On advanced solution strategies to overcome locking effects in strong discontinuity approaches

This paper is concerned with the analysis of locking effects resulting from different orientations of micro‐defects and those of the corresponding macro‐defects. Based on a mixed‐mode material model embedded within the framework of the strong discontinuity approach (SDA), the described locking effect is illustrated by means of a crack analysis of a notched concrete beam. To overcome the deficiency of the proposed finite element model, the original SDA is modified and extended. For that purpose, two different advanced numerical formulations are developed: a rotating crack approach and a multiple crack approach. Restricting the governing equations to the material point level, a standard return‐mapping procedure is applied to the algorithmic formulations of both models. The applicability and the performance of the proposed numerical implementations are investigated by means of a re‐analysis of the two‐dimensional notched concrete beam. Copyright © 2005 John Wiley & Sons, Ltd.     

A 3D anisotropic elastoplastic‐damage model using discontinuous displacement fields

This paper is concerned with the development of constitutive equations for finite element formulations based on discontinuous displacement fields. For this purpose, an elastoplastic continuum model (stress–strain relation) as well as an anisotropic damage model (stress–strain relation) are projected onto a surface leading to traction separation laws. The coupling of both continuum models and, subsequently, the derivation of the corresponding constitutive interface law are described in detail. For a simple calibration of the proposed model, the fracture energy resulting from the coupled elastoplastic‐damage traction separation law is computed. By this, the softening evolution is linearly dependent on the fracture energy. The second part of the present paper deals with the numerical implementation. Based on a local and incompatible additive split of the displacement field into a continuous and a discontinuous part, the parameters specifying the jump of the displacement field are condensed out at the material level without employing the standard static condensation technique. To reduce locking effects, a rotating localization zone formulation is applied. The applicability and the performance of the proposed numerical implementation is investigated by means of a re‐analysis of a two‐dimensional L‐shaped slab as well as by means of a three‐dimensional ultimate load analysis of a steel anchor embedded in a concrete block. Copyright © 2004 John Wiley & Sons, Ltd.     

Simulating the propagation of displacement discontinuities in a regularized strain‐softening medium

A numerical model is developed which allows the inclusion of displacement discontinuities in a strain‐softening medium, independent of the finite element mesh structure. Inelastic deformations develop in the continuum and, when a critical threshold of inelastic deformation is reached, a displacement discontinuity is inserted. Discontinuities are introduced using the partition of unity concept which allows discontinuous functions to be added to the standard finite element basis. It is shown that the introduction of displacement discontinuities at the later stages of the failure process can lead to a failure mode that is fundamentally different than that using a continuum model only. This combined continuum‐discontinuous model is better able to describe the entire failure process than a continuum or a discrete model alone and treats mode‐I and mode‐II failure in a unified fashion. Copyright © 2001 John Wiley & Sons, Ltd.     

Arbitrary discontinuities in space–time finite elements by level sets and X‐FEM

An enriched finite element method with arbitrary discontinuities in space–time is presented. The discontinuities are treated by the extended finite element method (X‐FEM), which uses a local partition of unity enrichment to introduce discontinuities along a moving hyper‐surface which is described by level sets. A space–time weak form for conservation laws is developed where the Rankine–Hugoniot jump conditions are natural conditions of the weak form. The method is illustrated in the solution of first order hyperbolic equations and applied to linear first order wave and non‐linear Burgers' equations. By capturing the discontinuity in time as well as space, results are improved over capturing the discontinuity in space alone and the method is remarkably accurate. Implications to standard semi‐discretization X‐FEM formulations are also discussed. Copyright © 2004 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.     

Mixed finite element method for saturated poroelastoplastic media at large strains

A mixed finite element for hydro‐dynamic analysis in saturated porous media in the frame of the Biot theory is proposed. Displacements, effective stresses, strains for the solid phase and pressure, pressure gradients, and Darcy velocities for the fluid phase are interpolated as independent variables. The weak form of the governing equations of coupled hydro‐dynamic problems in saturated porous media within the element are given on the basis of the Hu–Washizu three‐field variational principle. In light of the stabilized one point quadrature super‐convergent element developed in solid continuum, the interpolation approximation modes for the primary unknowns and their spatial derivatives of the solid and the fluid phases within the element are assumed independently. The proposed mixed finite element formulation is derived. The non‐linear version of the element formulation is further derived with particular consideration of pressure‐dependent non‐associated plasticity. The return mapping algorithm for the integration of the rate constitutive equation, the consistent elastoplastic tangent modulus matrix and the element tangent stiffness matrix are developed. For geometrical non‐linearity, the co‐rotational formulation approach is used. Numerical results demonstrate the capability and the performance of the proposed element in modelling progressive failure characterized by strain localization due to strain softening in poroelastoplastic media subjected to dynamic loading at large strain. Copyright © 2003 John Wiley & Sons, Ltd.     

