An assumed strain formulation of efficient solid triangular element for general shell analysis

An efficient assumed strain triangular solid element is developed for the analysis of plate and shell structures. The finite element formulation is based on the two‐field assumed strain formulation with two independent fields of assumed displacement and assumed strain. The assumed strain field is carefully selected to alleviate the shear locking effect without triggering undesirable spurious kinematic modes. The curvilinear surface of shell structures is modelled with flat facet elements to obviate the membrane locking effect. The patch tests are successfully passed, and numerical test involving various example problems demonstrates the validity and efficiency of the present element. Copyright © 2000 John Wiley & Sons, Ltd.     

A four-node solid shell element formulation with assumed strain

A set of four-node shell element models based on the assumed strain formulation is considered here. The formulation allows for changes in the shell thickness. As a result, the kinematics of deformation are described by purely vectorial variables, without using rotational angles. The present study investigates the use of bubble function displacements and the assumed strain field. Careful selection of the assumed strain terms generates an element whose order of numerical integration does not increase even when the bubble function displacements are added. Results for the four-node element without any bubble function terms show sensitivity to element distortion. Use of the bubble functions with a carefully chosen assumed strain field greatly improves element performance. © 1998 John Wiley & Sons, Ltd.     

Stress resultant geometrically non-linear shell theory with drilling rotations. Part III: Linearized kinematics

A consistent formulation of the geometrically linear shell theory with drilling rotations is obtained by the consistent linearization of the geometrically non-linear shell theory considered in Parts I and II of this work. It was also shown that the same formulation can be recovered by linearizing the governing variational principle for the three-dimensional geometrically non-linear continuum with independent rotation field. In the finite element implementation of the presented shell theory, relying on the modified method of incompatible modes, we were able to construct a four-node shell element which delivers a very high-level performance. In order to simplify finite element implementation, a shallow reference configuration is assumed over each shell finite element. This approach does not impair the element performance for the present four-node element. The results obtained herein match those obtained with the state-of-the-art implementations based on the classical shell theory, over the complete set of standard benchmark problems.     

An Assumed Strain Triangular Solid Element for Efficient Analysis of Plates and Shells with Finite Rotation

A simple triangular solid shell element formulation is developed for efficient analysis of plates and shells undergoing finite rotations. The kinematics of the present solid shell element formulation is purely vectorial with only three translational degrees of freedom per node. Accordingly, the kinematics of deformation is free of the limitation of small angle increments, and thus the formulation allows large load increments in the analysis of finite rotation. An assumed strain field is carefully selected to alleviate the locking effect without triggering undesirable spurious kinematic modes. In addition, the curved surface of shell structures is modeled with flat facet elements to obviate the membrane locking effect. Various numerical examples demonstrate the efficiency and accuracy of the present element formulation for the analysis of plates and shells undergoing finite rotation. The present formulation is attractive in that only three points are needed for numerical integration over an element.     

A nine node finite element for analysis of geometrically non-linear shells

A nine node finite element is presented for the analysis of thin shell structures undergoing large deflection. The finite element formulation is based on the concept of degenerate solid shell element and the Hellinger-Reissner principle with independent strain. Three versions of assumed independent strain are selected to suppress spurious kinematic modes. One version leads to a finite element model which is kinematically stable at element level while the other two give globally stable models. Numerical tests indicate that the finite element model which is stable at element level may reveal the locking effect in certain cases. However, the other two models are free of locking.     

An assumed strain triangular curved solid shell element formulation for analysis of plates and shells undergoing finite rotations

A formulation for 36‐DOF assumed strain triangular solid shell element is developed for efficient analysis of plates and shells undergoing finite rotations. Higher order deformation modes described by the bubble function displacements are added to the assumed displacement field. The assumed strain field is carefully selected to alleviate locking effect. The resulting element shows little effect of membrane locking as well as shear locking, hence, it allows modelling of curved shell structures with curved elements. The kinematics of the present formulation is purely vectorial with only three translational degrees of freedom per node. Accordingly, the present element is free of small angle assumptions, and thus it allows large load increments in the geometrically non‐linear analysis. Various numerical examples demonstrate the validity and effectiveness of the present formulation. Copyright © 2001 John Wiley & Sons, Ltd.     

