Enhanced Assumed Strain Method Research Articles

Mesh distortion insensitive and locking‐free Petrov–Galerkin low‐order EAS elements for linear elasticity

AbstractOne of the most successful mixed finite element methods in solid mechanics is the enhanced assumed strain (EAS) method developed by Simo and Rifai in 1990. However, one major drawback of EAS elements is the highly mesh dependent accuracy. In fact, it can be shown that not only EAS elements, but every finite element with a symmetric stiffness matrix must either fail the patch test or be sensitive to mesh distortion in bending problems (higher order displacement modes) if the shape of the element is arbitrary. This theorem was established by MacNeal in 1992. In the present work we propose a novel Petrov–Galerkin approach for the EAS method, which is equivalent to the standard EAS method in case of regular meshes. However, in case of distorted meshes, it allows to overcome the mesh‐distortion sensitivity without loosing other advantages of the EAS method. Three design conditions established in this work facilitate the construction of the element which does not only fulfill the patch test but is also exact in many bending problems regardless of mesh distortion and has an exceptionally high coarse mesh accuracy. Consequently, high quality demands on mesh topology might be relaxed.

Transformations and Enhancement Approaches for the Large Deformation EAS Method

AbstractThe computer simulation of large deformation solid mechanics requires highly efficient finite elements. One widely used method enabling the construction of such elements is the enhanced assumed strain (EAS) method (cf. [1]). Within that framework, many transformations for the enhanced as well as the compatible deformation gradient have been used in previous publications (see e.g. [2]). We propose a new frame‐invariant alternative which fulfills the patch test and exhibits superior bending performance. Finally, novel mixed finite element methods based on the combination of mixed finite elements for polyconvexity [3] with the EAS method are presented. It is shown that this combination provides a promising framework for novel finite element methods.

An improved enhanced solid shell element for static and buckling analysis of shell structures

This paper presents an improved higher order solid shell element for static and buckling analysis of laminated composite structures based on the enhanced assumed strain (EAS). The transverse shear strain is divided into two parts: the first one is independent of the thickness coordinate and formulated by the assumed natural strain (ANS) method; the second part is an enhancing part, which ensures a quadratic distribution through the thickness. This allows removing the shear correction factors and improves the accuracy of transverse shear stresses. In addition, volumetric locking is completely avoided by using the optimal parameters in the EAS method. The formulated finite element is implemented to study the static and buckling behavior of shell structures and to investigate the influence of some parameters on the buckling load. Comparisons of numerical results with those extracted from literature show the acceptable performance of the developed element.

적층된 복합 및 샌드위치 판 구조의 자유진동 해석을 위한 EAS 고체 유한요소

This study deals with an enhanced assumed strain (EAS) three-dimensional element for free vibration analysis of laminated composite and sandwich structures. The three-dimensional finite element (FE) formulation based on the EAS method for composite structures shows excellence from the standpoints of computational efficiency, especially for distorted element shapes. Using the EAS FE formulation developed for this study, the effects of side-to-thickness ratios, aspect ratios and ply orientations on the natural frequency are studied and compared with the available elasticity solutions and other plate theories. The numerical results obtained are in good agreement with those reported by other investigators. The new approach works well for the numerical experiments tested, especially for complex structures such as sandwich plates with laminated composite faces.

Optimal low-order fully integrated solid-shell elements

This paper presents three optimal low-order fully integrated geometrically nonlinear solid-shell elements based on the enhanced assumed strain (EAS) method and the assumed natural strain method for different types of structural analyses, e.g. analysis of thin homogeneous isotropic and multilayer anisotropic composite shell-like structures and the analysis of (near) incompressible materials. The proposed solid-shell elements possess eight nodes with only displacement degrees of freedom and a few internal EAS parameters. Due to the 3D geometric description of the proposed elements, 3D constitutive laws can directly be employed in these formulations. The present formulations are based on the well-known Fraeijs de Veubeke---Hu---Washizu multifield variational principle. In terms of accuracy as well as efficiency point of view, the choice of the optimal EAS parameters plays a very critical role in the EAS method, therefore a systematic numerical study has been carried out to find out the optimal EAS parameters to alleviate different locking phenomena for the proposed solid-shell formulations. To assess the accuracy of the proposed solid-shell elements, a variety of popular numerical benchmark examples related to element convergence, mesh distortions, element aspect ratios and different locking phenomena are investigated and the results are compared with the well-known solid-shell formulations available in the literature. The results of our numerical assessment show that the proposed solid-shell formulations provide very accurate results, without showing any numerical problems, for a variety of geometrically linear and nonlinear structural problems.

