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See detailThe smoothed extended finite element method
Natarajan, S.; Bordas, Stéphane UL; Minh, Q. D. et al

in Proceedings of the 6th International Conference on Engineering Computational Technology (2008)

This paper shows how the strain smoothing technique recently proposed by G.R.Liu [1] coined as smoothed finite element method (SFEM) can be coupled to partition of unity methods, namely extended finite ... [more ▼]

This paper shows how the strain smoothing technique recently proposed by G.R.Liu [1] coined as smoothed finite element method (SFEM) can be coupled to partition of unity methods, namely extended finite element method (XFEM) [2] to give birth to the smoothed extended finite element method (SmXFEM), which shares properties both with the SFEM and the XFEM. The proposed method suppresses the need to compute and integrate the derivatives of shape functions (which are singular at the tip in linear elastic fracture mechanics). Additionally, integration is performed along the boundary of the finite elements or smoothing cells and no isoparametric mapping is required, which allows elements of arbitrary shape. We present numerical results for cracks in linear elastic fracture mechanics problems. The method is verified on several examples and comparisons are made to the conventional XFEM. © 2008 Civil-Comp Press. [less ▲]

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See detailComparison of recently developed recovery type discretization error estimators for the extended finite element method
Ródenas, J. J.; Duflot, Marc; Bordas, Stéphane UL et al

in Schrefler, B A; Perego, U (Eds.) 8th World Congress on Computational Mechanics (WCCM8). 5th.European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2008) (2008)

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See detailDerivative recovery and a posteriori error estimate for extended finite elements
Bordas, Stéphane UL; Duflot, M.

in Computer Methods in Applied Mechanics & Engineering (2007), 196(35-36), 3381-3399

This paper is the first attempt at error estimation for extended finite elements. The goal of this work is to devise a simple and effective local a posteriori error estimate for partition of unity ... [more ▼]

This paper is the first attempt at error estimation for extended finite elements. The goal of this work is to devise a simple and effective local a posteriori error estimate for partition of unity enriched finite element methods such as the extended finite element method (XFEM). In each element, the local estimator is the L2 norm of the difference between the raw XFEM strain field and an enhanced strain field computed by extended moving least squares (XMLS) derivative recovery obtained from the raw nodal XFEM displacements. The XMLS construction is tailored to the nature of the solution. The technique is applied to linear elastic fracture mechanics, in which near-tip asymptotic functions are added to the MLS basis. The XMLS shape functions are constructed from weight functions following the diffraction criterion to represent the discontinuity. The result is a very smooth enhanced strain solution including the singularity at the crack tip. Results are shown for two- and three-dimensional linear elastic fracture mechanics problems in mode I and mixed mode. The effectivity index of the estimator is close to 1 and improves upon mesh refinement for the studied near-tip problem. It is also shown that for the linear elastic fracture mechanics problems treated, the proposed estimator outperforms one of the superconvergent patch recovery technique of Zienkiewicz and Zhu, which is only C0. Parametric studies of the general performance of the estimator are also carried out. © 2007 Elsevier B.V. All rights reserved. [less ▲]

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See detailA three-dimensional meshfree method for continuous multiple-crack initiation, propagation and junction in statics and dynamics
Rabczuk, Timon; Bordas, Stéphane UL; Zi, Goangseup

in Computational Mechanics (2007), 40(3), 473-495

This paper proposes a three-dimensional meshfree method for arbitrary crack initiation and propagation that ensures crack path continuity for non-linear material models and cohesive laws. The method is ... [more ▼]

This paper proposes a three-dimensional meshfree method for arbitrary crack initiation and propagation that ensures crack path continuity for non-linear material models and cohesive laws. The method is based on a local partition of unity. An extrinsic enrichment of the meshfree shape functions is used with discontinuous and near-front branch functions to close the crack front and improve accuracy. The crack is hereby modeled as a jump in the displacement field. The initiation and propagation of a crack is determined by the loss of hyperbolicity or the loss of material stability criterion. The method is applied to several static, quasi-static and dynamic crack problems. The numerical results very precisely replicate available experimental and analytical results. [less ▲]

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See detailThree-dimensional non-linear fracture mechanics by enriched meshfree methods without asymptotic enrichment
Bordas, Stéphane UL; Zi, G.; Rabczuk, T.

in Proceedings of the IUTAM Symposium on Discretization Methods for Evolving Discontinuities (2007)

