Quadrature Method Research Articles

Numerical solutions of the Benjamin–Bona–Mahony equation using the differential quadrature method with shifted legendre and generalized Laguerre polynomials

Numerical solutions of the Benjamin–Bona–Mahony equation using the differential quadrature method with shifted legendre and generalized Laguerre polynomials

Free vibration analysis of sandwich cylindrical shells with functionally graded carbon nanotube-reinforced composite face sheets using the differential quadrature (DQ) method

Free vibration analysis of sandwich cylindrical shells with functionally graded carbon nanotube-reinforced composite face sheets using the differential quadrature (DQ) method

Free and forced vibrations of circular plates made of functionally graded graphene origami-enabled auxetic metamaterials

Presented herein is a numerical investigation into the free and forced vibrational characteristics of circular plates made of functionally graded graphene origami-enabled auxetic metamaterials (FG-GOEAMs). The material properties of FG-GOEAM are estimated using genetic programing-assisted micromechanical models considering two graphene content distribution patterns. The effects of graphene content, size and folding degree on the material properties are taken into account by the models. The formulation of the paper is in the context of first-order shear deformation plate theory, and the governing equations of motion are achieved utilizing Hamilton’s principle. In the solution approach, the variational differential quadrature (VDQ) method is employed by which the variational statement of problem is directly derived and discretized using matrix differential and integral operators. Selected numerical results are presented to study the effects of graphene origami (GOri) content, folding degree, and distribution pattern on the free and forced vibration responses of FG-GOEAM circular plates with clamped and hinged boundary conditions.

Simulation of mixed-mode crack propagation in Mindlin plates by a hierarchical quadrature element method with minimal remeshing

Simulation of mixed-mode crack propagation in Mindlin plates by a hierarchical quadrature element method with minimal remeshing

Analysis and comparison of the effect of phase error on three quadrature suppression methods

Analysis and comparison of the effect of phase error on three quadrature suppression methods

Evaluation of fracture parameters for three-dimensional cracks by a hierarchical quadrature element method

Evaluation of fracture parameters for three-dimensional cracks by a hierarchical quadrature element method

On the electro-vibrations of piezoelectric structure rested on concrete auxetic foundation via Carrera unified formulation validated by machine learning algorithm

This study develops a comprehensive framework for the analysis of functionally graded piezoelectric plates (FGPPs) resting on a novel concrete auxetic viscoelastic foundation and subjected to airflow pressure. Employing the Carrera Unified Formulation (CUF) the mechanical and electrical behaviors of FGPPs are precisely modeled to capture intricate coupling effects. Maxwell’s equations are integrated within the formulation to account for electromagnetic formulations, enhancing the predictive accuracy for piezoelectric materials. The harmonic differential quadrature method (HDQM) is applied for numerical discretization, ensuring computational efficiency and accuracy in solving the governing equations. To validate the proposed model, results are rigorously benchmarked against published literature, demonstrating excellent agreement and ensuring reliability. Moreover, a deep fuzzy rough neural network (DFRNN) is introduced to extend the predictive capability, allowing for the exploration of nonlinear and complex dynamic behaviors under varied boundary and loading conditions. The DFRNN model effectively complements the analytical and numerical results, providing an innovative tool for robust simulations. This integrative approach not only advances the state-of-the-art in FGPP analysis but also introduces a versatile computational paradigm for multidisciplinary applications involving piezoelectric materials, airflow-structure interaction, and auxetic foundations. The study highlights the significant impact of coupling mechanical, electrical, and aerodynamic effects, paving the way for optimized designs in engineering applications.

Effect of surface elasticity on vibration frequencies and buckling loads of double nanobeams by differential quadrature method

The effect of the surface tension in a coupled nanobeam system is investigated with respect to the free vibrations and buckling of the system for simply supported (SS), simply supported-clamped (SC), and clamped-clamped (CC) boundary conditions. The constitutive relations are based on nonlocal theory of Euler–Bernoulli nanobeams. The problem formulation includes the effect of surface elasticity with the results given for circular cross sections. In studying the vibrations of the system, three cases of deflection modes are considered, namely, out-of-phase, in-phase, and one beam stationary with the results given in the form of figures and contour plots. The numerical results for the frequency and the buckling load are obtained by the method of differential quadrature. The method of solution is verified by applying the method of solution to results available in the literature. The effect of the coupling parameter and the nanobeam diameter on the frequency and the buckling load is investigated by means of counter plots. It is observed that the coupling parameter affects the frequency and the buckling load for low values of the nanobeam diameter. Furthermore, numerical results indicate that as the size of the nanobeams approaches nanoscale, surface effects become more prominent. Differences both in the frequency and the buckling load as compared to the local theory increase as the diameter and the coupling parameter values get smaller.

