The University of Texas at El Paso http://andrzej.pownuk.com P P - - PowerPoint PPT Presentation

Andrzej Pownuk, The University of Texas at El Paso http://andrzej.pownuk.com

1 2 3 4 5 6 7 8 9 10 11 12 13

1

P

2

P

3

P

14 15 L L L L L

No 1 2 3 4 5 6 7 8 ERROR % 10 9,998586 10,00184 10,00126 60,18381 11,67825 9,998955 31,8762 No 9 10 11 12 13 14 15 ERROR % 10,00126 11,67825 60,18381 9,998955 10,00184 10 9,998586

P P P =  

5% uncertainty

1 2 3 4 5 6 7 8 9 10 11 12 13

1

P

2

P

3

P

14 15 L L L L L

ax b =

Example

[1,2] [1,4] x =

? x =

= b x a

 http://en.wikipedia.org/wiki/Interval_finite_element

p p p p

Interval Random variable

1 p p −

1

1

... , ,

m i i i i

p p n p p     + + = =      

 

p p

( )

( )

( )

width n n p p n width = − =  p p

Intervals Random variables

2 2 n i i

n n     = = =



p

( ) ( )

width n n width =  p p

int int

( ) ( ) ( ) ( )

rand rand

width n n width n width n n width  = =  p p p p

Example n=100

int int

( ) ( ) 100 10 ( ) ( )

rand rand

width n n width width n n width  = = =  p p p p



• design value



• characteristic value

u d m

S S  =

d

S

c

S

 (i)

the possibility

• f

unfavourable deviation

• f

material strength from the

 characteristic value.  (ii)

the possibility

• f

unfavourable variation

• f

member sizes.

 (iii) the possibility of unfavourable reduction in

member strength due to

 fabrication and tolerances.  (iv) uncertainty in the calculation of strength of the

members.

d f c

F F  =



• design value



• characteristic value

d

F

c

F

( )

L Q T

R D L Q T        + + +

( )

,

i

L     

( )

( )

L Q T

R D L Q T       − + + + 

( )

i

L



 

f f

P P 

f

P  =

• probability of failure

 Probability of failure =  = (number of safe cases)/(number of all cases)

structure fail often = structure is not safe

( )

( ) 0 f X g x

P f x dx



= 

 Probabilistic methods

 How often the stucture fail?

 Non-probabilistic methods

(worst case design)

 How big force the structure is able to survive?

( )   ( )

( , ) 0

( , ) ,

f g x

P P g x f x dx



  



=  = 

( )

f f

P P    

 Elishakoff I., 2000, Possible limitations of

probabilistic methods in engineering. Applied Mechanics Reviews, Vol.53, No.2,pp.19-25

 Does God play dice?

1:09 18 /1 53

limit state uncertain limit state

1



2



crisp state uncertain state

19 /5 3

Set valued random variable Upper and lower probability

( )

n

R h h  →     :

( ) ( )  

   = A h P A Pl   :

( ) ( )  

A h P A Bel  =   :

( ) Pl P    

1:09 21/153

 Nested family of random sets

( ) ( ) ( )

N

h h h       ...

2 1

( ) ( )}

: {    h x P x

F

 =

x

( )

x

F



( )

1

 h

( )

2

 h

( )

3

 h

1

F

2

F

3

F

x

( )

x

F



 Fuzzy sets

−

F

+

F

− 1

F

+ 1

F



1

( )

x

F



− 

F

+ 

F

x

( )( ) ( )

( )

x y

F x f y x F f

 

=

=

:

sup

Extension tension principle le

( , )

F

g x  

 

: ( , ) 0, g x x x

 

   =  

( )  

max :

 

     = 

 http://andrzej.pownuk.com/fuzzy.htm  Fuzzy approach (the use of grades)

is similar to the concept of safety factor. Because of that fuzzy approach is very important.