This paper presents a three-dimensional, extrinsically enriched meshfree method for initiation, growth and coalescence of an arbitrary number of cracks in non-linear solids including large deformations ... [more ▼]

This paper presents a three-dimensional, extrinsically enriched meshfree method for initiation, growth and coalescence of an arbitrary number of cracks in non-linear solids including large deformations, for statics and dynamics. The novelty of the methodology fashioned in this work is that only an extrinsic discontinuous enrichment and no near-tip/asymptotic enrichment is required. Instead, a Lagrange multiplier field is added along the crack front to close the crack along the front. This decreases the computational cost and removes difficulties involved with a branch enrichment. Numerical examples treated include the pull-out of a reinforcement bar from a concrete block, and a Taylor bar impact with very large deformation and fragmentation. The results are compared to experimental results, and other simulations from the literature, which shows the robustness and accuracy of the method. © 2007 Springer. [less ▲]

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See detailAn extended finite element library
Bordas, Stéphane UL; Nguyen, P. V.; Dunant, C. et al

in International Journal for Numerical Methods in Engineering (2007), 71(6), 703-732

This paper presents and exercises a general structure for an object-oriented-enriched finite element code. The programming environment provides a robust tool for extended finite element (XFEM ... [more ▼]

This paper presents and exercises a general structure for an object-oriented-enriched finite element code. The programming environment provides a robust tool for extended finite element (XFEM) computations and a modular and extensible system. The programme structure has been designed to meet all natural requirements for modularity, extensibility, and robustness. To facilitate mesh-geometry interactions with hundreds of enrichment items, a mesh generator and mesh database are included. The salient features of the programme are: flexibility in the integration schemes (subtriangles, subquadrilaterals, independent near-tip, and discontinuous quadrature rules); domain integral methods for homogeneous and bi-material interface cracks arbitrarily oriented with respect to the mesh; geometry is described and updated by level sets, vector level sets or a standard method; standard and enriched approximations are independent; enrichment detection schemes: topological, geometrical, narrow-band, etc.; multi-material problem with an arbitrary number of interfaces and slip-interfaces; non-linear material models such as J2 plasticity with linear, isotropic and kinematic hardening. To illustrate the possible applications of our paradigm, we present 2D linear elastic fracture mechanics for hundreds of cracks with local near-tip refinement, and crack propagation in two dimensions as well as complex 3D industrial problems. Copyright © 2007 John Wiley & Sons, Ltd. [less ▲]

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See detailMechanical failure in microstructural heterogeneous materials
Bordas, Stéphane UL; Hoppe, R. H. W.; Petrova, S. I.

in Lecture Notes in Computer Science (2007), 4310 LNCS

Various heterogeneous materials with multiple scales and multiple phases in the microstructure have been produced in the recent years. We consider a mechanical failure due to the initiation and ... [more ▼]

Various heterogeneous materials with multiple scales and multiple phases in the microstructure have been produced in the recent years. We consider a mechanical failure due to the initiation and propagation of cracks in places of high pore density in the microstructures. A multi-scale method based on the asymptotic homogenization theory together with the mesh superposition method (s-version of FEM) is presented for modeling of cracks. The homogenization approach is used on the global domain excluding the vicinity of the crack where the periodicity of the microstructures is lost and this approach fails. The multiple scale method relies on efficient combination of both macroscopic and microscopic models. The mesh superposition method uses two independent (global and local) finite element meshes and the concept of superposing the local mesh onto the global continuous mesh in such a way that both meshes not necessarily coincide. The homogenized material model is considered on the global mesh while the crack is analyzed in the local domain (patch) which allows to have an arbitrary geometry with respect to the underlying global finite elements. Numerical experiments for biomorphic cellular ceramics with porous microstructures produced from natural wood are presented. [less ▲]

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See detailA simulation-based design paradigm for complex cast components
Bordas, Stéphane UL; Conley, James. G.; Moran, Brian et al

in Engineering with Computers (2007), 23(1), 25-37

This paper describes and exercises a new design paradigm for cast components. The methodology integrates foundry process simulation, non-destructive evaluation (NDE), stress analysis and damage tolerance ... [more ▼]

