Sheet metal forming and springback simulation by means of a new reduced integration solid-shell finite element technology

The paper deals with the validation of a recently proposed hexahedral solid-shell finite element in the field of sheet metal forming. Working with one integration point in the shell plane and an arbitrary number of integration points in thickness direction, highly non-linear stress states over the sheet thickness can be incorporated in an efficient way. In order to avoid volumetric locking and Poisson thickness locking at the level of integration points the enhanced assumed strain (EAS) concept with only one EAS degree-of-freedom is implemented. A key point of the formulation is the construction of the hourglass stabilization by means of different Taylor expansions. This leads to the advantage that the sensitivity with respect to mesh distortion is noticeably reduced. The hourglass stabilization includes the assumed natural strain (ANS) concept and a kind of B-Bar method. So transverse shear locking and volumetric locking are eliminated.The finite element formulation incorporates a finite strain material model for plastic anisotropy as well as non-linear (Armstrong–Frederick type) kinematic and isotropic hardening. In this context the plastic anisotropy can be modeled by representing the yield surface and the plastic flow rule as functions of so-called structural tensors. The integration of the evolution equations is performed by means of an exponential map exploiting the spectral decomposition. The element formulation and material model have been implemented into the commercial code ABAQUS/Standard by means of the UEL interface for user-defined elements. Using an implicit time integration scheme numerical results for classical deep drawing simulations as well as springback predictions are presented in comparison to experimental measurements.     

An efficient ABAQUS solid shell element implementation for low velocity impact analysis of FGM plates

The main objective of this paper is to develop a numerical model susceptible to solve the numerical locking problems that may appear when applying the conventional solid and shell finite elements of ABAQUS. This model is based on a hexahedral solid shell element. The formulation of this element relays on the combination of the enhanced assumed strain (EAS) and assumed natural strain (ANS) methods with modified First Shear Deformation Theory (FSDT). The developed element is implemented into the ABAQUS user element (UEL) interface. The performance of this element is demonstrated by different benchmark tests from the literature. Our contribution consists on applying a single solid shell element through the thickness direction to predict the low velocity impact behavior on functionally graded material (FGM) circular plates.

     

Numerical analysis of layered fiber composites accounting for the onset of delamination

Since delamination is a major failure mode of layered composites, predicting its initiation is essential for the design of composite structures. Evaluating delamination onset criteria based on stress–strength relations requires an accurate representation of the through-the-thickness stress distribution, which is delicate for thin shell-like structures. Therefore, in this paper, a solid-shell finite element is utilized, which allows for incorporating a fully three-dimensional, anisotropic, micro-mechanically motivated material model, still being suited for application to thin structures. Moreover, locking phenomena are cured by using both the enhanced assumed strain (EAS) and the assumed natural strain (ANS) concept, and numerical efficiency is ensured through reduced integration.     

A simple isoparametric three-node shell finite element

A three-node isoparametric shell finite element including membrane and bending effects is proposed. The element is based on the degenerated solid approach and uses an assumed strain method to avoid shear locking. An intermediate convected covariant frame is used in order to construct the modified shear strain interpolation matrix. Validation tests show that shear locking is avoided and that a reduced integration procedure can be used without any loss of accuracy which is useful for the numerical efficiency.     

Mixed meshless formulation for analysis of shell-like structures

An efficient mixed meshless computational method based on the Local Petrov–Galerkin approach for analysis of plate and shell structures is presented. A concept of a three-dimensional solid is applied allowing the use of complete three-dimensional constitutive equations, and exact shell geometry can be described. Discretization is carried out by using both the moving least square approximation and the polynomial functions. Independent field variables are the strain and stress tensor components expressed in terms of the nodal values, which are then replaced by the nodal displacements by using the independent displacement interpolation. A closed global system of equations with only nodal displacements as unknown variables is derived. The undesired locking phenomena are fully suppressed. The proposed mixed formulation is numerically more efficient than the available meshless fully displacement approach, as demonstrated by the numerical examples.     

基于假定应变法的协同转动四边形复合材料壳单元

为对复合材料层合板壳结构进行精确的大变形数值模拟,提出一种采用假定应变法的能分析层合结构大转动问题的协同转动四边形壳单元.该方法在建立有限元公式时引入假定应变法以克服膜闭锁和剪切闭锁的不利影响.与其他能分析大转动问题的复合材料壳单元相比,在新的协同转动框架中采用矢量型转动变量,可大大降低在非线性增量求解过程中更新转动变量的难度,且能得到对称的单元切线刚度矩阵,提高单元的计算效率.分析两个典型算例,并与其他学者的结果进行对比,结果表明在计算层合结构大转角问题时拥有较好的精度和收敛性.     

