Modeling of Uncertainty in Computational Mechanics

Andrzej Pownuk, The University of Texas at El Paso

http://andrzej.pownuk.com

FEM Models

Modeling errors

Approximation errors

Rounding errors

Uncertainty

Human errors

Errors in numerical computations

3

2 2

'' ''1 ( ')

y M MyEJ EJy

( )b

ii i

a

f x dx w f

Truss structure

1

23

4

5

67

8

91 0

111 2

1 3

1P 2P 3P

1 4

1 5

LLLL

L

Perturbated forces

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

No 9 10 11 12 13 14 15ERROR % 10,00126 11,67825 60,18381 9,998955 10,00184 10 9,998586

0P P P

5% uncertainty

1

23

4

5

67

8

91 0

111 2

1 3

1P 2P 3P

1 4

1 5

LLLL

L

Interval equations

ax bExample

[1, 2] [1,4]x

?x

bxa

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

Interval FEM

Differences between intervals and uniformly distributed random variables

p p p

p

Interval

Random variable 1p p

1

Differences between intervals and uniformly distributed random variables

1 ... , ,m i ii i

p p n p p p p

width n n p p n width p p

Intervals

Random variables

2 2n i

i

n n p

width n n width p p

Differences between intervals and uniformly distributed random variables

int int( ) ( )( ) ( )rand rand

width n n widthn

width n n width

p pp p

Example n=100

int int( ) ( )100 10

( ) ( )rand rand

width n n widthwidth n n width

p pp p

Partial safety factors for materials

- design value - characteristic value

ud

m

SS

dS

cS

Partial safety factor for materials, account for (i) the possibility of unfavourable deviation of

material strength from the characteristic value. (ii) the possibility of unfavourable variation of

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.

Partial safety factors for loads

d f cF F

- design value - characteristic value

dF

cF

Limit state is a function of safety factors

L Q TR D L Q T

, 0iL

0L Q TR D L Q T

Calibration of safety factors

0iL

0f fP P

fP - probability of failure

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

cases)

Probability of failure

structure fail often=

structure is not safe

( ) 0

f Xg 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?

Probabilistic vs nonprobabilistic methods

Probabilistic design

( , ) 0

( , ) 0 ,fg x

P P g x f x dx

0f fP 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

20/

153

More complicated safety conditions

lim it s ta te

uncerta in lim it s tate

1

2

crisp sta te

uncerta in sta te

21/

53

Uncertain parametersSet valued random variable

Upper and lower probability

nRhh :

AhPAPl :

AhPABel :

Design under uncertainty

0( )Pl P

1:09 23/153

Uncertain parameters

Nested family of random sets Nhhh ...21

}:{ hxPxF x

xF

1h

2h

3h

1F

2F

3F

0x

0xF

Uncertain parameters Fuzzy sets

0F

0F1F

1F

1

0

xF

F

F x

xy Fxfyx

Ff

:sup

Extension principle

Design with fuzzy parameters

( , ) 0Fg 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.

Fuzzy sets and probability

Alpha cut method

max : , F x x F F

, max : ( , ) R c p p c R

Interval risk curve for different uncertainty

Interval risk curve for different uncertainty

Interval risk curve for different uncertainty

Bayesian inference (subjective probability)

||

P E H P HP 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's theorem

Cox's theorem, named after the physicist Richard Threlkeld Cox, is a derivation of the laws of 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 of Bayesian probability. Other forms of Bayesianism, such as the subjective interpretation, are given other justifications.

Rough sets, soft sets(sets with features)

First described by Zdzisław I. Pawlak, is a formal approximation of a crisp set (i.e., conventional set) in terms of a pair of sets which give the lower and the upper approximation of the original 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.

Interval and set valued parameters

Interval parameters,p p p

( , ) 0g 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 of equations, Cambridge University Press, New York

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

Important papers

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

Important papers

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

Important papers

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.