This paper describes and exercises a new design paradigm for cast components. The methodology integrates foundry process simulation, non-destructive evaluation (NDE), stress analysis and damage tolerance simulations into the design process. Foundry process simulation is used to predict an array of porosity-related anomalies. The probability of detection of these anomalies is investigated with a radiographic inspection simulation tool (XRSIM). The likelihood that the predicted array of anomalies will lead to a failure is determined by a fatigue crack growth simulation based on the extended finite element method and therefore does not require meshing nor remeshing as the cracks grow. With this approach, the casting modeling provides initial anomaly information, the stress analysis provides a value for the critical size of an anomaly and the NDE assessment provides a detectability measure. The combination of these tools allows for accept/reject criteria to be determined at the early design stage and enables damage tolerant design philosophies. The methodology is applied to the design of a cast monolithic door used on the Boeing 757 aircraft. [less ▲]

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See detailArchitecture tradeoffs of integrating a mesh generator to partition of unity enriched object-oriented finite element software
Dunant, C.; Nguyen, V. P.; Belgasmia, M. et al

in European Journal of Computational Mechanics (2007), 16(2), 237-258

We explore the tradeoffs of using an internal mesher in a XFEM code. We show that it allows an efficient enrichement detection scheme, while retaining the ability to have welladapted meshes. We provide ... [more ▼]

We explore the tradeoffs of using an internal mesher in a XFEM code. We show that it allows an efficient enrichement detection scheme, while retaining the ability to have welladapted meshes. We provide benchmarks highlighting the considerable gains which can be expected from a well designed architecture. The efficiency of the proposed algorithm is shown by solving fracture mechanics problems of densely micro-cracked bodies including adaptive mesh refinement. [less ▲]

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See detailEnriched finite elements and level sets for damage tolerance assessment of complex structures
Bordas, Stéphane UL; Moran, B.

in Engineering Fracture Mechanics (2006), 73(9), 1176-1201

The extended finite element method (X-FEM) has recently emerged as an alternative to meshing/remeshing crack surfaces in computational fracture mechanics thanks to the concept of discontinuous and ... [more ▼]

The extended finite element method (X-FEM) has recently emerged as an alternative to meshing/remeshing crack surfaces in computational fracture mechanics thanks to the concept of discontinuous and asymptotic partition of unity enrichment (PUM) of the standard finite element approximation spaces. Level set methods have been recently coupled with X-FEM to help track the crack geometry as it grows. However, little attention has been devoted to employing the X-FEM in real-world cases. This paper describes how X-FEM coupled with level set methods can be used to solve complex three-dimensional industrial fracture mechanics problems through combination of an object-oriented (C++) research code and a commercial solid modeling/finite element package (EDS-PLM/ I-DEAS®). The paper briefly describes how object-oriented programming shows its advantages to efficiently implement the proposed methodology. Due to enrichment, the latter method allows for multiple crack growth scenarios to be analyzed with a minimal amount of remeshing. Additionally, the whole component contributes to the stiffness during the whole crack growth simulation. The use of level set methods permits the seamless merging of cracks with boundaries. To show the flexibility of the method, the latter is applied to damage tolerance analysis of a complex aircraft component. © 2006 Elsevier Ltd. All rights reserved. [less ▲]

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See detailMinimum energy multiple crack propagation. Part III: XFEM computer implementation and applications.
Sutula, Danas UL; Bordas, Stéphane UL

in Engineering Fracture Mechanics (n.d.)

The three-part paper deals with energy-minimal multiple crack propagation in a linear elastic solid under quasi-static conditions. The principle of minimum total energy, i.e. the sum of the potential and ... [more ▼]

The three-part paper deals with energy-minimal multiple crack propagation in a linear elastic solid under quasi-static conditions. The principle of minimum total energy, i.e. the sum of the potential and fracture energies, which stems directly from the Griffith's theory of cracks, is applied to the problem of arbitrary crack growth in 2D. The proposed formulation enables minimisation of the total energy of the mechanical system with respect to the crack extension directions and crack extension lengths to solve for the evolution of the mechanical system over time. The three parts focus, in turn, on (I) the theory of multiple crack growth including competing cracks, (II) the discrete solution by the extended finite element method using the minimum-energy formulation, and (III) the aspects of computer implementation within the Matlab programming language. The key contributions of Part-III of the three-part paper are as follows: (1) implementation of XFEM in Matlab with emphasis on the design of the code to enable fast and efficient computational times of fracture problems involving multiple cracks and arbitrary crack intersections, (2) verification of the minimum energy criterion and comparison with the maximum tension criterion via multiple benchmark studies, and (3) we propose a numerical improvement to the crack growth direction criterion that gives significant improvements in accuracy and convergence rates of the fracture paths, especially on coarse meshes. The comparisons of the fracture paths obtained by the maximum tension (or maximum hoop-stress) criterion and the energy minimisation approach via a multitude of numerical case studies show that both criteria converge to virtually the same fracture solutions albeit from opposite directions. In other words, it is found that the converged fracture path lies in between those obtained by each criterion on coarser meshes. Thus, a modified crack growth direction criterion is proposed that assumes the average direction of the directions obtained by the maximum tension and the minimum energy criteria. The numerical results show significant improvements in accuracy (especially on coarse discretisations) and convergence rates of the fracture paths. Finally, the open-source Matlab code, documentation, benchmarks and example cases are included as supplementary material. [less ▲]

