National Taiwan Ocean University Research Hubhttps://scholars.ntou.edu.twDSpace 數位典藏系統是用來獲取、儲存、索引、散佈數位研究資料。Tue, 25 Jun 2024 22:01:23 GMT2024-06-25T22:01:23Z501911Meshless generalized finite difference method with a domain-decomposition method for solving Helmholtz equation and its application to caisson resonance problemshttp://scholars.ntou.edu.tw/handle/123456789/23744標題: Meshless generalized finite difference method with a domain-decomposition method for solving Helmholtz equation and its application to caisson resonance problems
作者: Huang, Ji; Lyu, Hong-Guan; Chen, Jiahn-Horng; Fan, Chia-Ming
摘要: This paper aims at presenting a meshless Generalized Finite Difference Method (GFDM) for solving the Helmholtz equation. In particular, in order to tackle the problem where a fluid domain is divided into different parts by a thin-wall structure, a Domain-Decomposition Method (DDM) is introduced to prevent interactions between node pairs coming from different parts of the flow domain. Two rigorous benchmarks are performed to validate the convergence and accuracy of the present GFDM-DDM model. Furthermore, caisson resonance problems are also investigated and discussed based on the novel numerical tool. It is shown that the present GFDM-DDM model can provide satisfactory predictions for resonance modes in caisson resonance problems, and hence can be treated as a powerful numerical tool in ocean engineering design due to its accurate and efficient characteristics.
Sat, 28 Jan 2023 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/237442023-01-28T00:00:00ZA novel spatial-temporal radial Trefftz collocation method for the backward heat conduction analysis with time-dependent source termhttp://scholars.ntou.edu.tw/handle/123456789/24520標題: A novel spatial-temporal radial Trefftz collocation method for the backward heat conduction analysis with time-dependent source term
作者: Li, Mingjuan; Fu, Zhuojia; Xu, Wenzhi; Fan, Chia -Ming
摘要: In this paper, a novel spatial-temporal radial Trefftz collocation method (STRTCM) is proposed to solve 2D and 3D backward heat conduction equations with time-dependent source term. Unlike the traditional time-domain discretization strategies (Laplace/Fourier transformation and time-stepping methods), the proposed spatial-temporal meshless collocation method employs the derived semi-analytical spatialtemporal radial Trefftz basis function as basis function, which naturally satisfies the governing equation with zero source term in advance. Due to this property, only the final conditions and boundary conditions need to be discretized. For non-zero source term, the extended multiple reciprocity method (E-MRM) is introduced to transform the original governing equation with several specific source terms to the highorder governing equation without the source term. By deriving the high-order semi-analytical spatialtemporal radial Trefftz basis function, the STRTCM can also avoid the node discretization of governing equation. For solving the backward heat conduction equations, the present numerical scheme uses the final temperature distribution and the related boundary conditions to obtain the unknown initial temperature field. Finally, the accuracy and efficiency of the proposed method is numerically verified by four benchmark examples about backward heat conduction equations with time-dependent source term.(c) 2022 Elsevier Ltd. All rights reserved.
Thu, 10 Nov 2022 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/245202022-11-10T00:00:00ZDevelopments of Dynamic Shoreline Planform of Crenulate-Shaped Bay by a Novel Evolution Formulationhttp://scholars.ntou.edu.tw/handle/123456789/23654標題: Developments of Dynamic Shoreline Planform of Crenulate-Shaped Bay by a Novel Evolution Formulation
作者: Tao, Hung-Cheng; Hsu, Tai-Wen; Fan, Chia-Ming
摘要: In this paper, a simple evolution formulation, based on polar coordinate, is proposed to efficiently and accurately simulate the dynamic movement of sandy shoreline in a crenulate-shaped bay. The consistent actions by seasonal waves and swell usually result in severe erosion or deposition along sandy shoreline, so some mathematical formulations and numerical models, based on the concept of headland control, have been recently proposed to forecast and protect the shoreline planform. A simple and general formulation is derived in this study by considering the balance of longshore sediment transport, since the accurate prediction of dynamic movements of high planform-curvature shoreline in the shadow zone behind a headland is essential and critical to the headland control. The proposed formulation can be directly adopted to accurately and simply simulate the temporal variations of shoreline in both of the hooked zone, where is protected by a headland, and the unhooked zone, where is straight attacked by incident waves. Numerical results and comparisons of temporal variation of shoreline between two headlands are provided in this paper to demonstrate the accuracy and efficiency of the proposed evolution formulation. Besides, different time increments and different numbers of control volume are adopted to examine the merits of the proposed numerical model.
Tue, 01 Nov 2022 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/236542022-11-01T00:00:00ZThe localized method of fundamental solutions for 2D and 3D inhomogeneous problemshttp://scholars.ntou.edu.tw/handle/123456789/22017標題: The localized method of fundamental solutions for 2D and 3D inhomogeneous problems
作者: Zhang, Junli; Yang, Chenchen; Zheng, Hui; Fan, Chia-Ming; Fu, Ming-Fu
摘要: In this paper, the newly-developed localized method of fundamental solutions (LMFS) is extended to analyze multidimensional boundary value problems governed by inhomogeneous partial differential equations (PDEs). The LMFS can acquire highly accurate numerical results for the homogeneous PDEs with an incredible improvement of the computational speed. However, the LMFS cannot be directly used for inhomogeneous PDEs. Traditional two-steps scheme has difficulties in finding the particular solutions, and will lead to a loss of the accuracy and efficiency. In this paper, the recursive composite multiple reciprocity method (RC-MRM) is adopted to re-formulate the inhomogeneous PDEs to higher-order homogeneous PDEs with additional boundary conditions, which can be solved by the LMFS efficiently and accurately. The proposed combination of the RC-MRM and the LMFS can analyze inhomogeneous governing equation directly and avoid troublesome caused by the two-steps schemes. The details of the numerical discretization of the RC-MRM and the LMFS are elaborated. Some numerical examples are provided to demonstrate the accuracy and efficiency of the proposed scheme. Furthermore, some key factors of the LMFS are systematically investigated to show the merits of the proposed meshless scheme. (c) 2022 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.
Sat, 01 Oct 2022 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/220172022-10-01T00:00:00ZA space-time generalized finite difference method for solving unsteady double-diffusive natural convection in fluid-saturated porous mediahttp://scholars.ntou.edu.tw/handle/123456789/22080標題: A space-time generalized finite difference method for solving unsteady double-diffusive natural convection in fluid-saturated porous media
作者: Li, Po-Wei; Grabski, Jakub Krzysztof; Fan, Chia-Ming; Wang, Fajie
摘要: In this paper, the space-time generalized finite difference scheme is proposed to effectively solve the unsteady double-diffusive natural convection problem in the fluid-saturated porous media. In such a case, it is mathematically described by nonlinear time-dependent partial differential equations based on Darcy's law. In this work, the space-time approach is applied using a combination of the generalized finite difference, NewtonRaphson, and time-marching methods. In the space-time generalized finite difference scheme, the spatial and temporal derivatives can be performed using the technique for spatial discretization. Thus, the stability of the proposed numerical scheme is determined by the generalized finite difference method. Due to the property of this numerical method, which is based on the Taylor series expansion and the moving-least square method, the resultant matrix system is a sparse matrix. Then, the Newton-Raphson method is used to solve the nonlinear system efficiently. Furthermore, the time-marching method is utilized to proceed along the time axis after a numerical process in one space-time domain. By using this method, the proposed numerical scheme can efficiently simulate the problems which have an unpredictable end time. In this study, three benchmark examples are tested to verify the capability of the proposed meshless scheme.