( )

 

max : ,

 

    =  

F x

x F F

( )  

, max :( , )



  = 

R c p

p c R

( ) ( ) ( ) ( )

| | = P E H P H P H E P E

( )

P H

• is called the prior probability of H that was inferred

before new evidence, E, became available

( )

| P E H

• is called the conditional probability of seeing the evidence E

if the hypothesis H happens to be true. It is also called a likelihood function when it is considered as a function of H for fixed E.

( )

P E

• marginal probability

( )

| P H E

• is called the posterior probability of H given E.

 Cox

Cox's 's th theo eore rem, named after the physicist Richard Threlkeld Cox, is a derivation of the laws

• f

probability theory from a certain set of postulates. This derivation justifies the so-called "logical" interpretation of probability. As the laws of probability derived by Cox's theorem are applicable to any proposition, logical probability is a type

• f

Bayesian

• probability. Other forms of Bayesianism,

such as the subjective interpretation, are given other justifications.

 First described by Zdzisław I. Pawlak, is

a formal approximation of a crisp set (i.e., conventional set) in terms of a pair

• f sets which give the lower and the

upper approximation

• f

the

• riginal
• set. In the standard version of rough

set theory (Pawlak 1991), the lower- and upper-approximation sets are crisp sets, but in other variations, the approximating sets may be fuzzy sets.

, p p p    

( , ) g p  

 

: ( , ) 0, g p p   =   θ p

Design with the interval parameters

 R. E. Moore. Interval Analysis. Prentice-Hall,

Englewood Cliffs N. J., 1966

 Neumaier A., 1990, Interval methods for systems

• f equations, Cambridge University Press, New

York

 Ben-Haim Y., Elishakoff I., 1990, Convex Models

• f Uncertainty in Applied Mechanics. Elsevier

Science Publishers, New York

 Buckley J.J., Qy Y., 1990, On using a-cuts to

evaluate fuzzy equations. Fuzzy Sets and Systems, Vol.38,pp.309-312

 Köylüoglu H.U., A.S. Çakmak, and S. R. K.

Nielsen (1995). “Interval Algebra to Deal with Pattern Loading of Structural Uncertainties,” ASCE Journal of Engineering Mechanics, 11, 1149–1157

 Rump S.M., 1994, Verification methods for

dense and sparse systems of equations. J. Herzberger, ed., Topics in Validated

• Computations. Elsevier Science B.V.,pp.63-

135

 Muhanna in the paper Muhanna R.L., Mullen

R.L., Uncertainty in Mechanics Problems - Interval - Based Approach. Journal of Engineering Mechanics, Vol.127, No.6, 2001, 557-556

 E.Popova, On the Solution of Parametrised

Linear Systems. W. Kraemer, J. Wolff von Gudenberg (Eds.): Scientific Computing,Validated Numerics, Interval

• Methods. Kluwer Acad. Publishers, 2001, pp.

127-138.

 I. Skalna, A Method for Outer Interval Solution

• f Systems of Linear Equations Depending

Linearly on Interval Parameters, Reliable Computing, Volume 12, Number 2, April, 2006, Pages 107-120

 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, Stewart, 2001

Anti-optimisation of uncertain structures using interval analysis Computers and Structures Volume: 79, Issue: 4, February, 2001, pp. 421-430

 Pownuk A., Numerical solutions of fuzzy

partial differential equation and its application in computational mechanics, FuzzyPartial 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., Clouds, fuzzy sets and

probability intervals, Reliable Computing 10: 249–272, 2004

 http://andrzej.pownuk.com/IntervalEquation

s.htm

 http://webapp.math.utep.edu:8080/~andrzej

/php/ansys2interval/

 http://webapp.math.utep.edu/Pages/Interval

FEMExamples.htm

 http://calculus.math.utep.edu/IntervalODE-

1.0/default.aspx

 http://calculus.math.utep.edu/AdaptiveTaylo

rSeries-1.1/default.aspx

 http://andrzej.pownuk.com/silverlight/Vibrat

ionsWithIntervalParameters/VibrationsWithInt ervalParameters.html

ax b =

Example

[1,2] [1,4] x =

? x =

= b x a

[1,2] [1,4] x = [1,2] x =

because

[1,2][1,2] [1,4] =

[1,4] [1,4] x = [1,1] 1 x = =

because

[1,4] 1 [1,4]  =

[1,8] [1,4] x = ? x =

[1,2] [1,4] x = 1 ,4 2   =     x

because

{ : , [1,2], [1,4]} x ax b a b = =   x

      =              [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,

( )  

