Rational Approximation Method Research Articles

A Time-Domain Calculation Method for Gust Aerodynamics in Flight Simulation

Gusts have a significant impact on aircraft and need to be analyzed through flight simulations. The solution for time-domain gust aerodynamic forces stands as a pivotal stage in this process. With the increasing demand for flight simulations within gusty environments, traditional methods related to gust aerodynamics cannot fail to balance computational accuracy and efficiency. A method that can be used to quickly and accurately calculate the time-domain gust aerodynamic force is needed. This study proposes the fitting strip method, a gust aerodynamic force solution method that is suitable for real-time flight simulations. It only requires the current and previous gust information to calculate the aerodynamic force and is suitable for different configurations of aircraft and different kinds of gusts. Firstly, the fitting strip method requires the division of fitting strips and the calculation of the aerodynamic force under calibration conditions. In this study, the double-lattice method and computational fluid dynamics are used to calculate the aerodynamic force of the strips. Then, the amplitude coefficients and time-delay coefficients are obtained through a fitting calculation. Finally, the coefficients and gust information are put into the formula to calculate the gust aerodynamic force. An example of a swept wing is used for validation, demonstrating congruence between the computational results and experimental data across subsonic and transonic speeds, which proves the accuracy of the fitting strip method in both discrete gusts and continuous gusts. Compared with other methods, the fitting strip method uses the shortest time. Furthermore, the results of a calculation for normal-layout aircraft show that this method avoids the shortcomings of the rational function approximation method and is more accurate than the gust grouping method. Concurrently, gust aerodynamic force calculations were performed on aircraft with large aspect ratios and used in a real-time flight simulation.

On the Influence of Fractional-Order Resonant Capacitors on Zero-Voltage-Switching Quasi-Resonant Converters

This paper focuses on the influence of the fractional-order (FO) resonant capacitor on the zero-voltage-switching quasi-resonant converter (ZVS QRC). The FO impedance model of the capacitor is introduced to the circuit model of the ZVS QRC; hence, a piecewise smooth FO model is developed for the converter. Numerical solutions of the converter are obtained by using both the fractional Adams–Bashforth–Moulton (F-ABM) method and Oustaloup’s rational approximation method. In addition, the analytical solution of the converter is obtained by the Grünwald–Letnikov (GL) definition, which reveals the influence of the FO resonant capacitor on the zero-crossing point (ZCP) and resonant state of the converter. An experimental platform was built to verify the results of the theoretical analysis and numerical calculation.

Probabilistic approach of solving burnup problems

This paper presents a new approach to solve nuclear burnup problems using a probabilistic method. Unlike the traditional methods that rely on complex matrix exponential calculations, the proposed method tracks the transformation of nuclei over a period of time, enabling the estimation of nuclide concentrations. The method is implemented in a C++ program CNUCTRAN and verified against the Chebyshev Rational Approximation Method (CRAM). Three sample calculations are presented, showing a detailed comparison of results obtained using CNUCTRAN and CRAM48. The relative error analysis emphasizes the accuracy of the probabilistic method, with the most significant error observed below 0.001%. The computational efficiency is discussed in terms of CPU time, indicating that CNUCTRAN requires more CPU time than CRAM but with improved accuracy as the time step decreases. The numerical results for various burnup problems demonstrate the accuracy and efficiency of the probabilistic method, making it a promising alternative for simulating realistic nuclear transmutations.

Stability analysis and conserved quantities of analytic nonlinear wave solutions in multi-dimensional fractional systems

