Rotation errors in numerical manifold method and a correction based on large deformation theory

doi:10.1007/s11709-019-0535-5

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:

https://academic.hep.com.cn/fsce/EN/10.1007/s11709-019-0535-5
https://academic.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

small triangle block (see Fig. 1) (elastic plain strain)	value
edge length $a$	0.1 m
initial angular velocity $ω 0$	360 °/s
time step interval $Δ t$	0.01–0.0001s
total time $t max ?$	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 $n (1)$	$Δ t$ (s)	theoretical $α (°)$	volume error		angle error
test No.	total step $n (1)$	$Δ t$ (s)	theoretical $α (°)$	actual	Eq. (16)	angle error
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)

σ x

; (b)

τ x y

Fig.7 The period of high frequency is exactly 2

Δ t

Fig.8 After high frequency wave manually removed. (a)

σ x

vs time; (b)

σ x

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

1	G H Shi. Manifold method. In: Proceeding of the 1st International Forum on DDA Simulation of Discontinuous Media(ICADD-1). New Mexico: TSI Press, 1996, 52–204
2	G H Shi. Discontinuous deformation analysis: A new numerical model for the statics and dynamics of deformable block structures. Engineering Computations, 1992, 9(2): 157–168 https://doi.org/10.1108/eb023855
3	G W Ma, X M An, L He. The numerical manifold method: a review. International Journal of Computational Methods, 2010, 7(1): 1–32 https://doi.org/10.1142/S0219876210002040
4	I Babuska, J M. MelenkThe partition of unity method. International Journal for Numerical Methods in Engineering, 1997, 40(4): 727–758
5	G Chen, Y Ohnishi, T Ito. Development of high-order manifold method. International Journal for Numerical Methods in Engineering, 1998, 43(4): 685–712 https://doi.org/10.1002/(SICI)1097-0207(19981030)43:4<685::AID-NME442>3.0.CO;2-7
6	H D Su, X L Xie, Q Y Liang. Automatic programming for high-order numerical manifold method. In: Proceeding of the 6th International Conference Analysis of Discontinuous Deformation (ICADD-6). Trondheim: A. A. Balkema publishers, 2003, 153–160
7	T Strouboulis, I Babuska, K Copps. The design and analysis of the Generalized Finite Element Method. Computer Methods in Applied Mechanics and Engineering, 2000, 181(1–3): 43–69 https://doi.org/10.1016/S0045-7825(99)00072-9
8	X M An, L X Li, G W Ma, H H Zhang. Prediction of rank deficiency in partition of unity-based methods with plane triangular or quadrilateral meshes. Computer Methods in Applied Mechanics and Engineering, 2011, 200(5–8): 665–674 https://doi.org/10.1016/j.cma.2010.09.013
9	H Zheng, D Xu. New strategies for some issues of numerical manifold method in simulation of crack propagation. International Journal for Numerical Methods in Engineering, 2014, 97(13): 986–1010 https://doi.org/10.1002/nme.4620
10	G X Zhang, X Li, H F Li. Simulation of hydraulic fracture utilizing numerical manifold method. Science China. Technological Sciences, 2015, 58(9): 1542–1557 https://doi.org/10.1007/s11431-015-5901-5
11	Y Yang, X Tang, H Zheng, Q Liu, L He. Three-dimensional fracture propagation with numerical manifold method. Engineering Analysis with Boundary Elements, 2016, 72: 65–77 https://doi.org/10.1016/j.enganabound.2016.08.008
12	G W Ma, X M An, H H Zhang, L X Li. Modeling complex crack problems using the numerical manifold method. International Journal of Fracture, 2009, 156(1): 21–35 https://doi.org/10.1007/s10704-009-9342-7
13	H Su, Y Qi, Y Gong, J H Cui. Preliminary research of Numerical Manifold Method based on partly overlapping rectangular covers. In: DDA Commission of International Society for Rock Mechanics, Proceedings of the 11th International Conference on Analysis of Discontinuous Deformation (ICADD11). Fukuoka: CRC Press, 2013
14	Z Liu, H Zheng. Two-dimensional numerical manifold method with multilayer covers. Science China. Technological Sciences, 2016, 59(4): 515–530 https://doi.org/10.1007/s11431-015-5907-z
