基于Galerkin法弱形式的时间积分法

A TIME INTEGRATION METHOD BASED ON THE WEAK FORM GALERKIN METHOD

从加权残差法的角度出发,利用Galerkin法的弱形式建立了一种高精度的隐式时间积分法。在时间单元上用拉格朗日插值函数构造近似解。给出了使用线性、二次和三次插值函数构造的积分格式,它们的精度依次为二阶、四阶和六阶。求解时首先消去时间单元内部的未知量,有效地提高了计算效率。通过减缩积分,可将条件稳定的格式变为无条件稳定的格式。数值算例表明该方法的精度和效率明显高于Newmark法。

An implicit high-order accurate time integration method is presented based on the weak form Galerkin method. In each time element, approximate solution is constructed by the Lagrangian interpolation functions. Three formulations, which are two-, four-, six-order of accuracy, are obtained by using linear, quadratic and cubic Lagrangian interpolation functions. When solving the equations, unknown displacements in the time elements are eliminated first to make the method more effective. Stability analysis shows that the formulations are conditionally stable. By using reduced integration, three unconditionally stable formolations are obtained. Numerical examples are included to illustrate the behavior of these algorithms. The results show that their precision and efficiency are remarkably higher than those of the Newmark method.