6-node Triangular Element Research Articles

An implicit stabilized node-based smoothed finite element method for ultimate bearing capacity analysis of strip footing

The node-based smoothed finite element method (NS-FEM) has gained significant attention over the past decade. Nodal integration in NS-FEM, however, may lead to spurious oscillation of deformations of soil mass. To address the spurious oscillation of deformations associated with nodal integration, some stabilization techniques have been developed, but most of the stabilized NS-FEM have been implemented in the explicit finite element framework. The implicit framework, nonetheless, may offer some advantages including the high efficiency, good numerical accuracy, and suitability for handling large load increments and static problems. In this study, a novel stabilized NS-FEM by referring to the Taylor series expansion and incorporating the secant shear modulus, named sNS-FEM-KF(Gs), is developed in the implicit finite element framework. Based on the ultimate bearing capacity analysis of some strip footings, it is found that sNS-FEM-KF(Gs) may effectively eliminate the spurious oscillations observed in NS-FEM, and may resolve the overly-stiff behaviors of soil mass predicted by sNS-FEM-KF which is a reduced version of sNS-FEM-KF(Gs). Even compared to the standard finite element method with the 6-node triangular (T6) elements (i.e. FEM(T6)), sNS-FEM-KF(Gs) with linear 3-node triangular (T3) elements may outperform FEM(T6) in the investigated strip footing problems, indicating its good applicability in geotechnical analysis.

Application of improved Picard iteration method to simulate unsaturated flow and deformation in deformable porous media

In this paper, we investigate the effectiveness of the cascadic multigrid method applied to improved Picard iteration method for nonlinear problems arising in deforming variably saturated porous media. The finite element method with 6-node triangular elements is applied to discretize the space and obtain nonlinear algebraic equations. Then the nonlinear iterative method is used for iterative solution. Since the classical nonlinear Picard iteration method (PI) can be slow to converge, two improved Picard iteration methods are proposed. One is the improved Picard method with iterations k on the coarse grid and number of multiple grids m (PI-MG(m, k)), and the other is the improved Picard method based on the cascadic multigrid method without parameter k (PI-NMG(m)). Three numerical examples are given to verify the effectiveness of the improved method. Results indicate that the convergence rate of PI is the slowest (10–25 nonlinear iterations per time step), followed by PI-MG(m, k) (3–5 nonlinear iterations), and PI-NMG(m) is the fastest (stable 2 nonlinear iterations). The computational efficiency of PI-NMG(m) is improved by about 90% relative to PI and by about 70% relative to AR-PI. This method can be applied to large-scale numerical calculation of seepage and deformation in unsaturated deforming porous media.

An isoparametric element permutation method for railway tunnel–soil interaction modeling in train–track–tunnel–soil dynamic analysis

For evaluating the ground vibration involving metro tunnel operation, it is of necessity to extend the train-track-tunnel near-filed vibration to the train-track-tunnel-soil far-filed vibration in viewpoint of numerical analysis. It therefore becomes a key technology to achieve the tunnel-soil coupling simulation. In this work, a isoparametric element permutation method is proposed, where the 6-node triangular prismatic elements and 8-node spatial isoperimetric elements have been effectively integrated to achieve the finite elemental modelling of the soil system with the tunnel penetrating. The tunnel and soil subsystems are coupled by the spring-dashpot elements. Once the tunnel satisfies symmetrical property in geometry, the proposed method can be applied to build the tunnel-soil interaction by parametric finite element modelling. Through a combination of this proposed tunnel-soil interaction model and the train-track-tunnel dynamic model, a unified model for characterizing the train–track–tunnel–soil dynamic interaction analysis is constructed within the finite element framework. By comparing the simulation results to the measurements in site, the effectiveness of this model is validated, through which track, tunnel, and ground vibrations can be assessed in an entirely coupled way. Through the numerical studies, it is known that both the number of tunnel lines and the number of trains in operation and their running modes hold significant influence and show different vibration attenuation law on ground vibrations, and quantitative analysis can be achieved through this present work.

