A triangular finite element for thick slabs

Based upon the natural strain theory of Argyris et al., a triangular Finite Element for the analysis of plate bending problems including shear deformations is presented. The principal merit of the element is that it possesses a constant nodal valency and does not require interior nodes, perhaps the first triangular thick slab element to meet these requirements. The element is here applied to a number of thick slab problems and found to provide results in good agreement with those previously reported.     

Neurocomputing method based on structural finite element analysis of discrete model

Based on structural finite element analysis of discrete models, a neurocomputing strategy is developed in this paper. Dynamic iterative equations are constructed in terms of neural networks of discrete models. Determination of the iterative step size, which is important for convergence, is investigated based on the positive definiteness of the finite element stiffness matrix. Consequently, a method of choosing the step size of dynamic equations is proposed and the computational formula of the best step size is derived. The analysis of the computing model shows that the solution of finite element system equations can be obtained by the method of neural network computation efficiently. The proposed method can be used for parallel computation of structural finite element in a large-scale integrated circuit (LSI).     

Implementation of a C1 triangular element based on the P-version of the finite element method

The implementation of a computer code CONE (for C¹ continuity) based on the p-version of the finite element method is described. A hierarchic family of triangular finite elements of degree p ≥ 5 is used. This family enforces C¹-continuity across inter-element boundaries, and the code is applicable to fourth order partial differential equations in two independent variables, in particular to the biharmonic equation. Applications to several benchmark problems in plate bending are presented. Sample results are examined and compared both with theoretical predictions and with the computations of other programs. Significant improvements are shown for the results obtained using CONE.     

A high precision triangular laminated anisotropic cylindrical shell finite element

The stiffness matrix for a high precision triangular laminated anisotropic cylindrical shell finite element has been formulated and coded into a composite structural analysis program. The versatility of the element's formulation enables its use in the analysis of multilayered composite plate and cylindrical shell type structures taking into account actual lamination parameters. The example applications presented demonstrated that accurate predictions of stresses as well as displacements are obtained with modest number of elements.     

A high precision triangular laminated anisotropic shallow thin shell finite element

A high precision triangular laminated shallow thin shell finite element has been developed based on the classical lamination theory. The stiffness matrix is obtained explicitly by pre and post multiplying a few basic matrices. The formulation is almost an order of magnitude faster than those available for similar order elements. The numerical results of the example problems presented demonstrate that both displacements and stresses are predicted accurately with moderately coarse grids. A complete listing of FORTRAN subroutines is presented for users, to ease implementation of the algorithm.     

Topological refinement procedures for triangular finite element meshes

This paper presents a topological approach to improve the quality of unstructured triangular finite element meshes. Topological improvement procedures are presented both for elements that are interior to the mesh and for elements connected to the boundary. Optimal ordering of the topology improvement operations and their efficient implementation is also discussed. Several example meshes are included to demonstrate the effectiveness of the approach in improving element quality in a finite element mesh.     

Anisotropic mesh adaptation for finite volume and finite element methods on triangular meshes

The present paper deals with an anisotropic mesh adaptation (AMA) of triangulation which can be employed for the numerical solution various problems of physics. AMA tries to construct an optimal triangulation of the domain of computation in the sense that an “error” of the solution of the problem considered is uniformly distributed over the whole triangulation. First, we describe the main idea of AMA. We define an optimal triangle and an optimal triangulation. Then we describe the process of optimization of the triangulation and the complete multilevel computational process. We apply AMA to a problem of CFD, namely to inviscid compressible flow. The computational results for a channel flow are presented. Received: 11 December 1997 / Accepted: 16 February 1998     

A triangular bending element based on an energy-orthogonal free formulation

A new Kirchhoff triangular plate-bending element with nine degrees of freedom is derived. The assumed shape functions are nonconforming and consist of a complete quadratic expansion complemented by three cubic modes that are energy-orthogonal to the quadratic expansion. The derivation is based on the free formulation of Bergan and Nygård, which satisfies the conditions of invariance and the patch test. An explicit expression is obtained for the inverse transformation matrix required in this formulation, which makes the stiffness computations extremely efficient. The aspect ratio sensitivity of the element under cylindrical bending modes can be lessened by scaling the higher-order stiffness through coefficients determined by a superlinear patch test technique. Numerical experiments indicate that the new element outperforms previously derived 9-dof triangular elements based on displacement modes.     

