Probabilistic analysis is increasing in popularity and importance within engineering and the applied sciences. However, the stochastic perturbation technique is a fairly recent development and ...therefore remains as yet unknown to many students, researchers and engineers. Fields in which the methodology can be applied are widespread, including various branches of engineering, heat transfer and statistical mechanics, reliability assessment and also financial investments or economical prognosis in analytical and computational contexts.Stochastic Perturbation Method in Applied Sciences and Engineeringis devoted to the theoretical aspects and computational implementation of the generalized stochastic perturbation technique. It is based on any order Taylor expansions of random variables and enables for determination of up to fourth order probabilistic moments and characteristics of the physical system response.Key features:Provides a grounding in the basic elements of statistics and probability and reliability engineeringDescribes the Stochastic Finite, Boundary Element and Finite Difference Methods, formulated according to the perturbation method Demonstrates dual computational implementation of the perturbation method with the use of Direct Differentiation Method and the Response Function Method Accompanied by a website (www.wiley.com/go/kaminski) with supporting stochastic numerical softwareCovers the computational implementation of the homogenization method for periodic composites with random and stochastic material propertiesFeatures case studies, numerical examples and practical applicationsStochastic Perturbation Method in Applied Sciences and Engineeringis a comprehensive reference for researchers and engineers, and is an ideal introduction to the subject for postgraduate and graduate students.
In this paper theoretical formulation and computational implementation of the Stochastic perturbation-based Finite Element Method (SFEM) for uncertainty analysis in solid mechanics with symmetric ...non-Gaussian input parameters are presented. Theoretical foundations of the method are based on the general order Taylor expansions of all uncertain input parameters and state functions including even orders only. The first four probabilistic characteristics of the structural responses have been derived for symmetrical triangular and uniform probability distributions of random input including probability distribution truncation effect. The Stochastic Finite Element Method implementation has been completed for the displacement version of the FEM using statistically optimized nodal polynomial response bases, and their coefficients are determined using the Least Squares Method using the weighted and non-weighted schemes. Structural responses of several mechanical systems are analyzed using their basic probabilistic characteristics, which have been validated using the probabilistic semi-analytical approach, and also the crude Monte-Carlo simulation. A relatively good coincidence of three probabilistic numerical techniques confirms the applicability of the Stochastic perturbation-based Finite Element Method to study boundary and initial problems in mechanics with uncertainties having uniform and/or triangular probability distributions.
•New formulation of the iterative generalized stochastic perturbation technique.•Implementation of the Stochastic Finite Element Method for the triangular PDF.•Implementation of the Stochastic Finite Element Method for the input uniform PDF.•Comparative numerical analysis with Monte-Carlo simulation.•Comparative numerical analysis with the semi-analytical probabilistic method.
The main idea of this work is an application of probabilistic entropy and also relative entropy in the numerical analysis of uncertainty propagation in the homogenization of some composite materials. ...The homogenization method is based on the determination of deformation energy for the representative volume elements computed with the use of some specific finite element method experiments. Uncertainty propagation concerns material and geometrical design parameters of particulate composites and is performed thanks to an application of polynomial responses; probabilistic moments of the effective tensor are computed via the iterative generalized stochastic perturbation technique and the semi‐analytical probabilistic method. Probabilistic entropy is determined according to Shannon theory, while relative entropies equations employed here follow mathematical models created by Kullback & Leibler, Jeffreys, Hellinger, and Bhattacharyya. Deterministic analyses have been performed with the use of the system ABAQUS, while all the remaining procedures have been programmed in the computer algebra system MAPLE.
The main idea of this work is an application of relative entropy in the numerical analysis of probabilistic divergence between original material tensors of the composite constituents and its ...effective tensor in the presence of material uncertainties. The homogenization method is based upon the deformation energy of the representative volume elements for the fiber‐reinforced and particulate composites and uncertainty propagation begins with elastic moduli of the fibers, particles, and composite matrices. Relative entropy follows a mathematical model originating from Bhattacharyya probabilistic divergence and has been applied here for Gaussian distributions. The semi‐analytical probabilistic method based on analytical integration of polynomial bases obtained via the least squares method fittings enables for determination of the basic probabilistic characteristics of the effective tensor and the relative entropies. The methodology invented in this work may be extended toward other probability distributions and relative entropies, for homogenization of nonlinear composites and also accounting for some structural interface defects.
The main aim of this work is to deliver uncertainty propagation analysis for the homogenization process of fibrous metal matrix composites (MMCs). The homogenization method applied here is based on ...the comparison of the deformation energy of the Representative Volume Element (RVE) for the original and for the homogenized material. This part is completed with the use of the Finite Element Method (FEM) plane strain analysis delivered in the ABAQUS system. The probabilistic goal is achieved by using the response function method, where computer recovery with a few FEM tests enables approximations of polynomial bases for the RVE displacements, and further—algebraic determination of all necessary uncertainty measures. Expected values, standard deviations, and relative entropies are derived in the symbolic algebra system MAPLE; a few different entropy models have been also contrasted including the most popular Kullback–Leibler measure. These characteristics are used to discuss the influence of the uncertainty propagation in the MMCs’ effective material tensor components, but may serve in the reliability assessment by quantification of the distance between extreme responses and the corresponding admissible values.
