Triangular Shell Element Research Articles

A new triangular composite shell element with damping capability

This paper presents a new triangular composite shell element with damping capability. Formulation of the composite triangular shell element is based on stacking individual homogeneous triangular shell elements on top of each other. The homogeneous shell element is an assembly of a triangular membrane element with drilling degrees of freedoms and a plate element. Damping capability is provided by means of complex element stiffness matrix of individual flat layers of the composite element. These elements with damping capability allow modelling general structures with damping treatments. A few test cases are modelled using triangular finite element developed here and the results of the complex eigenvalue analyses are compared with those of the quadrilateral shell elements proposed recently. The results obtained using the presented triangular and previous quadrilateral composite elements are also compared with those based on modal strain energy method and experimental results. Comparisons of the experimental and the theoretical results confirm that the modal properties including modal damping levels of structures with damping treatments can be predicted with high accuracy using the proposed finite element.

Development of a strain-smoothed three-node triangular flat shell element with drilling degrees of freedom

This paper proposes a three-node triangular flat shell element based on the assumed natural deviatoric strain (ANDES) formulation. The free parameters included in the ANDES formulation are presented for better use of the proposed element in curved shell geometries. In order to further enhance the element behavior, a strain smoothing technique is applied to the derivation of the membrane stiffness of the proposed element. Furthermore, in order to avoid the drill rotation locking that is typically induced by the drilling degrees of freedom, a simple modification factor is introduced. Various benchmark examples are tested in order to verify the performance of the proposed element.

A continuum-based mixed shell element for shakedown analysis

The objective of this paper is to present a new triangular shell element for shakedown analysis. The element is based on a mixed variational formulation with displacement/velocity and stress fields independently interpolated. We also employ the continuum based (CB) approach which allows the definition of the yield function at the continuum level without any approximation in terms of generalized stresses. The formulation of the proposed element is presented in details after a brief review of relevant classical shakedown theory concepts. The performance of the element is assessed by means of a set of selected representative examples. The numerical tests include: (i) analyses of thin and thick-walled straight pipes under combined loads, (ii) the shakedown analysis of a pipe bend under internal pressure and in-plane bending and (iii) limit analysis of a cylinder–cylinder intersection subjected to bending and internal pressure. • A new CB mixed finite element is devised for limit and shakedown analysis of shells. • The use of generalized stresses in shakedown formulation is avoided with the element. • We focus on the ability of the element to provide accurate critical load domains. • Numerical tests confirm the accuracy and convergence of the element.

Polar decomposition based corotational framework for triangular shell elements with distributed loads

SUMMARYA polar decomposition based corotational formulation for deriving geometrically nonlinear triangular shell elements is proposed. This formulation is novel in two aspects. (1) Original formulas for the projector operator and its variation are presented, leading to simple algorithms for the computation of the nodal residual vector and of the consistent tangent stiffness tensor. (2) For the first time in the context of a corotational kinematic description, a rigorous treatment of distributed dead and follower loads is performed, thoroughly accounting for the various contributions entailed in the residual vector and in the tangent stiffness. Numerical simulations of popular benchmark problems are reported, showing the effectiveness of the proposed approach. An accessible and adaptable MATLAB toolkit implementing the present formulation is provided as supplementary material. Copyright © 2013 John Wiley & Sons, Ltd.

Explicit Dynamic Finite Element Method for Predicting Implosion/Explosion Induced Failure of Shell Structures

A simplified implementation of the conventional extended finite element method (XFEM) for dynamic fracture in thin shells is presented. Though this implementation uses the same linear combination of the conventional XFEM, it allows for considerable simplifications of the discontinuous displacement and velocity fields in shell finite elements. The proposed method is implemented for the discrete Kirchhoff triangular (DKT) shell element, which is one of the most popular shell elements in engineering analysis. Numerical examples for dynamic failure of shells under impulsive loads including implosion and explosion are presented to demonstrate the effectiveness and robustness of the method.

Study on the Explicit Formula of the Triangular Flat Shell Element Based on the Analytical Trial Functions for Anisotropy Material

This paper presents a novel way to formulate the triangular flat shell element. The basic analytical solutions of membrane and bending plate problem for anisotropy material are studied separately. Combining with the conforming displacement along the sides and hybrid element strategy, the triangular flat shell elements based on the analytical trial functions (ATF) for anisotropy material are formulated. By using the explicit integral formulae of the triangular element, the matrices used in proposed shell element are calculated efficiently. The benchmark examples showed the high accuracy and high efficiency.

