Element‐local level set method for three‐dimensional dynamic crack growth

An approximate level set method for three‐dimensional crack propagation is presented. In this method, the discontinuity surface in each cracked element is defined by element‐local level sets (ELLSs). The local level sets are generated by a fitting procedure that meets the fracture directionality and its continuity with the adjacent element crack surfaces in a least‐square sense. A simple iterative procedure is introduced to improve the consistency of the generated element crack surface with those of the adjacent cracked elements. The discrete discontinuity is treated by the phantom node method which is a simplified version of the extended finite element method (XFEM). The ELLS method and the phantom node technology are combined for the solution of dynamic fracture problems. Numerical examples for three‐dimensional dynamic crack propagation are provided to demonstrate the effectiveness and robustness of the proposed method. Copyright © 2009 John Wiley & Sons, Ltd.     

Non‐planar 3D crack growth by the extended finite element and level sets—Part II: Level set update

We present a level set method for treating the growth of non‐planar three‐dimensional cracks.The crack is defined by two almost‐orthogonal level sets (signed distance functions). One of them describes the crack as a two‐dimensional surface in a three‐dimensional space, and the second is used to describe the one‐dimensional crack front, which is the intersection of the two level sets. A Hamilton–Jacobi equation is used to update the level sets. A velocity extension is developed that preserves the old crack surface and can accurately generate the growing surface. The technique is coupled with the extended finite element method which approximates the displacement field with a discontinuous partition of unity. This displacement field is constructed directly in terms of the level sets, so the discretization by finite elements requires no explicit representation of the crack surface. Numerical experiments show the robustness of the method, both in accuracy and in treating cracks with significant changes in topology. Copyright © 2002 John Wiley & Sons, Ltd.     

Three‐dimensional non‐planar crack growth by a coupled extended finite element and fast marching method

A numerical technique for non‐planar three‐dimensional linear elastic crack growth simulations is proposed. This technique couples the extended finite element method (X‐FEM) and the fast marching method (FMM). In crack modeling using X‐FEM, the framework of partition of unity is used to enrich the standard finite element approximation by a discontinuous function and the two‐dimensional asymptotic crack‐tip displacement fields. The initial crack geometry is represented by two level set functions, and subsequently signed distance functions are used to maintain the location of the crack and to compute the enrichment functions that appear in the displacement approximation. Crack modeling is performed without the need to mesh the crack, and crack propagation is simulated without remeshing. Crack growth is conducted using FMM; unlike a level set formulation for interface capturing, no iterations nor any time step restrictions are imposed in the FMM. Planar and non‐planar quasi‐static crack growth simulations are presented to demonstrate the robustness and versatility of the proposed technique. Copyright © 2008 John Wiley & Sons, Ltd.     

Extended finite element method for three‐dimensional crack modelling

An extended finite element method (X‐FEM) for three‐dimensional crack modelling is described. A discontinuous function and the two‐dimensional asymptotic crack‐tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity. This enables the domain to be modelled by finite elements with no explicit meshing of the crack surfaces. Computational geometry issues associated with the representation of the crack and the enrichment of the finite element approximation are discussed. Stress intensity factors (SIFs) for planar three‐dimensional cracks are presented, which are found to be in good agreement with benchmark solutions. Copyright © 2000 John Wiley & Sons, Ltd.     

Extended finite element method coupled with face‐based strain smoothing technique for three‐dimensional fracture problems

In this work, the extended finite element method (XFEM) is for the first time coupled with face‐based strain‐smoothing technique to solve three‐dimensional fracture problems. This proposed method, which is called face‐based smoothed XFEM here, is expected to combine both the advantages of XFEM and strain‐smoothing technique. In XFEM, arbitrary crack geometry can be modeled and crack advance can be simulated without remeshing. Strain‐smoothing technique can eliminate the integration of singular term over the volume around the crack front, thanks to the transformation of volume integration into area integration. Special smoothing scheme is implemented in the crack front smoothing domain. Three examples are presented to test the accuracy, efficiency, and convergence rate of the face‐based smoothed XFEM. From the results, it is clear that smoothing technique can improve the performance of XFEM for three‐dimensional fracture problems. Copyright © 2015 John Wiley & Sons, Ltd.     

