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New Trends on Meshless Method and Numerical Analysis

Submission Deadline: 31 December 2024 View: 91 Submit to Special Issue

Guest Editors

Prof. Ji Lin, Hohai University, China
Prof. Fajie Wang, Qingdao University, China

Summary

In recent years, in the category of numerical analysis, the meshless method has witnessed a research boom to free engineers and scientists from the difficult task of mesh generation and to reduce mesh sensitivity of solutions. Meshless methods include the kernel methods, the moving least square method, the radial basis functions, etc, such as the well-known smoothed particle hydrodynamics, the diffuse element method, the element-free Galerkin, the method of fundamental solution, the reproducing kernel particle method. The meshless method has become an attractive alternative for problems in computational mechanics, computational physics, computational chemistry, computational biology, computational materials science, etc.

The main target of this special issue is to focus on the latest developments of meshless methods, such as theoretical analysis, applications, development of new methods, fast solution techniques, etc. The applications of advanced techniques such as isogeometric analysis, artificial intelligence, the physical-based numerical model, etc, in meshless methods are also welcomed.


Keywords

Meshless method, Boundary-type meshless method, Localized meshless method, Fast algorithm, Domain-type meshless method, Isogeometric analysis, Artificial intelligence, Physical-based numerical model, Theoretical analysis, Applications

Published Papers


  • Open Access

    ARTICLE

    Analysis of Extended Fisher-Kolmogorov Equation in 2D Utilizing the Generalized Finite Difference Method with Supplementary Nodes

    Bingrui Ju, Wenxiang Sun, Wenzhen Qu, Yan Gu
    CMES-Computer Modeling in Engineering & Sciences, DOI:10.32604/cmes.2024.052159
    (This article belongs to the Special Issue: New Trends on Meshless Method and Numerical Analysis)
    Abstract In this study, we propose an efficient numerical framework to attain the solution of the extended Fisher-Kolmogorov (EFK) problem. The temporal derivative in the EFK equation is approximated by utilizing the Crank-Nicolson scheme. Following temporal discretization, the generalized finite difference method (GFDM) with supplementary nodes is utilized to address the nonlinear boundary value problems at each time node. These supplementary nodes are distributed along the boundary to match the number of boundary nodes. By incorporating supplementary nodes, the resulting nonlinear algebraic equations can effectively satisfy the governing equation and boundary conditions of the EFK equation. More >

  • Open Access

    ARTICLE

    Composite Fractional Trapezoidal Rule with Romberg Integration

    Iqbal M. Batiha, Rania Saadeh, Iqbal H. Jebril, Ahmad Qazza, Abeer A. Al-Nana, Shaher Momani
    CMES-Computer Modeling in Engineering & Sciences, Vol.140, No.3, pp. 2729-2745, 2024, DOI:10.32604/cmes.2024.051588
    (This article belongs to the Special Issue: New Trends on Meshless Method and Numerical Analysis)
    Abstract The aim of this research is to demonstrate a novel scheme for approximating the Riemann-Liouville fractional integral operator. This would be achieved by first establishing a fractional-order version of the -point Trapezoidal rule and then by proposing another fractional-order version of the -composite Trapezoidal rule. In particular, the so-called divided-difference formula is typically employed to derive the -point Trapezoidal rule, which has accordingly been used to derive a more accurate fractional-order formula called the -composite Trapezoidal rule. Additionally, in order to increase the accuracy of the proposed approximations by reducing the true errors, we incorporate More >

  • Open Access

    ARTICLE

    Finite Difference-Peridynamic Differential Operator for Solving Transient Heat Conduction Problems

    Chunlei Ruan, Cengceng Dong, Zeyue Zhang, Boyu Chen, Zhijun Liu
    CMES-Computer Modeling in Engineering & Sciences, Vol.140, No.3, pp. 2707-2728, 2024, DOI:10.32604/cmes.2024.050003
    (This article belongs to the Special Issue: New Trends on Meshless Method and Numerical Analysis)
    Abstract Transient heat conduction problems widely exist in engineering. In previous work on the peridynamic differential operator (PDDO) method for solving such problems, both time and spatial derivatives were discretized using the PDDO method, resulting in increased complexity and programming difficulty. In this work, the forward difference formula, the backward difference formula, and the centered difference formula are used to discretize the time derivative, while the PDDO method is used to discretize the spatial derivative. Three new schemes for solving transient heat conduction equations have been developed, namely, the forward-in-time and PDDO in space (FT-PDDO) scheme,… More >

  • Open Access

    ARTICLE

    MPI/OpenMP-Based Parallel Solver for Imprint Forming Simulation

    Yang Li, Jiangping Xu, Yun Liu, Wen Zhong, Fei Wang
    CMES-Computer Modeling in Engineering & Sciences, Vol.140, No.1, pp. 461-483, 2024, DOI:10.32604/cmes.2024.046467
    (This article belongs to the Special Issue: New Trends on Meshless Method and Numerical Analysis)
    Abstract In this research, we present the pure open multi-processing (OpenMP), pure message passing interface (MPI), and hybrid MPI/OpenMP parallel solvers within the dynamic explicit central difference algorithm for the coining process to address the challenge of capturing fine relief features of approximately 50 microns. Achieving such precision demands the utilization of at least 7 million tetrahedron elements, surpassing the capabilities of traditional serial programs previously developed. To mitigate data races when calculating internal forces, intermediate arrays are introduced within the OpenMP directive. This helps ensure proper synchronization and avoid conflicts during parallel execution. Additionally, in… More >

