... ... ...
1 1 1 1 11
SLIDE 64
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 64/138
= Q u K
− = u K Q u K
i i i
h h h
=
m i i i
h u sign h u sign ,...,
1
S
i j i
C j = =
where ,
*
S S
)] ) 1 ( , ˆ ( ), , ˆ ( [ ˆ
* * * i i i
− = S h X S h X X
i j i i
C j X X = =
where , ˆ ˆ
*
SLIDE 65
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 65/138
Computational complexity
- f this algorithm
p - number of independent sign vectors .
* i
S
1+2p - system of equations.
] , 1 [ n p
n - number of degree of freedom.
] 2 1 , 2 1 [ n + +
- system of equations
SLIDE 66
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 66/138
Calculation of the solution between the nodal points
1
2 3 e
e
u1
e
u2
e
u3
e
u4
e
u5
e
u6
1
x
2
x
3
x
x
e e e
u x N x u ) ( ) ( =
SLIDE 67
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 67/138
) ( ) , ( ) , ( h u h x N h x u
e e e
=
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
=
m e e e
h u sign h u sign ) , ( , ... , ) , (
1
h x h x S
SLIDE 68
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 68/138
Calculation of extreme solutions between the nodal points.
1) Calculate sensitivity of the solution. (this procedure use existing results of the calculations)
=
m e e e
h u sign h u sign ) , ( , ... , ) , (
1
h x h x S
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.
SLIDE 69
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 69/138
Numerical example
Plane stress problem in theory of elasticity
1 2 3 4
q L L L
, E
SLIDE 70
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 70/138
Plane stress problem in theory of elasticity
, , 1,2 , , ) 1 ( 2 ) 1 ( 2
* * , ,
= = = = + − + + x t n x u u f u E u E
u
- mass density, E, - material constant,
- mass force.
f
= x u u ,
SLIDE 71
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 71/138
Finite element method
, =
d
e
T
B D B K
,
+ = dS d
T T
t N f N Q
Ku=Q
, ) ( ) ( u N u x x =
. ) ( ) (
j ij i
u x N x u =
SLIDE 72
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 72/138
1
2 3 e
e
u1
e
u2
e
u3
e
u4
e
u5
e
u6
1
x
2
x
3
x ,
2 1
= x x x
=
2 1
u u u
= =
6 5 4 3 2 1 3 2 1 3 2 1 2 1
) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( u u u u u u N N N N N N x u x u x x x x x x x u
u x N u ) ( ) ( = x
SLIDE 73
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 73/138
1
2 3 e
e
u1
e
u2
e
u3
e
u4
e
u5
e
u6
1
x
2
x
3
x
=
1 2 1 1 1
x x x =
2 2 2 1 2
x x x =
3 2 3 1 3
x x x
SLIDE 74
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 74/138
2 1
) ( x c x b a N
i i i i
+ + = x
ij j i
N = ) (x
3 2 3 1 2 2 2 1 1 2 1 1
1 1 1 x x x x x x =
− + − + − =
2 2 1 3 1 1 3 2 2 2 2 2 3 1 3 2 2 1 1
) ( ) ( ) ( x x x x x x x x x x N x
Etc.
SLIDE 75
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 75/138
=
1 3 2 3 1 2 2 2 2 1 1 1 2 3 2 2 2 1 1 3 1 2 1 1
x N x N x N x N x N x N x N x N x N x N x N x N B , =
d
e
T
B D B K
− − = 2 1 1 1 1
2
E D
SLIDE 76
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 76/138
1 2 3 4
q L L L
, E
Geometry of the problem
Fuzzy parameters:
4 3 2 1
, , , E E E E L q , ,
Real parameters:
SLIDE 77
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 77/138
Numerical data
L=1 [m],
, 1 = m kN q
. 3 . =
=0 =1
1
ˆ E
[189, 231] [GPa] 210 [GPa]
2
ˆ E
[189, 231] [GPa] 210 [GPa]
3
ˆ E
[189, 231] [GPa] 210 [GPa]
