Z. D. Han¹, S. N. Atluri¹

CMES-Computer Modeling in Engineering & Sciences, Vol.6, No.2, pp. 169-188, 2004, DOI:10.3970/cmes.2004.006.169

Abstract Three different truly Meshless Local Petrov-Galerkin (MLPG) methods are developed for solving 3D elasto-static problems. Using the general MLPG concept, these methods are derived through the local weak forms of the equilibrium equations, by using different test functions, namely, the Heaviside function, the Dirac delta function, and the fundamental solutions. The one with the use of the fundamental solutions is based on the local unsymmetric weak form (LUSWF), which is equivalent to the local boundary integral equations (LBIE) of the elasto-statics. Simple formulations are derived for the LBIEs in which only weakly-singular integrals are included for a simple numerical implementation.… More >

S. Choi¹, M.D. Marcozzi¹

CMES-Computer Modeling in Engineering & Sciences, Vol.5, No.3, pp. 201-212, 2004, DOI:10.3970/cmes.2004.005.201

Abstract The general intractability of derivative security pricing models to numerical techniques arguably remains one of the preeminant problems of mathematical finance. In particular, the valuations of such models may be represented as solutions of variational inequalities of evolutionary type typically characterized by their high number of degrees of freedom, unbounded domains, and asymptotic behavior. We consider the application of Multi-Quadratic Radial Basis Functions (MQ-RBF) to the problem of option pricing. More >

H.Q. Nguyen¹, C.-D. Tran¹, T. Tran-Cong¹

CMES-Computer Modeling in Engineering & Sciences, Vol.109-110, No.4, pp. 361-403, 2015, DOI:10.3970/cmes.2015.109.361

Abstract In this paper, an Integrated Radial Basis Function (IRBF)-based multiscale method is used to simulate the rheological properties of dilute fibre suspensions. For the approach, a fusion of the IRBF computation scheme, the Discrete Adaptive Viscoelastic Stress Splitting (DAVSS) technique and the Fibre Configuration Field has been developed to investigate the evolution of the flow and the fibre configurations through two separate computational processes. Indeed, the flow conservation equations, which are expressed in vorticity-stream function formulation, are solved using IRBF-based numerical schemes while the evolution of fibre configuration fields governed by the Jeffery’s equation is captured using the principle of… More >

H.Q. Nguyen¹, C.-D. Tran¹, T. Tran-Cong¹

CMES-Computer Modeling in Engineering & Sciences, Vol.105, No.5, pp. 399-439, 2015, DOI:10.3970/cmes.2015.105.399

Abstract In this paper, dynamic behaviours of dilute polymer solutions of various bead-spring chain models in shear flow are studied using a coarse-grained method based on the Integrated Radial Basis Function Networks (IRBFNs) and stochastic technique. The velocity field governed by the macroscopic conservation equations is determined by the IRBFN-based method, whereas the evolution of configurations of polymer chains governed by the diffusion stochastic differential equations are captured by the Brownian Configuration Field (BCF) approach. The system of micro-macro equations is closed by the Kramers’ expression, which allows for the determination of the polymer stresses in terms of BCF configurations. In… More >

C.M.T. Tien¹, N. Thai-Quang¹, N. Mai-Duy¹, C.-D. Tran¹, T. Tran-Cong¹

CMES-Computer Modeling in Engineering & Sciences, Vol.105, No.4, pp. 301-340, 2015, DOI:10.3970/cmes.2015.105.301

Abstract In this study, we present a numerical discretisation scheme, based on a direct fully coupled approach and compact integrated radial basis function (CIRBF) approximations, to simulate viscous flows in regular/irregular domains. The governing equations are taken in the primitive form where the velocity and pressure fields are solved in a direct fully coupled approach. Compact local approximations, based on integrated radial basis functions, over 3-node stencils are introduced into the direct fully coupled approach to represent the field variables. The present scheme is verified through the solutions of several problems including Poisson equations, Taylor-Green vortices and lid driven cavity flows,… More >

K. N. Grivas¹, M. G. Vavva¹, E. J. Sellountos², D. I. Fotiadis³, D. Polyzos^1,4

CMES-Computer Modeling in Engineering & Sciences, Vol.105, No.2, pp. 87-122, 2015, DOI:10.3970/cmes.2015.105.087

Abstract A simple Local Boundary Integral Equation (LBIE) method for solving the Fisher nonlinear transient diffusion equation in two dimensions (2D) is reported. The method utilizes, for its meshless implementation, randomly distributed nodal points in the interior domain and nodal points corresponding to a Boundary Element Method (BEM) mesh, at the global boundary. The interpolation of the interior and boundary potentials is accomplished using a Local Radial Basis Functions (LRBF) scheme. At the nodes of global boundary the potentials and their fluxes are treated as independent variables. On the local boundaries, potential fluxes are avoided by using the Laplacian companion solution.… More >

C.M.T. Tien¹, N. Thai-Quang¹, N. Mai-Duy¹, C.-D. Tran¹, T. Tran-Cong¹

CMES-Computer Modeling in Engineering & Sciences, Vol.104, No.6, pp. 425-469, 2015, DOI:10.3970/cmes.2015.104.425

Abstract In this paper, we propose a three-point coupled compact integrated radial basis function (CCIRBF) approximation scheme for the discretisation of second-order differential problems in one and two dimensions. The CCIRBF employs integrated radial basis functions (IRBFs) to construct the approximations for its first and second derivatives over a three-point stencil in each direction. Nodal values of the first and second derivatives (i.e. extra information), incorporated into approximations by means of the constants of integration, are simultaneously employed to compute the first and second derivatives. The essence of the CCIRBF scheme is to couple the extra information of the nodal first… More >

C.M.T. Tien¹, N. Pham-Sy¹, N. Mai-Duy¹, C.-D. Tran¹, T. Tran-Cong¹

CMES-Computer Modeling in Engineering & Sciences, Vol.104, No.4, pp. 251-304, 2015, DOI:10.3970/cmes.2015.104.251

Abstract This paper presents a high-order coupled compact integrated RBF (CC IRBF) approximation based domain decomposition (DD) algorithm for the discretisation of second-order differential problems. Several Schwarz DD algorithms, including one-level additive/ multiplicative and two-level additive/ multiplicative/ hybrid, are employed. The CCIRBF based DD algorithms are analysed with different mesh sizes, numbers of subdomains and overlap sizes for Poisson problems. Our convergence analysis shows that the CCIRBF two-level multiplicative version is the most effective algorithm among various schemes employed here. Especially, the present CCIRBF two-level method converges quite rapidly even when the domain is divided into many subdomains, which shows great… More >

Yin Xiao-Liang¹, Wu Yi-Zhong^1,2, Wan Li¹, Xiong Hui-Yuan³

CMES-Computer Modeling in Engineering & Sciences, Vol.103, No.6, pp. 429-462, 2014, DOI:10.3970/cmes.2014.103.429

Abstract The use of multi-dimensional global approximation for a complex black-box function (such as a simulation or an analysis model) is steadily growing in the past decade. It can be applied in many fields such as parameter experiment, sensibility analyses real-time simulation, and design/control optimization. However, the widespread use of approximation methods is hampered by the lack of the ability to approximate a complex simulation model which characterizes the dynamic feature with multiple inputs and multiple outputs (MIMO) in a large domain. In this paper, a novel global approximation method for simulation models based on the RBF response surface set is… 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 MLS approximation have kronecker… More >