Fractal boundaries of basin of attraction of Newton-Raphson method in helicopter trim

A key problem in helicopter aeroelastic analysis is the enormous computational time required for a numerical solution of the nonlinear system of algebraic equations required for trim, particularly when free wake models are used. Trim requires calculation of the main rotor and tail rotor controls and the vehicle attitude which leads to the six steady forces and moments about the helicopter center of gravity to be zero. An appropriate initial estimate of the trim state is needed for successful helicopter trim. This study aims to determine the control inputs that can have considerable effect on the convergence of trim solution in the aeroelastic analysis of helicopter rotors by investigating the basin of attraction of the nonlinear equations (set of initial guess points from which the nonlinear equations converge). It is illustrated that the three main rotor pitch controls of collective pitch, longitudinal cyclic pitch and lateral cyclic pitch have a significant contribution to the convergence of the trim solution. Trajectories of the Newton iterates are shown and some ideas for accelerating the convergence of a trim solution in the aeroelastic analysis of helicopters are proposed. It is found that the basins of attraction can have fractal boundaries. (C) 2010 Elsevier Ltd. All rights reserved.

Ancient Chinese algorithm: The Ying Buzu Shu (method of surplus and deficiency) vs Newton iteration method

Air exploratory discussion of an ancient Chinese algorithm, the Ying Buzu Shu, in about 2nd century BC, known as the rule of double false position in the West is given. In addition to pointing out that the rule of double false position is actually a translation version of the ancient Chinese algorithm, a comparison with well-known Newton iteration method is also made. If derivative is introduced, the ancient Chinese algorithm reduces to the Newton method. A modification of the ancient Chinese algorithm is also proposed, and some of applications to nonlinear oscillators are illustrated.

A new approach to the solution of the optimal power flow problem based on the modified Newton's method associated to an augmented Lagrangian function

This paper presents a new algorithm for optimal power flow problem. The algorithm is based on Newton's method which it works with an Augmented Lagrangian function associated with the original problem. The function aggregates all the equality and inequality constraints and is solved using the modified-Newton method. The test results have shown the effectiveness of the approach using the IEEE 30 and 638 bus systems.

Substitution-newton-Raphson method for the solution of electric network equations

The power flow problem, in transmission networks, has been well solved, for most cases, using Newton-Raphson method (NR) and its decoupled versions. Generally speaking, the solution of a non-linear system of equations refers to two methods: NR and Successive Substitution. The proposal of this paper is to evaluate the potential of the Substitution-Newton-Raphson Method (SNR), which combines both methods, on the solution of the power flow problem. Simulations were performed using a two-bus test network in order to observe the characteristics of these methods. It was verified that the NR is faster than SNR, in terms of convergence, considering non-stressed scenarios. For those cases where the power flow in the network is closed to the limits (stressed system), the SNR converges faster. This paper presents the power flow formulation of the SNR and describes its potential for its application in special cases such as stressed scenarios. © 2006 IEEE.

A Newton method for pose refinement of 3D models

Different optimization methods can be employed to optimize a numerical estimate for the match between an instantiated object model and an image. In order to take advantage of gradient-based optimization methods, perspective inversion must be used in this context. We show that convergence can be very fast by extrapolating to maximum goodness-of-fit with Newton's method. This approach is related to methods which either maximize a similar goodness-of-fit measure without use of gradient information, or else minimize distances between projected model lines and image features. Newton's method combines the accuracy of the former approach with the speed of convergence of the latter.

Jacobian-free Newton-Krylov methods with GPU acceleration for computing nonlinear ship wave patterns

The nonlinear problem of steady free-surface flow past a submerged source is considered as a case study for three-dimensional ship wave problems. Of particular interest is the distinctive wedge-shaped wave pattern that forms on the surface of the fluid. By reformulating the governing equations with a standard boundary-integral method, we derive a system of nonlinear algebraic equations that enforce a singular integro-differential equation at each midpoint on a two-dimensional mesh. Our contribution is to solve the system of equations with a Jacobian-free Newton-Krylov method together with a banded preconditioner that is carefully constructed with entries taken from the Jacobian of the linearised problem. Further, we are able to utilise graphics processing unit acceleration to significantly increase the grid refinement and decrease the run-time of our solutions in comparison to schemes that are presently employed in the literature. Our approach provides opportunities to explore the nonlinear features of three-dimensional ship wave patterns, such as the shape of steep waves close to their limiting configuration, in a manner that has been possible in the two-dimensional analogue for some time.

