Further developments and applications of the Green’s function method of sensitivity analysis in chemical kinetics

Efficient sensitivity analysis method for chaotic dynamical systems

Design sensitivity analysis of structures deals with the calculation of the response derivatives with respect to the design variables. These derivatives, called the sensitivity coefficients, are used in the solution of various problems. In design optimization, the sensitivity coefficients are often required to select a search direction. These coefficients are used also in generating approximations for the response of a modified system. In addition, the sensitivities are required for assessing the effects of uncertainties in the structural properties on the system response. Calculation of the sensitivities involves much computational effort, particularly in large structural systems with many design variables. As a result, there has been much interest in efficient procedures for calculating the sensitivity coefficients. Developments in methods for sensitivity analysis are discussed in many studies (e.g. Haug et. al. 1986; Haftka and Adelman, 1989; Haftka and Gurdal, 1993; van Keulen et. al. in press). Methods of sensitivity analysis for discretized systems can be divided into the following classes: a. Finite-difference methods, which are easy to implement but might involve numerous repeated analyses and high computational cost, particularly in problems with many design or response variables. In addition, finite-difference approximations might have accuracy problems. The efficiency can be improved by using fast reanalysis techniques. b. Analytical methods, which provide exact solutions but might not be easy to implement in some problems such as shape optimization. c. "Semi-analytical" methods, which are based on a compromise between finite-difference methods and analytical methods. These methods use finite-difference evaluation of the right-hand-side vector. They are easy to implement but might provide inaccurate results. In general, the following factors are considered in choosing a suitable sensitivity analysis method for a specific application: the accuracy of the calculations, the computational effort involved and the ease-of-implementation. The implementation effort is weighted against the performance of the algorithms as reflected in their computational efficiency and accuracy. The quality of the results and efficiency of the calculations are usually two conflicting factors. That is, higher accuracy is often achieved at the expense of more computational effort. Dynamic sensitivity analysis has been demonstrated by several authors. Using the mode superposition approach and assuming harmonic loading, the response sensitivities were evaluated by direct differentiation of the equations of motion in the generalized coordinates (Kramer and Grierson, 1989). In cases of earthquake loading the ground acceleration is usually given in discrete time steps, thus the loading is not given analytically. In several studies (Kim and Choi, 2000; van Keulen et. al, in press) the unconditionally stable implicit numerical equation was directly derived. It was found that the analysis equations and the sensitivity equations have the same left-hand side expression. Thus, it was possible to use the available factorized coefficient matrix. A numerical procedure was applied for calculation of the sensitivity of the response.

Perturbation mapping method for sensitivity analysis of three-dimensional cracks near a free surface

To estimate the uncertainty of the neutronic parameters it is required to calculate sensitivity coefficients to cross section changes. The sensitivity coefficients are usually calculated based on the diffusion theory. However, the accuracy of the sensitivity coefficients becomes doubtful because of the use of diffusion theory. The purpose of this paper is to derive the generalized perturbation theory using the transport theory, and to evaluate transport effect on the sensitivity coefficients. Sensitivity calculations were performed in 70 energy groups. The calculated sensitivity coefficients were compared with those obtained from direct transport calculations to check the accuracy of the present method, and those obtained from the generalized perturbation calculation based on the diffusion theory to investigate the transport effect on sensitivity coefficients. From the comparison, the difference was remarkably large for the neutronic parameters such as reaction rate distribution, reaction rate ratio, and reactivity worth. For example, the sensitivity coefficient of the 238U capture to 239Pu fission rate ratio in the lower core region of the under-moderated LWR core was large in the energy range 17.6–22.6 eV, and the sensitivity to 238U capture cross section calculated by the transport theory was 20% larger than that by the diffusion theory.

