A new nine‐node solid‐shell finite element using complete 3D constitutive laws

The solid‐shell element presented in this paper has nine nodes: eight are classically located at the apexes and are fitted with three translational DOFs whereas the ninth is sited at the center and is endowed with only one DOF; a displacement along the ‘thickness’ direction. Indeed, to be used for modeling thin structures under bending effects, this kind of finite element has a favored direction where several integration points are distributed. Besides, there is solely one ‘in‐plane’ quadrature point to avoid locking phenomena and prohibitive CPU costs for large nonlinear computations. Because a reduced integration is not enough to completely prevent transverse shear locking, a shear–strain field is assumed. Compared with the other eight‐node ‘solid‐shell' bricks, the presence of a supplementary node has a main aim: getting a linear normal strain component which, along with a full three‐dimensional constitutive strain–stress behavior, allows to achieve similar results in bending cases as those obtained with the usual plane stress state hypothesis. For that, the ninth node DOF plays the role of an extra parameter essential for a quadratic interpolation of the displacement in the thickness direction. The advantage is that this DOF has a physical meaning and, for instance, a strength equivalent to a normal pressure can be prescribed. With a suitable nodal numbering, the band width is not significantly increased and meshes can easily be generated because the extra nodes are always located at element centers. To emphasize the peculiar features of such an element, a set of examples (linear and nonlinear) is carried out. Numerous comparisons with other elements show pretty good results in bending dominating problems while adding the event of a normal stress component in sheet metal forming simulations with double side contact. Copyright © 2012 John Wiley & Sons, Ltd.     

An eight‐node hybrid‐stress solid‐shell element for geometric non‐linear analysis of elastic shells

This paper presents eight‐node solid‐shell elements for geometric non‐linear analysis of elastic shells. To subdue shear, trapezoidal and thickness locking, the assumed natural strain method and an ad hoc modified generalized laminate stiffness matrix are employed. A selectively reduced integrated element is formulated with its membrane and bending shear strain components taken to be constant and equal to the ones evaluated at the element centroid. With the generalized stresses arising from the modified generalized laminate stiffness matrix assumed to be independent from the ones obtained from the displacement, an extended Hellinger–Reissner functional can be derived. By choosing the assumed generalized stresses similar to the assumed stresses of a previous solid element, a hybrid‐stress solid‐shell element is formulated. Commonly employed geometric non‐linear homogeneous and laminated shell problems are attempted and our results are close to those of other state‐of‐the‐art elements. Moreover, the hybrid‐stress element converges more readily than the selectively reduced integrated element in all benchmark problems. Copyright © 2002 John Wiley & Sons, Ltd.     

A four‐node,shear‐deformable shell element developed via explicit Kirchhoff constraints

An efficient, four‐node quadrilateral shell element is formulated using a linear, first‐order shear deformation theory. The bending part of the formulation is constructed from a cross‐diagonal assembly of four three‐node anisoparametric triangular plate elements, referred to as MIN3. Closed‐form constraint equations, which arise from the Kirchhoff constraints in the thin‐plate limit, are derived and used to eliminate the degrees‐of‐freedom associated with the ‘internal’ node of the cross‐diagonal assembly. The membrane displacement field employs an Allman‐type, drilling degrees‐of‐freedom formulation. The result is a displacement‐based, fully integrated, four‐node quadrilateral element, MIN4T, possessing six degrees‐of‐freedom at each node. Results for a set of validation plate problems demonstrate that the four‐node MIN4T has similar robustness and accuracy characteristics as the original cross‐diagonal assembly of MIN3 elements involving five nodes. The element performs well in both moderately thick and thin regimes, and it is free of shear locking. Shell validation results demonstrate superior performance of MIN4T over MIN3, possibly as a result of its higher‐order interpolation of the membrane displacements. It is also noted that the bending formulation of MIN4T is kinematically compatible with the existing anisoparametric elements of the same order of approximation, which include a two‐node Timoshenko beam element and a three‐node plate element, MIN3. Copyright © 2000 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.     