Important papers

I. Skalna, A Method for Outer Interval Solution of Systems of Linear Equations Depending Linearly on Interval Parameters, Reliable Computing, Volume 12, Number 2, April, 2006, Pages 107-120

Important papers

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

Important papers

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

Important papers

Neumaier A., Clouds, fuzzy sets and probability intervals, Reliable Computing 10: 249–272, 2004

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

Important papers

http://webapp.math.utep.edu:8080/~andrzej/php/ansys2interval/

http://webapp.math.utep.edu/Pages/IntervalFEMExamples.htm

http://calculus.math.utep.edu/IntervalODE-1.0/default.aspx

http://calculus.math.utep.edu/AdaptiveTaylorSeries-1.1/default.aspx

http://andrzej.pownuk.com/silverlight/VibrationsWithIntervalParameters/VibrationsWithIntervalParameters.html

Examples

Interval equations

ax bExample

[1, 2] [1,4]x

?x

bxa

Algebraic solution

[1, 2] [1,4]x

[1, 2]x because

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

Algebraic solution

[1, 4] [1,4]x

[1,1] 1x because

[1, 4] 1 [1,4]

Algebraic solution

[1,8] [1,4]x

?x

United solution set

[1, 2] [1,4]x

1 ,42

xbecause

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

United solution set

[1,2][-1,1]

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

2

1

xx

1

2

3

3

3

3

)( BA ,hu ll

)( BA ,

Different solution sets

, : , ,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

Other solution sets

Parametric solution sets

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

Derivative

( ) ( ), ( )f x f x f x

'( ) min '( ), '( ) ,max '( ), '( )f x f x f x f x f x

What is the definition of the solution of differential equation?

Differentiation of the interval function

( )f x

( )f x

v ,x f x f x

v' min ' , ' ,max ' , 'x f x f x f x f x

What is the definition of the solution of differential equation?

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?

Other problems 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.

Methods of solutions

What does everyone have in mind?

Taylor expansion

1 10 0 1 10 01

,..., ,..., ...m m m mm

y yy p p y p p p p p pp p

0 10 0,..., my y p p

11

... mm

y yy p pp p

0 0, ,y y y y y y y

General optimization methods Gradient descent Interior point method Sequential quadratic programming Genetic algorithms …

Interval methodsand reliable computing

System of linear interval equations 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“,2006http://www.cs.cas.cz/~rohn/publist/handbook.zip

Oettli and Prager (1964)

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

Linear programming method

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

( )

( )

0

i

s

s

x

mid D rad x b

mid D rad x b

x

A A

A A

max

( )

( )

0

i

s

s

x

mid D rad x b

mid D rad x b

x

A A

A A

Rohn’s method

rc r c

r r

A mid D rad D

b mid D rad

A A

b b

1, 1,1,...,1 Tr J

rc rcconv conv

A,b A ,b

: , ,rc rc rc rconv conv x A x b r c J

A ,b

22 2 2n n n 2

2n nRohn’s method Combinatoric solution

Rohn sign-accord method (RSA) For every Select recommended

Solve If then register x and go to

next r Otherwise find Set and go to step 1.

c J 1rc sign mid b A

rc rA x b

r J

sign x c

min : j jk j sign x c

k kc c

SoftwareVERSOFT: Verification software in MATLAB / INTLAB

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 of 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.

Muhanna and Mullen

Fixed point theory Find the solution of Ax = b

◦ Transform into fixed point equation g(x) = xg (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

Fixed point theory Solve AX=b

◦ Brouwer’s fixed point theorem w/ Krawczyk’s operatorIf 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

Dependency problem

pu

ukkkkk 0

2

1

22

221

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

]11.1,91.0[]1.1,9.0[

11

11

ku

)tionoverestima(

]04.2 ,12.1[]2.2 ,8.1[]1.1 ,9.0[]2.2 ,8.1[]1.1 ,9.0[

2

21

21

kkkku

solution)exact (

]67.136.1[]2.2 ,8.1[

1]1.1 ,9.0[

111'21

2 , kk

u

Dependency problem

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

independent interval quantities having same bounds

21

21

kkkku

2

Naïve interval finite element Replace floating point arithmetic by interval

arithmetic Over-pessimistic result due to dependency

10

]2.2,8.1[]8.1,2.2[]8.1,2.2[]3.3,7.2[

2

1

uu

]5.137 ,5.134[

]112 ,110[

2

1

uu

Naïve solution Exact solution

]67.1 ,36.1[]11.1 ,91.0[

2

1

uu

Dependency problem How to reduce overestimation?

