Stretching and Condensing Tensors. Application to tensor and matrix equations

Since tensors are vectors, no matter their order and valencies, they can be represented in matrix form and in particular as column matrices of components with respect to their corresponding bases. This consideration allows using all the available tools for vectors and can be extremely advantageous to deal with some interesting problems in linear algebra, as solving tensor linear equations. Once, the tensors have been operated as vectors, they can be returned to their initial notation as tensors. This technique, that is specially useful for implementing computer programs, is illustrated by several examples of applications. In particular, some interesting tensor and matrix equations are solved. Several numerical examples are used to illustrate the proposed methodolog

Solving some special cases of monomial ratio equations appearing frequently in physical and engineering problems

We first show that monomial ratio equations are not only very common in Physics and Engineering, but the natural type of equations in many practical problems. More precisely, in the case of models involving scale variables if the used formulas are not of this type they are not physically valid. The consequence is that when estimating the model parameters we are faced with systems of monomial ratio equations that are nonlinear and difficult to solve. In this paper, we provide an original algorithm to obtain the unique solutions of systems of equations made of linear combinations of monomial ratios whose coefficient matrix has a proper null space with low dimension that permits solving the problem in a simple way. Finally, we illustrate the proposed methods by their application to two practical problems from the hydraulic and structural fields.Peer ReviewedPostprint (published version

An iterative method to obtain the specimen-independent three-parameter Weibull distribution of strength from bending tests

Brittle materials, such as glass and ceramics, usually present a large strength scatter. Among other probability distributions, the Weibull distribution is widely used to characterize their resistance. Often the two-parameter model is employed, omitting the consideration of a threshold stress, leading to a simplified estimation method. For the sake of generality the present work uses fracture data from bending tests to obtain a three-parameter Weibull distribution function valid for a uni-axially and uniformly tensioned material element. The variable stress state prevailing in the flexural specimen and the size effect are simultaneously accounted for by means of an iterative fitting procedure. The method is extended to account for bimodal flaw distributions, discriminating between edge and surface failure results based on experimental observation. Finally, the model is applied to simulated data sets, obtaining satisfactory results

Reliability analysis based on non-dimensional parameters

A reliability analysis method is proposed that starts with the identification of all variables involved. These are divided in three groups: (a) variables fixed by codes, as loads and strength project values, and their corresponding partial safety coefficients, (b) geometric variables defining the dimension of the main elements involved, (c) the cost variables, including the possible damages caused by failure, (d) the random variables as loads, strength, etc., and (e)the variables defining the statistical model, as the family of distribution and its corresponding parameters. Once the variables are known, the II-theorem is used to obtain a minimum equivalent set of non-dimensional variables, which is used to define the limit states. This allows a reduction in the number of variables involved and a better understanding of their coupling effects. Two minimum cost criteria are used for selecting the project dimensions. One is based on a bounded-probability of failure, and the other on a total cost, including the damages of the possible failure. Finally, the method is illustrated by means of an application

Probabilistic Safety Analysis of High Speed and Conventional Lines Using Bayesian Networks

[EN] A Bayesian network approach is presented for probabilistic safety analysis (PSA) of railway lines. The idea consists of identifying and reproducing all the elements that the train encounters when circulating along a railway line, such as light and speed limit signals, tunnel or viaduct entries or exits, cuttings and embankments, acoustic sounds received in the cabin, curves, switches, etc. In addition, since the human error is very relevant for safety evaluation, the automatic train protection (ATP) systems and the driver behavior and its time evolution are modelled and taken into account to determine the probabilities of human errors. The nodes of the Bayesian network, their links and the associated probability tables are automatically constructed based on the line data that need to be carefully given. The conditional probability tables are reproduced by closed formulas, which facilitate the modelling and the sensitivity analysis. A sorted list of the most dangerous elements in the line is obtained, which permits making decisions about the line safety and programming maintenance operations in order to optimize them and reduce the maintenance costs substantially. The proposed methodology is illustrated by its application to several cases that include real lines such as the Palencia-Santander and the Dublin-Belfast lines.Grande Andrade, Z.; Castillo Ron, E.; Nogal, M.; O'connor, A. (2016). Probabilistic Safety Analysis of High Speed and Conventional Lines Using Bayesian Networks. En XII Congreso de ingeniería del transporte. 7, 8 y 9 de Junio, Valencia (España). Editorial Universitat Politècnica de València. 362-371. https://doi.org/10.4995/CIT2016.2015.3428OCS36237

Conditionally specified distributions: an introduction

A bivariate distribution can sometimes be characterized completelybyproperties of its conditional distributions. The present article surveys available research in this area. Questions of compatibility of conditional specifications are addressed as are characterizations of distributions based on their having conditional distributions that are members of prescribed parametric families of distributions. The topics of compatibilityand near compatibilityof conditional distributions are discussed. Estimation strategies for conditionallyspecified distributions are summarized. Additionally, certain conditionally specified densities are shown to provide convenient flexible conjugate prior families in certain multiparameter Bayesian settings

An exponential family of Lorenz curves

A new method for building parametric-functional families of Lorenz curves, generated from an initial Lorenz curve (which satisfies some regularity conditions), is presented. The method is applied to the exponential family since they use the exponential Lorenz curves as their generating curves. Several properties of these families are analyzed, including the population function, inequality measures, and Lorenz orderings. Finally, an application is presented for data from various countries. The family is shown to perform well in fitting the data across countries. The results are very robust across data sources

A model allowing for the influence of geometry and stress in the assessment of fatigue data

Usually, the Wohler field of a material is obtained from fatigue lifetime data resulting from testing specimens of reduced size in the laboratory. This basic information finds subsequent application in lifetime prediction of larger structural and mechanical components. Thus, an important question arises: how can the S-N field be transformed into an ideal one referred to a characteristic size (length, area or volume) subjected to a constant stress distribution in order to achieve a safe structural integrity design? In this work, the influence of specimen geometry and variable stress state on the fatigue lifetime distribution for constant amplitude fatigue tests is investigated. An experimental program has been carried out with unnotched specimens of nominally the same material but differing in length, diameter, and shape. The experimental data is fitted to a newly developed fatigue model, capable of describing the S-N-field in a probabilistic manner accounting for both the specimen geometry and the variable stress state of the specimens. As the estimated Wohler field is referred to an elemental surface, loaded by a constant stress level Ds, the extrapolation of the fatigue resistance to different specimen geometries is possible. Additionally, problems encountered due to scatter of the material properties are discussed

Solving ordinary differential equations with range conditions. Applications

This paper introduces the problem of solving ordinary differential equations with extra linear conditions written in terms of ranges, and deals with the corresponding existence and uniqueness problems. Some methods for analyzing the existence of solutions and obtaining the set of all solutions, based on the theory of cones and polyhedra, are given. These solutions are found by first converting the problem to a system of linear algebraic equations and then applying the corresponding well-known theory for solving and discussing the existence and uniqueness of solutions of these systems. Finally, the methods are illustrated by their application to some practical examples of the beam problem