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  • Open Access

    ARTICLE

    Meshless Local Petrov-Galerkin (MLPG) Mixed Finite Difference Method for Solid Mechanics

    S. N. Atluri1, H. T. Liu2, Z. D. Han2

    CMES-Computer Modeling in Engineering & Sciences, Vol.15, No.1, pp. 1-16, 2006, DOI:10.3970/cmes.2006.015.001

    Abstract The Finite Difference Method (FDM), within the framework of the Meshless Local Petrov-Galerkin (MLPG) approach, is proposed in this paper for solving solid mechanics problems. A "mixed'' interpolation scheme is adopted in the present implementation: the displacements, displacement gradients, and stresses are interpolated independently using identical MLS shape functions. The system of algebraic equations for the problem is obtained by enforcing the momentum balance laws at the nodal points. The divergence of the stress tensor is established through the generalized finite difference method, using the scattered nodal values and a truncated Taylor expansion. The traction boundary conditions are imposed in… More >

  • Open Access

    ARTICLE

    Analysis of Elastodynamic Deformations near a Crack/Notch Tip by the Meshless Local Petrov-Galerkin (MLPG) Method

    R. C. Batra1, H.-K. Ching1

    CMES-Computer Modeling in Engineering & Sciences, Vol.3, No.6, pp. 717-730, 2002, DOI:10.3970/cmes.2002.003.717

    Abstract The Meshless Local Petrov-Galerkin (MLPG) method is used to analyze transient deformations near either a crack or a notch tip in a linear elastic plate. The local weak formulation of equations governing elastodynamic deformations is derived. It results in a system of coupled ordinary differential equations which are integrated with respect to time by a Newmark family of methods. Essential boundary conditions are imposed by the penalty method. The accuracy of the MLPG solution is established by comparing computed results for one-dimensional wave propagation in a rod with the analytical solution of the problem. Results are then computed for the… More >

  • Open Access

    ARTICLE

    Truly Meshless Local Petrov-Galerkin (MLPG) Solutions of Traction & Displacement BIEs

    Z. D. Han1, S. N. Atluri1

    CMES-Computer Modeling in Engineering & Sciences, Vol.4, No.6, pp. 665-678, 2003, DOI:10.3970/cmes.2003.004.665

    Abstract The numerical implementation of the truly Meshless Local Petrov-Galerkin (MLPG) type weak-forms of the displacement and traction boundary integral equations is presented, for solids undergoing small deformations. In the accompanying part I of this paper, the general MLPG/BIE weak-forms were presented [Atluri, Han and Shen (2003)]. The 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,… More >

  • Open Access

    ARTICLE

    Application of Meshless Local Petrov-Galerkin (MLPG) Method to Elastodynamic Problems in Continuously Nonhomogeneous Solids

    Jan Sladek1, Vladimir Sladek1, Chuanzeng Zhang2

    CMES-Computer Modeling in Engineering & Sciences, Vol.4, No.6, pp. 637-648, 2003, DOI:10.3970/cmes.2003.004.637

    Abstract A new computational method for solving transient elastodynamic initial-boundary value problems in continuously non-homogeneous solids, based on the meshless local Petrov-Galerkin (MLPG) method, is proposed in the present paper. The moving least squares (MLS) is used for interpolation and the modified fundamental solution as the test function. The local Petrov-Galerkin method for unsymmetric weak form in such a way is transformed to the local boundary integral equations (LBIE). The analyzed domain is divided into small subdomains, in which a weak solution is assumed to exist. Nodal points are randomly spread in the analyzed domain and each one is surrounded by… More >

  • Open Access

    ARTICLE

    A MLPG (LBIE) method for solving frequency domain elastic problems

    E. J. Sellountos1, D. Polyzos2

    CMES-Computer Modeling in Engineering & Sciences, Vol.4, No.6, pp. 619-636, 2003, DOI:10.3970/cmes.2003.004.619

