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Mathematics
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Recent Progresses in Localized Meshless Methods
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Recent Progresses in Localized Meshless Methods

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Engineering Mathematics".

Deadline for manuscript submissions: closed (30 June 2020) | Viewed by 8873

Special Issue Editor

Prof. Dr. Weichung Yeih

E-Mail Website
Guest Editor
Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung 20201, Taiwan
Interests: localized meshless method; nonlinear iteration; Trefftz method; inverse problem; construction and building materials
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Localized meshless methods, which can avoid mesh generation and yield the sparse leading coefficient matrix at the same time, have been developed for decades. As a result, localized meshless methods have been applied in various fields in engineering for solving linear PDE or nonlinear PDE, especially for large-scale applications. Some challenges still remain for these localized meshless methods, such as solving nonlinear problems, solving ill-posed inverse problems, application to mutiple-scale problems, optimal size for the bandwidth of the sparse matrix, etc. We welcome you to publish your work related to the recent progress in localized meshless methods in this Special Issue.

The purpose of this Special Issue is to gather a collection of articles reflecting the latest developments in different fields of localized meshless methods such as the meshless local Petrov–Galerkin method, the reproducing kernel particle method (RKPM), the smoothed-particle hydrodynamics method, the material point method, the generalized finite difference method, the localized radial basis function collocation method, the localized method of fundamental solutions, the localized Trefftz method, and the localized method of approximate particular solutions.

Prof. Dr. Weichung Yeih
Guest Editor

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

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Keywords

  • Localized meshless method
  • Sparse matrix
  • Local Petrov–Galerkin method
  • Reproducing kernel particle method
  • Smoothed-particle hydrodynamics
  • Material point method
  • Generalized finite difference method
  • Localized radial basis function collocation method
  • Localized method of fundamental solutions
  • Localized method of approximate particular solutions
  • Localized Trefftz method

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Published Papers (4 papers)

