Gaussian Quadrature Research Articles

Relay-Aided D2D MIMO Scheme (RAS) for Achieving Energy Efficiency in Satellite-Air-Ground Integrated Networks (SAGIN) Schéma D2D MIMO assisté par relais (RAS) pour atteindre l’efficacité énergétique dans les réseaux intégrés satellite-air-sol (SAGIN)

Space-air-ground integrated network (SAGIN), as a three-tiered architecture that assimilates satellite systems, aerial, and terrestrial communication networks, has become an intensive research domain in the present era of communications. SAGIN-based communication models are developed to enhance the user’s quality of experience (QoE). Besides providing noteworthy benefits in various applications and services, SAGIN has unprecedented challenges because of its self-organized, unpredictable, and heterogeneous nature. Relaying equipment in SAGIN can be a very low-orbit satellite, a base station (BS), and an unmanned vehicle assisting a pair of mobile users’ communications. Thus, developing a robust device-to-device (D2D) direct and relaying communication model concerning channel distribution is crucial. Based on this concern, this article proposes a relay-aided D2D multiple–input and multiple–output (MIMO) scheme (RAS) for enhancing the optimal energy efficiency (EE) as a function of spectral efficiency (SE). The proposed model derives a relay-based amplify-and-forward (AF) MIMO multihop communication system for implementation. The proposed computations of optimal EE and SE for D2D MIMO show that the approximation provided by a random matrix approximation is constrained to a specific signal-to-noise ratio (SNR) range when the optimal SE and EE are derived using Gaussian quadrature and a hypergeometric function.

Option Pricing Under the Normal SABR Model with Gaussian Quadratures

Option Pricing Under the Normal SABR Model with Gaussian Quadratures

Customization of the angular spectrum method for calculating the acoustic piston field transmitted through a solid plate using MATLAB

The angular spectrum (AS) model is customized to calculate the spatial acoustic pressure field generated by a piston source and transmitted through a steel plate immersed in water, for normal beam incidence. A MATLAB program is developed for this specific problem combining use of Gauss quadrature and a generalized Filon method. The program calculates the pressure wave number spectrum generated by the piston source and transforms the wave number spectrum into the spatial domain. Convergence analysis show that the MATLAB program is far more efficient than the more traditional approach of using the fast Fourier transform algorithm to transform the pressure wavenumber spectrum into the spatial domain. The MATLAB program is published here and free for others to use. The methods and MATLAB algorithms are obtained by•Converting the original 2D AS model to a 1D model using cylindrical coordinates.•Combining use of Gauss quadrature and a generalized Filon method for more accurate pressure calculations compared with use of the fast Fourier transform.•Introducing adaptive numerical integration algorithms in MATLAB and error control parameters which are easy to use.

An improved weak-form quadrature element (IWQE) method for static and dynamic analysis of non-homogeneous plane trusses

Based on Chebyshev interpolation and Gauss-Lobatto quadrature together with the variational principle, an improved weak-form quadrature element (IWQE) method is developed and further customized for analyzing plane truss structures. Compared to existing WQE method which adopts Lagrange interpolation and Gauss quadrature, this developed IWQE method has demonstrated robustness for elements with a large number of quadrature points. Compared to finite element (FE) method whose stiffness matrix of an element is positive and semi-definite, the stiffness matrix of the IWQE method of an element is always positive and definite. Therefore, the displacements of an element's internal nodes can be uniquely expressed by using the values of its end nodes only. Accordingly, the sizes of the assembled matrices only depend on the number of end nodes and the orders of the assembled matrices for the whole structure can be greatly reduced. Such attributes can substantially increase the computational efficiency. To validate the developed IWQE method, the static and dynamic analysis of a plane truss structure with non-homogeneous properties are taken as an example for case study. Compared to the other numerical methods, such as differential quadrature element (DQE), WQE and FE, this IWQE method developed in present work demonstrates better convergence, accuracy and robustness, providing a more efficient and powerful numerical tool for analyzing structures with non-homogeneous attributes.