4
ˆ E
[189, 231] [GPa] 210 [GPa]
SLIDE 78
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 78/138
Numerical results
=0 =1
1
ˆ y
[0.96749, 0.974493] [kPa] 0.971063 [kPa]
2
ˆ y
[1.02833, 1.02955] [kPa] 1.02894 [kPa]
3
ˆ y
[0.98086, 1.01719] [kPa] 0.999086 [kPa]
4
ˆ y
[0.982807, 1.01914] [kPa] 1.00091 [kPa]
Nr
, ˆ =
i
u
[m] Nr
, ˆ =
i
u
[m] Nr
, ˆ =
i
u
[m] 1 [0, 0] 5 [3.2517e-14,7.49058e-13] 9 [-1.5134e-12,1.0498e-12] 2 [0, 0] 6 [3.81132e-12, 4.692e-12] 10 [8.1381e-12,9.9465e-12] 3 [0, 0] 7 [-1.5243e-12,-4.9879e-13] 11 [-3.1758e-12,-1.7949e-13] 4 [0, 0] 8 [ 4.4199e-12, 5.4275e-12 ] 12 [8.7620e-12,1.0709e-11]
Fuzzy displacement Fuzzy stress
SLIDE 79
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 79/138
Numerical example Truss structure
SLIDE 80
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 80/138
Numerical example (truss structure)
= + conditions Boundary n dx du EA dx d
, ) , ( dx dx dv dx du EA v u a
L
=
..., ) ( + =
L
nvdx v l
) ( ) ( , v l u,v a V v =
SLIDE 81
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 81/138
1
P
2
P
3
P
P=10 [kN] Young’s modules the same like in previous example. L=1 [m]
3 . =
SLIDE 82
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 82/138
1 [ 3145.34, 4393.45 ] 21 [ -751.05, -742.133 ] 41 [ -194.644, -208.406 ] 61 [ 1686.62, 1641.68 ] 2 [ 1482.48, 1914.16 ] 22 [ 453.902, 470.55 ] 42 [ -2188.83, -2205.43 ] 62 [ 1528.04, 1545.77 ] 3 [ -172.138, -221.845 ] 23 [ -1417.47, -1433.55 ] 43 [ 275.268, 294.73 ] 63 [ -343.334, -358.339 ] 4 [ 164.454, 279.737 ] 24 [ 6437.89, 6417.04 ] 44 [ -7448.38, -7428.59 ] 64 [ 2470.18, 2524.72 ] 5 [ -958.619, -936.417 ] 25 [ -7444.75, -7432.58 ] 45 [ -194.644, -208.406 ] 65 [ -947.416, -949.597 ] 6 [ 2459.35, 2536.53 ] 26 [ -200.408, -202.065 ] 46 [ 6417.52, 6439.45 ] 66 [ 253.654, 185.319 ] 7 [ 1527.83, 1546.14 ] 27 [ -2196.2, -2197.33 ] 47 [ 451.658, 473.02 ] 67 [ 1683.18, 1701.27 ] 8 [ -343.544, -357.966 ] 28 [ 283.42, 285.763 ] 48 [ -1419.72, -1431.08 ] 68 [ -188.192, -202.832 ] 9 [ 1708.72, 1617.27 ] 29 [ 4020.01, 4013.59 ] 49 [ -738.486, -755.954 ] 69 [ 3683.74, 3761.16 ] 10 [ -840.883, -841.035 ] 30 [ -200.408, -202.065 ] 50 [ -166.773, -171.028 ] 11 [ 1132.62, 1189.25 ] 31 [ -9461.8, -9431.91 ] 51 [ 4242.96, 4244.56 ] 12 [ 1532.73, 1547.37 ] 32 [ 3589.87, 3583.79 ] 52 [ 1655.57, 1672.95 ] 13 [ -338.641, -356.736 ] 33 [ -3488.96, -3478.74 ] 53 [ -215.805, -231.149 ] 14 [ 3028.51, 2962.81 ] 34 [ 713.715, 704.035 ] 54 [ -266.518, -258.031 ] 15 [ -932.071, -929.76 ] 35 [ 4929.89, 4924.37 ] 55 [ -930.146, -931.887 ] 16 [ -278.358, -245.009 ] 36 [ 720.439, 696.638 ] 56 [ 3007.62, 2985.78 ] 17 [ 1656.79, 1671.62 ] 37 [ 3580.36, 3594.25 ] 57 [ 1531.23, 1549.04 ] 18 [ -214.586, -232.489 ] 38 [ -3482.95, -3485.36 ] 58 [ -340.144, -355.068 ] 19 [ 4264.06, 4221.36 ] 39 [ -9466.06, -9427.23 ] 59 [ 1144.66, 1176 ] 20 [ -169.222, -168.335 ] 40 [ 4010.55, 4024 ] 60 [ -839.969, -841.95 ]
Interval solution: axial force [N]
SLIDE 83
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 83/138
Truss structure (Second example)
SLIDE 84
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 84/138
P P
A Ei , ˆ
L
L
L
L
L n
SLIDE 85
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 85/138
Data
0, ], [ ] 231 , 189 [ ˆ = =
GPa E
], [ 0001 .
