A study on time schemes for DRBEM analysis of elastic impact wave

Abstract

The precise integration and differential quadrature methods are two new unconditionally stable numerical schemes to approximate time derivative with more than the second order accuracy. Recent studies showed that compared with the Houbolt and Newmark methods, they produced more accurate solutions with large time step for the problems where response is primarily dominated by low and intermediate frequency modes. This paper aims to investigate these time schemes in the context of the dual reciprocity BEM (DRBEM) formulation of various shockexcited scalar elastic wave problems, where high modes have important affect on traction response. The Houbolt method was widely recommended in many literatures for such DRBEM dynamic formulations. However, this study found that the damped Newmark algorithm was the most efficient and accurate for impact traction analysis in conjunction with the DRBEM. The precise integration and differential quadrature methods are shown inapplicable for such shock-excited problems due to the absence of numerical damping. On the other hand, we also found that to achieve the same order of accuracy, the differential quadrature method required much less computing effort than the precise integration method due to the use of the Bartels-Stewart algorithm solving the resulting Lyapunov matrix analogization equation.

A new linearized method for Benjamin‐Bona‐Mahony equations

“…(4.6) and the former transformation, we can obtain V(x, t kϩ1 ). From [18] we know that "The DQM holds the unconditionally stable merit for accuracy of order more than two." The trapezoid integration formula reaches the second-order accuracy.…”

Bending analysis of thin functionally graded plates using generalized differential quadrature method

“…Chen et al [31] proposed a special matrix product method to simplify the computer implementation and improved the efficiency of the DQ method, especially, for solving non-linear problems. Chen and Tanaka [32] extended the applications of DQ method to initial-value problems, where DQ method was used to approximate temporal derivatives. The first application of DQM for composite plates was carried out by Bert et al [33].…”

Localized triangular differential quadrature

“…Many references can be found in the two literature surveys of the method [3,4]. In addition, some other new development of the differential quadrature method can be found in the work of Chen et al [5,6]. With the domain decomposition technique, which was first incorporated into the differential quadrature method by Civan and Sliepcevich [7], the differential quadrature method can be used to deal with some problems with curvilinear boundaries and even irregular geometry [8,9].…”