Rational Approximation Method Research Articles

Study on a fractional-order controllers based on best rational approximation of fractional calculus operators

This paper presents a rational approximation method for fractional calculus operators <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo>±</mml:mo><mml:mi mathvariant="normal">α</mml:mi></mml:mrow></mml:msup></mml:math>in the given frequency range and the error, which is based on the best rational approximation definition. The fractional integral operator is selected as an example to describe the construction of the rational approximation functions. An application case (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>0.3</mml:mn></mml:mrow></mml:msup></mml:math>) is used to illustrate the effectiveness of the proposed method. The obtained approximation function in the frequency domain is a best rational approximation function, which can further improve the accuracy of the approximation without increasing the orders. On the basis of the presented rational approximation method, a rational approximation equation of a fractional-order PID controller is obtained. Finally, the method for analyzing the optimization and frequency characteristics of the fractional-order controller is implemented to demonstrate the good frequency characteristic and best structure. The results from theoretical analysis and experimental verification show that the proposed method provides a new design idea for the effective application of the fractional-order PID controller in engineering.

Calculating Time-Integral Quantities in Depletion Calculations

A method referred to as tally nuclides is presented for accurately and efficiently calculating the time-step averages and integrals of any quantities that are weighted sums of atomic densities with constant weights during the step. The method allows all such quantities to be calculated simultaneously as a part of a single depletion solution with existing depletion algorithms. Examples of results that can be extracted include step-average atomic densities and macroscopic reaction rates, the total number of fissions during the step, and the amount of energy released during the step. The method should be applicable with several depletion algorithms, and the integrals or averages should be calculated with an accuracy comparable to that reached by the selected algorithm for end-of-step atomic densities. The accuracy of the method is demonstrated in depletion calculations using the Chebyshev rational approximation method. As an example of a possible use, we demonstrate how the ability to calculate energy release in depletion calculations can be used to determine the accuracy of the normalization in a constant-power burnup calculation during the calculation without a need for a reference solution.

A Rational Approximation Method of Fractional Order Differentiators and Integrators Based On Joint Optimal Criterion

Huimin Zhao, Chen Guo, Wu Deng, Dongyan Li, Xinhua Yang and Li Zou Information Science and Technology College of Dalian Maritime University, Dalian 116026, China 2 Software Institute,Dalian Jiaotong University, Dalian 116028, China 3 Sichuan Provincial Key Lab of Process Equipment andControl (Sichuan University of Science and Engineering), Zigong 64300, China Artificial Intelligence Key Laboratory of Sichuan Province(Sichuan University of Science and Engineering) ,Zigong 64300, China E-mail: dw7689@163.com

Model-Based Vector-Fitting Method for Circuit Model Extraction of Coupled-Resonator Diplexers

In this paper, a novel rational function approximation method, namely, model-based vector fitting (MVF), is proposed for accurate extraction of the characteristic functions of a coupled-resonator diplexer with a resonant type of junction from noise-contaminated measurement data. MVF inherits all the merits of the vector-fitting (VF) method and can also stipulate the order of the numerator of the model. Thus, MVF is suitable for the high-order diplexer system identification problem against measurement noise. With the extracted characteristic functions, a three-port transversal coupling matrix of a diplexer can be synthesized. A matrix orthogonal transformation strategy is also proposed to transform the obtained transversal matrix to a target coupling matrix configuration, whose entries have one-to-one relationship with the physical tuning elements. The whole model extraction procedure is analytical and robust, and can be used in a computer-aided tuning (CAT) program for coupled-resonator diplexers. A practical tuning example of a diplexer with a common resonator is given in detail to demonstrate the effectiveness and the practical value of the proposed method.

Improving the Accuracy of the Chebyshev Rational Approximation Method Using Substeps

The Chebyshev rational approximation method (CRAM) for solving the decay and depletion of nuclides is shown to have a remarkable decrease in error when advancing the system with the same time step and microscopic reaction rates as the previous step. This property is exploited here to achieve high accuracy in any end-of-step solution by dividing a step into equidistant substeps. The computational cost of identical substeps can be reduced significantly below that of an equal number of regular steps, as the lower-upper decompositions for the linear solutions required in CRAM need to be formed only on the first substep. The improved accuracy provided by substeps is most relevant in decay calculations, where there have previously been concerns about the accuracy and generality of CRAM. With substeps, CRAM can solve any decay or depletion problem with constant microscopic reaction rates to an extremely high accuracy for all nuclides with concentrations above an arbitrary limit.

