Abstract
In this paper, we present a novel adaptive continuous–discontinuous approach for the analysis of phase field fracture. An initial trimmed hexahedral (TH) mesh is created by cutting a hexahedral background grid with the boundary of the solid domain. Octree-based adaptive mesh refinement is performed on the initial TH mesh based on an energy-based criterion to accurately resolve the damage evolution along the phase field crack. Critical damage isosurfaces of the phase field are used to convert fully developed phase field cracks into discontinuous discrete cracks. Mesh coarsening is also performed along the discontinuous discrete cracks to reduce the computational cost. Three-dimensional problems of quasi-brittle fracture are investigated to verify the effectiveness and efficiency of the present adaptive continuous–discontinuous approach for the analysis of phase field fracture.
Similar content being viewed by others
References
Francfort GA, Marigo JJ (1998) Revisiting brittle fracture as an energy minimization problem. J Mech Phys Solids 46:1319–1342. https://doi.org/10.1016/S0022-5096(98)00034-9
Griffits AA (1921) VI. The phenomena of rupture and flow in solids. Philos Trans R Soc Lond Ser A Contain Pap Math Phys Character 221:163–198. https://doi.org/10.1098/rsta.1921.0006
Miehe C, Welschinger F, Hofacker M (2010) Thermodynamically consistent phase-field models of fracture: variational principles and multi-field FE implementations. Int J Numer Methods Eng 83:1273–1311. https://doi.org/10.1002/nme.2861
Miehe C, Hofacker M, Welschinger F (2010) A phase field model for rate-independent crack propagation: robust algorithmic implementation based on operator splits. Comput Methods Appl Mech Eng 199:2765–2778. https://doi.org/10.1016/j.cma.2010.04.011
Ambati M, Gerasimov T, De Lorenzis L (2015) A review on phase-field models of brittle fracture and a new fast hybrid formulation. Comput Mech 55:383–405. https://doi.org/10.1007/s00466-014-1109-y
Jeong H, Signetti S, Han TS, Ryu S (2018) Phase field modeling of crack propagation under combined shear and tensile loading with hybrid formulation. Comput Mater Sci 155:483–492. https://doi.org/10.1016/j.commatsci.2018.09.021
Patil RU, Mishra BK, Singh IV (2018) An adaptive multiscale phase field method for brittle fracture. Comput Methods Appl Mech Eng 329:254–288. https://doi.org/10.1016/j.cma.2017.09.021
Patil RU, Mishra BK, Singh IV (2018) A local moving extended phase field method (LMXPFM) for failure analysis of brittle materials. Comput Methods Appl Mech Eng 342:674–709. https://doi.org/10.1016/j.cma.2018.08.018
Wu JY (2018) Robust numerical implementation of non-standard phase-field damage models for failure in solids. Comput Methods Appl Mech Eng 340:767–797. https://doi.org/10.1016/j.cma.2018.06.007
Kim H, Kim H (2021) A novel adaptive mesh refinement scheme for the simulation of phase-field fracture using trimmed hexahedral meshes. Int J Numer Methods Eng 122:1493–1512. https://doi.org/10.1002/nme.6587
Heister T, Wheeler MF, Wick T (2015) A primal-dual active set method and predictor-corrector mesh adaptivity for computing fracture propagation using a phase-field approach. Comput Methods Appl Mech Eng 290:466–495. https://doi.org/10.1016/j.cma.2015.03.009
Badnava H, Msekh MA, Etemadi E, Rabczuk T (2018) An h-adaptive thermo-mechanical phase field model for fracture. Finite Elem Anal Des 138:31–47. https://doi.org/10.1016/j.finel.2017.09.003
Nagaraja S, Elhaddad M, Ambati M et al (2019) Phase-field modeling of brittle fracture with multi-level hp-FEM and the finite cell method. Comput Mech 63:1283–1300. https://doi.org/10.1007/s00466-018-1649-7
Muixí A, Fernández-Méndez S, Rodríguez-Ferran A (2020) Adaptive refinement for phase-field models of brittle fracture based on Nitsche’s method. Comput Mech 66:69–85. https://doi.org/10.1007/s00466-020-01841-1
Klinsmann M, Rosato D, Kamlah M, McMeeking RM (2015) An assessment of the phase field formulation for crack growth. Comput Methods Appl Mech Eng 294:313–330. https://doi.org/10.1016/j.cma.2015.06.009
