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 the Matlab environment using Parallel… More >

N. Thai-Quang¹, N. Mai-Duy¹, C.-D. Tran¹, T. Tran-Cong^1,2

CMES-Computer Modeling in Engineering & Sciences, Vol.96, No.1, pp. 49-90, 2013, DOI:10.3970/cmes.2013.096.049

Abstract In this paper, we present a numerical scheme, based on the direct forcing immersed boundary (DFIB) approach and compact integrated radial basis function (CIRBF) approximations, for solving the Navier-Stokes equations in two dimensions. The problem domain of complicated shape is embedded in a Cartesian grid containing Eulerian nodes. Non-slip conditions on the inner boundaries, represented by Lagrangian nodes, are imposed by means of the DFIB method, in which a smoothed version of the discrete delta functions is utilised to transfer the physical quantities between two types of nodes. The velocities and pressure variables are approximated locally on Eulerian nodes using… More >

N. Pham-Sy¹, C.-D. Tran¹, T.-T. Hoang-Trieu¹, N. Mai-Duy¹, T. Tran-Cong¹

CMES-Computer Modeling in Engineering & Sciences, Vol.92, No.1, pp. 1-31, 2013, DOI:10.3970/cmes.2013.092.001

Abstract Compact Local Integrated Radial Basis Function (CLIRBF) methods based on Cartesian grids can be effective numerical methods for solving partial differential equations (PDEs) for fluid flow problems. The combination of the domain decomposition method and function approximation using CLIRBF methods yields an effective coarse-grained parallel processing approach. This approach has enabled not only each sub-domain in the original analysis domain to be discretised by a separate CLIRBF network but also compact local stencils to be independently treated. The present algorithm, namely parallel CLIRBF, achieves higher throughput in solving large scale problems by, firstly, parallel processing of sub-regions which constitute the… More >

E.J. Sellountos¹, D. Polyzos², S.N. Atluri³

CMES-Computer Modeling in Engineering & Sciences, Vol.89, No.6, pp. 513-551, 2012, DOI:10.3970/cmes.2012.089.513

Abstract A new meshless Local Boundary Integral Equation (LBIE) method for solving two-dimensional elastostatic problems is proposed. Randomly distributed points without any connectivity requirement cover the analyzed domain and Local Radial Basis Functions (LRBFs) are employed for the meshless interpolation of displacements. For each point a circular support domain is centered and a local integral representation for displacements is considered. At the local circular boundaries tractions are eliminated with the aid of companion solution, while at the intersections between the local domains and the global boundary displacements and tractions are treated as independent variables avoiding thus derivatives of LRBFs. Stresses are… More >

D.-A. An-Vo¹, C.-D. Tran¹, N. Mai-Duy¹, T. Tran-Cong¹

CMES-Computer Modeling in Engineering & Sciences, Vol.89, No.4, pp. 303-359, 2012, DOI:10.3970/cmes.2012.089.303

Abstract Many important engineering problems have multiple-scale solutions. Thermal conductivity of composite materials, flow in porous media, and turbulent transport in high Reynolds number flows are examples of this type. Direct numerical simulations for these problems typically require extremely large amounts of CPU time and computer memory, which may be too expensive or impossible on the present supercomputers. In this paper, we develop a high order computational method, based on multiscale basis function approach and integrated radialbasis- function (IRBF) approximant, for the solution of multiscale elliptic problems with reduced computational cost. Unlike other methods based on multiscale basis function approach, sets… More >

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

CMES-Computer Modeling in Engineering & Sciences, Vol.89, No.3, pp. 189-220, 2012, DOI:10.3970/cmes.2012.089.189

Abstract In this paper, the alternating direction implicit (ADI) method reported in [You(2006)] for the convection-diffusion equation is implemented in the context of compact integrated radial basis function (CIRBF) approximations. The CIRBF approximations are constructed over 3-point stencils, where extra information is incorporated via two forms: only nodal second-order derivative values (Scheme 1), and both nodal first- and second-order derivative values (Scheme 2). The resultant algebraic systems are sparse, especially for Scheme 2 (tridiagonal matrices). Several steady and non-steady problems are considered to verify the present schemes and to compare their accuracy with some other ADI schemes. Numerical results show that… More >

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

CMES-Computer Modeling in Engineering & Sciences, Vol.70, No.3, pp. 217-252, 2010, DOI:10.3970/cmes.2010.070.217

Abstract In this paper, one-dimensional integrated radial-basis-function networks (1D-IRBFNs) are introduced into the Galerkin and point-collocation formulations to simulate viscoelastic flows. The computational domain is represented by a Cartesian grid and IRBFNs, which are constructed through integration, are employed on each grid line to approximate the field variables including stresses in the streamfunction-vorticity formulation. Two types of fluid, namely Oldroyd-B and CEF models, are considered. The proposed methods are validated through the numerical simulation of several benchmark test problems including flows in a rectangular duct and in a corrugated tube. Numerical results show that accurate results are obtained using relatively-coarse grids. More >

N. Mai-Duy¹, T. Tran-Cong¹

CMES-Computer Modeling in Engineering & Sciences, Vol.51, No.3, pp. 213-238, 2009, DOI:10.3970/cmes.2009.051.213

Abstract This paper reports a new numerical scheme based on Cartesian grids and local integrated radial-basis-function networks (IRBFNs) for the solution of second-order elliptic differential problems defined on two-dimensional regular and irregular domains. At each grid point, only neighbouring nodes are activated to construct the IRBFN approximations. Local IRBFNs are introduced into two different schemes for discretisation of partial differential equations, namely point collocation and control-volume (CV)/subregion-collocation. Linear (e.g. heat flow) and nonlinear (e.g. lid-driven triangular-cavity fluid flow) problems are considered. Numerical results indicate that the local IRBFN CV scheme outperforms the local IRBFN point-collocation scheme regarding accuracy. Moreover, the former… More >

G. C. Bourantas¹, V. C. Loukopoulos²

CMES-Computer Modeling in Engineering & Sciences, Vol.86, No.4, pp. 275-300, 2012, DOI:10.3970/cmes.2012.086.275

Abstract This paper formulates a local Radial Basis Functions (LRBFs) collocation method for the numerical solution of the non-linear dispersive and dissipative KdV-Burger's (KdVB) equation. This equation models physical problems, such as irrotational incompressible flow, considering a shallow layer of an inviscid fluid moving under the influence of gravity and the motion of solitary waves. The local type of approximations used, leads to sparse algebraic systems that can be solved efficiently. The Inverse Multiquadrics (IMQ), Gaussian (GA) and Multiquadrics (MQ) Radial Basis Functions (RBF) interpolation are employed for the construction of the shape functions. Accuracy of the method is assessed in… More >

C.-D. Tran¹, N. Mai-Duy^1,1, K. Le-Cao¹, T. Tran-Cong¹

CMES-Computer Modeling in Engineering & Sciences, Vol.85, No.6, pp. 499-520, 2012, DOI:10.3970/cmes.2012.085.499

Abstract A numerical computation of continuum-microscopic model for visco-elastic flows based on the Integrated Radial Basis Function (IRBF) Control Volume and the Stochastic Simulation Techniques (SST) is reported in this paper. The macroscopic flow equations are closed by a stochastic equation for the extra stress at the microscopic level. The former are discretised by a 1D-IRBF-CV method while the latter is integrated with Euler explicit or Predictor-Corrector schemes. Modelling is very efficient as it is based on Cartesian grid, while the integrated RBF approach enhances both the stability of the procedure and the accuracy of the solution. The proposed method is… More >