Machine Learning Enhanced Boundary Element Method:Prediction of Gaussian Quadrature Points

This paper applies a machine learning technique to find a general and efficient numerical integration scheme for boundary element methods.A model based on the neural network multi-classification algorithmis constructed to find the minimum number of Gaussian quadrature points satisfying the given accuracy.The constructed model is trained by using a large amount of data calculated in the traditional boundary element method and the optimal network architecture is selected.The two-dimensional potential problem of a circular structure is tested and analyzed based on the determined model,and the accuracy of the model is about 90%.Finally,by incorporating the predicted Gaussian quadrature points into the boundary element analysis,we find that the numerical solution and the analytical solution are in good agreement,which verifies the robustness of the proposed method.

Explicit Gaussian Quadrature Rules for C1 Cubic Splines with Non-uniform Knot Sequences

This paper provides the explicit and optimal quadrature rules for the cubic C1 spline space,which is the extension of the results in Ait-Haddou et al.(J Comput Appl Math 290:543–552,2015)for less restricted non-uniform knot values.The rules are optimal in the sense that there exist no other quadrature rules with fewer quadrature points to exactly integrate the functions in the given spline space.The explicit means that the quadrature nodes and weights are derived via an explicit recursive formula.Numerical experiments and the error estimations of the quadrature rules are also presented in the end.

Convergence of Gaussian Quadrature Formulas for Power Orthogonal Polynomials

In classical theorems on the convergence of Gaussian quadrature formulas for power orthogonal polynomials with respect to a weight w on I （a,b）, a function G E S（w）= （f： fxlf（x）lw（x）dx 〈 ∞ satisfying the conditions G（2J）（x） ：〉 O, x E （a,b）, j = 0, 1 , and growing as fast as possible as x→ a- and x → b-, plays an important role. But to find such a function G is often difficult and complicated. This implies that to prove convergence of Gaussian quadrature formulas, it is enough to find a function G E S（w） with G ≥ 0 satisfying

Product Gaussian Quadrature on Circular Lunes

Resorting to recent results on subperiodic trigonometric quadrature,we provide three product Gaussian quadrature formulas exact on algebraic polynomials of degree n on circular lunes.The first works on any lune,and has n^(2)+O(n)cardinality.The other two have restrictions on the lune angular intervals,but their cardinality is n^(2)/2+O(n).

Approximate solution of Volterra-Fredholm integral equations using generalized barycentric rational interpolant

It is well-known that interpolation by rational functions results in a more accurate approximation than the polynomials interpolation.However,classical rational interpolation has some deficiencies such as uncontrollable poles and low convergence order.In contrast with the classical rational interpolants,the generalized barycentric rational interpolants which depend linearly on the interpolated values,yield infinite smooth approximation with no poles in real numbers.In this paper,a numerical collocation approach,based on the generalized barycentric rational interpolation and Gaussian quadrature formula,was introduced to approximate the solution of Volterra-Fredholm integral equations.Three types of points in the solution domain are used as interpolation nodes.The obtained numerical results confirm that the barycentric rational interpolants are efficient tools for solving Volterra-Fredholm integral equations.Moreover,integral equations with Runge’s function as an exact solution,no oscillation occurrs in the obtained approximate solutions so that the Runge’s phenomenon is avoided.

Modified precise time step integration method of structural dynamic analysis

The precise time step integration method proposed for linear time-invariant homogeneous dynamic systems can provide precise numerical results that approach an exact solution at the integration points. However, difficulty arises when the algorithm is used for non-homogeneous dynamic systems, due to the inverse matrix calculation and the simulation accuracy of the applied loading. By combining the Gaussian quadrature method and state space theory with the calculation technique of matrix exponential function in the precise time step integration method, a new modified precise time step integration method （e.g., an algorithm with an arbitrary order of accuracy） is proposed. In the new method, no inverse matrix calculation or simulation of the applied loading is needed, and the computing efficiency is improved. In particular, the proposed method is independent of the quality of the matrix H. If the matrix H is singular or nearly singular, the advantage of the method is remarkable. The numerical stability of the proposed algorithm is discussed and a numerical example is given to demonstrate the validity and efficiency of the algorithm.

