Gaussian Quadrature Research Articles

A binary-tree subdivision method for evaluation of singular integrals with discontinuous kernel in 3D BEM

PurposeA binary-tree subdivision method (BTSM) for numerical evaluation of weakly singular integrals with discontinuous kernel in the three-dimensional (3D) boundary element method (BEM) is presented in this paper.Design/methodology/approachIn this method, the singular boundary element is split into two sub-elements and subdivided recursively until the termination criterion is met and the subdivision is stopped. Then, the source point is surrounded by one or more spherical cavities determined by the discontinuous kernel function. The sub-elements located in spherical cavities will be eliminated, and the regular triangular or rectangle elements are employed to fill the spherical cavities.FindingsWith the proposed method, the obtained sub-elements are automatically refined as they approach the source point, and they are “good” in shape and size for standard Gaussian quadrature. Thus, the proposed method can be used to evaluate singular integrals owing discontinuous kernel function accurately for cases of different element shapes and various source point locations.Originality/valueNumerical examples show that the BTSM is suitable for planar and curved elements of arbitrary regular or irregular shape at various source point locations, and the results have much better accuracy and robustness than conventional subdivision method (CSM) when the kernel function is discontinuous.

Graph-accelerated non-intrusive polynomial chaos expansion using partially tensor-structured quadrature rules for uncertainty quantification

Recently, the graph-accelerated non-intrusive polynomial chaos (NIPC) method has been proposed for solving uncertainty quantification (UQ) problems. This method leverages the full-grid integration-based NIPC method to address UQ problems while employing the computational graph transformation approach, AMTC, to accelerate the tensor-grid evaluations. This method exhibits remarkable efficacy on a broad range of low-dimensional (three dimensions or less) UQ problems featuring multidisciplinary models. However, it often does not scale well with problem dimensions due to the exponential increase in the number of quadrature points when using the full-grid quadrature rule. To expand the applicability of this method to a broader range of UQ problems, this paper introduces a new framework for generating a tailored, partially tensor-structured quadrature rule to use with the graph-accelerated NIPC method. This quadrature rule, generated through the designed quadrature approach, possesses a tensor structure that is tailored for the computational model. The selection of the tensor structure is guided by an analysis of the computational graph, ensuring that the quadrature rule effectively capitalizes on the sparsity within the computational graph when paired with the AMTC method. This method has been tested on one 4D and one 6D UQ problem, both originating from aircraft design scenarios and featuring multidisciplinary models. Numerical results show that, when using with graph-accelerated NIPC method, our approach generates a partially tensor-structured quadrature rule that outperforms the full-grid Gauss quadrature and the designed quadrature methods (more than 40% reduction in computational costs) in both of the test problems.

Bending analysis of viscoelastic plates according to the Reissner’s theory using meshless local Petrov–Galerkin method and hereditary integral formulation

Purpose The purpose of this study is to apply the Meshless Local Petrov–Galerkin (MLPG) method to solve the bending problems of linear viscoelastic plates, considering Reissner’s theory.Design/methodology/approachThe weak formulation for the set of equations that govern Reissner’s plate theory is implemented in conjunction with the integral formulation applied to viscoelastic constitutive expressions. A meshless method based on the Moving Least Squares (MLS) approximation is considered in the numerical implementation. The final equation system is assembled by adopting simple and efficient schemes for numerical integration, considering a simplified formulation through centralization of the local interpolation domains and Gaussian quadrature at the same field point. The results obtained are compared with available solutions to demonstrate the efficiency of the proposed formulation.FindingsThe hereditary integral approach proved to be the most general way to analyze the viscoelastic problem, especially when applied together with the modified scheme for numerical integration. In addition, the variable changing technique is demonstrated to be an efficient formulation for solving shear-locking effects in thin plate problems.Originality/valueThe differential of the present study is related to the manner in which the properties of linear viscoelastic materials are considered in the formulation. Although most authors consider this point through the application of the correspondence principle, the present study works with a hereditary integral formulation. In addition, the variable changing technique is applied to solve the shear-locking effects, and an alternative approximation technique is considered to speed up the numerical integration process.