Response sensitivity of material and geometric nonlinear force‐based Timoshenko frame elements

Response sensitivity is an essential component to understanding the complexity of material and geometric nonlinear finite element formulations of structural response. The direct differentiation method (DDM), a versatile approach to computing response sensitivity, requires differentiation of the equations that govern the state determination of an element and it produces accurate and efficient results. The DDM is applied to a force‐based element formulation that utilizes curvature‐shear‐based displacement interpolation (CSBDI) in its state determination for material and geometric nonlinearity in the basic system of the element. The response sensitivity equations are verified against finite difference computations, and a detailed example shows the effect of parameters that control flexure–shear interaction for a stress resultant plasticity model. The developed equations make the CSBDI force‐based element available for gradient‐based applications such as reliability and optimization where efficient computation of response sensitivities is necessary for convergence of gradient‐based search algorithms. Copyright © 2016 John Wiley & Sons, Ltd.     

Numerical simulation of dynamic fracture using finite elements with embedded discontinuities

This paper presents the extension of some finite elements with embedded strong discontinuities to the fully transient range with the focus on dynamic fracture. Cracks and shear bands are modeled in this setting as discontinuities of the displacement field, the so-called strong discontinuities, propagating through the continuum. These discontinuities are embedded into the finite elements through the proper enhancement of the discrete strain field of the element. General elements, like displacement or assumed strain based elements, can be considered in this framework, capturing sharply the kinematics of the discontinuity for all these cases. The local character of the enhancement (local in the sense of defined at the element level, independently for each element) allows the static condensation of the different local parameters considered in the definition of the displacement jumps. All these features lead to an efficient formulation for the modeling of fracture in solids, very easily incorporated in an existing general finite element code due to its modularity. We investigate in this paper the use of this finite element formulation for the special challenges that the dynamic range leads to. Specifically, we consider the modeling of failure mode transitions in ductile materials and crack branching in brittle solids. To illustrate the performance of the proposed formulation, we present a series of numerical simulations of these cases with detailed comparisons with experimental and other numerical results reported in the literature. We conclude that these finite element methods handle well these dynamic problems, still maintaining the aforementioned features of computational efficiency and modularity.     

Frame‐indifferent beam finite elements based upon the geometrically exact beam theory

This paper describes a new beam finite element formulation based upon the geometrically exact beam theory. In contrast to many previously proposed beam finite element formulations the present discretization approach retains the frame‐indifference (or objectivity) of the underlying beam theory. The space interpolation of rotational degrees‐of‐freedom is circumvented by the introduction of a reparameterization of the weak form corresponding to the equations of motion of the geometrically exact beam theory. Copyright © 2002 John Wiley & Sons, Ltd.     

Iterative solvers for 3D linear and nonlinear elasticity problems: Displacement and mixed formulations

We present new iterative solvers for large‐scale linear algebraic systems arising from the finite element discretization of the elasticity equations. We focus on the numerical solution of 3D elasticity problems discretized by quadratic tetrahedral finite elements and we show that second‐order accuracy can be obtained at very small overcost with respect to first‐order (linear) elements. Different Krylov subspace methods are tested on various meshes including elements with small aspect ratio. We first construct a hierarchical preconditioner for the displacement formulation specifically designed for quadratic discretizations. We then develop efficient tools for preconditioning the 2 × 2 block symmetric indefinite linear system arising from mixed (displacement‐pressure) formulations. Finally, we present some numerical results to illustrate the potential of the proposed methods. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

A localized mixed‐hybrid method for imposing interfacial constraints in the extended finite element method (XFEM)

The paper proposes an approach for the imposition of constraints along moving or fixed immersed interfaces in the context of the extended finite element method. An enriched approximation space enables consistent representation of strong and weak discontinuities in the solution fields along arbitrarily‐shaped material interfaces using an unfitted background mesh. The use of Lagrange multipliers or penalty methods is circumvented by a localized mixed hybrid formulation of the model equations. In a defined region in the vicinity of the interface, the original problem is re‐stated in its auxiliary formulation. The availability of the auxiliary variable enables the consideration of a variety of interface constraints in the weak form. The contribution discusses the weak imposition of Dirichlet‐ and Neumann‐type interface conditions as well as continuity requirements not fulfilled a priori by the enriched approximation. The properties of the proposed approach applied to two‐dimensional linear scalar‐ and vector‐valued elliptic problems are investigated by studying the convergence behavior. Copyright © 2009 John Wiley & Sons, Ltd.     

