Gaussian Quadrature Method Research Articles

Gaussian quadrature method with exponential fitting factor for two-parameter singularly perturbed parabolic problem

The parabolic convection-diffusion-reaction problem is examined in this work, where the diffusion and convection terms are multiplied by two small parameters, respectively. The proposed approach is based on a fitted operator finite difference method. The Crank-Nicolson method on uniform mesh is utilized to discretize the time variables in the first step. Two-point Gaussian quadrature rule is used for further discretizing these semi-discrete problems in space, and the second order interpolation of the first derivatives is utilized. The fitting factor’s value, which accounts for abrupt changes in the solution, is calculated using the theory of singular perturbations. The developed scheme is demonstrated to be second-order accurate and uniformly convergent. The proposed method’s applicability is validated by two examples, which yielded more accurate results than some other methods found in the literatures.

Solute transport along a single channel with radial diffusion in the porous rock matrix: A simple analytical solution and the implementation of time domain random walk algorithm

Numerous field observations show that the channels in one fracture are narrow and the solute penetration depth might be larger than the width. For this case, the diffusion from a channel into the matrix is more realistic to be modeled as radial diffusion than one-dimensional. In present work, the single channel model with radial diffusion is revisited and a simple and robust analytical solution is developed. This solution takes a convolution form of two functions, in which different transport mechanisms are accounted for. The statistical interpretations of the two functions and the analytical solution aid to develop a simple Time Domain Random Walk (TDRW) algorithm and an extension is made to improve its accuracy, efficiency and applicability. To demonstrate the accuracy and efficacy of the extended algorithm, three groups of simulations are performed and it is found that the results of all approaches are identical. The TDRW algorithm, having the same performance as that of inverse Laplace transform solution, is superior to Gaussian quadrature method in computational time. However, due to Monte Carlo nature of the algorithm, the computational burden of TDRW algorithm is dependent on the number of particles applied, which also influences the calculation accuracy. Therefore, a trade-off between computational burden and calculation accuracy should always be made, once the TDRW algorithm is used.

3D large-scale forward modeling of gravitational fields using triangular spherical prisms with polynomial densities in depth

To take the sphericity of the Earth into account, tesseroids are often utilized as grid elements in large-scale gravitational forward modeling. However, such elements in a latitude–longitude mesh suffer from degenerating into poorly shaped triangles near poles. Moreover, tesseroids have limited flexibility in describing laterally variable density distributions with irregular boundaries and also face difficulties in achieving completely equivalent division over a spherical surface that may be desired in a gravity inversion. We develop a new method based on triangular spherical prisms (TSPs) for 3D gravitational modeling in spherical coordinates. A TSP is defined by two spherical surfaces of triangular shape, with one of which being the radial projection of the other. Due to the spherical triangular shapes of the upper and lower surfaces, TSPs enjoy more advantages over tesseroids in describing mass density with different lateral resolutions. In addition, such an element also allows subdivisions with nearly equal weights in spherical coordinates. To calculate the gravitational effects of a TSP, we assume the density in each element to be polynomial along radial direction so as to accommodate a complex density environment. Then, we solve the Newton’s volume integral using a mixed Gaussian quadrature method, in which the surface integral over the spherical triangle is calculated using a triangle-based Gaussian quadrature rule via a radial projection that transforms the spherical triangles into linear ones. A 2D adaptive discretization strategy and an extension technique are also combined to improve the accuracy at observation points near the mass sources. The numerical experiments based on spherical shell models show that the proposed method achieves good accuracy from near surface to a satellite height in the case of TSPs with various dimensions and density variations. In comparison with the classical tesseroid-based method, the proposed algorithm enjoys better accuracy and much higher flexibility for density models with laterally irregular shapes. It shows that to achieve the same accuracy, the number of elements required by the proposed method is much less than that of the tesseroid-based method, which substantially speeds up the calculation by more than 2 orders. The application to the tessellated LITHO1.0 model further demonstrates its capability and practicability in realistic situations. The new method offers an attractive tool for gravity forward and inverse problems where the irregular grids are involved.

Strengths Model for Youth (SM-Y) Case Management: Initial Findings on Youth Outcomes.