Bayesian hierarchical model for heterogeneity in the diagnosis and treatment of hepatocellular carcinoma (HCC) in the African continent

HCC prevalence and etiology differ between African regions. To better understand these differences, a survey was conducted to characterize the health care practices in HCC management in Africa. Due to limited number of respondents from certain countries in Africa and due to scarce availability of resources, the traditional method of adaptive quadrature produces unstable estimates in this complex setting with sparse multinomial outcomes and with sparse clustered random effects. We propose a Bayesian hierarchical model to address this challenge, apply it to the sparse real data captured in the survey, and validate the proposed method through simulations on synthetic data.

Nonlinear Thermomechanical Low-Velocity Impact Behaviors of Geometrically Imperfect GRC Beams.

This paper studies the thermomechanical low-velocity impact behaviors of geometrically imperfect nanoplatelet-reinforced composite (GRC) beams considering the von Kármán nonlinear geometric relationship. The graphene nanoplatelets (GPLs) are assumed to have a functionally graded (FG) distribution in the matrix beam along its thickness, following the X-pattern. The Halpin-Tsai model and the rule of mixture are employed to predict the effective Young modulus and other material properties. Dividing the impact process into two stages, the corresponding impact forces are calculated using the modified nonlinear Hertz contact law. The nonlinear governing equations are obtained by introducing the von Kármán nonlinear displacement-strain relationship into the first-order shear deformation theory and dispersed via the differential quadrature (DQ) method. Combining the governing equation of the impactor's motion, they are further parametrically solved by the Newmark-β method associated with the Newton-Raphson iterative process. The influence of different types of geometrical imperfections on the nonlinear thermomechanical low-velocity impact behaviors of GRC beams with varying weight fractions of GPLs, subjected to different initial impact velocities, are studied in detail.

Exploratory Study of a Green Function Based Solver for Nonlinear Partial Differential Equations

This work explores the numerical translation of the weak or integral solution of nonlinear partial differential equations into a numerically efficient, time-evolving scheme. Specifically, we focus on partial differential equations separable into a quasilinear term and a nonlinear one, with the former defining the Green function of the problem. Utilizing the Green function under a short-time approximation, it becomes possible to derive the integral solution of the problem by breaking it into three integral terms: the propagation of initial conditions and the contributions of the nonlinear and boundary terms. Accordingly, we follow this division to describe and separately analyze the resulting algorithm. To ensure low interpolation error and accurate numerical Green functions, we adapt a piecewise interpolation collocation method to the integral scheme, optimizing the positioning of grid points near the boundary region. At the same time, we employ a second-order quadrature method in time to efficiently implement the nonlinear terms. Validation of both adapted methodologies is conducted by applying them to problems with known analytical solution, as well as to more challenging, norm-preserving problems such as the Burgers equation and the soliton solution of the nonlinear Schrödinger equation. Finally, the boundary term is derived and validated using a series of test cases that cover the range of possible scenarios for boundary problems within the introduced methodology.

High order moment conserving method of classes in CFD code

Abstract Dispersed multiphase flows are widely present in the majority of chemical process industry equipment. For the purpose of design, optimization, and scale up of these industrial systems, robust, accurate and fast simulation methods are needed, especially when coupling computational fluid dynamics (CFD) with population balance models (PBM). In this work, the high-order moment conserving method of classes (HMMC) is implemented and tested within the commercial CFD software of ANSYS Fluent with two simple well-defined test cases and a realistic 3D simulation of rotating disc (RDC) extractor. Firstly, a zero-dimensional PBM with well-mixed assumption and no convection was solved for a single computational cell in Fluent to verify the implementation in comparison with a simple reactor model in MATLAB. Secondly, the convection was considered by solving a one-dimensional PBM for three computational cells side by side in Fluent and compared with an equivalent three staged reactor model in MATLAB. Eventually, realistic 3D RDC simulations were conducted to fully couple the HMMC and CFD in Fluent. The implementation of HMMC was compared to predictions with other state-of-the-art PBM solution methods, e.g., the quadrature method of moments (QMOM) and the fixed pivot technique (FPT) in Fluent and MATLAB platforms. The implementation of the HMMC in Fluent platform was straightforward with no additional challenges. The verification shows that the HMMC approach is robust and efficient for polydispersed multiphase CFD simulations.