2
m A =
], [ 1 m L =
1, ], [ ] 210 , 210 [ ˆ = =
GPa E
0, [kN], ] 11 , 9 [ ˆ = =
P 1. [kN], ] 10 , 10 [ ˆ = =
P
SLIDE 86
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 86/138
Time of calculation
Processor: AMD Duron 750 MHz RAM: 256 MB n DOF Elements Time 200 804 1000 00:02:38 300 1204 1500 00:08:56 400 1604 2000 00:20:46 500 2004 2500 00:39:45
SLIDE 87
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 87/138
Monotonicity tests
(point tests)
SLIDE 88
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 88/138
Monotone solutions. (Special case)
= =
j j jh
α h Q Ku ) ( R h Q
ij j j ij i
= , ) (h
const h
j nj j j
= = = α Q ...
1
SLIDE 89
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 89/138
K =
j
h
const h h h
j j j j
= = − =
− −
α K q K Q K u
1 1
) (h u u =
- linear function.
SLIDE 90
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 90/138
Natural interval extension
, ) (
2
x x x f − = x x x f ˆ ˆ ) ˆ ( ˆ
2 −
=
] 5 , 4 [ ] 1 , 2 [ ] 4 , 2 [ ] 2 , 1 [ ] 2 , 1 [ ] 2 , 1 [ ]) 2 , 1 ([ ˆ − = − + − = = − − − − = − f
− = − 2 , 4 1 ]) 2 , 1 ([ f
) ˆ ( ˆ ) ˆ ( x f x f
SLIDE 91
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 91/138
Monotonicity tests
=
− + =
m j j j j i i i
h h h h u h u h u
1 2
) ( ) ( ) ( ) ( h h h
=
− + =
m j j j j i i i
h h h h u h u h u
1 2
) ˆ ( ) ( ) ( ) ˆ ( ˆ h h h
If then function
) (h u u =
is monotone.
SLIDE 92
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 92/138
High order monotonicity tests
... ) )( ( ) ( 2 1 ) ( ) ( ) ( ) (
1 2 2
+ − − + − + =
= m j m j m k k k j j j i j j j i i i
h h h h h h u h h h h u h u h u h h h h
... ) ˆ ( ) ( ) ( ) ˆ ( ˆ
1 2
+ − + =
= m j j j j i i i
h h h h u h u h u h h h
If then function
) (h u u =
is monotone.
SLIDE 93
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 93/138
Numerical example
(Reinforced Concrete Beam)
Data Concrete Steel Geometry
MPa 10 1.3,1.5 E
4
MPa 10 2 . 2 , . 2 E
5
m 127 . a = MPa
ct =
3 . , 2 .
m 152 . b =
= 2
m 0.019 A =
Numerical result
m u x
4 2
10 200 . , 182 .
−
=0:
m u x
4 2
10 190 . , 190 .
−
=1:
SLIDE 94
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 94/138
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.
SLIDE 95
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 95/138
Taylor model
( )
) ˆ ( , ) ( ) ( ) (
1 =
= − + =
h h h h h mid h h h u u u
m i i i i
h
) (h u u
) ( ) ( ) ( ) (
− + = h h dh h du h u h u
h
SLIDE 96
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 96/138
( )
, ˆ ) ( ) ( ) ˆ ( ˆ ˆ
1 0 =
− + = =
m i i i i
h h h u u u u h h h
). ˆ ( ˆ
h u u
Approximate interval solution
SLIDE 97
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 97/138
Computational complexity
) (
h u
- 1 solution of
i
h u
)
( h
- the same matrix
1 - point solution
1 −
K
1 −
K
SLIDE 98
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 98/138
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
− − + − + =
i j j j i i j i i i i i L
h h h h h h u h h h u u u ) )( ( ) ( 2 1 ) ( ) ( ) ( ) ( h h h h
) ( ) ( h h u uL
h
+
h ) (
+
h u
) (
+
h uL
h
) (h u u =
) (h u u
L
=
SLIDE 99
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 99/138
Finite difference method h h h u h h u dx h du − − +
2 ) ( ) ( ) (
( )2
2 2
) ( ) ( 2 ) ( ) ( h h h u h u h h u dx h u d − + − +
) ( ) ( ) ( ) (
2 2
− + h h dx h u d dx h du dx h du
SLIDE 100
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 100/138
) ( ) ( ) ( ) (
2 2
= − +
h h dx h u d dx h du dx h du
) ( ) ( 2 ) ( ) ( ) ( ) ( ) (
2 2
h h u h u h h u h h h u h h u h dx h u d dx h du h h − + − + − − + − = − =
function is monotone.
