New crack‐tip elements for XFEM and applications to cohesive cracks

An extended finite element method scheme for a static cohesive crack is developed with a new formulation for elements containing crack tips. This method can treat arbitrary cracks independent of the mesh and crack growth without remeshing. All cracked elements are enriched by the sign function so that no blending of the local partition of unity is required. This method is able to treat the entire crack with only one type of enrichment function, including the elements containing the crack tip. This scheme is applied to linear 3‐node triangular elements and quadratic 6‐node triangular elements. To ensure smooth crack closing of the cohesive crack, the stress projection normal to the crack tip is imposed to be equal to the material strength. The equilibrium equation and the traction condition are solved by the Newton–Raphson method to obtain the nodal displacements and the external load simultaneously. The results obtained by the new extended finite element method are compared to reference solutions and show excellent agreement. Copyright © 2003 John Wiley & Sons, Ltd.     

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.     

Analysis of three‐dimensional crack initiation and propagation using the extended finite element method

An Erratum has been published for this article in International Journal for Numerical Methods in Engineering 2005, 63(8): 1228. We present a new formulation and a numerical procedure for the quasi‐static analysis of three‐dimensional crack propagation in brittle and quasi‐brittle solids. The extended finite element method (XFEM) is combined with linear tetrahedral elements. A viscosity‐regularized continuum damage constitutive model is used and coupled with the XFEM formulation resulting in a regularized ‘crack‐band’ version of XFEM. The evolving discontinuity surface is discretized through a C⁰ surface formed by the union of the triangles and quadrilaterals that separate each cracked element in two. The element's properties allow a closed form integration and a particularly efficient implementation allowing large‐scale 3D problems to be studied. Several examples of crack propagation are shown, illustrating the good results that can be achieved. Copyright © 2005 John Wiley & Sons, Ltd.     

Topology optimization using non‐conforming finite elements: three‐dimensional case

As in the case of two‐dimensional topology design optimization, numerical instability problems similar to the formation of two‐dimensional checkerboard patterns occur if the standard eight‐node conforming brick element is used. Motivated by the recent success of the two‐dimensional non‐conforming elements in completely eliminating checkerboard patterns, we aim at investigating the performance of three‐dimensional non‐conforming elements in controlling the patterns that are estimated overly stiff by the brick elements. To this end, we will investigate how accurately the non‐conforming elements estimate the stiffness of the patterns. The stiffness estimation is based on the homogenization method by assuming the periodicity of the patterns. To verify the superior performance of the elements, we consider three‐dimensional compliance minimization and compliant mechanism design problems and compare the results by the non‐conforming element and the standard 8‐node conforming brick element. Copyright © 2005 John Wiley & Sons, Ltd.     

Numerical method for the simulation of the Brownian dynamics of rod‐like microstructures with three‐dimensional nonlinear beam elements

In several fast‐growing research areas such as material science, macromolecular chemistry, bioengineering and biophysics, the dynamics of rod‐like microstructures such as polymers, rod‐like viruses, flagella of bacteria or thin glass fibers plays a crucial role. These microstructures are typically embedded into some viscous fluid and kept by stochastic thermal forces in an incessant motion, the so‐called Brownian dynamics. In this article, we introduce a novel approach wherein three‐dimensional finite beam elements can be employed for the computer simulation of the Brownian dynamics of rod‐like microstructures. This new approach is superior to state‐of‐the‐art methods by its transparent theoretical foundation and its remarkable efficiency. By means of numerical examples, we demonstrate the applicability of this approach to problems of biophysics and macromolecular chemistry. Copyright © 2012 John Wiley & Sons, Ltd.     

Universal three‐dimensional connection hexahedral elements based on hybrid‐stress theory for solid structures

High‐performance hybrid‐stress hexahedral solid elements are excellent choices for modeling joints, beams/columns walls and thick slabs for building structures if the exact geometrical representation is required. While it is straight‐forward to model beam–column structures of uniform member size with solid hexahedral elements, joining up beams and columns of various cross‐sections at a common point proves to be a challenge for structural modeling using hexahedral elements with specified dimensions. In general, the joint has to be decomposed into 27 smaller solid elements to cater for the necessary connection requirements. This will inevitably increase the computational cost and introduce element distortions when elements of different sizes have to be used at the joint. Universal connection hexahedral elements with arbitrary specified connection interfaces will be an ideal setup to connect structural members of different sizes without increasing the number of elements or introducing highly distorted elements. In this paper, the requirements and the characteristics of the hexahedral connection elements with 24 and 32 nodes will be discussed. Formulation of the connection elements by means of Hellinger–Reissner functional will be presented. The performance of connection elements equipped with different number of stress modes will be assessed with worked examples. Copyright © 2009 John Wiley & Sons, Ltd.     

