STRONG WEAK COUPLING FORM ELEMENT DIFFERENTIAL METHOD FOR SOLVING SOLID MECHANICS PROBLEMS
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摘要: 单元微分法是一种新型强形式有限单元法. 与弱形式算法相比, 该算法直接对控制方程进行离散, 不需要用到数值积分. 因此该算法有较简单的形式, 并且其在计算系数矩阵时具有极高的效率. 但作为一种强形式算法, 单元微分法往往需要较多网格或者更高阶单元才能达到满意的计算精度. 与此同时, 对于一些包含奇异点的模型, 如在多材料界面、间断边界条件、裂纹尖端等处, 传统单元微分法往往得不到较精确的计算结果. 为了克服这些缺点, 本文提出了将伽辽金有限元法与单元微分法相结合的强-弱耦合算法, 即整体模型采用单元微分法的同时, 在奇异点附近或某些关键部件采用有限元法. 该策略在保留单元微分法高效率与简洁形式等优点的同时, 确保了求解奇异问题的精度. 在处理大规模问题时, 针对关键部件采用有限元法, 其他部件采用单元微分法, 可以在得到较精确结果的同时, 极大提高整体计算效率. 在本文中, 给出了两个典型算例, 一个是具有切口的二维问题, 一个是复杂的三维发动机问题. 针对这两个问题, 分析了该耦合算法在求二维奇异问题和三维大规模问题时的精度与效率.Abstract: Element differential method (EDM) is a new strong-form finite element method. Compared with the weak form numerical methods, the method discretizes the governing equations directly and does not need any numerical integration. Therefore, the method has a relatively simple form, and it has high efficiency in calculating the coefficient matrix. But as a strong form method, more nodes or higher-order elements are needed to achieve a satisfactory calculation accuracy in the element differential method. At the same time, for some models containing singular points which occur on multi-material interfaces, abrupt changes in the boundary conditions, and especially at crack tips, accurate calculation results can not be obtained by the conventional element differential method. In order to overcome this weakness, a coupled method combining the element differential method and finite element method (FEM) is proposed in this paper. The main idea of the coupled method is that the finite element method is used around the singular points in the geometric model and the element differential method is selected at other parts. The strong weak coupling form not only retains the advantages of the element differential method but also ensures the accuracy of solving singular problems. At the same time, when dealing with large scale problems, the finite element method is selected for key components and the element differentiation method is used for other components. This treatment can not only obtain more accurate results but also can greatly improve the overall calculation efficiency for large scale 3D problem. In this paper, two typical examples are given, one is a two-dimensional problem with notch, and the other is a complex three-dimensional engine problem. Through the calculation and analysis of these two problems, the correctness, accuracy and efficiency of the proposed coupling method in solving two-dimensional singular problem and three-dimensional large-scale problem are proved.
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图 1 单元微分法和有限元法耦合形式的模型离散方案
Figure 1. Model discretization scheme in coupling form of element differential method and finite element method
图 7 两种方法的von-Mises stress应力云图
Figure 7. von-Mises stress cloud maps of the two methods
表 1 两种方法的具体刚度矩阵K及载荷向量b表达式
Table 1. The specific stiffness matrix K and load vector b expressions of the two methods
Stiffness matrix K Load vector b EDM $ \left\{\begin{array}{l}D_{i j k l}(\xi) \dfrac{\partial^{2} N_{\alpha}(\xi)}{\partial x_{j} \partial x_{l}} \quad \xi \in \Omega^{e} \\\displaystyle\sum_{e=1}^{N^{I}} \displaystyle\sum_{s=1}^{S^{e l}} D_{i j k l}\left(\xi^{e}\right) n_{j}^{s}\left(\xi^{e}\right) \dfrac{\partial N_{\alpha}\left(\xi^{e}\right)}{\partial x_{l}} \quad \xi^{e} \in \partial \Omega\end{array}\right.$ $\left\{\begin{array}{l}-f_{i} \quad \xi \in \Omega^{e} \\0 \quad \text { or } \displaystyle\sum_{e=1}^{N^{I}} \displaystyle\sum_{s=1}^{S^{e I}} t_{i}^{s}\left(\boldsymbol{\xi}^{e}\right), \quad \boldsymbol{\xi}^{e} \in \Gamma_{t}\end{array}\right. $ FEM $ \displaystyle\int_{\Omega^{\varepsilon}} \dfrac{\partial N_{*}}{\partial x_{j}} D_{i j k l} \dfrac{\partial N_{\alpha}}{\partial x_{l}} \mathrm{~d} \Omega$ $\displaystyle\int_{\Gamma^{\varepsilon}} N_{*} t_{i} \mathrm{~d} \Gamma+\displaystyle\int_{\Omega^{\varepsilon}} N_{*} f_{i} \mathrm{~d} \Omega $ 
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表 2 三种方法所得y方向最大位移结果对比
Table 2. Comparison of maximum displacement in Y direction obtained by three methods (mm)
Method FEM EDM EDM_FEM Max uy 0.061671 0.063748 0.061699
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表 3 三种方法组集方程及求解所用时间对比
Table 3. Time comparison of three methods (s)
Method Time of forming the matrix Time of solving equation Total time EDM 78 380 458 FEM 1096 382 1478 Coupled 280 387 667
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