Gauss-Legendre Quadrature Research Articles

Mixed boundary-value problems and dynamic impedance functions due to vibrations of a rigid disc on a thermoelastic transversely isotropic half-space

The mixed boundary-value problems and impedance functions due to forced vibrations of a rigid disc on a thermoelastic transversely isotropic half-space are presented analytically and numerically for the first time. Dynamic interaction problems of the disc with the thermoelastic half-space are solved by utilizing an indirect boundary-element method (BEM) which uses the dynamic Green's functions for uniform circular disc loads acting on the surface of a thermoelastic transversely isotropic half-space. The horizontal, vertical, rocking, and torsional vibrations of the disc are investigated and the results are presented in terms of several dual- and Fredholm- integral equations. For the four vibration modes, the related thermo-elastodynamic mixed boundary-value problems are solved and the contact normal and shear stresses, the resultant forces and moments acting on the disc as well as the impedance and compliance functions are derived analytically. By zeroing some parameters, we proved that the obtained analytical thermo-elastodynamic solutions are reduced to the corresponding elastodynamic results reported in the literature. The numerical procedures for the involved semi-infinite integrals of the kernel functions as well as the numerical quadrature methods for evaluating the impedance functions are further presented. The discretization method is required to be applied only to the interface of the rigid disc with the thermoelastic half-space. A computer program is coded in Mathematica where the singularities due to some branch points and the Rayleigh pole were treated. The Gauss-Legendre quadrature method is used for numerical evaluation of impedance functions, and its advantages are presented and compared with other methods. Some graphical results are presented for the dimensionless impedance functions versus the dimensionless frequency. Effects of thermal and anisotropic properties of the thermoelastic materials; and frequency of excitation on the impedance functions are presented and compared with the elastic materials. Numerical results indicate the significant effects of material anisotropy on the impedance functions.

A hamiltonian global nodal position finite element method for dynamics analysis of submarine cables

The dynamics of submarine cables can be expressed in Hamiltonian formalism, which can reduce the error accumulation over long-term numerical calculation. This paper proposes a Hamiltonian global nodal position finite element method to study the dynamic responses of submarine cables undergoing bending deformation and long-term large overall motions. The new global nodal position finite element discrete formulation is derived by Hamiltonian theory and Euler-Bernoulli beam theory, which takes the bending deformation of submarine cables into consideration. To verify the proposed method, the second-order Symplectic difference algorithm is built for numerical solution. Five-point Gauss-Legendre quadrature formulate is applied to calculate the hydrodynamic force. The calculation accuracy and efficiency of the proposed method are validated by the freefall cantilever test of submarine cable, the towing test of submarine cable with lumped mass, the motion simulation of submarine cable connected to the remotely operated vehicle (ROV) and the towing test of a submerged body. Compared with the existing Hamiltonian nodal position finite element method, all validations and results indicate that the proposed method can solve the motion of various submarine cables under different working conditions with higher accuracy and efficiency.

Barycentric interpolation collocation algorithm to solve fractional differential equations

Fractional equations have been paid much attention in recent years. Barycentric interpolation collocation algorithm (BICA) is proposed to solve the fractional differential equations in this manuscript. In order to calculate fractional derivatives in fractional differential equations, the Gauss quadrature formula with weights ρ(τ)=(t−τ)ξ−α is constructed and the error estimation of this quadrature formula is proved. The fractional differential term of equation is transformed into Riemann integral under Caputo definition. Barycentric interpolation is used to approximate the unknown function and matrix equation of transformed fractional differential equation is obtained by BICA. The error results show exponential convergence rate of Gauss quadrature formula with weights ρ(τ)=(t−τ)ξ−α by comparing with the Gauss–Legendre quadrature formula. In addition, several examples are given to solve the fractional differential equation.

