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This paper will try to overcome two difficulties encountered by the C0 three-node triangular element based on the displacement-based higher-order models. They are (i) transverse shear stresses computed from constitutive equations vanish at the clamped edges, and (ii) it is difficult to accurately produce the transverse shear stresses even using the integration of the three-dimensional equilibrium equation. Invalidation of the equilibrium equation approach ought to attribute to the higher-order derivations of displacement parameters involved in transverse shear stress components after integrating three-dimensional equilibrium equation. Thus, the higher-order derivatives of displacement parameters will be taken out from transverse shear stress field by using the three-field Hu–Washizu variational principle before the finite element procedure is implemented. Therefore, such method is named as the preprocessing method for transverse shear stresses in present work. Because the higher-order derivatives of displacement parameters have been eliminated, a C0 three-node triangular element based on the higher-order zig-zag theory can be presented by using the linear interpolation function. Performance of the proposed element is numerically evaluated by analyzing multilayered sandwich plates with different loading conditions, lamination sequences, material constants and boundary conditions, and it can be found that the present model works well in the finite element framework.

This paper presents a novel numerical model on unstructured grids for all-speed flows using the multimoment constrained finite volume method, where the point values at both the cell center and cell...

A fast computational code is presented that is dedicated for the elastic analysis of three-dimensional excavations and cracks in rocks. The problem is solved on the boundaries that are discretized with a new triangular leaf constant displacement discontinuity element with one collocation point. The creation of the new triangular element was inspired from Mindlin's special version of grade-2 or strain-gradient elasticity theory (second gradient of displacement, g2). This element is characterized by a much better measure of the average stress at the center of gravity of the triangular element compared to that of the classical elasticity element close to regions with stress or strain gradients (e.g. notches, cracks etc). In a verification stage, the accuracy of the computational algorithm for the pressurized penny-shaped and mixed-mode elliptical crack problems that have analytical solutions is demonstrated. More specifically, it is shown that the average error of the crack tip Stress Intensity Factor predicted by the gradient modified method for nine discretizations of varying density is around 3.5% with a maximum error of 5%, while the constant displacement discontinuity element displays errors varying around 14%. Moreover, the new method preserves the simplicity and hence the high speed of the constant displacement discontinuity with only one collocation point per element, but it is far more efficient compared to it, especially close to the crack tips and corners of excavations where the displacement and stress gradients are highest.

A brief review of the laminated shell analyses is presented. The related literatures demonstrate that most previous studies deal with the quadrilateral or flat triangular elements and not the curved triangular element. Based on the continuum mechanic’s theory, a 6-node triangular isoparametric element is formulated for geometrically nonlinear analysis of laminated shells. To prevent the membrane and shear locking effects, the suggested scheme uses assumed strains for in-plane and transverse shear strains. Large displacements and rotations in the nonlinear analysis are included via the total Lagrangian description. Several popular benchmark problems are solved by the Generalized Displacement Control Method. The numerical results obtained are in good agreement with those published by other investigators.

In this paper, effect of cavity on geogrid reinforced soil having cavity at certain depths are examined and analysis has been carried out for load bearing capacity and settlement of footing resting on geogrid reinforced surface. PLAXIS-2D (v8) finite element package has been used for which 15-noded a triangular element has been used. Outcomes obtained from the analytical study quite resemble with available literature results. A total of 52 cases has been analyzed to examine the impact of number of geogrid layers, spacing of subsequently layer, cavity location, size of the cavity 0.1B, 0.25B, and 0.5B and depth of footing for which M-C model is used to analyze the elastic failure. It can be observed that optimum depth of the first reinforcement bed was around 0.5 times footing breadth. Thus the optimum depth 0.5B was considered for different no of geogrid layers. The maximum improvement of load bearing capacity for reinforced without cavity of cohesive soil for N = 4 is 25.27% the percentage decrement of load bearing capacity for reinforcement with cavity size having diameter 0.1B, 0.25B, and 0.5B were 23.14%, 17.07% and 15.83% respectively. Due to this increasing diameter of cavity leads to observable decrease in the load carrying capacity of the foundation at constant depth of cavity to the footing breadth ratio.