A simple mixed finite element for static limit analysis

In this paper, we investigate the behavior of a simple mixed finite element for the limit analysis of plane structures. In particular, its ability to overcome incompressibility locking in plane strain situations is investigated. The element is constructed from a piecewise constant displacement field and a piecewise bilinear stress field, and is used within a mathematical programming based discrete representation of the classical static formulation. Several benchmark examples of both plane stress and plane strain situations are solved to illustrate the predictive accuracy and to assess the large-scale capability of the element. The results are compared with those obtained by a recent sophisticated enhanced strain mixed element formulation.     

Isoparametric axisymmetric shell elements with temperature gradients for heat conduction

This paper presents a finite element formulation for axisymmetric shell heat conduction where temperature gradients through the shell thickness are retained as primary nodal variables. The element geometry is constructed using the coordinates of the nodes lying on the middle surface of the shell and the middle surface nodal point normals. The element temperature field is approximated in terms of element approximation functions, the nodal temperature, and the nodal temperature gradients. The weak formulation of the two-dimensional Fourier heat conduction equation in cylindrical coordinate system is constructed. The finite element properties of the shell element are then derived using the weak formulation and the element temperature field approximation. The formulation permits linear temperature gradients through the shell thickness. Distributed heat flux as well as convective boundaries are permitted on all four faces of the element. Furthermore, the element can also have internal heat generation as well as orthotropic material properties. The superiority of the formulation in terms of efficiency and accuracy is demonstrated. Numerical examples are presented and a comparison is made with the theoretical results.     

Geometrically non-linear formulation for the three dimensional solid-shell transition finite elements

This paper presents a geometrically non-linear formulation using total lagrangian approach for the solid-shell transition finite elements. Such transition finite elements are necessary in geometrically non-linear analysis of structures modelled with three dimensional solid elements and the curved shell elements. These elements are an essential connecting link between the solid elements and the shell elements. The element formulation presented here is derived using the properties of the three dimensional solid elements and the curved shell elements. No restrictions are imposed on the magnitude of the nodal rotations. Thus the element formulation is capable of handling large rotations between two successive load increments. The element properties are derived and presented in detail. Numerical examples are also presented to demonstrate their behavior, accuracy and applications in three dimensional stress analysis.

It is shown that the selection of different stress and strain components at the integration points do not effect the overall linear response of the element. However, in geometrically non-linear applications it may be necessary to select appropriate stress and the strain components at the integration points for stable and converging element behavior. Numerical examples illustrate various characteristics of the element.     

A generalised technique for fracture analysis of cracked plates under combined tensile,bending and shear loads

The objective of this paper is to propose a generalized technique called numerically integrated modified virtual crack closure integral (NI-MVCCI) technique for fracture analysis of cracked plates under combined tensile, bending and shear loads. NI-MVCCI technique is used for post-processing the results of finite element analysis (FEA) for computation of strain energy release rate (SERR) components and the corresponding stress intensity factor (SIF) for cracked plates. NI-MVCCI technique has been demonstrated for 4-noded, 8-noded (regular and quarter-point) and 9-noded (regular and quarter-point) isoparametric plate finite elements. These elements are based on Mindlin’s plate theory that considers shear deformation. For all the elements, reduced integration/selective reduced integration techniques have been employed in the studies. In addition, for 9-noded element assumed shear interpolation functions have been used to overcome the shear locking problem. Numerical studies on fracture analysis of plates subjected to tension–moment and tension–shear loads have been conducted employing these elements. It is observed that among these elements, the 9-noded Lagrangian plate element with assumed shear interpolation functions exhibits better performance for fracture analysis of cracked plates.     

Geometrically nonlinear formulation for the axi-symmetric transition finite elements

This paper presents a geometrically nonlinear formulation for the axi-symmetric transition finite elements using total lagrangian approach. The basic element is formulated using properties of the axi-symmetric solids and the axi-symmetric shells. A novel feature of the formulation presented here is that the restriction on the magnitude of the rotations for the shell nodes of the transition element is eliminated. This is accomplished by retaining true nonlinear functions of nodal rotations in the definition of the element displacement field. Such transition elements are essential for geometrically nonlinear applications requiring both axi-symmetric solids and the axi-symmetric shells. They ensure proper connection of the axi-symmetric solid portion of the structure to the shell like portion of the structure. It is shown that the selection of different stress and strain components at the integration points does not effect the overall linear response of the element. However, in the geometrically nonlinear formulation, it is necessary to select appropriate components of the stresses and the strains at the integration point for accurate and converging element behavior. Numerical examples are presented to demonstrate such characteristics of the transition elements.     

