Finite Element Concept Research Articles

Analysis of the thermally stressed state of shells of arbitrary shape

Geometrically nonlinear relationships are derived for thin elastic shells subjected to nonuniform heating. A discrete system describing the thermally stressed state of a shell subject to significant shape variation is obtained using the concept of finite elements. The strain state is determined by the iteration method, based on computation of the first and second variations of the system's energy. Examples are cited for the calculation of a plate with an elliptical heating zone and a shell of canonical shape under nonstationary heating.

Geometric and Material Nonlinear Analysis of Thin‐Walled Beam‐Columns

In this paper, the effect of plastic area deformation after local yielding of thin‐walled beam‐columns is studied. Beyond the yield point of the material in the beam‐columns, the effective stress‐strain curve is determined by experiment. The solution method uses the finite element concept in conjunction with a finite segment method of cross section. Simpson's rule is used to calculate the elastic stiffness matrix for the integration over the member length. Nonlinear equations are solved by using incremental iterative techniques. Some numerical examples are given and compared with the experimental work. The results are satisfactory.

Polynomial Chaos in Stochastic Finite Elements

A new method for the solution of problems involving material variability is proposed. The material property is modeled as a stochastic process. The method makes use of a convergent orthogonal expansion of the process. The solution process is viewed as an element in the Hilbert space of random functions, in which a sequence of projection operators is identified as the polynomial chaos of consecutive orders. Thus, the solution process is represented by its projections onto the spaces spanned by these polynomials. The proposed method involves a mathematical formulation which is a natural extension of the deterministic finite element concept to the space of random functions. A beam problem and a plate problem are investigated using the new method. The corresponding results are found in good agreement with those obtained through a Monte-Carlo simulation solution of the problems.

A posteriori error estimation for the finite element and boundary element methods

An a posteriori error estimation technique is formulated for the finite element (FEM) and boundary element (BEM) methods. The evaluation of the error indicator is based on the projection process of element solutions. The error estimator is found by summing the error indicators. Global energy and L 2 error norms are employed for FEM and BEM, respectively. Example problems include three FEM problems and two BEM problems. It is shown how the one-dimensional FEM concepts can be easily applied to two-dimensional BEM.

Numerical simulation of flow processes

The numerical simulation of viscous flow problems is discussed by means of several examples pertaining to the flow in a two-roll mill, semi-conductor crystal growth, coextrusion and injection molding. The finite element concept and the method of weighted residuals are explained in some detail. The illustrations deal with two-dimensional, three-dimensional and transient flow problems.

Continuous full-field representation and differentiation of three-dimensional experimental vector data

A new numerical method is presented for representing and differentiating experimental data. It combines a smoothing technique with finite-element concepts. Discrete experimental vector data having up to at least three independent components can be processed to directly obtain smooth values of the vector function and its derivatives. Each of these vectors may contain several components. The isoparametric elements used have curve boundaries and can consequently fit an arbitrary geometry using only a few elements. As neither equilibrium nor any constitutive relationship is assumed, the concept has general applicability. The method is easy and expedient to use, and the illustrated results demonstrate its accuracy.

A conservative finite element method for heat conduction problems

AbstractThis paper is concerned with the development of a numerical model which is based on the direct integral balance of internal energy over an a priori defined subdomain, associated with any nodal point, and on the piecewise temperature and geometrical interpolation inherent in the isoparametric finite element concept. This discretization procedure, also called the conservative finite element method (CFEM), ensures local and global energy conservation, in spite of discretization errors, and preserves the major feature of the finite element technique, i.e. the versatility of its algorithm.The CFEM equations are first developed and then some features of the CFEM matrices and solutions are compared with relevant features of the finite element method based on the Galerkin orthogonalization process (GFEM).To confirm better accuracy of the numerical procedure thus developed, four examples of a comparative nature, dealing with simple configurations, are solved.Furthermore, the stability and oscillation characteristics of the CFEM and GFEM solutions are established by means of the von Neumann approach, in order to show less stringent stability requirements for the CFEM model.

A finite element formulation for multilayered and thick plates

A bilinear isoparametric finite element concept is used for the numerical analysis of multilayered plates. The underlying theory used allows for transverse shear and normal strains in each layer, thus extending the analysis to very thick plates and laminates. To illustrate the versatility of the multilayered element, three examples are presented and the results are compared with available exact solutions.

A simple interactive finite element processor for teaching purposes

The finite element method is now a standard numerical procedure for solving electric or magnetic fields. In this paper, we describe a simple interactive finite element processor used in a student environment to teach the finite element method, and its use in the design of electrical devices. This processor allows students to describe the main finite element concepts (mesh generation, region properties, boundary conditions) and to realise some numerical experiments in field computation. A particular attention is given to the influence of the finite elment discretization on the quality of the field solution. This processor is also a good introduction to CAD techniques. Students describe the device under study by means of interactive graphic procedures using a graphic display and digitizer. The canonical structure of the geometric and mesh data base is an example of the use of the modern concepts in data structures. The geometric structure chosen allows quick interactive modification of the representation of the device under study. Because this processor is especially built for teaching purposes, it is very easy to learn and to use. This point is very essential and will be developed in the full paper.

Large finite elements method for the solution of problems in the theory of elasticity

In contrast to conventional finite element (CFE) formulations, the large finite element (LFE) concept is based on subdividing the region under consideration into a small number of LFE and using in each of them an appropriate parametric displacement field such that the governing differential problem equations are satisfied a priori (Trefftz's method). Where relevant, known local solutions in the vicinity of a stress concentration or stress singularity are used as a convenient expansion basis. The boundary conditions, as well as the continuity across the interfaces, are implicitly imposed by an appropriate variational functional. The LFE concept attempts to combine the flexibility of the conventional FE method with the accuracy and high convergence rate of the Trefftz's method. The paper summarises the principal results obtained and shows that the practical efficiency of the LFE analysis is superior to a CFE solution, for both regular and singular problems.

