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See detailImproving the conditioning of XFEM/GFEM for fracture mechanics problems through enrichment quasi-orthogonalization
Agathos, Konstantinos; Bordas, Stéphane UL; Chatzi, Eleni

in Computer Methods in Applied Mechanics and Engineering (2018)

Partition of unity enrichment is known to significantly enhance the accuracy of the finite element method by allowing the incorporation of known characteristics of the solution in the approximation space ... [more ▼]

Partition of unity enrichment is known to significantly enhance the accuracy of the finite element method by allowing the incorporation of known characteristics of the solution in the approximation space. However, in several cases it can further cause conditioning problems for which a number of remedies have been proposed in the framework of the extended/generalized finite element method (XFEM/GFEM). Those solutions often involve significant modifications to the initial method and result in increased implementation complexity. In the present work, a simple procedure for the local near-orthogonalization of enrichment functions is introduced, which significantly improves the conditioning of the resulting system matrices, while requiring only minor modifications to the initial method. Although application to different types of enrichment functions is possible, the resulting scheme is specialized for the singular enrichment functions used in linear elastic fracture mechanics and tested through benchmark problems. [less ▲]

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See detailMultiple crack detection in 3D using a stable XFEM and global optimization
Agathos, Konstantinos; Chatzi, Eleni; Bordas, Stéphane UL

in Computational Mechanics (2018)

A numerical scheme is proposed for the detection of multiple cracks in three dimensional (3D) structures. The scheme is based on a variant of the extended finite element method (XFEM) and a hybrid ... [more ▼]

A numerical scheme is proposed for the detection of multiple cracks in three dimensional (3D) structures. The scheme is based on a variant of the extended finite element method (XFEM) and a hybrid optimizer solution. The proposed XFEM variant is particularly well-suited for the simulation of 3D fracture problems, and as such serves as an efficient solution to the so-called forward problem. A set of heuristic optimization algorithms are recombined into a multiscale optimization scheme. The introduced approach proves effective in tackling the complex inverse problem involved, where identification of multiple flaws is sought on the basis of sparse measurements collected near the structural boundary. The potential of the scheme is demonstrated through a set of numerical case studies of varying complexity. [less ▲]

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See detailStable 3D XFEM/vector-level sets for non-planar 3D crack propagation and comparison of enrichment schemes
Agathos, Konstantinos UL; Ventura, Giulio; Chatzi, Eleni et al

in International Journal for Numerical Methods in Engineering (2017)

We present a three-dimensional (3D) vector level set method coupled to a recently developed stable extended finite element method (XFEM). We further investigate a new enrichment approach for XFEM adopting ... [more ▼]

We present a three-dimensional (3D) vector level set method coupled to a recently developed stable extended finite element method (XFEM). We further investigate a new enrichment approach for XFEM adopting discontinuous linear enrichment functions in place of the asymptotic near-tip functions. Through the vector level set method, level set values for propagating cracks are obtained via simple geometrical operations, eliminating the need for solution of differential evolution equations. The first XFEM variant ensures optimal convergence rates by means of geometrical enrichment, i.e., the use of enriched elements in a fixed volume around the crack front, without giving rise to conditioning problems. The linear enrichment approach significantly simplifies implementation and reduces the computational cost associated with numerical integration. The two dicretization schemes are tested for different benchmark problems, and their combination to the vector level set method is verified for non-planar crack propagation problems. [less ▲]

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See detailWell Conditioned and Optimally Convergent Extended Finite Elements and Vector Level Sets for Three-Dimensional Crack Propagation
Agathos, Konstantinos UL; Ventura, Giulio; Chatzi, Eleni et al

Scientific Conference (2016, June)

A three-dimensional (3D) version of the vector level set method [1] is combined to a well conditioned and optimally convergent XFEM variant in order to deal with non-planar three dimensional crack ... [more ▼]

A three-dimensional (3D) version of the vector level set method [1] is combined to a well conditioned and optimally convergent XFEM variant in order to deal with non-planar three dimensional crack propagation problems. The proposed computational fracture method achieves optimal convergence rates by using tip enriched elements in a fixed volume around the crack front (geometrical enrichment) while keeping conditioning of the resulting system matrices in acceptable levels. Conditioning is controlled by using a three dimensional extension of the degree of freedom gathering technique [2]. Moreover, blending errors are minimized and conditioning is further improved by employing weight function blending and enrichment function shifting [3,4]. As far as crack representation is concerned, crack surfaces are represented by linear quadrilateral elements and the corresponding crack fronts by ordered series of linear segments. Level set values are obtained by projecting points at the crack surface and front respectively. Different criteria are employed in order to assess the quality of the crack representation. References [1] Ventura G., Budyn E. and Belytschko T. Vector level sets for description of propagating cracks in finite elements. Int. J. Numer. Meth. Engng. 58:1571-1592 (2003). [2] Laborde P., Pommier J., Renard Y. and Salaün M. High-order extended finite element method for cracked domains. Int. J. Numer. Meth. Engng. 64:354-381 (2005). [3] Fries T.P. A corrected XFEM approximation without problems in blending elements. Int. J. Numer. Meth. Engng. 75:503-532 (2008). [4] Ventura G., Gracie R. and Belytschko T. Fast integration and weight function blending in the extended finite element method. Int. J. Numer. Meth. Engng. 77:1-29 (2009). [less ▲]

