A contact algorithm for frictional crack propagation with the extended finite element method

We present an incremental quasi‐static contact algorithm for path‐dependent frictional crack propagation in the framework of the extended finite element (FE) method. The discrete formulation allows for the modeling of frictional contact independent of the FE mesh. Standard Coulomb plasticity model is introduced to model the frictional contact on the surface of discontinuity. The contact constraint is borrowed from non‐linear contact mechanics and embedded within a localized element by penalty method. Newton–Raphson iteration with consistent linearization is used to advance the solution. We show the superior convergence performance of the proposed iterative method compared with a previously published algorithm called ‘LATIN’ for frictional crack propagation. Numerical examples include simulation of crack initiation and propagation in 2D plane strain with and without bulk plasticity. In the presence of bulk plasticity, the problem is also solved using an augmented Lagrangian procedure to demonstrate the efficacy and adequacy of the standard penalty solution. Copyright © 2008 John Wiley & Sons, Ltd.     

Three‐dimensional finite elements with embedded strong discontinuities to model material failure in the infinitesimal range

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.     

A new method for modelling cohesive cracks using finite elements

A model which allows the introduction of displacements jumps to conventional finite elements is developed. The path of the discontinuity is completely independent of the mesh structure. Unlike so‐called ‘embedded discontinuity’ models, which are based on incompatible strain modes, there is no restriction on the type of underlying solid finite element that can be used and displacement jumps are continuous across element boundaries. Using finite element shape functions as partitions of unity, the displacement jump across a crack is represented by extra degrees of freedom at existing nodes. To model fracture in quasi‐brittle heterogeneous materials, a cohesive crack model is used. Numerical simulations illustrate the ability of the method to objectively simulate fracture with unstructured meshes. Copyright © 2001 John Wiley & Sons, Ltd.     

Stress‐hybrid quadrilateral finite element with embedded strong discontinuity for failure analysis of plane stress solids

A formulation of a quadrilateral finite element with embedded strong discontinuity, suitable for the material failure numerical analysis of plane stress solids, is presented. The kinematics of standard finite element is enhanced by displacement jumps that vary linearly along the embedded discontinuity line. They are described by four kinematic parameters that are related to four element separation modes. The modes are designed for no stress transfer over the discontinuity line at its fully softened (opened) state. As for the material, the bulk of the element is assumed to be elastic, and the softening plasticity, in terms of discontinuity tractions and displacement jumps, is assumed along the discontinuity line. The bulk stresses are described by the optimal five‐parameter interpolation. The combination of stress interpolation and enhanced kinematics yields simple form of the element stiffness matrix. To achieve efficient implementation, the stiffness matrix is statically condensed for both the enhanced kinematic parameters and the stress parameters. In a set of numerical examples, the performance of the derived element is illustrated. Obtained results are compared with some other representative embedded discontinuity quadrilateral elements (displacement‐based and enhanced assumed strain based). It turns out that the element performs very well. Copyright © 2013 John Wiley & Sons, Ltd.     

Autonomous decentralized finite element method and its applications

In this paper, a new solution procedure using the finite element technique in order to solve problems of structure analysis is proposed. This procedure is called the autonomous decentralized finite element method because it is based on the characteristic autonomy and decentralization in life or biological systems (life‐like approach). The fundamental approach is developed according to an idea of cellular automata manipulation by the new neighbourhood model. The finite element method with an algorithm of the relaxation method is adopted as the solution procedure in this approach. The proposed procedure demonstrates that it is a powerful means of numerical analysis for many kinds of structural problems that are structural morphogenesis, structural optimization and structural inverse problems. Our procedure is applied to numerical analysis of three simple plane models: (1) The structural shape analysis problem for the prescribed displacement mode of a truss structure, (2) An adaptive structure remodelling problem on an elastic continuum, (3) An identification problem of thermal conductivity on a continuum. The effectiveness and validity of our idea are shown from their numerical results. Copyright © 2003 John Wiley & Sons, Ltd.     

