Differential Quadrature Method Research Articles

AN INVESTIGATION ON NONLINEAR OPTION PRICING BEHAVIOURS THROUGH A NEW FRÉCHET DERIVATIVE-BASED QUADRATURE APPROACH

Abstract Complicated option pricing models attract much attention in financial industries, as they produce relatively better accurate values by taking into account more realistic assumptions such as market liquidity, uncertain volatility and so forth. We propose a new hybrid method to accurately explore the behaviour of the nonlinear pricing model in illiquid markets, which is important in financial risk management. Our method is based on the Newton iteration technique and the Fréchet derivative to linearize the model. The linearized equation is then discretized by a differential quadrature method in space and a quadratic trapezoid rule in time. It is observed through computations that the accurate solutions for the model emerge using very few grid points and time elements, compared with the finite difference method in the literature. Furthermore, this method also helps to avoid consideration of the convergence issues of the Newton approach applied to the nonlinear algebraic system containing many unknowns at each time step if an implicit method is used in time discretization. It is important to note that the Fréchet derivative supports to enhance the convergence order of the proposed iterative scheme.

Nonlinear Analysis of the Multi-Layered Nanoplates

This text investigates the bending/buckling behavior of multi-layer asymmetric/symmetric annular and circular graphene plates through the application of the nonlocal strain gradient model. Additionally, the static analysis of multi-sector nanoplates is addressed. By considering the van der Waals interactions among the layers, the higher-order shear deformation theory (HSDT), and the nonlocal strain gradient theory, the equilibrium equations are formulated in terms of generalized displacements and rotations. The mathematical nonlinear equations are solved utilizing either the semi-analytical polynomial method (SAPM) and the differential quadrature method (DQM). Also, the available references are used to validate the results. Investigations are conducted to examine the effect of small-scale factors, the van der Waals interaction value among the layers, boundary conditions, and geometric factors.

Size‐dependent buckling loads of non‐uniform Bernoulli–Euler beams with elastic boundary conditions and thermal effects based on two‐phase local/nonlocal elastic model

AbstractWe present predictive models of the size‐dependent buckling loads of non‐uniform Bernoulli–Euler beams under thermal effects based on the two‐phase local/nonlocal elastic model. The beam ends are assumed to be constrained by elastic springs with translational and rotational stiffness to simulate general boundary conditions. In contrast to most literature in this field, both the bending and thermal deformations of the beams are simultaneously considered to be two‐phase local/nonlocal of two phases, that is, the thermal effect is taken as equivalent to a size‐dependent thermal load. By using the fully equivalent differential form of the local/nonlocal equation with a set of constitutive boundary conditions, the problem is solved numerically with the aid of the generalized differential quadrature method (GDQM). Through conducting validation study, several parametric studies are given for examining the effects of the slope of beams’ thickness variation, nonlocal parameter, and elastically supported conditions on the buckling loads of non‐uniform beams. The results show that constrained stiffness has a drastic influence on the critical bucking loads of the beams. Furthermore, the consideration of the two‐phase thermal load will further reduce the actual buckling load of the beams.

A Robust and Versatile Numerical Framework for Modeling Complex Fractional Phenomena: Applications to Riccati and Lorenz Systems

The fractional differential quadrature method (FDQM) with generalized Caputo derivatives is used in this paper to show a new numerical way to solve fractional Riccati equations and fractional Lorenz systems. Unlike previous FDQM applications that have primarily focused on linear problems, our work pioneers the use of this method for nonlinear fractional initial value problems. By combining Lagrange interpolation polynomials and discrete singular convolution (DSC) shape functions with the generalized Caputo operator, we effectively transform nonlinear fractional equations into algebraic systems. An iterative method is then utilized to address the nonlinearity. Our numerical results, obtained using MATLAB, demonstrate the exceptional accuracy and efficiency of this approach, with convergence rates reaching 10−8. Comparative analysis with existing methods highlights the superior performance of the DSC shape function in terms of accuracy, convergence speed, and reliability. Our results highlight the versatility of our approach in tackling a wider variety of intricate nonlinear fractional differential equations.

