Computational Geomechanics. Manuel Pastor

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Название Computational Geomechanics
Автор произведения Manuel Pastor
Жанр Физика
Серия
Издательство Физика
Год выпуска 0
isbn 9781118535301



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(ii) bilinear for p: (c) (i) linear for u; (ii) linear for p: (d) (i) linear (with cubic bubble) for u; (ii) linear for Element (c) is not fully acceptable at incompressible–undrained limits.

      We shall return to this problem in Chapter 5 where a modification is introduced allowing the same interpolations to be used for both u and p.

      In that chapter, we shall discuss a possible amendment to the code permitting the use of identical u ‐p interpolation even in incompressible cases.

      We note that all computations start from known values of bold u overbar and bold p overbar w possibly obtained as the result of static computations by the same program in a manner which will be explained in the next Section 3.2.4. The incremental computation allows the various parameters, dependent on the solution history, to be updated.

      Thus, for known Δbold u overbar increment, the Δσ are evaluated by using an appropriate tangent matrix D and an appropriate stress integration scheme.

      Further, we note that if pw ≥ 0 (full saturation, described in Section 2.2 of the previous chapter), then we have

upper S Subscript w Baseline equals chi Subscript w Baseline equals 1

      and the permeability remains at its saturated value

k equals k Subscript normal upper S

      However, when negative pressures are reached, i.e. when pw < 0, the values of Sw, χw, and k have to be determined from appropriate formulae or graphs.

      3.2.4 General Applicability of Transient Solution (Consolidation, Static Solution, Drained Uncoupled, and Undrained)

      3.2.4.1 Time Step Length

      As explained in the previous Section 3.2.3, the computation always proceeds in an incremental manner and in the up form in general, the explicit time stepping is not used as its limitation is very serious. Invariably, the algorithm is applied here to the unconditionally stable, implicit form and the equation system given by the Jacobian of (3.46) with variables Δbold u overbar and Δbold p overbar needs to be solved at each time step.

      With unconditional stability of the implicit scheme, the only limitation on the length of the time step is the accuracy achievable. Clearly, in the dynamic earthquake problem, short time steps will generally be used to follow the time characteristic of the input motion. In the examples that we shall give later, we shall frequently use simply the time interval Δt = 0.02 s which is the interval used usually in earthquake records.

      However, once the input motion has ceased and its record no longer has to be followed, a much longer length of time step could be adopted. Indeed, after the passage of the earthquake, the remaining motion is caused by something resembling a consolidation process which has a slower response allowing longer time steps to be used.

      The length of the time step based on accuracy considerations was first discussed in Zienkiewicz et al. (1984), Zienkiewicz and Shiomi (1984), and, later, by Zienkiewicz and Xie (1991), and Bergan and Mollener (1985).

      The simplest process is that which considers the expansion for such a variable as u given by (3.43) and its comparison with a Taylor series expansion.

      (3.48)equation

      Using an approximation of this third derivative shown below

      (3.49)equation

      we have

      (3.50)e equals one sixth normal upper Delta t squared normal upper Delta ModifyingAbove u With two-dots Subscript n

      For a vector variable u, we must consider its L2 norm, i.e.

      (3.51)double-vertical-bar bold u double-vertical-bar Subscript 2 Baseline equals StartRoot bold u Superscript upper T Baseline bold u EndRoot e t c period

      and we can limit the error to

      (3.53)one sixth right double arrow StartFraction 3 beta 2 minus 1 Over 6 EndFraction

      Whatever the form of error estimator adopted, the essence of the procedure is identical and this is given by establishing a priori some limits or tolerance which must not be exceeded, and modifying the time steps accordingly.

      In the above, we have considered only the error in one of the variables, i.e. u but, in general, this suffices for quite a reasonable error control.

      The tolerance is conveniently chosen as some percentage η of the maximum value of norm ‖u2 recorded. Thus, we write

      (3.54)double-vertical-bar e double-vertical-bar Subscript 2 Baseline less-than eta double-vertical-bar bold u double-vertical-bar Subscript 2

      with some minimum specified.

      The time step can always be adjusted during the process of computation noting, however, not to change the length of the time step by more than a factor of 2 or ½, otherwise unacceptable oscillations may arise.

      3.2.4.2 Splitting or Partitioned Solution Procedures