Analysis of three‐dimensional fracture mechanics problems: A two‐scale approach using coarse‐generalized FEM meshes

This paper presents a generalized finite element method (GFEM) based on the solution of interdependent global (structural) and local (crack)‐scale problems. The local problems focus on the resolution of fine‐scale features of the solution in the vicinity of three‐dimensional cracks, while the global problem addresses the macro‐scale structural behavior. The local solutions are embedded into the solution space for the global problem using the partition of unity method. The local problems are accurately solved using an hp‐GFEM and thus the proposed method does not rely on analytical solutions. The proposed methodology enables accurate modeling of three‐dimensional cracks on meshes with elements that are orders of magnitude larger than the process zone along crack fronts. The boundary conditions for the local problems are provided by the coarse global mesh solution and can be of Dirichlet, Neumann or Cauchy type. The effect of the type of local boundary conditions on the performance of the proposed GFEM is analyzed. Several three‐dimensional fracture mechanics problems aimed at investigating the accuracy of the method and its computational performance, both in terms of problem size and CPU time, are presented. Copyright © 2009 John Wiley & Sons, Ltd.     

A well‐conditioned and optimally convergent XFEM for 3D linear elastic fracture

A variation of the extended finite element method for three‐dimensional fracture mechanics is proposed. It utilizes a novel form of enrichment and point‐wise and integral matching of displacements of the standard and enriched elements in order to achieve higher accuracy, optimal convergence rates, and improved conditioning for two‐dimensional and three‐dimensional crack problems. A bespoke benchmark problem is introduced to determine the method's accuracy in the general three‐dimensional 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 with standard extended finite element method, diminishing the number of iterations by almost one order of magnitude. Copyright © 2015 John Wiley & Sons, Ltd.     

Modeling crack discontinuities without element‐partitioning in the extended finite element method

In this paper, we model crack discontinuities in two‐dimensional linear elastic continua using the extended finite element method without the need to partition an enriched element into a collection of triangles or quadrilaterals. For crack modeling in the extended finite element, the standard finite element approximation is enriched with a discontinuous function and the near‐tip crack functions. Each element that is fully cut by the crack is decomposed into two simple (convex or nonconvex) polygons, whereas the element that contains the crack tip is treated as a nonconvex polygon. On using Euler's homogeneous function theorem and Stokes's theorem to numerically integrate homogeneous functions on convex and nonconvex polygons, the exact contributions to the stiffness matrix from discontinuous enriched basis functions are computed. For contributions to the stiffness matrix from weakly singular integrals (because of enrichment with asymptotic crack‐tip functions), we only require a one‐dimensional quadrature rule along the edges of a polygon. Hence, neither element‐partitioning on either side of the crack discontinuity nor use of any cubature rule within an enriched element are needed. Structured finite element meshes consisting of rectangular elements, as well as unstructured triangular meshes, are used. We demonstrate the flexibility of the approach and its excellent accuracy in stress intensity factor computations for two‐dimensional crack problems. Copyright © 2016 John Wiley & Sons, Ltd.     

A meshless method with enriched weight functions for fatigue crack growth

The meshless method is particularly appropriate to solve crack propagation problems. In this paper, the fatigue growth of cracks in two‐dimensional bodies is considered. The analysis is based upon Paris' equation. New enriched weight functions are introduced in the meshless method formulation to capture the singularity at the crack tip. Simple problems show the accuracy and efficiency of this method. Then, it is applied to fatigue analysis of single‐ and multi‐cracked bodies under mixed‐mode conditions. Copyright © 2004 John Wiley & Sons, Ltd.     

Boundary element analysis of three‐dimensional mixed‐mode thermoelastic crack problems using the interaction and energy domain integrals

A three‐dimensional boundary element method (BEM) implementation of the interaction integral methodology for the numerical analysis of mixed‐mode three‐dimensional thermoelastic crack problems is presented in this paper. The interaction integral is evaluated from a domain representation naturally compatible with the BEM, since stresses, strains, temperatures and derivatives of displacements and temperatures at internal points can be evaluated using the appropriate boundary integral equations. Several examples are analysed and the results compared with those available in the literature to demonstrate the efficiency and accuracy of the implementation to solve straight and curved crack‐front problems. Copyright © 2007 John Wiley & Sons, Ltd.     

