3-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.

Enriched Finite Element Method Based on Interpolation Covers for Structural Dynamics Analysis

This work proposes a novel enriched finite element method (E-FEM) for structural dynamics analysis. We developed the enriched 3-node triangular and 4-node tetrahedral displacement-based elements (T-elements). The standard linear shape functions of these T-elements were enriched using interpolation cover functions over each patch of elements. We also introduced and compared different orders of cover functions; higher-order functions obtained higher computational performance. Subsequently, the forced and free vibration analyses were performed on various typical numerical examples. The proposed enriched finite element method generated more precise numerical results and ensured faster convergence than the original linear elements.

Mechanical-electro-magnetic coupling enriched finite element method for static analysis of magneto-electro-elastic composites

Magneto-electro-elastic (MEE) composites have promising application prospects due to the rapid response to the excitation of external electric or magnetic fields. Finite element simulation of the multi-physics coupling problems in MEE-based structures is significant. The enriched finite element method (EFEM) is derived from the standard finite element method (FEM), with finite element shape functions improved by interpolation covers. Considering the complex mechanical-electro-magnetic multi-physics coupling mechanism of MEE-based structures, the mechanical-electro-magnetic coupling enriched finite element method (MEM-EFEM) was proposed. The method is well worked for investigating the mechanical-electro-magnetic multi-physics coupling issues to calculate the generalized displacement (displacement components, electric and magnetic potential) with 3-node triangular elements. In a set of numerical experiments involving the static analysis of MEE-based structures, MEM-EFEM outperforms the regular FEM. When it comes to the same solution accuracy level, the number of elements used by MEM-EFEM is lower than that of the standard FEM, suggesting that MEM-EFEM has better convergence and accuracy on this subject. Besides, the MEM-EFEM solution accuracy can be enhanced by mesh refinement. Overall, the MEM-EFEM has the potential to be used in engineering practice to solve the static responses of MEE-based structures on mechanical-electro-magnetic multi-physics coupling issues.

Regularized local damage model for quasi-brittle material

This paper presents an improved local damage model for quasi-brittle materials. A parameter, namely damage, in the range [0,1] is used to characterize the material from intact state to complete failure. Classical local damage model is known for low computational cost but it suffers from numerical issues such as difficult convergence and mesh-density-dependent results. Various non-local models have therefore been proposed, however the computational cost is increased, which hinders the applicability in practical problems. Furthermore, the width of the damage zone predicted by the non-local models is usually non-physically large. Here, the improvement is the introduction of the fracture energy and element-size into the calculation of damage parameters to mitigate the weakness of the local damage model, while keeping low computational cost. The employment of polygonal element is also proposed to utilize the advantage of high-accuracy, thus less number of elements (than the usual 3-node triangular or 4-node quadrilateral elements) can be used. The bi-energy norm-based equivalent strain is for the first time considered in a local damage model. This quantity is based on the maximum strain criterion, but with a modification to take into account the property of quasi-brittle materials such as concrete, limestone, etc., that load capacity in compression is higher than in tension. Accuracy and efficiency of the proposed model is investigated through comparison with reference results from experiments and other numerical methods.

A triangular inverse element coupling mixed interpolation of tensorial components technique for shape sensing of plate structure

This novel inverse element, named as iMITC3+, is proposed by coupling the inverse finite element method (iFEM) and mixed interpolation of tensorial components (MITC) technology for real-time reconstructing the structural deformation based on discrete strain measurement. This reconstructed method is established through combining the kinematics of enhanced 3-node triangular MITC element and least-squares variational function. Furthermore, the approach containing the cubic bubble function inside the element is less sensitive to deformation under bending-dominated and thus alleviates the shear locking for shape sensing of thin plate structure. In addition, the method is suitable for geometrically arbitrarily plate structures, due to that the rotation degree of freedom of the element is represented by the nodal director vector. The high accuracy and practicability of iMITC3+ has been verified by numerical and experimental results through analyses of plate model subjected to different loading.

