Study of flutter related computational procedures for minimum weight structural sizing of advanced aircraft, supplemental data

Computational aspects of (1) flutter optimization (minimization of structural mass subject to specified flutter requirements), (2) methods for solving the flutter equation, and (3) efficient methods for computing generalized aerodynamic force coefficients in the repetitive analysis environment of computer-aided structural design are discussed. Specific areas included: a two-dimensional Regula Falsi approach to solving the generalized flutter equation; method of incremented flutter analysis and its applications; the use of velocity potential influence coefficients in a five-matrix product formulation of the generalized aerodynamic force coefficients; options for computational operations required to generate generalized aerodynamic force coefficients; theoretical considerations related to optimization with one or more flutter constraints; and expressions for derivatives of flutter-related quantities with respect to design variables

Internal solitary waves in the ocean: Analysis using the periodic, inverse scattering transform

The periodic, inverse scattering transform (PIST) is a powerful analytical tool in the theory of integrable, nonlinear evolution equations. Osborne pioneered the use of the PIST in the analysis of data form inherently nonlinear physical processes. In particular, Osborne's so-called nonlinear Fourier analysis has been successfully used in the study of waves whose dynamics are (to a good approximation) governed by the Korteweg--de Vries equation. In this paper, the mathematical details and a new application of the PIST are discussed. The numerical aspects of and difficulties in obtaining the nonlinear Fourier (i.e., PIST) spectrum of a physical data set are also addressed. In particular, an improved bracketing of the "spectral eigenvalues" (i.e., the +/-1 crossings of the Floquet discriminant) and a new root-finding algorithm for computing the latter are proposed. Finally, it is shown how the PIST can be used to gain insightful information about the phenomenon of soliton-induced acoustic resonances, by computing the nonlinear Fourier spectrum of a data set from a simulation of internal solitary wave generation and propagation in the Yellow Sea.Comment: 10 pages, 4 figures (6 images); v2: corrected a few minor mistakes and typos, version accepted for publication in Math. Comput. Simu

A Derivative Free Hybrid Equation Solver by Alloying of the Conventional Methods

This paper pronounces a modified numerical scheme to the conventional formula of Newton-Raphson for solving the nonlinear and transcendental equations especiallythose which cannot be solved by the basic algebra. Finding the derivative of a function is difficult in some case of problems. The present formula is made with the target toaloof the need of obtaining the derivative of the function. Comparative analysis shows that the present method is faster than Newton-Raphson method, Adomian method,Rabolian method, Abbasbandy method, Basto method & Feng method. Iteration cost-effective parameters – number of iteration steps & the value of effective error is alsofound to be minimum than these methods

ON NEW HYBRID ROOT-FINDING ALGORITHMS FOR SOLVING TRANSCENDENTAL EQUATIONS USING EXPONENTIAL AND HALLEY'S METHODS

The objective of this paper is to propose two new hybrid root finding algorithms for solving transcendental equations. The proposed algorithms are based on the well-known root finding methods namely the Halley's method, regula-falsi method and exponential method. We show using numerical examples that the proposed algorithms converge faster than other related methods. The first hybrid algorithm consists of regula-falsi method and exponential method (RF-EXP). In the second hybrid algorithm, we use regula falsi method and Halley's method (RF-Halley). Several numerical examples are presented to illustrate the proposed algorithms, and comparison of these algorithms with other existing methods are presented to show the efficiency and accuracy. The implementation of the proposed algorithms is presented in Microsoft Excel (MS Excel) and the mathematical software tool Maple

AN IMPROVED ROOT LOCATION METHOD FOR FAST CONVERGENCE OF NON-LINEAR EQUATIONS

In this paper an improved root location method has been suggested for nonlinear equations f(x)=0. The proposed improved root location method is very much effective for solving nonlinear equations and several numerical examples associated with algebraic and transcendental functions are present in this paper to investigate the new method. Throughout the study we have proved that proposed method is cubically convergent.  All the results are executed on MATLAB 16 which has a machine precision of around . Key words: Newton’s method, Iterative method, third order convergent, Root finding methods

