Quadrature Method Research Articles

Mathematical Models of Wound Healing and Tissue Regeneration: Insights with Numerical Solution

Understanding and forecasting the dynamics of wound healing processes heavily relies on mathematical modeling. Researchers can understand the complicated connections between cells, tissues, and biochemical components involved in wound healing by using mathematical equations and computational simulations. One such mathematical model that has seeks the interest of researchers in exploring wound healing phenomenon is Fisher’s equation. In the present work, the equation is solved using differential quadrature method with trigonometric tension B-spline (TTBs) basis function. The obtained results are compared with the results obtained from earlier studies and the results are presented in form of tables and figures. The implementation of the particle swarm optimization algorithm has removes the dilemma of unknown parameter involvement in the TTBs and hence resulted in a novel technique to find the numerical results.

Analysis of natural vibration of truncated conical shells partially filled with fluid

AbstractIn this paper, we study the vibrational behavior of shells in the form of truncated cones containing an ideal compressible fluid. The sloshing effect on the free surface of the fluid is neglected. The dynamic behavior of the elastic structure is investigated based on the classical shell theory, the constitutive relations of which represent a system of ordinary differential equations written for new unknowns. Small fluid vibrations are described in terms of acoustic approximation using the wave equation for hydrodynamic pressure written in spherical coordinates. Its transformation into the system of ordinary differential equations is carried out by applying the generalized differential quadrature method. The formulated boundary value problem is solved by Godunov's orthogonal sweep method. Natural frequencies of shell vibrations are calculated using the stepwise procedure and the Muller method. The accuracy and reliability of the obtained results are estimated by making a comparison with the known numerical and analytical solutions. The dependencies of the lowest frequency on the fluid level and cone angle of shells under different combinations of boundary conditions (simply supported, rigidly clamped, and cantilevered shells) have been studied comprehensively. For conical straight and inverted shells, a numerical analysis has been performed to estimate the possibility of finding configurations at which the lowest natural frequencies exceed the corresponding values of the equivalent cylindrical shell.

Free vibration analysis of laminated anisotropic doubly-curved shell structures reinforced with three-phase polymer/CNT/fiber material

The present work investigates the vibrational response of laminated anisotropic doubly-curved shell structures reinforced with Carbon Nanotubes (CNTs) short fibers. The fundamental equations are derived from a curvilinear reference system of principal coordinates, employing the Equivalent Single Layer (ESL) methodology. Furthermore, a general variation in laminate thickness is considered. The unknown field variable is described using higher order theories through the unified formulation, taking into account a general interpolation of the unknown variables with Lagrange polynomials across the physical domain. The Hamiltonian Principle is adopted for the derivation of the dynamic equilibrium equations, which arediscretized numerically by using the Generalized Differential Quadrature (GDQ) method. In addition, an efficient isogeometric NURBS-based mapping is adopted for the distortion of the physical domain. The boundary conditions of the differential problem are modelled through a general distribution of linear elastic springs along the lateral surfaces of the three-dimensional doubly-curved solid. The theory is then applied to study the vibrational modes of structures with different curvatures and lamination schemes, taking into account a general distribution of CNTs along the thickness direction. The homogenized properties of CNTs layers are obtained from the Mori-Tanaka procedure, which accounts for the agglomeration effects of the dispersed nanofibers within the matrix. Unlike previous studies focusing on doubly-curved shells reinforced with composite materials and CNTs, the present formulation enables to derive the vibration characteristics of structures made of generally anisotropic materials with general orientations. Furthermore, the calibration of the parameters for the linear elastic springs’ distribution leads to general external constraints within a single element, thus reducing the computational cost of the problem.

