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Generating matching meshes for finite element analysis is not always a convenient choice, for instance, in cases where the location of the boundary is not known a priori or when the boundary has a complex shape. In such cases, enriched finite element methods can be used to describe the geometric features independently from the mesh. The Discontinuity‐Enriched Finite Element Method (DE‐FEM) was recently proposed for solving problems with both weak and strong discontinuities within the computational domain. In this paper, we extend DE‐FEM to treat fictitious domain problems, where the mesh‐independent boundaries might either describe a discontinuity within the object, or the boundary of the object itself. These boundaries might be given by an explicit expression or an implicit level set. We demonstrate the main assets of DE‐FEM as an immersed method by means of a number of numerical examples; we show that the method is not only stable and optimally convergent but, most importantly, that essential boundary conditions can be prescribed strongly.

The extended finite element method (XFEM) was introduced in 1999 to treat problems involving discontinuities with no or minimal remeshing through appropriate enrichment functions. This enables elements to be split by a discontinuity, strong or weak and hence requires the integration of discontinuous functions or functions with discontinuous derivatives over elementary volumes. Moreover, in the case of open surfaces and singularities, special, usually non-polynomial functions must also be integrated. A variety of approaches have been proposed to facilitate these special types of numerical integration, which have been shown to have a large impact on the accuracy and the convergence of the numerical solution. The smoothed extended finite element method (SmXFEM) [1], for example, makes numerical integration elegant and simple by transforming volume integrals into surface integrals. However, it was reported in [1, 2] that the strain smoothing is inaccurate when non-polynomial functions are in the basis. This is due to the constant smoothing function used over the smoothing domains which destroys the effect of the singularity. In this paper, we investigate the benefits of a recently developed Linear smoothing procedure [3] which provides better approximation to higher order polynomial fields in the basis. Some benchmark problems in the context of linear elastic fracture mechanics (LEFM) are solved to compare the standard XFEM, the constant-smoothed XFEM (Sm-XFEM) and the linear-smoothed XFEM (LSm-XFEM). We observe that the convergence rates of all three methods are the same. The stress intensity factors (SIFs) computed through the proposed LSm-XFEM are however more accurate than that obtained through Sm-XFEM. To conclude, compared to the conventional XFEM, the same order of accuracy is achieved at a relatively low computational effort.

We introduce a new methodology for modeling problems with both weak and strong discontinuities independently of the finite element discretization. At variance with the eXtended/Generalized Finite Element Method (X/GFEM), the new method, named the Discontinuity-Enriched Finite Element Method (DE-FEM), adds enriched degrees of freedom only to nodes created at the intersection between a discontinuity and edges of elements in the mesh. Although general, the method is demonstrated in the context of fracture mechanics, and its versatility is illustrated with a set of traction-free and cohesive crack examples. We show that DE-FEM recovers the same rate of convergence as the standard FEM with matching meshes, and we also compare the new approach to X/GFEM.

Summary This paper presents a multiscale finite element method with the embedded strong discontinuity model for the strain localization analysis of homogeneous and heterogeneous saturated porous media. In the proposed method, the strong discontinuities in both displacement and fluid flux fields are considered. For the localized fine element, the mathematical description and discrete formulation are built based on the so-called strong discontinuity approach. For the localized unit cell, numerical base functions are constructed based on a newly developed enhanced coarse element technique, that is, additional coarse nodes are dynamically added as the shear band propagating. Through the enhanced coarse element technique, the multiscale finite element method can well reflect the softening behavior at the post-localization stage. Furthermore, the microscopic displacement and pore pressure are obtained with the solution decomposition technique. In addition, a non-standard return mapping algorithm is given to update the displacement jumps. Finally, through three representative numerical tests comparing with the results of the embedded finite element method with fine meshes, the high efficiency and accuracy of the proposed method are demonstrated in both material homogeneous and heterogeneous cases. Copyright © 2017 John Wiley & Sons, Ltd.

Summary 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.

We apply an “immersed” finite element approach discussed in [1] for the 2D steady incompressible Stokes equations to a 2D fluid-structure interaction problem in a tube with a thin (i.e., codimension 1) hinged rigid leaflet sets in motion by an incompressible unsteady flow. In order to change the minimal possible fraction of elements we remesh only the triangles that are cut by the structure. Furthermore, with this approach essential constraints between the fluid and the solid may be strongly enforced in the finite element spaces and it is fairly easy to allow the fluid stress to be discontinuous across the structure. An obvious consequence of the proposed remeshing strategy is the presence of anisotropic triangles, for which it is known that the standard finite element method may be used (see, e.g., [2]). However, for mixed elements most of the inf-sup stability proofs require isotropic meshes and only few results are available on distorted triangles. In [1], the inf-sup stability of the Hood-Taylor element with the present method has been discussed. It was shown that instabilities may occur but also that adding a bubble in the velocity space stabilizes the element. In the present work we apply and extend the results presented in [1] to a 2D fluid-structure problem with a codimension 1 structure immersed in the fluid domain. As a consequence, it may be convenient to use mixed elements with a discontinuous pressure. Indeed, with such elements the discontinuity in the pressure field across the structure is straightforwardly enforced. We discuss the use of the P2/P0 and the P2-bubble/P1-discontinuous elements with the proposed remeshing algorithm. However, we also consider the elements in [1] with a discontinuous pressure is across the structure. Numerical tests show that the two elements with a discontinuous pressure suffer inf-sup stability issues in a more severe way than the Hood-Taylor element. The P2/P0 and P2-bubble/P1-discontinuous elements show spurious modes along the structure, and thus not only in corners as in the case of the Hood-Taylor element. On the contrary, the Hood-Taylor element with an additional bubble do not show any spurious modes in the performed tests. The deleterious effects of inf-sup instabilities are also observed in the conditioning of the linear system. Furthermore, in [3] a similar fluid-structure interaction algorithm to the one presented here is employed using the P2-bubble/P1-discontinuous element. However, a smoothing strategy is used such that all triangles are isotropic. Our results show in particular the necessity of such a smoothing with that element.