Parametric effects on embedded delamination buckling in composite structures using the EAS three-dimensional element

This study deals with parametric effects on bucking behaviors of laminated composite structures containing an embedded rectangular delamination using the enhanced assumed strain (EAS) three-dimensional element. The three-dimensional finite element (FE) formulation based on the EAS method for composite structures shows excellence from the standpoints of computational efficiency, especially for distorted element shapes. Using the EAS FE formulation developed for this study, the effects of embedded delamination sizes and ply orientations on the elastic buckling behaviors for various aspect ratios and width-to-thickness ratios are studied. The numerical results obtained are in good agreement with those reported by other investigators. Furthermore, the new results reported in this paper show the progression of local buckling and its influence on global buckling and vice versa. Key observation points are discussed and a brief design guideline is given.

Shear Locking in a Plane Elasticity Problem and the Enhanced Assumed Strain Method

The enhanced assumed strain (EAS) method is a popular tool for avoiding locking phenomena, e.g., a remedy for shear locking in plane elasticity. We consider bending-dominated problems on thin bodies which can be treated as beams and prove that the degree of approximation of the EAS method is at least as good as that of a beam model. The hypercircle method is combined with arguments of nonconforming methods.

A new axi-symmetric element for thin walled structures

A new axi-symmetric finite element for thin walled structures is presented in this work. It uses the solid-shell element’s concept with only a single element and multiple integration points along the thickness direction. The cross-section of the element is composed of four nodes with two degrees of freedom each. The proposed formulation overcomes many locking pathologies including transverse shear locking, Poisson’s locking and volumetric locking. For transverse shear locking, the formulation uses the selective reduced integration technique, for Poisson’s locking it uses the enhanced assumed strain (EAS) method with only one enhancing variable. The B-bar approach is used to eliminate the isochoric deformations in the hourglass field while the EAS method is used to alleviate the volumetric locking in the constant part of the deformation tensor. Several examples are shown to demonstrate the performance and accuracy of the proposed element with special focus on the numerical simulations for the beverage can industry.

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

AbstractIn 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.

Four‐node semi‐EAS element in six‐field nonlinear theory of shells

AbstractWe propose a new four‐node C0 finite element for shell structures undergoing unlimited translations and rotations. The considerations concern the general six‐field theory of shells with asymmetric strain measures in geometrically nonlinear static problems. The shell kinematics is of the two‐dimensional Cosserat continuum type and is described by two independent fields: the vector field for translations and the proper orthogonal tensor field for rotations. All three rotational parameters are treated here as independent. Hence, as a consequence of the shell theory, the proposed element has naturally six engineering degrees of freedom at each node, with the so‐called drilling rotation. This property makes the element suitable for analysis of shell structures containing folds, branches or intersections. To avoid locking phenomena we use the enhanced assumed strain (EAS) concept. We derive and linearize the modified Hu–Washizu principle for six‐field theory of shells. What makes the present approach original is the combination of EAS method with asymmetric membrane strain measures. Based on literature, we propose new enhancing field and specify the transformation matrix that accounts for the lack of symmetry. To gain knowledge about the suitability of this field for asymmetric strain measures and to assess the performance of the element, we solve typical benchmark examples with smooth geometry and examples involving orthogonal intersections of shell branches. Copyright © 2006 John Wiley & Sons, Ltd.

A new approach to reduce membrane and transverse shear locking for one-point quadrature shell elements: linear formulation

In the last decade, one-point quadrature shell elements attracted many academic and industrial researchers because of their computational performance, especially if applied for explicit finite element simulations. Nowadays, one-point quadrature finite element technology is not only applied for explicit codes, but also for implicit finite element simulations, essentially because of their efficiency in speed and memory usage as well as accuracy. In this work, one-point quadrature shell elements are combined with the enhanced assumed strain (EAS) method to develop a finite element formulation for shell analysis that is, simultaneously, computationally efficient and more accurate. The EAS method is formulated to alleviate locking pathologies existing in the stabilization matrices of one-point quadrature shell elements. An enhanced membrane field is first constructed based on the quadrilateral area coordinate method, to improve element's accuracy under in-plane loads. The finite element matrices were projected following the work of Wilson et al. (Numerical and Computer Methods in Structural Mechanics, Fenven ST et al. (eds). Academic Press: New York, 1973; 43–57) for the incompatible modes approach, but the present implementation led to more accurate results for distorted meshes because of the area coordinate method for quadrilateral interpolation. The EAS method is also used to include two more displacement vectors in the subspace basis of the mixed interpolation of tensorial components (MITC) formulation, thus increasing the dimension of the null space for the transverse shear strains. These two enhancing vectors are shown to be fundamental for the Morley skew plate example in particular, and in improving the element's transverse shear locking behaviour in general. Copyright © 2005 John Wiley & Sons, Ltd.

