An efficient co‐rotational formulation for curved triangular shell element

A 6‐node curved triangular shell element formulation based on a co‐rotational framework is proposed to solve large‐displacement and large‐rotation problems, in which part of the rigid‐body translations and all rigid‐body rotations in the global co‐ordinate system are excluded in calculating the element strain energy. Thus, an element‐independent formulation is achieved. Besides three translational displacement variables, two components of the mid‐surface normal vector at each node are defined as vectorial rotational variables; these two additional variables render all nodal variables additive in an incremental solution procedure. To alleviate the membrane and shear locking phenomena, the membrane strains and the out‐of‐plane shear strains are replaced with assumed strains in calculating the element strain energy. The strategy used in the mixed interpolation of tensorial components approach is employed in defining the assumed strains. The internal force vector and the element tangent stiffness matrix are obtained from calculating directly the first derivative and second derivative of the element strain energy with respect to the nodal variables, respectively. Different from most other existing co‐rotational element formulations, all nodal variables in the present curved triangular shell formulation are commutative in calculating the second derivative of the strain energy; as a result, the element tangent stiffness matrix is symmetric and is updated by using the total values of the nodal variables in an incremental solution procedure. Such update procedure is advantageous in solving dynamic problems. Finally, several elastic plate and shell problems are solved to demonstrate the reliability, efficiency, and convergence of the present formulation. Copyright © 2007 John Wiley & Sons, Ltd.     

A 9-node co-rotational quadrilateral shell element

A new 9-node co-rotational curved quadrilateral shell element formulation is presented in this paper. Different from other existing co-rotational element formulations: (1) Additive rotational nodal variables are utilized in the present formulation, they are two well-chosen components of the mid-surface normal vector at each node, and are additive in an incremental solution procedure; (2) the internal force vector and the element tangent stiffness matrix are respectively the first derivative and the second derivative of the element strain energy with respect to the nodal variables, furthermore, all nodal variables are commutative in calculating the second derivatives, resulting in symmetric element tangent stiffness matrices in the local and global coordinate systems; (3) the element tangent stiffness matrix is updated using the total values of the nodal variables in an incremental solution procedure, making it advantageous for solving dynamic problems. Finally, several examples are solved to verify the reliability and computational efficiency of the proposed element formulation.     

新型协同转动六节点三边形复合材料壳单元

为分析复合材料层合板壳结构,提出了一种协同转动六节点三边形复合材料曲壳单元。不同于现有的其它协同转动有限单元:1) 该单元中采用了增量可加的矢量型转动变量,因而在非线性增量求解过程中更新节点转动变量非常简单;2) 在计算应变能对局部节点变量的二阶偏微分时,微分的次序是可以交换的,并且通过链式微分计算应变能对整体节点变量的二阶偏微分时,微分的次序也是可以交换的,因此,得到的局部和整体坐标系下的切线刚度矩阵都是对称的;3) 在此有限单元公式中引入了混合公式法,以减轻膜闭锁和剪切闭锁的不利影响。对4个典型算例进行了分析,并与其他文献的结果进行对比,该文提出的单元的可靠性和计算效率得到了验证。     

A four‐node corotational quadrilateral elastoplastic shell element using vectorial rotational variables

A four‐node corotational quadrilateral elastoplastic shell element is presented. The local coordinate system of the element is defined by the two bisectors of the diagonal vectors generated from the four corner nodes and their cross product. This local coordinate system rotates rigidly with the element but does not deform with the element. As a result, the element rigid‐body rotations are excluded in calculating the local nodal variables from the global nodal variables. The two smallest components of each nodal orientation vector are defined as rotational variables, leading to the desired additive property for all nodal variables in a nonlinear incremental solution procedure. Different from other existing corotational finite‐element formulations, the resulting element tangent stiffness matrix is symmetric owing to the commutativity of the local nodal variables in calculating the second derivative of strains with respect to these variables. For elastoplastic analyses, the Maxwell–Huber–Hencky–von Mises yield criterion is employed, together with the backward‐Euler return‐mapping method, for the evaluation of the elastoplastic stress state; the consistent tangent modulus matrix is derived. To eliminate locking problems, we use the assumed strain method. Several elastic patch tests and elastoplastic plate/shell problems undergoing large deformation are solved to demonstrate the computational efficiency and accuracy of the proposed formulation. Copyright © 2013 John Wiley & Sons, Ltd.     