A new continuous-discontinuous strategy for the simulation of failure is presented. The continuous bulk is regularised by means of a gradient-enhanced damage model, where non-locality is introduced at the level of displacements. As soon as the damage parameter is close or equal to 1, a traction-free crack is introduced. To determine the direction of crack growth, a new criterion is proposed. In contrast to traditional techniques, where mechanical criteria are used to define the crack path, here, a geometrical approach is used. More specifically, given a regularised damage field D(x), we propose to propagate the discontinuity following the direction dictated by the medial axis of the isoline (or isosurface in 3D) D(x) = D*. The proposed approach is tested on different two-dimensional and three-dimensional examples that illustrate that this combined methodology is able to deal with damage growth and material separation. Copyright (c) 2014 John Wiley & Sons, Ltd.

SUMMARY This paper presents a FEM with mesh-separation-based approximation technique that separates a standard element into three geometrically independent elements. A dual mapping scheme is introduced to couple them seamlessly and to derive the element approximation. The novel technique makes it very easy for mesh generation of problems with complex or solution-dependent, varying geometry. It offers a flexible way to construct displacement approximations and provides a unified framework for the FEM to enjoy some of the key advantages of the Hansbo and Hansbo method, the meshfree methods, the semi-analytical FEMs, and the smoothed FEM. For problems with evolving discontinuities, the method enables the devising of an efficient crack-tip adaptive mesh refinement strategy to improve the accuracy of crack-tip fields. Both the discontinuities due to intra-element cracking and the incompatibility due to hanging nodes resulted from the element refinement can be treated at the elemental level. The effectiveness and robustness of the present method are benchmarked with several numerical examples. The numerical results also demonstrate that a high precision integral scheme is critical to pass the crack patch test, and it is essential to apply local adaptive mesh refinement for low fracture energy problems. Copyright © 2014 John Wiley & Sons, Ltd.

SUMMARY This paper presents an augmentation method that enables bilinear finite elements to efficiently and accurately account for arbitrary, multiple intra-elemental discontinuities in 2D solids. The augmented finite element method (A-FEM) employs four internal nodes to account for the crack displacements due to an intra-elemental weak or strong discontinuity, and it permits repeated elemental augmentation to include multiple interactive cracks. It thus enables a unified treatment of damage evolution from a weak discontinuity to a strong discontinuity, and to multiple interactive cohesive cracks, all within a single bilinear element that employs standard external nodal DoFs only. A novel elemental condensation procedure has been developed to solve the internal nodal DoFs as functions of the external nodal DoFs for any irreversible, piece-wise linear cohesive laws. It leads to a fully condensed elemental equilibrium equation with mathematical exactness, eliminating the need for nonlinear equilibrium iterations at elemental level. The new A-FEM's high-fidelity simulation capabilities to interactive cohesive crack formation and propagation in homogeneous, and heterogeneous solids have been demonstrated through several challenging numerical examples. It is shown that the proposed A-FEM, empowered by the new elemental condensation procedure, is numerically very efficient, accurate, and robust. Copyright © 2014 John Wiley & Sons, Ltd.

SUMMARY In the numerical analysis of 2‒D bimodular materials, strain discontinuity is problematic, and the traditional iterative algorithm is frequently unstable. This paper develops a stable algorithm for the large‒displacement and small‒strain analyses of 2‒D bimodular materials and structures. Geometrically nonlinear formulations are based on the co‒rotational approach. Using the parametric variational principle (PVP), a unified constitutive equation is created to resolve the problem induced by strain discontinuity in the local coordinate system. Because the traditional stress iteration is not required, the local linear stiffness matrix does not need to be updated when computing the global stiffness matrix and the nodal internal force vector. The nonlinear problem is ultimately transformed into a complementarity problem that is simply solved by combing the Newton–Raphson scheme and the mature quadratic programming algorithm. Numerical examples demonstrate that the PVP algorithm presents better convergence behavior than the traditional iterative algorithm. By incorporating the concept of material modification, the new algorithm is also be successfully extended to the wrinkling analysis of thin membranes. Copyright © 2014 John Wiley & Sons, Ltd.

Many computational problems incorporate discontinuities that evolve in time. The eXtendend Finite Element Method (XFEM) is able to represent discontinuities sharply on fixed arbitrary meshes, but numerical difficulties arise if these discontinuities move in time. We point out that this issue is crucial for interface problems with strongly discontinuous fields on fixed grids. A method using semi-Lagrangean techniques is proposed to adequately handle time integration based on finite difference schemes in the context of the XFEM. The basic idea is to adapt previous numerical solutions to the current interface position by tracking back virtual Lagrangean particles to their previous positions, where an appropriate solution can be extrapo- lated from a smooth field. Convergence properties of the proposed method in time and space are thoroughly studied for two one-dimensional model problems. Finally, the method is applied to the particularly challenging problem of premixed combustion, where the discontinuity appears at the flame front separating the burnt from the unburnt gases. A two-dimensional and a three-dimensional expanding flame demonstrates that the method is sufficiently accurate to retain the properties of the overall Nitsche-type formulation for interface problems with embedded strong discontinuities.