Studying the Effects of Negative Skin Friction on Single Piles in Basrah Governorate

The finite element method capable of simulating the behavior of deep foundations subjected to negative skin friction in Basrah soil is investigated. Single piles under drag forces are analyzed using the PLAXIS program with an axisymmetric model. Linear elastic, Soft Soil and Mohr-Coulomb constitutive relations are adopted, where higher order triangular element is chosen for pile and soil clusters. Both pile and soil are modeled using (15)-node triangular elements. Three sites in Basrah province (Umm Qasr Port, Khor Al-Zubair, and Shatt AlArab Hotel) were selected to perform this study. The soil profile and layer characteristics are obtained from the soil investigation reports. Where the negative skin friction is evaluated due to filling loads. It is Conclusion thatSmall relative displacements are necessary to activate the negative skin friction. The elastic shorting for pile effect negative skin friction, due to increase relative displacement. The elastic shorting of the driven pile is more than that of the bored pile due to the less cross-sectional area of the driven pile. The results revealed proportional relation between the developed drag forces and pile section dimensions, interface friction factor, and fill height, with a maximum effect on the section dimension and minimum effect on the interface factor. The locations of neutral points are not sensitive to the above-mentioned factors.

Free and Forced Vibration Analysis of Two-Dimensional Linear Elastic Solids Using the Finite Element Methods Enriched by Interpolation Cover Functions

In this paper, a novel enriched three-node triangular element with the augmented interpolation cover functions is proposed based on the original linear triangular element for two-dimensional solids. In this enriched triangular element, the augmented interpolation cover functions are employed to enrich the original standard linear shape functions over element patches. As a result, the original linear approximation space can be effectively enriched without adding extra nodes. To eliminate the linear dependence issue of the present method, an effective scheme is used to make the system matrices of the numerical model completely positive-definite. Through several typical numerical examples, the abilities of the present enriched three node triangular element in forced and free vibration analysis of two-dimensional solids are studied. The results show that, compared with the original linear triangular element, the present element can not only provide more accurate numerical results, but also have higher computational efficiency and convergence rate.

Nonlinear finite element analysis within strain gradient elasticity: Reissner-Mindlin plate theory versus three-dimensional theory

Nonlinear plate bending within Mindlin's strain gradient elasticity theory (SGT) is investigated by employing somewhat non-standard finite element methods. The main goal is to compare the bending results provided by the geometrically nonlinear three-dimensional (3D) theory and the geometrically nonlinear Reissner–Mindlin plate theory, i.e., the first-order shear deformation plate theory (FSDT), within the SGT. For the 3D theory, the nonlinear Green–Lagrange strain relations are adopted, while the von Kármán nonlinear strains are employed for the FSDT. The matrix-vector forms of the energy functionals are derived for both models. In order to perform the corresponding finite element discretizations, a quasi-C1-continuous 4-node tetrahedral solid element and a quasi-C1-continuous 6-node triangular plate element are employed for the 3D model and plate model, respectively. The first-order derivatives of the primal problem quantities are employed as additional nodal values to respond to the continuity requirements of class C1. A variety of computational results highlighting the differences between the 3D and FSDT models are given for two different plate geometries: a rectangular plate with a circular hole and an elliptical plate.

Convergence study on ultrasonic guided wave propagation modes in an axisymmetric cylindrical waveguide

Guided wave propagation and its dispersion phenomenon of infinite solid elastic rods are encountered in several applications including, mechanical and civil engineering fields. In this paper, the elastic stress wave propagation in the axisymmetric circular cross-section of a high strength steel wire with cylindrical waveguide is investigated using a semi-analytical finite element (SAFE) method. The error analyses are carried out on fundamental modes, namely, flexural longitudinal and torsional modes. The theoretical framework for finite element (FE) discretization is established for the cylindrical waveguide. A three-node triangular linear element is used for solving SAFE dispersion solutions such as the wavenumber-frequency curve, phase velocity, and group velocity curves. The convergence and accuracy of the method are analyzed by comparing it with the calculation results of the transcendental Pochhammer frequency equation, and the meshing criterion is proposed. The use of higher-order (quadratic) elements are proposed for lower computational burden and effective method for solving eigenvalue problems. The effect of using 6-node triangular quadratic elements in the SAFE method for improving frequency accuracy is discussed in a detailed manner by proposing the meshing criterion. The calculation accuracy of a quadratic element semi-analytical discretization exceeds that of a linear element discretization with four times its radial circumference. The statistical histograms are demonstrated to prove the results of the proposed semi-analytical discretization method which can be used for solving cylindrical waveguides.