Nonlinear analysis on electro-elastic coupling properties in bended piezoelectric semiconductor beams with variable cross section

In this paper, the effects of nonlinear constitutive relation on the electro-elastic coupling behaviors in non-uniform piezoelectric semiconductor beams are investigated. On the basis of three-dimensional theory for piezoelectric semiconductor and double power series, one-dimensional bending model is developed. In order to deal with the nonlinear problem, a differential quadrature method based iteration procedure is proposed. Comparing with the results from finite element method, the proposed method has good convergence and high accuracy. In numerical analysis, the effects of nonlinearity and non-uniformity in piezoelectric semiconductor are discussed thoroughly. It is found the introduction of nonlinearity induces the loss of symmetric for all electrical field quantities but little changes the mechanical quantities. While the reduction of width along the length direction produces larger mechanical and electrical responses. Importantly, the effects of nonlinearity on the electrical field quantities are more evident in a piezoelectric semiconductor beam with rapidly varied cross section. Besides, we designed the piezoelectric semiconductor beams subject to double shear forces and a beam with locally varied cross section. In the designed structures, the potential barriers and wells are realized, which provide new approaches adjusting the electro-elastic behaviors in piezoelectric semiconductors. The study in this paper could be the guidance designing high performance electronic devices.

Investigation of moving load-induced vibrations in layered functionally graded Terfenol-D beams: a differential quadrature method and analytical approach

The paper investigates the potential of the full-sinusoidal Fourier series as a solution form for the deflection of layered functionally graded beams associated with smart actuators, drawing on the fundamental principles of classical Fourier series theory. The current method simplifies the difficult beam problem to a set of linear algebraic equations by using the Duhamel integration technique and the orthogonality of the trial function. The study takes into account generalized boundary conditions and moving forces, which are seldom discussed in previous research. Under the action of a moving load, the boundary value problem for a functionally graded beam integrated with Terfenol-D is efficiently addressed using the approach of combined differential and integral quadrature. The generalized boundary conditions may be easily achieved by adjusting the stiffness of the restraining springs. The significant agreement between the differential quadrature solution and the Fourier series solution underlines the efficiency and accuracy of both methods. Furthermore, the influences of various crucial physical characteristics on the natural frequencies and the essential flow velocities are explored, including boundary stiffness, foundation parameters, and geometric parameters.

Vibration characteristics of a cylindrical sandwich shell with an FG auxetic honeycomb core and nanocomposite face layers subjected to supersonic fluid flow

In this paper, the flutter analysis is studied for a cylindrical sandwich shell subjected to external supersonic fluid flow. The sandwich shell consists of a re-entrant auxetic honeycomb (AH) core made of functionally graded materials (FGMs) which is covered with two polymeric face layers enriched with either carbon nanotubes (CNTs), graphene nanoplatelets (GNPs), or graphene oxide powders (GOPs). The volume fraction (percentage) of the ceramic phase in the functionally graded auxetic honeycomb (FGAH) core varies from zero at the inner surface to one at the outer one based on either power-law function (P-FGM), exponential function (E-FGM), or sigmoid function (S-FGM). It is assumed that the nanofillers are distributed uniformly inside the face layers. The first-order shear deformation theory (FSDT) and the piston theory are employed to provide the mathematical models of the shell and the aerodynamic pressure, respectively. The governing equations and boundary conditions are derived via Hamilton’s principle. An exact solution is performed in the circumferential direction via harmonic trigonometric functions (sine and cosine) and an approximate solution is performed in the axial direction utilizing the differential quadrature method (DQM). The natural frequencies and corresponding damping ratios are attained through the presented semi-analytical solution and the impacts of several factors on the natural frequencies and the critical aerodynamic pressure of the shell are examined. These factors the type of the FGM, the material index, and inclined angle of the cells in the FGAH core, the thickness of the FGAH core, the type and mass fraction of the nanofillers in the nanocomposite face layers, and boundary conditions.

