Local radial basis function–finite-difference method to simulate some models in the nonlinear wave phenomena: regularized long-wave and extended Fisher–Kolmogorov equations

In this investigation, we concentrate on solving the regularized long-wave (RLW) and extended Fisher–Kolmogorov (EFK) equations in one-, two-, and three-dimensional cases by a local meshless method called radial basis function (RBF)–finite-difference (FD) method. This method at each stencil approximates differential operators such as finite-difference method. In each stencil, it is necessary to solve a small-sized linear system with conditionally positive definite coefficient matrix. This method is relatively efficient and has low computational cost. How to choose the shape parameter is a fundamental subject in this method, since it has a palpable effect on coefficient matrix. We will employ the optimal shape parameter which results from algorithm of Sarra (Appl Math Comput 218:9853–9865, 2012). Then, we compare the approximate solutions acquired by RBF–FD method with theoretical solution and also with results obtained from other methods. The numerical results show that the RBF–FD method is suitable and robust for solving the RLW and EFK equations.

     

Explicit 3-D RKPM shape functions in terms of kernel function moments for accelerated computation

The construction of meshless shape functions is more time-consuming than evaluation of FEM shape functions. Therefore, it is of great importance to take measures to speed up the computation of meshless shape functions. 3-D meshless shape functions and their derivatives are, in the context of reproducing kernel particle method (RKPM), expressed explicitly in terms of kernel function moments for the very first time. This avoids solutions of linear algebraic equations and numerical inversions encountered in standard RKPM implementation, thus speeds up computation of meshless shape functions. A numerical test is performed in a hexahedral domain with the mere purpose of comparing the computation time for shape functions construction between the standard RKPM implementation and the enhanced procedure. Then two 3-D elastostatics numerical examples are presented, which demonstrate that the proposed unique treatment of RKPM shape functions is especially effective.     

On the optimal shape parameter for Gaussian radial basis function finite difference approximation of the Poisson equation

We investigate the influence of the shape parameter in the meshless Gaussian radial basis function finite difference (RBF-FD) method with irregular centres on the quality of the approximation of the Dirichlet problem for the Poisson equation with smooth solution. Numerical experiments show that the optimal shape parameter strongly depends on the problem, but insignificantly on the density of the centres. Therefore, we suggest a multilevel algorithm that effectively finds a near-optimal shape parameter, which helps to significantly reduce the error. Comparison to the finite element method and to the generalised finite differences obtained in the flat limits of the Gaussian RBF is provided.     

A MLPG(LBIE) approach in combination with BEM

The local boundary integral equation (LBIE) approach is a promising meshless method, recently proposed as an effective alternative to the boundary element method (BEM), for solving non-homogeneous, anisotropic and non-linear problems. Since the LBIE method utilizes in its weak form fundamental solutions as test functions, it can be considered as one of the six meshless local Petrov–Galerkin (MLPG) methods proposed by Atluri and coworkers. This explains the use of the initials MLPG(LBIE) in the title of the present paper. This work addresses a coupling of a new MLPG(LBIE) method, recently proposed by the authors for elastodynamic problems, and the BEM. Because both methods conclude to a final system of linear equations expressed in terms of nodal displacement and tractions, their combination is accomplished directly with no further transformations as it happens in other MLPG/BEM formulations as well as in typical hybrid finite element method/BEM schemes. The coupling approach is demonstrated for static and frequency domain elastodynamic problems. Three representative examples are provided in order to illustrate the achieved accuracy of the proposed here MLPG(LBIE)/BE methodology.     

