采用独立覆盖流形法(基于流形思想的“分区级数解”),提出在空间固定的网格中求解几何非线性问题的新方法:在当前构形中关注经过各空间点的物质点,通过级数“逆向追踪”物质点在上一时步的位置及其应力、速度等物理量,并采用最小二乘法形成新级数作为当前时步的初值,就可以在固定网格中求解拉格朗日型的控制方程;每步计算后更新材料体构形,即更新固定网格(独立覆盖)中的积分区域,以得到准确的材料边界;以覆盖合并方式处理边界网格中的“小块”问题,并通过“小块”实现新网格的信息传递。给出弹性体大变形、刚体旋转算例验证方法的有效性。新方法集合了拉格朗日法的跟踪物质点、控制方程简单、边界描述准确以及欧拉法的网格无扭曲的优点,避免了2种方法各自的缺陷,为下一步在固定网格中进行几何非线性的自适应分析打下基础。
Abstract
Using Manifold Method based on independent covers, or piecewise-defined series solutions based on manifold idea, a novel method of solving geometric nonlinear problems in fixed meshes is proposed. In the current configuration, the material point passing through each space point is paid attention to. Through the series solutions, the position of the material point in the previous step is traced backwards, and its stress, velocity and other physical quantities are obtained. A new series is formed as the initial values of the current time step using the least square method, hence the Lagrangian governing equation can be solved in fixed meshes. After each step, the material configuration is updated, that is, the integral region in the fixed meshes (independent covers) is updated so as to obtain the accurate material boundaries. The “small block” in the boundary meshes is processed by cover merging, and the information to the new mesh is transmitted through the “small block”. Some examples such as the large deformation of elastic body and the rotation of rigid body are given to verify the effectiveness of the method. The proposed method combines the advantages of Lagrangian method in tracking material points, simplicity of the governing equation, accurate boundary description, and of Eulerian method of undistorted meshes, and in the meantime avoids the defects of the two methods. The research finding lays a foundation for the next step adaptive analysis of solving geometric nonlinearity in fixed meshes.
关键词
几何非线性 /
拉格朗日描述 /
欧拉描述 /
数值流形方法 /
独立覆盖
Key words
geometric nonlinear problem /
Lagrangian description /
Eulerian description /
Numerical Manifold Method /
independent cover
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基金
中央级公益性科研院所基本科研业务费项目(CKSF2019394/GC)
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