[PPT] - Linear Dynamics of an Elastic Beam and Plate Under Moving Loads PowerPoint Presentation

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Linear Dynamics of an Elastic Beam and Plate Under Moving Loads with Uncertain Parameters Andrzej Pownuk The University of Texas at El Paso http://andrzej.pownuk.com

9/16/2011 1 http://andrzej.pownuk.com

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Outline of the presentation

Equations with the uncertain parameters and

their applications

New approach for the solution of the

equations with the interval parameters

Generalizations and conclusions

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Mathematical model of a machine

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2 2 * 3 1 3 3 1 1 3 * 1

1 ( , 0) 2 , , ,

i j i i j i j kl i j kl i j i i u i j j k j j i j j i i i i j

u f x t C u u x x u u x V n x x t x V V u u



      

= = = =

   + =       =        = +            =     =       = 

  

Such simulations are possible since early 1970s O.C. Zienkiewicz, Ivo M. Babuška, P.G. Ciarlet ...

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Beam model with the interval parameters

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2 2 2 4 4 2 2 2

( 0,) ( ,) ( 0,) ( ,) ( ( ) ( , 0) ( , 0) ( , ) , 0) , x w w w EJ q A x t w t w L t w t dx w L t x v x dx w x w v E q x A t     = −     =   =   =     = =   =   =          E q A

( ,) w x t x ( ) q q x = ( ) , ( ) E E x J J x = =

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Interval displacements

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Plate with the interval parameters

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* * 4 4 4 2 4 2 2 4 2 2 2 2 2 2 2 2 2

2 ( 0, ,) ( , ,) ( , 0,) ( , ,) ( 0, ,) ( , ,) ( , 0,) ( , ,) ( , , 0) ( , ) ( , , 0) ( , ) , , u u u u D q h x x y y t u y t u L y t u x t u x L t u y t x u L y t x u x t y u x L t y u x y u x y u x y v x y t E q h         + + = −             =  =   =   =   =    =   =   =  =   =    E q h                 

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Mathematical models

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physical problem mathematical models experiments predictions experimental results



expensive cheap

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Truss structure with uncertain forces

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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

8

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Perturbated forces

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P P P =  

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

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

9

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Uncertainty

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2 4 4 2 2 x x = = =

1 , 3 [ 3, 5] ? x x   =   =

Problem with real parameters Problem with interval parameters

2 1 , 3      4 3, 5     

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Algebraic Solution

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

because

[1,2] [1,2] [1,4]  =

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United Solution Set

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

   

1,4 1 ,4 1,2 2   = =     x

because

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

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Comparison of the solution sets

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

   

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

United Solution Set Algebraic Solution There are many ways how it is possible to extend equations with the real parameters into equations with the interval parameters.

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Stochastic differential equations

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' c

s

( ) ( 0) ( 0, 1 ) y p pt y p N  =  =   

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Interval equation

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' c

s

( ) ( 0) , y p pt y p p p  =  =        

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Solution set in 3D

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1 3 1 1 3 3 5 5 2 2 1 2 2 4 4 3 1 3 2 2

( ) : , , 2 , 2

i i i i i i

P P P P A E A u u u E E E P P P E A E P A E P u P A E                             = =                                 + − + −  + +  p

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Solution set in 3D

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( ) ( )

c

s

s i n , , , , ( 0) ( 0) 1 ( 0) dx pr pt dt dy pr pt dt dx p p p r r r p dt x y z  =    = −              =   =  =   = 

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Automatically generated test problems

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

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Automatically generated test problems

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DSL

(Domain Specific Languages)

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2D elasticity problem with the interval parameters

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Model Solution Mathematical model

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Adaptive Taylor series

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http://webapp.math.utep.edu/AdaptiveTaylorSeries-1.1/

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Adaptive Taylor series

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Tools which support my research

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Epistemic uncertainty

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This is a horse. Is this a horse?

H 

?

H 

H – set of horses

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Fuzzy sets

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( )

1

H

 =

( )

1

H

 =

( )

0. 5

H

 =

Fuzzy ≠ Probability H – set of horses

( )

H

 =

( )

0. 6

H

 =

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Fuzzy concept of safety

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 

( )

f f

P P g x P =  

m ax des i gn

P P  = ( ) g x  =

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Problems with binary logic

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Is it possible to find in the real world

statements which are absolutely true?

(L. Wittgenstein, Tractatus Logico-Philosophicus, Annalen der Naturphilosophie, 14, 1921)

, P Q P Q 

Modus ponens can be applied if and are true.

P Q 

Q

?

When modus ponens can be applied?

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Science

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Experiment Theory Mathematical model Simulations (predictions) Scientific hypothesis

HPC computing

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Mathematics and programming

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Mathematics Programming

mathematical method program results results

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Main problem

At this moment it is not possible perform

general mathematical research automatically without human input.

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mathematical method results YES NO

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Tools

Approach with tools

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Approach without tools

5 years of training Final result

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Mathematical tools

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Mathematica Matlab Octave Etc.

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Example: http://www.wolframalpha.com

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It is possible to calculate not only the result but also intermediate steps in the calculations

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Example: student’s tests

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1 000 000 pages

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Automated reports in Latex

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200 pages 550 pages

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Plate equation

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Plate equation

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Thank you

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I will be back …

with new results soon