The (3+1)-dimensional generalized nonlinear fractional Konopelchenko–Dubrovsky–Kaup–Kupershmidt [Formula: see text] model represents the propagation and interaction of nonlinear waves in complex multi-dimensional physical media characterized by anomalous dispersion and dissipation phenomena. By incorporating fractional derivatives, this model introduces non-locality and memory effects into the classical [Formula: see text] equations, commonly utilized in phenomena such as shallow water waves, nonlinear optics, and plasma physics. The fractional approach enhances mathematical representations, allowing for a more realistic depiction of the intricate behaviors observed in numerous modern physical systems. This study focuses on the development of accurate and efficient numerical techniques tailored for the computationally demanding [Formula: see text] model, leveraging the Khater II and generalized rational approximation methods. These methodologies facilitate stable time-integration, effectively addressing the model’s stiffness and multi-dimensional nature. Through numerical analysis, insights into the stability and convergence of the algorithms are gained. Simulations conducted validate the performance of these methods against established solutions while also uncovering novel capabilities for exploring complex wave dynamics in scenarios involving complete fractional formulations. The findings underscore the potential of integrating fractional calculus into higher-dimensional nonlinear partial differential equations, offering a promising avenue for advancing the modeling and computational analysis of complex wave phenomena across a spectrum of contemporary physical disciplines.

Aeroelasticity Analysis of Folding Wing with Structural Nonlinearity Using Modified DLM

The structural nonlinear aeroelastic analysis of folding wings is conducted by integrating the modified Doublet Lattice Method in this paper. Firstly, a modified Doublet Lattice Method is investigated to relax the aspect ratio limitation of aerodynamic elements, and its correctness is verified through numerical examples. Secondly, the rational function approximation method is discussed, and the modified Doublet Lattice Method is approximated using the minimum state method to obtain the unsteady aerodynamic forces in the time domain. Finally, the aeroelastic characteristics of a folding wing with free-play and friction nonlinearity are studied by integrating time-domain unsteady aerodynamics with the nonlinear structural dynamics model. The results show that the modified Doublet Lattice Method, which relaxes the aspect ratio limitation of aerodynamic grid elements, is feasible and effective; Friction can suppress the influence of free-play nonlinearity on aeroelastic response.

Second virial coefficient, Boyle temperature and equation of state of van Hove fluids with a downward concavity attractive parabolic-well

The second virial coefficient, the Boyle temperature and the equation of state of van Hove fluids with a relatively short ranged attractive parabolic-well of downward concavity are considered. The analytic second virial coefficient for this fluid is obtained explicitly and it is used to compute the Boyle temperature of the fluid as a function of the range of the potential. Further, an equation of state is derived using the second-order thermodynamic perturbation theory of Barker and Henderson in the macroscopic compressibility approximation, with the hard-sphere fluid being the reference fluid. For this latter we profit from the fully analytical expression of the radial distribution function, consistent with the Carnahan-Starling equation state, derived within the so-called rational function approximation method up to a range twice the size of the hard-core diameter. The results for the reduced pressure of the fluid as a function of the packing fraction and two values of the range of the potential well at different temperatures are compared with Monte Carlo simulation data. Estimates of the values of the critical temperature are also provided.

Application of the Gauss-Seidel Method to the Chebyshev Rational Approximation Method for Solving Nuclear Fuel Depletion Systems

The Chebyshev Rational Approximation Method (CRAM) has become one of the dominant methods for solving the Bateman equations for nuclear fuel depletion analysis. Since its introduction over a decade ago, several improvements have been made to CRAM improving its accuracy and reducing its run time. We analyzed its run time using two previously published methods for solving the CRAM system of equations, direct matrix inversion (DMI) and sparse Gaussian elimination (SGE), for depletion systems of varying numbers of nuclides to see how the two methods perform relative to one another. In addition to these two methods, we introduced the Gauss-Seidel (GS) method for solving the CRAM system of equations and compared its performance relative to DMI and SGE for depletion systems with varying numbers of nuclides. We demonstrated that for practical purposes, GS is faster than SGE and DMI and achieves a practical level of accuracy. All testing was performed using the CRAM implementation in the Griffin reactor physics analysis application.

