A post‐processing technique and an a posteriori error estimate for the newmark method in dynamic analysis

Abstract

SUMMARYIn this paper, we present a post-processing technique and an a posteriori error estimate for the Newmark method in structural dynamic analysis. By post-processing the Newmark solutions, we derive a simple formulation for linearly varied third-order derivatives. By comparing the Newmark solutions with the exact solutions expanded in the Taylor series, we achieve the local post-processed solutions which are of fifth-order accuracy for displacements and fourth-order accuracy for velocities in one step. Based on the post-processing technique, a posteriori local error estimates for displacements, velocities and, thus, also the total energy norm error estimate are obtained. If the Newmark solutions are corrected at each step, the post-processed solutions are of third-order accuracy in the global sense, i.e. one-order improvement for the original Newmark solutions is achieved. We also discuss a method for estimating the global time integration error. We find that, when the total energy norm is used, the sum of the local error estimates will give a reasonable estimate for the global error. We present numerical studies on a SDOF and a 2-DOF example in order to demonstrate the performance of the proposed technique.

A methodology for the formulation of error estimators for time integration in linear solid and structural dynamics

“…The result is compared to the error estimators of References [18,19], and to the local exact error. A first-order accurate method is being used for the time integration, and thus the local error estimates must exhibit second order accuracy.…”

“…This new estimator is compared to the ones proposed by Wiberg and Li [18] and Hulbert and Jang [19], two well-known and widely employed error estimators in elastodynamics, with similar formulation and performance as several others (e.g. References [16,20,22]).…”

Generalized Implicit Multi Time Step Integration for Nonlinear Dynamic Analysis

“…For solving this defect, method of ampli cation matrices is utilized for studying the stability conditions of G-IHOA, N-IHOA, and IHOA integrations [29]. It should be noted that the most common approach to verifying stability of step-by-step time integrations is performed by constructing the ampli cation matrix [30][31][32], dened for free vibration of a single degree of freedom system:…”

Adaptive superposition of finite element meshes in non‐linear transient solid mechanics problems

“…where u h is approximated by temporal postprocessing [20] of u h , and u is approximated by spatial postprocessing of u h using the superconvergent patch recovery technique [21,22]. Consequently, the relative local temporal and spatial error are defined as…”

“…The spatial error estimation was based on the superconvergent patch recovery technique of Zienkiewicz and Zhu [18,19], while the temporal error estimation was based on the assumption of linearly varying third-order time derivatives of the displacement field as discussed by Wiberg and Li [20]. Compared to conventional h-adaptive procedures [21,22], the s-adaptive procedure is faster and more efficient; these two attributes are extremely advantageous in computationally demanding non-linear transient problems.…”