Oscillatory crack tip triangular elements for finite element analysis of interface cracks

In this paper a special crack tip element has been developed in which displacements and stresses have the same behaviour as those of bi‐material interface cracks with open tips. The element degenerates into a traditional triangular quarter point element in cases of homogeneous cracks. An isoparametric co‐ordinate system (ρ, t) is defined in this study, and numerical techniques using these co‐ordinates to evaluate Jacobian matrices, shape function derivatives, and element stiffness matrices are developed. Also, equations calculating the complex stress intensity factor using displacements are obtained in this study. Numerical results are in good agreement with known analytical solutions in two examples. Copyright © 2003 John Wiley & Sons, Ltd.     

Automatic conversion of triangular finite element meshes to quadrilateral elements

A method is presented for the fully automatic conversion of a general finite element mesh containing triangular elements into a mesh composed of exclusively quadrilateral elements. The initial mesh may be constructed of entirely triangular elements or may consist of a mixture of triangular and quadrilateral elements. The technique used employs heuristic procedures and criteria to selectively combine adjacent triangular elements into quadrilaterals based on preestablished criteria for element quality. Additional procedures are included to eliminate isolated triangles. The methods operates completely without user intervention once the nodal co-ordinates and element connectivity of the original mesh are supplied.     

Hybrid-Trefftz six-node triangular finite element models for Helmholtz problem

In this paper, six-node hybrid-Trefftz triangular finite element models which can readily be incorporated into the standard finite element program framework in the form of additional element subroutines are devised via a hybrid variational principle for Helmholtz problem. In these elements, domain and boundary variables are independently assumed. The former is truncated from the Trefftz solution sets and the latter is obtained by the standard polynomial-based nodal interpolation. The equality of the two variables are enforced along the element boundary. Both the plane-wave solutions and Bessel solutions are employed to construct the domain variable. For full rankness of the element matrix, a minimal of six domain modes are required. By using local coordinates and directions, rank sufficient and invariant elements with six plane-wave modes, six Bessel solution modes and seven Bessel solution modes are devised. Numerical studies indicate that the hybrid-Trefftz elements are typically 50% less erroneous than their continuous Galerkin element counterpart.     

Efficient visualization of high‐order finite elements

A general method for the post‐processing treatment of high‐order finite element fields is presented. The method applies to general polynomial fields, including discontinuous finite element fields. The technique uses error estimation and h‐refinement to provide an optimal visualization grid. Some filtering is added to the algorithm in order to focus the refinement on a visualization plane or on the computation of one single iso‐zero surface. 2D and 3D examples are provided that illustrate the power of the technique. In addition, schemes and algorithms that are discussed in the paper are readily available as part of an open source project that is developed by the authors, namely Gmsh. Copyright © 2006 John Wiley & Sons, Ltd.     

Strain based triangular finite element for plate bending analysis

ABSTRACT

In this work, the formulation of a new triangular finite element is presented for static and free vibration of plate bending. The developed element which contains the three essential external degrees of freedom at each of the three corner nodes is based on the Reissner/Mindlin theory and the strain-based approach. This element is based on the linear variation of the three bending strains and constant transverse shear strains. The present element performances are evaluated through several tests related to moderated thick and thin plates with various shapes where it is found to be numerically more efficient than the bilinear element.     

Efficient numerical strategies for spectral stochastic finite element models

The use of spectral stochastic finite element models results in large systems of equations requiring specialized solution strategies. This paper discusses three different numerical algorithms for solving these large systems of equations. It presents a trade‐off of these algorithms in terms of memory usage and computation time. It also shows that the structure of the spectral stochastic stiffness matrix can be exploited to accelerate the solution process, while keeping the memory usage to a minimum. Copyright © 2005 John Wiley & Sons, Ltd.     

Analyses of magneto-electro-elastic plates using a higher order finite element model

In this paper is presented a higher-order model for static and free vibration analyses of magneto-electro-elastic plates, which allows the study of thin and thick plates. The finite element model is a single layer triangular plate/shell element with 24 degrees of freedom for the generalized mechanical displacements. Two degrees of freedom are introduced per each element layer, one corresponding to the electrical potential and the other for magnetic potential. Solutions are obtained for different laminations of the magneto-electro-elastic plate, as well as for the purely elastic plate as a special case. Results are compared with alternative models for static and free vibrations situations.     

