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 >

L.M.J.S. Dinis¹, R.M. Natal Jorge², J. Belinha³

CMES-Computer Modeling in Engineering & Sciences, Vol.44, No.1, pp. 1-34, 2009, DOI:10.3970/cmes.2009.044.001

Abstract The Natural Neighbour Radial Point Interpolation Method (NNRPIM) is extended to the large deformation analysis of non-linear elastic structures. The nodal connectivity in the NNRPIM is enforced using the Natural Neighbour concept. After the Voronoï diagram construction of the unstructured nodal mesh, which discretize the problem domain, small cells are created, the "influence-cells". These cells are in fact influence-domains entirely nodal dependent. The Delaunay triangles are used to create a node-depending background mesh used in the numerical integration of the NNRPIM interpolation functions. The NNRPIM interpolation functions, used in the Galerkin weak form, are constructed with the Radial Point Interpolators.… 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 >

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

CMES-Computer Modeling in Engineering & Sciences, Vol.38, No.3, pp. 263-284, 2008, DOI:10.3970/cmes.2008.038.263

Abstract We present a wavelet based adaptive scheme and investigate the efficiency of this scheme for solving nearly singular potential PDEs. Multiresolution wavelet analysis (MRWA) provides a firm mathematical foundation by projecting the solution of PDE onto a nested sequence of approximation spaces. The wavelet coefficients then were used as an estimation of the sensible regions for node adaptation. The proposed adaptation scheme requires negligible calculation time due to the existence of the fast Discrete Wavelet Transform (DWT). Certain aspects of the proposed adaptive scheme are discussed through numerical examples. It has been shown that the proposed adaptive scheme can detect… More >

Sirajul Haq¹, Siraj-ul-Islam², Arshed Ali³

CMES-Computer Modeling in Engineering & Sciences, Vol.38, No.1, pp. 1-24, 2008, DOI:10.3970/cmes.2008.038.001

Abstract In this paper we propose a meshfree technique for the numerical solution of the modified equal width wave (MEW) equation. Combination of collocation method using the radial basis functions (RBFs) with first order accurate forward difference approximation is employed for obtaining meshfree solution of the problem. Different types of RBFs are used for this purpose. Performance of the proposed method is successfully tested in terms of various error norms. In the case of non-availability of exact solution, performance of the new method is compared with the results obtained from the existing methods. Propagation of a solitary wave, interaction of two… More >

L. Mai-Cao¹, T. Tran-Cong²

CMES-Computer Modeling in Engineering & Sciences, Vol.31, No.3, pp. 157-188, 2008, DOI:10.3970/cmes.2008.031.157

Abstract This paper presents a new meshless numerical approach to solving a special class of moving interface problems known as the passive transport where an ambient flow characterized by its velocity field causes the interfaces to move and deform without any influences back on the flow. In the present approach, the moving interface is captured by the level set method at all time as the zero contour of a smooth function known as the level set function whereas one of the two new meshless schemes, namely the SL-IRBFN based on the semi-Lagrangian method and the Taylor-IRBFN scheme based on Taylor series… More >

G. Kosec¹, B. Šarler¹

CMES-Computer Modeling in Engineering & Sciences, Vol.25, No.3, pp. 197-208, 2008, DOI:10.3970/cmes.2008.025.197

Abstract This paper explores the application of the mesh-free Local Radial Basis Function Collocation Method (LRBFCM) in solution of coupled heat transfer and fluid flow problems in Darcy porous media. The involved temperature, velocity and pressure fields are represented on overlapping sub-domains through collocation by using multiquadrics Radial Basis Functions (RBF). The involved first and second derivatives of the fields are calculated from the respective derivatives of the RBF's. The energy and momentum equations are solved through explicit time stepping. The pressure-velocity coupling is calculated iteratively, with pressure correction, predicted from the local continuity equation violation. This formulation does not require… More >

P. Le¹, N. Mai-Duy², T. Tran-Cong³, G. Baker⁴

CMES-Computer Modeling in Engineering & Sciences, Vol.25, No.1, pp. 43-68, 2008, DOI:10.3970/cmes.2008.025.043

Abstract This paper describes an integrated radial basis function network (IRBFN) method for the numerical modelling of the dynamics of strain localization due to strain softening in quasi-brittle materials. The IRBFN method is a truly meshless method that is based on an unstructured point collocation procedure. We introduce a new and effective regularization method to enhance the performance of the IRBFN method and alleviate the numerical oscillations associated with weak discontinuity at the elastic wave front. The dynamic response of a one dimensional bar is investigated using both local and non-local continuum models. Numerical results, which compare favourably with those obtained… More >

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

CMES-Computer Modeling in Engineering & Sciences, Vol.24, No.1, pp. 61-80, 2008, DOI:10.3970/cmes.2008.024.061

Abstract The numerical solution of partial differential equations (PDEs) with Neumann boundary conditions (BCs) resulted from strong form collocation scheme are typically much poorer in accuracy compared to those with pure Dirichlet BCs. In this paper, we show numerically that the reason of the reduced accuracy is that Neumann BC requires the approximation of the spatial derivatives at Neumann boundaries which are significantly less accurate than approximation of main function. Therefore, we utilize boundary treatment schemes that based upon increasing the accuracy of spatial derivatives at boundaries. Increased accuracy of the spatial derivative approximation can be achieved by h-refinement reducing the… More >

E.J. Sellountos¹, A. Sequeira¹

CMES-Computer Modeling in Engineering & Sciences, Vol.23, No.2, pp. 127-148, 2008, DOI:10.3970/cmes.2008.023.127

Abstract In this work a hybrid velocity-vorticity scheme for the solution of the 2D Navier-Stokes equations is presented. The multi-region Local Boundary Integral Equation (LBIE) combined with Radial Basis Functions (RBF) interpolation is used for the solution of the kinematics and the multi-region BEM for the solution of the transport kinetics. The final system of equations is in band form for both methods. The issue of RBF discontinuities is resolved by constructing the RBF matrix locally in every region. The kinematics integral equation is used in three different forms, for coupling the velocity field on the boundary, on interior points and… More >