Highly Accurate Golden Section Search Algorithms and Fictitious Time Integration Method for Solving Nonlinear Eigenvalue Problems

This study sets up two new merit functions,which are minimized for the detection of real eigenvalue and complex eigenvalue to address nonlinear eigenvalue problems.For each eigen-parameter the vector variable is solved from a nonhomogeneous linear system obtained by reducing the number of eigen-equation one less,where one of the nonzero components of the eigenvector is normalized to the unit and moves the column containing that component to the right-hand side as a nonzero input vector.1D and 2D golden section search algorithms are employed to minimize the merit functions to locate real and complex eigenvalues.Simultaneously,the real and complex eigenvectors can be computed very accurately.A simpler approach to the nonlinear eigenvalue problems is proposed,which implements a normalization condition for the uniqueness of the eigenvector into the eigenequation directly.The real eigenvalues can be computed by the fictitious time integration method(FTIM),which saves computational costs compared to the one-dimensional golden section search algorithm(1D GSSA).The simpler method is also combined with the Newton iterationmethod,which is convergent very fast.All the proposed methods are easily programmed to compute the eigenvalue and eigenvector with high accuracy and efficiency.

Parameter optimization of gravity density inversion based on correlation searching and the golden section algorithm

For density inversion of gravity anomaly data, once the inversion method is determined, the main factors affecting the inversion result are the inversion parameters and subdivision scheme. A set of reasonable inversion parameters and subdivision scheme can, not only improve the inversion process efficiency, but also ensure inversion result accuracy. The gravity inversion method based on correlation searching and the golden section algorithm is an effective potential field inversion method. It can be used to invert 2D and 3D physical properties with potential data observed on flat or rough surfaces. In this paper, we introduce in detail the density inversion principles based on correlation searching and the golden section algorithm. Considering that the gold section algorithm is not globally optimized. we present a heuristic method to ensure the inversion result is globally optimized. With a series of model tests, we systematically compare and analyze the inversion result efficiency and accuracy with different parameters. Based on the model test results, we conclude the selection principles for each inversion parameter with which the inversion accuracy can be obviously improved.

Optimization of aero-engine pipeline for avoiding vibration based on length adjustment of straight-line segment

In the design and troubleshooting of aero-engine pipeline,the vibration reduction of the pipeline system is often achieved by adjusting the hoop layout,provided that the shape of pipeline remains unchanged.However,in reality,the pipeline system with the best antivibration performance may be obtained only by adjusting the pipeline shape.In this paper,a typical spatial pipeline is taken as the research object,the length of straight-line segment is taken as the design variable,and an innovative optimization method of avoiding vibration of aero-engine pipeline is proposed.The relationship between straight-line segment length and parameters that determine the geometric characteristics of the pipeline,such as the position of key reference points,bending angle,and hoop position,are derived in detail.Based on this,the parametric finite element model of the pipeline system is established.Taking the maximum first-order natural frequency of pipeline as the optimization objective and introducing process constraints and vibration avoidance constraints,the optimization model of the pipeline system is established.The genetic algorithm and the golden section algorithm are selected to solve the optimization model,and the relevant solution procedure is described in detail.Finally,two kinds of pipelines with different total lengths are selected to carry out a case study.Based on the analysis of the influence of straight-line segment length on the vibration characteristics of the pipeline system,the optimization methods developed in this paper are demonstrated.Results show that the developed optimization method can obtain the optimal single value or interval of the straight-line segment length while avoiding the excitation frequency.In addition,the optimization efficiency of the golden section algorithm is remarkably higher than that of the genetic algorithm for length optimization of a single straight-line segment.