New approach for accelerating the nonlinear Schwarz iterations

The vector Epsilon algorithm is an effective extrapolation method used for accelerating the convergence of vector sequences. In this paper, this method is used to accelerate the convergence of Schwarz iterative methods for stationary linear and nonlinear partial differential equations (PDEs). The vector Epsilon algorithm is applied to the vector sequences produced by additive Schwarz (AS) or restricted additive Schwarz (RAS) methods after discretization. Some convergence analysis is presented, and several test-cases of analytical problems are performed in order to illustrate the interest of such algorithm. The obtained results show that the proposed algorithm yields much faster convergence than the classical Schwarz iterations.

A Local Fractional Taylor Expansion and Its Computation for Insufficiently Smooth Functions

A general fractional Taylor formula and its computation for insufficiently smooth functions are discussed. The Aitken delta square method and epsilon algorithm are implemented to compute the critical orders of the local fractional derivatives, from which more critical orders are recovered by analysing the regular pattern of the fractional Taylor formula. The Richardson extrapolation method is used to calculate the local fractional derivatives with critical orders. Numerical examples are provided to verify the theoretical analysis and the effectiveness of our approach.

Multi-Objective Control Method in Structural Displacement Interval Reliability Based on Epsilon Algorithm

In this article, a kind of structural displacement controlled reliability method was presented. The interval extension of multi-objective control algorithm was applied to control the structural displacement reliability. The method has realized the goal of controlling the multiple static interval reliability indexes by the control of the structural interval parameters. In order to accelerate the speed of the structural reanalysis, the Epsilon algorithm was used in the process of the structural reanalysis when a wide range of modification happened in the interval parameters. This method can both get a satisfactory accuracy, and improve the speed of the reanalysis. Numerical examples show that the method is effective and feasible.

Multi-Objective Control Method in Structural Strain Interval Reliability Based on Epsilon Algorithm

In this article, a kind of structural strain controlled reliability method was presented. The interval extension of multi-objective control algorithm was applied to control the structural strain reliability. The method has realized the goal of controlling the multiple static interval reliability indexes by the control of the structural interval parameters. In order to accelerate the speed of the structural reanalysis, the Epsilon algorithm was used in the process of the structural reanalysis when a wide range of modification happened in the interval parameters. This method can both get a satisfactory accuracy, and improve the speed of the reanalysis. Numerical examples show that the method is effective and feasible.

Multi-Objective Control Method in Structural Stress Interval Reliability Based on Epsilon Algorithm

A kind of structural stress reliability control method was presented. In this article, the interval extension of multi-objective control algorithm was applied to control the structural stress reliability. By the control of the structural interval parameters, this method has realized the goal of controlling the multiple static interval reliability indexes. To insure the accuracy of the structural reanalysis, the Epsilon algorithm was used in the process of the structural reanalysis when a wide range of modification happened in the interval parameters. This method can both get a satisfactory accuracy, and improve the speed of the reanalysis. The results of the example further proved that the proposed method can be effectively applied in the multi-objective control of the structural stress interval reliability.

An Error Compensation Method for Structural Responses and Sensitivities

The computation of the responses and their design sensitivities play an essential role in structural analysis and optimization. Significant works have been done in this area. Modal method is one of the classical methods. In this study, a new error compensation method is constructed, in which the modal superposition method is hybrid with Epsilon algorithm for responses and their sensitivities analysis of undamped system. In this study the truncation error of modal superposition is expressed by the first L orders eigenvalues and its eigenvectors explicitly. The epsilon algorithm is used to accelerate the convergence of the truncation errors. Numerical examples show that the present method is validity and effectiveness.