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 引用本文: 刘华雩, 高效伟, 范伟龙. 分区有限线法及其在复合结构热应力分析中的应用. 力学学报, 2023, 55(6): 1394-1406.
Liu Huayu, Gao Xiaowei, Fan Weilong. Zonal finite line method and its applications in analyzing thermal stress of composite structures. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(6): 1394-1406.
 Citation: Liu Huayu, Gao Xiaowei, Fan Weilong. Zonal finite line method and its applications in analyzing thermal stress of composite structures. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(6): 1394-1406.

## ZONAL FINITE LINE METHOD AND ITS APPLICATIONS IN ANALYZING THERMAL STRESS OF COMPOSITE STRUCTURES

• 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. ZFLM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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