Quadrature Method Research Articles

A high-order generalised differential quadrature element method for simulating 2D and 3D incompressible flows on unstructured meshes

In this paper, a high-order generalised differential quadrature element method (GDQE) is proposed to simulate two-dimensional (2D) and three-dimensional (3D) incompressible flows on unstructured meshes. In this method, the computational domain is decomposed into unstructured elements. In each element, the high-order generalised differential quadrature (GDQ) discretisation is applied. Specifically, the GDQ method is utilised to approximate the partial derivatives of flow variables and fluxes with high-order accuracy inside each element. At the shared interfaces between different GDQ elements, the common flux is computed to account for the information exchange, which is achieved by the lattice Boltzmann flux solver (LBFS) in the present work. Since the solution in each GDQ element solely relies on information from itself and its direct neighbouring element, the developed method is authentically compact, and it is naturally suitable for parallel computing. Furthermore, by selecting the order of elemental GDQ discretisation, arbitrary accuracy orders can be achieved with ease. Representative incompressible flow problems, including 2D laminar flows as well as 3D turbulent simulations, are considered to evaluate the accuracy, efficiency, and robustness of the present method. Successful numerical simulations, especially for scale-resolving 3D turbulent flow problems, confirm that the present method is efficient and high-order accurate.

Differential quadrature lattice Boltzmann method for simulating fluid flows

In this study, a novel method for simulating incompressible flows using the lattice Boltzmann method is presented, aiming to improve robustness. This method is enhanced by the Qadyan numerical method. In the Qadyan method, Partial differential equations are solved using semi-discrete schemes combined with the differential quadrature method. To discretize the spatial derivatives of the lattice Boltzmann equation, the upwind differential quadrature method is employed, while the first-order forward difference scheme and/or fourth-order Runge-Kutta method handle the temporal term. The decision to exclude more complex methods stems from their requirement for refined computational grids. This work focuses on achievable results with similar types of uniform grids. The proposed scheme introduces and evaluates a novel method for solving both steady and unsteady problems. The present numerical method is validated by solving five benchmark problems, including heat diffusion on a slab, Stokes’ first problem, Taylor-Green vortex flow, flow around a flat plate, and flow in a lid-driven cavity. Reasonable agreements are obtained between the solutions obtained using this method and those from analytical/other numerical approaches. The Qadyan formulation delivers more accurate results compared to the finite-difference lattice Boltzmann method, (FDLBM), without requiring an increase in the number of grids. Notably, the novel Qadyan method achieves this increased accuracy without introducing additional complexity to the standard lattice Boltzmann method or resorting to non-uniform grids. This is possible due to the nature of the differential quadrature method, which utilizes more combinations of candidate stencils. The Qadyan method has a higher computational cost in comparison with the FDLBM.

Modelling of Cut-off Frequency of Armchair Graphene Nanoribbon Tunnel Field Effect Transistor Using Transfer Matrix Method

Graphene is considered as a promising two-dimensional material for electronic device applications due to its high charge carrier mobility and it is extremely thin in size. As a material, graphene has great potential for the development of high-speed nano electric devices. Graphene can be used as a channel material in Tunnel Field Effect Transistor (TFET). In this study, the cut-off frequency of the armchair graphene nanoribbon tunnel field effect transistor (AGNR-TFET) device was modelled quantum mechanically. The solution of the Dirac-like Hamiltonian and Poisson equations self-consistently is used to determine the potential profile of device and the transmittance is calculated using the transfer matrix method (TMM). The Landauer formulation with the help of Gauss Legendre Quadrature Method (GLQM) is used to calculate the tunnelling current and the cut-off frequency of AGNR-TFET. The calculation results showed that the cut-off frequency increased as the drain voltage increased, while the increase in channel length and the N-index graphene made the cut-off frequency decreased. Moreover, the thicker the oxide layer could increase the cut-off frequency and the higher the temperature on the device would reduce the cut-off frequency.