Numerical shape optimization based on meshless method and stochastic optimization technique

This paper puts forward a newer approach for structural shape optimization by combining a meshless method (MM), i.e. element-free Galerkin (EFG) method, with swarm intelligence (SI)-based stochastic ‘zero-order’ search technique, i.e. artificial bee colony (ABC), for 2D linear elastic problems. The proposed combination is extremely beneficial in structural shape optimization because MM, when used for structural analysis in shape optimization, eliminates inherent issues of well-known grid-based numerical techniques (i.e. FEM) such as mesh distortion and subsequent remeshing while handling large shape changes, poor accuracy due to discontinuous secondary field variables across element boundaries needing costly post-processing techniques and grid optimization to minimize computational errors. Population-based stochastic optimization technique such as ABC eliminates computational burden, complexity and errors associated with design sensitivity analysis. For design boundary representation, Akima spline interpolation has been used in the present work owing to its enhanced stability and smoothness over cubic spline. The effectiveness, validity and performance of the proposed technique are established through numerical examples of cantilever beam and fillet geometry in 2D linear elasticity for shape optimization with behavior constraints on displacement and von Mises stress. For both these problems, influence of a number of design variables in shape optimization has also been investigated.     

Application of radial basis functions to linear and nonlinear structural analysis problems

The basic characteristic of the techniques generally known as meshless methods is the attempt to reduce or even to eliminate the need for a discretization (at least, not in the way normally associated with traditional finite element techniques) in the context of numerical solutions for boundary and/or initial value problems.The interest in meshless methods is relatively new and this is why, despite the existence of various applications of meshless techniques to several problems of mechanics (as well as to other fields), these techniques are still relatively unknown to engineers. Furthermore, and compared to traditional finite elements, it may be difficult to understand the physical meaning of the variables involved in the formulations.As an attempt to clarify some aspects of the meshless techniques, and simultaneously to highlight the ease of use and the ease of implementation of the algorithms, applications are made, in this work, to structural analysis problems.The technique used here consists of the definition of a global approximation for a given variable of interest (in this case, components of the displacement field) by means of a superposition of a set of conveniently placed (in the domain and on the boundary) radial basis functions (RBFs).In this work various types of one-dimensional problems are analyzed, ranging from the static linear elastic case, free vibration and linear stability analysis (for a beam on elastic foundation), to physically nonlinear (damage models) problems. To further complement the range of problems analysed, the static analysis of a plate on elastic foundation was also addressed. Several error measures are used to numerically establish the performance of both symmetric and nonsymmetric approaches for several global RBFs. The results obtained show that RBF collocation leads to good approximations and very high convergence rates.     

Radial basis functions method for parabolic inverse problem

The radial basis functions (RBFs) method is employed to handle a class of multi-dimensional parabolic inverse problems. Because they are not modelled by classical parabolic initial-boundary value problems, theoretical behaviour and numerical approximation of these problems have been active areas of research. Based on the idea of RBF approximation, a fast and highly accurate meshless method is developed for solving an inverse problem with a control parameter. Moreover, with the meshless property, it can be used to handle multi-dimensional parabolic inverse problems defined on very complicated geometries.     

Adaptive WENO Methods Based on Radial Basis Function Reconstruction

We explore the use of radial basis functions (RBF) in the weighted essentially non-oscillatory (WENO) reconstruction process used to solve hyperbolic conservation laws, resulting in a numerical method of arbitrarily high order to solve problems with discontinuous solutions. Thanks to the mesh-less property of the RBFs, the method is suitable for non-uniform grids and mesh adaptation. We focus on multiquadric radial basis functions and propose a simple strategy to choose the shape parameter to control the balance between achievable accuracy and the numerical stability. We also develop an original smoothness indicator which is independent of the RBF for the WENO reconstruction step. Moreover, we introduce type I and type II RBF-WENO methods by computing specific linear weights. The RBF-WENO method is used to solve linear and nonlinear problems for both scalar and systems of conservation laws, including Burgers equation, the Buckley–Leverett equation, and the Euler equations. Numerical results confirm the performance of the proposed method. We finally consider an effective conservative adaptive algorithm that captures moving shocks and rapidly varying solutions well. Numerical results on moving grids are presented for both Burgers equation and the more complex Euler equations.     