Reliability-Optimal Pilot-Assisted Transmission for URLLC over Non-Reciprocal MISO Channels: TDD or FDD?

In this paper, we focus on the pilot-assisted transmission design for downlink URLLC over nonreciprocal channels, in which the multi-antenna controller sends mission-critical data signals to a singleantenna actuator. In this system, the prior knowledge of downlink channel state information(CSI) is a prerequisite for reliable data transmission. Generally, the acquisition of downlink CSI is completed either via the uplink pilot measurement exploiting channel reciprocity and time-division duplex(TDD) operation, or via the downlink pilot measurement with quantized feedback and frequency division duplex(FDD) operation. Inspired by this, we aim to investigate how the degree of channel non-reciprocity impacts the transmission reliability of our URLLC system, and the superiority between the TDD mode and FDD mode in terms of transmission reliability maximization. To describe the degree of reliability loss, we derive the closed-form approximations on the transmission error probability of URLLC in TDD and FDD modes, via leveraging the Gauss-Hermite and Gauss-Chebyshev quadrature rules. Following by the theoretical approximations, we demonstrate how to determine the optimal training pilot length and quantized feedback duration that maximize the transmission reliability under given latency constraint. Through numerical results,we validate the accuracy of theoretical approximations derived in this paper, and obtain some meaningful conclusions.

EMT simulation and effect of TTI anisotropic media in EMT signal

An axisymmetric finite difference method is employed for the simulations of electromagnetic telemetry in the homogeneous and layered underground formation.In this method,we defined the anisotropy property using extensive 2D conductivity tensor and solved it in the transverse magnetic mode.Significant simplification arises in the decoupling of the anisotropic parameter.The developed method is cost-efficient,more straightforward in modeling anisotropic media,and easy to be implemented.In addition,we solved the integral operation in the estimation of measured surface voltage using Gaussian quadrature technique.We performed a series of numerical modeling of EM telemetry signals in both isotropic and anisotropic models.Experiment with 2D tilt transverse isotropic media characterized by the tilt axis and anisotropy parameters shows an increase in the EMT signal with an increase in the angle of tilt of the principal axis for a moderate coefficient of anisotropy.We show that the effect of the tilt of the subsurface medium can be observed with sufficient accuracy and that it is an order of magnitude of 5 over the tilt of 90 degrees.Lastly,consistent results with existing field data were obtained by employing the Gaussian quadrature rule for the computation of surface measured signal.

Orthogonal Polynomials with Respect to Modified JacobiWeight and Corresponding Quadrature Rules of Gaussian Type

In this paper we consider polynomials orthogonal with respect to the linear functional L:P→C,defined on the space of all algebraic polynomials P by L[p]=∫_(-1)^(1)p(x)(1−x)^(α−1/2)(1+x)^(β−1/2)exp(iζx)dx,whereα,β>−1/2 are real numbers such thatℓ=|β−α|is a positive integer,andζ∈R\{0}.We prove the existence of such orthogonal polynomials for some pairs ofαandζand for all nonnegative integersℓ.For such orthogonal polynomials we derive three-term recurrence relations and also some differential-difference relations.For such orthogonal polynomials the corresponding quadrature rules of Gaussian type are considered.Also,some numerical examples are included.

An Extension of Chebyshev's Maximum Principle to Several Variables

In this article, we generalize Chebyshev＇s maximum principle to several variables. Some analogous maximum formulae for the special integration functional are given. A sufficient condition of the existence of Chebyshev＇s maximum principle is also obtained.

