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Modeling of Uncertainty in Computational Mechanics

Modeling of Uncertainty in Computational Mechanics. Andrzej Pownuk, The University of Texas at El Paso http://andrzej.pownuk.com . FEM Models. Errors in numerical computations. Modeling errors Approximation errors Rounding errors Uncertainty Human errors. Truss structure.

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Modeling of Uncertainty in Computational Mechanics

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  1. Modeling of Uncertainty in Computational Mechanics Andrzej Pownuk, The University of Texas at El Paso http://andrzej.pownuk.com

  2. FEM Models

  3. Errors in numerical computations • Modeling errors • Approximation errors • Rounding errors • Uncertainty • Human errors

  4. Truss structure

  5. Perturbated forces 5% uncertainty

  6. Interval equations Example

  7. Interval FEM • http://en.wikipedia.org/wiki/Interval_finite_element

  8. Differences between intervals and uniformly distributed random variables Interval Random variable

  9. Differences between intervals and uniformly distributed random variables Intervals Random variables

  10. Differences between intervals and uniformly distributed random variables Example n=100

  11. Partial safety factors for materials • - design value • - characteristic value

  12. 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.

  13. Partial safety factors for loads • - design value • - characteristic value

  14. Limit state is a function of safety factors

  15. Calibration of safety factors - probability of failure

  16. Probability of failure • Probability of failure = • = (number of safe cases)/(number of all cases) structurefail often = structure is not safe

  17. Probabilistic vs nonprobabilistic methods • Probabilistic methods • How often the stucture fail? • Non-probabilistic methods (worst case design) • How big force the structure is able to survive?

  18. Probabilistic design

  19. Elishakoff I., 2000, Possible limitations of probabilistic methods in engineering. Applied Mechanics Reviews, Vol.53, No.2,pp.19-25 • Does God play dice?

  20. More complicated safety conditions

  21. Uncertain parameters Set valued random variable Upper and lower probability

  22. Design under uncertainty

  23. Uncertain parameters • Nested family of random sets

  24. Uncertain parameters • Fuzzy sets Extension principle

  25. Design with fuzzy parameters

  26. Fuzzy sets and probability • 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.

  27. Alpha cut method

  28. Interval risk curve for different uncertainty

  29. Interval risk curve for different uncertainty

  30. Interval risk curve for different uncertainty

  31. Bayesian inference (subjective probability) - is called the prior probability of H that was inferred before new evidence, E, became available - is called the conditional probability of seeing the evidence Eif 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. - marginal probability - is called the posterior probability of H given E.

  32. 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.

  33. 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.

  34. Interval and set valued parameters

  35. Interval parameters Design with the interval parameters

  36. Important papers • 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

  37. 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

  38. 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

  39. 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,ValidatedNumerics, Interval Methods. Kluwer Acad. Publishers, 2001, pp. 127-138.

  40. 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

  41. 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

  42. 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

  43. Important papers • Neumaier A., Clouds, fuzzy sets and probability intervals, Reliable Computing 10: 249–272, 2004 • http://andrzej.pownuk.com/IntervalEquations.htm

  44. Examples • 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

  45. Interval equations Example

  46. Algebraic solution because

  47. Algebraic solution because

  48. Algebraic solution

  49. United solution set because

  50. United solution set

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