First-order Sensitivity Coefficients Research Articles

Discharge modeling and characteristic analysis of semi-circular side weir based on the soft computing method

Abstract In this study, a support vector machine (SVM) and three optimization algorithms are used to develop a discharge coefficient (Cd) prediction model for the semi-circular side weir (SCSW). After that, we derived the input and output parameters of the model by dimensionless analysis as the ratio of the flow depth at the weir crest point upstream to the diameter (h1/D), the ratio of main channel width to diameter (B/D), the ratio of side weir height to diameter (P/D), upstream of side weir Froude number (Fr), and Cd. The sensitivity coefficients for dimensionless parameters to Cd were calculated based on Sobol's method. The research shows that SVM and Genetic Algorithm (GA-SVM) have high prediction accuracy and generalization ability; the average error and maximum error were 0.08 and 2.47%, respectively, which were about 95.72 and 60.86% lower compared with the traditional empirical model. The first-order sensitivity coefficients S1 and global sensitivity coefficients Si of h1/D, B/D, P/D, and Fr were 0.35, 0.07, 0.13, and 0.02; 0.63, 0.25, 0.30, and 0.32, respectively. h1/D has a significant effect on Cd. In particular, when h1/D < 0.24 and 0.48 < Fr < 0.58, 0.67 < Fr < 0.72, the discharge capacity of the SCSW is relatively large.

Standard dynamic energy budget model parameter sensitivity

The Dynamic Energy Budget model describes energy and mass balance in living organisms. It has found many applications in biological and ecological sciences. The model parameters can be connected to a single underlying biological process, and this mechanistic approach is helpful in understanding how complex biological systems work. However, the large number of model parameters makes it difficult to estimate their value, especially from data limited only to adult growth and reproduction at abundant food and constant temperature. Therefore, in this study, the sensitivity of the model solution to primary parameter values and the sensitivity of parameters to the perturbation in the data were analyzed. It was shown that the first-order sensitivity coefficients are different for each of the analyzed primary parameters and depend on their values and configuration in the equation. The sensitivity of parameters to data changes across analyzed time intervals, reaching minima and maxima. Moreover, the influence of each data point is smaller with an increasing number of data points. The recognition of the impact of parameters on the model solution, as well as the identification of data points with the strongest influence on estimates, can be helpful in experimental design and evaluation of the model.

Sensitivity modeling of ozone and its precursors over the Chengdu metropolitan area

The comprehensive air quality model with extensions (CAMx)–high-order decoupled direct method (HDDM) was used to simulate ozone (O3)-NOx-VOCs sensitivity from June 1 to September 30 in 2019 over the Chengdu metropolitan area. In terms of the diurnal variations of O3 sensitivities to NOx and VOCs emissions, most cities in the Chengdu metropolitan area, except Ya'an, Suining and Ziyang, show a shift in O3 sensitivity from NOx-limited to VOC-limited from daytime to nighttime, when 8:00 and 19:00 were the cut-off points for the switch in O3-sensitive regimes. The average daily maximum 8-h O3 sensitivity to regional NOx and VOCs emissions show that the Chengdu metropolitan area is mainly NOx-limited. The O3 sensitivity to local emissions in urban areas of each city is VOC-limited or in a transition regime, while O3 sensitivities to overall regional emissions are prone to NOx-limited. Therefore, when proposing O3 control strategies, it is necessary to consider the different measures for the local and regional emissions. First-order sensitivity coefficients combined with the path integral method (PIM) are used to calculate the contributions of precursors' emissions to O3 concentrations in different sectors, revealing that mobile source accounted for 31.1–36.9%, followed by industrial source, with contribution of 22.2–30.5% to O3 concentrations. Finally, five emission reduction scenarios were designed with different NOx/VOC reduction ratios, and O3 concentration changes with different NOx/VOC reduction ratios were calculated based on O3 sensitivity coefficients. The results show that the influence of local emission reductions on the changes in O3 concentrations varies significantly among cities. The decrease in O3 concentrations reach maximum when the NOx/VOC reduction ratio is 3:1 in most of the cities, except Chengdu and Mianyang, with the highest O3 decrement occurs at the NOx/VOC reduction ratio of 1:3.

Sensitivity Analysis for Verification of an Anaerobic Digestion Model

In this article, the first-order sensitivity coefficients for a system of stiff nonlinear differential equations from the anaerobic digestion model are calculated. In this approach, the auxiliary equations used to calculate the sensitivity coefficients are solved separately from the model equations, using the same time steps and numerical approximations used in the calculation of the model solution by the Rosenbrock method of order four. The results show the importance of each reaction for each species involved in the model.