Calculation of strain-energy release rates with higher order and singular finite elements

A general finite element procedure for obtaining strain-energy release rates for crack growth in isotropic materials is presented. The procedure is applicable to two-dimensional finite element analyses and uses the virtual crack-closure method. The procedure was applied to non-singular 4-noded (linear), 8-noded (parabolic), and 12-noded (cubic) elements and to quarter-point and cubic singularity elements. Simple formulae for strain-energy release rates were obtained with this procedure for both non-singular and singularity elements. The formulae were evaluated by applying them to two mode I and two mixed mode problems. Comparisons with results from the literature for these problems showed that the formulae give accurate strain-energy release rates.     

Efficient implementation of high‐order finite elements for Helmholtz problems

Computational modeling remains key to the acoustic design of various applications, but it is constrained by the cost of solving large Helmholtz problems at high frequencies. This paper presents an efficient implementation of the high‐order finite element method (FEM) for tackling large‐scale engineering problems arising in acoustics. A key feature of the proposed method is the ability to select automatically the order of interpolation in each element so as to obtain a target accuracy while minimizing the cost. This is achieved using a simple local a priori error indicator. For simulations involving several frequencies, the use of hierarchical shape functions leads to an efficient strategy to accelerate the assembly of the finite element model. The intrinsic performance of the high‐order FEM for 3D Helmholtz problem is assessed, and an error indicator is devised to select the polynomial order in each element. A realistic 3D application is presented in detail to demonstrate the reduction in computational costs and the robustness of the a priori error indicator. For this test case, the proposed method accelerates the simulation by an order of magnitude and requires less than a quarter of the memory needed by the standard FEM. © 2016 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons Ltd.     

Construction of triangular finite element universal matrices

The various ‘universal’ matrices from which finite element matrices for triangular elements are assembled in many electromagnetics and acoustics problems, can all be derived from a basic set of three fundamental matrices. These represent, respectively, the metric of the linear manifold spanned by the triangle interpolation polynominals, the finite differentiation operator on that same manifold, and a product-embedding operator for the corresponding manifold for interpolation polynomials one order higher. Two of these have already been tabulated and published; the required method for computing the third is given in this paper, along with tables of low-order matrices.     

Composite plate analysis using a new discrete shear triangular finite element

This paper deals with the static and dynamic (free vibrations) analysis of plates built up with a symmetric series of orthotropic layers. The formulation of a new simple triangular finite element having three nodes and three degrees of freedom per node is presented. The element called DST (Discrete Shear Triangle) is free of shear locking and has a proper rank. It coincides with DKT (Discrete Kirchhoff Triangle) when the transverse shear effects are negligible. A large number of classical problems is considered to evaluate the performance of the element for the analysis of composite plates. Very satisfactory results are obtained for displacements, stresses and frequencies.     

Efficient implicit finite element analysis of sheet forming processes

The computation time for implicit finite element analyses tends to increase disproportionally with increasing problem size. This is due to the repeated solution of linear sets of equations, if direct solvers are used. By using iterative linear equation solvers the total analysis time can be reduced for large systems. For plate or shell element models, however, the condition of the matrix is so ill that iterative solvers do not reach the huge time‐savings that are realized with solid elements. By introducing inertial effects into the implicit finite element code the condition number can be improved and iterative solvers perform much better. An additional advantage is that the inertial effects stabilize the Newton–Raphson iterations. This also applies to quasi‐static processes, for which the inertial effects finally do not affect the results. The presented method can readily be implemented in existing implicit finite element codes. Industrial size deep drawing simulations are executed to investigate the performance of the recommended strategy. It is concluded that the computation time is decreased by a factor of 5 to 10. Copyright © 2003 John Wiley & Sons, Ltd.     