Development and verification of a new nuclear data processing code AXSP

The advanced reactor design needs an accurate cross-section generation code. In this study, a new nuclear data processing code AXSP is developed, and the method and performance of which are described. Compared with the NJOY program, the precision of the unresolved resonance processing module UnresXS has been significantly improved due to the adoption of a more accurate solution method and the consideration of in-sequence overlap integrals. The time consumption of PUnresXS has been decreased significantly due to an optimized sorting algorithm. At the same time, other modules of AXSP are relatively comprehensive. The function of resolved resonance cross-section reconstruction and linearization is the ReconXS module. The Doppler broadening module is BroadXS by using Gauss–Hermite quadrature and Gauss–Legendre quadrature from 0 K temperature pointwise cross section to any temperature which is defined by the user. The shielding factor in the unresolved resonance energy region is calculated by the UnresXS or the PUnresXS module, which are developed based on the Bondarenko method and the probability table method, respectively. The ACE formatted cross sections for the Monte Carlo code is processed by the ACEXS module, and the multigroup cross sections are generated by the GroupXS module. The cross sections processed by different modules were verified by the NJOY2016 code, and the multigroup cross sections were also verified by using the critical benchmarks. The multiplication factor difference between AXSP and NJOY2016 is less than 20 pcm. In addition to this, the ZPR6/7 fast reactor is used for ACE format library verification. The results show that the criticality calculated by AXSP has a good agreement with that of NJOY2016.

A numerical method based on the piecewise Jacobi functions for distributed-order fractional Schrödinger equation

In this work, the distributed-order time fractional version of the Schrödinger problem is defined by replacing the first order derivative in the classical problem with this kind of fractional derivative. The Caputo fractional derivative is employed in defining the used distributed fractional derivative. The orthonormal piecewise Jacobi functions as a novel family of basis functions are defined. A new formulation for the Caputo fractional derivative of these functions is derived. A numerical method based upon these piecewise functions together with the classical Jacobi polynomials and the Gauss–Legendre quadrature rule is constructed to solve the introduced problem. This method converts the mentioned problem into an algebraic problem that can easily be solved. The accuracy of the method is examined numerically by solving some examples.

Chelyshkov least squares support vector regression for nonlinear stochastic differential equations by variable fractional Brownian motion

The main aim of this study is to introduce an efficient method based on the Chelyshkov polynomials and least squares support vector regression (LS-SVR) for solving a class of nonlinear stochastic differential equations (SDEs) by variable fractional Brownian motion (VFBm). The derivative operational matrix and variable-order fractional integral operator of Chelyshkov polynomials (ChPs) are obtained. These operators, the standard Brownian motion with help of the Gauss–Legendre quadrature are applied for generating VFBm. We apply the Chelyshkov polynomials kernel and the collocation LS-SVR method for training the network. Then, the formulation of the scheme gives rise to an optimization problem. Finally, the classical optimization and Newton’s iterative scheme are used to train this problem. Moreover, we discuss convergence and error analysis of mentioned scheme. In the end, to reveal the superiority and efficiency of current paper, some test problems are applied.

3D inversion of gravity data with unstructured mesh and least-squares QR-factorization (LSQR)

Once inverting potential-field geophysical data an appropriate discretization of the model is required to accurately construct complicated geometries of the causative sources. Rectangular prisms (structured meshes) have limitations to recover and preserve the edges of real geological sources. Here an isoparametric finite-element (FE) methodology is used to design an unstructured mesh for use in 3D inverse modeling of gravity data. The calculation of the sensitivity kernel of the forward operator uses Gauss-Legendre quadrature rather than the analytic formulation. For the sake of instability of inversion operator in gravity data and to solve the Tikhonov norms term, the least-squares QR-factorization (LSQR) technique is used. This method is accompanied with weighted generalized cross-validation (WGCV) for selecting the optimum regularization parameter value. The depth weighting function is also incorporated in the formulation of the objective function to suppress the impact of shallow features and recover sources at an appropriate depth. The proposed algorithm was applied to noise-corrupted synthetic data along with a real case study where a gravity survey was used for iron exploration in Yazd province, central Iran. The obtained results of synthetic example indicated that the proposed 3D inversion method recovers the geometry and density contrast values which are similar to the true structures and its application on a real example data set recovers geologically reasonable complex structures. So, this modified algorithm is a type of smooth inversion for boundary detection and whole understanding of the physical property distribution of the subsurface.

