Quadrature Method Research Articles

Numerical investigation of temperature distribution on a tunnel boring machine disc cutter affected by geomechanically and excavation parameters

This study focuses on the thermal behavior of tunnel boring machines (TBMs) through an in-depth investigation into the temperature distribution of their disc cutters. Utilizing the differential quadrature method (DQM), the research conducts a comprehensive numerical analysis to assess the impact of excavation and geological parameters on disc cutter temperature and wear. The accuracy of the DQM model is validated against the finite difference method (FDM), demonstrating comparative results with reduced computational requirements. The findings indicate a significant correlation between disc cutter temperature and various factors, such as rotational speed, spacing, geological conditions, and material strength. Notably, increased spacing or cutter speed leads to higher temperatures and accelerated cutter wear. Moreover, geological factors, particularly rock strength, influence friction coefficients, affecting disc cutter temperatures significantly. For instance, even a slight increase in cutter spacing results in a substantial 65% rise in cutter consumption, underscoring the relevance of these findings for life cycle assessment (LCA) evaluations across diverse geological and environmental conditions in TBM operations.

Dual-functional perforated metamaterial plate for amplified energy harvesting of both acoustic and flexural waves

Metamaterials with defect states are commonly utilized in energy harvesting from acoustic and elastic waves due to their remarkable ability to localize waves. In this study, a novel piezoelectric energy harvester based on a perforated double-pillars metamaterial plate with a point defect, which offers the advantage of efficiently harvesting both ambient acoustic and flexural wave energy, is proposed. The proposed structure incorporates a locally resonant mechanism that enhances the concentration of vibration energy at the defect band frequency. Then, the amplified wave energy is efficiently converted into electrical energy by attaching a piezoelectric patch at the defect position. To initiate this investigation, the differential quadrature method is employed to theoretically estimate the boundary frequencies of the first band gap. Subsequently, this study investigates how holes size, electrical boundary conditions, and electrical circuit connections affect the energy harvesting of both acoustic and flexural waves. Correspondingly, the mechanisms behind their effects are thoroughly explained. Numerical results demonstrate that the wave energy localization performance can be remarkably amplified with an increasing holes radius, and the harvesters with the 1mm holes radius generate voltages that are approximately 2.12 and 1.44 times higher than those with the 0mm holes radius from acoustic and flexural waves, respectively. Furthermore, variations in the defect band frequency and corresponding wave localization behavior depend on the specific electrical boundary conditions and circuit connection approaches, ultimately leading to distinct results in the output electrical energy. These findings present valuable insights and guidelines for the development of high-performance electronic applications in the context of acoustic and elastic wave energy harvesting.

Refined parallel adaptive Bayesian quadrature for estimating small failure probabilities

Bayesian active learning methods have emerged for structural reliability analysis, showcasing more attractive features compared to existing active learning methods. The parallel adaptive Bayesian quadrature (PABQ) method, as a representative of them, allows to efficiently assessing small failure probabilities but faces the problem of empirically specifying several important parameters. The unreasonable parameter settings could lead to the inaccurate estimates of failure probability or the non-convergence of active learning. This study proposes a refined PABQ (R-PABQ) method by presenting three novel refinements to overcome the drawbacks of PABQ. Firstly, a sequential population enrichment strategy is presented and embedded into the importance ball sampling technique to solve the computer memory problem when involving large sample population. Secondly, an adaptive determination strategy for radius is developed to automatically adjust the sampling region during the active learning procedure. Lastly, an adaptive multi-point selection method is proposed to identify a batch of points to enable parallel computing. The effectiveness of the proposed R-PABQ method is demonstrated by four numerical examples. Results show that the proposed method can estimate small failure probabilities (e.g., 10−7∼10−9) with superior accuracy and efficiency over several existing active learning reliability methods.

Effect of porosity on the modal response of doubly-curved laminated shell structures made of functionally graded materials employing higher order theories

The manuscript investigates the modal response of laminated anisotropic doubly-curved shell structures of variable thickness made of Functionally Graded Materials (FGM), according to an efficient equivalent single layer strategy where the displacement field is described employing a condensed unified formulation, accounting for a higher order through-the-thickness expansion. A generalized power law distribution is adopted for the assessment of the FGM layers, whereas the presence of voids is considered starting from different homogenization assumptions. Unlike previous works which presented a variation of the homogenized material properties within the shell solid, in this paper the problem of material porosity is addressed, therefore a general distribution of the volume fraction of the constituent materials is considered, along with the presence of voids. To this end, both linear and trigonometric through-the-thickness distributions are here assumed. The fundamental governing equations are derived starting from a proper set of curvilinear principal coordinates, while a generalized set of blending functions accounts for arbitrarily shaped structures. Non-conventional boundary conditions are, here, modelled with a distribution of linear springs along the shell edges. The numerical implementation of the problem is performed with the generalized differential quadrature method. A systematic set of validating examples is presented, whose results are compared to predictions based on refined models, as well as some experimental evidence. After a validation step, an extensive parametric analysis points out the sensitivity of the porosity parameters on the modal response of structures with different curvatures and external constraints, with valid findings for engineering design purposes.

