Published 6 issues per year
ISSN Print: 1543-1649
ISSN Online: 1940-4352
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IMPROVED CRACK TIP ENRICHMENT FUNCTIONS AND INTEGRATION FOR CRACK MODELING USING THE EXTENDED FINITE ELEMENT METHOD
ABSTRACT
This paper focuses on two improvements of the extended finite element method (X-FEM) in the context of linear fracture mechanics. Both improve the accuracy and the robustness of the X-FEM. In a first contribution, a new enrichment strategy is proposed to take into account the singular stress field at the crack tip that is meant to replace the traditional four-crack-tip enrichment functions. The efficiency of the new approach is demonstrated on mesh convergence experiments for two-dimensional straight and curved crack problems, using first- and second-order shape functions, both in terms of convergence rates and in terms of condition number of the system to solve. The second contribution revisits the problem of the numerical integration of the stiffness operator when singular functions like the tip enrichment functions are used. An original algorithm to build accurate and fast integration rules for elements in the enrichment zone, touching the crack tip singularity, or not, is presented. The effects on convergence rate of the choice of the integration rule are illustrated on numerical examples.
-
Atluri, S. N., Kobayashi, A. S., and Nakagaki, M., An assumed displacement hybrid finite element model for linear fracture mechanics. DOI: 10.1007/BF00038893
-
Béchet, E., Minnebo, H., Moës, N., and Burgardt, B., Improved implementation and robustness study of the X-FEM method for stress analysis around cracks. DOI: 10.1002/nme.1386
-
Belytschko, T. and Fries, T., The generalized/extended finite element method: An overview of the method and its applications. DOI: 10.1002/nme.2914
-
Broberg, K. B., Cracks and Fractures.
-
Demmel, J. W., Eisenstat, S. C., Gilbert, J. R., Li, X. S., and Liu, J. W. H., A supernodal approach to sparse partial pivoting. DOI: 10.1137/S0895479895291765
-
Destuynder, P., Djaoua, M., and Lescure, S., Some remarks on elastic fracture mechanics (quelques remarques sur la mécanique de la rupture élastique).
-
Dréau, K., Chevaugeon, N., and Moës, N., Studied X-FEM enrichment to handle material interfaces with higher order finite element. DOI: 10.1016/j.cma.2010.01.021
-
Duarte, C., Babuška, I., and Oden, J., Generalized finite element methods for three-dimensional structural mechanics problems. DOI: 10.1016/S0045-7949(99)00211-4
-
Duarte, C. A., Hamzeh, O. N., Liszka, T. J., and Tworzydlo, W. W., A generalized finite element method for the simulation of three-dimensional dynamic crack propagation. DOI: 10.1016/S0045-7825(00)00233-4
-
Duffy, M. G., Quadrature over a pyramid or cube of integrands with a singularity at a vertex. DOI: 10.1137/0719090
-
Galassi, M., GNU Scientific Library Reference Manual (3rd ed.).
-
Geuzaine, C. and Remacle, J., Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities. DOI: 10.1002/nme.2579
-
Gosz, M., Dolbow, J., and Moran, B., Domain integral formulation for stress intensity factor computation along curved threedimensional interface cracks. DOI: 10.1016/S0020-7683(97)00132-7
-
Gosz, M. and Moran, B., An interaction energy integral method for the computation of mixed-mode stress intensity factors along non-planar crack fronts in three dimensions. DOI: 10.1016/S0013-7944(01)00080-7
-
Haasemann, G., Kästner, M., Prüger, S., and Ulbricht, V., Development of a quadratic finite element formulation based on the X-FEM and NURBS. DOI: 10.1002/nme.3120
-
Irwin, G. R., Analysis of stress and strains near the end of a crack traversing a plate.
-
Laborde, P., Pommier, J., Renard, Y., and Salaun, M., High-order extended finite element method for cracked domains. DOI: 10.1002/nme.1370
-
Legrain, G., Chevaugeon, N., and Dréau, K., High order X-FEM and levelsets for complex microstructures: Uncoupling geometry and approximation. DOI: 10.1016/j.cma.2012.06.001
-
Moës, N., Dolbow, J., and Belytschko, T., A finite element method for crack growth without remeshing. DOI: 10.1002/(SICI)1097-0207(19990910)46:1<131::AID-NME726>3.0.CO;2-J
-
Moës, N., Gravouil, A., and Belytschko, T., Non-planar 3D crack growth by the extended finite element and level sets. Part I: Mechanical model. DOI: 10.1002/nme.429
-
Moran, B. and Shih, C., Crack tip and associated domain integrals from momentum and energy balance. DOI: 10.1016/0013-7944(87)90155-X
-
Moran, B. and Shih, C., A general treatment of crack tip contour integrals. DOI: 10.1007/BF00276359
-
Mousavi, S. E. and Sukumar, N., Generalized duffy transformation for integrating vertex singularities. DOI: 10.1007/s00466-009-0424-1
-
Muskhelishvili, N. I., Some Basic Problems in the Mathematical Theory of Elasticity.
