On Drazin Spectral Equation for the Operator Products

2020 ◽  
Vol 14 (1) ◽  
Author(s):  
Kai Yan ◽  
Qingping Zeng ◽  
Yucan Zhu
Keyword(s):  
Spectral Equation

Effect of the Condition Number of the Inverse Fourier Transform Matrix on the Solution Behavior of the Time Spectral Equation System

2020 ◽  
Author(s):  
Sen Zhang ◽  
Dingxi Wang ◽  
Yi Li ◽  
Hangkong Wu ◽  
Xiuquan Huang
Keyword(s):  
Fourier Transform ◽  
Stability Analysis ◽  
Condition Number ◽  
Inverse Fourier Transform ◽  
Real Life ◽  
Equation System ◽  
Time Instant ◽  
Transform Matrix ◽  
Spectral Equation ◽  
The Impact

Abstract The time spectral method is a very popular reduced order frequency method for analyzing unsteady flow due to its advantage of being easily extended from an existing steady flow solver. Condition number of the inverse Fourier transform matrix used in the method can affect the solution convergence and stability of the time spectral equation system. This paper aims at evaluating the effect of the condition number of the inverse Fourier transform matrix on the solution stability and convergence of the time spectral method from two aspects. The first aspect is to assess the impact of condition number using a matrix stability analysis based upon the time spectral form of the scalar advection equation. The relationship between the maximum allowable Courant number and the condition number will be derived. Different time instant groups which lead to the same condition number are also considered. Three numerical discretization schemes are provided for the stability analysis. The second aspect is to assess the impact of condition number for real life applications. Two case studies will be provided: one is a flutter case, NASA rotor 67, and the other is a blade row interaction case, NASA stage 35. A series of numerical analyses will be performed for each case using different time instant groups corresponding to different condition numbers. The conclusion drawn from the two real life case studies will corroborate the relationship derived from the matrix stability analysis.

A Weighted Rearranged Spectral Equation Method for Structural Model Updating

1995 ◽  
Author(s):  
José Roberto F. Arruda ◽  
Carlson Antonio M. Verçosa
Keyword(s):  
Degrees Of Freedom ◽  
Structural Model ◽  
Model Updating ◽  
Structural Connectivity ◽  
Computational Cost ◽  
Force Balance ◽  
Dynamic Force ◽  
Stiffness Coefficients ◽  
Spring System ◽  
Spectral Equation

Abstract A new structural model updating method based on the dynamic force balance is presented. The method consists of rearranging the spectral equation so that measured modes and natural frequencies can be used to compute directly updated stiffness coefficients. The proposed method preserves both the structural connectivity and reciprocity, which translate into sparsity and symmetry of the stiffness matrix, respectively. Large changes in small-valued stiffness coefficients are avoided using parameter weighting in the rearranged spectral equation solution. It is shown that the proposed method produces results which are similar to the results obtained using Alvar Kabe’s method, with the advantages of simpler formulation and smaller computational cost. A simple example of an 8 degrees-of-freedom mass-spring system, originally used by Kabe to present his method, is used here to evaluate the proposed method.

Hyper-asymptotic expansions and instantons

2019 ◽  
pp. 471-492
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Quantum Mechanics ◽  
Asymptotic Expansions ◽  
Perturbative Expansion ◽  
Quantum Field ◽  
Wkb Method ◽  
Additional Information ◽  
Spectral Equation ◽  
Double Well Potential

Chapter 24 examines the topic of hyper–asymptotic expansions and instantons. A number of quantum mechanics and quantum field theory (QFT) examples exhibit degenerate classical minima connected by quantum barrier penetration effects. The analysis of the large order behaviour, based on instanton calculus, shows that the perturbative expansion is not Borel summable, and does not define unique functions. An important issue is then what kind of additional information is required to determine the exact expanded functions. While the QFT examples are complicated, and their study is still at the preliminary stage, in quantum mechanics, in the case of some analytic potentials that have degenerate minima (like the quartic double–well potential), the problem has been completely solved. Some examples are described in Chapter 24. There, the perturbative, complete, hyper–asymptotic expansion exhibits the resurgence property. The perturbative expansion can be related to the calculation of the spectral equation via the complex WKB method.

Canonical Spectral Equation

1995 ◽  
Vol 39 (4) ◽  
pp. 685-691 ◽  
Author(s):  
V. L. Girko
Keyword(s):  
Spectral Equation

Research on Inelastic Response Spectra for Asymmetric Plan Systems Subjected to Bi-Directional Earthquake Motions

2012 ◽  
Vol 166-169 ◽  
pp. 2332-2336
Author(s):  
Feng Wang ◽  
Hong Nan Li ◽  
Ting Hua Yi
Keyword(s):  
Soft Soil ◽  
Rotation Frequency ◽  
Response Spectra ◽  
Reduction Factor ◽  
Strength Reduction ◽  
Strength Reduction Factor ◽  
Inelastic Response ◽  
Surface Plasticity ◽  
Soil Site ◽  
Spectral Equation

The limitations of traditional inelastic response spectra are discussed. Considering a one-storey asymmetric plan system subjected to perpendicular bi-directional earthquake motions, the inelastic multi-dimensional strength reduction factor spectra is presented. The yield rule of the asymmetric plan system is determined by two-dimensional yield-surface plasticity function. The spectral equation is simplified by the relationship of strength reduction factors between x-direction and y-direction.The multi-dimensional spectra are analyzed based on 30 pair strong earthquake motion records for hard soil site, intermediate soil site and soft soil site. Analytic results shows that the strength reduction factor mean spectra for each soil site has its own characteristics, and the strength reduction factor spectra is affected strongly by ductility, normalized stiffness eccentricity, period and rotation frequency ratio.

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