Triangular Finite Element Model Research Articles

Error Analysis of Heat Conduction Partial Differential Equations using Galerkin’s Finite Element Method

The present work explores an error analysis of Galerkin finite element method (GFEM) for computing steady heat conduction in order to show its convergence and accuracy. The steady state heat distribution in a planar region is modeled by two-dimensional Laplace partial differential equations. A simple three-node triangular finite element model is used and its derivation to form elemental stiffness matrix for unstructured and structured grid meshes is presented. The error analysis is performed by comparison with analytical solution where the difference with the analytical result is represented in the form of three vector norms. The error analysis for the present GFEM for structured grid mesh is tested on heat conduction problem of a rectangular domain with asymmetric and mixed natural-essential boundary conditions. The accuracy and convergence of the numerical solution is demonstrated by increasing the number of elements or decreasing the size of each element covering the domain. It is found that the numerical result converge to the exact solution with the convergence rates of almost O(h²) in the Euclidean L2 norm, O(h²) in the discrete perpetuity L∞norm and O(h1) in the H1 norm.

An edge-based smoothed finite element method softened with a bubble function (bES-FEM) for solid mechanics problems

The edge-based smoothed finite element method (ES-FEM) was recently proposed to improve the performance of linearly triangular finite element models for mechanics problems. Such a good performance is attributed to the right amount softening induced by the edge-based smoothing operation. In this paper, we propose an improved formulation of the ES-FEM so that the bES-FEM can be further softened becoming volumetric locking free and hence works well also incompressible or nearly incompressible problems. The improved formulation uses the usual piecewise linear displacements but is supplemented with a cubic bubble function in triangular elements, which induces further softening to the bilinear form allowing the weakened weak (W2) procedure to search for a solution satisfying the divergence-free conditions. The smoothed strains are evaluated based on smoothing domains associated with edges of the adjacent elements as in the ES-FEM. The divergence-free condition of the bES-FEM is verified via detailed eigenvalue analyses. Several numerical examples are provided to show the effectiveness and reliability of the present method. We also show numerically that the present element is insensitive to mesh distortion and is superior to the bubble finite element (MINI element) in the incompressible limit.

Hybrid quadrilateral finite element models for axial symmetric Helmholtz problem

This paper is a continuation of the previous work in which six-node triangular finite element models for the axial symmetric Helmholtz problem are devised by using a hybrid functional and the spherical-wave modes [1]. The six-node models can readily be incorporated into the standard finite element program framework and are typically ∼50% less erroneous than their conventional or, equivalently, continuous Galerkin counterpart. In this paper, four-node and eight-node quadrilateral models are devised. Two ways of selecting the spherical-wave modes are attempted. In the first way, a spherical-wave pole is selected such that it is equal-distant from an opposing pair of element nodes. In the second way, the directions of the spherical-waves passing through the element origin are equal-spaced with one of the directions bisecting the two parametric axes of the element. Examples show that both ways lead to elements that yield very similar predictions. Furthermore, four-node and eight-node hybrid elements are typically ∼50% and ∼70% less erroneous than their conventional counterparts, respectively.

MULTI-FIELD THREE-NODE TRIANGULAR FINITE ELEMENT MODEL FOR HELMHOLTZ PROBLEM

In this paper, four three-node triangular finite element models which can readily be incorporated into the standard finite element program framework are devised via a multi-field variational functional for the bounded plane Helmholtz problem. In the models, boundary and domain fields are independently assumed. The former is constructed by nodal interpolation and the latter comprises nonsingular solutions of the Helmholtz equation. The equality of the two fields are enforced along the element boundary. Among the four devised models, the most accurate one is 1/3 to 1/2 less erroneous than the conventional single-field model in most examples.

Spherical-wave based triangular finite element models for axial symmetric Helmholtz problems

In this paper, six-node hybrid triangular finite element models are devised for axial symmetric Helmholtz problems. In the formulation, boundary and domain approximations to the Helmholtz field are defined for each element. While the boundary approximation is constructed by nodal interpolation, the domain approximation satisfies the Helmholtz equation and is composed of spherical waves with source points located along the axis of symmetry. To formulate rank sufficient six-node elements, a minimal of six wave modes from three source points are required. Two methods of selecting the source points are attempted. In the first method, the directions of the waves passing through the element are essentially parallel to the three lines connecting the parametric center of the element and its three corner (or side) nodes. In the second method, the directions are essentially equally spaced at 2 π/3 interval in the r– z-plane. For the attempted examples, the average error ratios of the proposed elements and the conventional element are around 50%.

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.

Analysis of Two-phase Fluid Flow through Anisotropic Media Using FEM-Permeability Tensor Transient Model

ABSTRACT Reservoir simulators generally employ a simple anisotropy model rather than using a permeability tensor model. However, this can be inaccurate for the fluid-flow calculation in a highly heterogeneous and anisotropic reservoir system having multiple permeability zones. In order to overcome such problems, we have first formulated the two-phase flow equation with permeability tensor, and developed a transient triangular finite element model (FEM) by using Galerkin's method with boundary element method (BEM) as a well model. The developed FEM permeability tensor model coupled with BEM technique for well model has been validated by comparing against the results from HYBOS FDM reservoir simulator. In order to examine the applicability of the developed model, a complex reservoir system having three different permeability zones has been established. As a result, it was found out that the conventional simple anisotropy model neglecting the effect of diagonal permeability yielded serious errors in interpreting the two-phase flow problems.