SUMMARY The most difficult aspect of modelling discontinuity on complicated domains is the mesh. In the extended finite element method (XFEM), the FE mesh need not conform to the internal boundaries. In the intrinsic XFEM, no additional unknowns are introduced at the nodes. Some of these discontinuities such as cracks, voids were developed in earlier papers. In the current work, the FE mesh is allowed to be independent of the rigid line (fixed DOFs path). Copyright © 2013 John Wiley & Sons, Ltd.

In this paper, we replace the asymptotic enrichments around the crack tip in the extended finite element method (XFEM) with the semi-analytical solution obtained by the scaled boundary finite element method (SBFEM). The proposed method does not require special numerical integration technique to compute the stiffness matrix and it improves the capability of the XFEM to model cracks in homogeneous and/or heterogeneous materials without a priori knowledge of the asymptotic solutions. A heaviside enrichment is used to represent the jump across the discontinuity surface. We call the method as the extended scaled boundary finite element method (xSBFEM). Numerical results presented for a few benchmark problems in the context of linear elastic fracture mechanics show that the proposed method yields accurate results with improved condition number. A simple MATLAB code is annexed to compute the terms in the stiffness matrix, which can easily be integrated in any existing FEM/XFEM code.

In this paper, a method to analyse and predict crack propagation in thin-walled structures subjected to large plastic deformations when loaded at high strain rates—such as impact and/or blast—has been proposed. To represent the crack propagation independently of the finite element discretisation, an extended finite element method based shell formulation has been employed. More precisely, an underlying 7-parameter shell model formulation with extensible directors has been extended by locally introducing an additional displacement field, representing the displacement discontinuity independently of the mesh. Of special concern in the paper has been to find a proper balance between, level of detail and accuracy when representing the physics of the problem and, on the other hand, computational efficiency and robustness. To promote computational efficiency, an explicit time step scheme has been employed, which however has been discovered to generate unphysical oscillations in the response upon crack propagation. Therefore, special focus has been placed to investigate these oscillations as well as to find proper remedies. The paper is concluded with three numerical examples to verify and validate the proposed model.Copyright © 2013 John Wiley & Sons, Ltd.

SUMMARY This paper extends hybrid equilibrium formulation concepts, previously used with success for planar problems, to the analysis of folded plates and curved shells. A 2D hybrid equilibrium flat shell quadrilateral element is formulated for linear analysis, where detailed consideration is given to the implication of slope discontinuity when the element is used for non-planar domains. Benchmark plate bending, folded plate and curved shell problems are modelled using equilibrium and conforming elements for comparison. In models of the latter two problems, torsional moments may be released along lines of slope discontinuity, and the effects of this assumption for the folded plate are studied by analysing a third type of model composed of 3D solid brick elements. The comparisons demonstrate an excellent performance from the new hybrid equilibrium analysis method for folded plates and curved shells. Copyright © 2013 John Wiley & Sons, Ltd.

SUMMARY The paper introduces a methodology for numerical simulation of landslides experiencing considerable deformations and topological changes. Within an interface capturing approach, all interfaces are implicitly described by specifically defined level-set functions allowing arbitrarily evolving complex topologies. The transient interface evolution is obtained by solving the level-set equation driven by the physical velocity field for all three level-set functions in a block Jacobi approach. The three boundary-coupled fluid-like continua involved are modeled as incompressible, governed by a generalized non-Newtonian material law taking into account capillary pressure at moving fluid–fluid interfaces. The weighted residual formulation of the level-set equations and the fluid equations is discretized with finite elements in space and time using velocity and pressure as unknown variables. Non-smooth solution characteristics are represented by enriched approximations to fluid velocity (weak discontinuity) and fluid pressure (strong discontinuity). Special attention is given to the construction of enriched approximations for elements containing evolving triple junctions. Numerical examples of three-fluid tank sloshing and air–water-liquefied soil landslides demonstrate the potential and applicability of the method in geotechnical investigations. Copyright © 2012 John Wiley & Sons, Ltd.

SUMMARY The embedding of discontinuities into finite elements has become a powerful technique for the simulation of fracture in a wide variety of mechanical problems. However, existing formulations still use non-conforming finite elements. In this manuscript, a new conforming formulation is proposed. The main properties of this formulation are as follows: (i) variational consistency; (ii) no limitations on the choice of the parent finite element; (iii) comprehensive kinematics of the discontinuity, including both rigid body motion and stretching; (iv) fully compatible enhanced kinematic field; (v) additional global DOFs located at the discontinuity; (vi) continuity of both jumps and tractions across element boundaries; and (vii) stress locking free. The performance of the proposed formulation is tested by means of academic and structural examples. The numerical results are compared with available experimental results and other numerical approaches, namely the generalized strong discontinuity approach and the generalized FEM/Extended FEM. Copyright © 2012 John Wiley & Sons, Ltd.

