On Shannon entropy computations in selected plasticity problems

Abstract

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. K E Y W O R D Sfinite element method, Monte-Carlo simulation, plasticity problems, probabilistic entropy, Shannon theory, stochastic perturbation technique INTRODUCTIONUncertainty quantification in mechanical problems is most often based on the expected values and the standard deviation (coefficient of variation) of extreme structural responses. Unfortunately, the second moment or coefficient varies for different probability distributions, so it cannot serve directly as a universal chaotic dispersion measure in 5128

Selected Problems of Random Free Vibrations of Rectangular Thin Plates with Viscoelastic Dampers

“…In the case of any mathematical difficulties with computer algebra integration, the iterative generalized stochastic perturbation technique implemented as the SFEM is recommended. Further uncertainty analysis may be alternatively completed by the application of probabilistic entropy or its relative version presented recently in the literature [ 17 , 18 ].…”

Stochastic spectral cell method for structural dynamics and wave propagations

“…This very small value ranges from 10 −6 to 10 −14 depending on the problem. Of course, the assigned small values do not need to be equal for Equations (10) to (13). For a partially filled cell, the integrand is discontinuous because 𝛼 can be equal to either 0 or 1.…”

“…From the mathematical viewpoint, the SFEM is known as a robust technique for the solution of stochastic partial differential equations. There are different variants of the SFEM such as the perturbation approach, 4 spectral decomposition approach, 2 and Monte Carlo simulation 3,5 which have been applied to numerous research fields consisting of earthquake engineering, [5][6][7][8][9] structural engineering, [10][11][12][13][14] mechanical engineering, [15][16][17] among others. 18 More recently, the stochastic spectral element method (SSEM) has been proposed as a computational framework based on the spectral decomposition.…”

Shannon Entropy in Stochastic Analysis of Some Mems

“…Shannon entropy is to be determined from its classical definition at any discrete-time moment τ. It yields [3,34]:…”

“…It is also a mathematical extension of its thermodynamic origin, invented by Boltzmann, and now it may serve as a universal measure of disorder in the given engineering system. The first method of its calculation was proposed by Claude Shannon [1,2], although the term "entropy" was introduced by Johann von Neumann; this has been used multiple times to analyse disorder in various engineering systems [3]. The original definition by Shannon includes a summation of probabilities of various admissible states of the given experiment multiplied with their logarithms; such an entropy was determined from the very beginning using various estimation methods [4].…”