Fractal patterns in fluid flows: A radial basis pseudospectral analysis of CuO-EG water nano, Saffman dusty, and plain Newtonian fluids in immiscible layers of a horizontal channel

Nano and dusty fluids provide new possibilities to improve energy transfer efficiencies. Motivated by immiscible fluids duct flow framework in the filtration of hydrocarbon materials, the objective of this manuscript is to present the numerical study of three immiscible fluids namely, copper oxide-ethylene glycol (CuO-EG) water nano, Saffman dusty, and plain Newtonian fluids in the horizontal channel. For the formulation of the mathematical model, the dusty fluid is positioned between the immiscible layers of the nanofluid and Newtonian fluid. The interface between the immiscible fluids is continuous and non-deformable in nature. The flow is driven by an externally time-dependent pressure gradient. The novel radial basis function pseudospectral (RBFPS) method is employed to solve the governed partial differential equations, and the resulting outcomes are compared with cubic B-spline differential quadrature and finite difference method. The impact of several crucial fluid parameters, comprising the volume percentage of CuO nanoparticles, the dust particle concentration parameter, and various others on velocity profiles, are presented graphically. It is observed that despite being present only in the lower zone, the CuO nanoparticles have an impact on the flow velocities in other zones also. The fluid and particle velocities decrease as the volume fraction of CuO increases.

Vibration characteristics of FG saturated porous annular plates integrated by piezoelectric patches on visco‐Pasternak foundation

AbstractThe vibrational behavior of a three‐layered annular plate is considered in the present study. The plate is composed of a functionally graded (FG) porous core which is saturated by fluid and two piezoelectric patches that are bonded to the core and are subjected to the electric field. The whole of the structure is also rested on visco‐Pasternak elastic foundation. The material properties of the FG porous core vary through the thickness direction based on different patterns which are called porosity distributions. Love‐Kirchhoff's hypothesis and Hamilton's principle is employed to extract the governing motion equations and boundary conditions and following it, they are solved by generalized differential quadrature method (GDQM) for various boundary conditions. After ensuring the validity of the results by comparing them with known data in the literature, the effect of the most important parameters on the results is considered. It is seen that the effect of the porosity coefficient on the natural frequencies is completely dependent on the pores’ distribution patterns. Also, increasing in externally applied voltage to the piezoelectric facesheets, leads the results to enhance. The outcomes of this work can help to design and create smart structures and systems such as sensors and actuators.

Thermal analysis for laminated plates with arbitrary supports under non-uniform temperature boundary conditions

This paper aims to investigate thermal behaviors of composite laminated plates with different supports under non-uniform temperature boundary conditions. The thermo-elastic solutions of temperature, displacements, and stresses for the laminated plate with arbitrary layer numbers and thickness are obtained based on the three-dimensional theory of thermoelasticity. By applying the Fourier law of heat conduction and the thermoelastic equations, the state-space equations are established by employing the temperature, heat flux, displacements, and stresses as state variables without the hypothesis of displacement form. The differential quadrature method is introduced to discretize the in-plane state variables. On the basis of the continuities of state variables at the interfaces in the laminated plate, the relationships of temperature, heat flux, displacements, and stresses between the top and bottom surfaces can be derived. By simultaneously considering temperature and mechanical boundary conditions applied to the surfaces of the laminated plate, the initial state variables for temperature, displacements, and stresses can be determined uniquely. The errors in the temperature, displacements and stresses resulting from the differential quadrature method can be eliminated by incrementally augmenting the number of sampling points in the convergence study. The accuracy of the present method is thoroughly verified by comparing the current results with both the finite element solutions and the findings reported in previous literature. Finally, the influences of surface temperature, length-to-thickness ratios, material properties, layer numbers, and support types for the distributions of the temperature field, displacements, and stresses in the laminated plate are discussed in detail.

