自由度具有物理含义的线性无关高阶数值流形法

发布时间：2018-04-11 04:25

本文选题：数值流形法 + 物理片　；参考：《岩土力学》2016年10期

【摘要】：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。
[Abstract]:The numerical manifold method (NMMM) is based on two sets of covering (mathematical and physical coverage) and contact loop. It can deal with the continuous and discontinuous deformation analysis problems in geotechnical engineering in a unified manner.Like other numerical methods based on the unit decomposition theory, NMM can freely increase the order of local displacement functions (polynomials) on a physical chip, thus significantly improving the calculation accuracy without encrypting the mesh.However, it is possible to make the total stiffness matrix singular and cause linear correlation problems.In order to solve this problem, a new local displacement function in the form of higher degree polynomial is introduced. Based on this, a new NMM solution system is established and applied to solve general elastic problems.The results show that the linear correlation problem is eliminated effectively, the NMMs with the first order polynomial are obtained higher accuracy than the traditional local displacement function, and the node stress is continuous.All the degrees of freedom defined on the physical sheet have a clear physical meaning, in which the 3rd ~ 5th is exactly the strain component at the interpolation point corresponding to the physical sheet. Therefore, the stress state here is obtained directly.This method can be easily extended to other numerical methods based on unit decomposition.
【作者单位】：长江科学院水利部岩土力学与工程重点实验室;中国科学院武汉岩土力学研究所岩土力学与工程国家重点实验室;
【基金】：国家自然科学基金(No.11502033,No.51579016,No.41272350,No.41672320) 国家重点基础研究发展计划(973)项目(No.2011CB710603)~~
【分类号】：O343;O302

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