Characterization of the angular coefficient method on 2D and 3D piecewise smooth boundaries

The angular coefficient method represents a valid and efficient strategy to estimate the distribution of molecules in ultra-high vacuum systems. The problem is described through a Fredholm integral equation of the second kind that is usually solved with standard numerical methods, e.g., the finite element method or the Nyström quadrature method. In this work, we aim to rigorously study the underlying integral equation in order to verify some fundamental mathematical properties and justify the application and behaviour of such numerical methods. In particular, we address to the general scenario where domains are not globally smooth. In such context, boundary corners entail poorly regular integral kernels, which in turn lead to non-Lipschitz solutions and require the adoption of non-standard analysis techniques. By introducing the concept of vacuum-connection, we can establish a methodology to prove the well-posedness of the underlying problem. Furthermore, the undermined regularity of the analytical solution and the consequent lower numerical convergence rate are proved analytically and verified through simple numerical tests.

A local differential quadrature method for the generalized nonlinear Schrödinger (GNLS) equation

A local differential quadrature method based on Fourier series expansion numerically solves the generalized nonlinear Schrodinger equation. For time integration, a Runge-Kutta fourth-order method is used. Matrix stability analysis is used to examine the method's stability. Three test problems involving the motion of a single solitary wave, the interaction of two solitary waves, and a solution that blows up in finite time, respectively, demonstrate the accuracy and efficiency of the provided method. Finally, the numerical results obtained from the presented method are compared with the exact solution and those obtained in earlier works available in the literature.

Nonlinear forced vibration of the FGM piezoelectric microbeam with flexoelectric effect

In this paper, based on the extended dielectric theory, Euler beams theory and von Karman’s geometric nonlinearity, a nonlinear FGM piezoelectric microbeam model is established with flexoelectric effect. The governing equations, initial conditions and boundary conditions are obtained by applying Hamilton’s principle and then solved by combining the differential quadrature method (DQM) and iteration method. The innovation of this paper is to construct a nonlinear forced vibration model of piezoelectric microbeams. The coupling response between the inverse flexoelectric effect and the inverse piezoelectric effect is investigated. Various effects are examined, including the functional gradient index m and transverse distributed load q affecting the distribution of electric potential. Results indicated that the functional gradient index m, beam thickness h, and span-length ratio L/h have a significant impact on the dimensionless deflection of the FGM microbeam. The influence of the flexoelectric effect on dimensionless deflection increases with the decrease of scale. In addition, transverse load q and the functional gradient index m also have a significant impact on the distribution of electric potential. This paper will provide useful theoretical guidance for the design of micro-sensors and micro-actuators.

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.

Nonlinear mechanical response of UHSS rectangular plate under thermo-mechanical-metallurgical coupling

The mechanical properties of ultrahigh-strength steel (UHSS) produced through the gradient thermoforming process (GTP) are significantly influenced by residual stress and deformation. However, the precise distribution of these factors during GTP remains unclear. To address this issue, a detailed thermal model is developed, incorporating key heat transfer mechanisms, including coupled heat conduction and radiation, using the method of separation of variables. Furthermore, the displacement and stress fields of UHSS rectangular plates are derived using the differential quadrature method (DQM), accounting for phase transitions and external load. The combination of the separation of variables and DQM methods significantly improves computational efficiency. Furthermore, this study presents the first analysis of the combined effects of thermal load, phase transitions, and external forces on UHSS rectangular plates. Finally, through a combination of theoretical analysis, experimental validation, and case studies, the influence of material properties, geometric parameters, and external load on the temperature, stress, and displacement fields is examined. The findings reveal that heating rate, phase transitions, and external load play a critical role in the stress and displacement distribution in UHSS plates during GTP. This research provides a theoretical foundation for optimizing GTP process parameters and material design, ultimately enhancing the quality and performance of UHSS components.

Dynamic Modeling and Vibration Characteristic Analysis of Fiber Woven Composite Shaft–Disk Rotor with Weight-Reducing Holes

In order to achieve the goal of a lightweight shaft–disk rotor, this paper applies the fiber woven composite material to the disk structure, and at the same time considers the design of the weight-reducing holes on the porous disk. It introduces the domain decomposition and coordinate mapping technology for this, and then establishes the dynamics model of the fiber woven composite material shaft–disk rotor. The model is based on the differential quadrature finite element method, which is suitable for fiber woven composite rotors with arbitrary complex hole patterns. The validity of the model is verified by comparing the results with the literature, finite element simulation, and experiments, and the mechanism of the influence of the material parameters and pore parameters on the vibration characteristics of the system is investigated, which provides the data support and theoretical basis for the analysis of the dynamics of the fiber woven composite rotor.

