Briti Sundar Deb¹, Srinivasan Natesan²

CMES-Computer Modeling in Engineering & Sciences, Vol.38, No.2, pp. 179-200, 2008, DOI:10.3970/cmes.2008.038.179

Abstract This paper presents an almost second--order uniformly convergent Richardson extrapolation method for convection- dominated coupled system of boundary value problems. First, we solve the system by using the classical finite difference scheme on the layer resolving Shishkin mesh, and then we construct the Richardson approximation solution using the solutions obtained on N and 2N mesh intervals. Second-order parameter--uniform error estimate is derived. The proposed method is applied to a test example for verification of the theoretical results for the case ε ≤ N⁻¹. More >

S.P. Hu¹, D.L. Young², C.M. Fan¹

CMES-Computer Modeling in Engineering & Sciences, Vol.24, No.1, pp. 1-20, 2008, DOI:10.3970/cmes.2008.024.001

Abstract In this paper, a novel numerical scheme called (FDMFS), which combines the finite difference method (FDM) and the method of fundamental solutions (MFS), is proposed to simulate the nonhomogeneous diffusion problem with an unsteady forcing function. Most meshless methods are confined to the investigations of nonhomogeneous diffusion equations with steady forcing functions due to the difficulty to find an unsteady particular solution. Therefore, we proposed a FDM with Cartesian grid to handle the unsteady nonhomogeneous term of the equations. The numerical solution in FDMFS is decomposed into a particular solution and a homogeneous solution. The particular solution is constructed using… More >

J.N. Johnson¹, J.M. Owen²

CMES-Computer Modeling in Engineering & Sciences, Vol.22, No.3, pp. 165-188, 2007, DOI:10.3970/cmes.2007.022.165

Abstract In this paper, we propose a Meshless Local Petrov-Galerkin method for studying the diffusion of a magnetic field within a non-magnetic (μ = μ₀) conducting medium with non-homogeneous and anisotropic electrical resistivity. We derive a local weak form for the magnetic diffusion equation and discuss the effects of different trial/test functions and nodal spacings on its solution. We then demonstrate that the method produces convergent results for several relevant one-dimensional test problems for which solutions are known. This method has the potential to be combined with other mesh-free methods such as Smoothed Particle Hydrodynamics (SPH) to solve problems in resistive… More >

A. J. Davies¹, D. Crann¹, S. J. Kane¹, C-H. Lai²

CMES-Computer Modeling in Engineering & Sciences, Vol.18, No.2, pp. 79-86, 2007, DOI:10.3970/cmes.2007.018.079

Abstract The solution process for diffusion problems usually involves the time development separately from the space solution. A finite difference algorithm in time requires a sequential time development in which all previous values must be determined prior to the current value. The Stehfest Laplace transform algorithm, however, allows time solutions without the knowledge of prior values. It is of interest to be able to develop a time-domain decomposition suitable for implementation in a parallel environment. One such possibility is to use the Laplace transform to develop coarse-grained solutions which act as the initial values for a set of fine-grained solutions. The… 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 >

Z. Chen¹, W. Hu¹, E.P. Chen²

CMES-Computer Modeling in Engineering & Sciences, Vol.1, No.4, pp. 57-62, 2000, DOI:10.3970/cmes.2000.001.509

Abstract To simulate the dynamic failure evolution without using nonlocal terms in the strain-stress space, a damage diffusion equation is formulated with the use of a combined damage/plasticity model that was primarily applied to the case of rock fragmentation. A vectorized model solver is developed for large-scale simulation. Two-dimensional sample problems are considered to illustrate the features of the proposed solution procedure. It appears that the proposed approach is effective in simulating the evolution of localization, with parallel computing, in a single computational domain involving different lower-order governing differential equations. More >

J. Lufrano¹, P. Sofronis¹

CMES-Computer Modeling in Engineering & Sciences, Vol.1, No.2, pp. 119-132, 2000, DOI:10.3970/cmes.2000.001.279

Abstract Transient hydrogen diffusion and hydride formation coupled with material deformation are studied in Zr-2.5Nb alloys used in the pressure tubes of CANDU nuclear generating stations. The energetics of the hydride formation is revisited and the terminal solid solubility of hydrogen in solution is defined on the basis of the total elastoplastic work done on the system by the forming hydride and the external loads. Probabilistic precipitation of hydride is modeled in the neighborhood of a crack tip under mode I plane strain loading and a uniform initial hydrogen concentration below the stress free terminal solid solubility. Finite element analysis is… 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 >

K. Garikipati¹, V.S. Rao², M.Y. Hao³, E. Ibok⁴, I. de Wolf⁵, R. W. Dutton⁶

CMES-Computer Modeling in Engineering & Sciences, Vol.1, No.1, pp. 65-84, 2000, DOI:10.3970/cmes.2000.001.065

Abstract This work is based upon a careful rendering of mechanics and mathematics to describe the phenomena that influence the stress engendered by the Shallow Trench Isolation process. The diffusion-reaction problem is posed in terms of fundamental mass balance laws. Finite strain kinematics is invoked to model the large expansion of SiO₂, dielectrics are modelled as viscoelastic solids and annealing-induced density relaxation of SiO₂ is incorporated as a history-dependent process. A levelset framework is used to describe the moving Si/SiO₂ interface. Sophisticated finite element methods are employed to solve the mathematical equations posed for each phenomenon. These include the incorporation of… More >

S. A. Mousavi¹, T. Yousefi², Z. Saghir³

FDMP-Fluid Dynamics & Materials Processing, Vol.13, No.4, pp. 213-220, 2017, DOI:10.3970/fdmp.2017.013.213

Abstract The main goal of this study was to investigate the thermal diffusion in ternary hydrocarbon mixtures composed of 1, 2, 3, 4 Tetrahydronaphtalene (THN)-Isobutylbenzene (IBB)-Dodecane (C12) with mass fractions of 80/10/10, 70/10/20, and 60/10/30 at mean temperature of 25 °C. Optical interferometry technique with Mach-Zehnder arrangement was used to conduct the experiments. The mixture was placed in a convectionless cell which was heated from above. The results for the mixture with mass fraction of 80/10/10 were in a good agreement with the corresponding benchmark values. Finally, the Soret coefficient for the other two mixtures have been proposed. More >