A coupled FEM/BEM approach and its accuracy for solving crack problems in fracture mechanics

Aour, Benaoumeur; Rahmani, O.; Naït-Abdelaziz, Moussa

doi:10.1016/j.ijsolstr.2006.08.001

Cited by 36 publications

(25 citation statements)

References 21 publications

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“…The potential energy of the coupled system can be written as:

{normalΠ}_{tot} = {normalΠ}_{F} + {normalΠ}_{B} + {normalΠ}_{I}

where Π t o t is the total energy of the whole system; Π F and Π B represent the potential energies of the FEM and BEM domains (excluding the interface), respectively; and Π I represents the potential energy components from the interface zone. For any elastic domain Ω, the potential energy Π Ω can be written as:

{normalΠ}_{normalΩ} = \frac{1}{2} \int_{normalΩ} σ_{ij} ε_{ij} d normalΩ - \int_{normalΓ} u_{i} t_{i} d normalΓ

Using the matrix form derived in Equations and can be written for the FEM domain as:

{normalΠ}_{F} = \frac{1}{2} {boldu}_{F}^{T} bold \bar{K} {boldu}_{F} - {boldu}_{F}^{T} boldP

In order to write Π B similarly, we first need to find the BEM stiffness matrix K B , which can be obtained by rearranging Equation to be in the following form:

[boldM] {[boldG]}^{- 1} [boldH] ({}, {boldu}_{B}) = [boldM] ({}, boldt)

where M is a matrix with similar structure as the standard consistent mass matrix, defined in terms of h i , the BEM 1D shape functions, as follows :

M_{ij} = \sum_{e = 1}^{N_{BE}} \int_{{normalΓ}^{e}} h_{i} h_{j} d normalΓ<...>

…”

Section: Coupling Proceduresmentioning

confidence: 99%

“…Coupled BEM‐FEM for the solution of non‐linear problems has been investigated by many researchers. Examples include elasto‐plasticity , fracture mechanics and contact problems , among other.…”

Section: Introductionmentioning

confidence: 99%

“…Coupled BEM-FEM for the solution of non-linear problems has been investigated by many researchers. Examples include elasto-plasticity [33][34][35][36][37], fracture mechanics [9,38] and contact problems [10,39,40], among other.Another interesting way of modelling material failure is by continuum damage mechanics, which leads to material non-linearity. The concept of damage mechanics assumes the presence of micro cracks and voids in the structure and therefore models the damaged zones by reducing the area of the structure in those regions [41].…”

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confidence: 99%

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Adaptive modeling of damage growth using a coupled FEM/BEM approach

Mobasher

Waisman

2015

Numerical Meth Engineering

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SUMMARYWe propose a coupled boundary element method (BEM) and a finite element method (FEM) for modelling localized damage growth in structures. BEM offers the flexibility of modelling large domains efficiently, while the non-linear damage growth is accurately accounted by a local FEM mesh. An integral-type nonlocal continuum damage mechanics with adapting FEM mesh is used to model multiple damage zones and follow their propagation in the structure. Strong form coupling, BEM hosted, is achieved using Lagrange multipliers. Because the non-linearity is isolated in the FEM part of the system of equations, the system size is reduced using Schur complement approach, then the solution is obtained by a monolithic Newton method that is used to solve both domains simultaneously. The coupled BEM/FEM approach is verified by a set of convergence studies, where the reference solution is obtained by a fine FEM. In addition, the method is applied to multiple fractures growth benchmark problems and shows good agreement with the literature.

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“…The potential energy of the coupled system can be written as:

{normalΠ}_{tot} = {normalΠ}_{F} + {normalΠ}_{B} + {normalΠ}_{I}

{normalΠ}_{normalΩ} = \frac{1}{2} \int_{normalΩ} σ_{ij} ε_{ij} d normalΩ - \int_{normalΓ} u_{i} t_{i} d normalΓ

Using the matrix form derived in Equations and can be written for the FEM domain as:

{normalΠ}_{F} = \frac{1}{2} {boldu}_{F}^{T} bold \bar{K} {boldu}_{F} - {boldu}_{F}^{T} boldP

In order to write Π B similarly, we first need to find the BEM stiffness matrix K B , which can be obtained by rearranging Equation to be in the following form:

