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Uncertainty quantification in mechanics
Modern composite structures have a wide spread in their failure stress. Advanced multiphysics codes can have a wide range of predicted behavior for nominally the same inputs. How do we certify the design of such structures or the accuracy of such codes?
The quantification of uncertainties in engineering design has garnered som interest in recent years. The most accurate method of quantifying the spread of outcomes of an experiment is the Monte Carlo approach. However, the cost of Monte Carlo simulations has caused most researchers to use some form of reliability analysis (no pun intended). Such approaches reduce the number of tests that are needed to quantify the behavior of a structure. However, some probability distribution function has to be assumed for the input parameters. An alternative is to solve a stochastic set of differential equations  often using the Stochastic finite element method.
A recent paper by Lucas, Owhadi, Ortiz takes a slightly different tack. They claim to provide tight upper bounds on the uncertainty through concentration of measure inequalities. Does anyone have a good idea of what these are and could they explain it to a lay audience? I'll also attempt to explain this idea as I learn more about it.
Links will be added as I learn about them.
 Biswajit
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This is a very good point! see an example on railways
There are many questions where this is entirely true!
FEM softwares, particularly commercial ones, promise a lot. Research is done on large scales, and engineers speak of "virtual prototyping" and "predictions". This is completely out of reality, particularly for fatigue!
See a recent paper I posted to to some forums in Railways  despite there have been large research projects, and big research centres have worked on railways corrugation for almost a century now, there is no understanding of the problem!
Rail corrugation, RCF, wear: a Tower of Babel?
The need for more collaboration, to prevent new disasters (perhaps in China) and win Global Challenges.
by M.Ciavarella
¹LMSEcole
Polytechnique, Palaiseaux, Paris. mciava@lms.polytechnique.fr
²Engineering Challenges, Cité
Universitaire, Paris, France. ciavarella@engineeringchallenges.eu
Railways: an established technology?
As recollected by Rod Smith recently (Smith, 2006), while some 20 or 30 years ago Railways were thought to be an old technology that had had its day, now their environmental advantages are now appreciated, and in many parts of the world, highspeed trains running on dedicated tracks are bringing high standards of speed, safety, comfort and punctuality to intercity ground transport. Europe is leading so far the field and for example, ALSTON proudly announces (IRJ Editorial March 2008 [1]) the world's newest and fastest highspeed train, was unveiled at Alstom's plant in La Rochelle, France, on February 5 by President Nicolas Sarkozy of France. Alstom's faith in its new train has already been rewarded with a contract for 25 trains from the world's first openaccess highspeed operator, reports David Briginshaw from France».
Fig.1
French President Sarkozy at the La Rochelle presentation, 5 Feb.2008
However, what is the real basic understanding of the railwheel contact, its induced damage, in the form of wear, RCF, corrugation, when only 8 years ago we had such a major disaster as the Hatfield accident, major not only for the 4 killed and the hundreds injured, but for the major shock it was for the industry. While on the technical side, the problem of gauge corner cracking (GCC) was in principle known and recommended checking procedures on tracks should perhaps have been sufficient to prevent the accident as declared in the court case by David Ventry, Former head of track at Railtrack [2], the following panic, which led to partial renationalization, suggests also that our knowledge is very limited on this problem. If knowledge were not limited, despite the resulting panic is much larger than justified if we look at the statistics of transport (Smith, 2003), some Engineer would have convinced the top managements involved, the politicians, and the public to take action immediately after the accident, and return rapidly to routine. Instead, already in May 2001 a record £534m loss was announced, and The Guardian in 2005 said « It is only now, nearly five years on, that punctuality has finally returned to its preHatfield level among passenger train companies «[3] saying «Hatfield changed the rail industry... forever ». Wow!
However, has technical knowledge changed forever, or just the companies involved, returning incidentally closer to the (national) structure they had before (what a funny backrevolution)? And is such a big mess compatible with a good knowledge of the technicalities of the rail fatigue, wear, and corrugation. We believe the answer to both questions is a firm: NO!
It is only because the knowledge of friction, Rail Contact Fatigue (RCF), wear, noise production, and all their interactions, is extremely weak, despite research so far.
Of course, Railways companies are reluctant to admit any lack of knowledge, big research programs have been established for decades, and huge research centers are at work, so it may sound a little destructive to say that our understand is extremely weak. Companies do make huge computational models today, with stateoftheart modeling of the contact (either elastic, or elastoplastic), taking into account of the complex dynamics of the vehicle (with sufficient
accurate description of the vehicle, the track, the support, etc.), although at this stage already some assumptions have to be made as the number of possible conditions starts to grow without limit.
Promising techniques are those using direct methods instead of full computation, as developed in Top Engineering Schools (DangVan, 2008, DangVan and Maitournam, 2003). But this is still not enough, as the prediction of the plastic response is not the prediction of fatigue life, and so far we haven't considered any fatigue and wear interaction, on interaction with more phenomena, usually studied by separate research departments, like « watertight
compartments » and this is not good. Only recently, we start to hear of « holistic approaches ».
But the truth is: we know very little about damage induced by railwheel contact. As recollected also in a 1998 lecture by Prof. Rod Smith of Imperial College, one of the leading of the authorities in the field, warned that rolling contact fatigue (RCF) is much less understood than fatigue in general, which incidentally has its origin when railways were born. While in fatigue in general, we have made considerable progress, mostly for constant amplitude loading acting on the same parts, under mainly tensile stresses, the railways contact is nothing of this sort. If different trains pass over different locations of the rail, and with different loads (actually each wheel will have different loading), it is almost impossible to predict the loads really occurring there, let alone «predict» fatigue damage, and even less likely, wear (perhaps in the presence of some lubricated contact) which in turn is related to fatigue since crack mouths are worn out so that wear may beneficial in the end. Vice versa, corrugations tend to make dynamic loads, and only recently it has been suggested that these dynamic loads should be used for the computation of fatigue life, and not the quasistatic ones as everybody does. So, really people within the companies themselves do not talk each other enough (a first reference to the Tower of Babel).
Moreover, Smith warns us that railway industry has no comparable extensive prototypical testing with respect to, for example, car manufacturing and aeronautical industries where ultimately fatigue testing in service is conducted on a large number of vehicles. So, like an Editorial of conservative The Engineer against the change from iron to steel in 1867 said «there is no steel rail now down in England or elsewhere which is not an experiment in itself. The progress of deterioration is slow and track may be serviceable for years, and then begin to fail without warning»: we suggest situation has not changed, nor is likely to change: strictly speaking there is no prototypical testing in this industry.
If this is the state of affairs, each time we change technology a little, either to increase the speed, or for example, to adapt a certain existing technology to existing tracks for example in China and India, then do we risk a Hatfield accident? My answer is: yes!
Hence, our provocative title. We clearly have the sensation of the Tower of Babel. In other words, there is no collaboration as a result of every company saying they know everything  as the Tower of Babel dedicated with a motive of making a 'celebrated name' for the
builders [4] [the Railways companies in this case], God seeing what the people were doing and sinning against him, confused
their languages and scattered the people throughout the earth!
Fig.2
 the Tower of Babel in the background of a depiction of the Hanging Gardens of Babylon by Martin Heemskerck. Is Railways research the modern attempt to build the Tower?
