Unconditionally Stable Explicit Algorithms For Structural Dynamic Time-history Analysis

Direct integration algorithm is a powerful numerical method to solve temporally discretized equations of motion for structural time-history analysis.On one hand,compared with implicit algorithm,explicit algorithm usually owns better computational efficiency but worse stability.Especially when the structure involves nonlinear behavior,the current explicit algorithms which are unconditionally stable in linear systems become merely conditionally stable.On the other hand,few algorithm has numerical damping and the method used to introduce numerical damping cannot be copied to other algorithms.To solve these two problems,the research conducted in this paper is as follows:(1)structural dynamic equation is rewritten as a first-order ordinary differential equation in state-space and its exact solution with implicit integral terms is derived by using integration factor method.Pade approximation is then used to provide an explicit approximation to the aforementioned implicit integral terms,and then a one-step explicit algorithm is proposed and referred as to Pade-based algorithm.This algorithm is second-order accurate for both displacement and velocity and unconditionally stable for linear system and nonlinear system.(2)we obtain the exact solution with implicit integral terms in similar way.Adams method,Pade approximation and Gauss numerical integration method are used to give explicit approximation to the implicit integral terms and then a multi-step explicit algorithm is proposed and referred to as Adams-based algorithm.This algorithm can be expressed in a general form with arbitrarily high order accuracy.The algorithm also is found to be unconditionally stable in both linear system and nonlinear system.Besides,the proposed algorithm can be A-stable or L-stable by changing the form of Pade approximation.(3)a general method is proposed to introduce controllable numerical damping based on general Pade approximation.The numerical damping of an algorithm can be controlled by changing the value of one parameter ?.If adjusted well,the spurious high-frequency component of the numerical solution can be dampen out while the accurate low-frequency component can be reserved.