The global approximate particular solution meshless method for two-dimensional linear elasticity problems

The two-dimensional linear elasticity equations are solved by the global method of approximate particular solution as a new meshless option to the conventional finite element discretization. The displacement components are approximated by a linear combination of the elasticity particular solutions and the stress tensor is obtained by differentiating the displacement expressions in terms of the particular solutions. The multiquadric radial basis function (RBF) is employed as the non-homogeneous term in the governing equation to compute the particular solutions. The cantilever beam and the infinite plate with a hole problem are solved to verify the implemented meshless method. For each situation, the trend of the root mean square error is assessed in terms of the shape parameter and the number of nodes. Unlike most of the RBF collocation strategies, it is found that numerical results are in good agreement with the analytical solutions for a wide range of shape parameter values.     

Smoothing large data sets using discrete thin plate splines

Traditional thin plate splines use radial basis functions and require the solution of a dense linear system of equations whose size is proportional to the number of data points. Instead of radial basis functions we present a method based on the use of polynomials with local support defined on finite element grids. This method is more efficient when dealing with large data sets as the resulting system of equations is sparse and its size depends only on the number of nodes in the finite element grid. Theory is developed for general d-dimensional data sets and model problems are presented in 3D to study the convergence behaviour.     

Nonlinear transient response of stiffened plates to air blast loading by a superelement approach

In this paper, new plate and stiffener beam elements are developed for the nonlinear dynamic analysis of stiffened plate structures subject to blast-type pressure waves. The displacement fields for the elements are represented by polynomial and analytical functions in both in-plane directions and they have been constructed so that only one element per bay or span is required to model the response. Geometric and material nonlinearities are included and the temporal equations are solved by the implicit Newmark beta method with Newton-Raphson sub-iteration. The new formulation has been tested on several numerical examples and the results obtained are compared with other available solutions. The present model is simple, requires much reduced storage and computing time and yet gives results suitable for practical design purposes.     

An Efficient Double Legendre Spectral Method for Parabolic and Elliptic Partial Differential Equations

We present a double Legendre spectral methods that allow the efficient approximate solution for the parabolic partial differential equations in a square subject to the most general inhomo-geneous mixed boundary conditions. The differential equations with their boundary and initial conditions are reduced to systems of ordinary differential equations for the time-dependent expansion coefficients. These systems are greatly simplified by using tensor matrix algebra, and are solved by using the step-by-step method. One numerical application of how to use these methods is described. Numerical results obtained compare favorably with those of the analytical solution. Accurate double Legendre spectral approximations for Poisson' and Helmholtz' equations are also noted.     

Buckling of thin-walled members by finite elements

We present a general two-dimensional thin plate/shell theory for the study of the elastic stability of sections formed by an assemblage of flat elements. These types of sections are generally analyzed by a one-dimensional model, but when, for instance, the width of the flanges is large or when there are openings in the section, the one-dimensional model cannot correctly predict the buckling load. We have developed a new finite element called DKM for the plate/shell model. We compare the result obtained with the present model with those of known analytical theories. The new model proves to be efficient and reliable. We also demonstrate that for wide plates, the analytical solution based on the one-dimensional model gives results slightly different from those of the proposed model.     

Asymmetric thermal buckling of temperature dependent annular FGM plates on a partial elastic foundation

In this investigation, the asymmetrical buckling behaviour of FGM annular plates resting on partial Winkler-type elastic foundation under uniform temperature elevation is investigated. Material properties of the plate are assumed to be temperature dependent. Each property of the plate is graded across the thickness direction using a power law function. First order shear deformation plate theory and von Kármán type of geometrical nonlinearity are used to obtain the equilibrium equations and the associated boundary conditions. Prebuckling deformations and stresses of the plate are obtained considering the deflection-less conditions. Only plates which are clamped on both inner and outer edges are considered. Applying the adjacent equilibrium criterion, the linearised stability equations are obtained. The governing equations are divided into two sets. The first set, which is associated with the in-contact region and the second set which is related to contact-less region. The resulting equations are solved using a hybrid method, including the analytical trigonometric functions through the circumferential direction and generalised differential quadratures method through the radial direction. The resulting system of eigenvalue problem is solved iteratively to obtain the critical conditions of the plate, the associated circumferential mode number and buckled shape of the plate. Benchmark results are given in tabular and graphical presentations dealing with critical buckling temperature and buckled shape of the plate. Numerical results are given to explore the effects of elastic foundation, foundation radius, plate thickness, plate hole size, and power law index of the graded plate. It is shown that, stiffness foundation, and radius of foundation may change the buckled shape of the plate in both circumferential and radial directions. Furthermore, as the stiffness of the foundation or radius of foundation increases, critical buckling temperature of the plate enhances.     