新型协同转动3节点三边形壳单元

发展了一种新型3节点三边形壳单元。计算单元在局部坐标系下的节点变量时,通过采用协同转动法,预先扣除节点整体变量中的刚体转动成分,从而简化了单元的计算公式。不同于现有的其他协同转动单元,在该单元中采用了增量可以直接累加的矢量型转动变量,单元的切线刚度矩阵可以通过直接计算能量泛函对节点变量的二阶偏微分得到,且对节点变量的偏微分次序是可以互换的,因而在局部和整体坐标系下都得到了对称的单元切线刚度矩阵。为消除单元中可能出现的闭锁现象,引入了MacNeal提出的线积分法,分别用沿单元边线方向的膜应变和剪切应变构造新的假定应变场。最后,通过对几个产生了大位移与大转角变形的板壳问题进行分析,检验了该单元的可靠性、计算精度和计算效率。     

A simplified co‐rotational method for quadrilateral shell elements in geometrically nonlinear analysis

This paper presents a simplified co‐rotational formulation for quadrilateral shell elements inheriting the merit of element‐independence from the traditional co‐rotational approach in literature. With the objective of application to nonlinear analysis of civil engineering structures, the authors further simplify the formulation of the geometrical stiffness using the small strain assumption, which is valid in the co‐rotational approach, with the warping effects considered as eccentricities. Compared with the traditional element‐independent co‐rotational method, the projector is neglected both in the tangent stiffness matrix and in the internal force vector for simplicity in formulation. Meanwhile, a quadrilateral flat shell element allowing for drilling rotations is adopted and incorporated into this simplified co‐rotational algorithm for geometrically nonlinear analysis involved with large displacements and large rotations. Several benchmark problems are presented to confirm the efficiency and accuracy of the proposed method for practical applications.     

Meshfree co‐rotational formulation for two‐dimensional continua

In this paper, a meshfree co‐rotational formulation for two‐dimensional continua is proposed. In a co‐rotational formulation, the motion of a body is separated into rigid motion and strain‐producing deformation. Traditionally, this has been done in the setting of finite elements for beams and shell‐type elements. In the present work every node in a meshfree discretized domain has its own co‐rotating coordinate system. Three key ingredients are established in order to apply the co‐rotational formulation: (i) the relationship between global and local variables, (ii) the angle of rotation of a typical co‐rotating coordinate system, and (iii) a variationally consistent tangent stiffness matrix. An algorithm for the co‐rotational formulation based on load control is provided. Maximum‐entropy basis functions are used to discretize the domain and stabilized nodal integration is implemented to construct the global system of equations. Numerical examples are presented to demonstrate the validity of the meshfree co‐rotational formulation. Copyright © 2009 John Wiley & Sons, Ltd.     

Bisector and zero‐macrospin co‐rotational systems for shell elements

A principal issue in any co‐rotational approach for large displacement analysis of plates and shells is associated with the specific choice of the local reference system in relation to the current deformed element configuration. Previous approaches utilised local co‐rotational systems, which are invariant to nodal ordering, a characteristic that is deemed desirable on several fronts; however, the associated definitions of the local reference system suffered from a range of shortcomings, including undue complexity, dependence on the local element formulation and possibly an asymmetric tangent stiffness matrix. In this paper, new definitions of the local co‐rotational system are proposed for quadrilateral and triangular shell elements, which achieve the invariance characteristic to the nodal ordering in a relatively simple manner and address the aforementioned shortcomings. The proposed definitions utilise only the nodal coordinates in the deformed configuration, where two alternative definitions, namely, bisector and zero‐macrospin definitions, are presented for each of quadrilateral and triangular finite elements. In each case, the co‐rotational transformations linking the local and global element entities are presented, highlighting the simplicity of the proposed approach. Several numerical examples are finally presented to demonstrate the effectiveness and relative accuracy of the alternative definitions proposed for the local co‐rotational system. Copyright © 2015 John Wiley & Sons, Ltd.     

A co-rotational formulation for 3D beam element using vectorial rotational variables

Based on a co-rotational framework, a 3-noded iso-parametric element formulation of 3D beam was presented, which was used for accurate modelling of frame structures with large displacements and large rotations. Firstly, a co-rotational framework was fixed at the internal node of the element, it translates and rotates with the node rigidly; then, vectorial rotational variables were defined, they are three smaller components of the cross-sectional principal vectors at each node, sometimes they represent different components of the cross-sectional principal vectors in incremental solution procedure so as to avoid the occurrence of ill-conditioned tangent stiffness matrix; thereafter, the internal force vector and tangent stiffness matrix in local system was derived from the strain energy of the element as its first partial derivative and second partial derivative with respect to local variables, respectively, and a symmetric tangent stiffness matrix was achieved; finally, several examples were analysed to illustrate the reliability and accuracy of this procedure.This work is supported by National Natural Science Foundation of China (50408022), and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry and Zhejiang Province     