Case management is a widely utilized service in both youth and adult outpatient community mental health settings. Despite its widespread use, previous findings suggest that youth case management often lacks empirically tested models or frameworks. This article presents the results of a pilot study that involved adapting the Strengths Model, an adult case management model, for the child and adolescent outpatient community mental health population. The newly adapted model, known as the Strengths Model for Youth (SM-Y), was implemented in an urban community mental health center across five different youth case management teams. To assess changes over time in youth receiving SM-Y case management, marginal maximum likelihood multilevel modeling with adaptive Gaussian quadrature methods was applied. The study focused on three domains: socialization, education, and hospitalization. Utilizing the logit link function and Bernoulli conditional distribution due to the binary nature of the outcome data, three individual trajectories were drawn for socialization, education, and hospitalization. Positive findings indicated increases in socialization and educational performance among children and adolescents.

Learning Fuel-Optimal Trajectories for Space Applications via Pontryagin Neural Networks

This paper demonstrates the utilization of Pontryagin Neural Networks (PoNNs) to acquire control strategies for achieving fuel-optimal trajectories. PoNNs, a subtype of Physics-Informed Neural Networks (PINNs), are tailored for solving optimal control problems through indirect methods. Specifically, PoNNs learn to solve the Two-Point Boundary Value Problem derived from the application of the Pontryagin Minimum Principle to the problem’s Hamiltonian. Within PoNNs, the Extreme Theory of Functional Connections (X-TFC) is leveraged to approximate states and costates using constrained expressions (CEs). These CEs comprise a free function, modeled by a shallow neural network trained via Extreme Learning Machine, and a functional component that consistently satisfies boundary conditions analytically. Addressing discontinuous control, a smoothing technique is employed, substituting the sign function with a hyperbolic tangent function and implementing a continuation procedure on the smoothing parameter. The proposed methodology is applied to scenarios involving fuel-optimal Earth−Mars interplanetary transfers and Mars landing trajectories. Remarkably, PoNNs exhibit convergence to solutions even with randomly initialized parameters, determining the number and timing of control switches without prior information. Additionally, an analytical approximation of the solution allows for optimal control computation at unencountered points during training. Comparative analysis reveals the efficacy of the proposed approach, which rivals state-of-the-art methods such as the shooting technique and the adaptive Gaussian quadrature collocation method.

Efficient and accurate numerical-projection of electromagnetic multipoles for scattering objects

In this paper, we develop an efficient and accurate procedure of electromagnetic multipole decomposition by using the Lebedev and Gaussian quadrature methods to perform the numerical integration. Firstly, we briefly review the principles of multipole decomposition, highlighting two numerical projection methods including surface and volume integration. Secondly, we discuss the Lebedev and Gaussian quadrature methods, provide a detailed recipe to select the quadrature points and the corresponding weighting factor, and illustrate the integration accuracy and numerical efficiency (that is, with very few sampling points) using a unit sphere surface and regular tetrahedron. In the demonstrations of an isotropic dielectric nanosphere, a symmetric scatterer, and an anisotropic nanosphere, we perform multipole decomposition and validate our numerical projection procedure. The obtained results from our procedure are all consistent with those from Mie theory, symmetry constraints, and finite element simulations.Graphical

Numerical Approach for Evaluation of Surface Integrals in Polygonal Domain by Gauss Legendre Quadrature and Generalized Gaussian Quadrature Method

We present a new approach for the numerical integration of arbitrary functions over polygonal region, by applying two kinds of quadrature method Gauss Legendre quadrature and Generalized Gaussian quadrature method, the polygonal region is divided into arbitrary triangles, the sides of each triangle is noted as equation of straight line by joining two end vertices, this approach is used to further reduces the integral equations, numerical integration of rational and irrational functions are approximated computationally, we illustrate several numerical examples to shows the accuracy of the present method

Adaptive sinh transformation Gaussian quadrature for 2D potential problems using deep learning

