Weiwu Jiang¹, Xiaowei Gao^1,2,*

CMES-Computer Modeling in Engineering & Sciences, Vol.140, No.1, pp. 41-76, 2024, DOI:10.32604/cmes.2024.048313

Abstract The collocation method is a widely used numerical method for science and engineering problems governed by partial differential equations. This paper provides a comprehensive review of collocation methods and their applications, focused on elasticity, heat conduction, electromagnetic field analysis, and fluid dynamics. The merits of the collocation method can be attributed to the need for element mesh, simple implementation, high computational efficiency, and ease in handling irregular domain problems since the collocation method is a type of node-based numerical method. Beginning with the fundamental principles of the collocation method, the discretization process in the continuous domain is elucidated, and how… More >

Zhuojia Fu^1,*, Qiang Xi², Wenzhi Xu¹

The International Conference on Computational & Experimental Engineering and Sciences, Vol.27, No.1, pp. 1-1, 2023, DOI:10.32604/icces.2023.09393

Abstract In the past few decades, although traditional computational methods such as finite element have been successfully used in many scientific and engineering fields, they still face several challenging problems such as expensive computational cost, low computational efficiency, and difficulty in mesh generation in the numerical simulation of wave propagation under infinite domain, large-scale-ratio structures, engineering inverse problems and moving boundary problems. This paper introduces a class of collocation discretization techniques based on physical-informed kernel function (PIKF) to efficiently solve the above-mentioned problems. The key issue in the physical-informed kernel function collocation methods (PIKFCMs) is to construct the related basis functions,… More >

A.Wakif^a,* , Z. Boulahia^a, A. Amine^b , I.L. Animasaun^c , M. I. Afridi^d, M. Qasim^d , R. Sehaqui^a

Frontiers in Heat and Mass Transfer, Vol.12, pp. 1-15, 2019, DOI:10.5098/hmt.12.3

Abstract The significance of an externally applied magnetic field and an imposed negative temperature gradient on the onset of natural convection in a thin horizontal layer of alumina-water nanofluid for various sizes of spherical alumina nanoparticles (e.g., 30 More >

Jingwen Ren¹, Hongwei Lin^1,2,*

CMES-Computer Modeling in Engineering & Sciences, Vol.136, No.3, pp. 2957-2984, 2023, DOI:10.32604/cmes.2023.025983

Abstract Isogeometric analysis (IGA) is introduced to establish the direct link between computer-aided design and analysis. It is commonly implemented by Galerkin formulations (isogeometric Galerkin, IGA-G) through the use of nonuniform rational B-splines (NURBS) basis functions for geometric design and analysis. Another promising approach, isogeometric collocation (IGA-C), working directly with the strong form of the partial differential equation (PDE) over the physical domain defined by NURBS geometry, calculates the derivatives of the numerical solution at the chosen collocation points. In a typical IGA, the knot vector of the NURBS numerical solution is only determined by the physical domain. A new perspective… More > Graphic Abstract

M. Dehghan¹, M. Abbaszadeh², A. Mohebbi³

CMES-Computer Modeling in Engineering & Sciences, Vol.107, No.6, pp. 481-516, 2015, DOI:10.3970/cmes.2015.107.481

Abstract In the current paper the two-dimensional time fractional Klein-Kramers equation which describes the subdiffusion in the presence of an external force field in phase space has been considered. The numerical solution of fractional Klein-Kramers equation is investigated. The proposed method is based on using finite difference scheme in time variable for obtaining a semi-discrete scheme. Also, to achieve a full discretization scheme, the Kansa's approach and meshless local Petrov-Galerkin technique are used to approximate the spatial derivatives. The meshless method has already proved successful in solving classic and fractional differential equations as well as for several other engineering and physical… More >

Jin Li^1,2, De-hao Yu^3,4

CMES-Computer Modeling in Engineering & Sciences, Vol.102, No.2, pp. 103-126, 2014, DOI:10.3970/cmes.2014.102.103