用于光滑/非光滑壳的稳定化新型协同转动4节点四边形壳单元

该文发展了一种适用于光滑壳和非光滑壳的新型协同转动4节点四边形壳单元。在单元中每个节点采用了3个平动自由度和2/3个矢量型转动自由度,每个光滑壳的节点或非光滑壳的非交界节点采用壳中性面法向矢量的2个最小分量作为矢量型转动变量,在非光滑壳中性面交界线上的节点采用3个矢量型转动变量,他们分别是节点定向矢量组中一个定向矢量的较小或最小分量和另一定向矢量的2个最小分量。在非线性增量求解过程中,这些矢量型转动变量可以采用简单的加法将增量累加到原矢量中直接进行更新,且采用了协同转动框架的单元在局部和整体坐标系下得到的切线刚度矩阵都是对称的,结构整体切线刚度矩阵可以采用一维线性存储,可节省大量的计算机存储资源和计算时间。为消除膜闭锁和剪切闭锁的不利影响,采用单点积分方案计算单元内力矢量和切线刚度矩阵,并借鉴Belytschko提出的物理稳定化零能模态控制法来消除可能出现的零能模态。通过对2个光滑壳和2个非光滑壳进行非线性分析,检验了单元的可靠性、计算效率与计算精度。     

An enhanced co‐rotational approach for large displacement analysis of plates

This paper presents a new co‐rotational approach for the large displacement analysis of plates employing 4‐noded quadrilateral flat shell elements. The proposed approach benefits from (i) a simple local co‐rotational system invariant to the element nodal ordering, (ii) the choice of the two smallest components of the nodal normal vector as global rotational degrees of freedom, and (iii) the use of hierarchic freedoms, that are unaffected by the co‐rotational transformations, for higher‐order accuracy. Important additional benefits that arise from the aforementioned features include symmetry of the tangent stiffness matrix and complete insensitivity of the large displacement transformations to the size of the incremental step. The applicability of the new approach to moderately thick as well as thin plates is illustrated by considering two alternative local formulations based on the Reissner–Mindlin and discrete Kirchhoff hypotheses. Several examples are finally presented which demonstrate the accuracy, step‐insensitivity and computational benefits of the proposed co‐rotational approach for large displacement analysis of plate structures. Copyright © 2005 John Wiley & Sons, Ltd.     

Geometrically nonlinear analysis of shells by quadrilateral flat shell element with drill,shear, and warping

In this paper, a four‐node quadrilateral flat shell element is proposed for geometrically nonlinear analysis based on updated Lagrangian formulation with the co‐rotational kinematics concept. The flat shell element combines the membrane element with drilling degrees of freedom and the plate element with shear deformation. By means of these linearized elements, a simplified nonlinear analysis procedure allowing for warping of the flat shell element and large rotation is proposed. The tangent stiffness matrix and the internal force recovery are formulated in this paper. Several classic benchmark examples are presented to validate the accuracy and efficiency of the proposed new and more proficient element for practical engineering analysis of shell structures. Copyright © 2016 John Wiley & Sons, Ltd.     

Mixed finite element method for coupled thermo‐hydro‐mechanical process in poro‐elasto‐plastic media at large strains

A mixed finite element for coupled thermo‐hydro‐mechanical (THM) analysis in unsaturated porous media is proposed. Displacements, strains, the net stresses for the solid phase; pressures, pressure gradients, Darcy velocities for pore water and pore air phases; temperature, temperature gradients, the total heat flux are interpolated as independent variables. The weak form of the governing equations of coupled THM problems in porous media within the element is given on the basis of the Hu–Washizu three‐filed variational principle. The proposed mixed finite element formulation is derived. The non‐linear version of the element formulation is further derived with particular consideration of the THM constitutive model for unsaturated porous media based on the CAP model. The return mapping algorithm for the integration of the rate constitutive equation, the consistent elasto‐plastic tangent modulus matrix and the element tangent stiffness matrix are developed. For geometrical non‐linearity, the co‐rotational formulation approach is utilized. Numerical results demonstrate the capability and the performance of the proposed element in modelling progressive failure characterized by strain localization and the softening behaviours caused by thermal and chemical effects. Copyright © 2005 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.     