In the boundary element method (BEM), the sinh transformation method is an effective method for evaluating nearly singular integrals, but a relationship between the integration accuracy and the number of Gaussian points is needed to achieve adaptive computation. Based on deep learning, we propose a novel integration scheme, adaptive sinh transformation Gaussian quadrature (ASTGQ), which can determine the number of Gaussian points according to the required accuracy. First, a large number of integration data samples of the sinh transformation method are generated in different cases, and the neural network is trained to establish the relationship between the number of Gaussian points and the integration accuracy. Then, based on the improved loss function and evaluation index, a better network model is obtained to ensure that the actual integration accuracy is slightly higher than the requirement of using the minimum Gaussian points. In this way, when the trained neural network is used in the sinh transformation method, the higher accuracy requirement can be met at a lower cost. Numerical examples demonstrate that, compared to the adaptive Gaussian quadrature (AGQ) method, the proposed scheme can significantly improve the computational efficiency when evaluating the nearly singular integrals for very thin coatings and other structures.

Integrated Soil–Pile Elements for Inelastic Design of Steel Piles Considering Lateral Soil–Pile Interactions

Steel piles are widely used in deep foundation systems to transfer significant structural loads to strong soil layers. A steel pile may buckle or fail by forming multiple plastic hinges at different locations. The pile’s plastic hinge distributions and failure mode are considerably affected by soil–pile interactions (SPIs). A proper consideration of SPIs is essential for the successful design of a steel pile foundation. In practice, empirical calculation or numerical modeling analysis methods are commonly utilized. Nevertheless, the conventional design methods are cumbersome in the modeling process and may be inaccurate in considering SPIs. This paper develops a new integrated soil–pile element to explicitly reflect SPIs and capture steel piles’ inelastic behaviors based on the plastic hinge method and the Euler–Bernoulli beam theory. Two virtual nonlinear springs, the concentrated rotational and distributed lateral springs, are used to simulate the plastic hinge deformations and the soil responses, respectively. The rotational and lateral springs are assigned at both ends and the Gauss integration points of an element, respectively. The Gaussian quadrature method is utilized to consider the soil reactions and generate the soil stiffness matrix. Based on the numerical integration and condensation methods, the virtual springs are directly integrated into the element formulation, simplifying the programming and practical modeling process. The updated Lagrangian approach is adopted for describing the element kinematic motions. The element tangent and secant relations between nodal forces and displacements are derived based on the potential energy principle. Several verification examples are presented to demonstrate the capability of the proposed element.

Estimation of contact stresses in EN31 rolling contact bearings for screw compressor using Gauss quadrature & statistical analysis

ABSTRACT The aim of this work is to estimate Hertzian contact stresses in EN31 alloy steel and to obtain the line of best fit using statistical analysis. In refrigerant screw compressors, EN31 is used as a material of construction for cylindrical roller bearings. The Hertzian contact with friction between the outer surface of the inner race of the rolling contact bearing & the outer surface of the cylindrical roller is simulated using pin-on-disk tribometer. Measurement of friction coefficient was carried out by sliding different cylinders of hardened EN31 material against the hardened EN31 disk of the pin-on-disk tribometer. A total 16 tests, each of 1 minute time duration, are performed for the estimation of friction coefficient with varying loads from 20 N to 100 N and speeds in the range of 50 RPM to 200 RPM. The magnitude of Von Mises stress induced in the inner race of the bearing is estimated using Gauss quadrature method with the help of a self-developed MS Excel program. Statistical analysis showed that the stress magnitude increases with the increase in the load & decreases with an increase in speed.

Accelerated estimation of sea-spray-mediated heat flux using Gaussian quadrature: case studies with a coupled CFSv2.0-WW3 system

Abstract. Sea-spray-mediated heat flux plays an important role in air–sea heat transfer. Heat flux integrated over the droplet size spectrum can simulate well the total heat flux induced by sea spray droplets. Previously, a fast algorithm of spray flux assuming single-radius droplets (A15) was widely used, as the full-size spectrum integral is computationally expensive. Based on the Gaussian quadrature (GQ) method, a new fast algorithm (SPRAY-GQ) of sea-spray-mediated heat flux is derived. The performance of SPRAY-GQ is evaluated by comparing heat fluxes with those estimated from the widely used A15. The new algorithm shows a better agreement with the original spectrum integral. To further evaluate the numerical errors of A15 and SPRAY-GQ, the two algorithms are implemented into the coupled Climate Forecast System model version 2.0 (CFSv2.0) and WAVEWATCH III (WW3) system, and a series of 56 d simulations in summer and winter are conducted and compared. The comparisons with satellite measurements and reanalysis data show that the SPRAY-GQ algorithm could lead to more reasonable simulation than the A15 algorithm by modifying air–sea heat flux. For experiments based on SPRAY-GQ, the sea surface temperature at middle to high latitudes of both hemispheres, particularly in summer, is significantly improved compared with the experiments based on A15. The simulation of 10 m wind speed and significant wave height at middle to low latitudes of the Northern Hemisphere after the first 2 weeks is improved as well. These improvements are due to the reduced numerical errors. The computational time of SPRAY-GQ is about the same as that of A15. Therefore, the newly developed SPRAY-GQ algorithm has potential to be used for the calculation of spray-mediated heat flux in coupled models.

