Time Integration in Discontinuous Deformation Analysis
Publication: Journal of Engineering Mechanics
Volume 130, Issue 3
Abstract
Discontinuous deformation analysis (DDA) is a discrete element method that was developed for computing large deformation in fractured rock masses. In this paper we present details of the DDA time integration scheme, where the acceleration is taken constant over the time step, equal to the acceleration at the end of the time step (“right Riemann”). The integration scheme has several advantages: (1) Self-starting, (2) accelerations never need to be computed which reduces implementation complexity, (3) unconditionally stable, and (4) dissipative, contains algorithmic damping which may be important considering the penalty formulation of DDA. However, the right Riemann scheme is implicit, requiring expensive factorization or iteration to solve the resulting system of equations, and is accompanied by a bifurcation in the spectrum when the time step is large with respect to the period. This bifurcation has important ramifications for controlling spurious resonance in DDA simulations due to linear scaling in system stiffness compared to cubic scaling of the system mass as the characteristic length of the domain increases.
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References
Bathe, K. J., and Wilson, E. L. (1976). Numerical methods in finite element analysis, 1st Ed., Prentice-Hall, Englewood Cliffs, N.J.
Demmel, J. W. (1997). Applied numerical linear algebra, Society for Industrial and Applied Mathematics, Philadelphia.
Dong, X., Wu, A., and Ren, F. (1996). “A preliminary application of discontinuous deformation analysis (DDA) to the Three Gorges Dam project.” Proc., 1st Int. Forum on Discontinuous Deformation Analysis (DDA) and Simulations of Discontinuous Media, Berkeley, Calif., M. R. Salami and D. Banks, eds., TSI, Albuquerque, N.M., 310–317.
Doolin, D. M. (2003). “Preliminary results unifying discontinuous deformation analysis (DDA) and the distinct element method (DEM),” Development and application of discontinuous modeling for rock engineering, Proc., 6th Int. Conf. on Analysis of Discontinuous Deformation (ICADD-6), Orkanger, Norway, October 5–8, M. Lu, ed., Balkema, Rotterdam, The Netherlands.
Doolin, D. M., and Sitar, N. (2002a). “Accuracy of the DDA method for frictionless impact.” Mining and tunnelling innovation and opportunity, R. Hammah, W. Bawden, J. Curran, and M. Telesnicki, eds., North American Rock Mechanics Symp.—Tunneling Association of Canada, Toronto, Balkema, Rotterdam, The Netherlands, 1289–1296.
Doolin, D. M., and Sitar, N.(2002b). “Displacement accuracy of discontinuous deformation analysis method applied to sliding block,” J. Eng. Mech., 128(11), 1158–1168.
Eaton, J. W. (2002). GNU Octave manual, Network Theory Ltd., Bristol, U.K.
Goudreau, G. L. (1970). “Evaluation of discrete methods for the linear dynamic response of elastic and viscoelastic solids,” PhD thesis, Univ. of California, Berkeley, Calif.
Hatzor, Y. H., Arzi, A. A., and Tsesarsky, M. (2002). “Realistic dynamic analysis of jointed rock slopes using DDA.” Stability of Rock Structures—Proc., 5th Int. Conf. on Analysis of Discontinuous Deformation, Abingdon, Y. H. Hatzor, ed., Balkema, Rotterdam, The Netherlands, 47–56.
Hilber, H. M. (1976). “Analysis and design of numerical integration methods in structural dynamics.” Technical Rep. No. EERC 76-29, Earthquake Engineering Research Center, Univ. of California, Berkeley, Calif.
Hughes, T. J. R. (1983). “Analysis of transient algorithms with particular reference to stability behavior.” Computational methods for transient analysis, T. Belytschko and T. J. R. Hughes, eds., Elsevier Science, Amsterdam.
Katona, M. G., and Zienkiewicz, O. C.(1985). “A unified set of single step algorithms.” Int. J. Numer. Methods Eng.,21, 1345–1359.
