scholarly journals Approach for Input Uncertainty Propagation and Robust Design in CFD Using Sensitivity Derivatives1

2001 ◽  
Vol 124 (1) ◽  
pp. 60-69 ◽  
Author(s):  
Michele M. Putko ◽  
Arthur C. Taylor , ◽  
Perry A. Newman ◽  
Lawrence L. Green
Keyword(s):  
Robust Optimization ◽  
Input Parameter ◽  
Uncertainty Propagation ◽  
Statistical Moment ◽  
Second Order ◽  
Statistical Moments ◽  
Approximate Methods ◽  
Mean Values ◽  
Sensitivity Derivatives ◽  
Input Variables

An implementation of the approximate statistical moment method for uncertainty propagation and robust optimization for quasi 1-D Euler CFD code is presented. Given uncertainties in statistically independent, random, normally distributed input variables, first-and second-order statistical moment procedures are performed to approximate the uncertainty in the CFD output. Efficient calculation of both first- and second-order sensitivity derivatives is required. In order to assess the validity of the approximations, these moments are compared with statistical moments generated through Monte Carlo simulations. The uncertainties in the CFD input variables are also incorporated into a robust optimization procedure. For this optimization, statistical moments involving first-order sensitivity derivatives appear in the objective function and system constraints. Second-order sensitivity derivatives are used in a gradient-based search to successfully execute a robust optimization. The approximate methods used throughout the analyses are found to be valid when considering robustness about input parameter mean values.

Sampling-Based Stochastic Sensitivity Analysis Using Score Functions for RBDO Problems With Correlated Random Variables

2010 ◽  
Author(s):  
Ikjin Lee ◽  
Kyung K. Choi ◽  
Yoojeong Noh ◽  
Liang Zhao ◽  
David Gorsich
Keyword(s):  
Sensitivity Analysis ◽  
Surrogate Model ◽  
Computational Cost ◽  
Random Variables ◽  
Score Function ◽  
Statistical Moment ◽  
Statistical Moments ◽  
Mean Values ◽  
Stochastic Sensitivity ◽  
Stochastic Sensitivity Analysis

This study presents a methodology for computing stochastic sensitivities with respect to the design variables, which are the mean values of the input correlated random variables. Assuming that an accurate surrogate model is available, the proposed method calculates the component reliability, system reliability, or statistical moments and their sensitivities by applying Monte Carlo simulation (MCS) to the accurate surrogate model. Since the surrogate model is used, the computational cost for the stochastic sensitivity analysis is negligible. The copula is used to model the joint distribution of the correlated input random variables, and the score function is used to derive the stochastic sensitivities of reliability or statistical moments for the correlated random variables. An important merit of the proposed method is that it does not require the gradients of performance functions, which are known to be erroneous when obtained from the surrogate model, or the transformation from X-space to U-space for reliability analysis. Since no transformation is required and the reliability or statistical moment is calculated in X-space, there is no approximation or restriction in calculating the sensitivities of the reliability or statistical moment. Numerical results indicate that the proposed method can estimate the sensitivities of the reliability or statistical moments very accurately, even when the input random variables are correlated.

Sampling-Based Stochastic Sensitivity Analysis Using Score Functions for RBDO Problems With Correlated Random Variables

2011 ◽  
Vol 133 (2) ◽  
Author(s):  
Ikjin Lee ◽  
K. K. Choi ◽  
Yoojeong Noh ◽  
Liang Zhao ◽  
David Gorsich
Keyword(s):  
Sensitivity Analysis ◽  
Surrogate Model ◽  
Computational Cost ◽  
Random Variables ◽  
Score Function ◽  
Statistical Moment ◽  
Statistical Moments ◽  
Mean Values ◽  
Stochastic Sensitivity ◽  
Stochastic Sensitivity Analysis

This study presents a methodology for computing stochastic sensitivities with respect to the design variables, which are the mean values of the input correlated random variables. Assuming that an accurate surrogate model is available, the proposed method calculates the component reliability, system reliability, or statistical moments and their sensitivities by applying Monte Carlo simulation to the accurate surrogate model. Since the surrogate model is used, the computational cost for the stochastic sensitivity analysis is affordable compared with the use of actual models. The copula is used to model the joint distribution of the correlated input random variables, and the score function is used to derive the stochastic sensitivities of reliability or statistical moments for the correlated random variables. An important merit of the proposed method is that it does not require the gradients of performance functions, which are known to be erroneous when obtained from the surrogate model, or the transformation from X-space to U-space for reliability analysis. Since no transformation is required and the reliability or statistical moment is calculated in X-space, there is no approximation or restriction in calculating the sensitivities of the reliability or statistical moment. Numerical results indicate that the proposed method can estimate the sensitivities of the reliability or statistical moments very accurately, even when the input random variables are correlated.