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See detail3D Crack Detection Using an XFEM Variant and Global Optimization Algorithms
Agathos, Konstantinos UL; Chatzi, Eleni; Bordas, Stéphane UL

Scientific Conference (2016, May)

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See detailStable 3D extended finite elements with higher order enrichment for accurate non planar fracture
Agathos, Konstantinos UL; Chatzi, Eleni; Bordas, Stéphane UL

in Computer Methods in Applied Mechanics & Engineering (2016), 306

An extended finite element method (XFEM) for three dimensional (3D) non-planar linear elastic fracture is introduced, which provides optimal convergence through the use of enrichment in a fixed area ... [more ▼]

An extended finite element method (XFEM) for three dimensional (3D) non-planar linear elastic fracture is introduced, which provides optimal convergence through the use of enrichment in a fixed area around the crack front, while also improving the conditioning of the resulting system matrices. This is achieved by fusing a novel form of enrichment with existing blending techniques. Further, the adoption of higher order terms of theWilliams expansion is also considered and the effects in the accuracy and conditioning of the method are studied. Moreover, some problems regarding the evaluation of stress intensity factors (SIFs) and element partitioning are dealt with. The accuracy and convergence properties of the method as well as the conditioning of the resulting stiffness matrices are investigated through the use of appropriate benchmark problems. It is shown that the proposed approach provides increased accuracy while requiring, for all cases considered, a reduced number of iterations for the solution of the resulting systems of equations. The positive impact of geometrical enrichment is further demonstrated in the accuracy of the computed SIFs where, for the examined cases, an improvement of up to 40% is achieved. [less ▲]

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See detailExtended Finite Element Method with Global Enrichment
Agathos, Konstantinos; Chatzi, Eleni; Bordas, Stéphane UL et al

Scientific Conference (2015, July)

A variant of the extended finite element method is presented which facilitates the use of enriched elements in a fixed volume around the crack front (geometrical enrichment) in 3D fracture problems. The ... [more ▼]

A variant of the extended finite element method is presented which facilitates the use of enriched elements in a fixed volume around the crack front (geometrical enrichment) in 3D fracture problems. The major problem associated with geometrical enrichment is that it significantly deteriorates the conditioning of the resulting system matrices, thus increasing solution times and in some cases making the systems unsolvable. For 2D problems this can be dealt with by employing degree of freedom gathering [1] which essentially inhibits spatial variation of enrichment function weights. However, for the general 3D problem such an approach is not possible since spatial variation of the enrichment function weights in the direction of the crack front is necessary in order to reproduce the variation of solution variables, such as the stress intensity factors, along the crack front. The proposed method solves the above problem by employing a superimposed mesh of special elements which serve as a means to provide variation of the enrichment function weights along the crack front while still not allowing variation in any other direction. The method is combined with special element partitioning algorithms [2] and numerical integration schemes [3] as well as techniques for the elimination of blending errors between the standard and enriched part of the approximation in order to further improve the accuracy of the produced results. Additionally, a novel benchmark problem is introduced which enables the computation of displacement and energy error norms as well as errors in the stress intensity factors for the general 3D case. Through this benchmark problem it is shown that the proposed method provides optimal convergence rates, improved accuracy and reduced computational cost compared to standard XFEM. [less ▲]

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See detailXFEM with global enrichment for 3D cracks
Agathos, Kostas; Chatzi, Eleni; Bordas, Stéphane UL et al

Scientific Conference (2015, June 17)

We present an extended finite element method (XFEM) based on fixed area enrichment which 1) suppresses the difficulties associated with ill-conditioning, even for "large" enrichment radii; 2) requires 50 ... [more ▼]

We present an extended finite element method (XFEM) based on fixed area enrichment which 1) suppresses the difficulties associated with ill-conditioning, even for "large" enrichment radii; 2) requires 50 times fewer enriched degrees of freedom (for a typical mesh) as the standard XFEM with geometrical enrichment (for the same or better accuracy level); 3) increases the accuracy level of the stress intensity factors and leads to "smooth" stress intensity variations along the crack front. [less ▲]

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See detailThree-Dimensional Crack Propagation with Global Enrichment XFEM and Vector Level Sets
Agathos, Konstantinos; Ventura, Giulio; Chatzi, Eleni et al

Scientific Conference (2015)

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See detailStable 3D extended finite elements with higher order enrichment for accurate non planar fracture
Agathos, Konstantinos; Chatzi, Eleni; Bordas, Stéphane UL

in Computer Methods in Applied Mechanics & Engineering (2015)

We present an extended finite element method (XFEM) for 3D nonplanar linear elastic fracture. The new approach not only provides optimal convergence using geometrical enrichment but also enables to ... [more ▼]

We present an extended finite element method (XFEM) for 3D nonplanar linear elastic fracture. The new approach not only provides optimal convergence using geometrical enrichment but also enables to contain the increase in conditioning number characteristic of enriched finite element formulations: the number of iterations to convergence of the conjugate gradient solver scales similarly to and converges faster than the topologically-enriched version of the standard XFEM. This has two advantages: (1) the residual can be driven to zero to machine precision for at least 50% fewer iterations than the standard version of XFEM; (2) additional enrichment functions can be added without significant deterioration of the conditioning. Numerical examples also show that our new approach is up to 40% more accurate in terms of stress intensity factors, than the standard XFEM. [less ▲]

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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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