Efficient analysis on buckling of FG-CNT reinforced composite joined conical–cylindrical laminated shells based on GDQ method under multiple loading conditions

This work presents an efficient analysis of buckling for functionally graded carbon nanotube reinforced composite (FG-CNTRC) joined conical–cylindrical laminated shells with lateral pressure and axial compressive load considering thermal effect. The governing equilibrium equation of the joined conical–cylindrical shell (CO-CYL) is determined using the first-order shear deformation theory (FSDT), Donnell kinematic assumptions, von-Kármán geometrical nonlinearity, and considering thermal effect. A semi-analytical solution, generalized differential quadrature (GDQ) method, and artificial springs are applied during the solution procedure. The comparison studies have demonstrated a remarkable 97% reduction in running time compared to the finite element method (FEM).

Model of branched crack and hole defects in finite plane

This paper addresses the issue of branched crack near hole defects within a finite plane. Based on the distributed dislocation method and the Gauss-Chebyshev quadrature method, it is possible to conveniently obtain the stress intensity factors of branched crack around single and multiple arbitrarily shaped hole defects, and to obtain the complete stress field of the interaction between branched cracks and hole defects. Additionally, it has been found that this technique can be used to address the problem of crack initiated from hole defect and has been validated through finite element analysis. The results show that finite boundary proportionate scaling has little impact on the direction of branching propagation; the biaxial load ratio is the crucial factor influencing the propagation angle. Under tension loading, even when the defect is far from the crack, it still significantly affects KI, while KII is only significantly influenced when the distance is small. Branched cracks with different branching angles exhibit a similar range of attraction to hole defects. The number of hole defects, their arrangement, and the distances between cracks and defects all influence the interactions between cracks and defects, as well as between hole defects themselves.

Coupled dynamic instability of graphene platelet-reinforced dielectric porous arches under electromechanical loading

Graphene platelet-reinforced dielectric porous (GPLRDP) arches are a kind of multifunctional structures, which can be used to design various dielectric resonators and soft robotic components. This work studies the coupled dynamic instability of GPLRDP arches under electromechanical loading. It is assumed that the presence of voids is uniformly distributed in the arch structure, while graphene platelets (GPLs) are uniformly dispersed inside the skeleton of such voids. The effective medium theory (EMT) is used to determine the effective dielectric permittivity and Young's modulus of such structures. Based on the Hamilton principle, the governing equations of this problem are formulated. The differential quadrature method (DQM) and the Bolotin method are then used for solving the arch system to identify the coupled dynamic instability region. In this study, we observe the coupling effect of parametric and forced resonance phenomena. A series of numerical experiments are also carried out to examine the influence of porosities, GPL weight fractions, boundary conditions, and electrical voltage levels on the critical excitation frequency and coupled dynamic instability behavior of GPLRDP arches. It is found that the coupled dynamic instability region becomes wider and shifts to low frequency as the effect of porosity increases. When the signal amplitude of AC frequency is within a certain range, there is an abrupt increase of the critical excitation frequency under a high level of DC voltage. The numerical results of this work demonstrate that the location and size of the coupled dynamic instability region can be actively controlled by adjusting the material and structural parameters.

Artificial neural networks coupled with numerical approach for the stability prediction of non-uniform functionally graded microscale cylindrical structures

Functionally graded materials (FGMs) have garnered significant attention in research due to their potential for creating innovative materials with diverse applications. This study focuses on investigating the behavior of steel-concrete functionally graded structures under axial loading, specifically exploring nonlinear buckling. The research considers cylindrical structures with simply-supported boundary conditions and analyzes four different configurations with varying thickness. The governing equations are derived using the modified couple stress theory, incorporating assumptions of nonlinear von-Kármán theory coupled with classical beam theory. Numerical solutions are obtained using the general differential quadrature (GDQ) method. A notable aspect of this work is the use of Artificial Neural Networks (ANN) to predict nonlinear buckling forces, greatly enhancing the accuracy and efficiency of the predictions. This approach offers a novel method for analyzing concrete structures. The results are validated by comparing them with data from other published works, using dimensionless parameters for consistent comparisons. The findings provide a comprehensive analysis of various parameters, including wall thickness function and porosity, on the buckling stability of the cylindrical structure. Both numerical and ANN approaches are employed to report the effects. The study concludes that the uniform thickness cylinder exhibits the highest resistance to axial loading, resulting in buckling.