Nuclear data uncertainty propagation analysis for depletion calculation in PWR and FR pin-cells

In order to assess the nuclear data uncertainty propagation in the depletion calculation, a computational code named SUNDEW has been developed based on the home-developed lattice code NECP-CACTI. In the SUNDEW, Generalized Perturbation Theory (GPT) is applied to calculate sensitivity coefficients of response function with respect to the nuclear cross sections. Method of Characteristics (MOC) is employed to solve the transport equation, adjoint and generalized adjoint transport equations. Chebyshev Rational Approximation Method (CRAM) is implemented to solve the depletion equation and adjoint depletion equation. The sensitivity coefficients of Keff and nuclide density with respect to the nuclear cross sections are verified by comparing with the results of direct perturbation calculation. The uncertainties on Keff and the nuclide density at different depletions, which are induced by the nuclear cross sections uncertainties, are analyzed based on ENDF/B-VII.1 covariance data for LWR and fast reactor pin-cells. The numerical results show that there are significant differences between LWR and fast reactor pin-cells. The differences are mainly caused by the differences of the sensitivity coefficients between the LWR and fast reactor pin-cells. In addition, to identify the cross section improvement priority for nuclides, reactions and energy ranges, the dominant contributors to Keff and nuclide density uncertainties are analyzed at different depletions.

Higher-Order Chebyshev Rational Approximation Method and Application to Burnup Equations

The burnup equations can, in principle, be solved by computing the exponential of the burnup matrix. However, the problem is extremely stiff, and the matrix exponential solution was long considered infeasible for entire burnup systems containing short-lived nuclides. After discovering that the eigenvalues of burnup matrices are confined to the vicinity of the negative real axis, the Chebyshev rational approximation method (CRAM) was introduced for solving the burnup equations and it was shown to be capable of providing accurate and efficient solutions without the need to exclude the short-lived nuclides. The main difficulty in using CRAM is determining the coefficients of the rational approximant for a given approximation order, with the previously published coefficients enabling only approximations up to order 16 for computing the matrix exponential. In this paper, a Remez-type method is presented for the computation of higher-order CRAM approximations. The optimal form of CRAM for the solution of burnup equations is discussed, and the method of incomplete partial fractions is proposed for this purpose. The CRAM coefficients based on this factorization are provided for approximation orders 4, 8, 12, …, 48. The accuracy of the method is demonstrated by applying it to large burnup and decay systems. It is shown that higher-order CRAM can be used to solve the burnup equations accurately for time steps of the order of 1 million years.

Nonlinear reanalysis for structural modifications based on residual increment approximations

This paper presents a reanalysis method for nonlinear problems. The main objective is to predict the changes in the state variables due to given design modifications, e.g. changes in the cross-sections, the geometry, material parameters etc. The approach is based on residual increment approximations, which are used within an iterative algorithm. The reanalysis method requires only the evaluation of residual vectors, which can be done very fast and efficient. Moreover, the validity of the approach is extended by using a rational approximation method. In contrast to other existing reanalysis methods, which are based on the evaluation of changed stiffness matrices, only residual vectors have to be computed and stored. The approach is general and can be applied to linear and nonlinear problems with different kind of design modifications. Furthermore, the proposed reanalysis method is easy to implement in existing finite element programs, because no derivatives with respect to the design variables are necessary. The capability of the proposed framework is demonstrated by means of several computational examples from nonlinear elasticity.

Numerical stability for nonlinear evolution equations

The paper deals with discretisation methods for nonlinear operator equations written as abstract nonlinear evolution equations. Brezis and Pazy showed that the solution of such problems is given by nonlinear semigroups whose theory was founded by Crandall and Liggett. By using the approximation theorem of Brezis and Pazy, we show the N-stability of the abstract nonlinear discrete problem for the implicit Euler method. Motivated by the rational approximation methods for linear semigroups, we propose a more general time discretisation method and prove its N-stability as well.

A method for including external feed in depletion calculations with CRAM and implementation into ORIGEN

A method for including external feed with polynomial time dependence in depletion calculations with the Chebyshev Rational Approximation Method (CRAM) is presented and the implementation of CRAM to the ORIGEN module of the SCALE suite is described. In addition to being able to handle time-dependent feed rates, the new solver also adds the capability to perform adjoint calculations. Results obtained with the new CRAM solver and the original depletion solver of ORIGEN are compared to high precision reference calculations, which shows the new solver to be orders of magnitude more accurate. Furthermore, in most cases, the new solver is up to several times faster due to not requiring similar substepping as the original one.

