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 >

N. Mai-Duy, T. Tran-Cong¹

CMES-Computer Modeling in Engineering & Sciences, Vol.4, No.1, pp. 85-102, 2003, DOI:10.3970/cmes.2003.004.085

Abstract This paper reports a mesh-free Indirect Radial Basis Function Network method (IRBFN) using Thin Plate Splines (TPSs) for numerical solution of Differential Equations (DEs) in rectangular and curvilinear coordinates. The adjustable parameters required by the method are the number of centres, their positions and possibly the order of the TPS. The first and second order TPSs which are widely applied in numerical schemes for numerical solution of DEs are employed in this study. The advantage of the TPS over the multiquadric basis function is that the former, with a given order, does not contain the adjustable shape parameter (i.e. the… 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 >

P. Le¹, N. Mai-Duy¹, T. Tran-Cong¹, G. Baker²

CMC-Computers, Materials & Continua, Vol.5, No.2, pp. 129-150, 2007, DOI:10.3970/cmc.2007.005.129

Abstract This paper presents a numerical simulation of the formation and evolution of strain localization in elasto-thermo-viscoplastic materials (adiabatic shear band) by the indirect/integral radial basis function network (IRBFN) method. The effects of strain and strain rate hardening, plastic heating, and thermal softening are considered. The IRBFN method is enhanced by a new coordinate mapping which helps capture the stiff spatial structure of the resultant band. The discrete IRBFN system is integrated in time by the implicit fifth-order Runge-Kutta method. The obtained results are compared with those of the Modified Smooth Particle Hydrodynamics (MSPH) method and Chebychev Pseudo-spectral (CPS) method. More >

L. Dong¹, S. N. Atluri²

CMC-Computers, Materials & Continua, Vol.24, No.1, pp. 61-104, 2011, DOI:10.3970/cmc.2011.024.061

Abstract A simple procedure to formulate efficient and stable hybrid/mixed finite elements is developed, for applications in macro- as well as micromechanics. In this method, the strain and displacement field are independently assumed. Instead of using two-field variational principles to enforce both equilibrium and compatibility conditions in a variational sense, the independently assumed element strains are related to the strains derived from the independently assumed element displacements, at a finite number of collocation points within the element. The element stiffness matrix is therefore derived, by simply using the principle of minimum potential energy. Taking the four-node plane isoparametric element as an… More >