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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.

Keywords numerical manifold method      rotation      large deformation      Green strain      open-close iteration     
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
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Ning ZHANG
Xu LI
Qinghui JIANG
Xingchao LIN
method reference configuration deformed configuration
FEM
(3 elements)
NMM
(1 element)
DDA
(1 element)
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
small triangle block (see Fig. 1) (elastic plain strain) value
edge length a 0.1 m
initial angular velocity ω0 360 °/s
time step interval Δt 0.01–0.0001s
total time tmax? 1 s
elasticity modulus E 10 Gpa
Poisson’s ratio μ 0.3
Tab.2  The settings of the rotation test
test No. total step n(1 ) Δt(s) theoretical α(°) volume error angle error
actual Eq. (16)
1 100 0.0100 3.6 39.4% 39.4% −8.40%
2 300 0.0333 1.2 13.1% 13.1% −3.10%
3 1000 0.0010 0.36 3.9% 3.9% −0.97%
4 3000 0.0033 0.12 1.3% 1.3% −0.33%
5 10000 0.0001 0.036 0.4% 0.4% −0.10%
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 2Δt
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
No. method strain newmark (β) numerical damping volume error (%) rotation angle error
T1 NMM small strain 1.0 yes 39 −8.40%
T2 NMM small strain 0.5 no 47 0.46%
T3 LDNMM green strain 1.0 yes <1e−6 −4.52%
T4 LDNMM green strain 0.5 no <1e−6 −0.033%
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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