Statistical and experimental methods in sensitivity analysis

Rockwell hardness testing is a widely used hardness test, and there are many scales depending on the indenter shape and test force for corresponding to various materials. The common Rockwell indenters are the diamond sphero-conical indenter and the tungsten carbide ball indenter. The influence of indenter shape error on Rockwell hardness has been acknowledged, as this shape impacts test results. To accurately estimate this uncertainty, it is essential to understand the sensitivity coefficient, which delineates the relationship between indenter shape and Rockwell hardness. Traditionally, this coefficient has been derived experimentally, but this approach faces challenges that are attributable to the nonuniformity of hardness test blocks and indenter shape measurement errors. This study introduces the estimation of sensitivity coefficients for Rockwell indenters using finite element analysis (FEA). The FEA method offers the advantage of being unaffected by the nonuniformity of test blocks and indenter shape measurement errors. We calculated the sensitivity coefficients by assessing the influence of variations in indenter angle and tip radius for diamond indenters, and indenter diameter for ball indenters, on hardness results through FEA. The sensitivity coefficients obtained via FEA for specific scales aligned with those from previous experimental studies, affirming the validity of FEA-derived coefficients for other scales. This study demonstrates that FEA is a useful tool for estimating the sensitivity coefficients of Rockwell indenters.

This paper is concerned with the shape sensitivity analysis of a class of boundary control, constrained, optimal control problems for parabolic systems. The notion of Euler and Lagrange derivatives of a boundary optimal control in the direction of a vector field is introduced. The derivatives are obtained in the form of optimal solutions of auxiliary optimal control problems. The method of sensitivity analysis used in this paper is based on related results on the differential stability of metric projections in Hilbert space onto a convex, closed subset, combined with the material derivative method of shape sensitivity analysis. Parabolic initial-boundary value problems with Dirichlet and Neumann boundary conditions are considered in this paper.

The present work belongs to a bigger project whose main goal is to study the technical feasibility of the application of a relatively new heat treating concept, called Quenching and Partitioning (Q&P), as an alternative to the processing of high strength ductile cast irons. The aim of the Q&P process is to obtain multiphase microstructures consisting of martensite and carbon enriched retained austenite. Martensite confers high strength, whereas austenite confers ductility. In the Q&P process, after total or partial austenitization of the alloy, the material is quenched in a quenching temperature T Q between the Ms and Mf temperatures to produce a controlled mixture of martensite and austenite. Next, at the partitioning step, the material is isothermally held at a either equal or higher temperature (so called partitioning temperature T P ) in order to promote the carbon diffusion (partitioning) from martensite to austenite. The present work focus on the study of phase transformations aspects -with emphasis on the microstructural evolution and kinetics of the reactions -of the Q&P process applied to a ductile cast iron alloy (Fe-3,47%C-2,47%Si-0,2%Mn). Heat treatments consisted of austenitization at 880 C for 30 min, followed by quenching at 140, 170, and 200 C and partitioning at 300, 375 e 450 C up to 2 h. The microstructural characterization was carried out by optical microscopy (OM), scanning electron microscopy (SEM), electron backscattered diffraction (EBSD), and electron probe microanalysis (EPMA). The kinetic analysis was studied by high resolution dilatometry tests and in situ X-ray diffraction using a synchrotron light source. Results showed that competitive reactions -bainite reaction and carbides precipitation in martensite -is unavoidable during the Q&P process. The bainite reaction kinetics is accelerated by the presence of martensite formed in the quenching step. The bainite reaction occurs at low temperatures without carbides precipitation and contributes to the carbon enrichment of austenite and its stabilization. Due to carbides precipitation in martensite, growth of bainitic ferrite is the main mechanism of carbon enrichment of austenite. Microsegregation inherited from the casting process is present in the heat treated material and affects the martensite distribution and the kinetics of the bainite reaction. In regions corresponding to eutectic cell boundaries less martensite is observed and the kinetics of bainite reaction is slower. The final microestructure produced by the Q&P process applied to the ductile cast iron consists of tempered martensite with carbides, bainitic ferrite, and carbon enriched austenite. Additionally, a computational model was developed to calculate the local kinetics of carbon redistribution during the partitioning step, considering the ef-fects of carbides precipitation and bainite reaction. The model showed that the kinetics of carbon partitioning from martensite to austenite is slower when the tempering carbides are more stable and that, when the carbides free energy is sufficiently low, the carbon diffuses from austenite to martensite. The model is not limited to the studied conditions and can be applied to the development of Q&P heat treatments to steels.