    Graphic Abstract

    MPI/OpenMP-Based Parallel Solver for Imprint Forming Simulation

  • Open Access

    ARTICLE

    Optimal Shape Factor and Fictitious Radius in the MQ-RBF: Solving Ill-Posed Laplacian Problems

    Chein-Shan Liu, Chung-Lun Kuo, Chih-Wen Chang
    CMES-Computer Modeling in Engineering & Sciences, Vol.139, No.3, pp. 3189-3208, 2024, DOI:10.32604/cmes.2023.046002
    (This article belongs to the Special Issue: New Trends on Meshless Method and Numerical Analysis)
    Abstract To solve the Laplacian problems, we adopt a meshless method with the multiquadric radial basis function (MQ-RBF) as a basis whose center is distributed inside a circle with a fictitious radius. A maximal projection technique is developed to identify the optimal shape factor and fictitious radius by minimizing a merit function. A sample function is interpolated by the MQ-RBF to provide a trial coefficient vector to compute the merit function. We can quickly determine the optimal values of the parameters within a preferred rage using the golden section search algorithm. The novel method provides the More >

  • Open Access

    ARTICLE

    Sub-Homogeneous Peridynamic Model for Fracture and Failure Analysis of Roadway Surrounding Rock

    Shijun Zhao, Qing Zhang, Yusong Miao, Weizhao Zhang, Xinbo Zhao, Wei Xu
    CMES-Computer Modeling in Engineering & Sciences, Vol.139, No.3, pp. 3167-3187, 2024, DOI:10.32604/cmes.2023.045015
    (This article belongs to the Special Issue: New Trends on Meshless Method and Numerical Analysis)
    Abstract The surrounding rock of roadways exhibits intricate characteristics of discontinuity and heterogeneity. To address these complexities, this study employs non-local Peridynamics (PD) theory and reconstructs the kernel function to represent accurately the spatial decline of long-range force. Additionally, modifications to the traditional bond-based PD model are made. By considering the micro-structure of coal-rock materials within a uniform discrete model, heterogeneity characterized by bond random pre-breaking is introduced. This approach facilitates the proposal of a novel model capable of handling the random distribution characteristics of material heterogeneity, rendering the PD model suitable for analyzing the deformation… More >

  • Open Access

    ARTICLE

    A Novel Method for Linear Systems of Fractional Ordinary Differential Equations with Applications to Time-Fractional PDEs

    Sergiy Reutskiy, Yuhui Zhang, Jun Lu, Ciren Pubu
    CMES-Computer Modeling in Engineering & Sciences, Vol.139, No.2, pp. 1583-1612, 2024, DOI:10.32604/cmes.2023.044878
    (This article belongs to the Special Issue: New Trends on Meshless Method and Numerical Analysis)
    Abstract This paper presents an efficient numerical technique for solving multi-term linear systems of fractional ordinary differential equations (FODEs) which have been widely used in modeling various phenomena in engineering and science. An approximate solution of the system is sought in the form of the finite series over the Müntz polynomials. By using the collocation procedure in the time interval, one gets the linear algebraic system for the coefficient of the expansion which can be easily solved numerically by a standard procedure. This technique also serves as the basis for solving the time-fractional partial differential equations More >

  • Open Access

    ARTICLE

    An Efficient Local Radial Basis Function Method for Image Segmentation Based on the Chan–Vese Model

    Shupeng Qiu, Chujin Lin, Wei Zhao
    CMES-Computer Modeling in Engineering & Sciences, Vol.139, No.1, pp. 1119-1134, 2024, DOI:10.32604/cmes.2023.030915
    (This article belongs to the Special Issue: New Trends on Meshless Method and Numerical Analysis)
    Abstract In this paper, we consider the Chan–Vese (C-V) model for image segmentation and obtain its numerical solution accurately and efficiently. For this purpose, we present a local radial basis function method based on a Gaussian kernel (GA-LRBF) for spatial discretization. Compared to the standard radial basis function method, this approach consumes less CPU time and maintains good stability because it uses only a small subset of points in the whole computational domain. Additionally, since the Gaussian function has the property of dimensional separation, the GA-LRBF method is suitable for dealing with isotropic images. Finally, a More >

    Graphic Abstract

    An Efficient Local Radial Basis Function Method for Image Segmentation Based on the Chan–Vese Model

  • Open Access

    ARTICLE

    An Effective Meshless Approach for Inverse Cauchy Problems in 2D and 3D Electroelastic Piezoelectric Structures

    Ziqiang Bai, Wenzhen Qu, Guanghua Wu
    CMES-Computer Modeling in Engineering & Sciences, Vol.138, No.3, pp. 2955-2972, 2024, DOI:10.32604/cmes.2023.031474
    (This article belongs to the Special Issue: New Trends on Meshless Method and Numerical Analysis)
    Abstract In the past decade, notable progress has been achieved in the development of the generalized finite difference method (GFDM). The underlying principle of GFDM involves dividing the domain into multiple sub-domains. Within each sub-domain, explicit formulas for the necessary partial derivatives of the partial differential equations (PDEs) can be obtained through the application of Taylor series expansion and moving-least square approximation methods. Consequently, the method generates a sparse coefficient matrix, exhibiting a banded structure, making it highly advantageous for large-scale engineering computations. In this study, we present the application of the GFDM to numerically solve More >

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