Transient analysis of composite plates by radial basis functions in a pseudospectral framework

This paper presents a study of the linear transient response of composite plates using radial basis functions and collocation method in a pseudospectral framework. The first-order shear deformation plate theory is used to define a set of algebraic equations from the equations of motion and boundary conditions. The transient analysis is performed by a Newmark algorithm. In order to assess the quality of the present numerical method, an analytical solution was also developed. Numerical tests on square and rectangular cross-ply laminated plates demonstrate that the present method produces highly accurate displacements and stresses when compared with the available results.     

The numerical solution of fractional differential equations using the Volterra integral equation method based on thin plate splines

Finding the approximate solution of differential equations, including non-integer order derivatives, is one of the most important problems in numerical fractional calculus. The main idea of the current paper is to obtain a numerical scheme for solving fractional differential equations of the second order. To handle the method, we first convert these types of differential equations to linear fractional Volterra integral equations of the second kind. Afterward, the solutions of the mentioned Volterra integral equations are estimated using the discrete collocation method together with thin plate splines as a type of free-shape parameter radial basis functions. Since the scheme does not need any background meshes, it can be recognized as a meshless method. The proposed approach has a simple and computationally attractive algorithm. Error analysis is also studied for the presented method. Finally, the reliability and efficiency of the new technique are tested over several fractional differential equations and obtained results confirm the theoretical error estimates.

     

A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions

The nonlinear sine-Gordon equation arises in various problems in science and engineering. In this paper, we propose a numerical scheme to solve the two-dimensional damped/undamped sine-Gordon equation. The proposed scheme is based on using collocation points and approximating the solution employing the thin plate splines (TPS) radial basis function (RBF). The new scheme works in a similar fashion as finite difference methods. Numerical results are obtained for various cases involving line and ring solitons.     

基于面积坐标和解析试函数的薄板元

为体现离散法与解析法的互补和渗透,构造基于第二类四边形面积坐标的广义协调薄板元AATF-BQ4;根据薄板理论的控制方程,采用Kirchhoff直法线假设求解基本解析解,并作为试函数构造单元AATF-BQ4.数值算例表明,单元AATF-BQ4具有较高的精度和较好的稳定性,适用于实际工程计算应用.     

The numerical solution of nonlinear generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations via the meshless method of integrated radial basis functions

In this paper, a numerical technique is proposed for solving the nonlinear generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations. The used numerical method is based on the integrated radial basis functions (IRBFs). First, the time derivative has been approximated using a finite difference scheme. Then, the IRBF method is developed to approximate the spatial derivatives. The two-dimensional version of these equations is solved using the presented method on different computational geometries such as the rectangular, triangular, circular and butterfly domains and also other irregular regions. The aim of this paper is to show that the integrated radial basis function method is also suitable for solving nonlinear partial differential equations. Numerical examples confirm the efficiency of the proposed scheme.

     

Dispersion free wave splittings for structural elements

Wave splittings are derived for three types of structural elements: membranes, Timoshenko beams, and Mindlin plates. The Timoshenko beam equation and the Mindlin plate equation are inherently dispersive, as is each Fourier component of the membrane equation in an angular decomposition of the field. The distinctive feature of the wave splittings derived in the present paper is that, in homogeneous regions, they transform the dispersive wave equations into simple one-way wave equations without dispersion. Such splittings have uses both for radial scattering problems in the 2D cases and for scattering problems in dispersive media. As an example of how the splittings may be applied, a direct scattering problem is solved for a membrane with radially varying density. The imbedding method is utilized, and agreement is obtained with an FE simulation.     