In recent years, we have reviewed many concepts which are believed to be the basis for the design against RCF. We have suggested also that the early investigation on the basic mechanisms of RCF, corrugation and wear were very naïf, and later proved of less use than what the early investigations would suggest. However, it was too late for the industry and the research centers to change approach, and so we have some inertia in finding a really good approach. For example, the concept of shakedown maps, and of ratcheting as the mechanism dominating in RCF fatigue, as suggested by Prof. Johnson in Cambridge often in collaboration with BR research, and Kapoor’s “critical ratchet strain” hypothesis, was reviewed critically in Ponter et al (2003), and Afferrante et al (2004), finding that the original Merwin and Johnson work was naïf in many respects, the experimental ratcheting rate being apparently similar to a very simple calculation for a double coincidence: first, the calculation using the simple perfectplasticity model was wrong by a large factor, and the yield strength in the material had been adapted a posteriori to find what seemed to be constant ratchet rate. Today, after 40 years, I cannot think of a simple calculation along these lines, and this is only for the basic problem. Forget about a realistic application.
Surprisingly, much simpler strategy would be to use design procedures similar to those used in gears design based on hardness (although of course the lubrication problem could be different) have, in the best of my knowledge, not been attempted by the Railways Industry. I would
not take a train if it were only based on these kind of calculations!
Rail corrugation.
Going to the point of rail corrugation, this phenomenon has been noticed at least for 100 years, but is still not fully understood (Grassie and Kalousek, 1993) particularly for shortpitch rail
corrugation ("roaring rails") in the range of 20÷80 mm wavelength. It has been the subject to extensive research programs both at single industries, in single Academia, and also of joint research efforts in the European Community, but (particularly short pitch one in the range 20÷80 mm) has been considered an enigma because measured corrugation wavelength did not relate well with wearinstability models for shortpitch rail corrugation ("roaring
rails") in the range of 20÷80 mm wavelength. Most available data seem to show a nonlinearly increasing wavelength with speed, i.e. an almost fixedwavelength and not fixedfrequency feature (see figure 3 adapted from figure 1 of Bhaskar et al. (1997), with quite a lot of data in BR old reports, David Harrison's Cambridge Ph.D. thesis and the Vancouver SkyTrain data – another case when change of technology in the lack of modeling capability led to very unexpected problems). Instead, most models generally obtain a resonance on the system, the most obvious being the vertical direction ones, the torsional ones of the wheel set, and the lateral ones. However, careful check of the phase between the possible corrugation ripple and the resulting dynamic amplification of load and of where needs to be done to suggest possible mechanisms.
Ideally, one would then compare between them to see most unstable under certain conditions (straight track, curve, etc.). It has been difficult to collect « clear evidence », and many tentative hypothesis have been advanced including very bizarre
ones, the majority of them contradict each other and are confusing.
Fig.3 – Some corrugation data in the « corrugation wavelengthtrain speed » plane from old BR report, from Cambridge group of Prof. Johnson in the late 1970s' and from the 1992 quite different Skytrain intercity track system show surprisingly
consistent results, not corresponding to any frequency in particular. Yet, while Prof. Johnson admit his attempts do not explain corrugation, many others have suggested mechanisms which correspond approximately to the 3 lines in the Figure!
Apart from the evidence collected from the beginning of 1900, up to the 1970's, it was the introduction of concrete sleepers and welded rails, when BR started to experience more serious corrugation problems at least on some lines [5], and accordingly seriously studied the problem (Frederick, 1986). Clearly, this is due to the change of the vertical receptance of the rail, and indeed, since then, use of resilient pads if often a
palliative solution (together with friction modifiers).
However, in the many attempts of modeling, BR's evidence is generally neglected, as it is probably disturbing to try to use it  except by Prof. Johnson from Cambridge who in fact remained always very negative in his conclusions in every paper, up to the late ones of 1997 (Bhaskar et al 1997). Others, like even his former student, Stuart Grassie, do not return to discuss these data and seem to change their mind various times never getting to a clear single model. In particular, Grassie & Johnson (1985) fail to justify corrugation, and Grassie & Kalousek (1993) admit it is not fully understood. However, suddenly Grassie (2005) says quite aggressively that Johnson's Cambridge group was following a « chimera » of a wavelengthfixing mechanics, but doesn't explain why, while he puts forward the work of Prof. Knothe (Hempelmann 1994, Hempelmann and Knothe, 1996) as to prove «definitive evidence» of shortpitch corrugation as being due to pinnedpinned resonance, which corresponds to a fixed frequency of about 1100 Hz (but see Fig.3?). Grassie & Edwards (2006) however returns to a model without pinnedpinned resonance!
However, Ciavarella and Barber (2008) recently suggested that most of the confusion around, and in particular the failure of the early models like Grassie and Johnson (1985) comes from erroneously assuming constant longitudinal creepage: when the inertia of the wheel and the rotational dynamics of the system are considered, a much better qualitative agreement with experimental evidence is immediately found. We decided to look at the simplest possible model, given we are not in a position to easily check results in the literature with
often sophisticated numerical simulations. To us, a lot more validation is needed for even the simplest of these numerical models, and our own efforts to produce clean and closed form solutions is also along the lines of extending the possibilities to check these models. In Ciavarella and Barber (2008), we assumed a simple full stick Winkler contact mechanics (but ongoing studies submitted for publication permit to say that not much is changed with other
approximate treatments considering full continuum solutions, 3D effects, and noncircular contact). To simplify the discussion even further from Ciavarella and Barber, we shall report the result of an extremely simplified Euler beam vertical receptance model (neglecting the effects of supports), which is valid in most cases in the frequency range greater than 500 Hz (and certainly with the vertical receptance from Bhaskar et al. (1997), i.e. a continuous support typical rail for Intercity Track  the paper refers to the Vancouver SkyTrain, but in part II seems to deal with BR track constants), of course under the assumption of neglecting pinnedpinned resonance due to discrete supports. This way, the vertical receptance high frequency tail has a single functional form, which depends on a single groups of parameters (mass per unit length, flexural rigidity of the rail), and therefore permits to present results of general validity in terms of frequency, naturally for this high frequency range.
This is extremely convenient, and perhaps corresponds also to the fact that the short pitch corrugation does not depend much on the particular system, as clearly seen in Fig.3, where, despite railways and intercity systems are included spanning very different technologies ( the old BR rails on wooden supports with unwelded rails, the newer continuously welded rails on concrete supports introduced in England by BR after 1966, the Vancouver Skytrain), the corrugation is seen to have more or less the same features. Hence, we produce results which are of interest for the frequency range f>450 Hz. In other cases, for particularly stiff pads for example, the frequency "crossover" may change, but these would be extreme cases which require separate investigations. By neglecting the effect of the pinnedpinned resonance at about 1 kHz, and also the difference between receptance midway between sleepers with that at a sleeper, we make the treatment extremely simple, a closed form equation, for the oscillatory part of the energy dissipation function (a complex function W1) in the contact area, in presence of an initial perturbation. In other words, if conditions are such that more energy is dissipated in the through of the contact, corrugation will continue, otherwise it will be suppressed. By plotting the minimum of the real part of W1 as a contour plot in the wavelengthspeed plane, only taking the negative part of the function, and dividing the contour lines from zero to the minimum in 10 contour levels, we obtain the results of fig.4.