-
Nagarajan, A. and Mukherjee, S., A mapping method for numerical evaluation of two-dimensional integrals with 1⁄<i>r</i> singularity. DOI: 10.1007/BF00370482
-
Nahta, R. and Moran, B., Domain integrals for axisymmetric interface crack problems. DOI: 10.1016/0020-7683(93)90049-D
-
Nakamura, T., Three-dimensional stress fields of elastic interface cracks. DOI: 10.1115/1.2897711
-
Nakamura, T. and Parks, D., Antisymmetrical 3-d stress field near the crack front of a thin elastic plate. DOI: 10.1016/0020-7683(89)90109-1
-
Nikishkov, G. and Atluri, S., Calculation of fracture mechanics parameters for an arbitrary three-dimensional crack, by the "equivalent domain integral" method. DOI: 10.1002/nme.1620240914
-
Osher, S. and Fedkiw, R., Level Set Methods and Dynamic Implicit Surfaces.
-
Park, K., Pereira, J. P., Duarte, C. A., and Paulino, G. H., Integration of singular enrichment functions in the generalized/extended finite element method for three-dimensional problems. DOI: 10.1002/nme.2530
-
Pereira, J. P., Duarte, C. A., Guoy, D., and Jiao, X., HP-generalized FEM and crack surface representation for non-planar 3D cracks. DOI: 10.1002/nme.2419
-
Prabel, B., Combescure, A., Gravouil, A., and Marie, S., Level set X-FEM non-matching meshes: Application to dynamic crack propagation in elastic-plastic media. DOI: 10.1002/nme.1819
-
Remacle, J., Chevaugeon, N., Marchandise, E., and Geuzaine, C., Efficient visualization of high-order finite elements. DOI: 10.1002/nme.1787
-
Rice, J., A path independent integral and the approximate analysis of strain concentration by notches and cracks. DOI: 10.1115/1.3601206
-
Sethian, J. A., Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science.
-
Shih, C. and Asaro, R., Elastic-plastic analysis of cracks on bimaterial interfaces: Part I — Small scale yielding. DOI: 10.1115/1.3173676
-
Solin, P., Segeth, K., and Dolezel, I., Higher-Order Finite Element Methods. Studies in Advanced Mathematics.
-
Stern, M., Becker, E., and Dunham, R., A contour integral computation of mixed-mode stress intensity factors. DOI: 10.1007/BF00032831
-
Sukumar, N., Chopp, D. L., Béchet, E., and Moës, N., Three-dimensional non-planar crack growth by a coupled extended finite element and fast marching method. DOI: 10.1002/nme.2344
-
Sukumar, N., Moës, N., Belytschko, T., and Moran, B., Extended finite element method for three-dimensional crack modelling. DOI: 10.1002/1097-0207(20000820)48:11<1549::AID-NME955>3.0.CO;2-A
-
Ferté G., Massin P., Moës N., Convergence analysis of linear or quadratic X-FEM for curved free boundaries, Computer Methods in Applied Mechanics and Engineering, 278, 2014. Crossref
-
Karoui A., Mansouri K., Renard Y., Arfaoui M., The eXtended finite element method for cracked hyperelastic materials: A convergence study, International Journal for Numerical Methods in Engineering, 100, 3, 2014. Crossref
-
Legrain Grégory, Moës Nicolas, Treatment of nearly-singular problems with the X-FEM, Advanced Modeling and Simulation in Engineering Sciences, 1, 1, 2014. Crossref
-
Song Gan, Waisman Haim, Lan Mengyu, Harari Isaac, Extraction of stress intensity factors from Irwin's integral using high-order XFEM on triangular meshes, International Journal for Numerical Methods in Engineering, 102, 3-4, 2015. Crossref
-
Sukumar N., Dolbow J. E., Moës N., Extended finite element method in computational fracture mechanics: a retrospective examination, International Journal of Fracture, 196, 1-2, 2015. Crossref
-
Woodward W. H., Utyuzhnikov S., Massin P., On the application of the method of difference potentials to linear elastic fracture mechanics, International Journal for Numerical Methods in Engineering, 103, 10, 2015. Crossref
-
Jaśkowiec Jan, Three-dimensional analysis of a cohesive crack coupled with heat flux through the crack, Advances in Engineering Software, 89, 2015. Crossref
