N. Mai-Duy¹, T. Tran-Cong¹, N. Phan-Thien²

CMES-Computer Modeling in Engineering & Sciences, Vol.79, No.3&4, pp. 223-236, 2011, DOI:10.3970/cmes.2011.079.223

Abstract This paper presents a new numerical technique for solving the evolution equations in molecular dynamics (MD). The variation of the MD system is represented by radial-basis-function (RBF) equations which are constructed using integrated multiquadric basis functions and point collocation. The proposed technique requires the evaluation of forces once per time step. Several examples are given to demonstrate the attractiveness of the present implementation. More >

D.-A. An-Vo¹, N. Mai-Duy¹, T. Tran-Cong¹

CMES-Computer Modeling in Engineering & Sciences, Vol.72, No.4, pp. 299-336, 2011, DOI:10.3970/cmes.2011.072.299

Abstract This paper presents a new control-volume discretisation method, based on Cartesian grids and integrated-radial-basis-function elements (IRBFEs), for the solution of second-order elliptic problems in one and two dimensions. The governing equation is discretised by means of the control-volume formulation and the division of the problem domain into non-overlapping control volumes is based on a Cartesian grid. Salient features of the present method include (i) an element is defined by two adjacent nodes on a grid line, (ii) the IRBF approximations on each element are constructed using only two RBF centres (a smallest RBF set) associated with the two nodes of… More >

Htike Htike¹, Wen Chen¹, Yan Gu¹, Junjie Yang¹

CMES-Computer Modeling in Engineering & Sciences, Vol.72, No.4, pp. 247-272, 2011, DOI:10.3970/cmes.2011.072.247

Abstract This paper makes the first attempt to employ the Radial Basis Function (RBF) interpolation in the material point method (MPM), in which the shape function is based on RBF and polynomial function and satisfies the partition of unity and possesses Delta-function property. It is worthy of stressing that the RBF interpolation has the merit of high smoothness and is very accurate and can easily be applied to the MPM framework for mapping information between moving particles, known as material point in the MPM, and background grids. The RBF-based MPM is designed to overcome the unphysical results, such as shear stress… More >

Phong B.H. Le¹, Timon Rabczuk², Nam Mai-Duy¹, Thanh Tran-Cong¹

CMES-Computer Modeling in Engineering & Sciences, Vol.66, No.1, pp. 25-52, 2010, DOI:10.3970/cmes.2010.066.025

Abstract A novel meshless method based on Radial Basis Function networks (RBFN) and variational principle (global weak form) is presented in this paper. In this method, the global integrated RBFN is localized and coupled with the moving least square method via the partition of unity concept. As a result, the system matrix is symmetric, sparse and banded. The trial and test functions satisfy the Kronecker-delta property, i.e. Φ_i(x_j) = δ_ij. Therefore, the essential boundary conditions are imposed in strong form as in the FEMs. Moreover, the proposed method is applicable to scattered nodes and arbitrary domains. The method is examined with… More >

Phong B.H. Le¹, Timon Rabczuk², Nam Mai-Duy¹, Thanh Tran-Cong¹

CMES-Computer Modeling in Engineering & Sciences, Vol.61, No.1, pp. 63-110, 2010, DOI:10.3970/cmes.2010.061.063

Abstract A novel approximation method using integrated radial basis function networks (IRBFN) coupled with moving least square (MLS) approximants, namely moving integrated radial basis function networks (MIRBFN), is proposed in this work. In this method, the computational domain Ω is divided into finite sub-domains Ω_I which satisfy point-wise overlap condition. The local function interpolation is constructed by using IRBFN supported by all nodes in subdomain Ω_I. The global function is then constructed by using Partition of Unity Method (PUM), where MLS functions play the role of partition of unity. As a result, the proposed method is locally supported and yields sparse… More >

Guangming Yao¹, Siraj-ul-Islam², Božidar Šarler²

CMES-Computer Modeling in Engineering & Sciences, Vol.59, No.2, pp. 127-154, 2010, DOI:10.3970/cmes.2010.059.127