We see that most data fall in the correct negative region of the function W1 indicating there is no enigma to explain this phenomenon. Also, for every given speed, we compute the local minimum of the function W1 to suggest the wavelength most likely to grow, and this is indicated by big dots. This defines (at least within the strong assumptions in the model) a frequency of corrugation nearly constant, not too far from the Winkler results of Ciavarella and Barber (2008), and close anyway to 500 Hz.
Fig.4 – Our model compares to the corrugation data from old BR report, from Cambridge group of Prof. Johnson in the late 1970s' and from the 1992 quite different Skytrain intercity track quite consistently. The shadowed area indicates expected poor predictions since the vertical receptance is simplified. Ellipticity of contact 2b = 3a; normal load P = 50 kN ; τ = 0.1. For more details, see
Afferrante et al (2008)
Fig.5 – Comparison between zero inertia (constant tangential force, full black circles) and infinite inertia (constant creepage, full black triangles), and a realist inertia (empty circles)
Other studies, omitted here for brevity, show that using the full receptance of the model, there are other minima of the dissipation function in the low frequency regime, and the tractive ratio τ=Q/μP may have an important rôle in deciding which regime is picked up as most fast growing. Also, the minimum of the real part of W1 decreases with the speed V almost linearly, but increases with the tractive ratio τ. Details are too complex to be shown here, see Afferrante et al (2008).
A comparison between constant creepage (large inertia) and constant tangential force (low inertia) is proposed in figure 5, to show that the assumptions of constant creepage, or constant tangential load, very common in the Literature, are dramatically wrong, and explains why the GrassieJohnson (1985) simplified model could not justify the observed corrugation, but also many other bizarre results in the models around. Notice not only the frequency is wrong, but also the modulus of the energy dissipation is erroneous, in particular, it is not suggesting corrugation for the constant creepage case (the Cambridge « enigma »), and it is over predicting corrugation in the constant tangential case. Recent examples assuming constant tangential load are Meehan et al. (2005), and Wu and Thompson, (2005).
Hence, we conclude that the original data collected by British Rail Research and Cambridge University, which are probably still the most reliable around, are very likely not explained at all in terms of pinnedpinned resonance, as it is clear from figure 3. Also, shortpitch corrugation appeared very strongly on Vancouver SkyTrain which has not a typical discrete support, and indeed did not change when the spacing of support changed. Indeed, we believe that the conclusions about pinnedpinned modes are affected by three unfortunate circumstances: first, the effect of parametric resonance attempt to introduce their effect, generally use local eigenvalue analysis, which means that they assume a steady state is reached like if the local receptance were valid for an infinite length. This exaggerates the amplification of normal load (or of lateral loads, or both) above sleepers easily by a factor between 2 and 5 (see Grassie et al 1982 one of the few papers which deals with this effect correctly). Also, models with continuum support have partly failed in quantitative prediction of wavelength only because of the erroneous tangential dynamics assumptions. Proper parametric resonance models appeared recently, but mostly they deal with noise rather than corrugation, and their effects are very difficult even to describe and to study.
Discussion
A general problem in RCF damage, wear and corrugation, was the unfortunate circumstance that there was never much discussion and comparison of models, perhaps for IPR reasons and fierce competition between railways. Also, many problems appeared "specific" and prone to various interpretations (the Tower of Babel punishment for having decided similar but slightly different systems in each country). To give an idea of the complexity of the problem which induces at time trivial errors, and on the other hand of the unfortunate lack of serious collaboration between industries and universities in the field, in the end of the 1990's, ERRI (European Railways Research Institute, which didn't survive very long) sponsored some research in order to make a comparison of predictive capabilities of the various models (Frederick' BR one, Berlin ones, Cambridge ones). This is hardly mentioned in the literature (perhaps for the prohibitive cost of these reports) but it is unfortunate, since they are extremely interesting and elucidating. In the report D185/RP1 (1993), various interesting facts are worth reporting:
b) There was a difference in the vertical receptance by a factor 10 or 20, which is certainly not a good start for any modeling!
In another such useful report (ERRID185WP2, 1997), not only more models are compared (Hempelmann's, Frederick in the extended 3D contact version of 1991, Bhaskar's model from Cambridge, a large creepage model from Clark at BR, and the Tassilly model by RATP and Vibratec), but an attempt is made to compare to experimental measurements on test sites. The 1993 findings were repeated; probably the 1500Hz regime difference was associated to difference in calculating wear in the contact area. The experimental validation was extremely confusing and even giving to the models the roughness input at the starting conditions, there was only vague qualitative agreement. But the most striking finding was that in the test site there was development of the 20mm roughness that everyone had considered to be "suppressed by contact filter"!
The risks of new Hatfields in China.
To make a long story short: we don’t know much about RCF, wear, corrugation and their interactions. If we go to China and use fast speed trains, even at slower speeds than what designed initially, but using the existing rail track, we don’t know nor we can predict the
outcome. More Hatfield will occur? Very likely, yes!
Way forward.
While we can make progress by more collaboration, the only solution is constant careful control of the lines, and trial and error in new design. Many privatizations in this business have been failures. In England, BR, before disappearing, had already given up the attempt to «understand» corrugation, for example, and moved later to «monitoring» and «corrective grinding». Most recently, after the Hatfield accident the companies responsible for the track are talking of «preventative grinding». The Dutch seem to have been between the first to use the idea of "preventative grinding" on rails within the first 6 months after their pose because "for reasons unknown" rail corrugation started only rarely on such grinded rails before they had seen significant traffic (Van der Bosch, 2002). Similar forms of "preventive grinding" or "regular grinding" is
also the routine solution for many railways (French never have admitted much corrugation problems probably because of a very serious such maintenance program), and while steels are harder and harder and wear in principle is reduced, regular grinding is nowadays conducted also as a means to reduce cracks anyway (Grassie and Baker, 2000) but this becomes then a matter hard to judge scientifically also because of lack of data and the presence obvious commercial, managerial and
political interests. This strategy is suggested more successful also by Zarembski (1997) and even convenient in Holland (van der Bosch, 2002), but it is not entirely clear why. Also, notice that grinding there is aimed not just at removing corrugation, but mostly avoid gauge corner cracking and in general contact fatigue, together with enhanced lubrication in certain areas (Middleton 2002). Speculations about future «proactive grinding» are not entirely well specified.
Holistic Approach?
A few people start to notice (Kuijpers et al 2007) that the «holistic approach» should be considered with a carefullytailored maintenance program that combines acoustic and preventative grinding. However, without a good model, how to do that? The problems of corrugation and RCF are related, although in not too clear form. First of all, wear and RCF are related in general, since removal of material also means removal of cracks. With harder and harder steels, wear becomes less crucial than RCF  In 1999 the European Rail Research Institute estimated the total cost of RCF to European railways at €300m a year. But it may be larger today. Too many and too independent efforts to generate models, many false beliefs, the difficulty to collect experimental evidence, and the lack of sufficient comparisons between models between different European and nonEuropean companies and Research Centers. A very useful exercise was done only by the European Railways Research Institute, which was perhaps useful, and hence was shut in 2004. ERRI did find some extremely surprising results, and obvious problems in comparing different models, and the latter with experiments in test sites. By working each with his own means, there has been no attempt of « collective knowledge » as it the Wikipedia times, this would be called. Hence, more than an enigma, corrugation was the result of a « Tower of Babel » approach.