Finite element corotational formulation for geometric nonlinear analysis of thin shells with large rotation and small strain

Based on the consistent symmetrizable equilibrated (CSE) corotational formulation, a linear triangular flat thin shell element with 3 nodes and 18° of freedom, constructed by combination of the optimal membrane element and discrete Kirchhoff triangle (DKT) bending plate element, was extended to the geometric nonlinear analysis of thin shells with large rotation and small strain. Through derivation of the consistent tangent stiffness matrix and internal force vector, the corotational nonlinear finite element equations were established. The nonlinear equations were solved by using the Newton-Raphson iteration algorithm combined with an automatic load controlled technology. Three typical case studies, i.e., the slit annular thin plate, top opened hemispherical shell and cylindrical shell, validated the accuracy of the formulation established in this paper.

Graph-Theoretic Approach for the Dynamic Simulation of Flexible Multibody Systems

This paper provides a general description of a variational graph-theoretic formulation for simulation of flexible multibody systems (FMSs) which includes a brief review of linear graph principles required to formulate this algorithm. The system is represented by a linear graph, in which nodes represent reference frames on flexible bodies, and edges represent components that connect these frames. The method is based on a simplistic topological approach which casts the dynamic equations of motion into a symmetrical format. To generate the equations of motion with elastic deformations, the flexible bodies are discretized using two types of finite elements. The first is a 2 node 3D beam element based on Mindlin kinematics with quadratic rotation. This element is used to discretize unidirectional bodies such as links of flexible systems. The second consists of a triangular thin shell element based on the discrete Kirchhoff criterion and can be used to discretize bidirectional bodies such as high-speed lightweight manipulators, large high precision deployable space structures, and micro/nano-electromechanical systems (MEMSs). Two flexible systems are analyzed to illustrate the performance of this new variational graph-theoretic formulation and its ability to generate directly a set of motion equations for FMS without additional user input.

A New FEM Approach for Simulation of Metal Foam Filled Tubes

The combination of thin metal cases or tubes with a filling made of metal foams is interesting and promising for many applications in mechanical engineering. Components made of an outer hollow thin compact metal structure and a cellular lightweight core are especially suited to energy absorption applications. In order to allow for an efficient product/process design with a concurrent engineering approach, reliable and computationally affordable finite element method (FEM) calculations are required by both product and process engineers. The structural performance of these complex composite parts must be numerically predicted, in order to find the optimal combination of outer structure and metal foam properties. While FEM simulation at large deformations of bending, crushing, etc. of thin sheets and tubes is state of the art, the accurate FEM simulation of the mechanical behavior of metal foams cannot be considered fully established. In this paper the three most common methods for FEM simulation of metal foam materials are discussed: (a) homogenization approach, (b) realistic reconstruction of tomographic data, and (c) repetition of standard unit cells. A new effective approach is proposed, suited for simulation of composite, metal foam filled, structures of realistic dimensions. The approach is based on meshing the metal foam by replicating a unit cell made of 32 triangular shell elements, and then by randomizing the nodal position in order to emulate the intrinsic homogeneity of foam morphology. The method is validated by means of different experimental tests. The results show that the proposed method correctly predicts the behavior of foam structures in axial compression. The method slightly overestimated the actual load registered in three point bending tests. Several improvements are described and discussed in the paper, such as randomization of nodal positions of the mesh, in order to reduce the overestimation of forces. An FEM approach for the simulation of large deformations of metal foam filled metal structure (e.g., tubes) suited for the design of realistic large dimensions structural components has been presented. The proposed method shows some innovative features with respect to the available scientific literature, such as a configuration based on octahedral unit cells with low number of triangular shell elements. Randomization of nodal positions of each unit cell has been implemented as a method for better representing the intrinsic variability of metal foams and for reducing the stiffness of the simulated structure.

Extended rotation-free shell triangles with transverse shear deformation effects

The paper extends recent work of the authors to include transverse shear effects on rotation-free triangular element for plates (Onate and Zarate in Int J Numer Methods Eng 83(2):196---227, 2010). Two new shell triangular elements are presented, the EBST+ and the EBST+1. Transverse shear deformation effects are important for thick shells, as well when the shell is laminated or formed by composite material. The ingredients for the element formulation are: a Hu-Washizu type mixed functional and linear interpolation for the displacement field. In both elements presented a finite volume approach is used for computing the bending moments and the curvatures over a patch of elements. The nodal translational degrees of freedom of the original enhanced basic shell triangle (EBST) are extended with the two shear deformation angles via two different approaches. The first one uses a linear interpolation of the rotation angles inside the element (EBST+) and the second one assumes a constant field for the rotation angles (EBST+1). For the thin shell case the shear angles vanish and the new elements reproduce the good behaviour of the original thin EBST element. As a consequence the elements can reproduce the solutions for thick to thin shells situations without exhibiting shear locking. The numerical solution for the thick shell case can be found iteratively starting from the deflection values for the Kirchhoff theory using the original thin EBST element. Examples of the good performance of the new rotation-free shell triangles are given.