, : , , x A b Ax b



=     =



A b A b

United solution set

( )  

, : , , x A b Ax b



=     =



A b A b

( )  

, : , , x A b Ax b



=     =



A b A b

Tolerable solution set Controllable solution set

http://www.ippt.gov.pl/~kros/pccmm99/10PSS.html

( ) ( ), ( ) f x f x f x   =  

   

'( ) min '( ), '( ) ,max '( ), '( ) f x f x f x f x f x   =  

What is is th the defini initio ion n of the solu lutio ion n

• f dif

iffer ferentia ential l equation? ation?

( ) f x ( ) f x

( ) ( ) ( )

v , x f x f x    

( ) ( ) ( )

 

( ) ( )

 

v' min ' , ' ,max ' , ' x f x f x f x f x    

Dubois D., Prade H., 1987, On Several Definition of the Differentiation of Fuzzy Mapping, Fuzzy Sets and Systems, Vol.24, pp.117-120

How about integral equations?



Modal interval arithmetic



Affine arithmetic



Constrain interval arithmetic



Ellipsoidal arithmetic



Convex models (equations with the ellipsoidal parameters)



General set valued arithmetic



Fuzzy relational equations



…. Etc.

( ) ( ) ( ) ( )

1 10 1 10 1

,..., ,..., ...

m m m m m

y y y p p y p p p p p p p p    + − + + −  

( )

10

,...,

m

y y p p =

1 1

...

m m

y y y p p p p      + +   

 

, , y y y y y y y     −  +   

 Gradient descent  Interior point method  Sequential quadratic programming  Genetic algorithms  …

 Endpoint combination method  Interval Gauss elimination  Interval Gauss-Seidel method  Linear programming method  Rohn method  Jiri Rohn, "A Handbook of Results on Interval

Linear Problems“,2006 http://www.cs.cas.cz/~rohn/publist/handboo k.zip

 W. Oettli, W. Prager. Compatibility of

approximate solution of linear equations with given error bounds for coefficients and right- hand sides. Numer. Math. 6: 405-409, 1964.

( ) ( ) ( ) ( )

( )

( ) x mid x rad rad x rad



  −  +



A,b A b A b

, A b   A b

 H.U. Koyluoglu, A. Çakmak, S.R.K. Nielsen.

Interval mapping in structural mechanics. In: Spanos, ed. Computational Stochastic

• Mechanics. 125-133. Balkema, Rotterdam

1995.

( )

( )

( )

( )

min ( ) ( )

i s s

x mid D rad x b mid D rad x b x   −    +      A A A A

( )

( )

( )

( )

max ( ) ( )

i s s

x mid D rad x b mid D rad x b x   −    +      A A A A

( ) ( ) ( ) ( )

rc r c r r

A mid D rad D b mid D rad = −    = +   A A b b

( )

1, 1,1,...,1

T

r J = − 

( ) ( )

rc rc

conv conv

 

=

 

A,b A ,b

( )  

: , ,

rc rc rc r

conv conv x A x b r c J



= = 



A ,b

2

2 2 2

n n n

 =

2

2n

n + Rohn’s method Combinatoric solution

 For every  Select recommended  Solve  If then register x and go to

next r

 Otherwise find  Set and go to step 1.

c J 

( )

( )

1 r

c sign mid b

−

= A

rc r

A x b =

r J 

( )

sign x c =

( )

 

min :

j j

k j sign x c = 

k k

c c = −

VERSO SOFT: FT: Veri rifi fica cati tion

• n software

are in MATLA LAB B / I INTLAB LAB http://uivtx.cs.cas.cz/~rohn/matlab/index.html

• Real data only: Linear systems (rectangular)
• Verified description of all solutions of a system of linear equations
• Verified description of all linear squares solutions of a system
• f linear equations
• Verified nonnegative solution of a system of linear inequalities
• VERLINPROG for verified nonnegative solution of a system of linear equations
• Real data only: Matrix equations (rectangular)
• See VERMATREQN for verified solution of the matrix equation A*X*B+C*X*D=F

(in particular, of the Sylvester or Lyapunov equation) Etc.