Finite Element Method For A Nonlocal Problem Of Kirchhoff Type

The nonlocal term in the nonlinear equations of Kirchhoff type causes difficulties when the equation is solved numerically by using the Newton-Raphson method. This is because the Jacobian of the Newton-Raphson method is full. In this article, the finite element system is replaced by an equivalent system for which the Jacobian is sparse. We derive quasi-optimal error estimates for the finite element method and demonstrate the results with numerical experiments.

Avaliação dos algoritmos de Picard-Krylov e Newton-Krylov na solução da equação de Richards

A engenharia geotécnica é uma das grandes áreas da engenharia civil que estuda a interação entre as construções realizadas pelo homem ou de fenômenos naturais com o ambiente geológico, que na grande maioria das vezes trata-se de solos parcialmente saturados. Neste sentido, o desempenho de obras como estabilização, contenção de barragens, muros de contenção, fundações e estradas estão condicionados a uma correta predição do fluxo de água no interior dos solos. Porém, como a área das regiões a serem estudas com relação à predição do fluxo de água são comumente da ordem de quilômetros quadrados, as soluções dos modelos matemáticos exigem malhas computacionais de grandes proporções, ocasionando sérias limitações associadas aos requisitos de memória computacional e tempo de processamento. A fim de contornar estas limitações, métodos numéricos eficientes devem ser empregados na solução do problema em análise. Portanto, métodos iterativos para solução de sistemas não lineares e lineares esparsos de grande porte devem ser utilizados neste tipo de aplicação. Em suma, visto a relevância do tema, esta pesquisa aproximou uma solução para a equação diferencial parcial de Richards pelo método dos volumes finitos em duas dimensões, empregando o método de Picard e Newton com maior eficiência computacional. Para tanto, foram utilizadas técnicas iterativas de resolução de sistemas lineares baseados no espaço de Krylov com matrizes pré-condicionadoras com a biblioteca numérica Portable, Extensible Toolkit for Scientific Computation (PETSc). Os resultados indicam que quando se resolve a equação de Richards considerando-se o método de PICARD-KRYLOV, não importando o modelo de avaliação do solo, a melhor combinação para resolução dos sistemas lineares é o método dos gradientes biconjugados estabilizado mais o pré-condicionador SOR. Por outro lado, quando se utiliza as equações de van Genuchten deve ser optar pela combinação do método dos gradientes conjugados em conjunto com pré-condicionador SOR. Quando se adota o método de NEWTON-KRYLOV, o método gradientes biconjugados estabilizado é o mais eficiente na resolução do sistema linear do passo de Newton, com relação ao pré-condicionador deve-se dar preferência ao bloco Jacobi. Por fim, há evidências que apontam que o método PICARD-KRYLOV pode ser mais vantajoso que o método de NEWTON-KRYLOV, quando empregados na resolução da equação diferencial parcial de Richards.

Newton`s iterates can converge to non-stationary points

In this note we discuss the convergence of Newton`s method for minimization. We present examples in which the Newton iterates satisfy the Wolfe conditions and the Hessian is positive definite at each step and yet the iterates converge to a non-stationary point. These examples answer a question posed by Fletcher in his 1987 book Practical methods of optimization.

Improving ultimate convergence of an augmented Lagrangian method

Optimization methods that employ the classical Powell-Hestenes-Rockafellar augmented Lagrangian are useful tools for solving nonlinear programming problems. Their reputation decreased in the last 10 years due to the comparative success of interior-point Newtonian algorithms, which are asymptotically faster. In this research, a combination of both approaches is evaluated. The idea is to produce a competitive method, being more robust and efficient than its `pure` counterparts for critical problems. Moreover, an additional hybrid algorithm is defined, in which the interior-point method is replaced by the Newtonian resolution of a Karush-Kuhn-Tucker (KKT) system identified by the augmented Lagrangian algorithm. The software used in this work is freely available through the Tango Project web page:http://www.ime.usp.br/similar to egbirgin/tango/.

Computational study of the step size parameter of the subgradient optimization method

The subgradient optimization method is a simple and flexible linear programming iterative algorithm. It is much simpler than Newton's method and can be applied to a wider variety of problems. It also converges when the objective function is non-differentiable. Since an efficient algorithm will not only produce a good solution but also take less computing time, we always prefer a simpler algorithm with high quality. In this study a series of step size parameters in the subgradient equation is studied. The performance is compared for a general piecewise function and a specific p-median problem. We examine how the quality of solution changes by setting five forms of step size parameter.