Inclusion Of Second-order Terms Research Articles

A second order SAP algorithm for risk and reliability based design optimization

This paper presents a decoupling approach for the efficient solution of Risk Optimization (RO) and Reliability Based Design Optimization (RBDO) problems. The proposed approach, SAP2nd, is a Sequential Approximate Programming (SAP) technique including second order terms obtained with the BFGS (Broyden-Fletcher-Goldfarb-Shanno) approximation for the Hessian. A first advantage of SAP2nd is that any reliability analysis method can be employed for the evaluation of the probabilities of failure and its sensitivities. Here, Polynomial Chaos Expansion (PCE) is employed for this purpose. Several benchmark problems are solved to study the efficiency, robustness and accuracy of SAP2nd. It is demonstrated that the inclusion of second order terms leads to: (i) a much more stable algorithm in comparison to a first order SAP algorithm, i.e. it was able to avoid convergence issues arising from cycling, and (ii) a more efficient algorithm since SAP2nd reduced the computational effort, in both RO and RBDO problems, when compared to the coupled PCE algorithm previously proposed by the authors. The use of PCE for the evaluation of the probabilities of failure and its sensitivities allowed SAP2nd to achieve much more accurate results when compared to FORM based approaches, requiring the same order of computational effort. Finally, SAP2nd using PCE for reliability and sensitivity analysis is well suited for RO and RBDO problems where the drawbacks of FORM based approaches prevail, especially cases with highly nonlinear limit state function and non Gaussian random variables.

An Accurate Third-Order Normal Form Approximation for Power System Nonlinear Analysis

The inclusion of higher-order terms in small-signal (modal) analysis has been an intensive research topic in nonlinear power system analysis. Inclusion of second-order terms with the method of normal forms (MNF) has been well developed and investigated, overcoming the linear conventional small-signal methods used in the power system control and stability analysis. However, application of the MNF has not yet been extended to include third-order terms in a mathematically accurate form to account for nonlinear dynamic stability and dynamic modal interactions. Due to the emergence of larger networks and long transmission line with high impedance, modern grids exhibit predominant nonlinear oscillations and existing tools have to be upgraded to cope with this new situation. In this paper, first, fundamentals of normal form theory along with a review of existing tools based on this theory is presented. Second, a new formulation of MNF based on a third-order transformation of the system's dynamic approximation is proposed and nonlinear indexes are proposed to make possible to give information on the contribution of nonlinearities to the system stability and on the presence of significant third-order modal interactions. The induced benefits of the proposed method are compared to those afforded by existing MNFs. Finally, the proposed method is applied to a standard test system, the IEEE 2-area 4-generator system, and results given by the conventional linear small signal and existing MNFs are compared to the proposed approach. The applicability of the proposed MNF to larger networks with more complex models has been evaluated on the New England–New York 16-machine 5-area system.

A comparative study of the first and second order theories and Goda's formula for wave-induced pressure on a vertical breakwater with irregular waves

The wave pressure intensity at the still water level exerted on a vertical type of breakwater with irregular waves is investigated in this study. During the study, a comparison is made between the first, second order theories, Goda's formula and the experimental data. The results of the comparison state that for smaller wave periods, first order term can adequately express the situation. When the wave period increases, the inclusion of second-order terms result in an overestimation of the pressures. For this purpose, a tentative equation as a function of wave steepness is proposed as a correction for second order terms. Also, the applicability of Goda's formula has been shown as a good estimator for maximum pressure values for all cases.

Stability of Shear Deformable Thin-Walled Space Frames and Circular Arches

A consistent finite-element formulation is presented in order to analyze the spatial stability of shear deformable thin-walled space frames and circular arches. The displacement field of unsymmetric thin-walled cross sections is introduced based on the inclusion of second-order terms of Rodriguez's finite rotations, and the potential energy corresponding to the semitangential moments is consistently derived. For finite-element analysis, cubic Hermitian polynomials including shear deformation effects due to flexural and restrained warping shear stress are utilized as shape functions. Load correction stiffness matrices for the different types of moments and off-axis loadings are taken into account. For spatial buckling behaviors of space frames and circular arches, finite-element solutions using the two-noded straight frame element are presented and compared with the analytical solutions and the results in the literature.

Minimum principle for Dirac scattering lengths.

A minimum principle for the nonrelativistic scattering length was derived some years ago under the assumption that only a finite number of discrete states exist below the scattering threshold. The variational bound is applicable even when the bound-state wave functions are imprecisely known--they need only be accurate enough to give binding in a Rayleigh-Ritz calculation. The method is generalized here to apply to potential scattering described by the Dirac equation. An apparent difficulty associated with the existence of a continuum of negative-energy states, that is, the problem of ``variational collapse,'' is removed through the inclusion of second-order terms in the variational expression involving matrix elements of the square of the Dirac Hamiltonian. In the course of the derivation a relativistic version of the Hylleraas-Undheim theorem is developed. Applications of this theorem are described that provide a sufficient condition for the existence of a given number of bound states, lower bounds on the energy eigenvalues, and a systematic procedure for improving the accuracy of trial bound-state wave functions. A very simple model calculation was performed to illustrate the minimum property and the stability of the numerical procedure.

