V. Panchore¹, R. Ganguli², S. N. Omkar³

CMES-Computer Modeling in Engineering & Sciences, Vol.104, No.5, pp. 353-373, 2015, DOI:10.3970/cmes.2015.104.353

Abstract Free vibration problem of a rotating Euler-Bernoulli beam is solved with a truly meshless local Petrov-Galerkin method. Radial basis function and summation of two radial basis functions are used for interpolation. Radial basis function satisfies the Kronecker delta property and makes it simpler to apply the essential boundary conditions. Interpolation with summation of two radial basis functions increases the node carrying capacity within the sub-domain of the trial function and higher natural frequencies can be computed by selecting the complete domain as a sub-domain of the trial function. The mass and stiffness matrices are derived More >

S.Yu. Reutskiy¹

CMES-Computer Modeling in Engineering & Sciences, Vol.99, No.4, pp. 327-349, 2014, DOI:10.3970/cmes.2014.099.327

Abstract The paper presents a new meshless numerical method for solving 2D and 3D boundary value problems (BVPs) with elliptic PDEs of general form. The coefficients of the PDEs including the main operator part are spatially dependent functions. The key idea of the method is the use of the basis functions which satisfy the homogeneous boundary conditions of the problem. This allows us to seek an approximate solution in the form which satisfies the boundary conditions of the initial problem with any choice of the free parameters. As a result we separate approximation of the boundary More >

M. Dehghan¹, M. Abbaszadeh², A. Mohebbi³

CMES-Computer Modeling in Engineering & Sciences, Vol.100, No.5, pp. 399-444, 2014, DOI:10.3970/cmes.2014.100.399

Abstract In this paper three numerical techniques are proposed for solving the system of N-coupled nonlinear Schrödinger (CNLS) equations. Firstly, we obtain a time discrete scheme by approximating the first-order time derivative via the forward finite difference formula, then for obtaining a full discretization scheme, we use the Kansa’s approach to approximate the spatial derivatives via radial basis functions (RBFs) collocation methodology. We introduce the moving least squares (MLS) approximation and radial point interpolation method (RPIM) with their shape functions, separately. It should be noted that the shape functions of RPIM unlike the shape functions of the… More >

N. Pham-Sy¹, C.-D. Tran¹, N. Mai-Duy¹, T. Tran-Cong¹

CMES-Computer Modeling in Engineering & Sciences, Vol.100, No.5, pp. 363-397, 2014, DOI:10.3970/cmes.2014.100.363

Abstract In this paper, a high performance computing method based on the Integrated Radial Basis Function (IRBF), Control Volume (CV) and Domain Decomposition technique for solving Partial Differential Equations is presented. The goal is to develop an efficient parallel algorithm based on the Compact Local IRBF method using the CV approach, especially for problems with non-rectangular domain. The results showed that the goal is achieved as the computational efficiency is quite significant. For the case of square lid driven cavity problem with Renoylds number 100, super-linear speed-up is also achieved. The parallel algorithm is implemented in More >

Soleiman Ghouhestani¹, Farzad Shahabian¹, Seyed Mahmoud Hosseini^2,3

CMES-Computer Modeling in Engineering & Sciences, Vol.100, No.4, pp. 295-321, 2014, DOI:10.3970/cmes.2014.100.295

Abstract In this paper, the meshless local Petrov-Galerkin (MLPG) method is exploited for dynamic analysis of functionally graded nanocomposite cylindrical layered structure reinforced by carbon nanotube subjected to mechanical shock loading. The carbon nanotubes (CNTs) are distributed across radial direction on thickness of cylinder, which can be simulated by linear and nonlinear volume fraction. Free vibration and elastic wave propagation are studied for various value of volume fraction exponent at various time intervals. The layered cylinder is assumed to be under axisymmetric and plane strain conditions. Four types of CNTs distributions including uniform and three kinds… More >