Nonlinear dynamics and forced vibrations of simply-supported fractional viscoelastic microbeams using a fractional differential quadrature method

In this paper, free and forced nonlinear vibrations of fractional viscoelastic microbeams are modeled based on Euler-Bernoulli theory, the Von Karman’s nonlinear strain relations, the modified couple stress theory (MCST), and the fractional Kelvin-Voigt viscoelastic model. In the present work, the nonlinear-fractional order governing equations are discretized in the space domain by Galerkin’s method. Two different approaches are introduced to solve the resulting nonlinear fractional-order Duffing equation in the time domain. In the first approach, a time marching fractional finite difference method is presented to compute the transient time response starting from the initial conditions. This approach is time consuming if the frequency-amplitude curves of steady state response are required since it provides just one point on the frequency-amplitude curve. More importantly, this approach usually converges only for the stable branch with smallest amplitude of frequency-amplitude curve. In the second approach, we introduce a novel fractional differential quadrature method (FDQM) to discretize the fractional Duffing equation and apply a pseudo-arc length algorithm to directly construct the frequency-amplitude curves. Effects of the fractional-order, linear and nonlinear viscoelasticity coefficients, viscous damping parameter, microstructure parameters, and the micro-beam thickness on the nonlinear dynamics of the viscoelastic micro-beam are analyzed numerically. Numerical results show that each of these parameters can change natural frequency and/or the damping behavior of the structure. The present model can be used for designing and analyzing the nonlinear microstructure viscoelastic beam under dynamic loads.

Large Deflection Analysis of Functionally Graded Beam by Using Combining Method

In this study, the large deflection behavior of a circular cross-section beam is examined using the Combining Method (CM). The CM is a numerical solution method that uses block diagrams in the Matlab-Simulink program and weighting coefficients in the Differential Quadrature Method (DQM). The beam material considered is Functionally Graded Material (FGM). Boundary conditions of the beam are taken as clamped-free (C-F) and a singular load is assumed to be applied from the free end of the beam. Geometric nonlinear analysis is performed while calculating the large deflection equations of the beam and performing numerical analysis. The effects of increasing the force applied to the beam, changing the beam cross-section in the longitudinal direction, and changing the material index of the FGM on the extreme deflection behavior of the beam were examined. For comparison purposes, the results obtained from CM are compared with the results obtained from both SolidWorks-Simulation and Ansys-Workbench programs. As a result of the analysis, increasing the applied force causes the x and y coordinates of the end point of the beam to decrease. The change in geometry and material index greatly affects the large deflection occurring in the beam.

Frequency Analysis of Asymmetric Circular Organic Solar Cells Embedded in an Elastic Medium under Hygrothermal Conditions

This research represents the first theoretical investigation about the vibration behavior of circular organic solar cells. Therefore, the vibration response of asymmetric circular organic solar cells that represent a perfect renewable energy source is demonstrated. For this purpose, the differential quadrature method (DQM) is employed. The organic solar cell is modeled as a laminated plate consisting of five layers of Al, P3HT:PCBM, PEDOT:PSS, ITO, and Glass. This cell is rested on a Winkler–Pasternak elastic foundation and assumed to be exposed to various types of hygrothermal loadings. There are three different kinds of temperature and moisture variations that are taken into account: uniform, linear, and nonlinear distribution throughout the cell’s thickness. The displacement field is presented based on a new inverse hyperbolic shear deformation theory considering only two unknowns. The motion equations including hygrothermal effect and plate–foundation interaction are established within the framework of Hamilton’s principle. The DQM is utilized to solve these equations. In order to ensure the accuracy of the proposed theory, the present results are compared with those reported by other higher-order theories. A comprehensive parametric illustration is conducted on the impacts of different parameters involving the geometrical configuration, elastic foundation parameters, temperature, and moisture concentration on the deduced eigenfrequency of the circular organic solar cells.

