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generalized finite difference method applied to solve seismic wave propagation problems. Examples of 3D simulations

MATHEMATICAL methodS IN THE APPLIED SCIENCES 2025年第7期48卷 7706-7726页

作者： Flores, Jesus Salete, Eduardo Benito, Juan Jose Vargas, Antonio M. Conde, Eduardo R. UNED Escuela Tecn Super Ingn Ind Madrid Spain UNED Fac Ciencias Madrid Spain

The simulation of seismic wave propagation generally requires dealing with complex tridimensional geometries that are irregular in shape and have non-uniform properties, features that make the application of the generalized finite difference method in this field interesting. This work continues the extensive developments by the research team focused on the simulation of seismic wave propagation in two-dimensional domains. In this new contribution, the general formulation and the treatment of free surface boundary conditions are extended for the three-dimensional case, and the results obtained from different examples are analyzed.

关键词： generalized finite difference method meshless method wave propagation

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generalized finite difference method based meshless analysis for coupled two-phase porous flow and geomechanics

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ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS 2023年 146卷 184-203页

作者： Liu, Yina Rao, Xiang Zhao, Hui Zhan, Wentao Xu, Yunfeng Liu, Yuan Yangtze Univ Sch Petr Engn Wuhan 430100 Peoples R China Key Lab Drilling & Prod Engn Oil & Gas Wuhan 430100 Hubei Peoples R China Yangtze Univ Cooperat Innovat Ctr Unconvent Oil & Gas Minist Educ & Hubei Prov Wuhan 430100 Peoples R China

This paper applies GFDM to analyze coupled two-phase flow and geomechanics for the first time, to our knowledge, is also the first meshless solver for two-phase fluid-solid coupling problems in porous media. Firstly, the implicit GFDM-based discrete schemes of the hyperbolic two-phase flow equation and elliptic stress equilibrium equation coupled by Biot's poroelastic theory are derived. Then the nonlinear solver based on Newton's method is used to solve the fluid pressure, water saturation, and displacement at each node simultaneously. Two technical details are found critical to the implementation of this method. The one is using different radii of the node influence domain to discretize the flow equation and the stress equilibrium equation respectively, and the other one is the selection criterion of the two radii of the node influence domain, which improves the implementation flexibility of the multi-physics coupling calculation, reduces the dissipation error of the calculation results of the convection-dominated water saturation distribution, and ensure a low computational cost. Several numerical test cases are implemented to analyze the computational efficiency and accuracy, and illustrate that GFDM can achieve good computational performance in case of regular domains, irregular domains, and different point clouds. Overall, this work may provide an important reference for building a GFDM-based general-purpose numerical simulator for multi-physics flow problems in porous media.

关键词： generalized finite difference method Meshless method Fluid -solid coupling problems Porous flow

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generalized finite difference method for solving the bending problem of variable thickness thin plate

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ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS 2022年 139卷 69-76页

作者： Liu, Fan Song, Lina Jiang, Maosheng Fu, Guangming Qingdao Univ Sch Math & Stat Qingdao 266071 Peoples R China China Univ Petr East China Minist Educ Key Lab Unconvent Oil & Gas Dev Qingdao 266580 Peoples R China

In this work, a meshless generalized finite difference method (GFDM), based on the Taylor series expansion and weighted moving-least square approximation, is proposed to solve the bending problem of variable thickness thin plate under various combinations of boundary conditions. The proposed method can treat the plates of complex shapes with an arbitrary continuous varying thickness in a simple and modular way. Besides, the method proposed a new strategy to create a well-determined linear system by introducing supplementary nodes on the boundary. Numerical examples are provided to demonstrate the accuracy and stability of the proposed method.

关键词： Meshless method generalized finite difference method Variable thickness thin plate

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generalized finite difference method for Solving Waveguide Eigenvalue Problems

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APPLIED COMPUTATIONAL ELECTROMAGNETICS SOCIETY JOURNAL 2022年第3期37卷 266-272页

作者： Xu, Hui Bao, Yang Nanjing Univ Posts & Telecommun Dept Elect & Opt Engn Nanjing 210000 Peoples R China Southeast Univ State Key Lab Millimeter Waves Nanjing 210096 Peoples R China

- The generalized finite difference method (GFDM) is a meshless method that has become popular in recent years. The basic theory underlying GFDM is to expand the point cluster function value at the center node by Taylor's formula and then obtain the best linear combinations of these function values to represent the derivative at the central node by the least square fitting technique. Subsequently, the minimized weighted error between the approximated value and the accurate value is obtained. This paper establishes the general steps for solving waveguide eigenvalue problems with GFDM. Excellent performance is shown by comparing the proposed method and other common solutions. The robustness of the proposed method is verified by calculating the cutoff wavenumbers of typical waveguides and the eccentric circular waveguide in different modes.