 Find the solution of Ax = b

• Transform into fixed point equation g(x) = x

g (x) = x – R (Ax – b) = Rb+ (I – RA) x (R nonsingular)

• Brouwer’s fixed point theorem

If Rb + (I – RA) X  int (X) then  x  X, Ax = b

 Solve AX

AX=b

• Brouwer’s fixed point theorem w/

Krawczyk’s operator

If R b + (I – RA) X  int (X) then (A, b)  X

• Iteration

 Xn+1= R b + (I – RA) εXn (for n = 0, 1, 2,…)  Stopping criteria: Xn+1  int( Xn )  Enclosure: (A, b)  Xn+1

        =                 − − + p u u k k k k k

2 1 2 2 2 2 1

k1 = [0.9, 1.1], k2 = [1.8, 2.2], p = 1.0

] 11 . 1 , 91 . [ ] 1 . 1 , 9 . [ 1 1

1 1

= = = k u ) tion

• verestima

( ] 04 . 2 , 12 . 1 [ ] 2 . 2 , 8 . 1 [ ] 1 . 1 , 9 . [ ] 2 . 2 , 8 . 1 [ ] 1 . 1 , 9 . [

2

=  + = + =

2 1 2 1

k k k k u solution) exact ( ] 67 . 1 36 . 1 [ ] 2 . 2 , 8 . 1 [ 1 ] 1 . 1 , 9 . [ 1 1 1 '

2 1 2

, = + = + = k k u

 Two k1: the same physical quantity  Interval arithmetic: treat two k1 as two

independent interval quantities having same bounds

2 1 2 1

k k k k u + =

2

 Replace floating point arithmetic by interval

arithmetic

 Over-pessimistic result due to dependency

        =                 − − − − 1 ] 2 . 2 , 8 . 1 [ ] 8 . 1 , 2 . 2 [ ] 8 . 1 , 2 . 2 [ ] 3 . 3 , 7 . 2 [

2 1

u u

        − − =         ] 5 . 137 , 5 . 134 [ ] 112 , 110 [

2 1

u u

Naïve solution Exact solution

        =         ] 67 . 1 , 36 . 1 [ ] 11 . 1 , 91 . [

2 1

u u

 How to reduce overestimation?

• Manipulate the expression to reduce multiple
• ccurrence
• Trace the sources of dependency

        =                 − − + p u u k k k k k

2 1 2 2 2 2 1

 Element-by-Element

• K: diagonal matrix, singular

p

L2, E2, A2 L1, E1, A1

p

 Element-by-element method

• Element stiffness:
• System stiffness:

) (

i i i

I K d K + = 

                +                     − − =             − − =

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 α α E E E E K

1 1 1 1

L A E L A E L A E L A E L A L A L A L A    

) ( d K + = I K 

 Lagrange Multiplier method

• With the constraints: CU – t = 0
• Lagrange multipliers: λ

        =                          =                 p λ u K C C t p u C C K

T T



 System equation: Ax

Ax = b b rewrite as:

        =                 p λ u K C CT         =                                 +         p λ u d k C C k

T

 

b x D = + ) ( S A 

 x

x = [u, λ]T, u is the displacement vector

 Calculate element forces

• Conventional FEM: F=k u ( overestimation)
• Present formulation: Ku

Ku = P – CT λ λ= = Lx, p = Nb P – CT λ = = p – CT L(x*n+1 + x0) P – CT λ = = Nb – CT L(Rb – RS Mn δ) P – CT λ = = (N – CT LR)b + CTLRS Mn δ