Effects of non-uniformity in thickness and volume fraction of nanofillers on the flutter characteristics of nanocomposite cantilever trapezoidal plates

The aim of this work is to investigate the flutter characteristics of nanocomposite cantilever trapezoidal plates with non-uniform thickness enriched with either carbon nanotubes (CNTs), graphene nanoplatelets (GNPs), or graphene oxide powders (GOPs) which are distributed functionally graded (FG) in the axial direction. It is assumed that the thickness of the plate and the volume fraction of the nanofillers vary in one direction from the wider clamped edge of the plate to the outer narrower free one. The modeling of the plate is done using the first-order shear deformation theory (FSDT) and the aerodynamic pressure generated by the aerodynamic pressure is modeled using the linear approximation of the piston theory. The material properties of the plate are calculated using the mixing rule (ROM) and the Halpin–Tsai model. The governing equations and boundary conditions at the clamped and free edges of the plate are derived via Hamilton’s principle. An approximate solution is applied using the differential quadrature method (DQM) to calculate the natural frequencies and the damping ratios of the plate. Numerical examples show that it is possible to find an optimal thickness variation profile that provides the greatest aeroelastic stability. It is concluded that by considering the same value for the mass fractions of the nanofillers, the highest aeroelastic stability can be attained by utilizing the GNPs as the reinforcers. It is found that to attain further improvement in aeroelastic stability, most nanofillers should be distributed near the clamped edge and away from the outer free edge.

Discussions on application of differential quadrature method for structural free vibration problems

A thorough examination of the use of the differential quadrature method has been presented, including an analysis of the weighting coefficient, several methods for implementing the boundary conditions, and the ultimate advantages it offers compared to other numerical techniques. The authors have detailed the technical techniques involved in this study, including the discretization of the computational domain, the generation of mass, damping, and stiffness matrices, and the subsequent partitioning of these matrices into submatrices. Exemplifications have been given for the differential quadrature (DQ) approaches that meet the clamped-spring support boundary requirement. This study on the principles of the differential quadrature technique will provide valuable insights for a research scholar, novice researcher, and academician in the field of computation.

Analysis of multilayered two-dimensional decagonal piezoelectric quasicrystal beams with mixed boundary conditions

Piezoelectric quasicrystals have a broad spectrum of promising applications and research value due to their unique atomic arrangement and multi-field coupling effects. This paper presents a mechanical analysis of the static bending and free vibration problems of two-dimensional multilayered decagonal piezoelectric quasicrystal laminated beams. Mechanical models of the beams are established, and the differential quadrature method in conjunction with the state-space method is utilized to obtain the solutions for the beams with mixed boundary conditions. We first investigated the static response and free vibration problem of quasicrystal laminated beams. By degrading them into crystalline materials, we verified the accuracy of the method by comparing the results with the existing ones. Then, the effects of different boundary conditions and slenderness ratios on the stresses, displacements, electric potentials, and electric displacements of quasicrystal laminated beams under normal mechanical loading and free vibration are investigated. Subsequently, the effects of different laying modes on the mechanical properties of the quasicrystal laminated beams when the material principal axis of the beam is changed are also analyzed.

Axially Non-Uniform Properties Effected Electro-Elastic Fields in Bended Piezoelectric Semiconductor Beams

This paper studied the effects of axially non-uniform material and non-uniform doping on the electro-elastic fields in bent piezoelectric semiconductor beams. The mathematical model is established based on the phenomenological theory of piezoelectric semiconductor and Euler’s beam theory. To solve the governing equations with variable coefficients, the differential quadrature method is adopted. As two representative structures, the pure bending beam and cantilever beam are studied, respectively. From the calculated results, it can be found the varied material properties changed the effective stiffness, as a result, all the field quantities are altered. Meanwhile, the symmetry of all field quantities in a pure bending beam is broken. Considering different Gaussian doping parameters [Formula: see text] and [Formula: see text], it can be observed that the distribution rule of perturbation carrier density becomes complex. The increment of [Formula: see text] broadens the width of optimum range-producing carriers, however, the increment of [Formula: see text] describes the moving of the optimum range from left to right. Importantly, the introduction of non-uniform doping breaks the limitation of perturbation carrier density in a cantilever beam with uniform material properties, the maximum value of perturbation carrier density does not appear at the fixed end only. The obtained results could be the guidance in designing high-performance electric devices.