Numerical solution of matrix exponential in burn-up equation using mini-max polynomial approximation

Nuclear fuel burn-up depletion calculations are essential to compute the nuclear fuel composition transition. In the burn-up calculations, the matrix exponential method has been widely used. In the present paper, we propose a new numerical solution of the matrix exponential, a Mini-Max Polynomial Approximation (MMPA) method. This method is numerically stable for burn-up matrices with extremely short half-lived nuclides as the Chebyshev Rational Approximation Method (CRAM), and it has several advantages over CRAM. We also propose a multi-step calculation, a computational time reduction scheme of the MMPA method, which can perform simultaneously burn-up calculations with several time periods. The applicability of these methods has been theoretically and numerically proved for general burn-up matrices. The numerical verification has been performed, and it has been shown that these methods have high precision equivalent to CRAM.

THE INVESTIGATION OF BURNUP CHARACTERISTICS USING THE SERPENT MONTE CARLO CODE FOR A SODIUM COOLED FAST REACTOR

In this research, we investigated the burnup characteristics and the conversion of fertile 232Th into fissile 233U in the core of a Sodium-Cooled Fast Reactor (SFR). The SFR fuel assemblies were designed for burning 232Th fuel (fuel pin 1) and 233U fuel (fuel pin 2) and include mixed minor actinide compositions. Monte Carlo simulations were performed using Serpent Code1.1.19 to compare with CRAM (Chebyshev Rational Approximation Method) and TTA (Transmutation Trajectory Analysis) method in the burnup calculation mode. The total heating power generated in the system was assumed to be 2000 MWth. During the reactor operation period of 600 days, the effective multiplication factor (keff) was between 0.964 and 0.954 and peaking factor is 1.88867.

Householder's approximants and continued fraction expansion of quadratic irrationals

There are numerous methods for rational approximation of real numbers. Continued fraction convergent is one of them and Newton's iterative method is another. Connections between these two approximation methods were discussed by several authors. Householder's methods are generalisation of Newton's method. In this paper, we will show that for these methods analogous connection with continued fractions hold.

Solving Linear Systems with Sparse Gaussian Elimination in the Chebyshev Rational Approximation Method

The Chebyshev Rational Approximation Method (CRAM) has recently been introduced by the authors to solve burnup equations, and the results have been excellent. This method has been shown to be capable of simultaneously solving an entire burnup system with thousands of nuclides both accurately and efficiently. The method was prompted by an analysis of the spectral properties of burnup matrices, and it can be characterized as the best rational approximation on the negative real axis. The coefficients of the rational approximation are fixed and have been reported for various approximation orders. In addition to these coefficients, implementing the method requires only a linear solver. This paper describes an efficient method for solving the linear systems associated with the CRAM approximation. The introduced direct method is based on sparse Gaussian elimination, where the sparsity pattern of the resulting upper triangular matrix is determined before the numerical elimination phase. The stability of the proposed Gaussian elimination method is discussed based on consideration of the numerical properties of burnup matrices. Suitable algorithms are presented for computing the symbolic factorization and numerical elimination in order to facilitate the implementation of CRAM and its adoption into routine use. The accuracy and efficiency of the described technique are demonstrated by computing the CRAM approximations for a large test case with 1606 nuclides.

Using Generalized Laguerre Polynomials to Compute the Matrix Exponential in Burnup Equations

Burnup calculations consider the time dependence of the material composition or isotope inventory, which has important influence on the neutronic properties of a nuclear reactor. An essential part of burnup calculations is to solve the burnup equations, which can be approximately treated as a first-order linear system and can be solved by means of matrix exponential methods. However, because of the large decay constants of short-lived nuclides, the coefficient matrix of the burnup equations has a large norm and a vast range of spectra. Consequently, it is quite difficult to directly compute the matrix exponential using conventional methods such as the truncated Taylor expansion and the Pade approximation. Recently, the Chebyshev rational approximation method (CRAM), which is based on rational functions on the complex plane, has shown the capability to deal with this problem. In this paper an alternative method based on the generalized Laguerre polynomials is proposed to compute the exponential of the burnup matrix. Against CRAM, the Laguerre polynomial approximation method (LPAM) has simple recursions for obtaining the coefficients in any order, and all the computations are real arithmetic. A point burnup case and a pin-cell burnup case are calculated for validation, and results show that LPAM is promising for burnup calculations.

