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In many problems involving the solution of a system of nonlinear equations, it is necessary to keep an approximation to the Jacobian matrix which is updated at each iteration. Computational experience indicates that the best updates are those that minimize some reasonable measure of the change to the current Jacobian approximation subject to the new approximation obeying a secant condition and perhaps some other approximation properties such as symmetry. All of the updates obtained thus far deal with updating an approximation to an nxn Jacobian matrix. In this thesis we consider extending most of the popular updates to the non-square case. Two applications are immediate: between-step updating of the approximate Jacobian of f(X,t) in a non-autonomous ODE system, and solving nonlinear systems of equations which depend on a parameter, such as occur in continuation methods. Both of these cases require extending the present updates to include the nx(n+l) Jacobian matrix, which is the issue we address here. Our approach is to stay with the least change secant formulation. Computational results for these new updates are also presented to illustrate their convergence behavior.

Submitted by Cátia Araújo (catia.araujo@unipampa.edu.br) on 2017-01-25T12:48:52Z No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Um modelo espectral para estimar a energia das ondas de gravidade na média atmosfera.pdf: 2073354 bytes, checksum: a85a3b8c026e5501cae61e1f337c03b3 (MD5) / Approved for entry into archive by Cátia Araújo (catia.araujo@unipampa.edu.br) on 2017-01-25T12:49:16Z (GMT) No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Um modelo espectral para estimar a energia das ondas de gravidade na média atmosfera.pdf: 2073354 bytes, checksum: a85a3b8c026e5501cae61e1f337c03b3 (MD5) / Made available in DSpace on 2017-01-25T12:49:16Z (GMT). No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Um modelo espectral para estimar a energia das ondas de gravidade na média atmosfera.pdf: 2073354 bytes, checksum: a85a3b8c026e5501cae61e1f337c03b3 (MD5) Previous issue date: 2014-06-05 / Ondas de gravidade atmosféricas são importantes no estudo da circulação e estrutura atmosférica planetária. Embora tenha efeito na baixa atmosfera, as maiores influências das ondas de gravidade ocorrem na média atmosfera, entre 10 e 100 km de altitude, devido ao decréscimo na densidade do ar e o crescimento da amplitude das ondas com a altitude. Neste trabalho, a equação para os espectros das ondas de gravidade é resolvida analiticamente pelo método da decomposição de Adomian. Como consequência, a natureza não linear do problema é preservada e os erros encontrados são devidos somente às parametrizações empregadas. Os resultados indicam que a solução da equação linearizada para o espectro das ondas de gravidade é uma boa aproximação somente para altitudes inferiores a 10 km, porque a linearização da equação conduz a uma solução que não descreve corretamente os espectros de energia cinética, comparando com a parametrização aplicada em simulações. / Atmospheric gravity waves are important in the study of circulation and mear structure of planetary atmosphere. Although it has effect in the lower atmosphere, the major influences of gravity waves occur in the middle atmosphere, between 10 and 100 km altitude due to the decrease in air density and growth of wave amplitude with altitude. In this work, the equation for the spectra of gravity waves is analytically solved by Adomian decomposition method. As a result, the nonlinear nature of the problem is preserved and errors encountered are only due to the parameterization employed only. The results indicate that the solution of the linear equation for the wave spectrum of gravity is a good approximation only for altitudes below 10 km, as the linearization of the equation leads to a solution that does not correctly describe the spectra of kinetic energy compared with the parameterization applied in simulations.