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.

A New Galerkin Quadrature Method Not Requiring a Matrix Inverse

We derive a new Galerkin quadrature (GQ) method for S n calculations that differs from the two methods preceding it in that a matrix inverse for an N d × N d matrix, where N d is the number of directions in the quadrature set, is no longer required. Galerkin quadrature methods are designed for calculations with highly anisotropic scattering. Such methods are not simply special angular quadratures but also are methods for representing the S n scattering source that offers several advantages relative to the standard scattering source representation when highly truncated Legendre cross-section expansions must be used. Galerkin quadrature methods are also useful when the scattering is moderately anisotropic, but the quadrature being used is not sufficiently accurate for the order of the scattering source expansion that is required. We derive the new method and present computational results showing that its performance for two challenging problems is comparable to those of the two GQ methods that preceded it.

Annular sector plates made of FG graphene origami-enabled auxetic metamaterial under bending and buckling loading: A VDQ-based approach

The behaviors of annular sector plates made of functionally graded graphene origami-enabled auxetic metamaterials (FG-GOEAMs) under the action of buckling and bending loads are investigated herein using a numerical approach. Multiple GOEAM layers with graphene origami (GOri) content are considered for the plate which changes in layer-wise patterns. Moreover, the material properties of the plate are calculated using novel genetic programming-assisted micromechanical models. In order to derive the governing equations, the Mindlin plate theory in conjunction with Hamilton’s principle is utilized within the framework of the variational differential quadrature (VDQ) method. The vector-matrix representation of the presented formulation can be beneficial from the viewpoint of implementing numerical approaches. The governing equations are then discretized and solved based on VDQ matrix differential and integral operators so as to obtain the critical buckling load and maximum lateral deflection of plates with different edge conditions. The effects of GOri content, folding degree and distribution pattern on the results are studied. It is revealed that increasing the GOri content leads to the negative Poisson’s ratio (NPR) effect which respectively reduces and increases the lateral deflection and the critical buckling load.

Aero-thermodynamic responses of a novel FGM sector disks using mathematical modeling and deep neural networks

Composite sector disks have extensive applications in aerospace industries, particularly when exposed to challenging conditions such as supersonic airflow and thermal environments. These applications leverage the superior properties of composite materials, including high strength-to-weight ratios, enhanced durability, and excellent thermal resistance, to meet the stringent requirements of aerospace operations. Multi-directional functionally graded (MD-FG) materials due to high-temperature resistance and other amazing properties in each direction have gotten plenty of attention recently. So, in this research, a thermoelasticity solution has been presented to study fundamental frequency traits of an MD-FG sector disk in supersonic airflow via both mathematics simulation and deep neural networks technique. For obtaining exact displacement fields, along with defining the changes of transverse shear strains along the system's thickness, the refined zigzag hypothesis is utilized. For obtaining the temperature-dependent equations, heat conduction relation and thermal boundary conditions of the MD-FG structure are presented. A coupled quasi-3D new refined theory (Q3D-NRT) and generalized differential quadrature method (GDQM) are presented for obtaining and solving the partial differential equations in the time-displacement domain. After obtaining the mathematics results, appropriate datasets are made for testing, training, and validation of the deep neural networks technique. Finally, the results have shown that aerodynamic pressure, temperature changes, Mach number, free stream speed, and air yaw angle have a major role in the stability/instability analyses of the thermally affected MD-FG sector disk in supersonic airflow. As an amazing outcome, increasing the sector angle, FG indexes, and temperature change lead to the reduction of the critical Mach number, and aerodynamic pressure associated with the flutter phenomenon.