15	Y Yang, H Zheng, M V Sivaselvan. A rigorous and unified mass lumping scheme for higher-order elements. Computer Methods in Applied Mechanics and Engineering, 2017, 319: 491–514 https://doi.org/10.1016/j.cma.2017.03.011
16	X An, G Ma, Y Cai, H Zhu. A new way to treat material discontinuities in the numerical manifold method. Computer Methods in Applied Mechanics and Engineering, 2011, 200(47–48): 3296–3308 https://doi.org/10.1016/j.cma.2011.08.004
17	Q Jiang, C Zhou, D Li. A three-dimensional numerical manifold method based on tetrahedral meshes. Computers & Structures, 2009, 87(13–14): 880–889 https://doi.org/10.1016/j.compstruc.2009.03.002
18	L He, G W Ma. Development of 3D numerical manifold method. International Journal of Computational Methods, 2010, 7(1): 107–129 https://doi.org/10.1142/S0219876210002088
19	Y Wu, G Chen, Z Jiang, L Zhang, H Zhang, F Fan, Z Han, Z Zou, L Chang, L Li. Research on fault cutting algorithm of the three-dimensional numerical manifold method. International Journal of Geomechanics, 2017, 17(5): E4016003 https://doi.org/10.1061/(ASCE)GM.1943-5622.0000655
20	C T Lin, B Amadei, S Sture, J Jung. Using an Augmented Lagrangian Method and Block Fracturing in the DDA Method (No. SAND-93-0817C; CONF-940506-3). Albuquerque, NM: Sandia National Labs, 1994
21	X Li, H Zheng. Condensed form of complementarity formulation for discontinuous deformation analysis. Science China Technological Sciences, 2015, 58(9): 1509–1519 https://doi.org/10.1007/s11431-015-5913-1
22	G Shi. Contact theory. Science China Technological Sciences, 2015, 58(9): 1450–1496 https://doi.org/10.1007/s11431-015-5814-3
23	S Z Lin, Y F Qi, H D Su. Formulation of high-order numerical manifold method and fast simplex integration based on special matrix operations. In: Proceeding of the 7th International Conference Analysis of Discontinuous Deformation (ICADD-7). Honolulu: CRC Press, 2005
24	X L Qu, G Y Fu, G W Ma. An explicit time integration scheme of numerical manifold method. Engineering Analysis with Boundary Elements, 2014, 48(6): 53–62 https://doi.org/10.1016/j.enganabound.2014.06.005
25	X L Qu, Y Wang, G Y Fu, G W Ma. Efficiency and accuracy verification of the explicit numerical manifold method for dynamic problems. Rock Mechanics and Rock Engineering, 2015, 48(3): 1131–1142 https://doi.org/10.1007/s00603-014-0613-x
26	C Y Koo, J C Chern. Modification of the DDA method for rigid block Problems. Journal of Rock Mechanics and Mining Sciences, 1998, 35(6): 683–693 https://doi.org/10.1016/S0148-9062(97)00319-7
27	M M MacLaughlin, N Sitar. Rigid body rotations in DDA. In: Proceedings of the 1st international forum on discontinuous deformation analysis (DDA) and simulation of discontinuous media. New Mexico: TSI Press, 1996
28	Y M Cheng, Y H Zhang. Rigid body rotation and block internal discretization in DDA analysis. International Journal for Numerical and Analytical Methods in Geomechanics, 2000, 24(6): 567–578 https://doi.org/10.1002/(SICI)1096-9853(200005)24:6<567::AID-NAG83>3.0.CO;2-N
29	H Fan, H Zheng, S He. S-R decomposition based numerical manifold method. Computer Methods in Applied Mechanics and Engineering, 2016, 304: 452–478 https://doi.org/10.1016/j.cma.2016.02.033
30	H Fan, H Zheng, J Zhao. Discontinuous deformation analysis based on strain-rotation decomposition. International Journal of Rock Mechanics and Mining Sciences, 2017, 92: 19–29 https://doi.org/10.1016/j.ijrmms.2016.12.003
31	W Wei, Q Jiang. A modified numerical manifold method for simulation of finite deformation problem. Applied Mathematical Modelling, 2017, 48: 673–687 https://doi.org/10.1016/j.apm.2017.04.026
32	K J Bathe, E Ramm, E L Wilson. Finite element formulations for large deformation dynamic analysis. International Journal for Numerical Methods in Engineering, 1975, 9(2): 353–386 https://doi.org/10.1002/nme.1620090207
33	K J Bathe. Finite Element Procedures. New Jersey: Prentice Hall, 1996
34	K Terada, M Asai, M Yamagishi. Finite cover method for linear and non-linear analyses of heterogeneous solids. International Journal for Numerical Methods in Engineering, 2003, 58(9): 1321–1346 https://doi.org/10.1002/nme.820