Alternative integration formulae for triangular finite elements

Some new formulae are presented for numerical integration over triangular regions. These formulae, which are triangularly symmetric, involve four, six, and seven integration points, respectively. They were derived by determining co-ordinates and weight factors for which polynomial functions of up to a certain degree are integrated exactly. The effectiveness of the formulae is demonstrated by applying them to the integration of five non-polynomial functions, two of which are singular within the region of integration. Results obtained with the new formulae are compared with results obtained with other, comparable formulae. It is seen that the new formulae are more accurate than some existing ones with the same numbers of points.     

A refined triangular plate bending finite element

The derivation of the stiffness matrix for a refined, fully compatible triangular plate bending finite element is presented. The Kirchhoff plate bending theory is assumed. Six parameters or degrees of freedom are introduced at each of the three corner nodes resulting in an 18 degree of freedom element. This refined element is found to give better results for displacements and particularly for internal moments than any plate bending element, regardless of shape, previously reported in the literature.     

Hierarchical high‐order conforming C1 bases for quadrangular and triangular finite elements

We present three new sets of C¹ hierarchical high‐order tensor‐product bases for conforming finite elements. The first basis is a high‐order extension of the Bogner–Fox–Schmit basis. The edge and face functions are constructed using a combination of cubic Hermite and Jacobi polynomials with C¹ global continuity on the common edges of elements. The second basis uses the tensor product of fifth‐order Hermite polynomials and high‐order functions and achieves global C¹ continuity for meshes of quadrilaterals and C² continuity on the element vertices. The third basis for triangles is also constructed using the tensor product of one‐dimensional functions defined in barycentric coordinates. It also has global C¹ continuity on edges and C² continuity on vertices. A patch test is applied to the three considered elements. Projection and plate problems with smooth fabricated solutions are solved, and the performance of the h‐ and p‐refinements are evaluated by comparing the approximation errors in the L₂‐ and energy norms. A plate with singularity is then studied, and h‐ and p‐refinements are analysed. Finally, a transient problem with implicit time integration is considered. The results show exponential convergence rates with increasing polynomial order for the triangular and quadrilateral meshes of non‐distorted and distorted elements. Copyright © 2016 John Wiley & Sons, Ltd.     

Behaviour of Lagrangian triangular mixed fluid finite elements

The behaviour of mixed fluid finite elements, formulated based on the Lagrangian frame of reference, is investigated to understand the effects of locking due to incompressibility and irrotational constraints. For this purpose, both linear and quadratic mixed triangular fluid elements are formulated. It is found that there exists a close relationship between the penalty finite element approach that uses reduced/selective numerical integration to alleviate locking, and the mixed finite element approach. That is, performing reduced/ selective integration in the penalty approach amounts to reducing the order of pressure interpolation in the mixed finite element approach for obtaining similar results. A number of numerical experiments are performed to determine the optimum degree of interpolation of both the mean pressure and the rotational pressure in order that the twin constraints are satisfied exactly. For this purpose, the benchmark solution of the rigid rectangular tank is used. It is found that, irrespective of the degree of mean and the rotational pressure interpolation, the linear triangle mesh, with or without central bubble function (incompatible mode), locks when both the constraints are enforced simultaneously. However, for quadratic triangle, linear interpolation of the mean pressure and constant rotational pressure ensures exact satisfaction of the constraints and the mesh does not lock. Based on the results obtained from the numerical experiments, a number of important conclusions are arrived at.     

A note on plate bending analysis using triangular finite element TUBA 3

In Reference 1 a mistake has been made by the derivation of equation (40). The correct expression is shown in the present note.     

High-order polynomial triangular finite elements for potential problems

An analytic derivation is given for high-accuracy triangular finite elements useful for numerical solution of field problems involving Laplace's, Poisson's, Helmholtz's, or related elliptic partial differential equations in two dimensions. General expressions are developed for complete polynomial fields of arbitrarily high order, and the method for obtaining element describing matrices is shown. These matrices can always be written in terms of trigonometric functions of the vertex angles, and the triangle area, multiplied by certain numerical coefficient matrices which are the same for any triangle. For polynomial fields up to fourth order, the numerical coefficient matrices are given, so that the element matrices for any triangle can be found easily. Use of these new elements is illustrated by a simple vibration problem.