A MLPG4 (LBIE) Formulation in Elastostatics

Very recently, Vavourakis, Sellountos and Polyzos (2006) ({CMES: Computer Modeling in Engineering {\&} Sciences, vol. 13, pp. 171--184}) presented a comparison study on the accuracy provided by five different elastostatic Meshless Local Petrov-Galerkin (MLPG) type formulations, which are based on Local Boundary Integral Equation (LBIE) considerations. One of the main conclusions addressed in this paper is that the use of derivatives of the Moving Least Squares (MLS) shape functions decreases the solution accuracy of any MLPG(LBIE) formulation. In the present work a new, free of MLS-derivatives and non-singular MLPG(LBIE) method for solving elastic problems is demonstrated. This is accomplished by treating displacements and stresses as independent variables through the corresponding local integral equations and considering nodal points located only internally and externally and not on the global boundary of the analyzed elastic structure. The MLS approximation scheme for the interpolation of both displacements and stresses is exploited. The essential displacement and traction boundary conditions are easily satisfied via the corresponding displacement and stress local integral equations. Representative numerical examples that demonstrate the achieved accuracy of the proposed MLPG(LBIE) method are provided.     

Pricing American options under jump-diffusion models using local weak form meshless techniques

Recently, several numerical methods have been proposed for pricing options under jump-diffusion models but very few studies have been conducted using meshless methods [R. Chan and S. Hubbert, A numerical study of radial basis function based methods for options pricing under the one dimension jump-diffusion model, Tech. Rep., 2010; A. Saib, D. Tangman, and M. Bhuruth, A new radial basis functions method for pricing American options under Merton's jump-diffusion model, Int. J. Comput. Math. 89 (2012), pp. 1164–1185]. Indeed, only a strong form of meshless methods have been employed in these lectures. We propose the local weak form meshless methods for option pricing under Merton and Kou jump-diffusion models. Predominantly in this work we will focus on meshless local Petrov–Galerkin, local boundary integral equation methods based on moving least square approximation and local radial point interpolation based on Wendland's compactly supported radial basis functions. The key feature of this paper is applying a Richardson extrapolation technique on American option which is a free boundary problem to obtain a fixed boundary problem. Also the implicit–explicit time stepping scheme is employed for the time derivative which allows us to obtain a spars and banded linear system of equations. Numerical experiments are presented showing that the presented approaches are extremely accurate and fast.     

A meshless local boundary integral equation method for solving transient elastodynamic problems

A new local boundary integral equation (LBIE) method for solving two dimensional transient elastodynamic problems is proposed. The method utilizes, for its meshless implementation, nodal points spread over the analyzed domain and employs the moving least squares (MLS) approximation for the interpolation of the interior and boundary variables. On the global boundary, displacements and tractions are treated as independent variables. The local integral representation of displacements at each nodal point contains both surface and volume integrals, since it employs the simple elastostatic fundamental solution and considers the acceleration term as a body force. On the local boundaries, tractions are avoided with the aid of the elastostatic companion solution. The collocation of the local boundary/volume integral equations at all the interior and boundary nodes leads to a final system of ordinary differential equations, which is solved stepwise by the -Wilson finite difference scheme. Direct numerical techniques for the accurate evaluation of both surface and volume integrals are employed and presented in detail. All the strongly singular integrals are computed directly through highly accurate integration techniques. Three representative numerical examples that demonstrate the accuracy of the proposed methodology are provided.     

Numerical determination of modal dispersion and AE signal characterization in waveguides through LBIE/BEM and time–frequency analysis

In the present paper a meshless local boundary integral equation (LBIE) method is employed for the numerical determination of dispersion curves in plate like waveguides. Next, the LBIE method is properly combined with the boundary element method (BEM) for the characterization of acoustic emission signals derived from nucleating surface-breaking and body cracks in plate-like waveguides being in contact with a fluid. The time history of the propagating guided waves is computed by means of either the LBIE method or the hybrid LBIE/BEM scheme and the representation of mode dispersion is accomplished through the Reassigned Smoothed Pseudo Wigner–Ville time–frequency analysis. The effectiveness of the method is demonstrated through representative numerical examples.     

Meshless local boundary integral equation (LBIE) method for the unsteady magnetohydrodynamic (MHD) flow in rectangular and circular pipes

The meshless local boundary integral equation (LBIE) method is given to obtain the numerical solution of the coupled equations in velocity and magnetic field for unsteady magnetohydrodynamic (MHD) flow through a pipe of rectangular and circular sections with non-conducting walls. Computations have been carried out for different Hartmann numbers and at various time levels. The method is based on the local boundary integral equation with moving least squares (MLS) approximation. For the MLS, nodal points spread over the analyzed domain, are utilized to approximate the interior and boundary variables. A time stepping method is employed to deal with the time derivative. Finally, numerical results are presented to show the behaviour of velocity and induced magnetic field.     

无网格局部边界元法弹性力学问题应用研究

无网格局部边界元法是一种真正的无网格方法。本文推导了弹性力学问题的局部边界积分方程，并且基于MLS近似方法实现了无网格离散，得出无网格局部边界元法的二维弹性力学问题的格式，推导了修正的基本解，并利用编制的计算程序，应用于实际算例。