Cracking elements method with 6-node triangular element

The cracking elements method (CEM) is a novel Galerkin-based numerical approach for simulating cracking and fracturing processes. It is a crack-opening approach that avoids precise descriptions of the mechanical states of crack tips and captures the initiations and propagations of multiple cracks without nodal enrichment or crack tracking. The CEM requires element types with nonlinear interpolation of the displacement field to avoid stress-locking. In the 2D condition, the 6-node triangular element (T6) and 8-node quadrilateral element (Q8) are potential candidates. However, despite the success of the formerly proposed CEM with Q8, the CEM with T6 showed considerable mesh dependencies. In this work, to solve this problem, the CEM with T6 is further investigated. The mesh dependencies are shown to be eliminated with simple modification to the real characteristic length of the T6 element in the CEM framework. Several numerical examples with regular and irregular Q8 and T6 mixed meshes are provided, indicating the effectiveness and robustness of this approach.

A Generation of Special Triangular Boundary Element Shape Functions for 3D Crack Problems

This paper focuses on tackling the two drawbacks of the dual boundary element method (DBEM) when solving crack problems with a discontinuous triangular element: low accuracy of the calculation of integrals with singularity and crack front element must be utilized to model the square-root property of displacement. In order to calculate the integrals with higher order singularity, the triangular elements are segmented into several subregions which consist of subtriangles and subpolygons. The singular integrals in those subtriangles are handled by the singularity subtraction technique in the integration space and can be regularized and accurately calculated. For the nearly singular integrals in those subpolygons, the element subdivision technique is employed to improve the calculation accuracy. In addition, considering the location of the crack front in the element, special crack front elements are constructed based on a 6-node discontinuous triangular element, in which the displacement extrapolation method is introduced to obtain the stress intensity factors (SIFs) without consideration of orthogonalization of the crack front mesh. Several numerical results are investigated to fully verify the validation of the presented approach.

Analysis of Magneto Rheological Fluid Journal Bearing

Magneto Rheological (MR) fluids are a class of smart materials where the shear stress is not directly proportional to rate of shear. The viscosity of fluid changes as magnetic field changes and hence this phenomenon is very useful in bearing-rotor system for attenuating the vibrations. In the present study the application of MR fluid as lubricant instead of Newtonian fluid in the journal bearing is explored through steady state, dynamic characteristics and stability. MR fluid film has been modeled as per Bingham rheological model. FEM with three node triangular elements has been used to solve the Reynolds equation both for the Newtonian fluid film and MR fluid film. The results show the load carrying capacity in the case of MR fluid journal bearing is higher than that of using the Newtonian fluid. The load carrying capacity increases with the increasing magnetic field for all eccentricity ratios. The results also show better stability of the bearing using MR fluid at higher eccentricity ratios. The unbalance response of the rotor mounted on the journal bearing using MR fluid is also estimated to be lower than that of with the Newtonian fluid.

Smoothed finite element approach for kinematic limit analysis of cohesive frictional materials

By employing different smoothed finite element (S-FE) methods, the kinematic limit analysis approach has been presented by using three noded triangular elements to solve plane strain and plane stress stability problems on basis of the Mohr-Coulomb yield criterion. By using the concept of duality in conic programming, the expressions for the power dissipation function and the associated constraints have been written entirely in terms of kinematic variables. The optimization formulation is framed as a second order cone programming problem. Based on the absolute maximum magnitude of the shear strain rate, an algorithm to update the mesh has been provided. Amongst the different approaches, the node-based S-FE formulation has been found to be very accurate as well as efficient to solve large scale stability problems. The selective edge-based and the edge-based with a bubble node approaches also generate quite accurate solutions though require a little more computational time.