-
Agathos Konstantinos, Chatzi Eleni, Bordas Stéphane P. A., Talaslidis Demosthenes, A well-conditioned and optimally convergent XFEM for 3D linear elastic fracture, International Journal for Numerical Methods in Engineering, 105, 9, 2016. Crossref
-
Vu M.N., Geniaut S., Massin P., Marigo J.J., Numerical investigation on corner singularities in cracked plates using the G-theta method with an adapted θ field, Theoretical and Applied Fracture Mechanics, 77, 2015. Crossref
-
Gupta V., Duarte C.A., Babuška I., Banerjee U., Stable GFEM (SGFEM): Improved conditioning and accuracy of GFEM/XFEM for three-dimensional fracture mechanics, Computer Methods in Applied Mechanics and Engineering, 289, 2015. Crossref
-
Jaśkowiec J., A model for heat transfer in cohesive cracks, Computers & Structures, 180, 2017. Crossref
-
Zhu Xiaohua, Liu Weiji, He Xianqun, The investigation of rock indentation simulation based on discrete element method, KSCE Journal of Civil Engineering, 21, 4, 2017. Crossref
-
Agathos Konstantinos, Chatzi Eleni, Bordas Stéphane P.A., Stable 3D extended finite elements with higher order enrichment for accurate non planar fracture, Computer Methods in Applied Mechanics and Engineering, 306, 2016. Crossref
-
Paul B., Ndeffo M., Massin P., Moës N., An integration technique for 3D curved cracks and branched discontinuities within the extended Finite Element Method, Finite Elements in Analysis and Design, 123, 2017. Crossref
-
Hattori G., Alatawi I. A., Trevelyan J., An extended boundary element method formulation for the direct calculation of the stress intensity factors in fully anisotropic materials, International Journal for Numerical Methods in Engineering, 109, 7, 2017. Crossref
-
Jin Yuan, Pierard Olivier, Wyart Eric, Béchet Eric, Crack Lip Contact Modeling Based on Lagrangian Multipliers with X-FEM, in Advances in Discretization Methods, 12, 2016. Crossref
-
Chin E. B., Lasserre J. B., Sukumar N., Modeling crack discontinuities without element‐partitioning in the extended finite element method, International Journal for Numerical Methods in Engineering, 110, 11, 2017. Crossref
-
Moës Nicolas, Dolbow J. E., Sukumar N., Extended Finite Element Methods, in Encyclopedia of Computational Mechanics Second Edition, 2017. Crossref
-
Agathos Konstantinos, Ventura Giulio, Chatzi Eleni, Bordas Stéphane P. A., Stable 3D XFEM/vector level sets for non-planar 3D crack propagation and comparison of enrichment schemes, International Journal for Numerical Methods in Engineering, 113, 2, 2018. Crossref
-
Chatzieleftheriou Stavros, Lagaros Nikos D., Smoothed reduction of fracture mechanics solutions to 1D cracked models, European Journal of Mechanics - A/Solids, 69, 2018. Crossref
-
Wang Yongxiang, Waisman Haim, Harari Isaac, Direct evaluation of stress intensity factors for curved cracks using Irwin's integral and XFEM with high-order enrichment functions, International Journal for Numerical Methods in Engineering, 112, 7, 2017. Crossref
-
Ndeffo M., Massin P., Moës N., Martin A., Gopalakrishnan S., On the construction of approximation space to model discontinuities and cracks with linear and quadratic extended finite elements, Advanced Modeling and Simulation in Engineering Sciences, 4, 1, 2017. Crossref
-
Wang Yongxiang, Waisman Haim, Material-dependent crack-tip enrichment functions in XFEM for modeling interfacial cracks in bimaterials, International Journal for Numerical Methods in Engineering, 112, 11, 2017. Crossref
-
Babuška Ivo, Banerjee Uday, Kergrene Kenan, Strongly stable generalized finite element method: Application to interface problems, Computer Methods in Applied Mechanics and Engineering, 327, 2017. Crossref
-