hp‐Generalized FEM and crack surface representation for non‐planar 3‐D cracks

A high‐order generalized finite element method (GFEM) for non‐planar three‐dimensional crack surfaces is presented. Discontinuous p‐hierarchical enrichment functions are applied to strongly graded tetrahedral meshes automatically created around crack fronts. The GFEM is able to model a crack arbitrarily located within a finite element (FE) mesh and thus the proposed method allows fully automated fracture analysis using an existing FE discretization without cracks. We also propose a crack surface representation that is independent of the underlying GFEM discretization and controlled only by the physics of the problem. The representation preserves continuity of the crack surface while being able to represent non‐planar, non‐smooth, crack surfaces inside of elements of any size. The proposed representation also provides support for the implementation of accurate, robust, and computationally efficient numerical integration of the weak form over elements cut by the crack surface. Numerical simulations using the proposed GFEM show high convergence rates of extracted stress intensity factors along non‐planar curved crack fronts and the robustness of the method. Copyright © 2008 John Wiley & Sons, Ltd.     

Stable 3D XFEM/vector level sets for non‐planar 3D crack propagation and comparison of enrichment schemes

We present a three‐dimensional 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, ie, 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, while providing nonoptimal convergence rates similar to standard finite elements. The 2 dicretization schemes are tested for different benchmark problems, and their combination to the vector level set method is verified for nonplanar crack propagation problems.     

Three‐dimensional brittle fracture: configurational‐force‐driven crack propagation

This paper presents a computational framework for quasi‐static brittle fracture in three‐dimensional solids. The paper sets out the theoretical basis for determining the initiation and direction of propagating cracks based on the concept of configurational mechanics, consistent with Griffith's theory. Resolution of the propagating crack by the FEM is achieved by restricting cracks to element faces and adapting the mesh to align it with the predicted crack direction. A local mesh improvement procedure is developed to maximise mesh quality in order to improve both accuracy and solution robustness and to remove the influence of the initial mesh on the direction of propagating cracks. An arc‐length control technique is derived to enable the dissipative load path to be traced. A hierarchical hp‐refinement strategy is implemented in order to improve both the approximation of displacements and crack geometry. The performance of this modelling approach is demonstrated on two numerical examples that qualitatively illustrate its ability to predict complex crack paths. All problems are three‐dimensional, including a torsion problem that results in the accurate prediction of a doubly‐curved crack. Copyright © 2013 John Wiley & Sons, Ltd.     

Three‐dimensional finite element implementation of the nonuniform transformation field analysis

In this paper aspects of the nonuniform transformation field analysis (NTFA) introduced by Michel and Suquet (Int. J. Solids Struct. 2003; 40 :6937–6955) are investigated for materials with three‐dimensional microtopology. A novel implementation of the NTFA into the finite element method (FEM) is described in detail, whereas the NTFA was originally used in combination with the fast Fourier transformation (FFT). In particular, the discrete equivalents of the averaging operators required for the preprocessing steps and an algorithm for the implicit time integration and linearization of the constitutive equations of the homogenized material are provided. To the authors knowledge this is the first implementation of the method for three‐dimensional problems. Further, an alternative mode identification strategy is proposed with the aim of small computational cost in combination with good efficiency. The new identification strategy is applied to three‐dimensional metal matrix composites in order to investigate its effective non‐linear behaviour. The homogenized material model is implemented into ABAQUS/STANDARD. Numerical examples at integration point level and in terms of structural problems highlight the efficiency of the method for three‐dimensional problems. Copyright © 2010 John Wiley & Sons, Ltd.     

A high‐order generalized FEM for through‐the‐thickness branched cracks

This paper presents high‐order implementations of a generalized finite element method for through‐the‐thickness three‐dimensional branched cracks. This approach can accurately represent discontinuities such as triple joints in polycrystalline materials and branched cracks, independently of the background finite element mesh. Representative problems are investigated to illustrate the accuracy of the method in combination with various discretizations and refinement strategies. The combination of local refinement at crack fronts and high‐order continuous and discontinuous enrichments proves to be an excellent combination which can deliver convergence rates close to that of problems with smooth solutions. Copyright © 2007 John Wiley & Sons, Ltd.     

A time‐dependent formulation of dual boundary element method for 3D dynamic crack problems

In this paper, the dual boundary element method in time domain is developed for three‐dimensional dynamic crack problems. The boundary integral equations for displacement and traction in time domain are presented. By using the displacement equation and traction equation on crack surfaces, the discontinuity displacement on the crack can be determined. The integral equations are solved numerically by a time‐stepping technique with quadratic boundary elements. The dynamic stress intensity factors are calculated from the crack opening displacement. Several examples are presented to demonstrate the accuracy of this method. Copyright © 1999 John Wiley & Sons, Ltd