An explicit method with improved stability property

Abstract

SUMMARYA new explicit method is presented. This method has an enhanced stability property when compared with the previously published explicit method (J. Eng. Mech. 2007; 133(7):748-760), which is unconditionally stable for linear elastic and any instantaneous stiffness softening systems based on linearized analysis whereas it is only conditionally stable for any instantaneous stiffness hardening system. In contrast to the previously published explicit method, the most important improvement of the new explicit method is that it can have unconditional stability for general instantaneous stiffness hardening systems in addition to linear elastic and instantaneous stiffness softening systems. Since these stability results are obtained from a linearized stability analysis they are applicable to the non-linear systems that satisfied the simplifications used for the analysis. This explicit method is also shown to be second-order accurate. It is computationally efficient in the solution of a general structural dynamic problem, where the total response is dominated by low-frequency modes, when compared with explicit methods, such as the Newmark explicit method and the family of explicit methods developed by Tamma et al. (Comput. Methods Appl. Mech. Eng. 2003; 192:257-290), and implicit methods, such as the constant average acceleration method. Some numerical examples are used to confirm the enhanced stability property and the efficiency in computing.

A One-Parameter Controlled Dissipative Unconditionally Stable Explicit Algorithm for Time History Analysis

“…0 are found for i = 1; 2; 3; ::: for a linear elastic system. The development details of this method are similar to those of the previously published algorithms [17,18] and, thus, they will not be elaborated on herein. In this derivation, the proof of convergence must be conducted rst.…”

“…The structure-dependent integration method was rst developed by Chang in 2002 [15]. Some integration methods of this type were also successfully developed by Chang [16][17][18][19][20][21][22] subsequently. These structure-dependent integration methods can simultaneously integrate the major advantages of the implicit and explicit algorithms, i.e.…”

A parameter to improve stability for a family of dissipative integration methods

“…In addition, it has been also shown that the numerical dispersion and energy dissipation characteristics of this family of methods are the same as those of the generalized-α method (Chung and Hulbert 1993). The most important property of KRM is controllable numerical dissipation when compared to the currently available structure-dependent integration methods (Chang 1997(Chang , 2002(Chang , 2007(Chang , 2009(Chang , 2010Chen and Ricles 2008) except for the Chang-α dissipative method (Chang 2014(Chang , 2015. The controllable numerical dissipation can be used to suppress the spurious oscillations of high frequency modes, while the low frequency modes can be integrated very accurately.…”

Comparison of capability of time integration methods in capturing dynamic loading

“…Note that the spectral decomposition technique to analytically obtain the numerical solution is generally not used for a general step-by-step integration procedure (Chang, 1997(Chang, , 2000(Chang, , 2002(Chang, , 2007(Chang, , 2009(Chang, , 2010Hilber et al, 1977;Newmark, 1959;Sha et al, 2003;Tamma et al, 2001Tamma et al, , 2003Zhou and Tamma, 2004). In fact, a computer program is usually applied to conduct the stepby-step integration to yield the numerical solution.…”