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 >

Arif Amirov¹, Fikret Gölgeleyen¹, Ayten Rahmanova²

The International Conference on Computational & Experimental Engineering and Sciences, Vol.12, No.4, pp. 125-136, 2009, DOI:10.3970/icces.2009.012.125

Abstract This paper has two purposes. The first is to prove the existence and uniqueness of the solution of an inverse problem for the general linear kinetic equation with a scattering term. The second one is to develop a numerical approximation method for the solution of this inverse problem for two dimensional case using finite difference method. More >

S.P. Hu¹, D.L. Young^1,2, C.M. Fan¹

The International Conference on Computational & Experimental Engineering and Sciences, Vol.6, No.1, pp. 25-50, 2008, DOI:10.3970/icces.2008.006.025

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 >

B. Mochnacki^*, Alicja Piasecka Belkhayat^†

Molecular & Cellular Biomechanics, Vol.10, No.3, pp. 233-244, 2013, DOI:10.3970/mcb.2013.010.233

Abstract Numerical analysis of heat transfer processes proceeding in a nonhomogeneous biological tissue domain is presented. In particular, the skin tissue domain subjected to an external heat source is considered. The problem is treated as an axially-symmetrical one (it results from the mathematical form of the function describing the external heat source). Thermophysical parameters of sub-domains (volumetric specific heat, thermal conductivity, perfusion coefficient etc.) are given as interval numbers. The problem discussed is solved using the interval finite difference method basing on the rules of directed interval arithmetic, this means that at the stage of FDM algorithm construction the mathematical manipulations… More >

C. Shu^1,2, W. X. Wu², C. M. Wang³

CMC-Computers, Materials & Continua, Vol.2, No.3, pp. 189-200, 2005, DOI:10.3970/cmc.2005.002.189

Abstract This paper demonstrates the application of a meshfree least square-based finite difference (LSFD) method for analysis of metallic waveguides. The waveguide problem is an eigenvalue problem that is governed by the Helmholtz equation. The second order derivatives in the Helmholtz equation are explicitly approximated by the LSFD formulations. TM modes and TE modes are calculated for some metallic waveguides with different cross-sectional shapes. Numerical examples show that the LSFD method is a very efficient meshfree method for waveguide analysis with complex domains. More >

Michely Laís de Oliveira^1,*, Marcio Augusto Villela Pinto², Simone de Fátima Tomazzoni Gonçalves², Grazielli Vassoler Rutz³

CMES-Computer Modeling in Engineering & Sciences, Vol.117, No.2, pp. 251-270, 2018, DOI:10.31614/cmes.2018.04237

Abstract Studies of problems involving physical anisotropy are applied in sciences and engineering, for instance, when the thermal conductivity depends on the direction. In this study, the multigrid method was used in order to accelerate the convergence of the iterative methods used to solve this type of problem. The asymptotic convergence factor of the multigrid was determined empirically (computer aided) and also by employing local Fourier analysis (LFA). The mathematical model studied was the 2D anisotropic diffusion equation, in which ε > 0 was the coefficient of a nisotropy. The equation was discretized by the Finite Difference Method (FDM) and Central… More >

X. X. Cui¹, X. Zhang^1,2, X. Zhou³, Y. Liu¹, F. Zhang¹

CMES-Computer Modeling in Engineering & Sciences, Vol.98, No.6, pp. 565-599, 2014, DOI:10.3970/cmes.2014.098.565

Abstract The material point method (MPM) discretizes the material domain by a set of particles, and has showed advantages over the mesh-based methods for many challenging problems associated with large deformation. However, at the same time, it requires more computational resource and has difficulties to construct high order scheme when simulating the fluid in high explosive (HE) explosion problems. A coupled finite difference material point (CFDMP) method is proposed through a bridge region to combine the advantages of the finite difference method (FDM) and MPM. It solves a 3D HE explosion and its interaction with the surrounding structures by dividing the… More >

M.D. Campos¹, E.C. Romão²

CMES-Computer Modeling in Engineering & Sciences, Vol.103, No.3, pp. 139-154, 2014, DOI:10.3970/cmes.2014.103.139

Abstract The objective of this paper aims to present a numerical solution of high accuracy and low computational cost for the three-dimensional Burgers equations. It is a well-known problem and studied the form for one and two-dimensional, but still little explored numerically for three-dimensional problems. Here, by using the High-Order Finite Difference Method for spatial discretization, the Crank-Nicolson method for time discretization and an efficient linearization technique with low computational cost, two numerical applications are used to validate the proposed formulation. In order to analyze the numerical error of the proposed formulation, an unpublished exact solution was used. More >

Po-Wei Li¹, Chia-Ming Fan^1,2, Chun-Yu Chen¹, Cheng-Yu Ku¹

CMES-Computer Modeling in Engineering & Sciences, Vol.101, No.5, pp. 319-350, 2014, DOI:10.3970/cmes.2014.101.319

Abstract A combination of the generalized finite difference method (GFDM), the implicit Euler method and the Newton-Raphson method is proposed to efficiently and accurately analyze the density-driven groundwater flows. In groundwater hydraulics, the problems of density-driven groundwater flows are usually difficult to be solved, since the mathematical descriptions are a system of time- and space-dependent nonlinear partial differential equations. In the proposed numerical scheme, the GFDM and the implicit Euler method were adopted for spatial and temporal discretizations of governing equations. The GFDM is a newly-developed meshless method and is truly free from time-consuming mesh generation and numerical quadrature. Based on… More >

Quan Zheng^1,2, Lei Fan¹, Xuezheng Li¹

CMES-Computer Modeling in Engineering & Sciences, Vol.100, No.6, pp. 445-461, 2014, DOI:10.3970/cmes.2014.100.445

Abstract In this paper, we construct a numerical method for one-dimensional Burgers’ equation in the unbounded domain by using artificial boundary conditions. The original problem is converted by Hopf-Cole transformation to the heat equation in the unbounded domain, the latter is reduced to an equivalent problem in a bounded computational domain by using two artificial integral boundary conditions, a finite difference method with discrete artificial boundary conditions is established by using the method of reduction of order for the last problem, and thereupon the numerical solution of Burgers’ equation is obtained. This artificial boundary method is proved and verified to be… More >