Finite-element Solution Of Problems Research Articles

NETWORK FORMULATION OF THE FINITE ELEMENT METHOD

The finite element method, which in recent years has proved to be a major contribution to many fields of engineering analysis (structures, heal conduction, fluid mechanics, etc.) is reformulated in network language in a manner which makes KVL, K.CL, and the incidence and loop matrices transparent. Solution of finite element problems often imposes significant demands upon computational resources. The network formulation described here makes it possible to apply well developed network analysis techniques for improved efficiency in finite element analysis.

The scattered decomposition for finite elements

The solution of finite element problems with irregular geometries on a parallel computer of the hypercube type (MIMD, distributed memory) is considered. The technique of scattering the decomposition is found to be easy to implement and to effectively load balance the computation.

Error-source effects in a high-accuracy optical finite-element processor

High-accuracy optical linear algebra processors are addressed with attention to three new aspects. These include: their application to the solution of finite-element problems; the first error-source models for component errors in such processors; and the first analysis of error sources in such processors.

Axisymmetric fracture mechanics analysis by the boundary integral equation method

The boundary integral equation (BIE) method of numerical stress analysis is shown to be a powerful technique for solving axisymmetric fracture mechanics problems of the types encountered in practice in pipes, tubes and pressure vessels. Comparisons with established analytical and finite element solutions for problems involving circumferential and penny-shaped cracks in round bars, spheres and hollow cylinders show that accurate values of stress intensity factors at the tips of cracks can be obtained by this method. The main advantage of the technique over the finite element method is the much reduced amount of labour required to prepare the mesh data, since in the BIE approach only the boundary of an axisymmetric component needs to be discretised into simple and relatively coarse line elements.

On the accuracy of finite element solutions of problems with traction boundary conditions

In the variational formulation of finite element models for mechanics problems, boundary conditions are classified as either essential or natural. The traction boundary conditions are identified as natural in many commonly-used finite element models. A valid finite element mode must satisfy explicitly the essential boundary conditions, since the latter are satisfied implicitly as a posteriori conditions of the formulation. Though assumed solutions can be arbitrary regarding natural boundary conditions, the arbitrariness can affect the accuracy of solutions, because the weighted error residuals of all the Euler equations of the variational principle should be minimized to obtain solutions. As a possible alternative to refining finite element meshes for improving the accuracy of solutions, an explicit enforcement of the natural boundary conditions can be considered. In this work, the nature of the implicit enforcement of the natural boundary conditions of a problem through a variational formulation is investigated. Systematic methods to enforce natural boundary conditions explicitly to improve the accuracy of solutions are developed for two finite element models. Consider a boundary value problem defined by VZu = f in a domain D and some boundary conditions at boundary B. The problem has a variational principle that the stationary condition of a scalar functional I with respect to an admissible solution u (which satisfies u = h on B, if such a condition exists) yields the interior equation and a natural boundary condition, au/an = g on B where n is the outward normal to B. The functional I is:

Solution of finite element problems without assembly using group theory

In this paper we wish to present a novel method to solve the monoenergetic diffusion equation. This works iteratively with decoupled equations at the single-element level. Group theory can be advantageously applied because Schur lemma enables the block diagonalization of the element matrix. Some preliminary results are presented for different degrees of approximation and different numbers of meshes.

Finite element analysis of cracked bodies to determine stress intensity factors

A method for the computation of the stress intensity and associated geometrical correction factors by use of the finite element analysis is presented. The lack of ability to represent crack tip stress conditions has been the shortcoming of the conventinal techniques in finite-element solutions of problems of cracked bodies. The proposed method provides the representation of crack tip stress conditions with elliptic displacement functions which lead to the direct computation of stress intensity factors. The method is applied to several crack configurations with relatively coarse finite-element networks and the accuracy is found to be satisfactory when compared with results of previous workers.