Abstract The generalized composite middle rectangle rule for the computation of Hilbert integral is discussed. The pointwise superconvergence phenomenon is presented, i.e., when the singular point coincides with some a priori known point, the convergence rate of the rectangle rule is higher than what is global possible. We proved that the superconvergence rate of the composite middle rectangle rule occurs at certain local coordinate of each subinterval and the corresponding superconvergence error estimate is obtained. By choosing the superconvergence point as the collocation points, a collocation scheme for solving the relevant Hilbert integral equation is presented and an error estimate is… More >

D. Ho-Minh¹, N. Mai-Duy¹, T. Tran-Cong¹

CMES-Computer Modeling in Engineering & Sciences, Vol.70, No.3, pp. 217-252, 2010, DOI:10.3970/cmes.2010.070.217

Abstract In this paper, one-dimensional integrated radial-basis-function networks (1D-IRBFNs) are introduced into the Galerkin and point-collocation formulations to simulate viscoelastic flows. The computational domain is represented by a Cartesian grid and IRBFNs, which are constructed through integration, are employed on each grid line to approximate the field variables including stresses in the streamfunction-vorticity formulation. Two types of fluid, namely Oldroyd-B and CEF models, are considered. The proposed methods are validated through the numerical simulation of several benchmark test problems including flows in a rectangular duct and in a corrugated tube. Numerical results show that accurate results are obtained using relatively-coarse grids. More >

Chih-Ping Wu^1,2, Jian-Sin Wang², Yung-Ming Wang²

CMES-Computer Modeling in Engineering & Sciences, Vol.52, No.1, pp. 1-38, 2009, DOI:10.3970/cmes.2009.052.001

Abstract A meshless collocation method based on the differential reproducing kernel (DRK) interpolation is developed for the three-dimensional (3D) coupled analysis of simply-supported, functionally graded (FG) piezoelectric hollow cylinders. The material properties of FG hollow cylinders are regarded as heterogeneous through the thickness coordinate, and then specified to obey an exponent-law dependent on this. In the present formulation, the shape function for the reproducing kernel (RK) interpolation function at each sampling node is separated into a primitive function possessing Kronecker delta properties and an enrichment function constituting reproducing conditions. By means of this DRK interpolation, the essential boundary conditions can be… More >

Chih-Ping Wu^1,2, Kuan-Hao Chiu², Yung-Ming Wang²

CMES-Computer Modeling in Engineering & Sciences, Vol.35, No.3, pp. 181-214, 2008, DOI:10.3970/cmes.2008.035.181

Abstract A mesh-free collocation method based on differential reproducing kernel (DRK) approximations is developed for the three-dimensional (3D) analysis of simply-supported, doubly curved functionally graded (FG) magneto-electro-elastic shells under the mechanical load, electric displacement and magnetic flux. The material properties of FG shells are firstly regarded as heterogeneous through the thickness coordinate and then specified to obey an identical power-law distribution of the volume fractions of the constituents. The novelty of the present DRK-based collocation method is that the shape functions of derivatives of reproducing kernel (RK) approximants are determined using a set of differential reproducing conditions without directly taking the… More >

Shih-Wei Yang¹, Yung-Ming Wang¹, Chih-Ping Wu^1,2, Hsuan-Teh Hu¹

CMES-Computer Modeling in Engineering & Sciences, Vol.60, No.1, pp. 1-40, 2010, DOI:10.3970/cmes.2010.060.001

Abstract A differential reproducing kernel (DRK) approximation-based collocation method is developed for solving ordinary and partial differential equations governing the one- and two-dimensional problems of elastic bodies, respectively. In the conventional reproducing kernel (RK) approximation, the shape functions for the derivatives of RK approximants are determined by directly differentiating the RK approximants, and this is very time-consuming, especially for the calculations of their higher-order derivatives. Contrary to the previous differentiation manipulation, we construct a set of differential reproducing conditions to determine the shape functions for the derivatives of RK approximants. A meshless collocation method based on the present DRK approximation is… More >