Research of mathematical models for minimizing power shortage with quadratic losses in power lines and with using network coefficients (sensitivity coefficients)

The article presents an analysis of the models for minimizing the power shortage used in adequacy assessment by the Monte Carlo method. We analysed two models: a model with quadratic losses in power lines and a model using network coefficients (first-order sensitivity coefficients), that are showing the dependence of the change in power flows through power lines on the balance of generation / consumption in the reliability zones. As a result of the analysis, a conclusion is made about the applicability of the investigated models for adequacy assessment in the long-term planning of the development of electric power systems.

Ambiguities in the Sensitivity and Uncertainty Analysis of Reactor Physics Problems Involving Constrained Quantities

The nuclear community relies heavily on computer codes both in research and in the operation of installations. The results of such computations are useful only if they are augmented with sensitivity and uncertainty studies. This technical note presents some theoretical considerations regarding traditional first-order sensitivity analysis and uncertainty quantification involving constrained quantities. The focus is on linear constraints, which are often encountered in reactor physics problems due to energy and angle distributions, or the correlation between the isotopic abundances of elements.A consistent theory is given for the derivation and interpretation of constrained first-order sensitivity coefficients; covariance matrix normalization procedures; their interrelation; and the treatment of constrained inputs with polynomial chaos expansion, which was the main motivation of this research. It is shown that if the covariance matrix violates the “generic zero column and row sum” condition, normalizing it is equivalent to constraining the sensitivities, but since both can be done in many ways, different sensitivity coefficients and uncertainties can be derived. This makes results ambiguous, underlining the need for proper covariance data. Furthermore, it is highlighted that certain constraining procedures can result in biased or unphysical uncertainty estimates. To confirm our conclusions, we demonstrate the presented theory on three analytical and two numerical examples including fission spectrum, isotopic distribution, and power distribution-related uncertainties, as well as the correlation between mass, volume, and density.

Path-integral method for the source apportionment of photochemical pollutants

Abstract. A new, path-integral method is presented for apportioning the concentrations of pollutants predicted by a photochemical model to emissions from different sources. A novel feature of the method is that it can apportion the difference in a species concentration between two simulations. For example, the anthropogenic ozone increment, which is the difference between a simulation with all emissions present and another simulation with only the background (e.g., biogenic) emissions included, can be allocated to the anthropogenic emission sources. The method is based on an existing, exact mathematical equation. This equation is applied to relate the concentration difference between simulations to line or path integrals of first-order sensitivity coefficients. The sensitivities describe the effects of changing the emissions and are accurately calculated by the decoupled direct method. The path represents a continuous variation of emissions between the two simulations, and each path can be viewed as a separate emission-control strategy. The method does not require auxiliary assumptions, e.g., whether ozone formation is limited by the availability of volatile organic compounds (VOCs) or nitrogen oxides (NOx), and can be used for all the species predicted by the model. A simplified configuration of the Comprehensive Air Quality Model with Extensions (CAMx) is used to evaluate the accuracy of different numerical integration procedures and the dependence of the source contributions on the path. A Gauss–Legendre formula using three or four points along the path gives good accuracy for apportioning the anthropogenic increments of ozone, nitrogen dioxide, formaldehyde, and nitric acid. Source contributions to these increments were obtained for paths representing proportional control of all anthropogenic emissions together, control of NOx emissions before VOC emissions, and control of VOC emissions before NOx emissions. There are similarities in the source contributions from the three paths but also differences due to the different chemical regimes resulting from the emission-control strategies.

Source Apportionment of the Anthropogenic Increment to Ozone, Formaldehyde, and Nitrogen Dioxide by the Path-Integral Method in a 3D Model.

The anthropogenic increment of a species is the difference in concentration between a base-case simulation with all emissions included and a background simulation without the anthropogenic emissions. The Path-Integral Method (PIM) is a new technique that can determine the contributions of individual anthropogenic sources to this increment. The PIM was applied to a simulation of O3 formation in July 2030 in the U.S., using the Comprehensive Air Quality Model with Extensions and assuming advanced controls on light-duty vehicles (LDVs) and other sources. The PIM determines the source contributions by integrating first-order sensitivity coefficients over a range of emissions, a path, from the background case to the base case. There are many potential paths, with each representing a specific emission-control strategy leading to zero anthropogenic emissions, i.e., controlling all sources together versus controlling some source(s) preferentially are different paths. Three paths were considered, and the O3, formaldehyde, and NO2 anthropogenic increments were apportioned to five source categories. At rural and urban sites in the eastern U.S. and for all three paths, point sources typically have the largest contribution to the O3 and NO2 anthropogenic increments, and either LDVs or area sources, the smallest. Results for formaldehyde are more complex.