The element is simple accurate and robust in the analysis of elastic plane problems.The displacement interpolation is enriched by a revised Allman interpolation.The choice of the displacement interpolation keeps out zero-energy modes.The assumed stress is important in reducing the over-stiffness of linear triangles. We propose a mixed smoothed finite element model for plane elasticity. Within each triangular element the displacement field is described by a revised Allman interpolation, while the stresses are assumed to be piece-wise constant on a background grid associated with the edges of the triangle. A straightforward implementation of the element, in order to make it easy to readily incorporate it in existing FE packages, is described. The numerical experiments show that the proposed model performs well and takes advantage of the enrichment of the displacement field. Moreover the drilling rotation parameters makes simple the extension to form shell elements.

Summary In this paper, a simple locking-free triangular plate element, labeled here as Mindlin-type triangular plate element with nine degrees of freedom (MTP9), is presented. The element employs an incompatible approximation with nine degrees of freedom (DOFs) independent of the nodes and the shape of the triangle to define the displacements u/v/w(which is similar to a general solid element) along the x/y/z axes. It is free of shear locking, has a proper rank, and provides stable solutions for thick and thin plates. Moreover, the paper provides a new way to develop simple and efficient locking-free thick–thin-plate/shell elements. A variety of numerical examples demonstrate the convergence, accuracy, and robustness of the present element MTP9. Copyright © 2016 John Wiley & Sons, Ltd.

A three-node triangular element fitted to the numerical manifold method with continuous nodal stress called Trig3-CNS (NMM) element for accurately modeling two-dimensional linear elastic fracture problems is presented. By adopting two cover systems, namely, the mathematical cover and physical cover, the numerical manifold method (NMM) could easily solve continuous and discontinuous problems in a unified way. In contrast to the three-node triangular element (Trig3), the Trig3-CNS element has higher order of approximations, much better accuracy and continuous nodal stress. Moreover, it is free from the “linear dependence” which otherwise cripples many of the partition of unity based methods with high order approximations. The purpose of the present work is to synergize the advantages of both the recently developed Trig3-CNS element and the NMM to precisely model two-dimensional linear elastic fracture problems. A number of numerical examples indicate the accuracy and robustness of the present Trig3-CNS (NMM) element.

Double set parameter nonconforming finite element method (FEM) for the two-sided displacement obstacle problem of clamped plate is considered. Firstly, a double set parameter nonconforming triangular element (named Veubeke–Zienkiewicz element) is introduced, which has the same global dimension as Zienkiewicz element, and can pass the generalized patch test for any mesh. Secondly, we skillfully construct an enriching operator for connecting the Veubeke–Zienkiewicz and the C 1 -conforming Bell element spaces. With the help of this operator and an auxiliary obstacle problem built as a bridge between the continuous and discrete obstacle problems, we present the optimal order error estimates in the broken energy norm and L ∞ norm, respectively. Finally, we establish a general framework of double set parameter nonconforming elements approximating for the two-sided displacement obstacle problem of clamped plate.

In this paper, two novel and efficient elements are proposed for analysis of anisotropic and symmetric laminated thick plates based on hybrid-Trefftz formulation. In this approach, two independent displacement fields are defined for internal and boundary of the elements. The internal field is selected in such a manner that the governing equation of thick plates could be satisfied. At the first step, the governing differential equations of anisotropic and symmetric laminated thick plates are derived. Then, the analytical solution of these equations is considered. By using these analytical functions, the internal displacement field of proposed elements is achieved. Also, the boundary field is related to the nodal degrees of freedom by the boundary interpolation functions. For this aim, the interpolation functions of tow-node laminated Timoshenko beam element are calculated for boundary field. By utilizing these displacement fields and hybrid-Trefftz functional, a triangular element THT-7 and quadrilateral element QHT-11 are proposed. These elements have 9 and 12 degrees of freedom, respectively. The results of the numerical testes are proven the absence of the shear locking and high accuracy and efficiency of the proposed elements for analysis of anisotropic and symmetric laminated plates.

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.

Heat-generating discs with point heat sinks on the rim are designed using the generating optimization method at both macroscale and microscale. The disc is constructed by elemental sectors with cooling channels extending inward from the rim. To formulate the initial structure for the generating optimization, an analytic analysis based on the constructal law and entransy theory is applied. It is the first time to analytically formulate the entransy dissipation rate on triangular element with variable but coupled lengths and widths. The analysis indicates that the initial structures for macro problems and micro problems are different. The macro disc should be divided into infinite elemental sectors, while the number of elemental sectors in the micro disc is limited owing to the size effect. Moreover, the difference between the macro problem and micro problem can also be observed in the final design. At the macroscale, tree-like constructal conduction networks are constructed using the generating optimization that exhibit better conduction performance than classic needle structures. At the microscale, the needle constructal conduction network is more capable of reducing thermal resistance. These findings are presented in numerical cases of generating optimization.

