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 >

Ali Shokri¹, Mehdi Dehghan¹

CMES-Computer Modeling in Engineering & Sciences, Vol.84, No.4, pp. 333-358, 2012, DOI:10.3970/cmes.2012.084.333

Abstract The Ginzburg-Landau equation has been used as a mathematical model for various pattern formation systems in mechanics, physics and chemistry. In this paper, we study the complex Ginzburg-Landau equation in two spatial dimensions with periodical boundary conditions. The method numerically approximates the solution by collocation method based on radial basis functions (RBFs). To improve the numerical results we use a predictor-corrector scheme. The results of numerical experiments are presented, and are compared with analytical solutions to confirm the accuracy and efficiency of the presented method. More >

T.-T. Hoang-Trieu¹, N. Mai-Duy¹, T. Tran-Cong¹

CMES-Computer Modeling in Engineering & Sciences, Vol.84, No.2, pp. 171-204, 2012, DOI:10.3970/cmes.2012.084.171

Abstract In this paper, new compact local stencils based on integrated radial basis functions (IRBFs) for solving fourth-order ordinary differential equations (ODEs) and partial differential equations (PDEs) are presented. Five types of compact stencils - 3-node and 5-node for 1D problems and 5×5-node, 13-node and 3×3 -node for 2D problems - are implemented. In the case of 3-node stencil and 3×3-node stencil, nodal values of the first derivative(s) of the field variable are treated as additional unknowns (i.e. 2 unknowns per node for 3-node stencil and 3 unknowns per node for 3×3-node stencil). The integration constants arising from the construction of… More >

Zhen Luo^1,2, Nong Zhang^1,3, Yu Wang¹

CMES-Computer Modeling in Engineering & Sciences, Vol.83, No.1, pp. 73-96, 2012, DOI:10.3970/cmes.2012.083.073

Abstract This paper aims to present a physically meaningful level set method for shape and topology optimization of structures. Compared to the conventional level set method which represents the design boundary as the zero level set, in this study the boundary is embedded into non-zero constant level sets of the level set function, to implicitly implement shape fidelity and topology changes in time via the propagation of the discrete level set function. A point-wise nodal density field, non-negative and value-bounded, is used to parameterize the level set function via the compactly supported radial basis functions (CSRBFs) at a uniformly defined set… More >

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 >

Liviu Marin¹

CMES-Computer Modeling in Engineering & Sciences, Vol.60, No.3, pp. 221-246, 2010, DOI:10.3970/cmes.2010.060.221

Abstract We investigate the implementation of the method of fundamental solutions (MFS), in an iterative manner, for the algorithm of Kozlov, Maz'ya and Fomin (1991) in the case of the Cauchy problem in two-dimensional isotropic linear elasticity. At every iteration, two mixed well-posed and direct problems are solved using the Tikhonov regularization method, while the optimal value of the regularization parameter is chosen according to the generalized cross-validation (GCV) criterion. An efficient regularizing stopping criterion is also presented. The iterative MFS algorithm is tested for Cauchy problems for isotropic linear elastic materials to confirm the numerical convergence, stability and accuracy of… More >

Liviu Marin¹

CMES-Computer Modeling in Engineering & Sciences, Vol.48, No.2, pp. 121-154, 2009, DOI:10.3970/cmes.2009.048.121

Abstract We investigate the numerical implementation of the alternating iterative algorithm originally proposed by ` 12 ` 12 `$12 `&12 `#12 `^12 `_12 `%12 `~12 *Kozlov91 in the case of the Cauchy problem for the two-dimensional Laplace equation using a meshless method. The two mixed, well-posed and direct problems corresponding to every iteration of the numerical procedure are solved using the method of fundamental solutions (MFS), in conjunction with the Tikhonov regularization method. For each direct problem considered, the optimal value of the regularization parameter is chosen according to the generalized cross-validation (GCV) criterion. An efficient regularizing stopping criterion which ceases… More >

Liviu Marin¹

CMES-Computer Modeling in Engineering & Sciences, Vol.46, No.3, pp. 221-254, 2009, DOI:10.3970/cmes.2009.046.221

Abstract We investigate the stable numerical reconstruction of an unknown portion of the boundary of a two-dimensional domain occupied by a functionally graded material (FGM) from a given boundary condition on this part of the boundary and additional Cauchy data on the remaining known portion of the boundary. The aforementioned inverse geometric problem is approached using the method of fundamental solutions (MFS), in conjunction with the Tikhonov regularization method. The optimal value of the regularization parameter is chosen according to Hansen's L-curve criterion. Various examples are considered in order to show that the proposed method is numerically stable with respect to… More >

Sirajul Haq^1,2, Siraj-Ul-Islam³, Marjan Uddin^1,4

CMES-Computer Modeling in Engineering & Sciences, Vol.44, No.2, pp. 115-136, 2009, DOI:10.3970/cmes.2009.044.115

Abstract A mesh free method for the numerical solution of the nonlinear Schrodinger (NLS) and coupled nonlinear Schrodinger (CNLS) equation is implemented. The presented method uses a set of scattered nodes within the problem domain as well as on the boundaries of the domain along with approximating functions known as radial basis functions (RBFs). The set of scattered nodes do not form a mesh, means that no information of relationship between the nodes is needed. Error norms L₂, L_∞ are used to estimate accuracy of the method. Stability analysis of the method is given to demonstrate its practical applicability. More >

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

CMES-Computer Modeling in Engineering & Sciences, Vol.41, No.3, pp. 215-242, 2009, DOI:10.3970/cmes.2009.041.215

Abstract A Meshless Local Petrov-Galerkin (MLPG) method based on Local Boundary Integral Equation (LBIE) techniques is employed here for the solution of transient elastic problems with damping. The Radial Basis Functions (RBF) interpolation scheme is exploited for the meshless representation of displacements throughout the computational domain. On the intersections between the local domains and the global boundary, tractions are treated as independent variables via conventional boundary interpolation functions. The MLPG(LBIE)/RBF method is applied to both transient and steady-state Fourier transform elastodynamic domains. In both cases the LBIEs employ the simple elastostatic fundamental solution instead of the complicated time and frequency dependent… More >