Analysis and modeling of two-dimensional piezoelectric semiconductor shell theory

In recent years, there has been a surge in research on piezoelectric semiconductors (PSs). However, studies specifically focused on two-dimensional PS structures remain scarce. In this paper, based on the principle of the virtual work method, the condition of charge continuity and the first-order thickness shear theory, we have established a two-dimensional model to investigate the electromechanical properties of two-dimensional PS shells. The basic field quantities, including the displacement, the electric potential and the carrier concentration perturbations were Fourier expanded along the width direction, and then the governing equations were successfully derived. The differential quadrature method was used to obtain numerical results for the PS shells with different loading modes and structural configurations. It was found that the inhomogeneous shear strain S13 is the key to the potential change in the PS shell structure. The emergence of the potential barriers and wells in the shell structure of PS is well explained. In addition, the position of the potential barriers can be modulated by varying the magnitude of the load, which provides a reference for the design of thin-film devices.

Spectral stability of periodic traveling wave solutions for a double dispersion equation

ABSTRACT In this article, we investigate the spectral stability of periodic traveling waves for a cubic-quintic and double dispersion equation. Using the quadrature method, we find explict periodic waves and we also present a characterization for all positive and periodic solutions for the model using the monotonicity of the period map in terms of the energy levels. The monotonicity of the period map is also useful to obtain the quantity and multiplicity of non-positive eigenvalues for the associated linearized operator and to do so, we use tools of the Floquet theory. Finally, we prove the spectral stability by analyzing the difference between the number of negative eigenvalues of a convenient linear operator restricted to the space constituted by zero-mean periodic functions and the number of negative eigenvalues of the matrix formed by the tangent space associated to the low order conserved quantities of the evolution model.

Localised wavenumber estimation and its application to waveguide discontinuities

Wave propagation methods have been widely investigated for the potential application to the detection and localisation of discontinuities in structural waveguides. In the low-mid frequency ranges, previous preliminary works that used the analytic signal theory have considered local wavenumber estimations based on the time domain for the purpose of localisation in mechanical structures. However, these have not fully exploited the potential of this approach. This is accomplished herein for a one-dimensional infinite-like waveguide by means of two methods applied in the spatial domain; namely the Hilbert transform via the wavenumber domain estimator and the Direct Quadrature method. Modifications are proposed to make these methods more suitable to the space-wavenumber application and for the purpose of localisation. A three-dimensional plot is proposed, where local divergences of the wavenumbers indicate the positions of the discontinuities for otherwise uniform waveguides. Numerical simulations and experiments show an accurate estimation of the local wavenumbers and validate the methods with the proposed modifications for the localisation of different types of discontinuities.

Transversely isotropic effects on the coupled thermo‐hydro‐mechanical performance for layered saturated media

AbstractIn this manuscript, a novel transformed differential quadrature solution to the coupled thermo‐hydro‐mechanical (THM) problem of layered transversely isotropic (TI) saturated media is proposed, accompanied by a sensitivity analysis of pertinent parameters. Initially, the THM governing equations that encompass the transverse isotropy characteristics of thermal, permeable, and mechanical properties are introduced. Subsequently, utilizing the transformed differential quadrature method (TDQM), the discretized algebraic equations in the transformed domain are derived. Boundary conditions related to stresses, displacements, and temperatures are defined leveraging integral transform techniques and the stress‐strain relationship, while algebraic equations for continuity conditions across adjacent layers are incorporated. The global matrix equation is assembled and solved. After verifying the proposed TDQM solution, a sensitivity analysis on the complex transversely isotropic parameters is conducted to identify the principal factors influencing THM coupling behaviors, thereby simplifying the parameters for discussion in practical engineering applications.

