# Frontiers of Structural and Civil Engineering

 Front. Struct. Civ. Eng.    2019, Vol. 13 Issue (5) : 1036-1053     https://doi.org/10.1007/s11709-019-0535-5
 RESEARCH ARTICLE
Rotation errors in numerical manifold method and a correction based on large deformation theory
Ning ZHANG1, Xu LI2(), Qinghui JIANG3, Xingchao LIN4
1. School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China
2. Key Laboratory of Urban Underground Engineering of Ministry of Education, Beijing Jiaotong University, Beijing 100044, China
3. School of Civil and Architectural Engineering, Wuhan University, Wuhan 430072, China
4. State Key Laboratory of Simulation and Regulation of Water Cycle in River Basin, Beijing 100038, China
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 Abstract Numerical manifold method (NMM) is an effective method for simulating block system, however, significant errors are found in its simulation of rotation problems. Three kinds of errors, as volume expansion, stress vibration, and attenuation of angular velocity, were observed in the original NMM. The first two kind errors are owing to the small deformation assumption and the last one is due to the numerical damping. A large deformation NMM is proposed based on large deformation theory. In this method, the governing equation is derived using Green strain, the large deformation iteration and the open-close iteration are combined, and an updating strategy is proposed. The proposed method is used to analyze block rotation, beam bending, and rock falling problems and the results prove that all three kinds of errors are eliminated in this method. Corresponding Author(s): Xu LI Just Accepted Date: 05 May 2019   Online First Date: 25 June 2019    Issue Date: 11 September 2019
 Cite this article: Ning ZHANG,Xu LI,Qinghui JIANG, et al. Rotation errors in numerical manifold method and a correction based on large deformation theory[J]. Front. Struct. Civ. Eng., 2019, 13(5): 1036-1053. URL: http://journal.hep.com.cn/fsce/EN/10.1007/s11709-019-0535-5 http://journal.hep.com.cn/fsce/EN/Y2019/V13/I5/1036
 Tab.1  Comparison of FEM, NMM, and DDA mesh for a same pentagon Fig.1  Triangle ABC rotates around vertex A Fig.2  The errors at different rotation angles. (a) Volume expansion phenomenon; (b) relationships between relative error and cumulative rotation angle Fig.3  The rigid rotation of the square. (a) Real rotation displacement; (b) approximation of small deformation assumption Tab.2  The settings of the rotation test Tab.3  The errors with different rotation angle in per step Fig.4  Linear relationship between the rotation errors and the incremental rotation angle $α$ Fig.5  The decreasing of angular velocity Fig.6  High frequency vibration phenomenon. (a) $σx$ ; (b) $τx y$ Fig.7  The period of high frequency is exactly 2$Δt$ Fig.8  After high frequency wave manually removed. (a) $σx$ vs time; (b) $σx$ vs rotation angle Fig.9  Flowchart of large deformation iteration in practical NMM simulations Tab.4  the results of 4 cases Fig.10  The results of new scheme vs original NMM Fig.11  The stress obtained by new scheme Fig.12  Cantilever beam. (a) Boundary conditions; (b) the 1st order NMM mesh Fig.13  The results of the static simulation. (a) Deformed beam; (b) iteration process Fig.14  Two methods offer two different traces Fig.15  The model of the sliding test Fig.16  The NMM mesh with 20 elements Fig.17  The sliding process of the sliding test Fig.18  Block volume vs time Fig.19  Block velocity vs time Fig.20  The energy loss during the sliding process
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