As is known to all, there always exists the numerical dispersive effects of the standard finite element (FE) for the wave propagation problems and the corresponding FE solutions are usually unreliable in relatively high frequency range. In this work, a coupled “FE-Meshfree” element based on triangular mesh is introduced to reduce the dispersion effects for wave propagation problems. In this coupled element, the standard FE nodal shape functions are combined with the meshfree nodal shape functions to give a new hybrid nodal shape functions. As a result, both the individual advantages of the FE technique and the meshfree technique are strengthened by the present hybrid method. Through the dispersion analysis for the wave propagation problems, it is found that this coupled “FE-Meshfree” element could significantly reduce the numerical dispersive effects and it also have a higher tolerance to the mesh distortion than the other existing elements, hence the present method is quite promising to handle the general wave propagation problems in practical engineering application.

A cell-based smoothed discrete shear gap method (CS-DSG3) using three-node triangular element was recently proposed to improve the effectiveness of the discrete shear gap method (DSG3) for static and vibration analyses of isotropic Mindlin plates and shells. In this study, the CS-DSG3 is further extended for static and free vibration responses of functionally graded shells. In the present method, the first-order shear deformation theory is used in the formulation owing to the simplicity and computational efficiency. Several numerical examples are provided to validate high reliability of the CS-DSG3 in comparison with other numerical methods.

A triangular plate-bending element with a new multi-resolution analysis (MRA) is proposed and a novel multiresolution element method is hence presented. The MRA framework is formulated out of a displacement subspace sequence whose basis functions are built out of scaling and shifting on the element domain of basic full node shape function. The basic full node shape function is constructed by means of extending the shape function triangle domain for a split node at the zero coordinates to the hexagon area enclosing the zero coordinates. As a result, a new split-full node notion is presented and a novel rational MRA concept together with the resolution level (RL) is constituted for the element. Via practical examples, it is found that the traditional triangular plate element and method is a mono-resolution one and also a special case of the proposed element and method. The meshing for the monoresolution plate element model is based on the empiricism while the RL adjusting for the multiresolution is laid on the rigorous mathematical basis. The analysis clarity of a plate structure is actually determined by the RL, not by the mesh. Thus, the accuracy of a structural analysis is replaced by the clarity, the irrational MRA by the rational and the mesh model by the RL that is the discretized model by the integrated. The continuous full node shape function unveils secrets behind assembling artificially of node-related items in global matrix formation by the conventional FEM., 21 pages,8figures. arXiv admin note: substantial text overlap with arXiv:1505.06822, arXiv:1404.1165, arXiv:1405.6775

Analytical and numerical results on transverse shear stresses of angle-ply composite plates subjected to hygrothermal loadings are scarce in literature. Thus, a higher order global-local model a priori satisfying the continuity condition of transverse shear stresses is proposed for hygrothermal analysis of angel-ply composite plates. Based on the higher order global-local model, a triangular element is developed by using the refined element approach. Comparing the results from available three-dimensional elasticity solution, the proposed model can produce promising transverse shear stresses directly from constitutive equations without any smoothing technique. By analyzing hygrothermal response of composite plates with general configurations, the effects of material properties, lamination angles, boundary conditions, transverse normal strain, and hygrothermal expansion coefficients on the hygrothermal response have also been studied. Differing from previous work, the corrected hygrothermal expansion coefficients are used for hygrothermal analysis of composite plates.

This paper presents an efficient procedure to reconstruct the thermodynamic properties of real fluids. An equation-of-state look-up table was constructed with adaptive triangular mesh, which not on...

This paper discusses the vibration and stability analysis of thick orthotropic plate structures using finite elements based on the hybrid-Trefftz formulation. While the formulation can be used for elements of arbitrary geometry, the paper concentrates on the use of a simple and robust triangular element. The key feature of the formulation is to use element interpolations that are consistent for all values of the plate thickness, including the limit when it goes to zero. This eliminates the locking problem automatically and ensures a robust approximation for thick and thin plates. Results for various problems are included to demonstrate the accuracy and efficiency of the element.