    Abstract A new meshless local Petrov-Galerkin (MLPG) method for solving two dimensional frequency domain elastodynamic problems is proposed. Since the method utilizes, in its weak formulation, either the elastostatic or the frequency domain elastodynamic fundamental solution as test function, it is equivalent to the local boundary integral equation (LBIE) method. Nodal points spread over the analyzed domain are considered and the moving least squares (MLS) interpolation scheme for the approximation of the interior and boundary variables is employed. Two integral equations suitable for the integral representation of the displacement fields in the local sub- domains are used. The first utilizes the… More >

  • Open Access

    ARTICLE

    Meshless Local Petrov-Galerkin and RBFs Collocation Methods for Solving 2D Fractional Klein-Kramers Dynamics Equation on Irregular Domains

    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 >

  • Open Access

    ABSTRACT

    Analysis of Reissner-Mindlin Orthotropic FGM Plates by the MLPG

    J. Sladek, V. Sladek1, P. Solek, P.H. Wen

    The International Conference on Computational & Experimental Engineering and Sciences, Vol.8, No.4, pp. 165-170, 2008, DOI:10.3970/icces.2008.008.165

    Abstract A meshless local Petrov-Galerkin (MLPG) method is applied to solve thermal bending problems described by the Reissner-Mindlin theory. Functionally graded material properties with continuous variation in the plate thickness direction are considered too. A weak formulation for the set of governing equations in the Reissner-Mindlin theory is transformed into local integral equations on local subdomains in the mean surface of the plate with using a unit test function. The meshless approximation based on the Moving Least-Squares (MLS) method is employed for the implementation. More >

  • Open Access

    ABSTRACT

    Progress in improving computational efficiency of MLPG_R method for nonlinear water waves

    Q.W. Ma

    The International Conference on Computational & Experimental Engineering and Sciences, Vol.19, No.1, pp. 15-16, 2011, DOI:10.3970/icces.2011.019.015

    Abstract Over the past several years, the research group led by the author has extended the MPLG (Meshless Local Petrov-Galerkin) method developed by Prof. Atluri and his group to model nonlinear water waves; and then made further development to produce a method called MLPG_R (Meshless Local Petrov-Galerkin method based on Rankine source solution) method. In order to improve the computational efficiency of the method for modelling nonlinear water waves, several techniques have been developed. They include (1) introduction of a weak form of governing equation that does not contain derivatives of unknown functions; (2) a new meshless interpolation method of a… More >

  • Open Access

    ABSTRACT

    A comparison of various basis functions to linear stability of circular jet using MLPG method

    M.L. Xie1

    The International Conference on Computational & Experimental Engineering and Sciences, Vol.14, No.3, pp. 85-90, 2010, DOI:10.3970/icces.2010.014.085

    Abstract Various basis function based on Fourier-Chebyshev Petrov-Galerkin spectral method is described for computation of temporal linear stability in a circular jet. Basis functions presented here are exponentially mapped Chebyshev functions. There is a linear dependence between the components of the vector field according to the perturbation continuum equation. Therefore, there are only two degrees of freedom. According to the principle of permutation and combination, the basis function has three basic forms, i.e., the radial, azimuthal or axial component is free. The results show that three eigenvalues for various cases are consistent, but the basis function in case I is preferable… More >

  • Open Access

    ABSTRACT

    A comparison of various basis functions to linear stability of circular jet using MLPG method

    Xie Ming-Liang1

    The International Conference on Computational & Experimental Engineering and Sciences, Vol.14, No.2, pp. 57-62, 2010, DOI:10.3970/icces.2010.014.057

    Abstract Various basis function based on Fourier-Chebyshev Petrov-Galerkin spectral method is described for computation of temporal linear stability in a circular jet. Basis functions presented here are exponentially mapped Chebyshev functions. There is a linear dependence between the components of the vector field according to the perturbation continuum equation. Therefore, there are only two degrees of freedom. According to the principle of permutation and combination, the basis function has three basic forms, i.e., the radial, azimuthal or axial component is free. The results show that three eigenvalues for various cases are consistent, but the basis function in case I is preferable… More >

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