Pure-positivity-preserving methods with an optimal sufficient CFL number for fifth-order MR-WENO schemes on structured meshes

In this paper, one-dimensional and two-dimensional pure-positivity-preserving (PPP) methods are proposed for fifth-order finite volume multi-resolution WENO (MR-WENO) schemes to solve some extreme problems on structured meshes. The MR-WENO spatial reconstruction procedures only require one five-cell, one three-cell, and one one-cell stencils for achieving uniform fifth-order accuracy in smooth regions and keeping essentially non-oscillatory property in non-smooth regions in one dimension. One redefines five new cell averages vectors after performing such spatial reconstructions and design one quartic polynomials vector and three quadratic polynomials vectors based on them. After that, a new detective process is used to examine the positivity of density and pressure of three quadratic polynomials vectors inside the whole target cell. If the negativity happens, a new compression limiter is carried out to enable the positivity of density and pressure of three quadratic polynomials vectors over the whole target cell and the positivity of density and pressure of one quartic polynomials vector at the midpoint of the target cell. It is a new way to design the positivity-preserving methods to keep fifth-order accuracy and the positivity over the target cell instead of only at some discrete Gauss-Lobatto quadrature points, since the precise minimum values of the density and pressure are now available. Then a theoretically proof is given to increase the optimal sufficient CFL number from 1/12 to 1/6 for the fifth-order WENO schemes. This methodology can be expanded to multi-dimensions easily. Unlike some classical positivity-preserving methods, the PPP methods could apply a special four-point Gauss-Lobatto quadrature formula or any other quadrature formulas on condition that their numerical precision is no smaller than four. Since the optimal CFL number of 1/6 is a sufficient but not necessary condition, the novelty PPP methods for fifth-order finite volume MR-WENO schemes with a larger practical CFL number of 0.6 are also available and robust enough when simulating some extreme problems without timely halving its value.

Mixed finite elements of higher-order in elastoplasticity

In this paper a higher-order mixed finite element method for elastoplasticity with linear kinematic hardening is analyzed. Thereby, the non-differentiability of the involved plasticity functional is resolved by a Lagrange multiplier leading to a three field formulation. The finite element discretization is conforming in the displacement field and the plastic strain but potentially non-conforming in the Lagrange multiplier as its Frobenius norm is only constrained in a certain set of Gauss quadrature points. A discrete inf-sup condition with constant 1 and the well posedness of the discrete mixed problem are shown. Moreover, convergence and guaranteed convergence rates are proved with respect to the mesh size and the polynomial degree, which are optimal for the lowest order case. Numerical experiments underline the theoretical results.

Static and Transient Response Analysis of Arbitrary Laminated Composite and Sandwich Shells by the FSDT Meshless Method

This paper, for the first time, presents a novel meshless method based on the first-order shear deformation theory (FSDT) and moving-least squares (MLS) approximation to carry out static and transient response analysis on arbitrary laminated and sandwich shell structures. The novelty of the method lies in the implementation of MLS in the curvilinear shell formulation, and the derivation of meshless governing equations on the static and transient response of arbitrary laminated and sandwich shells according to the principle of minimum potential energy and Hamilton’s principle, respectively. The concept of a Convected coordinate system is adopted to deal with arbitrary curvilinear surfaces. Both the position and displacement vectors of shells are approximated by MLS, which is conceptually the same procedure as that in the isoparametric finite element method (FEM). The numerical integration of the stiffness matrices is conducted by the simple Gaussian integral in the Convected coordinate system. Because the moving-least squares (MLS) approximation does not satisfy the Kronecker delta condition, the full transformation method is used to modify the meshless governing equations. Numerical examples are calculated to investigate the static and transient response analysis of laminated and sandwich structures with different geometrical shell shapes, including square plates, spherical shells, doubly-curved shells, and cylindrical shells. The numerical tests show that the proposed method is reliable, and robust and produces accurate results compared to the analytical and numerical solutions taken from the previous literature.

A nonuniform mesh method in the Floquet parameter domain for wave scattering by periodic surfaces

In this paper, we propose a new numerical method to simulate acoustic scattering problems in two‐dimensional periodic structures with non‐periodic incident fields. Applying the Floquet‐Bloch transform to the scattering problem yields a family of quasi‐periodic boundary value problems dependent on the Floquet‐Bloch parameter. Consequently, the solution of the original scattering problem is written as the inverse Floquet‐Bloch transform of the solutions to these boundary value problems. The key step in our method is the numerical approximation of this integral transform by a quadrature rule with a nonuniform choice of quadrature points adapted to the regularity of the family of quasi‐periodic solutions. This achieved by a graded subdivision of the full interval for the Floquet‐Bloch parameter and applying a Gauss‐Legrendre quadrature rule on each subinterval. We prove that the numerical method converges exponentially with respect to both the number of subintervals and the number of Gaussian quadrature points. Some numerical experiments are provided to illustrate the results.