Andreotti or Moro? For his final slide of the 19th Jenkin Lecture, Smith (2006) showed us a cautionary quotation from Aldo Moro, late Prime Minister of Italy, that: "There are two types of madmen: those who believe they are Napoleon and those who think they can sort out the railways."
Very true, but there is a misquote here, Rod, it was Andreotti not Moro!
References
L.Afferrante,
M. Ciavarella, G. Demelio, A reexamination of rolling contact
fatigue experiments by Clayton and Su with suggestions for surface
durability calculations, Wear 256 (2004) 329–334
L.Afferrante,
M.Ciavarella, Rail corrugation: “one hundred years of solitude”?
or a true enigma? Submitted 2008.
A.
Bhaskar, K. L. Johnson, G. D. Wood and J. Woodhouse, Wheelrail
dynamics with closely conformal contact Part 1: dynamic modelling and
stability analysis IMechE Proc Instn Mech Engrs Vol. 211 Part F
(1997), pp. 1126.
M.
Ciavarella and J.R.Barber, Influence of longitudinal creepage and
wheel inertia onshortpitch corrugation: a resonancefree mechanism
to explain the roaring rail phenomenon Proc. IMechE, Part J: J.
Engineering Tribology Vol. 222 (2008), pp. 111
Ky
Dang Van, Modelling of damage induced by contacts between solids,
Comptes Rendus Mécanique, 336 12, 2008, Pages 91101
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Dang Van and H. Maitournam, Steadystate flow in classical
elastoplasticity: application to repeated rolling and sliding
contact, J. Mech. Phys. Solids 41 (11) (1993), pp. 1691–1710.
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D185/RP1. Reduction of rail corrugation. Rail corrugation models 
comparison of results obtained using the Berlin Technical university
and British rail methods, 1993.
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O. Frederick, A rail corrugation theory. Proceedings of the 2nd
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Grassie and P Baker, Routine maintenance extends rail life and offers
longterm savings, Railway Gazette International, Feb 2000, pp.
8890.
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Grassie and K.L. Johnson, Periodic microslip between a rolling wheel
and a corrugated rail, Wear, Vol. 101 (1985), pp. 291305.
S.L.
Grassie and J. Kalousek, Rail corrugation: characteristics, causes
and treatments, J Rail Rapid Transit, Proc. Inst. Mech. Eng., Vol.
207F (1993), pp. 5768.
S.L.
Grassie R.W. Gregory D. Harrison and K.L. Johnson, The dynamic
response of railways track to high frequency vertical excitation, Int
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Grassie, Rail corrugation: advances in measurement, understanding and
treatment, Wear, Vol. 258 (78) (2005), pp. 12241234.
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Grassie and J.W. Edwards, Development of corrugation as a result of
varying normal load, 7th Int Conf on Contact Mechanics and Wear of
Rail/Wheel Systems (CM2006), Brisbane, Australia, September 2426,
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Hempelmann and K. Knothe, An Extended Linear Model for the Prediction
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Hempelmann, Short Pitch Corrugation on Railway Rails  A Linear
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Kuijpers, JC Schaffner M Bekooy, Grinding programme combines noise
reduction and preventative maintenance Railways Gazette 01 Jun 2007
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Meehan, W.J.T. Daniel and T. Campey,. Prediction of the growth of
weartype rail corrugation, Wear, Vol. 258 (78) (2005), pp.
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Middleton Railtrack invests in grinding capability Railways Gazette
01 Jan 2002
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open hearth steele, Technical Report IMDA257, BR Research, 1976.
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Ponter, L. Afferrante, M. Ciavarella, A note on Merwin’s
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321–328
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Smith, The 19th Jenkin Lecture, 23 September 2006: Railways: The
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A. Smith (2003) The wheelrail interface—some recent accidents,
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(10) , 901–907 doi:10.1046/j.14602695.2003.00701.x
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Van der Bosch, Les strategies de meulage des chemin de fer
neerlandais, Revue Generale des Chemins de Fer, nov., (2002), pp.
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Wu. and D.J. Thompson, An investigation into rail corrugation due to
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Zarembski, Intelligent use of rail grinding, Railways Gazette 01 Feb
1997
1
http://www.railjournal.com/altfeature1.shtml
2
Hatfield crash 'was preventable',
http://news.bbc.co.uk/2/hi/uk_news/4325731.stm
3http://www.guardian.co.uk/uk/2005/sep/06/hatfield.transport
4
Genesis 11:4
5
Pearce (1976) noticed only corrugation in the West Main line up to
the border of Scotland where Acid Bessemer steel left place to Open
Heart Steele, but not on the East main line, an enigma perhaps due
to different gauge.
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and validity of linear and nonlinear finite element models. The
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Upon completion of this course, attendees will be able to:
Describe the model validation paradigm of inputoutput effect
analysis, test/analysis correlation, and uncertainty analysis;
Summarize the history of finite element model test/analysis
correlation and updating;
Conduct code verification, solution verification, and solution
selfconvergence studies;
Describe the process for selecting and computing appropriate
response features from the model outputs;
Discuss current techniques for global sensitivity analysis and
parameter screening;
Explain the role of designs of experiments (DOE) and analysis of
variance (ANOVA) in model validation;
Define appropriate test/analysis correlation metrics for model
calibration and revision.
Upon completion of this course, attendees will have learned:
The latest techniques for evaluating the accuracy of computational
models over a range of parameter values;
How to design validation experiments that will help determine the
simulation range of validity;
How to calibrate model parameters to reflect the measured response
from experiments — even for nonlinear models!
Course Outline
Introduction
Historical overview
Feature extraction
Code verification and solution convergence
Uncertainty propagation and quantification
Local and global sensitivity analysis
Metamodeling, surrogate modeling
Design of validation experiment
Testanalysis correlation and validation metrics
Model revision and updating
Applications for linear and nonlinear structural dynamics
Course Instructor
Dr. François Hemez has been Technical Staff Member at the
Los Alamos National Laboratory (LANL) since 1997 where he is
contributing to the development of technology for solution
verification, model validation, uncertainty quantification, and
decisionmaking for engineering and weapon physics applications.
Before joining Los Alamos Dr. Hemez was a research associate of the
French National Center for Scientific Research, working in the area
of test analysis correlation and finite element model updating. He
has developed and taught undergraduate and graduate courses in
applied mathematics and structural dynamics. Dr. Hemez has taught
the firstever graduate course offered in a U.S. University on
uncertainty quantification and Verification and Validation
(V&V) at the University of California San Diego (Spring 2006).
Dr. Hemez is serving as chair of the Society for Experimental
Mechanics technical division on model validation and uncertainty
quantification (20052009). In 2005 Dr. Hemez received the Junior
Research Award of the European Association of Structural Dynamics;
in 2006 he received two U.S. Department of Energy Defense Programs
Awards of Excellence for applying V&V to programmatic work at
LANL. Dr. Hemez has authored over 200 peerreviewed publications,
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Sample course notes at www.ladynamics.com
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Course Outline (Somewhat More Detailed)
1. Introduction
Description of the model validation process
Definition of key terminology
Sources of modeling errors, examples of validation studies
2. Historical Overview
Discipline specific applications (aerospace, civil, automotive)
Linear test/analysis correlation methods
Examples in linear and nonlinear structural dynamics
3. Code and Calculation Verification
Code and solution verification
Asymptotic convergence, estimators of convergence
Extrapolation, solution error Ansatz, and selfconvergence
4. Extraction of Response Features
Desirable properties of response features
Classification and selection of features
Calculation of features, examples in linear and nonlinear dynamics
5. Overview of Probability and Statistics
Descriptive statistics and their properties
Statistical sampling, design of experiments (DOE)
Statistical testing, confidence intervals
6. Local and Global Sensitivity Analysis
Local sensitivity analysis, derivatives and finite differences
Global sensitivity analysis
Analysis of variance (ANOVA) and effect screening
7. Metamodeling
Overview of surrogate modeling, metamodel forms
Model regression and error estimation
Calibration, optimization, and sensitivity studies using
metamodels
8. Quantification of Uncertainty
Classification of measurement, simulation uncertainty
Forward propagation of uncertainty, sampling, DOE
Inverse propagation of uncertainty, optimization under uncertainty
9. Design of Validation Experiments
What makes validation experiments different from other tests?
