Rational Approximation Research Articles

Multivariate Rational Approximation of Functions with Curves of Singularities

Multivariate Rational Approximation of Functions with Curves of Singularities

A Rational Approximation for the Complete Elliptic Integral of the Second Kind Based on Padé approximant

A Rational Approximation for the Complete Elliptic Integral of the Second Kind Based on Padé approximant

Efficient Rational Approximation of Optical Response Functions with the AAA Algorithm (Laser Photonics Rev. 18(11)/2024)

Efficient Rational Approximation of Optical Response Functions with the AAA Algorithm (Laser Photonics Rev. 18(11)/2024)

Verification of direct coupling code system using FRENDY version 2 and GENESIS for light water reactor lattices

ABSTRACT This study newly established a direct coupling code system consisting of the nuclear data processing code FRENDY version 2, and the three-dimensional heterogeneous transport code GENESIS (FRENDY-V2/GENESIS) for easy implementation of the random-sampling-based uncertainty quantification considering the implicit effect due to nuclear cross-section (XS) perturbations. The multi-group macroscopic XSs prepared for GENESIS were generated by FRENDY version 2, where the Dancoff factor was calculated by the neutron current method. Then the background XSs were evaluated based on the Carlvik two-term rational approximation. The infinite multiplication factor (k-infinity) and the fission reaction rate distribution in UO2 and MOX lattice geometries were compared with MVP3 to verify the calculation accuracy of FRENDY-V2/GENESIS. The sensitivity analyses on the discretization conditions such as the ray tracing of the method of characteristics were also carried out. Through several comparisons between FRENDY-V2/GENESIS and MVP3, FRENDY-V2/GENESIS with the SHEM 361-group structure calculates the k-infinity within approximately 50 pcm and the fission reaction rate distribution within approximately 0.1% by the root mean square, respectively. Consequently, the applicability of FRENDY-V2/GENESIS was verified, and FRENDY-V2/GENESIS can be used to discuss the implicit effect due to multi-group XS perturbations.

Diophantine approximation by rational numbers of certain parity types

AbstractFor a given irrational number, we consider the properties of best rational approximations of given parities. There are three different kinds of rational numbers according to the parity of the numerator and denominator, say odd/odd, even/odd, and odd/even rational numbers. We study algorithms to find best approximations by rational numbers of given parities and compare these algorithms with continued fraction expansions.

Topology optimization based on the improved high-order RAMP interpolation model and bending properties research for curved square honeycomb sandwich structures

Curved honeycomb sandwich structures with larger surface areas effectively reduce the number of fasteners and connectors, resulting in weight reduction, cost savings, and reliability improvement. Square honeycombs exhibit higher in-plane tensile strength, and are more compatible with mechanical components. Topology optimization can realize the variable-density design of square honeycombs, enhancing the strength and stiffness of curved sandwich structures, but the high-order Rational Approximation of Material Properties (RAMP) interpolation model with a fast convergence rate has the relatively poor clarity and stability in topology boundaries. Hence, a variable-density topology optimization method based on the improved high-order RAMP model is developed for curved square honeycomb sandwich structures. The high-order RAMP interpolation model is improved by incorporating a minimum modulus term into the material interpolation function and employing the Bi-directional Evolutionary Structural Optimization (BESO) to refine the Optimality Criteria (OC) method. A functional relationship between the wall thickness of cross-shaped cells (i.e., simplified models of four adjacent square cells arranged in a cross) and the relative density of topology units is constructed using a density mapping method, and the topology optimization with the relative density of cross-shaped cells as the design variable is performed to minimize the compliance of the core. Three-point bending tests are conducted on 3D printed non-optimized and optimized curved honeycomb sandwich structures with different core material retention rates and panel thicknesses, and the experimental results are compared with those of simulations to explore the bending properties and failure behaviors. The results indicate that the modified high-order RAMP interpolation model enhances the clarity and stability of topology structures, the variable-density topology optimization significantly improves the bending properties of curved square honeycomb sandwich structures, and the experimental and numerical results are largely consistent.

Strange nonchaotic attractors and bifurcation of a four-wheel-steering vehicle system with driver steering control

In this paper, a dynamical model is developed for the four-wheel-steering(4WS)vehicles, with nonlinear lateral tyre forces and quasi-periodic disturbances from the steering mechanism and quasi-periodic pavement external deformation. Initially, the stability and Hopf bifurcation of the 4WS vehicle system without disturbance are analyzed employing the central manifold theory and projection method. The effects of parameters on the stability and the types of Hopf bifurcation for the vehicle system are also discussed. It is shown that both Hopf bifurcation and degenerate Hopf bifurcation occur in the vehicle system. The critical speed decreases with increased driver’s perceptual time delay and distance from the vehicle’s center of gravity to the front axles, while it increases with the control strategy coefficient. However, the increase of the frontal visibility distance of the driver leads to an initial increase of the critical speed, decreasing afterwards. Subsequently, the strange nonchaotic dynamical behaviour of the disturbed 4WS system is investigated. Several routes and mechanisms for the birth of strange nonchaotic attractors (SNAs) are identified, including the torus doubling bifurcation, the torus fractalization, and the loss of transverse stability of a torus. Finally, the nonchaotic property is verified through the calculation of the maximum Lyapunov exponent, and the strange property is characterized using power spectra and rational approximation.

