scholarly journals Comparing Two Strategies to Model Uncertainties in Structural Dynamics

2010 ◽  
Vol 17 (2) ◽  
pp. 171-186 ◽  
Author(s):  
Rubens Sampaio ◽  
Edson Cataldo
Keyword(s):  
Random Matrix ◽  
Degrees Of Freedom ◽  
Maximum Entropy Principle ◽  
Random Variable ◽  
Complex Structures ◽  
Random Parameters ◽  
Model Uncertainties ◽  
System A ◽  
Entropy Principle ◽  
Available Information

In the modeling of dynamical systems, uncertainties are present and they must be taken into account to improve the prediction of the models. Some strategies have been used to model uncertainties and the aim of this work is to discuss two of those strategies and to compare them. This will be done using the simplest model possible: a two d.o.f. (degrees of freedom) dynamical system. A simple system is used because it is very helpful to assure a better understanding and, consequently, comparison of the strategies. The first strategy (called parametric strategy) consists in taking each spring stiffness as uncertain and a random variable is associated to each one of them. The second strategy (called nonparametric strategy) is more general and considers the whole stiffness matrix as uncertain, and associates a random matrix to it. In both cases, the probability density functions either of the random parameters or of the random matrix are deduced from the Maximum Entropy Principle using only the available information. With this example, some important results can be discussed, which cannot be assessed when complex structures are used, as it has been done so far in the literature. One important element for the comparison of the two strategies is the analysis of the samples spaces and the how to compare them.

scholarly journals A LINK BETWEEN THE MAXIMUM ENTROPY APPROACH AND THE VARIATIONAL ENTROPY FORM

2011 ◽  
Vol 25 (22) ◽  
pp. 1821-1828 ◽  
Author(s):  
E. V. VAKARIN ◽  
J. P. BADIALI
Keyword(s):  
Probability Distribution ◽  
Maximum Entropy ◽  
Variational Formulation ◽  
Variational Approach ◽  
Maximum Entropy Principle ◽  
Random Variable ◽  
Entropy Measure ◽  
Universal Relation ◽  
Amount Of Information ◽  
Entropy Principle

The maximum entropy approach operating with quite general entropy measure and constraint is considered. It is demonstrated that for a conditional or parametrized probability distribution f(x|μ), there is a "universal" relation among the entropy rate and the functions appearing in the constraint. This relation allows one to translate the specificities of the observed behavior θ(μ) into the amount of information on the relevant random variable x at different values of the parameter μ. It is shown that the recently proposed variational formulation of the entropic functional can be obtained as a consequence of this relation, that is from the maximum entropy principle. This resolves certain puzzling points that appeared in the variational approach.

scholarly journals Molecular Extended Thermodynamics of Rarefied Polyatomic Gases with a New Hierarchy of Moments

Fluids ◽  
2021 ◽  
Vol 6 (2) ◽  
pp. 62
Author(s):  
Takashi Arima ◽  
Tommaso Ruggeri
Keyword(s):  
Degrees Of Freedom ◽  
Maximum Entropy Principle ◽  
Extended Thermodynamics ◽  
Moment Equations ◽  
Field Equations ◽  
Polyatomic Gases ◽  
Moment System ◽  
Rarefied Polyatomic Gases ◽  
Entropy Principle ◽  
Internal Degrees Of Freedom

The aim of this paper is to construct the molecular extended thermodynamics for classical rarefied polyatomic gases with a new hierarchy, which is absent in the previous procedures of moment equations. The new hierarchy is deduced recently from the classical limit of the relativistic theory of moments associated with the Boltzmann–Chernikov equation. The field equations for 15 moments of the distribution function, in which the internal degrees of freedom of a molecule are taken into account, are closed with the maximum entropy principle. It is shown that the theory contains, as a principal subsystem, the previously polyatomic 14 fields theory, and in the monatomic limit, in which the dynamical pressure vanishes, the differential system converges, instead of to the Grad 13-moment system, to the Kremer 14-moment system.