An efficient 3-node triangular plate element for static and dynamic analyses of microplates based on modified couple stress theory with micro-inertia

An efficient 3-node triangular plate element for static and dynamic analyses of microplates based on modified couple stress theory with micro-inertia

Two-phase PFEM with stable nodal integration for large deformation hydromechanical coupled geotechnical problems

The node-based strain smoothed particle finite element method (NS-PFEM) has a high level of computational efficiency. However, it has not yet been extended to hydromechanical coupling. This study presents the development of a two-phase one-point stable NS-PFEM (SNS-PFEM) for solving large deformation hydromechanical coupled geotechnical problems. First, the soil–water coupling is achieved by the v–w formulation based on Biot’s mixture theory, which automatically ensures the consistency between water pressure and stress that the former should be interpolated one order lower than that of velocity. Secondly, the node-based strain smoothing method with linear strain fields over the smoothed domain on 3-node triangular elements is adopted and implemented to obtain a stable nodal integration. Thirdly, a novel efficient water pressure stabilisation scheme based on an inverse distance interpolation algorithm is implemented to cure spurious spatial water pressure oscillations. The performance of the hydromechanical SNS-PFEM is primarily validated by both static and dynamic 1D consolidation and 2D wave propagation, and then applied to analyse the progressive landslide and the Selborne cutting slope involving large deformation. All results demonstrate that the proposed method is powerful and easily extensible for analysing large deformation hydromechanical problems in geotechnical engineering.

A semi-implicit characteristic-based polynomial pressure projection for FEM to solve incompressible flows

PurposeThe purpose of this paper is to investigate a novel stabilization scheme to handle convection and pressure oscillation in the process of solving incompressible laminar flows by finite element method (FEM).Design/methodology/approachThe semi-implicit stabilization scheme, characteristic-based polynomial pressure projection (CBP3) consists of the Characteristic-Galerkin method and polynomial pressure projection. Theoretically, the proposed scheme works for any type of element using equal-order approximation for velocity and pressure. In this work, linear 3-node triangular and 4-node tetrahedral elements are the focus, which are the simplest but most difficult elements for pressure stabilizations.FindingsThe present paper proposes a new scheme, which can stabilize FEM solution for flows of both low and relatively high Reynolds numbers. And the influence of stabilization parameters of the CBP3 scheme has also been investigated.Research limitations/implicationsThe research in this work is limited to the laminar incompressible flow.Practical implicationsThe verification and validation of the CBP3 scheme are conducted by several 2 D and 3 D numerical examples. The scheme could be used to deal with more practical fluid problems.Social implicationsThe application of scheme to study complex hemodynamics of patient-specific abdominal aortic aneurysm is also presented, which demonstrates its potential to solve bio-flows.Originality/valueThe paper simulated 2 D and 3 D numerical examples with superior results compared to existing results and experiments. The novel CBP3 scheme is verified to be very effective in handling convection and pressure oscillation.

Towards improving finite element solutions automatically with enriched 2D solid elements

In this paper, we propose an automatic procedure to improve the accuracy of finite element solutions using enriched 2D solid finite elements (4-node quadrilateral and 3-node triangular elements). The enriched elements can improve solution accuracy without mesh refinement by adding cover functions to the displacement interpolation of the standard elements. The enrichment scheme is more effective when used adaptively for areas with insufficient accuracy rather than the entire model. For given meshes, an error for each node is estimated, and then proper degrees of cover functions are applied to the selected nodes. A new error estimation method and cover function selection scheme are devised for the proposed adaptive enrichment scheme. Herein, we demonstrate the proposed enrichment scheme through several 2D problems.

The strain-smoothed 4-node quadrilateral finite element

Recently, the strain-smoothed element (SSE) method has been developed for 3-node triangular and 4-node tetrahedral solid elements. The method was also applied for enhancing the membrane performance of a 3-node triangular shell element (MITC3+ element). Using the SSE method, convergence behaviors of the finite elements were significantly improved without additional degrees of freedom. However, the application of the SSE method is limited to constant strain finite elements such as 3-node triangular and 4-node tetrahedral elements. In this paper, the SSE method is applied to a 4-node quadrilateral finite element. Doing so, the piecewise linear shape functions are employed instead of standard bilinear shape functions. The proposed strain-smoothed 4-node quadrilateral element passes all the basic tests, and shows a significantly improved accuracy in various numerical examples.