Hirshikesh H, Pramod ALN, Waisman H, Natarajan S (2021) Adaptive phase field method using novel physics based refinement criteria. Comput Methods Appl Mech Eng 383:113874. https://doi.org/10.1016/j.cma.2021.113874
Freddi F, Mingazzi L (2022) Mesh refinement procedures for the phase field approach to brittle fracture. Comput Methods Appl Mech Eng 388:114214. https://doi.org/10.1016/j.cma.2021.114214
Xu W, Li Y, Li H et al (2022) Multi-level adaptive mesh refinement technique for phase-field method. Eng Fract Mech. https://doi.org/10.1016/j.engfracmech.2022.108891
Xu W, Jiang D, Zhang C et al (2023) An adaptive mesh refinement strategy for 3D phase modeling of brittle fracture. Eng Fract Mech. https://doi.org/10.1016/j.engfracmech.2023.109241
Tian F, Tang X, Xu T et al (2019) A hybrid adaptive finite element phase-field method for quasi-static and dynamic brittle fracture. Int J Numer Methods Eng 120:1108–1125. https://doi.org/10.1002/nme.6172
Hirshikesh, Pramod ALN, Annabattula RK et al (2019) Adaptive phase-field modeling of brittle fracture using the scaled boundary finite element method. Comput Methods Appl Mech Eng 355:284–307. https://doi.org/10.1016/j.cma.2019.06.002
Aldakheel F, Hudobivnik B, Hussein A, Wriggers P (2018) Phase-field modeling of brittle fracture using an efficient virtual element scheme. Comput Methods Appl Mech Eng 341:443–466. https://doi.org/10.1016/j.cma.2018.07.008
Hirshikesh, Jansari C, Kannan K et al (2019) Adaptive phase field method for quasi-static brittle fracture using a recovery based error indicator and quadtree decomposition. Eng Fract Mech. https://doi.org/10.1016/j.engfracmech.2019.106599
Xing C, Yu T, Sun Y, Wang Y (2023) An adaptive phase-field model with variable-node elements for fracture of hyperelastic materials at large deformations. Eng Fract Mech. https://doi.org/10.1016/j.engfracmech.2023.109115
Assaf R, Birk C, Natarajan S, Gravenkamp H (2022) Three-dimensional phase-field modeling of brittle fracture using an adaptive octree-based scaled boundary finite element approach. Comput Methods Appl Mech Eng 399:115364. https://doi.org/10.1016/j.cma.2022.115364
Aldakheel F, Noii N, Wick T, Wriggers P (2021) A global–local approach for hydraulic phase-field fracture in poroelastic media. Comput Math Appl 91:99–121. https://doi.org/10.1016/j.camwa.2020.07.013
Sarkar S, Singh IV, Mishra BK (2020) Adaptive mesh refinement schemes for the localizing gradient damage method based on biquadratic-bilinear coupled-field elements. Eng Fract Mech. https://doi.org/10.1016/j.engfracmech.2019.106790
Chiaruttini V, Riolo V, Feyel F (2013) Advanced remeshing techniques for complex 3D crack propagation. In: 13th International conference on fracture 2013, ICF 2013 1, pp 547–555
Mediavilla J, Peerlings RHJ, Geers MGD (2006) A robust and consistent remeshing-transfer operator for ductile fracture simulations. Comput Struct 84:604–623. https://doi.org/10.1016/j.compstruc.2005.10.007
Eldahshan H, Alves J, Bouchard PO et al (2021) CIPFAR: a 3D unified numerical framework for the modeling of ductile fracture based on the phase field model and adaptive remeshing. Comput Methods Appl Mech Eng 387:114171. https://doi.org/10.1016/j.cma.2021.114171
Song JH, Wang H, Belytschko T (2008) A comparative study on finite element methods for dynamic fracture. Comput Mech 42:239–250. https://doi.org/10.1007/s00466-007-0210-x
Peerlings RHJ, Brekelmans WAM, De Borst R, Geers MGD (2000) Gradient-enhanced damage modelling of high-cycle fatigue. Int J Numer Methods Eng 49:1547–1569. https://doi.org/10.1002/1097-0207(20001230)49:12%3c1547::AID-NME16%3e3.0.CO;2-D
Tamayo-Mas E, Rodríguez-Ferran A (2015) A medial-axis-based model for propagating cracks in a regularised bulk. Int J Numer Methods Eng 101:489–520. https://doi.org/10.1002/nme.4757
Geelen RJM, Liu Y, Dolbow JE, Rodríguez-Ferran A (2018) An optimization-based phase-field method for continuous-discontinuous crack propagation. Int J Numer Methods Eng 116:1–20. https://doi.org/10.1002/nme.5911