Numerical simulation of residual stress and deformation in glass pressed parts

As an important glass manufacturing process, glass pressing has been an effective process with the advantages of mass‐production, net shape manufacturing, etc., in which the residual stress and deformation in glass pressed parts are key issues. Based on the geometric characteristics of glass pressed parts, the shear plate assumption is employed for residual stress analysis, together with an applicable linear viscoelastic material model. Based on the results of residual stress, the finite element method using a flat triangular shell element is introduced to compute the deformation. The shell element is constructed by combining a constant strain triangle (CST) membrane element and a discrete Kirchhoff triangle (DKT) plate bending element. The comparison shows that the predicted results agree well with the experimental data.

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

AbstractThe 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.

A surface model-based simulation for warpage of injection-molded parts

The mid-plane model for warpage simulation of injection-molded parts requires a mid-plane mesh whose transformation is considerably time consuming. To overcome this drawback, a surface model-based warpage simulation is presented, in which the part is represented as a perfect bonding of two half-thickness plates with their reference surfaces at the outer boundary of the part. The plates over the surface mesh are modeled as flat shell elements, and a new triangular flat shell element is developed which combines an Assumed Natural DEviatoric Strain (ANDES) based membrane component and a Refined Nonconforming Element Method (RNEM) based bending component. The bonding is accomplished by multipoint constraints and a Lagrange multiplier based elimination method is proposed for constraint application. The results show that compared with some popular shell elements, ANSYS, Moldflow and the experiments, the presented model exhibits a high performance in computation accuracy. POLYM. ENG. SCI., 2011. © 2011 Society of Plastics Engineers

Modal Analysis of Vehicle Body Based on a Novel Triangular Shell Element

As the existing triangular shell elements are difficult for simulation of vehicle body,a novel triangular shell element using edge-based smoothing operation is proposed for modal analysis of vehicle body parts.The formulation is based on the first order shear deformation theory,and a discrete shear gap method is employed to mitigate the shear locking.The domain is first discretized into a set of unstructured triangular meshes,and the integration domains associated with the edges of the triangles are then further formed.An edge local coordinate system is introduced for performing strain smoothing operations in each integration domain,and the system stiffness matrix is tuned effectively,which can obtain very accurate results.The discretized system equations are obtained by using the smoothed Galerkin weak form,and the stiffness and mass matrices are formulated.Modal analyses of complex vehicle body parts are studied and comparisons are made with the existing software and reference results,and which verify the effectiveness and high precision of the proposed algorithm.

Finite Element Analysis of Von-Mises Stress Distribution in a Spherical Shell of Liquified Natural Gas (Lng) Pressure Vessels

This research work investigated the modeling of Von Mises stress in LNG Spherical Carbon Steel Storage tank using assumed displacement Finite Element analysis based on shallow shell triangular elements. Using equations of elasticity, constant thickness carbon steel spherical storage tanks were subjected to different loading conditions. This paper stresses the need for proper definition of shallow element using sector angles to obtain the shallowness. The shallow spherical triangular element has five degrees of freedom at each of its corner node, which are the essential external degrees of freedom. The assumed displacement fields of these shallow triangular elements satisfied the exact requirement of rigid body modes of motion. The FORTRAN 90 programming language was used for the programme coding to solve finite element equations resulting from the model while Von Mises stresses distribution within the spherical storage tank shell subjected to different internal pressures were determined. The results showed that the use of non-shallow elements due to improper sector angles resulted in unreliable results while real shallow elements produced results that tallied with ASME Section VIII Div 1, Part UG values.

Deformable image registration of heterogeneous human lung incorporating the bronchial tree