Experiment design, instrumentation selection and placement
Replication, effect blocking, variable screening, experiment
uncertainty
10. Test/Analysis Correlation
Comparisons between predictions and measurements
Definitions of fidelity metrics, correlation metrics
How good is good enough?
11. Model Updating and Revision
Revising conceptual model forms and assumptions
Model parameter calibration, finite element model updating
Independent parameter assessment
12. Examples of Model Validation Studies
Explosively driven impulse on a threaded interface
Modal response, delamination damage of composite plates
Simulation of polymer foam impacts, shell buckling, Taylor anvil
impacts
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Date: October 27, 2008
Re: V&V NAFEMS course
Hi Mike,
Thanks for the pointer to the course.
The cost of the course is prohibitive. Also, I have decided not to travel to the US or Europe until the visa procedures become less insulting to my dignity. So any conferences/courses in those places are irrelevant to me. It will be more useful to me if you can point me to any courses in India.
 Biswajit
Well in the future there will be "virtual conferences"  anyway
You are quite rigth. One cannot go to all conferences! In the energy sector, I think there was recently a conference entirely on the web recently.
However, I pointed you to that NAFEMS organization, there are documents there, you can get in touch with people there, maybe you find something useful.
Keep searching.
I have noticed this post oscillates between highly abstract mathematics and more engineeringoriented one. It is difficult to conciliate.
Re: Highly abstract mathematics vs. more engineeringoriented
Hi Mike,
The problem of higly abstract math. vs engineering oriented approaches is one that I face on a regular basis. It takes a lot of effort to learn the math. But once I've learnt the math well enough, I realize that the papers that use the math could well have done without much of the general formalism. Often, I find that the results presented in mathematically couched papers may be trivial or irrelevant.
So I've decide to take the easy way out and ask experts on iMechanica to clear out the forest of ideas from measure theory and show me whether or not the paper in question presents a realistic aproach or not. I have my doubts after spending a lot of time on such buzzwords as spectral stochastic finite elements and polynomial chaos expansions  which are all very nice but essentially useless for practical problems.
On the other hand, give the fact that these problems are hard, the easiest way forward seems to be to learn the math and then use it properly.
 Biswajit
Probabilistic Approach in Mechanics
Hi Dr. Biswajit
As my research is related to uncertainity qualification, I can share some experiences that I had.
Uncertainity qualification is one of the most important issues in
structural mechanics now a days. The material property, Loading
condition and material response is not deterministic in reality.
Theoritically we can define a deterministic form of the governing
equation but we can convert this deterministic form into random form by
adding a white noise to the uncertain parameter such as modulus of
elasticity, load or displacements. For uncertainity assesment, there
are several approaches are followed now a days. If we take the
EulerBernoulli beam equation as an example; we can define the load or
modulus as a vector of Random variables or more generally Stochastic
Process. So, at any probability state the modulus or load changes with
respect to a independent random variable such as time. Once the beam
equation contains a Stochastic process, usually it can be called as
Stochastic Differential Equation (SDE). But in practice we use white
noise or Brownian Process as the stochsatic process. Suppose the
modulus is random, so we can write: E (deterministic or Predictable
Process) + White Noise (or Brownian Motion as Stochastic Process). It
is well known that the white noise is nowhere differentiable. We can
not use usual solution technique for PDE or ODE in general (The exception is there such as my graduate research where I used the formal solution technique for PDEs).The
most well known method is Ito's formula or Ito's Process. This is a
very wide topic. I can give a simple concept here. The differential
equation is converted into differential form rather than keeping in
derivative form. Hence Ito's form is deduced and we can use Ito's
method to solve the SDE of the beam. The solution is analytical. But
once we go for simulating the SDE, usually we have to go for
approximation method. The finite difference form of the SDE is quite
reknown in literature. However, as far as large structure is
concerned, these SDEs are used in Stochastic Finite Element Method
(SFEM). There we can use perturbation technique as well as other method
also used. The main idea behind SFEA is to find/estimate Cummulative
Distribution function of failure. Hence we can find reliability of the
structure. The stiffnes or load and strength are defined to be
modulated with white or Gaussian noise. Another straight forward
apporach in SFEA is Monte carlo Simulation. As we know the probability
distribution is required for Monte Carlo Simulation, it is kind of a
drawback, sometimes. Also the computation time is a big factor too. In
my research, what I realized that the distributaion function is not
well defined in most of the practicle applications. For these
situations, we can do several things to overcome the problems such as:

Approximating the physical parameters such as load or modulus by a
Random Walk model based on some criteria of Random Walk models. (High
computational cost)
 Estimating the probability
distribution based on several methods such as EM, Maximum likelihood
algorithm, Bayesian Inference, Hypothesis Testing, Nonparametric
methods etc. Some practical data such as histogram of the events or
random variables is required.
 Using
Nonparametric Monte Carlo approach such as Markov Chain Monte Carlo
method (see the paper by Geyer: "Practical Markov Chain Monte Carlo").
Besides
these there are other techiques are coming up in order to reduce
computational cost. Another important issue iis generating random
numbers, more generally generating psudorandom numbers. This issue
truely affects the computaion cost and accuracy of the Monte Carlo
simulation.
As far as validity of the outputs
from uncertainity qualification is concerned, you can not ensure 100%
of the validation. Because some cases may not arise in experiments or
testings. Usually reliability assessment is followed up, as the final
step in uncertainity qualification. Some hypothesis testing can be done
based on the experiments. The goal is to predict varience of the
outputs such as deflection of the beam at different locations as well
as covariance of the several outputs due to several probability states.
The uncertainity qualification is becoming a
very important too in Structural engineering. As I have noticed that
the Non Destructive Evaluation helps you to aquire some current status
of the structure or Part of the structure. We may be concerned with
damages, outside conditions, vibrations, microstructure change etc.
There is no way out to specify a global function about these
parameters. If 100 aircraft wings are manufactured, no one except GOD
can tell us when these wings are going to be failed. Hence the demand
for uncertainity qualification is increasing day by day. For further
reading I can recomend texts:
Probabilistic Methods in Structural Engineering: Giuliano Augusti, Alessandro Baratta, Fabio Casciati.
Stochastic Differential Equations: An Introduction with Applications by Bernt Karsten Øksendal (One of the best books on SDE I have ever seen)
Thanks
Rezwan
Re: Uncertainty quantification
Dear Rezwan,
I've attached the paper in question in my oiginal post. I am looking forward to your take on the proposed approach.
