A partial assumed strain formulation for triangular solid shell element

A mixed formulation to construct general shell elements based on the degenerated solid approach is presented. This mixed formulation can be understood as a modification of the discrete potential energy of the shell in which the membrane and shear energy terms are obtained as a combination of the usual terms, i.e., directly obtained from the displacement and geometry interpolation assumptions, and membrane and shear terms obtained from assumed strain fields. This energy splitting idea was first introduced in the context of the Naghdi shell theory by Arnold and Brezzi. This mixed formulation is used to study quadratic general triangular shell elements based on the degenerated solid approach. Although considerable success has been achieved in the formulation of reliable, free of locking, general quadrilateral shell elements, the attempts of formulating general triangular shell elements have not yet produced an element that satisfies the locking free requirements. Therefore, there is a great interest in investigating new approaches, as the one discussed in this paper, that can lead to triangular elements that do not suffer from locking difficulties. We present in this paper the formulation of specific elements and their numerical assessment through the use of well established benchmark shell problems.

Relevance. The use of the finite element method for determining the stressstrain state of thin-walled elements of engineering structures predetermines their discretization into separate finite elements. Splitting irregular parts of the structure is impossible without the use of triangular areas. The triangular elements of shell structures are joint in displacements and in their derivatives only at the nodal points. Therefore, ways to improve the compatibility conditions at the boundaries of triangular elements are relevant. Aims of research. The aim of the work is to improve the compatibility conditions at the boundaries of adjacent triangular elements based on equating the derivatives of normal displacements in the middle of the boundary sides. Methods. In order to improve the compatibility conditions at the boundaries of triangular elements in this work, the Lagrange functional is used with the condition of ensuring equality in the middle of the sides of adjacent elements derived from normal displacements in the directions of perpendiculars tangent to the middle surface of the shell. Results. Using the example of analysing an elliptical shell, the efficiency of using a joint triangular finite element is shown, whose stiffness matrix is formed in accordance with the algorithm outlined in this article.

The past few years have witnessed rapid advances in the integration of Computer-Aided Design (CAD) and Finite Element Analysis (FEA), such as the Isogeometric Analysis (IGA) method and the Integration of Computer Aided Design and Analysis (I-CAD-A) technique. Recent studies indicated that the geometry description used in the Absolute Nodal Coordinate Formulation (ANCF) is consistent with the geometry description in CAD systems, making it an attractive approach for I-CAD-A. ANCF as an accurate, non-incremental finite element method to deal with the dynamics of those systems subject to large motions and large deformations has developed towards practical application in the field of flexible multibody systems. The objective of this study is to enrich the family of ANCF elements with three new triangular shell elements: the thin triangular element, the low order thick triangular element and the high order thick triangular element, which are represented by Bezier triangles. The linear mapping relationship between the geometrical representation of the Bezier triangle and that of the proposed thin triangular element is established. The invertible property of the linear mapping relationship clearly indicates that the proposed thin triangular element can be also used to perform the I-CAD-A. The conditions of the approximate $G^{1}$ continuity between the proposed adjacent elements are also deduced in order to construct a smooth surface. Finally, several static and dynamic numerical examples are employed to validate the correctness and effectiveness of the proposed triangular shell elements.

The edge-based smoothing discrete shear gap method (ES-DSG3) using three-node triangular elements is combined with a C0-type higher-order shear deformation theory (HSDT) to give a new linear triangular plate element for static, free vibration, and buckling analyses of laminated composite plates. In the ES-DSG3, only the linear approximation is necessary, and the discrete shear gap method (DSG) for triangular plate elements is used to avoid the shear locking and spurious zero energy modes. In addition, the stiffness matrices are calculated relying on smoothing domains associated with the edges of the triangular elements through an edge-based strain smoothing technique. Using the C0-type HSDT, the shear correction factors in the original ES-DSG3 can be removed and replaced by two additional degrees of freedom at each node. The numerical examples demonstrated that the ES-DSG3 show remarkably excellent performance compared to several other published elements in the literature.