Abstract This paper focuses on the comparative study of global and local meshless methods based on collocation with radial basis functions for solving two dimensional initial boundary value diffusion-reaction problem with Dirichlet and Neumann boundary conditions. A similar study was performed for the boundary value problem with Laplace equation by Lee, Liu, and Fan (2003). In both global and local methods discussed, the time discretization is performed in explicit and implicit way and the multiquadric radial basis functions (RBFs) are used to interpolate diffusion-reaction variable and its spatial derivatives. Five and nine nodded sub-domains are used in the local support of… More >

E. J. Sellountos¹, A. Sequeira¹, D. Polyzos²

CMES-Computer Modeling in Engineering & Sciences, Vol.57, No.2, pp. 109-136, 2010, DOI:10.3970/cmes.2010.057.109

Abstract A new Local Boundary Integral Equation (LBIE) method is proposed for the solution of plane elastostatic problems. Non-uniformly distributed points taken from a Finite Element Method (FEM) mesh cover the analyzed domain and form background cells with more than four points each. The FEM mesh determines the position of the points without imposing any connectivity requirement. The key-point of the proposed methodology is that the support domain of each point is divided into parts according to the background cells. An efficient Radial Basis Functions (RBF) interpolation scheme is exploited for the representation of displacements in each cell. Tractions in the… More >

David Stevens¹, Henry Power^1,2

CMES-Computer Modeling in Engineering & Sciences, Vol.55, No.2, pp. 111-146, 2010, DOI:10.3970/cmes.2010.055.111

Abstract Problems involving nonlinear time-dependent heat conduction in materials which have temperature-dependent thermal properties are solved with a novel meshless numerical solution technique using multiquadric radial basis functions (RBFs). Unlike traditional RBF collocation methods, the local Hermitian interpolation (LHI) method examined here can be scaled to arbitrarily large problems without numerical ill-conditioning or computational cost issues, due to the presence of small overlapping interpolation systems which grow in number but not in size as the global dataset grows. The flexibility of the full-domain multiquadric collocation method to directly interpolate arbitrary boundary conditions is maintained, via the local interpolations. The Kirchhoff transformation… More >

Nicolas Ali Libre^1,2, Arezoo Emdadi², Edward J. Kansa^3,4, Mohammad Shekarchi², Mohammad Rahimian²

CMES-Computer Modeling in Engineering & Sciences, Vol.50, No.2, pp. 161-190, 2009, DOI:10.3970/cmes.2009.050.161

Abstract We present a wavelet based adaptive scheme and investigate the efficiency of this scheme for solving nearly singular potential PDEs over irregularly shaped domains. For a problem defined over Ω∈ℜ^d, the boundary of an irregularly shaped domain, Γ, is defined as a boundary curve that is a product of a Heaviside function along the normal direction and a piecewise continuous tangential curve. The link between the original wavelet based adaptive method presented in Libre, Emdadi, Kansa, Shekarchi, and Rahimian (2008, 2009) or LEKSR method and the generalized one is given through the use of simple Heaviside masking procedure. In addition… More >

D. Ho-Minh¹, N. Mai-Duy¹, T. Tran-Cong¹

CMES-Computer Modeling in Engineering & Sciences, Vol.44, No.3, pp. 219-248, 2009, DOI:10.3970/cmes.2009.044.219

Abstract This paper reports a new discretisation technique for the streamfunc -tion-vorticity-temperature (ψ−ω−T) formulation governing natural convection defined in 2D enclosured domains. The proposed technique combines strengths of three schemes, i.e. smooth discretisations (Galerkin formulation), powerful high-order approximations (one-dimensional integrated radial-basis-function networks) and pressure-free low-order system (ψ−ω−T formulation). In addition, a new effective way of deriving computational boundary conditions for the vorticity is proposed. Two benchmark test problems, namely free convection in a square slot and a concentric annulus, are considered, where a convergent solution for the former is achieved up to the Rayleigh number of 10⁸. More >