3D‐FEM formulations of limit analysis methods for porous pressure‐sensitive materials

The first purpose of this paper is the numerical formulation of the three general limit analysis methods for problems involving pressure‐sensitive materials, that is, the static, classic, and mixed kinematic methods applied to problems with Drucker–Prager, Mises–Schleicher, or Green materials. In each case, quadratic or rotated quadratic cone programming is considered to solve the final optimization problems, leading to original and efficient numerical formulations. As a second purpose, the resulting codes are applied to non‐classic 3D problems, that is, the Gurson‐like hollow sphere problem with these materials as matrices. To this end are first presented the 3D finite element implementations of the static and kinematic classic methods of limit analysis together with a mixed method formulated to give also a purely kinematic result. Discontinuous stress and velocity fields are included in the analysis. The static and the two kinematic approaches are compared afterwards in the hydrostatic loading case whose exact solution is known for the three cases of matrix. Then, the static and the mixed approaches are used to assess the available approximate criteria for porous Drucker–Prager, Mises–Schleicher, and Green materials. Copyright © 2013 John Wiley & Sons, Ltd.     

A highly efficient enhanced assumed strain physically stabilized hexahedral element

A method which combines the incompatible modes method with the physical stabilization method is developed to provide a highly efficient formulation for the single point eight‐node hexahedral element. The resulting element is compared to well‐known enhanced elements in standard benchmark type problems. It is seen that this single‐point element is nearly as coarse mesh accurate as the fully integrated EAS elements. A key feature is the novel enhanced strain fields which do not require any matrix inversions to solve for the internal element degrees of freedom. This, combined with the reduction of hourglass stresses to four hourglass forces, produces an element that is only 6.5 per cent slower than the perturbation stabilized single‐point brick element commonly used in many explicit finite element codes. Copyright © 2000 John Wiley & Sons, Ltd.     

A triangular hybrid equilibrium plate element of general degree

Hybrid stress‐based finite elements with side displacement fields have been used to generate equilibrium models having the property of equilibrium in a strong form. This paper establishes the static and kinematic characteristics of a flat triangular hybrid equilibrium element with both membrane and plate bending actions of general polynomial degree p. The principal characteristics concern the existence of hyperstatic stress fields and spurious kinematic modes. The former are shown to exist for p>3, and their significance to finite element analysis is reviewed. Knowledge of the latter is crucial to the determination of the stability of a mesh of triangular elements, and to the choice of procedure adopted for the solution of the system of equations. Both types of characteristic are dependent on p, and are established as regards their numbers and general algebraic forms. Graphical illustrations of these forms are included in the paper. Copyright © 2005 John Wiley & Sons, Ltd.     

Assessment of 4-node EAS–ANS shell elements for large deformation analysis

In this work we derive and mutually compare several 4-node shell hyperelastic finite elements for large deformation analysis. The elements are based on Reissner–Mindlin shell theory. They use the enhanced assumed strain (EAS) concept for enhancement of membrane and bending displacement-compatible strains and the assumed natural strain (ANS) concept for treatment of transverse shear strains. The elements differ from each other by the number of membrane and bending EAS parameters. An optimal number of these parameters is suggested on the basis of numerical results.     

EAS concept for higher‐order finite shell elements to eliminate volumetric locking

The deficiency of volumetric locking phenomena in finite elements using higher‐order shell element formulations based on Lagrangean polynomials and a linear finite shell kinematics cannot be avoided by the existent enhanced assumed strain (EAS) concept established for low‐order elements. In this paper a consistent modification of the EAS concept is proposed to extend its applicability to higher‐order shell elements. This modification, affecting the transversal normal strain for polynomial orders p>1, eliminates pathological modes caused by volumetric locking. The efficiency of the proposed extended EAS method is demonstrated by means of eigenvalue analyses and two representative numerical examples. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

A three‐dimensional finite element method with arbitrary polyhedral elements

The ‘variable‐element‐topology finite element method’ (VETFEM) is a finite‐element‐like Galerkin approximation method in which the elements may take arbitrary polyhedral form. A complete development of the VETFEM is given here for both two and three dimensions. A kinematic enhancement of the displacement‐based formulation is also given, which effectively treats the case of near‐incompressibility. Convergence of the method is discussed and then illustrated by way of a 2D problem in elastostatics. Also, the VETFEM's performance is compared to that of the conventional FEM with eight‐node hex elements in a 3D finite‐deformation elastic–plastic problem. The main attraction of the new method is its freedom from the strict rules of construction of conventional finite element meshes, making automatic mesh generation on complex domains a significantly simpler matter. Copyright © 2006 John Wiley & Sons, Ltd.     