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Research

19 pages, 6304 KiB  
Article
Detecting Inverse Boundaries by Weighted High-Order Gradient Collocation Method
by Judy P. Yang and Hon Fung Samuel Lam
Mathematics 2020, 8(8), 1297; https://doi.org/10.3390/math8081297 - 5 Aug 2020
Cited by 6 | Viewed by 1586
Abstract
The weighted reproducing kernel collocation method exhibits high accuracy and efficiency in solving inverse problems as compared with traditional mesh-based methods. Nevertheless, it is known that computing higher order reproducing kernel (RK) shape functions is generally an expensive process. Computational cost may dramatically [...] Read more.
The weighted reproducing kernel collocation method exhibits high accuracy and efficiency in solving inverse problems as compared with traditional mesh-based methods. Nevertheless, it is known that computing higher order reproducing kernel (RK) shape functions is generally an expensive process. Computational cost may dramatically increase, especially when dealing with strong-from equations where high-order derivative operators are required as compared to weak-form approaches for obtaining results with promising levels of accuracy. Under the framework of gradient approximation, the derivatives of reproducing kernel shape functions can be constructed synchronically, thereby alleviating the complexity in computation. In view of this, the present work first introduces the weighted high-order gradient reproducing kernel collocation method in the inverse analysis. The convergence of the method is examined through the weights imposed on the boundary conditions. Then, several configurations of multiply connected domains are provided to numerically investigate the stability and efficiency of the method. To reach the desired accuracy in detecting the outer boundary for two special cases, special treatments including allocation of points and use of ghost points are adopted as the solution strategy. From four benchmark examples, the efficacy of the method in detecting the unknown boundary is demonstrated. Full article
(This article belongs to the Special Issue Recent Progresses in Localized Meshless Methods)
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14 pages, 2043 KiB  
Article
Solving the Nonlinear Heat Equilibrium Problems Using the Local Multiquadric Radial Basis Function Collocation Method
by Weichung Yeih
Mathematics 2020, 8(8), 1289; https://doi.org/10.3390/math8081289 - 5 Aug 2020
Cited by 4 | Viewed by 1825
Abstract
In this article, the nonlinear heat equilibrium problems are solved by the local multiquadric (MQ) radial basis function (RBF) collocation method. The system of nonlinear algebraic equations is solved by iteration based on the residual norm-based algorithm, in which the direction of evolution [...] Read more.
In this article, the nonlinear heat equilibrium problems are solved by the local multiquadric (MQ) radial basis function (RBF) collocation method. The system of nonlinear algebraic equations is solved by iteration based on the residual norm-based algorithm, in which the direction of evolution is determined by a linear equation. In addition, the role of the collocation point and source point is clearly defined such that in our proposed method the field value of any interested point can be expressed. Six numerical examples are shown to check the performance of the proposed method. As the number of supporting points (mp) increases, the accuracy of numerical solution increases. Among all examples, mp = 50 can perform well. In addition, the selection of shape parameter, c, affects the accuracy. However, as c < 2 the maximum relative absolute error percentage is less than 1%. Full article
(This article belongs to the Special Issue Recent Progresses in Localized Meshless Methods)
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16 pages, 4337 KiB  
Article
Localized Boundary Knot Method for Solving Two-Dimensional Laplace and Bi-Harmonic Equations
by Jingang Xiong, Jiancong Wen and Yan-Cheng Liu
Mathematics 2020, 8(8), 1218; https://doi.org/10.3390/math8081218 - 24 Jul 2020
Cited by 9 | Viewed by 3079
Abstract
In this paper, a localized boundary knot method is proposed, based on the local concept in the localized method of fundamental solutions. The localized boundary knot method is formed by combining the classical boundary knot method and the localization approach. The localized boundary [...] Read more.
In this paper, a localized boundary knot method is proposed, based on the local concept in the localized method of fundamental solutions. The localized boundary knot method is formed by combining the classical boundary knot method and the localization approach. The localized boundary knot method is truly free from mesh and numerical quadrature, so it has great potential for solving complicated engineering applications, such as multiply connected problems. In the proposed localized boundary knot method, both of the boundary nodes and interior nodes are required, and the algebraic equations at each node represent the satisfaction of the boundary condition or governing equation, which can be derived by using the boundary knot method at every subdomain. A sparse system of linear algebraic equations can be yielded using the proposed localized boundary knot method, which can greatly reduce the computer time and memory required in computer calculations. In this paper, several cases of simply connected domains and multi-connected domains of the Laplace equation and bi-harmonic equation are demonstrated to evidently verify the accuracy, convergence and stability of this proposed meshless method. Full article
(This article belongs to the Special Issue Recent Progresses in Localized Meshless Methods)
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22 pages, 1692 KiB  
Article
Wave-Structure Interaction for a Stationary Surface-Piercing Body Based on a Novel Meshless Scheme with the Generalized Finite Difference Method
by Ji Huang, Hongguan Lyu, Chia-Ming Fan, Jiahn-Hong Chen, Chi-Nan Chu and Jiayang Gu
Mathematics 2020, 8(7), 1147; https://doi.org/10.3390/math8071147 - 14 Jul 2020
Cited by 7 | Viewed by 2086
Abstract
The wave-structure interaction for surface-piercing bodies is a challenging problem in both coastal and ocean engineering. In the present study, a two-dimensional numerical wave flume that is based on a newly-developed meshless scheme with the generalized finite difference method (GFDM) is constructed in [...] Read more.
The wave-structure interaction for surface-piercing bodies is a challenging problem in both coastal and ocean engineering. In the present study, a two-dimensional numerical wave flume that is based on a newly-developed meshless scheme with the generalized finite difference method (GFDM) is constructed in order to investigate the characteristics of the hydrodynamic loads acting on a surface-piercing body caused by the second-order Stokes waves. Within the framework of the potential flow theory, the second-order Runge-Kutta method (RKM2) in conjunction with the semi-Lagrangian approach is carried out to discretize the temporal variable of governing equations. At each time step, the GFDM is employed to solve the spatial variable of the Laplace’s equation for the deformable computational domain. The results show that the developed numerical method has good performance in the simulation of wave-structure interaction, which suggests that the proposed “RKM2-GFDM” meshless scheme can be a feasible tool for such and more complicated hydrodynamic problems in practical engineering. Full article
(This article belongs to the Special Issue Recent Progresses in Localized Meshless Methods)
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