ZONAL FINITE LINE METHOD AND ITS APPLICATIONS IN ANALYZING THERMAL STRESS OF COMPOSITE STRUCTURES
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摘要: 介绍一种全新的数值方法——分区有限线法 (zonal finite line method, ZFLM), 并将其应用于求解复合结构中的热应力问题. FLM是一种配点型的强形式算法, 对于每个配置点, 由过其的两条(二维问题)或三条(三维问题)线段建立一个交叉线系, 采用拉格朗日插值多项式对每条线段的坐标与物理量进行函数表征, 并用沿弧长方向求导法创建任意物理量对总体坐标的一阶偏导数解析计算式, 通过递推技术, 由一阶偏导数公式建立二阶偏导数计算式. 采用建立的偏导数计算式, 可直接由问题的控制微分方程及其边界条件建立离散的总体系统方程组. 为了建立高效的有限线法和能够求解复杂的由多种材料组成的复合结构问题, 提出一种分区计算方法, 即: 根据材料的不同或几何与载荷的不规则性, 将所分析的问题划分为若干个结构化计算区域, 在每个区域由插值函数自动产生一系列配置点, 并用有限线法建立每个配置点的离散方程. 对于区域间的公共节点, 由物理量的协调条件以及界面力的平衡条件建立界面节点代数方程; 对于几何不规则或载荷跳跃问题, 采用面力方程叠加法建立非规则节点的代数方程, 以提高计算结果的稳定性. 采用本文方法对二维/三维结构的热应力进行分析. 计算结果表明本文方法具有很好的精度, 且在边界上的应力更为精确, 应力集中的效果更为明显.Abstract: In this paper, we proposed a novel numerical method, Zonal Free Element Method (ZFLM), and used the proposed method to compute thermal stress in composite structures. ZFLM is a strong-form numerical method which solves the governing equations in differential form. For each node, we use two (two-dimension problems) or three (three-dimensions problems) lines to form a cross-line system. Then, we use Lagrange interpolating method to interpolate nodal coordinates and approximate the variables on each line. The gradients in the curvature direction are computed by the gradients of interpolating functions along the line. By a recursive procedure, the second or higher order of derivatives can be obtained by the expressions of the first order derivatives. Substituting the expressions of derivatives into the governing partial difference equations, we obtain the discretized linear system of equations. To solve the problem involving multiple kinds of composite structures efficiently, we use a zonal method. In the zonal method, we divide the computational domain into several regular zones by material types and geometric characteristics. We insert nodes in each zone by interpolating functions and use the finite line method to assemble the discretized governing equations at these nodes. For the nodes at the interfaces which are shared by two or more zones, the traction-equilibrium equations and the compatibility conditions of variables are used to construct the linear algebraic equations. For the irregular geometries and the nodes where the loads jump, we add up the traction-equilibrium equations of each neighbor faces in different directions to improve the robustness of the proposed method. We use the proposed method to solve several thermal stress problems in two- and three-dimension. The results of test cases indicate that the proposed method has good accuracy and a significant priority in problems involving stress concentration. Because the collocation method is used, the stress on the boundary is more accurate.
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Key words:
- finite line method /
- zonal method /
- collocation method /
- composite structure /
- thermal stress
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图 1 二维问题点系离散以及线系示意图
Figure 1. Schematic diagram of the point system discretization and line system in 2 D problems
图 3 一阶与二阶导数相关的节点示意图
Figure 3. Related nodes of the 1st and 2nd order partial derivatives
图 4 由大块单元产生配置点与线系
Figure 4. Generating collocation nodes and line systems from unit blocks
图 5 位于常规界面上的节点(只有2个区域共享)
Figure 5. Nodes located on the regular interface (shared by only two regions)
图 6 界面节点为4区域相接的角点
Figure 6. The interface node is a corner point shared by four regions
图 7 具有跳跃面力边界条件的区域分解示意图
Figure 7. Schematic of the regional decomposition with jump surface forces boundary conditions
图 8 具有跳跃面力边界条件的功能梯度薄板
Figure 8. Functional gradient laminates with jump surface force boundary conditions
图 9 功能梯度材料薄梁的2个计算域及其生成的配置点
Figure 9. Two computational domains of thin beams in functional gradient materials and the generated collocation nodes thereof
图 10 温度场计算中得到的温度场云图
Figure 10. Temperature contours obtained in the temperature field calculation
图 11 热应力分析中得到的变形图
Figure 11. The deformation diagram obtained in the thermal stress analysis
图 12 热应力分析中得到的von-Mises应力云图
Figure 12. Von-Mises stress contours obtained from the thermal stress analysis
图 16 L形复合结构及分区网格(密网格)
Figure 16. L-shaped composite structure and partitioned grid (dense grid)
图 19 L形结构底端位移曲线
Figure 19. Displacement curves at the bottom of the L-shaped structure
图 20 L形结构底端Mises应力曲线
Figure 20. Mises stress curves at the bottom of the L-shaped structure
图 22 厚壁圆筒的区域划分与线系连成的网格
Figure 22. Area division of thick-walled cylinders and the grid connected by line system
图 26 梯度材料圆筒x方向位移云图
Figure 26. Displacement clouds of gradient material cylinder in x-direction
图 29 不同网格计算出的圆筒应力与参考解误差
Figure 29. Errors of cylinder stresses calculated by different meshes from the reference solution
表 1 各层的厚度与物性参数
Table 1. Thickness and physical parameters of each layer
Layer H/mm $ \mathrm{\lambda }[ $$\lambda $ /(W·m−1·K−1) E/MPa ν $\alpha / {10^{ - 6}}$ 1 5.0 20.0 10.0 0.2 0.5 2 10.0 2.0 1.0 0.1 0.1 3 10.0 10.0 20.0 0.3 1.0 
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