A simple Galerkin meshless method, the Fragile Points method using point stiffness matrices, for <scp>2D</scp> linear elastic problems in complex domains with crack and rupture propagation

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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.

“…Substituting the test functions v andṽ into Equation (26), the formula can also be written in a collocation form: Or in a matrix form:…”

Section: The Weak Form 1 With Dirac Delta Function (Collocation Method)mentioning

confidence: 99%

Section: Trial Functions and Meshless Approximationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Guan

Atluri

2021

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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

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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.

An implicit Updated Lagrangian Fragile Points Method with a support domain refinement scheme for solving large deformation problems

Dai,

Ke,

et al. 2024

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Engineering structures may undergo large deformations, but simulating is still challenging for existing numerical methods, for example, the element‐based methods struggle from the mesh distortion and the strong form particle‐based methods exhibit tensile instability. This article aims on presenting a novel numerical method for large deformation simulations, which is named the implicit Updated Lagrangian Fragile Points Method (ULFPM). The point‐based nature of the proposed method ensures immunity to mesh distortion, and its stability is guaranteed by its foundation on weak form formulations. For verifications, the proposed implicit ULFPM has been applied to several large deformation problems. For all the examples, the proposed method is able to provide reliable results with a good agreement with the reference results. Moreover, the implicit ULFPM performs better than the Finite Element Method to some extent for extremely large deformation problems. It is also discussed that the accuracy of the implicit ULFPM is improved significantly by using the support domain refinement scheme. The examples indicate that the proposed method can serve as a reliable tool for predicting the large deformations of structures in engineering practices.