Click here

http://andrze tp://andrzej.po j.pownuk.com/ wnuk.com/

Click here

 Monotone solution  Non-monotone solution

1 1 2 2 2

1 1 1 1 p u p u p       =       −           +

1 1 1 2 2

, 2 2 p p p u u = + =

4 2

u p − =

2 2 1 2

, u p u p = − =

u p   

If then

,

m i n m ax

p p p p = =

u p   

If then

,

m i n m ax

p p p p = =

m ax

( ) , ( )

m i n

u u p u u p = =

analysis_type linear_static_functional_derivative parameter 1 [210E9,212E9] # E parameter 2 [0.2,0.4] # Poisson number parameter 3 0.1 # thickness parameter 4 [-3,-1] sensitivity # fy point 1 x 0 y 0 point 2 x 1 y 0 point 3 x 1 y 1 point 4 x 0 y 1 point 5 x -1 y 0 point 6 x -1 y 1 rectangle 1 points 1 2 3 4 parameters 1 2 3 rectangle 2 points 5 1 4 6 parameters 1 2 3 load constant_distributed_in_y_local_direction geometrical_object 1 fy 4 load constant_distributed_in_y_local_direction geometrical_object 2 fy 4 boundary_condition fixed point 1 ux boundary_condition fixed point 1 uy boundary_condition fixed point 2 ux boundary_condition fixed point 2 uy boundary_condition fixed point 5 ux boundary_condition fixed point 5 uy #Mesh

( ) ( )

min , u u C  =     =           

( )

min , u u =             

• r

A B A B   

 

, :   =   =         Ω

( )

C

xd x d  

 

 = 



Center of gravity

( )

2

I y d



 = 

Moment of inertia Solution of PDE

*

for int for u u q x t u u x   + =       =  

( )

( ) ( ) ( ) ( )

lim lim

 

 

+

→  →

 +  −  = = 

T

du u u d D x df f d

Examples

( ) ( ) 



 =  f L x d

( )

( ) df L x d  =

( )

C

xd x d  

 

 = 



( )

2 C

x xd dx d d   

 

 −  =      

 

 

2

( ) , 1,2 f x x x = 

( ) 2 [2,4] df x x dx = 

( )

2

1 1 f f x = = =

( )

2

2 4 f f x = = =

 

( ) , 1,4 f x f f    =  

( ) ( ) ( )

, f f df  +  −     

( ) ( )

f f  +   

( ) ( ) ( )

lim lim

  

  

+

→ →

 −  = du u u d df f d

( ) ( )

2

f L x d



   =    



( ) ( ) ( ) ( )

2 | | f f L x d L x 



   +  −       



 

1

, x x R  = 

1

( ) x x f x x −  = −

( ) ( )

1

1 1 1 1 2 1 1 1 1 1

( ) lim ( ) ( )

x

df f x x f x x x dx df d d x x x x x dx 

+

 →

+  − − = = = −   +  −  −

1

x x  = −

1

( ) x x f x x −  = −

( ) ( )

1 2 1

( ) lim ( ) ( )

x

df f x x f x dx x x df d d x x x x x dx 

+

 →

+  − − = = =   +  −  −

1

x x  = −

1 1

df df dx dx d d dx dx   

( )

( ) K p u Q p =

( )

( ) f p p = 

( ) ( )

( ) Q p K p u K p u p p p    = −   

( )

( ) p f p p p    =  

Formulation of the problem Calculation of the derivative

u u p p    =   

1 i i +

 =  + 

If

( )

, du   

then If

( )

, du   

then

1 i i +

 =  − 

f

• r

x  

 Coordinates of the nodes are the interval

numbers.

 Loads, material parameters (E, ) can be also

the interval numbers.

[ , ]

i i i

x x x 



[ , ] , [ , ] , [ , ] P P E E E P      

( ) ( , ) , ( ) ( , )

m i n m ax

u x u x u x u x =  = 

( ) { ( , ): [ , ] } x u x =       u

 Currend civil engineering codes are based on

worst case design concept, because of that it is possible to use presented approach in the framework of existing law.

 Using presented method it is possible

to solve engineering problems with interval parameters (interval parameters, functions, sets).