A Comparative Analysis of Rate of Convergence For Linear And Quadratic Approximations in N-R Method

Ndash Raphson (N-R) Method is commonly used in the solution of algebraic equations and transcendental equations. Using Taylorrsquos theorem for expansion of functions, generally expansion is truncated as linear approximation. In this research work, expansion of function is truncated as quadratic approximation and then a comparative analysis was done for linear and quadratic approximations.nbs

Selected fixed point problems and algorithms

technical reportWe present a new version of the almost optimal Circumscribed Ellipsoid Algorithm (CEA) for approximating fixed points of nonexpanding Lipschitz functions. We utilize the absolute and residual error criteria with respect to the second norm. The numerical results confirm that the CEA algorithm is much more efficient than the simple iteration algorithm whenever the Lipschitz constant is close to 1. We extend the applicability of the CEA algorithm to larger classes of functions that may be globally expanding, however are nonexpanding/contracting in the direction of fixed points. We also develop an efficient hyper-bisection/secant hybrid method for combustion chemistry fixed point problems

Bisection Method and Falsi Regulation Method to Determine The Roots of Polynomial Equations

Some simple polynomial equations can be solved by the remainder theorem, so there is no need for numerical methods to solve them, because the roots of equations are very easy to do using analytical methods, while there are some polynomial equations that are difficult and complex to find roots using analytical methods. In this literature review, researchers will use the bisection method and the false rule to find the roots of polynomial equations. Based on the steps or sequence of calculation of the polynomial roots of , using the bisection method, the author states that from the first step to the eleventh step, if the calculation continues then in the second step f(a)*f(c)>0 or away from zero as shown in table 1 above. The author states that if the twelfth step continues, then f(a)*f(c) will approach zero and it can be seen that there are looping process approaches resulting from f(a)*f(c). This research study concludes that the roots of the polynomial of , using the bisection method are 1.36474675. Based on the steps or sequence of calculating the roots of the polynomial of  on, using the false position method (false rule), the author states that from the first step to the 366th step it turns out that f(c)=0.003195 when c=1,365423447. Thus the polynomial roots of using the false position method (regulation false) are 1.365423447. Keywords: Roots of polynomial equations

Historical development of the BFGS secant method and its characterization properties

The BFGS secant method is the preferred secant method for finite-dimensional unconstrained optimization. The first part of this research consists of recounting the historical development of secant methods in general and the BFGS secant method in particular. Many people believe that the secant method arose from Newton's method using finite difference approximations to the derivative. We compile historical evidence revealing that a special case of the secant method predated Newton's method by more than 3000 years. We trace the evolution of secant methods from 18th-century B.C. Babylonian clay tablets and the Egyptian Rhind Papyrus. Modifications to Newton's method yielding secant methods are discussed and methods we believe influenced and led to the construction of the BFGS secant method are explored. In the second part of our research, we examine the construction of several rank-two secant update classes that had not received much recognition in the literature. Our study of the underlying mathematical principles and characterizations inherent in the updates classes led to theorems and their proofs concerning secant updates. One class of symmetric rank-two updates that we investigate is the Dennis class. We demonstrate how it can be derived from the general rank-one update formula in a purely algebraic manner not utilizing Powell's method of iterated projections as Dennis did it. The literature abounds with update classes; we show how some are related and show containment when possible. We derive the general formula that could be used to represent all symmetric rank-two secant updates. From this, particular parameter choices yielding well-known updates and update classes are presented. We include two derivations of the Davidon class and prove that it is a maximal class. We detail known characterization properties of the BFGS secant method and describe new characterizations of several secant update classes known to contain the BFGS update. Included is a formal proof of the conjecture made by Schnabel in his 1977 Ph.D. thesis that the BFGS update is in some asymptotic sense the average of the DFP update and the Greenstadt update