Giovanardi B, Scotti A, Formaggia L (2017) A hybrid XFEM—phase field (Xfield) method for crack propagation in brittle elastic materials. Comput Methods Appl Mech Eng 320:396–420. https://doi.org/10.1016/j.cma.2017.03.039
Ho-Nguyen-Tan T, Kim HG (2020) Numerical simulation of crack propagation in shell structures using interface shell elements. Comput Mech 66:537–557. https://doi.org/10.1007/s00466-020-01863-9
Hussein A, Hudobivnik B, Wriggers P (2020) A combined adaptive phase field and discrete cutting method for the prediction of crack paths. Comput Methods Appl Mech Eng. https://doi.org/10.1016/j.cma.2020.113329
Muixí A, Marco O, Rodríguez-Ferran A, Fernández-Méndez S (2021) A combined XFEM phase-field computational model for crack growth without remeshing. Comput Mech 67:231–249. https://doi.org/10.1007/s00466-020-01929-8
Bourdin B, Francfort GA, Marigo J-J (2008) The variational approach to fracture. J Elast 91:5–148. https://doi.org/10.1007/s10659-007-9107-3
Kim H-Y, Kim H-G (2020) Efficient isoparametric trimmed-hexahedral elements with explicit shape functions. Comput Methods Appl Mech Eng 372:113316. https://doi.org/10.1016/j.cma.2020.113316
Lorensen WE, Cline HE (1987) Marching cubes: a high resolution 3D surface construction algorithm. ACM SIGGRAPH Comput Graph 21:163–169. https://doi.org/10.1145/37402.37422
Sohn D, Han J, Cho YS, Im S (2013) A finite element scheme with the aid of a new carving technique combined with smoothed integration. Comput Methods Appl Mech Eng 254:42–60. https://doi.org/10.1016/j.cma.2012.10.014
Nguyen-Hoang S, Sohn D, Kim HG (2017) A new polyhedral element for the analysis of hexahedral-dominant finite element models and its application to nonlinear solid mechanics problems. Comput Methods Appl Mech Eng 324:248–277. https://doi.org/10.1016/j.cma.2017.06.014
Ho-Nguyen-Tan T, Kim HG (2018) A new strategy for finite-element analysis of shell structures using trimmed quadrilateral shell meshes: a paving and cutting algorithm and a pentagonal shell element. Int J Numer Methods Eng 114:1–27. https://doi.org/10.1002/nme.5730
Martin S, Kaufmann P, Botsch M et al (2008) Polyhedral finite elements using harmonic basis functions. Eurograph Symp Geom Process 27:1521–1529. https://doi.org/10.1111/j.1467-8659.2008.01293.x
Bishop JE (2014) A displacement-based finite element formulation for general polyhedra using harmonic shape functions. Int J Numer Methods Eng 97:1–31. https://doi.org/10.1002/nme.4562
Kim H, Sohn D (2015) A new finite element approach for solving three-dimensional problems using trimmed hexahedral elements. Int J Numer Methods Eng 102:1527–1553. https://doi.org/10.1002/nme.4850
Arriaga M, Waisman H (2018) Multidimensional stability analysis of the phase-field method for fracture with a general degradation function and energy split. Comput Mech 61:181–205. https://doi.org/10.1007/s00466-017-1432-1
Sohn D (2018) Periodic mesh generation and homogenization of inclusion-reinforced composites using an element-carving technique with local mesh refinement. Compos Struct 185:65–80. https://doi.org/10.1016/j.compstruct.2017.10.088
Nguyen SH, Sohn D, Kim H-G (2022) A novel hr-adaptive mesh refinement scheme for stress-constrained shape and topology optimization using level-set-based trimmed meshes. Struct Multidiscip Optim 65:71. https://doi.org/10.1007/s00158-021-03132-6
Morton DJ, Tyler JM, Dorroh JR (1995) A new 3D finite element for adaptive h-refinement in 1-irregular meshes. Int J Numer Methods Eng 38:3989–4008. https://doi.org/10.1002/nme.1620382306
Feder J (1980) Random sequential adsorption. J Theor Biol 87:237–254. https://doi.org/10.1016/0022-5193(80)90358-6
Acknowledgements
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2022R1I1A2053461).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Kim, HY., Kim, HG. An adaptive continuous–discontinuous approach for the analysis of phase field fracture using mesh refinement and coarsening schemes and octree-based trimmed hexahedral meshes. Comput Mech (2024). https://doi.org/10.1007/s00466-024-02472-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00466-024-02472-6