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
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