Collapse of thin shell structures—stress resultant plasticity modelling within a co‐rotated ANDES finite element formulation

Due to the very non‐linear behaviour of thin shells under collapse, numerical simulations are subject to challenges. Shell finite elements are attractive in these simulations. Rotational degrees of freedom do, however, complicate the solution. In the present study a co‐rotated formulation is employed. The deformation of the shell is decomposed in to a contribution from large rigid body rotation and a strain producing term. A triangular assumed strain shell finite element is used. Hence, a high performance elastic element is combined with the co‐rotated formulation. In the co‐rotated co‐ordinate system the plasticity is accounted for by a simplifyed Ilyushin stress resultant yield surface. The stress update is determined from the backward Euler difference, and a consistent geometrical and material tangent stiffness is derived. Comparison with other published analysis results show that the present formulation gives acceptable accuracy. Copyright © 1999 John Wiley & Sons, Ltd.     

Analysis of a rotation‐free 4‐node shell element

In this paper, a shell element for small and large deformations is presented based on the extension of the methodology to derive triangular shell element without rotational degrees of freedom (so‐called rotation‐free). As in our original triangular S3 element, the curvatures are computed resorting to the surrounding elements. However, the extension to a quadrilateral element requires internal curvatures in order to avoid singular bending stiffness. The quadrilateral area co‐ordinates interpolation is used to establish the required expressions between the rigid‐body modes of normal nodal translations and the normal through thickness bending strains at mid‐side. In order to propose an attractive low‐cost shell element, the one‐point quadrature is achieved at the centre for the membrane strains, which are superposed to the bending strains in the centred co‐rotational local frame. The membrane hourglass control is obtained by the perturbation stabilization procedure. Free, simply supported and clamped edges are considered without introducing virtual nodes or elements. Several numerical examples with regular and irregular meshes are performed to show the convergence, accuracy and the reasonable little sensitivity to geometric distortion. Based on an updated Lagrangian formulation and Newton iterations, the large displacements of the pinched hemispherical shell show the effectiveness of the proposed simplified element (S4). Finally, the deep drawing of a square box including large plastic strains with contact and friction completes the ability of the rotation‐free quadrilateral element for sheet‐metal‐forming simulations. Copyright © 2005 John Wiley & Sons, Ltd.     

A mixed tetrahedral element with nodal rotations for large‐displacement analysis of inelastic structures

A novel mixed four‐node tetrahedral finite element, equipped with nodal rotational degrees of freedom, is presented. Its formulation is based on a Hu–Washizu‐type functional, suitable to the treatment of material nonlinearities. Rotation and skew‐symmetric stress fields are assumed as independent variables, the latter entering the functional to impose rotational compatibility and suppress spurious modes. Exploiting the choice of equal interpolation for strain and symmetric stress fields, a robust element state determination procedure, requiring no element‐level iteration, is proposed. The mixed element stability is assessed by means of an original and effective numerical test. The extension of the present formulation to geometric nonlinear problems is achieved through a polar decomposition‐based corotational framework. After validation in both material and geometric nonlinear context, the element performances are investigated in demanding simulations involving complex shape memory alloy structures. Supported by the comparison with available linear and quadratic tetrahedrons and hexahedrons, the numerical results prove accuracy, robustness, and effectiveness of the proposed formulation. Copyright © 2016 John Wiley & Sons, Ltd.     

A co‐rotational grid‐based model for fabric drapes

Fabric drapes are typical large displacement, large rotation and small strain problems. Compared to conventional geometric non‐linear shell analyses, computational fabric drape analysis is particularly challenging due to the extremely weak bending rigidities of fabrics. Compared to continuum shell finite element methods, grid‐ or particle‐based methods appear to be more successful in high drapeability problems. The latter methods often resort to simple particle mechanics and formulate the elastic energy in terms of the inter‐particle distances and trigonometrical functions of the angles between the straight lines joining adjacent particles. In this paper, the co‐rotational approach and commonly employed assumptions for small strain problems in finite element analysis will be adopted to formulate the elastic energy. It will be seen that the internal force vector and the stiffness matrix are considerably simpler than other grid‐based models, yet the sparsity of the tangential stiffness matrix remains unchanged. A number of examples are considered and the predicted results are promising. Copyright © 2003 John Wiley & Sons, Ltd.     

A general high‐order finite element formulation for shells at large strains and finite rotations

For hyperelastic shells with finite rotations and large strains a p‐finite element formulation is presented accommodating general kinematic assumptions, interpolation polynomials and particularly general three‐dimensional hyperelastic constitutive laws. This goal is achieved by hierarchical, high‐order shell models. The tangent stiffness matrices for the hierarchical shell models are derived by computer algebra. Both non‐hierarchical, nodal as well as hierarchical element shape functions are admissible. Numerical experiments show the high‐order formulation to be less prone to locking effects. Copyright © 2003 John Wiley & Sons, Ltd.