Mathematical concept reliability

Laplace transforms, often used in the area of reliability to find solutions to first-order differential equations, were developed by the French mathematician named Pierre-Simon Laplace (1749-1827). Additional information on the history of mathematics and probability is available in References 1 and 2. This chapter presents basic mathematical concepts considered useful to understand subsequent chapters of this book. 

Thus, the main objective of this book is to combine safety and reliability in the oil and gas industry into a single volume, eliminate the need to consult a number of different and diverse sources in obtaining desired information, and provide up-to-date information on the subject. The somces of most of the material presented are given in the reference section at the end of each chapter. These will be useful to readers if they desire to delve deeper into a specific area. This book contains a chapter on mathematical concepts and another chapter on safety and reliability basics considered useful to imder-stand the contents of subsequent chapters. Furthermore, another chapter is devoted to methods considered useful to perform safety and reliability analyses in the oil and gas industry. 

Drawing together the latest research spread throughout the literature. Tmiisportation Systems Reliability and Safety eliminates the need to consult many different and diverse sources to obtain np-to-date information and research. It contains a chapter on mathematical concepts and another chapter on reliability and safety basics that form a foundation for understanding the contents of snhsecpient chapters. The hook also presents a chapter devoted to methods for performing transportation systems reliability and safety analysis. It includes a reference section at the end of each chapter for readers who wish to delve deeper into a specific area. 

The movement from the deterministic design criteria as described by equation 4.1 to the probability based one described by equation 4.2 has far reaching effects on design (Haugen, 1980). The particular change which marks the development of modern engineering reliability is the insight that probability, a mathematical theory, can be utilized to quantify the qualitative concept of reliability (Ben-Haim, 1994). 

A model may be defined as a device which behaves in a manner similar enough to some other system so that useful knowledge about the system may be gained from a study of the model. The concept of models in the form of laboratory or pilot plant equipment is certainly very familiar in chemical engineering. The usefulness of such models in predicting the operation of present or projected plant equipment is well appreciated. However, there are circumstances in which experimentation with physical models is not the best method of study. Frequently mathematical models are more convenient to use, lower in cost, and more reliable. 

To establish the molecular thermodynamic model for uniform systems based on concepts from statistical mechanics, an effective method by combining statistical mechanics and molecular simulation has been recommended (Hu and Liu, 2006). Here, the role of molecular simulation is not limited to be a standard to test the reliability of models. More directly, a few simulation results are used to determine the analytical form and the corresponding coefficients of the models. It retains the rigor of statistical mechanics, while mathematical difficulties are avoided by using simulation results. The method is characterized by two steps (1) based on a statistical-mechanical derivation, an analytical expression is obtained first. The expression may contain unknown functions or coefficients because of mathematical difficulty or sometimes because of the introduced simplifications. (2) The form of the unknown functions or unknown coefficients is then determined by simulation results. For the adsorption of polymers at interfaces, simulation was used to test the validity of the weighting function of the WDA in DFT. For the meso-structure of a diblock copolymer melt confined in curved surfaces, we found from MC simulation that some more complex structures exist. From the information provided by simulation, these complex structures were approximated as a combination of simple structures. Then, the Helmholtz energy of these complex structures can be calculated by summing those of the different simple structures. 

Prediction reliability relies on the structural interpolation concept. As for a classical mathematical function, predictions by interpolation are more reliable than by extrapolation. Moreover, regularity enhances the reliability of interpolated and, especially, extrapolated predictions. 

Even the concept of certainty factors is not derived from a formal mathematical basis it is the most commonly used method to describe uncertainty in expert systems. This is mainly due to the fact that certainty factors are easy to compute and can be used to effectively reduce search by eliminating branches with low certainty. However, it is difficult to produce a consistent and accurate set of certainty factors. In addition, they are not consistently reliable a certainty factor may produce results opposite to probability theory. 

An alternative to time-consuming risk assessments of chemical substances could be more reliable and advanced priority setting methods. Hasse Diagram Technique (HDT) and/or Multi-Criteria Analysis (MCA) provide an elaboration of the simple scoring methods. The present chapter evaluates HDT relative to two MCA techniques. The main methodological step in the comparison is the use of probability concepts based on mathematical tools such as linear extensions of partially ordered sets and Monte Carlo simulations. A data set consisting of 12 High Production Volume Chemicals (HPVCs) is used for illustration. 

While these two ciphers work, they are not reliable as they can be broken very easily. This is where mathematics gets involved. Mathematics can be used to create sophisticated ciphers that are virtually impossible to break. To begin this discussion, we will first see how a cipher, very similar to a plaintext shift or Caesar cipher, can be obtained mathematically by using some of the number theoretic concepts introduced earlier in this chapter. 

Reliability has sometimes been described as quality in the time dimension (RDG-376 1964) and a time oriented quality characteristic (Kapur 1986). The reliability characteristics of a product change with time. One of the characteristics is the concept of failure rate, which is defined mathematically later in this chapter. The failure rate, or the hazard rate, changes with the age or life of a product and has three distinct periods, as shown in Figures 3(a) and 3(7i). These three periods are described here (Kececioglu 1991). 

Marseguerra, M. et al. 1998. A concept paper on dynamic reliability via Monte Carlo simulation. Mathematics and Computers in Simulation 47 371-382. 

Today, various mathematics and probability concepts are being used to study various types of safety-related problems. For example, probability distributions are used to represent times to human error in performing various types of time-continuous tasks in the area of safety [3-7]. In addition, the Markov method is used to conduct human performance reliability analysis in regard to engineering systems safety [7-9]. 

Statistical assessment of time to failure is a basic topic in reliability engineering for which many mathematical tools have been developed. Evans, who also pioneered the mixed potential theory to explain basic corrosion kinetics, launched the concept of corrosion probability in relation to localized corrosion. According to Evans, an exact knowledge of corrosion rate was less important than the ascertainment of the statistical risk of its initiation [12]. The following examples illustrate the application of empirical modeling in two areas of high criticality. 

A model of a system is some form of mathematical representation that gives, in the final analysis, a relation between the inputs and outputs of the system. The simplest and least reliable are empirical models, which are based more or less on the black-box concept. As shown in Figure 1.1, an empirical model may have the following form ... 

When considering reaction paths on the PE surfaces of excited states, as required for the rationalization of photochemistry [4], two major additional complications arise. First, reliable ab initio energy calculations for excited states are typically much more involved than ground-state calculations. Secondly, multi-dimensional surface crossings are the rule rather than the exception for excited electronic states. The concept of an isolated Born-Oppenheimer(BO) surface, which is usually assumed from the outset in reaction-path theory, is thus not appropriate for excited-state dynamics. At surface crossings (so-called conical intersections [5-7]) the adiabatic PE surfaces exhibit non-differentiable cusps, which preclude the application of the established methods of mathematical reaction-path theory [T3]. As an alternative to non-differentiable adiabatic PE surfaces, so-called diabatic surfaces [8] may be introduced, which are smooth functions of the nuclear coordinates. However, the definition of these diabatic surfaces and associated wave functions is not unique and involves some subtleties [9-11]. 

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