Quadratic NMS Mindlin‐plate‐bending element

The development of a robust and efficient quadratic Mindlin‐plate‐bending elements mainly by the use of non‐conforming displacement modes is presented in this paper. A brief review on the previous efforts to develop efficient non‐conforming Mindlin plate bending elements is also given. The behaviour of the newly proposed plate element is further improved by the combined use of nonconforming displacement modes, the selectively reduced integration scheme, and the assumed shear strain fields. Thus, the newly developed element has been designated as ‘NMS‐8P’. The improvement achieved may be attributable to the fact that the merits of these improvement techniques are merged in the formation of the new element in a complementary manner. The proposed 8‐node element passes the patch tests, does not show spurious mechanism, and does not produce shear locking phenomena even with distorted meshes. It is also shown that the element produces reliable solutions through numerical tests for standard benchmark problems. Copypright © 1999 John Wiley & Sons, Ltd.     

An unsymmetric 8‐node hexahedral element with high distortion tolerance

Among all 3D 8‐node hexahedral solid elements in current finite element library, the ‘best’ one can produce good results for bending problems using coarse regular meshes. However, once the mesh is distorted, the accuracy will drop dramatically. And how to solve this problem is still a challenge that remains outstanding. This paper develops an 8‐node, 24‐DOF (three conventional DOFs per node) hexahedral element based on the virtual work principle, in which two different sets of displacement fields are employed simultaneously to formulate an unsymmetric element stiffness matrix. The first set simply utilizes the formulations of the traditional 8‐node trilinear isoparametric element, while the second set mainly employs the analytical trial functions in terms of 3D oblique coordinates (R, S, T). The resulting element, denoted by US‐ATFH8, contains no adjustable factor and can be used for both isotropic and anisotropic cases. Numerical examples show it can strictly pass both the first‐order (constant stress/strain) patch test and the second‐order patch test for pure bending, remove the volume locking, and provide the invariance for coordinate rotation. Especially, it is insensitive to various severe mesh distortions. Copyright © 2016 John Wiley & Sons, Ltd.     

A family of ANS four‐node exact geometry shell elements in general convected curvilinear coordinates

The non‐conventional exact geometry shell elements based on the Timoshenko–Mindlin kinematics with five displacement degrees of freedom are proposed. The term ‘exact geometry (EXG)’ reflects the fact that coefficients of the first and second fundamental forms of the reference surface and Christoffel symbols are taken exactly at every Gauss integration point. The choice of only displacements as fundamental shell unknowns gives an opportunity to derive strain–displacement relationships, which are invariant under rigid‐body shell motions in a convected curvilinear coordinate system. This paper presents a newly developed family consisting of three hybrid and one displacement‐based four‐node EXG shell elements. To avoid shear and membrane locking and have no spurious zero energy modes, the ANS concept is employed. The ANS interpolations satisfy exactly the plate compatibility equation for in‐plane strains. As a result, all EXG shell elements developed pass membrane and bending plate patch tests and exhibit a superior performance in the case of distorted coarse mesh configurations. Copyright © 2010 John Wiley & Sons, Ltd.     

Hybrid‐stress six‐node prismatic elements

This paper presents three novel hybrid‐stress six‐node prismatic elements. Starting from the element displacement interpolation, the equilibrating non‐constant stress modes for the first element are identified and orthogonalized with respect to the constant stress modes for higher computational efficiency. For the second element, the non‐constant stress modes are non‐equilibrating and chosen for the sake of stabilizing the reduced‐integrated element. The first two elements are intended for three‐dimensional continuum analysis with both passing the patch test for three‐dimensional continuum elements. The third element is primarily intended for plate/shell analysis. Shear locking is alleviated by a new assumed strain scheme which preserves the element accuracy with respect to the twisting load. Furthermore, the Poisson's locking along the in‐plane and out‐of‐plane directions is overcome by using the hybrid‐stress modes of the first element. The third element passes the patch test for plate/shell elements. Unless the element assumes the right prismatic geometry, it fails the patch test for three‐dimensional continuum elements. It will be seen that all the proposed elements are markedly more accurate than the conventional fully integrated element. Copyright © 2004 John Wiley & Sons, Ltd.     