A numerical method for structural uncertainty response computation

A novel numerical procedure, which realizes the stochastic analysis with dimensional reduction integration （DRI）, C-type Gram-Charlier （CGC） series, and finite element （FE） model, is proposed to assess the probability distribution of structural per- formance. From the relationship between the weighting function of orthogonal polynomial and probability density function （PDF） of random variable, the numerical integration formulas are derived for moment computation. Then, distribution of structural uncertainty response can be approximated by the CGC series with the calculated moments. Three engineering appli- cations for the distribution of, the maximum displacement of a ten-bar planer truss, natural frequency of an auto frame, and Von-Mises stress of a bending pipe, are employed to illustrate the computational efficiency and accuracy of the proposed methodology. As compared with plain Monte Carlo simulation （MCS）, the method can obtain the accurate distribution of structural performance. Especially at the tail region of cumulative distribution function （CDF）, results have shown the satisfy- ing estimators for small probabilities, say, Pc [104, 10-3]. That implies the method could be trusted for structural failure prob- ability prediction. As the computational efficiency is concerned, the procedure could save more than two orders of computational resources as compared with the direct numerical integration （NI） and MCS. Furthermore, realization of the procedure does not require computing the performance gradient or Hessian matrix with respect to random variables, or reshaping the finite element matrix as other stochastic finite element （SFE） codes. Therefore, it should be an efficient and reliable routine for uncertainty structural analysis.

A Uniqueness Theorem of a System of Nonlinear Equations and Its Applications

A uniqueness theorem of a solution of a system of nonlinear equations is given. Using this result uniqueness theorems for power orthogonal polynomials, for a Gaussian quadrature formula of an extended Chebyshev system, and for a Gaussian Birkhoff quadrature formula are easily deduced.

Comparison study of several numerical integration schemes used in calculations of density functional theory

Several numerical integration schemes for the evaluation of matrix elements in density functional theory calculations have been studied and compared by computational practice. The best scheme was found to be the combination of the atomic partition function proposed by Becke with the scaled generalized Gauss-Laguerre quadrature formula for radial integration suggested by Yang, which achieve the highest convergence rate to the numerical integration. With the same number of integration points, the accuracy of the calculated results by this scheme is higher by 1 to 2 orders of magnitudes than that by other schemes. The reason for achieving higher accuracy by this scheme has been proposed preliminarily.

A Spectral Time-Domain Method for Computational Electrodynamics

Ever since its introduction by Kane Yee over forty years ago,the finitedifference time-domain(FDTD)method has been a widely-used technique for solving the time-dependent Maxwell’s equations that has also inspired many other methods.This paper presents an alternative approach to these equations in the case of spatially-varying electric permittivity and/or magnetic permeability,based on Krylov subspace spectral(KSS)methods.These methods have previously been applied to the variable-coefficient heat equation and wave equation,and have demonstrated high-order accuracy,as well as stability characteristic of implicit timestepping schemes,even though KSS methods are explicit.KSS methods for scalar equations compute each Fourier coefficient of the solution using techniques developed by Golub and Meurant for approximating elements of functions of matrices by Gaussian quadrature in the spectral,rather than physical,domain.We show how they can be generalized to coupled systems of equations,such as Maxwell’s equations,by choosing appropriate basis functions that,while induced by this coupling,still allow efficient and robust computation of the Fourier coefficients of each spatial component of the electric and magnetic fields.We also discuss the application of block KSS methods to problems involving non-self-adjoint spatial differential operators,which requires a generalization of the block Lanczos algorithm of Golub and Underwood to unsymmetric matrices.

Ascent Trajectory Optimization for Air-Breathing Hypersonic Vehicles Based on IGS-MPSP

This paper proposes an improved Generalized Quasi-Spectral Model Predictive Static Programming(GS-MPSP)algorithm for the ascent trajectory optimization for hypersonic vehicles in a complex°ight environment.The proposed method guarantees the satisfaction of constraints related to the state and control vector while retaining its high computational e±ciency.The spectral representation technique is used to describe the control variables,which reduces the number of decision variables and makes the control input smooth enough.Through Taylor expansion,the constraints are transformed into an inequality containing only decision variables,such that it can be added into GS-MPSP framework.By Gauss quadrature collocation method,only a few collocation points are needed to solve the sensitivity matrix,which greatly accelerates the calculation.Subsequently,the analytical expression is obtained by combining the static optimization with the penalty function method.Finally,the simulation results demonstrate that the proposed improved GS-MPSP algorithm can achieve both high computational e±-ciency and high terminal precision under the constraints..