Stability analysis of ef Gaussian direct quadrature methods for Volterra integral equations

The present paper analyzes the numerical stability of exponentially fitted Gaussian direct quadrature methods for Volterra integral equations. By applying the exponential fitting technique, the coefficients of these methods are specially tuned on the problem. The analysis of numerical stability of these methods with respect to the classical basic test equation is carried out, which takes into account the dependence of the methods both on the stepsize and on the problem parameters. Several stability regions are shown and some numerical experiments confirm theoretical expectations.

Crack detection in arbitrary beam cross-sections using a new mass-spring system

This paper presents a new system of spring masses for crack detection in structures. For modeling cracks in arbitrary cross-sections, the Gaussian quadrature method is used to obtain the integral of the energy release rate over the cracked cross-section. In the proposed method, the system of spring masses is used to obtain the time histories of displacements, which are then combined into a relative time history by weighted summation. For detecting cracks, three springs and three equal masses with equal spacing from each other are placed on the beam, starting to move consecutively from one end. At each station, the time history of the system under excitation by an external load applied to the middle mass is extracted. The extracted data in each station shows the relative stiffness of that station compared to adjacent stations. The differences in system responses are used to estimate the position of cracks. The performance of the proposed method was evaluated using the Z24 bridge and a three-span concrete girder bridge in which a multiple-crack scenario was applied. These examples were used for evaluating the proposed system for the effect of parameters such as the weight and spacing of masses. The novelty of the presented method is to model cracks in arbitrary beam cross-sections and identify cracks in real bridge beams using a new effective mass-spring system that uses only three sensor data.

Analytical solution and flow topology in a lid-driven S-shaped cavity

In this study, the Stokes flow problem in an S-shaped double lid-driven cavity filled with fluid was analyzed. Side edges of the cavity were considered as immovable walls. The flow region was divided into two sub-regions, and the streamfunction in each sub-region was considered as an extension of Papkovich–Faddle eigenfunctions. Parameters in the analytical solution were obtained using biorthogonality conditions. The Newton iteration method was used to obtain the eigenvalues of the problem, and integrals were calculated with the Gaussian quadrature method. It was ensured that solutions made separately for the two sub-regions converge on the interface, which is the intersection of these sub-regions. The two parameters controlling the flow structure were determined as the speed ratio of movable lids (S) and the aspect ratio of the cavity (A). The effects of these parameters on flow structures were shown. New eddy formation mechanisms and bifurcations were observed in the cavity by keeping the speed ratio of the lids constant and slowly changing the aspect ratio.

Dynamic interactions, impedance functions and elastic wave propagation due to horizontal and vertical vibrations of a rigid foundation in a transversely isotropic elastic ground