Boundary elements for half‐space problems via fundamental solutions: A three‐dimensional analysis

An efficient solution technique is proposed for the three‐dimensional boundary element modelling of half‐space problems. The proposed technique uses alternative fundamental solutions of the half‐space (Mindlin's solutions for isotropic case) and full‐space (Kelvin's solutions) problems. Three‐dimensional infinite boundary elements are frequently employed when the stresses at the internal points are required to be evaluated. In contrast to the published works, the strongly singular line integrals are avoided in the proposed solution technique, while the discretization of infinite elements is independent of the finite boundary elements. This algorithm also leads to a better numerical accuracy while the computational time is reduced. Illustrative numerical examples for typical isotropic and transversely isotropichalf‐space problems demonstrate the potential applications of the proposed formulations. Incidentally, the results of the illustrative examples also provide a parametric study for the imperfect contact problem. Copyright © 2001 John Wiley & Sons, Ltd.     

Accuracy of Galerkin finite elements for groundwater flow simulations in two and three‐dimensional triangulations

In standard finite element simulations of groundwater flow the correspondence between hydraulic head gradients and groundwater fluxes is represented by the stiffness matrix. In two‐dimensional problems the use of linear triangular elements on Delaunay triangulations guarantees a stiffness matrix of type M. This implies that the local numerical fluxes are physically consistent with Darcy's law. This condition is fundamental to avoid the occurrence of local maxima or minima, and is of crucial importance when the calculated flow field is used in contaminant transport simulations or pathline evaluation. In three spatial dimensions, the linear Galerkin approach on tetrahedra does not lead to M‐matrices even on Delaunay meshes. By interpretation of the Galerkin approach as a subdomain collocation scheme, we develop a new approach (OSC, orthogonal subdomain collocation) that is shown to produce M‐matrices in three‐dimensional Delaunay triangulations. In case of heterogeneous and anisotropic coefficients, extra mesh properties required for M‐stiffness matrices will also be discussed. Copyright © 2001 John Wiley & Sons, Ltd.     

A novel three‐dimensional contact finite element based on smooth pressure interpolations

This article proposes a new three‐dimensional contact finite element which employs continuous and weakly coupled pressure interpolations on each of the interacting boundaries. The resulting formulation circumvents the geometric bias of one‐pass methods, as well as the surface locking of traditional two‐pass node‐on‐surface methods. A Lagrange multiplier implementation of the proposed element is validated for frictionless quasi‐static contact by a series of numerical simulations. Published in 2001 by John Wiley & Sons, Ltd.     

Delamination buckling and propagation analysis of honeycomb panels using a cohesive element approach

The cohesive element approach is proposed as a tool for simulating delamination propagation between a facesheet and a core in a honeycomb core composite panel. To determine the critical energy release rate (G _c) of the cohesive model, Double Cantilever Beam (DCB) fracture tests were performed. The peak strength (_c) of the cohesive model was determined from Flatwise Tension (FWT) tests. The DCB coupon test was simulated using the measured fracture parameters, and sensitivity studies on the parameters for the cohesive model of the interface element were performed. The cohesive model determined from DCB tests was then applied to a full-scale, 914×914 mm (36×36 in.) debond panel under edge compression loading, and results were compared with an experiment. It is concluded that the cohesive element approach can predict delamination propagation of a honeycomb panel with reasonable accuracy.     

Cracking node method for dynamic fracture with finite elements

A new method for modeling discrete cracks based on the extended finite element method is described. In the method, the growth of the actual crack is tracked and approximated with contiguous discrete crack segments that lie on finite element nodes and span only two adjacent elements. The method can deal with complicated fracture patterns because it needs no explicit representation of the topology of the actual crack path. A set of effective rules for injection of crack segments is presented so that fracture behavior beginning from arbitrary crack nucleations to macroscopic crack propagation is seamlessly modeled. The effectiveness of the method is demonstrated with several dynamic fracture problems that involve complicated crack patterns such as fragmentation and crack branching. Copyright © 2008 John Wiley & Sons, Ltd.     

A rate-dependent cohesive model for simulating dynamic crack propagation in brittle materials

Numerical investigations are conducted to simulate high-speed crack propagation in pre-strained PMMA plates. In the simulations, the dynamic material separation is explicitly modeled by cohesive elements incorporating an initially rigid, linear-decaying cohesive law. Initial attempts using a rate-independent cohesive law failed to reproduce available experimental results as numerical crack velocities consistently overestimate experimental observations. As proof of concept, a phenomenological rate-dependent cohesive law, which bases itself on the physics of microcracking, is introduced to modulate the cohesive law with the macroscopic crack velocity. We then generalize this phenomenological approach by establishing a rate-dependent cohesive law, which relates the traction to the effective displacement and rate of change of effective displacement. It is shown that this new model produces numerical results in good agreement with experimental data. The analysis demonstrates that the simulation of high-speed crack propagation in brittle structures necessitates the use of rate-dependent cohesive models, which account for the complicated rate-process of dynamic fracture at the propagating crack tip.     