Determination of coefficients of the crack tip asymptotic field by fractal hybrid finite elements

This paper applies the fractal finite element method (FFEM) together with 9-node Lagrangian hybrid elements to the calculation of linear elastic crack tip fields. An explicit stabilization scheme is employed to suppress the spurious kinematic modes of the sub-integrated Lagrangian element. An extensive convergence study has been conducted to examine the effects of the similarity ratio, reduced integration and the type of elements on the accuracy and stability of the numerical solutions. It is concluded that (i) a similarity ratio close to unity should be used to construct the fractal mesh, (ii) sub-integrated Lagrangian elements with occasionally unstable behaviour should not be used, and (iii) the good accuracy (with differences less than 0.5% with existing available solutions) and stability over a wide range of numerical tests support the use of fractal hybrid finite elements for determining crack tip asymptotic fields.     

Mixed variational principles and robust finite element implementations of gradient plasticity at small strains

This work outlines a theoretical and computational framework of gradient plasticity based on a rigorous exploitation of mixed variational principles. In contrast to classical local approaches to plasticity based on locally evolving internal variables, order parameter fields are taken into account governed by additional balance‐type PDEs including micro‐structural boundary conditions. This incorporates non‐local plastic effects based on length scales, which reflect properties of the material micro‐structure. We develop a unified variational framework based on mixed saddle point principles for the evolution problem of gradient plasticity, which is outlined for the simple model problem of von Mises plasticity with gradient‐extended hardening/softening response. The mixed variational structure includes the hardening/softening variable itself as well as its dual driving force. The numerical implementation exploits the underlying variational structure, yielding a canonical symmetric structure of the monolithic problem. It results in a novel finite element (FE) design of the coupled problem incorporating a long‐range hardening/softening parameter and its dual driving force. This allows a straightforward local definition of plastic loading‐unloading driven by the long‐range fields, providing very robust FE implementations of gradient plasticity. This includes a rational method for the definition of elastic‐plastic‐boundaries in gradient plasticity along with a post‐processor that defines the plastic variables in the elastic range. We discuss alternative mixed FE designs of the coupled problem, including a local‐global solution strategy of short‐range and long‐range fields. This includes several new aspects, such as extended Q1P0‐type and Mini‐type finite elements for gradient plasticity. All methods are derived in a rigorous format from variational principles. Numerical benchmarks address advantages and disadvantages of alternative FE designs, and provide a guide for the evaluation of simple and robust schemes for variational gradient plasticity. Copyright © 2013 John Wiley & Sons, Ltd.     

An evaluation of equivalent-single-layer and layerwise theories of composite laminates

A review of equivalent-single-layer and layerwise laminate theories is presented and their computational models are discussed. The layerwise theory advanced by the author is reviewed and a variable displacement finite element model and the mesh superposition techniques are described. The variable displacement finite elements contain several different types of assumed displacement fields. By choosing appropriate terms from the multiple displacement field, an entire array of elements with different orders of kinematic refinement can be formed. The variable kinematic finite elements can be conveniently connected together in a single domain for global-local analyses, where the local regions are modeled with refined kinematic elements. In the finite element mesh superposition technique an independent overlay mesh is superimposed on a global mesh to provide localized refinement for regions of interest regardless of the original global mesh topology. Integration of these two ideas yields a very robust and economical computational tool for global-local analysis to determine three-dimensional effects (e.g. stresses) within localized regions of interest in practical laminated composite structures.     

On transformations and shape functions for enhanced assumed strain elements

We summarize several previously published geometrically nonlinear EAS elements and compare their behavior. Various transformations for the compatible and enhanced deformation gradient are examined. Their effect on the patch test is one main concern of the work, and it is shown numerically and with a novel analytic proof that the improved EAS element proposed by Simo et al in 1993 does not fulfill the patch test. We propose a modification to overcome that drawback without losing the favorable locking-free behavior of that element. Furthermore, a new transformation for the enhanced field is proposed and motivated in a curvilinear coordinate frame. It is shown in numerical tests that this novel approach outperforms all previously introduced transformations.     

