# zbMATH — the first resource for mathematics

Fracture modeling using meshless methods and level sets in 3D: framework and modeling. (English) Zbl 1352.74312
Summary: In 3D fracture modeling, the complexity of the evolving crack geometry during propagation raises challenges in stress analysis because the accuracy of results mainly relies on the accurate description of the crack geometry. In this paper, a numerical framework is developed for 3D fracture modeling where a meshless method, the element-free Galerkin method, is used for stress analysis and level sets are used accurately to describe and capture crack evolution. In this framework, a simple and general formulation for associating the displacement jump in the field approximation with an arbitrary 3D curved crack surface is proposed. For accurate closure of the crack front, a tying procedure is extended to 3D from its original use in 2D in the previous paper by the authors. The benefits of level sets in improving the results accuracy and reducing the computational cost are explored, particularly in the model refinement and the confinement of the displacement jump. Issues arising in level sets updating are discussed and solutions proposed accordingly. The developed framework is validated with a number of 3D crack examples with reference solutions and shows strong potential for general 3D fracture modeling.

##### MSC:
 74R10 Brittle fracture 76S05 Flows in porous media; filtration; seepage 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 74S25 Spectral and related methods applied to problems in solid mechanics
##### Software:
XFEM; Gmsh; ToolboxLS
Full Text:
##### References:
 [1] Sukumar, Extended finite element method for three-dimensional crack modelling, International Journal for Numerical Methods in Engineering 48 pp 1549– (2000) · Zbl 0963.74067 [2] Stolarska, Modelling crack growth by level sets in the extended finite element method, International Journal for Numerical Methods in Engineering 51 pp 943– (2001) · Zbl 1022.74049 [3] Gravouil, Non-planar 3D crack growth by the extended finite element and level sets - part II: level set update, International Journal for Numerical Methods in Engineering 53 pp 2569– (2002) · Zbl 1169.74621 [4] Möes, Non-planar 3D crack growth by the extended finite element and level sets - part I: mechanical model, International Journal for Numerical Methods in Engineering 53 pp 2549– (2002) [5] Chopp, Fatigue crack propagation of multiple coplanar cracks with the coupled extended finite element/fast marching method, International Journal of Engineering Science 41 pp 845– (2003) · Zbl 1211.74199 [6] Sukumar, Extended finite element method and fast marching method for three-dimensional fatigue crack propagation, Engineering Fracture Mechanics 70 pp 29– (2003) · Zbl 1211.74199 [7] Areias, Analysis of three-dimensional crack initiation and propagation using the extended finite element method, International Journal for Numerical Methods in Engineering 63 pp 760– (2005) · Zbl 1122.74498 [8] Sukumar, Three-dimensional non-planar crack growth by a coupled extended finite element and fast marching method, International Journal for Numerical Methods in Engineering 76 pp 727– (2008) · Zbl 1195.76390 [9] Soparat, Analysis of cohesive crack growth by the element-free Galerkin method, Journal of Mechanics 24 pp 45– (2008) [10] Rannou, A local multigrid X-FEM strategy for 3D crack propagation, International Journal for Numerical Methods in Engineering 77 pp 581– (2009) · Zbl 1155.74414 [11] Zamani, Implementation of the extended finite element method for dynamic thermoelastic fracture initiation, International Journal of Solids And Structures 47 pp 1392– (2010) · Zbl 1193.74161 [12] Belytschko, A review of extended/generalized finite element methods for material modeling, Modelling and Simulation in Materials Science and Engineering 17 pp 043001– (2009) [13] Bordas, Three-dimensional crack initiation, propagation, branching and junction in non-linear materials by an extended meshfree method without asymptotic enrichment, Engineering Fracture Mechanics 75 pp 943– (2008) [14] Belytschko, Element-free Galerkin methods, International Journal for Numerical Methods in Engineering 37 pp 229– (1994) · Zbl 0796.73077 [15] Belytschko, Fracture and crack growth by element free Galerkin methods, Modelling and Simulation in Materials Science and Engineering 2 pp 519– (1994) [16] Organ, Continuous meshless approximations for nonconvex bodies by diffraction and transparency, Computational Mechanics 18 pp 225– (1996) · Zbl 0864.73076 [17] Belytschko, Element-free Galerkin methods for static and dynamic fracture, International Journal for Numerical Methods in Engineering 32 pp 2547– (1995) · Zbl 0918.73268 [18] Belytschko, Crack-propagation by element-free Galerkin methods, Engineering Fracture Mechanics 51 pp 295– (1995) [19] Rabczuk, Simulation of high velocity concrete fragmentation using SPH/MLSPH, International Journal For Numerical Methods In Engineering 56 (10) pp 1421– (2003) · Zbl 1106.74428 [20] Sukumar, An element-free Galerkin method for three-dimensional fracture mechanics, Computational Mechanics 20 pp 170– (1997) · Zbl 0888.73066 [21] Krysl, The element free Galerkin method for dynamic propagation of arbitrary 3D cracks, International Journal for Numerical Methods in Engineering 44 pp 767– (1999) · Zbl 0953.74078 [22] Brighenti, Application of the element-free Galerkin meshless method to 3D fracture mechanics problems, Engineering Fracture Mechanics 72 pp 2808– (2005) [23] Duflot, A meshless method with enriched weight functions for three-dimensional crack propagation, International Journal for Numerical Methods in Engineering 65 pp 1970– (2006) · Zbl 1114.74064 [24] Rabczuk, A three-dimensional meshfree method for continuous multiple-crack