Six-node Triangular Element Research Articles

Three new triangular thick plate/shell elements based on absolute nodal coordinate formulation

In this paper, three new triangular fully parameterized thick plate/shell elements are proposed based on the absolute nodal coordinate formulation. The volume parameters and corresponding independent volume gradient vectors are used to simplify the element construction. The node-wise structural parameter is adopted to circumvent the slope discontinuity issue associated with the volume gradient vectors. Two three-node elements inherit the interpolation strategy in the membrane direction of a well verified thin plate element. For the thickness direction, linear and quadratic interpolation are used in the three-node elements. Inspired by the quadrilateral plate elements with gradient vectors only along thickness direction, a six-node triangular element is presented with quadratic interpolation in both membrane and thickness direction. Three static analyses with large deformation and one dynamic analysis are conducted to evaluate the performance of the new elements. It concludes that the three-node element with linear interpolation in the thickness direction performs not well due to the incompatible/low interpolation in the thickness direction. The other two elements show good accuracy in all the numerical tests. Though the six-node element exhibits slightly higher accuracy than the quadratic three-node element, its DOFs are much more than the quadratic three-node element, which makes it less attractive.

A mixed selective edge-based smoothed PFEM with second-order cone programming for geotechnical large deformation analysis

Traditional particle finite element method (PFEM) with a six-node triangular (T6) element using an iteration algorithm often suffers from high computational costs and non-convergence problems. This paper proposes a novel dynamic implicit optimisation based smoothed PFEM for solving geotechnical large deformation problems. The governing equations are treated as an equivalent min–max optimisation that is recast as a second-order cone programming (SOCP) problem. Owing to the particularity of the min–max optimisation problem, which needs to simultaneously interpolate displacement and stress field, a novel mixed constant-stress selective edge-based smoothed strain element is proposed in which strain and stress are constant in the smoothing domain. The two-level mesh repartitioning scheme is implemented into the proposed method to overcome volumetric locking. Next, a mass lumping scheme using nodal integration is developed to improve computational efficiency. An efficient and simple variable mapping scheme is finally proposed and implemented into PFEM to tackle large deformation problems. Several classical statics and dynamics examples are used to validate the proposed method. All results demonstrate that the proposed method only using the lower-order element offers high accuracy, high efficiency, mesh insensitivity and is free of volumetric locking for geotechnical problems under both small and large deformation.

Peristaltic Phenomenon in an Asymmetric Channel Subject to Inclined Magnetic Force and Porous Space.

This research is engaged to explore biological peristaltic transport under the action of an externally applied magnetic field passing through an asymmetric channel which is saturated with porous media. The set of governing partial differential equations for the present peristaltic flow are solved in the absence of a low Reynolds number and long wavelength assumptions. The governing equations are to be solved completely, so that inertial effects can be studied. The numerical simulations and results are obtained by the help of a finite element method based on quadratic six-noded triangular elements equipped with a Galerkin residual procedure. The inertial effects and effects of other pertinent parameters are discussed by plotting graphs based on a finite element (FEM) solution. Trapped bolus is discussed using the graphs of streamlines. The obtained results are also compared with the results given in the literature which are highly convergent. It is concluded that velocity and the number of boluses is enhanced by an increase in Hartmann number and porosity parameter Increasing inertial forces increase the velocity of flow but increasing values of the porosity parameter lead to a decrease in the pressure gradient. The study elaborates that magnetic field and porosity are useful tools to control the velocity, pressure, and boluses in the peristaltic flow pattern.

An efficient and synchronous mesh refining–coarsening strategy in kinematic upper-bound limit analysis

This study presents an efficient and robust adaptive remeshing strategy in the kinematic upper-bound limit analysis in association with using six-node triangular elements and Second–Order Cone Programming (SOCP). In particular, a Synchronous Mesh Refining–Coarsening (SMRC) algorithm following a posterior plastic dissipation-based error indicator and the well-known Delaunay triangulation are implemented into the Upper Bound Finite Element Method (UBFEM). Detailed numerical implementation of the proposed method is presented, and its computational performance is demonstrated through several benchmark problems and comparisons with those solutions obtained using Pure Mesh Refinement (PMR) scheme. The proposed method, named UBFEM-SMRC, is proved to generate more rigorous solutions for all considered examples and significantly reduce the computational burdens of the UBFEM-PMR. Additionally, more clear adaptively refined mesh zones with intensive plastic dissipations, which can readily reproduce the potential failure mechanisms of geotechnical problems, are observed by successive performing of the proposed synchronous adaptive mesh refining–coarsening process.