Boundary point method for linear elasticity using constant and quadratic moving elements

Based on the boundary integral equations and stimulated by the work of Young et al. [J Comput Phys 2005;209:290–321], the boundary point method (BPM) is a newly developed boundary-type meshless method enjoying the favorable features of both the method of fundamental solution (MFS) and the boundary element method (BEM). The present paper extends the BPM to the numerical analysis of linear elasticity. In addition to the constant moving elements, the quadratic moving elements are introduced to improve the accuracy of the stresses near the boundaries in the post processing and to enhance the analysis for thin-wall structures. Numerical tests of the BPM are carried out by benchmark examples in the two- and three-dimensional elasticity. Good agreement is observed between the numerical and the exact solutions.     

A non-uniform Haar wavelet method for numerically solving two-dimensional convection-dominated equations and two-dimensional near singular elliptic equations

A non-uniform Haar wavelet based collocation method has been developed in this paper for two-dimensional convection dominated equations and two-dimensional near singular elliptic partial differential equations, in which traditional Haar wavelet method produces oscillatory solutions or low accurate solutions. The main idea behind the proposed method is to transform the computation of numerical solution of considered partial differential equations to computation of solution of a linear system of equations. This process is done by discretizing space variables with non-uniform Haar wavelets. To confirm efficiency of the proposed method seven benchmark problems are solved and the obtained results are compared with exact solutions and with local meshless methods, finite element method, finite difference method and polynomial collocation method. Numerical experiments show that the proposed method gives convincing results even in less number of collocation nodes.     

Finite element analysis of the in-plane behaviour of annular disks

In-plane analysis of annular disks using the finite element method is presented. A semi-analytical, one-dimensional finite element model is developed using a Fourier series approach to account for the circumferential behaviour. Using displacement functions which are exact solutions of the two dimensional elasticity plane stress problem, the shape functions, stiffness matrices and mass matrices corresponding to the 0th, 1st and nth harmonics are derived. To show the utility of this new element, example probelms have been solved and compared with the exact solution. The present element can be readily coded into any general purpose finite element program.     

A Numerically Efficient Dissipation-Preserving Implicit Method for a Nonlinear Multidimensional Fractional Wave Equation

The partition of unity (PU) method, performed with local radial basis function (RBF) approximants, has been proved to be an effective tool for solving large scattered data interpolation problems. However, in order to achieve a good accuracy, the question about how many points we have to consider on each local subdomain, i.e. how large can be the local data sets, needs to be answered. Moreover, it is well-known that also the shape parameter affects the accuracy of the local RBF approximants and, as a consequence, of the PU interpolant. Thus here, both the shape parameter used to fit the local problems and the size of the associated linear systems are supposed to vary among the subdomains. They are selected by minimizing an a priori error estimate. As evident from extensive numerical experiments and applications provided in the paper, the proposed method turns out to be extremely accurate also when data with non-homogeneous density are considered.     

基于径向基神经网络的语音识别技术

深入分析研究了径向基神经网络的优缺点,并对其进行了改进,分析讨论了语音识别研究中,径向基神经网络的设计原则以及特征参数等对语音识别结果的影响。将其应用于数字语音识别中,实验结果表明,基于改进型的径向基神经网络的语音识别方法有着较好的识别性能和应用效果。针对非特定人的孤立词识别,识别率可以达到90%以上。     

Modeling nonlinear waves in a numerical wave tank with localized meshless RBF method

This paper presents the numerical simulation of free surface waves in a 2D domain which represents a wave tank, using a localized approach of the meshless radial basis function (RBF) method. Instead of global collocation, the local approach breaks down the problem domain into subdomains, leading to a sparse global system matrix which is particularly advantageous in tackling the time consuming simulation process. Mixed Eulerian–Lagrangian approach is adopted for free surface tracking and fourth order Adams–Bashforth–Moulton scheme (ABM4) is used for time stepping. Both linear and nonlinear Stokes waves are simulated and compared to analytical solutions.     