 Biswajit
Concentration of Measure
Hi Dr. Biswajit
I have looked at the paper. At the first phase
I had some observations. I will present the second phase (more detail)
later on. Firstly, the term concentration of measure is a very
important from the perspective of functional analysis as well as probability theory. The idea is to sense the enrichment of any mathematical entity (dimension or variables) in a measure space such as Metric space, Hausdorff space and so on. Typically if we think to roll a dice once, the probability of getting side with "six" is 1/6. If we roll it twice this probability increases because the sample space is increased. Hence, if we roll the dice (say for an example) 10,000 times; definitely this probability of getting a "six" is increased quite highly. It is clear that the sample space gets big and the event "Getting the side with six" becomes almost certain when we roll the dice almost infinitely. Mathematically, according to measure theory the the measure of the probability in the probability space (the triplet, (Ω,F,P)) gets entrapped in a region/interval according to the concept of concentration of measure. On the other way we can say the concentration of the "measurable sets" get higher leads to very small deviation rather giving a densed interval or may be a point. Actually, in real life problems related to uncertainity, we look for the higher concentration of measurable sets in the probability (or measure) space. Now the question comes, should we do millions of experiments to converge to a almost certain output? Practically the answer is negetive. For that reason this paper was publised, I am working on the damage model and other researchers on uncertainity qualification too.
The authors focused on the parameter CF (Confedence factor) is the ratio of difference (may be eucledian distance or metric) "M" between mean,threshold of failure and the Uncertainity "U"; besed on concentration of measure. For several cases they calculated this term. They showed the demand of trials or experiments is expensive to extract a probability of failure, whereas it is welly managed by using concentration of measure inequalities.
The idea behind this paper is to use the consept of "concentration of measure" so that the model gets closer to less uncertainity as the number of trials (measure) is increased. But side by side a large number of trials is tedious and costly. The smaller failure tolerance "ε" leads to higher trials. The concentration of measure inequalities are giving an upper bound of the failure probability instead of doing a huge number of experiments.
They have defined, the probability measure of safe events "A" as:
P(Y to be safe) is not exactly "1" in real due to the fact that the probability space is not compact. Hence the failure probability P(Y fails) is defined under an upper bound or tolerance (ε) (eq 1,3 in the paper). So the concentration of measure inequalities actually gives the interval estimation in terms of inequality of the failure probability because the failure probability is supposed to be unknown in prior. The concentration of measure is nothing but a probability density itself in the form of inequality.And the final step looks for validation or activeness of the model. They showed this application of concentration of measure in some exact as well as nonexact scenarios. in some case the mean or mathematical expected value is known, in some cases it is unknown. They showed that the model can work for all of the cases. However, I will give more explaination after the weekend.
To go through the paper I would suggest to have some introductory reading on:
 Advanced Set theory (specially concept of supremum "sup" and infimum "inf")
 Measure theory (What is measure of a set? What happens when a set is not compact or
What is noncompact measure?)
 Some idea on some famous inequalities used in functional analysis such as:
Chernoff inequality,Hoeffding inequality,McDiarmid's inequality, etc.
I will send you the soft copies of some articles on concentration of measure inequality.
Thanks
Rezwan
Re: Concentration of Measure
1) I've attached a pdf version of a book (by Dubhasi and Panconesi) to the original post. The book discusses concentration of measure inequalities in an accessible manner.
2) Regarding the paper in question, the problem that they have considered is relatively simple. Very few uncertainty quantification exercises on realistic problems can be found in the literature  most papers deal with toy problems. Also, much of the recent literature suggests that we have to use thousands of processors to compute uncertainties accurately. Very few researchers have access to such huge computing resources.
I don't see the paper addressing a realistic problem or providing a realistic solution at this stage.
3) Also I wouldn't call the Chernoff inequality, Hoeffding inequality, or McDiarmid's inequality well known inequalities in functional analysis in general (though they are well known to probability theorists). For our readers, a starting point towards understanding these inequalities is Markov's inequality. See Eric Weisstein's Mathworld for a simple proof .
Re: Concentration of Measure
a) I can give another link: http://www.lsp.upstlse.fr/Proba_Winter_School/Lugosi.pdf.
b) Once you deal with probability theory in terms of measure theory, you can bring the concept of functional analysis. I agree that the Chernoff inequality, Hoeffding inequality, or McDiarmid's inequality etc are well known in probability theory, but if you want to deal with the inequalities, you can use some theories of functional analysis to deal with the probability because measure of a set is one kind of operator. Mathematically, probability in a sample space can be referded to probability space. Hence there are relations which can be easily seen. However, concentration of measure has a nice relation with Banach space. Theory of probability is one kind of application of functional analysis; in general.
c) Uncertainity qualification is not a straight forward approach. The most significant issue is defining the random variable and then defining the random process.
d) For further reading I can suggest:
Introduction to Probability Models: Sheldon M. Ross
Re: Concentration of Measure
Rezwan,
I've added Lugosi's PDF book to the main post.
At this stage I'm, like Mike, losing the thread because the discussion has become too general. Maybe I should make my questions a bit more specific.
In your comment (c) above you say that " The most significant issue is defining the random variable and then defining the random process".
Why is defining the random variable an issue? Don't we already know the variables which have known variability? The unknown uknowns can be ignored since we can't say anything useful about them. The problem that I see is that the epistemic (known) uncertainty cannot be quantified well enough. We cannot assign a probability distribution function based on the threefive tests that experimentalists perform for a particular configuration. And doing more tests is expensive and hence never done in real life. What should we do given these constraints?
Also, I don't understand what you mean by the random process in this context. Are you suggesting that the physics cannot be described by differential equations (with coefficients which are random variables)? Could you clarify?
 Biswajit
Re: Concentration of Measure
Hi Dr. Biswajit
Firstly, thanks for uploading the paper. I know abstract mathematics is not very clearly visible. I am kind of loosing the interest in this discussion too. Beacuse, the
question that was asked requires some specific approach and the comments regarding that are little offensive. Anyway, let us clarlify the questions that you have asked:
a) As I have mentioned, the significant issue is to define random variables has some "Practical Engineering", background. If we know a function has some certain number of variables and how they are related, life will become quite easier. But realistic modeling is not that straight forward. However, this issue is more significant when we deal with random variables. I will certainly not go for the abstract mathematical explaination. When we define a problem, firstly we define the variables. Similar thing happens in probabilistic modeling. The issues those arise are:
 classify the variables: which are known, which are unknown. you can not simply discard the unknown variables !!!! It contributes to the output. Once you discard such variables, you will get some garbage results and the blame will go to abstract mathematical techniques!!! You have to classify the variables which are significant, which are not. Mathematics is a tool but the success depends on how you utilize this tool.
 Now the unknown variables are to be predicted. If you really do not know which one is the unknown, you need to do the output analysis from some experiments. The paper on "concentration of measure" gives a concept about making a trade off between your experiments and objective. The pattern or output will give some estimation that how many variables that system can contain. If you suspect a parmater to be a variable but you do not have any information about it, you need to go for estimating prior information from different probabilistic inference techniques (such as Bayesian..). As we are dealing with a physical problem definitely the relevent parameters are to be suspected as random variables. If we are dealing with a fracture in polymer composite, definitely, the physical parameters such as crack growth, velocity, energy release rate, temperature etc are to be accounted. We can not bring "turbulance" by any chance !!!!