h h ˆ
If Monotonicity test based
- n finite difference method (1D)
SLIDE 101
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 101/138
) ˆ ( ) ( ) ( ) ˆ ( ˆ
2 2 ) 1 (
− + = = h h dx h u d dx h du dx h du u
, ˆ
) 1 (
u
Monotonicity test based on finite differences and interval extension (1D) then function is monotone. If
) (h u u =
SLIDE 102
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 102/138
( )
m i h h h h u h u
m j j j j i i
,..., 1 , ) ( ) (
1 * 2
= = − +
=
h h
m i h h
i i
,..., 1 , ˆ
*
=
Monotonicity test based
- n finite difference method
(multidimensional case)
SLIDE 103
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 103/138
We can check how reliable this method is.
h ˆ
*
h ) , ˆ (
*
h h
1 2 ˆ , *
2 1
) ˆ ( ) , ˆ ( h h h h h
h h h
− =
sup
SLIDE 104
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 104/138
( )
=
− + = =
m j j j j i i i i
h h h h u h u h u u
1 2 ) 1 (
ˆ ) ( ) ( ) ˆ ( ˆ h h h m i u i ,..., 1 , ˆ
) 1 (
=
In this procedure we don’t have to solve any equation.
Monotonicity test based on finite differences and interval extension (multidimensional case)
SLIDE 105
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 105/138
More reliable monotonicity test
dh h u d dh h u d h h
L L
) ˆ ( ˆ ) ˆ ~ ( ˆ ˆ ˆ ~
dh h u d L ) ˆ ~ ( ˆ
h
−
h
+
h
+
h
−
h
h
dh h du y ) ( = dh h u d
L
) ˆ ( ˆ
dh h u d L ) ˆ ~ ( ˆ
dh h du y
L
) ( =
SLIDE 106
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 106/138
Subdivision
dh h u d L ) ˆ ( ˆ
dh h u d L ) ˆ ( ˆ
1
dh h u d L ) ˆ ( ˆ
2
= h h h ˆ ˆ ˆ
2 1 1 12 11
ˆ ˆ ˆ
= h h h
dh h u d L ) ˆ ( ˆ
11
dh h u d L ) ˆ ( ˆ
12
12 122 121
ˆ ˆ ˆ
= h h h
SLIDE 107
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 107/138
If width of the interval i.e.
− +
− = h h h w ) ˆ (
is sufficiently small, then extreme values of the function u can be approximated by using the endpoints of given interval .
)}, ( ), ( {
+ − − =
h u h u min u )}. ( ), ( {
+ − + =
h u h u min u
h ˆ
h ˆ
SLIDE 108
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 108/138
Exact monotonicity tests based on the interval arithmetic
SLIDE 109
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 109/138
= h h h Q u h K ˆ ), ( ) ( ) ˆ ( ˆ ) ˆ ( ) ˆ ( ˆ ) ˆ ( ˆ
− = h u h K h Q u h K
i i i
h h h ) ˆ ( ) ˆ (
= h Q u h K
j i
h u
)
ˆ ( ˆ h
SLIDE 110
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 110/138
Numerical example
( ) ( ) ( )
T n
T T = ˆ ,..., ˆ ˆ
1
T
( ) ( ) ( ) ( ) ( )
− =
T K Q K T ˆ ˆ ˆ , ˆ ˆ λ λ hull
( ) ( ) ( ) ( )
= − = = = +
t 2 b 1 2 1
T r T R = r T r T α dr r dT
- λ
R r Q dr dT(r) rλ dr d r 1 R r R : : :
SLIDE 111
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 111/138
SLIDE 112
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 112/138
Sometimes system of algebraic equations is nonlinear. In this case we can apply interval Jacobean matrices.
) ( ) , ( h Q u u h K =
SLIDE 113
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 113/138
h u, F = ) (
m i h h
i i
1,..., , ) ( ) ( = = + h u, F u u h u, F
u F − =
+ − − − n n i n j n i n n n i j i j i
u F u F h F u F u F u F u F h F u F u F h u ... ... ... ... ... ... ... ... ... ... ...
1 1 11 1 1 1 1 1 1 1 1
SLIDE 114
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 114/138
u F F − =
+ −
) ,..., , , ,..., (
1 1 1 n i j i j i
u u h u u h u
const u u h u u sign const sign
n i j i
= =
+ −
) ,..., , , ,..., ( ,
1 1 1
F u F
const h u sign
j i
=
SLIDE 115
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 115/138
) ˆ ), ˆ ( , ( ˆ ) ), ( , ( , ˆ u h h u x F u h h u x F h h
. , ˆ A A A
Regular interval matrix
SLIDE 116
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 116/138
= h h h u F ˆ , ) , (
It can be shown that if the following interval Jacobean matrices are regular, then solutions of parameter dependent system of equations are monotone.
