Three-dimensional non-linear fracture mechanics by enriched meshfree methods without asymptotic enrichment

Bordas, Stéphane Pierre Alain; Zi, Goangseup; Rabczuk, Timon

doi:10.1007/978-1-4020-6530-9_2

Cited by 6 publications

(5 citation statements)

References 49 publications

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“…The 3 rd and 4 th order splines are the preferred types (see Equation (26) and Equation (27)). In order to vary the shape of the weight functions, we have composed the splines with the following power function…”

Section: Gfdmentioning

confidence: 99%

“…The exponential weight function presented in Equation 25is compared to the 3 rd and 4 th order splines, of equations (26) and (27) respectively. We have considered various combinations of shape parameters ( ) and exponents (α).…”

Section: Psementioning

confidence: 99%

“…For instance, methods such as XFEM enable the simulation of propagating discontinuities in the field variable or its derivative with minimal or no remeshing, whilst some versions of partition of unity methods require special treatment of boundary conditions. Some of the most exciting applications of partition of unity methods include fracture mechanics either as enriched finite elements [15][16][17][18][19][20][21][22] or as enriched meshfree methods [23][24][25][26][27][28][29]. Note that such partition of unity methods were also used to permit the implicit treatment of (evolving) discontinuities using level set methods, including an implicit description of the boundary of the computational domain [30,31].…”

Section: Introductionmentioning

confidence: 99%

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Taylor-Series Expansion Based Numerical Methods: A Primer, Performance Benchmarking and New Approaches for Problems with Non-smooth Solutions

Jacquemin

Tomar

Agathos

et al. 2019

Arch Computat Methods Eng

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We provide a primer to numerical methods based on Taylor series expansions such as generalized finite difference methods and collocation methods. We provide a detailed benchmarking strategy for these methods as well as all data files including input files, boundary conditions, point distribution and solution fields, so as to facilitate future benchmarking of new methods. We review traditional methods and recent ones which appeared in the last decade. We aim to help newcomers to the field understand the main characteristics of these methods and to provide sufficient information to both simplify implementation and benchmarking of new methods. Some of the examples are chosen within a subset of problems where collocation is traditionally known to perform sub-par, namely when the solution sought is non-smooth, i.e. contains discontinuities, singularities or sharp gradients. For such problems and other simpler ones with smooth solutions, we study in depth the influence of the weight function, correction function, and the number of nodes in a given support. We also propose new stabilization approaches to improve the accuracy of the numerical methods. In particular, we experiment with the use of a Voronoi diagram for weight computation, collocation method stabilization approaches, and support node selection for problems with singular solutions. With an appropriate selection of the above-mentioned parameters, the resulting collocation methods are compared to the moving least-squares method (and variations thereof), the radial basis function finite difference method and the finite element method. Extensive tests involving two and three dimensional problems indicate that the methods perform well in terms of efficiency (accuracy versus computational time), even for non-smooth solutions.

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Section: Gfdmentioning

confidence: 99%

Section: Psementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Taylor-Series Expansion Based Numerical Methods: A Primer, Performance Benchmarking and New Approaches for Problems with Non-smooth Solutions

Jacquemin

Tomar

Agathos

et al. 2019

Arch Computat Methods Eng

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“…Zi et al [18] investigated extended meshfree methods without branch enrichment for cohesive cracks. Bordas et al [19] proposed enriched meshfree methods without asymptotic enrichment for 3D nonlinear fracture mechanics. Rabczuk and Samaniego [20] worked on discontinuous modelling of shear bands using adaptive meshfree methods.…”

Section: Introductionmentioning

confidence: 99%

Error Estimate and Adaptive Refinement in Mixed Discrete Least Squares Meshless Method

Amani

Bagherzadeh

Rabczuk

2014

Mathematical Problems in Engineering

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The node moving and multistage node enrichment adaptive refinement procedures are extended in mixed discrete least squares meshless (MDLSM) method for efficient analysis of elasticity problems. In the formulation of MDLSM method, mixed formulation is accepted to avoid second-order differentiation of shape functions and to obtain displacements and stresses simultaneously. In the refinement procedures, a robust error estimator based on the value of the least square residuals functional of the governing differential equations and its boundaries at nodal points is used which is inherently available from the MDLSM formulation and can efficiently identify the zones with higher numerical errors. The results are compared with the refinement procedures in the irreducible formulation of discrete least squares meshless (DLSM) method and show the accuracy and efficiency of the proposed procedures. Also, the comparison of the error norms and convergence rate show the fidelity of the proposed adaptive refinement procedures in the MDLSM method.

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“…However, this method is not feasible when the crack front is curved in 3D. Alternatively the use of Lagrange multipliers is proposed in the same paper 15 and is later adopted in a number of subsequent papers 17–19 that develop meshless methods for 3D fracture modelling. However, this method involves additional unknowns that increases with the number of cracks.…”

Section: Introductionmentioning

confidence: 99%

Accurate fracture modelling using meshless methods, the visibility criterion and level sets: Formulation and 2D modelling

Zhuang

Augarde

Bordas

2011

Numerical Meth Engineering

Self Cite

102

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SUMMARYFracture modelling using numerical methods is well-advanced in 2D using techniques such as the extended finite element method (XFEM). The use of meshless methods for these problems lags somewhat behind, but the potential benefits of no meshing (particularly in 3D) prompt continued research into their development. In methods where the crack face is not explicitly modelled (as the edge of an element for instance), two procedures are instead used to associate the displacement jump with the crack surface: the visibility criterion and the diffraction method. The visibility criterion is simple to implement and efficient to compute, especially with the help of level set coordinates. However, spurious discontinuities have been reported around crack tips using the visibility criterion, whereas implementing the diffraction method in 3D is much more complicated than the visibility criterion. In this paper, a tying procedure is proposed to remove the difficulty with the visibility criterion so that crack tip closure can be ensured while the advantages of the visibility criterion can be preserved. The formulation is based on the use of level set coordinates and the element-free Galerkin method, and is generally applicable for single or multiple crack problems in 2D or 3D. The paper explains the formulation and provides verification of the method against a number of 2D crack problems.

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Three-dimensional non-linear fracture mechanics by enriched meshfree methods without asymptotic enrichment

Cited by 6 publications

References 49 publications

Taylor-Series Expansion Based Numerical Methods: A Primer, Performance Benchmarking and New Approaches for Problems with Non-smooth Solutions

Taylor-Series Expansion Based Numerical Methods: A Primer, Performance Benchmarking and New Approaches for Problems with Non-smooth Solutions

Error Estimate and Adaptive Refinement in Mixed Discrete Least Squares Meshless Method

Accurate fracture modelling using meshless methods, the visibility criterion and level sets: Formulation and 2D modelling

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