scholarly journals Price, Volatility and the Second-Order Economic Theory

2021 ◽  
Vol 10 (1) ◽  
pp. 139-165
Author(s):  
Victor Olkhov
Keyword(s):  
Economic Theory ◽  
Probability Measure ◽  
Price Volatility ◽  
Statistical Moment ◽  
Second Order ◽  
Statistical Moments ◽  
Probability Measures ◽  
Time Interval ◽  
The Mean ◽  
Averaging Time

We introduce the new price probability measure, which entirely depends on the probability measures of the value and the volume of the market trades. We define the nth statistical moment of the price as the ratio of the nth statistical moment of the value to the nth statistical moment of the volume of all trades performed during an averaging time interval Δ. The set of the price statistical moments determines the price characteristic function and its Fourier transform defines the price probability measure. The price volatility depends on the 1st and the 2nd statistical moments of the value and the volume of the trades. The prediction of the price volatility requires a description of the sums of squares of the value and the volume of the market trades during the interval Δ and we call it the second-order economic theory. To develop that theory, we introduce numerical continuous risk ratings and distribute the agents by the risk ratings as coordinates. Based on distributions of the agents by the risk coordinates, we introduce a continuous economic media approximation that describes the collective trades. The agents perform the trades under the action of their expectations. We model the mutual impact of the expectations and the trades and derive equations that describe their evolution. To illustrate the benefits of our approach, in a linear approximation we describe perturbations of the mean price, the mean square price and the price volatility as functions of the first and the second-degree trades’ disturbances.

Radiant Heat Transfer From Isothermal Dispersions With Isotropic Scattering

1967 ◽  
Vol 89 (4) ◽  
pp. 300-308 ◽  
Author(s):  
R. H. Edwards ◽  
R. P. Bobco
Keyword(s):  
Heat Transfer ◽  
Approximate Solutions ◽  
Radiant Heat ◽  
Second Order ◽  
Isotropic Scattering ◽  
Radiant Heat Transfer ◽  
Approximate Methods ◽  
First Order ◽  
Order Solution ◽  
Radiant Transfer

Two approximate methods are presented for making radiant heat-transfer computations from gray, isothermal dispersions which absorb, emit, and scatter isotropically. The integrodifferential equation of radiant transfer is solved using moment techniques to obtain a first-order solution. A second-order solution is found by iteration. The approximate solutions are compared to exact solutions found in the literature of astrophysics for the case of a plane-parallel geometry. The exact and approximate solutions are both expressed in terms of directional and hemispherical emissivities at a boundary. The comparison for a slab, which is neither optically thin nor thick (τ = 1), indicates that the second-order solution is accurate to within 10 percent for both directional and hemispherical properties. These results suggest that relatively simple techniques may be used to make design computations for more complex geometries and boundary conditions.

Probability Aspects in Foundation Plate Design

2016 ◽  
Vol 837 ◽  
pp. 64-67
Author(s):  
Katarina Tvrda
Keyword(s):  
Probabilistic Methods ◽  
Random Input ◽  
Mean Values ◽  
Software Model ◽  
Input Parameters ◽  
Distribution Parameters ◽  
Input Variables ◽  
Uncertain Input ◽  
Foundation Plate ◽  
Distribution Type

The probabilistic design analyses a plate involving uncertain input parameters. These input parameters (geometry, material properties, boundary conditions, etc.) are defined in the software model. The variations of input parameters are defined as random input variables and are characterized by their distribution type (Gaussian, lognormal, etc.) and by their distribution parameters (mean values, standard deviation, etc.). During a probabilistic analysis, software executes multiple analysis loops to compute the random output parameters as a function of the set of random input variables. The values for the input variables are generated either randomly (using Monte Carlo simulation) or as prescribed samples (using Response Surface Methods). In the conclusion, some results of these probabilistic methods are presented.

scholarly journals Estimation of Tramadol Hydrochloride in Bulk and Formulation by Second Order Derivative Area Under Curve UV-Spectrophotometric Methods.