A triangular finite element for thin plates and shells

A finite element modelling technique which utilizes a triangular element with 45 degrees-of-freedom and seven-point integration has been tested for analysis of thin plate and shell structures. The element is based on the degenerate solid shell concept and the mixed formulation with assumed independent inplane and transverse shear strains. Numerical result indicates effectiveness of the present modelling technique which features combined use of elements with kinematic modes and those without kinematic modes in an attempt to eliminate both locking and spurious kinematic modes at global structural level.     

A finite element formulation with stabilization matrix for geometrically non-linear shells

An assumed strain finite element formulation with a stabilization matrix is developed for analysis of geometrically non-linear problems of isotropic and laminated composite shells. The present formulation utilizes the degenerate solid shell concept and assumes an independent strain as well as displacement. The assumed independent strain field is divided into a lower order part and a higher order part. Subsequently, the lower order part is set equal to the displacement-dependent strain evaluated at the lower order integration points and the remaining higher order part leads to a stabilization matrix. The strains and the determinant of the Jacobian matrix are assumed to vary linearly in the thickness direction. This assumption allows analytical integration through thickness, independent of the number of plies. A nine-node element with a judiciously chosen set of higher order assumed strain field is developed. Numerical tests involving isotropic and composite shells undergoing large deflections demonstrate the validity of the present formulation.     

An assumed strain finite element model for large deflection composite shells

A nine node finite element model has been developed for analysis of geometrically non-linear laminated composite shells. The formulation is based on the degenerate solid shell concept and utilizes a set of assumed strain fields as well as assumed displacement Two different local orthogonal co-ordinate systems were used to maintain invariance of the element stiffness matrix. The formulation assumes strain and the determinant of the Jacobian matrix to be linear in the thickness direction. This allows analytical integration in the thickness direction regardless of ply layups. The formulation also allows the reference plane to be different from the shell midsurface. The results of numerical tests demonstrate the validity and the effectiveness of the present approach.     

A stabilized co‐rotational curved quadrilateral composite shell element

A new curved quadrilateral composite shell element using vectorial rotational variables is presented. An advanced co‐rotational framework defined by the two vectors generated by the four corner nodes is employed to extract pure element deformation from large displacement/rotation problems, and thus an element‐independent formulation is obtained. The present line of formulation differs from other co‐rotational formulations in that (i) all nodal variables are additive in an incremental solution procedure, (ii) the resulting element tangent stiffness is symmetric, and (iii) is updated using the total values of the nodal variables, making solving dynamic problems highly efficient. To overcome locking problems, uniformly reduced integration is used to compute the internal force vector and the element tangent stiffness matrix. A stabilized assumed strain procedure is employed to avoid spurious zero‐energy modes. Several examples involving composite plates and shells with large displacements and large rotations are presented to testify to the reliability, computational efficiency, and accuracy of the present formulation. Copyright © 2011 John Wiley & Sons, Ltd.     

A hybrid stress ANS solid‐shell element and its generalization for smart structure modelling. Part I—solid‐shell element formulation

In the recent years, solid‐shell finite element models which possess no rotational degrees of freedom and applicable to thin plate/shell analyses have attracted considerable attention. Development of these elements are not straightforward. Shear, membrane, trapezoidal, thickness and dilatational lockings must been visioned. In this part of this paper, a novel eight‐node solid‐shell element is proposed. To resolve the shear and trapezoidal lockings, the assumed natural strain (ANS) method is resorted to. The hybrid‐stress formulation is employed to rectify the thickness and dilatational locking. The element is computationally more efficient than the conventional hybrid elements by adopting orthogonal‐assumed stress modes and enforcing admissible sparsity in the flexibility matrix. Popular benchmark tests are exercised to illustrate the efficacy of the elements. In Part II of the paper, the element will be generalized for smart structure modelling by including the piezoelectric effect. Copyright © 2000 John Wiley & Sons, Ltd.     

A sixteen node shell element with a matrix stabilization scheme

A sixteen node shell element is developed using a matrix stabilization scheme based on the Hellinger-Reissner principle with independent strain. Initially the assumed independent strain is divided into a lower order part and a higher order part. The stiffness matrix corresponding to the lower order assumed strain is equivalent to the stiffness matrix of the assumed displacement model element with the reduced integration scheme. The spurious kinematic modes of the element are suppressed by introducing a stabilization matrix associated with a judiciously chosen set of higher order assumed strain fields. Numerical results show that this element is free of locking even for very thin plates and shells.     

