An efficient finite element formulation based on the strong discontinuity approach for the modeling of material failure

A new finite element formulation based on the strong discontinuity approach is proposed. Following [1], localized inelastic deformations are represented as surfaces of discontinuous displacements within the respective finite element by applying the Enhanced Assumed Strain (EAS) concept. However, and in contrast to previous publications, the jump part of the displacement field is not condensed out at the element level by employing static condensation. Instead, the L₂‐orthogonality condition is reformulated resulting in an equation which is formally identical to the necessary condition of yielding known from standard plasticity theory. Hence, it is possible to apply the return‐mapping algorithm to the numerical implementation. Only slight modifications of this by now classical algorithm are necessary. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Variational formulation of the material failure process in solids by embedded discontinuities model

This work presents a variational formulation of the material failure process, idealized as strain or displacement discontinuities, by weak, strong, or discrete embedded discontinuities into a continuum. It is shown that the solution of the proposed variational formulation may be approximated by different types of finite elements with embedded discontinuities. The developed displacement approximation of a finite element split by the discontinuity leads to a symmetric stiffness matrix, which considers not only the continuity of tractions but also the rigid body relative motions of the portions in which the element is split. The variational formulation of a continuum with more than one discontinuity in its interior is developed. It is shown that this formulation may lead to finite elements with embedded discontinuities that can be classified as displacement, force, mixed, and hybrid models. To show the effectiveness of the proposed formulation, the classical example of a bar under tension is solved using one and 2D finite element approximations. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Finite elements for symmetric tensors in three dimensions

We construct finite element subspaces of the space of symmetric tensors with square-integrable divergence on a three-dimensional domain. These spaces can be used to approximate the stress field in the classical Hellinger-Reissner mixed formulation of the elasticty equations, when standard discontinuous finite element spaces are used to approximate the displacement field. These finite element spaces are defined with respect to an arbitrary simplicial triangulation of the domain, and there is one for each positive value of the polynomial degree used for the displacements. For each degree, these provide a stable finite element discretization. The construction of the spaces is closely tied to discretizations of the elasticity complex and can be viewed as the three-dimensional analogue of the triangular element family for plane elasticity previously proposed by Arnold and Winther.

     

Numerical Simulation of Crack Growth

In recent years the X-FEM based on the partition of unity method and the strong discontinuity approach (SDA) have shown to be powerful tools to model crack growth. Both methods model the crack surface by introducing additional d.o.f.. In the X-FEM the nodes in the mesh around a crack are globally enhanced with new d.o.f. while in in the SDA the new d.o.f. are commonly introduced as internal ones. Thus the jump displacement fields are constant across elements. Therefore the d.o.f. can be condensed on element level which results in jumps in the displacement field at element edges. In this contribution the strong discontinuity approach is used approximating the displacement jump linearly across the crack length similar as e.g. in [3]. New additional nodes of the cracked elements that lie on the element edges are introduced but are not considered as internal nodes but remain global. Thus crack path continuity is automatically given. These global d.o.f. approximate the discontinuous part of the displacement field. The sum of the aforementioned part and the continuous displacement field represent the total displacement field including a possible jump. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Continuous finite element methods for Reissner-Mindlin plate problem

On triangle or quadrilateral meshes, two finite element methods are proposed for solving the Reissner-Mindlin plate problem either by augmenting the Galerkin formulation or modifying the plate-thickness. In these methods, the transverse displacement is approximated by conforming (bi)linear macroelements or (bi)quadratic elements, and the rotation by conforming (bi)linear elements. The shear stress can be locally computed from transverse displacement and rotation. Uniform in plate thickness, optimal error bounds are obtained for the transverse displacement, rotation, and shear stress in their natural norms. Numerical results are presented to illustrate the theoretical results.     

A fully variational strong discontinuity approach at finite strains

A novel, fully variational three-dimensional finite element formulation for the modeling of locally embedded strong discontinuities at finite strains is presented. The proposed numerical model is based on the Enhanced Assumed Strain concept with an additive decomposition of the displacement gradient into a conforming and an enhanced part. The discontinuous component of the displacement field which is associated with the failure in the modeled structure is isolated in the enhanced part of the deformation gradient. In contrast to previous works, a variational constitutive update is used. The internal variables are determined by minimizing a pseudo-elastic potential. The advantages of such a formulation are well known, e.g. the tangent stiffness matrix is symmetric, standard optimization algorithms can be applied and it represents a natural basis for error estimation and mesh adaption. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

板的低阶间断与连续有限元法统一分析

基于Reissner-Mindlin板问题的间断Galerkin有限元逼近, 建立了一个对挠度空间和角位移空间取连续或间断元都适用的低阶有限元离散格式. 取剪切力空间为分片常数元, 挠度空间和角位移空间无论取间断元还是连续元, 格式都是一致稳定的, 并给出了H¹范数估计及L²范数估计. 作为应用,对几类低阶有限元空间讨论. 结果表明, 该格式对常见的低阶有限元空间都适用, 并且若至少有一个元连续时, 该格式需要的空间比[1,2]中的都要简单.       

