2017 Volume 4 Issue 2 Pages 17-00040
A variety of computational methods that have a wide range of applications to nonlinear dynamical problems are presented in this review. Most of the methods are used in isolated ways in literature. The relationships underlying these methods and the distinctions between some methods are often not clarified. This may lead to confusion in understanding these methods, as well as unnecessary efforts in conducting further researches. In this paper, the methods are arranged in a unified manner. Four groups of distinct methods, namely the weighted residual methods, the finite difference methods, the asymptotic methods and the variational iteration-collocation methods, are introduced. The weighted residual methods comprise the collocation, the finite volume, the finite element, the boundary element methods, etc. Both the corresponding global and local methods are introduced. Depending on whether the problem is expressed in primal or mixed formulation, the weighted residual methods can be divided into primal or mixed methods. The finite difference methods are divided into explicit and implicit classes. Some commonly used methods such as the Runge-Kutta and Newmark method are introduced. The asymptotic methods are among the principal methods of nonlinear analyses. Some representative methods such as the perturbation method, Adomian decomposition method and variational iteration method are presented. The recently proposed variational iteration-collocation method is a kind of semi-analytical iteration method. It is applicable to strongly nonlinear problems and capable of achieving high accuracy and efficiency. All the general formulations of the aforementioned methods are derived. Some numerical examples are also used to illustrate these methods when necessary.