Finite element methods for the analysis of softening plastic hinges in beams and frames

This paper presents new finite element methods for the analysis of localized failures in plastic beams and frames in the form of plastic hinges. The hinges are modeled as discontinuities of the generalized displacements of the underlying Timoshenko beam/rod theory. Hinges accounting for a discontinuity in the transversal and longitudinal displacements and the rotation field are developed in this context. A multi-scale framework is considered in the incorporation of the dissipative effects of these discontinuities in the large-scale problem of a beam and a general frame. A localized softening cohesive law relating these generalized displacements with the stress resultants acting at the level of the cross section is effectively introduced in the frame response. The resulting models, referred to as localized models, are then able to capture the localized dissipation observed in the localized failures of these structural members, avoiding altogether the inconsistencies observed for classical models in the stress resultants with strain softening. The constructive approach followed in the development of these models leads naturally to the formulation of enhanced strain finite elements for their numerical approximation. In this context, we develop new finite elements incorporating the singular strains associated to the plastic hinges at the element level. A careful analysis is presented so the resulting finite elements avoid the phenomenon of stress locking, that is, an overstiff response in the softening of the hinge, not allowing for the full release of the stress. The accurate approximation of the kinematics of the hinges requires a strain enhancement linking the jumps in the deflection and the rotation fields, given the coupled definition of the transverse shear strain in these two fields. Different enhanced strain elements, involving different base finite elements and different enhancement strategies, are considered and analyzed in detail. Their performance are then compared in several representative numerical simulations. These analyses identify optimally enhanced finite elements for the accurate modeling the localized failures observed in common framed structures.Financial support for this research has been provided by the ONR under grant no. N00014-00-1-0306 with UC Berkeley. This support is gratefully acknowledged.     

Locking‐free discontinuous finite elements for the upper bound yield design of thick plates

This work investigates the formulation of finite elements dedicated to the upper bound kinematic approach of yield design or limit analysis of Reissner–Mindlin thick plates in shear‐bending interaction. The main novelty of this paper is to take full advantage of the fundamental difference between limit analysis and elasticity problems as regards the class of admissible virtual velocity fields. In particular, it has been demonstrated for 2D plane stress, plane strain or 3D limit analysis problems that the use of discontinuous velocity fields presents a lot of advantages when seeking for accurate upper bound estimates. For this reason, discontinuous interpolations of the transverse velocity and the rotation fields for Reissner–Mindlin plates are proposed. The subsequent discrete minimization problem is formulated as a second‐order cone programming problem and is solved using the industrial software package MOSEK . A comprehensive study of the shear‐locking phenomenon is also conducted, and it is shown that discontinuous elements avoid such a phenomenon quite naturally whereas continuous elements cannot without any specific treatment. This particular aspect is confirmed through numerical examples on classical benchmark problems and the so‐obtained upper bound estimates are confronted to recently developed lower bound equilibrium finite elements for thick plates. Copyright © 2015 John Wiley & Sons, Ltd.     

On the suppression of zero energy deformation modes

Based on the Hellinger-Reissner principle and the deformation energy due to assumed stresses and displacements, the problem of the kinematic deformation modes in assumed stress hybrid/mixed finite elements has been examined. Basic schemes are developed for the choice of assumed stress terms that will suppress all kinematic deformation modes. Quadrilateral membrane and axisymmetric elements, and three-dimensional hexahedral elements, are used to illustrate the suggested procedure.     

