A Study on the Convergence Analysis of the Inexact Simplified Jacobi–Davidson Method

The Jacobi–Davidson iteration method is very efficient in solving Hermitian eigenvalue problems. If the correction equation involved in the Jacobi–Davidson iteration is solved accurately, the simplified Jacobi–Davidson iteration is equivalent to the Rayleigh quotient iteration which achieves cubic convergence rate locally. When the involved linear system is solved by an iteration method, these two methods are also equivalent. In this paper, we present the convergence analysis of the simplified Jacobi–Davidson method and present the estimate of iteration numbers of the inner correction equation. Furthermore, based on the convergence factor, we can see how the accuracy of the inner iteration controls the outer iteration.

Nonlinear numerical model of plane frames considering semi-rigid connection and different beam theories

ABSTRACT This paper presents a numerical-computational model for frames with geometric nonlinear behavior, by the Finite Element Co-rotational method, considering the Euler-Bernoulli and Timoshenko beam theories. The connection between structural members is simulated by a null-length connection element, which considers the axial, translational and rotational stiffness. The nonlinear equations system that describes the structural problem is solved by the incremental and iterative procedure of Potra-Pták, with cubic convergence order, combined with the Linear Arc-Length path-following technique. The solution method algorithm is presented and the numerical examples are simulated with the free Scilab program. The numerical results show that the slenderness of the structure, geometric nonlinearity and semi-rigidity influence the behavior of the structure. Structural analysis and design procedures that consider these factors attains less conservative design thus obtaining more optimized structures.

Numerical-Computational Model for Nonlinear Analysis of Frames with Semirigid Connection

A numerical-computational model for static analysis of plane frames with semirigid connections and geometric nonlinear behavior is presented. The set of nonlinear equations governing the structural system is solved by the Potra–Pták method in an incremental procedure, with order of cubic convergence, combined with the linear arc-length path-following technique. The algorithm pseudo-code is presented, and the finite element corotational method is used for the discretization of the structures. The equilibrium paths with load and displacement limit points are obtained. The semirigidity is simulated by a linear connection element of null length, which considers the axial, tangential, and rotational stiffness. Nonlinear analyses of 2D frame structures are carried out with the free Scilab program. The results show that the Potra–Pták procedure can decrease the number of iterations and the computing time in comparison with the standard and modified Newton–Raphson iterative schemes. Also, the simulations show that the connection flexibility has a strong influence on the nonlinear behavior and stability of the structural systems.

Some novel Newton-type methods for solving nonlinear equations

The aim of this paper is to present a new nonstandard Newton iterative method for solving nonlinear equations. The convergence of the proposed method is proved and it is shown that the new method has cubic convergence. Furthermore, two new multi-point methods with sixth-order convergence, based on the introduced method, are presented. Also, we describe the basins of attraction for these methods. Finally, some numerical examples are given to show the performance of our methods by comparing with some other methods available in the literature

Trusses Nonlinear Problems Solution with Numerical Methods of Cubic Convergence Order

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.

A Filtered-Davidson Method for Large Symmetric Eigenvalue Problems

For symmetric eigenvalue problems, we constructed a three-term recurrence polynomial filter by means of Chebyshev polynomials. The new filtering technique does not need to solve linear systems and only needs matrix-vector products. It is a memory conserving filtering technique for its three-term recurrence relation. As an application, we use this filtering strategy to the Davidson method and propose the filtered-Davidson method. Through choosing suitable shifts, this method can gain cubic convergence rate locally. Theory and numerical experiments show the efficiency of the new filtering technique.