Thu, 01 Sep 2022 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/220802022-09-01T00:00:00ZNumerical analysis of two spheres falling side by sidehttp://scholars.ntou.edu.tw/handle/123456789/23103標題: Numerical analysis of two spheres falling side by side
作者: Chiu, Chia-Lin; Fan, Chia-Ming; Chu, Chia-Ren
摘要: This study numerically investigates the interaction between two side-by-side free-falling spheres in water. The predicted S-curve falling trajectories of the spheres were verified by laboratory experiments, and the attraction-repulsion of lateral force on the vertically falling spheres is investigated to give a better understanding of the evolution of the lateral force by excluding the effect of the transverse motion. The flow pattern around the spheres reveals that the wake flows play an essential role in the fluid-sphere interaction. Then, the two free-falling spheres with transverse motion are considered to examine the interacting force and the crosswise strength of wake flows. This study attempts to explore the wake flow as the possible mechanism for the S-curve falling trajectories of two spheres. Finally, the falling processes of the two spheres are simulated in a deeper tank to inspect the falling trajectories and the reappearing flow characteristics. Published under an exclusive license by AIP Publishing.
Fri, 01 Jul 2022 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/231032022-07-01T00:00:00ZThe localized method of fundamental solutions for 2D and 3D second-order nonlinear boundary value problemshttp://scholars.ntou.edu.tw/handle/123456789/22000標題: The localized method of fundamental solutions for 2D and 3D second-order nonlinear boundary value problems
作者: Zhao, Shengdong; Gu, Yan; Fan, Chia-Ming; Wang, Xiao
摘要: In this paper, a new framework for the numerical solutions of general nonlinear problems is presented. By employing the analog equation method, the actual problem governed by a nonlinear differential operator is converted into an equivalent problem described by a simple linear equation with unknown fictitious body forces. The solution of the substitute problem is then obtained by using the localized method of fundamental solutions, where the fictitious nonhomogeneous term is approximated using the dual reciprocity method using the radial basis functions. The main difference between the classical and the present localized method of fundamental solutions is that the latter produces sparse and banded stiffness matrix which makes the method very suitable for large-scale nonlinear simulations, since sparse matrices are much cheaper to inverse at each iterative step of the Newton's method. The present method is simple in derivation, efficient in calculation, and may be viewed as a completive alternative for nonlinear analysis, especially for large-scale problems with complex-shape geometries. Preliminary numerical experiments involving second-order nonlinear boundary value problems in both two- and three-dimensions are presented to demonstrate the accuracy and efficiency of the present method.
Wed, 01 Jun 2022 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/220002022-06-01T00:00:00ZLocalized Method of Fundamental Solutions for Two-Dimensional Inhomogeneous Inverse Cauchy Problemshttp://scholars.ntou.edu.tw/handle/123456789/21838標題: Localized Method of Fundamental Solutions for Two-Dimensional Inhomogeneous Inverse Cauchy Problems
作者: Zhang, Junli; Zheng, Hui; Fan, Chia-Ming; Fu, Ming-Fu
摘要: Due to the fundamental solutions are employed as basis functions, the localized method of fundamental solution can obtain more accurate numerical results than other localized methods in the homogeneous problems. Since the inverse Cauchy problem is ill posed, a small disturbance will lead to great errors in the numerical simulations. More accurate numerical methods are needed in the inverse Cauchy problem. In this work, the LMFS is firstly proposed to analyze the inhomogeneous inverse Cauchy problem. The recursive composite multiple reciprocity method (RC-MRM) is adopted to change original inhomogeneous problem into a higher-order homogeneous problem. Then, the high-order homogeneous problem can be solved directly by the LMFS. Several numerical experiments are carried out to demonstrate the efficiency of the LMFS for the inhomogeneous inverse Cauchy problems.
Sun, 01 May 2022 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/218382022-05-01T00:00:00ZMeshless Generalized Finite Difference Method for the Propagation of Nonlinear Water Waves under Complex Wave Conditionshttp://scholars.ntou.edu.tw/handle/123456789/21520標題: Meshless Generalized Finite Difference Method for the Propagation of Nonlinear Water Waves under Complex Wave Conditions
作者: Huang, Ji; Fan, Chia-Ming; Chen, Jiahn-Horng; Yan, Jin
摘要: The propagation of nonlinear water waves under complex wave conditions is the key issue of hydrodynamics both in coastal and ocean engineering, which is significant in the prediction of strongly nonlinear phenomena regarding wave-structure interactions. In the present study, the meshless generalized finite difference method (GFDM) together with the second-order Runge-Kutta method (RKM2) is employed to construct a fully three-dimensional (3D) meshless numerical wave flume (NWF). Three numerical examples, i.e., the propagation of freak waves, irregular waves and focused waves, are implemented to verify the accuracy and stability of the developed 3D GFDM model. The results show that the present numerical model possesses good performance in the simulation of nonlinear water waves and suggest that the 3D "RKM2-GFDM" meshless scheme can be adopted to further simulate more complex nonlinear problems regarding wave-structure interactions in ocean engineering.
Tue, 01 Mar 2022 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/215202022-03-01T00:00:00ZNumerical Simulation of the Time-Dependent Mild-Slope Equation by the Generalized Finite Difference Method (vol 178, pg 4401, 2021)http://scholars.ntou.edu.tw/handle/123456789/20229標題: Numerical Simulation of the Time-Dependent Mild-Slope Equation by the Generalized Finite Difference Method (vol 178, pg 4401, 2021)
作者: Zhang, Ting; Lin, Zhen-Huan; Lin, Chuan; Liang, Lin; Fan, Chia-Ming
Fri, 21 Jan 2022 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/202292022-01-21T00:00:00ZLocalized Method of Fundamental Solutions for Three-Dimensional Elasticity Problems: Theoryhttp://scholars.ntou.edu.tw/handle/123456789/17817標題: Localized Method of Fundamental Solutions for Three-Dimensional Elasticity Problems: Theory
作者: Gu, Yan; Fan, Chia-Ming; Fu, Zhuojia
摘要: A localized version of the method of fundamental solution (LMFS) is devised in this paper for the numerical solutions of three-dimensional (3D) elasticity problems. The present method combines the advantages of high computational efficiency of localized discretization schemes and the pseudo-spectral convergence rate of the classical MFS formulation. Such a combination will be an important improvement to the classical MFS for complicated and large-scale engineering simulations. Numerical examples with up to 100,000 unknowns can be solved without any difficulty on a personal computer using the developed methodologies. The advantages, disadvantages and potential applications of the proposed method, as compared with the classical MFS and boundary element method (BEM), are discussed.
Wed, 01 Dec 2021 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/178172021-12-01T00:00:00ZNumerical Simulation of the Time-Dependent Mild-Slope Equation by the Generalized Finite Difference Methodhttp://scholars.ntou.edu.tw/handle/123456789/18191標題: Numerical Simulation of the Time-Dependent Mild-Slope Equation by the Generalized Finite Difference Method
作者: Zhang, Ting; Lin, Zhen-Huan; Lin, Chuan; Liang, Lin; Fan, Chia-Ming
摘要: The time-dependent mild-slope equation (MSE) is a second-order hyperbolic equation, which is adopted to consider the irregularity of waves. For the difficulty of directly solving the partial derivative terms and the second-order time derivative term, a novel mesh-free numerical scheme, based on the generalized finite difference method (GFDM) and the Houbolt finite difference method (HFDM), is developed to promote the precision and efficiency of the solution to time-dependent MSE. Based on the local characteristics of the GFDM, as a new domain-type meshless method, the linear combinations of nearby function values can be straightforwardly and efficiently implemented to compute the partial derivative term. It is worth noting that the application of the HFDM, an unconditionally stable finite difference time marching scheme, to solve the second-order time derivative term is critical. The results obtained from four examples show that the propagation of waves can be successfully simulated by the proposed numerical scheme in complex seabed terrain. In addition, the energy conversion of waves in long-distance wave propagation can be accurately captured using fast Fourier transform (FFT) analysis, which investigates the energy conservation in wave shoaling problems.
