Purpose

Various time integration methods and time finite element methods have been developed to obtain the responses of structural dynamic problems, but the accuracy and computational efficiency of them are sometimes not satisfactory. The purpose of this paper is to present a more accurate and efficient formulation on the basis of the weak form quadrature element method to solve linear structural dynamic problems.

Design/methodology/approach

A variational principle for linear structural dynamics, which is inspired by Noble's work, is proposed to develop the weak form temporal quadrature element formulation. With Lobatto quadrature rule and the differential quadrature analog, a system of linear equations is obtained to solve the responses at sampling time points simultaneously. Computation for multi-elements can be carried out by a time-marching technique, using the end point results of the last element as the initial conditions for the next.

Findings

The weak form temporal quadrature element formulation is conditionally stable. The relation between the normalized length of element and the suggested number of integration points in one element is given by a simple formula. Results show that the present formulation is much more accurate than other time integration methods and its dissipative property is also illustrated.

Originality/value

The weak form temporal quadrature element formulation provides a choice with high accuracy and efficiency for solution of linear structural dynamic problems.

中文翻译：

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线性结构动力学的弱形式时间正交单元公式

目的

已经开发了各种时间积分方法和时间有限元方法来获得结构动力问题的响应，但它们的准确性和计算效率有时并不令人满意。本文的目的是在弱形式正交元法的基础上提出一种更准确、更有效的公式来解决线性结构动力问题。

设计/方法/方法

受 Noble 工作启发的线性结构动力学变分原理被提出来开发弱形式时间正交元素公式。使用 Lobatto 正交规则和微分正交模拟，获得了一个线性方程组来同时求解采样时间点的响应。多元素的计算可以通过时间推进技术进行，使用最后一个元素的终点结果作为下一个元素的初始条件。

发现

弱形式时间正交元素公式是条件稳定的。单元的归一化长度与一个单元中建议的积分点数之间的关系由一个简单的公式给出。结果表明，本公式比其他时间积分方法准确得多，并且还说明了其耗散特性。

原创性/价值

弱形式时间正交单元公式为线性结构动力学问题的求解提供了一种高精度和高效率的选择。