Complete stress field in sandwich panels with a new triangular finite element of single-layer theory

An alternative to conventional three-dimensional solid elements or elements based on the layerwise (zig-zag) theory is an element based on a single-layer plate theory in which the weighted-average field variables capture the panel displacement and stress fields. This study presents a new triangular finite element for modeling thick sandwich panels based on a {3, 2}-order single-layer plate theory. It utilizes seven weighted-average field variables arising from the cubic and quadratic representations of the in-plane and transverse displacement fields, respectively. In order to satisfy the C¹ interelement continuity requirement, this triangular sandwich element is developed by utilizing the hybrid energy functional.     

Fully discrete finite element method based on pressure stabilization for the transient Stokes equations

In this work, a new fully discrete stabilized finite element method is studied for the two-dimensional transient Stokes equations. This method is to use the difference between a consistent mass matrix and underintegrated mass matrix as the complement for the pressure. The spatial discretization is based on the P₁–P₁ triangular element for the approximation of the velocity and pressure, the time discretization is based on the Euler semi-implicit scheme. Some error estimates for the numerical solutions of fully discrete stabilized finite element method are derived. Finally, we provide some numerical experiments, compared with other methods, we can see that this novel stabilized method has better stability and accuracy results for the unsteady Stokes problem.     

Aspects of a simple triangular plate bending finite element

This paper discusses an early phase of a study whose aim is the development of economical, yet still accurate, finite elements for dynamic analyses. Transverse shear deformation is utilized to construct a nine-degree-of-freedom plate bending element. Numerical difficulties, not unexpected, occur. A device for overcoming these difficulties is applied.     

A finite element method for Stokes equation using discrete singularity expansion

The numerical method presented in this paper for Stokes equation with corner singularity includes mainly two steps. Firstly, we solve a simple eigenvalue problem, which is one dimension less than the original problem, to obtain the discrete expansion of the singularity near the corner. Secondly, we combine the approximation of the singularity and standard finite element basis functions to construct special finite element spaces, and solve the original problem in the special spaces on a conventional mesh. The numerical examples show the effectiveness of this method.     

Computer modelling and finite element analysis of spiral triangular strands

In this paper a new mathematical geometric model of spiral triangular wire strands with a construction of (3 + 9) and (3 + 9 + 15) wires is proposed and an accurate computational two-layered triangular strand 3D solid modelling, which is used for a finite element analysis, is presented. The present geometric model fully considers the spatial configuration of individual wires in the strand. The three dimensional curve geometry of wires axes in the individual layers of the triangular strand consists of straight linear and helical segments. The derived mathematical representation of this curve is in the form of parametric equations with variable input parameters which facilitate the determination of the centreline of an arbitrary circular wire of the right and left hand lay triangular one and two-layered strands. Derived geometric equations were used for the generation of accurate 3D geometric and computational strand models. The correctness of the derived parametric equations and performance of the generated strand model are controlled by visualizations. The 3D computational model was used for a finite element behaviour analysis of the two-layered triangular strand subjected to tension loadings. Illustrative examples are presented to highlight the benefits of the proposed geometric parametric equations and computational modelling procedures by using the finite element method.     

A novel discrete image encryption algorithm based on finite algebraic structures

Multimedia Tools and Applications - The arithmetic properties of a finite field have a remarkable impact on the security features of symmetric and asymmetric cryptosystems. Conventionally, in...     

BatTri: A two-dimensional bathymetry-based unstructured triangular grid generator for finite element circulation modeling

A brief summary of Delaunay unstructured triangular grid refinement algorithms, including the recent “off-centers” method, is provided and mesh generation requirements that are imperative to meet the criteria of the circulation modeling community are defined. A Matlab public-domain two-dimensional (2-D) mesh generation package (BatTri) based on these requirements is then presented and its efficiency shown through examples. BatTri consists of a graphical mesh editing interface and several bathymetry-based refinement algorithms, complemented by a set of diagnostic utilities to check and improve grid quality. The final output mesh node locations, node depths and element incidence list are obtained starting from only a basic set of bathymetric data. This simple but efficient setup allows fast interactive mesh customization and provides circulation modelers with problem-specific flexibility while satisfying the usual requirements on mesh size and element quality. A test of the “off-centers” method performed on 100 domains with randomly generated coastline and bathymetry shows an overall 25% reduction in the number of elements with only slight decrease in element quality. More importantly, this shows that BatTri is easily upgradeable to meet the future demands by the addition of new grid generation algorithms and Delaunay refinement schemes as they are made available.     