Mon, 01 Nov 2021 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/181912021-11-01T00:00:00ZAnalysis of in-plane crack problems using the localized method of fundamental solutionshttp://scholars.ntou.edu.tw/handle/123456789/18192標題: Analysis of in-plane crack problems using the localized method of fundamental solutions
作者: Gu, Yan; Golub, Mikhail, V; Fan, Chia-Ming
摘要: In this paper, the localized method of fundamental solutions (LMFS), a recently developed meshless collocation method, is applied to the numerical solution of problems with cracks in linear elastic fracture mechanics. The main idea of the LMFS is to divide the entire computational domain into a set of overlapping sub-domains, and in each sub-domain, the classical MFS formulation and the moving least squares (MLS) technique are applied to form the corresponding local system of equations. The LMFS will finally generate a banded and sparse matrix system which makes the method very attractive for large-scale engineering applications. To deal with inplane crack problems, an enriched LMFS approach is proposed by combining the LMFS formulation for linear elasticity problems and a set of enrichment functions which take into account the asymptotic behavior of the near-tip displacement and stress fields. The enriched LMFS can significantly improve the computational accuracy of the calculation of stress intensity factor (SIF) of the cracked materials, even with a very coarse LMFS node distribution. Several benchmark numerical examples are presented to illustrate the accuracy and efficiency of the proposed method.
Fri, 15 Oct 2021 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/181922021-10-15T00:00:00ZA novel space-time generalized FDM for dynamic coupled thermoelasticity problems in heterogeneous plateshttp://scholars.ntou.edu.tw/handle/123456789/18187標題: A novel space-time generalized FDM for dynamic coupled thermoelasticity problems in heterogeneous plates
作者: Lei, Jun; Wei, Xun; Wang, Qin; Gu, Yan; Fan, Chia-Ming
摘要: A novel space-time generalized finite difference method based on the direct space-time discretization technique is developed for solving dynamic coupled thermoelasticity problems. By considering the time scale as an additional space dimension, the spatial and temporal domains are simultaneously discretized. In our numerical implementation, the velocity is introduced as an additional unknown field quantity for dealing with the inertia item appearing in elastodynamic equations. Some dynamic coupled thermoelasticity problems in homogeneous or heterogeneous plates under different loading cases are numerically analyzed by this method. The accuracy of the present method is verified by comparison with analytical solutions or other numerical results.
Tue, 05 Oct 2021 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/181872021-10-05T00:00:00ZLocalized singular boundary method for solving Laplace and Helmholtz equations in arbitrary 2D domainshttp://scholars.ntou.edu.tw/handle/123456789/17527標題: Localized singular boundary method for solving Laplace and Helmholtz equations in arbitrary 2D domains
作者: Wang, Fajie; Chen, Zengtao; Li, Po-Wei; Fan, Chia-Ming
摘要: In this research, the localized singular boundary method (LSBM) is proposed to solve the Laplace and Helmholtz equations in 2D arbitrary domains. In the traditional SBM, the resultant matrix system is a dense matrix, and it is unsuited for solving the large-scale problems. As a localized domain-type meshless method, a local subdomain for every node can be composed by its own and several nearest nodes. To each of the subdomains, the SBM formulation is applied to derive an implicit expression of the variable at each node in conjunction with the moving least-square approximation. To satisfy the boundary conditions at every boundary node and the governing equation at every node, a sparse linear algebraic system can be obtained. Thus, the numerical solutions at all nodes can be achieved by solving it. Compared with the traditional SBM, the LSBM involves only the origin intensity factor on a circular boundary associated with Dirichlet boundary conditions. It can also effectively avoid the boundary layer effect in the conventional SBM. Furthermore, the proposed LSBM requires less memory storage and computational cost due to the sparse and banded matrix system. Several numerical examples are tested to verify the accuracy and stability of the proposed LSBM.
Sun, 01 Aug 2021 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/175272021-08-01T00:00:00ZA generalized finite difference method for solving elasticity interface problemshttp://scholars.ntou.edu.tw/handle/123456789/17500標題: A generalized finite difference method for solving elasticity interface problems
作者: Xing, Yanan; Song, Lina; Fan, Chia-Ming
摘要: In this paper, a generalized finite difference method (GFDM) is proposed to solve the elasticity interface problem. This method turns the original elasticity interface problem to be some coupled non-interface subproblems. A large sparse matrix can be yielded by those subproblems. Since this method makes the interface become a part of boundaries of subproblems, it can deal well with problems with complex geometrical interfaces. Moreover, this method can also deal well with the interface conditions with derivatives, because the GFDM uses a linear summation of nearby nodal values to express the derivatives of unknown variables. Numerical examples are provided to verify the accuracy and stability of the proposed method for elasticity interface problems. They show that the H-1 error of the method has the almost same convergence rate as the L-2 error and the size of jumps in the interface conditions only has a little effect on the stability of the proposed method.
Thu, 01 Jul 2021 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/175002021-07-01T00:00:00ZFracture mechanics analysis of bimaterial interface cracks using the generalized finite difference methodhttp://scholars.ntou.edu.tw/handle/123456789/17393標題: Fracture mechanics analysis of bimaterial interface cracks using the generalized finite difference method
作者: Jiang, Songwei; Gu, Yan; Fan, Chia-Ming; Qu, Wenzhen
摘要: This paper makes the first attempt to apply the generalized finite difference method (GFDM), a recently developed meshless collocation method, for fracture mechanics analysis of dissimilar elastic materials with interfacial cracks. The main idea of the GFDM is to divide the entire computational domain into a set of overlapping small subdomains in which the local Taylor series expansion and moving-least square approximation are employed for generating the local systems of linear equations. Since the method is meshless and no element connectivity is needed, the burdensome remeshing procedures associated with the finite element method (FEM) is avoided. The multi-domain GFDM technique is used to handle the non-homogeneity of the cracked dissimilar materials. The displacement extrapolation method (DEM), which avoids the direct calculation of the oscillatory near-tip displacement and stress fields, is employed to compute the complex stress intensity factors (SIFs) for cracked composite bimaterials. Several representative numerical examples are presented and discussed to demonstrate that the present method is highly accurate and relatively robust for interface crack analysis of composite bimaterials.
Tue, 01 Jun 2021 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/173932021-06-01T00:00:00ZLocalized Chebyshev collocation method for solving elliptic partial differential equations in arbitrary 2D domainshttp://scholars.ntou.edu.tw/handle/123456789/17387標題: Localized Chebyshev collocation method for solving elliptic partial differential equations in arbitrary 2D domains
作者: Wang, Fajie; Zhao, Qinghai; Chen, Zengtao; Fan, Chia-Ming
摘要: In this paper, a novel collocation method is presented for the efficient and accurate evaluation of the two-dimensional elliptic partial differential equation. In the new method, the physical domain is discretized into a series of overlapping small (local) subdomains, and in each of the subdomain, a localized Chebyshev collocation method is applied in which the unknown functions at every node can be computed by using a linear combination of unknowns at its near-by nodes. The Chebyshev polynomials employed here can provide the spectral accuracy of new approach. The concept of the local subdomain is introduced to derive a sparse system, which ensures the feasibility for large-scale simulation. This paper aims at proposing a new method to solve general partial differential equations accurately and efficiently. Several numerical examples including Poisson equation, Helmholtz-type equation and transient heat conduction equation are provided to demonstrate the validity and applicability of the proposed method. Numerical experiments indicate that the localized Chebyshev collocation method is very promising for the efficient and accurate solution of large-scale problems. (C) 2020 Elsevier Inc. All rights reserved.
Sat, 15 May 2021 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/173872021-05-15T00:00:00ZNumerical solutions of two-dimensional Laplace and biharmonic equations by the localized Trefftz methodhttp://scholars.ntou.edu.tw/handle/123456789/17340標題: Numerical solutions of two-dimensional Laplace and biharmonic equations by the localized Trefftz method
作者: Liu, Yan-Cheng; Fan, Chia-Ming; Yeih, Weichung; Ku, Cheng-Yu; Chu, Chiung-Lin
摘要: In this paper, the localized Trefftz method (LTM) is proposed to accurately and efficiently solve two-dimensional boundary value problems, governed by Laplace and biharmonic equations, in complex domains. The LTM is formed by combining the classical indirect Trefftz method and the localization approach, so the LTM, free from mesh and numerical quadrature, has great potential for solving large-scale problems. For problems in multiply-connected domains, the solutions expressions in the proposed LTM is much simpler and more compact than that in the conventional indirect Trefftz method due to the localization concept and the overlapping subdomains. In the proposed LTM, both of the interior nodes and boundary nodes are required and the algebraic equation at each node, represents the satisfaction of governing equation or boundary condition, can be derived by implementing the Trefftz method at every subdomain. By enforcing the satisfaction of governing equations at every interior node and of boundary conditions at every boundary node, a sparse system of linear algebraic equations can be yielded. Then, the numerical solution of the proposed LTM can be efficiently obtained by solving the sparse system. Several numerical examples in simply-connected and multiply-connected domains are provided to demonstrate the accuracy and simplicity of the proposed LTM. Besides, the extremely-accurate solutions of the LTM are simultaneously demonstrated. (C) 2020 Elsevier Ltd. All rights reserved.