Mean‐strain 10‐node tetrahedron with energy‐sampling stabilization

In this study, a new mean‐strain 10‐node tetrahedral element is developed using energy‐sampling stabilization. The proposed 10‐node tetrahedron is composed of several four‐node linear tetrahedral elements, four tetrahedra in the corners and four tetrahedra that tile the central octahedron in three possible sets of four‐node linear tetrahedra, corresponding to three different choices for the internal diagonal. The assumed strains are calculated from mean ‘basis function gradients.’ The energy‐sampling technique introduced previously for removing zero‐energy modes in the mean‐strain hexahedron is adapted for the present element: the stabilization energy is evaluated on the four‐corner tetrahedra. The proposed element naturally leads to a lumped‐mass matrix and does not have unphysical low‐energy vibration modes. For simplicity, we limit our developments to linear elasticity with compressible and nearly incompressible material. The numerical tests demonstrate that the present element performs well compared with the classical 10‐node tetrahedral elements for shell and plate structures, and nearly incompressible materials. Copyright © 2016 John Wiley & Sons, Ltd.     

A new reduced integration solid‐shell element based on EAS and ANS with hourglass stabilization

In this paper, a novel reduced integration eight‐node solid‐shell finite element formulation with hourglass stabilization is proposed. The enhanced assumed strain method is adopted to eliminate the well‐known volumetric and Poisson thickness locking phenomena with only one internal variable required. In order to alleviate the transverse shear and trapezoidal locking and correct rank deficiency simultaneously, the assumed natural strain method is implemented in conjunction with the Taylor expansion of the inverse Jacobian matrix. The projection of the hourglass strain‐displacement matrix and reconstruction of its transverse shear components are further employed to avoid excessive hourglass stiffness. The proposed solid‐shell element formulation successfully passes both the membrane and bending patch tests. Several typical examples are presented to demonstrate the excellent performance and extensive applicability of the proposed element. Copyright © 2015 John Wiley & Sons, Ltd.     

Patch‐averaged assumed strain finite elements for stress analysis

A finite element model for linear‐elastic small deformation problems is presented. The formulation is based on a weighted residual that requires a priori the satisfaction of the kinematic equation. In this approach, an averaged strain‐displacement matrix is constructed for each node of the mesh by defining an appropriate patch of elements, yielding a smooth representation of strain and stress fields. Connections with traditional and similar procedure are explored. Linear quadrilateral four‐node and linear hexahedral eight‐node elements are derived. Various numerical tests show the accuracy and convergence properties of the proposed elements in comparison with extant finite elements and analytic solutions. Specific examples are also included to illustrate the ability to resist numerical locking in the incompressible limit and insensitive response in the presence of shape distortion. Furthermore, the numerical inf‐sup test is applied to a selection of problems to show the stability of the present formulation. Copyright © 2012 John Wiley & Sons, Ltd.     

Efficient and accurate multilayer solid‐shell element: non‐linear materials at finite strain