A generalized differential quadrature approach to the modelling of heat transfer in non-similar flow with nonlinear convection

This study aims to investigate heat transfer and entropy production in the non-similar flow of an incompressible fluid, with nonlinear convection and viscous dissipation. To obtain precise solutions, the Sparrow-Quack-Boerner local non-similarity method is implemented. The non-similarity arises from the nonlinear convection term present in the momentum equation. The non-similarity arises from the nonlinear convection term present in the momentum equation. Consequently, the inclusion of non-similarity terms into the energy equation is achieved through the interconnection of momentum and energy equations. Entropy generation is explored by employing the second law of thermodynamics. Numerical results from several truncation levels are presented in tabular form, and equations for both first and second-level truncations are generated. The one- and two-equation models are solved by implementing the Generalized Differential Quadrature Method (GDQM). Comparison with a second level of truncation reveals significantly greater inaccuracies in numerical results obtained from the first level. This discrepancy arose due to the omission of non-similar terms in the primary governing equations at the first truncation stage. The validity and accuracy of the derived numerical solutions are further demonstrated by the application of the GDQM and the midpoint method with Richardson extrapolation. Additionally, a plot of the numerical data produced from the second level of truncations is presented together with a discussion of the various physical factors. Increasing the mixed convection parameter accelerates fluid velocity, also when the viscous dissipation parameter increases, temperature and the Bejan number increase. The Prandtl number and the thickness of the thermal boundary layer are observed to be inversely related. Moreover, the quantity of entropy generated at the stretching surface and inside the boundary layer rises in proportion to the increases in the Prandtl and Eckert numbers.

Vibration analysis of a sandwich Timoshenko beam reinforced by GOAM/CNT with various boundary conditions using VIM

In the present study, a sandwich beam reinforced by carbon nanotubes (CNTs)/graphene origami auxetic metamaterials (GOAM) and porous cores is studied. Various distributions of CNTs in facesheets and porosity in the core are stated in terms of thickness. The equations of motion for a sandwich beam are derived and solved under various boundary conditions using variational iteration method (VIM), in which, it is a powerful technique with higher convergence with the computational method that gives analytical response to a linear problem. The effect of the types of CNT distributions, porosity coefficient, types of porous cores, porosity parameter, weight fraction and folding degree of GOAM, different geometric dimensions and different B.C.’s on the natural frequency is investigated. The obtained results are compared with the other literature, that there is a good agreement between them. On the other hands, the maximum error percentage of the present work (VIM) with Civalek and Kiracioglu [66] (Discrete singular convolution method) for the first five frequencies is equal to 0.3% according to Table 2, while in Table 3, the maximum error percentage of the present work (VIM) with Yas and Samadi [67] (differential quadrature method) for the first frequency with various boundary conditions and nanocomposite material is equal to 0.22%. In this regard, the present study can be a suitable platform to find the natural frequency and prevent the phenomenon of resonance and increase the life of the structure. Moreover, it is seen that with enhancing of weight fraction of GOAM, frequency increases because increasing the stiffness of structures. The importance of this research is to extract the formulation of sandwich beam structure for GOAM under different boundary conditions and compare the results between them by repeating the changes.

Temporal High-Order Accurate Numerical Scheme for the Landau–Lifshitz–Gilbert Equation

In this paper, a family of temporal high-order accurate numerical schemes for the Landau–Lifshitz–Gilbert (LLG) equation is proposed. The proposed schemes are developed utilizing the Gauss–Legendre quadrature method, enabling them to achieve arbitrary high-order time discretization. Furthermore, the geometrical properties of the LLG equation, such as the preservation of constant magnetization magnitude and the Lyapunov structure, are investigated based on the proposed discrete schemes. It is demonstrated that the magnetization magnitude remains constant with an error of (2p+3) order in time when utilizing a (2p+2)th-order discrete scheme. Additionally, the preservation of the Lyapunov structure is achieved with a second-order error in the temporal step size. Numerical experiments and simulations effectively verify the performance of our proposed algorithm and validate our theoretical analysis.