Thu, 15 Apr 2021 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/173402021-04-15T00:00:00ZLocalized method of fundamental solutions for two-dimensional anisotropic elasticity problemshttp://scholars.ntou.edu.tw/handle/123456789/17323標題: Localized method of fundamental solutions for two-dimensional anisotropic elasticity problems
作者: Liu, Q. G.; Fan, C. M.; Sarler, B.
摘要: The purpose of the presented paper is to develop further the Localized Method of Fundamental Solutions (LMFS) for solving two-dimensional anisotropic elasticity problems. The computational domain is divided into overlapping subdomains. In the LMFS, the classical Method of Fundamental Solutions (MFS) is employed in each of these local subdomains to get an expression of the solution for the main node of this subdomain. The expression is structured by a linear combination of the solutions of the other nodes in this subdomain. Displacement or traction boundary conditions are satisfied at the boundary nodes. The solution is calculated from an equation set formed by the boundary conditions for the boundary nodes and expressions in the subdomain for the interior nodes. The presented three numerical examples demonstrate that the novel method is suitable for solving large-scale problems, and especially, the problems with complicated domains.
Thu, 01 Apr 2021 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/173232021-04-01T00:00:00ZEstimation of Tumor Characteristics in a Skin Tissue by a Meshless Collocation Solverhttp://scholars.ntou.edu.tw/handle/123456789/17325標題: Estimation of Tumor Characteristics in a Skin Tissue by a Meshless Collocation Solver
作者: Fu, Zhuo-Jia; Chu, Wen-Hui; Yang, Min; Li, Po-Wei; Fan, Chia-Ming
摘要: This paper aims to noninvasively estimate the sizes and locations of tumors via the surface temperature in the skin tissue. The famous 2D Pennes bioheat transfer equation is used to describe the heat transfer behavior in the skin tissue, which is solved by the recently-developed meshless generalized finite difference method (GFDM) in the proposed solver. The hybrid optimization algorithm based on genetic algorithm (GA) and Levenberg-Marquardt algorithm (LM) is introduced to estimate the sizes and locations of tumors. The efficiency of the proposed GA-LM-GFDM solver is verified under several benchmark examples. Numerical investigation shows that the tumor characteristics can still be accurately estimated with the contaminated temperature data measured on the skin surface.
Thu, 01 Apr 2021 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/173252021-04-01T00:00:00ZImprovement of generalized finite difference method for stochastic subsurface flow modelinghttp://scholars.ntou.edu.tw/handle/123456789/17208標題: Improvement of generalized finite difference method for stochastic subsurface flow modeling
作者: Chen, Shang-Ying; Hsu, Kuo-Chin; Fan, Chia-Ming
摘要: Uncertainty is embedded in groundwater flow modeling because of the heterogeneity of hydraulic conductivity and the scarcity of measurements. To quantify the uncertainty of the modeled hydraulic head, this study proposes an improved version of the meshless generalized finite difference method (GFDM) for solving the statistical moment equation (ME). The proposed GFDM adopts a new support sub-domain for calculating the derivative of the head to improve accuracy. Synthetic fields are applied to validate the proposed method. The proposed GFDM outperforms the conventional GFDM in terms of accuracy based on a comparison with the results of the finite difference method. The ME-GFDM scheme is shown to be 2.6 times faster than Monte Carlo simulation with comparable accuracy. The ME-GFDM is versatile in that it easily handles irregular domains, allows the node location and number to be changed, and allows the sequential addition of new data without remeshing, which is required for traditional mesh-based methods. (C) 2020 Elsevier Inc. All rights reserved.
Mon, 15 Mar 2021 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/172082021-03-15T00:00:00ZLocalized method of fundamental solutions for two- and three-dimensional transient convection-diffusion-reaction equationshttp://scholars.ntou.edu.tw/handle/123456789/17196標題: Localized method of fundamental solutions for two- and three-dimensional transient convection-diffusion-reaction equations
作者: Liu, Shuainan; Li, Po-Wei; Fan, Chia-Ming; Gu, Yan
摘要: This paper investigates the use of the localized method of fundamental solutions (LMFS) for the numerical solution of general transient convection-diffusion-reaction equation in both two-(2D) and three-dimensional (3D) materials. The method is developed as a generalization of the author's earlier work on Laplace's equation to transient convection-diffusion-reaction equation. The popular Crank-Nicolson (CN) time-stepping technology is adopted to perform the temporal simulations. The LMFS approach is then introduced for solving the resulting inhomogeneous boundary value problems, where a pseudo-spectral Chebyshev collocation scheme (CCS) is employed for the approximation of the corresponding particular solutions. As compared with the classical MFS and boundary element method (BEM), the present CN-CCS-LMFS approach produces sparse and banded stiffness matrix which makes the method possible to perform large-scale dynamic simulations. Several benchmark numerical examples are presented to demonstrate the efficiency and feasibility of the present method.
Mon, 01 Mar 2021 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/171962021-03-01T00:00:00ZLocal non-singular knot method for large-scale computation of acoustic problems in complicated geometrieshttp://scholars.ntou.edu.tw/handle/123456789/17172標題: Local non-singular knot method for large-scale computation of acoustic problems in complicated geometries
作者: Yue, Xingxing; Wang, Fajie; Li, Po-Wei; Fan, Chia-Ming
摘要: This paper presents a local non-singular knot method (LNKM) to accurately solve the large-scale acoustic problems in complicated geometries. The LNKM is a domain-type meshless collocation method, which relies only on scattered nodes. Firstly, a series of subdomains corresponding to every nodes can be searched based on the Euclidean distance between nodes. To each subdomain, a small linear system can be yielded by using the non-singular general solutions of Helmholtz-type equations. Secondly, the unknown variables at every nodes can be explicitly expressed by the function values at their corresponding supporting nodes. Finally, a large sparse system of linear equations is formed and solved to obtain the numerical solutions of physical quantities at every nodes. The proposed LNKM is mathematically simple, numerically accurate, and more applicable to the large-scale computation. Four numerical examples conform its effectiveness and accuracy for the large-scale computation of Helmholtz-type equations in complicated geometries. (C) 2021 Elsevier Ltd. All rights reserved.
Mon, 15 Feb 2021 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/171722021-02-15T00:00:00ZOn the propagation of nonlinear water waves in a three-dimensional numerical wave flume using the generalized finite difference methodhttp://scholars.ntou.edu.tw/handle/123456789/1196標題: On the propagation of nonlinear water waves in a three-dimensional numerical wave flume using the generalized finite difference method
作者: Ji Huang; Chi-Nan Chu; Chia-Ming Fan; Jiahn-Horng Chen; Hongguan Lyu
摘要: Nonlinear water waves are common physical phenomena in the field of coastal and ocean engineering, which plays a critical role in the investigation of hydrodynamics regarding offshore and deep-water structures. In the present study, a three-dimensional (3D) numerical wave flume (NWF) is constructed to simulate the propagation of nonlinear water waves. On the basis of potential flow theory, the second-order Runge-Kutta method (RKM2) combining with a semi-Lagrangian approach is carried out to discretize the temporal variable of the 3D Laplace’s equation. For the spatial variables, the generalized finite difference method (GFDM) is adopted to solve the governing equations for the deformable computational domain at each time step. The upstream condition is considered as a wave-making boundary with imposing horizontal velocity while the downstream condition as a wave-absorbing boundary with a pre-defined sponge layer to deal with the phenomenon of wave reflection. Three numerical examples are investigated and discussed in detail to validate the accuracy and stability of the developed 3D GFDM-based NWF. The results show that the newly-proposed numerical method has good performance in the prediction of the dynamic evolution of nonlinear water waves, and suggests that the novel 3D “RKM2-GFDM” meshless scheme can be employed to further investigate more complicated hydrodynamic problems in practical applications.
