线性无关高阶数值流形法

Linearly independent higher-order numerical manifold method

对高阶数值流形方法来说,若采用1阶局部位移函数显然提高了计算精度,但又不可避免地使总体刚度矩阵亏秩,出现线性相关问题。针对这种情况,提出局部位移函数采用1阶泰勒展开形式,使得定义在物理覆盖上的自由度具有明确的物理意义。当基函数所对应自由度取为应变分量时,定义物理覆盖为PC-u-ε型;使用局部坐标系下的应力分量来代替应变分量,进而发展了PC-u-σ型。这样方便了位移和应力边界条件的施加。数值算例表明,PC-u-ε型显著地减少了亏秩数;PC-u-σ型的施加完全地消除了亏秩数,同样保持了很高的计算精度。

The adoption of the first-order local approximation functions has improved the accuracy, but it has also led to rank deficiency of the global stiffness matrix for the higher-order numerical manifold method, meaning the existence of the linear dependence. The first-order Taylor’s expansions with regard to the interpolation point are adopted as the local displacement functions, which makes the degrees of freedom defined on the physical cover have definite physical meanings. Then the first-order partial differential derivatives are expressed by the strain components, leading to the PC-u-ε. The strain components are further replaced by the stress components in the local framework, creating the PC-u-σ. In this way, both the displacement and the stress boundary conditions are easily applied. Numerical examples show that the PC-u-ε alone significantly causes a drastic decrease in rank deficiency, while deploying the PC-u-σ along the stress boundary completely eliminates the rank deficiency and retains higher accuracy.