Free vibration analysis of moderately thick functionally graded plates by local Kriging meshless method

This paper mainly Presents free vibration analyses of metal and ceramic functionally graded plates with the local Kriging meshless method. The Kriging technique is employed to construct shape functions which possess Kronecker delta function property and thus make it easy to implement essential boundary conditions. The eigenvalue equations of free vibration problems are based on the first-order shear deformation theory and the local Petrov–Galerkin formulation. The cubic spline function is used as the weight function which vanishes on internal boundaries of local quadrature domains and hence simplifies the implementation. Convergence studies are conducted to examine the stability of the present method. Three types of functionally graded plates – square, skew and quadrilateral plates – are considered as numerical examples to demonstrate the versatility of the present method for free vibration analyses.     

Transient heat conduction analysis in functionally graded materials by the meshless local boundary integral equation method

Advanced computational method for transient heat conduction analysis in continuously nonhomogeneous functionally graded materials (FGM) is proposed. The method is based on the local boundary integral equations with moving least square approximation of the temperature and heat flux. The initial-boundary value problem is solved by the Laplace transform technique. Both Papoulis and Stehfest algorithms are applied for the numerical Laplace inversion to obtain the time-dependent solutions. Numerical results are presented for a finite strip and a hollow cylinder with an exponential spatial variation of material parameters.     

Analysis of meshless local radial point interpolation (MLRPI) on a nonlinear partial integro-differential equation arising in population dynamics

In this paper the meshless local radial point interpolation (MLRPI) method is applied to simulate a nonlinear partial integro-differential equation arising in population dynamics. This PDE is a competition model in which similar individuals are competing for the same resources. It is a kind of reaction–diffusion equation with integral term corresponding to nonlocal consumption of resources. In MLRPI method, it does not require any background integration cells so that all integrations are carried out locally over small quadrature domains of regular shapes, such as circles or squares in two dimensions and spheres or cubes in three dimensions. The point interpolation method is proposed to construct shape functions using the radial basis functions. A one-step time discretization method is employed to approximate the time derivative. To treat the nonlinearity, a simple predictor–corrector scheme is performed. Also the integral term, which is a kind of convolution, is treated by the cubic spline interpolation. The numerical studies on sensitivity analysis and convergence analysis show that our approach is stable. Finally, two numerical examples are presented showing the behavior of the solution and the efficiency of the proposed method.     

Solution of the second-order one-dimensional hyperbolic telegraph equation by using the dual reciprocity boundary integral equation (DRBIE) method

In this paper, we use a numerical method based on the boundary integral equation (BIE) and an application of the dual reciprocity method (DRM) to solve the second-order one space-dimensional hyperbolic telegraph equation. Also the time stepping scheme is employed to deal with the time derivative. In this study, we have used three different types of radial basis functions (cubic, thin plate spline and linear RBFs), to approximate functions in the dual reciprocity method (DRM). To confirm the accuracy of the new approach and to show the performance of each of the RBFs, several examples are presented. The convergence of the DRBIE method is studied numerically by comparison with the exact solutions of the problems.     

On the convergence analysis,stability, and implementation of meshless local radial point interpolation on a class of three‐dimensional wave equations

In this article, the meshless local radial point interpolation method is applied to analyze three space dimensional wave equations of the form subject to given initial and Dirichlet boundary conditions. The main difficulty of the great number of methods in full 3‐D problems is the large computational costs. In meshless local radial point interpolation method, it does not require any background integration cells, so that all integrations are carried out locally over small quadrature domains of regular shapes such as circles or squares in two dimensions and spheres or cubes in three dimensions. The point interpolation method with the help of radial basis functions is proposed to construct shape functions that have Kronecker delta function property. A weak formulation with the Heaviside step function converts the set of governing equations into local integral equations on local subdomains. A two‐step time discretization method is employed to evaluate the time derivatives. This suggests Crank‐Nicolson technique to be applied on the right hand side of the equation. The convergence analysis and stability of the method are fully discussed. Three illustrative examples are presented, and satisfactory agreements are achieved. It is shown theoretically that the proposed method is unconditionally stable for the second example whereas it is not for the first and third ones. Copyright © 2015 John Wiley & Sons, Ltd.