Shot-noise limited optical hybrid based on fused fiber couplers

We describe a fiber-based coherent receiver topology which utilizes intrinsic phase shifts from fiber couplers to enable instantaneous quadrature projection with shot-noise limited signal-to-noise ratio (SNR). Fused 3 × 3 fiber couplers generate three phase-shifted signals simultaneously that can be combined with quadrature projection methods to detect magnitude and phase unambiguously. We present a novel, to the best of our knowledge, differential detection topology which utilizes a combination of 3 × 3 and 2 × 2 couplers to enable quadrature projection with fully differential detection. We present a mathematical analysis of this 3 × 3 differential detection topology, extended methods for signal calibration, and SNR analysis. We characterize the SNR advantage of this approach and demonstrate a sample application illustrating simultaneous magnitude and phase imaging of a chrome-on-glass test chart.

Computations of J-integral and T*-integral in elastic–plastic fracture by the quadrature element method

In this paper, J-integral and T*-integral in elastic–plastic fracture are computed by the quadrature element method (QEM). Since high stress gradients exist in the immediate vicinity of the crack front, computing these integrals accurately is not a trivial task. The QEM facilitates the construction of arbitrarily high-order elements, which effectively capture the stress gradients, making it particularly suitable for this computation. After performing an elastoplastic stress–strain analysis of the fracture problem, the stress and deformation states are further formulated to compute the integrals. The integrals are formulated in several forms, including the contour and domain forms, as well as the incremental and total forms. Classical numerical examples are solved to demonstrate the effectiveness and high accuracy of the method. Overall, the equivalent domain integral (EDI) form outperforms the contour integral form because errors at the local contour have less effect on the integral. Moreover, the QEM yields more accurate solutions than the popular finite element method (FEM) that employs low-order finite elements and shares the same number of degrees of freedom. Additionally, it is found that the EDI form solution is not very sensitive to the specific type of the S function. Therefore, a linear function is suggested for convenience.

Dynamic analysis of double cracked bi-directional functionally graded nanobeam using the differential quadrature method

This study investigates the free vibration behavior of a double cracked nanobeam composed of bi-directional functionally graded material. The analysis incorporates Eringen’s nonlocal elasticity theory and the Euler–Bernoulli theory. The material properties are considered to vary in both the thickness and length directions. The cracked nanobeam is modeled as a series of interconnected sub-beams, with rotational springs placed at the cracked sections. This modeling approach accounts for the discontinuities in rotational displacement resulting from bending, which is directly related to the bending moment transmitted by the cracked section. The problem is solved using the differential quadrature method, which approximates the derivatives of the field quantities by employing a weighted linear sum of the nodal values. By doing so, the problem is transformed into a linear algebraic system. Various supporting cases are examined, and a parametric study is conducted to analyze the impact of the axial and transverse gradient indices, nonlocal parameter, and crack severity on the obtained results.

Approximation of Any Particle Size Distribution Employing a Bidisperse One Based on Moment Matching

Dispersed phases like colloidal particles and emulsions are characterized by their particle size distribution. Narrow distributions can be represented by the monodisperse distribution. However, this is not the case for broader distributions. The so-called quadrature methods of moments assume any distribution as a bidisperse one in order to solve the corresponding population balance. The generalization of this approach (i.e., approximation of the actual particle size distribution according to a bidisperse one) is proposed in the present work. This approximation helps to compress the amount of numbers for the description of the distribution and facilitates the calculation of the properties of the dispersion (especially convenient in cases of complex calculations). In the present work, the procedure to perform the approximation is evaluated, and the best approach is found. It was shown that the approximation works well for the case of a lognormal distribution (as an example) for a moments order from 0 to 2 and for dispersivity up to 3.

Kuramoto-Sivashinsky equation: Numerical solution using two quintic B-splines and differential quadrature method

This paper aims to obtain the numerical solution of a one-dimension fourth-order Kuramoto-Sivashinsky (KS) equation, which has application in the context of flame propagation. This study combines the implementation of two new mechanisms quintic Uniform Algebraic Hyperbolic (QUAH) tension B-spline and quintic Uniform Algebraic Trigonometric (QUAT) tension B-spline with a renowned method known as Differential Quadrature Method (DQM). DQM is used to approximate the derivatives. Then, to obtain the weighting coefficients both quintic splines QUAH and QUAT are implemented with DQM and the original boundary value problem was then transformed into an ordinary differential equation. The Runge-Kutta 43 technique was used to solve the resultant ODE. With this approach, the outcomes are precise and close to the exact solution. The differential quadrature technique has a significant advantage over the previous approaches because it prevents the perturbation in order to find the better results for the given nonlinear equations. The accuracy of the suggested method is then illustrated numerically resolving four test problems and calculating the error norms L2 and L∞. Calculated results are displayed graphically and in tabular format for quick and simple access. This paper introduces an original work, to the best of the author’s knowledge. The obtained solution and graphical observations show that QUAH and QUAT tension B-splines with DQM is a strong and dependable methods for estimating nonlinear problem solutions. Quintic B-splines are higher-order interpolation method compared to lower-degree B-splines, which results in more accurate approximations, and provides smooth and continuous representations of functions. Moreover, DQM when combined with quintic B-splines, can help reduce numerical dispersion, and can be applied to a wide range of problems.

