Abstract
The second order equations are the form more widely used both in solids and in soil dynamics. This Chapter is devoted to present the Discretization techniques which are used. The solution strategy consists on (i) perform a discretization in space from which a system of ordinary differential equations (ODE) is obtained, and (ii) discretize this system in time. An important aspect is that of imposing suitable boundary conditions. Here we will present a simple technique which can be used for elastic or viscoplastic materials assuming that the wave is planar in the neighbourhood of the boundary. Finally, we will present some applications to cyclic and dynamic problems.
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Bibliography
I. Babuška. The finite element method with Lagrange multipliers. Numerische Mathematik, 20:179–192, 1973.
K. J. Bathe. Finite Element Procedures. Prentice Hall, 1996.
F. Brezzi. On the existence, uniqueness and approximations of saddle point problems arising from Lagrange multipliers. RAIRO, 8-R2:129–151, 1974.
F. Brezzi and J. Pitkaranta. On the stabilization of finite element approximations of the Stokes problem. In W. Hackbusch, editor, Efficient solutions of elliptic problems, Notes on Numerical Fluid Mechanics, volume 10, pages 11–19, Wiesbaden, 1984. Vieweg.
A. J. Chorin. Numerical solution of incompressible flow problems. Studies in Numerical Analysis 2. 1968.
R. Codina, M. Vázquez, and O. C. Zienkiewicz. A fractional step method for compressible flows: boundary conditions and incompressible limit. In M. Morandi Cecchi, K. Morgan, J. Periaux, B. A. Schrefler, and O. C. Zienkiewicz, editors, Proc. Int. Conf. on Finite Elements in Fluids-New trends and applications, pages 409–418, Venezia, Italy, October 1995.
M. Hafez and M. Soliman. Numerical solution of the incompressible Navier-Stokes equations in primitive variables on unstaggered grids. In Proc. AIAA Conf., volume 91-1561-CP, pages 368–379, 1991.
T. J. R. Hughes. The Finite Element Method. Linear static and Dynamic Finite Element Analysis. Prentice Hall, London, 1987.
T. J. R. Hughes, L. P. Franca, and M. Balestra. A new finite element formulation for fluid dynamics. V. Circumventing the Babuska-Brezzi condition: A stable Petrov-Galerkin formulation of the Stokes problem accommodating equal order interpolation. Comp. Meth. Appl. Mech. Eng., 59:85–99, 1986.
M. Kawahara and K. Ohmiya. Finite element analysis of density flow using velocity correction method. Int. J. Num. Meth. Fluids, 5:981–993, 1985.
S. López Querol and R. Blázquez. Liquefaction and cyclic mobility model for saturated granular media. Int. J. Num. Anal. Meth. Geomech., 30: 413–439, 2006.
S. López Querol, J. A. Fernández Merodo, P. Mira, and M. Pastor. Numerical modelling of dynamic consolidation on granular soils. Int. J. Num. Anal. Meth. Geomech., 32(12):1431–1457, 2007.
M. Mabssout, I. Herreros, and M. Pastor. Wave propagation and localization problems in saturated viscoplastic geomaterials. Int. J. Num. Meth. Engng., 68(4):425–447, 2006.
P. Mira, M. Pastor, T. Li, and X. Liu. A new stabilized enhanced strain element with equal order of interpolation for soil consolidation problems. Comp. Meth. Appl. Mech. Engrg., 192:4257–4277, 2003.
J. L. Pan and A. R. Selby. Simulation of dynamic compaction of loose granular soils. Advances in Engineering Software, 33:631–640, 2002.
M. Pastor, M. Quecedo, and O. C. Zienkiewicz. A mixed displacementpressure formulation for numerical analysis of plastic failure. Computers and Structures, 62:13–23, 1996.
M. Pastor, T. Li, and J. A. Fernández Merodo. Stabilized finite elements for harmonic soil dynamics problems near the undrained-incompressible limit. Soil Dynamics and Earthquake Engineering, 16:161–171, 1997.
M. Pastor, T. Li, X. Liu, and O. C. Zienkiewicz. Stabilized low-order finite elements for failure and localization problems in undrained soils and foundations. Comp. Meth. Appl. Mech. Engng., 174:219–234, 1999a.
M. Pastor, O. C. Zienkiewicz, T. Li, L. Xiaoqing, and M. Huang. Stabilized finite elements with equal order of interpolation for soil dynamics problems. Archives of Computational Methods in Engineering, 6(1):3–33, 1999b.
M. Quecedo, M. Pastor, and O. C. Zienkiewicz. Application of a fractional step method to localization problems. Computers and Structures, 74: 535–545, 2000.
G. E. Schneider, G. D. Raithby, and M. M. Yovanovich. Finite element analysis of incompressible flow incorporating equal order pressure and velocity interpolation. In K. Morgan C. Taylor and C. Brebbia, editors, Numerical Methods for Laminar and Turbulent Flow, Plymouth, 1978. Pentech Press.
O. C. Zienkiewicz and R. L. Taylor. The Finite Element Method. Butterworth-Heinmann, 5th edition, 2000.
O. C. Zienkiewicz and Wu. Incompressibility without tears! How to avoid restrictions of mixed formulations. Int. J. Num. Meth. Engng., 32:1184–1203, 1991.
O. C. Zienkiewicz, C. T. Chang, and P. Bettess. Drained, undrained, consolidating dynamic behaviour assumptions in soils. Géotechnique, 30: 385–395, 1980.
O. C. Zienkiewicz, S. Qu, R. L. Taylor, and S. Nakazawa. The patch test for mixed formulations. Int. J. Num. Meth. Engng., 23:1873–1883, 1986.
O. C. Zienkiewicz, M. Huang, Wu, and S. Wu. A new algorithm for the coupled soil-pore fluid problem. Shock and Vibration, 1:3–14, 1993.
O. C. Zienkiewicz, M. Huang, and M. Pastor. Computational soil dynamics — A new algorithm for drained and undrained conditions. In H. J. Siriwardne and M. M. Zaman, editors, Comp. Meth. Adv. Geomechanics, pages 47–59, Balkema, 1994.
O. C. Zienkiewicz, J. Rojek, R. L. Taylor, and M. Pastor. Triangles and tetrahedra in explicit dynamic codes for solids. Int. J. Num. Meth. Engng., 43:565–583, 1998.
O. C. Zienkiewicz, A. H. C. Chan, M. Pastor, B. A. Shrefler, and T. Shiomi. Computational Geomechanics. J. Wiley and Sons, Chichester, 1999.
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Pastor, M., Fernández Merodo, J.A., Mira, P., López Querol, S., Herreros, I., Mabssout, M. (2012). Discretization Techniques for Transient, Dynamic and Cyclic Problems in Geotechnical Engineering: Second Order Equation. In: Di Prisco, C., Wood, D.M. (eds) Mechanical Behaviour of Soils Under Environmentally Induced Cyclic Loads. CISM Courses and Lectures, vol 534. Springer, Vienna. https://doi.org/10.1007/978-3-7091-1068-3_6
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DOI: https://doi.org/10.1007/978-3-7091-1068-3_6
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