Thu, 01 Oct 2020 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/11962020-10-01T00:00:00ZA localized meshless collocation method for bandgap calculation of anti-plane waves in 2D solid phononic crystalshttp://scholars.ntou.edu.tw/handle/123456789/1177標題: A localized meshless collocation method for bandgap calculation of anti-plane waves in 2D solid phononic crystals
作者: Zhuo-Jia Fu; Ai-Lun Li; Chuanzeng Zhang; Chia-Ming Fan; Xiao-Ying Zhuang
摘要: In this paper, a localized meshless collocation method, the generalized finite difference method (GFDM), is first applied to calculate the bandgaps of anti-plane transverse elastic waves in 2D solid phononic crystals with square and triangular lattice. The corresponding theoretical consistency analysis of the GFDM is given. The universal algorithm for the uniform/scattered node generation in the GFDM is presented. In comparison with the traditional plane wave expansion (PWE) method and Pressure Acoustics Module in COMSOL software, the proposed GFDM can provide the similar accurate results with less computational times for calculating the band structures of the simple/complicated shape scatterers in the square/triangular lattice. Three influence factors (Filling fractions (Ff), rotation angles (Ra) and arm widths (Aw) in the unit-cell) of the bandgap properties in 2D phononic crystals are numerically discussed.
Thu, 01 Oct 2020 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/11772020-10-01T00:00:00ZTopology optimization of steady-state heat conduction structures using meshless generalized finite difference methodhttp://scholars.ntou.edu.tw/handle/123456789/1267標題: Topology optimization of steady-state heat conduction structures using meshless generalized finite difference method
作者: Qinghai Zhao; Chia-Ming Fan; Fajie Wang; Wenzhen Qu
摘要: This paper proposes the topology optimization for steady-state heat conduction structures by incorporating the meshless-based generalized finite difference method (GFDM) and the solid isotropic microstructures with penalization interpolation model. In the meshless GFDM numerical scheme, the explicit formulae of the partial differential equation are expressed by the Taylor series expansions and the moving-least squares approximations to address the required partial derivatives of unknown nodal variables. With the relative density of meshless GFDM node as the design variable, the implementation of the topology optimization is formulated involving the minimization of heat potential capacity as the objective function under node number constraint. Moreover, sensitivity of the objective function is derived based on the adjoint method, and sensitivity filtering subsequently suppresses the checkerboard pattern. Next, the update of design variables at each iteration is solved by the optimality criteria method. At last, several numerical examples are illustrated to demonstrate the validity and feasibility of the proposed method.
Thu, 01 Oct 2020 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/12672020-10-01T00:00:00ZA novel space-time generalized FDM for transient heat conduction problemshttp://scholars.ntou.edu.tw/handle/123456789/1205標題: A novel space-time generalized FDM for transient heat conduction problems
作者: Jun Lei; Qin Wang; Xia Liu; Yan Gu; Chia-Ming Fan
摘要: In this paper, a novel space-time generalized finite difference method (GFDM) is proposed for solving transient heat conduction problems by integrating a direct space-time discretization technique into the meshless GFDM. The spatial and temporal dimensions are simultaneously discretized by randomly distributed nodes in the coupled space-time continuum. Transient heat conduction in homogenous and heterogeneous materials are analyzed by this novel meshless space-time GFDM. Examples involving multidimensional spatiotemporal domains and various complex boundary conditions are studied and discussed. The high accuracy and efficiency of this new algorithm are verified by comparing with analytical and other numerical results.
Thu, 01 Oct 2020 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/12052020-10-01T00:00:00ZGeneralized finite difference method for solving stationary 2D and 3D Stokes equations with a mixed boundary conditionhttp://scholars.ntou.edu.tw/handle/123456789/1221標題: Generalized finite difference method for solving stationary 2D and 3D Stokes equations with a mixed boundary condition
作者: Lina Song; Po-Wei Li; Yan Gu; Chia-Ming Fan
摘要: In the present work, a generalized finite difference method (GFDM), a meshless method based on Taylor-series approximations, is proposed to solve stationary 2D and 3D Stokes equations. To overcome the troublesome pressure oscillation in the Stokes problem, a new simple formulation of boundary condition for the Stokes problem is proposed. This numerical approach only adds a mixed boundary condition, the projections of the momentum equation on the boundary outward normal vector, to the Stokes equations, without any other change to the governing equations. The proposed formulation can be easily discretized by the GFDM. The GFDM is evolved from the Taylor series expansions and moving-least squares approximation, and the derivative expressed of unknown variables as linear combinations of function values of neighboring nodes. Numerical examples are utilized to verify the feasibility of the proposed GFDM scheme not only for the Stokes problem, but also for more involved and general problems, such as the Poiseuille flow, the Couette flow and the Navier–Stokes equations in low-Reynolds-number regime. Moreover, numerical results and comparisons show that using the GFDM to solve the proposed formulation of the Stokes equations is more accurate than the classical formulation of the pressure Poisson equation.
Tue, 01 Sep 2020 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/12212020-09-01T00:00:00ZA Localized Space-Time Method of Fundamental Solutions for Diffusion and Convection-Diffusion Problemshttp://scholars.ntou.edu.tw/handle/123456789/1234標題: A Localized Space-Time Method of Fundamental Solutions for Diffusion and Convection-Diffusion Problems
作者: Fajie Wang; Chia-Ming Fan; Chuanzeng Zhang; Ji Lin
摘要: In this paper, a localized space-time method of fundamental solutions (LSTMFS) is proposed to solve the diffusion and convection-diffusion problems. The proposed LSTMFS only requires some arbitrarily-distributed nodes inside the space-time domain and along its boundary. The local subdomain corresponding to each node can firstly be determined based on the Euclidean distance between the nodes. Then, the variable at each node can be expressed as a linear combination of variables at its supporting nodes. By solving a resultant sparse system, the variable at any node in the considered space-time domain can be obtained. Compared with the traditional space-time method of fundamental solutions, the proposed LSTMFS is more suitable for solving large-scale and long-time diffusion problems. Furthermore, the LSTMFS without temporal-difference is simple, accurate and easy-to-implement due to its semi-analytical and meshless features. Numerical experiments, including diffusion and convection-diffusion problems, confirm the validity and accuracy of the proposed LSTMFS.
Sat, 01 Aug 2020 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/12342020-08-01T00:00:00ZAnalysis of an augmented moving least squares approximation and the associated localized method of fundamental solutionshttp://scholars.ntou.edu.tw/handle/123456789/1218標題: Analysis of an augmented moving least squares approximation and the associated localized method of fundamental solutions
作者: Wenzhen Qu; Chia-Ming Fan; Xiaolin Li
摘要: The localized method of fundamental solutions (LMFS) is an efficient meshless collocation method that combines the concept of localization and the method of fundamental solutions (MFS). The resultant system of linear algebraic equations in the LMFS is sparse and banded and thus, drastically reduces the storage and computational burden of the MFS. In the LMFS, the moving least square (MLS) approximation, based on fundamental solutions, is used to construct approximate solution at each node. In this paper, this fundamental solutions-based MLS approximation, named as an augmented MLS (AMLS) approximation, is generalized to any point in the computational domain. Computational formulas, theoretical properties and error estimates of the AMLS approximation are derived. Then, taking Laplace equation as an example, this paper sets up a framework for the theoretical error analysis of the LMFS. Finally, numerical results are presented to verify the efficiency and theoretical results of the AMLS approximation and the LMFS. Convergence and comparison researches are conducted to validate the accuracy, convergence and efficiency.