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See detailMinimum energy multiple crack propagation Part I: Theory.
Sutula, Danas UL; Bordas, Stéphane UL

in Engineering Fracture Mechanics (n.d.)

The three-part paper deals with energy-minimal multiple crack propagation in a linear elastic solid under quasi-static conditions. The principle of minimum total energy, i.e. the sum of the potential and ... [more ▼]

The three-part paper deals with energy-minimal multiple crack propagation in a linear elastic solid under quasi-static conditions. The principle of minimum total energy, i.e. the sum of the potential and fracture energies, which stems directly from the Griffith's theory of cracks, is applied to the problem of arbitrary crack growth in 2D. The proposed formulation enables minimisation of the total energy of the mechanical system with respect to the crack extension directions and crack extension lengths to solve for the evolution of the mechanical system over time. The three parts focus, in turn, on (I) the theory of multiple crack growth including competing cracks, (II) the discrete solution by the extended finite element method using the minimum-energy formulation, and (III) the aspects of computer implementation within the Matlab programming language. The key contributions of Part-I of this three-part paper are: (1) formulation of the total energy functional governing multiple crack behaviour, (2) three solution methods to the problem of competing crack growth for different fracture front stabilities (e.g. stable, unstable, or a partially stable configuration of crack tips), and (3) the minimum energy criterion for a set of crack tip extensions is posed as the criterion of vanishing rotational dissipation rates with respect to the rotations of the crack extensions. The formulation lends itself to a straightforward application within a discrete framework for determining the crack extension directions of multiple finite-length crack tip increments, which is tackled in Part-II, using the extended finite element method. In Part-III, we discuss various applications and benchmark problems. The open-source Matlab code, documentation, benchmark/example cases are included as supplementary material. [less ▲]

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See detailMinimum energy multiple crack propagation. Part II: Discrete Solution with XFEM.
Sutula, Danas UL; Bordas, Stéphane UL

in Engineering Fracture Mechanics (n.d.)

The three-part paper deals with energy-minimal multiple crack propagation in a linear elastic solid under quasi-static conditions. The principle of minimum total energy, i.e. the sum of the potential and ... [more ▼]

The three-part paper deals with energy-minimal multiple crack propagation in a linear elastic solid under quasi-static conditions. The principle of minimum total energy, i.e. the sum of the potential and fracture energies, which stems directly from the Griffith's theory of cracks, is applied to the problem of arbitrary crack growth in 2D. The proposed formulation enables minimisation of the total energy of the mechanical system with respect to the crack extension directions and crack extension lengths to solve for the evolution of the mechanical system over time. The three parts focus, in turn, on (I) the theory of multiple crack growth including competing cracks, (II) the discrete solution by the extended finite element method using the minimum-energy formulation, and (III) the aspects of computer implementation within the Matlab programming language. This Part-II of our three-part paper examines three discrete solution methods for solving fracture mechanics problems based on the principle of minimum total energy. The discrete solution approach is chosen based on the stability property of the fracture configuration at hand. The first method is based on external load-control. It is suitable for stable crack growth and stable fracture configurations. The second method is based on fractured area-control. This method is applicable to stable or unstable fracture growth but it is required that the fracture front be stable. The third solution method is based on a gradient-descent approach. This approach can be applied to arbitrary crack growth problems; however, the gradient-descent formulation cannot be guaranteed to yield the optimal solution in the case of competing crack growth and an unstable fracture front configuration. The main focus is on the gradient-descent solution approach within the framework of the extended finite element discretisation. Although a viable solution method is finally proposed for resolving competing crack growth in the case of an unstable fracture front configuration, the method is not implemented within the present XFEM code but rather exists as a separate proof-of-concept algorithm that is tested against several fabricated benchmark problems. The open-source Matlab code, documentation and example cases are included as supplementary material. [less ▲]

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See detailSolving the stochastic Burgers equation with a sensitivity derivative-driven Monte Carlo method
Hauseux, Paul UL; Hale, Jack UL; Bordas, Stéphane UL

Software (n.d.)