On the dynamics and stability of size-dependent symmetric FGM plates with electro-elastic coupling using meshless local Petrov-Galerkin method

In present study, the equations of motion of sandwich symmetric functionally graded size-dependent plates integrated with piezoelectric layers are derived using Hamilton’s variational principle (with modified Lagrangian) based on a higher-order nonlocal strain gradient theory in conjunction with Reddy’s third-order shear deformation plate kinematics. For the first time, a set of sixth-order partial differential equations are derived and solved by meshless local Petrov-Galerkin method to investigate the bifurcation buckling and free vibration of plates under electromechanical in-plane forces with considered nanostructure and various boundary conditions. The shifted and scaled moving least-squares approach is utilized for approximation of unknown variables, and the Gauss weight function is used as test function for deriving local weak form of governing equations. The Gauss-Legendre quadrature is employed for numerical evaluation of weak forms. A power-law distribution and a half cosine variation are applied to model the variation of materials properties and electric potentials, respectively. The obtained results are verified with well-known solutions in the literature. The effect of nonlocal parameter, material length scale parameter, power-law index, predefined electric field, axial compressive and tensile forces, volume ratio of FG core and piezoelectric layers, and stiffness of Pasternak foundation on the critical buckling loads and natural frequencies are presented and discussed comprehensively. The results obtained can be used in the analysis, design and control of composite nanostructures that are widely used in MEMS and NEMS devices.

On the competition between adhesive and surface effects in the nanocontact properties of an exponentially graded coating

This paper aims at investigating the adhesive nanocontact properties between a rigid cylindrical punch and an exponentially graded coating perfectly bonded with a homogeneous half-plane substrate. The interface of nanocontact is modeled with the Maugis–Dugdale adhesive contact model and the Steigmann–Ogden surface mechanical theory. Under plane strain assumption, governing equations and boundary conditions of the nanocontact problem are converted into triple integral equations. They are numerically solved for the determination of nanocontact length, pressure distribution and the application range of the adhesive zone, by the use of Gauss–Chebyshev quadrature, Gauss–Legendre quadrature and a self-designed iterative algorithm. The method of solution and numerical algorithm are first validated against literature results. Extensive parametric studies are then conducted with respect to the adhesive energy density, Tabor’s parameter, surface material constants and inhomogeneity index of the exponentially graded coating. Based on these results, the relationship between adhesive and surface effects can be identified as competitive. Both are of great importance in the determination of nanocontact behaviors of graded materials and structures. In addition, the Maugis–Dugdale adhesive contact model is found to converge to the Johnson–Kendall–Roberts model under large Tabor’s parameters. Large indenters, high adhesive energy densities, soft substrates and small adhesive cut-off distances are all helpful to the transition. The current work provides a generalized solution framework to the nanocontact problems of graded materials and structures in the simultaneous presence of adhesive and surface effects.

A new operational vector approach for time‐fractional subdiffusion equations of distributed order based on hybrid functions

This research study deals with the numerical solutions of linear and nonlinear time‐fractional subdiffusion equations of distributed order. The main aim of our approach is based on the hybrid of block‐pulse functions and shifted Legendre polynomials. We produce a novel and exact operational vector for the fractional Riemann–Liouville integral and use it via the Gauss–Legendre quadrature formula and collocation method. Consequently, we reduce the proposed equations to systems of equations. The convergence and error bounds for the new method are investigated. Six problems are tested to confirm the accuracy of the proposed approach. Comparisons between the obtained numerical results and other existing methods are provided. Numerical experiments illustrate the reliability, applicability, and efficiency of the proposed method.

Solution of optimal control problems governed by volterra integral and fractional integro-differential equations

In this manuscript, we investigate two categories of optimal control problems (OCPs), OPCs via fractional Volterra integro-differential equations and Volterra integral equations. Touchard wavelets as an appropriate class of bases are defined to develop a new hybrid scheme for the considered problems. To this approach, Riemann–Liouville fractional integral operator (RLFIO) of Touchard wavelets is achieved exactly using the Hypergeometric functions. Next, by approximating the fractional derivative of the state variables and control variables using the mentioned wavelet functions, applying RLFIO, collocation method, and Gauss–Legendre quadrature formula, the considered problems are inserted into systems of algebraic equations, which can be solved using “FindRoot” package in Mathematica software. Numerical results are presented that validate the theory and show the effectiveness of the established technique.