Transition finite elements with temperature and temperature gradients as primary variables for axisymmetric heat conduction

This paper presents finite element formulation for a special class of elements referred to as “transition finite elements” for axisymmetric heat conduction. The transition elements are necessary in applications requiring the use of both axisymmetric solid elements and axisymmetric shell elements. The elements permit transition from the solid portion of the structure to the shell portion of the structure. A novel feature of the formulation presented here is that nodal temperatures as well as nodal temperature gradients are retained as primary variables. The weak formulation of the Fourier heat conduction equation is constructed in the cylindrical co-ordinate system (r, z). The element geometry is defined in terms of the co-ordinates of the nodes as well as the nodal point normals for the nodes lying on the middle surface of the element. The element temperature field is approximated in terms of element approximation functions, nodal temperatures and the nodal temperature gradients. The properties of the transition elements are then derived using the weak formulation and the element temperature approximation. The formulation presented here permits linear temperature distribution through the element thickness. Convective boundaries as well as distributed heat flux is permitted on all four faces of the element. Furthermore, the element formulation also permits distributed heat flux and orthotropic material behaviour. Numerical examples are presented, first to illustrate the accuracy of the formulation and second to demonstrate its usefulness in practical applications. Numerical results are also compared with the theoretical solutions.     

Study of trial functions in shell triangular finite elements of quadratic parametric representation

A study is made of trial functions in a curved triangular finite element for the solution of problems in first approximation shell theory. The element geometry is in quadratic parametric representation of the real shell surface while the displacement and stress function trial functions are in cubic parametric representation. Special attention is given to inextensional bending and homogeneous membrane states. Numerical examples are provided for specimen elements with positive, zero and negative Gaussian curvature.Vector equations are listed for the underlying shell theory together with Fortran subroutines for the element geometry, kinematics and statics.     

Three dimensional solid-shell transition finite elements for heat conduction

This paper presents a finite element formulation for a special class of finite elements referred to as ‘Solid-Shell Transition Finite Elements’ for three dimensional heat conduction. The solid-shell transition elements are necessary in applications requiring the use of both three dimensional solid elements and the curved shell elements. These elements permit transition from the solid portion of the structure to the shell portion of the structure. A novel feature of the formulation presented here is that nodel temperatures as well as nodal temperature gradients are retained as primary variables. The element geometry is defined in terms of coordinates of the nodes as well as the nodal point normals for the nodes lying on the middle surface of the element. The temperature field with the element is approximated in terms of element approximation functions, nodal temperatures and nodal temperature gradients. The properties of the transition element are then derived using the weak formulation (or the quadratic functional) of the Fourier heat conduction equation in the Cartesian coordinate system and the element temperature approximation. The formulation presented here permits linear temperature distribution in the element thickness direction.

Convective boundaries as well as distributed heat flux is permitted on all six faces of the elements. Furthermore, the element formulation also permits internal heat generation and orthotropic material behavior. Numerical examples are presented firstly to illustrate the accuracy of the formulation and secondly to demonstrate its usefulness in practical application. Numerical results are also compared with the theoretical solutions.     

A general isoparametric finite element program SDRC SUPERB

SDRC SUPERB is a general purpose finite element program that performs linear static, dynamic and steady state heat conduction analyses of structures made of isotropic and/or orthotropic elastic materials having temperature dependent properties. The finite element library of SUPERB contains isoparametric plane stress, plane strain, flat plate, curved shell, solid type curved shell and solid elements in addition to conventional beam and spring elements. Linear, quadratic and cubic interpolation functions are available for all isoparametric elements. Independent parameters such as displacements and temperatures are obtained from SUPERB using the stiffness method of analysis. The remaining dependent parameters, such as stresses and strains, are evaluated at element gauss points and extrapolated to nodal locations. Averaged values are given as final output. The graphic capabilities of SUPERB consists of geometry and distorted geometry plotting, and stress, strain and temperature contouring. Contours are plotted at user defined cutting planes for solids and at top, middle or bottom surfaces for plate and shell types of structures.In the first part of this paper, the program capabilities of SUPERB are summarized. Extrapolation techniques used for determining dependent nodal parameters and for contour plotting are explained in the second part of the paper. Behavior of standard, wedge and transition type isoparametric elements and the effect of interpolation function orders on accuracy are discussed in the third part. The results of several illustrative problems are included.     