Integration of a Gaussian quadrature grid discretization approach with a generalized stiffness reduction method and a parallelized direct solver for 3-D frequency-domain seismic wave modelling in viscoelastic anisotropic media

SUMMARY We integrate three advanced numerical techniques—Gaussian quadrature grid (GQG) discretization, a new generalized stiffness reduction method and the latest version of an efficient parallelized direct solver to achieve accurate 3-D frequency-domain seismic wave modelling in viscoelastic anisotropic media. A GQG is employed to sample and interpolate both model parameters and wavefield quantities as well as to fit with arbitrary free-surface topography and subsurface interfaces of a geological model. A new version of the generalized stiffness reduction method is utilized to effectively remove the artificial boundary edge effects for which the common perfectly matched layer method fails. The most recent version of a multifrontal massively parallel direct solver is applied to tackle the notoriously expensive computation of frequency-domain 3-D wave modelling. We validate the 3-D modelling by comparing with the exact solutions for homogeneous viscoelastic isotropic, vertically transversely isotropic and orthorhombic media. All the results show very close matches between the numerical and analytical solutions. Then, we investigate the computational efficiency of the parallelized direct solver, compare its performance using different ordering schemes, in-core and out-of-core factorization modes and the block low-rank approximation in the factorization for different grid sizes. Our modelling results show that the ordering scheme of the so-called ‘Metis’ is the best for reducing computer memory and run time, and the parallelized direct solver is remarkably faster than iterative solvers for similar workloads but at the expense of higher memory requirements. The out-of-core factorization mode can effectively reduce the memory cost without a compromising on run time. The block low-rank approximation is able to significantly reduce the run time in both the factorization and solving process (up to 56 per cent in total), but will increase the memory cost when using the out-of-core factorization mode. Efficient application of this parallel direct solver should use ‘Metis’ as the ordering scheme and select the out-of-core factorization mode without the block low-rank approximation as the best scheme to save the memory cost, or the in-core factorization mode with the block low-rank approximation for the fastest computation. Finally, we demonstrate the excellent applicability of the 3-D wave modelling scheme for a practical and complex heterogeneous geological model.

Gaussian‐like measurement likelihood based particle filter for extended target tracking

AbstractExtended target tracking based on the star‐convex model is a multi‐dimensional nonlinear estimation problem. To approximate the shape of an extended target, the star‐convex model requires a higher‐order Fourier series expansion, which will increase the dimensionality of the extended state, so that the nonlinear filters under Gaussian assumptions cannot converge to the optimal state estimation. In this paper, a novel particle filter algorithm based on Gaussian‐like measurement likelihood (GL) is proposed. As latent variables with high uncertainty in the measurement model, the scattering centres are modelled as a Gaussian‐like probability distribution characterising the target's shape. Then the measurement noise is integrated into the Gaussian‐like probability distribution via Gaussian quadrature to obtain the accurate measurement likelihood. Moreover, particle swarm optimization is employed to reduce computational complexity. It enables particles to move into the vicinity of measurements, thereby increasing the diversity of the particles belonging to the target's shape. The simulation results show that the GL can improve the accuracy of shape estimation under high measurement noise and low measurement rates. Furthermore, the proposed particle filter can track the extended target effectively with an aeroplane‐like shape.

A comparison study of the efficiency and accuracy of IEFG in solving elasticity problems using different nodal integration schemes

The interpolation property (Kronecker delta property) is a favorable feature for the shape functions of meshless methods. The meshless methods with interpolation property can directly impose the essential boundary conditions of problems, which saves computing resources. The interpolating element-free Galerkin method (IEFG) based on the improved interpolating moving least-squares (IMLS) method is an efficient meshless method with high accuracy, which changes the disadvantage that the traditional element-free Galerkin method (EFG) does not have interpolation property. To achieve higher efficiency, direct nodal integration (DNI) is introduced into IEFG in this paper. DNI is simple in execution and cheaper in computing resources, compared with Gauss integration (GI), but it will produce stability issues and sub-optimal convergence. To eliminate the instability of DNI, this paper adopts three improved direct nodal integration schemes, residual balance direct nodal integration (RDNI), naturally stabilized direct nodal integration (NDNI), and modified direct nodal integration (MDNI). Through three 2D elasticity numerical examples, the meshless method combining IEFG with improved direct nodal integration schemes is verified and proved to be efficient, and the effectiveness of several nodal integration schemes in improving integration efficiency and retaining integration stability is compared.

