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

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 http://andrzej.pownuk.com 1

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 9/16/2011 http://andrzej.pownuk.com 2

Mathematical model of a machine     2 u 3  i j +  =   i f  i  2 x  t = j 1 j  3 3    =  i j kl C  i j kl = = k 1 j 1       u u 1  = +  j   i     i j 2 x x    j i  =   * u u x , V  i i u  3   =   *  n t x , V  i j j i  = j 1  =  u u ( , x 0) , x V  i i Such simulations are possible since early 1970s O.C. Zienkiewicz, Ivo M. Babuška , P.G. Ciarlet ... 9/16/2011 http://andrzej.pownuk.com 3

• Beam model with the interval parameters    4 2 w w = −   EJ q A   4 2 x t  =  w ( 0,) t 0 = q q x ( )  = w L t ( ,) 0   2 w ( 0,) t = = = E E x J ( ) , J x ( )  0  2 dx x  2 w L t ( ,)  = 0 w x t ( ,)  2 dx  = w x ( , 0) w ( ) x  0   w ( , x 0) = =  v x ( , 0) v ( x )  0 t      E E , q q , A A  http://andrzej.pownuk.com 4

Interval displacements 9/16/2011 http://andrzej.pownuk.com 5

• Plate with the interval parameters        4 4 4 2 u u u u + + = −   D  2  q h      4 2 2 4 2 x x y y t     = u ( 0, ,) y t 0  =  u L y t ( , ,) 0  = u x ( , 0,) t 0   = u x L t ( , ,) 0  2 u  = ( 0, ,) y t 0   2 x    2 u L y t  = ( , ,) 0   2 x   2 u x  = ( , 0,) t 0  2  y   2 u x L t  = ( , ,) 0   2 y  = * u x y ( , , 0) u x y ( , )    u x y ( , , 0) = * v x y ( , )   t      E E , q q , h h  http://andrzej.pownuk.com 6

Mathematical models physical problem mathematical models cheap expensive experimental results predictions  experiments 9/16/2011 http://andrzej.pownuk.com 7

Truss structure with uncertain forces P P P 3 1 2 1 14 4 9 10 5 15 3 13 8 L 2 11 6 12 7 L L L L 3/17/2011 http://andrzej.pownuk.com 8

Perturbated forces =   5% uncertainty P P P 0 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 3 1 2 1 14 4 9 10 5 15 3 13 8 L 2 11 6 7 12 L L L L 3/17/2011 http://andrzej.pownuk.com 9

Uncertainty Problem with real parameters = 2 x 4 4 = = x 2         2 1 , 3 4 3, 5   2 Problem with interval parameters   = 1 , 3 x [ 3, 5]   = x ? 9/16/2011 http://andrzej.pownuk.com 10

Algebraic Solution = [1,2] x [1,4] = x [1,2] because  = [1,2] [1,2] [1,4] 9/16/2011 11

United Solution Set x = [1,2] [1,4]     1,4 1 ,4 = =     x   1,2 2 because = =   { : x ax b a , [1,2], b [1,4]} x 9/16/2011 http://andrzej.pownuk.com 12

Comparison of the solution sets x = [1,2] [1,4]     1,4 1 =  = =  [1,2] ,4    x x   1,2 2 Algebraic Solution United Solution Set There are many ways how it is possible to extend equations with the real parameters into equations with the interval parameters. 9/16/2011 http://andrzej.pownuk.com 13

Stochastic differential equations  = y ' p c os ( ) pt  =  y ( 0) 0  p N ( 0, 1 )  9/16/2011 http://andrzej.pownuk.com 14

Interval equation  = y ' p c os ( ) pt  =  y ( 0) 0      p p p ,   9/16/2011 http://andrzej.pownuk.com 15

Solution set in 3D     + P P 2 2 P P     + + 1 1 1 3    A E A E A E    3 3 5 5 2 2 u     1     P       = = −   u ( ) p  u 2 : E E E , , P P P ,          2 i i i i i i A E     u 4 4       + 3 P P    −  1 3    A E      2 2 9/16/2011 http://andrzej.pownuk.com 16

Solution set in 3D  dx ( ) = pr c os pt  dt   dy ( ) = − pr s i n pt  dt        dx  , p p p , , r r r ,     = p  dt  = x ( 0) 0   = y ( 0) 1  = z ( 0) 0  9/16/2011 http://andrzej.pownuk.com 17

Automatically generated test problems http://webapp.math.utep.edu/Pages/IntervalFEMExamples.htm 9/16/2011 http://andrzej.pownuk.com 18

Automatically generated test problems DSL (Domain Specific Languages) 9/16/2011 http://andrzej.pownuk.com 19

2D elasticity problem with the interval parameters Model Solution Mathematical model 9/16/2011 http://andrzej.pownuk.com 20

Adaptive Taylor series http://webapp.math.utep.edu/AdaptiveTaylorSeries-1.1/ 9/16/2011 http://andrzej.pownuk.com 21

Adaptive Taylor series http://andrzej.pownuk.com/silverlight/VibrationsWithIntervalParameters/VibrationsWithIntervalParameters.html 9/16/2011 http://andrzej.pownuk.com 22

Tools which support my research 9/16/2011 http://andrzej.pownuk.com 23

Epistemic uncertainty H – set of horses This is a horse. Is this a horse? ?   H H 9/16/2011 http://andrzej.pownuk.com 24

Fuzzy sets H – set of horses ( )  = 0 H ( )  = 1 H ( )  = 0. 5 H ( )  = 1 H ( )  = 0. 6 H Fuzzy ≠ Probability 9/16/2011 http://andrzej.pownuk.com 25

Fuzzy concept of safety P =  m ax P des i gn =  g x ( )   =   0 P P g x ( ) 0 P f f 9/16/2011 http://andrzej.pownuk.com 26

Problems with binary logic • 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 Q Modus ponens can be applied if and are true.  P Q P , When modus ponens Q can be applied? 9/16/2011 http://andrzej.pownuk.com 27

Science Experiment Theory Mathematical model HPC computing Scientific hypothesis Simulations (predictions) 9/16/2011 http://andrzej.pownuk.com 28

Mathematics and programming Mathematics Programming mathematical method program results results 9/16/2011 http://andrzej.pownuk.com 29

Main problem • At this moment it is not possible perform general mathematical research automatically without human input. NO mathematical method results YES 9/16/2011 http://andrzej.pownuk.com 30

Tools • Approach without tools • Approach with tools 5 years of training Final result 9/16/2011 http://andrzej.pownuk.com 31

Mathematical tools Mathematica Matlab Octave Etc. 9/16/2011 http://andrzej.pownuk.com 32

Example: http://www.wolframalpha.com It is possible to calculate not only the result but also intermediate steps in the calculations 9/16/2011 http://andrzej.pownuk.com 33

Example: student’s tests 1 000 000 pages 9/16/2011 http://andrzej.pownuk.com 34

Automated reports in Latex 200 pages 550 pages 9/16/2011 http://andrzej.pownuk.com 35

Plate equation 9/16/2011 http://andrzej.pownuk.com 36

Plate equation 9/16/2011 http://andrzej.pownuk.com 37

Thank you I will be back … with new results soon 9/16/2011 http://andrzej.pownuk.com 38