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 More >

Giorgio Pini¹, Annamaria Mazzia¹, Flavio Sartoretto²,

CMES-Computer Modeling in Engineering & Sciences, Vol.36, No.1, pp. 43-64, 2008, DOI:10.3970/cmes.2008.036.043

Abstract Meshless methods have been explored in many 2D problems and they have been shown to be as accurate as Finite Element Methods (FEM). Compared to the extensive literature on 2D applications, papers on solving 3D problems by meshless methods are surprisingly few. Indeed, a main drawback of these methods is the requirement for accurate cubature rules. This paper focuses on the so called Meshless Local Petrov Galerkin (MLPG) methods. We show that accurate solutions of 3D potential problems can be attained, provided suitable cubature rules are identified, sparse data structures are efficiently stored, and strategies 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 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 More >

Arezoo Emdadi¹, Edward J. Kansa², Nicolas Ali Libre^1,3, Mohammad Rahimian¹, Mohammad Shekarchi¹

CMES-Computer Modeling in Engineering & Sciences, Vol.25, No.1, pp. 23-42, 2008, DOI:10.3970/cmes.2008.025.023

Abstract We present a new method based upon the paper of Volokh and Vilney (2000) that produces highly accurate and stable solutions to very ill-conditioned multiquadric (MQ) radial basis function (RBF) asymmetric collocation methods for partial differential equations (PDEs). We demonstrate that the modified Volokh-Vilney algorithm that we name the improved truncated singular value decomposition (IT-SVD) produces highly accurate and stable numerical solutions for large values of a constant MQ shape parameter, c, that exceeds the critical value of c based upon Gaussian elimination. 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 More >

P.H. Wen¹, M.H. Aliabadi²

Structural Durability & Health Monitoring, Vol.3, No.2, pp. 107-120, 2007, DOI:10.3970/sdhm.2007.003.107

Abstract In the last decade, meshless methods for solving differential equations have become a promising alternative to the finite element and boundary element methods. Based on the variation of potential energy, the element-free Galerkin method is developed on the basis of finite element method by the use of radial basis function interpolation. An enriched radial basis function is formulated to capture the stress singularity at the crack tip. The usual advantages of finite element method are retained in this method but now significant improvement of accuracy. Neither the connectivity of mesh in the domain by the More >

P. H. Wen¹, Y. C. Hon²

CMES-Computer Modeling in Engineering & Sciences, Vol.21, No.3, pp. 177-192, 2007, DOI:10.3970/cmes.2007.021.177

Abstract In this paper, we perform a geometrically nonlinear analysis of Reissner-Mindlin plate by using a meshless collocation method. The use of the smooth radial basis functions (RBFs) gives an advantage to evaluate higher order derivatives of the solution at no cost on extra-interpolation. The computational cost is low and requires neither the connectivity of mesh in the domain/boundary nor integrations of fundamental/particular solutions. The coupled nonlinear terms in the equilibrium equations for both the plane stress and plate bending problems are treated as body forces. Two load increment schemes are developed to solve the nonlinear More >

S.Y. Wang^1,2, K.M. Lim^2,3, B.C. Khoo^2,3, M.Y. Wang⁴

CMES-Computer Modeling in Engineering & Sciences, Vol.21, No.1, pp. 1-40, 2007, DOI:10.3970/cmes.2007.021.001

Abstract The level set method is a numerical technique for simulating moving interfaces. In this paper, an unconditionally BIBO (Bounded-Input-Bounded-Output) time-stable consistent meshfree level set method is proposed and applied as a more effective approach to simultaneous shape and topology optimization. In the present level set method, the meshfree infinitely smooth inverse multiquadric Radial Basis Functions (RBFs) are employed to discretize the implicit level set function. A high level of smoothness of the level set function and accuracy of the solution to the Hamilton-Jacobi partial differential equation (PDE) can be achieved. The resulting dynamic system of… More >

S.Y. Wang^1,2, M.Y. Wang³

CMES-Computer Modeling in Engineering & Sciences, Vol.13, No.2, pp. 119-148, 2006, DOI:10.3970/cmes.2006.013.119

Abstract In this paper, an implicit free boundary parametrization method is presented as an effective approach for simultaneous shape and topology optimization of structures. The moving free boundary of a structure is embedded as a zero level set of a higher dimensional implicit level set function. The radial basis functions (RBFs) are introduced to parametrize the implicit function with a high level of accuracy and smoothness. The motion of the free boundary is thus governed by a mathematically more convenient ordinary differential equation (ODE). Eigenvalue stability can be guaranteed due to the use of inverse multiquadric… More >