Mathematical model building

In this part the course is motivated. It is indicated how incorrect models appear in some books and research articles, which produce an important damage to society. This fact is illustrated with five examples that provide a first look to the problems deal with in the course. Next, the aims of the course are listed, including the detection of physical, dimensional and extreme value model inconsistencies and how to solve then. Finally, the three dimensional homogeneity rules given by Sonin are presented and discussed.

In this block we introduce the Buckingham theorem, which is essential to derive as simple as possible models and models having sense. Unfortunately, this theorem is not known by a large number of people involved in modeling problems, when this theorem should be included in the high school material, due to its importance. A detailed description of the steps to be followed to derive a model is included, incorporating an explanation of how to obtain dimensionless ratios and how to make dimensional analysis tables, which are important in the application of the theorem to modelling. The theorem is illustrated with several examples, including the Sonin Ball example, the simply supported beam, the flexible retaining wall, the crack grow model and some counter-examples. For the latter, the problem and resulting inconveniences of using these models are discussed.

We dedicate this second block to the model consistency problem. We start with the problem of physical consistency by indicating that not all formulas are valid to represent a physical variable when written in terms of other basic variables. We formulate a consistency condition in terms of a functional equation and show the only valid form. Second, we discuss probability consistency and show that some proposed models are contradictory from the point of vie of probability theory. We define operation stable consistency and suggest the use of reproductive models when possible. Finally, we deal with extreme value consistency, providing some rules for dsatisfying this condition and indicating some common errors found in the literature when using extreme value models in engineering practice.

This third block is dedicated to the implications of the consistency constraints on the statistical families of distributions used when building stochastic models. In particular we show that some families cannot be used unless the random variable is dimensionless. In particular the dimensionless ratios provided by the Bauckingham theorem arise as natural candidates for these distributions. We also deal with how to define multivariate models and present underdetermined, overdetermined and strictly determined methods, indicating about some warnings when using these alternatives. In particular, we suggest the use of Bayesian networks as the best way of defining multivariate models, because of the fact that they always satisfy consistency and have a clear physical interpretation.

In this fourth blok we describe some engineering applications. We start with a fatigue model and derive, step by step, the model without arbitrary assumptions. In this case a comaptibility assumption, the weakest link principle and extreme value theory are sufficient to derive the functional form of the S-N curve and the Weibull model. When incorporating min and max stresses, another compatibility condition allows to extend the previous model. Next, we derive a crack-.grow model, using also a similar compatibility condition and extreme value analysis. Finally, both models are linked by using the failure curve, providing a very interesting link, not previosly obtained, in which we can move from S-N curves to derive the crack-grow curve model and vice versa.

This block is dedicated to provide the readers with some references

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