Two-scale concurrent optimization of composites with elliptical inclusions under microstress constraints within the FE2 framework

In order to avoid the adverse effects of structural damage caused by stress concentration, the optimization problem considering stress constraints is very important in structural design. This paper aims at introducing microstress constraints within a multilevel finite element (FE2) analysis framework to perform the two-scale concurrent optimization and reduce the stress concentration while exhibiting better overall stiffness properties for composites with elliptical inclusions. In order to do that, at the macroscale, relying on the solid isotropic material with penalization (SIMP), a density-based optimization algorithm is raised for a maximum stiffness design and free material distribution optimization under a volume constraint, these displacement solutions and material distributions at the macroscopic structural scale provide kinematic constraints for microscale optimization. At the microscale, the parameterized composite microstructure (inclusion orientation and aspect ratio) is optimized under the principal strain direction constraints of the macroscale element’s Gauss integration point. The optimized microstructure in turn updates the macroscopic effective stress directly from the volume average of the microscopic stress field. The clustering strategy is used in order to improve the computation efficiency of the optimization. The novelty of the work lies in the use of microstress constraints to solve the concurrent optimization problems of two-phase composites within the FE2 framework. The effectiveness of the proposed concurrent optimization approach is validated through three numerical examples.

Shifted Gegenbauer-Gauss Collocation Method for Solving Fractional Neutral Functional-Differential Equations with Proportional Delays

In this paper, the shifted Gegenbauer-Gauss collocation (SGGC) method is applied to fractional neutral functional-differential equations with proportional delays. The technique we have used is based on shifted Gegenbauer polynomials and Gauss quadrature integration. The shifted Gegenbauer-Gauss method reduces solving the generalized fractional pantograph equation fractional neutral functional-differential equations to a system of algebraic equations. Reasonable numerical results are obtained by selecting few shifted Gegenbauer-Gauss collocation points. Numerical results demonstrate its accuracy, and versatility of the proposed techniques.

Computational Approach to Solving a Layered Behaviour Differential Equation with Large Delay Using Quadrature Scheme

This paper deals with the computational approach to solving the singularly perturbed differential equation with a large delay in the differentiated term using the two-point Gaussian quadrature. If the delay is bigger than the perturbed parameter, the layer behaviour of the solution is destroyed, and the solution becomes oscillatory. With the help of a special type mesh, a numerical scheme consisting of a fitting parameter is developed to minimize the error and to control the layer structure in the solution. The scheme is studied for convergence. Compared with other methods in the literature, the maximum defects in the approach are tabularized to validate the competency of the numerical approach. In the suggested technique, we additionally focused on the effect of a large delay on the layer structure or oscillatory behaviour of the solutions using a special form of mesh with and without a fitting parameter. The effect of the fitting parameter is demonstrated in graphs to show its impact on the layer of the solution.

Towards improving the 2D-MITC4 element for analysis of plane stress and strain problems

In this paper, we improve the performance of the 2D-MITC4 element for analysis of plane stress and plane strain problems. While the element shows excellent convergence behavior in regular meshes, it exhibits accuracy loss in distorted meshes, as many other elements do. We focus on improving performance in distorted meshes for general use in engineering practice. We simplify the assumed strain field of the original 2D-MITC4 element and introduce the geometry dependent Gauss integration scheme, in which integration points vary according to element geometry distortion. Consequently, a new 2D-MITC4 element is developed. The new 2D-MITC4 element passes the basic tests: patch, zero energy mode, and isotropy tests. Through various numerical examples, improved convergence behaviors of the new element are demonstrated, especially in distorted meshes.

Finite element geotechnical analysis incorporating deep learning-based soil model

Owing to the complicated mechanical behaviors of soils, their constitutive models often involve obscureformulations and suffer from poor applicability in engineering practice. In this study, a novel framework for the finite element (FE) analysis of geotechnical engineering problems is proposed, in which a deep learning (DL) model is employed to depict the constitutive behaviors of soils, circumventing the difficulties associated with conventional approaches. The DL model can incorporate different neural network architectures and is trained with stress–strain data, obtained either experimentally or numerically, before being integrated into the FE solver for analyzing various boundary value problems (BVPs). During the FE solution, the DL model receives strains at the Gauss integration points and returns the predicted stresses to advance the computation. The applicability and capacity of the framework were demonstrated by analyzing three BVPs, in which different geometries, meshes, and boundary conditions were considered. It was shown that the framework is capable of reproducing satisfactory solutions without resorting to any constitutive theory. Furthermore, the use of the DL model not only avoids the stress integration of the conventional FE analysis, but also leads to better computational efficiency.