Wave Forces on Offshore Structures: Nonlinear Wave Diffraction by Large Cylinders

A nonlinear theory of wave diffraction is presented and used to evaluate the forces exerted On a cylinder of large diameter. A perturbation technique has been used to solve the problem with the inclusion of second-order terms. Analytical solutions are expressed in the form of an integral, and a numerical technique is applied to solve the resulting integral equation with satisfaction of all necessary hydrodynamic boundary conditions including surface and radiation boundary conditions. Predictions of the present analysis are compared with the experimental data. The comparison shows excellent agreement.

Least Squares Adjustment with Relatively Large Observation Errors, Inaccurate Initial Approximations, or Both

This paper concerns least squares adjustment problems in which the condition equations are non-linear in both the observations and the adjustment parameters (‘unknowns’). We present a solution algorithm which takes into account the second (as well as the first) order derivatives in the adjustment residuals (observation errors) and the corrections to the initially available approximations to the adjustment parameters. The inclusion of second order terms in the adjustment residuals is necessary whenever the adjustment residuals themselves cannot be regarded as negligible as compared to the adjustment parameters, that is, whenever the linearized condition equations would no longer be practically equivalent to the rigorous ones. The conventional solution techniques would then not lead to the ‘best’ estimators for the adjustment parameters in the sense of least squares. (An example for this situation is the determination of the orbit of a narrow visual double. The errors in the measured distance may then be relatively large compared to the angular size of the semi major axis.) The consideration of second order terms in the parameter corrections is completely independent of the second order terms in the adjustment residuals. It may be of advantage when the first approximation values for the adjustment parameters are so bad that the usually employed Newton-type iterations converge either not at all or only very badly. The inclusion of the second order terms will speed up the convergence toward the definitive values, and will sometimes bring about convergence when a strictly linear approach would be divergent.

Amplitude modulation of electromagnetic waves by acoustic waves in a dispersive plasma: modified theory

Modified analysis of the two cases, namely, (i) the modulation of an electromagnetic wave propagating through a plasma disturbed by an acoustic wave, and (ii) the modulation of the waves transmitted and reflected from the boundary between the plasma and free-space, has been given separately. In contrast to earlier work, the present analysis does not neglect the second order terms in the current density expression; it is seen that this inclusion of second order terms alters the modulation significantly. Numerical calculations have been made to study the variation of modulation with various parameters.

Jahn-Teller coupling and the photoelectron spectrum of cyclopropane

Reparameterized CNDO calculations on the ground and first excited states of the cyclopropane cation show that both states are subject to significant Jahn-Teller distortions. First and second order vibronic coupling parameters derived from the calculations are used to predict the shape of spectra which would be observed for electronic transitions to these degenerate states. Good agreement with the observed photoelectron spectrum is found. The inclusion of second order terms increases the complexity of the spectra but does not greatly alter the shape of the band contours.

Suboptimal state estimation for continuous-time nonlinear systems from discrete noisy measurements

This paper presents the derivation of the dynamical equations of a second-order filter which estimates the states of a non-linear plant on the basis of discrete noisy measurements. The filter equations contain terms involving the second-order partial derivatives of the plant and output equations. Simulation results are presented which yield a comparison of the performance of the first-versus the second-order filter when applied to a nonlinear third-order system. The results indicate that the inclusion of second-order terms can markedly improve the filter performance.

Deduction of the Zero Differential Overlap Approximation from an Orthogonal Atomic Orbital Basis

The validity of the zero differential overlap approximation is investigated by means of a systematic expansion in terms of a parameter of the order of magnitude of a typical overlap integral between neighboring atoms. The terms of the Fock operator in a basis of orthogonal atomic orbitals can be arranged after different orders of the parameter. It is demonstrated that, by inclusion of second-order terms, the conditions imposed by the zero differential overlap approximation are automatically fulfilled. Moreover, all the basic integrals are shown to be transferable in the first order and all but W in the second order. The connection with the ω technique in the Hückel method is also given. A numerical application has been made to ethylene and benzene.

A note on the tangential transfer of energy between wind and waves

A mathematical treatment of the transfer of energy between wind and waves by tangential friction forces is given, consistent to the second degree of approximation. The results, when compared with the basic theory previously developed, show that the inclusion of second order terms results in a more consistent theory.