SUMMARY Noise reduction for passengers' comfort in transport industry is now an important constraint to be taken into account during the design process. This process involves to study several configurations of the structures immersed in a given acoustic cavity in the context of an optimization, uncertainty, or reliability study for instance. The finite element method may be used to model this coupled fluid–structure problem but needs an interface conforming mesh for each studied configuration that may become time consuming. This work aims at avoiding this remeshing step by using noncompatible meshes between the fluid and the structures. The immersed structures are supposed to be thin shells and are localized in the fluid domain by a signed distance level-set. To take into account the pressure discontinuity from one side of the structures to the other one, the fluid pressure approximation is enriched according to the structures positions by a Heaviside function using a partition of unity strategy (extended finite element method). The same fluid mesh of the empty cavity is then used during the whole parametric study. The method is implemented for a three-dimensional fluid and tested on academic examples before being applied to an industrial-like case. Copyright © 2012 John Wiley & Sons, Ltd.

SUMMARY In this paper, we study the efficient numerical integration of functions with sharp gradients and cusps. An adaptive integration algorithm is presented that systematically improves the accuracy of the integration of a set of functions. The algorithm is based on a divide and conquer strategy and is independent of the location of the sharp gradient or cusp. The error analysis reveals that for a C0 function (derivative discontinuity at a point), a rate of convergence of n + 1 is obtained in Rn. Two applications of the adaptive integration scheme are studied. First, we use the adaptive quadratures for the integration of the regularized Heaviside function—a strongly localized function that is used for modeling sharp gradients. Then the adaptive quadratures are employed in the enriched finite element solution of the all-electron Coulomb problem in crystalline diamond. The source term and enrichment functions of this problem have sharp gradients and cusps at the nuclei. We show that the optimal rate of convergence is obtained with only a marginal increase in the number of integration points with respect to the pure finite element solution with the same number of elements. The adaptive integration scheme is simple, robust, and directly applicable to any generalized finite element method employing enrichments with sharp local variations or cusps in n-dimensional parallelepiped elements. Copyright © 2012 John Wiley & Sons, Ltd.

SUMMARY This paper presents new three-dimensional finite elements with embedded strong discontinuities in the small strain infinitesimal range. The goal is to model localized surfaces of failure in solids, such as cracks at fracture, through enhancements of the finite elements that capture the propagating discontinuities of the displacement field in the element interiors. In this way, such surfaces of discontinuity can be sharply resolved in general meshes not necessarily related to the detailed geometry of the surface, unknown a priori. An important issue is also the consideration of general finite element formulations in the developments (e.g., basic displacement-based, mixed or enhanced assumed strain finite element formulations), as needed to optimally resolve the continuum problem in the bulk. The actual modeling of the discontinuity effects, including the incorporation of the cohesive law defining the discontinuity constitutive response, is carried out at the element level with the proper enhancement of the discrete strain field of the element. The added elemental degrees of freedom approximate the displacement jumps associated with the discontinuity and are defined independently from element to element, thus allowing their static condensation at the element level without affecting the global mechanical problem in terms of the number and topology of the global degrees of freedom. In fact, this global-local structure of the finite element methods developed in this work arises naturally from a multi-scale characterization of these localized solutions, with the discontinuities understood to appear in the small scales, thus leading directly to these computationally efficient numerical methods for their numerical resolution, easily incorporated to an existing finite element code. The focus in this paper is on the development of finite elements incorporating a linear interpolation of the displacement jumps in the general three-dimensional setting. These interpolations are shown to be necessary for hexahedral elements to avoid the so-called stress locking that occurs with simpler constant approximations of the jumps (namely, a spurious transfer of stresses across the discontinuity not allowing its full release and, hence, resulting in an overstiff or locked numerical solution). The design of the new finite elements is accomplished in this work by a direct identification of the separation modes to be incorporated in the discrete strain field of the element, rather than from an assumed discontinuous interpolation of the displacements, assuring with this approach their locking-free response by design. An additional issue addressed in the paper is the geometric characterization and propagation of the discontinuity surfaces in the general three-dimensional setting of interest here. The paper includes a series of numerical simulations illustrating and evaluating the properties of the new finite elements. Copyright © 2012 John Wiley & Sons, Ltd.

SUMMARY The modelization of bending plates with through-the-thickness cracks is investigated. We consider the Kirchhoff–Love plate model, which is valid for very thin plates. Reduced Hsieh–Clough–Tocher triangles and reduced Fraejis de Veubeke–Sanders quadrilaterals are used for the numerical discretization. We apply the eXtended Finite Element Method strategy: enrichment of the finite element space with the asymptotic bending singularities and with the discontinuity across the crack. The main point, addressed in this paper, is the numerical computation of stress intensity factors. For this, two strategies, direct estimate and J-integral, are described and tested. Some practical rules, dealing with the choice of some numerical parameters, are underlined. Copyright © 2012 John Wiley & Sons, Ltd.