 Then you have to classify the independent and dependentrandom variables. If you assume that you have two variables which are not totally independent and you assume the probability distribution (not the conditional probability) is normal keeping in your mind that they are independent, I think the output will give some unrealistic results. So, it is also an important issue.
 If you classify the independent or dependent variables; now they relation among them is to be determined. And so on.
So, it is not very easy that we think. However, besides defining the random variables, we need to define the behavir of these random variables. Suppose, if we design an aircraft wing and keep the loads as a well defined sinusoidal functions; I think this concept is rather unrealistic than defining the loads as random functions. Moreover, if the random variables follow a nongaussian distribution, we should not deal the stochastic process containing those random variables with gaussian distribution. There are several things to be considered to make your solution "Realistic".
Random processes itself gets involved in a differential equation describing a problem. That why it becomes stochastic or random differential equation. I discussed this in earlier post by giving an simplest example of EulerBernoulli Beam. But, the random process gets involved in the ODE or PDE shuld be well understood. Random process can be anything. If my random process is not Brownian motion, I can not include it in my governing ODE or PDE.
However, the question that you posted in this discussion demands abstract mathematical explainations. Because, (as I mentioned earlier) "concentration of measure" is not a well visible term, it originates from the concept of measure theory and functional analysis. So in order to keep the relevency of your question I followed the abstract mathematical explaination. Without this type of explaination I think you can not deal with it. It is not like, I draw a picture and that reveals the whole idea. You can not say any technique in the world as a perfect one. We have limitations. Classical mechanics can not work for electrons. But we can not call it useless. People realized this and quantum mechanics came out.
The paper on "Concentration of measure" that you posted, the discussion started based on that paper. I am not reviewing the paper, so I should not make any comments about its realistic or nonrelistic outcomes. As you mentioned earlier, most of the papers with abstract mathematics are nonrealistic, I expect the example or explaination that leads you give such comments.
Re: Concentration of Measure
Hi Rezwan,
I think you've misunderstood what I meant to say in my previous comment. When I say "losing the thread" I mean that I am not being able to follow the thread your argument.
Let me give you an example. You have said in one of your comments that "However, concentration of
measure has a nice relation with Banach space. Theory of probability
is one kind of application of functional analysis; in general."
That comment is a general statement of things that most of us in iMechanica probably know. However, the implication of the first statement is not clear. What is the relation to Banach space and why is that interesting to a non mathematician? That general statement does not help to clarify in my mind what the shortcomings of the concentration of measure approach are.
I seek a physical and clear understanding of the underlying ideas. Even though I have a first year math graduate level understanding of functional analysis and measure theory  I don't understand them well enough to readily connect new aspects of to physical problems.
Also, I think you misunderstood what I mean by "unknown unknowns". Because of the fact that they are unknown, you can't identify what these unknown variables are!
What you have talked about in your comment is about "known unknowns" (in Rumsfeld's terms) rather than "uknown uknowns". Bayesian priors can be used to take care of "unknown unknowns" but it's not clear at all how such priors should be calculated even for relatively simple engineering problems. If I have not understood you correctly, please explain again.
You also say " I am not reviewing the paper, so I should not make any comments about its realistic or nonrelistic outcomes." But comments about those outcomes are precisely what I'm looking for! In my current position I have to rapidly evaluate papers for their usefulness or otherwise to my work. My personal evaluation has been that the concentration of measure work is essentially useless for practical purposes (as of now). I would like to make sure that I have understood the material correctly and am not making a wrong judgement call.
Also, thanks for taking the time to discuss the matter and providing pointers.
 Biswajit
Re: Concentration of measure
Hi Dr Biswajit
Thanks for the reply. I am sorry if my comments sounded rude. It is nice to share and discuss about some topic. That is why the blog imechanica for. We can share knowledge and experiences. Definitely it is my pleasure to discuss about a topic. However, lets discuss a more about the topic. Actually the relation of concentration of mesure with Banach space is given nicely in the book: The Concentration of Measure Phenomenon: Michel Ledoux
In the first chapter, the term "Concentration function" is used. The relation with Banach space is discussed in the third chapter. But I did not talk about this book because it is totally abstract. That is why I just left a comment. The relation comes between these two terms by eigenvalues.
I understood the question regarding unknown unknown. Actually in terms of mathematics you can call them "blind terms". As I told you, if you deal with a physical problem there should be some parameters that you can define from the physical phenomena. But, the blind terms are determind in a different way. I will not say that these techniques are sufficient. Because in the researh on "experimental data analysis" people are still working.In order to find out blind terms, firsly we need to see an important thing: I have some known parameters, how they affect the output function? Say we know some "n" parameters from the physical phenomena. But these are may not be enough. Some extra parameters are there which we do not know. For these cases you can use several (usual and well known) methods such as:
ANOVA, ANCOVA, mulifactor experimental design, Bayesian estimation, Bayesian Network, Principle component analysis, Clustering, Decision tree, Hidden Markov model etc.
Unfortunately, these methods sometimes do not work for some specific data. Then some other techniques in data classification literature are used. (such as Independent component analysis or blind decomposition, and lots more).
Using ANOVA, ANCOVA,MANOVA or experimental design you may find answers of your questions:
How many parameters are affecting the output?, Which of them are highly correlated?, and What is the pattern of correlation? etc. From this analysis you may have an estimation that "n" parameters are known but some extra parameters can be there. (Just for an example: may be you know "n" parameters,but among them "m" parameters are providing equvalent bahavior, hence you can replace these "m" parameters by one parameter, so you may have nm+1 parameters indeed , so you may need actually some more parameters in the experiments). Way may not know what parameter it is but we may suspect that some extra parameters are weakly active. This is just an example. In some cases the blind parameters can be easily discarded. Geometrically, the region that we are dealing is a circular disk (number of dimensions:2) cut from a soild sphere (number of dimensions:3).
I would suggest you to go through a paper "Comparative study of deconvolution algorithms with applications in
nondestructive testing" Nandi, A.K.; Mampel, D.; Roscher, B. it is kind of old work but some idea can be generated. Now these techniques are quite popular in digital image processing, pattern recognition etc. They detect actual signals from blurred signal using these techniques. Also please have a look on the website: http://www.statsoft.com/textbook/stexdes.html. You may consult a book also: Design of Experiments for Engineers and Scientists: Jiju Antony. Hidden markov model is also very active one in predicting unknown unknowns (Answering kind of : Is there any unknown paramter here?). You may also see the book: Inference in Hidden Markov Models: Olivier Cappe, Eric Moulines, Tobias Ryden.
However, these methods need data which comes from experiments. To get meaningful output higher number of experiment is prefered when the parameters are random. The use of concentration of measure can be a way to get the successful experimental dataset without doing a large number of experiment. Sometimes you may know the mean, sometimes you may not know the mean of the output. They showed the applicability of the theory of "concentration of measure" in these cases.
I agree that the paper on concentration of measure did not give very well visible and practical solution of a real life problem. But I would appreciate that they have used the idea for using the concept of concentration of measure. According to my understanding and knowlege on this term, it has a great potential of clustering the probability. Ususlally, all the above methods that I mentioned, they work on classiying or grouping the parameters but concentration of measure can classify the probability of a a parameter arround a physical value. I would say the better analogy (weak analogy though) is with: estimating range for mean or standard deviation done in statistical analysis.
Let me know about further questions or comments.
Rezwan
ok, but be warned that statistics is often a false solution!