◦ Manipulate the expression to reduce multiple occurrence

◦ Trace the sources of dependency

pu

ukkkkk 0

2

1

22

221

Present formulation Element-by-Element

◦ K: diagonal matrix, singular

p

L2, E2, A2L1, E1, A1

p

Present formulation Element-by-element method

◦ Element stiffness:

◦ System stiffness:

)( iii IK dK

1

1

1

11

1

11

1

11

1

11

1

1

1

1

1

1

1

1

1 00

1001

αα

EE

EE

K11

11

LAE

LAE

LAE

LAE

LA

LA

LA

LA

)( dK IK

Present formulation Lagrange Multiplier method

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

000p

λuK

CC

tpu

CCK TT

Present formulation System equation: Ax = b

rewrite as:

00p

λuK

CCT

000

0000

0p

λudk

CCk T

bxD )( SA

Present formulation x = [u, λ]T, u is the displacement vector Calculate element forces

◦ Conventional FEM: F=k u ( overestimation)◦ Present formulation: Ku = P – CT λ

λ= Lx, p = NbP – 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

http://en.wikipedia.org/wiki/File:Truss1FemModel.gif

http://en.wikipedia.org/wiki/File:Truss1results.gif

Click here

Web applicationhttp://andrzej.pownuk.com/

Click here

Truss structure with interval parameters

Monotonicity of the solution Monotone solution

Non-monotone solution

1 1 2

2 2

1 11 1

pu pu p

1 11 2 2, 2 2

p ppu u

42 0u p

2 21 2,u p u p

Interval solution for monotone functions(gradient descent method)

0up

If then ,min maxp pp p

0up

If then ,min maxp pp p

max( ), ( )minu u p u u p

Monotonicity of the solution

Plane stress

Interval solutionVerification of the results by using search method with 3 intermediate points

Interval solution

Interval solution

Interval solution

Interval solution

2D problem with the interal parameters and interval functions parameters

Scripting languageanalysis_type linear_static_functional_derivative

parameter 1 [210E9,212E9] # Eparameter 2 [0.2,0.4] # Poisson numberparameter 3 0.1 # thicknessparameter 4 [-3,-1] sensitivity # fy

point 1 x 0 y 0point 2 x 1 y 0 point 3 x 1 y 1 point 4 x 0 y 1 point 5 x -1 y 0point 6 x -1 y 1 rectangle 1 points 1 2 3 4 parameters 1 2 3rectangle 2 points 5 1 4 6 parameters 1 2 3

load constant_distributed_in_y_local_direction geometrical_object 1 fy 4load constant_distributed_in_y_local_direction geometrical_object 2 fy 4

boundary_condition fixed point 1 uxboundary_condition fixed point 1 uyboundary_condition fixed point 2 uxboundary_condition fixed point 2 uyboundary_condition fixed point 5 uxboundary_condition fixed point 5 uy

#Mesh

Shape optimization

min

0

,

u u

C

min

,

u u

or

Interval setA B A B

, : Ω

Set dependent functions

C

xdx

d

Center of gravity

2I y d

Moment of inertia

Solution of PDE

*

for int

for

uu q xt

u u x

Topological derivative

00lim lim

T

duu u d D xdff

d

Topological derivativeExamples

( ) ( )

f L x d ( )df L xd

C

xdx

d

2C

x xddx

dd

Extreme values of montone functions

2( ) , 1, 2f x x x

( ) 2 [2,4]df x xdx

21 1f f x

22 4f f x

( ) , 1, 4f x f f

Topological derivative and monotonicity

, 0f f df

f f

Parametric representation

00

lim lim

duu u d

dffd

Results which are independent of parameterization

2

f L x d

2 | |f f L x d L x

Results which are dependent on parameterization

0 1,x x R

0

1 0

( ) x xfx x

1

1 1 1 0120

1 1 1 1 0

1

( )lim

( ) ( )x

dff x x f x x xdxdf

dd x x x x xdx

1 0x x

Results which are dependent on parameterization

0

1 0

( ) x xfx x

0

0 0 0 0 120

0 0 0 1 0

0

( )lim

( ) ( )x

dff x x f x dx x xdf

dd x x x x xdx

1 0x x

01

1 0

dfdfdxdx

d ddx dx

Computational method

( )K p u Q p

( )f p p

( )

Q p K puK p up p p

( )pf pp p

Formulation of the problem

Calculation of the derivative

Computational methodu

u p

p

1i i If , 0du then

If , 0du then 1i i

for x

Coordinates of the nodes are the interval numbers.

Loads, material parameters (E, ) can be also the interval numbers.

Data for Calculations

[ , ]i i ix x x

[ , ], [ , ], [ , ]P P E E EP

Interval solution

( ) ( , ), ( ) ( , )min maxu x u x u x u x

( ) { ( , ) : [ , ]}x u x u

Interval von Mises stress

Truss structure with the uncertain geometry

Conclusions 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).