A Gradient Stable Node-Based Smoothed Discrete Shear Gap Method for Analysis of Reissner–Mindlin Plates

In this paper, a gradient stable node-based smoothed discrete shear gap method (GS-DSG) using 3-node triangular elements is presented for Reissner–Mindlin plates in elastic-static, free vibration, and buckling analyses fields. By applying the smoothed Galerkin weak form, the discretized system equations are obtained. In order to carry out the smoothing operation and numerical integration, the smoothing domain associated with each node is defined. The modified smoothed strain with gradient information is derived from the Hu–Washizu three-field variational principle, resulting in the stabilization terms in the system equations. The stabilized discrete shear gap method is also applied to avoid transverse shear-locking problem. Several numerical examples are provided to illustrate the accuracy and effectiveness. The results demonstrate that the presented method is free of shear locking and can overcome the temporal instability issues, simultaneously obtaining excellent solutions.

A sampling point quadrature method for 3-node triangular element for the evaluation of element stiffness matrix in finite element analysis

Recently, many literature studies have focused on the development on new elements in finite elements. This paper aims to develop a new quadrature for the 3-node triangular element for the purpose of evaluation of element stiffness matrix. The analysis of triangular element is usually done in a quadrilateral element by dividing the quadrilateral element into two. The edge sampling point quadrature is a mimics of Gauss numerical integration scheme. This sampling integration scheme consists of five sampling points and weights where four sampling points are at the edge and one at the center of the element. Accuracy of results, convergence of the results and stability of values have been tested using the standard benchmarked problems defined by various research studies.

Static analysis of Reissner-Mindlin plates using ES+NS-MITC3 elements

In this research, the smoothed finite element methods (S-FEM) based on the edge-based (ES) and node-based (NS) approaches are combined to develop for the 3-node triangular plate element which uses the mixed interpolation of tensorial components (MITC3) technique to remove the shear-locking phenomenon. This approach is based on the βFEM in which the parameter β is used to tune the contribution ratio of the edge-based and node-based smoothed domains. The strain fields of the proposed ES+NS-MITC3 element are smoothed on a part of the edge-based domains and the other on the node-based domains which are respectively defined by elements sharing common edges and common nodes. The ES+NS-MITC3 element passes the patch test and is employed to statically analyze some benchmark Reissner-Mindlin plates, including square and rhombus ones. Numerical results show that, in both thin and thick plates the ES+NS-MITC3 element can give results better than similar elements using the ES-FEM or NS-FEM only.

The strain-smoothed MITC3+ shell finite element

In this paper, we propose the strain-smoothed MITC3+ shell finite element. The membrane strain field of the continuum mechanics based 3-node triangular shell element (MITC3+) is smoothed using the recently developed strain-smoothed element (SSE) method. The strain-smoothed MITC3+ shell element passes basic tests (patch, isotropy and zero energy mode tests) and shows significantly improved membrane behavior in various numerical examples. The major advantage of the SSE method is that no additional degree of freedom is required for solution improvement.

A novel node-based smoothed finite element method with linear strain fields for static, free and forced vibration analyses of solids

This paper presents a novel node-based smoothed finite element method (NS-FEM) with a higher order strain field. It uses piecewise linear displacements on 3-node triangular elements, together with strain field which is also linear but over the node-based smoothed domains. This high-order strain NS-FEM is fundamentally different from the standard FEM that uses piecewise linear displacement and constants strain fields. In our high-order strain NS-FEM, the smoothed strains are expressed with complete order of polynomials which have coefficients, linear and high-order terms. This is also different from the smoothed strain used in the existing standard NS-FEM which is a constant obtained by a generalized smoothing technique. The unknown coefficients in assumed strain functions can be uniquely determined by linearly independent weight functions. This is because the smoothed strain and compatible strain within a local region are equal in an integral sense when weighted by continuous functions. We present, with proofs on convergence, two versions of high-order strain NS-FEMs which are termed as: NS-FEM-1 that uses 1st order smoothed strains and NS-FEM-2 that uses 2nd order smoothed strains. The new developed high-order strain NS-FEM is applied for static, free and forced vibration analyses of solids, and our numerical results support our theorems.

High-performance unsymmetric 3-node triangular membrane element with drilling DOFs can correctly undertake in-plane moments

PurposeThis paper aims to propose a simple but robust three-node triangular membrane element with rational drilling DOFs for efficiently analyzing plane problems.Design/methodology/approachThis new element is developed within the general framework of unsymmetric FEM. The element test functions are determined by using a conforming displacement field which is slightly different with the classical Allman’s interpolations, while a self-equilibrated stress field formulated based on the analytical airy stress solutions is adopted as the trial functions. To ensure the correctness between the drilling DOFs and the true rotations in elasticity, reasonable constraints are introduced through the penalty function method. Moreover, the special quadrature strategy is used for operating related integrations for future enrichment of element behavior.FindingsNumerical benchmark tests reveal that this new triangular membrane element has exceptional prediction capabilities. In particular, this element can correctly reproduce a rigid body rotation motion and correctly undertake the external in-plane twisting moments; thus, it is a reasonable choice for being used to formulate flat shell elements or to be connected with other kind of elements with physical rotational DOFs.Originality/valueThis work provides a new approach for developing high-performance lower-order elements with simple formulations and good numerical accuracies.