Application of Bernoulli matrix method for solving two-dimensional hyperbolic telegraph equations with Dirichlet boundary conditions

The present article is devoted to develop a new approach and methodology to find the approximate solution of second order two-dimensional telegraph equations with the Dirichlet boundary conditions. We first transform the telegraph equations into equivalent partial integro-differential equations (PIDEs) which contain both initial and boundary conditions and therefore can be solved numerically in a more appropriate manner. Operational matrices of integration and differentiation of Bernoulli polynomials together with the completeness of these polynomials are used to reduce the PIDEs into the associated algebraic generalized Sylvester equations which can be solved by an efficient Krylov subspace iterative (i.e., BICGSTAB) method. The efficiency of the proposed method has been confirmed with several test examples and it is clear that the results are acceptable and found to be in good agreement with exact solutions. We have compared the numerical results of the proposed method with radial basis function method and differential quadrature method. Also, the method is simple, efficient and produces very accurate numerical results in considerably small number of basis functions and hence reduces computational effort. Moreover, the technique is easy to apply for multidimensional problems.     

Mesh deformation based on radial basis function interpolation

A new mesh movement algorithm for unstructured grids is developed which is based on interpolating displacements of the boundary nodes to the whole mesh with radial basis functions (RBF’s). A small system of equations, only involving the boundary nodes, has to be solved and no grid-connectivity information is needed. The method can handle large mesh deformations caused by translations, rotations and deformations, both for 2D and 3D meshes. However, the performance depends on the used RBF. The best accuracy and robustness with the highest efficiency are obtained with a C² continuous RBF with compact support, closely followed by the thin plate spline. The deformed meshes are suitable for flow computations as is shown by performing calculations around a NACA-0012 airfoil.     

Minimal realisations of odd transfer functions for first-degree nD systems

ABSTRACT

Minimal realisation problems of odd transfer functions for first-degree (multi-linear) nD single-input single-output discrete systems have been studied, but it has not been well solved. This paper provides a new, different method to solve absolutely minimal realisation problems. By methods of limits and algebraic techniques, without using the symbolic approach by Gröbner basis, the requirements of absolutely minimal realisation are transformed into a system of equations represented by the determinants. Since the equations for first-degree 2D systems are solvable by quadratic equations and the conditions for higher-dimensional realisations can be expressed by the results of 2D systems, the absolutely minimal realisations for nD systems can be found by using the realisations of n(n ? 1)/2 2D systems. Furthermore, the conditions for existence and construction of the absolutely minimal realisation for the lack of items and not missing two cases are derived from the Pfaffian function of the skew-symmetric matrix. Finally, two numerical examples for 3D and 4D systems are presented to illustrate the basic ideas as well as the effectiveness of the proposed procedure.     

Analysis of axially temperature-dependent functionally graded carbon nanotube reinforced composite plates

This article presents a comprehensive analysis to investigate the static buckling stability and static deflection of axially single-walled (SW) functionally graded (FG) carbon nanotubes reinforced composite (CNTRC) plates with temperature-dependent material properties and graded by different functions, for the first time. The distribution of the carbon nanotubes is described by two functions, one for the x-direction CNTs distribution (CNTRC-x plate) and another for z-direction CNTs distribution (CNTRC-z plate). The graduation functions of CNTs are unidirectional (UD CNTRC), FG-X CNTRC, FG-O CNTRC, and FG-V CNTRC. The extended rule of mixture and the molecular dynamics simulations are exploited to evaluate the equivalent mechanical properties of FG-CNTRC plate. Equilibrium equations are formulated using principal of Hamilton and solved analytically using Galerkin method to cover various boundary conditions. New higher order shear deformation theory is proposed. The numerical results gained by the proposed solution are verified by comparing with those of published ones. Numerical results present influences of gradation function, inhomogeneity parameters, aspect ratio, thickness ratio, boundary conditions and temperature on the static buckling and deflection of FG-CNTRC plate using modified higher order shear theories.