Two-step Monte Carlo sensitivity analysis of alpha- and gamma-eigenvalues with the differential operator sampling method

The response and operation of a heat flux gage is studied using sensitivity analysis. Sensitivity analysis is the process by which one determines the sensitivity of a model output to changes in the model parameters. This process uses sensitivity coefficients, which are defined as partial derivatives of field variables (e.g. temperature) with respect to model parameters (e.g. thermal properties and boundary conditions). Computing sensitivity coefficients, in addition to the response of a heat flux gage, aids in identifying model parameters that significantly impact the temperature response. A control volume finite element based code is used to implement numerical sensitivity coefficient calculations, allowing general problems to be studied. Sensitivity coefficients are discussed for the well known Gardon gage.

Sensitivity analysis of CHF parameters under flow instability by using a neural network method

In this study, two-phase production history data is incorporated into reservoir characterization by inverse modeling. As a first step, an analytical method has been developed to compute sensitivity coefficients for two-phase flow conditions. These equations are independent of flow simulators, and can be incorporated to any finite difference or streamline simulators. They only require one simulation run to compute the sensitivities for all time steps. In this study, we have chosen streamline simulation, which is order of magnitude faster than finite difference simulations. In the second step, a dual loop iteration technique is employed which uses conjugate gradient relaxation in order to bypass the explicit construction of the sensitivity coefficient matrix and the inversion of the product of sensitivity coefficient matrix and its transpose. Thus, a substantial reduction in the computational cost for solving the inverse problem is achieved, including the help of streamline simulation. Finally, this technique was validated using synthetic as well as field cases. The field case study was conducted on a sandstone reservoir. P. 151

Sensitivity coefficients are calculated, for the simple thickness modes of quartz plates, due to perturbations of the elastic, piezoelectric, and dielectric material values. These are given in tabular form for various cuts of practical and analytic interest. Also shown are a number of sensitivity coefficient surfaces using the two plate orientational angles as variables.

To materialize actual designs in computer-aided design, we presented a method to determine structural dimensions in order to make all design requirements accomplish their respective objective values, using the attainability and simultaneous equations with several unknowns. However, most of the designs have no theoretical design formula or sensitivity coefficients. This paper presents an approximation method to identify sensitivity coefficients by trial production evaluations of the designs. The coefficient matrix is specified between k th revised values of the attainabilities and k th revised values of the design variables, and made into a square or a triangular matrix by neglecting insignificant variables. The matrix is solved iteratively until all coefficients converge. This paper also presents a multiobjective satisfactory design using the identified coefficients to prove the effectiveness of the identification, comparing the identified coefficients with the theoretical ones.

The standardized FAO56 Penman-Monteith model, which has been the most reasonable method in both humid and arid climatic conditions, provides reference evapotranspiration (ETo) estimates for planning and efficient use of agricultural water resources. And sensitivity analysis is important in understanding the relative importance of climatic variables to the variation of reference evapotranspiration. In this study, a non-dimensional relative sensitivity coefficient was employed to predict responses of ETo to perturbations of four climatic variables in the Ejina oasis northwest China. A 20-year historical dataset of daily air temperature, wind speed, relative humidity and daily sunshine duration in the Ejina oasis was used in the analysis. Results have shown that daily sensitivity coefficients exhibited large fluctuations during the growing season, and shortwave radiation was the most sensitive variable in general for the Ejina oasis, followed by air temperature, wind speed and relative humidity. According to this study, the response of ETo can be preferably predicted under perturbation of air temperature, wind speed, relative humidity and shortwave radiation by their sensitivity coefficients.