Abstract
In this article, third- and fourth-order accurate explicit time integration methods are developed for effective analyses of various linear and nonlinear dynamic problems stated by second-order ordinary differential equations in time. Two sets of the new methods are developed by employing the collocation approach in the time domain. To remedy some shortcomings of using the explicit Runge-Kutta methods for second-order ordinary differential equations in time, the new methods are designed to introduce small period and damping errors in the important low-frequency range. For linear cases, the explicitness of the new methods is not affected by the presence of non-diagonal damping matrix. For nonlinear cases, the new methods can handle velocity dependent problems explicitly without decreasing order of accuracy. The new methods do not have any undetermined algorithmic parameters. Improved numerical solutions are obtained when they are applied to various linear and nonlinear problems.
Highlights
Step-by-step direct time integrations are dominantly used for transient analyses of linear and nonlinear dynamic problems described by second-order ordinary differential equations in time
Explicit methods are more frequently used in analyses of large systems, such as the wave propagation and impact problems, where sizes of optimal time steps are slightly smaller than critical time steps of explicit methods
6 Concluding remarks The new third- and fourth-order explicit methods presented in this paper could provide accurate numerical solutions when applied to various linear and nonlinear test problems
Summary
Step-by-step direct time integrations are dominantly used for transient analyses of linear and nonlinear dynamic problems described by second-order ordinary differential equations in time. Numerous implicit [5-9] and explicit [10-16] time integration methods have been introduced to effectively analyze challenging dynamic problems. The Soares method can be used for the analysis of wave propagation problems, but it becomes an only first-order accurate implicit method in the presence of non-diagonal damping matrix. The Kim and Lee (KL) method can include a full range of dissipative cases and remain as a second-order explicit method in the presence of a nondiagonal damping matrix. Numerical solutions of the RK methods may become inaccurate by the excessive numerical damping if large time steps are used For this reason, the RK methods were not recommended for the analysis of structural dynamics, while many second-order explicit methods gained popularity. The single-degree-of-freedom case of the linear structural dynamics equation given in Eq(3) is used in the procedures of the development
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