The boundary element-free method for 2D interior and exterior Helmholtz problems

The boundary element-free method (BEFM) is developed in this paper for numerical solutions of 2D interior and exterior Helmholtz problems with mixed boundary conditions of Dirichlet and Neumann types. A unified boundary integral equation is established for both interior and exterior problems. By using the improved interpolating moving least squares method to form meshless shape functions, mixed boundary conditions in the BEFM can be satisfied directly and easily. Detailed computational formulas are derived to compute weakly and strongly singular integrals over linear and higher order integration cells. Three numerical integration procedures are developed for the computation of strongly singular integrals. Numerical examples involving acoustic scattering and radiation problems are presented to show the accuracy and efficiency of the meshless method.     

The meshless approach for solving 2D variable-order time-fractional advection–diffusion equation arising in anomalous transport

In this paper, with the aim of extending an elegant and straightforward numerical approximation to describe one of the most common physical phenomena has been undertaken. In this regard, the generalization of advection–diffusion equation, namely, the time-fractional advection–diffusion equation with understanding sense variable-order fractional derivative, is taken into consideration. An efficient and accurate approach is relying on the Kansa scheme and finite difference method to provide a mathematical framework to treat the spatial discretization and temporal term, respectively. The meshless collocation approach is utilized for interior scattered points and those on the boundary. Thus, the problem under consideration is reduced to a system of linear algebraic equations. The use of the radial basis function as shape function brings many advantages for proposal numerical method in terms of improved accuracy by setting an appropriate shape parameter and applied for solving high-dimensional models without extra cost. The validity and accuracy of the proposed approach is investigated by four various examples involving three benchmark examples and a practical application of pollution transfer phenomena.

     

An efficient algorithm for finite-difference modeling of mixed-boundary-value elastic problems

The paper describes an efficient numerical scheme for the solution of displacements and stresses in mixed-boundary-value elastic problems of solid mechanics. A new variable reduction scheme is used to develop the computational model. Solution of both two- and three-dimensional problems of linear elasticity is considered in the present paper. In the present approach, the problems are formulated in terms of a potential function, defined in terms of the displacement components. Compared to the conventional computational approaches, the present scheme is capable of providing numerical solution of higher accuracy with less computational effort. Application of the present finite-difference modeling scheme is demonstrated through the solutions of a number of practical stress problems of interest, and the results are compared with those obtained by the standard method of solution. The comparison of the results establishes the rationality as well as suitability of the present variable reduction scheme.     

Three-Dimensional Integral Mathematical Models of the Dynamics of Thick Elastic Plates

The complex of problems related to constructing three-dimensional field of elastic dynamic displacements of flat elastic plate with arbitrary boundary-edge surface is solved. It is assumed that boundary condition of the plate is given in terms of powerful perturbation factors or displacement vector function. Problems solutions are based on classical Lame equations of spatial theory of elasticity under root-mean-square consistency of the solution with corresponding external-dynamic observations of the plate. The accuracy of such consistency is estimated. The uniqueness conditions for the solution of the considered problems are formulated.     

A priori error estimates for explicit finite element for linear elasto-dynamics by Galerkin method and central difference method

The explicit finite element method for transient dynamics of linear elasticity is formulated by using Galerkin method for space and the central difference method for time. An a priori error estimate is derived and the optimal rate of convergence for displacement similar to the linear elliptic problem is found. The error estimation is extended to velocity, internal (strain) energy and kinetic energy for engineering applications. The approximation error of initial data is analyzed. The error estimate is refined for a class of engineering applications with zero initial deformation and initial force. Examples of a 1-D rod axial vibration and a 2-D plate in-plane vibration are solved using linear elements as verification.     

基于Matlab的径向基函数数值计算软件包

为推动基于无网格方法的计算软件的发展,介绍基于Matlab自主开发的径向基函数（Radial Basis Function,RBF）数值计算软件包,阐述软件的理论基础、设计思路以及该软件包的功能和特点,并结合边界节点法（Boundary Knot Method,BKM）的数值实例给出软件的使用过程．该软件包可以根据不同的数学物理模型选择合适的数值算法来求解多种实际物理问题,也可对不同数值算法得到的结果进行比较．最后,总结应用Matlab进行数值计算软件开发的优缺点．