An improved numerical method for lunar dust transport during electric curtain-traveling wave removal in the lunar environment

The law of dust removal by electric curtain in the lunar environment is considerably different from that in Earth's environment. This paper proposes an improved calculation method, which successfully simulates the kinematics of lunar dust transport during removal by electric curtain-traveling waves in the lunar environment. The numerical formula of the dielectrophoretic force component is derived based on a six-node triangular (T6) finite element method (FEM) in electromagnetics. A discrete element method (DEM) based on the Johnson–Kendall–Roberts soft-sphere model is chosen to simulate lunar dust particle transport under the action of the electric curtain dust removal system in the lunar environment. The validity, accuracy, and reliability of the present DEM algorithm are verified by comparing the experimental and numerical results. The final dust removal rate is positively correlated with frequency when the surface energy between the particles and insulating film is γf≥0.0075 J/m2. The final dust removal rate is negatively related to the surface energy between the particles and the film but positively related to the applied voltage and the charge of the particles. This research is of great significance to the mitigation of the lunar dust pollution problem that the crewed lunar exploration project will face in the future.

Microarchitecture-dependent nonlinear bending analysis for cellular plates with prismatic corrugated cores via an anisotropic strain gradient plate theory of first-order shear deformation

This study focuses on the microarchitecture-dependent nonlinear bending behavior of cellular plates with equitriangularly prismatic microarchitectures by adopting a dimensionally and constitutively reduced strain gradient plate model. The strain energy formulation is based on the dimension reduction of the first-order shear deformation plate theory along with von Kármán’s nonlinear strain relations and anisotropic strain gradient theory. The classical and higher-order constitutive parameters are obtained according to the recently published homogenization results for a corresponding linear plate model. The corresponding finite element simulations, numerically solving the anisotropic strain gradient plate problems, rely on a nonstandard, higher-order, six-node triangular element showing good convergence properties. Comparisons between the proposed (2D) strain gradient shear deformation plate model and the corresponding (3D) detailed full-field reference models demonstrate for a variety of cellular plate structures that the accuracy of the proposed approach is at a very good level with relatively low computational costs. A diverse set of numerical examples is provided in order to investigate the size-dependent nonlinear structural response of cellular plates having different numbers of microarchitectural layers, midsurface shapes and boundary conditions.

Improved shell element for geometrically non-linear analysis of thin-walled structures

A six-node triangular shell element was developed to study the geometrically non-linear behaviour of open-section members. The present formulation is free from shear and membrane locking due to use of mixed interpolation of tensorial components scheme. Due to model thin-walled section, an improvement is considered to update the normal vectors of elements which are located at the angle between flange and web of section. In other words, an average of director vectors is employed for these elements in updating formulation. Moreover, the total Lagrangian formulation is employed to consider large displacement and rotation. An incremental iterative technique named generalised displacement control method is utilised to solve the governing equation. Several numerical examples are studied to illustrate the accuracy and predictability of the proposed formulation.

Application of the particle finite element method for large deformation consolidation analysis

Purpose Large deformation problems are frequently encountered in various fields of geotechnical engineering. The particle finite element method (PFEM) has been proven to be a promising method to solve large deformation problems. This study aims to develop a computational framework for modelling the hydro-mechanical coupled porous media at large deformation based on the PFEM. Design/methodology/approach The PFEM is extended by adopting the linear and quadratic triangular elements for pore water pressure and displacements. A six-node triangular element is used for modelling two-dimensional problems instead of the low-order three-node triangular element. Thus, the numerical instability induced by volumetric locking is avoided. The Modified Cam Clay (MCC) model is used to describe the elasto-plastic soil behaviour. Findings The proposed approach is used for analysing several consolidation problems. The numerical results have demonstrated that large deformation consolidation problems with the proposed approach can be accomplished without numerical difficulties and loss of accuracy. The coupled PFEM provides a stable and robust numerical tool in solving large deformation consolidation problems. It is demonstrated that the proposed approach is intrinsically stable. Originality/value The PFEM is extended to consider large deformation-coupled hydro-mechanical problem. PFEM is enhanced by using a six-node quadratic triangular element for displacement and this is coupled with a four-node quadrilateral element for modelling excess pore pressure.