A probabilistic mesoscale damage detection in polycrystals using a random matrix approach

2013 ◽  
Vol 24 (8) ◽  
pp. 1007-1017 ◽  
Author(s):  
Arash Noshadravan ◽  
Roger Ghanem
Keyword(s):  
Damage Detection ◽  
Random Matrix ◽  
Matrix Theory ◽  
Maximum Entropy Principle ◽  
Coarse Scale ◽  
Predictive Tool ◽  
Random Matrix Model ◽  
Damage Prognosis ◽  
Probabilistic Description ◽  
Entropy Principle

This article is concerned with a probabilistic mesoscale damage detection in polycrystals. For this purpose, we make use of a stochastic model describing the linear elasticity matrix of material at the mesoscale. The model is constructed using a maximum entropy principle and random matrix theory and allows one to directly construct a probabilistic model for the system random matrices characterizing the constitutive behavior of the system. First, the theoretical framework and upscale scheme in the construction of the model are briefly reviewed. For each case of healthy and damaged materials, where the damage is introduced in the form of intergranular microcavities, the random matrix model is calibrated by performing simulations on an ensemble of statistical volume elements of microstructure. The calibrated models are then used in a simple coarse-scale simulation in order to explore the sensitivity of the model in detecting the location of mesoscale damages. The result shows that in most cases, one can identify the location of cracks by comparing the probabilistic description of a suitable response quantity of interest predicted for both healthy and damaged systems. Such a probabilistic description is suitable for detecting signature of fine-scale defects where the consequences are reflected at the coarse scale in the form of random fluctuations around the mean behavior. The model can be used as a predictive tool in the context of structural health monitoring and damage prognosis of metallic systems.

Use in Probabilistic Design of the Maximum Entrophy Distribution Based on Ranked Data

1980 ◽  
Vol 102 (3) ◽  
pp. 460-468
Author(s):  
J. N. Siddall ◽  
Ali Badawy
Keyword(s):  
Probability Distribution ◽  
Maximum Entropy ◽  
Probability Distributions ◽  
Maximum Entropy Principle ◽  
Random Variable ◽  
Probabilistic Design ◽  
Entropy Principle ◽  
Almost All ◽  
New Distribution

A new algorithm using the maximum entropy principle is introduced to estimate the probability distribution of a random variable, using directly a ranked sample. It is demonstrated that almost all of the analytical probability distributions can be approximated by the new algorithm. A comparison is made between existing methods and the new algorithm; and examples are given of fitting the new distribution to an actual ranked sample.

Random Matrix Approach: Toward Probabilistic Formulation of the Manipulator Jacobian

2014 ◽  
Vol 137 (3) ◽  
Author(s):  
Javad Sovizi ◽  
Sonjoy Das ◽  
Venkat Krovi
Keyword(s):  
Gaussian Distribution ◽  
Random Matrix ◽  
Matrix Theory ◽  
Jacobian Matrix ◽  
Random Perturbation ◽  
Maximum Entropy Principle ◽  
Robotic Systems ◽  
Entropy Measure ◽  
Probabilistic Formulation ◽  
Available Information

In this paper, we formulate the manipulator Jacobian matrix in a probabilistic framework based on the random matrix theory (RMT). Due to the limited available information on the system fluctuations, the parametric approaches often prove to be inadequate to appropriately characterize the uncertainty. To overcome this difficulty, we develop two RMT-based probabilistic models for the Jacobian matrix to provide systematic frameworks that facilitate the uncertainty quantification in a variety of complex robotic systems. One of the models is built upon direct implementation of the maximum entropy principle that results in a Wishart random perturbation matrix. In the other probabilistic model, the Jacobian matrix is assumed to have a matrix-variate Gaussian distribution with known mean. The covariance matrix of the Gaussian distribution is obtained at every time point by maximizing a Shannon entropy measure (subject to Jacobian norm and covariance positive semidefiniteness constraints). In contrast to random variable/vector based schemes, the benefits of the proposed approach now include: (i) incorporating the kinematic configuration and complexity in the probabilistic formulation; (ii) achieving the uncertainty model using limited available information; (iii) taking into account the working configuration of the robotic systems in characterization of the uncertainty; and (iv) realizing a faster simulation process. A case study of a 2R serial manipulator is presented to highlight the critical aspects of the process.

scholarly journals Uncertainties in Drill Dynamics: The Role of a Random Weight-on-Hook

2009 ◽  
Author(s):  
T. G. Ritto ◽  
C. Soize ◽  
R. Sampaio
Keyword(s):  
Drilling Fluid ◽  
Maximum Entropy Principle ◽  
Beam Theory ◽  
Random Variable ◽  
Drill String ◽  
Rock Interaction ◽  
Entropy Principle ◽  
Random Weight ◽  
Supporting Force