Energy driven crack propagation at finite strains based on the embedded strong discontinuity approach

New advances in three-dimensional finite element modeling of crack propagation at finite strains are presented. The proposed numerical model is based on the Enhanced Assumed Strain concept. The enhanced part of the deformation gradient is associated with a displacement discontinuity. In contrast to previous works, a new, energy based criterion for crack propagation is presented. The necessity for a tracking algorithm for the crack path is avoided by using more than one discontinuity within each finite element. This leads to a strictly local formulation, i.e., no information about the neighboring elements are required. Further advantages of such a formulation are a symmetric tangent stiffness matrix and the reduction of locking effects. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Normal and tangential continuous elements for least‐squares mixed finite element methods

We consider a finite element discretization of the primal first‐order least‐squares mixed formulation of the second‐order elliptic problem. The unknown variables are displacement and flux, which are approximated by equal‐order elements of the usual continuous element and the normal continuous element, respectively. We show that the error bounds for all variables are optimal. In addition, a field‐based least‐squares finite element method is proposed for the 3D‐magnetostatic problem, where both magnetic field and magnetic flux are taken as two independent variables which are approximated by the tangential continuous and the normal continuous elements, respectively. Coerciveness and optimal error bounds are obtained. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004.     

Advanced finite element formulations for modeling thin piezoelectric structures

This contribution is concerned with mixed finite element formulations for modeling piezoelectric beam and shell structures. Due to the electromechanical coupling, specific deformation modes are joined with electric field components. In bending dominated problems incompatible approximation functions of these fields cause incorrect results. These effects occur in standard finite element formulations, where interpolation functions of lowest order are used. A mixed variational approach is introduced to overcome these problems. The mixed formulation allows for a consistent approximation of the electromechanical coupled problem. It utilizes six independent fields and could be derived from a Hu-Washizu variational principle. Displacements, rotations and the electric potential are employed as nodal degrees of freedom. According to the Timoshenko theory (beam) and the Reissner-Mindlin theory (shell), the formulations account for constant transversal shear strains. To incorporate three dimensional constitutive relations all transversal components of the electric field and the strain field are enriched by mixed finite element interpolations. Thus the complete piezoelectric coupling is appropriately captured. The common assumption of vanishing transversal stress and dielectric displacement components is enforced in an integral sense. Some numerical examples will demonstrate the capability of the presented finite element formulation. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Thermo-mechanical modeling of turbine blades taking into account structural defects

In turbine blades of aero-engines typical defects are cracks due to high mechanical and thermal loads. The extended finite element method (XFEM) is used for simulations of fracture mechanics problems with cracks. Discontinuities in the displacement and temperature field are allowed and the crack opening displacement and crack tip stress field are reproduced accurately. Since crack closure and non-physical penetration of the crack surfaces may occur under certain load conditions, it becomes necessary to enforce the non-penetration condition for crack surfaces. This contact formulation is assumed to be frictionless. The node-to-segment approach proposed in [3] is extended to ten-node tetrahedral elements with quadratic shape functions. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A type of new posteriori error estimators for stokes problems

1. IntroductionIn the numerical approximation of PDE, it is often very importals to detect regionswhere the accuracy of the numerical solution is degraded by local singularities of the solutionof the continuous problem such as the singularity near the re-entrant corller. An obviousremedy is to refine the discretization in the critical regions, i.e., to place more gridpointswhere the solution is less regular. The question is how to identify these regions automdticallyand how to determine a goo…     

Derivation of the consistent flexibility matrix for geometrically nonlinear Timoshenko frame finite element

A consistent flexibility matrix is presented for a large displacement equilibrium-based Timoshenko beam–column element. This development is an improvement and extension to Neuenhofer–Filippou [1] (1998. ASCE J. Struct. Eng. 124, 704–711) for geometrically nonlinear Euler–Bernoulli force-based beam element. In order to find weak form compatibility and strong form equilibrium equations of the beam, the Hellinger–Reissner potential is expressed. During the formulation process, an extended displacement interpolation technique named curvature/shearing based displacement interpolation (CSBDI) is proposed for the strain–displacement relationship. Finally, the extended CSBDI technique is validated for geometric nonlinear examples and accuracy of the method is investigated concluding improved convergence rates with respect to the general finite element formulation. Also it is seen that the use of force based formulation removes shear locking effects. The results demonstrate considerable accuracy even in presence of high axial loading in comparison with the displacement based approach.     

应力协调迭代法在高速冲击动力有限元中的应用

在EPIC[2][3],NONSAP[4]等弹塑性撞击动力有限元程序中,有一个共同的弱点是都采取了静力有限元方法,把位移函数用线性插值表示.单元之间应力是非协调的.因此应用虚功原理的基础不合理.为了克服以上困难,本文引入一个新的方法,即协调应力迭代法.实例表明,这种方法在冲击动力有限元计算中是稳定和精确的,同时具有减小单元刚度的作用.     