A finite‐strain gradient‐inelastic beam theory and a corresponding force‐based frame element formulation

This paper extends the gradient‐inelastic (GI) beam theory, introduced by the authors to simulate material softening phenomena, to further account for geometric nonlinearities and formulates a corresponding force‐based (FB) frame element computational formulation. Geometric nonlinearities are considered via a rigorously derived finite‐strain beam formulation, which is shown to coincide with Reissner's geometrically nonlinear beam formulation. The resulting finite‐strain GI beam theory: (i) accounts for large strains and rotations, unlike the majority of geometrically nonlinear beam formulations used in structural modeling that consider small strains and moderate rotations; (ii) ensures spatial continuity and boundedness of the finite section strain field during material softening via the gradient nonlocality relations, eliminating strain singularities in beams with softening materials; and (iii) decouples the gradient nonlocality relations from the constitutive relations, allowing use of any material model. On the basis of the proposed finite‐strain GI beam theory, an exact FB frame element formulation is derived, which is particularly novel in that it: (a) expresses the compatibility relations in terms of total strains/displacements, as opposed to strain/displacement rates that introduce accumulated computational error during their numerical time integration, and (b) directly integrates the strain‐displacement equations via a composite two‐point integration method derived from a cubic Hermite interpolating polynomial to calculate the displacement field over the element length and, thus, address the coupling between equilibrium and strain‐displacement equations. This approach achieves high accuracy and mesh convergence rate and avoids polynomial interpolations of individual section fields, which often lead to instabilities with mesh refinements. The FB formulation is then integrated into a corotational framework and is used to study the response of structures, simultaneously accounting for geometric nonlinearities and material softening. The FB formulation is further extended to capture member buckling triggered by minor perturbations/imperfections of the initial member geometry.     

An efficient time‐domain perfectly matched layers formulation for elastodynamics on spherical domains

Many practical applications require the analysis of elastic wave propagation in a homogeneous isotropic media in an unbounded domain. One widely used approach for truncating the infinite domain is the so‐called method of perfectly matched layers (PMLs). Most existing PML formulations are developed for finite difference methods based on the first‐order velocity‐stress form of the elasticity equations, and they are not straight‐forward to implement using standard finite element methods (FEMs) on unstructured meshes. Some of the problems with these formulations include the application of boundary conditions in half‐space problems and in the treatment of edges and/or corners for time‐domain problems. Several PML formulations, which do work with FEMs have been proposed, although most of them still have some of these problems and/or they require a large number of auxiliary nodal history/memory variables. In this work, we develop a new PML formulation for time‐domain elastodynamics on a spherical domain, which reduces to a two‐dimensional formulation under the assumption of axisymmetry. Our formulation is well‐suited for implementation using FEMs, where it requires lower memory than existing formulations, and it allows for natural application of boundary conditions. We solve example problems on two‐dimensional and three‐dimensional domains using a high‐order discontinuous Galerkin (DG) discretization on unstructured meshes and explicit time‐stepping. We also study an approach for stabilization of the discrete equations, and we show several practical applications for quality factor predictions of micromechanical resonators along with verifying the accuracy and versatility of our formulation. Copyright © 2014 John Wiley & Sons, Ltd.     

A multiscale finite element method for modeling fully coupled thermomechanical problems in solids

This article proposes a two‐scale formulation of fully coupled continuum thermomechanics using the finite element method at both scales. A monolithic approach is adopted in the solution of the momentum and energy equations. An efficient implementation of the resulting algorithm is derived that is suitable for multicore processing. The proposed method is applied with success to a strongly coupled problem involving shape‐memory alloys.Copyright © 2012 John Wiley & Sons, Ltd.     