Wang Yongxiang, Cerigato Chiara, Waisman Haim, Benvenuti Elena, XFEM with high-order material-dependent enrichment functions for stress intensity factors calculation of interface cracks using Irwin’s crack closure integral, Engineering Fracture Mechanics, 178, 2017. Crossref
-
Agathos Konstantinos, Ventura Giulio, Chatzi Eleni, Bordas Stéphane P. A., Well Conditioned Extended Finite Elements and Vector Level Sets for Three-Dimensional Crack Propagation, in Geometrically Unfitted Finite Element Methods and Applications, 121, 2017. Crossref
-
Agathos Konstantinos, Bordas Stéphane P.A., Chatzi Eleni, Improving the conditioning of XFEM/GFEM for fracture mechanics problems through enrichment quasi-orthogonalization, Computer Methods in Applied Mechanics and Engineering, 346, 2019. Crossref
-
Wang Yongxiang, Waisman Haim, An arc-length method for controlled cohesive crack propagation using high-order XFEM and Irwin’s crack closure integral, Engineering Fracture Mechanics, 199, 2018. Crossref
-
Salzman Alexis, Moës Nicolas, Chevaugeon Nicolas, On use of the thick level set method in 3D quasi-static crack simulation of quasi-brittle material, International Journal of Fracture, 202, 1, 2016. Crossref
-
Agathos Konstantinos, Chatzi Eleni, Bordas Stéphane P.A., A unified enrichment approach addressing blending and conditioning issues in enriched finite elements, Computer Methods in Applied Mechanics and Engineering, 349, 2019. Crossref
-
Egger Adrian, Pillai Udit, Agathos Konstantinos, Kakouris Emmanouil, Chatzi Eleni, Aschroft Ian A., Triantafyllou Savvas P., Discrete and Phase Field Methods for Linear Elastic Fracture Mechanics: A Comparative Study and State-of-the-Art Review, Applied Sciences, 9, 12, 2019. Crossref
-
Yadav Aanchal, Godara R.K., Bhardwaj Gagandeep, A review on XIGA method for computational fracture mechanics applications, Engineering Fracture Mechanics, 230, 2020. Crossref
-
Geelen Rudy, Plews Julia, Tupek Michael, Dolbow John, An extended/generalized phase‐field finite element method for crack growth with global‐local enrichment, International Journal for Numerical Methods in Engineering, 121, 11, 2020. Crossref
-
Feulvarch Eric, Lacroix Rémi, Deschanels Hubert, A 3D locking-free XFEM formulation for the von Mises elasto-plastic analysis of cracks, Computer Methods in Applied Mechanics and Engineering, 361, 2020. Crossref
-
Rao S.S. Durga, Raju Sethuraman, Orthogonalized Generalized Iso-Geometric Analysis (OGIGA) and its applications to problems of fracture mechanics, Computer Methods in Applied Mechanics and Engineering, 372, 2020. Crossref
-
Crusat L., Carol I., Garolera D., Application of configurational mechanics to crack propagation in quasi-brittle materials, Engineering Fracture Mechanics, 241, 2021. Crossref
-
Bryant Eric C., Sun WaiChing, Phase field modeling of frictional slip with slip weakening/strengthening under non-isothermal conditions, Computer Methods in Applied Mechanics and Engineering, 375, 2021. Crossref
-
Choi Habeun, Cui Huiru, Park Kyoungsoo, Evaluation of stress intensity factor for arbitrary and low-quality meshes using virtual grid-based stress recovery (VGSR), Engineering Fracture Mechanics, 263, 2022. Crossref
-
Liao Minmao, Zhang Pan, Multiple Crack Propagation and Coalescence in Finite Elements with Minimal Local Remeshing Using the Subregion Generalized Variational Principle, Journal of Engineering Mechanics, 146, 10, 2020. Crossref
-
de Oliveira Thaianne S., Barros Felício B., Fonseca Gabriela M., Pitangueira Roque L.S., 2-D fracture mechanics problems by SGFEM, Engineering Analysis with Boundary Elements, 108, 2019. Crossref
-
Agathos Konstantinos, Dodwell Tim, Chatzi Eleni, Bordas Stéphane P.A., An adapted deflated conjugate gradient solver for robust extended/generalised finite element solutions of large scale, 3D crack propagation problems, Computer Methods in Applied Mechanics and Engineering, 395, 2022. Crossref