Decoupled direct sensitivity analysis of regional ozone pollution over the Pearl River Delta during the PRIDE-PRD2004 campaign

Abstract High-order decoupled direct method is applied to investigate O3-precursor sensitivity during the Program of Regional Integrated Experiments of Air Quality over the Pearl River Delta (PRD) Region, China in October 2004 (PRIDE-PRD2004). First-order sensitivity coefficients of O3 show a NOx-inhibited chemistry along the polluted plumes in central and southern PRD areas, and a NOx-limited condition over larger areas of the western, eastern and northern PRD. A nonlinearity ratio derived from O3 sensitivity coefficients clearly describes the nonlinear feature of O3 chemistry over PRD. The distributions of NOx- and VOC-sensitive regimes are determined by near-ground flow conditions, and O3 sensitivity is found to shift from VOC-sensitive conditions to NOx-sensitive conditions from morning to afternoon in central and western areas of the PRD. Elevated surface O3 around mid-afternoon in October mainly occurred in VOC-sensitive regions or those NOx-limited areas in close vicinity of the VOC-sensitive regions. Ratios of H2O2/HNO3 and peroxides/HNO3 for indicating O3-precursor sensitivity are evaluated and the transition values of 0.45 for H2O2/HNO3 and 0.80 for peroxides/HNO3 were found, with an overall accuracy of about 92% for sensitivity identification at elevated O3 hours in afternoons during the campaign.

Using Orthogonal Arrays in the Sensitivity Analysis of Computer Models

We consider a class of input sampling plans, called permuted column sampling plans, that are popular in sensitivity analysis of computer models. Permuted column plans, including replicated Latin hypercube sampling, support estimation of first-order sensitivity coefficients, but these estimates are biased when the usual practice of random column permutation is used to construct the sampling arrays. Deterministic column permutations may be used to eliminate this estimation bias. We prove that any permuted column sampling plan that eliminates estimation bias, using the smallest possible number of runs in each array and containing the largest possible number of arrays, can be characterized by an orthogonal array of strength 2. We derive approximate standard errors of the first-order sensitivityindices for this sampling plan. We give two examples demonstrating the sampling plan, behavior of the estimates, and standard errors, along with comparative results based on other approaches.

A direct method of calculating sensitivity coefficients of chemical kinetics

In this paper, a new direct method of calculating the first-order sensitivity coefficients using sparse matrix technology to chemical kinetics is presented. The Gear type procedure is used to integrate a model equation and its coupled auxiliary sensitivity coefficient equations. Because the Jacobian matrix of the model equation is the same as that of the sensitivity coefficient equation with respect to a parameter, it is only necessary to triangularize the matrix related to the Jacobian matrix of the model equation. The FORTRAN subroutines of the model equation, the sensitivity coefficient equations, and their Jacobian analytical expressions are generated automatically from chemical mechanism. This method greatly increases the efficiency of computation by taking advantage of the fact that the auxiliary equations for different sensitivity coefficients are linear and quite similar. Two sets of chemical reactions are used to illustrate this approach: oxidation of formaldehyde in the presence of carbon monoxide and photo-oxidation of dimethyl sulfide. The accuracy and computational efficiency of the new direct method is demonstrated by comparing the results from the new direct method and from the indirect method.

Neural Estimation of First-Order Sensitivity Coefficients: Application to the Control of a Simulated Pressurized Water Reactor

In real, complex plants, a sensitivity analysis of the effects that variations in the plant inputs and design parameters have on the outputs is of great importance both from the point of view of productivity and of safety. To a first approximation, sensitivity analysis consists of estimating the partial derivatives of the outputs with respect to the varied quantities. These derivatives cannot be obtained on the real plant directly since the effects of all the involved variables are intermixed. Therefore, one has to resort to suitable computational models and algorithms.A new neural network approach that aims at creating a differentiable copy of the plant is proposed. A feature of the method is that the data for network training are collected with the system in nominal operation: This represents, indeed, a fundamental constraint for all risky plants, for which unrestrained playing is definitely not recommended. The sensitivity coefficients (partial derivatives) thereby obtained are applied for the regulation of the reactivity of a simulated pressurized water reactor in response to changes in the electric load at the power grid, so as to maintain the average temperature of the water in the reactor core at a constant value.