Stable parameters identification for rational approximation of Single degree of freedom frequency response function of semi‐infinite medium

AbstractThe rational approximation is an important tool for establishing a dynamic analysis model in time domain for semi‐infinite water and soil. It accurately represents the interaction between the semi‐infinite medium and an engineering structure. Current research on rational approximation method focuses mainly on the establishment of time‐domain analysis models. However, the stability, accuracy, and calculation efficiency of the rational approximation model cannot be guaranteed simultaneously. Based on linear system control theory, this work decomposes the continuous‐time rational approximation (CRA) and the discrete‐time rational approximation (DRA) into a combination of first‐ and second‐order subsystems, and the stability boundaries for the identified parameters are established according to the stability conditions of each subsystem. A genetic algorithm and a sequential quadratic programming algorithm are used to construct a parameter identification method that can ensure stability in the time domain. Through parameter identification of complex frequency response functions, it is demonstrated that the method proposed in this work can guarantee the stability and accuracy simultaneously, and the calculation efficiency is also substantially improved over that of the existing methods.

Generalized method for rational approximation of SISO/MIMO fractional-order systems using squared magnitude function

A new method for the rational approximation of stable/unstable single-input, single-output (SISO)/multiple-input, multiple-output (MIMO) fractional-order systems is proposed. The objective of the proposed algorithm is to obtain an integer-order approximant of an SISO/MIMO fractional-order system. The developed method utilizes the concept of matching of an appropriate number of approximate generalized time moments and approximate generalized Markov parameters of squared magnitude function of fractional-order system to those of approximant. The proposed method preserves the stability/instability property and minimum phase/non-minimum phase characteristics of fractional-order system in the approximant. The method also incorporates a provision for matching the steady-state response of the approximant to that of fractional-order system. Numerical examples consider three cases of approximation, while fractional-order system has the characteristics of (a) stable non-minimum phase SISO, (b) stable non-minimum phase MIMO, and (c) unstable SISO which are presented to validate the efficiency of the proposed method.

Predicting wing-pylon-nacelle configuration flutter characteristics using adaptive continuation method

In this study, the aeroelastic response of a wing-pylon-nacelle system in subsonic and low supersonic flow regimes is analyzed using the continuation method in conjunction with an adaptive step size control algorithm. Idealizing the pylon and nacelle as a point mass, the computed effects of a standard structural analysis of the wing together with the pylon and nacelle are compared with those of a clean wing to build a reduced-order model for analysis. The aerodynamic forces relating to different reduced frequencies are assessed using the Doublet Lattice Method (DLM) in the subsonic flow regime and supersonic lifting surface theory relying on the unsteady linearized small-disturbance potential flow model in the low supersonic flow regime. The Rational Function Approximation (RFA) method is then utilized for the state-space formulation of the system equations, appended with the continuation method for flutter prediction. Thereafter, the linearized aeroelastic equations are resolved using the continuation method with adaptive step size, the results of which are matched with those obtained from the traditional p-k method to emphasize that the continuation method exhibits a distinct advantage in achieving better accuracy in estimating the flutter speed and identifying the “mode switching” phenomenon.

Error analyses and evaluation for solving burnup equations with MMPA method

The Mini-Max Polynomial Approximation (MMPA) is a new method to solve burnup equations, which has attracted much attention due to its advantages of good computational accuracy and efficiency. Understanding its error mechanism is significant for its application in practical burnup calculations. In this research, a mathematical model is proposed to clarify the error mechanism of the MMPA method for the burnup calculation. Then the coefficients of the MMPA method with different orders are given by the Remez algorithm. The errors of the MMPA method with different orders in burnup calculation are compared and analyzed, especially in single-step burnup calculation and infinite-step burnup calculation. Three cases are selected to verify the correctness of the mathematical error model, including fresh fuel burnup, depleted fuel burnup and decay calculation. The results show that when the order is greater than 36, the calculation accuracy of the MMPA method is less than the numerical noise of double precision calculation. As the order increases, the step length sensitivity of the MMPA method decreases. Finally, the computational accuracy and computational efficiency of the MMPA method with different orders are compared with those of the 16th order Chebyshev Rational Approximation Method (CRAM). The accuracy of the 16th order CRAM is between the 32nd and 36th order MMPA method. And the computation time of MMPA methods below order 76 is less than that of the 16th order CRAM. The theoretical analysis results of the mathematical error model are consistent with the numerical results. Then several best practices for solving burnup equations using the MMPA method are presented: (1) It is recommended to use the MMPA method with orders above 36 and below 76, whose accuracy and efficiency are better than 16th order CRAM in different degrees. (2) In multi-physics coupled computation which may require a strategy of large step length calculation, increasing the order of the MMPA method can not only improve the computational accuracy but also improve the computational efficiency.