I wish this of stochastic ODE were really a solution!
Often you simply are requested even MORE information on the input and your model, than if you had a deterministic model. Now, you don't have only to know your input value, but its distribution.... You can easily see how this produces nice mathematics, but not a response to a real problem in mechanics, i.e. when you DON'T have a model....
In fatigue, you simply do NOT know this "equation" for the behaviour of material, and sometimes it is even already difficult to predict the elastoplastic response without many FITTING parameters.
So I don't want to sound destructive, but this is even more smoke in the eyes, in most cases. "Smoke and mirrors" as the English say.... This is to clarify the issue. If you really have THE model, then statistics ODE is interesting.
Good question
Hi Dr. Mike
It is nice that you have figured out a great question that I often face. To analyze this issue, firstly, a question may come why we look for nondeterministic approach? If deterministic model cay say everthing or if a physical phenomena can be modeled by a well defined deterministic equation, we do not need the uncertainity assessment. For an example, classical mechanics, theory of elasticity and plasticity, fracture mechanics etc are built up on deterministic concept. Suppose; a structure is made by aluminum. Now the question comes can we gurantee that the aluminum based structure dose not have any microdefects or damages? If not can we say; if we make 10 similar(100%) structures by same material, there is always perfectly a fixed number of microdefects or damages. If we need to find the answer (Removing the smoke) we may go for experiments. Now a question arises, how many samples are perfectly enough to give answer of the above question? And if we take a certain number of samples; say N samples, can we say that (N+1)th sample will have same number of microdefect or no microdefect at all? It is quite relevent that a single macrocrack can be dealt by conventioanal fracture mechanics. It is quite relevent to deal a macrocrack by Paris equation. But we can't model the microdefects, inclusions by the conventional franture mechanics. Because, how can we know the stress intensity factor for each of the microdefect with unknown orientations, unknown growth pattern; among unknown number of microdefects? Well, now the question comes, convensional elasticity or plasticity do not work at this microlevel; what is the way to get rid of it? What we see from curve fitting; is the outside behavior of the material which is nothing but the over all response of these uncertain issues. A large macrocrack is a combined effect of large number of microcracks and their growth. Any one can predict the crack growth pattern of a big Vnotched crack. But can anyone say the pattern of bifurcation or branching of the small cracks at the tip of Vnotch?
These above questions remain unanswered, if you still stay with the thought with deterministic approach. So, when a structure goes under operation, a typical issue comes, if modulus "E" is the average elastic moduli, what is the variation pattern, and how it will vary. We do not have a perfect answer. But we have to survive. Hence probabilistic modeling comes.We are not sure about the variation of the "E", we may add some uncertain errors to "E", which is relevent but not perfect. So mathematics calls it "NOISE". Life becomes more beautiful if there is no noise in the system. But we have no option. Uncertainity is a pattern of the nature.
As far as knowing the number of parameters, probability distrubutuion etc are concerned, a big thrust in research on mathematical statistics, stochastic process, probabilistic modeling is still going on to figure out the issue. We have nonparametric approach that I mentioned before which is developed in order to deal the problem regarding probability distrubution. Extracting the effective parameters is a serious issue. Beyesian modeling, Spectral analysis and so on are focusing on this issue. But what ever we do, still noise will remain. And as long as noise remains we need "uncertainity qualification".
Of course, some problems seem to be intrinsically statistics
One of the first examples and indeed the original interest of Weibull, was ceramic materials.
In that case, you find in experiment so much a deviation from a single "strength", that you need a distribution to model it.
The next step is that you recognize the shape of the distribution, and in some cases it is "mild" like a Gaussian, which means that more or less you are happy with the mean (and perhaps the standard deviation) as information to carry over.
However, in other cases you have a "wild" distribution, like a power law  this is the case of many critical phenomena in nature, there is a huge literature, Mandelbrot, Sornette, to name but a few (the denomination "mild" vs "wild" is indeed Mandelbrot's recent contribution in finance).
For a wild distribution, the mean means little! You need to carry over some analysis also with the tails.
To return to Weibull statistics, it is not precisely power law, nor is a Gaussian, so it is somewhere intermediate. You are supposed to measure the strength, and as I said, having a brittle ceramic means you need MORE tests to make the distribution, than if you have tensile test of a steel, which is quite "deterministic", i.e. it has a narrow Gaussian distribution of which mean is ok, and you can even forget standard deviation.
The next problem becomes that people think the property of the statistical ceramic material can be measured easily and are "material properties". This is NOT entirely true, I wish life were easy....
Is Weibull’s modulus really a material constant? Example case with interacting collinear cracks
International Journal of Solids and Structures, Volume 43, Issue 17, August 2006, Pages 51475157
L. Afferrante, M. Ciavarella, E. Valenza
Anyway, the question becomes long. Speaking of Weibull, however, I suggest to read this remarkable paper using some analogies between "failure" of material, and human death.
A statistical analogy between collapse of solids and death of living organisms: Proposal for a ‘law of life’
Medical Hypotheses, Volume 69, Issue 2, 2007, Pages 441447
Nicola M. Pugno
Enough for now! I wait for more input to give more output...
Thanks for the papers.
Thanks for the papers. Weibull distribution is not only a statistical tool. The most important issue in theoretical modeling is the correct model for correct case. It is often seen that the physical phenomena is thought to be a deterministic one and suddenly people use statistcs to overcome some defeciencies. As I am working on composites, actually I saw the use of Weibull distribution in several wroks. People tried to define the strength of the fibermatrix interface in terms of Weibull distribution based on some test. The important thing is to be noted: this is not exactly a statistical modeling. Can you send me a paper which accounts the "Goodeness of Fit" in approximating the strength with Weibull distribution? However, this is not probabilistic modeling. Stochastic modeling or Probabilistic modeling has a visible difference with fitting a data set to Weibull distrubution; this is rather statistical data analysis.It does not account the actual phenomena happening inside the material. The uncertainity of a material is more significant at meso/micro level rather than at macro level; I mentioned before. As far as thin gaussian distribution (mild) is concern; it is good to approximate the distribution with Laplacian distribution.
Mathematically, a term "almost certain" has a difference with "certain".We can not say a random variable with probability of 0.999999...99, a deterministic variable. The risk is : (10.999999...99).
Statistical approach is not the same as probabilistic approach. As long as you deal with probabilistic or statistical approach, you can not expect 100% perfect output. That is why we have "mean;median and standard deviation". Probabilistic modeling or statistical modeling makes you get agree at the beginning when you defined a random variable having mean "μ" and standard deviation "σ".
I don't follow here.But anyway let's move on.You pose a problem?
michele ciavarella
www.micheleciavarella.it
I remain very "uncertain" about this "uncertain mechanics" ! ;)
michele ciavarella
www.micheleciavarella.it
Uncertainity qualification of aircraft structures
Please look at the paper:
A probabilistic approach to fatigue risk assessment in aerospace components : G. Cavallini , R. Lazzeri
It is a nice source of idea on typical uncertainity qualification method in structural mechanics. The schematic diagram (figure1 in the paper) can give a nice instant idea about formal uncertainity qualification method.
Re: Uncertainty and Monte Carlo
Cosma Shalizi has a nice intro to Monte Carlo methods and pointers to other sources in his Notebooks. See http://cscs.umich.edu/~crshalizi/notebooks/montecarlo.html.
 Biswajit