Topology optimization of structures with anisotropic plastic materials using enhanced assumed strain elements

The focus of this paper is on consistent and accurate adjoint sensitivity analyses for structural topology optimization with anisotropic plastic materials under plane strain conditions. In order to avoid the locking issue, the Enhanced Assumed Strain (EAS) elements are adopted in the finite element discretization, and the anisotropic Hoffman plasticity model, which can simulate the strength differences in tension and compression, is incorporated within the framework of density-based topology optimization. The path-dependent sensitivity analysis is presented wherein the enhanced element parameters are consistently incorporated in the constraints. The objective of topology optimization is to maximize the plastic work. Several numerical examples are presented to show the effectiveness of the proposed framework. The results illustrate that the optimized topologies are highly dependent on the plastic anisotropic material properties.     

Time integration in the context of energy control and locking free finite elements

Summary In the present paper two main research areas of computational mechanics, namely the finite element development and the design of time integration algorithms are reviewed and discussed with a special emphasis on their combination. The finite element techniques are designed to prevent locking and the time integration schemes to guarantee numerical stability in non-linear elastodynamics. If classical finite element techniques are used, their combination with time integration schemes allow to avoid any modifications on the element or algorithmic level. It is pointed out, that on the other hand Assumed Stress and Enhanced Assumed Strain elements have to be modified if they are combined with energy conserving or decaying time integration schemes, especially the Energy-Momentum Method in its original and generalized form. The paper focusses on the necessary algorithmic formulation of Enhanced Assumed Strain elements which will be developed by the reformulation of the Generalized Energy-Momentum Method based on a classical one-field functional, the extension to a modifiedHu-Washizu three-field functional including enhanced strains and a suitable time discretization of the additional strain terms. The proposed method is applied to non-linear shell dynamics using a shell element which allows for shear deformation and thickness change, and in which the Enhanced Assumed Strain Concept is introduced to avoid artificial thickness locking. Selected examples illustrate the locking free and numerically stable analysis.     

Improvements in the membrane behaviour of the three node rotation-free BST shell triangle using an assumed strain approach

In this paper an assumed strain approach is presented in order to improve the membrane behaviour of a thin shell triangular element. The so called Basic Shell Triangle (BST) has three nodes with only translational degrees of freedom and is based on a Total Lagrangian Formulation. As in the original BST element the curvatures are computed resorting to the surrounding elements (patch of four elements). Membrane strains are now also computed from the same patch of elements which leads to a non-conforming membrane behaviour. Despite this non-conformity the element passes the patch test. Large strain plasticity is considered using a logarithmic strain–stress pair. A plane stress behaviour with an additive decomposition of elastic and plastic strains is assumed. A hyperplastic law is considered for the elastic part while for the plastic part an anisotropic quadratic (Hill) yield function with non-linear isotropic hardening is adopted. The element, termed EBST, has been implemented in an explicit (hydro-)code adequate to simulate sheet-stamping processes and in an implicit static/dynamic code. Several examples are given showing the good performance of the enhanced rotation-free shell triangle.     

Membrane locking and assumed strain shell elements

An explanation of membrane locking behaviour in shell elements and also the use of reduced and selective integration is described. To overcome the conflict between the locking and mechanism problems the author proposed the degenerated shell elements with assumed transverse shear and membrane strains. The location of sampling points for the assumed strain fields is given in the present work. In the formulation of the new elements, assumed transverse shear strains in the natural coordinate system are used to overcome the shear locking problem. Also, assumed membrane strains in the orthogonal curvilinear coordinate system are applied to avoid membrane locking behaviour. Some numerical tests are presented to illustrate the good performance of the assumed strain shell elements.     

Element free method for static and free vibration analysis of spatial thin shell structures

The implementation of the element free Galerkin method (EFG) for spatial thin shell structures is presented in this paper. Both static deformation and free vibration analyses are considered. The formulation of the discrete system equations starts from the governing equations of stress resultant geometrically exact theory of shear flexible shells. Moving least squares approximation is used in both the construction of shape functions based on arbitrarily distributed nodes as well as in the surface approximation of general spatial shell geometry. Discrete system equations are obtained by incorporating these interpolations into the Galerkin weak form. The formulation is verified through numerical examples of static stress analysis and frequency analysis of spatial thin shell structures. For static load analysis, essential boundary conditions are enforced through penalty method and Lagrange multipliers while boundary conditions for frequency analysis are imposed through a weak form using orthogonal transformation techniques. The EFG results compare favorably with closed-form solutions and that of finite element analyses.