A high precision coupled bending–extension triangular finite element for laminated plates

It is well recognized that the estimation of interlaminar stresses and strain energy release rates is important in designing laminated composite panels. Generally coupled bending–extension finite elements are necessary to study laminates to include the effects of coupling and/or combined transverse and extensional loads. Such elements are normally formulated adapting the classical theory of bending and extension. While the classical laminated plate theory of bending has provision to obtain interlaminar stresses due to transverse loading, it is necessary to include certain higher order terms in the extensional theory in order to obtain the interlaminar stresses due to inplane loads. A high precision triangular element based on a theory which includes both the bending and extension with necessary higher order terms is presented in this paper. The performance of this element is validated with the aid of examples. Numerical results for displacements in symmetric and unsymmetric laminates under bending loads have been given. Numerical results for interlaminar stresses in symmetric and unsymmetric laminates have been given for the well-known benchmark problem of a coupon with free edges. Strain energy release rate components at the delamination tip in coupons with unsymmetric sublaminates have been given. The effects of delamination length and location on the components of the strain energy release rate have been studied. Results indicated that with the use of this element, the interlaminar stresses can be estimated reasonably accurately, over a major part of the laminate except in a small local region close to the free edge. Global–local analysis with three-dimensional elements in the local region, is suggested to obtain local stresses more accurately. Interlaminar stresses at the boundary of a hole in a perforated plate under extension have been obtained to illustrate the use of the present element in a global–local analysis strategy.     

Shot peening simulation using discrete and finite element methods

Modeling shot peening process is very complex as it involves the interaction of metallic surfaces with a large number of shots of very small diameter. Conventionally such problems are solved using the finite element software (such as ABAQUS) to predict the stresses and strains. However, the number of shots involved and the number of elements required in a real-life components for a 100% coverage that lasts a considerable duration of peening make such an approach impracticable. Ideally, a method that is suitable for obtaining residual compressive stresses (RCS) and the amount of plastic deformations with the least computational effort seems a dire need.In this paper, an attempt has been made to address this issue by using the discrete element method (DEM) in combination with the finite element method (FEM) to obtain reasonably accurate predictions of the residual stresses and plastic strains. In the proposed approach, the spatial information of force versus time from the DEM simulation is utilized in the FE Model to solve the shot peening problem as a transient problem. The results show that the RCS distribution obtained closely matches with that of the computationally intensive direct FEM simulation. It has also been established, in this paper, that this method works well even in the situations where the robust unit cell approaches are found to be difficult to handle.     

On a posteriori error estimates for the linear triangular finite element

Based on equilibration of side fluxes, an a posteriori error estimator is obtained for the linear triangular element for the Poisson equation, which can be computed locally. We present a procedure for constructing the estimator in which we use the Lagrange multiplier similar to the usual equilibrated residual method introduced by Ainsworth and Oden. The estimator is shown to provide guaranteed upper bound, and local lower bounds on the error up to a multiplicative constant depending only on the geometry. Based on this, we give another error estimator which can be directly constructed without solving local Neumann problems and also provide the two-sided bounds on the error. Finally, numerical tests show our error estimators are very efficient.     

Mixed finite element formulations with volume bubble functions for triangular elements

This paper deals with volume bubble functions for mixed finite triangular elements in geometrically linear elasticity. In two different versions these functions are used in order to enrich the displacement field and the enhanced strain field, respectively. Appropriate conditions for satisfaction of the patch test are verified. In the numerical example, firstly the patch test is satisfied. Secondly, simulations of Cook’s membrane problem demonstrate that the proposed formulations avoid locking and reduce stress oscillations for incompressible materials.