The dual reciprocity method applied to free vibrations of 2D structures using compact supported radial basis functions

This paper presents a new boundary element application for free vibration analysis of 2D elastic structures. The dual reciprocity method is applied using four compact supported radial basis functions for approximating the domain inertia terms. The eigen-problem of displacement is then solved considering the traction contribution by means of static condensation. The formulation is also extended to consider additional internal nodes to improve accuracy. Three numerical problems are studied to demonstrate the validity and accuracy of the developed formulation. The results are compared to those obtained from analytical and other numerical solutions. A parametric study is set up to demonstrate the effect of the compact support radius on the final results and on the sparsity of system matrices.     

SOLVING NONLINEAR DIFFERENTIAL EQUATIONS OF MECHANICS WITH THE BOUNDARY ELEMENT METHOD AND RADIAL BASIS FUNCTIONS

The Boundary Element Method is a very effective method for solving linear differential equations. To use it also in the consideration of non-linear problems some different procedures were developed, among them the dual reciprocity method and the particular integral method. Both procedures use interpolation conditions for the approximation with radial basis functions. In this paper a method is presented which avoids problems connected with interpolation by means of quasi-interpolation. It is possible to solve differential equations of the kind Δ^mu=p(u) approximately; the application to two non-linear problems of plate theory yield good results. Hints to a theoretical examination of the method including sufficient conditions for feasibility and convergence are given. © 1997 by John Wiley & Sons, Ltd.     

基于径向基函数的自适应网格方法

本文给出了一种基于径向基函数的自适应网格方法．该方法利用网格依赖方法的解与径向基函数插值解的信息来细化或粗化网格,充分利用了径向基函数计算格式简单、节点配置灵活的优点与网格依赖方法的稳健性．提出的算法很容易编程实现．数值算例表明该算法可以在解变化剧烈的区域加密网格,在解变化平缓的地方粗化网格,从而在保证相同数值求解精度的情况下,能够极大地节省计算量．     

A local RBF collocation method for band structure computations of 2D solid/fluid and fluid/solid phononic crystals

In this paper, an efficient local radial basis function collocation method (LRBFCM) is presented for computing the band structures of the two‐dimensional (2D) solid/fluid and fluid/solid phononic crystals. Both systems of solid scatterers embedded in a fluid matrix (solid/fluid phononic crystals) and fluid scatterers embedded in a solid matrix (fluid/solid phononic crystals) are investigated. The solid–fluid interactions are taken into account by properly formulating and treating the continuity/equilibrium conditions on the solid–fluid interfaces, which require an accurate computation of the normal derivatives of the displacements and the pressure on the fluid–solid interfaces and the unit‐cell boundaries. The developed LRBFCM for the mixed wave propagation problems in 2D solid/fluid and fluid/solid phononic crystals is validated by the corresponding results obtained by the finite element method (FEM). To the best knowledge of the authors, the present LRBFCM has yet not been applied to the band structure computations of 2D solid/fluid and fluid/solid phononic crystals. For different lattice forms, scatterers' shapes, acoustic impedance ratios, and material combinations (solid scatterers in fluid matrix or fluid scatterers in solid matrix), numerical results are presented and discussed to reveal the efficiency and the accuracy of the developed LRBFCM for calculating the band structures of 2D solid/fluid and fluid/solid phononic crystals. A comparison of the present numerical results with that of the FEM shows that the present LRBFCM is much more efficient than the FEM for the band structure computations of the considered 2D solid/fluid and fluid/solid phononic crystals. Copyright © 2016 John Wiley & Sons, Ltd.     

Two-dimensional stress analysis of functionally graded solids using the MLPG method with radial basis functions

The meshless local Petrov–Galerkin (MLPG) method is used for analysing two-dimensional (2D) static and dynamic deformations of functionally graded materials (FGMs) with material response modelled as either linear elastic or as linear viscoelastic. The multiquadric radial basis function (RBF) is employed to approximate the trial solution. Results are computed with two different choices of test functions, namely a fourth-order spline weight function, and a Heaviside step function, each having a compact support. No background mesh is used to numerically evaluate integrals appearing in the weak formulation of the problem, thus the method is truly meshless. A benefit of using RBFs is that they possess the Kronecker delta property; thus it is easy to satisfy essential boundary conditions. For five problems, the computed results are found to match well with those either from their analytical solutions or numerical solutions of other researchers who employed different algorithms. For a dynamic problem, the Laplace-transform technique is utilised. The numerical examples illustrate that displacements and stress distributions in a structure made of an FGM differ considerably from those at the corresponding points in the same structure made of a homogeneous material. Thus, the inhomogeneity in material properties can be exploited to optimise stress distribution, minimise deflection and reduce the maximum stress.     

A Galerkin-reproducing kernel method: Application to the 2D nonlinear coupled Burgers' equations

The paper introduces a Galerkin method in the reproducing kernel Hilbert space. It is implemented as a meshless method based on spatial trial spaces spanned by the Newton basis functions in the “native” Hilbert space of the reproducing kernel. For the time-dependent PDEs it leads to a system of ordinary differential equations. The method is used for solving the 2D nonlinear coupled Burgers' equations having Dirichlet and mixed boundary conditions. The numerical solutions for different values of Reynolds number (Re) are compared with analytical solutions as well as other numerical methods. It is shown that the proposed method is efficient, accurate and stable for flow with reasonably high Re in the case of Dirichlet boundary conditions.