Parabolic Interpolation Method Research Articles

Preliminary design optimization of a steam generator

In the present work, a procedure for preliminary design optimization of a steam generator has been developed with the objective of minimizing the weight of the generator. Some real engineering constraints are considered in the problem formulation. A method of evaluating the objective function and constraints of the problem is presented. The problem has been solved numerically by using the exterior SUMT in which the Powell unconstrained minimization technique improved by Sargent with the parabolic interpolation method of one-dimensional minimization, is employed. The results of the optimum design and a sensitivity analysis conducted about the optimum point have been reported.

Co-rotational formulation for geometric nonlinear analysis of doubly symmetric thin-walled beams

A doubly symmetric thin-walled beam element with open section is derived using co-rotational (CR) total Lagrangian (TL) formulation. The effects of deformation-dependent third-order terms of element nodal forces on the buckling load and post-buckling behavior are investigated. All coupling among bending, twisting, and stretching deformations for beam element is considered by consistent second-order linearization of the fully geometrically nonlinear beam theory. However, all third-order terms of nodal forces, which are relevant to the twist rate, rate of twist rate and curvature of the beam axis are also considered. An incremental-iterative method based on the Newton–Raphson method combined with constant arc length of incremental displacement vector is employed for the solution of nonlinear equilibrium equations. The zero value of the tangent stiffness matrix determinant of the structure is used as the criterion of the buckling state. A parabolic interpolation method of the arc length is used to find the buckling load. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed element and to investigate the effect of third-order terms of element nodal forces on the buckling load and post-buckling behavior of doubly symmetric thin-walled beams.

A co-rotational formulation for thin-walled beams with monosymmetric open section

A consistent co-rotational total Lagrangian finite element formulation and numerical procedure for the geometric nonlinear buckling and postbuckling analysis of thin-walled beams with monosymmetric open section is presented. The element developed here has two nodes with seven degrees of freedom per node. The element nodes are chosen to be located at the shear centers of the end cross-sections of the beam element and the shear center axis is chosen to be the reference axis. The deformations of the beam element are described in the current element coordinate system, which is constructed at the current configuration of the beam element. In element nodal forces, all coupling among bending, twisting, and stretching deformations of the beam element is considered by consistent second-order linearization of the fully geometrically nonlinear beam theory. However, the third-order term of the twist rate of the beam axis is considered in element nodal forces. An incremental-iterative method based on the Newton–Raphson method combined with constant arc length of incremental displacement vector is employed for the solution of nonlinear equilibrium equations. The zero value of the tangent stiffness matrix determinant of the structure is used as the criterion of the buckling state. A parabolic interpolation method of the arc length is used to find the buckling load. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.

Application of Two Modified Fast Fourier Transform Algorithms in Power System Harmonic Analysis

Abstract The fast Fourier transform (FFT) algorithm used in harmonic analysis is first studied. Then, two modified FFT algorithms - LFFT based on linear interpolation method and PFFT based on parabolic interpolation method are derived. Typical examples show that both the LEFT and PFFT clearly increase analysis precision, and reduce the demand to sampling frequency thus greatly enhance computing speed. These two algorithms have been successfully applied in harmonic analysis of power system.

A successive bounding method to find the exact eigenvalues of transcendental stiffness matrix formulations

AbstractAn alternative algorithm for finding exact natural frequencies or buckling loads from the transcendental, e.g. dynamic, stiffness matrix method is presented in this paper and evaluated by using the plate assembly testbed program VICONOPT. The method is based on the bounding properties of the eigenvalues provided by either linear or quadratic matrix pencils on the exact solutions of the transcendental eigenvalue problem. The procedure presented has five stages, including two accuracy checking stages which prevent unnecessary calculations. Numerical tests on buckling of general anisotropic plate assemblies show that significant time savings can be achieved compared with an earlier multiple determinant parabolic interpolation method.

More efficient use of determinants to solve transcendental structural eigenvalue problems reliably

The authors have previously presented the ‘multiple determinant parabolic interpolation method’ for solving the transcendental eigenvalue problems which arise when the ‘exact’ member equations obtained by solving the governing differential equations of members are used to find the eigenvalues of structures. For non-coincident eigenvalues this method achieves substantial time savings over bisection and is faster than other competing methods. The present paper extends the method to groups of coincident eigenvalues, for which it achieves comparable time savings while retaining the efficiency of the original method for other problems including those with close eigenvalues.

Reliable use of determinants to solve non‐linear structural eigenvalue problems efficiently

AbstractA ‘multiple determinant parabolic interpolation method’ is described and evaluated, principally by using a plane frame test‐bed program. It is intended primarily for solving the transcendental eigenvalue problems arising when the ‘exact’ member equations obtained by solving the governing differential equations of members are used to find the eigenvalues (i.e. critical buckling loads or undamped natural frequencies) of structures. The method has five stages which together ensure successful convergence on the required eigenvalues in all circumstances. Thus, whenever checks indicate its suitability, parabolic interpolation is used to obtain eigenvalues more rapidly than would the popular bisection alternative. The checks also ensure a wise choice of the determinant used by the interpolation. The determinants available are all usually zero at eigenvalues, and comprise the determinant of the overall stiffness matrix Kn and the determinants which result, with negligible extra computation, from effectively considering all except the last m (m=1, 2,…, n−1) freedoms to which Kn corresponds as internal substructure freedoms. Tests showed time savings compared to bisection of 31 per cent when finding non‐coincident eigenvalues to relative accuracy ϵ = 10−4, increasing to 62 per cent when ϵ = 10−8. The tests also showed time savings of about 10 per cent compared with an earlier Newtonian approach. The method requires no derivatives and its use in the widely available space frame program BUNVIS‐RG has demonstrated how easily it can replace bisection, which was used in the earlier program BUNVIS.