A novel triangular boundary crack front element for 3D crack problems based on 8-node serendipity element

This paper developed a novel 6-node triangular boundary crack front element for 3D crack problems. The developed element method was constructed based on an 8-node serendipity element and a distance function that defines the distance from arbitrary point inside the element to the crack front. The specially constructed shape function captures the square-root asymptotic behavior of the distribution of the displacement nearby the crack front. In order to employ the crack front element in the dual boundary element method, conversion from conforming triangular crack front elements to non-conforming triangular ones was performed through a matrix transformation. In the implementation of dual boundary element method, a displacement extrapolation method is adopted to compute the stress intensity factor at the crack front. Finally, three examples were presented to verify the efficiency of our proposed crack front element. In all the numerical examples, the presented crack front element exhibits high accuracy and efficiency in the examples.

Evaluation of dynamic characteristics of liquid sloshing in sloped bottom tanks

Sloshing characteristics in partially filled slo-ped bottom container is investigated numerically. Galer-kin finite element method is used to derive the modal equation of motion for sloshing in 2D tank. The dynamic charcateristics of liquid sloshing in V-shaped bottom and the trapezoid tank are the critical investigations in the present work under the action of periodic ground motion. The liquid domain in V-shaped bottom and the trapezoid tank is discretized with a combination of three node triangular and four-node quadrilateral elements for this analysis. The reliability of the present numerical method is verified by comparing the results with the available analytical and experimental results for $$30^\circ $$ trapezoid tank and $$20^\circ $$ V-shaped bottom tank. Further, a parametric study on the V-shaped bottom and trapezoid tank is carried out to determine it’s dynamic behavior. The dynamic response of trapezoid tank under horizontal harmonic motion is manifested in terms of slosh wave displacement, base shear force at tank bottom and overturning base moment which remained limited for sloped bottom container.

Nonlinear analysis of FG-sandwich plates and shells

This study is dedicated to develop a mixed interpolated formulation for nonlinear analysis of plates and shells. Using Equivalent Single Layer (ESL) theory and the rule of mixture scheme, the authors present a formulation for analyzing Functionally Graded (FG) sandwich structures. To incorporate large displacements and rotations, and alleviate the shear and membrane locking phenomena, a mixed interpolation of strain fields, including in-plane and shear strains, is utilized. Analyses are based on an isoparamertic 6-node triangular shell element. There are five degrees of freedom, including three displacements and two rotations at each node of this element. In order to validate the proposed formulation, some well-known benchmark problems are solved. Besides, the obtained responses are compared to the other available solutions, separately. Moreover, two other practical structures, including FG-sandwich plate and curved shallow panel, are considered to indicate the accuracy and high performance of the authors' approach.

Thermal Analysis of Viscoelastic Propellant Grains with Developed Axisymmetric Finite Elements using Herrmann Formulation

Solid Rocket Motor (SRM) is developed based on casting method where solid propellant grains are cast into a composite or metallic casing. Generally, SRMs are exposed to extreme loading scenarios during storage, transportation, and firing, leading to cracks in the solid propellants. In this paper, Computational Finite Element Analysis is performed with developed 8 node quadrilateral, 9 node quadrilateral and 6 node triangular elements using Herrmann formulation to analyze stress and strain variations in the head and mid segments of the SRM subjected to thermal loading. The obtained results are compared with commercially available Finite element software

Efficient hypersingular line and surface integrals direct evaluation by complex variable differentiation method

We present an efficient numerical scheme to evaluate hypersingular integrals appeared in boundary element methods. The hypersingular integrals are first separated into regular and singular parts, in which the singular integrals are defined as limits around the singularity and their values are determined analytically by taking the finite part values. The remaining regular integrals can be evaluated using rational interpolatory quadrature or complex variable differentiation (CVDM) for the regular function when machine precision like accuracy is required. The proposed method is then generalised for evaluating hypersingular surface integrals, in which the inner integral is treated as the hypersingular line integral via coordinate transformations. The procedure is implemented into 8-node rectangular boundary element and 6-node triangular element for numerical evaluation. Finally, several numerical examples are presented to demonstrate the efficiency of the present method. To the best of our knowledge, the proposed method is more accurate, faster and more generalised than other methods available in the literature to evaluate hypersingular integrals.