Burnup calculations using serpent code in accelerator driven thorium reactors

Abstract In this study, burnup calculations have been performed for a sodium cooled Accelerator Driven Thorium Reactor (ADTR) using the Serpent 1.1.16 Monte Carlo code. The ADTR has been designed for burning minor actinides, mixed 232Th and mixed 233U fuels. A solid Pb-Bi spallation target in the center of the core is used and sodium as coolant. The system is designed for a heating power of 2000 MW and for an operation time of 600 days. For burnup calculations the Advanced Matrix Exponential Method CRAM (Chebyshev Rational Approximation Method) and different nuclear data libraries (ENDF7, JEF2.2, JEFF3.1.1) were used. The effective multiplication factor change from 0.93 to 0.97 for different nuclear data libraries during the reactor operation period.

An Efficient Method for Approximation of Non Rational Transfer Functions

A method for rational approximation of linear fractional order systems (LFOS) is presented in the present paper. The method is computationally efficient, flexible and effective, as is illustrated by numerous examples. The proposed approach can also be used as an intermediate stage in designing indirect discrete rational approximations.

Use of squared magnitude function in approximation and hardware implementation of SISO fractional order system

This paper uses squared magnitude function and genetic algorithm (GA) to propose a generalized method for rational approximation of stable minimum phase fractional order transfer functions (FOTF). The unknown coefficients of an approximant are obtained by equating the squared magnitude of the approximant with that of the FOTF at frequency points (over a specified bandwidth) that are chosen optimally by using GA. Two FOTFs have been approximated by using the proposed method and one of the approximants has been realized in hardware using available discrete circuit components. The results of approximation and circuit realization strongly demonstrate the application of this work in approximation and/or realization of fractional order systems specifically, fractional order controller and filter of higher order.

Pole-zero modeling of the middle ear based on acoustic reflectance measurements

Fitting poles and zeros to complex acoustic reflectance (CAR) data using a “rational approximation method” [Gustavsen and Semlyen (1999)] allows for a precise parameterization of complex real-ear measurements. CAR is measured using a foam-tipped probe sealed in the ear canal, containing a microphone and receiver (i.e., MEPA3 system, Mimosa Acoustics). From the complex pressure response to a broadband stimulus, the acoustic impedance and reflectance of the middle ear can be calculated as functions of frequency. The goal of this work is to establish a quantitative connection between the fitted pole-zero locations and underlying physical properties of the CAR and impedance of the middle ear. It was found that (1) the contribution of the ear canal may be approximated as the lossless all-pass component of the factored reflectance fit, (2) individual CAR magnitude variations for normal middle ears in the 1 to 4 kHz range give rise to closely placed pole-zero pairs, and (3) the locations of the poles and zeros in the s-plane may differ between normal and pathological middle ears. Pole-zero fitting allows for concise characterization of individual CAR measurements, providing a foundation for modeling individual and pathological variations of middle ears.

Development of the point-depletion code DEPTH

The burnup analysis is an important aspect in reactor physics, which is generally done by coupling of transport calculations and point-depletion calculations. DEPTH is a newly-developed point-depletion code of handling large burnup depletion systems and detailed depletion chains. For better coupling with Monte Carlo transport codes, DEPTH uses data libraries based on the combination of ORIGEN-2 and ORIGEN-S and allows users to assign problem-dependent libraries for each depletion step. DEPTH implements various algorithms of treating the stiff depletion systems, including the Transmutation trajectory analysis (TTA), the Chebyshev Rational Approximation Method (CRAM), the Quadrature-based Rational Approximation Method (QRAM) and the Laguerre Polynomial Approximation Method (LPAM). Three different modes are supported by DEPTH to execute the decay, constant flux and constant power calculations. In addition to obtaining the instantaneous quantities of the radioactivity, decay heats and reaction rates, DEPTH is able to calculate the integral quantities by a time-integrated solver. Through calculations compared with ORIGEN-2, the validity of DEPTH in point-depletion calculations is proved. The accuracy and efficiency of depletion algorithms are also discussed. In addition, an actual pin-cell burnup case is calculated to illustrate the DEPTH code performance in coupling with the RMC Monte Carlo code.