# Andrzej Pownuk, The University of Texas at El Paso http://andrzej.pownuk.com

Post on 18-Dec-2015

213 views

Category:

## Documents

Embed Size (px)

TRANSCRIPT

• Slide 1
• Andrzej Pownuk, The University of Texas at El Paso http://andrzej.pownuk.com
• Slide 2
• Slide 3
• Modeling errors Approximation errors Rounding errors Uncertainty Human errors
• Slide 4
• Slide 5
• 5% uncertainty
• Slide 6
• Example
• Slide 7
• http://en.wikipedia.org/wiki/Interval_finite_element http://en.wikipedia.org/wiki/Interval_finite_element
• Slide 8
• Interval Random variable
• Slide 9
• Intervals Random variables
• Slide 10
• Example n=100
• Slide 11
• - design value - characteristic value
• Slide 12
• (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.
• Slide 13
• - design value - characteristic value
• Slide 14
• Slide 15
• - probability of failure
• Slide 16
• Probability of failure = = (number of safe cases)/(number of all cases) structure fail often = structure is not safe
• Slide 17
• Probabilistic methods How often the stucture fail? Non-probabilistic methods (worst case design) How big force the structure is able to survive?
• Slide 18
• Slide 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?
• Slide 20
• 1:09 20 /1 53
• Slide 21
• 21 /5 3 Set valued random variable Upper and lower probability
• Slide 22
• Slide 23
• 1:0923/153 Nested family of random sets
• Slide 24
• Fuzzy sets Extension principle
• Slide 25
• Slide 26
• http://andrzej.pownuk.com/fuzzy.htm 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.
• Slide 27
• Slide 28
• Slide 29
• Slide 30
• Slide 31
• - is called the prior probability of H that was inferred before new evidence, E, became availableprior probability - 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.conditional probabilitylikelihood function - marginal probability - is called the posterior probability of H given E.posterior probability
• Slide 32
• 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. Richard Threlkeld CoxprobabilityBayesian probabilitysubjective interpretation
• Slide 33
• First described by Zdzisaw 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.crisp setfuzzy sets
• Slide 34
• Slide 35
• Design with the interval parameters
• Slide 36
• 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
• Slide 37
• Kyloglu H.U., A.S. akmak, and S. R. K. Nielsen (1995). Interval Algebra to Deal with Pattern Loading of Structural Uncertainties, ASCE Journal of Engineering Mechanics, 11, 11491157
• Slide 38
• 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
• Slide 39
• 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.
• Slide 40
• 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
• Slide 41
• 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
• Slide 42
• 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
• Slide 43
• Neumaier A., Clouds, fuzzy sets and probability intervals, Reliable Computing 10: 249272, 2004 http://andrzej.pownuk.com/IntervalEquation s.htm http://andrzej.pownuk.com/IntervalEquation s.htm
• Slide 44
• http://webapp.math.utep.edu:8080/~andrzej /php/ansys2interval/ http://webapp.math.utep.edu:8080/~andrzej /php/ansys2interval/ http://webapp.math.utep.edu/Pages/Interval FEMExamples.htm http://webapp.math.utep.edu/Pages/Interval FEMExamples.htm http://calculus.math.utep.edu/IntervalODE- 1.0/default.aspx http://calculus.math.utep.edu/IntervalODE- 1.0/default.aspx http://calculus.math.utep.edu/AdaptiveTaylo rSeries-1.1/default.aspx http://calculus.math.utep.edu/AdaptiveTaylo rSeries-1.1/default.aspx http://andrzej.pownuk.com/silverlight/Vibrat ionsWithIntervalParameters/VibrationsWithInt ervalParameters.html http://andrzej.pownuk.com/silverlight/Vibrat ionsWithIntervalParameters/VibrationsWithInt ervalParameters.html
• Slide 45
• Example
• Slide 46
• because
• Slide 47
• Slide 48
• Slide 49
• Slide 50
• Slide 51
• United solution set Tolerable solution set Controllable solution set
• Slide 52
• Slide 53
• http://www.ippt.gov.pl/~kros/pccmm99/10PSS.html
• Slide 54
• Slide 55
• Slide 56
• What is the definition of the solution of differential equation?
• Slide 57
• Slide 58
• 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?
• Slide 59
• 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.
• Slide 60
• Slide 61
• Slide 62
• Slide 63