Extending GPU-accelerated Gaussian integrals in the TeraChem software package to f type orbitals: Implementation and applications.

The increasing availability of graphics processing units (GPUs) for scientific computing has prompted interest in accelerating quantum chemical calculations through their use. However, the complexity of integral kernels for high angular momentum basis functions often limits the utility of GPU implementations with large basis sets or for metal containing systems. In this work, we report the implementation of f function support in the GPU-accelerated TeraChem software package through the development of efficient kernels for the evaluation of Hamiltonian integrals. The high efficiency of the resulting code is demonstrated through density functional theory (DFT) calculations on increasingly large organic molecules and transition metal complexes, as well as coupled cluster singles and doubles calculations on water clusters. Preliminary investigations into Ni(I) catalysis with DFT and the photochemistry of MnH(CH3) with complete active space self-consistent field are also carried out. Overall, our GPU-accelerated software appears to be well-suited for fast simulation of large transition metal containing systems, as well as organic molecules.

Research on signal enhancement and flow rate calculation methods of multi-channel ultrasonic flowmeter

Abstract In the field of drilling, accurate measurement of mud flow helps detect potential leaks or abnormal situations, thereby improving overall safety and preventing accidents from occurring. Traditional contact flow meters require breaking the pipeline before installation, making them difficult to install in underground environments. In addition, the complex composition of underground fluids and their corrosive nature can also lead to unsatisfactory measurement results. Therefore, as a non-invasive flow measurement tool, ultrasonic flowmeter is a better choice. However, the ultrasonic echo signals of current ultrasonic flow meters are susceptible to external interference in actual testing, resulting in a decrease in signal quality and making it impossible to perform accurate detection. This article focuses on the research of ultrasound echo signal processing model, designs a four channel ultrasound transceiver model, improves the wavelet denoising algorithm, designs signal reconstruction and baseline correction processing methods, and proposes a pipeline flow calculation algorithm based on Gaussian numerical integration. This method can greatly improve the quality of ultrasound echo signals and the average accuracy of flow measurement. Finally, relevant testing experiments were conducted, and the results of the final performance measurement showed that the method proposed in this paper improved the average accuracy of the measurement by 5.07%, and the relative error remained at around 2%, with a significant improvement in error rate. Therefore, the method proposed in this article has extremely high accuracy in mud flow measurement.

Multimode solitons in optical fibers: a review

This review describes recent theoretical and experimental advances in the area of multimode solitons, focusing primarily on multimode fibers. We begin by introducing the basic concepts such as the spatial modes supported by a multimode fiber and the coupled mode equations for describing the different group delays and nonlinear properties of these modes. We review several analytic approaches used to understand the formation of multimode solitons, including those based on the 3D+1 spatiotemporal nonlinear Schrödinger equation (NLSE) and its approximate 1D+1 representation that has been found to be highly efficient for studying the self-imaging phenomena in graded-index multimode fibers. An innovative Gaussian quadrature approach is used for faster numerical simulations of the 3D+1 NLSE. The impact of linear mode coupling is discussed in a separate section using a generalized Jones formalism because of its relevance to space-division multiplexed optical communication systems. The last section is devoted to the relevant experimental studies involving multimode solitons.

Computation and Applications of Gaussian Integrals in Mathematics and Applied Sciences

Gaussian integrals, particularly those involving the function e−x2 , play a central role in various fields, ranging from physics to finance. This paper explores the computation of Gaussian integrals, beginning with the fundamental integral in the interval −∞ to ∞, which yields π . The author extends this analysis to integrals involving powers of x multiplied by e−x2 , showing their relevance in calculating moments in quantum mechanics and probability theory. Additionally, the author discusses the applications of multivariate Gaussian integrals in machine learning and statistical mechanics, where they are key to solving problems in high-dimensional spaces. Practical examples are provided from quantum mechanics (path integrals), statistical mechanics (partition functions), finance (option pricing models), and machine learning (Gaussian processes). Through these examples, the paper highlights the versatility and universality of Gaussian integrals as essential tools in both theoretical and applied contexts. The integral’s widespread applicability reflects its importance in connecting mathematical theory with real-world phenomena. This work highlights the role played by the Gaussian integral.