Finite Rotation Piezoelectric Exact Geometry Solid-Shell Element with Nine Degrees of Freedom per Node

This paper presents a robust non-linear piezoelectric exact geometry (EG) four-node solid-shell element based on the higher-order 9-parameter equivalent single-layer (ESL) theory, which permits one to utilize 3D constitutive equations. The term EG reflects the fact that coefficients of the first and second fundamental forms of the reference surface are taken exactly at each element node. The finite element formulation developed is based on a new concept of interpolation surfaces (I-surfaces) inside the shell body. We introduce three I-surfaces and choose nine displacements of these surfaces as fundamental shell unknowns. Such choice allows us to represent the finite rotation piezoelectric higher-order EG solid-shell element formulation in a very compact form and to utilize in curvilinear reference surface coordinates the strain-displacement relationships, which are objective, that is, invariant under arbitrarily large rigid-body shell motions. To avoid shear and membrane locking and have no spurious zero energy modes, the assumed displacement-independent strain and stress resultant fields are introduced. In this connection, the Hu-Washizu variational equation is invoked. To implement the analytical integration throughout the element, the modified ANS method is applied. As a result, the present finite rotation piezoelectric EG solid-shell element formulation permits the use of coarse meshes and very large load increments.     

A new efficient approach to the formulation of mixed finite element models for structural analysis

A new mixed finite element formulation is developed based on the Hellinger-Reissner principle with independent strain. By dividing the assumed strain into its lower order and higher order parts, the new formulation can be made much more efficient than the conventional mixed formulation. In addition the present new approach provides an alternative way of introducing a stabilization matrix to suppress undesirable kinematic modes.     

A hybrid stress nine-node degenerated shell element for geometric nonlinear analysis

A stabilized nine-node degenerated shell element, previously derived for linear analysis by the admissible matrix formulation, is extended to geometrically nonlinear analysis. The assumed stress modes pertinent to stabilization matrix are contravariant in nature and their energy products with the displacement-derived covariant strain can be programmed without resorting to numerical integration and the Gram-Schmidt orthogonalization. Numerical tests show that its accuracy is virtually identical to the uniformly reduced integration element.     

带有沙漏控制的相对自由度壳元

针对一点积分的八节点相对自由度壳单元存在的沙漏现象，提出采用拟应变法解决该问题的方法，并对锁死问题进行研究。给出了带有沙漏控制的八节点相对自由度壳元内的坐标、位移插值公式，推导了拟应变的表达式，通过Hu-Washizu变分原理，建立了有限元求解方程。利用Wilson非协调位移模式，单元的计算精度得到了明显改善。算例表明：基于八节点相对自由度壳单元，本文给出的沙漏控制算法能够有效的解决线性静力问题，并且具有较高的计算精度。     

Finite rotation FE-simulation and active vibration control of smart composite laminated structures

The present article focuses on the nonlinear finite element simulation and control of large amplitude vibrations of smart piezolaminated composite structures. Full geometrically nonlinear finite rotation strain–displacement relations and Reissner–Mindlin first-order shear deformation hypothesis to include the transverse shear effects are considered to derive the variational formulation. A quadratic variation of electric potential is assumed in transverse direction. An assumed natural strain method for the shear strains, an enhanced assumed strain method for the membrane strains and an enhanced assumed gradient method for the electric field is incorporated to improve the behavior of a four-node shell element. Numerical simulations presented in this article show the accurate prediction capabilities of the proposed method, especially for structures undergoing finite deformations and rotations, in comparison to the results obtained by simplified nonlinear models available in references and also with those obtained by using the C3D20RE solid element for piezoelectric layers in the Abaqus code.     

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.     

基于几何非线性不协调元变分原理的圆柱壳非线性初始稳定性分析

本文根据几何非线性不协调元增量变分原理,按严格的壳体方程,建立了高精度的圆柱壳几何非线性20参数矩形精化不协调元RCSR4,并用于圆柱壳非线性初始稳定性分析。计算结果表明,该方法收敛性良好。