Nonlinear vibration analysis of carbon nanotube reinforced composite plane structures

This paper is dedicated to nonlinear static and free vibration analysis of Uniform Distributed Carbon Nanotube Reinforced Composite (UD-CNTRC) structures under in-plane loading. The authors have suggested an efficient six-node triangular element. Mixed Interpolation of Tensorial Components (MITC) approach is employed to alleviate the membrane locking phenomena. Moreover, the behavior of the well-known LST element is considerably improved by applying an additional linear interpolation on the strain fields. Based on the rule of mixture, the properties of CNTRC are obtained. In this study, only the uniform distributed CNTs are employed through the thickness direction of element. To achieve the natural frequencies and shape modes, the eigenvalue problem is also solved. Using Total Lagrangian Principles, large amplitude free vibration is considered based on the first normalized mode shape of structure. Different well-known plane problem benchmarks and some proposed ones are studied to validate the accuracy and capability of authors' formulations. In addition, the effects of length to the height ratio of beam, CNT's characteristics, support conditions and normalized amplitude parameter on the linear and nonlinear vibration parameters are investigated.

Incompatible Graded Finite Elements for Analysis of Nonhomogeneous Materials

A six-node incompatible graded finite element is developed and studied. Such element is recommended for use since it is more accurate than four-node compatible element and more efficient than eight-node compatible element in two-dimensional plane elasticity. This paper presents comparison between six-node incompatible (QM6) and four-node compatible (Q4) graded elements. Numerical solution is obtained from abaqus using UMAT capability of the software and exact solution is provided as reference for comparison. A graded plate with exponential and linear gradation subjected to traction and bending load is considered. Additionally, three-node triangular (T3) and six-node triangular (T6) graded elements are compared to QM6 element. Incompatible graded element is shown to give better performance in terms of accuracy and computation time over other element formulations for functionally graded materials (FGMs).

A high-order control volume finite element method for thermoelastic analysis of functionally graded solids with mixed grids

In this article, a new two-dimensional control volume finite element method has been developed for thermoelastic analysis in functionally graded materials. A nine-node quadrilateral element and a six-node triangular element are employed to deal with the mixed-grid problem. The unknown variables and material properties are defined at the node. The high-order shape functions of six-node triangular and nine-node quadrilateral element are employed to obtain the unknown variables and their derivatives. In addition, the material properties in functionally graded structure are also modeled by applying the high-order shape functions. The capabilities of the presented method to heat conduction problem, elastic problem, and thermoelastic problem have been validated. First, the defined location of material properties is found to be important for the accuracy of the numerical results. Second, the presented method is proven to be efficient and reliable for the elastic analysis in multi-phase materials. Third, the presented method is capable of high-order mixed grids. The memory and computational costs of the presented method are also compared with other numerical methods.

High-order quasi-conforming triangular Reissner-Mindlin plate element

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.

Geometrically nonlinear analysis of FG doubly-curved and hyperbolical shells via laminated by new element

An isoparametric six-node triangular element is utilized for geometrically nonlinear analysis of functionally graded (FG) shells. To overcome the shear and membrane locking, the element is improved by using strain interpolation functions. The Total Lagrangian formulation is employed to include the large displacements and rotations. Finding the nonlinear behavior of FG shells via laminated modeling is also the goal. A power function is employed to formulate the variation of elastic modulus through the thickness of shells. The results are presented in two ways, including the general FGM formulation and the laminated modeling. The equilibrium path is obtained by using the Generalized Displacement Control Method. Some popular benchmarks, including hyperbolical shell structures are solved to declare the correctness and accuracy of proposed formulations.

A triangular shell element for geometrically nonlinear analysis

In this paper, a geometrically nonlinear formulation for a six-node triangular shell element is proposed. Total Lagrangian formulation is utilized to consider large displacements and rotations in the shell analysis. To avoid shear and membrane locking, a proper interpolation function for the strain field is implemented. Both algorithm and flowchart of the nonlinear solution, which are utilized in the author’s computer program, are presented. To validate the suggested formulation, several popular benchmark problems are solved. Moreover, the obtained results are compared with those of the other well-known elements. Findings demonstrate the ability of the suggested shell element.

Geometrically nonlinear thermomechanical analysis of shell-like structures

ABSTRACTIn this study, a locking-free six-node triangular shell element is proposed to nonlinear analysis of shell structures under the effects of both thermal and mechanical loads. Total Lagrangian formulation is used to consider large deflections and rotations. Besides, mixed interpolation of tensorial components is applied in the relations to deviate shear and membrane locking. The effects of temperature on the nonlinear deflections of shells are studied in several numerical examples. Moreover, analytical solution will be proposed for large deflection analysis of cantilever and simply supported beam under the thermal loads. To validate the proposed formulation, the finite element analysis results are compared by analytical solutions. Based on the obtained results of linear and nonlinear problems, it will be depicted that the authors’ scheme is capable for thermomechanical analysis of shell structures accurately.