On a time-discrete convolution—space collocation BEM for the numerical solution of two-dimensional wave propagation problems in unbounded domains

Abstract For the numerical solution of a classical two-dimensional Dirichlet exterior problem for the wave equation in the time domain, we consider a space-time boundary integral equation approach, based on its discretization by means of the coupling of a time discrete convolution quadrature of order 2, with the classical continuous piecewise linear boundary element method. The latter is either the Galerkin method or the collocation one, this being computationally much cheaper. For the efficient evaluation of most of the integrals generated by these two boundary element methods, but also for the subsequent analysis, we first define a new Gaussian quadrature. After recalling that stability and convergence have been proved for the Galerkin method, while for the collocation one we only have numerical evidences of these properties, to (partially) fill this gap, we first note that collocation can be obtained directly from the Galerkin system, after discretizing its inner product integral by using the composite trapezoidal rule, which in this case turns out to be the Gaussian rule mentioned above with $1$ node. Then, we show that for all (positive) time step-sizes $\varDelta _t$, when we let $h\rightarrow 0$, independently from $\varDelta _t$, the matrix of the linear system produced by the collocation method, properly normalized, converges, in the $\infty $-norm, to the corresponding Galerkin matrix. The behavior of the associated absolute and relative errors in the $\infty $-norm, as $h\rightarrow 0$, are $O\big (h^{\frac {5}{3}}+\varDelta _t^{-(1+\varepsilon )}h^{\frac {5+\varepsilon }{2}}[|\!\ln \varDelta _t|+|\!\ln h|]\big )$ and $O\big (h^{\frac {2}{3}}/|\!\ln \varDelta _t|+\varDelta _t^{-(1+\varepsilon )}h^{\frac {3+\varepsilon }{2}}[1+|\!\ln h|/|\!\ln \varDelta _t|]\big )$, respectively, with $\varepsilon>0$ as close as one likes to $0$ and the $O(\cdot )$ constant independent from $\varDelta _t, h$. Some possible extensions of the investigation we carried out are also mentioned.

Estimating the trace of matrix functions with application to complex networks

The approximation of trace(f(Ω)), where f is a function of a symmetric matrix Ω, can be challenging when Ω is exceedingly large. In such a case even the partial Lanczos decomposition of Ω is computationally demanding and the stochastic method investigated by Bai et al. (J. Comput. Appl. Math. 74:71–89, 1996) is preferred. Moreover, in the last years, a partial global Lanczos method has been shown to reduce CPU time with respect to partial Lanczos decomposition. In this paper we review these techniques, treating them under the unifying theory of measure theory and Gaussian integration. This allows generalizing the stochastic approach, proposing a block version that collects a set of random vectors in a rectangular matrix, in a similar fashion to the partial global Lanczos method. We show that the results of this technique converge quickly to the same approximation provided by Bai et al. (J. Comput. Appl. Math. 74:71–89, 1996), while the block approach can leverage the same computational advantages as the partial global Lanczos. Numerical results for the computation of the Von Neumann entropy of complex networks prove the robustness and efficiency of the proposed block stochastic method.

Meshless numerical solution for nonlocal integral differentiation equation with application in peridynamics

The mid-point quadrature rule is often used to solve the nonlocal integral differentiation equations, in which a central material point is used to calculate the integral of the subdomain in the meshless geometry. However, it can only achieve the linear approximation and the first order convergence rate, and it is only accurate enough when the entire cell overlaps with the neighborhood of a material point. In this paper, to achieve the higher precision and the higher rate of convergence speed, a novel coupling method of reproducing kernel particle method and Gaussian-Legendre quadrature scheme is proposed to solve the nonlocal integral differentiation equations in the meshless geometry, in which the mid-point quadrature scheme is replaced with Gauss quadrature scheme to calculate the integral of the subdomain in the meshless geometry, the reproducing kernel particle method is employed to construct the shape functions for material points in the nonlocal model, and to approximate the value of the Gaussian quadrature node. Moreover, the meshless solution for the peridynamic equation is derived detailedly based on the proposed method. Compared with the traditional meshless solution for the nonlocal equations, the proposed solution has higher precision and higher rate of convergence speed in different problems. The numerical results demonstrate improved results in both mathematics and mechanical problems using the proposed method.