Gaussian quadrature formulae are strongly asymptotically optimal for a class of infinitely differentiable functions

This paper investigates the optimal quadrature formulae of a class F∞ of infinitely differentiable functions on [−1,1]. We obtain the strong equivalences of the optimal worst case errors of F∞ for standard information and Hermite data. We proved that the Gaussian quadrature formulae are strongly asymptotically optimal.

A computationally efficient Element Edge point numerical integration scheme in the meshless method framework for solving fracture problems

The fracture problem is modeled using the Radial Point Interpolation Meshless Method (RPIM) to solve for displacements. Further, the stresses and Stress Intensity Factor (SIF) are calculated using obtained displacements. An effective numerical integral quadrature called the Element Edge point(EE) scheme is used as an alternative to conventional Gauss Quadrature to improve computational efficiency. A comparative study based on two numerical integration schemes, the Element Edge point scheme and Gauss Quadrature, is conducted on four benchmark problems of thick, cracked plates owing to plane strain conditions. The study reveals that the proposed Element Edge point (EE) scheme is computationally efficient and works well in the meshless method framework.

Three-dimensional buckling analysis of stiffened plates with complex geometries using energy element method

A novel numerical method, energy element method (EEM), is proposed for the three-dimensional (3D) buckling analysis of stiffened plates with complex geometries. The problem is formulated in a cuboidal domain, and any complex geometric stiffened plate is modeled by assigning cutouts within the cuboidal domain. The stiffened plate is considered as an energy body and is discretized using Gauss points with variable stiffness properties to simulate its energy distribution. Incorporating the extended interval integration, Gauss quadrature, variable stiffness properties, and Chebyshev polynomials, the strain energy of stiffened plates with complex geometries can be numerically simulated by putting the stiffness and thickness of Gauss points in the cutouts to zero in the cuboidal domain. Using the principle of minimum potential energy and Ritz solution procedure, the deformation and buckling behaviors of stiffened plates with complex geometries can be captured. As a result of the new formulations in EEM, new standard energy functionals and solving procedures have been developed. In addition, Gauss points are generated within the energy elements accounting for the geometric boundaries of the stiffened plate, which are characterized by level set functions. EEM is employed to investigate complex-shaped stiffened plates with straight or curvilinear stiffeners, and the results are compared to those obtained using FEM or mesh-free method. The precision, generalization, and stability of EEM are demonstrated.

Numerical Integration Techniques: A Comprehensive Review

Numerical integration is a fundamental concept in computational mathematics and plays a crucial role in various scientific and engineering disciplines. This paper provides a comprehensive review of numerical integration techniques, their applications, comparative analysis, and conclusions. The discussed methods include the trapezoidal rule, Simpson’s rule, Gaussian quadrature, and Widdle’s method methods. The accuracy, efficiency, and limitations of each method are evaluated through theoretical analysis and practical examples.

Numerical evaluation of orientation averages and its application to molecular physics.

In molecular physics, it is often necessary to average over the orientation of molecules when calculating observables, in particular when modeling experiments in the liquid or gas phase. Evaluated in terms of Euler angles, this is closely related to integration over two- or three-dimensional unit spheres, a common problem discussed in numerical analysis. The computational cost of the integration depends significantly on the quadrature method, making the selection of an appropriate method crucial for the feasibility of simulations. After reviewing several classes of spherical quadrature methods in terms of their efficiency and error distribution, we derive guidelines for choosing the best quadrature method for orientation averages and illustrate these with three examples from chiral molecule physics. While Gauss quadratures allow for achieving numerically exact integration for a wide range of applications, other methods offer advantages in specific circumstances. Our guidelines can also be applied to higher-dimensional spherical domains and other geometries. We also present a Python package providing a flexible interface to a variety of quadrature methods.