[boldM] {[boldG]}^{- 1} [boldH] ({}, {boldu}_{B}) = [boldM] ({}, boldt)

where M is a matrix with similar structure as the standard consistent mass matrix, defined in terms of h i , the BEM 1D shape functions, as follows :

M_{ij} = \sum_{e = 1}^{N_{BE}} \int_{{normalΓ}^{e}} h_{i} h_{j} d normalΓ<...>

…”

Section: Coupling Proceduresmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Adaptive modeling of damage growth using a coupled FEM/BEM approach

Mobasher

Waisman

2015

Numerical Meth Engineering

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“…This approach has been assessed by the evaluation of Stress Intensity Factors (SIF) using two examples of fracture mechanics, i.e., centre-cracked plate, (b) cracks emanating from a circular hole [8] . The prediction of fatigue crack path was applied on tensile specimen with holes.…”

Section: Introductionmentioning

confidence: 99%

Finite Element Analysis of the Crack Propagation for Solid Materials

Souiyah

Muchtar²,

Alshoaibi

et al. 2009

American J. of Applied Sciences

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Problem statement:The use of fracture mechanics techniques in the assessment of performance and reliability of structure is on increase and the prediction of crack propagation in structure play important part. The finite element method is widely used for the evaluation of SIF for various types of crack configurations. Source code program of two-dimensional finite element model had been developed, to demonstrate the capability and its limitations, in predicting the crack propagation trajectory and the SIF values under linear elastic fracture analysis. Approach: Two different geometries were used on this finite element model in order, to analyze the reliability of this program on the crack propagation in linear and nonlinear elastic fracture mechanics. These geometries were namely; a rectangular plate with crack emanating from square-hole and Double Edge Notched Plate (DENT). Where, both geometries are in tensile loading and under mode I conditions. In addition, the source code program of this model was written by FORTRAN language. Therefore, a Displacement Extrapolation Technique (DET) was employed particularly, to predict the crack propagations directions and to, calculate the Stress Intensity Factors (SIFs). Furthermore, the mesh for the finite elements was the unstructured type; generated using the advancing front method. And, the global h-type adaptive mesh was adopted based on the norm stress error estimator. While, the quarterpoint singular elements were uniformly generated around the crack tip in the form of a rosette. Moreover, make a comparison between this current study with other relevant and published research study. Results: The application of the source code program of 2-D finite element model showed a significant result on linear elastic fracture mechanics. Based on the findings of the two different geometries from the current study, the result showed a good agreement. And, it seems like very close compare to the other published results. Conclusion: A developed a source program of finite element model showed that is capable of demonstrating the SIF evaluation and the crack path direction satisfactorily. Therefore, the numerical finite element analysis with displacement extrapolation method, had been successfully employed for linear-elastic fracture mechanics problems.

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“…With the EFG method, a growing crack can be modeled simply by extending the surfaces that correspond to the crack without the need for remeshing. The concept of extending surfaces within the domain to represent crack propagation is actually similar to the treatment of cracks in the boundary element method [12][13][14][15][16]. Several extensions of the EFG method to model crack propagation have been proposed [17][18][19][20][21][22][23][24][25][26][27][28].…”

Section: Introductionmentioning

confidence: 99%

Analysis of Cohesive Crack Growth by the Element-Free Galerkin Method

Soparat

Nanakorn

2008

J. mech.

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In this study, the element-free Galerkin (EFG) method is extended to include nonlinear behavior of cohesive cracks in 2D domains. A cohesive curved crack is modeled by using several straight-line interface elements connected to form the crack. The constitutive law of cohesive cracks is considered through the use of these interface elements. The stiffness equation of the domain is constructed by directly including, in the weak form of the global system equation, a term related to the energy dissipation along the interface elements. The constitutive law of cohesive cracks can then be considered directly and efficiently by using this energy term. The validity and efficiency of the proposed method are discussed by using problems found in the literature. The proposed method is found to be an efficient method for simulating propagation of cohesive cracks.

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A coupled FEM/BEM approach and its accuracy for solving crack problems in fracture mechanics

Cited by 36 publications

References 21 publications

Adaptive modeling of damage growth using a coupled FEM/BEM approach

Adaptive modeling of damage growth using a coupled FEM/BEM approach

Finite Element Analysis of the Crack Propagation for Solid Materials

Analysis of Cohesive Crack Growth by the Element-Free Galerkin Method

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