Consistent use of finite elements in time and the performance of various recurrence schemes for the heat diffusion equation

AbstractThe use of the finite element concept in the time domain has resulted in a reinterpretation of many previously recorded recurrence formulae and the discovery of new ones. An important new feature common to all such schemes is the fact that a weighted forcing term is used. This precludes the application of unrealistic forcing terms resulting in a direct finite difference derivation and ‘smooths’ oscillation due to sudden changes of boundary condition. This problem is discussed in the context of various two‐ and three‐level time schemes.

Element spaces and their separation for determining bounds to eigenvalues in vibration problems

This paper introduces the concept of convergent eigenvalue transformations and elaborates on its use with the finite element method for obtaining close upper and lower bounds to the eigenvalues of continuous systems in free vibration. It is shown that, when incorporated with the finite element concept, convergent eigenvalue transformations attain a unique form defining an infinite number of convergent elements, whose totality is referred to as an element space. Criteria are derived for the determination of the elements which give upper and lower bounds to the eigenvalues and their applicability is discussed with reference to some vibration, problems of beams. The analysis presupposes the knowledge of a convergent element but creates useful possiblities such as the improvement of known bounds, generation of bounds that are opposite to the known bounds and assessment of the prevalent error laws.

Full-field numerical differentiation

The numerical representation and differentiation of experimental information is extended to full-field capability involving two independent variables. Bicubic spline, regression analysis and finite-element concepts are employed. Smoothing of the experimental data is accomplished mathematically. The technique are demonstrated by analysing loaded plates.

A least squares finite element approach to unsteady gas dynamics

AbstractA numerical technique for solving unsteady gas dynamic equations is presented. The technique is based on least squares finite element concepts with elements that are constructed in both space and time. Both linear and quadratic interpolation is used on individual elements. The technique is tested against a problem whose solution is known so that numerical accuracy can be ascertained.

Development of a finite dynamic element for free vibration analysis of two‐dimensional structures

AbstractThe paper develops an efficient free‐vibration analysis procedure of two‐dimensional structures. This is achieved by employing a discretization technique based on a recently developed concept of finite dynamic elements, involving higher order dynamic correction terms in the associated stiffness and inertia matrices. A plane rectangular dynamic element is developed in detail. Numerical solution results of free‐vibration analysis presented herein clearly indicate that these dynamic element combined with a suitable quadratic matrix eigenproblem solution technique effect a most economical and efficient solution for such an analysis when compared with the usual finite element method.

A drafting program for isoparametric finite elements

AbstractConsiderable effort has been invested lately in the application of isoparametric finite elements for numerical solution of a wide range of applied mechanics problems. In fact, several general purpose computer programs are now available which are based upon such finite elements. In the present paper, a new application of the isoparametric finite element concept is introduced which significantly extends its usefulness for many practical structural configurations. In this application, final working or architectural drawings of the structure are made from the same (or similar) finite element model as was utilized in a structural integrity analysis. The hardware necessary to produce such drawing, a computer driven plotter or automated drafting machine, is available commercially or through most data centres, and the software concepts required are described herein.

Finite Element Simulation of Rotor-Bearing Systems With Internal Damping

The implementation of finite element simulations for the study of rotor dynamic systems has been the subject of recent publications. Since the finite element offers obvious modeling advantages, particularly in modeling large-scale systems, this study extends the linear finite element concept to provide a detailed evaluation of damped rotor stability. In this work the effects of both internal viscous and hysteretic damping have been incorporated into the finite element model. Both produce circulatory terms in the generalized equations of motion which encourages the destabilization of this nonconservative system. Results are presented for both hysteretic and viscous forms of damping. Both forms of internal damping destabilize the rotor system and induce nonsynchronous forward precession. The stabilizing effects of anisotropic bearing stiffness and external damping are also demonstrated.

On a finite dynamic element method for free vibration analysis of structures

This paper explores the concept of finite dynamic elements involving higher order dynamic correction terms in the associated stiffness and mass matrices. Such matrices are then developed for a rectangular prestressed membrane element. Next, efficient analysis techniques for the eigenproblem solution of the resulting quadratic matrix equations are described in detail. These are followed by suitable numerical examples which indicate that employment of such dynamic elements in conjunction with an efficient quadratic matric solution technique will result in a most significant economy in the free vibration analysis of structures.

A finite element analysis of shocks and finite-amplitude waves in one-dimensional hyperelastic bodies at finite strain

The general theory of shock and acceleration waves in isotropic, incompressible, hyperelastic solids is used in conjunction with the concept of finite elements to construct discrete models of highly nonlinear wave phenomena in elastic rods. A numerical integration scheme which combines features of finite elements and the Lax-Wendroff method is introduced. Numerical calculations of the critical time for shock formulation are given. Numerical results obtained from representative cases are discussed.

Finite-element model for possible isostatic rebound in the Grand Canyon

In parts of the Grand Canyon a sinuous anticlinal structure in the Muav Limestone trends along the Colorado River and dips away from the river course. If the structure was there before the canyon was created, it appears that parts of the Colorado River channel coincide with the axis of the anticline. A model study based on the concept of finite elements shows that a possible mechanism for the folding of the Muav Limestone is elastic rebound caused by relief from overburden pressure as material was eroded to form the canyon.