A drill-string is a slender structure that turns and drills into the rock in search of oil. This paper aims to investigate uncertainties on the weight-on-hook, which is the supporting force exerted by the hook at the top. In a drilling operation there are three parameters that can be continuously controlled: (1) the weight-on-hook, (2) the drilling fluid inlet velocity and (3) the rotational speed of the rotary table. The idea is to understand how the performance of the system (which is measured by the rate-of-penetration) if affected when the weight-on-hook is considered uncertain. A numerical model is developed using the Timoshenko beam theory and discretized by means of the Finite Element Method. The nonlinear dynamics presented includes bit-rock interaction, fluid-structure interaction and impacts. The probability theory is used to model the uncertainties. To construct the probability density function of the random variable, the Maximum Entropy Principle is employed, so that the probability distribution is derived from the mechanical properties of the weight-on-hook.

Reversibly Sampling Conformations and Binding Modes Using Molecular Darting

2020 ◽  
Author(s):  
Samuel C. Gill ◽  
David Mobley
Keyword(s):  
Monte Carlo ◽  
Ligand Binding ◽  
Binding Sites ◽  
Degrees Of Freedom ◽  
Dynamics Simulation ◽  
Hiv Integrase ◽  
Binding Modes ◽  
System A ◽  
Multiple Binding ◽  
Internal Degrees Of Freedom

<div>Sampling multiple binding modes of a ligand in a single molecular dynamics simulation is difficult. A given ligand may have many internal degrees of freedom, along with many different ways it might orient itself a binding site or across several binding sites, all of which might be separated by large energy barriers. We have developed a novel Monte Carlo move called Molecular Darting (MolDarting) to reversibly sample between predefined binding modes of a ligand. Here, we couple this with nonequilibrium candidate Monte Carlo (NCMC) to improve acceptance of moves.</div><div>We apply this technique to a simple dipeptide system, a ligand binding to T4 Lysozyme L99A, and ligand binding to HIV integrase in order to test this new method. We observe significant increases in acceptance compared to uniformly sampling the internal, and rotational/translational degrees of freedom in these systems.</div>

An entropy concentration theorem: applications in artificial intelligence and descriptive statistics

1990 ◽  
Vol 27 (2) ◽  
pp. 303-313 ◽  
Author(s):  
Claudine Robert
Keyword(s):  
Artificial Intelligence ◽  
Maximum Entropy ◽  
Large Deviation ◽  
Maximum Entropy Principle ◽  
Statistical Modelling ◽  
Linear Constraints ◽  
Descriptive Statistics ◽  
Entropy Principle ◽  
Maximum Entropy Distribution ◽  
Mathematical Justification

The maximum entropy principle is used to model uncertainty by a maximum entropy distribution, subject to some appropriate linear constraints. We give an entropy concentration theorem (whose demonstration is based on large deviation techniques) which is a mathematical justification of this statistical modelling principle. Then we indicate how it can be used in artificial intelligence, and how relevant prior knowledge is provided by some classical descriptive statistical methods. It appears furthermore that the maximum entropy principle yields to a natural binding between descriptive methods and some statistical structures.

Selection of Degrees of Freedom for Dynamic Analysis

1987 ◽  
Vol 109 (1) ◽  
pp. 65-69 ◽  
Author(s):  
K. W. Matta
Keyword(s):  
Finite Element ◽  
Dynamic Analysis ◽  
Degrees Of Freedom ◽  
General Purpose ◽  
Component Mode Synthesis ◽  
Complex Structures ◽  
Large Complex ◽  
Static Condensation ◽  
Basis Vectors ◽  
Selection Of

A technique for the selection of dynamic degrees of freedom (DDOF) of large, complex structures for dynamic analysis is described and the formulation of Ritz basis vectors for static condensation and component mode synthesis is presented. Generally, the selection of DDOF is left to the judgment of engineers. For large, complex structures, however, a danger of poor or improper selection of DDOF exists. An improper selection may result in singularity of the eigenvalue problem, or in missing some of the lower frequencies. This technique can be used to select the DDOF to reduce the size of large eigenproblems and to select the DDOF to eliminate the singularities of the assembled eigenproblem of component mode synthesis. The execution of this technique is discussed in this paper. Examples are given for using this technique in conjunction with a general purpose finite element computer program GENSAM[1].

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