The free vibration analysis of ring-stiffened FG conical sandwich shells with auxetic and non-auxetic honeycomb cores

In this paper, the natural frequencies of a ring-stiffened conical sandwich shell with a functionally graded (FG) honeycomb core are calculated. The core can be either a non-auxetic honeycomb (NAH) or an auxetic honeycomb (AH). The honeycomb core and face layers are fabricated from metal-ceramic functionally graded material (FGM) in which the volume fraction of the ceramic increases from zero at the inner surface of the shell to one at its outer surface based on either power-law function (P-FGM), sigmoid function (S-FGM), or exponential function (E-FGM). The sandwich shell is modeled based on Murakami’s zig-zag theory and the intermediate ring support is modeled as a rigid structure. Hamilton’s principle is employed to derive the governing equations, compatibility conditions, and boundary conditions. A semi-analytical solution is presented which includes an exact solution in the circumferential direction followed by an approximate numerical solution via the differential quadrature method (DQM) in the meridional direction. It is concluded that for each vibrational mode, optimal values can be found for the inclined angle, core-to-shell thickness ratio, and location of the ring support which bring about the highest natural frequency. The presented work is the first theoretical work associated with the free vibration analysis of a ring-stiffened FG conical sandwich shell with a honeycomb core which provides a reduction in the mass and improvement in the thermo-mechanical characteristics, simultaneously. Such properties make such a structure widely used in the aerospace industry. Considering the importance of the dynamic characteristics of aerospace structures, the results of this research can be used by these industries in the analysis and optimal design of the main body of airplanes and missiles.

In-plane vibration analysis of elastically restrained FGM skew plates using variational differential quadrature method

This work presents an accurate in-plane vibration analysis of functionally graded material (FGM) skew plates with elastically restrained boundaries using the variational differential quadrature method (VDQM). The weak form of the governing equations is derived by integrating two-dimensional elasticity theory with Hamilton's principle. The differential and integral operators are directly converted into matrix forms, thereby removing the necessity for higher-order derivative approximations in the displacement field. Transformation matrices are then developed for both interior and boundary nodes to link the governing equations with the boundary conditions, leading to the formulation of the vibration eigenvalue equations for FGM skew plates. Various factors, including aspect ratios, skew angles, boundary restraints, and gradient indices, are considered to investigate the in-plane vibration mode characteristics of FGM skew plates. Detailed solution procedures are provided, and numerical examples using the proposed solutions indicate that the VDQM exhibits superior numerical convergence and stability compared to other existing methods. The study also investigates the influence of highly skewed plates (75°), where stress singularities arise at the corners. This aspect is crucial for in-plane vibration analysis and has garnered limited attention in the existing literature. Furthermore, the dynamic analysis of FGM skew plates reveals a coupling between normal and tangential vibration modes.

Advanced dynamic thermal vibration of thick magnetostrictive FGM plates-cylindrical shells in unsteady yawed supersonic flow

The thermal vibration of magnetostrictive functionally graded material (FGM) thick plates-cylindrical shells in advanced dynamic unsteady yawed supersonic flow with nonlinear third-order piston theory is studied by using numerical method. The effects of yawed angle, controlled gain, and third-order shear deformation theory (TSDT) on the thermal stress and center displacement are analyzed. The nonlinear coefficient of TSDT and advanced shear correction provide the main effects on the time response values of displacements and stresses. Also the controlled gain provides the additional effects on the transient response values to control the displacements and stresses into smaller values. The novelties of the present article are the advanced shear coefficient included with TSDT displacement, velocity feedback control law, and nonlinear third-order piston theory used to obtain the thermal vibration modeling equation and the simulation results for FGM plates-shells in nonlinear unsteady yawed supersonic flow. The FGM structures are usually used for high temperature area to provide better resistance in the deflection and stress. The generalized differential quadrature (GDQ) methods used in the numerical calculation for getting the simulation results for FGM plates-shells in nonlinear unsteady yawed supersonic flow.

BS-CP: Efficient streaming Bayesian tensor decomposition method via assumed density filtering.

Tensor data is common in real-world applications, such as recommendation system and air quality monitoring. But such data is often sparse, noisy, and fast produced. CANDECOMP/PARAFAC (CP) is a popular tensor decomposition model, which is both theoretically advantageous and numerically stable. However, learning the CP model in a Bayesian framework, though promising to handle data sparsity and noise, is computationally challenging, especially with fast produced data streams. The fundamental problem addressed by the paper is mainly tackles the efficient processing of streaming tensor data. In this work, we propose BS-CP, a quick and accurate structure to dynamically update the posterior of latent factors when a new observation tensor is received. We first present the BS-CP1 algorithm, which is an efficient implementation using assumed density filtering (ADF). In addition, we propose BS-CP2 algorithm, using Gauss-Laguerre quadrature method to integrate the noise effect which shows better empirical result. We tested BS-CP1 and BS-CP2 on generic real recommendation system datasets, including Beijing-15k, Beijing-20k, MovieLens-1m and Fit Record. Compared with state-of-the-art methods, BS-CP1 achieve 31.8% and 33.3% RMSE improvement in the last two datasets, with a similar trend observed for BS-CP2. This evidence proves that our algorithm has better results on large datasets and is more suitable for real-world scenarios. Compared with most other comparison methods, our approach has demonstrated an improvement of over 10% and exhibits superior stability.