PurposeA higher-order Reissner-Mindlin plate element method is presented based on the framework of assumed stress quasi-conforming method and Hellinger-Reissner variational principle. A novel six-node triangular plate element is proposed by utilizing this method for the static and free vibration analysis of Reissner-Mindlin plates.Design/methodology/approachFirst, the initial assumed stress field is derived by using the fundamental analytical solutions which satisfy all governing equations. Then the stress matrix is treated as the weighted function to weaken the strain-displacement equations after the strains are derived by using the constitutive equations. Finally, the arbitrary order Timoshenko beam function is adopted as the string-net functions along each side of the element for strain integration.FindingsThe proposed element can pass patch test and is free from shear locking and spurious zero energy modes. Numerical tests show that the element can give high-accurate solutions, good convergence and is a good competitor to other models.Originality/valueThis work gives new formulations to develop high-order Reissner-Mindlin plate element, and the new strategy exhibits advantages of both analytical and discrete methods.

The paper offers a new flat shell finite element. It results from the combination of a membrane and a bending elements, both founded on the strain based formulation. The used membrane element is the T43_Eq. It is a triangular element with four nodes, three nodes at the vertices and the fourth one at the barycenter. Each node has three degrees of freedom, two translations and one rotation around the normal. The coefficients related to the degrees of freedom at the internal node are subsequently removed from the element stiffness matrix by using the static condensation procedure. The interpolation functions of strain, displacements and stresses fields are developed from equilibrium conditions. Then, the plate element used for the construction of the shell element is a triangular four-node thin plate element based on: the deformation approach, the four fictitious node, the static condensation and the analytic integration. The shell element result of this combination is robust, competitive and efficient. The shell element result of this combination is robust, competitive and efficient.

Multilayered triangular and quadrilateral flat shell elements based on the Refined Zigzag Theory

A combined boundary-finite element procedure for dynamic analysis of plates with fluid and foundation interaction considering free surface effect

SummaryA 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.

Hybrid strain based geometrically nonlinear laminated composite triangular shell finite elements

Research Article Variational boundary element acoustic modelling over mixed quadrilateral-triangular element meshes Ahlem Alia 1 *, Mhamed Souli 1, Fouad Erchiqui 2 1LML, Boulevard Paul Langevin, Cite Scientifique, 59655 Villeneuve d'Ascq Cedex, France 2Universite du Quebec en Abitibi-Temiscamingue, 445 boul. de l'Universite, Rouyn-Noranda QC, Que., Canada J9X5E4 email: Ahlem Alia (Ahlem.Alia@ed.univ-lille1.fr) *Correspondence to Ahlem Alia, LML, Boulevard Paul Langevin, Cite Scientifique, 59655 Villeneuve d'Ascq Cedex, France Keywords variational indirect boundary element method • singularity • triangles • polar co-ordinate transformation Abstract The variational indirect boundary element method is widely used in many acoustical problems such as simulation of scattering and radiation phenomena. Although it has many advantageous features in dealing with those problems, it suffers from the singularity problem when the double surface integral is done over the same element. In this paper, a straightforward technique for computing the singularity 1/R over triangular element is presented. It is based on a generalized polar co-ordinates transformation to transform the triangular master element into a square one. The singularity factor is taken as weight function. Consequently, a single straightforward implementation without any special treatment of singularity is possible for meshes involving both quadrilateral and triangular elements of higher order. The efficiency of the proposed method is demonstrated for some numerical examples.

Meshfree-enriched simplex elements with strain smoothing for the finite element analysis of compressible and nearly incompressible solids

On the choice of the linear element for corotational triangular shells

A parallel finite element procedure for contact-impact problems using edge-based smooth triangular element and GPU

Hybrid stress-based finite elements with side displacement fields have been used to generate equilibrium models having the property of equilibrium in a strong form. This paper establishes the static and kinematic characteristics of a flat triangular hybrid equilibrium element with both membrane and plate bending actions of general polynomial degree p. The principal characteristics concern the existence of hyperstatic stress fields and spurious kinematic modes. The former are shown to exist for p>3, and their significance to finite element analysis is reviewed. Knowledge of the latter is crucial to the determination of the stability of a mesh of triangular elements, and to the choice of procedure adopted for the solution of the system of equations. Both types of characteristic are dependent on p, and are established as regards their numbers and general algebraic forms. Graphical illustrations of these forms are included in the paper. Copyright © 2005 John Wiley & Sons, Ltd.