Rational Chebyshev Research Articles

Extracting non-equilibrium material properties in visco-hyperelastic polymers based on analytical stress solution

This study extracts non-equilibrium material properties in the visco-hyperelastic polymers using an analytical stress solution. At first, the analytical stress solution of visco-hyperelastic polymers is obtained from finite-strain creep test data based on the generalized Maxwell’s rheological model. The solution uses the low-computational-cost exponential generalized pseudospectral method and analytically brings the stress components in terms of time and unknown non-equilibrium material properties of Maxwell’s elements. The solution invokes the Ogden hyperelastic model for the elastic element of the generalized Maxwell’s model. The exponential generalized pseudospectral method, based on exponential–rational Lagrange functions suitable for the semi-infinite time interval [Formula: see text] and the rational Chebyshev roots as the collocation points, is applied to derive the stress components. The non-equilibrium properties are derived by minimizing a residual, which is obtained by making the derivatives of the residual in terms of the non-equilibrium properties as zero. Notably, the proposed analytical method requires a small number of exponent–rational Lagrange basis functions and collocation points. Besides, the proposed method does not require integration, discretization, or linearization. Two case studies are considered: the moderate-stretched ETFE polymer with the neo-Hookean model as the elastic element and the large-stretched rubber with the six-parameter Ogden hyperelastic model as the elastic branch. The results are validated using uniaxial creep modeling in ANSYS and available reports in the literature.

Solving a class of Thomas–Fermi equations: A new solution concept based on physics-informed machine learning

This paper presents a novel physics-informed machine learning approach, designed to approximate solutions to a specific category of Thomas–Fermi differential equations. To tackle the inherent intricacies of solving Thomas–Fermi equations, we employ the quasi-linearization technique, which transforms the original non-linear problem into a series of linear differential equations. Our approach utilizes collocation least-squares support vector regression, leveraging fractional Chebyshev functions for finite interval simulations and fractional rational Chebyshev functions for semi-infinite intervals, enabling precise solutions for linear differential equations across varied spatial domains. These selections facilitate efficient simulations across both finite and semi-infinite intervals. Key contributions of our approach encompass its versatility, demonstrated through successful approximation of solutions for diverse Thomas–Fermi problem types, including those featuring non-local integral terms, Bohr radius boundary conditions, and isolated neutral atom boundary conditions defined on semi-infinite domains. Furthermore, our method exhibits computational efficiency, surpassing classical collocation methods by solving a sequence of positive definite linear equations or quadratic programming problems. Notably, our approach showcases precision, as evidenced by experiments, including the attainment of the initial slope of the renowned Thomas–Fermi equation with an impressive 39-digit precision.

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.

Asymptotics of Chebyshev Rational Functions with Respect to Subsets of the Real Line

Asymptotics of Chebyshev Rational Functions with Respect to Subsets of the Real Line

The Riesz–Zygmund Sums of Fourier–Chebyshev Rational Integral Operators and Their Approximation Properties

The Riesz–Zygmund Sums of Fourier–Chebyshev Rational Integral Operators and Their Approximation Properties

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.

Numerically efficient full-vectorial rational Chebyshev pseudo-spectral modal analysis for optical Waveguides

In this paper, an efficient full-vectorial modal analysis based on the rational Chebyshev pseudo-spectral method (V-RCPSM) is introduced to analyze 3 dimensional (3D) structures that are invariant along one spatial variable. Such structures are essential in silicon photonics and plasmonics applications where permittivity profiles with high-index contrast need precise treatment of the interface boundary conditions. Besides, such structures are open in general. Hence, good domain truncation is important. Our method handles these challenges via hybrid usage of the domain decomposition technique where the electromagnetic field is expanded in terms of Chebyshev functions in homogeneous regions, while the rational Chebyshev functions are used for semi-infinite homogeneous domains. The boundary conditions are rigorously imposed along the interfaces, a step that maintains the known exponential convergence rate of Chebyshev functions. Chebyshev functions have the ability to capture the correct rapid variation of the electromagnetic fields at the interfaces of the high-index-contrast waveguides using only a few basis functions; a critical feature for accurate mode computation. To show the accuracy and efficiency of our new approach, we studied rib and plasmonic waveguides and compared the results with those obtained using other full-vectorial approaches such as the finite elements method (FEM). Our developed approach has achieved a huge reduction in computational resources over the FEM.

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.