Analysis of Vibration Characteristics for Rotating Braided Fiber-Reinforced Composite Annular Plates with Perforations

In the current study, a comprehensive numerical model for analyzing the vibrational characteristics of braided fiber-reinforced composite (BFRC) rotating annular plate with perforations under diverse boundary constraints was introduced. This model employs the differential quadrature finite element method (DQFEM), which was developed based on the first-order shear deformation theory (FSDT) and coordinate transformation approach. The BFRC material, specifically a two-dimensional biaxial orthogonal fabric, was utilized to fabricate the annular plate with two distinct types of holes: circular and sector-shaped. The model’s convergence, accuracy, numerical stability, and reliability were confirmed through comparative assessments utilizing data from the literature, from ABAQUS software, and from experimental findings. The analysis focuses on studying the influences of structural properties, material parameters, and boundary restraints on the frequencies of vibration for BFRC rotating annular plates with holes. This theoretical model helps provide scientific basis and technical guidance for the stability and lightweight design of rotating annular plates, such as rotor structures in aircraft engines.

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.

Nonlinear generalized piezothermoelasticity of spherical vessels made of functionally graded piezoelectric materials

The present study investigates the thermoelastic response of a heterogeneous piezoelectric sphere under thermal shock loading. Boundary conditions as well as loading are considered as symmetric; thus, the response of the sphere is expected to be symmetric. All of the properties of the thick-walled sphere, including mechanical, electrical and thermal properties, are considered dependent on the radial position, except for the relaxation time, which is considered a constant value along the radius. The governing equations of the sphere have been derived under heterogeneous anisotropic assumptions. The general form of the second law of thermodynamics, which is nonlinear in nature, and is called nonlinear energy equation is used. The number of the established equations is three, which includes the motion equation, energy equation and Maxwell electrostatic equation of Maxwell. These equations are obtained in terms of radial displacement, temperature difference and electric potential. The energy equation is derived based on Lord and Shulman theory with a single relaxation time. In the next step, by introducing dimensionless variables, the governing equations are provided in dimensionless presentation. Then these equations have been discretized using generalized differential quadrature method. Also, in order to follow the solution of the equations in time domain, Newmark method has been used. Since the system of equations is nonlinear, Picard algorithm is applied as a predictor-corrector mechanism to solve the nonlinear system of equations. Then numerical results are presented to investigate the propagation of mechanical, thermal and electric waves inside the heterogeneous sphere and also their reflection from the outer surface of the sphere. By examining the results, it can be seen that mechanical and thermal waves propagate with a limited speed, while the speed of electric wave propagation is infinite.

Free vibration and supersonic flutter analyses of a sandwich cylindrical shell with CNT-reinforced honeycomb core integrated with piezoelectric layers

In this research, the free vibration and supersonic flutter analyses of a cylindrical sandwich shell with a reinforced honeycomb core (RHC) integrated with piezoelectric face sheets are investigated. The core is a honeycomb structure enriched with fibers and carbon nanotubes (CNTs). The layers of the shell are modeled based on the Cooper-Naghdi shell theory and the continuity conditions between the layers are considered. The aerodynamic pressure of the supersonic flow is modeled using the linear piston theory. Hamilton’s principle is used to derive the governing equations. A semi-analytical solution is presented including an exact solution in the circumferential direction followed by a numerical solution in the longitudinal direction via the differential quadrature method (DQM). The effects of several parameters on the frequencies, the corresponding damping coefficients, and the critical aerodynamic pressure are investigated. It is concluded that improving the stiffness of the honeycomb core with the CNTs does not necessarily improve the aeroelastic stability of the shell. The presented results can be used in the design and analysis of aerospace structures.

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.

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.

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.

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.

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.

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.