Low-Memory Krylov Subspace Methods for Optimal Rational Matrix Function Approximation

We describe a Lanczos-based algorithm for approximating the product of a rational matrix function with a vector. This algorithm, which we call the Lanczos method for optimal rational matrix function approximation (Lanczos-OR), returns the optimal approximation from a given Krylov subspace in a norm depending on the rational function's denominator, and can be computed using the information from a slightly larger Krylov subspace. We also provide a low-memory implementation which only requires storing a number of vectors proportional to the denominator degree of the rational function. Finally, we show that Lanczos-OR can be used to derive algorithms for computing other matrix functions, including the matrix sign function and quadrature based rational function approximations. In many cases, it improves on the approximation quality of prior approaches, including the standard Lanczos method, with little additional computational overhead.

Gradyan İniş Algoritması Kullanarak CFE Tabanlı Yaklaşık Kesirli Dereceli Türev Modellerinin Zaman Cevabının İyileştirilmesi İçin Hibrit Yaklaşım Yöntemi

Due to its high computational complexity, fractional order (FO) derivative operators have been widely implemented by using rational transfer function approximation methods. Since these methods commonly utilize frequency domain approximation techniques, their time responses may not be prominent for time-domain solutions. Therefore, time response improvements for the approximate FO derivative models can contribute to real-world performance of FO applications. Recent works address the hybrid use of popular frequency-domain approximation methods and time-domain approximation methods to deal with time response performance problems. In this context, this study presents a hybrid approach that implements Continued Fraction Expansion (CFE) method as frequency domain approximation and applies the gradient descent optimization (GDO) for step response improvement of the CFE-based approximate model of FO derivative operators. It was observed that GDO can fine-tune coefficients of CFE-based rational transfer function models, and this hybrid use can significantly improve step and impulse responses of CFE-based approximate models of derivative operators. Besides, we demonstrate analog circuit realization of this optimized transfer function model of the FO derivative element according to the sum of low pass active filters in Multisim and Matlab simulation environments. Performance improvements of hybrid CFE-GDO approximation method were demonstrated in comparison with the stand-alone CFE method.

Practical methods for GPU-based whole-core Monte Carlo depletion calculation

Several practical methods for accelerating the depletion calculation in a GPU-based Monte Carlo (MC) code PRAGMA are presented including the multilevel spectral collapse method and the vectorized Chebyshev rational approximation method (CRAM). Since the generation of microscopic reaction rates for each nuclide needed for the construction of the depletion matrix of the Bateman equation requires either enormous memory access or tremendous physical memory, both of which are quite burdensome on GPUs, a new method called multilevel spectral collapse is proposed which combines two types of spectra to generate microscopic reaction rates: an ultrafine spectrum for an entire fuel pin and coarser spectra for each depletion region. Errors in reaction rates introduced by this method are mitigated by a hybrid usage of direct online reaction rate tallies for several important fissile nuclides. The linear system to appear in the solution process adopting the CRAM is solved by the Gauss-Seidel method which can be easily vectorized on GPUs. With the accelerated depletion methods, only about 10% of MC calculation time is consumed for depletion, so an accurate full core cycle depletion calculation for a commercial power reactor (BEAVRS) can be done in 16 h with 24 consumer-grade GPUs.

Product integration rules by the constrained mock-Chebyshev least squares operator

In this paper we consider the problem of the approximation of definite integrals on finite intervals for integrand functions showing some kind of “pathological” behavior, e.g. “nearly” singular functions, highly oscillating functions, weakly singular functions, etc. In particular, we introduce and study a product rule based on equally spaced nodes and on the constrained mock-Chebyshev least squares operator. Like other polynomial or rational approximation methods, this operator was recently introduced in order to defeat the Runge phenomenon that occurs when using polynomial interpolation on large sets of equally spaced points. Unlike methods based on piecewise approximation functions, mainly used in the case of equally spaced nodes, our product rule offers a high efficiency, with performances slightly lower than those of global methods based on orthogonal polynomials in the same spaces of functions. We study the convergence of the product rule and provide error estimates in subspaces of continuous functions. We test the effectiveness of the formula by means of several examples, which confirm the theoretical estimates.