RATIONAL AND GENERALISED RATIONAL CHEBYSHEV APPROXIMATION PROBLEMS AND THEIR APPLICATIONS

RATIONAL AND GENERALISED RATIONAL CHEBYSHEV APPROXIMATION PROBLEMS AND THEIR APPLICATIONS

Barycentric rational collocation method for semi-infinite domain problems

<abstract><p>The barycentric rational collocation method for solving semi-infinite domain problems is presented. Following the barycentric interpolation method of rational polynomial and Chebyshev polynomial, matrix equation is obtained from discrete semi-infinite domain problem. Truncation method and transformation method are presented to solve linear and nonlinear differential equation defined on the semi-infinite domain problems. At last, three numerical examples are presented to valid our theoretical analysis.</p></abstract>

Numerical solution of fractional Volterra integral equations based on rational Chebyshev approximation

Numerical solution of fractional Volterra integral equations based on rational Chebyshev approximation

Methods and performance of a GPU-based pinwise two-step nodal code VANGARD

The methods and performance of a GPU-based pinwise two-step nodal core calculation code VANGARD are presented which is targeted for use in the commercial nuclear design analyses. In order to realize high-speed core cycle depletion with a single consumer-grade GPU mounted on ordinary personal computers, various elaborated computational methods are introduced as follows. The computationally intense pinwise nodal calculation is optimized by adaptive employment of low-order expansion. The two-level surface current correction method is used for updating pinwise incoming partial currents within the assembly-level coarse mesh finite difference framework. A cross section data compression technique is introduced to fit the enormous pinwise group constant data into the limited GPU memory. A massively parallelized depletion scheme based on the Chebyshev Rational Approximation Method is implemented. The neighbor-informed burnup correction method is developed to ensure the accuracy of the pinwise depletion for gadolinia-bearing pins. Performance examinations are performed for the realistic APR1400 and AP1000 core problems to demonstrate that three-dimensional pinwise cycle depletion can be completed within 3 minutes with high accuracy and resolution.

Introduction of the Adding and Doubling Method for Solving Bateman Equations for Nuclear Fuel Depletion

This paper introduces and evaluates the Adding and Doubling Method (ADM) for solving the Bateman equations for depletion systems with varying numbers of nuclides and compares it to the Chebyshev Rational Approximation Method (CRAM), both implemented in the reactor physics analysis application Griffin. ADM, when applied to the Crank-Nicolson Finite Difference method, can produce results comparable in accuracy and precision to CRAM with comparable run times for systems with 35 or 297 nuclides. For systems with more than 300 nuclides, the matrix-matrix operations required by ADM are significantly more costly than the matrix-vector operations required by CRAM, making CRAM the more efficient method for systems with large numbers of nuclides. ADM is an accurate method that maintains other advantages over CRAM in that it does not depend on pre-generated coefficients or require complex number operations. ADM also manages to outperform CRAM by a factor of more than 250 in terms of run time for depletion systems that require multiple Bateman solves while the depletion matrix and time step size remain constant over all depletion intervals.

Investigation and improvement of the Mini-Max Polynomial Approximation method for solving burnup equations

In this work, a new efficient solution algorithm named Combined Matrix Iterative Algorithm based on the Mini-Max Polynomial Approximation (MMPA) method is proposed for burnup calculations. As a new matrix exponential method in recent years, MMPA method has good numerical stability and computational accuracy, and all calculation are real number operations. As the same with Chebyshev Rational Approximation Method (CRAM) which is widely and successfully used in solving burnup equations, the method can also directly deal with rigid and sparse burnup matrices. Compared with the traditional solution algorithms based on MMPA method, the Combined Matrix Iterative Algorithm (MMPA-CMIA) proposed in this paper can effectively reduce the computational complexity and improve the computational efficiency after equivalently transforming the expression of the MMPA method. Then the MMPA-CMIA is used to theoretically demonstrate and analyze the solution efficiency of random matrices with different orders which has the similar distribution characteristics to burnup matrices. Compared with the 16-order CRAM method and the other two traditional solution algorithms based on MMPA method, the algorithm shows good acceleration effect and computational precision. Finally, the algorithm is used in the actual burnup benchmarks to verify and validate the accuracy and efficiency. The burnup matrices of different orders are constructed through detailed burnup chain and simplified burnup chain. The results show that the algorithm has good accuracy and numerical stability. And comparing with 16-order CRAM, the algorithm can save about 30% of computing time for solving burnup equations. It shows that the algorithm is a potentially powerful method for accelerating the solution of large-scale burnup calculations.