A method that combines high-order triangular element and PO method over electrically large conducting targets illuminated by a complex source beam (CSB) is developed in this paper. High-order triangular element is used to subdivide the surface of target and PO result is obtained to investigate the scattering properties. This method has the merit of being very accurate and efficient, and its computation cost does not increase with frequency. Numerical examples are also given in this paper to demonstrate the efficiency of this method.

To analyze thin bending plates, a triangular and a quadrilateral element are presented in this paper. These suggested elements are formulated based on the hybrid-Trefftz method. The triangular element is named as THT-15, and it has 6 nodes and 18 degrees of freedom. The quadrilateral element is composed of 8 nodes and 24 degrees of freedom, and it is denoted as QHT-23. Two independent fields are introduced: one within the element and the other on the edges of the element. The internal field satisfies the governing equation of the thin plates. The boundary field is related to the nodal degrees of freedom by shape functions. For better capability, the shape functions of a 3 node Euler–Bernoulli beam are used for each edge of the element. The order of these functions for the deflection, rotation and torsion fields is equal to five, four and two, respectively. By depicting these fields in general coordinates, x–y , shape functions for the elements׳ edges are obtained. Several numerical tests are performed to assess the robustness of the suggested elements. The findings demonstrate the high accuracy of the proposed elements in analyzing the thin bending plates.

To analyze the plane problem with irregular mesh and complicated geometry, it is helpful to utilize the triangular element. In this study, several optimization criteria will be elaborated. By utilizing these provisions and satisfying the equilibrium conditions, a novel triangular element, named SST, is developed. To demonstrate the high accuracy and efficiency of the new element, a variety of structures will be solved. The findings will prove that the presented element has a low sensitivity to the geometric distortion. Moreover, the parasitic shear error will not arise when this element is employed. In addition to these, the proposed element is rotational invariant. Comparison studies will reveal that the SST element is more robust than the other well-known triangular ones.

The strain formulation approach improves accuracy and removes complications, such as shear parasitic errors and sensitivity to mesh distortions. For analyzing plane stress and strain problems, two new strain-based triangular elements are proposed. Both compatibility and equilibrium conditions are imposed to these elements. Contrary to the quadrilateral shape, triangular element facilitates proper meshing of various geometries. To formulate these elements, the linear strain field is assumed. The first element is a five-node triangular element, in which each node has two degrees of freedom. In the second one, which is a four-node element, drilling degrees of freedom are added to improve applicability of the element for bending problems. Various numerical examples and patch tests verify high accuracy and efficiency of the suggested elements in comparison with the other existing plane elements.

The mechanism of mountain forming is based on the phenomenon of the formation of compressive normal horizontal stresses exceeding gravitational. These stresses are due to contact friction between the layers of rocks. To reveal the mechanism of mountain forming, the vector method of the body equilibrium on the path of maximum effective tangential stresses of a triangular element is used, taking into account the external and internal forces acting on it. It is shown that the total vector of these forces, taking into account internal and contact friction, is directed against the gravitational force and generates mountain forming. A generalization of experimental observations on the elevation of rock structures and anomalous manifestations of rock pressure is given.

Small elements of circular, square, triangular and thin-strip cross-sections are used to suppress vortex shedding from a rectangular cylinder of stream-wise to transverse scale ratio L / B =3.0 at Reynolds numbers in the range of Re= V ∞ B / ν =75–130, where V ∞ is the on-coming velocity of the stream, and ν is the kinematic viscosity. The relative transverse dimension of the small element b / B is fixed at 0.2. The results of numerical simulation and visualization experiment show that, vortex shedding from both sides of the cylinder can be suppressed and the fluctuating drag and lift of the cylinder can be greatly reduced, if the element is placed in a certain region referred to as the effective zone. Comparisons at a specific Reynolds number indicate that the square element produces the largest size of the effective zone, whereas the triangular element yields the smallest. Results also show that the effective zone for the square element shrinks with increasing Re and disappears at Re>130. Independent of element cross-section shape and Reynolds number, the center of the effective zone is always at X / B =2.5–3.0 and Y / B ≈1.0. The mechanism of the suppression is discussed from the view points of velocity profile stability and stress distribution.