NUMERICAL SOLUTION OF DISTRIBUTED ORDER INTEGRO-DIFFERENTIAL EQUATIONS

In this paper, a numerical algorithm is presented to obtain approximate solution of distributed order integro-differential equations. The approximate solution is expressed in the form of a polynomial with unknown coefficients and in place of differential and integral operators, we make use of matrices that we deduce from the shifted Legendre polynomials. To compute the numerical values of the polynomial coefficients, we set up a system of equations that tallies with the number of unknowns, we achieve this goal through the Legendre–Gauss quadrature formula and the collocation technique. The theoretical aspects of the error bound are discussed. Illustrative examples are included to demonstrate the validity and applicability of the method.

Performance of the CUSUM Control Chart Using Approximation to ARL for Long- Memory Fractionally Integrated Autoregressive Process with Exogenous Variable

The cumulative sum (CUSUM) control chart is a well-known process monitoring tool that is sensitive to small-to-moderate changes in process parameters. In this paper, we propose an approximated average run length (ARL) method based on the numerical integral equation (NIE) method for monitoring the mean of a long-memory autoregressive fractionally integrated process with an exogenous variable (ARFIX) running on a CUSUM control chart. The approximated ARL based on the NIE method is realized by solving a system of linear equations and integration based on the partitioning and summation of the area under the curve of a function derived by using the Gauss-Legendre quadrature. In a comparative study with the ARL based on analytical formulas, the proposed approximated ARL method could detect shifts of various sizes in the process mean of an ARFIX process running on a CUSUM control chart. In addition, the proposed method was compared with their analytical formulas in terms of the relative percentage change (r%) to verify the accuracy of the ARL results. The results revealed that the ARL results obtained from the NIE method are an approach to analytical formulas with r% of less than 0.25. Hence, the NIE method is very accurate and in excellent agreement with the analytical formulas approach. Apparently, the NIE method is an alternative as efficiently as the analytical method for this situation. It also performed well in comparison with the approximated ARL for the same process running on an exponentially weighted moving average control chart. In addition, real datasets are also used to demonstrate the efficacy of the proposed method.

Numerical integration of locally peaked bivariate functions

The aim of this paper is to compare the relative accuracies between deterministic and stochastic methods for solving bounded integrals numerically to observe which methods tend to function well and converge to a small amount of error based on computational resources. For the deterministic method, the Gauss-Legendre quadrature method has been selected and for the stochastic method, the Monte Carlo integration has been selected. For each case, the number of variables will be adjusted to observe the effect on error.

AN EFFECTIVE AND SIMPLE SCHEME FOR SOLVING NONLINEAR FREDHOLM INTEGRAL EQUATIONS

In this paper, a simple scheme is constructed for finding approximate solution of the nonlinear Fredholm integral equation of the second kind. To this end, the Lagrange interpolation polynomials together with the Gauss-Legendre quadrature rule are used to transform the source problem to a system of nonlinear algebraic equations. Afterwards, the resulting system can be solved by the Newton method. The basic idea is to choose the Lagrange interpolation points to be the same as the points for the Gauss-Legendre integration. This facilitates the evaluation of the integral part of the equation. We prove that the approximate solution converges uniformly to the exact solution. Also, stability of the approximate solution is investigated. The advantages of the method are simplicity, fastness and accuracy which enhance its applicability in practical situations. Finally, we provide some test examples.

A generalization of Müntz-Legendre polynomials and its implementation in optimal control of nonlinear fractional delay systems

A hybrid of Müntz-Legendre polynomials (MLPs) and block-pulse functions (BPFs) is defined and carried out to analyze nonlinear fractional optimal control problems consisting of multiple delays. Instead of using the Caputo fractional derivative, an alternative fractional derivative operator is utilized. The primary optimization problem reduces to an alternative optimization one involving unknown parameters. For this purpose, the fractional Legendre-Gauss quadrature formula is utilized for approximating the associated cost functional and the fractional Legendre-Gaussian nodes are taken as the collocation points. Some new aspects of the proposed basis are demonstrated to verify the effectiveness of the proposed approach based on the Müntz-Legendre basis (MLB) against the classical orthogonal bases. The simulation results certify the feasibility and reliability of the proposed method. Numerous fractional control problems, e.g., bang-bang controls and control systems with any irregularities in the structure of control input can be handled successfully by employing the new fractional basis.

