自由度具有物理含义的线性无关高阶数值流形法

A linearly independent high-order numerical manifold method with physically meaningful degrees of freedom

数值流形方法（NMM）基于两套覆盖（数学和物理覆盖）和接触环路而建立,能够统一地处理岩土工程中的连续和非连续变形分析问题。与其他基于单位分解理论的数值方法一样,NMM可以自由地提高物理片上局部位移函数（多项式）的阶次,从而在不加密网格的情况下显著地提高计算精度,但有可能会使总体刚度矩阵奇异,产生线性相关问题。针对这种情况,引入了一种新的高次多项式形式的局部位移函数,在此基础上,建立了新的NMM求解体系,并应用于求解一般的弹性力学问题。结果表明：它有效地消除了线性相关问题;较之传统局部位移函数取一次多项式的NMM,达到了更高的精度;节点应力是连续的;定义在物理片上的所有自由度都具有明确的物理含义,其中第3~5个刚好是物理片所对应插值点处的应变分量,因此,直接获得此处的应力状态。该方法可以很容易地推广到其他基于单位分解的数值方法中。

Numerical manifold method（NMM） is established based on the two cover systems（including mathematical cover and physical cover） and the contact loop, and can be used to solve the continuous and discontinuous deformation problems in the geotechnical engineering in a unified way. Similar to other numerical methods based on the partition of unity（PU） theory, the NMM can also improve the computational accuracy by increasing the orders of local displacement functions freely without mesh refinement, though this may cause the global stiffness matrix singular, leading to the linear dependence issue. In this study, a new local displacement function of high-order polynomials is proposed. The new function is applied to solve the general elastic problems. The results show that the linear dependence is eliminated. Compared with the traditional NMM based on the first order polynomials, higher precision is reached. Stresses at nodes are continuous. All the degrees of freedom defined on a physical patch are physically meaningful, with the third to fifth simply being the strain components at the interpolation point of the patch. As a result, the stresses at the interpolation points can be directly obtained. The proposed procedure can be easily extended to other PU-based methods.