( )
h h u F
ˆ
, ˆ ˆ
( )
) ,..., , , ,..., ( ˆ , ˆ ˆ
1 1 1 n j j i
u u h u u
+ −
h u F
SLIDE 117
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 117/138
Numerical example
P P P P L H H
1
q
2
q
3
q
4
q
5
q
6
q
7
q
8
q
9
q
10
q
11
q
12
q
Uncertain parameters: E,A,J.
SLIDE 118
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 118/138
Equilibrium equations
- f rod structures
) (
2 2 2 2
x q dx u d EJ dx d =
, ) , (
2 2 2 2
=
L
dx dx v d dx u d EJ v u a
... ) ( + =
L
qvdx v l
) ( ) ( , v l u,v a V v =
SLIDE 119
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 119/138
], [ ] 220 , 210 [ GPa E
], [ 12 0.055 , 12 05 .
4 4 4
m J
], [ ] 0.055 , [0.05
2 2 2
m A
L=H=1 [m], P=1 [kN].
1
q [m]
2
q [m]
3
q
4
q
[m]
5
q
[m]
6
q
− i
q
0.035716 0.000008
- 0.011230
0.035716
- 0.000021
- 0.011230
+ i
q
0.037414 0.000009
- 0.010718
0.037414
- 0.000017
- 0.010718
7
q [m]
8
q
[m]
9
q
10
q
[m]
11
q
[m]
12
q
− i
q
0.082163 0.00009
- 0.007494
0.082163
- 0.000033
- 0.007494
+ i
q
0.086067 0.000010
- 0.007151
0.086067
- 0.000026
- 0.007151
SLIDE 120
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 120/138
Optimization methods
SLIDE 121
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 121/138
= =
+ −
h h h f h u L h h h f h u L ˆ ) ( ) , ( ˆ ) ( ) , (
i i i i
u max u u min u = =
+ −
h h h Q u h K h h h Q u h K ˆ ) ( ) ( , ˆ ) ( ) (
i i i i
u max u u min u
SLIDE 122
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 122/138
These methods can be applied to the very wide intervals
. ˆ h
Function
) (h u u =
doesn't have to be monotone.
SLIDE 123
Andrzej Pownuk http://zeus.polsl.gliwice.pl/~pownuk 123/138
Numerical example
= = = = = 2 3 , ) ( , 2 3 , 2 ), (
2 2 2 2 2 2 2 2
L u dx d dx u d L u L u x q dx u d EJ dx d
q
L 2
L
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Numerical data
2 3 , 2 dla 128 48 2 48 9 24 1 EJ 1 2 0, dla 128 48 24 1 1 ) (
4 3 3 4 4 3 4
+ − − − + − = L L x qL x qL L x qL qx L x ql x ql qx EJ x u
Analytical solution
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05 . 15 . 1 0 037 . 0 022 . y x ( ) x
q
L 2
L
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Other methods and applications
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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
Iterative methods
= h h h Q u h K ˆ ), ( ) (
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k k ij ij
h C K = ) (h
k k j j
h C Q = ) (h
Inner solution Outer solution
) ( ) (
ˆ ˆ ˆ
i OUT i INNER
u u u u u ˆ ˆ
) (
→
i OUT
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Valliappan S., Pham T.D., 1993, Fuzzy Finite Element Analysis
- f a Foundation on Elastic Soil Medium.
International Journal for Numerical and Analytical Methods in Geomechanics, Vol.17, s.771-789 The authors were solved some special fuzzy partial differential equations using only endpoints of given intervals. In some cases we can prove, that the solution can be calculated using only endpoints of given intervals.
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Load combinations in civil engineering
Many existing civil engineering programs can calculate extreme solutions
- f 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).
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Fuzzy eigenvalue problem
( )
) ( ) ( = − h K h M det
} ˆ )), ( ) ( det( : {
) ( ) (
− = h h h K h M
i i
} : { ) | (
) ( ) ( i i F
sup
=
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Upper probability
- f the stability
) | ( } ) {Re(
) ( : ) ( i F i
sup Pl =
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))} ( ˆ ( : { )) ˆ ( ( 0 =
H u u P H u u Pl
Random set Monte Carlo simulations
In some cases we cannot apply fuzzy sets theory to solution of this problem.
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Conclusions
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Conclusions
1) Calculation of the solutions
- f 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
- f computational mechanics.
(thousands degree of freedom)
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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
- f 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)
- ptimization method can be applied.
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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.