The analytical and numerical treatments of horizontal and vertical forced time-harmonic vibrations of a rigid circular foundation in a transversely isotropic elastic half-space ground are presented. A complete discussion on the boundary conditions and elastic wave propagation due to horizontal and vertical vibrations of the foundation on the half-space is presented and a comparison is made with the full-space problem. With the aid of appropriate dynamic Green’s functions reported in the literature, the horizontal and vertical vibrations of the foundation are treated analytically, and the results are expressed in terms of the solutions of several Fredholm integral equations. For these two vibration modes, the relations for the contact normal and shear stresses, the resultant forces acting on the foundation, and the dimensionless impedance and compliance functions are given. To validate the analytical formulations, the results obtained in this paper are reduced to static responses for transversely isotropic half-space, and to dynamic responses for isotropic half-space; and compared with the results reported in the literature. An efficient numerical procedure is presented to evaluate the semi-infinite integrals appeared in the analytical solutions utilizing some features of the Mathematica software. The Fredholm integral equations and impedance and compliance functions are solved and evaluated numerically using the Gaussian quadrature method. Some graphical results are provided to present the impedance and compliance functions for different transversely isotropic materials. The effects of material anisotropy, as well as the frequency of excitation on the responses are discussed. The differences between vibrations of rigid foundations in the half-space and full-space mediums are emphasized in view of the Rayleigh wave propagation. Moreover, the difficulties related to numerical procedures needed for handling the strong singularities due to propagation of Rayleigh waves in the half-space media are presented.

Skill assessment of different quadruplet wave-wave interaction formulations in the WAVEWATCH-III model with application to the Gulf of Mexico

In this study, the Gaussian Quadrature Method (GQM) method was implemented in the WAVEWATCH-III model and used along with embedded formulations for estimating the nonlinear wave interactions. The results of DIA, GMD, GQM and WRT methods were compared for ideal test cases. The GQM was in good agreement with the exact WRT method while its medium resolution conjuration was ∼10 times faster than WRT. The GMD was more than 50 times faster than GQM but it could not reproduce WRT results for slanting fetch condition. Two different packages named ST3 and ST6, were employed for calculating the wind input and energy dissipation formulations over the Gulf of Mexico. The high quality ERA5 wind data from August to September in 2017 blended with Holland parametric model were used to run the wave model. The simulation period includes both fair weather condition and extreme events of Hurricanes Harvey and Irma. The performance of wave model using different nonlinear wave interaction terms was assessed against bulk wave parameters measured by in-situ buoys as well as altimeter-derived data by introducing a new error index. The general features of in-situ directional wave spectra were well captured by all DIA, GMD, GQM and WRT methods; however, the spectra produced by GMD and GQM were closer to the results by WRT method. Interestingly, the DIA method with calibrated whitecap dissipation term outperformed other methods in reproducing in-situ data (i.e. wave bulk parameters and wave spectra) during both fair weather and extreme events; indicating overfitting in the white capping or/and wind input terms in the wave model.

Equivalent dynamic stiffnesses and 3D wave propagations of a transversely isotropic elastic ground in rocking and torsional interactions with a harmonically loaded rigid foundation

An advanced theoretical formulation along with a robust numerical evaluation method are employed to study the 3 D wave propagations and equivalent dynamic stiffnesses of a transversely isotropic elastic ground in rocking and torsional interactions with a harmonically loaded rigid foundation. Using appropriate dynamic Green’s functions presented in the literature, the rocking and torsional harmonic oscillations of the foundation are formulated analytically in terms of several Fredholm integral equations of the second kind. Then the dynamic stiffnesses and flexibilities are presented analytically for rocking and torsional oscillations. To validate the analytical results, they are reduced to two simpler cases reported in the literature: the static problem and the case of isotropic materials. The semi-infinite integrals appeared in the solutions are evaluated numerically by utilizing the contour integration and residue theorem along with several commands of the Mathematica software. A complete discussion on the elastic wave propagations due to oscillations of the foundation are presented and a detailed comparison is performed between different waves propagate within the half-space and full-space elastic grounds including the P, SV, SH and the Rayleigh waves. Our results showed that due to rocking oscillation of the foundation, the Rayleigh wave does not propagate in the full-space medium but it propagates in the half-space medium. However due to torsional oscillation, the Rayleigh wave does not propagate neither in the full-space nor in the half-space elastic medium. In addition, due to rocking oscillation, the body waves including P, SV and SH propagate within both half-space and full-space elastic mediums; but due to torsional oscillation, only SH-wave propagates within both full-space and half-space elastic mediums. In some tables, numerical values and mathematical formulas for the dimensionless wave numbers and wave velocities are provided for the P, SV, SH and the Rayleigh waves. The Gaussian quadrature method is implemented for definite integrals appeared in the dynamic stiffnesses. Some graphical figures are provided to represent the dimensionless rocking and torsional dynamic stiffnesses and flexibilities versus the dimensionless frequency for different transversely isotropic materials. The effects of anisotropy of materials and the frequency of excitations on the responses are discussed. The results presented in this paper including the dynamic stiffnesses and wave propagations have direct applications in dynamic soil-structure interaction analyses in civil engineering.