The linear stress triangular equilibrium model in plane elasticity

Advantage is taken of oblique coordinates to give a revised theory of the triangular equilibrium finite element model with linearly varying stresses for problems in plane linear elasticity. This new formulation has considerably improved computational efficiency over earlier derivations because equilibrium of the stresses within the element is enforced explicitly without recourse to the more expensive numerical methods used previously. The equilibrium model provides upper bounds to the elastic strain energy of a body under applied loading; it is the dual of a triangular finite element model with linear strains obtained from compatible quadratic displacements which provides corresponding lower bounds.

A finite element procedure for plane strain metal flow within specified plastic boundaries

In this paper a new set of governing equations are developed for analyzing two-dimensional plane strain plasticity problems. Two sets of stress components are introduced—one set satisfying the equilibrium equations and the second ensuring that the yield criterion is satisfied. The differences between these two sets of stress components constitute the approximation errors. Using the method of least squares a simple triangular finite element model is developed by means of which the differences between the two sets of stress components are minimized. The formulation and the element developed are verified by analyzing three metal forming examples. Results computed with the proposed method correlate well with the exact solution of one example, and with the classical slip-line fields solutions of the other two examples.

Computationally Efficient Algorithms for Parameter Estimation and Uncertainty Propagation in Numerical Models of Groundwater Flow

Finite difference and finite element methods are frequently used to study aquifer flow; however, additional analysis is required when model parameters, and hence predicted heads are uncertain. Computational algorithms are presented for steady and transient models in which aquifer storage coefficients, transmissivities, distributed inputs, and boundary values may all be simultaneously uncertain. Innovative aspects of these algorithms include a new form of generalized boundary condition; a concise discrete derivation of the adjoint problem for transient models with variable time steps; an efficient technique for calculating the approximate second derivative during line searches in weighted least squares estimation; and a new efficient first‐order second‐moment algorithm for calculating the covariance of predicted heads due to a large number of uncertain parameter values. The techniques are presented in matrix form, and their efficiency depends on the structure of sparse matrices which occur repeatedly throughout the calculations. Details of matrix structures are provided for a two‐dimensional linear triangular finite element model.

A Symmetric Rail Shear Test for Mode II Fracture Toughness (GIIC) of Composite Materials—Finite Element Analysis

Using a high precision triangular finite element model, the elastic stress field in a symmetric rail shear test specimen is determined. The results are graphically presented and discussed. A simple finite element technique for energy release rate analysis of the pre-cracked test specimen is presented. Numerical results showing the effects of material properties and crack length are tabulated as normalized stress-intensity fac tors. Based on this numerical study it is fair to say that the symmetric rail shear test with some modifications is a viable method for shear characterization of composite materials in general and for GIIC determination in particular. Final acceptance of the test method, however, awaits experimental verification.

A shear‐flexible triangular finite element model for laminated composite plates

AbstractFormulation and numerical evaluation of a shear‐flexible triangular laminated composite plate finite element is presented in this paper. The element has three nodes at its vertices, and displacements and rotations along with their first derivatives have been chosen as nodal degrees‐of‐freedom. Computation of element matrices is highly simplified by employing a shape function subroutine, and an optimal numerical integration scheme has been used to improve the performance. The element has satisfactory rate of convergence and acceptable accuracy with mesh refinement for thick as well as thin plates of both homogeneous isotropic and laminated anisotropic materials. The numerical studies also suggest that reliable prediction of the behaviour of laminated composite plates necessitates the use of higher order shear‐flexible finite element models, and the proposed finite element appears to have some advantages over available elements.

Improved finite element models for the large displacement bending and post buckling analysis of thin plates

Triangular finite element models are presented for the large displacement bending and postbuckling analysis of thin plates. The formulation is based upon a general variational theory which offers a wide scope for the selection of valid finite element trial functions. This makes it feasible to model the particular physical actions crucial to the successful calculation of large displacement behaviour. In this paper it is demonstrated by numerical examples that a notable improvement in performance is achieved, without a significant increase in computational effort, when simple inextensional bending deformations are included explicitly in the finite element trial functions.

Finite element analysis of free convection flows

Two dimensional natural convection problems governed by the continuity, momentum and energy balance equations and simplified through the Boussinesque approximation for buoyancy forces, are studied. By using a stream function the continuity equation is satisfied exactly and the remaining equations are expressed in terms of temperature and vorticity functions. A simple triangular finite element model is then developed and it is employed to analyse two examples, under varying conditions. For the cases where results (experimental and numerical) obtained by other authors are available a comparison is made. This comparison exhibits good agreement.

A mixed finite element formulation for plate problems

AbstractA mixed triangular finite element model has been developed for plate bending problems in which effects of shear deformation are included. Linear distribution for all variables is assumed and the matrix equation is obtained through Reissner's variational principle. In this model, interelement compatibility is completely satisfied whereas the governing equations within the element are satisfied ‘in the mean’. A detailed error analysis is made and convergence of the scheme is proved. Numerical examples of thin and moderately thick plates are presented.