To investigate the effect of the bronchial tree on the accuracy of biomechanical-based deformable image registration of human lungs. Three dimensional finite element models have been developed using four dimensional computed tomography image data of ten lung cancer patients. Each model is built of a body, left and right lungs, tumor, and bronchial trees. Triangular shell elements are used for the bronchial trees while tetrahedral elements are used for other components. Hyperelastic material properties based on experimental investigation on human lungs are used for the lung parenchyma. Different material properties are assigned for the bronchial tree using five values for the modulus of elasticity of 0.01, 0.12, 0.5, 10, and 18 MPa. Lungs are modeled to slide inside chest cavities by applying frictionless contact surfaces between each lung and corresponding chest cavity. The accuracy of the models is examined using an average of 40 bronchial bifurcation points identified on inhale and exhale images. Relative accuracy is evaluated by comparing the displacement of all nodes within the lungs as well as the dosimetric difference at the exhale position predicted by the model. There is no significant effect of bronchial tree on the model accuracy based on the bifurcation points analysis. However, on the local level, using an average of 38 000 nodes, there is a maximum difference of 8.5 mm in the deformation of the bronchial trees, as the modulus of elasticity of the bronchial trees increases from 0.01 to 18 MPa; however, more than 96% of nodes are within a 2.5 mm difference in each direction. The average dose difference at the predicted exhale position is less than 35 cGy between the models. The bronchial tree has little effect on the global deformation and the accuracy of deformable image registration of lungs. Hence, the homogenous model is a reasonable assumption. Since there are some local deformation differences between nodes as the material properties of the bronchial tree change that may affect the accuracy of dosimetric results, heterogeneity may be required for a smaller scale modeling of lungs.

Triangular higher-order element for better prediction of stress resultants and stresses in plated and shell structures

Although they furnish accurate displacements, conventional displacement-based lower order finite elements fail to predict accurate stress resultants and stresses in certain classes of plate and shell problems that involve free edges, steep stress gradients and singularities. In order to tackle such problems, a triangular higher-order shell element based on the nodal basis approach has been developed. The nodes of the element are located at optimal points and its more superior shape functions derived from orthogonal Proriol polynomials. To illustrate the improved performance of the higher-order element as compared to commonly used lower order shell elements in predicting the variations of stress resultants and stresses, three example problems involving a simply supported skew plate, a corner supported square plate, and a clamped cylindrical shell are solved. The stress resultants and the stresses furnished by the higher-order element for the problems considered are found to be accurate with the satisfactio...

The quadratic MITC plate and MITC shell elements in plate bending

The analysis of plates can be achieved using the quadratic MITC plate or MITC shell elements. The plate elements have a strong mathematical basis and have been shown to be optimal in their convergence behavior, theoretically and numerically. The shell elements have not (yet) been analyzed mathematically in depth for their rates of convergence, with the plate/shell thickness varying, but have been shown numerically to perform well. Since the shell elements are general and can be used for linear and nonlinear analyses of plates and shells, it is important to identify the differences in the performance of these elements when compared to the plate elements. We briefly review the quadratic quadrilateral and triangular MITC plate and shell elements and study their performances in linear plate analyses.

On the decomposition of generalized eigenproblems for the free vibration analysis of cyclically symmetric finite element models

AbstractFree vibration analysis is a major part of any dynamic analysis. Natural frequencies and related mode shapes may be obtained from free vibration analysis as the solutions of generalized eigenproblems. Although the eigensolutions of large‐scale structures require large computational efforts, these solutions may be achieved simply for symmetric structures. We present an efficient method for the decomposition of generalized eigenproblems related to finite element models with cyclic symmetry (having nodes at the axis of symmetry) into eigensubproblems with significantly smaller dimensions. This decomposition is obtained by block diagonalization of a matrix with a special pattern known as a block circulant, using the concept of the Kronecker product and similarity transformations. The proposed method is applied to three finite element models discretized by triangular and four‐node quadrilateral plate and shell elements, and its efficiency, accuracy and simplicity are evaluated. Copyright © 2009 John Wiley & Sons, Ltd.

Biomechanical Effects of Fixed Functional Appliance on Craniofacial Structures

To evaluate displacement and stress distribution on craniofacial structures associated with fixed functional therapy. A finite element model of the human skull was constructed from sequential computed tomography images at 2-mm intervals using a dry adult human skull. In this study, linear, four-nodal, tetramesh and triangular shell elements were used with six degrees of freedom at each of their unstrained nodes: three translations (x, y, and z) and three rotations (around the x-, y-, and z-axes). The entire mandible moved anteroinferiorly. Maximum displacement was observed in the parasymphyseal and midsymphyseal regions. The pterygoid plate was displaced in a posterosuperior direction. The anteroinferior displacement of the mandibular dentition was most pronounced in the incisor region, while the maxillary dentition was displaced posterosuperiorly. The entire dentition experienced tensile stress except for the maxillary posterior teeth. Tensile stress was also demonstrated at point A, the pterygoid plates, and the mandible, and minimal compressive stress was demonstrated at anterior nasal spine. Maximum tensile stress and von Mises stresses occurred in the condylar neck and head. Fixed functional therapy causes a posterosuperior displacement of the maxillary dentition and pterygoid plate and thus can contribute to the correction of Class II malocclusion. The displacement was more pronounced in the dentoalveolar region as compared to the skeletal displacement. All dentoalveolar structures experienced tensile stress, except for anterior nasal spine and the maxillary posterior teeth.