Geometrically nonlinear electro-thermo-static analysis of piezoelectric/CNT/GPL/fibre/polymer sandwich panels with double curvature resting on elastic foundation

Evaluation of the electro-thermo-static behaviour of the shell panels of double curvature using an analytical method is carried out in this paper. The proposed panels are made of a hybrid composite, and considered to be rested on the Winkler-Pasternak elastic foundation. The hybrid composite shell contains two thin piezoelectric sheets at its bottom and top surfaces, and some plies made of carbon fibres and polymeric resin which is reinforced by the randomly oriented nanocomposites. The employed nanocomposites for some layers are carbon nanotubes (CNTs) and for some other are graphene nanoplatelets (GPLs). The governing nonlinear system of differential equations of orthotropic doubly-curved shell panels are developed based on the first-order transverse shear deformation theory (FSDT) which incorporates moderately thick and also thin shells. The Gauss Quadrature method is applied to obtain the forces and moments due to the thermo-electrical loads, and the Galerkin method is employed to solve the developed system of equations which leads to a closed-form load-deformation relationship (equilibrium path of the shell). The novelty of the present study can be expressed in the solution and employed hybrid composite materials. First, by employing the proposed closed-form solution no time consuming computations are required. On the other hand, the effects of the piezoelectric and two different nanocomposite materials on the thermo-electro-mechanical nonlinear static behaviour of the hybrid composite shells can be evaluated. The accuracy of the present method is validated by comparing the achieved results for the some benchmark problems with those are existent in the literature. The impacts of various electro-thermo-mechanical and geometrical characteristics of the proposed shells on their equilibrium paths are also considered by carrying out an extended parametric study.

A Blend of Analytical and Numerical Methods to Compute Orthogonal Image Moments over a Unit Disk

Accurate image moment computation is critical because they are used in a variety of fields, including image reconstruction and object recognition. Orthogonal polynomials are frequently used to compute moments due to their numerous intersecting and important theoretical properties. In polar coordinate systems, orthogonal polynomials such as the Polar Harmonic Fourier Transform (PHFT) and Polar Harmonic Transformation (PHT) are defined. However, the images are defined by a Cartesian coordinate system. To obtain image moments, a double integration over a unit circle over the product of the PHFT or PHT and the image function must be performed. The choice of double integration techniques and domain for the double integral has a significant effect on the precision of the computed moments. We have proposed a method for using the entire unit circle as an integration domain in this study, for image reconstruction. We used the PHFT and PHT to apply this technique to computing moments for the reconstruction. The proposed method outperforms other state-of-the-art methods, including Gaussian quadrature numerical integration method (GQM) and zeroth order approximation (ZOA) on variety of scenarios using three benchmarked image sets. We have demonstrated experimentally that this technique significantly improves the accuracy of image moments in higher order moments, as measured by reduced mean squared error (MSE). Additionally, the proposed method significantly improved the moment’s rotational (an improvement of 1-3% as compared to GQM and an improvement of 37-93% as compared to ZOA) and scaling invariance (an improvement of 0-500% as compared to GQM and an improvement of 3-7000% as compared to ZOA).

Numerical integration of neutrosophic valued function by Gaussian quadrature methods

In this article, different types of Gaussian quadrature methods have been presented to find the numerical integration of a neutrosophic valued function. A new definition of the distance between two neutrosophic number has been defined and it has been proved that the distance and the set of all neutrosophic number form a complete metric space. Also, the definition of neutrosophic continuity on a closed-bounded interval has been defined in the sense of (alpha ,beta ,gamma )-cut. This is the first time, when the Gauss–Legendre integration, Gauss–Chebyshev integration and Gauss–Laguerre integration rule have been discussed in neutrosophic environment. In the first test example, the comparison between one-point, two-point and three-point Gauss–Legendre integration rules have been presented in terms of tables and figures. Also, in the second example, the comparison between one-point, two-point Gauss–Chebyshev and Gauss–Laguerre integration rules have been discussed.