T.A. Elgohary^1,2, L. Dong³, J.L. Junkins^2,4, S.N. Atluri^1,4

CMES-Computer Modeling in Engineering & Sciences, Vol.100, No.3, pp. 249-275, 2014, DOI:10.3970/cmes.2014.100.249

Abstract In this study, we consider Initial Value Problems (IVPs) for strongly nonlinear dynamical systems, and study numerical methods to analyze short as well as long-term responses. Dynamical systems characterized by a system of second-order nonlinear ordinary differential equations (ODEs) are recast into a system of nonlinear first order ODEs in mixed variables of positions as well as velocities. For each discrete-time interval Radial Basis Functions (RBFs) are assumed as trial functions for the mixed variables in the time domain. A simple collocation method is developed in the time-domain, with Legendre-Gauss-Lobatto nodes as RBF source points… More >

T.A. Elgohary¹, L. Dong², J.L. Junkins³, S.N. Atluri⁴

CMES-Computer Modeling in Engineering & Sciences, Vol.100, No.1, pp. 59-84, 2014, DOI:10.3970/cmes.2014.100.059

Abstract In this study, we consider ill-posed time-domain inverse problems for dynamical systems with various boundary conditions and unknown controllers. Dynamical systems characterized by a system of second-order nonlinear ordinary differential equations (ODEs) are recast into a system of nonlinear first order ODEs in mixed variables. Radial Basis Functions (RBFs) are assumed as trial functions for the mixed variables in the time domain. A simple collocation method is developed in the time-domain, with Legendre-Gauss-Lobatto nodes as RBF source points as well as collocation points. The duffing optimal control problem with various prescribed initial and final conditions,… More >

Peter L. Bishay¹, Satya N. Atluri¹

CMC-Computers, Materials & Continua, Vol.33, No.1, pp. 19-62, 2013, DOI:10.3970/cmc.2013.033.019

Abstract In this paper, 2D and 3D Multiphysics Voronoi Cells (MVCs) are developed, for the Direct Mesoscale Numerical Simulation (DMNS) of the switching phenomena in ferroelectric polycrystalline materials. These arbitrarily shaped MVCs (arbitrary polygons in 2D, and arbitrary polyhedrons in 3D with each face being an arbitrary polygon) are developed, based on assuming radial basis functions to represent the internal primal variables (mechanical displacements and electric potential), and assuming linear functions to represent the primal variables on the element boundaries. For the 3D case, the linear functions used to represent the primal variables on each of… More >

L. Godinho¹, D. Dias-da-Costa²

CMES-Computer Modeling in Engineering & Sciences, Vol.96, No.5, pp. 293-316, 2013, DOI:10.3970/cmes.2013.096.293

Abstract Transient heat conduction problems can be dealt with using different numerical approaches. In some recent papers, a strategy to tackle these problems using a frequency domain formulation has been presented and successfully applied associated to methods such as the BEM. Here a formulation of the meshless local Petrov-Galerkin (MLPG) is developed and presented to allow the analysis of such problems. The proposed formulation makes use of the RBF-based version of the MLPG and employs the Heaviside step function as the test function, leading to the so-called MLPG(5). In addition, the method is associated with a More >

P.H. Wen¹, X.J. Huang¹, M.H. Aliabadi²

CMES-Computer Modeling in Engineering & Sciences, Vol.96, No.3, pp. 199-225, 2013, DOI:10.3970/cmes.2013.096.199

Abstract In this paper, a Local Integral Equation Method (LIEM) is presented for solving two-dimensional nonlocal elasticity problems . The approach is based on the Eringen’s model with LIEM and the interpolation using the radial basis functions to obtain the numerical solutions for 2D problems. A weak form for the set of governing equations with a unit test function is transformed into the local integral equations. The meshless approximation technique with radial basis functions is employed for the implementation of displacements. A set of the local domain integrals is obtained in analytical form for the local More >