SUMMARY A novel approach based on a combination of isogeometric analysis (IGA) and extended FEM is presented for fracture analysis of structures. The extended isogeometric analysis is capable of an efficient analysis of general crack problems using nonuniform rational B-splines as basis functions for both the solution field approximation and the geometric description, and it can reproduce crack tip singular fields and discontinuity across a crack. IGA has attracted a lot of interest for solving different types of engineering problems and is now further extended for the analysis of crack stability and propagation in two-dimensional isotropic media. Concepts of the extended FEM are used in IGA to avoid the necessity of remeshing in crack propagation problems and to increase the solution accuracy around the crack tip. Crack discontinuity is represented by the Heaviside function and isotropic analytical displacement fields near a crack tip are reproduced by means of the crack tip enrichment functions. Also, the Lagrange multiplier method is used to impose essential boundary conditions. Moreover, the subtriangles technique is utilized for improving the accuracy of integration by the Gauss quadrature rule. Several two-dimensional static and quasi-static crack propagation problems are solved to demonstrate the efficiency of the proposed method and the results of mixed-mode stress intensity factors are compared with analytical and extended FEM results. Copyright © 2011 John Wiley & Sons, Ltd.

In this paper, the Polytope Finite Element Method is employed to model an embedded interface through the body, independent of the background FEM mesh. The elements that are crossed by the embedded interface are decomposed into new polytope elements which have some nodes on the interface line. The interface introduces discontinuity into the primary variable (strong) or into its derivatives (weak). Both strong and weak discontinuities are studied by the proposed method through different numerical examples including fracture problems with traction-free and cohesive cracks, and heat conduction problems with Dirichlet and Dirichlet–Neumann types of boundary conditions on the embedded interface. For traction-free cracks which have tip singularity, the nodes near the crack tip are enriched with the singular functions through the eXtended Finite Element Method. The concept of Natural Element Coordinates (NECs) is invoked to drive shape functions for the produced polytopes. A simple treatment is proposed for concave polytopes produced by a kinked interface and also for locating crack tip inside an element prior to using the singularity enrichment. The proposed method pursues some implementational details of eXtended/Generalized Finite Element Methods for interfaces. But here the additional DOFs are constructed on the interface lines in contrast to X/G-FEM, which attach enriched DOFs to the previously existed nodes. Copyright © 2011 John Wiley & Sons, Ltd.

The extended finite element method (XFEM) is further improved for fracture analysis of composite laminates containing interlaminar delaminations. New set of bimaterial orthotropic enrichment functions are developed and utilized in XFEM analysis of linear-elastic fracture mechanics of layered composites. Interlaminar crack-tip enrichment functions are derived from analytical asymptotic displacement fields around a traction-free interfacial crack. Also, heaviside and weak discontinuity enrichment functions are utilized in modeling discontinuous fields across interface cracks and bimaterial weak discontinuities, respectively. In this procedure, elements containing a crack-tip or strong/weak discontinuities are not required to conform to those geometries. In addition, the same mesh can be used to analyze different interlaminar cracks or delamination propagation. The domain interaction integral approach is also adopted in order to numerically evaluate the mixed-mode stress intensity factors. A number of benchmark tests are simulated to assess the performance of the proposed approach and the results are compared with available reference results. Copyright © 2011 John Wiley & Sons, Ltd.

This paper is concerned with the effective numerical implementation of the adaptive dual boundary-element method (DBEM), for two-dimensional potential problems. Two boundary integral equations, which are the potential and the flux equations, are applied for collocation along regular and degenerate boundaries, leading always to a single-region analysis. Taking advantage on the use of non-conforming parametric boundary-elements, the method introduces a simple error estimator, based on the discontinuity of the solution across the boundaries between adjacent elements and implements the p, h and mixed versions of the adaptive mesh refinement. Examples of several geometries, which include degenerate boundaries, are analyzed with this new formulation to solve regular and singular problems. The accuracy and efficiency of the implementation described herein make this a reliable formulation of the adaptive DBEM.

A conservative high-order Godunov-type scheme is presented for solving the balance laws of the 1D shallow water equations (SWE). The scheme adopts a finite element Runge–Kutta (RK) discontinuous Galerkin (DG) framework. Based on an overall third-order accurate formulation, the model is referred to as RKDG3. Treatment of topographic source term is built in the DG approximation. Simplified formulae for initializing bed data at a discrete level are derived by assuming a local linear bed function to ease practical flow simulations. Owing to the adverse effects caused by using an uncontrolled global limiting process in an RKDG3 scheme (RKDG3-GL), a new conservative RKDG3 scheme with user-parameter-free local limiting method (RKDG3-LL) is designed to gain better accuracy and conservativeness. The advantages of the new RKDG3-LL model are demonstrated by applying to several steady and transient benchmark flow tests with irregular (either differentiable or non-differentiable) topography. Copyright © 2010 John Wiley & Sons, Ltd.

Complex dispersion relation calculations with the applications to absorptive photonic crystals

We propose a discontinuous Galerkin discretization scheme for the forward modelling in optical diffusion tomography in highly scattering media to facilitate dynamic mesh adaptation for complex domains. In addition, the numerical method is also shown to effectively deal with inhomogeneities in optical properties and refractive index mismatch at material interfaces. The accuracy of the method is demonstrated over a model concentric spherical layers problem where the discontinuous Galerkin method is compared against an analytical solution. Copyright (C) 2010 John Wiley & Sons, Ltd.

