Tsung-Han Li^1,*, Chia-Ming Fan¹, Po-Wei Li²

The International Conference on Computational & Experimental Engineering and Sciences, Vol.32, No.1, pp. 1-1, 2024, DOI:10.32604/icces.2024.012120

Abstract The generalized finite difference method (GFDM), which cooperated with the fictitious-nodes technique, is proposed in this study to accurately analyze three-dimensional boundary value problems, governed by high-order partial differential equations. Some physical applications can be mathematically described by boundary value problems governed by high-order partial differential equations, but it is non-trivial to analyze the high-order partial differential equations by adopting conventional mesh-based numerical schemes, such as finite difference method, the finite element method, etc. In this study, the GFDM, a localized meshless method, is proposed to accurately and efficiently solve boundary value problems governed by… More >

Bingrui Ju^1,2, Wenxiang Sun², Wenzhen Qu^1,2,*, Yan Gu²

CMES-Computer Modeling in Engineering & Sciences, Vol.141, No.1, pp. 267-280, 2024, DOI:10.32604/cmes.2024.052159 - 20 August 2024

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 >

Ziqiang Bai¹, Wenzhen Qu^2,*, Guanghua Wu^3,*

CMES-Computer Modeling in Engineering & Sciences, Vol.138, No.3, pp. 2955-2972, 2024, DOI:10.32604/cmes.2023.031474 - 15 December 2023

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 >

Po-Wei Li¹, Chia-Ming Fan^1,2, Chun-Yu Chen¹, Cheng-Yu Ku¹

CMES-Computer Modeling in Engineering & Sciences, Vol.101, No.5, pp. 319-350, 2014, DOI:10.3970/cmes.2014.101.319

Abstract A combination of the generalized finite difference method (GFDM), the implicit Euler method and the Newton-Raphson method is proposed to efficiently and accurately analyze the density-driven groundwater flows. In groundwater hydraulics, the problems of density-driven groundwater flows are usually difficult to be solved, since the mathematical descriptions are a system of time- and space-dependent nonlinear partial differential equations. In the proposed numerical scheme, the GFDM and the implicit Euler method were adopted for spatial and temporal discretizations of governing equations. The GFDM is a newly-developed meshless method and is truly free from time-consuming mesh generation… More >

Chun Yang^1,2, Dalin Tang^2,3 Satya Atluri⁴

CMES-Computer Modeling in Engineering & Sciences, Vol.57, No.1, pp. 51-76, 2010, DOI:10.3970/cmes.2010.057.051

Abstract Cardiovascular disease (CVD) is becoming the number one cause of death worldwide. Atherosclerotic plaque rupture and progression are closely related to most severe cardiovascular syndromes such as heart attack and stroke. Mechanisms governing plaque rupture and progression are not well understood. A computational procedure based on three-dimensional meshless generalized finite difference (MGFD) method and serial magnetic resonance imaging (MRI) data was introduced to quantify patient-specific carotid atherosclerotic plaque growth functions and simulate plaque progression. Participating patients were scanned three times (T1, T2, and T3, at intervals of about 18 months) to obtain plaque progression data.… More >

J.J. Benito¹, F. Ureňa², L. Gavete³, B. Alonso³

CMES-Computer Modeling in Engineering & Sciences, Vol.38, No.1, pp. 39-58, 2008, DOI:10.3970/cmes.2008.038.039

Abstract One of the most universal and effective methods, in wide use today, for solving equations of mathematical physics approximately is the finite difference method (FDM). The Generalized finite difference method (GFDM) is evolved fron classical (FDM), which can be applied over general or irregular clouds of points.
This paper starts by showing the GFDM. In this paper, this meshless method is used for solving second-order partial (pde's) with constant coefficients in any type of domain. The method gives the values of derivatives in the nodes using the direct application of the formulae in differences obtained.
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Chun Yang¹, Dalin Tang², Chun Yuan³, William Kerwin², Fei Liu³, Gador Canton³, Thomas S. Hatsukami^3,4, Satya Atluri⁵

CMES-Computer Modeling in Engineering & Sciences, Vol.28, No.2, pp. 95-108, 2008, DOI:10.3970/cmes.2008.028.095

Abstract Atherosclerotic plaque rupture and progression have been the focus of intensive investigations in recent years. Plaque rupture is closely related to most severe cardiovascular syndromes such as heart attack and stroke. A computational procedure based on meshless generalized finite difference (MGFD) method and serial magnetic resonance imaging (MRI) data was introduced to quantify patient-specific carotid atherosclerotic plaque growth functions and simulate plaque progression. Participating patients were scanned three times (T₁,T₂, and T₃, at intervals of about 18 months) to obtain plaque progression data. Vessel wall thickness (WT) changes were used as the measure for plaque progression.… More >