Finite block method in fracture analysis with functionally graded materials

Li, J.; Liu, J.Z.; Korakianitis, Theodosios; Wen, P.H.

doi:10.1016/j.enganabound.2017.05.012

Cited by 25 publications

(9 citation statements)

References 20 publications

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“…can be approximated by using Lagrange polynomials as [42]: The first order derivative of u with respect to can be represented as:…”

Section: Two-dimensional Casementioning

confidence: 99%

“…nodes whose distribution are selected to be Chebyshev's roots as: [44] and numerical solutions by FBM [42], FEM [43] and EFG [45][46].…”

Section: Example 61: 2d Beammentioning

confidence: 99%

“…Another advantage of the block method is the satisfaction of continuity at the interface of neighboring blocks. The analysis of elastic fracture mechanics in finite regions by strong form meshless finite block method had already been performed in [42]. In the present paper, the elastic and crack problems in unbounded domains are investigated also.…”

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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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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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“…can be approximated by using Lagrange polynomials as [42]: The first order derivative of u with respect to can be represented as:…”

Section: Two-dimensional Casementioning

confidence: 99%

“…nodes whose distribution are selected to be Chebyshev's roots as: [44] and numerical solutions by FBM [42], FEM [43] and EFG [45][46].…”

Section: Example 61: 2d Beammentioning

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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“…Researchers are actively engaged in an integral transform to develop the subject as it is applicable in describing and analyzing physical systems 1–17 . Laplace transforms (Debnath 10 ) is one of them due to its importance in the signal processing system (Andrews and Phillip 18 ) for solving the dynamic and delayed differential equations 7,19–21 . Jackson 22 introduced q‐calculus, and now a day's q‐calculus got much significance in different branches of science and engineering.…”

Section: Introductionmentioning

confidence: 99%

On a system of q‐modified Laplace transform and its applications

Kılıçman

Sinha

Panda

2021

Math Methods in App Sciences

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“…The split equilibrium problem (1)-(2) constitute a pair of equilibrium problems where is the generalization of split feasibility problems. Some iterative methods have been rapidly established for solving these problems (see [1][2][3][4][5][6][7][8][9][10]).…”

Section: Introductionmentioning

confidence: 99%

Strong convergent iterative techniques for 2-generalized hybrid mappings and split equilibrium problems

Zhao

Liang

Liu

2019

Filomat

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In this paper, we suggest a new iterative scheme for finding a common element of the set of solutions of a split equilibrium problem and the set of fixed points of 2-generalized hybrid mappings in Hilbert spaces. We show that the iteration converges strongly to a common solution of the considered problems. A numerical example is illustrated to verify the validity of the proposed algorithm. The results obtained in this paper extend and improve some known results in the literature.

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Finite block method in fracture analysis with functionally graded materials

Cited by 25 publications

References 20 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

On a system of q‐modified Laplace transform and its applications

Strong convergent iterative techniques for 2-generalized hybrid mappings and split equilibrium problems

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