The direct approximation of strong form using radial basis functions (RBFs), commonly called the radial basis collocation method (RBCM), has been recognized as an effective means for solving boundary value problems. Nevertheless, the non-compactness of the RBFs precludes its application to problems with local features, such as fracture problems, among others. This work attempts to apply RBCM to fracture mechanics by introducing a domain decomposition technique with proper interface conditions. The proposed method allows (1) natural representation of discontinuity across the crack surfaces and (2) enrichment of crack-tip solution in a local subdomain. With the proper domain decomposition and interface conditions, exponential convergence rate can be achieved while keeping the discrete system well-conditioned. The analytical prediction and numerical results demonstrate that an optimal dimension of the near-tip subdomain exists. The effectiveness of the proposed method is justified by the mathematical analysis and demonstrated by the numerical examples. Copyright © 2010 John Wiley & Sons, Ltd.

We present a meso-scale model for failure of heterogeneous quasi-brittle materials. The model problem of heterogeneous materials that is addressed in detail is based on two-phase 3D representation of reinforced heterogeneous materials, such as concrete, where the inclusions are melt within the matrix. The quasi-brittle failure mechanisms are described by the spatial truss representation, which is defined by the chosen Voronoi mesh. In order to explicitly incorporate heterogeneities with no need to change this mesh, some bar elements are cut by the phase-interface and must be split into two parts. Any such element is enhanced using both weak and strong discontinuities, based upon the Incompatible Mode Method. Furthermore, a dedicated operator split solution procedure is proposed to keep local any additional computation on elements with embedded discontinuities. The results for several numerical simulations are presented to illustrate the capabilities of the proposed model to provide an excellent representation of failure mechanisms for any different macroscopic loading path. Copyright © 2010 John Wiley & Sons, Ltd.

In this paper a new time discontinuous Galerkin (TDG) formulation for non-linear elastodynamics is presented. The new formulation embeds an energy correction which ensures truly energy decaying, thus allowing to achieve unconditional stability that, as shown in the paper, is not guaranteed by the classical TDG formulation. The resulting method is simple and easily implementable into existing finite element codes. Moreover, it inherits the desirable higher-order accuracy and high-frequency dissipation properties of the classical formulation. Numerical results illustrate the very good performance of the proposed formulation. Copyright © 2010 John Wiley & Sons, Ltd.

A general GFEM/XFEM formulation is presented to solve two-dimensional problems characterized by C0 continuity with gradient jumps along discrete lines, such as those found in the thermal and structural analysis of heterogeneous materials or in line load problems in homogeneous media. The new enrichment functions presented in this paper allow solving problems with multiple intersecting discontinuity lines, such as those found at triple junctions in polycrystalline materials and in actively cooled microvascular materials with complex embedded networks. We show how the introduction of enrichment functions yields accurate finite element solutions with meshes that do not conform to the geometry of the discontinuity lines. The use of the proposed enrichments in both linear and quadratic approximations is investigated, as well as their combination with interface enrichment functions available in the literature. Through a detailed convergence study, we demonstrate that quadratic approximations do not require any correction to the method to recover optimal convergence rates and that they perform better than linear approximations for the same number of degrees of freedom in the solution of these types of problems. In the linear case, the effectiveness of correction functions proposed in the literature is also investigated. Copyright © 2009 John Wiley & Sons, Ltd.

A method for treating fluid-structure interaction of fracturing structures under impulsive loads is described. The coupling method is simple and does not require any modifications when the structure fails and allows fluid to flow through openings between crack surfaces. Both the fluid and the structure are treated by meshfree methods. For the structure, a Kirchhoff-Love shell theory is adopted and the cracks are treated by introducing either discrete (cracking particle method) or continuous (partition of unity-based method) discontinuities into the approximation. Coupling is realized by a master-slave scheme where the structure is slave to the fluid. The method is aimed at problems with high-pressure and low-velocity fluids, and is illustrated by the simulation of three problems involving fracturing cylindrical shells coupled with fluids.

A method for coarse graining of microcrack growth to the macroscale through the multiscale aggregating discontinuity (MAD) method is further developed. Three new features are: (1) methods for treating nucleating cracks, (2) the linking of the micro unit cell with the macroelement by the hourglass mode, and (3) methods for recovering macrocracks with variable crack opening. Unlike in the original MAD method, ellipticity is not retained at the macroscale in the bulk material, but we show that the element stiffness of the bulk material is positive definite. Several examples with comparisons with direct numerical simulations are given to demonstrate the effectiveness of the method. Copyright © 2009 John Wiley & Sons, Ltd.

This contribution presents a hierarchical variational multiscale framework to model propagating discontinuities at finite strains. Thereby the deformation map is decomposed into coarse-scale and fine-scale displacements, which results in a decoupled system of coarse-scale and fine-scale equations. Both are solved numerically by means of the finite element method whereby crack propagation is taken into account at the fine scale. Growing cracks are numerically handled by the introduction of discontinuous elements. A locality assumption on the fine-scale solution and an adaptive scheme to resize the fine-scale domain are introduced and demonstrated to increase the efficiency of the method. Copyright © 2009 John Wiley & Sons, Ltd.

We extend the finite cover method (FCM), which is known as a generalized finite element method, to crack simulations for quasi-brittle heterogeneous solids by using only a regular structured mathematical mesh. The proposed version of the FCM is powered by a method of modeling arbitrary numbers of physical cover layers. This ‘multi-cover layer modeling’ relies crucially on not only the introduction of multiple layers of physical covers, each of which is associated with a single physical domain, but also the identification of the values of level-set functions for discontinuities, which can be evaluated from their digital image information. Several numerical examples are presented to demonstrate the successful mesh-based mesh-free analyses of heterogeneous solids with arbitrarily evolving cracks and interfacial debonding thanks to the introduction of multiple cover layers. Copyright © 2009 John Wiley & Sons, Ltd.