We present in this paper an efficient and accurate low‐order solid‐shell element formulation for analyses of large deformable multilayer shell structures with non‐linear materials. The element has only displacement degrees of freedom (dofs), and an optimal number of enhancing assumed strain (EAS) parameters to pass the patch tests (both membrane and out‐of‐plane bending) and to remedy volumetric locking. Based on the mixed Fraeijs de Veubeke‐Hu‐Washizu (FHW) variational principle, the in‐plane and out‐of‐plane bending behaviours are improved and the locking associated with (nearly) incompressible materials is avoided via a new efficient enhancement of strain tensor. Shear locking and curvature thickness locking are resolved effectively by using the assumed natural strain (ANS) method. Two non‐linear 3‐D constitutive models (Mooney–Rivlin material and hyperelastoplastic material at finite strain) are applied directly without requiring the enforcement of the plane‐stress assumption. In particular, we give a simple derivation for the hyperelastoplastic model using spectral representations. In addition, the present element has a well‐defined lumped mass matrix, and provides double‐side contact surfaces for shell contact problems. With the dynamics referred to a fixed inertial frame, the present element can be used to analyse multilayer shell structures undergoing large overall motion. Numerical examples involving static analyses and implicit/explicit dynamic analyses of multilayer shell structures with both material and geometric non‐linearities are presented, and compared with existing results obtained from other shell elements and from a meshless method. It is shown that elements that did not pass the out‐of‐plane bending patch test could not provide accurate results, as compared to the present element formulation, which passed the out‐of‐plane bending patch test. The present element proves to be versatile and efficient in the modelling and analyses of general non‐linear composite multilayer shell structures. Copyright © 2005 John Wiley & Sons, Ltd.     

A reduced integration solid‐shell finite element based on the EAS and the ANS concept—Geometrically linear problems

In this paper a new reduced integration eight‐node solid‐shell finite element is presented. The enhanced assumed strain (EAS) concept based on the Hu–Washizu variational principle requires only one EAS degree‐of‐freedom to cure volumetric and Poisson thickness locking. One key point of the derivation is the Taylor expansion of the inverse Jacobian with respect to the element center, which closely approximates the element shape and allows us to implement the assumed natural strain (ANS) concept to eliminate the curvature thickness and the transverse shear locking. The second crucial point is a combined Taylor expansion of the compatible strain with respect to the center of the element and the normal through the element center leading to an efficient and locking‐free hourglass stabilization without rank deficiency. Hence, the element requires only a single integration point in the shell plane and at least two integration points in thickness direction. The formulation fulfills both the membrane and the bending patch test exactly, which has, to the authors' knowledge, not yet been achieved for reduced integration eight‐node solid‐shell elements in the literature. Owing to the three‐dimensional modeling of the structure, fully three‐dimensional material models can be implemented without additional assumptions. Copyright © 2009 John Wiley & Sons, Ltd.     

A stabilized one‐point integrated quadrilateral Reissner–Mindlin plate element

A new quadrilateral Reissner–Mindlin plate element with 12 element degrees of freedom is presented. For linear isotropic elasticity a Hellinger–Reissner functional with independent displacements, rotations and stress resultants is used. Within the mixed formulation the stress resultants are interpolated using five parameters for the bending moments and four parameters for the shear forces. The hybrid element stiffness matrix resulting from the stationary condition can be integrated analytically. This leads to a part obtained by one‐point integration and a stabilization matrix. The element possesses a correct rank, does not show shear locking and is applicable for the evaluation of displacements and stress resultants within the whole range of thin and thick plates. The bending patch test is fulfilled and the computed numerical examples show that the convergence behaviour is better than comparable quadrilateral assumed strain elements. Copyright © 2004 John Wiley & Sons, Ltd.     

A 3‐node flat triangular shell element with corner drilling freedoms and transverse shear correction

The formulation, implementation and testing of simple, efficient and robust shell finite elements have challenged investigators over the past four decades. A new 3‐node flat triangular shell element is developed by combination of a membrane component and a plate bending component. The ANDES‐based membrane component includes rotational degrees of freedom, and the refined nonconforming element method‐based bending component involves a transverse shear correction. Numerical examples are carried out for benchmark tests. The results show that compared with some popular shell elements, the present one is simple but exhibits excellent all‐around properties (for both membrane‐and bending‐dominated situations), such as free of aspect ratio locking, passing the patch test, free of shear locking, good convergence and high suitability for thin to moderately thick plates. The developed element has already been adopted in a warpage simulation package for injection molding. Copyright © 2011 John Wiley & Sons, Ltd.     