Relaxed Nonlinear Vector Fitting for Calculation of Rational Approximation of Systems Defined by Input/Output Responses in the Frequency Domain

Relaxed Nonlinear Vector Fitting for Calculation of Rational Approximation of Systems Defined by Input/Output Responses in the Frequency Domain

The Gonchar–Chudnovskies conjecture and a functional analogue of the Thue–Siegel–Roth theorem

This article examines the Gonchar–Chudnovskies conjecture about the limited size of blocks of diagonal Padé approximants of algebraic functions. The statement of this conjecture is a functional analogue of the famous Thue–Siegel–Roth theorem. For algebraic functions with branch points in general position, we will show the validity of this conjecture as a consequence of recent results on the uniform convergence of the continued fraction for an analytic function with branch points. We will also discuss related problems on estimating the number of “spurious” (“wandering”) poles for rational approximations (Stahl’s conjecture), and on the appearance and disappearance of defects (Froissart doublets).

Rational approximation of golden angles: Accelerated reconstructions for radial MRI.

To develop a generic radial sampling scheme that combines the advantages of golden ratio sampling with simplicity of equidistant angular patterns. The irrational angle between consecutive spokes in golden ratio-based sampling schemes enables a flexible retrospective choice of temporal resolution, while preserving good coverage of k-space for each individual bin. Nevertheless, irrational increments prohibit precomputation of the point-spread function (PSF), can lead to numerical problems, and require more complex processing steps. To avoid these problems, a new sampling scheme based on a rational approximation of golden angles (RAGA) is developed. The theoretical properties of RAGA sampling are mathematically derived. Sidelobe-to-peak ratios (SPR) are numerically computed and compared to the corresponding golden ratio sampling schemes. The sampling scheme is implemented in the BART toolbox and in a radial gradient-echo sequence. Feasibility is shown for quantitative imaging in a phantom and a cardiac scan of a healthy volunteer. RAGA sampling can accurately approximate golden ratio sampling and has almost identical PSF and SPR. In contrast to golden ratio sampling, each frame can be reconstructed with the same equidistant trajectory using different sampling masks, and the angle of each acquired spoke can be encoded as a small index, which simplifies processing of the acquired data. RAGA sampling provides the advantages of golden ratio sampling while simplifying data processing, rendering it a valuable tool for dynamic and quantitative MRI.

Hybrid differential evolution algorithm for Falkner-Skan flow with rotation

A hybrid differential evolution algorithm is used in this work to study the rotating transport of Falkner-Skan flow. The problem is modeled as an equivalent optimization problem by using Padé rational approximation functions. The primary model of the governing partial differential equations is imposed to subjugate the error between profiles. A hybrid evolutionary algorithm based on differential evolution and a convergent version of Nelder-Mead direct search algorithm is employed to perform global exploratory search along with an enhanced exploitation to improve the accuracy of the proposed solution scheme. The resulting scheme is named as evolutionary Padé approximation (EPA) scheme. The performance of the proposed EPA scheme on the Falkner-Skan boundary value problem is investigated by considering various values of the rotation parameters. The developed optimizer in EPA scheme was able to minimize the residuals up to10−10. Results are displayed graphically in order to study the effect of various types of parameters. EPA scheme determined that angular velocity increases or decreases accordingly as fluid parameter (β) and the rotation parameter (λ) but shows inverse behavior with respect to power law index(n). Similarly, the response of velocity profile along y−axis was decreasing function of β as well as n but increasing function of λ. The performance of the proposed EPA scheme has been demonstrated by comparing results with a hybrid neural network scheme and found in excellent agreement.

A rational‐Chebyshev projection method for nonlinear eigenvalue problems

AbstractThis article describes a projection method based on a combination of rational and polynomial approximations for efficiently solving large nonlinear eigenvalue problems. In a first stage the nonlinear matrix function under consideration is approximated by a matrix polynomial in . The error resulting from this polynomial approximation is in turn approximated by rational functions with the help of the Cauchy integral formula. The two approximations are combined and a linearization is performed. A key ingredient of the proposed approach is a projection method that uses subspaces spanned by vectors of the same dimension as that of the original problem instead of that of the linearized problem. A procedure is also presented to automatically select shifts and to partition the region of interest into a few subregions. This allows to subdivide the problem into smaller subproblems that are solved independently. The accuracy of the proposed method is theoretically analyzed and its performance is illustrated with a few test problems that have been discussed in the literature.