Transverse free vibration analysis of thin sectorial plates by the weak form quadrature element method

The weak form quadrature element method is applied to free vibration analysis of thin sectorial plates with arbitrary vertex angles and boundary conditions. To tackle the strong stress singularity around the vertex, analytical displacement descriptions are introduced into the inner sectorial subdomain, while the outer annular subdomain is modeled by a single weak form quadrature thin plate element. The continuity on the interface between the two subdomains is enforced afterwards. Eventually, a generalized eigenvalue formulation is established after introducing Hamilton’s principle. The first six non-dimensional frequency parameters for various vertex angles and boundary conditions are obtained and compared with available results. Several typical free vibration modes are plotted. The accuracy, convergence rate, and computational cost of the present formulation are discussed at length.

Multiple Access Design for Symbiotic Radios: Facilitating Massive IoT Connections With Cellular Networks

Symbiotic radio (SR) has emerged as a spectrum-and energy-efficient paradigm to support massive Internet of Things (IoT) connections. Two multiple access schemes are proposed in this paper to facilitate massive IoT connections using the cellular network based on the SR technique, namely, the simultaneous access (SA) scheme and the selection diversity access (SDA) scheme. In the SA scheme, the base station (BS) transmits information to the receiver while multiple IoT devices transmit their information simultaneously by passively backscattering the BS signal to the receiver, while in the SDA scheme, only the IoT device with the strongest backscatter link transmits information. In both of the schemes, the receiver jointly decodes the information from the BS and IoT devices. To evaluate the above two schemes, the closed-form expressions of the ergodic rates in high signal-to-noise ratio (SNR) regimes and outage probabilities for cellular and IoT transmissions are derived by using extreme value theory, generalized-K distribution approximation and Gaussian-Chebyshev quadrature methods. Finally, numerical results are provided to verify the theoretical analysis and compare the proposed multiple access schemes. When the number of IoT devices is small, the SDA scheme is more appealing since it can significantly reduce the computational complexity while achieving equivalent performance to the SA scheme. When the number of IoT devices is large, the SA scheme is preferable since it guarantees a significantly better rate performance and a lower outage probability.

A normal-mode approach for high-speed rarefied plane Couette flow

Based on gas kinetic theory, a linear stability analysis method for low-speed rarefied flows was developed by Zou et al. [“A new linear stability analysis approach for microchannel flow based on the Boltzmann Bhatnagar–Gross–Krook equation,” Phys. Fluids 34, 124114 (2022) and “A novel linear stability analysis method for plane Couette flow considering rarefaction effects,” J. Fluid Mech. 963, A33 (2023)]. In the present study, we extended the method to high-speed rarefied flows using the Bhatnagar–Gross–Krook model. The Chebyshev spectral method is employed to discretize physical space, and the Gauss–Hermite and fourth-order Newton–Cotes quadrature methods are used to discretize velocity space. The fourth-order Newton–Cotes quadrature method was found to have sufficient accuracy for the stability analysis, laying the foundation for future research on hypersonic flows. The stability analysis of compressible rarefied Couette flow showed that acoustic modes are reflected between the wall and the relative sonic line, and the variation in their phase speed and growth rate with the wavenumber is not affected by the Mach number (Ma) and the Knudsen number (Kn). Increasing Kn has a stabilizing effect on both the acoustic and viscous modes, but as Ma increases, the attenuation rate of each mode's growth rate gradually decreases. In subsonic and sonic flows, the least stable viscous mode dominates in the case of small numbers. As Kn increases, the viscous mode gradually dominates over all wavenumber ranges considered in subsonic flow. In sonic flow, mode 1 is dominant in the region beyond the range of small wavenumbers. In supersonic flow, mode 2 is the least stable in the large wavenumber ranges, while mode 1 is the least stable in other wavenumber ranges. At a fixed wavenumber, as Kn increases, the decay rate of the growth rate of mode 2 is the highest. Additionally, under different Knudsen numbers, the growth rates of mode 1, mode 2, and the least stable viscous mode monotonically increase with an increase in Ma, with mode 2 showing the most significant increase.

Evaluation of the 1-D hyperbolic quadrature method of moments for non-equilibrium flows

Evaluation of the 1-D hyperbolic quadrature method of moments for non-equilibrium flows