A second-order perturbation method for fuzzy eigenvalue problems

2016 ◽  
Vol 33 (2) ◽  
Author(s):  
Mengwu Guo ◽  
Hongzhi Zhong ◽  
Kuan You
Keyword(s):  
Perturbation Method ◽  
Perturbation Methods ◽  
Eigenvalue Problems ◽  
Second Order ◽  
Order Perturbation ◽  
First Order ◽  
Alpha Level ◽  
Basic Parameters ◽  
Fuzzy Eigenvalues ◽  
Derivatives Of

Purpose For eigenvalue problems containing uncertain inputs characterized by fuzzy basic parameters, first-order perturbation methods have been developed to extract eigen-solutions, but either the result accuracy or the computational efficiency of these methods is less satisfactory. This paper presents an efficient method for estimation of fuzzy eigenvalues with high accuracy. Design/methodology/approach Based on the first order derivatives of eigenvalues and modes with respect to the fuzzy basic parameters, expressions of the second order derivatives of eigenvalues are formulated. Then a second-order perturbation method is introduced to provide more accurate fuzzy eigenvalue solutions. Only one eigenvalue solution is sought for the perturbed formulation, and quadratic programming is performed to simplify the alpha-level optimization. Findings Fuzzy natural frequencies and buckling loads of some structures are estimated with good accuracy, illustrating the high computational efficiency of the proposed method. Originality/value Up to the second order derivatives of the eigenvalues with respect to the basic parameters are represented in functional forms, which are used to introduce a second-order perturbation method for treatment of fuzzy eigenvalue problems. The corresponding alpha-level optimization is thus simplified into quadratic programming. The proposed method provides much more accurate interval solutions at alpha-cuts for the membership functions of fuzzy eigenvalues. Analogously, third- and higher-order perturbation methods can be developed for more stringent accuracy demands or for the treatment of stronger nonlinearity. The present work can be applied to realistic structural analysis in civil engineering, especially for those structures made of dispersed materials such as concrete and soil.

Improving efficiency of semi‐direct møller–plesset second‐order perturbation methods through oversubscription on multiple nodes

2019 ◽  
Vol 40 (24) ◽  
pp. 2146-2157 ◽  
Author(s):  
Ellie L. Fought ◽  
Vaibhav Sundriyal ◽  
Masha Sosonkina ◽  
Theresa L. Windus
Keyword(s):  
Perturbation Methods ◽  
Second Order ◽  
Order Perturbation ◽  
Moller Plesset

Second Order Perturbation Analysis of a Forced Nonlinear Mathieu Equation

2012 ◽  
Author(s):  
Venkatanarayanan Ramakrishnan ◽  
Brian F. Feeny
Keyword(s):  
Perturbation Analysis ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Mathieu Equation ◽  
Second Order ◽  
Order Perturbation ◽  
Cubic Nonlinearity ◽  
First Order ◽  
Order Expansion ◽  
Direct Forcing

The present study deals with the response of a forced nonlinear Mathieu equation. The equation considered has parametric excitation at the same frequency as direct forcing and also has cubic nonlinearity and damping. A second-order perturbation analysis using the method of multiple scales unfolds numerous resonance cases and system behavior that were not uncovered using first-order expansions. All resonance cases are analyzed. We numerically plot the frequency response of the system. The existence of a superharmonic resonance at one third the natural frequency was uncovered analytically for linear system. (This had been seen previously in numerical simulations but was not captured in the first-order expansion.) The effect of different parameters on the response of the system previously investigated are revisited.

scholarly journals An inverse problem on determining second order symmetric tensor for perturbed biharmonic operator

2021 ◽  
Author(s):  
Sombuddha Bhattacharyya ◽  
Tuhin Ghosh
Keyword(s):  
Inverse Problem ◽  
Vector Field ◽  
Symmetric Matrix ◽  
Symmetric Tensor ◽  
Second Order ◽  
Order Perturbation ◽  
Biharmonic Operator ◽  
First Order ◽  
Dirichlet To Neumann Map ◽  
First Order Perturbation

AbstractThis article offers a study of the Calderón type inverse problem of determining up to second order coefficients of higher order elliptic operators. Here we show that it is possible to determine an anisotropic second order perturbation given by a symmetric matrix, along with a first order perturbation given by a vector field and a zero-th order potential function inside a bounded domain, by measuring the Dirichlet to Neumann map of the perturbed biharmonic operator on the boundary of that domain.

Reexamination of the Symmetry Rule for Bond Distortions in Conjugated Hydrocarbons

2002 ◽  
Vol 2002 (8) ◽  
pp. 372-373
Author(s):  
Masahiro Kataoka
Keyword(s):  
Perturbation Method ◽  
Electron Correlation ◽  
Correlation Effect ◽  
Second Order ◽  
Order Perturbation ◽  
Mo Calculations ◽  
Geometrical Structures ◽  
Electron Correlation Effect ◽  
Conjugated Hydrocarbons

Pariser–Parr–Pople-type SCF MO calculations with the electron correlation effect given by the Brillouin-Wigner second-order perturbation method with Epstein–Nesbet energy denominators and by the Pople–Seeger–Krishnan correction show that the symmetry rule for bond distortions in conjugated hydrocarbons is effective for predicting the geometrical structures of [4 n+2]annulenes.

Analytic First-Order Derivatives of Partially Contracted N-Electron Valence State Second-Order Perturbation Theory (PC-NEVPT2)

2019 ◽  
Author(s):  
Yoshio Nishimoto
Keyword(s):  
Perturbation Theory ◽  
Electron Correlation ◽  
Valence State ◽  
Second Order ◽  
Order Perturbation ◽  
Order Perturbation Theory ◽  
Transition Energies ◽  
First Order ◽  
Static Electron ◽  
Second Order Perturbation Theory

A balanced treatment of dynamic and static electron correlation is important in computational chemistry, and multireference perturbation theory (MRPT) is able to do this at a reasonable computational cost. In this paper, analytic first-order derivatives, speci cally gradients and dipole moments, are developed for a particular MRPT method, state-specific partially contracted n-electron valence state second-order perturbation theory (PC-NEVPT2). Only one linear equation needs to be solved for the derivative calculation if the Z-vector method is employed, which facilitates the practical application of this approach. Comparison of the calculated results with experimental geometrical parameters of O<sub>3</sub> indicates excellent agreement, although the calculated results for O<sub>3</sub><sup>-</sup> are slightly outside the experimental error bars. The 0-0 transition energies of various methylpyrimidines and trans-polyacetylene are calculated by performing geometry optimizations and seminumerical second-order geometrical derivative calculations. In particular, the deviations of 0-0 transition energies of trans-polyacetylene from experimental values are consistently less than 0.1 eV with PC-NEVPT2, indicating the reliability of the method. These results demonstrate the importance of adding dynamic electron correlation on top of methods dominated by static electron correlation and of developing analytic derivatives for highly accurate methods.

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