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In this paper, an explicit analytical method called the variational iteration method is presented for solving the second-order singular initial value problems of the Lane–Emden type. In addition, the local convergence of the method is discussed. It is often useful to have an approximate analytical solution to describe the Lane–Emden type equations, especially in the case that the closed-form solutions do not exist at all. This convince us that an effective improvement of the method will be useful to obtain a better approximate analytical solution. The improved method is then treated as a local algorithm in a sequence of intervals. Besides, an adaptive version is suggested for finding accurate approximate solutions of the nonlinear Lane–Emden type equations. Some examples are given to demonstrate the efficiency and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

The failure probability function (FPF) expresses the probability of failure as a function of the distribution parameters associated with the random variables of a reliability problem. Knowledge on this FPF is of much relevance for reliability sensitivity analysis and reliability-based design optimisation. However, its calculation is usually a challenging task. Therefore, this paper presents an efficient approach for estimating the FPF based on an adaptive strategy and a combination algorithm. The proposed approach involves three basic elements: (1) a Weighted Importance Sampling approach, which allows determining local FPF estimates; (2) an adaptive strategy for determining at which realisations of the distribution parameters it is necessary to perform local FPF estimation; and (3) an optimal combination algorithm, which allows to aggregate local FPF estimations together to form a global estimate of the FPF. Test and practical examples are presented to demonstrate the efficiency and feasibility of the proposed approach. • A global approach for failure probability function estimation is proposed. • It utilises weighted importance sampling to efficiently obtain local estimators. • It includes an adaptive active strategy to guide the local estimations. • It adopts an optimal combination algorithm to combine the local FPF estimators. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, an adaptive trust region filter algorithm for unconstrained optimization is presented. The global convergence results of the algorithm are established and the numerical results show that the new algorithm is efficient in practical computation. [Copyright &y& Elsevier]

A lower bound (LB) axisymmetric limit analysis formulation is presented for Mohr-Coulomb materials using finite elements and semidefinite programming (SDP). The details of the application of SDP to the axisymmetric LB finite element method for Mohr-Coulomb materials are described in this paper. Through the use of the SDP, the axisymmetric Mohr-Coulomb yield criterion can be directly applied in its native form in terms of principal stresses without any smoothing of the parent yield surface, and no assumption needs to be made about σ θ. A node-stress-based adaptive mesh algorithm is also introduced into the proposed method to obtain an accurate high-accuracy LB solution. The proposed method is then applied to estimate the collapse load of circular footings and the stability of an unlined vertical circular excavation. The results are also compared with those reported in the literature; the results obtained from the proposed method are accurate and can provide failure patterns with greater clarity. [ABSTRACT FROM AUTHOR]