A spent nuclear fuel source term calculation code BESNA with a new modified predictor-corrector scheme

This paper introduces a new point depletion-based source term calculation code named BESNA (Bateman Equation Solver for Nuclear Applications), which is aimed to estimate nuclide inventories and source terms from spent nuclear fuels. The BESNA code employs a new modified CE/CM (Constant Extrapolation – Constant Midpoint) predictor-corrector scheme in depletion calculations for improving computational efficiency. In this modified CE/CM scheme, the decay components leading to the large norm of the depletion matrix are excluded in the corrector, and hence the corrector calculation involves only the reaction components, which can be efficiently solved with the Talyor Expansion Method (TEM). The numerical test shows that the new scheme substantially reduces computing time without loss of accuracy in comparison with the conventional scheme using CRAM (Chebyshev Rational Approximation Method), especially when the substep calculations are applied. The depletion calculation and source term estimation capability of BESNA are verified and validated through several problems, where results from BESNA are compared with those calculated by other codes as well as measured data. The analysis results show the computational efficiency of the new modified scheme and the reliability of BESNA in both isotopic predictions and source term estimations.

On the Numerical Solution of Volterra-Fredholm Integro-Differential Equations via Rational Chebyshev Spectral Collocation Approach

On the Numerical Solution of Volterra-Fredholm Integro-Differential Equations via Rational Chebyshev Spectral Collocation Approach

A Comparison Study of Numerical Techniques for Solving Ordinary Differential Equations Defined on a Semi-Infinite Domain Using Rational Chebyshev Functions

A rational Chebyshev (RC) spectral collocation technique is considered in this paper to solve high-order linear ordinary differential equations (ODEs) defined on a semi-infinite domain. Two definitions of the derivative of the RC functions are introduced as operational matrices. Also, a theoretical study carried on the RC functions shows that the RC approximation has an exponential convergence. Due to the two definitions, two schemes are presented for solving the proposed linear ODEs on the semi-infinite interval with the collocation approach. According to the convergence of the RC functions at the infinity, the proposed technique deals with the boundary value problem which is defined on semi-infinite domains easily. The main goal of this paper is to present a comparison study for differential equations defined on semi-infinite intervals using the proposed two schemes. To demonstrate the validity of the comparisons, three numerical examples are provided. The obtained numerical results are compared with the exact solutions of the proposed problems.

Linear Barycentric Rational Method for Solving Schrodinger Equation

A linear barycentric rational collocation method (LBRCM) for solving Schrodinger equation (SDE) is proposed. According to the barycentric interpolation method (BIM) of rational polynomial and Chebyshev polynomial, the matrix form of the collocation method (CM) that is easy to program is obtained. The convergence rate of the LBRCM for solving the Schrodinger equation is proved from the convergence rate of linear barycentric rational interpolation. Finally, a numerical example verifies the correctness of the theoretical analysis.

On one generalization of the Hermite quadrature formula

In this paper we propose a new approach to the construction of quadrature formulas of interpolation rational type on an interval. In the introduction, a brief analysis of the results on the topic of the research is carried out. Most attention is paid to the works of mathematicians of the Belarusian school on approximation theory – Gauss, Lobatto, and Radau quadrature formulas with nodes at the zeros of the rational Chebyshev – Markov fractions. Rational fractions on the segment, generalizing the classical orthogonal Jacobi polynomials with one weight, are defined, and some of their properties are described. One of the main results of this paper consists in constructing quadrature formulas with nodes at zeros of the introduced rational fractions, calculating their coefficients in an explicit form, and estimating the remainder. This result is preceded by some auxiliary statements describing the properties of special rational functions. Classical methods of mathematical analysis, approximation theory, and the theory of functions of a complex variable are used for proof. In the conclusion a numerical analysis of the efficiency of the constructed quadrature formulas is carried out. Meanwhile, the choice of the parameters on which the nodes of the quadrature formulas depend is made in several standard ways. The obtained results can be applied for further research of rational quadrature formulas, as well as in numerical analysis.

Matrix computational collocation approach based on rational Chebyshev functions for nonlinear differential equations

In this work, a numerical technique for solving general nonlinear ordinary differential equations (ODEs) with variable coefficients and given conditions is introduced. The collocation method is used with rational Chebyshev (RC) functions as a matrix discretization to treat the nonlinear ODEs. Rational Chebyshev collocation (RCC) method is used to transform the problem to a system of nonlinear algebraic equations. The discussion of the order of convergence for RC functions is introduced. The proposed base is specified by its ability to deal with boundary conditions with independent variable that may tend to infinity with easy manner without divergence. The technique is tested and verified by two examples, then applied to four real life and applications models. Also, the comparison of our results with other methods is introduced to study the applicability and accuracy.