In this paper, equilateral triangular microstrip star array antenna was designed for wireless communication applications. The antennas were modeled using microstrip lines, MTEEs and S parameter data for star arrayed triangular element .The data is extracted from the momentum simulation and combined with the microstrip transmission line. The properties of antennas such as bandwidth, gain, cross polar isolation and the return loss have been simulated. Since the resonant frequency is of these antenna is 2.0GHz, these antenna is suitable for wireless network.

A cell-based smoothed discrete shear gap method (CS-DSG3) for static and free vibration analyses of Reissner–Mindlin shells is formulated by combining the cell-based strain smoothing technique with the discrete shear gap method (DSG3) using three-node triangular elements. In the CS-DSG3, each triangular element will be divided into three sub-triangles, and in each sub-triangle, the stabilized DSG3 is used to compute the strains and to avoid the transverse shear locking. Then the strain smoothing technique on whole of the triangular element is used to smooth the strains on these three sub-triangles. The CS-DSG3 hence not only overcomes the drawback of the DSG3 but also improves the accuracy as well as the stability of the DSG3. The numerical examples demonstrated that the CS-DSG3 is free of shear locking and achieves the high accuracy compared to other existing shell elements.

Hierarchical bases of arbitrary order for (div)-conforming triangular and tetrahedral elements are constructed with the goal of improving the conditioning of the mass and stiffness matrices. For the basis with the triangular element, it is found numerically that the conditioning is acceptable up to the approximation of order four, and is better than a corresponding basis in the dissertation by Sabine Zaglmayr [High Order Finite Element Methods for Electromagnetic Field Computation, Johannes Kepler Universität, Linz, 2006]. The sparsity of the mass matrices from the newly constructed basis and from the one by Zaglmayr is similar for approximations up to order four. The stiffness matrix with the new basis is much sparser than that with the basis by Zaglmayr for approximations up to order four. For the tetrahedral element, it is identified numerically that the conditioning is acceptable only up to the approximation of order three. Compared with the newly constructed basis for the triangular element, the sparsity of the mass matrices from the basis for the tetrahedral element is relatively sparser.

In fracture simulation, how to model the pre-existing cracks and simulate their propagation without remeshing is an important topic. The newly developed triangular element partition method (TEPM) provides an efficient approach to this problem. It firstly meshes the cracked body regardless of the geometry integrity of the interesting object with triangular elements. After the meshing procedure is completed, some elements are intersected by cracks. For the element intersected by a crack, the TEPM takes the element partition technique to incorporate the discontinuity into the numerical model without any interpolation enrichment. By this approach, the TEPM can simulate fracture without mesh modification. In the TEPM, all the cracked elements are treated as the usual partitioned elements in which the crack runs through. The virtual node pairs (the intersection points of crack faces and elements) at the opposite faces of the crack move independently. Their displacements are respectively determined by their neighbor real nodes (nodes formatted in the original mesh scheme) at the same side of the crack. However, among these cracked elements, the element containing a crack tip, referred to as the crack tip element thereafter, behaves differently from those cut through by the crack. Its influence on the singular field at the vicinity of the fracture tip becomes increasingly significant with the element size increasing. In the crack tip element, the virtual node pair at the crack tip move consistently before fracture occurs while the virtual node pair separate and each virtual node moves independently after the fracture propagates. Accordingly, the crack tip element is automatically transformed into the usual partitioned element. In the present paper, the crack tip element is introduced into the TEPM to account for the effect of the crack tip. Validation examples indicate that the present method is almost free from the element size effect. It can reach the same precision as the conventional finite element method under the same meshing scheme. But the TEPM is much more efficient and convenient than the conventional finite element method because the TEPM avoids the troubles that the conventional finite element method suffers, e.g., the meshing problem of cracked body, modification of mesh scheme, etc. Though the extended finite element method can also avoid these troubles, it introduces extra degrees of freedom due to node interpolation enrichment. Due to the simplicity of the present TEPM, it is believed that its perspective should be highly inspiring.

We derive generic formulae for all possible C1 continuous polynomial interpolations for triangular elements, by considering individual shape functions, without the need to prescribe the type of the degrees of freedom in advance. We then consider the possible ways in which these shape functions can be combined to form finite elements with given properties. The simplest case of fifth-order polynomial functions is presented in detail, showing how two existing elements can be obtained, as well as two new elements, one of which shows good numerical behaviour in numerical tests.