This paper studies the geometrically non-linear bending behavior of functionally graded beams subjected to buckling loads using the finite element method. The computational model is based on an improved first-order shear deformation theory for beams with five independent variables. The abstract finite element formulation is derived by means of the principle of virtual work. High-order nodal-spectral interpolation functions were utilized to approximate the field variables which minimizes the locking problem. The incremental/iterative solution technique of Newton's type is implemented to solve the nonlinear equations. The model is verified with benchmark problems available in the literature. The objective is to investigate the effect of volume fraction variation in the response of functionally graded beams made of ceramics and metals. As expected, the results show that transverse deflections vary significantly depending on the ceramic and metal combination. / Revisión por pares

The author examines nonlinear aeroelastic responses of air vehicle systems. Herein, the governing equations for a cantilevered configuration are developed and the methods of analysis are explored. Based on the developed nonlinear bending-bending-torsion equations, internal resonance, which is possible in future air vehicles, and the possible cause of limit cycle oscillations of aircraft wings with stores are investigated. The nonlinear equations have three types of nonlinearities caused by wing flexibility, store geometry and aerodynamic stall, and retain up to third-order nonlinear terms. The internal resonance conditions are examined by the Method of Multiple Scales and demonstrated by time simulations. The effect of velocity change for various physical parameters and stiffness ratio is investigated through bifurcation diagrams derived from Poinar´e maps. The dominant factor causing limit cycle oscillations is the stiffness ratio between in-plane and out-of-plane motion.

The author examines nonlinear aeroelastic responses of air vehicle systems. Herein, the governing equations for a cantilevered configuration are developed and the methods of analysis are explored. Based on the developed nonlinear bending-bending-torsion equations, internal resonance, which is possible in future air vehicles, and the possible cause of limit cycle oscillations of aircraft wings with stores are investigated. The nonlinear equations have three types of nonlinearities caused by wing flexibility, store geometry and aerodynamic stall, and retain up to third-order nonlinear terms. The internal resonance conditions are examined by the Method of Multiple Scales and demonstrated by time simulations. The effect of velocity change for various physical parameters and stiffness ratio is investigated through bifurcation diagrams derived from Poinar´e maps. The dominant factor causing limit cycle oscillations is the stiffness ratio between in-plane and out-of-plane motion.

Dauguma praktinių mokslinių, įvairių taikymų uždavinių yra spendžiami skaitiniais metodais. Šie metodai aktualūs ir mokymo aspektu. Įvairiose aukštosiose mokyklose studentai, studijuodami skaitinių metodų teoriją, sprendžia praktines užduotis, taigi tenka arba patiems programuodami algoritmus, arba naudoti kompiuterines programas. Šiame darbe nagrinėjami universalių kompiuterinių matematinių sistemų – MATLAB IR MATHCAD, kaip vieno svarbiausių skaitiniais metodais sprendžiamų uždavinių priemonės. Taip pat nagrinėjamas Excel programos panaudojimas netiesinių lygčių sprendimui skaitiniais metodais. Darbe išnagrinėti netiesinių lygčių sprendimo metodai, tokie kaip paprastųj����� iteracijų, pusiaukirtos, Niutono ir kirstinių metodai, jų realizacija kiekvienoje iš nurodytų programų. Pateikiant kiekvieną iš šių temų yra trumpai aprašyti patys metodai, lyginamos kiekvienos kompiuterinės programos galimybės spręsti uždavinius pagal šiuos metodus, jų funkcijų aprašymas, taikymai ir sprendimo pavyzdžiai. Darbe pristatyta sukurta kompiuterinė mokymo priemonė, skirta netiesinių lygčių sprendimo skaitiniais metodais įsisavinimui. / The most practical, scientific and various applied tasks are solved by using numerical methods. These methods are urgent in teaching aspect, too. The students of various higher schools are solving practical tasks when studying theory of numerical methods, therefore they have to programme algorithms by themselves or use computer programs. In this work there are analyzed universal computer algebra systems – Matlab and Mathcad – as the one of the most important measures of solving tasks by using numerical methods. Also there is analyzed the use of Excel program for the reason how to solve the nonlinear equation by using numerical methods. In the work there are analyzed methods of nonlinear equations solving, such as simple iterative, bisection, Newton and secant methods and their realization in each referred program.. There are described the very methods when presenting every those themes and comparised the possibilities of each computer program for the reason how to solve the tasks by using these methods. There is given description of methods functions, application and examples of solving. In the work there is presented created computer training appliance for the reason of mastering with nonlinear equation solving by using numerical methods.