Strong tangential discontinuity modeling of shear bands using the extended finite element method

A method is developed for modeling of shear band with strong tangential discontinuity by means of cohesive surfaces within the extended finite element method (XFEM). A rate-independent non-associated plasticity model is incorporated along the strong discontinuity to consider the highly localized regions. Once the localization is occurred, tangential enrichment degrees of freedom are added to the localized elements, and the discontinuity is captured regardless of mesh resolution and alignment. By introducing the tangential enrichment function, the discontinuity is only imposed in the tangential direction, while the continuity across the shear band is automatically fulfilled. Adopting bilinear quadrilateral elements within the context of XFEM allows for the plastic deformation of shear band to be obtained with quadratic distribution within an enriched element. Since the strong discontinuity approach is employed, the singularity of strain field at the position of displacement jump is attained through a Dirac delta distribution. By means of this singularity, the cohesive shear traction is derived for the J2 plasticity model and is applied onto the band interfaces in order to reproduce the dissipative mechanism of the band. Several numerical examples are analyzed to assess the accuracy and robustness of the proposed approach.     

An augmented Lagrangian quasi-Newton solver for constrained nonlinear finite element applications

A solution scheme is presented for constrained non-linear equations of evolution that result, for example, from the finite element discretization of mechanical contact problems. The algorithm discussed utilizes a quasi-Newton non-linear equation solving strategy, with constraints enforced by an augmented Lagrangian iteration procedure. Through presentation of a simple model problem and its generalization, it is shown that the iterations associated with both the quasi-Newton algorithm and the augmentation procedure can be interwoven to produce a highly efficient and robust solution strategy.     

On the strong discontinuity approach in finite deformation settings

Taking the strong discontinuity approach as a framework for modelling displacement discontinuities and strain localization phenomena, this work extends previous results in infinitesimal strain settings to finite deformation scenarios. By means of the strong discontinuity analysis, and taking isotropic damage models as target continuum (stress–strain) constitutive equation, projected discrete (tractions–displacement jumps) constitutive models are derived, together with the strong discontinuity conditions that restrict the stress states at the discontinuous regime. A variable bandwidth model, to automatically induce those strong discontinuity conditions, and a discontinuous bifurcation procedure, to determine the initiation and propagation of the discontinuity, are briefly sketched. The large strain counterpart of a non‐symmetric finite element with embedded discontinuities, frequently considered in the strong discontinuity approach for infinitesimal strains, is then presented. Finally, some numerical experiments display the theoretical issues, and emphasize the role of the large strain kinematics in the obtained results. Copyright © 2003 John Wiley & Sons, Ltd.     

An immersed finite element method with integral equation correction

We propose a robust immersed finite element method in which an integral equation formulation is used to enforce essential boundary conditions. The solution of a boundary value problem is expressed as the superposition of a finite element solution and an integral equation solution. For computing the finite element solution, the physical domain is embedded into a slightly larger Cartesian (box‐shaped) domain and is discretized using a block‐structured mesh. The defect in the essential boundary conditions, which occurs along the physical domain boundaries, is subsequently corrected with an integral equation method. In order to facilitate the mapping between the finite element and integral equation solutions, the physical domain boundary is represented with a signed distance function on the block‐structured mesh. As a result, only a boundary mesh of the physical domain is necessary and no domain mesh needs to be generated, except for the non‐boundary‐conforming block‐structured mesh. The overall approach is first presented for the Poisson equation and then generalized to incompressible viscous flow equations. As an example of fluid–structure coupling, the settling of a heavy rigid particle in a closed tank is considered. Copyright © 2010 John Wiley & Sons, Ltd.     

An orthogonal residual procedure for non-linear finite element equations

A general and robust solution procedure for non-linear finite element equations with limit points is developed. At each equilibrium iteration the magnitude of the load is adjusted such that the residual force is orthogonal to the current displacement increment from the last equilibrium state. The method implements the physical condition that the orthogonal residual force will neither increase nor decrease the magnitude of the current displacement increment vector. The orthogonality condition is formulated directly in terms of conjugate variables and therefore does not contain any scaling parameters. Passage of load and displacement limit points is discussed as well as the relation to line search, minimum residual, and are-length methods. The method is illustrated by two examples.     

Penalty-function iterative procedures for mixed finite element formulations

Iterative methods for solving mixed finite element equations that correct displacement and stress unknowns in ‘staggered’ fashion are attracting increased attention. This paper looks at the problem from the standpoint of allowing fairly arbitrary approximations to be made on both the stiffness and compliance matrices used in solving for the corrections. The resulting iterative processes usually diverge unless stabilized with Courant penalty terms. An iterative procedure previously constructed for equality-constrained displacement models is recast to fit the mixed finite element formulation in which displacements play the role of Lagrange multipliers. The penalty function iteration is shown to reduce to an ordinary staggered stress-displacement iteration if the weight is set to zero. Convergence conditions for these procedures are stated and the potentially troublesome effect of prestress modes noted.     