Time‐domain BEM for three‐dimensional site response analysis of topographic structures

This paper presents a formulation of a time‐domain three‐dimensional boundary element method for site response analysis of topographic structures. The boundary element algorithm that uses the presented time‐convoluted traction kernels is applied to site response analyses of topographic structures. The seismic responses of canyon and ridge subjected to incident P and S waves are analyzed to demonstrate the accuracy of the kernels and the applicability of the presented boundary element algorithm for site response analysis of topographic structures. Seismic response analyses of three‐dimensional Gaussian‐shaped ridges show that the three‐dimensional axisymmetric ridge has a more amplification potential compared with three‐dimensional non‐axisymmetric elongated and two‐dimensional ridges, if the ridge is impinged by incident waves with wavelength of about the ridge's width. Copyright© 2009 John Wiley & Sons, Ltd.     

Theory and numerics of three‐dimensional beams with elastoplastic material behaviour

A theory of space curved beams with arbitrary cross‐sections and an associated finite element formulation is presented. Within the present beam theory the reference point, the centroid, the centre of shear and the loading point are arbitrary points of the cross‐section. The beam strains are based on a kinematic assumption where torsion‐warping deformation is included. Each node of the derived finite element possesses seven degrees of freedom. The update of the rotational parameters at the finite element nodes is achieved in an additive way. Applying the isoparametric concept the kinematic quantities are approximated using Lagrangian interpolation functions. Since the reference curve lies arbitrarily with respect to the centroid the developed element can be used to discretize eccentric stiffener of shells. Due to the implemented constitutive equations for elastoplastic material behaviour the element can be used to evaluate the load‐carrying capacity of beam structures. Copyright © 2000 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.     

Mesh refinement strategies without mapping of nonlinear solutions for the generalized and standard FEM analysis of 3‐D cohesive fractures

A robust and efficient strategy is proposed to simulate mechanical problems involving cohesive fractures. This class of problems is characterized by a global structural behavior that is strongly affected by localized nonlinearities at relatively small‐sized critical regions. The proposed approach is based on the division of a simulation into a suitable number of sub‐simulations where adaptive mesh refinement is performed only once based on refinement window(s) around crack front process zone(s). The initialization of Newton‐Raphson nonlinear iterations at the start of each sub‐simulation is accomplished by solving a linear problem based on a secant stiffness, rather than a volume mapping of nonlinear solutions between meshes. The secant stiffness is evaluated using material state information stored/read on crack surface facets which are employed to explicitly represent the geometry of the discontinuity surface independently of the volume mesh within the generalized finite element method framework. Moreover, a simplified version of the algorithm is proposed for its straightforward implementation into existing commercial software. Data transfer between sub‐simulations is not required in the simplified strategy. The computational efficiency, accuracy, and robustness of the proposed strategies are demonstrated by an application to cohesive fracture simulations in 3‐D. Copyright © 2016 John Wiley & Sons, Ltd.     

A meshless method with enriched weight functions for three‐dimensional crack propagation

The propagation of cracks in three dimensions is analysed by a meshless method. The cracks are modelled by a set of triangles that are added when the propagation occurs. Since the method is meshless, no remeshing of the domain is necessary during the propagation. To avoid using a large number of degrees of freedom, the stress singularity along the front of the cracks is taken into account by an enrichment of the shape functions of the meshless method by means of appropriate weight functions. This enrichment technique is an extension of the technique that proved to be successful in two dimensions in a previous paper. Several algorithms for efficiently implementing the meshless method in three dimensions are detailed. The accuracy of the enrichment is first assessed on simple examples and some results of non‐planar propagation of multiple cracks are then presented. Copyright © 2005 John Wiley & Sons, Ltd.     

Finite volume analysis of dynamic fracture phenomena – II. A cohesive zone type methodology

A truly predictive dynamic fracture model would require detailed information about the local fracture process. For instance, recent experimental evidence has conclusively demonstrated that rapid crack propagation (RCP) in brittle polymers such as PMMA is accompanied by the nucleation, growth, interaction and coalescence of microcracks and the advancing macroscopic fracture. Some insights into this phenomenon are offered and a novel computational model of this aspect of dynamic fracture is proposed. The procedure is based upon local material (cohesive) strength considerations. Here, the initial development of such procedures is presented. Application is at this stage restricted to single crack problems so that the effects of various geometrical, but more importantly, cohesive parameters on predicted fractures may be examined. Extension of the cohesive computational procedures of this work to multiple crack problems is proposed to be straightforward and without the need for extensive reconstructing of the computational procedures outlined.