Use of sensitivity analysis methods in the modelling of electrochemical transients.

Abstract We propose a systematic and mathematically rigorous procedure of performing the expansion or reduction of kinetic models arising in the area of electrochemical transient modelling. The procedure is based on the use of sensitivity analysis and involves calculation of first-order sensitivity coefficients of the transient curves, with respect to the parameters of various candidate or inherent model components (e.g. rate constants of elementary reactions in the reaction mechanisms). By confronting the sensitivity coefficients with the deviations of simulated transients from experimental curves, decisions can be taken as to whether a particular model component should be added to or removed from the model. The utility of this procedure is verified by its application to selected cyclic voltammetric data for two example electrochemical problems: anodic oxidation of 2, 2′, 3, 3′, 4, 4′, 5, 5′-octamethyl-1,1′-bipyrrole in dichloromethane at a Pt stationary disk electrode (model expansion from E rev to E rev C irr and cathodic reduction of bis([2 2 ] paracyclophane) Ru(II) bis(tetrafluoroborate) in propylene carbonate at a glassy carbon stationary disk electrode (model reduction from E rev E rev -DISP to E rev E rev ). The procedure may become part of a wider, largely automated computer-aided strategy of developing kinetic models in electrochemistry.

Reduced basis technique for evaluating the sensitivity of the nonlinear vibrational response of composite plates

A reduced basis technique and a computational procedure are presented for generating the nonlinear vibrational response, and evaluating the first-order sensitivity coefficients of composite plates (derivatives of the nonlinear frequency with respect to material and geometric parameters of the plate). The analytical formulation is based on a form of the geometrically nonlinear shallow shell theory with the effects of transverse shear deformation, rotatory inertia and anisotropic material behavior included. The plate is discretized by using mixed finite element models with the fundamental unknowns consisting of both the nodal displacements and the stress-resultant parameters of the plate. The computational procedure can be conveniently divided into three distinct steps. The first step involves the generation of various-order perturbation vectors, and their derivatives with respect to the material and lamination parameters of the plate, using Linstedt-Poincaré perturbation technique. The second step consists of using the perturbation vectors as basis vectors, computing the amplitudes of these vectors and the nonlinear frequency of vibration, via a direct variational procedure. The third step consists of using the perturbation vectors, and their derivatives, as basis vectors and computing the sensitivity coefficients of the nonlinear frequency via a second application of the direct variational procedure. The effectiveness of the proposed technique is demonstrated by means of numerical examples of composite plates.

Nonlinear Vibrations of Thin-Walled Composite Frames

A reduced basis technique and a computational procedure are presented for generating the nonlinear vibrational response, and evaluating the first-order sensitivity coefficients of thin-walled composite frames. The sensitivity coefficients are the derivatives of the nonlinear frequency with respect to the material and lamination parameters of the frame. A mixed formulation is used with the fundamental unknowns consisting of both the generalized displacements and stress resultants in the frame. The flanges and webs of the frames are modeled by using geometrically nonlinear two-dimensional shell and plate finite elements. The computational procedure can be conveniently divided into three distinct steps. The first step involves the generation of various-order perturbation vectors, and their derivatives with respect to the material and lamination parameters of the frame, using the Linstedt–Poincaré perturbation technique. The second step consists of using the perturbation vectors as basis vectors, computing the amplitudes of these vectors and the nonlinear frequency of vibration, via a direct variational procedure. The third step consists of using the perturbation vectors, and their derivatives, as basis vectors and computing the sensitivity coefficients of the nonlinear frequency via a second application of the direct variational procedure. Numerical results are presented for semicircular thin-walled frames with I and J sections, showing the convergence of the nonlinear frequency and the sensitivity coefficients obtained by both the reduced-basis and perturbation techniques.