Kottenstette, J. T. (1999). “DDA analysis of the RCC modification for Pueblo Dam.” Proc., 3rd Int. Conf. on Analysis of Discontinuous Deformation—From Theory to Practice (ICADD-3), Vail, Colo., June 3–4, B. Amadei, ed., American Rock Mechanics Association, Alexandria, Va., 127–132.
Lanczos, C. (1970). The variational principles of mechanics, 4th ed., University of Toronto Press, Toronto.
Law, H. K., and Lam, I. P.(2003). “Evaluation of seismic perfomance for tunnel retrofit project.” J. Geotech. Geoenviron. Eng., 129(7), 575–589.
MacLaughlin, M. M., Doolin, D. M., and Berger, E. A. (2003). “Ten years of DDA validation.” Development and application of discontinuous modeling for rock engineering, Proc., 6th Int. Conf. on Analysis of Discontinuous Deformation (ICADD-6), Orkanger, Norway, October 5–8, M. Lu, ed., Balkema, Rotterdam, The Netherlands.
Meriam, J. L., and Kraige, L. G. (1992). Engineering mechanics, 3rd Ed., Wiley, New York.
Newmark, N. M., and Rosenblueth, E. (1971). Fundamentals of earthquake engineering, Prentice–Hall, Englewood Cliffs, N.J.
O’Sullivan, C., and Bray, J. D. (2001). “A comparative evaluation of two approaches to discrete element modelling particulate media.” Proc., 4th Int. Conf. on Analysis of Discontinuous Deformation, N. Bićanić, ed., University of Glasgow, 97–110.
Scheldt, T., Lu, M., and Myrvang, A. (2002). “Numerical analysis of Gjovik cavern: a comparison of continuous and discontinuous results by using and DDA.” Stability of rock structures, Proc., 5th Int. Conf. on Analysis of Discontinuous Deformation, Y. H. Hatzor, ed., Abingdon, Balkema, Rotterdam, The Netherlands, 125–132.
Shi, G.-H. (1988). “Discontinuous deformation analysis—A new model for the statics and dynamics of block systems,” PhD thesis, Univ. of California, Berkeley, Calif.
Shi, G.-H. (1993). Block system modeling by discontinuous deformation analysis, Computation Mechanics, Southampton, U.K.
Shi, G.-H. (1996). “Discontinuous deformation analysis of block systems.” Tech. Rep., U.S. Army Engineer Waterways Experiment Station, Vicksburg, Miss.
Wang, C.-Y., Chuang, C.-C., and Sheng, J. (1996). “Time integration theories for the DDA method with finite element meshes.” Proc., 1st Int. Forum on Discontinuous Deformation Analysis (DDA) and Simlations of Discontinuous Media, Berkeley, CA, M. R. Salami and D. Banks, eds., TSI, Albuquerque, N.M., 263–287.
Wood, W. L. (1990). Practical time-stepping schemes, Oxford University Press, New York.
Zhu, W., Zhang, Q., and Jing, L. (1999). “Stability analysis of the ship-lock slopes of the Three Gorges project by three-dimensional FEM and DEM techniques.” Proc., 3rd Int. Conf. on Analysis of Discontinuous Deformation—From theory to practice (ICADD-3), Vail, Colo., June 3–4, B. Amadei, ed., American Rock Mechanics Association, Alexandria, Va., 263–272.
Zienkiewicz, O. C., and Taylor, R. L. (2000). The finite element method: The basis, Vol. 1, 5th Ed., Butterworth-Heinemann, Linacre House, Jordan Hill, Oxford, U.K.
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Information
Published In
Journal of Engineering Mechanics
Volume 130 • Issue 3 • March 2004
Pages: 249 - 258
Copyright
Copyright © 2004 American Society of Civil Engineers.
History
Received: Mar 28, 2003
Accepted: Aug 18, 2003
Published online: Feb 19, 2004
Published in print: Mar 2004
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