A finite element model of localized deformation in frictional materials taking a strong discontinuity approach

A finite element model of localized deformation in frictional materials taking a strong discontinuity approach is presented. A rate-independent, non-associated, strain-softening Drucker–Prager plasticity model is formulated in the context of strong discontinuities and implemented along with an enhanced quadrilateral element within the framework of an assumed enhanced strain finite element method. For simple model problems such as uniform compression, the strong discontinuity approach has been shown to lead to mesh-independent finite element solutions when localized deformation is present. In this paper, a finite element analysis of localized deformation occurring in a more complex model problem of slope stability is conducted in a nearly mesh-independent manner. The effect of dilatancy on the orientation of slip lines is demonstrated for the slope stability problem.     

Advanced Shell Element Formulations for Coupled Electromechanical Systems

With the significantly increasing applications of smart structures, piezoelectric material is widely used in branches of engineering sciences. Normally, the Finite Element Method is employed in the numerical analysis of these structures [2]. In this contribution, in order to avoid the locking effects and zero energy modes, the Assumed Natural Strain (ANS) Method [4] is implemented into four‐node piezoelectric shallow shell elements, by using the two‐field variational formulation in which displacements and electric potentials serve as independent variables and the three‐field variational formulation in which the dielectric displacement is taken as an independent variable additionally [3]. Moreover, a quadratic variation of the electric potential through the thickness direction is applied in the two‐field formulation. Numerical examples of piezoelectric sensors and actuators are presented, showing the behaviour of the shell elements by using different hybrid finite element formulations. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A combined nodal continuous-discontinuous finite element formulation for the Maxwell problem

Continuous Galerkin formulations are appealing due to their low computational cost, whereas discontinuous Galerkin formulation facilitate adaptative mesh refinement and are more accurate in regions with jumps of physical parameters. Since many electromagnetic problems involve materials with different physical properties, this last point is very important. For this reason, in this article we have developed a combined cG-dG formulation for Maxwell’s problem that allows arbitrary finite element spaces with functions continuous in patches of finite elements and discontinuous on the interfaces of these patches. In particular, the second formulation we propose comes from a novel continuous Galerkin formulation that reduces the amount of stabilization introduced in the numerical system. In all cases, we have performed stability and convergence analyses of the methods. The outcome of this work is a new approach that keeps the low CPU cost of recent nodal continuous formulations with the ability to deal with coefficient jumps and adaptivity of discontinuous ones. All these methods have been tested using a problem with singular solution and another one with different materials, in order to prove that in fact the resulting formulations can properly deal with these problems.     

Application of the discontinuous Galerkin finite element method in small deformation regimes

In this paper, a finite element formulation is defined in the framework of the discontinuous Galerkin method. Discontinuous Galerkin (dG) methods are classically used in fluid mechanics, however recently their application in solid mechanics has become more vivid among scientists. Of special interest is their application in elliptic problems with constraints such as incompressibility which leads to volumetric locking phenomenon and also in some structural models of shells, plates and beams with compatibility constraints, which brings about shear locking [1]. While classical standard Galerkin methods must be continuous, dG methods can be applied for discontinuities across element boundaries, where a jump of a value (displacement) can be observed. In the present work, a dG method is applied to a linear elastic bar, where a weak discontinuity is allowed in the bar. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finite element analysis of pressure vessel using beam on elastic foundation analysis

In this study, we first applied the variation principle to derive a new finite element method (FEM) based on the theory of beam on elastic foundation using line element. The derived FEM was then applied to solve, for the first time, the pressure vessel problems with uniform thickness. Our FEM results, obtained even by using only one line element, agreed exactly with the available closed-form solution, confirming the validity and computing efficiency of our finite element formulation. Moreover, we have applied our new FEM to solve pressure vessel problems with non-uniform thickness where no exact analytical solution is known to exist. The distributions of discontinuity stress in the cylindrical part were obtained. We found that shear force and bending moment were indeed discontinuous at the geometrically discontinuous juncture, due to the bending rigidity and elastic constant change by the non-uniform thickness. Finally, the case of discontinuity stresses in a bimetallic joint was also studied. The locations of maximum shear force and bending moment were found to be affected by the bending rigidity of the material.     

A quadrilateral plate bending element based on deformation modes

In this paper, a quadrilateral element is proposed for the analysis of thin plate bending. This element is non-conforming and consists of four-nodes and twelve degrees of freedom. A third-order field for the element displacement is written in terms of the deformation modes. Moreover, the rotational fields are obtained by utilizing the first-order Jacobean matrix. All interpolation functions are explicitly found by the presented formulation. The stiffness matrix of the element is then computed by using these functions. Finally, the accuracy of the suggested element is evaluated by solving some thin plate bending structures. Numerical findings reveal the new quadrilateral element MKQ12 is robust and accurate for analysis of thin plates.