Chein-Shan Liu¹, Yung-Wei Chen², Jiang-Ren Chang²

The International Conference on Computational & Experimental Engineering and Sciences, Vol.3, No.2, pp. 93-100, 2007, DOI:10.3970/icces.2007.003.093

Abstract A new collocation method is developed here to solve the elliptic boundary value problems with singularities. Specifically, we consider the Motz problem as a test of the performance of the new method, which is found accurate and effective. More >

WU XueHong¹, SHEN ShengPing², TAO WenQuan^1,3

CMES-Computer Modeling in Engineering & Sciences, Vol.22, No.1, pp. 65-76, 2007, DOI:10.3970/cmes.2007.022.065

Abstract Meshless local Petrov-Galerkin collocation method is applied to compute two-dimensional heat conduction problems in irregular domain. By taking the Dirac's Delta function as the test function, the local domain integration is avoided. The essential boundary conditions can be implemented easily in this method. A case that has analytical solution shows the present method can obtain desired accuracy and efficient. Two cases in engineering are computed to validate the approach by comparing the present method with the finite volume method (FVM) solutions obtained from a commercial CFD package FLUENT 6.3. The results show that the present More >

Chein-Shan Liu¹

CMES-Computer Modeling in Engineering & Sciences, Vol.21, No.1, pp. 53-66, 2007, DOI:10.3970/cmes.2007.021.053

Abstract A newly modified Trefftz method is developed to solve the exterior and interior Dirichlet problems for two-dimensional Laplace equation, which takes the characteristic length of problem domain into account. After introducing a circular artificial boundary which is uniquely determined by the physical problem domain, we can derive a Dirichlet to Dirichlet mapping equation, which is an exact boundary condition. By truncating the Fourier series expansion one can match the physical boundary condition as accurate as one desired. Then, we use the collocation method and the Galerkin method to derive linear equations system to determine the More >

V. Kozulić¹, H. Gotovac¹, B. Gotovac¹

CMC-Computers, Materials & Continua, Vol.6, No.2, pp. 51-70, 2007, DOI:10.3970/cmc.2007.006.051

Abstract In this paper, we present a multi-resolution adaptive algorithm for solving problems described by partial differential equations. The technique is based on the collocation method using Fup basis functions, which belong to a class of Rvachev's infinitely differentiable finite functions. As it is possible to calculate derivation values of Fup basis functions of high degree in a precise yet simple way, so it is possible to efficiently apply strong formulation procedures. The mesh free method developed in this work is named Adaptive Fup Collocation Method (AFCM). The distribution of collocation points within the observed area… More >

S. Chantasiriwan¹

CMES-Computer Modeling in Engineering & Sciences, Vol.15, No.3, pp. 137-146, 2006, DOI:10.3970/cmes.2006.015.137

Abstract The multiquadric collocation method is the collocation method based on radial basis function known as multiquadrics. It has been successfully used to solve several linear and nonlinear problems. Although fluid flow problems are among problems previously solved by this method, there is still an outstanding issue regarding the influence of the free parameter of multiquadrics (or the shape parameter) on the performance of the method. This paper provides additional results of using the multiquadric collocation method to solve the lid-driven cavity flow problem. The method is used to solve the problem in the stream function-vorticity More >

S. N. Atluri¹, H. T. Liu², Z. D. Han²

CMES-Computer Modeling in Engineering & Sciences, Vol.14, No.3, pp. 141-152, 2006, DOI:10.3970/cmes.2006.014.141

Abstract The Meshless Local Petrov-Galerkin (MLPG) mixed collocation method is proposed in this paper, for solving elasticity problems. In the present MLPG approach, the mixed scheme is applied to interpolate the displacements and stresses independently, as in the MLPG finite volume method. To improve the efficiency, the local weak form is established at the nodal points, for the stresses, by using the collocation method. The traction boundary conditions are also imposed into the stress equations directly. It becomes very simple and straightforward to impose various boundary conditions, especially for the high-order PDEs. Numerical examples show that More >

Vladimir Sladek¹, Jan Sladek¹, Chuanzeng Zhang²

CMC-Computers, Materials & Continua, Vol.4, No.3, pp. 177-188, 2006, DOI:10.3970/cmc.2006.004.177

Abstract This paper concerns the stability, convergence of accuracy and cost efficiency of four various formulations for solution of boundary value problems in non-homogeneous elastic solids with functionally graded Young's modulus. The meshless point interpolation method is employed with using various basis functions. The interaction among the elastic continuum constituents is considered in the discretized formulation either by collocation of the governing equations or by integral satisfaction of the force equilibrium on local sub-domains. The exact benchmark solutions are used in numerical tests. More >

B. Šarler¹

CMES-Computer Modeling in Engineering & Sciences, Vol.7, No.2, pp. 185-194, 2005, DOI:10.3970/cmes.2005.007.185

Abstract This paper explores the application of the mesh-free radial basis function collocation method for solution of heat transfer and fluid flow problems. The solution procedure is represented for a Poisson reformulated general transport equation in terms of a-symmetric, symmetric and modified (double consideration of the boundary nodes) collocation approaches. In continuation, specifics of a primitive variable solution procedure for the coupled mass, momentum, and energy transport representing the natural convection in an incompressible Newtonian Bussinesq fluid are elaborated. A comparison of different collocation strategies is performed based on the two dimensional De Vahl Davis steady More >

A. La Rocca¹, H. Power¹, V. La Rocca², M. Morale²

CMC-Computers, Materials & Continua, Vol.2, No.4, pp. 239-250, 2005, DOI:10.3970/cmc.2005.002.239

Abstract This work presents a meshless numerical scheme for the solution of time dependent non linear heat transfer problems in terms of a radial basis function Hermite collocation approach. The proposed scheme is applied to foodstuff's samples during freezing process; evaluation of the time evolution of the temperature profile along the sample, as well as at the core, is carried out. The moving phase-change zone is identified in the domain and plotted at several timesteps. The robustness of the proposed scheme is tested by a comparison of the obtained numerical results with those found using a More >