Optimal solid shells for non-linear analyses of multilayer composites. I. Statics

We present in this paper a simple low-order solid-shell element formulation – having only displacement degrees of freedom (dofs), i.e., without rotational dofs – that has an optimal number of parameters to pass the patch tests, and thus allows for efficient and accurate analyses of large deformable multilayer shell structures using elements at extremely high aspect ratio. The formulation of this element is based on the mixed Fraeijs de Veubeke–Hu–Washizu (FHW) variational principle leading to a novel enhancing strain tensor (EAS method) that renders the computation particularly efficient, with improved in-plane and out-of-plane bending behavior (Poisson thickness locking), especially in refined analyses of composite structures involving a large number of high aspect-ratio layers. We also review the equivalence between various choices of the enhancing strains in tensor form, and point out the relative efficiency of these choices. We discuss the enhanced assumed strain (EAS) formulations based on both the Green–Lagrange strain and the displacement gradient (the companion paper), point out the pitfalls in each approach, e.g., not passing the patch test, and the possibility and the method to remedy the problem. Shear locking and curvature thickness locking are treated using the assumed natural strain (ANS) method. The element passes the patch tests (both membrane and out-of-plane bending). We provide an optimal combination of the ANS method and the minimal number of EAS parameters required to pass the out-of-plane bending patch test. Numerical examples involving static analyses of multilayer shell structures having a large range of element aspect ratios are presented. Finally, we note that the topic in this paper is a fitting dedication to Professor Ekkehard Ramm, who has made important pioneering contributions in this research direction.

Development of shear locking‐free shell elements using an enhanced assumed strain formulation

AbstractThe degenerated approach for shell elements of Ahmad and co‐workers is revisited in this paper. To avoid transverse shear locking effects in four‐node bilinear elements, an alternative formulation based on the enhanced assumed strain (EAS) method of Simo and Rifai is proposed directed towards the transverse shear terms of the strain field. In the first part of the work the analysis of the null transverse shear strain subspace for the degenerated element and also for the selective reduced integration (SRI) and assumed natural strain (ANS) formulations is carried out. Locking effects are then justified by the inability of the null transverse shear strain subspace, implicitly defined by a given finite element, to properly reproduce the required displacement patterns. Illustrating the proposed approach, a remarkably simple single‐element test is described where ANS formulation fails to converge to the correct results, being characterized by the same performance as the degenerated shell element. The adequate enhancement of the null transverse shear strain subspace is provided by the EAS method, enforcing Kirchhoff hypothesis for low thickness values and leading to a framework for the development of shear‐locking‐free shell elements. Numerical linear elastic tests show improved results obtained with the proposed formulation. Copyright © 2001 John Wiley & Sons, Ltd.

A deformation dependent stabilization technique, exemplified by EAS elements at large strains

Stabilized finite element methods have been developed mainly in the context of Computational Fluid Dynamics (CFD) and have shown to be able to add stability to previously unstable formulations in a consistent way. In this contribution a deformation dependent stabilization technique, conceptually based on the above mentioned developments in the CFD area, is developed for Solid Mechanics to cure the well-known enhanced assumed strain (EAS) method from artificial instabilities (hourglass modes) that have been observed in the range of large compressive strains. In investigating the defect of the original formulation the dominating role of the kinematic equation as cause for the instabilities is revealed. This observation serves as key ingredient for the design of the stabilizing term, introduced on the level of the variational equation. A proper design for the stabilization parameter is given based on a mechanical interpretation of the underlying defect as well as of the stabilizing action. This stabilizing action can be thought of an additional constraint, introduced into the reparametrized Hu–Washizu functional in a least-square form, together with a deformation dependent stabilization parameter. Numerical examples show the capability of this approach to effectively eliminate spurious hourglass modes, which otherwise may appear in the presence of large compressive strains, while preserving the advantageous features of the EAS method, namely the reduction of the stiffness for an `in-plane bending' mode, i.e. when plane stress elements are used in a bending situation.