This paper describes the application of the so-called hybridizable discontinuous Galerkin (HDG) method to linear elasticity problems. The method has three significant features. The first is that the only globally coupled degrees of freedom are those of an approximation of the displacement defined solely on the faces of the elements. The corresponding stiffness matrix is symmetric, positive definite, and possesses a block-wise sparse structure that allows for a very efficient implementation of the method. The second feature is that, when polynomials of degree k are used to approximate the displacement and the stress, both variables converge with the optimal order of k+1 for any k⩾0. The third feature is that, by using an element-by-element post-processing, a new approximate displacement can be obtained that converges at the order of k+2, whenever k⩾2. Numerical experiments are provided to compare the performance of the HDG method with that of the continuous Galerkin (CG) method for problems with smooth solutions, and to assess its performance in situations where the CG method is not adequate, that is, when the material is nearly incompressible and when there is a crack. Copyright © 2009 John Wiley & Sons, Ltd.

Discontinuous Galerkin (DG) methods provide a means of weakly enforcing the continuity of the unknown-field derivatives and have particular appeal in problems involving high-order derivatives. This feature has previously been successfully exploited (Comput. Methods Appl. Mech. Eng. 2008; 197:2901–2929) to develop a formulation of linear Kirchhoff–Love shells considering only the membrane and bending responses. In this proposed one-field method—the displacements are the only unknowns, while the displacement field is continuous, the continuity in the displacement derivative between two elements is weakly enforced by recourse to a DG formulation. It is the purpose of the present paper to extend this formulation to finite deformations and non-linear elastic behaviors. While the initial linear formulation was relying on the direct linear computation of the effective membrane stress and effective bending couple-stress from the displacement field at the mid-surface of the shell, the non-linear formulation considered implies the evaluation of the general stress tensor across the shell thickness, leading to a reformulation of the internal forces of the shell. Nevertheless, since the interface terms resulting from the discontinuous Galerkin method involve only the resultant couple-stress at the edges of the shells, the extension to non-linear deformations is straightforward. Copyright © 2008 John Wiley & Sons, Ltd.

This paper presents the particle discretization scheme (PDS) to analyze brittle failure of solids. The scheme uses characteristic functions of Voronoi and Delaunay tessellations to discretize a function and its derivatives, respectively. A discretized function has numerous discontinuities so that these discontinuities are utilized as a candidate of crack path segment in modeling propagating cracks, without making any extra computation to accommodate new displacement discontinuities. When the scheme is implemented to a finite element method (FEM), the resulting stiffness matrix coincides with the one that is obtained by using linear elements. The accuracy of computing a stress intensity factor at a crack tip is examined. It is shown that the accuracy is better than that of a FEM with linear elements when the rotational degree of freedom is included in discretizing displacement functions. Three three-dimensional growing crack problems are solved by means of the PDS and the results are presented. Copyright © 2009 John Wiley & Sons, Ltd.

Explicit schemes are known to provide less numerical diffusion in solving the advection–diffusion equation, especially for advection-dominated problems. Traditional explicit schemes use fixed time steps restricted by the global CFL condition in order to guarantee stability. This is known to slow down the computation especially for heterogeneous domains and/or unstructured meshes. To avoid this problem, local time stepping procedures where the time step is allowed to vary spatially in order to satisfy a local CFL condition have been developed. In this paper, a local time stepping approach is used with a numerical model based on discontinuous Galerkin/mixed finite element methods to solve the advection–diffusion equation. The developments are detailed for general unstructured triangular meshes. Numerical experiments are performed to show the efficiency of the numerical model for the simulation of (i) the transport of a solute on highly unstructured meshes and (ii) density-driven flow, where the velocity field changes at each time step. The model gives stable results with significant reduction of the computational cost especially for the non-linear problem. Moreover, numerical diffusion is also reduced for highly advective problems. Copyright © 2009 John Wiley & Sons, Ltd.

Methods for treating fracture in composite material by the extended finite element method with meshes that are independent of matrix/fiber interfaces and crack morphology are described. All discontinuities and near-tip enrichments are modeled using the framework of local partition of unity. Level sets are used to describe the geometry of the interfaces and cracks so that no explicit representation of either the cracks or the material interfaces are needed. Both full 12 function enrichments and approximate enrichments for bimaterial crack tips are employed. A technique to correct the approximation in blending elements is used to improve the accuracy. Several numerical results for both two-dimensional and three-dimensional examples illustrate the versatility of the technique. The results clearly demonstrate that interface enrichment is sufficient to model the correct mechanics of an interface crack. Copyright © 2008 John Wiley & Sons, Ltd.

Traditional smeared orthotropic models display an unacceptable dependence of the solution on the alignment of the mesh, which usually manifests as stress locking. A solution for this drawback is proposed in this paper by adopting the concept of embedded inelastic strains, rather than displacement jumps, and by linking the structure of the inelastic strain to the geometry of the cracked element. The resulting model, applicable to linear 3-noded triangles, is formulated as a non-symmetric orthotropic local damage constitutive model, with the softening modulus regularized according to the material fracture energy and the element size. Analytical and numerical results show that this approach is effective in removing the locking problem as well as efficient from the computational point of view.

