J. Sladek¹, V. Sladek¹, S.N. Atluri²

CMES-Computer Modeling in Engineering & Sciences, Vol.2, No.4, pp. 423-434, 2001, DOI:10.3970/cmes.2001.002.423

Abstract A new meshless method for solving stationary thermoelastic boundary value problems is proposed in the present paper. The moving least square (MLS) method is used for the approximation of physical quantities in the local boundary integral equations (LBIE). In stationary thermoelasticity, the temperature and displacement fields are uncoupled. In the first step, the temperature field, described by the Laplace equation, is analysed by the LBIE. Then, the mechanical quantities are obtained from the solution of the LBIEs, which are reduced to elastostatic ones with redefined body forces due to thermal loading. The domain integrals with temperature gradients are transformed to… More >

Jianming Zhang, Zhenhan Yao¹

CMES-Computer Modeling in Engineering & Sciences, Vol.2, No.3, pp. 307-318, 2001, DOI:10.3970/cmes.2001.002.307

Abstract The Meshless Local Boundary Integral Equation (MLBIE) method is a truly meshless one as it does not need a 'finite element or boundary element mesh', either for variable interpolation or for 'energy' integration. The Boundary Node Method (BNM) further reduces the dimensionality of the problem by one, i.e. it only requires nodes constructed on the surface. However, the BNM is not truly meshless, as a background mesh is needed for boundary integration; and the MLBIE does not have the advantage of reduced dimensionality as the BNM. A new Regular Hybrid Boundary Node method based on a modified functional and the… 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 >

Xiang-YangLi¹, Shang-Hua Teng², Peng-Jun Wan³

CMES-Computer Modeling in Engineering & Sciences, Vol.2, No.1, pp. 97-116, 2001, DOI:10.3970/cmes.2001.002.097

Abstract To conduct numerical simulations by finite element methods, we often need to generate a high quality mesh, yet with a smaller number of elements. Moreover, the size of each of the elements in the mesh should be approximately equal to a given size requirement. Li et al. recently proposed a new method, named biting, which combines the strengths of advancing front and sphere packing. It generates high quality meshes with a theoretical guarantee. In this paper, we show that biting squares instead of circles not only generates high quality meshes but also has the following advantages. It is easier to… More >

M.A. Golberg¹, C.S. Chen²

CMES-Computer Modeling in Engineering & Sciences, Vol.2, No.1, pp. 87-96, 2001, DOI:10.3970/cmes.2001.002.087

Abstract The purpose of this paper is to develop a highly efficient mesh-free method for solving nonlinear diffusion-reaction equations in R^d, d=2, 3. Using various time difference schemes, a given time-dependent problem can be reduced to solving a series of inhomogeneous Helmholtz-type equations. The solution of these problems can then be further reduced to evaluating particular solutions and the solution of related homogeneous equations. Recently, radial basis functions have been successfully implemented to evaluate particular solutions for Possion-type equations. A more general approach has been developed in extending this capability to obtain particular solutions for Helmholtz-type equations by using polyharmonic spline… 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 >

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 >

C.J. Wordelman, N.R. Aluru, U. Ravaioli¹

CMES-Computer Modeling in Engineering & Sciences, Vol.1, No.1, pp. 121-126, 2000, DOI:10.3970/cmes.2000.001.121

Abstract This paper describes the application of the meshless Finite Point (FP) method to the solution of the nonlinear semiconductor Poisson equation. The FP method is a true meshless method which uses a weighted least-squares fit and point collocation. The nonlinearity of the semiconductor Poisson equation is treated by Newton-Raphson iteration, and sparse matrices are employed to store the shape function and coefficient matrices. Using examples in two- and three-dimensions (2- and 3-D) for a prototypical n-channel MOSFET, the FP method demonstrates promise both as a means of mesh enhancement and for treating problems where arbitrary point placement is advantageous, such… More >

Jacek Narski^1,2, Marco Picasso¹

FDMP-Fluid Dynamics & Materials Processing, Vol.3, No.1, pp. 49-64, 2007, DOI:10.3970/fdmp.2007.003.049

Abstract An adaptive phase field model for the solidification of binary alloys in three space dimensions is presented. The fluid flow in the liquid due to different liquid/solid densities is taken into account. The unknowns are the phase field, the alloy concentration and the velocity/pressure in the liquid. Continuous, piecewise linear finite elements are used for the space discretization, a semi-implicit scheme is used for time discretization. An adaptive method allows the number of degrees of freedom to be reduced, the mesh tetrahedrons having high aspect ratio whenever needed. Numerical results show that our method is effective and allows to perform… More >