关键词： Cutoff wavenumber generalized finite difference method meshless method waveguide eigen-value problem

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generalized finite difference method (GFDM) based analysis for subsurface flow problems in anisotropic formation

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ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS 2022年 140卷 48-58页

作者： Zhan, Wentao Rao, Xiang Zhao, Hui Zhang, Hairong Hu, Siwei Dai, Weixin Yangtze Univ Hubei Key Lab Drilling & Prod Engn Oil & Gas Wuhan 430100 Peoples R China Zhanjiang Bay Laborary Southern Marine Sci & Engn Guangdong Laborary Zhanjiang 524000 Peoples R China Yangtze Univ Western Res Inst Karamay 834000 Peoples R China Yangtze Univ Sch Petr Engn Wuhan 430100 Peoples R China

This paper applies the meshless generalized finite difference method (GFDM) to analyze the subsurface flow problem in anisotropic formations for the first time, and develops the treatment methods of anisotropic permeability tensor in continuous function and discrete distribution expressions respectively. In particular, in case of the common discrete distribution expression of anisotropic permeability tensor in reservoir simulation, this paper suggests combining general difference operators and harmonic average scheme to simply and efficiently obtain the meshless discrete scheme of the anisotropic flow equation. Three numerical examples of steady and transient flow in the rectangular computational domain, transient flow with complex boundary, and steady flow with strongly discontinuous anisotropic permeability are used to verify the high accuracy and good convergence of GFDM to handle anisotropic flow problems. Compared with the traditional mesh-based reservoir numerical simulation method that replies on laborious gridding to discretize the reservoir domain and complicated multi-point flux approximation (MPFA) to handle discrete distribution anisotropic permeability tensor, the meshless GFDM can be more practically applied to the anisotropic flow problems in the reservoir domain with complex geometry and complex boundary conditions. Besides, it is found that the radius of the influence domain has little impact on the accuracy of steady flow. For the transient flow, it generally holds that the larger the radius of the influence domain, the lower the calculation accuracy in case of Cartesian collocation. In sum, this work provides an efficient and simple meshless solver to handle anisotropic flow problems in porous media under GFDM framework, which reveals the great application potential of GFDM in reservoir numerical simulation.

关键词： Meshless methods generalized finite difference method Subsurface flow Anisotropic formation Reservoir simulation

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Application of generalized finite difference method for Nonlinear Analysis of the Electrothermal Micro-Actuator

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MICROMACHINES 2025年第3期16卷 325页

作者： Chen, Hao Kong, Xiaoyu Sun, Xiangdong Chen, Mengxu Yuan, Haiyang Lishui Univ Engn Sch Lishui 323000 Peoples R China Cent South Univ State Key Lab Precis Mfg Extreme Serv Performance Changsha 410083 Peoples R China Nanjing Univ Sci & Technol Sch Mech Engn Nanjing 210094 Peoples R China Xian Univ Technol Sch Elect Engn Xian 710048 Peoples R China Zhenhang Ind Grp Co Ltd Lishui 323000 Peoples R China

In this work, the generalized finite difference method (GFDM), a popular meshless numerical method, is employed for predicting the thermal and mechanical behavior of an electrothermal micro-actuator. Based on the concept of GFDM and discretization on the computational domain, the discrete forms of the thermal and mechanical governing equations are derived, respectively. With the help of the incremental load method, the discrete form from the electrothermal analysis is solved precisely and the temperature distribution is obtained. Meanwhile, combining this approach with the discrete control equation derived from the natural boundary condition, its displacement is also evaluated. The convergence of the temperature by different iterative methods is tested and compared. The computational stability and efficiency (CPU time) in these two analyses are also given in this study. To further investigate the accuracy of the solutions, experiments to capture temperature and FEM analysis are conducted. Regardless of the imperfect boundary condition, the temperature distribution calculated by the GFDM shows great agreement with that obtained by experiment and FEM. A similar phenomenon can be also found in the comparison between the displacements evaluated by the GFDM and FEM, respectively.

关键词： generalized finite difference method meshless electrothermal actuator incremental load method

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Analysis of Extended Fisher-Kolmogorov Equation in 2D Utilizing the generalized finite difference method with Supplementary Nodes

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Computer Modeling in Engineering & Sciences 2024年第10期141卷 267-280页

作者： Bingrui Ju Wenxiang Sun Wenzhen Qu Yan Gu Key Laboratory of Road Construction Technology and Equipment of MOE Chang’an UniversityXi’an710018China School of Mathematics and Statistics Qingdao UniversityQingdao266071China

In this study,we propose an efficient numerical framework to attain the solution of the extended Fisher-Kolmogorov(EFK)*** temporal derivative in the EFK equation is approximated by utilizing the Crank-Nicolson *** temporal discretization,the generalized finite difference method(GFDM)with supplementary nodes is utilized to address the nonlinear boundary value problems at each time *** supplementary nodes are distributed along the boundary to match the number of boundary *** incorporating supplementary nodes,the resulting nonlinear algebraic equations can effectively satisfy the governing equation and boundary conditions of the EFK *** demonstrate the efficacy of our approach,we present three numerical examples showcasing its performance in solving this nonlinear problem.