2015 ◽  
Vol 1 (7) ◽  
pp. 308
Author(s):  
Rekha Sudam Kharat
Keyword(s):  
Area Under Curve ◽  
Pharmaceutical Formulations ◽  
Second Order ◽  
Order Derivative ◽  
Tramadol Hydrochloride ◽  
Pharmaceutical Dosage Forms ◽  
Mean Values ◽  
Spectrophotometric Methods ◽  
Significant Difference

Simple, fast and reliable spectrophotometric methods were developed for determination of Tramadol Hydrochloride in bulk and pharmaceutical dosage forms. The solutions of standard and the sample were prepared in Distilled Water. The quantitative determination of the drug was carried out using the second order Derivative Area under Curve method values measured at 272-280nm. Calibration graphs constructed at their wavelengths of determination were linear in the concentration range of Tramadol Hydrochloride using 2-10?g/ml (r=0.9925) for second order Derivative Area under Curve spectrophotometric method. All the proposed methods have been extensively validated as per ICH guidelines. There was no significant difference between the performance of the proposed methods regarding the mean values and standard deviations. The developed methods were successfully applied to estimate the amount of Tramadol Hydrochloride in pharmaceutical formulations.

scholarly journals Estimation of Ofloxacin in Bulk and Formulation by Second Order DerivativeUV-Spectrophotometric Methods

2015 ◽  
Vol 1 (5) ◽  
pp. 217
Author(s):  
Shivaji Shinde ◽  
Santosh Jadhav ◽  
Rekha Kharat ◽  
Afaque Ansari ◽  
Ashpak Tamboli
Keyword(s):  
Area Under Curve ◽  
Pharmaceutical Formulations ◽  
Second Order ◽  
Order Derivative ◽  
Pharmaceutical Dosage Forms ◽  
Mean Values ◽  
Spectrophotometric Methods ◽  
Ich Guidelines ◽  
Significant Difference

Simple, fast and reliable spectrophotometric methods were developed for determination of Ofloxacin in bulk and pharmaceutical dosage forms. The solutions of standard and the sample were prepared in Methanol. The quantitative determination of the drug was carried out using the second order Derivative Area under Curve method values measured at 295-301nm. Calibration graphs constructed at their wavelengths of determination were linear in the concentration range of Ofloxacin using 2-10?g/ml (r=0.9947) for second order Derivative Area under Curve spectrophotometric method. All the proposed methods have been extensively validated as per ICH guidelines. There was no significant difference between the performance of the proposed methods regarding the mean values and standard deviations. The developed methods were successfully applied to estimate the amount of Ofloxacin in pharmaceutical formulations.

Multiobjective Collaborative Robust Optimization With Interval Uncertainty and Interdisciplinary Uncertainty Propagation

2008 ◽  
Vol 130 (8) ◽  
Author(s):  
M. Li ◽  
S. Azarm
Keyword(s):  
Robust Optimization ◽  
Probability Distributions ◽  
Uncertainty Propagation ◽  
Multiple Objectives ◽  
Interval Uncertainty ◽  
Optimization Approach ◽  
Solution Approach ◽  
Time In Literature ◽  
Collaborative Robust Optimization ◽  
First Time

We present a new solution approach for multidisciplinary design optimization (MDO) problems that, for the first time in literature, has all of the following characteristics: Each discipline has multiple objectives and constraints with mixed continuous-discrete variables; uncertainty exists in parameters and as a result, uncertainty propagation exists within and across disciplines; probability distributions of uncertain parameters are not available but their interval of uncertainty is known; and disciplines can be fully (two-way) coupled. The proposed multiobjective collaborative robust optimization (McRO) approach uses a multiobjective genetic algorithm as an optimizer. McRO obtains solutions that are as best as possible in a multiobjective and multidisciplinary sense. Moreover, for McRO solutions, the variation of objective and/or constraint functions can be kept within an acceptable range. McRO includes a technique for interdisciplinary uncertainty propagation. The approach can be used for robust optimization of MDO problems with multiple objectives, or constraints, or both together at system and subsystem levels. Results from an application of McRO to a numerical and an engineering example are presented. It is concluded that McRO can solve fully coupled MDO problems with interval uncertainty and obtain solutions that are comparable to a single-disciplinary robust optimization approach.

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