Nonlinear Vibrations of Thin-Walled Composite Frames

A reduced basis technique and a computational procedure are presented for generating the nonlinear vibrational response, and evaluating the first-order sensitivity coefficients of thin-walled composite frames. The sensitivity coefficients are the derivatives of the nonlinear frequency with respect to the material and lamination parameters of the frame. A mixed formulation is used with the fundamental unknowns consisting of both the generalized displacements and stress resultants in the frame. The flanges and webs of the frames are modeled by using geometrically nonlinear two-dimensional shell and plate finite elements. The computational procedure can be conveniently divided into three distinct steps. The first step involves the generation of various-order perturbation vectors, and their derivatives with respect to the material and lamination parameters of the frame, using the Linstedt–Poincare perturbation technique. The second step consists of using the perturbation vectors as basis vectors, computing the amplitudes of these vectors and the nonlinear frequency of vibration, via a direct variational procedure. The third step consists of using the perturbation vectors, and their derivatives, as basis vectors and computing the sensitivity coefficients of the nonlinear frequency via a second application of the direct variational procedure. Numerical results are presented for semicircular thin-walled frames with I and J sections, showing the convergence of the nonlinear frequency and the sensitivity coefficients obtained by both the reduced-basis and perturbation techniques.

Hybrid analytical technique for evaluating the sensitivity of the nonlinear vibrational response of beams

A three-step hybrid analytical technique is presented for evaluating the nonlinear vibrational response as well as the first-order sensitivity coefficients of thin-walled beams (derivatives of the nonlinear frequency with respect to material and geometric parameters of the beam). The first step involves the generation of various-order perturbation functions, and their derivatives with respect to the material and geometric parameters of the beam, using the Linstedt-Poincaré perturbation technique. The second step consists of using the perturbation functions as coordinate (or approximation) functions and then computing the amplitudes of these functions and the nonlinear frequency via a direct variational procedure. The third step consists of using the perturbation functions, and their derivatives, as coordinate functions and computing the sensitivity coefficients of the nonlinear frequency via a second application of the direct variational procedure. The analytical formulation is based on a form of the geometrically nonlinear beam theory with the effects of in-plane inertia, rotatory inertia, transverse shear deformation, and cross-sectional warping included. The effectiveness of the proposed technique is demonstrated by means of a numerical example of thin-walled beam with a doubly symmetric I-section.

Application of Sensitivity Analysis to Premixed Hydrogen-Air Flames

In this paper an efficient method for calculating the sensitivity coefficients of a system of nonlinear two-point boundary value problems is applied to a set of freely propagating, atmospheric pressure, premixed, hydrogen-air flames. The procedure utilizes the Jacobian matrix in Newton's method to generate efficiently first-order sensitivity coefficients. In our analysis we illustrate the effect of variations in the reaction rates and the transport coefficients on the adiabatic flame speed, the temperature and the species profiles. We discuss how the existence of scaling and self-similarity relations among the sensitivity coefficients can imply that relatively simple behavior can exist in complex flames. We also make specific predictions regarding the response of the flame thickness to parametric and species flux disturbances. The sensitivity analysis presented in the paper can also be applied to other more complicated systems. In this way the important kinetic and transport processes in such flames can be determined accurately and efficiently.

Algorithm 658

ODESSA is a package of FORTRAN routines for simultaneous solution of ordinary differential equations and the associated first-order parametric sensitivity equations, yielding the ODE solution vector y _( t ) and the first-order sensitivity coefficients with respect to equation parameters p _, ∂ y _( t )/∂ p _. ODESSA is a modification of the widely disseminated initial-value solver LSODE, and retains many of the same operational features. Standard program usage and optional capabilities, installation, and verification considerations are addressed herein.

The decoupled direct method for calculating sensitivity coefficients in chemical kinetics

A version of the direct method for calculating first-order sensitivity coefficients is extended to nonlinear, time-dependent models defined by stiff differential equations. In this approach the auxiliary equations for the sensitivity coefficients are solved separately from the model equations. Accuracy and stability are maintained by using exactly the same time steps and numerical approximations in calculating the sensitivities as are used in calculating the model solution. The decoupling procedure also greatly increases the efficiency of the method by taking advantage of the fact that the auxiliary equations for different sensitivity coefficients are quite similar. The decoupled direct method is applied to stiff chemical mechanisms for the oxidation of hydrocarbons in the atmosphere, the pyrolysis of ethane, and the oxidation of formaldehyde in the presence of carbon monoxide. Sensitivity coefficients are also calculated for the three mechanisms by a method employing Green’s function and by actually varying the input parameters. Based on these results, the decoupled direct method has advantages in simplicity, stability, accuracy, efficiency, storage requirements, and program size over other methods, including those using Green’s function. Specifically, the decoupled direct method is as much as a factor of 6 more efficient than a recent version of the Green’s function method. Extensions of the decoupled direct method are also discussed.