A fundamental problem investigated in this paper is computational homogenization of two-component particulate and fibrous composites with material characteristics treated as symmetric but ...non-Gaussian random processes. Homogenization method is based on equity of deformation energies for the real multi-component and the homogenized Representative Volume Elements (RVE) of the given composite structure. The iterative generalized stochastic perturbation technique is used to provide the additional Taylor expansions about expected values and to finally derive analytical formulas for the basic probabilistic characteristics of the effective tensor components. These expansions use polynomial response functions relating this tensor components with real material characteristics determined through the series of the Finite Element Method homogenization tests and the Least Squares Method. Applicability of this stochastic strategy and its Stochastic Finite Element Method implementation are verified for two well-known probability distributions, i.e. the uniform and the triangular one, to model uncertainty in Young modulus of some composite components. Validation of this approach is carried out using two alternative methods – statistical one known as the Monte-Carlo simulation as well as the semi-analytical one using the same polynomial responses as the perturbation approach, but finally processed using classical analytical probability integrals. This study generally confirms applicability of the proposed probabilistic method to homogenize composite structures with linear elastic and isotropic components exhibiting uniform and triangular uncertainties in their elastic properties.
This paper considers the problem of determining probabilistic entropy fluctuations, which are important for understanding uncertainty propagation in mechanical systems in the elasto‐plastic regime. ...Probabilistic entropy is conceptualized based on an initial definition by Shannon, which demands discrete representation of the uncertainty source. Numerical analysis is performed using the Response Function Method with polynomial bases. Coefficients are found and order optimization is completed using polynomial interpolations or the Least Squares Method. Approximations are based on the Finite Element Method. Local polynomial bases enable nonlinear increment analysis, and allow for a given degree of freedom in the FEM model to be described as a function of a random input parameter. Academic FEM software and the ABAQUS system were used for numerical experiments. Polynomial approximations, probabilistic moment computations, and statistical entropy estimations were programmed in the symbolic algebra package MAPLE. Transformation of the input probability density into the output function was performed using the Monte‐Carlo simulation algorithm for statistically optimized polynomial bases of extreme displacement functions. Two computational examples are given to demonstrate probabilistic entropy fluctuations for a small statically indeterminate aluminum truss structure and also for practical engineering case study of the steel round bar under uniform tensile stress. In these examples, some material and geometrical uncertainties distributed according to Gaussian, triangular, uniform as well as lognormal distributions were analyzed. The presented approach could be used for constitutive models of solids, computational fluid dynamics, and in other discrete numerical methods.
Przedmiotem rozważań jest zagadnienie szczególnych reżimów ochronnych prawa administracyjnego jako kwalifikowanego typu specjalnego porządku regulacji administracyjnoprawnej obowiązującej w ...sytuacjach zagrożeń dla istotnych dóbr prawnych o charakterze kolektywnym (np. bezpieczeństwa publicznego lub zdrowia publicznego). Celem analizy jest przedstawienie relacji tego rodzaju reżimów do stanów nadzwyczajnych (stanu wojennego, stanu wyjątkowego, stanu klęski żywiołowej) oraz konstytucyjnych granic ustawowego ograniczania wolności i praw jednostek w okresie obowiązywania lub stosowania norm należących do tych reżimów. Szczególna uwaga została poświęcona zagadnieniom reżimów związanych z wystąpieniem epidemii i zagrożenia epidemicznego oraz zdarzenia radiacyjnego. W opracowaniu uwzględniono także problematykę nowych kategorii reżimów szczególnych (kwalifikowanych stref ochronnych na terenie obowiązywania stanów epidemii lub zagrożenia epidemicznego oraz obszaru obowiązywania zakazu przebywania w strefie nadgranicznej). Rozważania bazują na metodzie uogólniającej analizy formalno-dogmatycznej regulacji normatywnej z wykorzystaniem prawniczych konstrukcji teoretycznych.
Cellular materials are fundamental elements in civil engineering, known for their porous nature and lightweight composition. However, the complexity of its microstructure and the mechanisms that ...control its behavior presents ongoing challenges. This comprehensive review aims to confront these uncertainties head-on, delving into the multifaceted field of cellular materials. It highlights the key role played by numerical and mathematical analysis in revealing the mysterious elasticity of these structures. Furthermore, the review covers a range of topics, from the simulation of manufacturing processes to the complex relationships between microstructure and mechanical properties. This review provides a panoramic view of the field by traversing various numerical and mathematical analysis methods. Furthermore, it reveals cutting-edge theoretical frameworks that promise to redefine our understanding of cellular solids. By providing these contemporary insights, this study not only points the way for future research but also illuminates pathways to practical applications in civil and materials engineering.
