Vladimir Sladek¹, Jan Sladek¹, Chuanzeng Zhang²

CMC-Computers, Materials & Continua, Vol.4, No.3, pp. 177-188, 2006, DOI:10.3970/cmc.2006.004.177

Abstract This paper concerns the stability, convergence of accuracy and cost efficiency of four various formulations for solution of boundary value problems in non-homogeneous elastic solids with functionally graded Young's modulus. The meshless point interpolation method is employed with using various basis functions. The interaction among the elastic continuum constituents is considered in the discretized formulation either by collocation of the governing equations or by integral satisfaction of the force equilibrium on local sub-domains. The exact benchmark solutions are used in numerical tests. More >

A.I.Tolstykh, D.A. Shirobokov¹

CMES-Computer Modeling in Engineering & Sciences, Vol.7, No.2, pp. 207-222, 2005, DOI:10.3970/cmes.2005.007.207

Abstract A way of using RBF as the basis for PDE's solvers is presented, its essence being constructing approximate formulas for derivatives discretizations based on RBF interpolants with local supports similar to stencils in finite difference methods. Numerical results for different types of elasticity equations showing reasonable accuracy and good$h$-convergence properties of the technique are presented. Applications of the technique to problems with non-self-adjoint operators (like those for the Navier-Stokes equations) are also considered. More >

L. Mai-Cao¹, T. Tran-Cong²

CMES-Computer Modeling in Engineering & Sciences, Vol.7, No.2, pp. 149-172, 2005, DOI:10.3970/cmes.2005.007.149

Abstract The Indirect Radial Basis Function Network (IRBFN) method has been reported to be a highly accurate tool for approximating multivariate functions and solving elliptic partial differential equations (PDEs). The present method is a truly meshless method as defined in [\citet *{Atluri_Shen_02a}]. A recent development of the method for solving transient problems is presented in this paper. Two numerical schemes combining the IRBFN method with different time integration techniques based on either fully or semi-discrete framework are proposed. The two schemes are implemented making use of Hardy's multiquadrics (MQ) and Duchon's thin plate splines (TPS). Some More >

Y. C. Hon¹, L. Ling², K. M. Liew³

CMC-Computers, Materials & Continua, Vol.2, No.1, pp. 39-50, 2005, DOI:10.3970/cmc.2005.002.039

Abstract In this paper we investigate a thermal driven Micro-Electrical-Mechanical system which was originally designed for inkjet printer to precisely deliver small ink droplets onto paper. In the model, a tiny free-ended beam of metal bends and projects ink onto paper. The model is solved by using the recently developed radial basis functions method. We establish the accuracy of the proposed approach by comparing the numerical results with reported experimental data. Numerical simulations indicate that a light (low composite mass) beam is more stable as it does not oscillate much. A soft (low rigidity) beam results More >

S. N. Atluri¹, Z. D. Han¹, A. M. Rajendran²

CMES-Computer Modeling in Engineering & Sciences, Vol.6, No.6, pp. 491-514, 2004, DOI:10.3970/cmes.2004.006.491

Abstract The Meshless Finite Volume Method (MFVM) is developed for solving elasto-static problems, through a new Meshless Local Petrov-Galerkin (MLPG) ``Mixed'' approach. In this MLPG mixed approach, both the strains as well as displacements are interpolated, at randomly distributed points in the domain, through local meshless interpolation schemes such as the moving least squares(MLS) or radial basis functions(RBF). The nodal values of strains are expressed in terms of the independently interpolated nodal values of displacements, by simply enforcing the strain-displacement relationships directly by collocation at the nodal points. The MLPG local weak form is then written… More >

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 More >

B. Amaziane¹, A. Naji², D. Ouazar³

CMC-Computers, Materials & Continua, Vol.1, No.2, pp. 117-128, 2004, DOI:10.3970/cmc.2004.001.117

Abstract In this paper, a meshless method based on Radial Basis Functions (RBF) is coupled with genetic algorithms for parameter identification to some selected groundwater flow applications. The treated examples are generated by the diffusion equation with some specific boundary conditions describing the groundwater fluctuation in a leaky confined aquifer system near open tidal water. To select the best radial function interpolation and show the powerful of the method in comparison to domain based discretization methods, Multiquadric (MQ), Thin-Plate Spline (TPS) and Conical type functions are investigated and compared to finite difference results or analytical one. More >

S. N. Atluri¹, Z. D. Han¹, S. Shen¹

CMES-Computer Modeling in Engineering & Sciences, Vol.4, No.5, pp. 507-518, 2003, DOI:10.3970/cmes.2003.004.507

Abstract The general Meshless Local Petrov-Galerkin (MLPG) type weak-forms of the displacement & traction boundary integral equations are presented, for solids undergoing small deformations. These MLPG weak forms provide the most general basis for the numerical solution of the non-hyper-singular displacement and traction BIEs [given in Han, and Atluri (2003)], which are simply derived by using the gradients of the displacements of the fundamental solutions [Okada, Rajiyah, and Atluri (1989a,b)]. By employing the various types of test functions, in the MLPG-type weak-forms of the non-hyper-singular dBIE and tBIE over the local sub-boundary surfaces, several types of… More >

M.A. Golberg¹, C.S. Chen²

CMES-Computer Modeling in Engineering & Sciences, Vol.2, No.1, pp. 87-96, 2001, DOI:10.3970/cmes.2001.002.087

Abstract The purpose of this paper is to develop a highly efficient mesh-free method for solving nonlinear diffusion-reaction equations in R^d, d=2, 3. Using various time difference schemes, a given time-dependent problem can be reduced to solving a series of inhomogeneous Helmholtz-type equations. The solution of these problems can then be further reduced to evaluating particular solutions and the solution of related homogeneous equations. Recently, radial basis functions have been successfully implemented to evaluate particular solutions for Possion-type equations. A more general approach has been developed in extending this capability to obtain particular solutions for Helmholtz-type equations More >