FINITE ELEMENT BASED PREDICTION OF DEFORMATION IN SHEET METAL FORMING PROCESS

In this paper the sheet forming process of cylindrical drawpieces was sim-ulated based on the finite element method by the explicit approach in the presence of contact conditions with isotropic and anisotropic friction. The experimental and numerical results obtained in the Abaqus finite element (FE) based program are presented. The aim of the experimental study is to analyse material behaviour under deformation and in addition to use the results to verify numerical simulation results. It was found that, although, the anisotropy of resistance to friction affects the height of ears, the influence of the friction formulation is relatively small in comparison with material anisotropy. The study indicates that FE analysis with 3-node triangular shell element S3R elements ensures the best approximation of the numerical results to the real process when both material and friction anisotropy are taken into account.

A new strain smoothing method for triangular and tetrahedral finite elements

This paper presents a simple and effective strain smoothing method (the strain-smoothed element method) for 3-node triangular and 4-node tetrahedral solid finite elements. While piecewise constant strain fields are constructed through smoothing domains in previous strain smoothing methods, in the proposed method, linear strain fields are constructed within finite elements using constant strains of neighboring finite elements. When the new strain smoothing method is adopted, the triangular and tetrahedral solid finite elements pass all the basic tests; patch, isotropy and zero energy mode tests, and show improved predictive capability. The formulation of the proposed method is presented in detail. Through various numerical examples, we demonstrate the accuracy improvement achieved.

A New Approach to Automatic Generation of All Graded Triangular and Quadrilateral Finite Element Mesh Over Polygonal Domains

This paper presents a new automesh generation method for a convex and cracked convex polygonal domains. We first decompose the polygon into simple sub regions in the shape of triangles. These simple regions are then triangulated to generate a mesh of graded 3-node triangular elements. We propose then an automatic 6-node triangluar conversion scheme by inserting midside nodes to these triangles. Each isolated 6-node triangle is split into four triangles according to the usual scheme,that is,by joining the midside nodes by straight lines. To preserve the mesh conformity a similar procedure is also applied to every triangle of the domain to fully discretize the given convex or cracked convex polygonal domain into all triangles, thus propagating a uniform or graded refinement. The quadrangulation of graded 3-node linear triangles is done by inserting three midside nodes and a centroidal node.Then each graded triangle is split into three quadrilaterals by joining the centroid to the midside nodes. This simple method generates a high quality mesh whose elements confirm well to the requested shape by refining the problem domain. Examples are presented to illustrate the simplicity and efficiency of the new mesh generation method for standard and arbitrary shaped polygonal domains. We have appended MATLAB programs which incorporate the mesh generation scheme developed in this paper. These programs provide valuable output on the nodal coordinates ,element connectivity and graphic display of the all triangular and quadrangular meshes for application to finite element analysis.

A Three-Node Triangular Element with Continuous Nodal Stress (Trig3-CNS) for Geometry Nonlinear Solid Mechanics Problems

This paper investigates the performance of the three-node triangular element with continuous nodal stress (Trig3-CNS) for geometry nonlinear solid mechanic problems. This Trig3-CNS element was recently proposed to improve accuracy of the finite element method (FEM). By synergizing the individual strengths of meshfree method and FEM, the Trig3-CNS element achieves higher accuracy and convergence rate. Furthermore, Trig3-CNS presents high tolerance to mesh distortion. Therefore, it is potentially useful for geometry nonlinear solid mechanics problems in which mesh distortion takes place. Compared with the traditional hybrid “FE-Meshfree” elements, Trig3-CNS naturally processes CNS without requiring any extra operation in post-processing. Numerical tests in the present work show that for geometry nonlinear analysis, the results of the Trig3-CNS element are better than the 3-node triangular element (Trig3) and 4-node isoparametric quadrilateral element (Quad4). In addition, the performance of Trig3-CNS is comparable to that of traditional hybrid “FE-Meshfree” elements.