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  • Open Access


    Research Advances on the Collocation Methods Based on the PhysicalInformed Kernel Functions

    Zhuojia Fu1,*, 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 >

  • Open Access



    A.Wakifa,* , Z. Boulahiaa, A. Amineb , I.L. Animasaunc , M. I. Afridid, M. Qasimd , R. Sehaquia

    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 >

  • Open Access


    New Perspective to Isogeometric Analysis: Solving Isogeometric Analysis Problem by Fitting Load Function

    Jingwen Ren1, Hongwei Lin1,2,*

    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

    New Perspective to Isogeometric Analysis: Solving Isogeometric Analysis Problem by Fitting Load Function

  • Open Access


    Meshless Local Petrov-Galerkin and RBFs Collocation Methods for Solving 2D Fractional Klein-Kramers Dynamics Equation on Irregular Domains

    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 >

  • Open Access


    Collocation Methods to Solve Certain Hilbert Integral Equation with Middle Rectangle Rule

    Jin Li1,2, De-hao Yu3,4

    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 >

  • Open Access


    Galerkin/Collocation Methods Based on 1D-Integrated-RBFNs for Viscoelastic Flows

    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 >

  • Open Access


    A DRK Interpolation-Based Collocation Method for the Analysis of Functionally Graded Piezoelectric Hollow Cylinders under Electro-Mechanical Loads

    Chih-Ping Wu1,2, 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 >

  • Open Access


    A Mesh-Free DRK-Based Collocation Method for the Coupled Analysis of Functionally Graded Magneto-Electro-Elastic Shells and Plates

    Chih-Ping Wu1,2, 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 >

  • Open Access


    A Meshless Collocation Method Based on the Differential Reproducing Kernel Approximation

    Shih-Wei Yang1, Yung-Ming Wang1, Chih-Ping Wu1,2, 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 >

  • Open Access


    Adaptive Support Domain Implementation on the Moving Least Squares Approximation for Mfree Methods Applied on Elliptic and Parabolic PDE Problems Using Strong-Form Description

    G. C. Bourantas1, E. D. Skouras2,3,4, G. C. Nikiforidis1

    CMES-Computer Modeling in Engineering & Sciences, Vol.43, No.1, pp. 1-26, 2009, DOI:10.3970/cmes.2009.043.001

    Abstract The extent of application of meshfree methods based on point collocation (PC) techniques with adaptive support domain for strong form Partial Differential Equations (PDE) is investigated. The basis functions are constructed using the Moving Least Square (MLS) approximation. The weak-form description of PDEs is used in most MLS methods to circumvent problems related to the increased level of resolution necessary near natural (Neumann) boundary conditions (BCs), dislocations, or regions of steep gradients. Alternatively, one can adopt Radial Basis Function (RBF) approximation on the strong-form of PDEs using meshless PC methods, due to the delta function behavior (exact solution on nodes).… More >

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