The eXtended finite element method for cracked hyperelastic materials: A convergence study

The present work aims to look into the contribution of the extended finite element method for large deformation of cracked bodies in plane strain approximation. The unavailability of sufficient mathematical tools and proofs for such problem makes the study exploratory. First, the asymptotic solution is presented. Then, a numerical analysis is realized to verify the pertinence of solution given by the asymptotic procedure, because it serves as an eXtended finite element method enrichment basis. Finally, a convergence study is carried out to show the contribution of the exploitation of such method. Copyright © 2014 John Wiley & Sons, Ltd.     

Failure analysis of natural gas buried X65 steel pipeline under deflection load using finite element method

A 3D parametric finite element model of the pipeline and soil is established using finite element method to perform the failure analysis of natural gas buried X65 steel pipeline under deflection load. The pipeline is assumed to be loaded in a parabolic deflection displacement along the axial direction. Based on the true stress–strain constitutive relationship of X65 steel, the elastic–plastic finite element analysis employs the arc-length algorithm and non-linear stabilization algorithm respectively to simulate the strain softening properties of pipeline after plastic collapse. Besides, effects of the soil types and model sizes on the maximum deflection displacement of pipeline are investigated. The proposed finite element method serves as a base available for the safety design and evaluation as well as engineering acceptance criterion for the failure of pipeline due to deflection.     

Numerical study on deformations in a cracked viscoelastic body with the extended finite element method

Deformations such as crack opening and sliding displacements in a cracked viscoelastic body are numerically investigated by the extended finite element method (XFEM). The solution is carried out directly in time domain with a mesh not conforming to the crack geometry. The generalized Heaviside function is used to reflect the displacement discontinuity across a crack surface while the basis functions extracted from the viscoelastic asymptotic fields are used to manifest the gradient singularity at a crack tip. With these treatments, the XFEM formulations are derived in an incremental form. In evaluating the stiffness matrix, a selective integration scheme is suggested for problems with high Poisson ratios often encountered in viscoelastic materials over different element types in the XFEM. Numerical examples show that the crack opening displacement and crack sliding displacement are satisfactory.     

Strong discontinuities in antiplane/torsional problems of computational failure mechanics

This paper considers the analysis of localized failures and fracture of solids under antiplane conditions. We consider the longitudinal cracking of shafts in torsion, with the crack propagating through the cross section, besides pure antiplane problems (that is, with loading perpendicular to the plane of analysis). The main goal is the theoretical characterization and the numerical resolution of strong discontinuities in this setting, that is, discontinuities of the antiplane displacement field modeling the cracks. A multi-scale framework is considered, by which the discontinuities are treated locally in the (global) antiplane mechanical boundary value problem of interest, incorporating effectively the contributions of the discontinuities to the failure of the solid. We can identify among these contributions, besides the change in the stiffness of the solid or structural member, the localized energy dissipation associated to the cohesive law governing the physical response of the discontinuity surfaces. A main outcome of this approach is the development of new finite elements with embedded discontinuities for the antiplane problem that capture these solutions, and physical effects, locally at the element level. This local structure allows the static condensation at the element level of the degrees of freedom considered in the approximation of the antiplane displacement jumps along the discontinuity. In this way, the new elements result not only in a cost efficient computational tool of analysis for these problems, but also in a technique that can be easily incorporated in an existing finite element code, while resolving objectively those physical dissipative effects along the localized surfaces of failure. We develop, in particular, quadrilateral finite elements with the embedded discontinuities exhibiting constant and linear approximations of the displacement jumps, showing the superior performance of the latter given the stress locking associated with quadrilateral elements with constant jumps only. This limitation manifests itself in spurious transfers of stresses across the discontinuities, leading to severe oscillations in the stress field and an overall excessively stiff solution of the problem. These features are illustrated with several numerical examples, including convergence tests and validations with analytical results existing in the literature, showing in the process the treatment of characteristic situations like snap-backs, commonly encountered in the modeling of these structural members at failure.     