Improved graded finite elements with applications in simulating non-homogeneous elastic, piezoelectric and magneto-electro-elastic materials

Graded finite elements (GFEs) provide a promising way for simulating functionally graded materials. Nevertheless, the existing GFE method takes the conventional isoparametric transform functions as a unified representation for the material non-homogeneity in each element regardless of the practical global distribution forms of material properties. This inevitably leads to a certain difference between the local formulation of material property variation in the elements and the corresponding global gradation patterns. In order to eliminate this difference, an improved GFE algorithm is proposed in the present article. The property distribution in the element is formulated by substituting the isoparametric transform of the coordinates directly into the corresponding global gradation functions. Therefore, the local property distribution is always consistent with the global one. Both the improved six-node triangular elements (T6) and eight-node quadrilateral elements (Q8) are developed for non-homogeneous elastic, piezoelectric and magneto-electro-elastic materials, respectively. Exact solutions in three special cases are presented to make comparison with the numerical results, and the accuracy of the proposed algorithm is verified. It is demonstrated that the improved GFE is an effective method for the numerical simulation of functionally graded materials.

Efficient C0 Finite Element Formulation for Thermal Stress Analysis of Laminated Composite and Sandwich Plates

AbstractBy introducing thermal expansion coefficients in the transverse displacement field, this paper proposes an approach to improve the performance of the global-local higher-order theory for thermal stress analysis of laminated composite plates. Transverse normal deformation has been considered in the proposed model, whereas the additional displacement variables have not increased because thermal loads could be included in the generalized force vector. The proposed model a priori satisfies the continuity conditions of transverse shear stresses at interfaces, and the number of displacement variables involved in the present model is independent of the number of layers. The derivatives of transverse displacement have been eliminated from the displacement field so that the C0 interpolation functions are only required for the finite-element implementation. Based on the proposed model, a six-node triangular element is proposed for thermal expansion problems of laminated composite and sandwich plates. Numeri...

Numerical Investigation of Process Parameters on External Inversion of Thin-Walled Tubes

In this paper, a computational study of development of external inversion mode of deformation of round-metallic aluminum tubes over a fixed profile die is presented. Inversion mode of deformation is analyzed in detail by using a finite element code FORGE2. The proposed finite element model for this purpose idealizes the deformation as axisymmetric. Six-noded triangular elements are used to discretize the domain. The material is modeled as rigid-viscoplastic. Typical variations of the equivalent strain rate and equivalent strain along the length of the deforming tube are studied to predict the development of inversion mode of deformation. The influence of the friction present at the contact interface between the tube and the die is also examined to suggest a successful inversion of tube. Energy absorbed in overcoming the frictional stresses between the tube-die interfaces is compared with the total energy required in the inversion of tube. A few predicted results which include the geometry of inverted tube load-compression variation during inversion process are compared with their experimental counterparts to validate the computational model.

Hybrid quadrilateral finite element models for axial symmetric Helmholtz problem

This paper is a continuation of the previous work in which six-node triangular finite element models for the axial symmetric Helmholtz problem are devised by using a hybrid functional and the spherical-wave modes [1]. The six-node models can readily be incorporated into the standard finite element program framework and are typically ∼50% less erroneous than their conventional or, equivalently, continuous Galerkin counterpart. In this paper, four-node and eight-node quadrilateral models are devised. Two ways of selecting the spherical-wave modes are attempted. In the first way, a spherical-wave pole is selected such that it is equal-distant from an opposing pair of element nodes. In the second way, the directions of the spherical-waves passing through the element origin are equal-spaced with one of the directions bisecting the two parametric axes of the element. Examples show that both ways lead to elements that yield very similar predictions. Furthermore, four-node and eight-node hybrid elements are typically ∼50% and ∼70% less erroneous than their conventional counterparts, respectively.

A study on inversion of metallic thin-walled conical shells

The paper presents an experimental and computational study of the inversion process of metallic thin-walled conical shells subjected to axial compression. The study mainly focuses on two aspects: firstly, to optimise the semi-apical angle of a cone to get a constant collapse load and secondly, to study the mechanics of the development of the inversion mode of collapse. Axial compression of aluminium thin-walled conical shells of semi-apical angles between 8° and 11° was performed on a universal testing machine Instron. An axisymmetric finite element computational model of the development of the inversion mode of collapse is also presented and analysed, using a non-linear finite element code FORGE2. Six-noded triangular elements are used for discretising the deforming shell domain. The material of the shells is idealised as rigid viscoplastic. Experimental and computed results are compared to validate the computational model. On the basis of the obtained results, the development of the axisymmetric inversion mode of collapse is presented, analysed and discussed.