Wed, 01 Jul 2020 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/12182020-07-01T00:00:00ZWave-Structure Interaction for a Stationary Surface-Piercing Body Based on a Novel Meshless Scheme with the Generalized Finite Difference Methodhttp://scholars.ntou.edu.tw/handle/123456789/1197標題: Wave-Structure Interaction for a Stationary Surface-Piercing Body Based on a Novel Meshless Scheme with the Generalized Finite Difference Method
作者: Ji Huang; Hongguan Lyu; Chia-Ming Fan; Jiahn-Hong Chen; Chi-Nan Chu; Jiayang Gu
摘要: 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. View Full-Text
Keywords: wave-structure interaction; nonlinear water waves; surface-piercing body; meshless method; generalized finite difference method
Wed, 01 Jul 2020 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/11972020-07-01T00:00:00ZGeneralized finite difference method for three-dimensional eigenproblems of Helmholtz equationhttp://scholars.ntou.edu.tw/handle/123456789/22374標題: Generalized finite difference method for three-dimensional eigenproblems of Helmholtz equation
作者: Zhang, Juan; Shuy, Rong-Juin; Chu, Chiung-Lin; Fan, Chia-Ming
摘要: In this paper, a meshless numerical procedure, based on the generalized finite difference method (GFDM) is proposed to efficiently and accurately solve the three-dimensional eigenproblems of the Helmholtz equation. The eigenvalues and eigenvectors are very important to various engineering applications in three-dimensional acoustics, optics and electromagnetics, so it is essential to develop an efficient numerical model to analyze the three-dimensional eigenproblems in irregular domains. In the GFDM, the Taylor series and the moving-least squares method are used to derive the expressions at every node. By enforcing the satisfactions of governing equation at interior nodes and boundary conditions at boundary nodes, the resultant system of linear algebraic equations can be expressed as the eigenproblems of matrix and then the eigenvalues and eigenvectors can be efficiently acquired. In this paper, four numerical examples are provided to validate the accuracy and simplicity of the proposed numerical scheme. Furthermore, the numerical results are compared with analytical solutions and other numerical results to verify the merits of the proposed method.(c) 2022 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.
Mon, 01 Jun 2020 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/223742020-06-01T00:00:00ZA meshless collocation scheme for inverse heat conduction problem in three-dimensional functionally graded materialshttp://scholars.ntou.edu.tw/handle/123456789/1195標題: A meshless collocation scheme for inverse heat conduction problem in three-dimensional functionally graded materials
作者: Wen Hu; Yan Gu; Chia-Ming Fan
摘要: This short communication documents the first attempt to apply the generalized finite difference method (GFDM) for inverse heat conduction analysis of functionally graded materials (FGMs). The fact that the GFDM is a meshless collocation method makes it particularly attractive in solving problems with complex geometries and high dimensions. By employing the Taylor series expansion and the moving least-squares technique, the method produces sparse and banded matrix which makes it possible to perform large-scale simulations. Three benchmark examples are provided to demonstrate the accuracy and adaptability of the GFDM approach in solving the inverse Cauchy problems. The convergence and stability of the method with respect to the amount of noise added into the input data are analyzed.
Fri, 01 May 2020 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/11952020-05-01T00:00:00ZSolving Boussinesq equations with a meshless finite difference methodhttp://scholars.ntou.edu.tw/handle/123456789/1264標題: Solving Boussinesq equations with a meshless finite difference method
作者: Ting Zhang; Zhen-Huan Lin; Guan-Yi Huang; Chia-Ming Fan; Po-Wei Li
摘要: This paper mainly focus on presenting a newly-developed meshless numerical scheme, named the generalized finite difference method (GFDM), to efficiently and accurately solve the improved Boussinesq-type equations (BTEs). Based on the improved BTEs, the wave propagated over a flat or irregular bottom topography is described as a two-dimensional horizontal problem with nonlinear water waves. The GFDM and the 2nd-order Runge-Kutta method (RKM) were employed for spatial and temporal discretizations for this problem, respectively. The ramping function and the sponge layer, combing in this proposed scheme, were adopted for incident and outgoing waves, respectively. As one of domain-type meshless methods, GFDM can improve the numerical efficiency due to avoiding time-consuming meshing generation and numerical quadrature. Furthermore, the partial derivatives of Boussinesq equations can be transformed as linear combinations of nearby function values by the moving-least-squares method of the GFDM, simplifying the numerical procedures. Specifically, GFDM is suitable for complex fluid field with some irregular boundaries because of the flexible distribution of nodes. Four numerical examples were selected to verify the accuracy and applicability in the improved BTEs of the proposed meshless scheme. The results were compared with other numerical predictions and experimental observations and good agreements were depicted.
Sat, 01 Feb 2020 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/12642020-02-01T00:00:00ZLocalized MFS for the inverse Cauchy problems of two-dimensional Laplace and biharmonic equationshttp://scholars.ntou.edu.tw/handle/123456789/1233標題: Localized MFS for the inverse Cauchy problems of two-dimensional Laplace and biharmonic equations
作者: Fajie Wang; Chia-Ming Fan; Qingsong Hua; Yan Gu
摘要: This paper makes a first attempt to use a new localized method of fundamental solutions (LMFS) to accurately and stably solve the inverse Cauchy problems of two-dimensional Laplace and biharmonic equations in complex geometries. The LMFS firstly divides the whole physical domain into several small overlapping subdomains, and then employs the traditional method of fundamental solutions (MFS) formulation in every local subdomain for calculating the unknown coefficients on the local fictitious boundary. After that, a sparse linear system is formed by using the governing equation for interior nodes and the nodes on under-specified boundary, and by using the given boundary conditions for the nodes on over-specified boundary. Finally, the numerical solutions of the inverse problems can be obtained by solving the resultant sparse system. Compared with the traditional MFS with the “global” boundary discretization, the LMFS requires less computational cost, which may make the LMFS suitable for solving large-scale problems. Numerical experiments demonstrate the validity and accuracy of the proposed LMFS for the inverse Cauchy problems of two-dimensional Laplace and biharmonic equations with noisy boundary data.
Wed, 01 Jan 2020 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/12332020-01-01T00:00:00ZLocalized method of fundamental solutions for three-dimensional inhomogeneous elliptic problems: theory and MATLAB codehttp://scholars.ntou.edu.tw/handle/123456789/1185標題: Localized method of fundamental solutions for three-dimensional inhomogeneous elliptic problems: theory and MATLAB code
作者: Yan Gu; Chia-Ming Fan; Wenzhen Qu; Fajie Wang; Chuanzeng Zhang
摘要: In this paper we investigate the application of the localized method of fundamental solutions (LMFS) for solving three-dimensional inhomogeneous elliptic boundary value problems. A direct Chebyshev collocation scheme (CCS) is employed for the approximation of the particular solutions of the given inhomogeneous problem. The Gauss–Lobatto collocation points are used in the CCS to ensure the pseudo-spectral convergence of the method. The resulting homogeneous equations are then calculated by using the LMFS. In the framework of the LMFS, the computational domain is divided into a set of overlapping local subdomains where the traditional MFS formulation and the moving least square method are applied. The proposed CCS-LMFS produces sparse and banded stiffness matrix which makes the method possible to perform large-scale simulations on a desktop computer. Numerical examples involving Poisson, Helmholtz as well as modified-Helmholtz equations (with up to 1,000,000 unknowns) are presented to illustrate the efficiency and accuracy of the proposed method.
Sun, 01 Dec 2019 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/11852019-12-01T00:00:00ZAnalysis of three-dimensional heat conduction in functionally graded materials by using a hybrid numerical methodhttp://scholars.ntou.edu.tw/handle/123456789/1219標題: Analysis of three-dimensional heat conduction in functionally graded materials by using a hybrid numerical method
作者: Wenzhen Qu; Chia-Ming Fan; Yaoming Zhang
摘要: A hybrid numerical method is developed for three-dimensional (3D) heat conduction in functionally graded materials (FGMs) by integrating the advantages of the generalized finite difference method (GFDM) and Krylov deferred correction (KDC) technique. The temporal direction of the problems is first discretized by applying the KDC approach for high-accuracy results, which yields a partial differential equation (PDE) boundary value problem at each time step. The corresponding PDE boundary value problem is then solved by introducing the meshless GFDM that has no requirement of time-consuming meshing generation and numerical integration for 3D problems with complex geometries. Numerical experiments with four types of material gradations are provided to verify the developed combination scheme, and numerical results demonstrate that the method has a great potential for 3D transient heat conduction in FGMs especially for those in a long-time simulation.