initiation, propagation and junction in statics and dynamics, Computational Mechanics 40 pp 473– (2007) · Zbl 1161.74054 [25] Bordas, An extended finite element library, International Journal for Numerical Methods in Engineering 71 pp 703– (2007) · Zbl 1194.74367 [26] Rabczuk, A three-dimensional large deformation meshfree method for arbitrary evolving cracks, Computer Methods in Applied Mechanics and Engineering 196 pp 2777– (2007) · Zbl 1128.74051 [27] Rabczuk, On three-dimensional modelling of crack growth using partition of unity methods, Computers & Structures 88 pp 1391– (2010) [28] Rabczuk, Simulations of instability in dynamic fracture by the cracking particles method, Engineering Fracture Mechanics 76 pp 730– (2009) [29] Song, A comparative study on finite element methods for dynamic fracture, Computational Mechanics 42 pp 239– (2008) · Zbl 1160.74048 [30] Duflot, A study of the representation of cracks with level sets, International Journal for Numerical Methods in Engineering 70 pp 1261– (2007) · Zbl 1194.74516 [31] Zhuang, Accurate fracture modelling using meshless methods and level sets: formulation and 2D modelling, International Journal for Numerical Methods in Engineering 86 pp 249– (2011) · Zbl 1235.74346 [32] Nayroles, Generalizing the finite element method: diffuse approximation and diffuse elements, Computational Mechanics 10 pp 307– (1992) · Zbl 0764.65068 [33] Zhuang, Aspects of the use of orthogonal basis functions in the element-free Galerkin method, International Journal for Numerical Methods in Engineering 81 pp 366– (2010) · Zbl 1183.74376 [34] Fleming, Enriched element-free Galerkin methods for crack tip fields, International Journal for Numerical Methods in Engineering 40 pp 1483– (1997) [35] Daux, Arbitrary branched and intersecting cracks with the extended finite element method, International Journal for Numerical Methods in Engineering 48 pp 1741– (2000) · Zbl 0989.74066 [36] Duflot, A meshless method with enriched weight functions for fatigue crack growth, International Journal for Numerical Methods in Engineering 59 pp 1945– (2004) · Zbl 1060.74664 [37] Fries, The intrinsic partition of unity method, Computational Mechanics 40 pp 803– (2007) · Zbl 1162.74049 [38] Zhuang, On error control in the element-free Galerkin method, Engineering Analysis with Boundary Elements 36 pp 351– (2012) [39] Sussman, A level set approach for computing solutions to incompressible two-phase flow, Journal of Computational Physics 114 pp 146– (1994) · Zbl 0808.76077 [40] Osher, Level Set Methods and Dynamic Implicit Surfaces (2003) [41] Sethian, Level Sets Methods and Fast Marching Methods (1996) · Zbl 0852.65055 [42] Mitchell I A toolbox of level set methods (version 1.1) Technical Report 2007 [43] Liu, Weighted essentially nonoscillatory schemes, Journal of Computational Physics 115 pp 200– (1994) [44] Griffith, Phenomena of rupture and flow in solids, Philosophical Transactions of the Royal Society of London, Series A 221 pp 163– (1921) [45] Peng, A PDE-based fast local level set method, Journal of Computational Physics 155 pp 410– (1999) · Zbl 0964.76069 [46] Adalsteinsson, The fast construction of extension velocities in level set methods, Journal of Computational Physics 148 pp 2– (1999) · Zbl 0919.65074 [47] Russoa, A remark on computing distance functions, Journal of Computational Physics 163 pp 51– (2000) [48] Burchard, Motion of curves in three spatial dimensions using a level set approach, Journal of Computational Physics 170 pp 720– (2001) · Zbl 0991.65077 [49] Houlsby, A tying scheme for imposing displacement constraints in finite element analysis, Communications in Numerical Methods in Engineering 16 pp 721– (2000) · Zbl 0993.74065 [50] Hou, A hybrid method for moving interface problems with application to the hele-shaw flow, Journal of Computational Physics 134 pp 236– (1997) · Zbl 0888.76067 [51] Fleming M The element-free Galerkin method for fatigue and quasi-static fracture PhD Thesis 1997 [52] Lee, An improved crack analysis technique by element-free Galerkin method with auxiliary supports, International Journal for Numerical Methods in Engineering 56 pp 1291– (2003) · Zbl 1106.74426 [53] Ventura, A vector level set method and new discontinuity approximations for crack growth by EFG, International Journal for Numerical Methods in Engineering 54 pp 923– (2002) · Zbl 1034.74053 [54] Duflot, A truly meshless Galerkin method based on a moving least squares quadrature, Communications in Numerical Methods in Engineering 18 pp 441– (2002) · Zbl 1008.74082 [55] Gosz, Domain integral for stress intensity factor computation along curved three-dimensional interface cracks, International Journal of Solids and Structures 35 pp 1763– (1997) · Zbl 0936.74057 [56] Rigby, Decomposition of the mixed-mode J-integral revisited, International Journal of Solids and Structures 35 pp 2073– (1998) · Zbl 0933.74005 [57] Gosz, An interaction energy integral method for computation of mixed-mode stress intensity factors along non-planar crack fronts in three dimensions, Engineering Fracture Mechanics 69 pp 299– (2002) [58] Chang, Stress intensity factor computation along a non-planar curved crack in three dimensions, International Journal of Solids and Structures 44 pp 371– (2007) · Zbl 1119.74017 [59] Anderson, Fracture Mechanics : Fundamentals and Applications (1995) [60] Stern, A contour integral computation of mixed-mode stress intensity factors, International Journal of Fracture 12 pp 359– (1976) [61] Geuzaine, Gmsh: a three-dimensional finite element mesh generator with built-in pre-processing and post-processing facilities, International Journal for Numerical Methods in Engineering 79 pp 1309– (2009) · Zbl 1176.74181 [62] Murakami, The Stress Intensity Factors Handbook (1987)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.