Estimation of the surface impedance based on the discrete complex image source method and array processing

This paper introduces a method for estimating the sound absorption coefficient of a large panel of absorbing material from measurements using a microphone array. The method relies upon describing the pressure field as the set of monopoles obtained when the reflection coefficient is represented as the Laplace Transform of an image source distribution. This formulation results in a well-behaved integral equation, which is discretized using Gaussian quadrature to yield an inverse problem. It is possible to determine the regularized solution to the inverse problem and reconstruct the surface impedance of the sample by conducting measurements of the sound pressure at multiple points of observation. This sound field representation is adequate for sound sources in the close range of the material and the array. The paper presents simulated and experimental investigations to delineate the efficacy of the proposed measurement technique against the well-known image source method, thereby highlighting its significant potential to advance acoustic characterization methodologies.

A High-Order Well-Balanced Discontinuous Galerkin Method for Hyperbolic Balance Laws Based on the Gauss-Lobatto Quadrature Rules

In this paper, we develop a high-order well-balanced discontinuous Galerkin method for hyperbolic balance laws based on the Gauss-Lobatto quadrature rules. Important applications of the method include preserving the non-hydrostatic equilibria of shallow water equations with non-flat bottom topography and Euler equations in gravitational fields. The well-balanced property is achieved through two essential components. First, the source term is reformulated in a flux-gradient form in the local reference equilibrium state to mimic the true flux gradient in the balance laws. Consequently, the source term integral is discretized using the same approach as the flux integral at Gauss-Lobatto quadrature points, ensuring that the source term is exactly balanced by the flux in equilibrium states. Our method differs from existing well-balanced DG methods for shallow water equations with non-hydrostatic equilibria, particularly in the aspect that it does not require the decomposition of the source term integral. The effectiveness of our method is demonstrated through ample numerical tests.

A High-Precision Inverse Finite Element Method for Shape Sensing and Structural Health Monitoring

In the contemporary era, the further exploitation of deep-sea resources has led to a significant expansion of the role of ships in numerous domains, such as in oil and gas extraction. However, the harsh marine environments to which ships are frequently subjected can result in structural failures. In order to ensure the safety of the crew and the ship, and to reduce the costs associated with such failures, it is imperative to utilise a structural health monitoring (SHM) system to monitor the ship in real time. Displacement reconstruction is one of the main objectives of SHM, and the inverse finite element method (iFEM) is a powerful SHM method for the full-field displacement reconstruction of plate and shell structures. However, existing inverse shell elements applied to curved shell structures with irregular geometry or large curvature may result in element distortion. This paper proposes a high-precision iFEM for curved shell structures that does not alter the displacement mode of the element or increase the mesh and node quantities. In reality, it just modifies the methods of calculation. This method is based on the establishment of a local coordinate system on the Gaussian integration point and the subsequent alteration of the stiffness integration. The results of numerical examples demonstrate that the high-precision iFEM is capable of effectively reducing the displacement difference resulting from inverse finite element method reconstruction. Furthermore, it performs well in practical engineering applications.

A continuous-discontinuous coupling computational method for multi-material mixtures

Considering the important role of particle-reinforced composites (PRCs), and to accurately assess the mechanical properties of PRC with arbitrarily complex internal structures, this paper proposes a new continuous-discontinuous coupled computational method, i.e., the multi-material integrated analysis (MIA) method, by combining the advantages of the discontinuous deformation analysis (DDA) method, the numerical manifold method (NMM), and the meshless method, for the continuous-discontinuous dual state of this kind of materials when they are subjected to forces. The core idea of this method is by establishing a single physical cover (PC) on each particle and defining a unified and piecewise approximation function over it to achieve displacement coordination between the particles and the matrix. In this model, the particles can maintain the contact and separation characteristics like the blocks, while the matrix is able to achieve the bonding function by overlapping different covers. Two numerical integration schemes, namely single-point Gaussian integration and Monte Carlo integration, are proposed to address the integration issues in stiffness matrix calculations. Meanwhile by considering the similarity between NMM and meshless method interpolation function construction, this paper combines circular coverage and 0th-order moving least squares to construct the interpolation function to form the displacement approximation function for the whole solution space. Moreover, the method of lattice retrieval (also Nearest Neighbor Search, NNS) is invoked for particle contact determination to improve the efficiency of contact and coverage relationship determination. The validity of the model is verified by taking asphalt mixture, a typical PRC, as a demonstrative case study. The results show that the MIA method solves the one-sided problem that the existing algorithms usually only consider the continuous or discontinuous properties of PRC, and provides a new method for comprehensively analyzing the micromechanical response of PRC and solving its large deformation problem.