MC-VMD-CNN-BiLSTM short-term wind power prediction considering rolling error correction

Abstract Wind energy is a clean and renewable source that has the potential to alleviate the global fossil fuel crisis and environmental pollution by generating electricity. However, accurately predicting wind energy output remains challenging due to its inherent uncertainty. To enhance the accuracy of wind power prediction, a short-term wind power forecasting method for power systems, MC-VMD-CNN-BiLSTM, is proposed, which considers error rolling correction. The method begins with feature selection and outlier handling using the quadrature method. Then, wind power data is decomposed into multiple sub-sequences using the Variational Mode Decomposition (VMD) technique to reduce the raw volatility of wind power. Then, a Convolutional Neural Network (CNN) followed by a Bidirectional Long Short-Term Memory (BiLSTM) model is used for wind power prediction. Finally, the proposed method utilizes the Monte Carlo method for rolling error correction by using known errors from previous time frames to correct subsequent predictions. The MC-VMD-CNN-BiLSTM proposed in this paper considering error rolling correction is compared with ELM, SVM, PSO-BP and ARIMA models through an example analysis of the data of a city, and the proposed model in this paper reduces 61.78%, 50.35%, 62.30% and 73.05% in the NRMSE index in the spring as an example, respectively. The results show that the prediction model proposed in this paper has higher prediction accuracy compared with the traditional prediction model.

Free vibration of functionally graded graphene platelets reinforced porous composite plate with a diamond-shaped hole and two cracks

This paper investigates the free vibration of the functionally graded graphene platelets reinforced porous composite (FG-GPLRPC) plate with a diamond-shaped hole and two cracks. Based on the Mindlin-Reissner plate theory and Hamilton principle, the governing equations and boundary conditions are derived. These equations are discretized using the generalized differential quadrature finite element method (GDQFEM), and the natural frequency is obtained by numerical solutions. The effects of the hole size, crack length and crack angle on the natural frequency are discussed. The results show that the natural frequency decreases with increasing crack angle and crack length.

Discretization of Continuous Spaces using Barycentric Subdivision Method with Metric Space Constraints on the Nearest Neighbour Edges

The Rao-Wilton-Glisson (RWG) is a commonly used basis function in the numerical solution of the electric field integral equation (EFIE) using the Method of Moments (MoM) and Galerkin approach. This method relies on triangular patches to approximate the surface current. Traditionally, barycentric subdivision of a primary triangle into n-sub-triangles has been used with RWG basis function to solve the EFIE using MoM. This paper presents a method of approximating a surface using triangular patches by sub-dividing a primary triangle into (2n-1) subtriangles. which creates a denser mesh than the widely used nine (9) points quadrature method. The structure is approximated by small square patches, which are further sub-divided into two primary triangles. By applying barycentric sub-division, the primary triangles are decomposed into sub-triangles to create a mesh over the surface of the structure. Using graph theory, the triangular meshes are defined by the function, G (V, E), where V and E are the vertices and edges of the triangles in the mesh space, respectively. The connectivity matrix of shared edges is found by imposing a constraint on the edge length using metric spaces. This method approximates a square patch with 32 scalene triangles and shows that it can be used to reconstruct equilateral patches and doubly split double ring into mesh structures.

Dynamic buckling active control of FGM spherical shell with piezoelectric sensor and actuator layers

The mechanical snap-through phenomenon represents one of the buckling modes inherent in shallow shells, exemplifying the shell's reaction to external loads, which can potentially have detrimental effects on structural performance. This article delves into the study of this phenomenon within the context of spherical shells. Leveraging functionally graded material (FGM) in the main layer of this structure proves instrumental in enhancing the shell's performance against rapid mechanical loads. Furthermore, the strategic placement of piezoelectric layers, both symmetrically and asymmetrically, coupled with the implementation of a proportional and derivative control system, facilitates the control of the shell's response. The formulation of displacement field and electric potential is grounded in the first-order shear deformation theory (FSDT) and second-order distribution, respectively. The strong form of nonlinear dynamic and electrostatic equations is derived using Hamilton's principle. Solving these equations is achieved through the implementation of the generalized differential quadrature (GDQ) method in the spatial domain and Newmark in the time domain. Additionally, the Newton-Raphson method is employed to tackle the inherent nonlinearity of the problem. The Budiansky's criterion is employed for the assessment of the load necessary for the buckling of the shell. Following the validation of results against numerical, analytical, and laboratory outcomes, a set of parametric results is presented. These results underscore the remarkable impact of the control system in mitigating shell deformation under time-dependent mechanical loads, as well as in postponing the occurrence of the snap-through phenomenon and damping the vibrations.

