Nonlinear evaluations of unconditionally stable explicit algorithms

                                                 Shuenn-Yih Chang

Department of Civil Engineering, Taipei University of Technology, NTUT Box 2653, Chinese Taipei 106

Abstract: Two explicit integration algorithms with unconditional stability for linear elastic systems have been
successfully developed for pseudodynamic testing. Their numerical properties in the solution of a linear elastic system have
been well explored and their applications to the pseudodynamic testing of a nonlinear system have been shown to be feasible.
However, their numerical properties in the solution of a nonlinear system are not apparent. Therefore, the performance of both
algorithms for use in the solution of a nonlinear system has been analytically evaluated after introducing an instantaneous
degree of nonlinearity. The two algorithms have roughly the same accuracy for a small value of the product of the natural
frequency and step size. Meanwhile, the first algorithm is unconditionally stable when the instantaneous degree of nonlinearity
is less than or equal to 1, and it becomes conditionally stable when it is greater than 1. The second algorithm is conditionally
stable as the instantaneous degree of nonlinearity is less than 1/9, and becomes unstable when it is greater than 1. It can have
unconditional stability for the range between 1/9 and 1. Based on these evaluations, it was concluded that the fi rst algorithm
is superior to the second one. Also, both algorithms were found to require commensurate computational efforts, which are
much less than needed for the Newmark explicit method in general structural dynamic problems.

Keywords: explicit integration algorithms; unconditional stability; pseudodynamic algorithm; nonlinear system;
instantaneous degree of nonlinearity


กก