In memory of Peter Carr

Demonstrates Peter den Iseger�s paper that combines Gaussian quadrature with Fast Fourier Transforms to produce Laplace Transform inversion that is extremely fast and approaching machine precision

A two-level nesting smoothed extended meshfree method for static and dynamic fracture mechanics analysis of orthotropic materials

This paper presents a two-level nesting smoothed extended mesh-free method (S-XMM-N) for the static and dynamic fracture mechanics analysis in orthotropic materials. To capture the jump of displacement fields across the crack surface, as well as singularities of stress fields in the vicinity of the crack tip, both Heaviside function and asymptotic solutions from the orthotropic fracture modeling are incorporated into the radial point interpolation shape functions associated with the partition of unity approach. To improve the accuracy and accelerate the computation of the extended mesh-free method, the two-level nesting smoothing integration scheme based on the first and second-level triangular smoothing sub-domains is developed to perform the stiffness integration. Compared to the extended mesh-free method using the conventional Gaussian quadrature, the presented S-XMM-N uses even less integration sampling points. At the same time, the complex and time-consuming derivative calculation of extended shape functions is completely avoided. This leads to the dramatically improvement of efficiency. Moreover, the presented method gives very exactly numerical solutions by optimally combining the contributions from the two-level nesting smoothing sub-domains based on the Richardson extrapolation method. We use the interaction integral method in conjunction with the asymptotic near crack tip fields of orthotropic materials to extract the static and dynamic stress intensity factors. The numerical solutions obtained from the S-XMM-N are further compared with reference solutions from the literature to verify the accuracy of the proposed method.

A new method for nonlinear state estimation problem

In this paper, a new filtering technique to solve a nonlinear state estimation problem has been developed with the help of the Gaussian integral. It is well known that for a nonlinear system, the prior and the posterior probability density functions (pdfs) are non-Gaussian in nature. However, in this work, they are assumed to be Gaussian; subsequently, the mean and the covariance are calculated. In the proposed method, nonlinear functions of process dynamics and measurements are expressed in a polynomial form with the help of the Taylor series expansion. In order to calculate the prior and the posterior mean and covariance, the functions are integrated over the Gaussian pdf with the Gaussian integral. The performance of the proposed method is tested on three nonlinear state estimation problems. The simulation results show that the proposed filter provides more accurate results than other existing deterministic sample point filters such as the cubature Kalman filter, the unscented Kalman filter, and the Gauss-Hermite filter.

Application of the gradient theory to interface crack between two dissimilar dielectric materials

In the present paper, the interface crack between two dissimilar dielectric materials under a mechanical load is investigated with including flexoelectricity effects. Flexoelectricity is a size dependent electro-mechanical coupling phenomenon, where the electric polarization is induced by a strain gradient in dielectrics. The strain gradients may potentially break the inversion symmetry in centrosymmetric crystals and polarization is observed even in all dielectric materials. The polarization is proportional to the strain gradients in the direct flexoelectricity. Layered composite structures are frequently utilized in microelectronics. Due to a poor adhesion of protection layer and basic material, the interface crack can be created there and for the prediction of failure of these structures it becomes essential to investigate distribution of the interfacial stress and strain fields. Governing equations in the gradient theory contain higher-order derivatives than in the standard continuum mechanics. Therefore, areliable computational tool is required to solve these boundary-value problems. The mixed finite element method (FEM) is developed, where the standard C0 continuous finite elements are utilized for independent approximations of displacements and strains. The constraints between the displacement gradients and strains are satisfied by collocation at Gaussian integration points inside elements. In numerical examples, a parametric study is performed with respect to flexoelectric and elastic coefficients for both material regions. The influence of these parameters on the crack opening displacement is discussed.

A machine learning-based atomistic-continuum multiscale technique for modeling the mechanical behavior of Ni3Al

In this paper, a machine learning-based atomistic-continuum multiscale method is developed to model the mechanical behavior of Ni-based superalloys. The kinematic and energetic consistency principles are employed to link the atomistic and continuum scales. The Ni-based superalloys with different void defects are proposed, and the effect of various parameters is investigated to distinguish the proper atomistic RVE in the multiscale analysis. The dataset of stress-strain samples is generated using the molecular dynamics analysis under various loading conditions that is divided into several groups according to strain values by the K–means algorithm to enhance the regression accuracy, in which a feedforward neural network is trained for each group. In order to optimize the unknown parameters in neural networks, the Bayesian regularization approach is employed through the optimization process. The material properties of the coarse-scale domain required for the multiscale analysis, i.e., the stress tensor and tangent modulus, are then derived from the trained neural networks for each Gauss integration point. Finally, several numerical examples are solved to illustrate the capability of the proposed machine learning-based multiscale technique. The results of the multiscale method are compared with those of molecular dynamics analysis that shows the capability of the proposed computational algorithm in capturing the phenomena, such as the rupture and stress concentration in Ni-based superalloys