Selective smoothed finite element methods for extremely large deformation of anisotropic incompressible bio‐tissues

Two dynamic selective smoothed FEM (selective S‐FEM) are proposed for analysis of extremely large deformation of anisotropic incompressible bio‐tissues using the simplest four‐node tetrahedron elements. In the present two Selective S‐FEMs, the method that consists of the face‐based smoothed FEM (FS‐FEM) used for the deviatoric part of deformation and the node‐based smoothed FEM (NS‐FEM) used for the volumetric part is called FS/NS‐FEM; another method that replaces the deviatoric part of deformation in the first one by the edge‐based smoothed FEM (3D‐ES‐FEM) is call 3D‐ES/NS‐FEM. Both selective S‐FEMs can achieve outstanding accuracy, and stability of volumetric locking free. This is because the NS‐FEM offers an ‘overly‐soft’ feature (in contrast to the standard FEM ‘overly‐stiff’ model), which can be used to effectively mitigate the volumetric locking, and on the other hand, the 3D‐ES‐FEM and FS‐FEM produce close to exact stiffness for the discretized model leading to accurate solution. Numerical examples are presented to examine the performance of the selective S‐FEM methods, including soft bio‐tissues that may be isotropic, transversely isotropic, and anisotropic arterial layered materials. The present methods are found having good accuracy and performance. The examples also demonstrate that the proposed methods are very robust and possess remarkable capabilities of handling element distortion, which is very useful for simulating soft materials including bio‐tissues. Copyright © 2014 John Wiley & Sons, Ltd.     

A new shell formulation using complete 3D constitutive laws

This paper should not be only regarded as a presentation of new shell elements but rather as a methodology which can be applied to most classical shell elements and has two aims: Achieving the same results in bending cases while breaking from plane stress state hypothesis and adding a normal stress component for process simulations such as hydro‐forming, hemming, sheet metal forming with bottoming, flanging, incremental forming and so on. Owing to the non‐linear applications quoted before, only shell elements with one integration point on the mid‐plane are selected: Triangles that are naturally constant strain elements and reduced integration quadrilaterals. The method mainly consists of adding a central node at the center (of gravity for a triangle) with two degrees of freedom: Two translations normal to the mid‐surface for which one corresponds to the bottom surface (‘lower skin’ of the shell) and the other to the top surface (‘upper skin’ of the shell). Then a full 3D constitutive strain–stress behavior can be used. For triangles in bending state—either based on Kirchhoff's or on Mindlin's assumptions—, it is shown that the results are exactly the same as those given by the initial formulation of these elements using a plane stress hypothesis. For quadrilaterals, the results are slightly different but many numerical examples—including non‐linear computations—prove that those differences are not significant. Copyright © 2010 John Wiley & Sons, Ltd.     

Non‐linear analysis of structures using high performance hybrid elements

In this work, we describe the formulation and implementation for stress‐based hybrid elements for conducting non‐linear analysis of elastic structures. The motivation behind developing these elements is that they should be as simple to use as standard displacement‐based isoparametric brick elements, but at the same time, be relatively immune to the shortcoming that these elements suffer from, namely, ‘locking’ problems which occur when they are used to model plate/shell geometries, almost incompressible materials or when the elements are distorted, and so on. The formulation is based on a two‐field mixed variational principle. Numerical examples are presented to demonstrate the excellent performance of the proposed elements on a variety of challenging problems involving very large deformations, buckling, mesh distortions, almost incompressible materials, etc. Copyright © 2006 John Wiley & Sons, Ltd.     

Corotational non‐linear analysis of thin plates and shells using a new shell element

A new three‐node triangular shell element is developed by combining the optimal membrane element and discrete Kirchhoff triangle (DKT) plate bending element, and is modified for laminated composite plates and shells so as to include the membrane‐bending coupling effect. Using appropriate shape functions for the bending and membrane modes of the element, the ‘inconsistent’ stress stiffness matrix is formulated and the tangent stiffness matrix is determined. Non‐linear analysis of thin‐walled structures with geometric non‐linearity is conducted using the corotational method. The new element is thoroughly tested by solving few popular benchmark problems. The results of the analysis are compared with those obtained using existing membrane elements. Copyright © 2006 John Wiley & Sons, Ltd.