A study of development mode of collapse with variation of strains and stresses during compression of metallic shells having dome-cone shape

The present paper deals with the experimental and computational analysis of the deformation behaviour of the metallic thin walled aluminium shells having dome-cone combined geometry. The specimens were tested on a universal testing machine INSTRON under axial compression to identify their modes of collapse and to study the associated energy absorption capacity. In experiments all the shell specimens were found to collapse with the formation of an axisymmetric mode of collapse due to development of the associated plastic Zones in dome and cones.A Finite Element computational model of development of the axisymmetric mode of collapse is presented and analysed, using a non-linear finite element code FORGE2. The proposed finite element model for this purpose idealizes the deformation as axisymmetric. Six noded triangular elements were used to discretize the domain. The material of the shell specimens was idealized as rigid visco-plastic. Experimental and computed results of the deformed shapes and their corresponding load-compression and energy-compression curves were presented and compared to validate the computational model. Typical variations of equivalent strain, equivalent strain rate and nodal velocity distribution are presented to help in predicting the mode of collapse. On the basis of the obtained results development of the axisymmetric mode of collapse has been presented, analysed and discussed.

A Study on Analysis of Collapse of Metallic Shells having Dome-cone Shape

The present paper deals with the experimental and computational analysis of the deformation behaviour of the metallic thin walled aluminium shells having dome-cone combined geometry. The specimens were tested on a universal testing machine INSTRON under axial compression to identify their modes of collapse and to study the associated energy absorption capacities. In experiments all the shell specimens were found to collapse with the formation of an axisymmetric mode of collapse due to development of the associated plastic zones in dome and cones.A Finite Element computational model of development of the axisymmetric mode of collapse is presented and analysed, using a non-linear finite element code FORGE2. The proposed finite element model for this purpose idealizes the deformation as axisymmetric. Six noded triangular elements were used to discretize the domain. The material of the shell specimens was idealized as rigid visco-plastic. Experimental and computed results of the deformed shapes and their corresponding load-compression and energy-compression curves were presented and compared to validate the computational model. Typical variations of equivalent strain and equivalent strain rate are presented to help in predicting the mode of collapse. On the basis of the obtained results development of the axisymmetric mode of collapse has been presented, analysed and discussed.

Nonlinear Analysis of Twisted Wind Turbine Blade

AbstractModal analysis is developed in this paper in order to study the dynamic characteristics of rotating segmented blades assembled with spar. Accordingly, a three dimensional finite element model was built using the three node triangular shell element DKT18, which has six degrees of freedom, to model the blade and the spar structures. This study covers the effect of rotation speed and geometrically nonlinear problems on the vibration characteristics of rotating blade with various pretwist angles. Likewise, the effect of the spar in the blade is taken into consideration. The equation of motion for the finite element model is derived by using Hamilton's principle, while the resulting nonlinear equilibrium equation is solved by applying the Newmark method combined with the Newton Raphson schema. Results show that the natural frequencies increase by taking account of the spar, they are also proportional to the angular rotation speed and influenced by geometric nonlinearity and pretwist angle.

Stress recovery techniques for natural element method in 2-D solid mechanics

This paper is concerned with the stress recovery for the natural element method in which the problem domain is discretized with Delaunay triangles and the structural behavior is approximated with Laplace interpolation functions. Basically, the global and local patch recovery techniques based on the L2-projection method are adopted. For the local patch recovery, the local element patches are defined by the supports of each Laplace interpolation function. For the comparison purpose, the local stress recovery is also performed using Lagrange-type basis functions that are used for 3- and 6-node triangular elements. The stresses that are recovered by the present global and local recovery techniques are compared each other and compared with the available analytic solution, in terms of their spatial distributions and the convergence rates. As well, the dependence of the recovered stress field on the type of test basis functions that are used forbnov-Galerkin (BG) and Petrov-Galerkin (PG) natural element methods is also investigated.