     

Application of thin plate splines for solving a class of boundary integral equations arisen from Laplace's equations with nonlinear boundary conditions

This article describes a technique for numerically solving a class of nonlinear boundary integral equations of the second kind with logarithmic singular kernels. These types of integral equations occur as a reformulation of boundary value problems of Laplace's equations with nonlinear Robin boundary conditions. The method uses thin plate splines (TPSs) constructed on scattered points as a basis in the discrete collocation method. The TPSs can be seen as a type of the free shape parameter radial basis functions which establish effective and stable methods to estimate an unknown function. The proposed scheme utilizes a special accurate quadrature formula based on the non-uniform Gauss–Legendre integration rule for approximating logarithm-like singular integrals appeared in the approach. The numerical method developed in the current paper does not require any mesh generations, so it is meshless and independent of the geometry of the domain. The algorithm of the presented scheme is accurate and easy to implement on computers. The error analysis of the method is provided. The convergence validity of the new technique is examined over several boundary integral equations and obtained results confirm the theoretical error estimates.     

A novel radial basis function method for 3D linear and nonlinear advection diffusion reaction equations with variable coefficients

The radial basis function method for 3D advection diffusion reaction equations with variable coefficients is presented. The proposed method implements the linear combination of radial basis functions which impose boundary conditions in advance, and thus such a combination with weighted parameters can be used to construct the final approximation. Furthermore, the weighted parameters are solved by substituting the approximation into governing equations. This method leads to crucial improvements in the feasibility and accuracy which can now be easily applied to general 3D nonlinear problems through linearized techniques. Finally, accuracy and efficiency of the proposed method are verified by several examples.

     

MELM-GRBF: A modified version of the extreme learning machine for generalized radial basis function neural networks

In this paper, we propose a methodology for training a new model of artificial neural network called the generalized radial basis function (GRBF) neural network. This model is based on generalized Gaussian distribution, which parametrizes the Gaussian distribution by adding a new parameter τ. The generalized radial basis function allows different radial basis functions to be represented by updating the new parameter τ. For example, when GRBF takes a value of τ=2, it represents the standard Gaussian radial basis function. The model parameters are optimized through a modified version of the extreme learning machine (ELM) algorithm. In the methodology proposed (MELM-GRBF), the centers of each GRBF were taken randomly from the patterns of the training set and the radius and τ values were determined analytically, taking into account that the model must fulfil two constraints: locality and coverage. An thorough experimental study is presented to test its overall performance. Fifteen datasets were considered, including binary and multi-class problems, all of them taken from the UCI repository. The MELM-GRBF was compared to ELM with sigmoidal, hard-limit, triangular basis and radial basis functions in the hidden layer and to the ELM-RBF methodology proposed by Huang et al. (2004) [1]. The MELM-GRBF obtained better results in accuracy than the corresponding sigmoidal, hard-limit, triangular basis and radial basis functions for almost all datasets, producing the highest mean accuracy rank when compared with these other basis functions for all datasets.     

Coupled electrostatic and mechanical FEA of a micromotor

The electrostatic forces occurring in a novel double stator axial-drive variable capacitance micromotor (VCM) are studied as a function of rotor-stator overlap, applied voltage, rotor support morphology, and rotor thickness. Analytical equations are developed using parallel plate assumptions, and results are compared with those obtained with 3D Finite Element Analysis (FEA) for tangential, axial, and radial electrostatic forces. The influence of the axial forces on the rotor deflections are studied using iterative indirect coupled field analysis, where the axial forces obtained from the electrostatic 3D FE model are iteratively applied to a structural FE model until stable rotor deflections are obtained. It was found that the axial forces, taking the rotor deflection into account, are twice as high as those obtained by analytical evaluation neglecting rotor deflections and about 70 times higher than the radial forces at a typical operating voltage of 100 V. Inclusion of bushing supports results in lower axial forces and decreases the influence of rotor tilt. Tangential forces likely to be exerted on the rotor at start-up are also examined and compared with analytical predictions. The study demonstrates that FEA provides more accurate results than analytical equations due to the geometry and field simplifications assumed in the latter