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.

中文翻译：

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关于选定塑性问题中的香农熵计算

本文考虑了确定概率熵涨落的问题，这对于理解弹塑性区域机械系统中的不确定性传播很重要。概率熵是基于香农的初始定义概念化的，它需要不确定源的离散表示。使用基于多项式的响应函数方法进行数值分析。使用多项式插值或最小二乘法找到系数并完成阶数优化。近似值基于有限元方法。局部多项式基支持非线性增量分析，并允许将 FEM 模型中的给定自由度描述为随机输入参数的函数。数值实验采用学术有限元软件和ABAQUS系统。多项式近似、概率矩计算和统计熵估计在符号代数包 MAPLE 中编程。输入概率密度到输出函数的转换是使用蒙特卡罗模拟算法进行的，用于极端位移函数的统计优化多项式基。给出了两个计算示例，以证明小型静态不定铝桁架结构的概率熵涨落以及均匀拉应力下钢圆棒的实际工程案例研究。在这些示例中，分析了根据高斯分布、三角形分布、均匀分布和对数正态分布的一些材料和几何不确定性。