基于Hamilton变分原理求解有限域波动问题的时空元方法

Space-time element method for solving finite domain wave problem based on Hamilton variational principle

传统的波动数值模拟方法多假定时间和空间是相互独立的,这样的假定具有明确的物理意义且广泛应用于实践。但是,时空分离的处理无法体现时空存在的固有联系,造成时间离散的精度损失。本文基于传统哈密顿原理,构造了与一维有限域波动方程等价时空泛函"弱形式",单元采用一阶精度时空坐标同时插值的方式,分别建立高斯及结点积分准则的两种计算模型,比较了这两种数值离散方法在波动数值模拟时,计算结果的精度及特征。计算结果表明了结点积分单元模型较高斯积分单元模型具有更好的表现,且结点积分模型具有更合理的物理意义。

The assumption in traditional numerical simulation method for time and space are independent of each other. This assumption has clearly physical meaning and is widely applied in practice. However,since this treatment can not reflect the inherent relation between time and space variables,the time discrete precision will be lost.Based on Hamilton variational principle,the weak form equivalent to one-dimensional finite domain wave equation is founded. Using the first order time and space interpolation foundation,two numerical models relying on Gauss and Node integral criterion respectively are presented for wave problems. The accuracy and calculation characteristics are compared between the two numerical discretization methods. The calculation results show that the node integral unit model is better in performance than the Gauss integral element model.