×

A parallel scalable domain decomposition preconditioner for elastic crack simulation using XFEM. (English) Zbl 07768037

Summary: In this article, a parallel overlapping domain decomposition preconditioner is proposed to solve the linear system of equations arising from the extended finite element discretization of elastic crack problems. The algorithm partitions the computational mesh into two types of subdomains: the regular subdomains and the crack tip subdomains based on the observation that the crack tips have a significant impact on the convergence of the iterative method while the impact of the crack lines is not that different from those of regular mesh points. The tip subdomains consist of mesh points at crack tips and all neighboring points where the branch enrichment functions are applied. The regular subdomains consist of all other mesh points, including those on the crack lines. To overcome the mismatch between the number of subdomains and the number of processor cores, the proposed method is divided into two steps: solve the crack tip problem and then the regular subdomain problem during each iteration. The proposed method was used to develop a parallel XFEM package which is able to test different types of iterative methods. To achieve good parallel efficiency, additional methods were introduced to reduce communication and to maintain the load balance between processors. Numerical experiments indicate that the proposed method significantly reduces the number of iterations and the total computation time compared to the classical methods. In addition, the method scales up to 8192 processor cores with over 70% parallel efficiency to solve problems with more than \(2\times 1{0}^8\) degrees of freedom.
{© 2022 John Wiley & Sons Ltd.}

MSC:

65Nxx Numerical methods for partial differential equations, boundary value problems
74Sxx Numerical and other methods in solid mechanics
65Fxx Numerical linear algebra

Software:

