Weiran Yuan¹, Pu Chen^1,2, Kaishin Liu^1,3

CMES-Computer Modeling in Engineering & Sciences, Vol.17, No.2, pp. 115-134, 2007, DOI:10.3970/cmes.2007.017.115

Abstract In this paper we propose a direct solution method for the quasi-unsymmetric sparse matrix (QUSM) arising in the Meshless Local Petrov-Galerkin method (MLPG). QUSM, which is conventionally treated as a general unsymmetric matrix, is unsymmetric in its numerical values, but nearly symmetric in its nonzero distribution of upper and lower triangular portions. MLPG employs trial and test functions in different functional spaces in the local domain weak form of governing equations. Consequently the stiffness matrix of the resultant linear system is a QUSM. The new solver for QUSM conducts a two-level unrolling technique for LDU factorization method and can be… More >

Q.W. Ma¹

CMES-Computer Modeling in Engineering & Sciences, Vol.9, No.2, pp. 193-210, 2005, DOI:10.3970/cmes.2005.009.193

Abstract Recently, the MLPG (Meshless Local Petrov-Galerkin Method) method has been successfully extended to simulating nonlinear water waves [Ma, (2005)]. In that paper, the author employed the Heaviside step function as the test function to formulate the weak form over local sub-domains, acquiring an expression in terms of pressure gradient. In this paper, the solution for Rankine sources is taken as the test function and the local weak form is expressed in term of pressure rather than pressure gradient. Apart from not including pressure gradient, velocity gradient is also eliminated from the weak form. In addition, a semi-analytical technique is developed… More >

U. Andreaus^1,3, R.C. Batra², M. Porfiri^{2, 3}

CMES-Computer Modeling in Engineering & Sciences, Vol.9, No.2, pp. 111-132, 2005, DOI:10.3970/cmes.2005.009.111

Abstract Structural health monitoring techniques based on vibration data have received increasing attention in recent years. Since the measured modal characteristics and the transient motion of a beam exhibit low sensitivity to damage, numerical techniques for accurately computing vibration characteristics are needed. Here we use a Meshless Local Petrov-Galerkin (MLPG) method to analyze vibrations of a beam with multiple cracks. The trial and the test functions are constructed using the Generalized Moving Least Squares (GMLS) approximation. The smoothness of the GMLS basis functions requires the use of special techniques to account for the slope discontinuities at the crack locations. Therefore, a… More >

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

CMES-Computer Modeling in Engineering & Sciences, Vol.10, No.1, pp. 1-12, 2005, DOI:10.3970/cmes.2005.010.001

Abstract A nonlinear formulation of the Meshless Local Petrov-Galerkin (MLPG) finite-volume mixed method is developed for the large deformation analysis of static and dynamic problems. In the present MLPG large deformation formulation, the velocity gradients are interpolated independently, to avoid the time consuming differentiations of the shape functions at all integration points. The nodal values of velocity gradients are expressed in terms of the independently interpolated nodal values of displacements (or velocities), by enforcing the compatibility conditions directly at the nodal points. For validating the present large deformation MLPG formulation, two example problems are considered: 1) large deformations and rotations of… More >

J. Sorić¹, Q. Li², T. Jarak¹, S.N. Atluri²

CMES-Computer Modeling in Engineering & Sciences, Vol.6, No.4, pp. 349-358, 2004, DOI:10.3970/cmes.2004.006.349

Abstract An efficient meshless formulation based on the Local Petrov-Galerkin approach for the analysis of shear deformable thick plates is presented. Using the kinematics of a three-dimensional continuum, the local symmetric weak form of the equilibrium equations over the cylindrical shaped local sub-domain is derived. The linear test function in the plate thickness direction is assumed. Discretization in the in-plane directions is performed by means of the moving least squares approximation. The linear interpolation over the thickness is used for the in-plane displacements, while the hierarchical quadratic interpolation is adopted for the transversal displacement in order to avoid the thickness locking… More >

Q. Li¹, S. Shen¹, Z. D. Han¹, S. N. Atluri¹

CMES-Computer Modeling in Engineering & Sciences, Vol.4, No.5, pp. 571-586, 2003, DOI:10.3970/cmes.2003.004.571

Abstract In this paper, a truly meshless method, the Meshless Local Petrov-Galerkin (MLPG) Method, is developed for three-dimensional elasto-statics. The two simplest members of MLPG family of methods, the MLPG type 5 and MLPG type 2, are combined, in order to reduce the computational requirements and to obtain high efficiency. The MLPG5 method is applied at the nodes inside the 3-D domain, so that any domain integration is eliminated altogether, if no body forces are involved. The MLPG 2 method is applied at the nodes that are on the boundaries, and on the interfaces of material discontinuities, so that the boundary… More >

L. F. Qian^1,2, R. C. Batra³, L. M. Chen¹

CMES-Computer Modeling in Engineering & Sciences, Vol.4, No.5, pp. 519-534, 2003, DOI:10.3970/cmes.2003.004.519

Abstract We use a meshless local Petrov-Galerkin (MLPG) method to analyze three-dimensional infinitesimal elastodynamic deformations of a homogeneous rectangular plate subjected to different edge conditions. We employ a higher-order plate theory in which both transverse shear and transverse normal deformations are considered. Natural frequencies and the transient response to external loads have been computed for isotropic and orthotropic plates. Computed results are found to agree with those obtained from the analysis of the 3-dimensional problem either analytically or by the finite element method. 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 MLPG/BIEs are formulated, while also… More >

Z.Tang, S. Shen, S.N. Atluri¹

CMES-Computer Modeling in Engineering & Sciences, Vol.4, No.1, pp. 177-196, 2003, DOI:10.3970/cmes.2003.004.177

Abstract A meshless numerical implementation is reported of the 2-D Fleck-Hutchinson phenomenological strain-gradient theory, which fits within the framework of the Toupin-Mindlin theories and deals with first-order strain gradients and the associated work-conjugate higher-order stresses. From a mathematical point of view, the two-dimensional Toupin-Mindlin strain gradient theory is a generalization of the Poisson-Kirchhoff plate theories, involving, in addition to the fourth-order derivatives of the displacements, also a second-order derivative. In the conventional displacement-based approaches in FEM, the interpolation of displacement requires C$^{1}$ --continuity (in order to ensure convergence of the finite element procedure for 4$^{th}$ order theories), which inevitably involves very… More >

L. F. Qian^1,3, R. C. Batra², L. M. Chen³

CMES-Computer Modeling in Engineering & Sciences, Vol.4, No.1, pp. 161-176, 2003, DOI:10.3970/cmes.2003.004.161

Abstract We use two meshless local Petrov-Galerkin formulations, namely, the MLPG1 and the MLPG5, to analyze infinitesimal deformations of a homogeneous and isotropic thick elastic plate with a higher-order shear and normal deformable plate theory. It is found that the two MLPG formulations give results very close to those obtained by other researchers and also by the three-dimensional analysis of the problem by the finite element method. More >