M. Dehghan¹, M. Abbaszadeh², A. Mohebbi³

CMES-Computer Modeling in Engineering & Sciences, Vol.107, No.6, pp. 481-516, 2015, DOI:10.3970/cmes.2015.107.481

Abstract In the current paper the two-dimensional time fractional Klein-Kramers equation which describes the subdiffusion in the presence of an external force field in phase space has been considered. The numerical solution of fractional Klein-Kramers equation is investigated. The proposed method is based on using finite difference scheme in time variable for obtaining a semi-discrete scheme. Also, to achieve a full discretization scheme, the Kansa's approach and meshless local Petrov-Galerkin technique are used to approximate the spatial derivatives. The meshless method has already proved successful in solving classic and fractional differential equations as well as for several other engineering and physical… More >

N. Mai-Duy¹, T. Tran-Cong¹

The International Conference on Computational & Experimental Engineering and Sciences, Vol.14, No.2, pp. 51-56, 2010, DOI:10.3970/icces.2010.014.051

Abstract This paper presents a preconditioning scheme to improve the condition number of integrated radial-basis-function (RBF) matrices in solving large-scale 2D elliptic problems. The problem domain is discretised using a Cartesian grid, over which integrated RBF networks are employed to represent the field variable. The present preconditioner is constructed from 1D integrated RBF networks along grid lines. Test problems defined on rectangular and non-rectangular domains are employed to study the performance of the scheme. 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 example problems are solved by… 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 for the equilibrium equations over… 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 for a simple numerical implementation.… 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. Through two sample problems in… More >

G. Kosec¹, B. Šarler^1,2

CMC-Computers, Materials & Continua, Vol.26, No.3, pp. 227-254, 2011, DOI:10.3970/cmc.2011.026.227

Abstract This paper introduces an effective H-adaptive upgrade to solution of the transport phenomena by the novel Local Radial Basis Function Collocation Method (LRBFCM). The transport variable is represented on overlapping 5-noded influence-domains through collocation by using multiquadrics Radial Basis Functions (RBF). The involved first and second derivatives of the variable are calculated from the respective derivatives of the RBFs. The transport equation is solved through explicit time stepping. The H-adaptive upgrade includes refinement/derefinement of one to four nodes to/from the vicinity of the reference node. The number of the nodes added or removed depends on the topology of the reference… More >

R. Chen¹, X. Han^1,2, J. Liu¹, W. Zhang¹

CMC-Computers, Materials & Continua, Vol.25, No.2, pp. 135-158, 2011, DOI:10.3970/cmc.2011.025.135

Abstract In this paper, a computational inverse technique is presented to determine the constitutive parameters of concrete based on the penetration experiments. In this method, the parameter identification problem is formulated as an inverse problem, in which the parameters of the constitutive model can be characterized through minimizing error functions of the penetration depth measured in experiments and that computed by forward solver LS-DYNA. To reduce the time for forward calculation during the inverse procedure, radial basis function approximate model is used to replace the actual computational model. In order to improve the accuracy of approximation model, a local-densifying method combined… More >

Y. Sadeghi Ferezghi¹, M.R. Sohrabi¹, S.M Mosavi Nezhad ^{2, *}

CMES-Computer Modeling in Engineering & Sciences, Vol.113, No.4, pp. 497-520, 2017, DOI:10.3970/cmes.2017.113.497

Abstract In this paper, dynamic behavior of non-symmetric Functionally Graded (FG) cylindrical structure under shock loading is carried out. Dynamic equations in the polar coordinates are drawn out using Meshless Local Petrov-Galerkin (MLPG) method. Nonlinear volume fractions are used for radial direction to simulate the mechanical properties of Functionally Graded Material (FGM). To solve dynamic equations of non-symmetric FG cylindrical structure in the time domain, the MLPG method are combined with the Laplace transform method. For computing the inverse Laplace transform in the present paper, the Talbot algorithm for the numerical inversion is used. To verify the obtained results by the… More >