PETSc
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] TianW, LiP, DongY, LuZ, LuD. Numerical simulation of sequential, alternate and modified zipper hydraulic fracturing in horizontal wells using XFEM. J Pet Sci Eng. 2019;183:106251.
[2] SuoY, ChenZ, RahmanSS, YanH. Numerical simulation of mixed‐mode hydraulic fracture propagation and interaction with different types of natural fractures in shale gas reservoirs. Environ Earth Sci. 2020;79:1‐11.
[3] DongY, TianW, LiP, ZengB, LuD. Numerical investigation of complex hydraulic fracture network in naturally fractured reservoirs based on the XFEM. J Nat Gas Sci Eng. 2021;96:104272.
[4] MoësN, DolbowJ, BelytschkoT. A finite element method for crack growth without remeshing. Int J Numer Methods Eng. 1999;46(1):131‐150. · Zbl 0955.74066
[5] BelytschkoT, BlackT. Elastic crack growth in finite elements with minimal remeshing. Int J Numer Methods Eng. 1999;45(5):601‐620. · Zbl 0943.74061
[6] JiangS, DuC, GuC, ChenX. XFEM analysis of the effects of voids, inclusions and other cracks on the dynamic stress intensity factor of a major crack. Fatigue Fract Eng Mater Struct. 2014;37(8):866‐882.
[7] WangZ, YuT, BuiTQ, et al. Numerical modeling of 3‐D inclusions and voids by a novel adaptive XFEM. Adv Eng Softw. 2016;102:105‐122.
[8] SteinN, DöllingS, ChalkiadakiK, BeckerW, WeißgraeberP. Enhanced XFEM for crack deflection in multi‐material joints. Int J Fract. 2017;207(2):193‐210.
[9] LiuP, LuoY, KangZ. Multi‐material topology optimization considering interface behavior via XFEM and level set method. Comput Methods Appl Mech Eng. 2016;308:113‐133. · Zbl 1439.74290
[10] FahsiA, Soulaı̈maniA. Numerical investigations of the XFEM for solving two‐phase incompressible flows. Int J Comput Fluid Dyn. 2017;31(3):135‐155. · Zbl 07518296
[11] SauerlandH, FriesTP. The stable XFEM for two‐phase flows. Comput Fluids. 2013;87:41‐49. · Zbl 1290.76073
[12] ChahineE, LabordeP, RenardY. Crack tip enrichment in the XFEM using a cutoff function. Int J Numer Methods Eng. 2008;75(6):629‐646. · Zbl 1195.74167
[13] MalekanM, BarrosFB. Well‐conditioning global-local analysis using stable generalized/extended finite element method for linear elastic fracture mechanics. Comput Mech. 2016;58(5):819‐831. · Zbl 1398.74367
[14] WenL, TianR. An extra dof‐free and well‐conditioned XFEM. Proceedings of the 5th International Conference on Computational Methods; 2014:437.
[15] LangC, MakhijaD, DoostanA, MauteK. A simple and efficient preconditioning scheme for heaviside enriched XFEM. Comput Mech. 2014;54(5):1357‐1374. · Zbl 1311.74124
[16] SharmaA, VillanuevaH, MauteK. On shape sensitivities with heaviside‐enriched XFEM. Struct Multidiscip Optim. 2017;55(2):385‐408.
[17] MenouillardT, SongJH, DuanQ, BelytschkoT. Time dependent crack tip enrichment for dynamic crack propagation. Int J Fract. 2010;162(1‐2):33‐49. · Zbl 1425.74442
[18] ElguedjT, GravouilA, MaigreH. An explicit dynamics extended finite element method. Part 1: mass lumping for arbitrary enrichment functions. Comput Methods Appl Mech Eng. 2009;198(30‐32):2297‐2317. · Zbl 1229.74128
[19] FriesTP, BelytschkoT. The intrinsic XFEM: a method for arbitrary discontinuities without additional unknowns. Int J Numer Methods Eng. 2006;68(13):1358‐1385. · Zbl 1129.74045
[20] ZhaoJ, HouY, SongL. Modified intrinsic extended finite element method for elliptic equation with interfaces. J Eng Math. 2016;97(1):147‐159. · Zbl 1360.65287
[21] WuJY, QiuJF, NguyenVP, MandalTK, ZhuangLJ. Computational modeling of localized failure in solids: XFEM vs PF‐CZM. Comput Methods Appl Mech Eng. 2019;345:618‐643. · Zbl 1440.74349
[22] TianR, WenL, WangL. Three‐dimensional improved XFEM (IXFEM) for static crack problems. Comput Methods Appl Mech Eng. 2019;343:339‐367. · Zbl 1440.74444
[23] TianR, WenL. Improved XFEM an extra‐dof free, well‐conditioning, and interpolating XFEM. Comput Methods Appl Mech Eng. 2015;285:639‐658. · Zbl 1423.74926
[24] TranT, StephanEP. An overlapping additive Schwarz preconditioner for boundary element approximations to the Laplace screen and Lamé crack problems. J Numer Math. 2004;12(4):311‐330. · Zbl 1069.65132
[25] SvolosL, Berger‐VergiatL, WaismanH. Updating strategy of a domain decomposition preconditioner for parallel solution of dynamic fracture problems. J Comput Phys. 2020;422:109746. · Zbl 07508378
[26] Berger‐VergiatL, WaismanH, HiriyurB, TuminaroR, KeyesD. Inexact Schwarz‐algebraic multigrid preconditioners for crack problems modeled by extended finite element methods. Int J Numer Methods Eng. 2012;90(3):311‐328. · Zbl 1242.74094
[27] WaismanH, Berger‐VergiatL. An adaptive domain decomposition preconditioner for crack propagation problems modeled by XFEM. Int J Multisc Comput Eng. 2013;11(6):633‐654.
[28] ChenX, CaiXC. Effective two‐level domain decomposition preconditioners for elastic crack problems modeled by extended finite element method. Commun Comput Phys. 2020;28(4):1561‐1584. · Zbl 1473.65294
[29] HowellP, KozyreffG, OckendonJ. Applied Solid Mechanics. Cambridge University Press; 2009. · Zbl 1153.74003
[30] GrisvardPG. Elliptic Problems in Non Smooth Domains. SIAM; 1985.
[31] FriesTP. A corrected XFEM approximation without problems in blending elements. Int J Numer Methods Eng. 2008;75(5):503‐532. · Zbl 1195.74173
[32] SihGC. Mechanics of Fracture Initiation and Propagation: Surface and Volume Energy Density Applied as Failure Criterion. Springer Science & Business Media; 2012.
[33] SutulaD, KerfridenP, vanDamT, BordasSP. Minimum energy multiple crack propagation. Part‐II: discrete solution with XFEM. Eng Fract Mech. 2018;191:225‐256.
[34] FriesTP. Overview and comparison of different variants of the XFEM. Proc Math Mech. 2014;14(1):27‐30.
[35] FriesTP, BelytschkoT. The extended/generalized finite element method: an overview of the method and its applications. Int J Numer Methods Eng. 2010;84(3):253‐304. · Zbl 1202.74169
[36] CaiXC, SarkisM. A restricted additive Schwarz preconditioner for general sparse linear systems. SIAM J Sci Comput. 1999;21(2):792‐797. · Zbl 0944.65031
[37] ToselliA, WidlundO. Domain Decomposition Methods‐Algorithms and Theory. Springer Science & Business Media; 2006.
[38] BalayS, AbhyankarS, AdamsM, et al. PETSc users manual revision 3.3. Sports Med. 2013;23(46):21‐30.
[39] TarancónJ, VercherA, GinerE, FuenmayorF. Enhanced blending elements for XFEM applied to linear elastic fracture mechanics. Int J Numer Methods Eng. 2009;77(1):126‐148. · Zbl 1195.74199
[40] MenkA, BordasSPA. A robust preconditioning technique for the extended finite element method. Int J Numer Methods Eng. 2011;85(13):1609‐1632. · Zbl 1217.74128
[41] SaadY. Iterative Methods for Sparse Linear Systems. SIAM; 2003. · Zbl 1031.65046
[42] HiptmairR, Jerez‐HanckesC, Urzúa‐TorresC. Mesh‐independent operator preconditioning for boundary elements on open curves. SIAM J Numer Anal. 2014;52(5):2295‐2314. · Zbl 1310.65155
[43] BrownPN, VassilevskiPS, WoodwardCS. On mesh‐independent convergence of an inexact Newton-multigrid algorithm. SIAM J Sci Comput. 2003;25(2):570‐590. · Zbl 1049.65115
[44] OliveiraSLGD, CarvalhoC, OsthoffC. The effect of symmetric permutations on the convergence of a restarted GMRES solver with ILU‐type pre‐conditioners. Proceedings of the 2019 Winter Simulation Conference (WSC); 2019:3219‐3230.
[45] BehrischM, BachB, Henry RicheN, SchreckT, FeketeJD. Matrix reordering methods for table and network visualization. Comput Graph Forum. 2016;35(3):693‐716.
[46] LiivI. Seriation and matrix reordering methods: an historical overview. Stat Anal Data Mining ASA Data Sci J. 2010;3(2):70‐91. · Zbl 07260234
[47] NapovA. Conditioning analysis of incomplete Cholesky factorizations with orthogonal dropping. SIAM J Matrix Anal Appl. 2013;34(3):1148‐1173. · Zbl 1314.65045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.