A recent approach to fracture modeling has combined the extended finite element method (XFEM) with cohesive zone models. Most studies have used simplified enrichment functions to represent the strong discontinuity but have lacked an analytical basis to represent the displacement gradients in the vicinity of the cohesive crack. In this study enrichment functions based upon an existing analytical investigation of the cohesive crack problem are proposed. These functions have the potential of representing displacement gradients in the vicinity of the cohesive crack and allow the crack to incrementally advance across each element. Key aspects of the corresponding numerical formulation and enrichment functions are discussed. A parameter study for a simple mode I model problem is presented to evaluate if quasi-static crack propagation can be accurately followed with the proposed formulation. The effects of mesh refinement and mesh orientation are considered. Propagation of the cohesive zone tip and crack tip, time variation of the cohesive zone length, and crack profiles are examined. The analysis results indicate that the analytically based enrichment functions can accurately track the cohesive crack propagation of a mode I crack independent of mesh orientation. A mixed mode example further demonstrates the potential of the formulation. Copyright © 2008 John Wiley & Sons, Ltd.

An evanescent wave occurs when a propagating incident wave impinges on an interface between two fluid, solid, or fluid–solid media at a subcritical angle. Mathematical properties of such a wave make it difficult to capture with standard finite element discretization schemes. For this reason, the discontinuous enrichment method (DEM) developed in (Comput. Methods Appl. Mech. Eng. 2001; 190:6455–6479; Comput. Methods Appl. Mech. Eng. 2003; 192:1389–1419; Comput. Methods Appl. Mech. Eng. 2003; 192:3195–3210; Int. J. Numer. Meth. Engng 2004; 61:1938–1956; Wave Motion 2004; 39(4):307–317; Int. J. Numer. Meth. Engng 2006; 66:2086–2114; Int. J. Numer. Meth. Engng 2006; 66:796–815) is extended here to the solution of a class of three-dimensional evanescent wave problems in the frequency domain. To this effect, new DEM elements for three-dimensional elastodynamic problems are first proposed. Then, these and other DEM elements previously developed for the efficient solution of the Helmholtz problem are further enriched with free-space solutions of model evanescent wave problems, in order to achieve high accuracy at practical mesh resolution for fluid–fluid and fluid–solid applications. The performance of the extended DEM elements is reported to be better than that of its basic Helmholtz and Navier counterparts and superior to that achieved by the classical high-order polynomial finite element method. Copyright © 2008 John Wiley & Sons, Ltd.

In this paper, we develop a discontinuous Galerkin method on triangular meshes to solve the reactive dynamic user equilibrium model for pedestrian flows. The pedestrian density in this model is governed by the conservation law in which the flow flux is implicitly dependent on the density through the Eikonal equation. To solve the Eikonal equation efficiently at each time level, we use the fast sweeping method. Two numerical examples are then used to demonstrate the effectiveness of the algorithm. Copyright © 2008 John Wiley & Sons, Ltd.

The extended finite element method (XFEM) enables local enrichments of approximation spaces. Standard finite elements are used in the major part of the domain and enriched elements are employed where special solution properties such as discontinuities and singularities shall be captured. In elements that blend the enriched areas with the rest of the domain problems arise in general. These blending elements often require a special treatment in order to avoid a decrease in the overall convergence rate. A modification of the XFEM approximation is proposed in this work. The enrichment functions are modified such that they are zero in the standard elements, unchanged in the elements with all their nodes being enriched, and varying continuously in the blending elements. All nodes in the blending elements are enriched. The modified enrichment function can be reproduced exactly everywhere in the domain and no problems arise in the blending elements. The corrected XFEM is applied to problems in linear elasticity and optimal convergence rates are achieved. Copyright © 2007 John Wiley & Sons, Ltd.

In this work, we consider the local discontinuous Galerkin (LDG) method applied to second-order elliptic problems arising in the modeling of single-phase flows in porous media. It has been recently proven that the spectral condition number of the stiffness matrix exhibits an asymptotic behavior of O(h -2 ) on structured and unstructured meshes, where h is the mesh size. Thus, efficient preconditioners are mandatory. We present a semi-algebraic multilevel preconditioner for the LDG method using local Lagrange-type interpolatory basis functions. We show, numerically, that its performance does not degrade, or at least the number of iterations increases very slowly, as the number of unknowns augments. The preconditioner is tested on problems with high jumps in the coefficients, which is the typical scenario of problems arising in porous media.

This paper presents a computational technique based on the extended finite element method (XFEM) and the level set method for the growth of biofilms. The discontinuous-derivative enrichment of the standard finite element approximation eliminates the need for the finite element mesh to coincide with the biofilm–fluid interface and also permits the introduction of the discontinuity in the normal derivative of the substrate concentration field at the biofilm–fluid interface. The XFEM is coupled with a comprehensive level set update scheme with velocity extensions, which makes updating the biofilm interface fast and accurate without need for remeshing. The kinetics of biofilms are briefly given and the non-linear strong and weak forms are presented. The non-linear system of equations is solved using a Newton–Raphson scheme. Example problems including 1D and 2D biofilm growth are presented to illustrate the accuracy and utility of the method. The 1D results we obtain are in excellent agreement with previous 1D results obtained using finite difference methods. Our 2D results that simulate finger formation and finger-tip splitting in biofilms illustrate the robustness of the present computational technique. Copyright © 2007 John Wiley & Sons, Ltd.