Polynomial and rational convergence rates for Laplace problems on planar domains

Laplace problems on planar domains can be solved by means of least-squares expansions associated with polynomial or rational approximations. Here it is shown that, even in the context of an analytic domain with analytic boundary data, the difference in convergence rates may be huge when the domain is non-convex. Our proofs combine the theory of the Schwarz function for analytic continuation, potential theory for polynomial and rational approximation rates and the theory of crowding of conformal maps.

Piecewise nonlinear approximation for non-smooth functions

Piecewise affine or linear approximation has garnered significant attention as a technique for approximating piecewise-smooth functions. In this study, we propose a novel approach: piecewise non-linear approximation based on rational approximation, aimed at approximating non-smooth functions. We introduce a method termed piecewise Padé Chebyshev (PiPC) tailored for approximating univariate piecewise smooth functions. Our investigation focuses on assessing the effectiveness of PiPC in mitigating the Gibbs phenomenon during the approximation of piecewise smooth functions. Additionally, we provide error estimates and convergence results of PiPC for non-smooth functions. Notably, our technique excels in capturing singularities, if present, within the function with minimal Gibbs oscillations, without necessitating the explicit specification of singularity locations. To the best of our knowledge, prior research has not explored the use of piecewise non-linear approximation for approximating non-smooth functions. Finally, we validate the efficacy of our methods through numerical experiments, employing PiPC to reconstruct a non-trivial non-smooth function, thus demonstrating its capability to significantly alleviate the Gibbs phenomenon.

Unsteady Lifting-Line Theory for Camber Morphing Wings State-Space Aeroelastic Modeling

A novel frequency-domain analytical-numerical model for the aerodynamic solution of camber morphing wings is developed. The model combines an unsteady lifting-line formulation with Küssner–Schwarz aerodynamic theory to provide the pressure distribution and thus the transcendental aerodynamic matrix that relates the generalized aerodynamic forces to the Lagrangian coordinates of a given wing structural dynamics model. The state-space form of the aerodynamic loads is obtained from the rational matrix approximation of the aerodynamic matrix. This, combined with the wing structural dynamics operator, can readily provide the state-space form of the aeroelastic problem. To assess the accuracy of the proposed aerodynamic solution method, numerical investigations are performed. These consist of its application to several conventional wing configurations and wings with camber morphing, followed by the comparisons of the corresponding solutions with the predictions given by a well-validated panel-method solver for potential flows. These results are shown to be in very good agreement, thus demonstrating that the proposed approach can capture the effects due to wing tapering, sweep angle, and camber morphing, while requiring a remarkably lower computational effort. The excellent accuracy of a few-pole finite-state approximation of the aerodynamic matrix is finally proven for a swept wing with camber morphing.

Efficient Rational Approximation of Optical Response Functions with the AAA Algorithm

AbstractA theoretical framework for the rational approximation of optical response functions in resonant photonic systems is introduced. The framework is based on the AAA algorithm and further allows to solve the underlying nonlinear eigenproblems and to efficiently model sensitivities. An adaptive sampling strategy exploits the predominance of resonances in the physical response. A chiral metasurface is investigated and it is shown that the chiroptical response on parameter variations can be accurately modeled in the vicinity of the relevant resonance frequencies.

Reduced-order helicopter model identification from closed-loop data

This paper presents the development of an algorithm for open-loop transfer functions identification and reduced-order modelling of a dynamical system, from closed-loop data with highly correlated inputs. Suitable for aeronautical applications (particularly for intrinsically unstable vehicles such as helicopters), it allows the identification of aircraft transfer functions from the knowledge of the time histories of arbitrary external inputs and the corresponding actuated controls and responses. After a numerical verification of the proposed approach considering a simple analytical aircraft dynamical system, it is successfully applied to a realistic engineering problem consisting of identifying the transfer functions of the AW-09 helicopter that relate pilot inputs to vehicle attitude and kinematics. It is shown that helicopter responses to arbitrary pilot inputs predicted by the reduced-order model representation of the helicopter dynamics based on the rational approximation of the identified transfer functions are in good agreement with those determined through the high-fidelity nonlinear flight dynamics solver used to evaluate the closed-loop database.

Computation of polynomial and rational approximations in complex domains by the $$\tau $$-method

Computation of polynomial and rational approximations in complex domains by the $$\tau $$-method

A Rational Approximation of the Two-Term Machin-like Formula for π

A Rational Approximation of the Two-Term Machin-like Formula for π

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.