Two‐ and three‐dimensional transition element families for adaptive refinement analysis of elasticity problems

In this paper, new enhanced assumed strain (EAS) and hybrid stress transition element families are developed for 2D and 3D adaptive refinement analysis of elasticity problems. The EAS element families are based on some existing incompatible transition element families. By using the EAS method and the previous incompatible modes, the B ‐matrix columns associated with the EAS modes can be directly designed such that their domain integrals vanish automatically and they can be computed more efficiently. For 2D hybrid stress transition element families, it is possible to derive different stress fields that lead to rank‐sufficient transition elements. However, the task becomes intractable for 3D hybrid stress transition elements in which many combinations of mid‐side and mid‐face nodes are possible. This paper proposes to use hybrid stress transition element families in which the assumed stress fields are linearly complete. The new 2D element family is more accurate than the 2D rank‐sufficient element family. The new 3D element family is more accurate than the one with additional bilinear stress modes. Numerical examples reveal that the most accurate transition element families are the newly developed hybrid stress families followed by the EAS families, the incompatible families and then the compatible families. Copyright © 2008 John Wiley & Sons, Ltd.     

梯度复合材料热传导分析的梯度单元法

给出了一种适用于梯度复合材料热传导分析的梯度单元, 采用细观力学方法描述材料变化的热物理属性, 通过线性插值和高阶插值温度场分别给出了4节点和8节点梯度单元随空间位置变化的热传导刚度矩阵。推导了在温度梯度载荷和热流密度载荷作用下, 矩形梯度板的稳态温度场和热通量场精确解。基于该精确解对比了连续梯度模型和传统的离散梯度模型的热传导有限元计算结果, 验证了梯度单元的有效性, 并讨论了相关参数对梯度单元的影响。结果表明, 梯度单元和均匀单元得到的温度场基本一致; 当热载荷垂直于材料梯度方向时, 梯度单元能够给出更加精确的局部热通量场; 当热载荷平行于材料梯度方向时, 4节点梯度单元性能恶化, 8节点梯度单元和均匀单元的计算结果与精确解吻合很好。     

A new mass lumping scheme for the mixed hybrid finite element method

Diffusion‐type partial differential equation is a common mathematical model in physics. Solved by mixed finite elements, it leads to a system matrix which is not always an M‐matrix. Therefore, the numerical solution may exhibit unphysical results due to oscillations. The criterion necessary to obtain an M‐matrix is discussed in details for triangular, rectangular and tetrahedral elements. It is shown that the system matrix is never an M‐matrix for rectangular elements and can be an M‐matrix for triangular an tetrahedral elements if criteria on the element's shape and on the time step length are fulfilled. A new mass lumping scheme is developed which leads to a less restrictive criterion: the discretization must be weakly acute (all angles less than π/2) and there is no constraint on the time step length. The lumped formulation of mixed hybrid finite element can be applied not only to triangular meshes but also to more general shape elements in two and three dimensions. Numerical experiments show that, compared to the standard mixed hybrid formulation, the lumping scheme avoids (or strongly reduce) oscillations and does not create additional numerical errors. Copyright © 2006 John Wiley & Sons, Ltd.     

A Simple Procedure to Develop Efficient & Stable Hybrid/Mixed Elements,and Voronoi Cell Finite Elements for Macro- & Micromechanics

A simple procedure to formulate efficient and stable hybrid/mixed finite elements is developed, for applications in macro- as well as micromechanics. In this method, the strain and displacement field are independently assumed. Instead of using two-field variational principles to enforce both equilibrium and compatibility conditions in a variational sense, the independently assumed element strains are related to the strains derived from the independently assumed element displacements, at a finite number of collocation points within the element. The element stiffness matrix is therefore derived, by simply using the principle of minimum potential energy. Taking the four-node plane isoparametric element as an example, different hybrid/mixed elements are derived, by adopting different element strain field assumptions, and using different collocation points. These elements are guaranteed to be stable. Moreover, the computational efficiency of these elements is far better than that for traditional hybrid/mixed elements, or even better than primal finite elements, because the strain field is expressed analytically as simple polynomials (whereas, in isoparametric displacement-based element, the strain field is far more complicated), with nodal displacements as unknowns. The essential idea is thereafter extended to Voronoi cell finite elements, for micromechanical analysis of materials. Neither these four-node hybrid/mixed elements nor the Vonroni cell finite elements need to satisfy the equilibrium conditions a priori, making them suitable for extension to geometrically nonlinear and dynamical analyses. Various numerical experiments are conducted using these new elements, and the results are compared to those obtained by using traditional hybrid/mixed elements and primal finite elements. Performances of the different elements are compared in terms of efficiency, stability, invariance, locking, sensitivity to mesh distortion, and convergence rates.