A new meshless “fragile points method” and a local variational iteration method for general transient heat conduction in anisotropic nonhomogeneous media. Part I: Theory and implementation

Guan, Yongqiang; Grujicic, Rade; Wang, Xuechuan; Dong, Leiting; Atluri, Satya N.

doi:10.1080/10407790.2020.1747278

Cited by 12 publications

(33 citation statements)

References 41 publications

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“…The material is functionally graded (FG), 39 with the following material properties: These examples have also been studied using the LBIE method, 40 the MLPG method, [41][42][43] the meshless point interpolation method (PIM), 1 and the Galerkin FPM. 27 These previous studies present consistent solutions. Therefore, in this article, the numerical results of the Galerkin FPM and the exact solutions 1 are exploited as benchmarks.…”

Section: Anisotropic Nonhomogeneous Examples In a Square Domainsupporting

confidence: 69%

“…In the current work, we employ the backward Euler (BE) scheme and a local variational iteration method (LVIM). The formulation and numerical implementation of the LVIM are presented in our previous works 27,36 and thus omitted here. Note that the LVIM can be remarkably more efficient than the classic Backward Euler scheme, especially for problems with a sparse Jacobian matrix.…”

Section: Time Discretization Methodsmentioning

confidence: 99%

“…27,28 This article is a follow-up of the previous study. 27,28 Several improved versions of the FPM are carried out in this article, based on Petrov-Galerkin formulations. When the test functions are chosen as Dirac delta function, Heaviside step function, and local fundamental solutions (assuming locally homogeneous material properties), integrals in the subdomains or on the local boundaries may vanish or become pure contour integrals.…”

Section: Introductionmentioning

confidence: 92%

“…If the test function v is selected to have the same polynomial shape as the trial function u h , the Galerkin FPM is achieved. 27 For isotropic problems, the Galerkin FPM leads to symmetric and sparse matrices. Since there exist only the first derivatives of u and v, linear trial and test functions are sufficient, and thus the local approximation calculations are heavily reduced.…”

Section: 2mentioning

confidence: 99%

“…As can be seen, the result of all the methods shows great consistency with the exact solution, as well as the numerical results achieved in previous literatures. 27,37,38 In Table 2, 1 and 2 denote the nondimensional penalty parameters used in the FPM and PG-FPMs. The recommended ranges of 1 and 2 will be given in section 8.…”

Section: Isotropic Homogeneous Examplesmentioning

confidence: 99%

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Meshless fragile points methods based on Petrov‐Galerkin weak‐forms for transient heat conduction problems in complex anisotropic nonhomogeneous media

Guan

Atluri

2021

Numerical Meth Engineering

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Three kinds of fragile points methods based on Petrov-Galerkin weak-forms (PG-FPMs) are proposed for analyzing heat conduction problems in nonhomogeneous anisotropic media. This is a follow-up of the previous study on the original FPM based on a symmetric Galerkin weak-form. The trial function is piecewise-continuous, written as local Taylor expansions at the fragile points. A modified radial basis function-based differential quadrature (RBF-DQ) method is employed for establishing the local approximation. The Dirac delta function, Heaviside step function, and the local fundamental solution of the governing equation are alternatively used as test functions. Vanishing or pure contour integral formulation in subdomains or on local boundaries can be obtained. Extensive numerical examples in 2D and 3D are provided as validations. The collocation method (PG-FPM-1) is superior in transient analysis with arbitrary point distribution and domain partition. The finite volume method (PG-FPM-2)shows the best efficiency, saving 25% to 50% computational time comparing with the Galerkin FPM. The singular solution method (PG-FPM-3) is highly efficient in steady-state analysis. The anisotropy and nonhomogeneity give rise to no difficulties in all the methods. The proposed PG-FPM approaches represent an improvement to the original Galerkin FPM, as well as to other meshless methods in earlier literature.

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Section: Anisotropic Nonhomogeneous Examples In a Square Domainsupporting

confidence: 69%

Section: Time Discretization Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 92%

Section: 2mentioning

confidence: 99%

Section: Isotropic Homogeneous Examplesmentioning

confidence: 99%

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Meshless fragile points methods based on Petrov‐Galerkin weak‐forms for transient heat conduction problems in complex anisotropic nonhomogeneous media

Guan

Atluri

2021

Numerical Meth Engineering

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A Fragile Points Method, with an interface debonding model, to simulate damage and fracture of U‐notched structures

Wang

Shen

et al. 2022

Numerical Meth Engineering

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Notched components are commonly used in engineering structures, where stress concentration may easily lead to crack initiation and development. The main goal of this work is to develop a simple numerical method to predict the strength and crack‐growth‐path of U‐notched specimens made of brittle materials. For this purpose, the Fragile Points Method (FPM), as previously proposed by the authors, has been augmented by an interface debonding model at the interfaces of the FPM domains, to simulate crack initiation and development. The formulations of FPM are based on a discontinuous Galerkin weak form where point‐based piece‐wise‐continuous polynomial test and trial functions are used instead of element‐based basis functions. In this work, the numerical fluxes introduced across interior interfaces between subdomains are postulated as the tractions acting on the interface derived from an interface debonding model. The interface damage is triggered when the numerical flux reaches the interface strength, and the process of crack‐surface separation is governed by the fracture energy. In this way, arbitrary crack initiation and propagation can be naturally simulated without the need for knowing the fracture‐patch before‐hand. Additionally, a small penalty parameter is sufficient to enforce the weak‐form continuity condition before damage initiation, without causing problems such as artificial compliance and numerical ill‐conditioning. As validations, the proposed FPM method with the interface debonding model is used to predict fracture strength and crack‐growth trajectories of U‐notched structures made of brittle materials, which is useful but challenging in engineering structural design practices.

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A Meshless Method of Solving Three-Dimensional Nonstationary Heat Conduction Problems in Anisotropic Materials

et al. 2021

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A new meshless “fragile points method” and a local variational iteration method for general transient heat conduction in anisotropic nonhomogeneous media. Part I: Theory and implementation

Cited by 12 publications

References 41 publications

Meshless fragile points methods based on Petrov‐Galerkin weak‐forms for transient heat conduction problems in complex anisotropic nonhomogeneous media

Meshless fragile points methods based on Petrov‐Galerkin weak‐forms for transient heat conduction problems in complex anisotropic nonhomogeneous media

A Fragile Points Method, with an interface debonding model, to simulate damage and fracture of U‐notched structures

A Meshless Method of Solving Three-Dimensional Nonstationary Heat Conduction Problems in Anisotropic Materials

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