关键词： generalized finite difference method nonlinear extended Fisher-Kolmogorov equation Crank-Nicolson scheme

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A high order generalized finite difference method for solving the anisotropic elliptic interface problem in static and moving systems

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2024年 166卷 1-23页

作者： Xing, Yanan Zheng, Haibiao East China Normal Univ Sch Math Sci Key Lab MEA Minist EducShanghai Key Lab PMMP Shanghai 200241 Peoples R China

In this paper, a high order meshless numerical scheme based on the generalized finite difference method (GFDM) is proposed to analyze the anisotropic elliptic interface problem. The GFDM has an advantage in dealing with some anisotropic elliptic problems with complex interfaces, interface conditions with the jump of derivatives. Four problems with eight examples are provided to verify the existence of the good performance of the GFDM for anisotropic elliptic interface problems, including that the simplicity, accuracy, and stability in static and moving systems. Especially, GFDM has the tolerance of the complexity of interface shape, the interface movement and the large jump.

关键词： High order meshless method generalized finite difference method Anisotropic elliptic problem Interface problem Moving interface problem

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Three-dimensional elastodynamic analysis employing the generalized finite difference method with arbitrary-order accuracy

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2024年 174卷 151-166页

作者： Sun, Wenxiang Qu, Wenzhen Gu, Yan Zhao, Shengdong Qingdao Univ Sch Math & Stat Qingdao 266071 Peoples R China Qingdao Univ Inst Mech Multifunct Mat & Struct Qingdao 266071 Peoples R China

This study introduces an efficient numerical methodology for the analysis of three-dimensional (3D) elastodynamics, featuring high-order precision in the temporal and spatial domains. In the temporal discretization process using the Krylov deferred correction (KDC) technique, the second-order time derivative is treated as a new variable in the governing equations. Spectral integration is then employed to mitigate the instability associated with numerical differentiation operators. Additionally, an improved numerical implementation of boundary conditions based on time integration is incorporated into the KDC approach. The boundary value problems at time nodes resulting from the above discretization process are resolved by employing generalized finite difference method (GFDM), providing the flexibility to choose the Taylor series expansion order. We present four numerical examples to indicate the performance of the developed method in the accuracy and stability. The obtained numerical results are meticulously compared with either analytical solutions or those calculated using COMSOL software.

关键词： Elastodynamic problem Krylov deferred correction method Long-time simulations Meshless method generalized finite difference method

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Simulation of heat transfer in Poiseuille pipe flow via generalized finite difference method with a space stepping algorithm

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2024年 163卷 136-145页

作者： Hong, Yongxing Lin, Ji Cheng, Alexander H. D. Wang, Yanjie Hohai Univ Dept Mech & Elect Engn Changzhou 213022 Peoples R China Changzhou Univ Jiangsu Prov Engn Res Ctr High Level Energy & Powe Changzhou 213064 Peoples R China Minist Educ Engn Res Ctr Dredging Technol Changzhou 213022 Peoples R China Hohai Univ Coll Mech & Mat Int Ctr Simulat Software Engn & Sci State Key Lab Hydrol Water Resources & Hydraul Eng Nanjing 211100 Peoples R China Univ Mississippi Dept Civil Engn Oxford MS USA

The aim of this paper is to propose an efficient numerical scheme to deal with heat transfer problems in pipe flow with a large length/diameter ratio. The generalized finite difference method (GFDM) is combined with a space stepping algorithm (SSA) for the solution. The SSA divides the solution domain into a number of overlapping sections in the length direction of the pipe. Due to the large Peclet number, the heat transport process is dominated by advection, which allows an approximate boundary condition to be applied at the downstream cross section. The problem is then solved section by section. Using the uniform distributed points, the solution matrix for each section does not change, and only the right hand side needs to be refreshed for the changing boundary conditions. This leads to a highly efficient scheme for problems with a large pipe length. To show the accuracy of the numerical scheme, a problem with available analytical solution is studied. Then the method is applied to problems of single pipe with different pipe wall temperature and flow Reynolds number, and concentric annular pipe. Numerical results confirm the stability and efficiency of the GFDM-SSA. The method can be applied to many real world transient heat transfer problems in which the pipe is heated or cooled along its length with arbitrary wall temperature for heat exchange and other purposes.

关键词： Heat transfer generalized finite difference method Space stepping algorithm Poiseuille pipe flow Upwind scheme

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