I. S. Raju¹, D. R. Phillips²

CMES-Computer Modeling in Engineering & Sciences, Vol.4, No.1, pp. 141-160, 2003, DOI:10.3970/cmes.2003.004.141

Abstract An accurate and yet simple Meshless Local Petrov-Galerkin (MLPG) formulation for analyzing beam problems is presented. In the formulation, simple weight functions are chosen as test functions as in the conventional MLPG method. Linear test functions are also chosen, leading to a variation of the MLPG method that is computationally efficient compared to the conventional implementation. The MLPG method is evaluated by applying the formulation to a variety of patch tests, thin beam problems, and problems with load discontinuities. The formulation successfully reproduces exact solutions to machine accuracy when higher order power and spline functions are chosen as test functions… More >

Satya N. Atluri¹, Shengping Shen¹

CMES-Computer Modeling in Engineering & Sciences, Vol.3, No.1, pp. 11-52, 2002, DOI:10.3970/cmes.2002.003.011

Abstract A comparison study of the efficiency and accuracy of a variety of meshless trial and test functions is presented in this paper, based on the general concept of the meshless local Petrov-Galerkin (MLPG) method. 5 types of trial functions, and 6 types of test functions are explored. Different test functions result in different MLPG methods, and six such MLPG methods are presented in this paper. In all these six MLPG methods, absolutely no meshes are needed either for the interpolation of the trial and test functions, or for the integration of the weak-form; while other meshless methods require background cells.… More >

Y. T. Gu, G. R. Liu¹

CMES-Computer Modeling in Engineering & Sciences, Vol.2, No.4, pp. 463-476, 2001, DOI:10.3970/cmes.2001.002.463

Abstract A meshless method for the analysis of Kirchhoff plates based on the Meshless Local Petrov-Galerkin (MLPG) concept is presented. A MLPG formulation is developed for static and free vibration analyses of thin plates. Local weak form is derived using the weighted residual method in local supported domains from the 4th order partial differential equation of Kirchhoff plates. The integration of the local weak form is performed in a regular-shaped local domain. The Moving Least Squares (MLS) approximation is used to constructed shape functions. The satisfaction of the high continuity requirements is easily met by MLS interpolant, which is based on… More >

H.-K. Ching, R. C. Batra¹

CMES-Computer Modeling in Engineering & Sciences, Vol.2, No.2, pp. 273-290, 2001, DOI:10.3970/cmes.2001.002.273

Abstract It is shown that the MLPG method augmented with the enriched basis functions and either the visibility criterion or the diffraction criterion successfully predicts the singular stress fields near a crack tip. Results are presented for a single edge-cracked plate and a double edge-cracked plate with plate edges parallel to the crack axis loaded in tension, the single edge-cracked plate with one plate edge parallel to the crack axis clamped and the other loaded by tangential tractions, and for a double edge-notched plate with the side between the notches loaded in compression. For the first three problems, the stress intensity… More >

H. Lin, S.N. Atluri¹

CMES-Computer Modeling in Engineering & Sciences, Vol.2, No.2, pp. 117-142, 2001, DOI:10.3970/cmes.2001.002.117

Abstract The truly Meshless Local Petrov-Galerkin (MLPG) method is extended to solve the incompressible Navier-Stokes equations. The local weak form is modified in a very careful way so as to ovecome the so-called Babus^~ka-Brezzi conditions. In addition, The upwinding scheme as developed in Lin and Atluri (2000a) and Lin and Atluri (2000b) is used to stabilize the convection operator in the streamline direction. Numerical results for benchmark problems show that the MLPG method is very promising to solve the convection dominated fluid mechanics problems. More >

H.-G. Kim, S. N. Atluri¹

CMES-Computer Modeling in Engineering & Sciences, Vol.1, No.3, pp. 11-32, 2000, DOI:10.3970/cmes.2000.001.313

Abstract The truly meshless local Petrov-Galerkin (MLPG) method holds a great promise in solving boundary value problems, using a local symmetric weak form as a natural approach. In the present paper, in the context of MLPG and the meshless interpolation of a moving least squares (MLS) type, a method which uses primary and secondary nodes in the domain and on the global boundary is introduced, in order to improve the accuracy of solution. The secondary nodes can be placed at any location where one needs to obtain a better resolution. The sub-domains for the shape functions in the MLS approximation are… More >

A.J. Yiotis¹, J.T. Katsikadelis¹

CMES-Computer Modeling in Engineering & Sciences, Vol.1, No.2, pp. 95-104, 2000, DOI:10.3970/cmes.2000.001.255

Abstract The Analog Equation Method is applied to the static and dynamic analysis of thin cylindrical shell panels. The Fl\"{u}gge theory is adopted. The three displacement components are established by solving two membrane and one plate bending problems under the same boundary conditions subjected to "appropriate'' (equivalent) fictitious loads. Numerical results are presented which illustrate the efficiency and the accuracy of the proposed method. More >

H. Lin, S.N. Atluri¹

CMES-Computer Modeling in Engineering & Sciences, Vol.1, No.2, pp. 45-60, 2000, DOI:10.3970/cmes.2000.001.205

Abstract Due to the very general nature of the Meshless Local Petrov-Galerkin (MLPG) method, it is very easy and natural to introduce the upwinding concept (even in multi-dimensional cases) in the MLPG method, in order to deal with convection-dominated flows. In this paper, several upwinding schemes are proposed, and applied to solve steady convection-diffusion problems, in one and two dimensions. Even for very high Peclet number flows, the MLPG method, with upwinding, gives very good results. It shows that the MLPG method is very promising to solve the convection-dominated flow problems, and fluid mechanics problems. More >

D.V. Jayalakshmamma¹, Dinesh P.A.², M. Sankar³, D.V. Ch,rashekhar⁴

FDMP-Fluid Dynamics & Materials Processing, Vol.10, No.3, pp. 343-357, 2014, DOI:10.3970/fdmp.2014.010.343

Abstract We examine the motion of the two concentric immiscible liquid spheres with different viscosities in an electrically conducting fluid in the presence of transverse magnetic field. The inner sphere is assumed to move at a constant velocity. The Stoke’s equation along with the Lorentz force is considered to model the resulting fluid flow, analytical solutions being obtained by the similarity solution method in terms of modified Bessel’s functions. Streamlines related to the fluid circulation in the annulus between the two liquid spheres and inside the inner liquid sphere are presented for different combinations of the governing non-dimensional parameters. More >

K.Y. Liu^1,2,3, S.Y. Long^1,2,4, G.Y. Li¹

CMC-Computers, Materials & Continua, Vol.7, No.1, pp. 43-58, 2008, DOI:10.3970/cmc.2008.007.043

Abstract A meshless local Petrov-Galerkin method (MLPG) [[Atluri and Zhu (1998)] for the analysis of cracks in isotropic functionally graded materials is presented. The meshless method uses the moving least squares (MLS) to approximate the field unknowns. The shape function has not the Kronecker Delta properties for the trial-function-interpolation, and a direct interpolation method is adopted to impose essential boundary conditions. The MLPG method does not involve any domain and singular integrals to generate the global effective stiffness matrix if body force is ignored; it only involves a regular boundary integral. The material properties are smooth functions of spatial coordinates and… More >