Sun, 01 Dec 2019 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/12192019-12-01T00:00:00ZAnalysis of three-dimensional interior acoustic fields by using the localized method of fundamental solutionshttp://scholars.ntou.edu.tw/handle/123456789/1217標題: Analysis of three-dimensional interior acoustic fields by using the localized method of fundamental solutions
作者: Wenzhen Qu; Chia-Ming Fan; Yan Gu; Fajie Wang
摘要: The traditional method of fundamental solutions has a full interpolation matrix, and thus its solution is computationally expensive, especially for large-scale problems with complicated domains. In this paper, we make a first attempt to apply the localized method of fundamental solutions for analysis of 3D interior acoustic fields. The present method first divides the whole computational domain into some overlapping subdomains, and then expresses physical variables as linear combinations of the fundamental solution in each subdomain. Finally, the method forms a sparse and banded system matrix by satisfying both governing equations at interior nodes and boundary conditions at boundary nodes. We provide four numerical experiments to verify the accuracy and the stability of the method. Comparisons of numerical results and computational time are also made between the present method, the method of fundamental solutions, and the COMSOL software.
Sun, 01 Dec 2019 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/12172019-12-01T00:00:00ZA Spacetime Meshless Method for Modeling Subsurface Flow with a Transient Moving Boundaryhttp://scholars.ntou.edu.tw/handle/123456789/1199標題: A Spacetime Meshless Method for Modeling Subsurface Flow with a Transient Moving Boundary
作者: Cheng-Yu Ku; Chih-Yu Liu; Jing-En Xiao; Wei-Chung Yeih; Chia-Ming Fan
摘要: In this paper, a spacetime meshless method utilizing Trefftz functions for modeling subsurface flow problems with a transient moving boundary is proposed. The subsurface flow problem with a transient moving boundary is governed by the two-dimensional diffusion equation, where the position of the moving boundary is previously unknown. We solve the subsurface flow problems based on the Trefftz method, in which the Trefftz basis functions are obtained from the general solutions using the separation of variables. The solutions of the governing equation are then approximated numerically by the superposition theorem using the basis functions, which match the data at the spacetime boundary collocation points. Because the proposed basis functions fully satisfy the diffusion equation, arbitrary nodes are collocated only on the spacetime boundaries for the discretization of the domain. The iterative scheme has to be used for solving the moving boundaries because the transient moving boundary problems exhibit nonlinear characteristics. Numerical examples, including harmonic and non-harmonic boundary conditions, are carried out to validate the method. Results illustrate that our method may acquire field solutions with high accuracy. It is also found that the method is advantageous for solving inverse problems as well. Finally, comparing with those obtained from the method of fundamental solutions, we may obtain the accurate location of the nonlinear moving boundary for transient problems using the spacetime meshless method with the iterative scheme.
Sun, 01 Dec 2019 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/11992019-12-01T00:00:00ZLocalized method of fundamental solutions for interior Helmholtz problems with high wave numberhttp://scholars.ntou.edu.tw/handle/123456789/1216標題: Localized method of fundamental solutions for interior Helmholtz problems with high wave number
作者: Qu Wenzhen; Chia-Ming Fan; Gu Yan
摘要: This paper introduces a localized version of the method of fundamental solutions (MFS) named as the localized MFS (LMFS) for two-dimensional (2D) interior Helmholtz problems with high wave number. Due to its full interpolation matrix, the traditional MFS is low-efficiency to solve the above-mentioned problem that requires a large number of boundary nodes for obtaining availably numerical results. For the LMFS, the computational domain is first divided into some overlap subdomains based on the distributed nodes. In each subdomain, physical variables are then represented as linear combinations of the fundamental solution of the governing equation as same as in the traditional MFS. A sparse and banded system matrix is finally formed for the LMFS by satisfying Helmholtz equation and boundary conditions, and thus the developed method is inherently efficient for large-scale problems. Three numerical examples are provided to verify the accuracy and the stability of the LMFS.
Tue, 01 Oct 2019 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/12162019-10-01T00:00:00ZThe method of two-point angular basis function for solving Laplace equationhttp://scholars.ntou.edu.tw/handle/123456789/1202標題: The method of two-point angular basis function for solving Laplace equation
作者: Chung-Lun Kuo; Wei-Chung Yeih; Cheng-Yu Ku; Chia-Ming Fan
摘要: In this paper, an approach to improve the method of angular basis function (MABF) proposed by Young et al. (2015) is proposed. Instead of using lnr in the method of fundamental solution (MFS), the MABF adopts θ to construct the solution. However, since the nature of θ introduces multiple values along the branch cut such that to avoid the branch cut passing through the domain is an important issue (Li et al., 2018). Noticing this difficulty, Alves et al. (2018) first proposed a remedy which used a pair of two points to restrict the discontinuity appearing only along the line segment between two points, and they named this approach as cracklets. In this article, the two-point angular basis function (cracklets) is investigated deeply. We explain why for a multiply connected domain with a logarithm singular solution the cracklets will encounter failure. To resolve this difficulty, one can adopt the proposed method (cracklets) with the MFS or one can use domain decomposition method to separate the domain into several singly connected domains. Seven numerical examples are provided to show the validity of this method, and examples for dealing with the multiply connected domain are focused to support our claims.
Sun, 01 Sep 2019 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/12022019-09-01T00:00:00ZLocalized method of fundamental solutions for large-scale modelling of three-dimensional anisotropic heat conduction problems - Theory and MATLAB codehttp://scholars.ntou.edu.tw/handle/123456789/1184標題: Localized method of fundamental solutions for large-scale modelling of three-dimensional anisotropic heat conduction problems - Theory and MATLAB code
作者: Yan Gu; Chia-Ming Fan; WenZhen Qu; Fajie Wang
摘要: The method of fundamental solutions (MFS) belongs to the family of meshless boundary collocation methods and now has been successfully tried for many kinds of engineering problems. The traditional MFS based on the “global” boundary discretization, however, leads to dense and non-symmetric coefficient matrices that, although smaller in sizes, require huge computational cost to compute the system of equations using direct solvers. Such an approach will be arduous, time consuming and computationally expensive for analyzing large-scale problems. In the present work, a localized version of the MFS, named as the localized MFS (LMFS), is proposed for large-scale modelling of three-dimensional (3D) anisotropic heat conduction problems. In the LMFS, the computational domain can be divided into small subdomains with a simple geometry such as circle and/or sphere. To each of the subdomains, the MFS formulation is applied and the unknown coefficients on the local simple geometric boundary are approximated by the moving least square (MLS) method. The satisfactions of governing equations at interior points and boundary conditions at boundary nodes lead to a sparse and banded system matrix. Numerical examples with up to 1,000,000 unknowns are solved successfully using the developed LMFS code. The advantages, disadvantages and potential applications of the proposed method, as compared with the traditional MFS and boundary element method (BEM), are discussed. Finally, a fast, reliable and self-contained MATLAB code is provided in the part of Supplementary Materials of the paper.
Thu, 01 Aug 2019 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/11842019-08-01T00:00:00ZA direct Chebyshev collocation method for the numerical solutions of three-dimensional Helmholtz-type equationshttp://scholars.ntou.edu.tw/handle/123456789/1140標題: A direct Chebyshev collocation method for the numerical solutions of three-dimensional Helmholtz-type equations
作者: Bai Zi-Qiang; Gu Yan; Fan Chia-Ming
摘要: In this study, a new framework for the numerical solutions of inhomogeneous Helmholtz-type equations on three-dimensional (3D) arbitrary domains is presented. A Chebyshev collocation scheme (CCS) is introduced for the efficient and accurate approximation of particular solution for the given 3D boundary value problem. We collocate the numerical scheme at the Gauss–Lobatto nodes to ensure the pseudo-spectral convergence of the Chebyshev interpolation. After the particular solution is evaluated, the introduced CCS is coupled with a two-stage and one-stage numerical schemes to evaluate the final solutions of the given problem. In the two-stage approach, the given inhomogeneous problem is converted to a homogeneous equation and then the boundary-type methods, such as the method of fundamental solutions (MFS), can be used to evaluate the resulting homogeneous solutions. In the one-stage scheme, by imposing the boundary conditions directly to the CCS procedure, the final solutions of the given inhomogeneous problem can be obtained straightforward without the need of using the MFS or other boundary methods to find the homogeneous solution. Two benchmark numerical examples in both smooth and piecewise smooth 3D geometries are presented to demonstrate the applicability and efficiency of the proposed method.