Effective 3D failure simulations by combining the advantages of embedded Strong Discontinuity Approaches and classical interface elements

An efficient numerical framework suitable for three-dimensional analyses of brittle material failure is presented. The proposed model is based on an (embedded) Strong Discontinuity Approach (SDA). Hence, the deformation mapping is elementwise additively decomposed into a conforming, continuous part and an enhanced part associated with the kinematics induced by material failure. To overcome locking effects and to provide a continuous crack path approximation, the approach is extended and combined with advantages known from classical interface elements. More precisely, several discontinuities each of them being parallel to a facet of the respective finite element are considered. By doing so, crack path continuity is automatically fulfilled and no tracking algorithm is necessary. However, though this idea is similar to that of interface elements, the novel SDA is strictly local (finite element level) and thus, it does not require any modification of the global data structure, e.g. no duplication of nodes. An additional positive feature of the advocated finite element formulation is that it leads to a symmetric tangent matrix. It is shown that several simultaneously active discontinuities in each finite element are required to capture highly localized material failure. The performance and predictive ability of the model are demonstrated by means of two benchmark examples.     

A mixed finite element for the nonlinear analysis of in-plane loaded masonry walls

A mixed membrane eight-node quadrilateral finite element for the analysis of masonry walls is presented. Assuming that a nonlinear and history-dependent 2D stress-strain constitutive law is used to model masonry material, the element derivation is based on a Hu-Washizu variational statement, involving displacement, strain, and stress fields as primary variables. As the behavior of masonry structures is often characterized by strain localization phenomena, due to strain softening at material level, a discontinuous, piecewise constant interpolation of the strain field is considered at element level, to capture highly nonlinear strain spatial distributions also within finite elements. Newton's method of solution is adopted for the element state determination problem. For avoiding pathological sensitivity to the finite element mesh, a novel algorithm is proposed to perform an integral-type nonlocal regularization of the constitutive equations in the present mixed formulation. By the comparison with competing serendipity displacement-based formulation, numerical simulations prove high performances of the proposed finite element, especially when coarse meshes are adopted.     

基于参变量变分原理的三维Cosserat体模型弹塑性分析与应变局部化模拟

基于参变量变分原理,该文发展了三维Cosserat连续体模型弹塑性有限元分析的二次规划算法。由于Cosserat连续体模型的本构方程中存在材料内尺度参数,该模型可以解决经典连续介质理论在分析应变软化问题时病态的有限元网格依赖性问题。数值结果表明所发展的三维Cosserat连续介质弹塑性有限元模型可以有效的模拟应变局部化现象并且该算法具有很好的数值稳定性,同时获得的数值结果具有良好的非网格依赖性。     

On the enhancement of low‐order mixed finite element methods for the large deformation analysis of diffusion in solids

A stabilized scheme is developed for mixed finite element methods for strongly coupled diffusion problems in solids capable of large deformations. Enhanced assumed strain techniques are employed to cure spurious oscillation patterns of low‐order displacement/pressure mixed formulations in the incompressible limit for quadrilateral elements and brick elements. A study is presented that shows how hourglass instabilities resulting from geometrically nonlinear enhanced assumed strain methods have to be distinguished from pressure oscillation patterns due to the violation of the inf‐sup condition. Moreover, an element formulation is proposed that provides stable results with respect to both types of instabilities. Comparisons are drawn between material models for incompressible solids of Mooney–Rivlin type and models for standard diffusion in solids with incompressible matrices such as polymeric gels. Representative numerical examples underline the ability of the proposed element formulation to cure instabilities of low‐order mixed formulations. Copyright © 2015 John Wiley & Sons, Ltd.     

Velocity‐based formulations for standard and quasi‐incompressible hypoelastic‐plastic solids

We present three velocity‐based updated Lagrangian formulations for standard and quasi‐incompressible hypoelastic‐plastic solids. Three low‐order finite elements are derived and tested for non‐linear solid mechanics problems. The so‐called V‐element is based on a standard velocity approach, while a mixed velocity–pressure formulation is used for the VP and the VPS elements. The two‐field problem is solved via a two‐step Gauss–Seidel partitioned iterative scheme. First, the momentum equations are solved in terms of velocity increments, as for the V‐element. Then, the constitutive relation for the pressure is solved using the updated velocities obtained at the previous step. For the VPS‐element, the formulation is stabilized using the finite calculus method in order to solve problems involving quasi‐incompressible materials. All the solid elements are validated by solving two‐dimensional and three‐dimensional benchmark problems in statics as in dynamics. Copyright © 2016 John Wiley & Sons, Ltd.