A generalized finite element method with global-local enrichment functions for confined plasticity problems

The main feature of partition of unity methods such as the generalized or extended finite element method is their ability of utilizing a priori knowledge about the solution of a problem in the form of enrichment functions. However, analytical derivation of enrichment functions with good approximation properties is mostly limited to two-dimensional linear problems. This paper presents a procedure to numerically generate proper enrichment functions for three-dimensional problems with confined plasticity where plastic evolution is gradual. This procedure involves the solution of boundary value problems around local regions exhibiting nonlinear behavior and the enrichment of the global solution space with the local solutions through the partition of unity method framework. This approach can produce accurate nonlinear solutions with a reduced computational cost compared to standard finite element methods since computationally intensive nonlinear iterations can be performed on coarse global meshes after the creation of enrichment functions properly describing localized nonlinear behavior. Several three-dimensional nonlinear problems based on the rate-independent J ₂ plasticity theory with isotropic hardening are solved using the proposed procedure to demonstrate its robustness, accuracy and computational efficiency.     

Modelling of muscle behaviour by the finite element method using Hill's three-element model

We present a numerical algorithm for the determination of muscle response by the finite element method. Hill's three-element model is used as a basis for our analysis. The model consists of one linear elastic element, coupled in parallel with one non-linear elastic element, and one non-linear contractile element connected in series. An activation function is defined for the model in order to describe a time-dependent character of the contractile element with respect to stimulation. Complex mechanical response of muscle, accounting for non-linear force–displacement relation and change of geometrical shape, is possible by the finite element method. In an incremental-iterative scheme of calculation of equilibrium configurations of a muscle, the key step is determination of stresses corresponding to a strain increment. We present here the stress calculation for Hill's model which is reduced to the solution of one non-linear equation with respect to the stretch increment of the serial elastic element. The muscle fibers can be arbitrarily oriented in space and we give a corresponding computational procedure of calculation of nodal forces and stiffness of finite elements. The proposed computational scheme is built in our FE package PAK, so that real muscles of complex three-dimensional shapes can be modelled. In numerical examples we illustrate the main characteristic of the developed numerical model and the possibilities of solution of real problems in muscle functioning. © 1998 John Wiley & Sons, Ltd.     

A simple finite element for beams on elastic foundations

A simple "routine" beam on elastic foundation finite element using a polynomial displacement function has been developed which yields acceptably accurate deflection, shear and bending moment values for prismatic or non-prismatic beams of elastic material resting on foundations with varying or nonlinear subgrade reactions. Limited extension of the formulation to an "exact" finite element using the exact displacement function of a beam on elastic foundation has also been carried out. The subgrade is represented by a non-homogeneous solid medium to include nonlinear parameters if required. The iterative solution is extended to cases where the beam may uplift because the foundation is a no tension material. The model is also suitable for calculating the elastic deflections, membrane. and bending stress resultants for axisymmetrically loaded variable thickness shells of revolution. A computer program called FEBEF [finite element: beam on elastic foundation] incorporating the routine finite element has been prepared for the solution of beams on elastic foundations and axi symmetrically loaded shells of revolution.     

A geometric nonlinear triangular finite element with an embedded discontinuity for the simulation of quasi-brittle fracture

This paper is aimed to model the appearance and evolution of discrete cracks in quasi-brittle materials using triangular finite elements with an embedded interface in a geometric nonlinear setting. The kinematics for the discontinuous displacement field is presented and the standard variational formulation with respect to the reference configuration is extended to a body with an internal discontinuity. Special attention is paid to the algorithmic treatment. The discontinuity is modeled by additional global degrees of freedom and the continuity of the displacements across the element boundaries is enforced. Finally, representative numerical examples for mode-I and mixed-mode fracture, namely a tension test, different three-point bending tests and a single edge notched beam with structured and unstructured finite element meshes are discussed to study the evolving crack pattern and to show the ability of the model.     

Linear smoothed extended finite element method

The extended finite element method 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. 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 (XFEM), for example, makes numerical integration elegant and simple by transforming volume integrals into surface integrals. However, it was reported in the literature that the strain smoothing is inaccurate when non‐polynomial functions are in the basis. In this paper, we investigate the benefits of a recently developed Linear smoothing procedure which provides better approximation to higher‐order polynomial fields in the basis. Some benchmark problems in the context of linear elastic fracture mechanics are solved and the results are compared with existing approaches. We observe that the stress intensity factors computed through the proposed linear smoothed XFEM is more accurate than that obtained through smoothed XFEM.