Finite block method for interface cracks

Li, M.; Meng, Lan; Hinneh, P.; Wen, P.H.

doi:10.1016/j.engfracmech.2016.02.015

Cited by 20 publications

(14 citation statements)

References 30 publications

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“…This section is a brief introduction of the strong form FBM for one-and two-dimensional problems using Lagrange series of interpolation. More details can be referred to [36][37][38][39][40].…”

Section: Strong-form Finite Block Methodsmentioning

confidence: 99%

“…which can be found in previous works of strong form FBM [36][37][38][39][40]. The corresponding partial differentials are given in appendix.…”

Section: Mapping and Coordinate Transformation 31 Finite Blocksmentioning

confidence: 99%

“…At corner joint (Figure 12), the following displacement continuity conditions and point equilibrium equations should be satisfied: Figure 14 and 15, respectively. Numerical results calculated by BEM [38] with 64 boundary elements are shown also as a reference. It can be observe that results produced by irregularly distributed nodes are better than those by uniformly distributed ones.…”

Section: Example 61: 2d Beammentioning

confidence: 99%

“…The finite block method (FBM) proposed recently by Li and Wen et al was a strong form approach and had been successfully applied to deal with heat conduction [36], contact [37] and fracture mechanics [38]. The main idea of the FBM is solving PDEs in large global domain with blocks which can be mapped into normalized sub-domains.…”

mentioning

confidence: 99%

“…Because the edge of infinite block at reachable location is straight, all the mapping introduced in section 2.3 can be employed. In order to capture the stress concentration better, the coordinate of nodes in mapped domains are26 selected to be Chebyshev's roots as given in Eq (38)…”

mentioning

confidence: 99%

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Finite and infinite block Petrov–Galerkin method for cracks in functionally graded materials

Wen

2019

Applied Mathematical Modelling

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This paper presents a novel weak-form block Petrov-Galerkin method (BPGM) for linear elastic and crack problems in functionally graded materials with bounded and unbounded problem domains. The main idea of this approach is to combine the meshless local Petrov-Galerkin method with block method. Once the problem domain is discretized into several sub-regions, named blocks, which can be mapped into normalized square domains. The weak-form Petrov-Galerkin method and polynomial series of interpolations are employed in each block. The computational efficiency is rigorously examined against the strong-form finite block method, the finite element technique and meshless approaches. Numerical results demonstrate that the BPGM possess the following important properties: (1) only a few blocks are required for calculating problems in unbounded regions which saves tedious work of meshing; (2) the displacements and stresses are continuous at the interfaces of neighboring blocks;(3) due to the use of weak formulation, the continuity requirements of the approximation functions are reduced and numerical solutions are stable; (4) because of using Lagrange polynomial interpolation, highly accurate results can be obtained with a small amount of nodes in each block.Emails: liyan001de@gmail.com 2 IntroductionMany engineering problems can be modeled as an extension of either finite or semi-infinite regions. The analytical solutions of these models are often difficult to obtain due to the complexity of material properties and irregular geometries.Numerical techniques are developed to deal with these problems which are modeled by partial differential equations (PDEs), among which the widely used approaches are finite difference method (FDM) [1], finite element method (FEM) [2] and boundary element method (BEM) [3]. The FDM and FEM are mesh dependent and become computationally time demanding on generation of high quality meshes for efficient solutions. The use of fundamental solutions or Green's functions in BEM provides more accurate and efficient approximation. However, it is usually difficult to obtain the fundamental solutions in closed forms, especially for nonhomogeneous and anisotropic materials. In recent decades, meshless techniques were developed to solve practical problems and had drawn the attention of many investigators, including the smooth particle hydrodynamics method [4] , the diffuse element method [5], the element-free Galerkin method (EFG) [6] and the reproducing kernel particle method [7]. Based on the weak Petrov-Galerkin formulation, Atluri and his co-workers developed the meshless local Petrov-Galerkin method (MLPG) with moving least square approximation for problems in arbitrary geometries [8]. Sladek et al. [9] introduced the local boundary integral equation method for boundary value problems in anisotropic non-homogeneous media. Among these meshless approaches, the radial basis functions (RBFs) method [10] was a truly meshless collocation method using in particular the multiquadric shape function [11][12] for...

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Section: Strong-form Finite Block Methodsmentioning

confidence: 99%

“…which can be found in previous works of strong form FBM [36][37][38][39][40]. The corresponding partial differentials are given in appendix.…”

Section: Mapping and Coordinate Transformation 31 Finite Blocksmentioning

confidence: 99%

Section: Example 61: 2d Beammentioning

confidence: 99%

mentioning

confidence: 99%

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confidence: 99%

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Finite and infinite block Petrov–Galerkin method for cracks in functionally graded materials

Wen

2019

Applied Mathematical Modelling

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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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Prediction of 3D grinding temperature field based on meshless method considering infinite element

Wang

et al. 2018

Int J Adv Manuf Technol

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Finite block method for interface cracks

Cited by 20 publications

References 30 publications

Finite and infinite block Petrov–Galerkin method for cracks in functionally graded materials

Finite and infinite block Petrov–Galerkin method for cracks in functionally graded materials

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

Prediction of 3D grinding temperature field based on meshless method considering infinite element

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