Mon, 01 Jul 2019 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/11402019-07-01T00:00:00ZLocalized method of fundamental solutions for large-scale modeling of two-dimensional elasticity problemshttp://scholars.ntou.edu.tw/handle/123456789/1186標題: Localized method of fundamental solutions for large-scale modeling of two-dimensional elasticity problems
作者: Yan Gu; Chia-Ming Fan; Rui-Ping Xu
摘要: The traditional method of fundamental solutions (MFS) based on the “global” boundary discretization leads to dense and non-symmetric coefficient matrices that, although smaller in sizes, require huge computational cost to compute the system of equations using direct solvers. In this study, a localized version of the MFS (LMFS) is proposed for the large-scale modeling of two-dimensional (2D) elasticity problems. In the LMFS, the whole analyzed domain can be divided into small subdomains with a simple geometry. To each of the subdomain, the traditional MFS formulation is applied and the unknown coefficients on the local geometric boundary can be calculated by the moving least square method. The new method yields a sparse and banded matrix system which makes the method very attractive for large-scale simulations. Numerical examples with up to 200,000 unknowns are solved successfully using the developed LMFS code.
Mon, 01 Jul 2019 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/11862019-07-01T00:00:00ZThe MAPS based on trigonometric basis functions for solving elliptic partial differential equations with variable coefficients and Cauchy-Navier equationshttp://scholars.ntou.edu.tw/handle/123456789/1232標題: The MAPS based on trigonometric basis functions for solving elliptic partial differential equations with variable coefficients and Cauchy-Navier equations
作者: Dan Wang; C.S. Chen; C.M. Fan; Ming Li
摘要: In this paper, we extended the method of approximate particular solutions (MAPS) using trigonometric basis functions to solve two-dimensional elliptic partial differential equations (PDEs) with variable-coefficients and the Cauchy–Navier equations. The new approach is based on the closed-form particular solutions for second-order differential operators with constant coefficients. For the Cauchy–Navier equations, a reformulation of the equations is required so that the particular solutions for the new differential operators are available. Five numerical examples are provided to demonstrate the effectiveness of the proposed method.
Wed, 01 May 2019 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/12322019-05-01T00:00:00ZLocalized method of fundamental solutions for solving two-dimensional Laplace and biharmonic equationshttp://scholars.ntou.edu.tw/handle/123456789/1163標題: Localized method of fundamental solutions for solving two-dimensional Laplace and biharmonic equations
作者: C.M. Fan; Y.K. Huang; C.S. Chen; S.R. Kuo
摘要: The localized method of fundamental solutions (LMFS) is proposed in this paper for solving two-dimensional boundary value problems, governed by Laplace and biharmonic equations, in complicated domains. Traditionally, the method of fundamental solutions (MFS) is a global method and the resultant matrix is dense and ill-conditioned. In this paper, it is the first time that the LMFS, the localized version of the MFS, is proposed. In the LMFS, the solutions at every interior node are expressed as linear combinations of solutions at some nearby nodes, while the numerical procedures of MFS are implemented within every local subdomain. The satisfactions of governing equation at interior nodes and boundary conditions at boundary nodes can yield a sparse system of linear algebraic equations, so the numerical solutions can be efficiently acquired by solving the resultant sparse system. Six numerical examples are given to demonstrate the effectiveness of the proposed LMFS.
Mon, 01 Apr 2019 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/11632019-04-01T00:00:00ZA novel space-time meshless method for solving the backward heat conduction problemhttp://scholars.ntou.edu.tw/handle/123456789/1200標題: A novel space-time meshless method for solving the backward heat conduction problem
作者: Cheng-Yu Ku; Chih-Yu Liu; Wei-Chung Yeih; Chein-Shan Liu; Chia-Ming Fan
摘要: This paper presents a novel space–time meshless method for solving the backward heat conduction problem (BHCP). A numerical approximation is obtained using the Trefftz basis function of the heat equation. The Trefftz method, which differs from conventional collocation methods based on a set of unstructured points in space, is used in this study to collocate boundary points in the space–time coordinate system such that the initial and boundary conditions can both be treated as boundary conditions on the space–time domain boundary. Because the solution in time on the boundary of the domain is unknown, the BHCP can be transformed into an inverse boundary value problem. The numerical solution is obtained by superpositioning the Trefftz base functions that automatically satisfy the governing equation. The validity of the proposed method is established for several test problems, including the one-dimensional BHCP and two-dimensional BHCP. The accuracy of the proposed method is compared with that of a conventional time-marching scheme based on the finite difference method. The results demonstrate that highly accurate numerical solutions can be obtained and errors may not accumulate over the entire time domain. Moreover, the boundary data on the inaccessible boundary can be recovered even when the partial data on the final time boundary are absent.
Fri, 01 Mar 2019 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/12002019-03-01T00:00:00ZNumerical solutions of waves-current interactions by generalized finite difference methodhttp://scholars.ntou.edu.tw/handle/123456789/1156標題: Numerical solutions of waves-current interactions by generalized finite difference method
作者: Chia-Ming Fan; Chu Chi-Nan; Božidar Šarler; Li Tsung-Han
摘要: In this paper, a meshless numerical wave flume, based on the generalized finite difference method (GFDM), is adopted to accurately and efficiently simulate the interactions of water waves and current. The GFDM, a newly-developed meshless method, is truly free from mesh generation and numerical quadrature. The proposed meshless numerical wave flume is the combination of the GFDM, the second-order Runge–Kutta method, the semi-Lagrangian approach, the sponge layer and the ramping function. The problems of wave-current interactions in flumes with horizontal and inclined bottoms are accurately and stably investigated by the proposed meshless scheme, respectively. The changes of waveform can be obviously found, while the cases of coplanar, opposing and no currents are stably simulated. Besides, the distribution of steady current in the flume with inclined bottom, which is governed by an inverse Cauchy problem, is acquired by the GFDM in a stable manner. Numerical results of wave-current interactions are compared with other solutions to verify the accuracy of the proposed meshless scheme. Additionally, different parameters of the proposed meshless numerical scheme are examined to validate the consistency and stability of the proposed numerical wave flume for solutions of wave-current interactions.
Fri, 01 Mar 2019 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/11562019-03-01T00:00:00ZOn modeling subsurface flow using a novel hybrid Trefftz-MFS methodhttp://scholars.ntou.edu.tw/handle/123456789/1201標題: On modeling subsurface flow using a novel hybrid Trefftz-MFS method
作者: Cheng-Yu Ku; Jing-En Xiao; Chih-Yu Liu; Chia-Ming Fan
摘要: In this paper, a novel hybrid Trefftz–MFS method for modeling subsurface flow problems is presented. The proposed method constructs its nonsingular basis function as a series using T functions from the Trefftz method instead of using the singular solution in the method of fundamental solutions (MFS). Numerical solutions are approximated by superpositioning basis functions that are expressed in terms of many source points. Because of the use of the nonsingular basis function, the position of the source point in the proposed method is not sensitive to the results; thus, it resolves a major issue in the MFS for finding a satisfactory location for the source point. Additionally, the order of the nonsingular basis function can be reduced, because the addition theorem is used for approximating the solution for many source points. Consequently, the ill-posedness resulting from the adoption of higher order terms for the solution with only one source point in the collocation Trefftz method (CTM) is mitigated. The proposed method is validated for several test problems. Application examples are also performed. The results reveal that this novel method provides a promising solution that combines the benefits of the CTM and the MFS, such as the boundary collocation only and high accuracy. Moreover, the limitations to the practical application of the CTM and the MFS can be overcome using the proposed method.
Fri, 01 Mar 2019 00:00:00 GMThttp://scholars.ntou.edu.tw/handle/123456789/12012019-03-01T00:00:00Z