Chein-Shan Liu¹

CMES-Computer Modeling in Engineering & Sciences, Vol.30, No.2, pp. 65-76, 2008, DOI:10.3970/cmes.2008.030.065

Abstract Trefftz method (TM) is one of widely used meshless numerical methods in elliptic type boundary value problems, of which the approximate solution is expressed as a linear combination of T-complete bases, and the unknown coefficients are determined from boundary conditions by solving a linear equations system. However, the accuracy of TM is severely limited by its ill-conditioning. This paper is a continuation of the work of Liu (2007a). The collocation TM is modified and applied to the direct and inverse problems of biharmonic equation in a simply connected plane domain. Due to its well-conditioning of the resulting linear equations system,… More >

Chein-Shan Liu¹

CMES-Computer Modeling in Engineering & Sciences, Vol.28, No.2, pp. 77-94, 2008, DOI:10.3970/cmes.2008.028.077

Abstract The method of fundamental solutions (MFS) is a truly meshless numerical method widely used in the elliptic type boundary value problems, of which the approximate solution is expressed as a linear combination of fundamental solutions and the unknown coefficients are determined from the boundary conditions by solving a linear equations system. However, the accuracy of MFS is severely limited by its ill-conditioning of the resulting linear equations system. This paper is motivated by the works of Chen, Wu, Lee and Chen (2007) and Liu (2007a). The first paper proved an equivalent relation of the Trefftz method and MFS for circular… 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 Fourier coefficients. Here, the factor… More >

Chein-Shan Liu ¹

CMES-Computer Modeling in Engineering & Sciences, Vol.20, No.2, pp. 111-122, 2007, DOI:10.3970/cmes.2007.020.111

Abstract A highly accurate new solver is developed to deal with interior and exterior mixed-boundary value problems for two-dimensional Laplace equation, including the singular ones. To promote the present study, we introduce a circular artificial boundary which is uniquely determined by the physical problem domain, and derive a Dirichlet to Robin mapping on that artificial circle, which is an exact boundary condition described by the first kind Fredholm integral equation. As a consequence, we obtain a modified Trefftz method equipped with a characteristic length factor, ensuring that the new solver is stable because the condition number can be greatly reduced. Then,… More >

H.B. Chen¹, D.J. Fu¹, P.Q. Zhang¹

CMES-Computer Modeling in Engineering & Sciences, Vol.20, No.2, pp. 85-98, 2007, DOI:10.3970/cmes.2007.020.085

Abstract Researches today show that, both approximation and dispersion errors are encountered by classical Galerkin FEM solutions for Helmholtz equation governing the harmonic wave propagation, which leads to numerical inaccuracies especially for high wave number cases. In this paper, Local Boundary Integral Equation Method (LBIEM) is firstly implemented to solve the boundary value problem of Helmholtz equation. Then the regularized LBIE is proposed to overcome the singularities of the boundary integrals in the LBIEM. Owing to the advantages of the Moving Least Square Approximation (MLSA), the frequency-dependent basis functions modified by the harmonic wave propagation solutions are easily adopted instead of… More >

D.L. Young¹, K.H. Chen², J.T. Chen³, J.H. Kao⁴

CMES-Computer Modeling in Engineering & Sciences, Vol.19, No.3, pp. 197-222, 2007, DOI:10.3970/cmes.2007.019.197

Abstract A boundary-type method for solving the Laplace problems using the modified method of fundamental solutions (MMFS) is proposed. The present method (MMFS) implements the singular fundamental solutions to evaluate the solutions, and it can locate the source points on the real boundary as contrasted to the conventional MFS, where a fictitious boundary is needed to avoid the singularity of diagonal term of influence matrices. The diagonal term of influence matrices for arbitrary domain can be novelly determined by relating the MFS with the indirect BEM and are also solved for circular domain analytically by using separable kernels and circulants. The… More >

Yu.A. Melnikov, M.Yu. Melnikov¹

CMES-Computer Modeling in Engineering & Sciences, Vol.2, No.2, pp. 291-306, 2001, DOI:10.3970/cmes.2001.002.291

Abstract The use of potential (integral) representations is studied when computing Green's functions for boundary value problems stated for Laplace and biharmonic equations over regions of complex configuration in two dimensions. The emphasis is on the non-traditional potentials, whose observation and source points occupy different sets. Such potentials reduce the original boundary value problems to functional (integral) equations with smooth kernels. Special integral representations are studied, the ones whose kernels are built not of the fundamental solutions of governing differential equations but of the Green's functions for simply shaped regions, which are associated with boundary value problems under consideration. Such integral… More >

Hayder A. Abdulbari ^1,2, Hassan D. Mahammed¹, Z. Hassan, Wafaa K. Mahmood³

FDMP-Fluid Dynamics & Materials Processing, Vol.12, No.2, pp. 86-101, 2016, DOI:10.3970/fdmp.2016.012.086

Abstract An experimental investigation was carried out to study the response of a turbulent pressure drop fluctuations to longitudinal groove riblets, involved two configurations being triangular and spaced triangular grooves with height 600, 800, 1000 μm and peak to peak spacing 1000 μm and 2000 μm respectively. Experiments were therefore performed at free stream velocity up to 0.44 m/sec, which were corresponding to Reynolds number (Re) 53000. The development of the obtained turbulent layer downstream of the grooves was then compared with the results from the corresponding smooth-wall case. To conclude, the effect of the spaced triangular riblets on the turbulent… More >

Y.H. Liu¹, X.X. Xue², J.M.Shen¹

FDMP-Fluid Dynamics & Materials Processing, Vol.11, No.2, pp. 197-204, 2015, DOI:10.3970/fdmp.2015.011.195

Abstract Blends of Polyethylene (PE) and modified-oil shale ash (MOSA) with different fractions of MOSA were prepared by the melting blend method. The effects of MOSA content on the rheological and mechanical properties of the blend were properly assessed via direct experimental analysis (more precisely, all rheological measurements were performed using a laboratory-scale XSS-300 torque rheometer with single screw extruder; the temperatures were maintained at 170°C, 180°C and 190°C under continuous extrusion). The prepared samples were observed to display a shear-thinning behaviour. Moreover, with increasing the MOSA content, we found the yield strength of the blends to increase, while its elongation… More >

D.V. Jayalakshmamma¹, Dinesh P.A.², M. Sankar³, D.V. Ch,rashekhar⁴

FDMP-Fluid Dynamics & Materials Processing, Vol.10, No.3, pp. 343-357, 2014, DOI:10.3970/fdmp.2014.010.343

Abstract We examine the motion of the two concentric immiscible liquid spheres with different viscosities in an electrically conducting fluid in the presence of transverse magnetic field. The inner sphere is assumed to move at a constant velocity. The Stoke’s equation along with the Lorentz force is considered to model the resulting fluid flow, analytical solutions being obtained by the similarity solution method in terms of modified Bessel’s functions. Streamlines related to the fluid circulation in the annulus between the two liquid spheres and inside the inner liquid sphere are presented for different combinations of the governing non-dimensional parameters. More >