Efficient One-Step Second Derivative Block Numerical Scheme for Solution of first-Order Integro-differential Equations

In this work, a new class of one-step second derivative block hybrid numerical scheme is developed for the numerical solution of first-order integro-differential equations with the aid of Simpson's 1/3 quadrature method. The collocation techniques is used for the derivation of this new block method which gives high order of accuracy with very low error constants and its zero stable and convergent. The results from the numerical solution of the examples drawn from literature shows its efficiency in terms of the small errors obtained.

Lax-Wendroff Flux Reconstruction on adaptive curvilinear meshes with error based time stepping for hyperbolic conservation laws

Lax-Wendroff Flux Reconstruction (LWFR) is a single-stage, high order, quadrature free method for solving hyperbolic conservation laws. This work extends the LWFR scheme to solve conservation laws on curvilinear meshes with adaptive mesh refinement (AMR). The scheme uses a subcell based blending limiter to perform shock capturing and exploits the same subcell structure to obtain admissibility preservation on curvilinear meshes. It is proven that the proposed extension of LWFR scheme to curvilinear grids preserves constant solution (free stream preservation) under the standard metric identities. For curvilinear meshes, linear Fourier stability analysis cannot be used to obtain an optimal CFL number. Thus, an embedded-error based time step computation method is proposed for LWFR method which reduces fine-tuning process required to select a stable CFL number using the wave speed based time step computation. The developments are tested on compressible Euler's equations, validating the blending limiter, admissibility preservation, AMR algorithm, curvilinear meshes and error based time stepping.

Snap-through of functionally graded graphene origami-enabled auxetic metamaterial doubly curved nonlinear shells

The lightweight design of thin-walled curved structures as spherical shells are frequently implemented in architecture, aerospace, mechanical, automotive, nuclear, and defense structures. Under the transverse loads, shells may largely deform and hence snap from one equilibrium position to the other. Thus, this work aims to develop a mathematical model and computational solution to investigate the nonlinear bending and snap-through behavior of doubly curved auxetic metamaterial shell subjected to transversal loading, for the first time. The shell is composed of several layers through the shell thickness, each layer is manufactured from a copper (Cu) matrix reinforced with a specified weight fraction of graphene origami auxetic metamaterial (GOAM). The mechanical properties of the GOAM shell panel are presented and described by functions of the GOAM volume fraction and folding degree. Three types of GOAM-distributions are considered, which are U-type, X-type, and O-type. The theoretical framework of Kirchhoff–Love hypotheses for thin shells and von Karman type nonlinearity are used to derive the governing equations. The differential quadrature method (DQM) is implemented to discretize the space domain and convert the nonlinear partial differential equation to nonlinear algebraic equations in terms of displacement field. An efficient incremental iterative procedure is developed to solve the nonlinear equations and predict the snap-through behavior. A model conversion and validation with isotropic spherical shell is considered. Several numerical results are conducted considering the effects of changing GOAM content, distribution pattern, folding degree, and shell curvature and thickness. For panels exhibiting snap-through behavior, increasing the shell curvature leads to higher snap-through limiting load; however, the instability gap is enlarged.

Advanced Dynamic Vibration Control Algorithms of Materials Terfenol-D Si3N4 and SUS304 Plates/Cylindrical Shells with Velocity Feedback Control Law

A numerical, generalized differential quadrature (GDQ) method is presented on applied heat vibration for a thick-thickness magnetostrictive functionally graded material (FGM) plate coupled with a cylindrical shell. A nonlinear c1 term in the z axis direction of a third-order shear deformation theory (TSDT) displacement model is applied into an advanced shear factor and equation of motions, respectively. The equilibrium partial differential equation used for the thick-thickness magnetostrictive FGM layer plate coupled with the cylindrical shell under thermal and magnetostrictive loads can be implemented into the dynamic GDQ discrete equations. Parametric effects including nonlinear term coefficient of TSDT displacement field, advanced nonlinear varied shear coefficient, environment temperature, index of FGM power law and control gain on displacement, and stress of the thick magnetostrictive FGM plate coupled with cylindrical shell are studied. The vibrations of displacement and stress can be controlled by the control gain algorithms in velocity feedback control law.