FE Discretization Research Articles

An assessment of some preconditioning techniques in shell problems

SUMMARY Preconditioned Krylov subspace methods have proved to be eAcient in solving large, sparse linear systems in many areas of scientific computing. The success of these methods in many cases is due to the existence of good preconditioning techniques. In problems of structural mechanics, like the analysis of heat transfer and deformation of solid bodies, iterative solution of the linear equation system can result in a significant reduction of computing time. Also many preconditioning techniques can be applied to these problems, thus facilitating the choice of an optimal preconditioning on the particular computer architecture available. However, in the analysis of thin shells the situation is not so transparent. It is well known that the stiAness matrices generated by the FE discretization of thin shells are very ill-conditioned. Thus, many preconditioning techniques fail to converge or they converge too slowly to be competitivewith direct solvers. In this study, the performance of some general preconditioning techniques on shell problems is examined. #1998 John Wiley & Sons, Ltd.

A fast interface solver for the biharmonic Dirichlet problem on polygonal domains

In this paper we propose and analyze an efficient discretization scheme for the boundary reduction of the biharmonic Dirichlet problem on convex polygonal domains. We show that the biharmonic Dirichlet problem can be reduced to the solution of a harmonic Dirichlet problem and of an equation with a Poincare-Steklov operator acting between subspaces of the trace spaces. We then propose a mixed FE discretization (by linear elements) of this equation which admits efficient preconditioning and matrix compression resulting in the complexity \(\log \varepsilon^{-1} O ( N \log^qN)\). Here \(N\) is the number of degrees of freedom on the underlying boundary, \(\varepsilon > 0\) is an error reduction factor, \(q = 2\) or \(q = 3\) for rectangular or polygonal boundaries, respectively. As a consequence an asymptotically optimal iterative interface solver for boundary reductions of the biharmonic Dirichlet problem on convex polygonal domains is derived. A numerical example confirms the theory.

SPECTRAL FINITE ELEMENT ANALYSIS OF THE VIBRATION OF STRAIGHT FLUID-FILLED PIPES WITH FLANGES

A spectral finite element formulation for the analysis of stationary vibration of straight fluid-filled pipes is introduced. Element formulations for flanges and rigid masses attached to the pipe are also presented. In the spectral finite element formulation, the base functions are frequency-dependent solutions to the local equations of motion. The formulation is valid for arbitrarily long pipes and losses may be distributed in the system and may vary with frequency. The solutions of the equations of motion are expressed in terms of exponential functions, describing propagation in the waveguide, together with corresponding cross-sectional mode shapes. These solutions are found by using an FE discretization of the cross-sectional motion. To increase the numerical efficiency, methods for using FE shape functions with higher order polynomials are developed. The numerical accuracy is investigated by comparisons with results achieved with an exact formulation. It is found that, for frequencies of interest in many engineering problems, pipes may be modelled by using only one element to describe the fluid motion. The vibrations of a simple pipe structure with an infinite pipe, a flange and a small rigid mass are calculated. Just below the cut-on frequency of a shell mode, the stiffness controlled shell mode and the rigid mass may resonate, resulting in high vibration levels concentrated near the mass.

Static behaviour simulation of a bridge structure by an FE-PC model

Static behaviour simulation of a highway bridge is presented for integrity assessment of an existing structure. The assessment scheme uses a combined FE analysis and in situ traffic load testing for establishing the current material characteristics. The coarse-mesh accurate hierarchic hybrid FE models used in the analysis enable simple but accurate FE discretization of the structure with a few hundred nodes, which is subjected to the p-method of refinements for convergence. The simplicity of the proposed FE modelling makes PC based high speed FE analysis possible for routine structural integrity assessment and maintenance. A typical standard reinforced concrete bridge structure is analysed to demonstrate the effectiveness of the method. It is shown that the present FE-PC model verifies the test measurements accurately, validating reliability of the assessment.

Solution of 3D-Laplace and Helmholtz equations in exterior domains using hp-infinite elements

This work is devoted to a convergence study for infinite element discretizations for Laplace and Helmholtz equations in exterior domains. The proposed approximation applies to separable geometries only, combining an hp FE discretization on the boundary of the domain with a spectral-like representation (resulting from the separation of variables) in the ‘radial’ direction. The presentation includes a convergence proof for the Laplace equation and a stability analysis for the variational formulation of the Helmholtz equation in weighted Sobolev spaces. The theoretical investigations are verified and illustrated with numerical examples for the exterior spherical domain.