Path-following methods for Kuhn-Tucker curves by an active index set strategy

Abstract The path-following scheme in Loisel and Maxwell (SIAM J Matrix Anal Appl 39(4):1726–1749, 2018) is adapted to efficiently calculate the dispersion relation curve for linear surface waves on an arbitrary vertical shear current. This is equivalent to solving the Rayleigh stability equation with linearized free-surface boundary condition for each sought point on the curve. Taking advantage of the analyticity of the dispersion relation, a path-following or continuation approach is adopted. The problem is discretized using a collocation scheme, parametrized along either a radial or angular path in the wave vector plane, and differentiated to yield a system of ODEs. After an initial eigenproblem solve using QZ decomposition, numerical integration proceeds along the curve using linear solves as the Runge–Kutta $$F(\cdot )$$ F ( · ) function; thus, many QZ decompositions on a size 2N companion matrix are exchanged for one QZ decomposition and a small number of linear solves on a size N matrix. A piecewise interpolant provides dense output. The integration represents a nominal setup cost whereafter very many points can be computed at negligible cost whilst preserving high accuracy. Furthermore, a two-dimensional interpolant suitable for scattered data query points in the wave vector plane is described. Finally, a comparison is made with existing numerical methods for this problem, revealing that the path-following scheme is the most competitive algorithm for this problem whenever calculating more than circa 1,000 data points or relative normwise accuracy better than $$10^{-4}$$ 10 - 4 is sought.

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Trusses Nonlinear Problems Solution with Numerical Methods of Cubic Convergence Order

TEMA (São Carlos) ◽

10.5540/tema.2018.019.01.161 ◽

2018 ◽

Vol 19 (1) ◽

pp. 161 ◽

Cited By ~ 4

Author(s):

Luiz Antonio Farani de Souza ◽

Emerson Vitor Castelani ◽

Wesley Vagner Inês Shirabayashi ◽

Angelo Aliano Filho ◽

Roberto Dalledone Machado

Keyword(s):

Nonlinear Behavior ◽

Path Following ◽

Nonlinear Problems ◽

Displacement Curve ◽

Equilibrium Path ◽

Cubic Convergence ◽

Structural Problems ◽

Nodal Coordinates ◽

Newton Raphson ◽

Path Following Methods

A large part of the numerical procedures for obtaining the equilibrium path or load-displacement curve of structural problems with nonlinear behavior is based on the Newton-Raphson iterative scheme, to which is coupled the path-following methods. This paper presents new algorithms based on Potra-Pták, Chebyshev and super-Halley methods combined with the Linear Arc-Length path-following method. The main motivation for using these methods is the cubic order convergence. To elucidate the potential of our approach, we present an analysis of space and plane trusses problems with geometric nonlinearity found in the literature. In this direction, we will make use of the Positional Finite Element Method, which considers the nodal coordinates as variables of the nonlinear system instead of displacements. The numerical results of the simulations show the capacity of the computational algorithm developed to obtain the equilibrium path with force and displacement limits points. The implemented iterative methods exhibit better efficiency as the number of time steps and necessary accumulated iterations until convergence and processing time, in comparison with classic methods of Newton-Raphson and Modified Newton-Raphson.

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