Construction and Experimental Realization of the Fractional-Order Transformer by Oustaloup Rational Approximation Method

In previous studies, there were few application circuits involving fractional-order mutual-coupling circuits. Therefore, based on the fractional mutual coupling circuit, the Oustaloup rational approximation method is proposed to construct fractional transformers. Firstly the fractional-order transformer circuit model is derived based on the fractional-order coupled inductor. And the frequency characteristics of the impedance of the fractional-order transformer under open-circuit and load conditions are analyzed. Then different orders of fractional transformers are constructed to verify the rationality of the proposed circuit by simulation. Finally, two fractional-order transformers with different orders constructed, and simulations and experiments show that the constructed circuit has good electrical approximation performance.

A Comparison Study of Time-Domain Computation Methods for Piecewise Smooth Fractional-Order Circuit Systems

The role of fractional calculus in circuit systems has received increased attention in recent years. In order to evaluate the effectiveness of time-domain calculation methods in the analysis of fractional-order piecewise smooth circuit systems, an experimental prototype is developed, and the effects of three typical calculation methods in different test scenarios are compared and studied in this paper. It is proved that Oustaloup’s rational approximation method usually overestimates the peak-to-peak current and brings in the pulse–voltage phenomenon in piecewise smooth test scenarios, while the results of the two iterative recurrence-form numerical methods are in good agreement with the experimental results. The study results are dedicated to provide a reference for efficiently deploying calculation methods in fractional-order piecewise smooth circuit systems. Some quantitative analysis results are concluded in this paper.

Evaluating Quantities of Interest Other Than Nuclide Densities in the Bateman Equations

Traditionally, analysts solve the Bateman depletion equations to calculate the nuclide number density (NND) of each nuclide since these densities impact other reactor parameters, such as reactivity, as they change. Many quantities of interest, such as radiation damage, are calculated using simple integration methods, assuming that the NNDs are constant over a given depletion interval. However, the NNDs are time dependent, which can be accurately represented only by the Bateman depletion equations. We propose that these quantities can be calculated simultaneously with the NNDs within the Bateman depletion equations, preserving the coupled nature of these quantities to the time-dependent NNDs. We implemented this functionality in Griffin, demonstrating that only minor code modifications were necessary in order to accommodate an evaluation of these quantities in the Bateman depletion equations. The Chebyshev Rational Approximation Method was used to successfully solve for these additional quantities in the Bateman depletion equations. For radiation damage, the results calculated by Griffin were very accurate, differing by less than 2.5% from an analytical benchmark. For other quantities, the discrepancy between quantities calculated by the Bateman depletion equations versus those calculated by the Forward Euler method exceeded 10% for decay energy and 2% for fissions per initial heavy metal atom and kinetic energy released per unit mass when few depletion intervals were used. As the number of depletion intervals increased, both methods began to converge as expected.

Construction of a new smooth support vector machine model and its application in heart disease diagnosis.

Support vector machine (SVM) is a new machine learning method developed from statistical learning theory. Since the objective function of the unconstrained SVM model is a non-smooth function, a lot of fast optimization algorithms can't be used to find the solution. Firstly, to overcome the non-smooth property of this model, a new padé33 approximation smooth function is constructed by rational approximation method, and a new smooth support vector machine model (SSVM) is established based on the smooth function. Then, by analyzing the performance of the smooth function, we find that the smooth precision is significantly higher than existing smooth functions. Moreover, theoretical and rigorous mathematical analyses are given to prove the convergence of the new model. Finally, it is applied to the heart disease diagnosis. The results show that the Padé33-SSVM model has better classification capability than existing SSVMs.

A Newton method for best uniform rational approximation

A Newton method for best uniform rational approximation