Optimized Phonon Band Discretization Scheme for Efficiently Solving the Nongray Boltzmann Transport Equation

Abstract The phonon Boltzmann transport equation (BTE) is an important tool for studying the nanoscale thermal transport. Because phonons have a large spread in their properties, the nongray (i.e., considering different phonon bands) phonon BTE is needed to accurately capture the nanoscale transport phenomena. However, BTE solvers generally require large computational cost. Nongray modeling imposes significant additional complexity on the numerical simulations, which hinders the large-scale modeling of real nanoscale systems. In this work, we address this issue by a systematic investigation on the phonon band discretization scheme using real material properties of four representative materials, including silicon, gallium arsenide, diamond, and lead telluride. We find that the schemes used in previous studies require at least a few tens of bands to ensure the accuracy, which requires large computational costs. We then propose an improved band discretization scheme, in which we divide the mean free path domain into two subdomains, one on either side of the inflection point of the mean free path accumulated thermal conductivity, and adopt the Gauss–Legendre quadrature for each subdomain. With this scheme, the solution of the phonon BTE converges (error < 1%) with less than ten phonon bands for all these materials. The proposed scheme allows significantly reducing the time and memory consumption of the numerical BTE solver, which is an important step toward large-scale phonon BTE simulations for real materials.

ECCPA: Calculation of classical and quantum cross sections for elastic collisions of charged particles with atoms

ECCPA: Calculation of classical and quantum cross sections for elastic collisions of charged particles with atoms

Gravity field forward modelling using tesseroids accelerated by Taylor series expansion and symmetry relations

SUMMARY In this study, we developed a new method that can significantly accelerate the forward modelling of gravity fields generated by large-scale tesseroids while keeping the computational accuracy as high as possible. The cost of the high efficiency is that the method only works under the assumptions that (1) all tesseroids in the same latitude band have the same horizontal dimension, (2) the computation points are located at the same surface level and aligned with the horizontal centres of tesseroids and (3) each tesseroid has a constant or linearly varying density. The new method first integrates the kernel function of the Newton’s volume integral analytically in the radial direction to eliminate its dependence on the vertical dimension of the tesseroid, and then expands the integrated kernel function into a Taylor series up to a certain order. Because the Taylor series expansion term of the integrated kernel function is an odd or even function of the difference between the longitudes of the tesseroid and computation point, there exist shifting or swapping symmetry relations among the gravity field of tesseroids. Consequently, the shifting or swapping symmetry is extended to the tesseroids with unequal vertical dimensions. Numerical experiments using the spherical shell model are conducted to verify the effectiveness of the new method. The results show that the computational speed of the new method is about 30 times faster than that of the traditional method, which employs the Gauss–Legendre quadrature rule and a 2-D adaptive subdivision approach, while keeping almost the same computational accuracy. When applying the new method to an ice shell with unequal thicknesses, the results reveal that the relative errors of calculating V, Vz and Vzz are smaller than 10−8, 10−6 and 10−4, respectively if the Taylor series expansion is truncated at order 4, while the computational time consumed by the new method is about 7 times less than that of the traditional method. Finally, the influence of the truncation order on the computational accuracy and the strategies for dividing the latitude band into several parts to further improve the accuracy are discussed.

A fast algorithm for two-dimensional distributed-order time-space fractional diffusion equations

In this paper, a fast algorithm is proposed for solving distributed-order time-space fractional diffusion equations. Integral terms in time and space directions are discretized by the Gauss-Legendre quadrature formula. The Caputo fractional derivatives are approximated by the exponential-sum-approximation method, and the finite difference method is applied for spatial approximation. The coefficient matrix of the discretized linear system is symmetric positive definite and possesses block-Toeplitz-Toeplitz-block structure. The preconditioned conjugate gradient method with a block-circulant-circulant-block preconditioner is employed to solve the linear system. Theoretically, the stability and convergence of the proposed scheme are discussed. Numerical experiments are carried out to demonstrate the effectiveness of the scheme.