35	A Hansbo, P Hansbo. A finite element method for the simulation of strong and weak discontinuities in solid mechanics. Computer Methods in Applied Mechanics and Engineering, 2004, 193(33–35): 3523–3540 https://doi.org/10.1016/j.cma.2003.12.041
36	T Rabczuk, G Zi, A Gerstenberger, W A Wall. A new crack tip element for the phantom-node method with arbitrary cohesive cracks. International Journal for Numerical Methods in Engineering, 2008, 75(5): 577–599 https://doi.org/10.1002/nme.2273
37	T Chau-Dinh, G Zi, P S Lee, T Rabczuk, J H Song. Phantom-node method for shell models with arbitrary cracks. Computers & Structures, 2012, 92–93(3): 242–256 https://doi.org/10.1016/j.compstruc.2011.10.021
38	C Y Wang, C Chuang, J Sheng. Time integration theories for the DDA method with finite element meshes. In: Proceedings of the 1st International Forum on Discontinuous Deformation Analysis (DDA) and Simulations of Discontinuous Media. Albuquerque, NM: TSI Press, 1996
39	D M Doolin, N Sitar. Time integration in discontinuous deformation analysis. Journal of Engineering Mechanics, 2004, 130(3): 249–258 https://doi.org/10.1061/(ASCE)0733-9399(2004)130:3(249)
40	Q Jiang, Y Chen, C Zhou, M R Yeung. Kinetic energy dissipation and convergence criterion of discontinuous deformations analysis (DDA) for geotechnical engineering. Rock Mechanics and Rock Engineering, 2013, 46(6): 1443–1460 https://doi.org/10.1007/s00603-012-0356-5
41	C A Tang, S B Tang, B Gong, H M Bai. Discontinuous deformation and displacement analysis: From continuous to discontinuous. Science China. Technological Sciences, 2015, 58(9): 1567–1574 https://doi.org/10.1007/s11431-015-5899-8
42	C Antoci, M Gallati, S Sibilla. Numerical simulation of fluid-structure interaction by SPH. Computers & Structures, 2007, 85(11–14): 879–890 https://doi.org/10.1016/j.compstruc.2007.01.002
43	T Rabczuk, G Zi, S Bordas, H Nguyen-Xuan. A simple and robust three-dimensional cracking-particle method without enrichment. Computer Methods in Applied Mechanics and Engineering, 2010, 199(37–40): 2437–2455 https://doi.org/10.1016/j.cma.2010.03.031
44	H Ren, X Zhuang, Y Cai, T Rabczuk. Dual-horizon peridynamics. International Journal for Numerical Methods in Engineering, 2016, 108(12): 1451–1476 https://doi.org/10.1002/nme.5257
45	H Ren, X Zhuang, T Rabczuk. Dual-horizon peridynamics: A stable solution to varying horizons. Computer Methods in Applied Mechanics and Engineering, 2017, 318: 762–782 https://doi.org/10.1016/j.cma.2016.12.031
46	O C Zienkiewicz, R L Taylor, D D Fox. The finite element method for Solid & Structural Mechanics. Singapore: Elsevier, 2015
47	W Jiang, H Zheng. An efficient remedy for the false volume expansion of DDA when simulating large rotation. Computers and Geotechnics, 2015, 70: 18–23 https://doi.org/10.1016/j.compgeo.2015.07.008
48	K E Bisshopp, D C Drucker. Large deflection of cantilever beams. Mathematics and Fly Fishing, 1945, 3(3): 272–275
49	R Macciotta, C D Martin, D M Cruden, M Hendry, T Edwards. Rock fall hazard control along a section of railway based on quantified risk. Assessment and Management of Risk for Engineered Systems and Geohazards, 2017, 11(3): 272–284 https://doi.org/10.1080/17499518.2017.1293273
50	T Rabczuk, T Belytschko. Cracking particles: A simplified meshfree method for arbitrary evolving cracks. International Journal for Numerical Methods in Engineering, 2004, 61(13): 2316–2343 https://doi.org/10.1002/nme.1151
51	T Rabczuk, T Belytschko. A three-dimensional large deformation meshfree method for arbitrary evolving cracks. Computer Methods in Applied Mechanics and Engineering, 2007, 196(29–30): 2777–2799 https://doi.org/10.1016/j.cma.2006.06.020
52	T Rabczuk, G Zi, A Gerstenberger, W A Wall. A new crack tip element for the phantom-node method with arbitrary cohesive cracks. International Journal for Numerical Methods in Engineering, 2008, 75(5): 577–599 https://doi.org/10.1002/nme.2273

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