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See detailBayesian inference for the stochastic identification of elastoplastic material parameters: Introduction, misconceptions and insights
Rappel, Hussein UL; Beex, Lars UL; Hale, Jack UL et al

E-print/Working paper (n.d.)

We discuss Bayesian inference (BI) for the probabilistic identification of material parameters. This contribution aims to shed light on the use of BI for the identification of elastoplastic material ... [more ▼]

We discuss Bayesian inference (BI) for the probabilistic identification of material parameters. This contribution aims to shed light on the use of BI for the identification of elastoplastic material parameters. For this purpose a single spring is considered, for which the stress-strain curves are artificially created. Besides offering a didactic introduction to BI, this paper proposes an approach to incorporate statistical errors both in the measured stresses, and in the measured strains. It is assumed that the uncertainty is only due to measurement errors and the material is homogeneous. Furthermore, a number of possible misconceptions on BI are highlighted based on the purely elastic case. [less ▲]

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See detailA two-dimensional isogeometric boundary element method for linear elastic fracture: a path towards damage tolerance analysis without meshing
Peng, Xuan; Atroshchenko, Elena; Kulasegaram, Sivakumar et al

Report (n.d.)

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See detailAn extended finite element method with smooth nodal stress
Peng, Xuan; Kulasegaram, Sivakumar; Bordas, Stéphane UL et al

Report (n.d.)

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See detailA well-conditioned and optimally convergent XFEM for 3D linear elastic fracture
Agathos, Konstantinos; Chatzi, Eleni; Bordas, Stéphane UL et al

in International Journal for Numerical Methods in Engineering (n.d.)

A variation of the extended finite element method for 3D fracture mechanics is proposed. It utilizes global enrichment and point-wise as well as integral matching of displacements of the standard and ... [more ▼]

A variation of the extended finite element method for 3D fracture mechanics is proposed. It utilizes global enrichment and point-wise as well as integral matching of displacements of the standard and enriched elements in order to achieve higher accuracy, optimal convergence rates and improved conditioning for two and three dimensional crack problems. A bespoke benchmark problem is introduced to determine the method's accuracy in the general 3D case where it is demonstrated that the proposed approach improves the accuracy and reduces the number of iterations required for the iterative solution of the resulting system of equations by 40% for moderately refined meshes and topological enrichment. Moreover, when a fixed enrichment volume is used, the number of iterations required grows at a rate which is reduced by a factor of 2 compared to standard XFEM, diminishing the number of iterations by almost one order of magnitude. [less ▲]

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See detailLinear smoothed extended finite element method
Murugesan; Natarajan, Sundararajan; Gadyam, Palani et al

E-print/Working paper (n.d.)

The extended finite element method (XFEM) was introduced in 1999 to treat problems involving discontinuities with no or minimal remeshing through appropriate enrichment functions. This enables elements to ... [more ▼]

The extended finite element method (XFEM) was introduced in 1999 to treat problems involving discontinuities with no or minimal remeshing through appropriate enrichment functions. This enables elements to be split by a discontinuity, strong or weak and hence requires the integration of discontinuous functions or functions with discontinuous derivatives over elementary volumes. Moreover, in the case of open surfaces and singularities, special, usually non-polynomial functions must also be integrated.A variety of approaches have been proposed to facilitate these special types of numerical integration, which have been shown to have a large impact on the accuracy and convergence of the numerical solution. The smoothed extended finite element method (SmXFEM) [1], for example, makes numerical integration elegant and simple by transforming volume integrals into surface integrals. However, it was reported in [1, 2] that the strain smoothing is inaccurate when non-polynomial functions are in the basis. This is due to the constant smoothing function used over the smoothing domains which destroys the effect of the singularity. In this paper, we investigate the benefits of a recently developed Linear smoothing procedure [3] which provides better approximation to higher order polynomial fields in the basis. Some benchmark problems in the context of linear elastic fracture mechanics (LEFM) are solved to compare the standard XFEM, the constant-smoothed XFEM (Sm-XFEM) and the linear-smoothed XFEM (LSm-XFEM). We observe that the convergence rates of all three methods are the same. The stress intensity factors (SIFs) computed through the proposed LSm-XFEM are however more accurate than that obtained through Sm-XFEM. To conclude, compared to the conventional XFEM, the same order of accuracy is achieved at a relatively low computational effort. [less ▲]

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