An efficient and accurate hybrid weak-form meshless method for transient nonlinear heterogeneous heat conduction problems

Xu, Bing‐Bing; Gao, Xiao‐Wei; Cui, Miao

doi:10.1007/s00366-020-01050-7

Cited by 11 publications

(7 citation statements)

References 49 publications

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“…Easy to find that the strong‐form methods are more flexible. But as discussed in some meshless References 18 and 19, the accuracy and stability of the strong‐form methods are not ideal. In order to obtain better stability and higher accuracy, the local weak‐form formulation is constructed in this article.…”

Section: Zonal Free Element Methodsmentioning

confidence: 99%

“…[8][9][10][11] Motivated by the goal of efficient and robust discretization schemes in computational solid mechanics, the virtual element method has been proposed 12,13 and developed for nonlinear problems. 14 To avoid the mesh generation in the above mesh-based methods, the mesh-free methods (MFMs) [15][16][17][18][19] have been proposed and developed rapidly. As an important class of mesh-free methods, strong-form methods often have greater efficiency and flexibility because it does not need any integration.…”

Section: Introductionmentioning

confidence: 99%

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Petrov–Galerkin zonal free element method for 2D and 3D mechanical problems

Gao

2023

Numerical Meth Engineering

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In this article, a novel weak‐form zonal Petrov–Galerkin free element method is proposed for two‐ and three‐dimensional linear mechanical problems. By absorbing the advantages of finite block method and strong‐form finite element method, the block mapping technique is used in the free element method. Combining the characteristics of the meshless local Petrov–Galerkin method, the local Petrov–Galerkin formulation based on the zonal free element method is formed at last. Besides, the local integral domain selected in the local collocation element is circular or spherical to simplify programming. The transformation of the local integral domain between the physical and normalized spaces is given for two‐ and three‐dimensional problems. The comparison of accuracy and convergence between the new proposed Petrov–Galerkin method and the conventional methods is carried out. Some challenging examples including fracture mechanics problems and a complex 3D problem are given to validate the convergence and accuracy of the proposed method.

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Section: Zonal Free Element Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Petrov–Galerkin zonal free element method for 2D and 3D mechanical problems

Gao

2023

Numerical Meth Engineering

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“…In particular, some phenomena involving materials with memory can be described by the boundary value problems of nonlocal VIDE in [1][2][3][4]46]. This kind of equation often appears in the form of convolution in practical application, and has been widely used in many scientific fields such as wave propagation theory, physical heat conduction and super-flow theory [30,43,44].…”

Section: Introductionmentioning

confidence: 99%

Numerical analysis of a nonlocal Volterra integro-differential equation

Zhu,

Dong,

Zheng

et al. 2024

Preprint

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We study the numerical approximation to a nonlocal Volterra integro-differential equation, in which the integral term is the convolution product of a positive-definite kernel and a nonlocal peridynamic differential operator. Compared with the classical differential operators, the nonlocal peridynamic differential operators describe, e.g., discontinuities, and have domonstrated more widespread applications. The equation is discretized in space by the Galerkin finite element method, and we accordingly prove its error estimate. We then discretize the equation in time by the backward Euler method, and a positive quadrature rule is combined to approximate the convolution term. The convergence rate of the fully-discrete finite element scheme is proved, and numerical experiments are carried out to substantiate the theoretical findings.

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“…Xu et al [10] proposed a hybrid method combining the Galerkin free element method (GFREM) and the local radial point interpolation method (LRPIM) for solving steady and transient heat conduction problems with temperature-dependent thermophysical properties in heterogeneous media. In simulation of heat conduction with temperature-dependent physical properties and boundary conditions, the general finite element method (FEM) instead of the conventional FEM is introduced by Yao et al [11].…”

Section: Introductionmentioning

confidence: 99%

A Mixed Pseudo-spectral FFT-FE Method for Asymmetric Nonlinear Heat Transfer of a Functionally Graded Hollow Cylinder

2023

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In this article, a novel spectral method based on the integral transform and finite element (FE) method is introduced for nonlinear thermal analysis of a hollow cylinder under asymmetric boundary excitations. The material properties are temperature-dependent and vary in terms of spatial coordinates. This dependency makes the problem to be nonlinear. The intended nonlinear heat conduction equation is discretized using finite elements in the radial direction. Fast Fourier transform (FFT) technique with the uniform distribution of the harmonics in the circumferential direction, is used to discretize the periodic domain and boundary conditions. The use of the FFT algorithm is accompanied by a significant save in computational times and efforts. In such problems, the Pseudo-spectral technique, as an evolved model of the spectral method, is utilized whenever the material properties vary in terms of the periodic variables or there exists a nonlinear term. The convolution sum technique is appropriately used to transform the nonlinear terms in the Fourier space. Thermal boundary conditions at the inner surface of the cylinder are considered in asymmetrical form. In compliance with the other analytical and numerical solutions, the present mixed-method benefits from the fast rate of convergence and high accuracy.

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An efficient and accurate hybrid weak-form meshless method for transient nonlinear heterogeneous heat conduction problems

Cited by 11 publications

References 49 publications

Petrov–Galerkin zonal free element method for 2D and 3D mechanical problems

Petrov–Galerkin zonal free element method for 2D and 3D mechanical problems

Numerical analysis of a nonlocal Volterra integro-differential equation

A Mixed Pseudo-spectral FFT-FE Method for Asymmetric Nonlinear Heat Transfer of a Functionally Graded Hollow Cylinder

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