Proposed Step Size Research Articles

User friendly FE Formulation for anisotropic distortional hardening model based on non-associated flow plasticity and its application to springback prediction

Based on non-associated flow plasticity, a newly developed anisotropic distortional hardening model developed by Hu and Yoon [15] is implemented in finite element analysis in a user-friendly manner. The derivatives of complex hardening models are calculated using the Finite Difference Method (FDM), which is much more convenient than using the analytical derivatives. To further improve the accuracy of the proposed method, the step size analysis in FDM is performed by analyzing the derivative formation. To evaluate the accuracy and computational efficiency of a proposed step size for FDM, single element simulations are performed under different loading paths. It has been found that the maximum absolute error of the flow curves between the simulation and the theoretical result is less than 0.3%. The U-bending tests for DP600 and TRIP1180 are used to verify the ability of the distortional hardening model for springback prediction. The simulation result of the strain hardening model is in good agreement with the experiment. The computational efficiency is also increased by 24% due to the improved convergence rate.

New Step Size Control Algorithm for Semi-Implicit Composition ODE Solvers

Composition is a powerful and simple approach for obtaining numerical integration methods of high accuracy order while preserving the geometric properties of a basic integrator. Adaptive step size control allows one to significantly increase the performance of numerical integration methods. However, there is a lack of efficient step size control algorithms for composition solvers due to some known difficulties in constructing a low-cost embedded local error estimator. In this paper, we propose a novel local error estimator based on a difference between the semi-implicit CD method and semi-explicit midpoint methods within a common composition scheme. We evaluate the performance of adaptive composition schemes with the proposed local error estimator, comparing it with the other state-of-the-art approaches. We show that composition ODE solvers with the proposed step size control algorithm possess higher numerical efficiency than known methods, by using a comprehensive set of nonlinear test problems.

New adaptive Barzilai--Borwein step size and its application in solving large-scale optimization problems

We propose a new adaptive and composite Barzilai–Borwein (BB) step size by integrating the advantages of such existing step sizes. Particularly, the proposed step size is an optimal weighted mean of two classical BB step sizes and the weights are updated at each iteration in accordance with the quality of the classical BB step sizes. Combined with the steepest descent direction, the adaptive and composite BB step size is incorporated into the development of an algorithm such that it is efficient to solve large-scale optimization problems. We prove that the developed algorithm is globally convergent and it R-linearly converges when applied to solve strictly convex quadratic minimization problems. Compared with the state-of-the-art algorithms available in the literature, the proposed step size is more efficient in solving ill-posed or large-scale benchmark test problems. doi:10.1017/S1446181118000263

NEW ADAPTIVE BARZILAI–BORWEIN STEP SIZE AND ITS APPLICATION IN SOLVING LARGE-SCALE OPTIMIZATION PROBLEMS

We propose a new adaptive and composite Barzilai–Borwein (BB) step size by integrating the advantages of such existing step sizes. Particularly, the proposed step size is an optimal weighted mean of two classical BB step sizes and the weights are updated at each iteration in accordance with the quality of the classical BB step sizes. Combined with the steepest descent direction, the adaptive and composite BB step size is incorporated into the development of an algorithm such that it is efficient to solve large-scale optimization problems. We prove that the developed algorithm is globally convergent and it R-linearly converges when applied to solve strictly convex quadratic minimization problems. Compared with the state-of-the-art algorithms available in the literature, the proposed step size is more efficient in solving ill-posed or large-scale benchmark test problems.

Analysing the effects of various switching probability characteristics in flower pollination algorithm for solving unconstrained function minimization problems

Due to their unique offerings, bio-inspired algorithms have become popular in problem solving. Flower pollination algorithm (FPA), which is relatively a new member of this family, is shown to be one promising algorithm and this optimizer is still open to possible enhancements. One of the reasons that adds to the popularity of FPA is indeed the simplicity in implementation. It has two basic procedures, namely global and local pollination, which correspond to global and local search, respectively. Moreover, a single parameter, referred to as switching probability, puts control on these search procedures. Thus, the mentioned switching probability actually defines the search characteristics throughout generations, which directly affects the success of FPA. Accordingly, the present work analyses the effects of various switching probability characteristics, including exponentially, linearly and sawtooth changing patterns. This is the main motivation of the present study. Secondarily, a systematically intensifying step size procedure, which is commonly ignored by most of the stochastic search algorithms, is adopted along with these strategies. The aim of the proposed step size function is to encourage a more intensified search towards the end, while providing a more diversified search at the initialization stage to avoid local optima and premature convergence. Thus, more promising results might be obtained. All developed modifications are tested by using well-known unconstrained function minimization problems. As demonstrated by several nonparametric statistical tests, results point out significant improvements over the standard FPA.

A band-dependent variable step-size sign subband adaptive filter

This letter proposes a band-dependent variable step-size sign subband adaptive filter using the concept of mean-square deviation (MSD) minimization. Since it is difficult to obtain the value of the MSD accurately, the proposed step size is derived by minimizing the upper bound of the conditional MSD with given input. By assigning the different step size in each band, the filter performance can be improved. Moreover, we suggest the estimation method of the measurement-noise variance in an impulsive-noise environment, because the proposed algorithm needs the measurement-noise variance to calculate the step size. The reset algorithm is also applied for maintaining the filter performance when a system change occurs suddenly. Simulation results demonstrate that the proposed algorithm performs better than the existing algorithms in aspects of the convergence rate and the steady-state estimation error.

Method of Successive Weighted Averages (MSWA) and Self-Regulated Averaging Schemes for Solving Stochastic User Equilibrium Problem

Although stochastic user equilibrium (SUE) problem has been studied extensively in the past decades, the solution convergence of SUE is generally quite slow because of the use of the method of successive averages (MSA), in which the auxiliary flow pattern generated at each iteration contributes equally to the final solution. Realizing that the auxiliary flow pattern is in fact approaching to the solution point when the iteration number is large, in this paper, we introduce the method of successive weighted averages (MSWA) that includes a new step size sequence giving higher weights to the auxiliary flow patterns from the later iterations. We further develop a self-regulated averaging method, in which the step sizes are varying, rather than fixed, depending on the distance between intermediate solution and auxiliary point. The proposed step size sequences in both MSWA and self-regulated averaging method satisfy the Blum theorem, which guarantees the convergence of SUE problem. Computational results demonstrate that convergence speeds of MSWA and self-regulated averaging method are much faster than those of MSA and the speedup factors are in a manner of magnitude for high accuracy solutions. Besides SUE problem, the proposed methods can also be applied to other fixed-point problems where MSA is applicable, which have wide-range applications in the area of transportation networks.

An error estimate and step size control method for nonlinear solution techniques

The condition number of the tangent stiffness matrix is shown to reflect the error encountered in the solution of nonlinear structural systems. Due to the linearization process in the incremental solution, two sources of error exist: (1) the error associated with the solution of the linearized set of equations; and (2) the error associated with the unbalanced force residual. While the condition number itself reflects the first error, its change from step to step represents the second error. Being an expensive quantity to compute, only an estimate of the condition number that is based on the L 1-norm of the tangent stiffness matrix has been effectively used in guiding the solution algorithm in adjusting the step size according to the degree of nonlinearity of the system. It is also used to identify the stable from the unstable paths of the response curve. Several examples are solved to show the applicability of the proposed step size control with different solution techniques.