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.
Key words
geometric nonlinear problem /
Lagrangian description /
Eulerian description /
Numerical Manifold Method /
independent cover
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