Three-dimensional elasticity solutions for doubly-curved composite shells by extended differential quadrature method

This paper presents the three-dimensional (3D) stress analysis of doubly-curved composite shells with general boundary conditions using the strong sampling surfaces (SaS) formulation. The SaS method is based on the choice of SaS parallel to the middle surface and located at Chebyshev polynomial nodes to introduce the displacements of these surfaces as unknown functions that leads to a non-conventional shell formulation, in which strain–displacement and stress–strain relationships are represented in terms of SaS displacements. This is due to the use of Lagrange polynomials in approximating displacements, strains and stresses in the thickness direction. The outer surfaces are not included into a set of SaS that makes it possible to uniformly minimize the error ca used by Lagrange interpolation. The strong SaS formulation, based on direct integration of elasticity equilibrium equations in the thickness direction by the extended differential quadrature (EDQ) method, can be applied efficiently for high-precision calculations of doubly-curved composite shells with clamped and free edges. This is because in the SaS/EDQ formulation, displacements, strains and stresses of SaS are interpolated in a rectangular domain specified in a curvilinear coordinate system using a Chebyshev-Gauss-Lobatto grid and Lagrange polynomials are also used as basis functions. The proposed approach deals with equilibrium equations in terms of SaS stresses, avoiding the integration of second order differential equations in terms of SaS displacements, that greatly simplifies the implementation of the EDQ method.

RFOR-DQHFEM: Hybrid relaxed first-Order reliability and differential quadrature hierarchical finite element method for multi-physics reliability analysis of conical shells

In this current work, a hybrid reliability analysis and theoretical frequency technique are suggested for the reliability response of conical shells. Two levels of analyses are proposed as the main loop of the reliability method for finding the failure probability and the second level applied in the main loop for giving the performance function of frequency applied in conical shell structures with multi-physics vibration analysis. A dynamical adjusting procedure is proposed for computing the relaxed factor using the enough descent condition inside the reliability method. The superior convergence rate is considered for selecting the relaxed factor of the proposed first-order reliability method named RFORM. An elastic-electro-mechanical model based on the Higher-Order Shear Deformation Theory (HSDT) is extended for frequency analysis of conical shells. The innovative numerical procedure named Differential Quadrature Hierarchical Finite Element Method (DQHFEM) as a robust framework for giving the vibration behavior of studied mechanical structures is applied for solving motion equations. The developing DQHFEM and RFORM are applied for the laminated, nanocomposite, and piezoelectric conical shell structures with multi-source uncertainties. Increasing the volume percentage of nanoparticles from 0% to 10% significantly enhances the reliability index, with carbon nanoparticles showing a 132% increase, silica nanoparticles showing a 97% increase, and other nanoparticles showing an approximate 40% increase. Also, as moisture content increases from 0% to 30%, the reliability index for a thickness-to-large-radius ratio of 0.2 drops by about five times. Excessive moisture levels (above 20%) result in a negative reliability index, indicating a hazardous condition.

Vibration analysis of shear actuated bi-directional FGP beam using generalized differential quadrature method

The primary objective of this study is to exhibit the shear mode vibrations of a bidirectional functionally graded piezoelectric beam. All material properties are assumed to vary bidirectional in accordance with an exponential law. Using Timoshenko beam theory, the governing equations and boundary conditions are derived using the Hamilton principle. The equations of motion are then solved using the generalized differential quadrature method. Since no previous solutions existed for the current problem, the obtained results are compared with the findings from a simplified case in previous research. The effects of exponent parameters, slenderness ratios, and boundary conditions on natural frequencies of the bidirectional FGP beam are discussed in detail.